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Demo Video Transcripts
00:00:30.930 -- OK guys, this is lecture 8C441.
00:00:36.670 -- I think we started problem two last time, right?
00:00:41.780 -- And I'm not. I remember is that we already almost done with
00:00:47.420 -- that. But let's just finalize it again. This is problem tool in
00:00:53.060 -- hand out #2.
00:00:56.140 -- So this is the problem here.
00:00:59.010 -- I think we got. We calculated the if we go to the most.
00:01:05.460 -- We calculated the location of the neutral axis and
00:01:10.041 -- that was 6.78.
00:01:13.120 -- Inches and then we calculated the tracking moment of inertia
00:01:18.570 -- or the correct moment of inertia which is 4067.
00:01:24.740 -- And then we just applied the.
00:01:28.670 -- Basic basic equation from the mechanics of
00:01:32.709 -- materials which has.
00:01:35.960 -- This form here F sub C = M * X divided by the correct
00:01:42.904 -- moment of inertia and we calculated the stresses here to
00:01:47.864 -- be about 1400 peace sign.
00:01:51.440 -- Right?
00:01:53.290 -- So the last thing that we need to do is to calculate the
00:01:56.709 -- stress in the steel and the stresses in the steel as well
00:01:59.865 -- or the stress of the steel F South. This is the steel
00:02:03.021 -- stress.
00:02:06.600 -- We can still use the same formula from the mechanics of
00:02:10.835 -- materials, but what we have to do is we have to multiply that
00:02:15.840 -- by the model ratio and so that should be N times the moment
00:02:20.845 -- times the distance from the neutral axis to the centroid of
00:02:25.080 -- the steel which is.
00:02:27.430 -- Do you minus X? That should be divided by the correct
00:02:30.862 -- moment of inertia.
00:02:33.160 -- Just to make it clear here guys.
00:02:36.820 -- We do have.
00:02:38.840 -- The assumption that at this stage of loading.
00:02:44.390 -- The stress distribution, if you check the handout.
00:02:48.710 -- That's the cross section of the beam.
00:02:51.790 -- And we had, I think the width of the beam is 12 inches.
00:02:56.030 -- And the total depth here is 20, right?
00:03:00.290 -- So this is that's the dimension and the first step. The first
00:03:04.250 -- step that we did last time is to locate the neutral axis to the
00:03:08.870 -- neutral axis somewhere here like
00:03:10.520 -- that. And the distance or the location of the neutral axis
00:03:14.940 -- which is X is measured from the compression side, assuming that
00:03:18.625 -- the main steel is down here,
00:03:20.635 -- similar to. The figures shown on the handout right so
00:03:27.556 -- this distance here. That's the 6.78 inches, right?
00:03:34.350 -- And based on the bending theory.
00:03:37.090 -- We assume that the stress distribution at this stage is
00:03:42.360 -- linear like that.
00:03:51.190 -- So the dashed line below the neutral axis means that
00:03:55.490 -- concrete. These dash line here means that concrete is
00:03:59.360 -- ignored, so there is no stresses in the concrete.
00:04:05.540 -- So that's here no.
00:04:08.130 -- I will just. Say no concrete stresses, and this is the
00:04:12.816 -- stress in the concrete on the compression side which is F
00:04:17.293 -- sub C and we calculated the FC here from again the mechanics
00:04:22.177 -- of materials equation is 1400 PS I so this is 14 one 400 PS
00:04:27.875 -- I so the maximum the maximum compressive stress is located
00:04:31.945 -- on the top surface of the section, right?
00:04:36.690 -- And then the stresses or the stresses is usually decreases
00:04:41.460 -- when we. Approach the neutral axis till we have a zero stress
00:04:46.890 -- zero strain at this point.
00:04:49.550 -- And then stresses will be.
00:04:52.620 -- Converted from compression to tension, so anything below the
00:04:56.607 -- neutral axis is under tension and the maximum steel stress
00:05:01.037 -- which is F Subs. Here will be the modular ratio, and that's
00:05:06.353 -- given as nine times the moment which is already given as well.
00:05:14.770 -- Now at 70, so that's 70. Kept foot again. We are working in
00:05:20.698 -- pounds and inches, so that should be multiplied by 1000.
00:05:26.590 -- Times 12 to have it an pound
00:05:29.740 -- inch. And that should be multiplied by the distance from
00:05:34.275 -- the neutral axis to the centroid of the steel, which is this
00:05:38.775 -- distance. So the distance from here to here. This is guys the
00:05:43.275 -- distance D -- X.
00:05:46.490 -- So the depth of the beam that's.
00:05:49.770 -- Given in the figure, which is 17 inches minus X, which is 6.78
00:05:56.803 -- inches and that should be divided by the correct moment of
00:06:02.754 -- inertia, which is 4067.
00:06:06.690 -- So if we calculate that in peace I so the steel stress here would
00:06:12.486 -- be about 19,000 peace sign.
00:06:24.240 -- Yes, So what is the again D is the depth the depth is measured
00:06:30.988 -- from the compression side.
00:06:33.530 -- To the centroid of the steel here.
00:06:37.550 -- So this distance here, that's D.
00:06:42.510 -- And the total height of the section H. That's 20, but the
00:06:46.974 -- depth is 17.
00:06:50.280 -- This makes sense, so now this is the stresses or the stress
00:06:53.712 -- distribution based on that stage or based on that
00:06:56.286 -- applied moment. Now let me ask you a question here.
00:07:00.180 -- What will happen if we increase at the moment the value of the
00:07:03.911 -- moment given is about 70 Kip
00:07:05.633 -- foot. Right?
00:07:08.880 -- So if we increase this M this.
00:07:12.390 -- Moment here what would happen to FC&F Steel will go up right
00:07:17.298 -- anwer this moment usually increases when we increase the
00:07:20.979 -- load applied to the beam. That makes sense. So when we
00:07:25.478 -- increase the load moments will be increased. Stresses in
00:07:29.159 -- concrete and stress in the steel will be increased.
00:07:34.520 -- Til the whole failure of the beam, right. And then we so
00:07:40.427 -- based on that we will go over the ultimate strength limit
00:07:44.794 -- state. But before going to the ultimate strength we just solve
00:07:49.161 -- another problem here guys. And the same handout before we move
00:07:53.528 -- on to another problem you have. Do you have any questions here?
00:08:00.690 -- Yes and represent.
00:08:03.060 -- M and this is the modular issue. Again, this model ratio
00:08:08.100 -- and. This represents the elastic modulus of the steel divided by
00:08:13.560 -- the elastic modulus of concrete and why you are doing that, or
00:08:18.780 -- why you are using NB cause this kind of analysis in concrete
00:08:24.000 -- sections based on something called the transformed area
00:08:27.480 -- method. So we convert or we
00:08:30.090 -- transfer everything. Into an equivalent concrete make sense.
00:08:34.450 -- Instead of dealing with concrete and steel, we.
00:08:39.320 -- We say that no, we will transfer everything, convert everything
00:08:42.820 -- to an equivalent concrete section. That's the reason that
00:08:45.970 -- if you check the.
00:08:48.850 -- If you check the hand out the figure that is drawn on page
00:08:54.804 -- tool, I have something like this you have.
00:09:01.840 -- Did you see the guys?
00:09:04.360 -- So this is the concrete.
00:09:07.120 -- On the compression side, and we mentioned that the concrete
00:09:10.600 -- and attention is ignored, so we converted all the steel on
00:09:14.428 -- the tension side to an equivalent concrete section.
00:09:20.100 -- And you see that on the finger. So this is the.
00:09:24.450 -- End times area steam.
00:09:29.220 -- So that's the distance X. Again, this distance here is
00:09:32.840 -- the D -- X which is from the neutral axis. This line here
00:09:37.546 -- represents the neutral axis to the centroid of this team.
00:09:43.060 -- So let's go to problem. Problem Three is a straight forward. We
00:09:47.908 -- can go over that quickly. So again in problem 3.
00:09:53.020 -- We need to determine the allowable bending moment
00:09:55.356 -- that may be applied to the beam of example tool. So
00:09:58.568 -- we use these numbers here.
00:10:02.840 -- If the allowable stresses is 1350PSI for concrete in
00:10:07.367 -- compression and 20,000 piece I.
00:10:11.660 -- For that, enforcing steel in tension so it's the same thing,
00:10:16.302 -- just problem Series A straightforward business that.
00:10:19.256 -- Let's assume guys that I that we have some limiting values for
00:10:24.320 -- stresses. We call it the
00:10:26.430 -- allowable stresses. So what is the maximum? Let's say that the
00:10:30.845 -- maximum allowable stresses for concrete in compression is.
00:10:34.110 -- 1350 PS I and the maximum allowable stresses for the
00:10:40.090 -- steel intention is about 20,000 pieces.
00:10:46.790 -- So can we use these two numbers to find the?
00:10:52.610 -- Global moment.
00:10:56.190 -- So the allowable moment means that the maximum moment that
00:11:00.330 -- should be applied to that beam without exceeding this allowable
00:11:04.470 -- stresses. Right, which is the same.
00:11:11.050 -- Same equations now if we.
00:11:16.150 -- And also, given that OK, so the moment equation, same
00:11:19.920 -- thing it's.
00:11:21.970 -- FC. Times I sub CR which is the correct moment of
00:11:27.980 -- inertia divided by Y. So this is the just rearranging
00:11:32.270 -- the equation from the previous problem. Same
00:11:35.273 -- equation. So we do have FC. This is the allowable
00:11:39.563 -- compressive stresses which is 1350 that's given.
00:11:44.410 -- Times the correct moment of inertia, which already
00:11:48.234 -- calculated in problem 2 and that was 46, four 067.
00:11:53.880 -- And that should be divided by the value of Y.
00:11:57.840 -- Which is.
00:12:00.000 -- Distance from neutral. The distance from the neutral axis
00:12:04.680 -- to the compression side, which is 6.6 point in 76.7.
00:12:11.340 -- So that will bring us up tool.
00:12:17.920 -- A big number.
00:12:20.180 -- Which which is if you divide the whole thing.
00:12:24.995 -- Let's divide the whole thing guys by 1000 * 12
00:12:30.345 -- again to convert it to Capen foot. I think that
00:12:35.695 -- will be 67.5 foot.
00:12:45.270 -- So.
00:12:48.010 -- To make sense so that this is the moment this is the
00:12:51.694 -- global moment based on.
00:12:54.400 -- The allowable compressive stresses in the concrete.
00:12:58.400 -- Now we can repeat the same equation from problem tool
00:13:02.525 -- here for the steel. So the moment equation based on the
00:13:06.650 -- steel stress that will be what will be again F sub S.
00:13:11.750 -- Times the correct moment of
00:13:13.790 -- inertia. Divided by the modular ratio times D -- X.
00:13:20.580 -- Again, this is the same equation that we just used in
00:13:24.749 -- problem 2, but just rearranging the equation so
00:13:27.781 -- the steel stress that's given.
00:13:30.870 -- As 20,000, which is the level steel stress 20,000 piece I.
00:13:36.970 -- Times the correct moment of inertia, which is a constant
00:13:43.470 -- number 4067 inch 4.
00:13:48.550 -- Divided by the moderation which is 9.
00:13:53.880 -- Times D -- X, which is the depth of the beam 17.
00:13:58.980 -- Minus 6.78.
00:14:03.360 -- Anne. Again, if you divide the whole thing by 1000.
00:14:10.600 -- By 12 that will give us a
00:14:14.597 -- hard 73. .7 keep foot.
00:14:20.830 -- So we do have two moments now to a level moments. One is
00:14:24.613 -- calculated based on the.
00:14:26.460 -- Compressive stresses in the concrete and the 2nd is
00:14:30.240 -- calculated based on the tensile stress in the steel right.
00:14:35.910 -- So this is the number 67 point.
00:14:39.740 -- Five and the second moment is 73.7 and I think the allowable
00:14:44.432 -- one will be which one?
00:14:47.910 -- Smaller, right? So that world controls.
00:14:53.760 -- Windows so that will control 67.5. That will be the
00:14:59.490 -- moment controls the.
00:15:04.980 -- Beam.
00:15:07.130 -- Makes sense, yes.
00:15:10.340 -- So the first moment equation you said why yes, but you just use
00:15:14.786 -- the X value from the past. That is basically the same thing. Yes
00:15:19.232 -- Simpson something so that the genetic equation in mechanics of
00:15:22.652 -- materials says M y /, y, right or myo over I. Sorry so M Y / I
00:15:28.466 -- so this Y the definition of this wine. Concrete is the distance
00:15:32.570 -- from the neutral axis to the compression side which is X OK.
00:15:38.620 -- OK questions.
00:15:44.590 -- So let's go to problem 4 then.
00:15:47.790 -- Which is.
00:15:52.550 -- That is a bit interesting here to have it.
00:15:58.150 -- So for problem 4.
00:16:01.030 -- We have just a weird section.
00:16:07.480 -- So we have a market section guys like this.
00:16:23.860 -- So.
00:16:25.970 -- And we do have steel bars down here, so that's the
00:16:29.523 -- tension side.
00:16:32.580 -- Um?
00:16:35.820 -- The total width here is 18.
00:16:39.800 -- That's.
00:16:44.580 -- So.
00:16:46.920 -- 6 inches each and the height of this notch here is.
00:16:52.580 -- About 6 inches.
00:16:55.260 -- Um?
00:16:57.340 -- So the model ratio is given the value of N is 8.
00:17:02.440 -- And the moment the applied moment to that beam is about
00:17:06.477 -- 110 kept foot.
00:17:10.830 -- So we need to find the game. The bending stresses in the.
00:17:16.730 -- Concrete and steel.
00:17:19.490 -- So the challenge here will be locating the
00:17:23.418 -- neutral axis, right?
00:17:26.780 -- How?
00:17:31.150 -- How, how, how, how we find the neutral axis location here?
00:17:37.970 -- And the moment of inertia for each square and then translating
00:17:41.919 -- it to no first week before, before, before we find the
00:17:45.868 -- moment of inertia, we have to find the neutral axis location.
00:17:49.817 -- We cannot find the moment of inertia without knowing the
00:17:53.407 -- location of the neutral axis. So we need to locate Mr. NA.
00:18:00.680 -- And to do that?
00:18:03.320 -- We have
00:18:07.520 -- two options or two scenarios options, scenarios, right?
00:18:10.832 -- Because we don't know if the neutral axis will be located
00:18:15.386 -- over here within the notch, right or outside here.
00:18:21.960 -- Makes sense, so we have two scenarios. Either the neutral
00:18:24.900 -- axis located. Over the neutral axis will be less than the six
00:18:29.590 -- inches. The height of this launch, or it will be greater
00:18:33.242 -- than the six inches, so that.
00:18:36.410 -- The easiest way to do that is just assume one scenario and
00:18:39.578 -- see. If the scenario is achieved so you are correct. If not, we
00:18:44.534 -- have to go to the other one. So in other words, what we can do,
00:18:49.664 -- let's assume that the neutral axis is located outside the
00:18:53.084 -- match like that. So in this case this distance here that's our X,
00:18:57.530 -- which is, I think, drawn in the figure. But just in case. So
00:19:01.976 -- this is the X value. Now to find the neutral Axis location X.
00:19:07.460 -- We have to take the first moment of area about that line to be 0.
00:19:13.550 -- So what about this? Can we guys? You know guys that
00:19:19.182 -- this area here?
00:19:22.500 -- Has nothing right? This is void.
00:19:26.090 -- So the first moment of area what we can do is
00:19:29.335 -- we can assume the whole.
00:19:32.010 -- The first moment of area of the whole compression block here,
00:19:36.564 -- OK, which will be what will be B again. B is the width here.
00:19:44.580 -- Times X. Times X / 2.
00:19:54.150 -- So B * X This is the area of the concrete rectangle.
00:20:00.280 -- Above the neutral axis.
00:20:02.470 -- Times X / 2 because we're taking the moment of that area about
00:20:07.514 -- the neutral axis.
00:20:09.300 -- Two more easily calculate the area, multiply the area where
00:20:12.650 -- distance and the distance is X / 2 because we measure distances
00:20:16.670 -- from. Centroid the centroid of that shape, which is so
00:20:21.936 -- the centroid of everything here guys.
00:20:27.010 -- These block here are these box here the centroid is at the
00:20:30.214 -- middle which is X at distance X
00:20:32.083 -- / 2 right? Makes sense.
00:20:35.160 -- So this is the X / 2.
00:20:39.510 -- Minus now we need to subtract
00:20:42.258 -- the. The void.
00:20:47.650 -- OK, which will be what?
00:20:50.910 -- 6 * X -- 6 So that any of that voyante is.
00:20:56.950 -- 6 by 666 by 6 right, because this is 6
00:21:00.710 -- inches, this is 6 inches, but that should
00:21:03.718 -- be 6 * 6 times.
00:21:07.520 -- The distance from the centroid of that void.
00:21:12.040 -- Which is here. To the neutral axis so that distances.
00:21:19.080 -- X -- 3.
00:21:22.130 -- Three yes X -- 3 because this is
00:21:24.562 -- 6 right guys? So make sense. So this distance here I will
00:21:29.512 -- just draw an error here. So that's X -- 3.
00:21:34.580 -- So that should be multiplied by X -- 3.
00:21:40.660 -- And then.
00:21:43.440 -- Another negative sign.
00:21:47.210 -- Will take the first moment of area of the steel.
00:21:51.720 -- About the neutral axis.
00:21:55.370 -- So the first moment of area of
00:21:56.980 -- this deal will be. The area of the steel. Sorry N times the
00:22:01.523 -- area of the steel because we need to transform this steel to
00:22:05.087 -- an equivalent concrete. So multiply that by N so that's
00:22:08.836 -- N times a sub S which is area of the steel.
00:22:14.170 -- Times the distance from the centroid of the steel bars.
00:22:18.850 -- To the neutral axis, which is this distance.
00:22:23.750 -- This is D -- X.
00:22:27.130 -- So that's times D -- X that should be 0, so makes sense.
00:22:33.640 -- So if we do that, just let's plug numbers here, the width B
00:22:40.062 -- is 18 * X.
00:22:42.680 -- Times X / 2 -- 36 * X -- 3.
00:22:51.550 -- Minus N, which is given as eight times the area of the steel. And
00:22:57.192 -- if you look at the figure, the area of the steel is given as 4
00:23:03.237 -- #10 four bars number 10 which is 5.06 square inches times D -- X
00:23:08.879 -- D is the depth.
00:23:11.500 -- What is the depth guys? Can you see that in front of
00:23:14.224 -- you 23 -- X?
00:23:16.440 -- Yeah, so the depth is.
00:23:19.130 -- 23 inches, can you see that?
00:23:22.130 -- Minus X = 0.
00:23:25.080 -- So have a nice equation here and you know that
00:23:27.910 -- you're expert in math.
00:23:30.590 -- It was your magic Calculator to find what is X.
00:23:36.740 -- So X here will be.
00:23:39.940 -- 9.32 inches, which is a good sign.
00:23:46.370 -- Why it's a good sign?
00:23:50.060 -- It's outside, avoid yes, because we assumed at the beginning that
00:23:54.383 -- the neutral X is larger than the six inches depth is away from
00:23:59.492 -- the void. Based on that
00:24:02.435 -- assumption. The exact solution is 9.32, which is verifying what
00:24:07.570 -- we're what we have assumed to.
00:24:10.770 -- Our scenario is good makes sense.
00:24:14.280 -- So from here guys, once we have the neutral axis questions about
00:24:17.892 -- this, yeah, probably know if our assumption is bad. If it's
00:24:21.203 -- negative or if it's just smaller now this more if it's 4 inches.
00:24:25.116 -- So in this case that means that we have to go back and repeat
00:24:29.330 -- everything. That's a good question. Makes sense guys so
00:24:32.039 -- again. This is now this is good, right?
00:24:38.020 -- Now if it's bad.
00:24:44.890 -- Which is again or correct.
00:24:48.270 -- FX for some reason 3 inches, so that's bad. So what should we
00:24:54.549 -- do? We have to neglect all of that and start over from
00:25:00.345 -- scratch, assuming that the neutral axis whoops.
00:25:05.910 -- The neutral axis is somewhere
00:25:07.775 -- here. And then you have to repeat the process to
00:25:10.727 -- find what is the exact X.
00:25:19.470 -- Are you following them here?
00:25:22.340 -- OK.
00:25:25.560 -- OK, so we have X which is good 9.32 Now the second stage step
00:25:30.908 -- is to find.
00:25:33.670 -- The moment of inertia. What is the correct moment of inertia
00:25:37.784 -- and at 12?
00:25:39.660 -- Have some.
00:25:42.230 -- Computational effort here to find it, but in
00:25:46.182 -- order to small guys so.
00:25:49.850 -- Step #2
00:25:53.020 -- why find the?
00:25:56.910 -- Cracked moment of inertia. So the correct moment of inertia
00:25:59.780 -- Now will be a challenge. How can we find it?
00:26:12.620 -- Let me draw this again here.
00:26:16.930 -- So.
00:26:19.940 -- This is the neutral axis, right?
00:26:25.510 -- So we need to find the moment of
00:26:27.134 -- inertia of two things. For the concrete and the compression
00:26:30.492 -- side and for the steel and attention side. So for the
00:26:33.979 -- concrete and the compression side we have a very weird shape
00:26:37.466 -- because we do have a void here. So we have many different ways
00:26:41.587 -- to do it OK.
00:26:43.870 -- We know that this distance now is X.
00:26:47.270 -- Which is 9.32 inches.
00:26:50.720 -- We know that the width here of.
00:26:54.420 -- Of this
00:26:56.250 -- port, 6 inches. Same thing here.
00:27:01.140 -- 6 inches So what we can do is we can divide that weird shape
00:27:06.954 -- into subdivisions or some small shapes to find the moment of
00:27:10.716 -- inertia of each.
00:27:13.220 -- OK, So what we can do guys?
00:27:17.140 -- Let's do this so that's the fairest shape here.
00:27:21.820 -- Or the 1st part. This is the second part.
00:27:26.080 -- And that's the third part. So this is part one. This
00:27:30.821 -- is Part 2 and.
00:27:33.960 -- This is Part 3.
00:27:36.650 -- So whenever you have a very weird shape like that, the
00:27:40.236 -- easiest way is to divide it into small rectangles, because we
00:27:43.822 -- know the moment of an edge of
00:27:46.104 -- each rectangle is. BH cubed over.
00:27:51.040 -- No. Yes, I know, but this is about the centroid, but is for
00:27:56.600 -- our case is BH cubed over 3.
00:28:00.830 -- Do you understand why correct?
00:28:03.350 -- No.
00:28:05.830 -- Yes no.
00:28:08.040 -- Why it's over 3? Again, we mentioned that last time.
00:28:12.780 -- Side note.
00:28:15.830 -- So the BH cubed over 12. This is the moment of inertia when the
00:28:22.130 -- neutral axis is passing through the centroid of the area.
00:28:27.470 -- So these BHQ over 12 is the moment of inertia about this
00:28:32.282 -- line, which is passing through
00:28:34.287 -- the centroid. But if we.
00:28:39.560 -- If we do have the same rectangle, if we need to find a
00:28:42.810 -- moment of inertia of a
00:28:44.060 -- rectangular section. About a line passing through its lower
00:28:48.327 -- edge like this.
00:28:51.120 -- Note the centroid, so that will be BH cubed over three.
00:28:55.652 -- This makes sense.
00:29:01.350 -- Wake up.
00:29:04.610 -- So here we have.
00:29:07.220 -- Oh well, here we have.
00:29:10.630 -- What is the first moment of inertia? What is the moment of
00:29:14.122 -- inertia of the first part then?
00:29:18.210 -- Six times so B is 6 inches, right? So six times.
00:29:24.100 -- The height which is 9.32 cubed over.
00:29:31.620 -- 3.
00:29:34.200 -- Over 3 * 2.
00:29:36.700 -- Because area one or part one is similar to Part 2 makes sense.
00:29:43.440 -- OK. Plus the moment of inertia of the small part,
00:29:48.837 -- which is part number three, we know that this width is.
00:29:56.280 -- That's six inches, and we know the height as well this.
00:30:00.250 -- Height is what is 9.32 -- 6,
00:30:03.659 -- right? So that will be 3.3 two? Yeah that will be 3.
00:30:11.390 -- That would be 3.32 inches.
00:30:15.160 -- So from here, the moment of inertia of this small part here
00:30:20.632 -- will be the width, which is 6 times the height which is 3.32
00:30:26.560 -- ^3 / 3 as well.
00:30:29.990 -- This makes sense. Again, this tool because we have two
00:30:33.500 -- similar parts which is part one and Part 2, and this term
00:30:37.712 -- is for part number 3.
00:30:40.990 -- Plus the moment of inertia of their enforcing steel, which is.
00:30:47.790 -- Lying here in the lower side.
00:30:52.750 -- And that should be an N, which is the molar ratio that's eight
00:30:58.379 -- times the steel area which is.
00:31:02.460 -- Five point 5.06.
00:31:05.880 -- So this is the in value. This is the area of the steel times the
00:31:11.520 -- distance from the centroid of
00:31:13.400 -- the steel. Through the neutral axis, which is.
00:31:19.470 -- 9.3 Two yes D -- X which is 23 -- 9.32.
00:31:28.660 -- So this is 23 -- 9.32 ^2.
00:31:35.970 -- Squared
00:31:38.390 -- OK. Because you know the problem with this concrete calculations.
00:31:43.139 -- If you forget the square here, everything down here will be
00:31:47.330 -- missed. Will be missed, right
00:31:50.074 -- so? Please be focused with her with us. If so, this is the
00:31:55.874 -- moment of inertia that we should have and that will be about
00:32:00.578 -- 10,887 inch 4.
00:32:03.940 -- So at this stage, once you have
00:32:07.580 -- them. Moment of inertia. And once you have the location of
00:32:12.720 -- the neutral axis, we can easily move on to find the stresses at
00:32:17.530 -- any. Location across the section that we have so.
00:32:26.020 -- To find the stresses again will recall the mechanics of
00:32:30.190 -- materials equation F sub C will be the moment.
00:32:35.310 -- Times our why?
00:32:37.790 -- Which is equivalent to X to M * X divided by the correct moment
00:32:42.816 -- of inertia. This is the equation to find the concrete stress, and
00:32:47.124 -- we do have the moment because
00:32:49.278 -- that's given. 110
00:32:54.900 -- kept foot, so this 110 should be multiplied by again 1000 * 12.
00:33:03.140 -- Times the distance X which is the neutral axis, which is 9.32.
00:33:10.700 -- Divided by the correct moment of inertia, which we just
00:33:15.030 -- calculated the 10,800.
00:33:18.530 -- 87 that will give us like 1130 P sign.
00:33:26.550 -- So about 11130, pyside, that's the stress in the concrete. And
00:33:31.676 -- for the steel.
00:33:34.280 -- It's the equation. It's N times
00:33:37.730 -- the moment. Times the distance D
00:33:41.562 -- -- X. Divided by the correct moment of inertia.
00:33:47.550 -- So again, repeating that N is 8.
00:33:52.250 -- The moment is 110.
00:33:55.420 -- Times 12,000.
00:34:00.780 -- Times D -- X, which is 23 -- 9.32.
00:34:07.500 -- That's divided by 10,887. So if you simplify that, I
00:34:14.680 -- think we'll have about 13,000.
00:34:20.180 -- 269 PS I.
00:34:25.080 -- So these are the stresses in the concrete.
00:34:29.940 -- And in the steel at the extreme.
00:34:36.360 -- Favor so.
00:34:42.990 -- This makes sense here guys.
00:34:54.060 -- So going back to this figure here.
00:35:02.750 -- Are you done this part?
00:35:07.540 -- So stressing concrete is about
00:35:09.730 -- 11:30. Steel is 13,000.
00:35:13.790 -- So if I ask you to draw the stress distribution here, so
00:35:17.834 -- that should be the stress distribution. Again similar to
00:35:20.867 -- what we did last time. We do have a triangle like this.
00:35:26.800 -- And the maximum stress in the concrete is in the top surface
00:35:31.120 -- on the compression side, which is 1130 P sign.
00:35:36.220 -- And concrete on the tension site
00:35:38.836 -- is ignored. And the maximum stress on the steel level, which
00:35:44.291 -- is down here.
00:35:47.160 -- That is 1113 thousand 269 peace sign.
00:35:57.310 -- So that's the stress distribution still.
00:36:01.540 -- Perfect linear noise.
00:36:07.200 -- We'd like just to look at this figure and just have some
00:36:12.084 -- conclusions here so.
00:36:15.990 -- From C 357 guys, you remember that you know the target
00:36:20.731 -- compressive strength for normal concrete at 28 days was what?
00:36:25.860 -- Roughly.
00:36:29.650 -- 4000 something like that, right? This for normal concrete that we
00:36:33.610 -- use for bridge decks like 4000. PS. I so.
00:36:37.970 -- F prime C at 28 days.
00:36:43.890 -- This should be the target. This is a very well known number
00:36:48.596 -- in the in the outside the field. The 4000 piece sign.
00:36:53.790 -- Let's assume that this concrete that has been used in this
00:36:58.135 -- section has a compressive strength at 28 days equals 4000,
00:37:02.085 -- right? Now when the moment applied when the moment.
00:37:10.620 -- When the moment of 110 kept foot.
00:37:17.570 -- Is applied to that section. How much concrete stress we got.
00:37:22.890 -- 1130 So FC we got
00:37:26.430 -- 11. 30 or 1100 thirties makes sense.
00:37:32.440 -- So this is the maximum compressive
00:37:34.804 -- stresses on the concrete when the
00:37:37.168 -- moment was 110.
00:37:39.990 -- The question now is.
00:37:42.790 -- What is the relationship between the 100 so that 1130 piece I
00:37:47.086 -- compared with the 4000 peace
00:37:48.876 -- sign? Is it like less than half equals half
00:37:52.564 -- of their value or what?
00:37:56.960 -- It's 11:30 is less than half of the 4000 is right, so
00:38:04.832 -- when groups when not if when?
00:38:10.580 -- When F sub C, which is the 11:30 equals
00:38:16.691 -- oh sorry less than .5 F prime C. The
00:38:22.802 -- target at 20 days.
00:38:26.690 -- OK.
00:38:29.770 -- Stress is for the stress distribution.
00:38:38.130 -- As assumed to be linear.
00:38:45.390 -- So as long as.
00:38:47.360 -- The compressive stress is less than 50% of the 28
00:38:53.230 -- days compressive strength.
00:38:57.230 -- The stress distribution is assumed to be linear
00:39:00.110 -- over the cross section.
00:39:02.930 -- If this number, which is F sub
00:39:05.380 -- C. Exceeds 50%
00:39:11.490 -- of the 4000.
00:39:13.770 -- The stress distribution will be
00:39:16.635 -- nonlinear. Because after that number after that, sorry after
00:39:20.806 -- that threshold value which is
00:39:22.616 -- the 50%. Concrete the concrete section will be having major
00:39:27.764 -- cracks and this major cracks will produce non linearity in
00:39:32.194 -- the concrete behavior.
00:39:34.660 -- And in that stage.
00:39:38.350 -- The actual stress distribution will be not linear. It will be a
00:39:43.438 -- nonlinear system distribution, which will be. Other would be
00:39:47.254 -- our topic here so.
00:39:49.690 -- If we go back to the screen.
00:39:52.310 -- So which? Is showing like this?
00:40:00.050 -- So. So once.
00:40:02.740 -- FC exceeds point 5F.
00:40:06.710 -- Prime, see.
00:40:09.810 -- We now entering the ultimate flexural strength stage of
00:40:14.697 -- the concrete section and in that stage.
00:40:19.620 -- And that's the image we do have the stress distribution
00:40:22.910 -- groups. Can you see that the stress distribution now became
00:40:26.200 -- nonlinear? So this is just a 3D thing. Just to make sure to
00:40:30.477 -- visualize to make sure that you understand this. That's
00:40:33.438 -- the width of the section. That's the height this is the
00:40:37.057 -- C value or the.
00:40:39.930 -- The location of the neutral axis. So in the uncorrect
00:40:43.380 -- stage we named the location of the Neutral X as an X.
00:40:47.520 -- Once we jump into the ultimate stage now we will
00:40:50.970 -- call it C and the strip the stress distribution now is a
00:40:55.110 -- parabolic or has a public shape which is not linear,
00:40:58.560 -- and in this case.
00:41:01.640 -- The analysis will be a little bit different, but again.
00:41:06.570 -- As you know, the ACI dimeric and concrete Institute committee
00:41:11.540 -- knows that civil engineers are
00:41:14.025 -- lazy, so. And you know, we know that we are.
00:41:19.790 -- Very strong math, right? So they switch it or we made the life
00:41:25.406 -- more easier for us.
00:41:27.700 -- So as long as the stresses or the stress distribution is
00:41:32.034 -- nonlinear and has a public **** like that we have, we can assume
00:41:37.156 -- it to be or to have an equivalent stress equivalent
00:41:41.096 -- rectangular stress block similar to the one that is shown here.
00:41:45.430 -- So in other words, once this is the actual stress distribution
00:41:49.764 -- for get it, which is a public complicated shape for get it,
00:41:54.492 -- and then we will assume that the
00:41:57.250 -- section. OPS the section. We will have a rectangular
00:42:01.474 -- equivalent stress block like
00:42:03.326 -- that. So go back going back to it again. Sorry this is the
00:42:08.326 -- cross section. I think you know you're familiar with it.
00:42:11.606 -- Now this is the strain distribution. Hope So what you
00:42:14.886 -- can conclude here that.
00:42:17.370 -- Regardless of the loading stage.
00:42:20.580 -- The strain distribution is assumed linear.
00:42:24.850 -- On correct, correct fully cracked ultimate stage. The
00:42:27.650 -- strain distribution is linear, but for the stress the situation
00:42:31.150 -- is different. So for the stress distribution as you can see this
00:42:35.350 -- is the parabolic shape and we do have the compression force on
00:42:39.550 -- the compression side. This is the tension force and retention
00:42:43.050 -- side that is complicated for us. So we will replace these public
00:42:47.250 -- with an equivalent stress block. 2 main important things that you
00:42:51.100 -- must understand when we talk about Ultimate stage ultimate
00:42:54.250 -- strength. That means that the concrete, which is the maximum
00:42:59.592 -- maximum stresses and concrete will start to fail. So the ACI.
00:43:05.410 -- Put a threshold of the ultimate failure strain, so once you hear
00:43:13.258 -- that the concrete strain reaches
00:43:16.528 -- 0.003. That means concrete died.
00:43:22.050 -- Recent.
00:43:24.330 -- When the steel reaches the steel strain reaches the yield strain.
00:43:30.800 -- That means that steel is filled.
00:43:34.920 -- So in conclusion, here concrete fails at a strain equals 0.003.
00:43:41.910 -- Steel fields at a strain equal to the yield strain.
00:43:48.210 -- So these two failure failure thresholds or values are.
00:43:54.260 -- Or done or made for the design purpose, so makes sense.
00:44:01.960 -- When we talk about design.
00:44:04.030 -- You have to memorize these two numbers. However in the lab.
00:44:09.580 -- You should remember that beam.
00:44:11.910 -- That I showed you guys in the lab when we start pushing the
00:44:16.694 -- beam to the limit, the concrete strain will exceed .00 three and
00:44:21.110 -- the steel strain will exceed the yield strength at the final
00:44:25.158 -- filter stage. But we cannot do that in design and design. We
00:44:29.574 -- have to be very conservative right? To make sure that the
00:44:33.622 -- beam or the element the concrete element will not reach the
00:44:37.670 -- ultimate stage, because if it if that element reaches that, so.
00:44:41.920 -- Everything will fail immediately, right guys? So we
00:44:44.920 -- have to have a very safety factor here, and that's based on
00:44:49.420 -- the values that the ACI specified. So this is the actual
00:44:53.545 -- stress distribution. This is the equivalent stress block. We
00:44:57.472 -- assume that the actual neutral axis has a.
00:45:01.410 -- Annotation of C. Here. Once we transfer that to the
00:45:05.160 -- equivalence, replug the neutral X is location will be.
00:45:09.560 -- Or will equal to a?
00:45:13.970 -- OK, so this a this is the neutral axis location the new
00:45:18.770 -- one. What is the relationship
00:45:20.770 -- between A&C? A equals another factor called beta 1 * C.
00:45:27.630 -- So be ready that because it will be exposed to about 1000 factors
00:45:32.531 -- from then on. So beta one. That's the factor that we
00:45:37.055 -- must consider this beta one depends on the compressive
00:45:40.448 -- strength of concrete. Is normal concrete high strength, ultra
00:45:43.841 -- high performance? All a that's will be shown here. So based on
00:45:48.365 -- the concrete compressive strength, you can determine what
00:45:51.381 -- is the value of beta one going back to hear the maximum
00:45:55.905 -- concrete stress that is limited
00:45:57.790 -- for design. Is 0.85 times.
00:46:01.490 -- The FC prime don't left .8 Zero Point 8 five times.
00:46:07.570 -- Is it if Ramsey lifsey prime?
00:46:11.950 -- FC Prime FC prime.
00:46:15.640 -- Fusion FC Prime so .5 so the FC prime that you got from the
00:46:20.960 -- machine in the lab which is 4000 PS I we will multiply that by
00:46:26.280 -- .85 to have the maximum compressive stress limit for
00:46:29.700 -- design. Makes sense.
00:46:33.480 -- Break so.
00:46:35.750 -- So it's you guys on Friday. I haven't hand out here. Please
00:46:40.178 -- take a copy that will use it next week and maybe Friday and
00:46:44.975 -- it's already posted too similar.
00:46:48.500 -- Thank you.
00:00:24.990 -- Hi welcome everybody to our 27th class. I guess I'm not even sure
00:00:28.916 -- if we're In Sync on the web.
00:00:33.100 -- So.
00:00:34.900 -- We second here less than we just about finished up the
00:00:41.467 -- wireless network section, except we didn't quite.
00:00:46.770 -- We didn't. We kind of hated for the last couple of slides and I
00:00:50.998 -- would like to just catch up there where we left off last
00:00:54.622 -- time hoops with the last slide. So in our in our hierarchy here.
00:00:59.970 -- Well.
00:01:02.180 -- Here.
00:01:03.740 -- So where we're sitting there at 802 eleven, which was a MIMO
00:01:09.656 -- MIMO scenario and.
00:01:12.140 -- In if you have now this, this capability can use different
00:01:17.520 -- type of strategies. You might use actually know aggregation
00:01:21.102 -- and remember aggregation means putting together or we can have
00:01:25.082 -- where we basically put a bunch of frames and we put a common
00:01:30.256 -- header in there.
00:01:32.030 -- And when you do that and it's called a MSUD. So it's basically
00:01:37.802 -- an aggregation where you put in these packets here, right
00:01:42.686 -- next to each other with one Mac header. So the advantage of such
00:01:48.458 -- a thing here is of course that.
00:01:53.560 -- You have less overhead, so the frame overhead compared to its
00:01:57.201 -- alternative, which would be like where you send it with their own
00:02:01.173 -- Mac overhead so we can look at this here and you will see that
00:02:05.807 -- the contribution of overhead of course here is much higher due
00:02:09.448 -- to that, but everything has a pro Ana con, so everything has
00:02:13.420 -- an advantage and a disadvantage from a overhead POV. This is
00:02:17.061 -- actually much better because you have only one header and then
00:02:20.702 -- this case works. Then we have
00:02:22.688 -- four protocol units. Data units, on the other hand, if we do have
00:02:28.270 -- a corruption, then we have an issue because that would mean
00:02:32.318 -- that every Mindy header actually has the ECS the see the error
00:02:36.734 -- correction code over the entire the CRC, and to say it has the
00:02:41.518 -- CRC over the entire. In this case 4 frame of four subframe
00:02:45.934 -- frame. Meaning if one goes bad it has to reset the whole thing.
00:02:51.450 -- And so that is not the case down here, where now, however, you
00:02:55.688 -- have to carry the burden of.
00:02:58.890 -- Of the different units with their own headers, so that's
00:03:03.290 -- essentially what we have, and then you can have, like
00:03:07.690 -- aggregation of multiple scenario would have like. In this case
00:03:12.090 -- two of those packed together with one physical header. So
00:03:16.490 -- these are the different options that we have on the plate. So
00:03:21.770 -- now this was for 802 eleven North, but it also transfers
00:03:26.610 -- into a 211.
00:03:29.020 -- At 802 eleven AC and in AC we looked at it. I mean, we had
00:03:35.065 -- several advantages in AC. First of all, we had a much bigger
00:03:39.901 -- bandwidth. We had 160 megahertz versus the small.
00:03:44.150 -- 40 megahertz that error to leaven in had we have more. My
00:03:49.514 -- more capability up to 8 doesn't have to be, but up to
00:03:54.878 -- 8 different antennas, and we had a modulation that went
00:03:59.348 -- from 64 quam for the North to 256 quam for the AC. So All in
00:04:06.053 -- all, that is where the big improvements occur.
00:04:13.990 -- So the advantage is now that if you have such a scenario, we can
00:04:19.380 -- have actually group, so we can have different constituents or
00:04:23.230 -- multiple users there. I can have a multi user MIMO where I'm
00:04:27.850 -- talking to you with one antenna. I'm talking to you with another
00:04:32.470 -- antenna and that is a very different scenario in a straight
00:04:36.705 -- out scenario where we have one antenna. Because I can first
00:04:40.940 -- talk to you and then I can talk
00:04:44.020 -- to you. But here I can actually split up the streams in
00:04:48.770 -- multiplayer games that are based now on antennas and that becomes
00:04:52.818 -- a variable. Very powerful means. So I might have. Now I'm sending
00:04:57.234 -- something to you and I will use my antennas to have different
00:05:01.650 -- data stream two antennas directed to one link to more
00:05:05.330 -- antennas, one each going to another device. So that is the
00:05:09.378 -- kind of advantage that you would have from, let's say, a router
00:05:13.794 -- point of view.
00:05:15.190 -- The router that is capable of AC can make these decisions, so if
00:05:20.572 -- you have like 4 different or eight different people in a
00:05:25.126 -- place, you can have 8 ongoing communications, otherwise they
00:05:28.852 -- would not be on this at the same time. So that's a huge advantage
00:05:34.648 -- of this multi user MIMO capability of AC.
00:05:40.310 -- So if everybody I know if everybody gets it here, but
00:05:44.281 -- essentially now if I'm sending it on the downlink. I
00:05:47.891 -- mean I'm sending it towards you here, I could use now
00:05:51.862 -- different antenna and do it all at the same time. So that
00:05:56.194 -- is quite some difference in mentality. There the other
00:05:59.443 -- tool of an NI don't think can do that. I'm not 100% sure
00:06:04.136 -- but I don't think with the four in Tennessee.
00:06:10.050 -- So the only disadvantage than this you have to send them as
00:06:13.878 -- individual frames. He cannot aggregated. You cannot aggregate
00:06:16.430 -- him in the fashion that we had here, so this won't work.
00:06:20.850 -- Which is the low overhead, so we have to go with this right
00:06:23.606 -- here with the higher overhead.
00:06:26.510 -- No, but that's essentially it. So NAD now, remember AD. The
00:06:30.437 -- biggest thing we should remember if anything at all like we were
00:06:34.721 -- in a complete different League here we're switching now from
00:06:38.291 -- the five Giga Hertz to the 60 Giga Hertz Band and that gives
00:06:42.932 -- you a huge advantage of.
00:06:46.070 -- Of bandwidth. The one thing is you will have very little
00:06:51.031 -- contention there at the moment and this is in devices will
00:06:55.200 -- trickle in that take over there. So that's one advantage. There
00:06:59.369 -- is little contention, whereas the 2.4 giga Hertz range is very
00:07:03.538 -- occupied. Everybody runs on 2.4 less people but more and more
00:07:07.707 -- run on five, and I don't have a single device that can run on
00:07:13.013 -- 60. At this point I think so that, but that's where things
00:07:17.561 -- are moving so.
00:07:19.160 -- Not everything is always great,
00:07:21.900 -- why? I mean, what are the disadvantages well?
00:07:26.820 -- You go to higher goal with your frequencies. The more lost you
00:07:32.436 -- have to endure, so that is one thing. So higher losses there
00:07:38.052 -- and multipath is also an issue. Multipath losses are a big
00:07:43.200 -- issue. Remember multipath is when a signal bounces off
00:07:47.412 -- somewhere. And obviously I get now are reflection. I get the
00:07:52.560 -- original one. I get the
00:07:54.900 -- reflection but. The reflection can interfere with my primary or
00:08:00.606 -- original signal. And that can cause big problems. So it's not
00:08:05.486 -- just that I get an echo, but the echo messes up my original 1 as
00:08:10.346 -- well. So that is a typical example of multi loss problems
00:08:13.910 -- here. The next thing that is a little bit of a pain is you go
00:08:18.770 -- that high with the frequencies and that will not be able to go
00:08:22.982 -- through objects anymore. So that is a big problem. For example of
00:08:26.870 -- fear and building. Or if you want to go through buildings.
00:08:32.840 -- We just ran some examples.
00:08:38.880 -- Just as a little side note here, we ran just week
00:08:45.930 -- ago, so we're an example where.
00:08:49.670 -- We're having a huge building
00:08:51.235 -- here. So this is a big building and we're trying out some
00:08:55.543 -- collision. Avoidance scenarios were.
00:09:00.100 -- In 811 Piso, there's another 802 eleven standard. This one
00:09:03.780 -- happens to be also in the five Giga Hertz range, but it's only
00:09:08.564 -- for vehicles, and we were trying to test the impact of.
00:09:13.390 -- These applications where this vehicle comes and if this one
00:09:17.100 -- doesn't stop, there should be a alert that tells her if you mean
00:09:21.923 -- watch out, you're on collision course with some other vehicle,
00:09:25.633 -- so that was the thing and we're testing. For example, in this
00:09:30.085 -- case, how much buildings would affect such measurements? And we
00:09:33.795 -- were in the five Giga Hertz Band, 5 GHz band and this year
00:09:38.618 -- is roughly what did we have? The 150 hundred 50 meter or
00:09:43.070 -- something like that? If I got the exact dimensions?
00:09:46.810 -- And it turned out that in the five Giga Hertz Band we had
00:09:50.177 -- fairly good communication, even though there's no line of sight.
00:09:52.767 -- You don't need that made with cell phones. You don't need line
00:09:55.875 -- of sight to the tower. I mean, we're in the building here. It
00:09:59.242 -- works just fine. I'm.
00:10:01.960 -- But if you were and we could run this very nicely. So we're
00:10:06.068 -- driving, measuring, logging, everything an yeah work just
00:10:08.596 -- fine. Incidentally, as we're doing it, one of people that
00:10:11.756 -- there was a car ahead of us and that car introduced exactly what
00:10:15.864 -- we're testing for, 'cause that car just about there was a girl
00:10:19.656 -- there on her phone, but she was paying attention. Run the stop
00:10:23.448 -- sign right in front of me. So unfortunately it didn't have a
00:10:27.240 -- webcam that would have been funnier than heck to show at a
00:10:31.032 -- conference. So like you were testing for this application,
00:10:33.876 -- and guess what happened?
00:10:35.200 -- The exact case that we tried to test for.
00:10:38.520 -- But if you were to experience, expect that experience.
00:10:41.445 -- Experiment now in the 60 Giga Hertz range the reliability
00:10:44.695 -- would definitely much different, 'cause the communications would
00:10:47.295 -- have been much more Hanford by the building here, which happens
00:10:50.870 -- to be in our case the Wallace complex. We use that as an
00:10:55.095 -- example. If we had been NYC with there being cooler 'cause we had
00:10:59.320 -- bigger buildings. But in Moscow we don't have big buildings.
00:11:04.780 -- So that can be a problem here. So these millimeter wavelengths
00:11:09.367 -- and remember higher frequency. The shorter the wavelength.
00:11:13.420 -- They have issues they don't like to go through objects anymore
00:11:17.061 -- and that becomes a real problem because that now has more of a
00:11:21.364 -- flavor of line of sight. That's the big problem that we run
00:11:25.336 -- into. So not everything is always perfect and this is
00:11:29.773 -- definitely one of them, where it's a compromise space.
00:11:35.160 -- So the biggest ones here turn out to be well here we have a
00:11:40.788 -- MIMO antenna conversion here. We have single one, but we
00:11:44.808 -- have a huge number of channels due to a big bandwidth at the
00:11:50.034 -- 60 giga Hertz Band.
00:11:52.900 -- Then they show you in the book a nice example here of
00:11:57.256 -- justice, different physical layers for us, I mean, for
00:12:00.523 -- the most users, the only thing that matters is the
00:12:04.153 -- bitrate. Here they care about how fast is it? So we see
00:12:08.509 -- here basically what the speeds are for.
00:12:12.890 -- 480 two 1180 depending on what the physical layer is, and so we
00:12:17.895 -- can just see what happens here. The modulation. Incidentally,
00:12:21.360 -- now you probably see what all is involved. We haven't really
00:12:25.595 -- talked about some of the flavors. You know, π / 2
00:12:29.830 -- lalalala so, but you probably get the feeling of what that
00:12:34.065 -- might be. Phase shift keying involved. There's a shift there.
00:12:37.915 -- When we looked at phase shift keying of 0 and 180, we.
00:12:42.970 -- Looked at when you do it differently, let's say like 90,
00:12:46.391 -- then 270 and so on. So it should give you an idea what's
00:12:50.434 -- happening here, so hopefully when you look at it gives an
00:12:53.855 -- idea you might have to kind of go back and say that guys not
00:12:58.209 -- forgotten what was the difference between this and
00:13:00.697 -- that. No.
00:13:03.660 -- So this obviously is a frequency division multiplexing and 16
00:13:07.610 -- qualm just means we have these different levels there to
00:13:11.560 -- differentiate with. That's the big one here, so anyway.
00:13:15.680 -- Just kind of cool to see the differences here.
00:13:22.680 -- So that's on the AD.
00:13:26.310 -- So then what we kind of just hand waved was the excess an
00:13:31.159 -- privacy information and anybody that had configured
00:13:33.770 -- wireless router would kind of have an idea what's involved
00:13:37.500 -- there. The first thing is to establish station. So if you
00:13:41.603 -- want to hook up to station you need to have access to it. So
00:13:46.825 -- if you're up to VandalWeb gold for the first time, well
00:13:51.301 -- it asks you for information and then it comes in there.
00:13:57.050 -- Once you're on there, well, then you expect and it turns out to
00:14:01.821 -- be the case here too that there will be some encryption in place
00:14:06.592 -- and that is now something that's not the access itself, but that
00:14:10.996 -- deals with the privacy and so.
00:14:14.330 -- The different type of author mean authorizations
00:14:17.473 -- authentication schemes that are offered, and the main thing for
00:14:21.963 -- us is just to see what we have and the typical ones are the
00:14:28.249 -- authentication itself. Do it to password. You can normally pick
00:14:32.739 -- the parameters for that.
00:14:35.140 -- You can use further.
00:14:38.450 -- Authorization by implementing things like lists of specific
00:14:41.066 -- devices that you allow and which ones you do not allow. That is
00:14:45.317 -- much more useful in an office environment, not at University
00:14:48.587 -- environment, 'cause that would be a maintenance nightmare or
00:14:51.530 -- maintaining a list where every day somebody comes and says like
00:14:55.127 -- I got rid of this machine. Now I have another machine and can you
00:14:59.705 -- lock the new Mac address? That would be really not a practical
00:15:03.629 -- thing, but in certain environments I would strongly
00:15:06.245 -- suggest doing exactly that,
00:15:07.553 -- meaning only. Those that have a particular MAC address can
00:15:11.436 -- actually join such a device, and the others don't. So just
00:15:15.528 -- because somebody would give you a password then to have access
00:15:19.620 -- to the note would not get you on there because you also would
00:15:24.456 -- have to be in the Mac list birth specifically. I mean there by
00:15:29.292 -- specifically allowing you to do that, it can become a bummer for
00:15:33.756 -- consumers if they have a guest network, because once
00:15:37.104 -- established, that's a list.
00:15:38.950 -- The guest network typically also likes to access such a list, and
00:15:42.514 -- that means now you have many kind of just give somebody the
00:15:46.078 -- password. So like hey, the guests are Hello World and then
00:15:49.345 -- it will tell you, well you know but I don't know you mattress. I
00:15:53.503 -- can't let you in here.
00:15:55.850 -- It's not like I'm VandalWeb guest. If you have your goal,
00:16:00.158 -- vandals password there when you can go on and there's no more
00:16:04.466 -- questions asked other than that, and that makes a lot of sense
00:16:08.774 -- from a privacy point of view. Of course we want to have
00:16:13.082 -- encryption involved, because otherwise if you sit there and
00:16:16.313 -- you run an unencrypted connection, all you need is a
00:16:19.903 -- packet sniffer and you can get the information. So we've tried
00:16:23.852 -- that before when we used to have a class here.
00:16:27.910 -- That already class called 421, which was the data
00:16:30.826 -- communications lab. So it was a one hour lap. We ran with the
00:16:35.038 -- Klausner. I got rid of it because it turned out to be too
00:16:39.250 -- much work for one credit hour class for me and the
00:16:42.814 -- infrastructure also was a little bit difficult. But let's take a
00:16:46.378 -- look at this here. So we had. Let's say we have a device here.
00:16:53.420 -- And we have now information that goes. He ran. It could
00:16:58.458 -- be, it could be.
00:17:01.680 -- On wireless this could be wireless. Here it could be
00:17:05.370 -- wired, you know it could be wired here, so we might have
00:17:09.798 -- wired notes here. Or maybe I have a router here like an 802
00:17:14.595 -- eleven type thing. So Whoops 802 eleven X or some whatever
00:17:18.654 -- you might have here. So now you're sitting there.
00:17:23.550 -- On a smart phone or what have you here?
00:17:28.670 -- So if you were to establish, let's say we pick the no.
00:17:34.800 -- Your other device on your laptop and your
00:17:37.848 -- establish a telnet connection. Telnet is
00:17:40.134 -- not, is not.
00:17:44.040 -- Tell that it's not encrypted.
00:17:46.970 -- And you do this in your favorite coffee shop that uses
00:17:50.567 -- no encryption or at the airport or in a hotel where
00:17:54.164 -- you just go on and there's no encryption, and you know when
00:17:58.088 -- that is because there's no encryption key there, you just
00:18:01.358 -- see the little symbols here for wireless, but you don't
00:18:04.628 -- see anything next to it, so you know this is an open
00:18:08.552 -- wireless an. Typically it tells you it's a non secure
00:18:11.822 -- line, so if you sit there and you run a packet sniffer.
00:18:20.910 -- And you can download those free from the Internet and it may be
00:18:24.784 -- there everywhere we use them always. When we run experiments.
00:18:28.660 -- You can just never packet an. You can get everything out, so
00:18:32.476 -- we used to do that in the networking lab, which we did
00:18:36.292 -- not. On Wi-Fi wireless, but we did it here on such a
00:18:40.108 -- scenario, and so I asked them to students. So like OK, you
00:18:43.924 -- type in your password you establish from here as a
00:18:47.104 -- server. So this is server and this is a client, so I'm
00:18:50.920 -- trying to establish telnet from here to there.
00:18:54.930 -- So now you start sitting there is typing telnet and then you
00:18:58.470 -- type your password and you would think like OK so I should get
00:19:02.305 -- now. The password in the packet. Well that happens to not be the
00:19:06.140 -- case because we are so slow in typing that it will already send
00:19:09.975 -- that packet with like 1 character at a time just because
00:19:13.220 -- we are so slow compared to the
00:19:15.285 -- data rate. So what you would have to do now on a sniffer.
00:19:20.060 -- You would, whether it's wireless or where it makes the
00:19:22.990 -- difference. Here you would target a machine. I target this
00:19:25.920 -- particular phony and I said I want to have only the packets
00:19:29.436 -- that are related to this phone that I'm trying to bring in, and
00:19:33.245 -- then I look at those packets a string him up here.
00:19:37.440 -- And then between those packets I will find the telnet. I mean
00:19:41.280 -- maybe T in here and El in here and letter at a time and then
00:19:46.080 -- we'll get the password. I will
00:19:48.000 -- get the password. And it will however only show up if I
00:19:52.353 -- capture all of the packets from this unit. Here, it's not as one
00:19:56.084 -- might think, like here is the packet that has the password.
00:19:59.241 -- No, it's just. Too slow, we're just too slow typing by the time
00:20:03.410 -- we're finally done, each packet gets sent out by itself.
00:20:08.000 -- And so, in an unencrypted environment is a real
00:20:12.360 -- problem. If however, you were sitting in this coffee shop and
00:20:17.156 -- now you use SSH instead of telnet, then actually this will
00:20:21.952 -- not be a problem because now your traffic itself with SSH is
00:20:27.184 -- encrypted and you can run a packet sniffer and all you get
00:20:32.416 -- is rip gibberish.
00:20:34.290 -- You cannot really use that, so that is the good part. The bad
00:20:38.684 -- part is that often when it comes, for example to web
00:20:42.402 -- context content. We don't really know is this not encrypted or
00:20:47.098 -- not. It's not always obvious whether it is encrypted or not,
00:20:51.300 -- like you're sitting in an Internet cafe at the McDonald's
00:20:55.120 -- in Paris or who knows were and you're trying to access your
00:20:59.704 -- account. Is this safe or not?
00:21:03.110 -- You know, and then the question really depends on the
00:21:06.320 -- application support for encryption, yes or no. If you
00:21:09.209 -- do have it, yes, if you don't have it, no. But how do you
00:21:13.703 -- know? And sometimes it's worth sniffing your own connections
00:21:16.592 -- to see how exposed you are.
00:21:19.680 -- I've done that not too long ago with the Mail
00:21:23.320 -- client on a machine where we tried it out and sure
00:21:27.324 -- enough, despite me thinking we had everything
00:21:29.872 -- set up right.
00:21:32.120 -- We did not have it right and there was a setting wrong. I
00:21:35.526 -- thought it was unencrypted, turned out not to be encrypted
00:21:38.146 -- so we could see the password
00:21:39.718 -- flying by. And I thought, like, woah, woah. Woah. How
00:21:44.310 -- would I have known that an so if you exercise this somewhere
00:21:48.210 -- where they have a sniffer running like at the airport, how
00:21:51.785 -- do I know not there's nobody there that actually just sits
00:21:55.360 -- there praying on just careless or ignorant users that don't
00:21:58.610 -- really understand it and run it. So that is a problem. So the
00:22:02.835 -- original able to 11 came with web that turned out to not be
00:22:07.060 -- that great of a thing. An incidentally if you own an old
00:22:10.960 -- machine like an old cell phone
00:22:12.910 -- and I. Old iPad or what might be what you might have. It might
00:22:18.382 -- just basically allow you only to use these old, not so secure.
00:22:23.840 -- Encryption methods and it might not let you in. For example, if
00:22:26.984 -- I use web here at the University, this I know can't do
00:22:30.128 -- that. So if I don't have the
00:22:34.034 -- WPA. Encryption I cannot get on at all, so they were not allowed
00:22:39.380 -- that. So one has to be aware of that. Different levels of
00:22:43.340 -- encryption. That's essentially all I wanted to discuss here. I
00:22:46.640 -- hope you have a feeling for things that didn't make us any
00:22:50.600 -- experts on wireless, but I hope you get to feeling for it.
00:22:55.530 -- There are other wireless definitions there.
00:23:00.570 -- Like I said, we work with 802 Eleven P at the moment, which is
00:23:05.694 -- something like 802, eleven, or 811 a, but it operates for it's
00:23:10.086 -- designed for vehicle and networks and some of the
00:23:13.380 -- parameters are a little bit different. Some of the
00:23:16.674 -- parameters like the it's the back of the medium access
00:23:20.334 -- parameters are different, so their minor changes, but enough
00:23:23.628 -- to make a difference there.
00:23:28.230 -- Any questions to any of the wireless stuff?
00:23:38.140 -- Alright.
00:23:40.390 -- When with it.
00:23:43.120 -- We're ready to go for the next one here, which turns out to be.
00:23:47.770 -- The Internet protocol, so we're going very. We're
00:23:51.250 -- almost there. I'm almost at the upper level now.
00:23:56.610 -- We have one more level to go through, and that's done the
00:24:01.338 -- transport transport layer, but here we're looking at the
00:24:04.884 -- Internet protocol. IP would be the Classic One that we have,
00:24:09.218 -- but Internet Protocol it doesn't have to be IP, we're just
00:24:13.552 -- talking about in general.
00:24:16.930 -- So before we go here, let's just look at a couple of notations
00:24:21.550 -- or not connotations like couple of terminologies that they use
00:24:24.850 -- here. I mean one over the communication network. While we
00:24:28.150 -- know what that is, a bunch of notes there, I mean for so to
00:24:32.770 -- provide data transfer among devices attached to the network.
00:24:35.740 -- Of course we have a network of notes. That's all we have. So
00:24:40.030 -- the network communication network that facilitates that
00:24:42.340 -- might be in the ether. Might it be using wire? What have you?
00:24:47.240 -- Then Internet here. That is a collection of.
00:24:52.680 -- In the networks interconnected with bridges and routers and
00:24:56.496 -- this and that. So it's a big mess. A big spaghetti of.
00:25:02.250 -- Smaller, bigger spaghetti of devices, links, etc networks.
00:25:10.660 -- And then we have intranet and this is actually when we scale
00:25:15.472 -- it down. Intranet Internet is bigger in between, intranet is
00:25:19.482 -- inside, so here it's like an Internet used by single
00:25:23.492 -- organization. So we're basically running this thing on our own
00:25:27.502 -- and typically it's like for World Wide Web so.
00:25:32.690 -- So it's like in self, perhaps even self contained Internet, so
00:25:37.794 -- it's like a part of it. A smaller contained portion of the
00:25:43.362 -- Internet, Internet. The big thing, intranet small single
00:25:47.074 -- organization. Child childish subset. That's what this would
00:25:51.919 -- be. Then we have separate
00:25:54.454 -- subnetworks. If you have a big network, sometimes she only
00:25:58.671 -- interested in a small portion of that and we would treat it as a
00:26:03.557 -- subnetwork. It means, like some constituent network of an
00:26:06.698 -- Internet, meaning a smaller segment in our organization.
00:26:09.490 -- Here for example, we have different subnetworks. CS has a
00:26:12.980 -- couple of so then if you're on these networks more than you're
00:26:17.168 -- in this CSS network.
00:26:19.170 -- And we see later on how one would treat the addressing of
00:26:23.646 -- the network itself, including the subnetworks. There's more
00:26:26.630 -- effect. You probably all have seen in your configurations that
00:26:30.360 -- sometimes you see there is a subnet mask and the subnet mask
00:26:34.836 -- typically is a mechanism to say, like all the bits that are one
00:26:39.685 -- later on, get XI mean ended with the real address to identify.
00:26:44.161 -- That's the network address and all the ones that have zero.
00:26:48.264 -- That turns out to be.
00:26:50.320 -- The host address, so we see that
00:26:52.917 -- later on. So then we have our end systems like my laptop, my
00:26:58.284 -- tablet, here that would be an end system at the moment. So
00:27:02.412 -- that is just something to be attached and we have
00:27:05.852 -- intermediate systems that would be some kind of a switch or
00:27:09.636 -- something that can hook up.
00:27:12.170 -- Two or more networks, and so device used to connect two
00:27:16.350 -- networks and permit communication between these
00:27:18.630 -- things and we might have them as bridges or routers where the
00:27:23.190 -- bridge is considered to be a lower level. It doesn't
00:27:26.990 -- translate, it doesn't do much thinking, it just forwards.
00:27:30.410 -- That's all they do and then the router is considered like a
00:27:34.970 -- bridge on steroids. I mean it would be a router that actually
00:27:39.530 -- looks at things can do certain
00:27:41.810 -- manipulation. Both of them will send traffic through in One
00:27:46.040 -- Direction if needed or not.
00:27:48.590 -- But this one here operates at Layer 3, where this one
00:27:52.231 -- operates at layer 2.
00:27:54.600 -- So layer two is a lower layer in the OS I hierarchy. Remember we
00:27:58.604 -- start out with the physical layer and work our way up.
00:28:02.490 -- So this is at a lower layer. The higher the layer, the more
00:28:06.767 -- knowledge you can apply, for example error checking
00:28:09.399 -- fragmentation, D. Later on we see that when taking something
00:28:12.689 -- apart of putting it back together all of that.
00:28:17.110 -- Should not be confused with.
00:28:20.310 -- Terms like I have a layer 2 switch or have a layer 3 switch
00:28:24.398 -- etc. That means simply.
00:28:26.320 -- From a device point of view, how far do you use hardware control?
00:28:31.820 -- And I can run a Linux box and we've done that as a router.
00:28:36.370 -- Press your Linux box as a
00:28:38.122 -- router. But I can also get a layer 3 switch that has this
00:28:43.466 -- hardware encoded and it will be so much faster than my Linux box
00:28:48.185 -- running as a router.
00:28:51.340 -- So the question is almost how much how much visibility you
00:28:54.442 -- have in terms of the smarts and how much of that is implemented
00:28:58.108 -- in hardware that determines the quality of the speed or the.
00:29:01.970 -- Typically the price tag also of these devices.
00:29:07.240 -- Alright, so these are the standard terms that the book
00:29:10.450 -- uses. And we would have no applications sitting like this,
00:29:15.599 -- so here would be the classic connection between two hosts A&B
00:29:20.670 -- and they have some applications that now interface with ports.
00:29:26.220 -- Like if it's a project would be port 80 if it was a Mail program
00:29:30.480 -- as a port 25.
00:29:32.260 -- So whatever port you might have, so these applications now want
00:29:35.934 -- to connect to some application on the other side. That's
00:29:39.274 -- typically what we have.
00:29:41.900 -- Sending something and let's say I do an FTP which is on port 20
00:29:46.226 -- and 21 if I recall right? So we would basically establish a
00:29:49.934 -- connection from one year to the other one. This is the server
00:29:53.642 -- and this is now a client, so I would say like I want to
00:29:57.968 -- download this file and
00:29:59.204 -- essentially these two. Logically, connect and request,
00:30:02.158 -- like I say one download so they would say OK here it comes. This
00:30:08.276 -- is simplified language here.
00:30:10.640 -- So. The application still themselves would now have these
00:30:15.252 -- access points which are ports, and then they would form a TCP
00:30:19.680 -- packet. Whether TCP packet. Now what could be UDP packet as
00:30:23.739 -- well. The main thing is that transfer transport layer
00:30:27.060 -- protocol we will put that and box it into the IP packet and
00:30:31.857 -- that's where our attention is right now. The IEP IP packet is
00:30:36.285 -- the one that has the IP address of the destination and we have
00:30:41.082 -- to find our way from here to that other host.
00:30:44.970 -- And we don't even know what's all in between in this
00:30:48.215 -- particular toy example here, there are two networks in
00:30:50.870 -- between, so we come from the host that lives in this
00:30:54.115 -- network. We go through router that will realize based on the
00:30:57.360 -- adressing, that I use in the information up here.
00:31:01.450 -- It will. It will figure out that it has to forward it to that
00:31:07.050 -- site here switching over.
00:31:09.400 -- The IP packet of course will be then brought down to the lower
00:31:13.248 -- link control to the Mac layer and then it will back. Here it
00:31:17.096 -- will look at it, receive it, bring it up, and will look
00:31:20.648 -- inside to peak. Where is it going to? So it has to look
00:31:24.496 -- into the header of the IP packet to make that decision.
00:31:29.140 -- So this is a router that has to know the IP address. Otherwise
00:31:33.066 -- we can do it and then there has to be a routing table here so
00:31:37.596 -- that it will find out OK. This particular traffic. Now I'm
00:31:40.918 -- sending out on that physical port here Becausw the
00:31:43.636 -- destination is attached to this port and there might be more
00:31:46.958 -- networks in between. This would
00:31:48.976 -- be a very simple. Where we have one, we have
00:31:53.814 -- a two hop scenario here.
00:31:57.230 -- Two edges in the graph.
00:32:00.950 -- So these are logical connections and these are
00:32:04.830 -- physical connections here and again from here to here we
00:32:09.680 -- have logical connections, so that's how that would work.
00:32:14.045 -- So the first question is, should IP be connection
00:32:18.410 -- oriented or connectionless?
00:32:21.050 -- And here's an example WHI perhaps we don't want to have
00:32:26.528 -- connection oriented traffic, because if we have now.
00:32:31.300 -- The application going from one machine 8 to be which is
00:32:35.040 -- far away. For example, an award to use connection
00:32:38.100 -- oriented and would first have to basically request a
00:32:41.160 -- connection, then the other side. Like Yup, I can. I can
00:32:44.900 -- do that, so there would be an accept. Now would
00:32:48.300 -- transfer my file.
00:32:51.160 -- You know one of the time here and then you have multiple
00:32:55.624 -- exchanges with acknowledgements and then I would terminate this
00:32:58.972 -- and finish this up and that would basically use up quite a
00:33:03.808 -- bit of time. Time goes in this direction and oops, the root
00:33:08.272 -- problem is here in the establishment of the connection,
00:33:11.620 -- the acceptance, determination etc in the middle. Here we don't
00:33:15.340 -- have a problem.
00:33:17.900 -- But this of course is now connection a.
00:33:22.650 -- An established connection, so we don't want to deal with the
00:33:26.632 -- overhead. Another part is the whole idea of the Internet in
00:33:30.614 -- its own right. Originally this came of course from a defense
00:33:34.596 -- point of view. DARPA, DARPA was the one that actually started
00:33:38.578 -- this whole thing. I mean, that was under defense thinking, and
00:33:42.560 -- it came in a time where people were afraid of the Cold War and
00:33:47.628 -- they would say like we don't want to have a network that
00:33:51.972 -- looks like this here.
00:33:55.490 -- Let's say I have here. Here is Seattle and here in New York.
00:34:03.010 -- If I had such a network here.
00:34:05.860 -- According to a military thinker, they would say, well, all you
00:34:09.941 -- need to do is bomb one of these links here and that's the that's
00:34:15.135 -- the end of the communication between those two. So the idea
00:34:19.216 -- was let's establish a network that is interconnected and
00:34:22.555 -- really has bunch of links there. We don't even know what all is
00:34:27.378 -- available here. So we have a lot of these things. Now it becomes
00:34:32.201 -- much more difficult to corrupt such a network by physically.
00:34:36.090 -- Damage in the system so you can drop a couple of bomb Siri can
00:34:40.024 -- blow this thing up. You can blow that edge up and you have a
00:34:43.958 -- fairly high resilience there.
00:34:45.720 -- It's not necessarily optimized.
00:34:48.680 -- To be cake connected.
00:34:51.540 -- Remember K connected means there are at least K disjoint
00:34:54.760 -- paths between any two. We don't have to have that, but
00:34:58.302 -- from an attacker POV, if I saw something, I would say like
00:35:02.166 -- what are the two weakest connected ones, the minimum?
00:35:06.250 -- Cut set.
00:35:08.580 -- Where the weights are equal, for example, and that would be my
00:35:12.144 -- most efficient attack point.
00:35:15.120 -- Find the one in this particular case here, while if
00:35:18.450 -- we're interested from here to here, where do I attack? Well,
00:35:22.113 -- it would be. Basically I take this no doubt in that no
00:35:26.109 -- doubt, and things are done.
00:35:32.680 -- Well, this is silly exactly because it's so, so we only are
00:35:36.880 -- two connected here. I can send it like this or can send it out
00:35:41.780 -- like this. These are vertex. These are vertex disjoint paths.
00:35:46.110 -- One here.
00:35:49.330 -- And one goes here.
00:35:53.030 -- So these are vertex disjoint
00:35:55.420 -- paths. Edge disjoint would be different, but edge disjoint is
00:35:59.829 -- a difficult thing to me. Can knockout an edge alright? Like a
00:36:03.921 -- backhoe is typically the cost for that. You have construction
00:36:07.331 -- project and somebody ***** into data cable. That's what would
00:36:10.741 -- happen here. But if a router goes out, all the links attached
00:36:14.833 -- to the router route. So this is this is a weaker thing. So at
00:36:19.607 -- the time the thinking was like let's come up with a highly
00:36:23.699 -- dynamic quickly changing very easily kind of figure big.
00:36:27.210 -- Mash big mash or should you smash 'cause that's normally
00:36:31.160 -- considered like a certain structure, but something
00:36:33.925 -- that's difficult to knockout? That was the overall
00:36:37.085 -- motivation, and that would mean such idea here may not be
00:36:41.430 -- it. It might be much better to set it up in a way that these
00:36:47.355 -- things right themselves.
00:36:49.590 -- Then we have flexibility, send it there and see how you make it
00:36:53.646 -- to the other side, regardless of
00:36:55.518 -- something failing perhaps. And that was it. So initially
00:37:00.451 -- developed by DARPA, the D for defense, of course.
00:37:05.320 -- So it was a defense project and the Internet protocol IP is
00:37:11.776 -- basically the outcome of this whole mess, so.
00:37:18.840 -- So now if and that uses connectionless.
00:37:22.910 -- Connection this communication. What does it mean? What we set
00:37:25.980 -- something out there packet and say go and make it to your
00:37:29.971 -- destination off you go.
00:37:32.110 -- So there are huge advantages to
00:37:34.174 -- that. For example.
00:37:38.050 -- Flexible something changes. This router goes out. This
00:37:40.810 -- router goes out due to maintenance or what? Who knows
00:37:44.260 -- what? Maybe they have a thunderstorm. They lost power.
00:37:48.550 -- So we can write differently.
00:37:52.210 -- Robust. Against. All sorts of things.
00:37:58.950 -- Typically I mean flexibility would be also like there's a
00:38:02.260 -- quicker link. There's a found link that is much less
00:38:05.570 -- utilized so I can go there rather than where there's a
00:38:09.211 -- lot of contention.
00:38:11.700 -- Robust means new links fail. The systems fail, we can do it and.
00:38:19.780 -- The other thing is, there's no overhead. There's no other.
00:38:23.590 -- There's no handshaking going on, their data crams IP is a
00:38:27.781 -- datagram thing. The datagram is the postcard. You drop it off
00:38:31.972 -- and you think it makes it there, but there's no guarantee
00:38:36.163 -- datagrams have no guarantee. It's the postcard of data
00:38:39.592 -- communication. You send it there and off it goes. We know also
00:38:44.164 -- that we have TCP and UDP.
00:38:48.560 -- And those are TCP is actually reliable and UDP is also
00:38:52.784 -- datagram. So we would have now day to cram in a datagram the IP
00:38:58.160 -- packet. The datagram in the datacom? Then you
00:39:01.019 -- send it off to China.
00:39:03.260 -- The reason we have of course TNTS UDP to have a port number
00:39:08.187 -- because our IP packet has no clue what application this is.
00:39:12.356 -- It doesn't know this is for.
00:39:15.570 -- The browser this is FTP. This is this or that you know. So the
00:39:19.000 -- port number has to be part of it, and that's at the higher
00:39:22.185 -- level. We stick that into the IP packet, but the upper layer
00:39:26.666 -- would have to deal, for example with reliability issues and the
00:39:30.252 -- only one we have there is TCP.
00:39:33.170 -- You DPS no reliability. It's also a data crime.
00:39:37.250 -- So that is the. This is advantage. It's not reliable.
00:39:41.848 -- There's no guaranteed delivery. There's no guaranteed order of
00:39:45.610 -- delivery, meaning I sent three things out. The last one might
00:39:50.208 -- arrive 1st, and so that becomes securing problem for the two
00:39:54.806 -- sides. If I send something from here to there, they can go do.
00:40:00.320 -- Different route so I don't know what they take. There's no
00:40:03.598 -- guarantee an I might be waiting here for awhile till that second
00:40:07.174 -- packet arrives because it's worse. The second packet,
00:40:09.558 -- whereas the second packet and it might not make it.
00:40:14.010 -- In which case then later on the layer above would have to say
00:40:18.014 -- like what the heck is going on? Where is my packet? So that
00:40:22.018 -- would be now where reliability would be based on TCP and TCP is
00:40:26.022 -- a protocol that actually does do reliability. There is a
00:40:29.102 -- handshake and if I send something in order to hear from
00:40:32.490 -- you, I assume you didn't get it and I would send it again.
00:40:37.840 -- Whether you never got it or whether your acknowledge was
00:40:40.420 -- lost, for me, there's no difference in that. All I can
00:40:43.258 -- say is I didn't hear back from
00:40:45.064 -- you. So that's the unreliable
00:40:47.912 -- part here. So now this would be the typical example of a
00:40:53.250 -- configuration and the packets that are being used here. For
00:40:57.010 -- example here I have some station that looks up to a router, then
00:41:01.898 -- it goes through some frame relay wide area network to another
00:41:06.034 -- router to another station, and in this case here.
00:41:11.260 -- We want to have a TCP package. Let's say we do an FTP transfer.
00:41:16.730 -- That would use TCP, whereas if you were to use, let's say,
00:41:20.738 -- WhatsApp or something like that audio video that is most likely
00:41:24.412 -- UDP, because So what a frame missed. No big deal. But if a
00:41:28.754 -- bit flips in a file executor that we transfer, that would be
00:41:32.762 -- a disaster. So from here it wants to talk to this site here
00:41:37.104 -- and it does so by now. Sending an IP packet to the lower link
00:41:41.780 -- control, which then brings it to the Mac layer, which using the
00:41:45.788 -- physical layer brings it out to the neighboring device which is.
00:41:49.590 -- In this case, this particular router in here you would go up
00:41:53.682 -- to ask the question, well, what's the IP address and I have
00:41:57.774 -- to unpack all the way to here. Remember the structure is.
00:42:02.180 -- The TCP packet.
00:42:05.410 -- Here with its header.
00:42:07.890 -- Get stuffed into, let's say the IP packet.
00:42:13.230 -- So now this is a header. This is now going in here.
00:42:17.970 -- Run and they should try the other direction here, so this
00:42:21.908 -- is now exactly this thing here and then. This one here gets
00:42:26.204 -- down until we're at the Mac layer. So until I'm down here
00:42:30.500 -- at the Mac layer.
00:42:33.620 -- Where is Santas across on the other side. In the router I have
00:42:37.494 -- to 1st get this the lower link control data units routes. So
00:42:41.070 -- this is my link control. Then I open up in a find the IP to get
00:42:45.838 -- the IP address here. So this is where. Whoops, I'm certainly in
00:42:49.414 -- the header here, so this is I'm interested in the address.
00:42:55.450 -- So we need to know the address where this is going,
00:42:58.981 -- the destination address and that would now be.
00:43:02.750 -- Then we would consult or their router would consult its routing
00:43:05.588 -- table to decide where does it go out and it would say OK out of
00:43:09.458 -- this link here. This might be rather than as many ports 32
00:43:12.554 -- ports or what have you, I don't
00:43:14.360 -- know. Then it would go to the next router. On this end here.
00:43:20.820 -- Same thing again. They would have to unpack it all the way to
00:43:24.395 -- here to find out what's the address and it would forward it
00:43:27.695 -- here. That's why if you think of the devices themselves, if you
00:43:30.995 -- do that a lot.
00:43:32.790 -- Like a gateway does or high in high throughput router, you
00:43:36.541 -- better get something that does most of the work in hardware
00:43:40.292 -- rather than software. If you configure your Linux system like
00:43:43.702 -- your laptop running Linux as a router for this you will not get
00:43:48.135 -- any glory for being a high speed
00:43:50.522 -- network. 'cause that would all be under software control. You
00:43:54.132 -- need hardware controller so the higher level you can buy, the
00:43:57.190 -- faster it would work. So there would be essentially happening
00:43:59.970 -- over and over and over until this system. Here a would talk
00:44:03.306 -- to the one in China which would be. So we're constantly go up to
00:44:07.198 -- the next one. Look at the table, go up next one, look at the
00:44:11.090 -- table. What we want to have of course, and then it will go to
00:44:14.982 -- some links that will either use satellite to go over the
00:44:18.040 -- continent or it will go through a cable in the ocean. There will
00:44:21.654 -- be a long link somewhere.
00:44:23.810 -- And the cable will be just about like 1 cable in reality. Of
00:44:27.892 -- course that cable segments that have to be powered because
00:44:31.032 -- there's no fiber optics that can send something over that length
00:44:34.486 -- without nothing coming out. So every so often you have to have
00:44:38.254 -- a repeat are station and you have companies that have big
00:44:41.708 -- boats that are traded at Mastec that would go and maintain those
00:44:45.476 -- huge cables and there would be there will be a link somewhere
00:44:49.244 -- where that cable will be part of the Lingard will be 1 edge in
00:44:53.640 -- that link. It will be a fast edge, so from a router POV
00:44:57.805 -- that will be attached to very fast device.
00:45:03.010 -- So connectionless is what we have here and connectionless has
00:45:09.100 -- this. Great thing flexibel
00:45:12.426 -- robust. And it does not do any additional overhead,
00:45:17.116 -- nor handshaking involves, so very little overhead.
00:45:21.620 -- So now the big problems when it comes to Internetworking would
00:45:25.228 -- be like how do I route the
00:45:27.524 -- packets? And how do I find my way through from here to China?
00:45:32.466 -- Great RFC? Or is it 1088? I think it's the one that gives
00:45:36.730 -- the tutorial on how this works.
00:45:39.960 -- Take a look at that one. That's the only one that I know that's
00:45:43.124 -- not a sleeping pill. That's actually kind of fun to read.
00:45:45.610 -- All the other RFC's are like Reading man pages for
00:45:47.870 -- entertainment. Then you have to have a mental problem to do
00:45:52.100 -- that, but not really, but you know what I mean. They're
00:45:56.096 -- not very entertaining and RFC's are not entertaining. I think
00:45:59.426 -- it's a 1088 that one actually because it's a tutorial of how
00:46:03.422 -- it would find its way. The first issue that we run into is like
00:46:08.084 -- the lifetime of a datagram. How long should it be defined and
00:46:12.080 -- the problem will be that there is a potential for example to
00:46:16.076 -- get a datagram into a loop
00:46:18.074 -- situation for example. It could, it might loop around in such a
00:46:22.095 -- loop, here where. We send it from here to there that goes
00:46:26.641 -- from here to there goes back here and then pretty soon we're
00:46:30.229 -- back here and we're whipping around the loop and there has to
00:46:33.817 -- be a way to kill it off so there will be a lifetime defined.
00:46:38.760 -- And if you exceed your lifetime, the router that sees a packet
00:46:42.636 -- with that light would simply just ignore it, destroy the
00:46:45.866 -- package. Therefore, I mean so you don't have it then.
00:46:49.650 -- Fragmentation reassembly because a lot of times we sent something
00:46:52.890 -- and we might not be able to keep the length of the packet we
00:46:57.426 -- might have to actually chop up the packet. It depends on what
00:47:01.314 -- we have in terms of the neighborhood network, what
00:47:04.230 -- technology we have, what capabilities etc. And so we
00:47:07.146 -- might have to shrink it down, cut it in half. For example we
00:47:11.358 -- call that fragmentation and at one point you have to reassemble
00:47:14.922 -- that as well and their issues that need to we need to look at
00:47:19.458 -- there. And would be error control and flow control,
00:47:22.202 -- and that's where we start next time. Have a nice day
00:47:24.996 -- and I see you on Wednesday.
00:00:21.550 -- OK, a couple things as we get started. The first one is we
00:00:26.828 -- have the last lab assignment.
00:00:31.670 -- And so this one is going to be a bus differential protection lab
00:00:35.349 -- and so the on campus students is pretty much going to be a
00:00:39.028 -- similar setup to what you did before. You just need to read
00:00:42.424 -- through this and then work with the TA. As far as if you're
00:00:46.103 -- going to, I think you all of you have groups that you've been
00:00:49.782 -- doing the labs with the TA. If you want to stick with those
00:00:53.461 -- groups in those times. If you wanted to negotiate a different
00:00:56.574 -- time, then you just need to communicate with him about that.
00:01:02.040 -- Until you have a system and you're going to look at fault
00:01:05.628 -- at a couple of different places, this is actually
00:01:08.319 -- should be a little bit shorter than the last, quite a bit
00:01:11.907 -- shorter than the last lab.
00:01:14.920 -- And so you're really just going to look at several
00:01:17.710 -- different cases.
00:01:19.710 -- Look at the behavior with this.
00:01:22.940 -- The Engineering Outreach Lab is going to be similar.
00:01:26.730 -- So this is just the description of the entering outreach lab.
00:01:31.580 -- And so it's a little bit more complicated system, but it's
00:01:34.616 -- still the same basic idea.
00:01:36.980 -- And also you have some information about the CT
00:01:40.410 -- ratio that's was used for this.
00:01:44.480 -- And then this is using that.
00:01:47.590 -- Relay model that the differential relay model we
00:01:50.462 -- talked about. So again this is a low impedance restrained
00:01:54.052 -- differential element, so it's not. It's not a high
00:01:57.283 -- impedance differential element.
00:02:01.530 -- If anyone has fair time and wants to create their own
00:02:05.369 -- creative all the create this, it wouldn't be that hard to
00:02:09.208 -- create a lab for the restraint for the high impedance
00:02:12.698 -- differential elements. We just haven't had a chance to put
00:02:16.188 -- together the simulation files.
00:02:18.800 -- So anyway, it's the same idea you read in the data files.
00:02:24.470 -- Very similar to the handout that we talked about with the lecture
00:02:28.310 -- last week. All of this stuff we're reading the comtrade file,
00:02:31.830 -- and so where this really starts to differ a little bit is
00:02:35.670 -- towards the end of it. Once we've got the phasers, so we've
00:02:39.510 -- got the things where we're looking at the voltages in the
00:02:43.030 -- currents, and then we have the operating restraint current, and
00:02:46.230 -- so one thing that's different from the hand out before is now
00:02:50.070 -- the. In this case, there's no.
00:02:53.660 -- Nothing where you put in a multiplier to imitate
00:02:56.297 -- saturation. The simulation data that you're using for this now
00:02:59.227 -- actually has saturation in it.
00:03:01.850 -- And the case is that you'll be doing for the on campus
00:03:05.450 -- students in the lab. You're actually going to be doing
00:03:08.450 -- these with an RTS simulation instead of using the model
00:03:11.450 -- power system, and so that the RTS will have setae. Models
00:03:14.750 -- that include saturation, but you're still going to be
00:03:17.450 -- setting the actual physical relay.
00:03:21.310 -- And then one of the things that this is going to show is the
00:03:25.664 -- basically the how they operate. Quantity changes. So basically
00:03:28.463 -- as it reads through samples, this thing is moving and then it
00:03:32.195 -- works its way up and then it has some final value it goes to and
00:03:36.860 -- so you can as you look at these different cases once you enter
00:03:40.903 -- the slope setting you can actually look at a little bit
00:03:44.324 -- how the how the value evolves and when you look at the case
00:03:48.367 -- with the saturation you can actually see how it.
00:03:51.300 -- Now the saturation changes what it's what the relay
00:03:54.171 -- element is seeing too, and so this was a case for an
00:03:57.999 -- internal fault, so it grows quickly.
00:04:02.860 -- So any questions about that?
00:04:09.170 -- Hey are there any questions from the last lecture?
00:04:12.680 -- Yeah, so in the last lecture when you talk about the high
00:04:16.832 -- impedance plus differential protection, you mentioned that
00:04:19.254 -- for an external fault. Once one of the see T starts to saturate
00:04:23.752 -- it will dive deeper into saturation, right? So my
00:04:26.866 -- question is how will that?
00:04:29.160 -- To how will that city begin to saturate? Like because?
00:04:33.660 -- The currents are all balanced, right? I mean based on the
00:04:37.972 -- culture of slow, so part of it's too far into this into the
00:04:43.460 -- hand out so.
00:04:47.760 -- That's the internal fault. So for the external fault part of
00:04:51.148 -- it's going to be the case that.
00:04:54.690 -- We've got this one. This is 1 heck external fault, right? So
00:04:58.338 -- this is seeing the current from all of the other feeders or
00:05:01.986 -- other lines going through it, and so depending on what the
00:05:05.330 -- burden is for this one.
00:05:07.800 -- Oh that 'cause there's going to be?
00:05:12.460 -- The relay and the and some of the winding resistance is going
00:05:16.084 -- to be dominant. Burden that affect saturation in this one in
00:05:19.406 -- a lot of ways.
00:05:21.210 -- So if this one, if there's a fault with a lot of DC offset,
00:05:25.088 -- especially then this one is going to start to saturate.
00:05:27.858 -- 'cause this is seeing the most current. I thought there is only
00:05:31.182 -- one button then that's the one at the end. Well, remember that
00:05:34.506 -- the burden and we look at ACT when we look at burden.
00:05:40.260 -- Mr Lead wire.
00:05:48.530 -- So the first thing we're going to have is the CT winding
00:05:51.338 -- resistance. And it's so. So in this case the Siti
00:05:54.649 -- winding resistance is going to be the most significant
00:05:57.088 -- one, because once we get to the terminals of the see T.
00:06:07.470 -- We're basically connecting each of the CTS.
00:06:12.750 -- In parallel on the secondary side, right and then once
00:06:16.320 -- they once we have this parallel combination, then
00:06:19.176 -- that's going. Then we have the rest of the lead wire.
00:06:25.200 -- And we have the relay out here.
00:06:30.540 -- But there's the secondary current on the secondary
00:06:33.404 -- winding, and the CT is still going to see.
00:06:37.340 -- All that current, right? The current when they sum
00:06:40.328 -- to 0 between.
00:06:44.320 -- We put in a third CT just to kind of.
00:06:49.030 -- Illustrate this a little bit more.
00:06:58.680 -- When I talk about connecting them together right, this is
00:07:01.940 -- where they sum to 0, right? So if it's if it's an
00:07:05.852 -- external fault.
00:07:12.350 -- So let's say that this is the one with.
00:07:18.870 -- The external fault, right? So that's going to have.
00:07:23.100 -- Let's say we have current going this way and this one. Each of
00:07:26.948 -- these are going to have their share feeding it right, so this
00:07:30.500 -- one is going to be the sum of this plus this and so at this
00:07:34.940 -- point here. They're going to sum
00:07:37.224 -- to 0. But this one, each one of these is going to have its own
00:07:42.072 -- fault current share the fault current, it's it's
00:07:44.628 -- carrying. It's going to go
00:07:46.048 -- through this resistance. And so basically what's going to
00:07:49.888 -- drive that start driving in this one in the saturation is
00:07:53.936 -- going to be a combination of the voltage drop across this
00:07:57.984 -- plus the ACE asymmetric current due to the DC offset.
00:08:03.130 -- Remember that as we talked about with on the BH
00:08:07.030 -- characteristic, the DC offset is shifting you in One
00:08:10.540 -- Direction and the BH characteristic.
00:08:17.450 -- And so when we look at this.
00:08:21.080 -- So under normal conditions.
00:08:23.650 -- It's going to be doing something like this, right? And
00:08:26.760 -- if we have a fault with no set without significant saturation?
00:08:31.480 -- It's going to be doing some like this, and so if we have well
00:08:36.324 -- size CTS we may only see behavior that looks like this.
00:08:40.730 -- But for a bus situation, sometimes it's hard to get
00:08:44.380 -- around that, but if we add.
00:08:47.510 -- The.
00:08:52.590 -- The DC offset.
00:08:54.830 -- I did not draw that very well, sorry. So we may start out with
00:09:00.248 -- something like this. Then the
00:09:02.183 -- next cycle. It's going to be working like this and it's going
00:09:06.480 -- to be following that DC offset, so it's going to push it into
00:09:10.302 -- saturation. Discuss. The flux loops are being pushed this way
00:09:13.242 -- by the DC offset.
00:09:15.870 -- And in some cases with a combination of the of a large
00:09:20.286 -- current and going through this resistance in a DC
00:09:23.598 -- offset, this one may start to go into saturation an.
00:09:29.390 -- Lessina cycle.
00:09:31.800 -- Possibly quite a bit less in the cycle.
00:09:35.790 -- And so that's why that's why even though you on the surface,
00:09:39.606 -- you would say that there shouldn't be much voltage across
00:09:42.786 -- this, because these current sum to zero and the voltage drop
00:09:46.284 -- across this should normally be negligible. But what's going to
00:09:49.464 -- happen is that the combination of that fault current going
00:09:52.644 -- through this winding resistance and the DC offset starts this
00:09:55.824 -- one into saturation. And then that mismatch current through.
00:10:00.130 -- That saturation goes through this, and because of that
00:10:03.577 -- compensating resistor that's going to drive this voltage up.
00:10:08.730 -- But because this is the one that's already starting to
00:10:12.290 -- saturate and has a lower impedance than it's, it's
00:10:15.494 -- going to tend to make this voltage collapse and keep
00:10:19.054 -- these from rising.
00:10:28.560 -- Like I said, it's not. That's a very good question. 'cause it's
00:10:32.280 -- there's a lot of things that are not intuitively obvious when we
00:10:36.000 -- look at the high impedance bus
00:10:37.860 -- differential. Because we're basically using something that's
00:10:42.662 -- inherently nonlinear to work.
00:10:57.580 -- Any other questions for my son?
00:11:06.790 -- OK, so then we're going to start on. Next, we're going to start
00:11:11.223 -- talking bout transformer protection and I talked to I did
00:11:14.633 -- a very quick introduction to some of the some of the issues
00:11:18.725 -- and the difference.
00:11:20.960 -- Things were gonna talk about.
00:11:21.870 -- We're going to talk about. Fall protection of the
00:11:25.190 -- transformer itself for faults inside the transformer.
00:11:29.680 -- And then we're also going to look at protecting the
00:11:32.850 -- transformer, firm external conditions, and
00:11:34.435 -- this can include faults external to the
00:11:36.654 -- transformer. Boy, the transformer is carrying
00:11:38.556 -- the fault currents that goes that go to it.
00:11:47.680 -- And then there are Transformers introduce a number of unique
00:11:51.640 -- challenges that we'll talk about as we go through this.
00:11:56.550 -- So in some ways it will start out looking at a concept similar
00:12:01.308 -- to what we looked at with the bus protection. So we're going
00:12:05.700 -- to a lot of the internal fault protection for Transformers.
00:12:09.360 -- Starts with the idea of restrained low impedance
00:12:12.288 -- differential element, so it's kind of build time. We start. I
00:12:16.314 -- started with the bus protection.
00:12:29.630 -- And so one of the things that the bear in mind as we talk
00:12:35.748 -- about transformer protection is when we talk about bus
00:12:39.681 -- protection. Fast protection has a bus fault or misoperation
00:12:43.614 -- where a bus gets tripped when it shouldn't can have very severe
00:12:48.858 -- operational. Consequences for our power system. So bus faults
00:12:52.556 -- are actually fairly rare.
00:12:54.760 -- Fat faults that cause were and the bigger concern is as
00:12:59.028 -- generally going to be external faults that caused the bus
00:13:02.908 -- protection to miss operate.
00:13:06.160 -- And so that's why the restrained differential element, the high
00:13:09.640 -- impedance differential element, have so there so much efforts
00:13:12.772 -- gone into developing and optimizing those at the relay
00:13:15.904 -- vendors is because they are very high consequences operationally
00:13:19.036 -- to the system in the short term.
00:13:24.130 -- Transformer failures, on the other hand.
00:13:44.050 -- Can have longer time consequences.
00:13:54.730 -- And that's because there are longer replacement times.
00:13:59.700 -- And in most cases, if an internal fault happens in a
00:14:04.560 -- transformer.
00:14:06.370 -- There is a good chance that it's going to evolve to the point
00:14:10.348 -- where it's not something that's very simply repaired. In some
00:14:13.408 -- cases there are still a number of cases where they're caught
00:14:16.774 -- fast enough, or it could be repaired simply, but if it gets
00:14:20.446 -- to severe faults and you'll have a fire in the transformer, then
00:14:24.118 -- it can be very severe.
00:14:27.950 -- And so there are a number of things. The number of strategies
00:14:32.750 -- that try to minimize the impact of transformer faults.
00:14:49.760 -- So one of the big ones is finding ways to reduce the
00:14:53.252 -- likelihood of them happening.
00:15:05.320 -- And so part of what a lot of this comes down to is.
00:15:10.900 -- Track external events.
00:15:31.620 -- And it's really the life of the installation. That's a
00:15:34.010 -- big issue.
00:15:35.620 -- So one of the things that I mentioned is that we have two
00:15:39.741 -- directions. We're gonna go to, and they actually are related to
00:15:43.228 -- each other. So one of the big things that is a has a
00:15:47.349 -- consequense for Transformers is.
00:16:06.620 -- Meeting of the installation will have a big impact on
00:16:09.700 -- how the life or how long that installation is going
00:16:12.780 -- to be good.
00:16:23.030 -- Transient overvoltages is another another issue.
00:16:44.770 -- So what are some of the things that are going to
00:16:47.168 -- cause a transformer? Cause heating in a transformer?
00:16:51.350 -- So let's think about a transformer for a second
00:16:53.690 -- we have.
00:16:56.970 -- So I'm just going to draw a single phase core.
00:17:01.570 -- So as we've talked about where we have a single phase core
00:17:05.410 -- and have the low voltage winding on the inside, an will
00:17:08.930 -- have a higher voltage winding wrapped around the outside of
00:17:12.130 -- it, right? And then we'll take those out to the bushings.
00:17:16.690 -- And as I mentioned earlier, we don't. You don't see a
00:17:20.397 -- transformer core just sitting out open in the air, right?
00:17:24.360 -- And so usually this is going to be.
00:17:32.140 -- In a tank.
00:17:37.710 -- Anna's tank is going to be.
00:17:45.730 -- Filled with oil, right? So usually it's going to be some
00:17:48.271 -- sort of a dielectric oil.
00:18:02.600 -- Is also used as a coolant.
00:18:08.650 -- And so you may look at a name plate for a transformer, an it
00:18:13.914 -- may say that you have a transformer that's rated at 15
00:18:18.050 -- MVA, 20 MVA.
00:18:20.300 -- 25 NBA
00:18:23.920 -- so why would why would there be 3 MVA ratings for the
00:18:27.712 -- same transformer?
00:18:34.120 -- Different cooling stages. It's different cooling stages, so
00:18:37.424 -- this is going to be.
00:18:40.720 -- Basically, entirely passive cooling.
00:18:45.100 -- So there is going to be there will be radiator fins or on the
00:18:49.510 -- side of this case on the side of
00:18:52.030 -- that tank. This is going to be.
00:19:03.070 -- Going to be pumps used to circulate oil to cool the
00:19:06.029 -- transformer or cool the oil so it's going to circulate because
00:19:08.988 -- there are going to be.
00:19:11.010 -- Different spots in the winding that are hot spots said certain
00:19:14.156 -- certain points are going to be
00:19:15.872 -- hotter than others. And so if you don't circulate the coolant,
00:19:19.424 -- there will be a little bit of natural convection, but you're
00:19:22.262 -- going to. Those hot spots are not going to be cooled as well.
00:19:26.580 -- And then this is going to be pumps.
00:19:31.090 -- Plus
00:19:32.920 -- running cooling fans that are blowing error basically across
00:19:36.997 -- the radiator so that the radiator works more efficiently.
00:19:45.810 -- So depending in some cases people will just run these
00:19:49.220 -- all the time. In some cases they'll based on the load
00:19:52.971 -- conditions, they'll start and stop this equipment.
00:19:56.890 -- And if you have a transformer that's always lightly loaded,
00:19:59.290 -- they may not. Run it as. Run to run them very much at all.
00:20:15.100 -- So other things that could cause heating.
00:20:23.270 -- So I want to be carrying harmonic currents.
00:20:38.460 -- Do you know external loads?
00:20:48.380 -- So for example, if we have a transformer that one of
00:20:52.153 -- the loads.
00:20:54.110 -- Is.
00:20:59.190 -- A dialed dialed rectifier.
00:21:04.740 -- And then we have a voltage source converter.
00:21:09.580 -- Anyway, have an induction motor.
00:21:17.060 -- If.
00:21:19.260 -- This doesn't have any compensation.
00:21:28.570 -- The current strong by this drive are going to look
00:21:30.800 -- something like this.
00:21:34.920 -- And so this is going to have 5, seven, 1113 and
00:21:39.463 -- basically multiples of 6 plus or minus one.
00:21:47.670 -- Is there going to have other loads here? But this
00:21:50.100 -- transformer is going to be carrying this current plus
00:21:52.287 -- whatever loads are here.
00:21:57.000 -- And carrying those harmonic currents increases Eddy current
00:22:00.808 -- losses in the transformer core.
00:22:04.480 -- And so that the transformer is going to run hotter.
00:22:23.280 -- And so they actually you can actually get.
00:22:27.470 -- K factor rated.
00:22:35.920 -- So basically these K factors are more of a derating factor.
00:22:41.170 -- And so if you have, if you know you're going to be supplying
00:22:45.642 -- harmonic loads, you can buy a transformer that has basically
00:22:49.082 -- an extra factor in its MVA rating to be able to deal with
00:22:53.554 -- harmonics. If you're not, if you don't have a transformer
00:22:58.028 -- that has any K rating an you start supplying harmonics,
00:23:01.848 -- then usually you can. There's there are formulas from the
00:23:05.668 -- IEEE standards that talked about how you derate the
00:23:09.106 -- transformer, so instead of being a 15 MVA transformer, it
00:23:12.926 -- may actually be a 12 MVA transformer due to the extra
00:23:17.128 -- heating from the harmonics.
00:23:19.820 -- And so when someone buys a transformer, usually you're.
00:23:24.470 -- Part of the data for when you sign the contract with the
00:23:28.019 -- supplier and stuff like that is saying well, this is. This has a
00:23:31.568 -- 30 year design life for this as a 25 year design life.
00:23:35.850 -- If you routinely overheat the transformer, you may take years
00:23:39.750 -- off of that life.
00:23:41.870 -- So we had an outreach student awhile back that worked at an
00:23:46.334 -- industrial facility that was basically with zinc smelter.
00:23:50.010 -- And so they had a lot of very large rectifier loads and so
00:23:54.924 -- they had Trent. They bought Transformers that had.
00:23:59.340 -- 30 year old designlife
00:24:01.880 -- Then they push them kind of right. It may be a slightly
00:24:07.520 -- beyond their NBA ratings.
00:24:10.240 -- And then they gave this heavy harmonic loading. So they
00:24:13.060 -- lasted about 10 years.
00:24:18.790 -- An that fit and when I say lasted about 10 years, they had
00:24:24.237 -- a fault, and so if I did so by heating the insulation, you end
00:24:30.103 -- up causing the you decrease the lifespan of the installation and
00:24:34.712 -- your moral an it's more likely to fail by having our fault. And
00:24:40.159 -- so that's why this external event, external condition stuff
00:24:44.349 -- matters from the from the transformer Protection POV.
00:24:52.670 -- So transformer protection will usually track the loading on a
00:24:56.900 -- transformer an if the transformer is overloaded, and
00:25:00.284 -- then there are formulas you can use to figure out how much
00:25:05.360 -- that's affected the life.
00:25:10.980 -- And so some other things that will go into this are going
00:25:13.776 -- to be over excitation.
00:25:23.720 -- So on a transformer over excitation basically means
00:25:26.400 -- a steady state.
00:25:34.200 -- However, voltage that means you're partially saturating.
00:25:57.850 -- Angene why the transformer is going to produce more
00:26:01.478 -- harmonics because of this? Because this is a steady state
00:26:05.438 -- sinusoidal condition, these will be only odd harmonics.
00:26:11.110 -- And often the 5th harmonic is usually going to be the one
00:26:14.674 -- that's used as sort of the main detection detector for that.
00:26:21.140 -- But again, because you're saturating the core.
00:26:26.070 -- What does that? What does it mean when you saturate
00:26:28.960 -- the core more deeply?
00:26:35.050 -- More excited, you have more expectations, well over
00:26:37.938 -- expectations. We have more expectation right? But what
00:26:40.826 -- losses go up?
00:26:44.480 -- The winding losses go up, or so we're going to increase
00:26:50.200 -- hysteresis losses.
00:26:54.180 -- Remember, hysteresis losses are basically proportional to
00:26:56.672 -- the area of the hysteresis loop it follows, so if you're
00:27:00.588 -- over exciting the transformer, your loop has a bigger bigger
00:27:04.148 -- area, so the losses are going to be higher.
00:27:24.780 -- Another one that's a big factor are through faults, which means
00:27:29.345 -- that the transformer.
00:27:52.160 -- So basically, one of the things that also gets tracked is how
00:27:56.120 -- many, how many faults is this transformer supplied? What is
00:27:59.420 -- the magnitude of the fault
00:28:01.070 -- current bin? Because. Oh through fault can cause very substantial
00:28:05.092 -- heating. It may not. It's not going to last very long, but
00:28:08.764 -- it's going to take a long time. It's going to take awhile quite
00:28:12.742 -- awhile for the transformer to cool down from that.
00:28:37.870 -- So even frequent large motor starting or if the transformer
00:28:42.020 -- is supplying current to energize other Transformers.
00:28:48.500 -- So for example when.
00:28:53.020 -- I think their procedures have changed a little bit, but at
00:28:56.771 -- Grand Coulee there's a pumped hydro storage facility that
00:28:59.840 -- has very large synchronous Motors. They generally only
00:29:02.568 -- start those Motors once a day because the thermal shock on
00:29:06.319 -- the Motors every time they start them is so much that
00:29:10.070 -- they can't start them more often.
00:29:14.040 -- They redid that facility.
00:29:17.610 -- And within the last.
00:29:20.130 -- Eight years, so I think they've redone it, so
00:29:23.019 -- it's not quite as harsh.
00:29:26.260 -- But so basically all of these things get tracked.
00:29:45.910 -- They predict lifespan loss and we're going to. We're going to
00:29:48.814 -- come back and talk about the over some of these issues and
00:29:51.982 -- how and how this factors into the transformer protection later
00:29:54.622 -- in the course. I want to talk about internal faults. First,
00:29:57.526 -- we're going to come back to
00:29:59.110 -- this. That a good resource for this. Our textbook does a pretty
00:30:03.945 -- good job with this, but also the IEEE 30 C 3791.
00:30:08.770 -- Also another good one for this and or there's some
00:30:11.730 -- other references. We'll talk about a little bit later.
00:30:21.310 -- And So what I want to start talking about is now protection.
00:30:27.370 -- For internal faults.
00:30:33.170 -- And will be going through this over the next couple
00:30:35.320 -- of lectures.
00:30:45.020 -- And so I guess that's one other sort of structural
00:30:47.990 -- thing. When we look at.
00:30:51.370 -- Large Transformers again.
00:31:13.460 -- I felt it evolved to the point where there's
00:31:15.485 -- a fire can cause long.
00:31:19.490 -- As I said, long repair times.
00:31:23.550 -- And so some of the things that you'll see in a substation, for
00:31:29.205 -- example for large transfer transmission substations
00:31:31.815 -- especially often you'll see single phase Transformers used,
00:31:35.295 -- and so you'll see.
00:31:40.310 -- Three single phase units, and actually they are often going
00:31:43.810 -- to be 3 winding Transformers as we talked about earlier in
00:31:47.660 -- the semester.
00:31:49.800 -- And so they're going to have their own individual tanks.
00:32:00.700 -- And when you look at the substation.
00:32:04.490 -- You'll see a wall that's been placed.
00:32:09.980 -- Between the Transformers.
00:32:13.120 -- So what's the purpose of that wall?
00:32:16.130 -- Prevent fire from cleaning, so these are.
00:32:20.530 -- Firewalls raise more of the archaic usage of the term
00:32:24.070 -- instead of the one that's now everyone uses when they talk
00:32:27.964 -- about software.
00:32:29.990 -- And so this is basically if this one has a fault, and as
00:32:34.072 -- a fire, the idea is that this is that this is going
00:32:37.840 -- to basically make it less likely for any for the heat
00:32:41.294 -- in the flames to get to this transformer, so it fails to.
00:32:49.730 -- And a lot of utilities will
00:32:52.352 -- have. A limited number of spare Transformers that they
00:32:56.392 -- can put in to replace a failed transformer.
00:33:00.350 -- So.
00:33:03.620 -- This was probably almost 15 years ago. Now there was a
00:33:08.328 -- transformer fault at a 500 kva. Think it's a 500KV substation in
00:33:13.464 -- the Southwest. An they did not have.
00:33:18.530 -- Firewalls between the single phase transformer, so they lost
00:33:22.364 -- all three phases. They had their spares close enough that it
00:33:27.050 -- actually scorched the paint off of the tanks, but they actually
00:33:31.736 -- didn't lose the spares.
00:33:36.820 -- But because they lost all three and they only had three spares,
00:33:41.392 -- then they had to scramble to try to get spares from other people.
00:33:46.345 -- And I know that one of the utilities in the northwest
00:33:50.536 -- sentence pairs and they had all sorts of issues because these
00:33:54.727 -- were 500 kva Transformers, Oran, high MVA ratings. Just
00:33:58.156 -- transporting them was difficult.
00:34:04.410 -- And I think even transporting the spares
00:34:06.867 -- took like several months.
00:34:17.190 -- So then actually one of the things that the.
00:34:21.070 -- US Department of Energy in the Department of Homeland
00:34:24.814 -- Security been working on in the last several years, is
00:34:28.974 -- basically trying to form a kind of a national database
00:34:33.134 -- of transformer spares and also trying to increase the
00:34:36.878 -- inventory of spares so that if there is something like.
00:34:42.950 -- High energy electromagnetic pulse from a nuclear weapon or a
00:34:47.550 -- major Geo Geo magnetic.
00:34:50.070 -- A disturbance for the gym geomagnetically induced currents
00:34:53.454 -- caused transformer failures that they've got something that they
00:34:57.261 -- can go to restore power in some
00:35:00.222 -- areas quickly. Relatively quickly.
00:35:05.270 -- OK, so let's now start talking a little bit more about the
00:35:08.798 -- Internal fault protection.
00:35:16.340 -- Really, the first line for this is going to be
00:35:19.390 -- differential protection.
00:35:27.600 -- So as I said, much like what we were just talking
00:35:31.285 -- about with the.
00:35:33.640 -- Boss protection for the restrained low impedance
00:35:38.078 -- differential protection.
00:35:42.100 -- So let's start out looking at a transformer that.
00:35:47.170 -- We have a YY connection.
00:35:51.350 -- And so, let's say it's.
00:35:55.330 -- 3:45 KV. 2.
00:36:00.110 -- 138 KV.
00:36:07.700 -- And so for the moment, let's just say it's a.
00:36:11.870 -- 2 winding Transformers. So we're going to have
00:36:14.102 -- three leads coming out.
00:36:34.210 -- Now I have see T is on each phase and will just look at one
00:36:38.110 -- phase for the moment.
00:36:47.100 -- And so we start out saying, OK, well, this looks a lot
00:36:50.436 -- like what we talked about when we anytime we talked
00:36:53.216 -- about differential protection. So we're going to
00:36:55.162 -- have current if we have current going this way.
00:37:02.630 -- Then we're going to have.
00:37:06.340 -- Secondary current. That's going to circulate like this, and.
00:37:12.870 -- I op should be about 0, right? That would be. That's
00:37:17.666 -- what we would expect.
00:37:23.200 -- Now, unlike the virus protection, we've got a number
00:37:27.574 -- of factors that complicate this.
00:37:44.360 -- So what do you think? Some of the complicating factors
00:37:46.660 -- might be?
00:37:49.540 -- Configuration. Well, let's say they will stick with the
00:37:53.020 -- YY for the moment.
00:37:56.010 -- If it's why Delta that, that will add, that will be the next
00:37:59.195 -- challenge, will talk about after we finish this one.
00:38:03.450 -- CD accuracy. Find CD accuracy.
00:38:07.640 -- So ciety accuracy, but there's actually something
00:38:09.831 -- before that. One is going to be the CT ratios.
00:38:37.200 -- So we may not get apart. We may not get a perfect
00:38:40.284 -- cancellation of.
00:38:42.410 -- So let's say that just for making this easier, let's say
00:38:46.172 -- that this was a 2 to one ratio.
00:38:54.770 -- So let's say that this was 500KV and this was 250KV just
00:38:58.598 -- for nice numbers. Even though the 2:50 is not something
00:39:01.788 -- you'd run across much.
00:39:04.660 -- Then we would say OK. Well then this. Let's say that
00:39:07.608 -- this is 1000 to one CT and this is going to be what?
00:39:15.290 -- Or 1000 to 5C T, and that's what would this
00:39:17.760 -- would need to be then.
00:39:24.010 -- Remember, this is.
00:39:26.200 -- Two to one is the effective voltage transformation
00:39:28.680 -- ratio, so the current goes the opposite, right?
00:39:32.170 -- So so this one would need to have 500 to 5 setes.
00:39:37.110 -- So that would be one that would be an example of a
00:39:39.894 -- good cancellation. So let's say that this was.
00:39:44.450 -- 500KV to 250KV.
00:39:50.810 -- And the cities were.
00:39:53.330 -- 1000 to 5
00:39:56.690 -- in. 500 to 5 so that's something that you could pretty easily.
00:40:00.320 -- Fine cities.
00:40:03.260 -- To cancel that right?
00:40:06.340 -- If we look at 3:45 to 138.
00:40:13.080 -- That's not going to be so easy to find CTS that give
00:40:16.572 -- you a good cancellation on that. So even if this was
00:40:19.773 -- even if these were still.
00:40:22.920 -- Thousands of five.
00:40:27.930 -- This would need to be basically 1000 times.
00:40:33.640 -- 38 / 345.
00:40:37.240 -- To five.
00:40:43.830 -- And chances are that's not going to be a nice stock
00:40:47.108 -- number that you're going to be able to buy in. SNS ET.
00:40:56.510 -- And so it's one that we're we'll talk about a solution for that,
00:41:01.424 -- but this is basically going to
00:41:03.692 -- be. Having
00:41:06.760 -- taps on the relay.
00:41:10.600 -- So watch mechanical relays. What they had was they had multiple
00:41:13.625 -- tap points where you could
00:41:15.000 -- connect. The inputs from the transformer for the differential
00:41:19.010 -- and you could partly correct for that mismatch to a degree you
00:41:23.690 -- couldn't. You could not connect 4 correct for it perfectly, but
00:41:27.980 -- you could. You could go a long ways towards correcting it.
00:41:33.160 -- What we'll see in probably not today. We may. I don't know if
00:41:37.697 -- we get to the example today, what you'll see in
00:41:41.187 -- microprocessor relays now that's just a number, so it's just a
00:41:45.026 -- scaling factor, so you can. So basically you as you enter the
00:41:49.214 -- stuff into the relay for setting it, you're entering the
00:41:52.704 -- information so the relay calculates that tap and you
00:41:55.845 -- don't even have to answer. Calculate it yourself so you say
00:41:59.684 -- OK, here is the MVA rating. Here's the voltage rating.
00:42:03.680 -- And then at the relay says OK and this is the rated
00:42:06.980 -- current and just basically calculates it for you.
00:42:11.830 -- And then you also put the seat. The actual CT ratios 'cause it
00:42:15.444 -- puts that in as a correction to.
00:42:27.670 -- Another thing you'll see in a lot of large power Transformers
00:42:31.080 -- is they have taps, right?
00:42:34.410 -- So we may see.
00:42:37.830 -- 500KV to 250KV.
00:42:42.510 -- Anne, this could be we could
00:42:46.392 -- have. Plus 2 1/2% + 5%
00:42:58.620 -- And these could also have some different apps. So if
00:43:01.930 -- you start putting.
00:43:04.060 -- If you and so in some cases, these maybe.
00:43:08.340 -- For lower power ones, these may be on load. Tap
00:43:11.695 -- changing Transformers where they can be changed. In other cases
00:43:14.745 -- the transformer has to be D energized for crew to come in
00:43:18.405 -- and change that tag.
00:43:22.870 -- What what is that tap change due to the differential current?
00:43:32.540 -- You just change the ratio of the transformer, right? So you've
00:43:36.984 -- gone to the effort of correcting for compensating for this, this
00:43:41.428 -- ratio and the CT ratios. Now you just threw that off because you
00:43:46.680 -- changed the transfer. The power transformation ratio by 2 1/2%.
00:43:59.070 -- Then another one would be.
00:44:25.660 -- The transformer is always going to draw some magnetizing current
00:44:28.480 -- if it's energized right.
00:44:32.250 -- And this is something that's.
00:44:34.160 -- Going into the transformer and not coming out.
00:44:44.190 -- And as we talked about last time, this might be 2 to 4%,
00:44:48.948 -- maybe 5% of the rated current.
00:45:07.230 -- It will be higher if the transformer is over excited.
00:45:13.210 -- So there's really two things that you need to look at with
00:45:16.414 -- over. Excitation is going to be.
00:45:18.830 -- If the over excitation is severe enough and last long enough you
00:45:23.114 -- want to trip the transformer.
00:45:25.910 -- But you don't want to trip it because you think it's an
00:45:29.414 -- internal fault, so you don't want to trip at the instant it
00:45:32.918 -- happens. So there's some tradeoffs on that, and the
00:45:36.460 -- harmonic content of that's going to be a factor in how
00:45:39.595 -- the relay responds to it.
00:45:44.120 -- Now there's another issue that you have to worry about
00:45:46.480 -- with magnetizing current.
00:45:51.280 -- What would that be?
00:45:58.370 -- So we have magnetizing inrush current.
00:46:09.250 -- So if you energize a transformer.
00:46:23.570 -- You're going to see a current that's going to
00:46:25.568 -- start out looking like this.
00:46:28.260 -- And it may take a second or two
00:46:31.508 -- to. One at one to two seconds to get down to the normal
00:46:36.314 -- magnetizing current.
00:46:40.990 -- So are people familiar? Why Transformers exhibit
00:46:43.867 -- this behavior?
00:46:52.120 -- So it goes down, it goes back to our hysteresis characteristic.
00:46:57.400 -- So the transformer is going to when it's operating is
00:47:00.650 -- going to be.
00:47:03.580 -- Following something that looks like this, right? So if this is
00:47:07.507 -- B versus H.
00:47:10.670 -- This is proportional to voltage. This is proportional to current.
00:47:15.790 -- So every time you go through a sinusoidal cycle, it's going to
00:47:18.982 -- trace this curve, right?
00:47:22.010 -- And so when you deenergize the transformer, you deenergize
00:47:26.042 -- nearer at a current 0, right?
00:47:29.630 -- And so when the current goes to zero, you're going to be
00:47:32.654 -- somewhere up here. And so there's going to be some trapped
00:47:36.706 -- flux on the core.
00:47:38.830 -- When it's deenergized and depending on where you were in
00:47:42.030 -- that hysteresis cycle, when the breaker contact cleared or what
00:47:45.230 -- the power factor of the current
00:47:47.150 -- was. Usually the final invoice and normal routine operation
00:47:51.948 -- when I want to Transformers.
00:47:54.940 -- D energize you open one side, then you open the other ones
00:47:59.476 -- you're interrupting, basically just magnetizing current with
00:48:02.122 -- the final. The energizing of the transformer.
00:48:06.540 -- When you re energize it.
00:48:09.140 -- How is voltage related to flux in a transformer?
00:48:13.830 -- So V is equal to NDF DT, right? So the flux in the voltage or 90
00:48:19.014 -- degrees out of phase with each other. But you can so that the
00:48:23.226 -- voltage here at some point in a sinusoidal voltage waveform you
00:48:26.790 -- can map that the flux when you energize it. So when you're when
00:48:31.002 -- you close a circuit breaker, there's going to be some
00:48:34.242 -- basically effective flux that you're you're trying to impose
00:48:37.158 -- on that core. So if you're lucky and you and you pose a circuit
00:48:41.694 -- breaker in the effective flux for the point on waiver, you're
00:48:45.258 -- closing. It's about what you trapped on the core.
00:48:48.680 -- Then there's not really going to draw any current.
00:48:53.430 -- If you're unlucky and you had trap works up here and you're
00:48:56.562 -- closed when you're somewhere down like this, now the
00:48:58.911 -- transformer is going to draw a lot of current to try to
00:49:02.043 -- equalize that flux. And after magnetizing inrush current.
00:49:06.320 -- And it's very nonlinear current.
00:49:09.000 -- And so this has a lot of harmonic content. The
00:49:12.210 -- generally it's going to be dominated by second and
00:49:15.099 -- then 5th and so on. But it's going to have more
00:49:18.630 -- even harmonics where the over excitation is only
00:49:21.198 -- going to be odd.
00:49:25.610 -- How's the modern steels that they're using in newer
00:49:29.615 -- Transformers? Do not have a sharper second harmonic
00:49:32.839 -- characteristic. They still draw big magnetizing currents, but
00:49:35.135 -- now there's not as clear a second harmonic, and we'll talk
00:49:38.292 -- about some of the issues with that later in the.
00:49:42.650 -- Not this, not later today, but next week or
00:49:45.570 -- the week after next.
00:49:49.040 -- So you've got these very large currents again, they're just
00:49:51.940 -- going into the transformer.
00:49:57.860 -- And so you know, if you're doing
00:49:59.764 -- a normal. Registration of the transformer. Not something
00:50:02.364 -- following like Re closing in a fault. You might have this side
00:50:06.312 -- open and you energize this side and so now you're seeing current
00:50:10.260 -- San people have measured currents as high as 15 per unit.
00:50:16.260 -- If there are a lot of lights, limits that is partly whether
00:50:19.596 -- the surrounding power system can supply that much current.
00:50:22.098 -- If there's too much impedance in the power system that won't
00:50:25.156 -- supply it.
00:50:28.120 -- And so you're doing. You have a differential element. You're
00:50:31.140 -- going to see. Let's say it's something more normal, like 5 to
00:50:34.764 -- 7 per unit for a second.
00:50:37.980 -- So in electromechanical relays.
00:50:41.570 -- One of the things that they did initially was basically turn off
00:50:46.274 -- the differential element until the inrush current period was
00:50:49.802 -- over. They still had issues where if you had two
00:50:53.199 -- Transformers that were close together and you energized one
00:50:55.638 -- when the other one was on, sometimes you had a sympathetic
00:50:58.619 -- trip of the different of the differential element for the one
00:51:01.600 -- that was already energized.
00:51:07.180 -- Professor, I have a question on this one, so
00:51:09.952 -- there is no saturation really, it's just the.
00:51:13.740 -- The core trying to reach that
00:51:15.876 -- flux level. But there's no saturation, so as.
00:51:21.150 -- It face it, it started has sort of a saturation effect because
00:51:24.966 -- of where it pushes the flux, but there really isn't any true
00:51:28.782 -- saturation of the core in this.
00:51:31.560 -- So why isn't it sinusoidal?
00:51:35.990 -- So when you think about the iron in the core right, you
00:51:40.423 -- basically have a bunch of magnetic domains that want to be
00:51:44.174 -- in random directions, right? So let's say that because of the
00:51:47.925 -- trap flux, they're all pointing
00:51:49.630 -- this direction. And for the inrush you're trying to flip
00:51:53.712 -- them all to go back. Basically you want the flux to go this
00:51:58.249 -- way, so you need to flip all
00:52:00.692 -- these domains. And.
00:52:03.920 -- They don't, simply.
00:52:06.420 -- Follow a nice thing in sinusoidal behavior as they flip
00:52:09.250 -- on this. So there's some resistance. I'm really
00:52:12.652 -- oversimplifying this, but basically it's it's a
00:52:15.186 -- magnetic. The nonlinear magnetic behavior of the core
00:52:18.082 -- that keeps it from looking sinusoidal.
00:52:25.980 -- And this harmonic, and So what we're going to see in a little
00:52:30.426 -- bit, is that to try to minimize
00:52:32.820 -- this effect. The second harmonic is often used as a
00:52:37.252 -- as a signature, so if the second harmonics above a
00:52:40.942 -- certain threshold.
00:52:43.030 -- Then it's got the relay will block the differential
00:52:46.189 -- element, so you can either do harmonic blocking or harmonic
00:52:49.699 -- restraint, which is basically making the slope steeper.
00:52:53.590 -- Now, this raises an interesting thing. From a relay point of
00:52:57.572 -- view. We talked about digital filters, right? So here we
00:53:01.192 -- talked about second harmonic. I talked about fifth Harmonic when
00:53:04.812 -- I talked about over excitation detecting over excitation.
00:53:09.520 -- So remember what we talked about with digital filters? If
00:53:12.270 -- we're using cosine filters.
00:53:14.730 -- Well, the is the what is a cosine filter due to harmonics.
00:53:19.866 -- What's the gain about cosine filter 0, right? So the relay
00:53:24.574 -- needs a separate.
00:53:26.820 -- Cosign filter that if you want to measure second harmonic or
00:53:30.582 -- you want to measure 5th harmonic or any of the others, you need
00:53:35.028 -- to have some separate filter elements that are going
00:53:38.448 -- to calculate those.
00:53:40.160 -- Because the normal cosine filter using for your protection
00:53:43.400 -- calculations is going to have a gain of zero and block those.
00:53:49.450 -- And when you start getting up to 5th or 7th, now you're
00:53:52.450 -- starting to get up to the range where the low pass filters,
00:53:55.450 -- anti aliasing filters also going to have an effect on
00:53:57.950 -- them.
00:54:03.460 -- So when you talk about residual magnetism, why doesn't it die
00:54:07.387 -- out? So if I'm.
00:54:09.370 -- I'm switching off or closing opening the breaker in front of
00:54:13.286 -- the transformer at equals to zero. Eventually the residual
00:54:16.490 -- magnetism should die out, right? If I'm not energizing it back in
00:54:20.762 -- let's say days or weeks. So does it die out and not? It does
00:54:25.746 -- decay OK, so basically it's a it's a thermal process. So
00:54:29.662 -- basically these are going to try to randomize if the car is warm
00:54:34.290 -- when you demagnetize it, then they tend to randomize faster
00:54:37.850 -- than if the core is cool as the core as a transformer cools that
00:54:42.834 -- slows down the rate.
00:54:44.460 -- That randomization OK, but even if it's gone to zero an you
00:54:48.900 -- closing your somewhere up here still we're going to have
00:54:52.970 -- some issues on that.
00:54:57.930 -- Awhile back, well actually one of the Masters students here who
00:55:01.989 -- works at Sweitzer. Now guy named Doug Taylor looked at using a DC
00:55:06.786 -- source to preflex the transformer so you could put
00:55:10.476 -- the trap flux at a known at a known point and then if you have
00:55:16.011 -- Breakers with individual phase control then you can control
00:55:19.332 -- when you close them.
00:55:22.220 -- They also are using variations of that an like.
00:55:28.760 -- There's been a lot of stuff looking at that in Europe, for
00:55:32.324 -- example, in some of the offshore wind farms where they basically
00:55:35.591 -- are in a system that can't supply that magnetizing current
00:55:38.561 -- to magnetize the core, because there isn't a source strong
00:55:41.531 -- enough to provide it out there.
00:55:44.090 -- And so they want to be able to close the Transformers with no
00:55:48.276 -- inrush. And so rather than pre flexing the cores, they're
00:55:52.692 -- looking at trying trying to dissipate the flux in the
00:55:56.960 -- core so that they can bring it to zero, and then they do
00:56:02.004 -- individual phase control on the Breakers to minimize the inrush.
00:56:07.670 -- Also the whole pre fluxing minimize trying to get the
00:56:10.730 -- known side of inrush makes a big difference. If you have a
00:56:14.402 -- five legged core versus the three legged core.
00:56:18.190 -- So when you see the anticipated, basically they figure out at
00:56:21.644 -- what time or what voltage at what point in the voltage the
00:56:25.412 -- breaker was opened, and then based on that they calculate the
00:56:28.866 -- residual magnetism and the decay, and then they open
00:56:31.692 -- individual phases at different times. Or they close them, they
00:56:34.832 -- close them at specific times. OK, so the Breakers are always
00:56:38.286 -- going to try to open it. A natural current 0. Sure, an
00:56:42.054 -- there are actually some big problems if you don't open it in
00:56:45.822 -- natural current 0, because then you can get very big.
00:56:49.350 -- Transient response if you do a current shopping.
00:56:53.590 -- So the parasitic capacitance of the winding will interact
00:56:56.560 -- with the magnetizing branch, and you can see like 2 / 2
00:57:00.520 -- per unit voltage.
00:57:03.600 -- Even if you're chopped like half an amp.
00:57:11.700 -- That's a topic more for you. See 524 though.
00:57:20.290 -- OK, so any other questions related to the magnetizing.
00:57:24.950 -- Current behavior.
00:57:27.630 -- So these are all things that need to be accounted for in
00:57:31.758 -- creating the differential element an in setting like
00:57:34.510 -- the slope and the minimum operate current.
00:57:39.090 -- The other one to look at is going to be the transformer
00:57:42.342 -- phase shift.
00:57:49.760 -- So I started out drawing a YY transformer.
00:58:00.000 -- So the other thing we have to look at is Delta Y.
00:58:04.150 -- Or why Delta Transformers?
00:58:22.310 -- And so in North America there's an ANSI IEEE standard so that
00:58:27.926 -- the phase shift is generally very predictable, right?
00:58:33.330 -- And what's the standard?
00:58:37.720 -- Sorry. The high side is leading by $30.
00:58:59.020 -- So V line the neutral in the high voltage side leads
00:59:01.902 -- vilanda neutral in the low voltage side by 30 degrees.
00:59:06.370 -- The Power systems textbook I used when I was an undergrad
00:59:10.055 -- gave the impression that whenever you had a Y Delta
00:59:13.405 -- transformer or the Y side always led the Delta side by 30 degrees
00:59:17.760 -- because the author in.
00:59:20.620 -- All the cases he had run across the Y side was always
00:59:24.328 -- a high voltage transformer, 'cause he'd always worked in
00:59:27.109 -- transmission and never worked in distribution.
00:59:38.430 -- And so. So one of the effects were going to have
00:59:42.274 -- obviously is the 30 degree phase shift this also.
00:59:58.400 -- The Delta Y connection also
00:59:59.820 -- impacts the. Turns ratios right. So now you've got this other
01:00:03.574 -- sqrt 3 that gets put in there in addition to having.
01:00:11.110 -- The voltage transformation ratio.
01:00:14.910 -- That sqrt 3 shows up in the current so that reflects
01:00:18.320 -- back to the CTS.
01:00:23.620 -- And let's say that we have a Delta Y grounded transformer.
01:00:28.640 -- So this side.
01:00:41.830 -- When we're measuring the phase currents, there's going to be 0
01:00:45.537 -- sequence current on this side, but there won't be on this side.
01:00:53.200 -- And so even some Even so, one of the things that you have to be
01:00:57.550 -- careful of his solutions to try to fix this phase shift.
01:01:01.490 -- And fix this also after account for this. So I said that they
01:01:05.871 -- are one of the solutions that people did for less mechanical
01:01:09.578 -- relays. Had to have an extra step added to it because of
01:01:14.094 -- the zero sequence kind.
01:01:26.250 -- So if we have a transformer.
01:01:47.130 -- So we can look at the CTS.
01:01:51.140 -- So for electromechanical relays.
01:02:00.880 -- The common solution in this for this was going to.
01:02:06.860 -- To use the CT connections to help cancel for the cancel this.
01:02:12.520 -- And so.
01:02:16.250 -- So one option.
01:02:31.830 -- Would be to connect the CTS on the Y grounded side in Delta.
01:02:38.340 -- And the CTS and the Delta side and Y.
01:02:57.440 -- You need to make sure you connect the Delta properly to
01:03:01.092 -- cancel the shift. But So what that means is that the that the.
01:03:07.110 -- Phase currents that the Delta phase currents.
01:03:12.580 -- Well, include the zero sequence current that's going
01:03:15.148 -- to circulate in that Delta, but then the line currents
01:03:18.358 -- coming off the Delta which go to the differential relay
01:03:21.568 -- will not have.
01:03:23.640 -- That current
01:03:29.600 -- morning your device is running low on memory.
01:03:37.470 -- So one of my colleagues has a sledgehammer. He brings the
01:03:40.737 -- class for people whose cell phones make noise during class.
01:03:47.100 -- The new phone is trying to shut it down.
01:03:52.290 -- And so this is so, you still will run across substations that
01:03:57.018 -- have the CTS wired this way from the electromechanical relays.
01:04:03.920 -- And then a second option.
01:04:16.440 -- Would be the connect.
01:04:18.660 -- This it is an Y and this it isn't Delta.
01:04:24.240 -- So yes, there's a problem with this one, right?
01:04:30.810 -- So now the.
01:04:35.270 -- The differential element on this, the current that goes to
01:04:38.140 -- the differential an element from this side, it's going to include
01:04:41.297 -- zero sequence current. The one in this one won't, right.
01:04:45.970 -- So this one is going to need.
01:04:55.020 -- So basically this one needed an auxiliary set of current
01:04:58.210 -- Transformers that would block the zero sequence current by
01:05:01.081 -- basically circulating it in the auxiliary Transformers and
01:05:03.633 -- not have a go to the differential element.
01:05:26.570 -- So now if you go to a substation where it's new
01:05:31.553 -- construction and it's designed not anticipating
01:05:34.271 -- that there's going to be microprocessor relays
01:05:37.442 -- protecting this.
01:05:47.680 -- Now the seats are going to be why on both sides and there
01:05:51.632 -- will be a ground reference in the seat path.
01:06:19.860 -- And it will also the CTA will basically perform calculations.
01:06:24.340 -- To compensate for the phase shift an it's going to
01:06:28.920 -- perform another calculation to remove I 0.
01:06:35.480 -- And these are actually going to be matrix multiplications.
01:06:48.380 -- So I have a handout that.
01:06:51.350 -- Maybe I will pass it out today. You need to
01:06:53.950 -- remember to bring it.
01:06:57.250 -- Don't be sorry.
01:07:13.600 -- And so.
01:07:18.360 -- This first calculation is basically.
01:07:23.660 -- Typical calculation that you would see.
01:07:27.220 -- Done in the relay.
01:07:29.840 -- For the.
01:07:32.120 -- As an intermediate step for going to the
01:07:35.264 -- differential element.
01:07:37.540 -- So you're gonna have.
01:07:40.570 -- You're going to have the primary currents. Then they're going to
01:07:44.112 -- be divided by the current
01:07:45.722 -- transform transformation ratio. Remember, these are
01:07:48.686 -- why connected.
01:07:54.000 -- And then there's also going to be this tap calculation, and
01:07:57.744 -- the other hand out goes into more detail about the how this
01:08:01.488 -- tap is calculated. And then there's going to be a correction
01:08:06.020 -- matrix, so the correction matrix the output is going to be the
01:08:09.920 -- secondary current with the phase and zero sequence correction.
01:08:16.110 -- And so the current from both windings are going to. So this
01:08:20.190 -- is actually. This would be the primary side, and then we're
01:08:23.930 -- going to secondary sidewinding. So this is actually.
01:08:27.470 -- The power transformer primary.
01:08:53.070 -- And then the correction matrix, or a number of correction matrix
01:08:58.240 -- we can do. And so when I say matrix zero, that is using the
01:09:04.820 -- IC Clock terminology. So if we think about o'clock, we're going
01:09:09.990 -- to have 12369, etc and then 12.
01:09:13.890 -- 12 is also equal to 0, right?
01:09:19.820 -- And so if we have a Y connection with, if you say
01:09:24.212 -- that we have basically our phase, a voltage is going to
01:09:28.238 -- be here at an angle of 90 degrees. That's our zero
01:09:32.264 -- position.
01:09:37.340 -- And so the Matrix Zero is assuming we have a Y
01:09:40.783 -- connection and we're not trying to do any reversal of
01:09:43.913 -- the voltages, so this will be just the identity matrix.
01:09:53.370 -- And then where matrix one is the one o'clock position and this is
01:09:59.129 -- one that in.
01:10:00.540 -- South America is often referred to as the DAB and
01:10:03.490 -- this would be a Delta.
01:10:08.250 -- AV connection so that means that the first winding of the
01:10:11.583 -- Delta is connected from A to B. The second line will be to
01:10:15.522 -- see the third one will be see to a. This gives you remember
01:10:19.461 -- North America. You're limited to either plus 30 degrees or
01:10:22.491 -- minus 30 degrees when you're going from Y to Delta. So all
01:10:26.127 -- we care about in North America is going to be the D1
01:10:29.763 -- in the D11 connection.
01:10:33.240 -- And then we have the D11 connection, and so if we
01:10:37.398 -- compare these all it's doing is exchanging
01:10:40.044 -- which rows are have the.
01:10:43.410 -- Then have the different column combinations.
01:10:47.840 -- And so, well, we'll talk about this a little bit more, applying
01:10:52.172 -- it in the other example.
01:10:54.970 -- And then, as I mentioned, we have that we need that zero
01:10:58.054 -- sequence removal matrix too.
01:11:03.900 -- And so that's what this one does.
01:11:08.460 -- And so this is mathematically reproducing
01:11:10.410 -- the effect of the current circulating in the Delta.
01:11:20.750 -- Anworth this what this is coming from?
01:11:24.700 -- A very good reference for summarizing this is.
01:11:31.420 -- A paper that was written by.
01:11:35.230 -- I group from Basler Electric John Horack.
01:11:37.659 -- Actually, I have a link to on their class links web
01:11:41.476 -- page. I have a link to webpage it he's got put
01:11:45.293 -- together an extensive web page was protective
01:11:47.722 -- relaying. Related links.
01:11:51.260 -- And so I did not. I gave you copy. It's, uh, some of the
01:11:54.676 -- pages from this paper. I have links to the whole paper on
01:11:57.604 -- the course web page. That's the on campus students. There
01:12:00.044 -- were some of the pages that I'm going to talk to talk
01:12:02.972 -- about today and next time.
01:12:06.810 -- So this is just showing sort of the connection information
01:12:10.070 -- as a reference for the rest of this paper.
01:12:16.410 -- So.
01:12:18.450 -- He has uppercase letters to indicate the primary lowercase
01:12:22.635 -- to do the secondary.
01:12:25.520 -- And then he has the third of the terminal ends an the.
01:12:31.060 -- So this would be the polarity end of the wine,
01:12:33.656 -- and this is the nonpolarity end of the winding.
01:12:40.150 -- And so. This is one of the things that you go through.
01:12:45.030 -- You're going to find different people in different places, use
01:12:48.580 -- somewhat different notation so we see UV WABC.
01:12:52.070 -- And so on.
01:12:59.290 -- And so if we wanted to build a YY transformer in a typical
01:13:05.166 -- North American connection so when we see the W1W 2W3, those
01:13:10.138 -- are referring to the winding.
01:13:14.330 -- The windings of the six windings that produced the
01:13:17.372 -- three phase transformer.
01:13:21.590 -- And then it's not very obvious, but these are his
01:13:24.910 -- polarity marks for those windings.
01:13:28.910 -- And so H1X1 this is high voltage. This is
01:13:31.664 -- low voltage and so on.
01:13:34.330 -- And so mapping these this is how they would map.
01:13:39.870 -- Tell the two winding sets.
01:13:47.510 -- And so winding one and winding 4 on the same course.
01:13:50.282 -- So these two are going to be in phase with each other.
01:13:55.700 -- And so you can use this to build the diagram for how
01:13:59.168 -- the transformer ones relate to how the windings relate
01:14:01.769 -- to each other.
01:14:06.900 -- And so then he goes on to look at.
01:14:15.830 -- So the basically the Y zero is the one that's most
01:14:19.669 -- common in North America.
01:14:24.100 -- And so we can look at things that change polarities by so
01:14:27.556 -- the Y four is now we're shifting things down to the
01:14:30.724 -- 4:00 o'clock by putting winding one connected to Phase
01:14:33.316 -- V.
01:14:35.850 -- White and then we can just look at all these different
01:14:39.546 -- combinations. WHI Six is just reversing the polarity so the
01:14:42.906 -- polarity marks reversed unwinding one.
01:14:47.440 -- And so this is another one that is more of an industrial
01:14:51.076 -- power systems one, but you'll sometimes see Transformers
01:14:53.500 -- with wired opposite of the polarity marks.
01:14:57.580 -- Then he goes through the same thing with Delta windings.
01:15:02.450 -- So the. And so next time we'll go back and look at
01:15:06.270 -- this in terms of a Y Delta transformer. How we do the
01:15:09.054 -- plus 30 if the Y is a high side, how we do the minus 30?
01:15:12.534 -- If the why is the low side?
01:15:17.010 -- And so this paper goes on to kind of lead into deriving
01:15:21.402 -- those connection matrices.
01:15:24.560 -- And so we'll finish talking about this paper next time, and
01:15:28.014 -- then we'll talk about the.
01:15:31.130 -- Example handout so that we're going to apply these
01:15:34.622 -- connection matrices to measurements for a fault.
01:15:38.450 -- We can look at an internal fault or an external fault. We
01:15:42.458 -- can also look at what happens if somebody accidentally left
01:15:45.798 -- ascete shorted in the substation and how that plays
01:15:48.804 -- through these connection matrices.
01:15:51.560 -- So with that, well, any questions before we stop.
01:15:55.730 -- OK, and just a reminder for the outreach students.
01:15:58.115 -- There is no class on campus next week, so there will be
01:16:01.295 -- no new lectures for a week.
01:16:05.650 -- OK, that's all done.
00:00:29.460 -- Hi, welcome back.
00:00:33.550 -- So we're going to resume chapter two. We are in the section on
00:00:39.738 -- project management planning tools and the next thing I
00:00:44.022 -- wanted to talk about was sipoc diagrams. And really, there's
00:00:48.782 -- this one and one other slide coming up here, which probably.
00:00:54.840 -- I mean I I would characterize them as a project management
00:00:59.405 -- planning tool, although they're really most relevant if you're
00:01:03.140 -- doing process improvement. And again, many of us as a part of
00:01:08.120 -- our role as a project manager have some element of process
00:01:12.685 -- improvement that has to be done. Anna Sipoc diagram might be
00:01:17.250 -- something you would use and this is basically where OK, let's.
00:01:23.170 -- Um?
00:01:25.970 -- This is where you would basically identify these
00:01:29.314 -- dimensions of your process. You want to look at suppliers inputs
00:01:33.912 -- to the process, what the process itself is, what are the outputs
00:01:38.928 -- and who are the customers. So in this case this is a process for
00:01:44.780 -- making pizza, so you know it looks at our suppliers are
00:01:49.378 -- inputs or process our outputs in our customers. You can read
00:01:53.976 -- those you know and maybe.
00:01:56.160 -- We're doing this because our we've been getting.
00:02:02.040 -- You know complaints about how long it takes to make pizzas in
00:02:06.936 -- our particular business, and we might want to take a look at how
00:02:12.240 -- can we improve that? And you want to kind of take this broad
00:02:17.544 -- perspective so you're not necessarily honing in on
00:02:20.808 -- something which maybe isn't going to solve your problem? It
00:02:24.888 -- may be an issue, but it might not be related to the particular
00:02:30.192 -- metric you're trying to solve, so it's a good way.
00:02:34.400 -- To tackle process improvement I you know I'll be honest in
00:02:40.516 -- research and development. We didn't really use sipoc
00:02:44.964 -- diagrams, or I hadn't seen amused. But when I I did a
00:02:51.636 -- about 18 month rotation into our customer service business and
00:02:57.196 -- they they always had teams who were doing process improvements.
00:03:04.180 -- Particularly within call centers, and they use sipoc
00:03:08.180 -- diagrams. You know it was amazing what they what they
00:03:13.180 -- did with these as a method to truly understand where
00:03:18.180 -- to focus their efforts.
00:03:22.470 -- Racy, racy diagram. You kind of look at this and
00:03:25.600 -- say, well, is that really a project management tool?
00:03:29.530 -- We will hit on this a little more when we talk about
00:03:35.326 -- communication, which is, I think in the leading chapter, but a
00:03:40.639 -- raci diagram is a very important tool to have if you work in any
00:03:47.401 -- kind of environment that has more than one team in more than
00:03:53.197 -- a handful of people, because it helps you identify who's
00:03:58.027 -- responsible for particular sets
00:03:59.959 -- of work. Who is accountable?
00:04:03.880 -- And by that I mean who's making decisions and who has ultimate
00:04:08.740 -- ownership, who's consulted? So who are stakeholders in the
00:04:12.385 -- process and who might need to be consulted before you make a
00:04:17.245 -- decision or take some action and who just needs to be informed
00:04:22.105 -- and? You know an example. If you work in a team where maybe
00:04:28.100 -- you're part of a matrix organization and we'll talk
00:04:31.880 -- about that in our next chapter on organizing. But say you have
00:04:36.920 -- multiple teams that are a part of a project.
00:04:41.330 -- Um? You want to make sure you're very clear about who's
00:04:47.120 -- doing what to get pieces of the project done, in particular for
00:04:52.184 -- a matrix. It's also very important to understand who's
00:04:55.982 -- making the final decision, because everyone might think
00:04:59.358 -- they're making the decision right there. They are managing a
00:05:03.578 -- team. Why aren't they responsible? Well, in fact, if
00:05:07.376 -- you're part of a matrix organization, you may have a a
00:05:12.018 -- program manager or.
00:05:13.370 -- A project management organization who does in fact
00:05:16.770 -- have the final authority on the work that gets done. People who
00:05:21.870 -- are informed might be the managers above you. You've taken
00:05:26.120 -- some course of action and it was clear you had the ability to
00:05:31.645 -- make that decision, but it's good to let other people know
00:05:36.320 -- who might. Maybe just be interested or who may need to
00:05:42.153 -- take other action based on something you do, and so they
00:05:47.070 -- might be in inform you can find.
00:05:51.420 -- Lots of examples on line for how you might fill that out, but
00:05:56.685 -- it's a good tool to get clarity and alignment within a project.
00:06:03.340 -- Risk analysis.
00:06:06.440 -- You know, again, we've probably all done risk analysis at some
00:06:12.347 -- level. I just, you know, pulled in this example where it's
00:06:18.254 -- basically identified 10 risks that have been deemed to be
00:06:23.624 -- project risks. It talks about the worst case scenario, what
00:06:28.994 -- happens in case of that coming to bear, and then you basically
00:06:35.438 -- do a qualitative and
00:06:37.586 -- quantitative. Assessment and ultimately come up with the risk
00:06:42.148 -- rating. You can come up with much simpler ways of looking at.
00:06:48.040 -- You could identify your risk. Basically make an assessment of
00:06:52.950 -- the likelihood of it happening, and then maybe you do some
00:06:58.351 -- assessment of what's the impact and then basically multiply
00:07:02.770 -- those together and that's your risk assessment. You can make it
00:07:08.171 -- as complicated. Or as simple as needed. The point here
00:07:12.574 -- though is every project that you manage. You should at
00:07:16.754 -- least do a very high level risk analysis, typically as a
00:07:21.352 -- part of a you know if you're following some kind of a
00:07:26.368 -- structured project management lifecycle.
00:07:29.290 -- When you're doing your initial project planning, you would
00:07:33.232 -- likely do a very high level risk analysis and then have.
00:07:38.860 -- You know, figure out what your cadence is for going back and
00:07:44.920 -- assessing where things are. Have new risks, come up,
00:07:49.970 -- etc. The you don't want to just put a lot of effort into
00:07:55.670 -- doing a risk analysis and then and then never come back
00:07:59.520 -- around to in fact evaluating it. They can be very helpful
00:08:03.370 -- in helping you mitigate issues that may come up.
00:08:08.990 -- A quality management plan is another example of a project
00:08:13.070 -- management tool you might use. If you're in the quality area or
00:08:17.966 -- if you have any responsibilities for quality and you know this is
00:08:22.862 -- something very simple which is looking at what's the particular
00:08:26.942 -- characteristic you're looking at. Why is it important? How are
00:08:31.022 -- you going to test for quality? Who's going to do it, and then
00:08:36.326 -- simply a status?
00:08:39.040 -- My guess is most businesses probably have a you know more
00:08:43.803 -- specific template you might use as a part of a quality
00:08:48.566 -- management plan. But again, the point here is.
00:08:53.060 -- Always be thinking about that.
00:08:56.750 -- Even you know we all have a need to be delivering the
00:09:02.431 -- highest quality and most value we can of whatever we do for our
00:09:08.112 -- business. And so you want to be thinking about how can I, you
00:09:13.793 -- know what's important for me in my team in order to deliver on
00:09:19.474 -- that high quality. So this is an example of that. Another quality
00:09:24.718 -- tool is a failure. Modes,
00:09:26.903 -- effects analysis. And this again, is where you're really
00:09:32.193 -- looking at. Different in this particular case, we're
00:09:36.585 -- looking at different process steps and identifying
00:09:40.428 -- potential failure modes.
00:09:43.460 -- What are the effects of those modes? Assessing severity? How
00:09:47.870 -- frequently is it likely to occur, etc. And ultimately,
00:09:51.839 -- you're going to come up with an overall risk priority number,
00:09:56.690 -- and I have seen these use
00:09:59.336 -- pretty. Sensibly in various research and development type
00:10:03.988 -- teams. And there are good.
00:10:07.450 -- You know fairly simple way to do a pretty in depth analysis and
00:10:12.845 -- get an understanding of where in fact you might be want to. You
00:10:18.240 -- might want to be investing effort in order to prevent some
00:10:22.805 -- issues from happening.
00:10:26.740 -- Dmax
00:10:29.110 -- define measure, analyze, improve, control is.
00:10:33.660 -- Probably a process improvement approach. You might be familiar
00:10:37.449 -- with if you've done that as a part of your role and again.
00:10:44.350 -- You know, when I was working in R&D we were doing lots of
00:10:50.122 -- process improvement. We probably weren't as rigorous as we could
00:10:54.562 -- have been at using something like Demac as a model for doing
00:10:59.890 -- our process improvement, but it's a good approach to
00:11:03.886 -- methodically walk through a process improvement approach. It
00:11:07.438 -- can be for a very simple improvement in each of the steps
00:11:12.766 -- might be quite short.
00:11:15.670 -- But it helps you think.
00:11:19.220 -- I guess more completely about all the elements of the
00:11:24.240 -- problem in what you're trying to do to improvement, so
00:11:29.260 -- definitely worth looking into if you have an element of
00:11:34.280 -- process improvement in your job and it's something that's
00:11:38.798 -- talked about pretty extensively in the process
00:11:42.312 -- improvement class.
00:11:47.130 -- So wrapping up the discussion on action planning, you know just a
00:11:52.470 -- couple of comments that I thought were worth including.
00:11:56.475 -- You know. Oftentimes when we're managers, we think it's our job
00:12:01.370 -- to do all the planning and it is, you know, it is the role of
00:12:08.045 -- the technology and engineering managers to do the planning. But
00:12:12.495 -- be sure to involve the people who do the work.
00:12:17.020 -- In the planning where you can now you don't want to go to
00:12:22.077 -- extremes. I was talking to a friend of mine who works at a
00:12:27.134 -- very large company who's in the midst of, I guess a very
00:12:31.802 -- horrendous product release and everybody is getting really
00:12:34.914 -- nervous that they're going to be late and so every day.
00:12:40.230 -- The senior vice president calls every single engineer into a
00:12:45.140 -- meeting at 7:00 AM to walk through their action planning
00:12:50.050 -- for the day. Now, do you think that's really productive? The
00:12:55.451 -- answer is no, because a it's people are, you know, people who
00:13:01.343 -- can are quitting because they're there. It's ridiculous, you
00:13:05.762 -- know. So that's an example where there's people at too high of
00:13:11.654 -- levels. Involved in the planning with the doers. That's not the
00:13:16.110 -- intent here, but the intent is if I'm a project manager and I'm
00:13:21.193 -- planning the next project, it would behooves me to have a
00:13:25.494 -- session with the engineers at some point. Not that you
00:13:29.404 -- necessarily want to ask them to sit with you for two days to do
00:13:34.878 -- all of your scheduling, but you probably want to have a.
00:13:40.510 -- You're kind of a validation
00:13:43.590 -- that. That you're on track because a you want them to buy
00:13:48.525 -- into that plan. If you're expecting them to deliver it.
00:13:52.990 -- Similarly, if you're doing strategic planning, if you're
00:13:57.110 -- more senior executive and you're doing strategic planning, always
00:14:01.745 -- involve your staff in that you know that's a great opportunity
00:14:07.410 -- for a. Regular, you know, a quarterly staff offsite to not
00:14:13.952 -- only build and foster teamwork among the team, but.
00:14:19.410 -- Drive good alignment on that strategic plan because
00:14:22.530 -- ultimately the people in your team are the ones who are going
00:14:27.210 -- to have to do the work, so use those planning.
00:14:32.220 -- Opportunities as a way to drive alignment.
00:14:36.310 -- Use computer based tools when you have access to them, and
00:14:41.403 -- again similarly to don't go crazy involving people. Don't go
00:14:46.033 -- crazy with it mean there's some really incredible tools out
00:14:50.663 -- there to do scheduling and things like that, but if you
00:14:55.756 -- have, you know 1000 or 2000 tasks in a schedule is just too
00:15:01.775 -- unwieldy to manage, so use them when they make sense
00:15:06.405 -- Alternatively. Use simple tools when they make sense.
00:15:10.980 -- If you're doing software development, you know everybody
00:15:14.940 -- is familiar with Agile there is.
00:15:19.200 -- Kind of an element of agile for very simple projects where you
00:15:23.820 -- can basically use a con Bon bored. So if you're fixing
00:15:28.055 -- defects for example in a product, it's very easy to use a
00:15:32.675 -- con Bon bored to show how you know when the defect gets
00:15:37.295 -- accepted into the system, who's working on it when it's done,
00:15:41.530 -- when it's tested, when it's been deployed to a customer, for
00:15:45.765 -- example. That's a very visual way. You don't need a very
00:15:50.000 -- complex. Tool to track that work, but the visual kambam
00:15:54.290 -- board is a good way to keep everybody up to date. So figure
00:15:59.269 -- out what you need and don't don't apply technology where you
00:16:03.482 -- don't need to.
00:16:06.430 -- Make sure you're looking at risks in doing contingency
00:16:10.084 -- planning where you need to, and you might have to go back and
00:16:15.362 -- iterate on the planning process. You may say you do your planning
00:16:20.234 -- as you know your project manager. You do some planning,
00:16:24.294 -- you have a review, say with your team and there were some things
00:16:29.572 -- that you missed. Well, you gotta go back and iterate. It's not
00:16:34.444 -- you don't need to feel like.
00:16:37.040 -- Iteration is a bad thing because it's an opportunity
00:16:40.532 -- to get things right.
00:16:43.550 -- So I think those are some things
00:16:45.496 -- that. That you can keep in mind.
00:16:49.330 -- Hey, the last couple of topics are issuing policies and
00:16:53.990 -- basically documenting procedures, and I think
00:16:56.786 -- typically when we hear oh gosh, you know I have to do I have
00:17:03.310 -- to generate policies that can take a very negative connotation
00:17:07.970 -- in really policy czar directives intended to address repetitive
00:17:12.164 -- questions, issues of general concern, and really to drive
00:17:16.358 -- equity across your workforce. So here's some good examples.
00:17:21.090 -- Hiring and firing guidelines. You want to make sure that
00:17:25.700 -- you've got strong policy's for expectations around hiring, and
00:17:29.849 -- also around terminating people. You know, it's your it would be
00:17:34.920 -- a very uncomfortable environment if there were no guidelines for
00:17:39.530 -- how people were terminated.
00:17:42.120 -- Equal opportunity policies might be an example. Performance
00:17:47.760 -- appraisals are something that.
00:17:52.420 -- You are necessary in the workplace and you want to be
00:17:57.073 -- able to do those consistently. You might be in
00:18:00.880 -- a business where a drug policy or drug testing is
00:18:05.110 -- mandatory.
00:18:07.090 -- So you know, these are some examples of things where you're
00:18:11.875 -- really trying to.
00:18:14.680 -- Make sure there's equity and address repetitive concerns.
00:18:19.008 -- Policies are there to save management time. No, they're
00:18:23.877 -- not intended to generate lots more work.
00:18:29.400 -- They are intended to capture it. You know, the experience and
00:18:35.263 -- past learning of the company and hopefully facilitate delegation
00:18:40.060 -- if there are clear policies in place, then for example, if I'm
00:18:46.456 -- a senior level executive and there are clear policies around
00:18:51.786 -- travel expenses and trip reports, I could perhaps
00:18:56.050 -- delegate the ability or delegate the responsibility to my
00:19:00.847 -- administrative assistant. To look at those and approve them,
00:19:05.030 -- for example. That might be, that might be something.
00:19:10.350 -- If that's allowed in your particular
00:19:12.636 -- business, but basically you're trying to figure
00:19:15.303 -- out a way to be consistent on things
00:19:18.351 -- that are going to come up over and over again.
00:19:25.340 -- Policies will apply uniformly to all employees. They should be
00:19:30.930 -- pretty permanent. You don't want to be changing policy's real
00:19:36.520 -- frequently, and hopefully they foster corporate objectives. You
00:19:40.992 -- know you don't want to have policies that really are in
00:19:47.141 -- conflict with things.
00:19:51.100 -- Things that are valued at the corporate level, and so I think
00:19:55.780 -- you need to think about when you need to have policy's.
00:20:00.420 -- You might have policies about working at home. That's probably
00:20:04.720 -- the one that has come up several times through the course of my
00:20:10.310 -- career. I can remember when working at home or remote, you
00:20:15.040 -- know. Being a remote worker located in a different geography
00:20:20.542 -- just wasn't an accepted Norm, and I can remember the first
00:20:26.020 -- time we had to address this was we had a very Senior High
00:20:32.494 -- performing engineer. Needed to move to Wyoming because of some
00:20:37.780 -- family things with his wife and.
00:20:41.500 -- So we you know the question was do we let him resign or do we?
00:20:47.480 -- Basically, craft a policy about a remote worker and so we did
00:20:54.152 -- and it was interesting because that got tested.
00:21:01.080 -- Over and over again in terms of people you know other people
00:21:05.064 -- wanting to take advantage of that, and it was interesting
00:21:08.384 -- because you know what? If you have somebody who comes in,
00:21:12.036 -- wants to be a remote worker, but there may be some kind of middle
00:21:16.684 -- of the road performer, well, how do you know? Then you have to
00:21:21.000 -- start thinking about. Do you have to create a policy that so
00:21:24.984 -- regimented in terms of if you come with the request to work at
00:21:29.300 -- home? You need to be?
00:21:31.550 -- In the you know, whatever top two tiers of performance you
00:21:36.686 -- know etc., etc.
00:21:40.230 -- Think we tried to have a policy that was more general.
00:21:46.710 -- Probably the biggest challenge we had was when we started
00:21:50.680 -- working with teams in other geographies where suddenly you
00:21:54.253 -- know we worked a lot with India and that was not a commonplace
00:21:59.414 -- thing to have people working at home and but then they started
00:22:04.178 -- raising that with their management and it was
00:22:07.354 -- interesting because then when I had my assignment in Singapore,
00:22:11.324 -- that was probably one of the first policy things we had to
00:22:16.088 -- come up with was.
00:22:17.800 -- What are we going to do? How are we going to create a work at
00:22:23.650 -- home policy for an environment that historically did not permit
00:22:27.550 -- that? So again, you know.
00:22:30.060 -- That's something that came up many years ago, and it's evolved
00:22:34.614 -- overtime. I think in general, when I was when I retired from
00:22:39.582 -- HP, we were going back to a policy of everyone being back on
00:22:44.964 -- site so things can swing pretty radically and come full
00:22:49.518 -- circle based on the needs of the business. I think that's the
00:22:54.486 -- main thing you have to keep in mind is you may create a policy.
00:23:02.020 -- If the business needs change, you may have to go back and
00:23:06.400 -- revisit that policy and there's nothing wrong with doing that.
00:23:11.820 -- Procedures, it's kind of the same, you know, we think about,
00:23:16.275 -- oh, brother, you know I have to follow a set of procedures to
00:23:21.540 -- doing something, and it's really you're trying to standardize
00:23:25.185 -- work that benefits from.
00:23:28.560 -- Procedures, because you're doing it over and over again, you've
00:23:32.670 -- got or you're.
00:23:34.890 -- You have some kind of certification, perhaps that
00:23:37.810 -- is dependent on having a procedure to ensure that
00:23:41.095 -- work is done a certain way. Or maybe you have a say to
00:23:45.840 -- health and safety thing where certain types of
00:23:48.760 -- manufacturing wastes have to be disposed in a certain way
00:23:52.410 -- and you need to follow procedures in order to
00:23:55.695 -- ensure health and safety of.
00:23:59.040 -- The workforce and.
00:24:02.560 -- So again, depending on the type of work you're doing, the
00:24:07.301 -- procedures you're involved with are going to be quite different.
00:24:11.611 -- If you're in an R&D team, the product management lifecycle is
00:24:16.352 -- a procedure that establishes and
00:24:18.507 -- standardizes how. The work is
00:24:21.446 -- going to. Or the steps if you will. At a high level the
00:24:27.586 -- work is going to follow and what is going to happen at each of
00:24:32.878 -- those checkpoint or handoff process is that would be a
00:24:37.036 -- procedure if you're working in.
00:24:40.210 -- You know a part of the business where you're installing devices
00:24:45.072 -- or you're in your field. Engineer installing devices at
00:24:49.050 -- customer sites. It's important you have an installation manual
00:24:53.028 -- so you can follow the appropriate steps for ensuring
00:24:57.006 -- that things are done appropriately. So again, it's
00:25:00.542 -- not to create a bunch of overhead and procedures for
00:25:04.962 -- every single thing you do, but it is important to.
00:25:10.590 -- Make sure that when you need a procedure, you get 1 written
00:25:16.026 -- appropriately. Actually this here we go. You want to.
00:25:20.930 -- Preserve the best way to get the work done. So how can
00:25:24.182 -- you be efficient?
00:25:26.460 -- It can help you know. Oftentimes, procedures are one
00:25:29.997 -- of the outcomes of a process improvement approach. You want
00:25:33.927 -- to ensure that you have standardized action you want to
00:25:37.857 -- simplify things, and in particular it's a way to save
00:25:41.787 -- some of your corporate memory. How do things get done? What's
00:25:46.110 -- the right way to do things? What's the procedure for testing
00:25:50.433 -- your device now? It doesn't matter if somebody leaves the
00:25:54.363 -- company, you know how things get
00:25:56.721 -- done. Because you have that documented in the form of a
00:26:00.770 -- procedure. So again, you don't want to overdo it, but you want
00:26:05.812 -- to have good procedures when they make sense.
00:26:10.940 -- When you want to develop a procedure, again, concentrate
00:26:15.251 -- on the critical work. Look at the inputs and outputs of
00:26:20.520 -- what's happening. You might even use a sipoc diagram as
00:26:25.310 -- input to detailing or developing a new procedure.
00:26:29.142 -- You need to talk about or think about the
00:26:33.453 -- characteristics.
00:26:35.620 -- Proposed the procedures and then figure out the regular timeframe
00:26:40.290 -- that you're going to come back and review. These probably most
00:26:45.427 -- important is making sure that the people who are involved in
00:26:50.564 -- doing the procedure have an opportunity to give input before
00:26:55.234 -- you go develop something in handed off to them and inspect.
00:27:00.371 -- Expect them to do it I ideally you'd like to have their input.
00:27:06.510 -- In the creation of the procedure in some way,
00:27:10.164 -- certainly you want to have the review of people who are going
00:27:15.036 -- to have to execute the procedure before you turn them
00:27:19.096 -- loose.
00:27:24.630 -- We talked about different types of planning. We talked, we
00:27:28.350 -- started out with some discussion on strategic planning.
00:27:32.300 -- How do we figure out what are the right things to do in our
00:27:36.612 -- business and then? As we transition into operation
00:27:39.725 -- planning, what are some of the tools to help us get things done
00:27:43.898 -- the right way? Just some things to keep in mind.
00:27:50.730 -- Validate your assumptions. You're going to want to go out
00:27:55.160 -- there, and even if you're planning a project that's a
00:27:59.590 -- follow on project that you've done five times, something will
00:28:04.020 -- be different, so be sure to make sure you're getting appropriate
00:28:08.893 -- information. You're doing some of that forecasting. You're
00:28:12.437 -- looking at alternatives, but really validating that the
00:28:15.981 -- assumptions you're making are
00:28:17.753 -- correct. From a people perspective, involve the
00:28:21.938 -- right people.
00:28:24.250 -- One of the things that.
00:28:27.140 -- You know? Is important is consider what we used to call it
00:28:32.860 -- the with them. What's in it for me. For all stakeholders
00:28:37.458 -- involved in your planning so involved the people are going to
00:28:42.474 -- do the work. If you're making. If you're planning some things
00:28:47.072 -- that are going to be done differently, you know, introduce
00:28:51.252 -- those changes in a way that maybe you can't avoid resistance
00:28:55.850 -- but you manage it and.
00:28:58.530 -- Will in the chapter on leading will talk a little bit
00:29:03.054 -- about John Carter's eight step change management approach.
00:29:06.620 -- This is a perfect opportunity for where if you're doing some
00:29:11.504 -- planning, that's going to
00:29:13.280 -- involve. Someone elses work being done a different way?
00:29:17.756 -- Don't discard the need to do some active change management
00:29:21.626 -- and at a minimum this consideration of what's in it
00:29:25.496 -- for me for all your stakeholders will help you
00:29:28.979 -- think through that.
00:29:31.250 -- Be sure to understand the benefit versus the cost. You may
00:29:36.398 -- come up with a great plan to do, you know, some great product,
00:29:42.482 -- but. Is the benefit there? Is it going to cost so much that you
00:29:49.085 -- know you're never going to recoup what you've put into it?
00:29:53.430 -- You really have to think about benefits versus costs. Make sure
00:29:57.775 -- when you're doing your planning
00:29:59.750 -- have. A series of small steps along the way. This allows you
00:30:05.036 -- to get some small wins. It also allows you to make course
00:30:09.788 -- corrections if you do a project management plan that goes
00:30:13.748 -- basically from investigation to and say you have one task which
00:30:18.104 -- is develop the product and then your product is done, your
00:30:22.460 -- opportunity for making midcourse corrections is not very good in
00:30:26.420 -- that case, so you need to figure out what's that right level of.
00:30:31.860 -- Um?
00:30:34.160 -- What's the right level you need to break that work
00:30:37.740 -- down such that you have the control you need, and
00:30:41.320 -- in particular the ability to make these corrections.
00:30:46.160 -- You want to be anticipating changes in future conditions,
00:30:49.742 -- and again, this is where you may be thinking about
00:30:53.722 -- contingencies, and you may have to apply a formal change
00:30:57.702 -- management process if needed. And Lastly, of course, make
00:31:01.284 -- sure you get the commitment of the resources you need to
00:31:05.662 -- achieve the objectives. It's great to have a wonderful
00:31:09.244 -- plan, but if you don't have the ability to deliver on it,
00:31:14.020 -- then that.
00:31:16.040 -- Is very discouraging for people overtime.
00:31:22.030 -- I think wrapping up, then, you know, planning. I think it's
00:31:26.848 -- probably fairly obvious to all of us we plan in every part
00:31:32.542 -- of our lives really, but it is a very important function in
00:31:37.798 -- engineering management and technology management and the
00:31:40.864 -- key activities we talked about were the need to forecast action
00:31:45.682 -- planning. Of course, related to both strategic planning and
00:31:49.624 -- tactical planning, issuing policies and establishing
00:31:52.252 -- procedures. You know, oftentimes we think that forecasting and in
00:31:57.313 -- particular strategic planning, are only activities by the high
00:32:01.696 -- level executives. In my, you know, kind of my opinion is
00:32:07.053 -- don't discount those activities at any management level in the
00:32:11.923 -- organization, because if you're if you understand what the
00:32:16.306 -- strategic plan is at the top levels of your business,
00:32:21.176 -- ideally. Eat their cascaded to each level so each level then
00:32:27.150 -- could take those objectives in based on the work they are
00:32:32.562 -- responsible for. Create their key objectives that link to the
00:32:37.482 -- overall objectives above them and then ultimately if you take
00:32:42.402 -- that to the you know kind of the final step. Each individual on
00:32:48.798 -- your team hopefully has a set of
00:32:52.242 -- performance objectives. Ideally they can see within
00:32:55.435 -- their performance objectives how they fit
00:32:57.889 -- within the context of the team and how the work they
00:33:02.388 -- are doing is going to contribute to the success
00:33:06.069 -- of the team's objectives. The teams objectives.
00:33:10.010 -- Hopefully are linked to the team or manager above them,
00:33:15.390 -- etc and so it really allows clear line of sight from every
00:33:21.846 -- single person in your business or team up to the high levels
00:33:28.302 -- of the organization and.
00:33:32.270 -- My my personal opinion is that every single manager
00:33:36.437 -- should take the time to do that at the level that's
00:33:41.530 -- appropriate for where their team fits in the
00:33:45.234 -- organization.
00:33:46.850 -- And then I think, Lastly operational planning, you know.
00:33:50.970 -- Really forms the basis for much of what we do.
00:33:55.400 -- And so you need to figure out what are the tools that are
00:33:58.871 -- important for you to do.
00:34:01.960 -- Here's just an example of you know how you might have to think
00:34:06.926 -- a little bit strategically, and this was question 2.2 at the
00:34:11.128 -- back of the textbook and it
00:34:13.420 -- says. So the company has always been focused on the
00:34:18.105 -- high quality, high priced end of the market.
00:34:22.500 -- Now, market intelligence indicates that some competitors
00:34:26.399 -- are planning to enter the low price, low quality into the
00:34:32.526 -- market. What would you do?
00:34:38.660 -- It's an interesting question because from a strategy
00:34:44.460 -- perspective you probably have focused on.
00:34:50.170 -- Well, you obviously have focused on the high end element of the
00:34:55.090 -- market. Probably everything in your company is structured
00:34:58.370 -- around that. You certainly want to figure out how to protect
00:35:04.074 -- that Mitch if you will, but likely if you do nothing.
00:35:09.820 -- Your business will slowly be eroded by.
00:35:15.220 -- People who are anticipating this kind of low, low price, low
00:35:19.972 -- quality product by the competition and there's a number
00:35:23.860 -- of options you could explore.
00:35:26.890 -- You could really look at the option of partnering with
00:35:32.160 -- someone and you know, importing a low price, low quality
00:35:37.430 -- product, perhaps you.
00:35:40.640 -- Label it as you know you work with somebody by the technology
00:35:45.416 -- and label it as your own.
00:35:49.650 -- That would certainly be a way to quickly get a product into the
00:35:55.448 -- market with the least amount of investment necessary. Of course,
00:35:59.908 -- you know the downside of that is if it really is low quality and
00:36:06.152 -- your brand has been all about high quality, what does that do
00:36:11.504 -- to your customer base? They may not be accepting of that, so you
00:36:17.302 -- have to think through.
00:36:19.810 -- That may be a really good thing to do, but what are
00:36:23.830 -- the implications? So there you would probably need to
00:36:26.845 -- do some scenario planning and think through that you
00:36:29.860 -- could certainly.
00:36:33.610 -- Follow the competition more closely and perhaps start
00:36:37.418 -- preparing to take your product. You know, kind of downmarket
00:36:42.178 -- some. That obviously takes a much bigger investment and takes
00:36:46.938 -- a longer period of time.
00:36:52.620 -- That might be a way to get started on this notion of having
00:36:58.002 -- a second brand if you will within your business. So you
00:37:02.556 -- could still maintain that high price, high quality brand and
00:37:06.696 -- basically Re brand of product line that's targeted at a lower
00:37:11.250 -- end of the market.
00:37:15.260 -- Yeah, I think the point is though, you probably can't.
00:37:18.870 -- You know doing nothing is probably a recipe for
00:37:22.119 -- failure. So in a case like that, you need to think
00:37:26.090 -- through.
00:37:27.600 -- From a strategic planning process, what are your options?
00:37:32.082 -- What makes sense and they can
00:37:35.070 -- range from? Investing in new product development for that low
00:37:39.856 -- end of the product line, recognizing that takes a long
00:37:43.426 -- time. You can do nothing at the other end of the spectrum, which
00:37:49.104 -- probably is going to be.
00:37:52.510 -- It's going to impact your business overtime or you
00:37:55.876 -- come up with something in the middle. Which is this
00:37:59.616 -- idea of partnering with somebody. And each of those
00:38:02.982 -- will have pros and cons and benefits and risks, and that
00:38:07.096 -- would be an assessment you have to make.
00:38:12.090 -- So I think what you can see and will see this probably in every
00:38:17.928 -- chapter in the textbook.
00:38:20.730 -- Engineering management or technology management is usually
00:38:24.335 -- not very black and white.
00:38:28.130 -- There is always this kind of, typically a Gray, you know a
00:38:33.314 -- Gray area in the middle, and that's where we want to take
00:38:38.498 -- advantage of all the tools we have available to us. You want
00:38:43.682 -- to certainly apply critical thinking as you're looking at
00:38:47.570 -- homework assignments that are case studies. There's typically
00:38:51.026 -- not going to be necessarily a right and wrong answer.
00:38:56.730 -- So what's going to be important is are you able to think through
00:39:02.099 -- and analyze the particular situation and use the tools at
00:39:06.229 -- hand to come up with some possible options? So don't get
00:39:10.772 -- hung up on.
00:39:13.250 -- So you know I have to do a case study and it's going to be. It
00:39:16.770 -- has to be. You know, if I don't get this right answer, I'm not
00:39:21.339 -- going to get 100%. That's not really the case. There's going
00:39:24.518 -- to be a lot of flexibility. The main thing is to think
00:39:27.986 -- critically and apply the tools that you have at hand.
00:39:32.010 -- So with that next, the next lecture we will talk
00:39:38.090 -- about Chapter 3, which is focused on organizing and.
00:39:45.870 -- Will look at a number of different organization
00:39:48.742 -- structures when you might use them. Some of the pros
00:39:52.332 -- and cons, and so I think it will be an interesting
00:39:56.281 -- discussion. So thanks bye.
00:00:30.200 -- Yes.
00:00:33.030 -- So today we will continue discussion about the
00:00:35.870 -- modified, all the method and Runge Kutta methods. So we
00:00:39.420 -- will talk about the formulas and then accuracy and so on.
00:00:43.325 -- So I give you hand out and the problem. I'll use it
00:00:47.585 -- today so that we can cover a little bit faster. And then
00:00:51.845 -- I'll spend time on other material. OK, so.
00:00:58.460 -- In there you remember in all this method in order to go from
00:01:03.751 -- point X&YN to point XN plus one 1 + 1, essentially with another
00:01:09.042 -- next index, we only use information from the previous
00:01:12.705 -- point. So in a modified or leave use information from 2 points
00:01:17.589 -- and we use oil as step to go to the point X N + 1 NU N +
00:01:24.915 -- 1. This is predicted point.
00:01:27.950 -- And then be available slope at the predicted point and we use a
00:01:33.059 -- slope at initial, not initial. But the point that we start
00:01:37.382 -- start from and then we average these slopes defined slope
00:01:41.312 -- alone, which we find essentially construct line right tangent
00:01:44.849 -- line and then we find approximation at the next step.
00:01:48.779 -- So I also wrote this method last
00:01:51.530 -- time. So you can either define predictor which is the Oilers
00:01:57.000 -- step and then this is slope at.
00:02:01.370 -- .1 right and here we have slope at .2 and then we average slopes
00:02:07.082 -- and this is how we find the next. The next point all we can
00:02:12.794 -- write down these slopes explicitly. So K1 is a slope at
00:02:17.282 -- point. XNYN and then we use it to March to find point you and
00:02:22.907 -- plus one. Then we find K2 slope at the second point and then we
00:02:27.793 -- take every to the slopes defined. And if you don't want
00:02:31.632 -- to use K1K2 and just write this in terms of an even without you
00:02:36.518 -- N + 1, then you just write explicitly all the expressions
00:02:40.357 -- for for you and plus one. So this is a first step. Predictor
00:02:44.894 -- does not change and in the second step in the character.
00:02:48.970 -- You have your own plus one equals UN plus H / 2, so you
00:02:53.702 -- take average. This is your slope at point XYN. This is your point
00:02:58.096 -- and you X N + 1 right here. This is your predicted point. U N + 1
00:03:03.842 -- essentially just written
00:03:04.856 -- explicitly. OK.
00:03:09.510 -- Modified the oldest method uses two term approximation from the
00:03:13.990 -- Taylor series right. The constant term and the linear
00:03:18.022 -- term. The modified Euler method uses. Also next terms uses
00:03:22.502 -- quadratic term in the Taylor expansion, so if we go back to
00:03:27.878 -- their tail expansion then modified Euler will use up to
00:03:32.358 -- age squared term. So this means that the first time that you
00:03:37.734 -- neglect will be proportional to
00:03:39.974 -- H cube. Right next will be age to the 4th. Each of the 5th and
00:03:45.450 -- if H is small then this will be a dominant term. So air local
00:03:50.070 -- error over one step will be proportional to H cube.
00:03:54.290 -- And then you find cumulative error after multiple steps right
00:03:58.000 -- after. If you're going from zero to X final, then the error will
00:04:02.823 -- be proportional to age squared, so similar usually you lose one
00:04:06.904 -- order when you sum the errors you find cumulative error. So
00:04:10.985 -- since modified term all this but it matches the 1st three terms
00:04:15.437 -- in the Taylor series up to and including termination squared,
00:04:19.147 -- the local area is proportional to each cube, but the cumulative
00:04:23.228 -- error is proportional to age
00:04:25.083 -- squared. So if air is proportional to H squared
00:04:28.998 -- and instead of H, you take H / 2, what would happen
00:04:32.910 -- with the error?
00:04:35.610 -- As will decrease by approximately 1 force, right? So
00:04:38.265 -- if you see so, this is a way how you can check that your method
00:04:42.690 -- is quadratic. So so this means that your method is quadratic,
00:04:45.935 -- so your error is proportional to each squared. Let's say you
00:04:49.180 -- write a program and how would you verify that? Yes, the method
00:04:52.720 -- is programmed correctly. So what you can do you take you take a
00:04:56.555 -- test problem for which you know exact solution, so you can look
00:05:00.095 -- at the error because error would be the difference between exact
00:05:03.340 -- solution and numerical solution. So you go from.
00:05:06.430 -- Initial time to some final time final point, and you compute
00:05:11.457 -- solution at the final point.
00:05:14.530 -- And you look at the error right? And then you decrease error by
00:05:18.391 -- half and look how the error will change. So if error bill
00:05:21.955 -- decreased by by half, this means that you have a linear method.
00:05:26.250 -- If it decreased by quarter, than its accuracy is quadratic.
00:05:32.310 -- OK. So this is a way to verify that your program is is correct,
00:05:37.448 -- and then once you verify your code then you can change
00:05:41.034 -- equation. You can change function, then you can more or
00:05:44.294 -- less thing that your program is reliable, computes correctly, so
00:05:47.554 -- this is what happens with the error. Is H decreased by half
00:05:51.466 -- then the arrabelle because by by a factor of four and just for
00:05:55.704 -- comparison. Again for all this method it's a linear
00:05:58.638 -- convergence. So if you decrease age by half your error will also
00:06:02.550 -- decrease approximately by half.
00:06:05.710 -- OK.
00:06:11.410 -- And so essentially we know the methods we just. I can just
00:06:15.442 -- rewrite it may be in the way that is more convenient for for
00:06:19.810 -- programming. So if we want to solve initial value problem with
00:06:23.506 -- some initial condition. So what do we need? We need initial
00:06:27.202 -- condition right? So X not, why not? We also know we need to
00:06:31.570 -- know the step size and how many steps we have to perform right
00:06:35.938 -- function F is known. So once you have equation you can find
00:06:39.970 -- function F so again.
00:06:41.420 -- Then before you need to compute 'cause you have some homework
00:06:46.051 -- that you have to actually implement by hand or using
00:06:50.261 -- Calculator. So write down the formulas before you substitute
00:06:54.050 -- values right so?
00:06:56.320 -- You can, we can use either write this in terms of predictor
00:07:00.256 -- corrector or we can use this slopes K1K2 to write the method
00:07:04.192 -- so XN plus 1 = X N plus H. So every time you increment by H
00:07:09.440 -- right and also we can write
00:07:11.408 -- that. H is X final minus X starting divided by number of
00:07:17.326 -- steps right or number of steps is X final minus 0 / H right?
00:07:23.150 -- So if if you know number of steps you know initial point
00:07:28.142 -- terminal point then you can find step size or vice versa.
00:07:32.718 -- If you know step size you can find number of steps.
00:07:39.730 -- OK, predict this step is just the oldest method.
00:07:43.490 -- Right and then corrector? So predicted allows you to find
00:07:46.940 -- this predictive point you N + 1 and then corrector will find
00:07:51.080 -- slopes at both points and average them to find exponent.
00:07:55.210 -- OK, and again Alternatively this is using the K1K2 and but
00:07:59.687 -- essentially the same.
00:08:01.550 -- OK, so whatever way you prefer, you can use.
00:08:08.170 -- OK, any questions here.
00:08:14.070 -- So let's look at the example.
00:08:16.890 -- So in this example you have to implement modified order in.
00:08:23.520 -- And solve the problem in 2
00:08:24.972 -- steps. So equation is Y prime equals X + y -- 1 squared.
00:08:30.250 -- Initial condition by 0 = 2. So find Y at. So you start from X
00:08:35.575 -- equals. O you go to X = 0.2 in two steps means that step
00:08:42.728 -- step sizes. 0.1 right again, it's a 0.2, so H is 0.2
00:08:49.380 -- -- 0 / / 2 zero point 1 which is written here.
00:08:56.710 -- Initial condition X00Y0 stole from here number of steps two
00:09:02.050 -- and then H you find.
00:09:06.260 -- Their function function F function F is the right inside
00:09:09.760 -- of your equation.
00:09:12.550 -- OK, and I know it's tempting to write down right away their
00:09:17.230 -- solutions, but take some time. Just write down the formulas in
00:09:21.520 -- terms of X&YN, it's easier than to substitute. I mean, if you
00:09:26.200 -- program something then you just program with indices and then it
00:09:30.490 -- computable repeat, write your computations. But when you do by
00:09:34.390 -- hand then you have to keep track of X0X1Y0Y1 and then here you
00:09:39.460 -- have you also UN to worry about.
00:09:43.700 -- So you write down the formula. So this is your next.
00:09:46.850 -- Approximation of X. This is your predicted value just
00:09:50.540 -- using the Euler's method, because this is your function
00:09:54.230 -- F at X&YN and then.
00:09:57.420 -- This is your next approximation.
00:10:00.110 -- By using the previous and the average of slopes.
00:10:04.030 -- OK.
00:10:06.520 -- So for all this method to go from one point to another, you
00:10:11.109 -- do one is 1 stage method because you only use one point for the
00:10:16.051 -- modified order, it is 2 stage because you have predictor an
00:10:19.934 -- you have character. So each step has two parts.
00:10:24.160 -- OK.
00:10:26.920 -- So if we take so here, we have N equals.
00:10:33.010 -- Zero, so when N = 0, I have X 1 = X O plus H. We find 0.1,
00:10:40.462 -- which is what supposed to be predicted point Yuan Yuan plus
00:10:45.016 -- one. Will you one and then it's Y0 plus HX0Y0 and you substitute
00:10:50.398 -- values you get 2.1. So this is your predicted value and then
00:10:55.366 -- you can use it in the next stage
00:10:58.678 -- defined. Correction, OK, so this is your essentially. This is the
00:11:02.650 -- same as what you have here.
00:11:05.450 -- So it might be more beneficial to use key one key two if you
00:11:10.014 -- want to reduce time on writing because you have to rewrite
00:11:13.600 -- this. And this is your slope at the predicted point. Again, just
00:11:17.512 -- write down X0Y0X1U one before you substitute values, because I
00:11:20.772 -- mean you see that becomes messy.
00:11:29.940 -- OK, so then we substitute values and we obtain approximation. So
00:11:33.845 -- so we did two stages, but this is the first step.
00:11:38.670 -- OK, it's not 2 steps first step. So now we use N = 1 and
00:11:45.525 -- this allows us to find X2U2 and Y2. So X2 is exam plus H, so
00:11:52.380 -- we have you too is a prediction using the Oilers step from Point
00:11:58.321 -- X one U-1 and then you do is correction with average of
00:12:03.805 -- slopes. Again as you see, right down X one U1X1X2U2 and so on.
00:12:09.880 -- And then approximate and then substitute values.
00:12:16.130 -- So finally so this is our approximation of a solution at
00:12:19.639 -- 0.2, and again this is not exact value, right? It's only
00:12:23.148 -- approximation because we use out of infinitely many terms in the
00:12:26.657 -- Taylor series, we only use 3.
00:12:29.290 -- So H is finite, right? So definitely we have an error. OK,
00:12:33.442 -- so schematically what is going on here? You start. Your initial
00:12:37.248 -- condition was at 02 right? This is your point.
00:12:42.180 -- Predictor brings you to point X one U-1.
00:12:47.560 -- You find slope at this point at X1. You want you find slope at
00:12:54.224 -- X0Y0. You average corrector gives you point X1Y1.
00:12:59.420 -- This is your first step, but
00:13:01.412 -- still stages. Then again from point X1 U one you find
00:13:06.648 -- predictor X2U2 right YouTube means has index as Y two. So
00:13:11.510 -- please different letter. But it is the same index and then
00:13:16.372 -- you've added slopes at X 11X2U2 average them and this
00:13:20.792 -- gives you correct correction point X2Y two again two stage
00:13:25.212 -- but it's one step.
00:13:31.180 -- OK.
00:13:33.760 -- Any questions here?
00:13:36.930 -- So example have either Euler or modified Euler method to
00:13:40.870 -- implement by hand, which means the step size will be generously
00:13:45.204 -- large, maybe like one or something that doesn't require
00:13:48.750 -- because you cannot use calculators for the test there
00:13:52.296 -- 'cause I don't know which device you bring mini. Something
00:13:56.236 -- computer that has access online and so on. So the algebra will
00:14:00.964 -- be simple enough that you can do
00:14:03.722 -- by hand. But for me, even if you have to perform 2 steps.
00:14:08.650 -- I need to see that yes, you know what is initial
00:14:11.411 -- condition. What is the next point and so on. So it will
00:14:14.423 -- not be a lot of steps, but at most Euler or modified Euler.
00:14:18.810 -- OK, your homework has more steps to perform, so you're welcome to
00:14:23.358 -- use whatever calculators computers to get the values, but
00:14:26.769 -- you have to write down. Then you can probably minimize number of
00:14:31.317 -- things that you write.
00:14:33.600 -- OK, your project Modeler project is based on
00:14:36.888 -- implementing these methods actually not implementing.
00:14:39.354 -- Using them to solve problems because the
00:14:42.231 -- programs functions are available on the course
00:14:45.108 -- websites. You just have to.
00:14:49.070 -- Maybe on Monday I'll bring the laptop so I'll show you where
00:14:52.646 -- files are and how to use them.
00:14:57.500 -- OK so next method.
00:15:00.250 -- To consider is so called 1st order on the quota method.
00:15:06.320 -- And the idea here is the falling. So we saw from their
00:15:10.880 -- modified all the method that if we use information from two
00:15:15.060 -- points then we get more accurate
00:15:17.340 -- approximation. Right, so can we use more points to get the even
00:15:22.577 -- more accuracy and the question the answer is yes. So in this
00:15:27.101 -- case we use four points.
00:15:29.560 -- So we go from .1.
00:15:32.980 -- 2.2 Essentially this is your order step. We get point .2.
00:15:38.645 -- Then we use this slope K2 to go to .3.
00:15:44.670 -- We use the .3 slope. Do you go to .4 and then we take weighted
00:15:50.790 -- average of the slopes at this
00:15:53.238 -- point? OK, so OK.
00:15:57.250 -- Um?
00:16:00.400 -- So which points we use? We use
00:16:03.529 -- point X. We use point in the middle of this interval at X N +
00:16:08.925 -- H / 2 and here we have two points to use and we also use
00:16:13.050 -- point at X = N + 1.
00:16:16.350 -- So do we? Do we give the same weight essentially the sum of
00:16:21.316 -- slopes over 4? No, we give twice more weight at points
00:16:25.518 -- in the middle.
00:16:36.200 -- And this is last page that you have an I I did not print. I
00:16:42.095 -- have a few more pages, but.
00:16:45.910 -- I'll explain what we have here. So if you have.
00:16:51.170 -- Probably let let me use, maybe maybe maybe this so you don't
00:16:55.094 -- have this page, but this is a recap of the last page, so you
00:16:59.672 -- have to want to solve the 1st order equation with some given
00:17:03.596 -- initial. So I'll bring a copy of
00:17:05.885 -- this next time. So what you do you find the slope at .1. This
00:17:11.178 -- is where you start.
00:17:13.540 -- Then you match half step to .2 using this slope.
00:17:19.110 -- So you have you have X N + H over to you. This is your X
00:17:25.462 -- displacement an in. Why you do Oilless step with step size H of
00:17:30.623 -- it but slow K1.
00:17:33.320 -- So once you have this point, you use this point
00:17:37.380 -- to evaluate slope.
00:17:39.970 -- So I compute slope K2 and I find
00:17:43.858 -- .3. By marching again from KXAN half step and using alone Def
00:17:50.700 -- line with slope Cato.
00:17:53.650 -- OK, this gives me point X 3.3, so from .3 then we match full
00:18:00.090 -- step to find point for using Slope case 3.
00:18:05.020 -- Once you have all these four slopes, you have weighted
00:18:08.630 -- average so you have you give weight 1 to the first point and
00:18:13.323 -- to the last point, but two weights to the .3 and two and
00:18:18.016 -- three. So overall you have for slopes six slopes. So you divide
00:18:22.348 -- age by 6.
00:18:24.130 -- So this is your average weighted slope.
00:18:28.090 -- OK, and then you can write this slope like even if you don't
00:18:32.640 -- know this. So you use information from four points. OK
00:18:36.140 -- to find, so this is a full stage
00:18:38.940 -- method. Anne.
00:18:43.250 -- In order to go from X&YN 2 X N + 1 one plus one, it is still
00:18:49.098 -- using only one previous point, right essentially, but it does
00:18:52.538 -- it in four in four stages.
00:18:55.260 -- OK.
00:19:01.760 -- OK, So what I can say here is there wrong accoutre
00:19:08.756 -- force order matches there?
00:19:14.230 -- The local error in their own decoder 1st order method is of
00:19:18.406 -- order H as a power 5.
00:19:21.980 -- OK, but when you find cumulative error then you lose one order
00:19:27.476 -- and then overall the error is.
00:19:31.300 -- Proportional to H is about four and you can. You can appreciate
00:19:35.116 -- it if H is let's say 0.01 to 10 to the point is the power of
00:19:40.204 -- negative one right? All this method will have error also of
00:19:43.702 -- the order of 10 to the minus
00:19:45.928 -- one. Right modified order will have error to the order 10 to
00:19:50.999 -- the minus. Two but longer code will have error of the order 10
00:19:56.205 -- to the minus four right? So you see that it's occasionally.
00:20:00.180 -- Logic difference in the in the accuracy. So all this method in
00:20:03.744 -- order to get the same accuracy.
00:20:06.480 -- You need to use smaller H. Ruby code allows you to use larger
00:20:11.992 -- step size. Because the the error is small and So what you save,
00:20:17.542 -- you save the number of steps. But again, remember that one
00:20:21.634 -- step of the longer quota has
00:20:23.866 -- four stages. So at each stage you have to evaluate function
00:20:28.565 -- and function evaluation may be consuming, so that's so. That's
00:20:32.015 -- why it's not very cheap method because at every step you have
00:20:36.155 -- four function evaluations.
00:20:39.020 -- OK.
00:20:40.850 -- How do we check that method is first order accurate? If we
00:20:45.686 -- decrease H by half, their level decreased by a factor of.
00:20:55.410 -- If H is replaced with H / 2, so the arrabelle decreased
00:20:59.622 -- by a factor of.
00:21:03.430 -- 22 to the power. 416 right so this is, you see, is a
00:21:08.929 -- significant difference between this method and that method OK?
00:21:14.650 -- Which method you would like to use if you have
00:21:17.570 -- to solve your problem?
00:21:22.740 -- So you have a choice. You have three methods and you have to
00:21:27.095 -- implement MCF thread programs, foiler for modified or Lefranc
00:21:30.110 -- equal to which method you would start with.
00:21:33.900 -- If you want to solve the problem that you don't know
00:21:36.595 -- solution about anything about.
00:21:39.660 -- Probably oil it while it's easy to implement, its lately least
00:21:43.037 -- accurate, but it's easy to implement, and for example, if
00:21:46.107 -- you programmed at an, you see that it doesn't work. Maybe
00:21:49.484 -- there is no point of investing time, right? But if you know
00:21:53.168 -- that yes solution exists, an that gives you what you need,
00:21:56.545 -- you can start with all the method just to get a feeling of
00:22:00.536 -- what solution is going to do. But then if you need to have
00:22:04.527 -- more accuracy, or let's say if you have to compute for long
00:22:08.211 -- time and maybe. Many points then you probably would use on
00:22:12.492 -- GeForce order method. Matlab in fact has so called variable
00:22:15.782 -- Force 5th order method ricotta which allows us to change the
00:22:19.401 -- step size depending on the estimate of the error. So they
00:22:23.020 -- have some estimate of the error in air is small. Then
00:22:26.639 -- you can use largest largest step. If estimate becomes
00:22:29.600 -- large then you decrease the time step so it's not
00:22:32.890 -- constant, is not the same method that would be
00:22:35.851 -- considered here.
00:22:37.550 -- OK, I mean whatever Matlab built-in function solver.
00:22:42.500 -- OK, so an example and I'll have this available on the course
00:22:47.564 -- website and then I'll give you a hand out next time just to show
00:22:53.472 -- you what what is going on in this ricotta method. So if we
00:22:58.958 -- want to solve this initial value problem starting from .12 and
00:23:03.600 -- finding oh at 1.4 in two steps using force ordering decoder
00:23:08.242 -- method, so two steps means that.
00:23:11.460 -- What is H we go from 1 to 1.4.
00:23:15.900 -- Each is.
00:23:19.750 -- So age is 1.4 -- 1 / / 2, so this will give us.
00:23:27.190 -- Zero Point 4 / 2 will be 0.2, right? So this is your step size
00:23:32.515 -- capital N number of steps is 2 inside of each step. How many
00:23:37.130 -- stages do you have?
00:23:39.330 -- Four stages right? So 4th function evaluations. So for
00:23:42.480 -- each stage you have to write K1K2K3K four and then the
00:23:46.330 -- weighted average to find next
00:23:48.080 -- approximation. So K1K2K64 will be different for
00:23:51.738 -- inside of each step.
00:23:55.390 -- OK, so H with no envy, no initial condition. X Zero is
00:23:59.626 -- one, XY0 is 2 OK, what is a function function is X + sqrt y.
00:24:04.921 -- This is your function F so F of XNYN is X N + sqrt y N.
00:24:12.780 -- OK, and then you carefully substitute these values, right?
00:24:16.263 -- I mean it's OK for demonstration purposes, so you probably want
00:24:20.520 -- to have this done by computer right? Unless function is simple
00:24:24.777 -- that you can, you can do it. OK, so gave one is a slope at first
00:24:30.969 -- point. In this case at X0Y0, right? You find Cato is you
00:24:35.613 -- March, you replace X with 0 + H to point in between and Y zero.
00:24:41.418 -- You follow The Cave one slope.
00:24:45.130 -- Right, so this is your X value. This is your why value once you
00:24:48.882 -- have them, you substitute them in the function, so you replace
00:24:51.830 -- X with this. Why is that?
00:24:54.140 -- Annual value it so this gives you slope K2 then use K2 here to
00:24:59.768 -- find .3 again. X is just half step away while zero plus K 2 *
00:25:05.798 -- H / 2 This is your ex. This is your Y value you put in the
00:25:12.230 -- function you evaluate. Finally K 4 you much full step.
00:25:16.890 -- Use slope case 3. This is your X value. This is your.
00:25:20.694 -- Why will you find slope K 4? You take weighted average.
00:25:24.181 -- You get next approximation.
00:25:31.380 -- OK, so now what you found you found.
00:25:37.130 -- X1 is 1.2 and Y one is 2.5201.
00:25:45.570 -- So now you use this.
00:25:47.540 -- To do another step so we have two steps here to do.
00:25:51.350 -- Right, so we have this and then again K1K2K3K four. But now
00:25:56.990 -- instead of X0Y0 you have X1Y1.
00:26:00.690 -- Just indexes shifted and so on, so I'll have this online and
00:26:05.034 -- I'll bring this on Monday.
00:26:09.020 -- OK, any are there any questions yes.
00:26:14.720 -- This is based on.
00:26:17.670 -- Next one you just. Right, you found this one from the previous
00:26:22.220 -- right step and then you just keep it the same, but you keep
00:26:26.640 -- adding. So what I do OK, I have formulas dependent on X&YN
00:26:30.720 -- right? So here I had to use.
00:26:34.040 -- My end was zero.
00:26:37.050 -- So I replace end with zero everywhere before I try to
00:26:41.285 -- compute anything. So in the next stage I have to use N equals.
00:26:46.910 -- 1.
00:26:48.560 -- OK so I replace.
00:26:51.160 -- SNV X1 Y end with Y1 and similarly everything else but
00:26:56.011 -- K1K2K3 will be different now from the previous case from the
00:27:00.862 -- previous step. So I have F of X1Y1 compared to.
00:27:06.930 -- F of X0Y0 I have for K2 I have F of X1 plus HY one plus K 1 * H
00:27:14.250 -- / 2 I have here with HO, but this key one and escape one of
00:27:19.740 -- the same. OK, so at every state at every step you
00:27:24.894 -- K1K2K3K four will be different, so he probably
00:27:28.142 -- technically we have to write down another index an, but
00:27:32.202 -- it just will increase. It will be very cumbersome. So
00:27:36.262 -- so all slopes are different. So for each step you
00:27:40.322 -- recompute your slopes.
00:27:44.870 -- OK, that's why.
00:27:47.280 -- Write this before you implement your substitute values.
00:27:52.260 -- OK, right X 0X1YY1Y2 and so on.
00:28:00.600 -- This will not be on the test.
00:28:04.540 -- OK, but it is in the homework so so you have to do it.
00:28:10.850 -- OK, any other questions?
00:28:16.150 -- So more about numerical methods. So we teach a
00:28:20.236 -- course which is now taught between three department's
00:28:23.868 -- mathematics, physics, and engineering is typically
00:28:26.592 -- chemical genius teaching and then so this method are
00:28:30.678 -- studied in more details, but not only this, but also
00:28:35.218 -- root, finding methods, argon values, eigenvectors,
00:28:37.942 -- solving linear systems. So maybe I should write so.
00:28:47.210 -- More about.
00:28:58.720 -- Anne.
00:29:01.050 -- 428 and there's also so this physics for 28 and engineering.
00:29:07.850 -- So it is the same course. I mean, of course the also
00:29:12.602 -- graduate version.
00:29:15.760 -- 529 I think and physics.
00:29:20.070 -- 528 So it's slightly dependants who is teaching, but we cover
00:29:24.437 -- the same material, so professors from different departments POV
00:29:28.010 -- alternate, but we have the same syllabus to follow.
00:29:36.920 -- No, normally you choose whatever flavor you want on
00:29:40.664 -- your transcript, but that's the only difference.
00:29:46.810 -- OK questions.
00:29:51.800 -- So.
00:29:53.810 -- I'll start Chapter 3, which is linear equations of
00:29:57.689 -- higher order.
00:30:16.410 -- So far we've dealt only with first order linear equations,
00:30:20.940 -- but we will look at their methods that will allow us to
00:30:26.376 -- solve equations of high order and linear equations do not
00:30:30.906 -- require. Coefficients to be constantly constant, but we will
00:30:35.444 -- for simplicity we will start with questions of miss
00:30:39.656 -- constantly efficients. OK, so let's just recall the definition
00:30:43.868 -- of the linear equation of ends order so linear.
00:30:51.720 -- And order.
00:30:54.120 -- Differential equation. Has function derivative, second
00:30:58.768 -- derivative, and so on up the derivative order NPL linearly in
00:31:04.675 -- the equation so?
00:31:07.440 -- Hey Ann.
00:31:13.720 -- Plus a N -- 1.
00:31:21.670 -- Loss etc plus a 2X.
00:31:25.550 -- D2Y T X ^2.
00:31:29.040 -- Plus a one of X.
00:31:34.190 -- Plus a 0 times function Y
00:31:37.382 -- equals. Some function that does not depend on why.
00:31:44.150 -- So remember.
00:31:46.460 -- How, how, how we define linear function we defined in a
00:31:50.673 -- function is a X + B right? So your independent variable AP is
00:31:55.652 -- linearly means raised to the power one. So now in the linear
00:32:00.248 -- differential equation you have the same but for the function
00:32:04.078 -- derivative, second derivative and up to the ends of the
00:32:07.908 -- derivative. These are the functions of X only.
00:32:11.540 -- Right then they don't involve why dependence are
00:32:14.716 -- of X is right inside.
00:32:18.020 -- Can be 00 but linearity means that you don't have y ^2.
00:32:22.604 -- Don't have y * Y prime and so on so so they appear linearly
00:32:27.952 -- same way as X appears in the linear function.
00:32:32.770 -- In this case, we multiply by constant in the equation. In
00:32:36.268 -- the case of, the equation, coefficients can be functions
00:32:39.130 -- of X at most.
00:32:41.960 -- OK. So if.
00:32:46.070 -- Oldest coefficients.
00:32:51.390 -- Constants.
00:32:57.190 -- Then we have equations with constant coefficients.
00:33:00.960 -- Then differential equation is.
00:33:05.460 -- A linear.
00:33:08.260 -- Differential equation with.
00:33:13.190 -- Constant.
00:33:18.560 -- Coefficients. And these are, these equations are
00:33:23.016 -- typically easier to solve. Otherwise equation has
00:33:26.103 -- variable coefficients.
00:33:36.630 -- This differential equation is.
00:33:41.680 -- Linear, viz.
00:33:48.200 -- Variable coefficients.
00:33:53.330 -- OK.
00:33:55.070 -- If you have a linear equation an if right hand
00:33:59.190 -- side is identically zero, then we have linear
00:34:02.486 -- homogeneous equation and in fact homogeneous equation
00:34:05.370 -- only can be introduced for linear equations. I mean
00:34:09.078 -- sometimes can be introduced for nonlinear, but typical
00:34:12.374 -- is for linear equations.
00:34:15.830 -- Then
00:34:19.440 -- linear differential equation.
00:34:22.060 -- Is homogeneous.
00:34:29.190 -- Otherwise.
00:34:35.270 -- Linear differential equation is.
00:34:42.490 -- Nonhomogeneous
00:34:47.710 -- let's look at some examples that we've just trying to classify
00:34:51.593 -- and then to analyze the order if it is linear. If it is
00:34:56.182 -- homogeneous or non homogeneous.
00:35:01.010 -- So Y double prime plus X y = 0. So what is the
00:35:05.716 -- order of this equation?
00:35:09.740 -- 2nd order.
00:35:12.000 -- Is it linear or nonlinear?
00:35:16.800 -- Linear right? Because XY is multiplied by a function of
00:35:21.040 -- XY, double prime is multiplied by one. So linear is a
00:35:25.704 -- sensitive linear. Is it homogeneous or non
00:35:28.672 -- homogeneous?
00:35:32.050 -- Homogeneous because there is no function that only depends on X
00:35:36.428 -- rated 0 so homogeneous.
00:35:42.100 -- Coefficients are constant or variable.
00:35:46.930 -- Variable because we have X right? So this.
00:35:55.180 -- Variable coefficients. OK.
00:35:59.730 -- What about this equation?
00:36:03.530 -- X ^2 y double prime minus two XY prime plus Y to the XY equals
00:36:10.640 -- two X -- 1.
00:36:13.550 -- Order
00:36:15.760 -- 2nd. Is it linear or nonlinear?
00:36:25.740 -- OK, so we have Y times each of the XY prime times minus two XY
00:36:30.630 -- double prime times X squared. We have termed it depend on on why
00:36:34.868 -- is it in this form?
00:36:38.870 -- That you have derivatives multiplied by at most
00:36:41.350 -- functions of X.
00:36:43.760 -- Yes, so it is linear, right?
00:36:46.730 -- Is it homogeneous since it is linear or not homogeneous?
00:36:51.860 -- None, because we have to explain this one.
00:36:56.590 -- So, nonhomogeneous? And coefficients are variable
00:37:00.892 -- variable right? Because we have functions so this.
00:37:07.660 -- Variable coefficients.
00:37:12.280 -- OK, next example.
00:37:15.440 -- Is 2 Y triple prime minus three Y prime plus seven Y equals
00:37:22.499 -- luxury four X ^2 -- 1?
00:37:26.490 -- OK, the order of the equation is 3 third order.
00:37:35.940 -- Is it linear or nonlinear?
00:37:42.440 -- Huh?
00:37:43.970 -- Linear or nonlinear?
00:37:47.530 -- Why is it nonlinear?
00:37:52.610 -- We have function multiplied by 7 derivative multiplied by
00:37:57.002 -- negative three, so the order to multiply by two.
00:38:03.220 -- Linear.
00:38:07.420 -- What is in here an is 3.
00:38:10.950 -- Look for linear equation. You have function multiplied by at
00:38:14.430 -- most, so this this may be
00:38:16.518 -- constant. Or maybe some function of X. This functional effects
00:38:20.159 -- may be nonlinear, but we look at the look at the YY prime Y
00:38:24.373 -- double prime up to the highest order derivative, not in terms
00:38:27.684 -- of X in terms of Y.
00:38:30.880 -- OK. So equation is.
00:38:34.260 -- Linear.
00:38:36.670 -- Since it is linear, is it homogeneous or homogeneous?
00:38:41.910 -- Non, because of the logarithm of X ^2. So nonhomogeneous
00:38:46.250 -- and coefficients are.
00:38:48.580 -- Constant rate with constant coefficients.
00:38:54.990 -- OK and last example.
00:38:59.640 -- White triple prime my plus 2Y double prime
00:39:04.216 -- minus y * Y prime +7.
00:39:08.920 -- The order is.
00:39:11.750 -- So the order. 3rd order.
00:39:16.710 -- Linnaean olenia.
00:39:21.560 -- Nonlinear because we have y * y prime right nonlinear.
00:39:28.920 -- We cannot say if it is ominous nonhomogeneous becausw we don't
00:39:33.980 -- have linearity to say this.
00:39:38.200 -- OK.
00:39:40.660 -- So big chunk of this course will be devoted on the 2nd order well
00:39:45.924 -- probably not sister going to order, so essentially it's
00:39:49.308 -- easier probably to solve 2nd order equations, especially when
00:39:52.692 -- you consider with variable coefficients. But the method
00:39:55.700 -- that we will develop for equations with constant
00:39:58.708 -- coefficients can be easy.
00:40:00.470 -- Applied to the 2nd order first Order 5th order intense order I
00:40:06.086 -- will have 19th order example to consider. So yes.
00:40:15.070 -- It is defined only for linear for linear equations, so.
00:40:21.530 -- I've seen some definitions that say if identical is zero
00:40:24.950 -- solution satisfies equation, then you can think of this as
00:40:28.370 -- homogeneous. In this case it won't be because if you have
00:40:32.132 -- zero then this is 0. This is non 0 but typically homogeneous is
00:40:36.578 -- only for linear equations because you have some relation
00:40:39.656 -- to linear algebra. So linear systems, linear equations so
00:40:42.734 -- that that's the reason. So once you may have a question on the
00:40:47.180 -- test to classify equation equations and then so similar
00:40:50.258 -- like like we we've done here.
00:40:52.400 -- You look at the order if it is linear then you can think
00:40:56.716 -- it's homogeneous, nonhomogeneous, but if it's
00:40:58.708 -- not linear then you just stop.
00:41:01.720 -- OK.
00:41:06.820 -- OK, so let's start with second order linear
00:41:10.396 -- homogeneous equations so.
00:41:15.190 -- So we consider 2nd.
00:41:19.420 -- Modern.
00:41:24.600 -- Linear homogeneous differential equations.
00:41:30.280 -- We will first address the problem when we have none of
00:41:34.031 -- them, we have homogeneous equation. Once we know how
00:41:37.100 -- to solve homogeneous then we will study how to solve
00:41:40.510 -- nonhomogeneous equations because there are different
00:41:42.556 -- methods how to address this problem. OK, so in general,
00:41:45.966 -- if you have second order linear equation then you can
00:41:49.376 -- write it in just using some coefficients which are
00:41:52.445 -- functions of X.
00:41:55.200 -- A1 of X.
00:41:57.440 -- Divide the X + A zero XY homogeneous. This means very
00:42:03.776 -- inside is 0.
00:42:15.190 -- And so let's look at example and then we will try to establish
00:42:20.546 -- some properties of solutions to the homogeneous equations.
00:42:25.750 -- So example is.
00:42:38.500 -- Let's let's let's do 2 examples, so example a.
00:42:43.530 -- X ^2 D two YG X ^2 -- 2 X divided X.
00:42:51.680 -- Plus plus two y = 0.
00:42:54.760 -- So you can see it is second order, right?
00:42:58.620 -- It is linear 'cause you have y * 2 divided you exams minus 2X and
00:43:03.480 -- this is also linear term and it is not just homogeneous because
00:43:07.368 -- there is no function that only depends on X and not multiplied
00:43:11.256 -- by wire derivative and.
00:43:13.310 -- My first statement is that the X ^2.
00:43:17.680 -- Is a solution of this equation.
00:43:21.770 -- How do we? How do we verify that this function is a solution?
00:43:27.300 -- We have the substitute and check if you get identity right. OK,
00:43:30.840 -- So what do we have? If X squared is a solution, what is the
00:43:34.970 -- derivative of this solution?
00:43:37.500 -- 2X and 2nd derivative will be 2, so we have X ^2 * 2 -- 2
00:43:44.572 -- X times. Two X + 2 times function. So do we have 0?
00:43:51.430 -- We have two X ^2 -- 4 X squared plus two X squared
00:43:55.863 -- right, so cancels so 0 = 0. So this means that X squared
00:44:00.296 -- is a solution of the equation. What happens if we?
00:44:05.020 -- Multiply this function by by constant.
00:44:09.370 -- By some arbitrary constant.
00:44:13.130 -- The claim is that this is also a solution.
00:44:18.740 -- So C One is an arbitrary constant.
00:44:25.870 -- Indeed.
00:44:27.990 -- 1st Order derivative will be 2 C 1X and 2nd order derivative will
00:44:32.839 -- be 2C1, right?
00:44:35.290 -- So we have X ^2 * 2 C 1 -- 2 X times 2C. One X
00:44:43.162 -- + 2 * y C One X ^2.
00:44:48.260 -- C1 is present in all the terms, right and otherwise
00:44:51.760 -- you have two X ^2 -- 4 X squared X squared, so so this
00:44:56.660 -- is also zero. So again, if you take a solution of a
00:45:00.860 -- linear homogeneous equation multiplied by arbitrary
00:45:02.960 -- constant, you still get the solution, so this will be
00:45:06.460 -- still a solution.
00:45:08.850 -- So similarly.
00:45:11.370 -- And you can verify that X is a solution.
00:45:18.240 -- The first derivative is.
00:45:20.930 -- One second derivative is 0, right? So we have X ^2 * 0
00:45:26.819 -- plus. I'm sorry minus.
00:45:32.040 -- Minus two X * 1 + 2 times function you can see that
00:45:37.318 -- this is 0.
00:45:40.060 -- And if I multiply this solution by an arbitrary constant, I also
00:45:44.452 -- get a solution.
00:45:49.450 -- Let's say C 2 * X is a solution.
00:45:55.110 -- And we can verify this by substitute and so again, second
00:45:58.883 -- derivative will be 0, so we have X, y ^2 * 0 -- 2 X times C 2
00:46:05.057 -- + 2 * C Two X.
00:46:07.860 -- Zero and finally, if you consider linear combination of
00:46:12.225 -- these two functions.
00:46:14.860 -- In linear combination is you multiply function by constant by
00:46:19.410 -- different constant and you add
00:46:21.685 -- so C1. X ^2 + C two X is.
00:46:27.720 -- Also a solution.
00:46:33.150 -- OK, let's let's verify, because probably those cases are easy to
00:46:36.560 -- see. This one is a little bit tricky. OK, so we have X squared
00:46:40.900 -- times second derivative. What is the 2nd derivative here?
00:46:45.460 -- To see one right plus zero.
00:46:48.830 -- Minus two X times first order
00:46:51.908 -- derivative 2C1X. Plus C2.
00:46:56.020 -- And plus two times functions, so C One X ^2.
00:46:59.830 -- Plus it 2X.
00:47:02.870 -- Do we have here?
00:47:06.240 -- So if I look at terms with C1.
00:47:10.270 -- I have two X ^2 -- 4 X squared, two X squared, they cancel.
00:47:18.040 -- In terms with C2.
00:47:21.470 -- Minus two XY2 plus to exit to
00:47:24.837 -- also cancel. Right, and this is here.
00:47:29.070 -- So 0 = 0.
00:47:36.510 -- So what we showed here is that if you have linear homogeneous
00:47:41.178 -- equation an if you have solutions, you form linear
00:47:44.679 -- combination. So you multiply by constants and you add and you
00:47:48.958 -- have you keep them arbitrary. Then result is also a solution
00:47:53.237 -- to this equation.
00:48:01.930 -- So maybe just another example be.
00:48:05.740 -- G2Y G X ^2 +
00:48:09.486 -- 3. Divide X + 2 * y = 0 again. This is second
00:48:17.322 -- order equation. Linear homogeneous coefficients are.
00:48:22.880 -- Constant variable so 2nd order.
00:48:29.390 -- Linear homogeneous.
00:48:34.050 -- With constant coefficients.
00:48:39.860 -- And the claims here are that E to the minus X is a solution. So
00:48:44.615 -- at this point I'm not saying how we find them, we will. We will
00:48:49.053 -- know this soon, but let's just just check. So if you have it to
00:48:53.491 -- the minus X derivative will be minus E to the minus X second
00:48:57.612 -- derivative will be with the plus sign, right? So you have either
00:49:01.416 -- the minus X + 3 * E to the minus X minus sign plus two times
00:49:06.488 -- function E to the minus X.
00:49:10.010 -- You get 0 right, and similarly if you multiply by constant.
00:49:16.900 -- Is a solution.
00:49:20.190 -- That I will not verify, but you can see that this is also
00:49:23.284 -- straightforward to do.
00:49:27.080 -- And then another solution here available is E to the minus, 2X
00:49:32.864 -- is a solution.
00:49:38.360 -- And if we multiply by constant, it is a -- 2. X is a solution.
00:49:45.230 -- And finally, linear combination is.
00:49:50.340 -- Also a solution.
00:49:53.260 -- OK.
00:49:56.540 -- So so the result is how much time do I have left?
00:50:05.830 -- One minute. OK, so I'll I'll write the just result
00:50:09.690 -- so theorem.
00:50:12.320 -- So principle.
00:50:15.760 -- Of linear superposition?
00:50:22.460 -- It only works for linear homogeneous equations, so given.
00:50:28.790 -- 2nd order equation.
00:50:39.160 -- 2nd order.
00:50:42.280 -- Linear.
00:50:44.550 -- Homogeneous equation.
00:50:51.200 -- If. Why one of XY2 of X?
00:50:56.650 -- Our solutions.
00:51:01.250 -- Of this differential equation.
00:51:05.810 -- Then
00:51:08.050 -- their linear combination.
00:51:15.580 -- C1Y one of X + y two Y2FX is also solution
00:51:21.212 -- of the same equation.
00:51:37.400 -- See once you're here.
00:51:41.630 -- C1C2 are arbitrary constants.
00:51:50.360 -- And similar result holds 4th order equations, right? So this
00:51:54.430 -- doesn't change. OK, so I guess I'm out of time, any questions?
00:52:01.690 -- OK, thank you and drive safely.
00:00:28.270 -- Alright, welcome class. Today I'll we've got. It's one of my
00:00:33.242 -- favorite lectures is today, so I'm happy to share that joy with
00:00:38.666 -- all of you for class today. Alright, before we get going you
00:00:44.090 -- got your homeworks back just so that you know there are some
00:00:49.514 -- issues. A number of issues arose in this homework assignment, so
00:00:53.720 -- in problem 5 two finding the velocity downstream of the shock
00:00:57.724 -- appeared to be a problem. I think we talked a little bit
00:01:02.092 -- about that last time problem 5 seven finding the velocity
00:01:05.732 -- downstream of the shock appeared to be a problem as well, and
00:01:10.100 -- then finding the change in pressure in problem 20 was a
00:01:14.104 -- problem. Please the solutions are all on baby learn. Go look
00:01:18.108 -- at that and make sure that you can get those assignments done.
00:01:22.630 -- OK, I think for probably 5 two that a lot of you solve that as
00:01:26.590 -- a moving shock problem, and that's going to screw you up
00:01:29.494 -- from the get go. So it's just a stationary shock, that is, that
00:01:32.926 -- is a bread and butter. Normal
00:01:34.510 -- shock problem. OK, so make sure that again, compare
00:01:38.642 -- your solutions your homework with with the solutions that
00:01:42.476 -- are available online and make sure that you can solve
00:01:46.736 -- the problems correctly. OK, alright anybody go to the go
00:01:50.996 -- to career fair yesterday.
00:01:54.000 -- OK, engineers were in huge demand. I walked through, talk
00:01:58.920 -- to some of the employers navair. We have sent a number of
00:02:04.824 -- students to navair. They're looking for 250 engineers.
00:02:10.730 -- So if you're looking for work, it's a great place to
00:02:14.173 -- go, and you can apply gas dynamics to it. OK, you get
00:02:17.929 -- to work at airplanes to work on jets. I mean, how cool is
00:02:21.998 -- that? OK, so I've got just a little bit of information
00:02:25.441 -- here. If you are interested, come on by after class, OK?
00:02:31.020 -- Let's get on. Let's get on here. Alright we are going to be
00:02:35.622 -- reviewing oblique shockwaves, and then we're going to see some
00:02:39.162 -- applications of those shockwaves. We're going to see
00:02:41.994 -- why, why airplanes are designed the way they are now that we
00:02:46.242 -- know about oblique shockwaves here. OK, we're going to learn
00:02:49.782 -- about supersonic diffusers, and that's just a fancy name for an
00:02:53.676 -- inlet to a jet aircraft. That's all it is. OK, so we're going to
00:02:58.632 -- learn about jet inlets today.
00:03:00.550 -- We learn about reflected shockwaves and you have a
00:03:04.087 -- homework problem on reflected shock waves and then we'll talk
00:03:08.017 -- a little bit about the differences between subsonic and
00:03:11.554 -- supersonic aerodynamics. OK, so before we get going a couple of
00:03:15.877 -- principles I want you to keep in your noggins. What we talk about
00:03:20.986 -- some of our material today. OK,
00:03:23.344 -- First off. The change of entry is equal to that of the negative
00:03:29.051 -- natural log of the stagnation pressure ratio. OK, so if you
00:03:33.022 -- want to minimize the losses, if you want S 2 -- S one to be
00:03:38.437 -- close to 0 or as small as possible, you want this ratio
00:03:42.769 -- here on the right hand side to be as close to one as possible.
00:03:47.823 -- Can peanut to ever be greater than peanut one.
00:03:55.460 -- Because then what would happen to S 2 -- S one? What
00:03:58.820 -- happens and what that would violate what law?
00:04:02.550 -- Second law of Thermo dynamics.
00:04:05.170 -- If Pete shot 2 or greater than peanut one, that would be. That
00:04:10.084 -- would be negative.
00:04:12.000 -- Change there due decreasing the entropy. OK, not going to
00:04:15.740 -- happen. Alright, second thing to keep in mind we haven't. We
00:04:20.390 -- haven't developed this relationship, but we will
00:04:22.826 -- later on a semester. But I want you to keep this in mind
00:04:27.350 -- now and that is that the thrust generated an engine is
00:04:31.178 -- related to the stagnation pressure.
00:04:34.310 -- OK, the higher the stagnation pressure, the greater the thrust
00:04:38.460 -- that you can develop.
00:04:41.550 -- OK so again keep that in mind.
00:04:45.250 -- Stagnation, that the decrease in stagnation pressure is related
00:04:48.382 -- to the losses in the flow.
00:04:51.090 -- The thrust is related to the stagnation pressure. OK here we
00:04:55.248 -- go. Oblique shockwaves. This is just a review from what we did
00:04:59.784 -- last time, so.
00:05:01.790 -- We talked about. In fact, this is a hint for the first problem
00:05:05.755 -- we've talked about mock waves. That was the 2nd second lecture
00:05:09.110 -- I think second or third lecture in class and a mock wave occurs
00:05:13.075 -- when there is just a very small disturbance. We could measure
00:05:16.430 -- the angle and once we measure the angle we could calculate
00:05:19.785 -- with the model number was OK. That's a hint for one of your
00:05:23.750 -- homework problems. OK, now instead of just having a small
00:05:28.140 -- disturbance, just like a little Nick in the wall, now there's
00:05:32.375 -- going to be a large wedge or large angle right here. OK, when
00:05:37.380 -- that occurs, we have an oblique shockwave that forms, so flow
00:05:41.615 -- comes down. This way it has to be supersonic. Makes this turn
00:05:46.235 -- through this turning angle Delta. It creates a shock,
00:05:49.700 -- creates an oblique shock that has an angle of beta associated
00:05:53.935 -- with it right there.
00:05:56.110 -- OK, so again the turning angle Delta, the shock angle, beta,
00:06:01.962 -- animac angle, mu.
00:06:04.730 -- Make sure we've got those down there. OK, very good. An oblique
00:06:08.714 -- shock forms when a flow turns into itself, so you have a flow
00:06:13.030 -- that's coming down this way, and it's kind of like blocking it
00:06:17.014 -- off. OK, so flow comes in gets directed up into itself. An
00:06:20.998 -- oblique shockwave forms were going to talk next week about
00:06:24.318 -- what happens when you have a supersonic flow and the flow
00:06:27.970 -- turns away from itself.
00:06:30.170 -- OK, for now the flow is turning into itself. An
00:06:34.110 -- oblique shockwave forms, so here's something that's very
00:06:37.262 -- important to know. After that oblique shockwave, then
00:06:40.414 -- the flow follows the wall, so we have here flow that's
00:06:44.748 -- coming out in this duct. Supersonic flow makes it
00:06:48.294 -- turn Delta right here, creates an oblique shockwave
00:06:51.446 -- that has an angle beta here and the direction of the
00:06:55.780 -- flow is with the wall.
00:06:58.900 -- OK, yes.
00:07:05.750 -- The Mach angle is going to be somewhere likely in between
00:07:09.281 -- those right there. OK, so usually in oblique shock,
00:07:13.356 -- oblique shock problems, we don't use the mock angle very much.
00:07:19.170 -- OK so I just included it in this just so that you saw in fact,
00:07:23.325 -- the reason it's the dotted line is it's an imaginary angle. In
00:07:26.649 -- this problem, just so that you could distinguish the two other
00:07:29.696 -- angles from that. That's the only reason why it's there, so
00:07:32.743 -- we're primarily going to be concerned with the turning angle
00:07:35.513 -- and the shock angle.
00:07:37.330 -- OK, alright so here's a duct again. The flow follows the wall
00:07:41.398 -- in that direction. Here we have flow in. This wedge comes down
00:07:45.466 -- this way. Supersonic flow gets turned up at this angle Delta.
00:07:49.195 -- The flow follows the wall so we get the direction of the flow
00:07:53.602 -- there. This total angle right here is called the included
00:07:56.992 -- angle of the wedge.
00:07:59.190 -- Gate we're going to talk later on today about what happens when
00:08:03.186 -- you have a wedge like that at an angle of attack and see some of
00:08:08.181 -- the differences there. That's coming later on today. This just
00:08:11.511 -- review from last time. OK, we talked last time and showed how
00:08:15.507 -- the turning angle Delta, the shock angle, beta, and the
00:08:18.837 -- upstream Mach number M someone are related to each other
00:08:22.167 -- through this oblique shock equation. OK, so in any kind of
00:08:25.830 -- problem you're going to get two of those three values.
00:08:29.230 -- You'll get Delta. You'll get beta or M's of 1. Any two of
00:08:33.624 -- those, and if you know two, you can use the relation to
00:08:37.680 -- get three there OK. Appendix three. We talked about the
00:08:41.060 -- oblique shock chart. That's this right here. Here we have
00:08:44.440 -- the shock angle. Here is the turning angle right here, and
00:08:48.158 -- each one of those lines corresponds to a Mach number.
00:08:52.600 -- OK, so three variables here. That's all related through this
00:08:57.210 -- relationship again man use com problem. OK, it's going to make
00:09:02.281 -- your life much easier. Solvable shot problems. OK, alright or
00:09:06.891 -- your app? OK, good.
00:09:11.900 -- Right also recall that for every for every Mach number and
00:09:16.828 -- turning angle combination, so you have an upstream Mach number
00:09:21.308 -- Anna turning angle combination. There is a maximum turning angle
00:09:25.788 -- associated with that that maximum turning angle
00:09:28.924 -- corresponds to whether or not you have an attached oblique
00:09:33.404 -- shockwave right across here, or a detached shockwave. OK, if the
00:09:38.332 -- turning angle is greater than the maximum turning angle.
00:09:42.430 -- For that Mach number, then you have a detached shockwave. OK,
00:09:46.071 -- it's going to behave like a normal shock right up there if
00:09:50.043 -- the turning angles less than the maximum turning angle you have
00:09:53.684 -- an attached shockwave. How do you know what that maximum
00:09:56.994 -- turning angle is? Well, if you
00:09:58.980 -- go up here. Each one of these lines corresponds to a Mach
00:10:03.215 -- number. This let's if we look at 2.2 right here. Here's the line
00:10:07.700 -- from lot number of 2.2. This maximum point right there? The
00:10:11.495 -- tip of that little thumb that comes out there. If you go down.
00:10:16.950 -- You can read off what the maximum turning angle is going
00:10:21.064 -- to be. The turning angle is greater than that. Then what
00:10:25.178 -- is 2? That's about 25 or so degrees right there. The
00:10:29.292 -- turning angle is greater than 25 degrees from lot #22 you
00:10:33.406 -- have a detached shockwave.
00:10:36.350 -- OK.
00:10:39.160 -- That's a review from last time. Any questions?
00:10:43.410 -- Any questions at all? Yes.
00:10:48.860 -- Could you speak a little louder so I can hear very
00:10:50.598 -- well?
00:10:54.770 -- OK, so the very first slide right there. OK, the thrust is
00:11:00.158 -- proportional to the stagnation
00:11:01.954 -- pressure. So the higher the stagnation pressure, the higher
00:11:05.830 -- the thrust, the lower the stagnation pressure, the lower
00:11:08.665 -- the thrust. OK.
00:11:12.750 -- Good, any other questions?
00:11:16.350 -- OK, not not everything. Fly straight and level.
00:11:21.600 -- OK, so you could have say a wedge like we have here, so the
00:11:26.822 -- wedge is going to have some included angle right here, but
00:11:30.925 -- now this wedge, the wedge itself is bent down a little bit.
00:11:36.800 -- OK, so that means, so that means we're going to have different
00:11:41.528 -- conditions on the top and on the bottom of our wedge, right here.
00:11:47.250 -- OK, remember the flow follows the wall, so we have a flow
00:11:51.582 -- direction here and a flow direction here. Let's just let's
00:11:55.192 -- just draw this out here so we can see get an idea for what's
00:12:00.246 -- going on. OK, so let's say that we have a wedge here.
00:12:06.910 -- OK it has.
00:12:09.370 -- This angle Delta, so a total included angle of 2D right
00:12:14.122 -- across there we have flow that comes down this way some
00:12:18.874 -- supersonic flow. And the shockwave forms here. Anna
00:12:23.336 -- Shockwave forms here.
00:12:25.360 -- OK.
00:12:27.030 -- Nothing big there. Could you solve that problem if you knew
00:12:29.197 -- what the Mach number in the
00:12:30.379 -- turning angle was? Yeah, OK, look at com property. You get
00:12:33.832 -- the shock angle you could figure out what the pressure in each
00:12:36.880 -- one of those regions are.
00:12:39.610 -- And the directions of the flow. You could figure out what the
00:12:43.162 -- temperature is. All sorts of stuff. So now let's take this
00:12:46.418 -- wedge here and. Turn it up a little bit this way.
00:12:52.620 -- OK, so now we're going to have an angle of attack on this
00:12:57.924 -- wedge here. So now all exaggerate my turning angle.
00:13:01.596 -- So let's say that looks like this and like this, and the
00:13:06.492 -- flow comes this way. So the wedge still has a total
00:13:10.980 -- included angle of 2D. But now here is the centerline of the
00:13:15.876 -- wedge.
00:13:17.620 -- It makes an angle of attack.
00:13:20.610 -- Alpha
00:13:24.430 -- everybody see that.
00:13:27.700 -- OK, immediately, what could you see? What's going to be
00:13:30.570 -- different on the top part of the wedge compared to the bottom
00:13:34.014 -- part? What's different geometrically?
00:13:38.940 -- I heard somebody say someone.
00:13:44.040 -- Less flow.
00:13:46.830 -- OK, that might be the case. There's only going to be
00:13:50.680 -- differences. What's the turning angle for the flow on the top
00:13:54.530 -- compared to the floor on the
00:13:56.630 -- bottom? What's the turning angle?
00:14:00.610 -- So this half angle here is is
00:14:03.795 -- Delta. Right that half angle there is Delta, so knowing
00:14:07.531 -- that the half angles Delta and you have an angle of attack
00:14:10.879 -- Alpha, what's the turning angle there at the top?
00:14:16.590 -- It should be smaller.
00:14:18.900 -- Delta. Minus Alpha.
00:14:24.390 -- Right?
00:14:26.250 -- What's the turning angle going to be on the bottom?
00:14:31.610 -- Little bit here. How do we do it this way? So here is that
00:14:35.796 -- wedge it's coming in this way. So now the turning angle top
00:14:39.384 -- and bottom is Delta OK if it's just going in horizontally
00:14:42.673 -- here I take that angle down, has an angle of attack Alpha.
00:14:46.261 -- So what's the turning angle here at the bottom?
00:14:52.800 -- Here we go here. The turning
00:14:55.416 -- angles Delta. By now add.
00:14:59.570 -- An angle attack Alpha, what's the? What's the
00:15:01.770 -- turning angle on the bottom?
00:15:04.500 -- Delta plus Alpha.
00:15:06.820 -- So now I'm going to have a shockwave here and a shockwave
00:15:11.692 -- here, but this turning angle here at the bottom is going to
00:15:16.564 -- be Alpha plus Delta.
00:15:19.630 -- And here on the top, it's going to be.
00:15:23.780 -- Alpha minus depends on which one of those two is larger, but you
00:15:27.589 -- see, there's going to be different between those two
00:15:30.226 -- angles there. OK, so now let's think about this. You can you
00:15:33.742 -- see clearly, can you see clearly that there's going to be a
00:15:37.258 -- larger turning angle at the bottom and at the top you buy
00:15:40.774 -- that? I mean, if I kept this angle of attack this way,
00:15:45.395 -- eventually that's going to have no shockwave on top.
00:15:48.910 -- When Alpha is equal to Delta, when that turning angle is the
00:15:52.006 -- same as the wedge angle, there's no. There's no big shock that
00:15:55.102 -- forms, it's just a straight
00:15:56.392 -- line. OK, so now keep that in mind. So which
00:16:00.337 -- of those two shockwaves is going to be stronger?
00:16:08.500 -- Which is going to have the
00:16:10.666 -- highest change. Across there, what's going what, what? What do
00:16:14.858 -- you see with the flow going on there? Have a have a higher
00:16:19.421 -- turning there at the bottom.
00:16:22.230 -- Stronger shock. And the bottom everybody see that? OK, think of
00:16:26.998 -- it this way, you are turning that air in the higher
00:16:30.650 -- direction, you're deflecting it more on the bottom then you are
00:16:34.302 -- the top. OK, so there's going to be a stronger shock.
00:16:39.130 -- On the bottom.
00:16:41.670 -- And this is going to be a weaker shock here at the top.
00:16:46.670 -- OK, where is going? Where is the highest pressure going to be?
00:16:54.980 -- Top or on the bottom. The highest change in pressure on
00:16:58.687 -- the top is if we call this region one and this region 2 and
00:17:03.405 -- this region 3. Where's the higher pressure going to be in
00:17:07.112 -- region 2 or region three there?
00:17:09.920 -- Region 2.
00:17:12.000 -- That's why it's a stronger shock. There's a larger increase
00:17:15.510 -- in the pressure, so here P2.
00:17:18.380 -- Is going to be greater than P1 everybody by that.
00:17:23.310 -- K. Aerodynamicists what's that going to generate?
00:17:30.450 -- You have a wedge. Here you have an angle of attack right
00:17:33.906 -- here. The pressure on the bottom is higher on the is the
00:17:37.362 -- pressure on the bottom is higher than the pressure on
00:17:40.242 -- the top. What do you get lift?
00:17:43.660 -- That's supersonic aerodynamics, right there in a nutshell.
00:17:47.680 -- There's no nice curved airfoils.
00:17:51.800 -- OK, with this wedge right here, just just giving it an angle of
00:17:56.701 -- attack in a supersonic flow, you're going to generate a
00:18:00.471 -- higher pressure on the bottom then you will on the top and you
00:18:05.372 -- generate lift. Out of that?
00:18:09.570 -- OK. Next week, we'll talk more about supersonic airfoils
00:18:13.434 -- because there's one other. There's one other key ingredient
00:18:16.116 -- that we need to know more about that, but this is the crux right
00:18:20.288 -- here. OK, just because of that pressure difference because of
00:18:23.697 -- the way that you have a stronger shock on the bottom
00:18:26.370 -- then you have on the top, you're going to get higher
00:18:29.043 -- pressure on the bottom and lift.
00:18:33.590 -- That's supersonic aerodynamics right there, OK questions.
00:18:39.810 -- Alright.
00:18:41.650 -- Let's talk about diffusers. Now diffuser that is just a
00:18:45.160 -- fancy name for a jet inlet for the intake. That's where
00:18:49.021 -- the air comes into the airplane. Let's see now what
00:18:52.531 -- we can do in our now. Now that we know about oblique
00:18:56.743 -- shockwaves and normal shocks, let's apply those to see if
00:19:00.253 -- we can figure out why diffusers are designed the
00:19:03.412 -- way they are here. OK, again, here's the equation that we
00:19:07.273 -- started out class with.
00:19:10.200 -- That to minimize the change in entropy, we want to keep peanut
00:19:14.988 -- two as high as possible. OK, in supersonic aerospace design
00:19:18.978 -- that's called pressure recovery. We want to recover as much
00:19:22.968 -- pressure as possible here. OK, we want to keep peanut to as
00:19:27.756 -- high as possible here, OK?
00:19:30.640 -- So let's look at a couple of jet inlet designs and see how the
00:19:35.792 -- stagnation pressure is going to be here. OK, good, so we're
00:19:39.840 -- going to look at two inlets.
00:19:42.930 -- Air is coming into both of those inlets out of Mach
00:19:46.131 -- number of 2.5.
00:19:48.080 -- OK, in the first inlet design all we have is a normal shock.
00:19:54.510 -- OK, what is the stagnation pressure loss going to be in
00:19:58.019 -- that normal shock? OK.
00:20:00.340 -- Then we're going to have we're going to have air that
00:20:04.311 -- comes in again. Same lot number 2.5. Everything is the
00:20:07.921 -- same. It's going to go through a turn of 18 degrees.
00:20:12.930 -- So it's going to create an oblique shock. That's going to
00:20:16.263 -- slow it down, and then it goes through a normal shock.
00:20:20.720 -- OK, now you might ask, why does it have to be a normal shock at
00:20:25.550 -- the end there? So for jet turbine airplanes, the air inlet
00:20:29.092 -- always has to be subsonic.
00:20:31.720 -- OK, if you have jet turbines, OK, compressor blades that are
00:20:35.350 -- spinning that air has to go into that turban, sub sonically.
00:20:38.980 -- Otherwise you get shock waves that form along the blade, and
00:20:42.610 -- that's not a very good process. OK, so the idea in the jet
00:20:46.900 -- aircraft supersonic air comes in, you slow it down, recover as
00:20:50.530 -- much pressure as possible that sends it through the jet turbine
00:20:54.160 -- and then it goes out the
00:20:56.140 -- exhaust. OK, that's that's the process here. Alright, let's
00:21:00.488 -- look at this.
00:21:02.860 -- Get your books out.
00:21:07.380 -- And we have our very first inlet design right here, and let's see
00:21:13.893 -- what's going on here. OK, we have an inlet.
00:21:19.020 -- To this aircraft looks like
00:21:20.845 -- this. Looks like this here comes in at a Mach number of two
00:21:28.330 -- point 5K and as it enters this inlet here a normal shock.
00:21:35.800 -- Forms right at the inlet.
00:21:38.020 -- OK.
00:21:40.550 -- Good. We have that.
00:21:44.610 -- Will have that P1 is equal to 70 kilopascals and that T one is
00:21:49.902 -- equal to 200 degrees Kelvin.
00:21:52.880 -- OK, we want to know.
00:21:56.420 -- How much stagnation pressure is lossed in this process? Right
00:22:00.720 -- here? OK, so you can go to your normal shock tables. So go to
00:22:06.740 -- your normal shock tables and figure out for this inlet
00:22:11.040 -- design, what peanut too.
00:22:13.470 -- Over P, not one, is equal to.
00:22:21.160 -- What do you get?
00:22:26.200 -- 24 nine. 0.499 can we round that up to .5?
00:22:32.970 -- OK, that says that you have already lossed half the
00:22:39.180 -- available thrust.
00:22:42.040 -- In this design.
00:22:44.360 -- It's not very good.
00:22:46.970 -- That's terrible. OK, all the available thrust that you could
00:22:51.050 -- to generate to make your plane go faster. You've lost half of
00:22:55.250 -- it just because it sends through a normal shock.
00:22:59.940 -- OK, not good.
00:23:01.900 -- Not good, OK? Let's look at the second design now, here. So
00:23:06.460 -- let's go to the computer so we can see that again. So here now
00:23:11.780 -- we have a wedge right here. Flow comes in supersonically. There
00:23:15.960 -- is an oblique shock that's formed, and then a normal shock
00:23:20.140 -- that's formed here on the inside. OK, this normal shock,
00:23:23.940 -- right? There is called the terminal shock. It's the very
00:23:27.740 -- last shockwave in the system.
00:23:30.830 -- OK, remember as the air crosses it goes through the inlet an
00:23:34.634 -- approaches the engine. It has to
00:23:36.536 -- be subsonic. OK, so it goes. It's subsonic across a normal
00:23:41.039 -- shockwave. That's how it finally is going to slow down here. OK,
00:23:45.371 -- so let's calculate this problem right here, OK?
00:23:49.250 -- So we have a wedge. Looks like this and we have the top of
00:23:56.586 -- the inlet right there here.
00:24:00.630 -- OK, the flow comes in at a Mach number of 2.5 and there is an
00:24:07.605 -- oblique shock. That forms right across there and then a normal
00:24:13.078 -- shock right down here.
00:24:16.400 -- OK, we were given in this problem. This has a turning
00:24:21.834 -- angle. Delta is 18 degrees.
00:24:25.040 -- OK, let's we have three different flow regimes here, so
00:24:29.800 -- let's call this region 1.
00:24:32.980 -- Call this region 2 and Region 3 right there.
00:24:40.950 -- OK.
00:24:44.430 -- Are we good? Are we good with the visual here?
00:24:48.220 -- Pretty straightforward problem. We are now two separate
00:24:51.948 -- shockwaves. OK, so now you want to get your app out here for M1
00:24:58.472 -- equal to 2.5 and Delta of 18 degrees right there tell me what
00:25:04.530 -- you get. First off for M2 and then tell me what P not 2 /
00:25:11.520 -- P nought one is.
00:25:17.040 -- See if it comes with my numbers here.
00:25:20.840 -- Yep, 1.739 is the Mach number in region 2 peanut
00:25:26.720 -- two over peanut one.
00:25:30.750 -- 1.
00:25:32.240 -- 8870.88 seventh OK, I got 7 as well. I thought I heard you say
00:25:38.372 -- 6.8877 really that's one part out of 88,000. I'm not going to
00:25:43.628 -- worry about that too much. OK, we go there.
00:25:48.520 -- OK, you could go com prop gives you that you could go to the
00:25:52.300 -- charts and get the same thing. No big deal. Now what's the next
00:25:55.810 -- step in the problem?
00:25:57.580 -- What do we do?
00:26:00.860 -- It's a normal shock problem.
00:26:03.140 -- OK, so when we get to more complicated problems, we're
00:26:06.580 -- going to use the same process. You just go through a serially.
00:26:10.708 -- What's going on here? It's an oblique shock problem and then a
00:26:14.836 -- normal shock problem, but the Mach number in region 2 is
00:26:18.620 -- different than the model number in region 1, so you just have to
00:26:23.092 -- keep that into consideration there. OK, so for now we know
00:26:26.876 -- what the model number in Region 2 is. So from the normal shot
00:26:31.348 -- tables now. Read off
00:26:34.622 -- M3. And read off now Peanut three over peanut two.
00:26:40.574 -- What do you get?
00:26:48.350 -- What's M3 go into the normal shock tables for an M2 that you
00:26:52.198 -- could say that's one point 7 four just to read it off there?
00:26:56.990 -- What do you get?
00:27:01.730 -- 0.63 should it be subsonic?
00:27:06.860 -- Better be 'cause it's going across a normal shockwave right
00:27:09.980 -- there. What is peanut three over Peanut 2 for that normal shot?
00:27:18.500 -- Say that again.
00:27:20.480 -- .8 Four K 0.84.
00:27:26.690 -- OK, so we're going to look at some trends here before we
00:27:31.718 -- calculate our final number 'cause we want to know what
00:27:35.908 -- peanut three is compared to peanut one? That's our. That's
00:27:40.098 -- our stagnation. Pressure loss from this region to this region
00:27:44.288 -- right here. Look at look at these two numbers right here for
00:27:49.316 -- model number of 1.739. You lose
00:27:51.830 -- 16%. Of your stagnation
00:27:54.416 -- pressure. If you go up from 1.742 amount number 2.5, you
00:28:00.560 -- lose half. Of your stagnation pressure.
00:28:04.960 -- OK, let's look at those trends here.
00:28:09.680 -- OK, if you look at the normal shock tables right here.
00:28:15.730 -- Here is Peanut peanut, two over peanut one right there
00:28:20.070 -- for this mock number. Notice that to a mock number, let's
00:28:24.844 -- just say right here at amount number of 1.58 you lose 10%
00:28:30.052 -- of your stagnation pressure.
00:28:32.900 -- OK, for model number of 1.2 you only lose 1%.
00:28:39.190 -- Of your stagnation pressure. OK, if you go to a Mach number
00:28:45.430 -- of two right. Over here you lose
00:28:49.070 -- 28%. Of your stagnation
00:28:51.705 -- pressure. Everybody see what those values are coming from.
00:28:55.460 -- So. You're an aircraft designer. What should you do?
00:29:02.410 -- What should you do?
00:29:06.830 -- How about if you take this right here? Shouldn't you slow
00:29:10.350 -- that flow down as much as possible? It's going to have
00:29:13.870 -- to go through a normal shock.
00:29:17.110 -- But then you want that normal shock right there to
00:29:19.910 -- be as weak as possible.
00:29:22.460 -- Everybody see that. OK, so let's calculate now Peanut 20 excuse
00:29:28.168 -- me peanut three over peanut one that is the stagnation pressure
00:29:33.459 -- loss from this region to this region right here. OK so.
00:29:40.270 -- Peanut 3 / P nought one is equal to P not 3 / P
00:29:47.452 -- not two that's this.
00:29:50.370 -- Multiplied by peanut two over peanut one.
00:29:57.110 -- We know those two values .8.
00:30:00.330 -- 877 times .84.
00:30:07.840 -- I get a value of 0.745.
00:30:18.740 -- Compare values again. What have you done to your aircraft?
00:30:25.130 -- You have improved the pressure recovery by 25 points.
00:30:33.850 -- Instead of losing half of it, instead of losing half of your
00:30:37.786 -- stagnation pressure, now you lose just one quarter of it.
00:30:41.890 -- And what have you done?
00:30:44.210 -- All you did was put a little wedge at the inlet right there.
00:30:51.440 -- OK, This is why I wanted you to keep in your noggins. Delta S
00:30:56.410 -- changing entropy over R is equal to minus natural log of the
00:31:00.670 -- stagnation pressure loss.
00:31:02.590 -- Stagnation pressure is directly related to the thrust, so now
00:31:06.890 -- you have made your aircraft.
00:31:09.930 -- Actually 50% better you've improved in performance.
00:31:14.170 -- And all you did was put a wedge here before that terminal shock
00:31:19.110 -- is the only thing you did.
00:31:22.570 -- And you can get more thrust out
00:31:23.592 -- of your play. Is that cool or what?
00:31:26.920 -- OK, let's see. Here's an F-14 in low, by the way. Actually,
00:31:31.240 -- before we get to that before we get to that here.
00:31:36.750 -- Let's go to let's go to the overhead here. Could you see
00:31:41.526 -- that a natural extension to this problem might be if now you have
00:31:46.700 -- a wedge and you have a 9 degree
00:31:49.884 -- turn. And then, and another nine degree turn here and then.
00:31:57.270 -- It goes to the inlet right here. What have you got?
00:32:02.140 -- M1 is equal to 2.5. You'll have an oblique shock that
00:32:06.144 -- forms there.
00:32:08.700 -- The flow follows the wall. You have another oblique shock here.
00:32:15.650 -- Your total turning angle is 18 degrees, just like you did
00:32:19.236 -- before, but now you send it through two oblique shockwaves,
00:32:22.496 -- not just one. And then you have your terminal shock right there.
00:32:27.110 -- OK, as it goes into the inlet right here. So you have
00:32:30.446 -- region 1, Region 2, Region 3, Region 4. How would you solve
00:32:33.782 -- that problem?
00:32:37.080 -- Same way that we did the first problem. It's an oblique shock
00:32:40.488 -- across here with the turning angle of nine degrees and some
00:32:43.612 -- Mach number. Then over here you have a new model number M2
00:32:47.020 -- Anna Divina, same turning angle of 90 degrees there and
00:32:49.860 -- then you have a normal shock.
00:32:52.990 -- OK, in this design here Peanut 4 divided by peanut one is
00:33:00.550 -- equal to 0.79.
00:33:03.680 -- So you've improved the performance from 70 from losing
00:33:08.891 -- 25% to losing 21%.
00:33:12.880 -- OK.
00:33:14.970 -- Alright, now we can look at F-14
00:33:18.309 -- Tomcat here. Here is that inlet OK.
00:33:24.060 -- Uh, let's see what you can see here. OK, so actually, actually
00:33:27.852 -- here we'll look at it from here and then we'll see the picture.
00:33:31.960 -- So this is an F-14 Tomcat. We saw this in class last time
00:33:36.068 -- here. Here are the inlets here. Actually, if you look in the
00:33:39.860 -- overhead here, there's an inlet and there's an inlet there.
00:33:43.020 -- Notice it has a sharp leading
00:33:44.916 -- edge. OK, if you look at it on the side right there.
00:33:50.410 -- OK, there it has a very sharp leading edge. If you look
00:33:55.078 -- inside right here, zoom in. I'll just a little bit more.
00:33:59.357 -- You can see there's a hinge there and a hinge there.
00:34:06.080 -- So what this airplane is designed to do is that depending
00:34:10.469 -- on the speed, this inlet right here can change angles.
00:34:16.310 -- OK, and and what is going to be The upshot of changing the
00:34:20.561 -- angles? What is that? What is the trying to do?
00:34:25.230 -- Recover the pressure.
00:34:28.310 -- So now let's go look at the computer. You can get a better
00:34:32.301 -- idea of what it looks like here. There you can see a hinge Anna
00:34:36.599 -- plate right there. Here you can see another hinge and another
00:34:40.960 -- plate right there. So in this aircraft that's going to produce
00:34:44.645 -- three oblique shockwaves at the inlet, one at the very front. So
00:34:48.665 -- you can't see it. But that's right that first inlet. This is
00:34:52.685 -- where the second oblique shockwave forms. Here's where
00:34:55.365 -- the third oblique shockwave forms. And then inside here is
00:34:58.715 -- where the normal shock form. So very nice everybody see that we
00:35:02.735 -- can see it both at the inset right, right there in the upper
00:35:07.090 -- right corner. Thanks in the control room. That's very good,
00:35:10.878 -- OK? So good again, depending on the speed of the aircraft and
00:35:16.140 -- its conditions, it automatically adjusts the angles there.
00:35:20.410 -- To maximize the pressure recovery and generate the most
00:35:23.209 -- stress in the airplane.
00:35:25.910 -- Is that cool or what? That's awesome. That is awesome
00:35:29.210 -- engineering design right there. OK, so here. Here's a little
00:35:32.510 -- schematic of the inlets here. Here, the first one we see what
00:35:36.470 -- the flow pattern looks like for subsonic speeds, so obviously
00:35:39.770 -- there's going to be no shock waves that form there, so the
00:35:43.730 -- plane has to take off and ask to fly subsonic Lee at some point.
00:35:48.350 -- So it comes through here. It's got a little. It's got a little
00:35:52.640 -- bleed or there at the top. Air comes in this way and it goes
00:35:57.260 -- and notice here. It's always a
00:35:59.240 -- subsonic diffuser. Because you want the air going into the
00:36:02.668 -- engine and the jet turbine engine here to be subsonic. OK,
00:36:06.364 -- when it gets to transonic speeds, so that's from a model
00:36:10.060 -- number of .8 to about 1.2 ish in that range. OK, notice that you
00:36:14.764 -- have a normal shock that forms right there in the front, or
00:36:18.796 -- you're worried about the losses across that normal shock.
00:36:22.990 -- Because why Jesus? Why is that?
00:36:29.130 -- The upstream lot number here is really close to one and so the
00:36:33.745 -- stagnation pressure losses for Mott numbers close to one are
00:36:37.295 -- don't say negligible, but are pretty small, so not to worry
00:36:41.200 -- about that, OK?
00:36:43.700 -- In and out at supersonic speeds, look at this. There is an
00:36:48.068 -- oblique shockwave right there. There is an oblique shockwave
00:36:51.344 -- right there. There is an oblique shockwave right there and here
00:36:55.348 -- you can see the actuators. This is what moves those plates into
00:36:59.716 -- position to get the get the best pressure recovery and then
00:37:03.720 -- finally at this point is the terminal or the normal shockwave
00:37:07.724 -- right there behind that shock the flow subsonic and it goes
00:37:11.728 -- into the diffuser right here.
00:37:16.760 -- So now you could look at a jet aircraft and say I know why
00:37:19.406 -- it's designed that way.
00:37:21.270 -- To maximize stagnation pressure to maximize the pressure
00:37:25.382 -- recovery. OK, unfortunately, this plane does not fly anymore,
00:37:30.008 -- but just to let you know, there are aircraft companies are
00:37:35.662 -- looking to redesign or to remake this aircraft. In some sense,
00:37:41.316 -- this is a supersonic transport designed to fly from either
00:37:46.456 -- France or London, Paris or London across the Atlantic Ocean
00:37:51.596 -- supersonically. Has a cruising speed of Mach number
00:37:55.373 -- of two and land in New York OK. If you are ever in
00:38:01.340 -- Seattle, there's the Museum of Flight just South of
00:38:05.471 -- Seattle off of off of Hwy 5. I think there.
00:38:11.840 -- And there's a super Sonic.
00:38:14.670 -- There's there's this airplane there you can go check out OK,
00:38:19.730 -- look at the intakes here at the
00:38:22.950 -- bottom. OK, notice also even before we get there, notice the
00:38:28.090 -- Delta wing shape right there, because when this is flying
00:38:32.130 -- supersonically you want that conical you want the airplane to
00:38:36.170 -- sit inside that conical shockwave right there. Then as
00:38:39.806 -- it goes down inside here, notice the angle of those inlets right
00:38:44.654 -- there. Let's take a look and see what those inlets look like
00:38:49.502 -- right here. So you can see says danger. It's got hinges here.
00:38:54.620 -- And here as well.
00:38:57.370 -- So is this plane screws in over the Atlantic Ocean, creating
00:39:01.165 -- shockwaves? You remember the shockwave sounded like.
00:39:04.390 -- That's why it doesn't fly over the continental United States
00:39:07.750 -- flies over the ocean. OK, that those plates can go up and down
00:39:12.118 -- depending on its speed and the other conditions of the flight
00:39:15.814 -- there. To maximize the pressure recovery before it goes into the
00:39:19.510 -- engine there. OK fact, here's a schematic of that of that.
00:39:23.206 -- What's called the shock train? OK, so here's the first
00:39:26.566 -- shockwave that's an oblique shockwave. You know what? You
00:39:29.590 -- know, what oblique shockwaves are. So you can look at this
00:39:33.286 -- intelligently. Here's the 2nd.
00:39:34.700 -- Oblique shockwave right here. These are actually compression
00:39:37.756 -- waves and we'll talk about these next week, right here and
00:39:41.958 -- then another shockwave here. So 1/2 compression, 3 right there
00:39:45.778 -- and then finally the terminal shock there on the inside.
00:39:49.598 -- That's the normal shock. Downstream of that it's
00:39:52.654 -- subsonic flow and goes into the engine.
00:39:57.790 -- OK, and again you can adjust.
00:40:00.320 -- That ramp right there to maximize the pressure recovery.
00:40:05.120 -- Good, here is a MIG fighter aircraft, so this is a this is a
00:40:10.776 -- Russian Russian jet. It was used after World War Two during the
00:40:15.624 -- Korean War. OK, but notice it's design here. There's a spike
00:40:20.068 -- that sits there right in the center so the air comes in and
00:40:25.320 -- hits that spike. And what's going to form when it hits the
00:40:30.168 -- spike? An oblique shockwave. Actually, in this case of
00:40:33.804 -- conical shock, but.
00:40:35.090 -- Not a normal shot. OK can you see? Yeah you can see
00:40:39.074 -- on the screen there can you see this line right there?
00:40:44.140 -- What do you think that is?
00:40:47.670 -- An angle change.
00:40:50.040 -- So it comes in and the initial change in the flow is half
00:40:54.408 -- whatever that conical angle is right there. Then it hits this
00:40:58.104 -- an another oblique shockwave forms follows the wall, and it
00:41:01.464 -- eventually goes into its inlet right here and down inside
00:41:04.824 -- there's going to be normal shock.
00:41:08.700 -- OK. Here's another one right here. This is from a. I think
00:41:14.908 -- this is an F-104 Starfighter, so again, an older supersonic jet
00:41:19.506 -- fighter here. You can see the inlet right here, Annacone.
00:41:24.480 -- OK, so the flow goes over this and forms a conical shockwave.
00:41:28.032 -- Here it's probably a little hard to see on your on your on
00:41:31.880 -- the TV screens there, but you see these two little Nicks
00:41:35.136 -- right up there. OK, this cone. In fact you could see it right
00:41:38.984 -- there from this line that cone can move in and out.
00:41:45.180 -- So again, depending on the speed of the aircraft that cone is
00:41:49.476 -- going to go out or kind of go in to generate whatever sort of
00:41:54.488 -- oblique shockwave you need to maximize the pressure recovery.
00:41:57.710 -- OK, this is an older fighter aircraft in the 1960s and so
00:42:02.006 -- that moving the moving inlet technology isn't quite as mature
00:42:05.586 -- as it was for the SST, and some of the other aircraft. OK, but
00:42:10.598 -- the same principle holds. It adjusts this cone, moves in and
00:42:14.536 -- out so that.
00:42:15.760 -- So that the flow field generates the maximum amount of pressure
00:42:20.380 -- recovery in the airplane, Sir.
00:42:24.030 -- Square.
00:42:28.080 -- It depends. OK, so that's that's a great question. It
00:42:31.390 -- depends on where you put the inlets. Usually on the side
00:42:35.031 -- of the aircraft, right? Here, 'cause the flows coming right
00:42:38.341 -- down the this part of the fuselage. It tends to hug the
00:42:42.313 -- wall a little bit better. Then you haven't inlet here.
00:42:47.170 -- That that really isn't going to work for the F-14 Tomcat because
00:42:50.698 -- of the armaments and other stuff that you had on the side there.
00:42:54.520 -- So depends on the design. Depends on the design there and
00:42:57.754 -- how you want your fuselages to
00:42:59.518 -- work there. OK, any questions?
00:43:04.170 -- OK, this is a
00:43:08.162 -- subsonic turbofan. Jet inlet.
00:43:13.490 -- Can you tell the difference between that inlet and the other
00:43:16.548 -- inlets that we saw before?
00:43:18.940 -- OK, are shockwaves going to form here?
00:43:23.530 -- Negative. OK, So what you see here are nice curved surface
00:43:27.710 -- is you don't see any sharp surfaces on a subsonic
00:43:31.510 -- aircraft. OK, nice curved surface. Is there? These
00:43:34.550 -- compressor blades spin this way and compress the air as it
00:43:38.730 -- goes inside and comes out the engine. There's already coming
00:43:42.530 -- in subsonic Lee.
00:43:44.800 -- OK, so that's the difference between a subsonic inlet and
00:43:50.530 -- a supersonic inlet.
00:43:53.990 -- OK, let's just hypothesize here. Let's say that we use
00:43:57.790 -- this plane in a fly supersonically. What's going
00:44:00.830 -- to form on the outside right there?
00:44:05.640 -- Shock waves normal shocks, oblique shocks. What do
00:44:07.976 -- you think?
00:44:10.770 -- OK, since that surrounded surface right there, it's not
00:44:13.443 -- a sharp surface. You're likely going to get a
00:44:16.116 -- detached bow shock that's there. And what kind of
00:44:18.789 -- losses are associated with that?
00:44:21.710 -- Huge.
00:44:23.390 -- OK, we've just shown here that in oblique shockwave the losses
00:44:26.195 -- are less than for a normal shot.
00:44:28.930 -- OK. Good alright?
00:44:34.390 -- Questions on this?
00:44:37.270 -- We'll talk more. We'll talk more about it. OK, alright, you're
00:44:41.615 -- going to have a homework problem or some homework problems
00:44:45.565 -- related to reflected shockwaves. OK, so let's look at this. Let's
00:44:49.910 -- look at this particular diagram
00:44:51.885 -- right here. So we have flow that's in a duct.
00:44:56.706 -- Duct is coming down here has a Mach number of one.
00:45:01.403 -- It's coming down Supersonically and now
00:45:03.965 -- here we have an angle change of 12 degrees.
00:45:08.840 -- OK, so what's going to happen? Well, add this for this Mach
00:45:12.368 -- number and this turning angle of 12 degrees you're going to get a
00:45:16.190 -- shock angle. And the flow turns Y.
00:45:21.340 -- The flow follows the wall.
00:45:25.050 -- OK, so it's going to come down this way and it sees this little
00:45:28.858 -- change right there and it changes direction, so it's going
00:45:31.578 -- down horizontally and then it turns up 12 degrees.
00:45:35.290 -- OK. Then the flow comes this way right here, and then it sees
00:45:41.310 -- this wall on the top side right
00:45:43.830 -- there. So now the flow is going to change directions again.
00:45:50.270 -- So it's coming down.
00:45:52.620 -- It goes up 12 degrees. It's going to hit the wall on the top
00:45:56.134 -- that's parallel to the bottom wall on the other side. Here,
00:45:58.895 -- it's going to hit that wall and
00:46:00.652 -- change directions again. So now this direction of M3 is the same
00:46:05.413 -- as M1. The directions the same.
00:46:08.890 -- How would you solve that problem?
00:46:15.480 -- Could you solve the problem from Region 1 to region 2?
00:46:20.110 -- You know in one you know the turning angle. You could get the
00:46:23.711 -- shock angle and you could get the Mach number right there.
00:46:27.490 -- In order to see that.
00:46:29.570 -- OK, how would you solve it from Region 2 to region 3?
00:46:35.010 -- Same thing.
00:46:37.270 -- However, however, are the turning angles the same?
00:46:43.260 -- Turning angles are the same, what's different?
00:46:48.640 -- Get shock angles gonna be different? How does
00:46:51.016 -- M2 compared to M1?
00:46:53.850 -- Smaller.
00:46:56.800 -- OK, so you solve it from Region 1 to region 2.
00:47:00.672 -- From this Mach number M1 and that turning angle.
00:47:05.060 -- Then you're going to calculate or determine what your M2 is.
00:47:09.100 -- And then solve this turning angle problem right here with
00:47:12.900 -- M2, not M1.
00:47:14.930 -- M2 with the turning angle of 12 degrees and it comes
00:47:18.590 -- out this way right here.
00:47:22.260 -- Everybody see that.
00:47:26.100 -- That's all reflected shock is all. Reflected shock is, it's
00:47:29.490 -- just, it's just another in a series of oblique shockwaves as
00:47:33.219 -- it comes down the duck there.
00:47:36.080 -- Nothing more than that. OK, and all because of the changes here
00:47:40.892 -- in the angles.
00:47:43.790 -- OK.
00:47:45.780 -- I believe that you have a homework problem, will just
00:47:49.960 -- outline it here. OK, you have a homework problem.
00:47:55.230 -- Let's see here.
00:47:57.330 -- Have a duct comes down here and
00:48:00.032 -- there's some. Change here Mitch was at 5 degrees.
00:48:06.100 -- We talked about this in class. I think this is 5 degree change
00:48:10.247 -- here. OK then you have another
00:48:12.161 -- wall. Here, and this is bent down at 2 degrees.
00:48:22.770 -- 2 degrees right there.
00:48:24.960 -- OK, so you have a flow.
00:48:28.100 -- Coming down this way, M1, what's going to form first?
00:48:34.060 -- What do you get? 'cause of this turn right up there?
00:48:39.150 -- An oblique shot.
00:48:41.360 -- So I'm going to be shocked that comes here.
00:48:44.960 -- OK, and this region 1 this is Region 2 right here. Given
00:48:49.124 -- that turn and that model number, can you find the
00:48:52.594 -- pressure and all the good stuff in region two there?
00:48:58.020 -- It's just an oblique shock problem. OK, now remember the
00:49:02.180 -- flow follows the wall.
00:49:04.750 -- So now it's coming down here at 5 degrees.
00:49:08.290 -- OK, coming in this direction.
00:49:10.440 -- However. This turn, this wall at the bottom is not.
00:49:16.700 -- 5 degrees.
00:49:19.680 -- OK.
00:49:22.000 -- Let's do a hypothetical. If this wall did come down at 5 degrees.
00:49:29.360 -- What would form from region 2 to this region down here?
00:49:33.910 -- Actually nothing.
00:49:37.550 -- 'cause it's going to go down this duck and turn this way.
00:49:41.650 -- You could have a normal shock sometime later on there, but you
00:49:44.134 -- all you do is turn in the flow.
00:49:46.860 -- OK, if this were five degrees. However, now this angle is 2
00:49:52.872 -- degrees from the horizontal.
00:49:55.800 -- So guess what's going to form.
00:49:59.270 -- No big shockwave.
00:50:00.590 -- Right here.
00:50:04.110 -- OK.
00:50:06.530 -- I will leave it up to your geometrical Wiles
00:50:09.959 -- to figure out what this turning angle has to be.
00:50:15.750 -- Actually kind of hinted on how you could solve it. OK, the
00:50:19.698 -- turning angle is what you're going to figure out. OK, this
00:50:23.317 -- turn here is 2 degrees. What is the turning angle of that flow?
00:50:30.880 -- Think about it, yes.
00:50:36.130 -- What is the form here? Because this is where it's being turned.
00:50:43.390 -- OK, so it's coming OK. So the reason the reason it's attached
00:50:46.762 -- here 'cause the flow comes down and that's where it turns.
00:50:50.780 -- OK, so in this problem it's just kind of hypothetical that had
00:50:54.416 -- this turn right. Here we have all the conditions such that
00:50:57.749 -- this oblique shockwave comes down right to that point.
00:51:02.210 -- OK, now it's being turned again.
00:51:06.130 -- From here. Again, if this wall were turned down 5 degrees, it
00:51:10.540 -- would be parallel with the one at the top. It's only turned
00:51:14.308 -- down 2 degrees.
00:51:16.480 -- So that means there's an angle change there. An since
00:51:19.360 -- there is an angle change right there, and oblique
00:51:21.952 -- shockwaves going to form.
00:51:25.180 -- That makes sense. You still look.
00:51:28.740 -- How about how about since since we're talking about fine
00:51:33.030 -- degrees, let's blow up the picture a little bit. OK, let's
00:51:37.749 -- say this makes a turn of of, let's say, 20 degrees.
00:51:43.660 -- K and this makes a turn of five degrees. Let's just say so this
00:51:50.310 -- is 20 degrees.
00:51:53.010 -- So we have an oblique shockwave that's here OK. Can
00:51:55.750 -- you see now? The flow follows the wall. It's going to come
00:51:59.038 -- down this direction right here, but what does it see
00:52:01.778 -- when it hits that wall?
00:52:05.170 -- Can you see how it's getting
00:52:06.370 -- bent up a little bit? OK, so this was 20 degrees. We said
00:52:10.610 -- that this is 5 degrees right here, so it's running into
00:52:14.185 -- this wall so it has to change directions.
00:52:18.030 -- That's why I emphasized the flow
00:52:19.572 -- follows the wall. Comes down here now it's going to get bent.
00:52:24.990 -- Here, if this were 20 degrees
00:52:27.810 -- down here. If this bottom were bent at 20 degrees in, that flow
00:52:31.563 -- would just follow the wall all
00:52:32.721 -- the way through. But now it's only turned 5 degrees though,
00:52:36.032 -- so there's an angle change and you create an oblique
00:52:38.722 -- shockwave and the flow is going to follow this wall.
00:52:42.520 -- So I've exaggerated the angles here.
00:52:46.330 -- But it's the same principle about what's going on in that
00:52:48.574 -- part of the problem.
00:52:50.620 -- Good good.
00:52:53.310 -- Anybody else?
00:52:55.490 -- OK, don't let reflected shocks get to you. It's nothing but a
00:53:00.074 -- series of single oblique shockwaves, so if you can
00:53:03.512 -- solve one oblique shockwave, you just add on whatever the
00:53:07.332 -- turns are and keep going from there. It's all it is OK.
00:53:11.916 -- Don't let it intimidate you.
00:53:15.230 -- OK, let's see. We got here. Let's look at now the inlets
00:53:22.334 -- to the SR-71.
00:53:25.560 -- OK, here we go right here and notice that these inlets are
00:53:32.196 -- spiked. OK, so you see this spike right here. This is where
00:53:38.151 -- the air enters OK, and guess what those spikes move in and
00:53:42.963 -- out depending on the speed of
00:53:45.369 -- the aircraft. To recover as much stagnation pressure as possible.
00:53:51.030 -- OK, let's see what the shock train looks like for this. In
00:53:55.650 -- fact, we'll do that little inset trick again in the room. There.
00:54:00.270 -- Let's go to the computer. OK, and here is what that here's
00:54:04.890 -- what the inlet to the SR-71 looks like. Check this out. Here
00:54:09.510 -- is the first oblique shockwave that's formed because of the
00:54:13.360 -- nose cone. OK, so it comes down this way. It follows the wall
00:54:20.088 -- comes in here now this is the first. This is the cowl lip
00:54:26.172 -- right there, so there's a second oblique shockwave and then this
00:54:31.320 -- inlet is designed to go through now a number of reflected
00:54:36.468 -- shocks, 123456 oblique
00:54:37.872 -- shockwaves. Before it reaches its terminal shock, right
00:54:40.942 -- there and then, the flow subsonic all the way through
00:54:43.802 -- there. Why is that?
00:54:47.590 -- What's that going really fast? OK, this this
00:54:52.510 -- particular design? This aircraft recovers. I think
00:54:56.815 -- it's 97% of the stagnation pressure.
00:55:03.140 -- It cruises that amount number of
00:55:04.886 -- three. Remember the problem that we had earlier in class? We had
00:55:09.650 -- a model number 2.5. It just went through a normal shock. It lost
00:55:14.070 -- half of its stagnation pressure through this design right here.
00:55:17.470 -- Sending it through all these small oblique shockwaves. OK,
00:55:20.530 -- this small weak shockwave oblique shockwave. Then it
00:55:23.250 -- recovers more that pressure, it just sends it through a bunch of
00:55:27.330 -- 'em before it goes through that final normal shock, which is a
00:55:31.410 -- very weak and or shockwave.
00:55:34.230 -- So this particular inlet design then slows the fluid down the
00:55:38.927 -- air down very efficiently.
00:55:41.330 -- It loses 3% of the stagnation pressure.
00:55:45.970 -- That's awesome. That is awesome. OK, and again as this is taking
00:55:50.960 -- off and landing and cruising up to speed that cone goes in and
00:55:55.601 -- out there again to maximize the pressure recovery as it goes
00:55:59.528 -- through those. Parts right there.
00:56:03.290 -- I love that, OK?
00:56:07.030 -- Will talk about ramjet engines now before we talk about ramjet
00:56:11.584 -- engines. Let's talk, let's just talk general engine like
00:56:15.310 -- internal combustion engines first, so we have some. We have
00:56:19.450 -- some members of the snowmobile team here is that right? OK,
00:56:24.004 -- what is what is the pressure ratio in your combustion engine?
00:56:28.558 -- So pressure ratio. So that means the air goes into the piston
00:56:33.526 -- cylinder head. There piston
00:56:35.182 -- compresses it. So the pressure gets high. What does that
00:56:38.986 -- increase in pressure? What is it about? About 10K so air comes in
00:56:43.367 -- an atmospheric pressure. You compress it to get high pressure
00:56:46.737 -- so you have fuel that's in there air at high pressure you ignite
00:56:51.118 -- it so you have high temperature, high pressure air that expands
00:56:54.825 -- the piston goes down right? So that's that's the compression
00:56:58.195 -- ratio in an aircraft engine will be there. OK, so now let's think
00:57:02.576 -- about that problem. Let's think about that problem an for. Right
00:57:06.283 -- now we're going to hypothetical. Let's pretend no shocks.
00:57:09.410 -- Let's just pretend no shocks. OK, no shocks here. So I have
00:57:14.138 -- air that's coming this way and it goes into an engine right
00:57:18.866 -- here. And let's just say.
00:57:22.230 -- Let's just say that I want to have that same pressure
00:57:26.850 -- recovery here P.
00:57:29.660 -- If I write it down, I kind of give it away here. I'll just
00:57:34.280 -- call it P2 over P1, no shocks. Remember the hypothetically no
00:57:37.910 -- shockwave here, so the air comes in and that pressure goes up by
00:57:42.200 -- 10. So we want that increasing pressure to be 10. OK, let's
00:57:46.160 -- just say that no shocks, so remember what happens to the
00:57:49.790 -- speed of the flow of the pressure goes up? What happens
00:57:53.420 -- to the speed the flow goes down? OK, so let's look at now. This
00:57:58.040 -- air that comes in.
00:57:59.440 -- It has some static atmospheric pressure. It's going to slow
00:58:02.870 -- down before it gets into the
00:58:04.928 -- compressor blades. Right here. OK, well actually no no.
00:58:08.617 -- Compressor blades here. It's just going to come down and it's
00:58:12.148 -- going to slow down and let's just say that the Mach number is
00:58:16.321 -- about 0. Let's just say so. We slow it down a bunch there. What
00:58:20.258 -- kind of pressure are we talking about there for the model number
00:58:22.586 -- 0? Stagnation, so look up in your tables. Look
00:58:27.046 -- up in your tables for P, not over P equal to 10.
00:58:33.800 -- What do you get?
00:58:37.010 -- Peanut over P is equal to 10, so you go to the isentropic tables.
00:58:48.610 -- 2.16 is perfect.
00:58:52.210 -- So now think about this engine design. Think about this. So M1
00:58:57.166 -- is 2, what do we say? 2.16 that we said 2.16. OK so think about
00:59:03.361 -- this. You could have an engine design where you take high speed
00:59:08.317 -- air that goes into it and all you have to do is slow it down.
00:59:15.790 -- And what happens to the pressure?
00:59:18.830 -- Goes up yeah supersonic flow that enters this and there's
00:59:22.310 -- this right here you don't have to have any compressor blades.
00:59:26.138 -- You don't have to have a piston to compress the air. All you're
00:59:30.662 -- doing is slowing the air down.
00:59:34.510 -- By slowing it down.
00:59:36.550 -- From what number of two .16 to about zero could be a model
00:59:40.190 -- number .1 whatever it's going to be slightly you get a
00:59:43.270 -- pressure increase of 10.
00:59:45.740 -- OK, that is essentially how now we can go to our
00:59:51.306 -- computer here. That is how a ramjet engine works.
00:59:57.240 -- It has no moving parts.
01:00:01.640 -- There are no compressor blades, there are no Pistons.
01:00:05.250 -- All it does is take air that comes in right here. Obviously
01:00:11.154 -- high speed air slows it down that's slowing down. Is the
01:00:16.566 -- compression process.
01:00:19.020 -- You inject fuel into it, combust it, and comes out
01:00:21.800 -- the nozzle right there.
01:00:25.930 -- That's what are antigens?
01:00:28.290 -- No moving parts. What's the downside to it?
01:00:32.870 -- How does it have to work?
01:00:35.270 -- You already have to be traveling at a Mach number of two.
01:00:39.440 -- OK, ramjet engine doesn't work when you're on the runway an you
01:00:42.980 -- accelerate down and go up.
01:00:45.080 -- That's all subsonic flow. You have to get that pressure
01:00:49.000 -- increase by the speed of the air going into that to get it up
01:00:54.488 -- there. OK, ramjet engines typically function at a Mach
01:00:58.016 -- number of about 3:00, so if you're looking here in our
01:01:02.328 -- tables here for a model number of three, look at what that
01:01:07.032 -- pressure increase would be about 36 times. That's assuming no
01:01:10.952 -- shocks, so assuming no shocks was obviously in this problem,
01:01:14.872 -- they're going to be shocked
01:01:16.832 -- there. OK, so for a ramjet engine to work, you already
01:01:22.730 -- have to be flying at supersonic speeds.
01:01:27.800 -- Happens is that they'll have a little rocket boost, or, uh,
01:01:31.727 -- it's usually a rocket boost. Or actually, in the case of
01:01:35.654 -- this aircraft that this does have, this does have ramjet
01:01:39.224 -- engines in it, so there's a jet turbine engine here, and
01:01:43.151 -- it actually ducks the flow around that engine and turns
01:01:46.721 -- it into a ramjet engine, flying fast enough.
01:01:50.900 -- OK, so this is a combination jet turbine and ramjet engine.
01:01:56.125 -- There OK, obviously we want to recover as much pressure as
01:02:01.350 -- possible so here.
01:02:04.250 -- Then yeah, right there. OK, so let's look at here's another
01:02:09.904 -- diagram of it. Here's the cone here on the inside. Notice that
01:02:16.072 -- you see oblique shockwaves. There an look at the shock train
01:02:21.726 -- 1234556 oblique shockwaves, so it's slowing it down through a
01:02:26.866 -- series of oblique shockwaves. Before the terminal shock right
01:02:31.492 -- there. So now it's subsonic
01:02:34.062 -- flow. Then in that subsonic flow, you inject the fuel. You
01:02:37.730 -- combust it. Here the flame holders and then it comes out
01:02:40.766 -- the backside right there.
01:02:43.760 -- So it uses the fact that you transforms high speed flow
01:02:47.511 -- with low pressure, high speed, low pressure to low
01:02:50.580 -- speed, high pressure flow, just by slowing it down.
01:02:55.170 -- OK, good, that's how ramjet engine works. A scramjet engine
01:02:59.110 -- is a supersonic combustion ramjet engine that's the SC in
01:03:03.050 -- the scramjet, so now it slows it down, but there is no terminal
01:03:08.172 -- shock. OK, you have supersonic combustion that occurs in this
01:03:12.112 -- region and goes out the back.
01:03:15.430 -- For scramjet engine, you typically flying around 7:00 AM
01:03:19.111 -- on #7 or 8.
01:03:23.120 -- OK, we'll talk about that in
01:03:25.346 -- class. Alright.
01:03:29.020 -- Let's let's now talk a little bit about the difference between
01:03:33.145 -- subsonic and supersonic aerodynamics. OK, so here is
01:03:36.145 -- this is this is a subsonic airfoil right here at the top
01:03:40.645 -- and and we have a supersonic airfoil here on the bottom, can
01:03:45.145 -- you see the difference between those two? This is kind of fat
01:03:49.645 -- in the front, got a rounded edge? Kind of hard to see the
01:03:54.520 -- rounded edge, but here's the supersonic airfoil down there.
01:03:57.895 -- Thin with a sharp leading edge.
01:04:00.830 -- OK, this is at subsonic speeds here at transonic speeds. Now
01:04:05.274 -- shockwaves begin to form here at the top because you've got this
01:04:10.122 -- rounded front here and this thin supersonic airfoil, oblique
01:04:13.758 -- shockwaves start to form OK, it's in the transonic regime,
01:04:17.798 -- but now look here in the supersonic regime. Notice that
01:04:21.838 -- there is a bow shock that forms in front of the airfoil.
01:04:27.260 -- And now for supersonic airflow. You see these oblique shockwaves
01:04:30.660 -- that form. OK.
01:04:33.930 -- Let's talk about this. So here we have you have a subsonic.
01:04:42.130 -- Airfoil looks something like that, and here we have a
01:04:46.630 -- supersonic airfoil thin and sharp right there. OK, subsonic.
01:04:53.020 -- Super Sonic
01:04:56.470 -- K alright.
01:04:59.660 -- This is this is a great airfoil in subsonic flow.
01:05:04.690 -- Pilots, what kind of airfoil is that in subsonic
01:05:07.543 -- flow right there?
01:05:09.800 -- Andrew.
01:05:11.690 -- It's not. It's awful, right?
01:05:15.500 -- OK, so subsonic flow this kind of what you learned in is what
01:05:19.933 -- you learned in your influence classes. You know you get a nice
01:05:24.025 -- smooth flow that's over there. It flows faster on the top then
01:05:28.117 -- on the bottom, so you have a lower pressure region on the top
01:05:32.550 -- so it generates lift. Here this airfoil is awful subsonic Lee.
01:05:38.290 -- OK, so now let's go. Let's go. This is a subsonic airfoil and
01:05:42.788 -- supersonic airfoil, and here both Mach numbers are less than
01:05:46.248 -- one lot number less than one right here. And this is just
01:05:50.400 -- kind of flow over this way. Now let's let's beef up things here.
01:05:56.050 -- Mott numbers greater than one lot. Numbers greater
01:05:58.810 -- than one. We have a nice, beautiful rounded airfoil
01:06:01.915 -- right there in a supersonic flow. What's going to form?
01:06:06.780 -- What's going to form right up there?
01:06:09.310 -- A detached bow shock or it's going to be pretty
01:06:13.250 -- much normal shock.
01:06:16.630 -- Right across here and right across there, what forms and
01:06:21.430 -- supersonic flow in our supersonic airfoil right here?
01:06:26.780 -- What forms? Oblique shocks
01:06:31.050 -- here. In here.
01:06:34.520 -- OK.
01:06:36.070 -- Can you see the conundrum?
01:06:38.930 -- Between subsonic and supersonic.
01:06:43.060 -- Airfoil or aerodynamic design? What is good subsonic Lee?
01:06:49.830 -- Is awful supersonically
01:06:53.520 -- OK, this is one reason why airplanes in the 1940s.
01:06:57.390 -- Propeller driven aircraft couldn't break the sound
01:07:00.099 -- barrier.
01:07:01.910 -- OK, that they would use subsonic airfoils. That's what they knew.
01:07:07.150 -- They started getting up into the transonic regime and all of a
01:07:11.734 -- sudden these bow shocks form the drag went Sky high. That's what
01:07:16.318 -- that's what the sound barrier is is extreme increase in the drag
01:07:20.902 -- propeller driven aircraft couldn't overcome that. So now
01:07:23.958 -- we have a supersonic airfoil design like we see here, which
01:07:28.160 -- behaves awfully. Terribly in
01:07:31.124 -- subsonic design. Or in subsonic flows I should say, but
01:07:36.008 -- beautifully in supersonic flows.
01:07:39.650 -- So now. Aircraft designers have to balance those two out when
01:07:44.556 -- you're making aircraft. If you fly supersonically if you if you
01:07:48.450 -- fly supersonic fighter jet, do you ever fly subsonic Lee in it?
01:07:52.698 -- You gotta take off, you have to land and I presume you want to
01:07:57.654 -- land safely, right?
01:07:59.610 -- Yeah, you know. Depends on the pilot I guess. OK, so that
01:08:04.206 -- means. So that means the aircraft designer has to take a
01:08:08.419 -- supersonic airfoil and make it
01:08:10.334 -- function. For the time that that plane is in flight, subsonic
01:08:14.100 -- Lee. We see that.
01:08:18.050 -- And there are tricks and will show some tricks that that
01:08:22.340 -- designers use. In fact, one of
01:08:24.680 -- 'em is. You can kind of see
01:08:27.635 -- here. That in the root of this airplane, right here, it
01:08:32.190 -- has a very thick airfoil right there in there, but
01:08:35.590 -- when it expands out, if you look at the wings, the wings
01:08:39.670 -- are actually pretty thin.
01:08:42.910 -- OK, so this plane actually uses a lot of the fuselage and put it
01:08:47.446 -- this way. Uses a lot of this fuselage. That flat area right
01:08:51.334 -- there to generate lift when it's taking off and landing. You see
01:08:55.222 -- how that see how that works there and then. As it goes
01:08:59.110 -- supersonic, it comes back here. It uses these airfoils that are
01:09:02.674 -- actually pretty thin, not as soon as other aircraft OK, but
01:09:06.238 -- you see then the conundrum that designers have to work with when
01:09:10.126 -- they develop airplanes like
01:09:11.422 -- that. Let's look at the SR-71 and check out this
01:09:15.407 -- wing right here.
01:09:17.720 -- OK, put in the background. Look at that. That is really thin.
01:09:21.344 -- That's a very sharp edge by the way. The Museum of Flight has an
01:09:25.572 -- SR-71, so when you're in Seattle you can look at all these
01:09:29.196 -- airplanes OK, this has very thin wing. It's kind of hard to
01:09:32.820 -- notice, but do you see that there's a little kind of a
01:09:36.444 -- divot? A little curve right
01:09:37.954 -- there? You see that.
01:09:41.120 -- Right there.
01:09:43.380 -- And there this is the aerodynamicists answer to the
01:09:46.719 -- subsonic supersonic flow conundrum. Here it's got a
01:09:49.687 -- little curve, so it has a little curvature in that airflow right
01:09:54.139 -- there so that it can take off and land somewhat safely. You
01:09:58.591 -- get better control on that.
01:10:00.820 -- It's not a straight flat airfoil as it goes across there.
01:10:04.620 -- OK, same thing on the other side you can see you can kind of see
01:10:09.375 -- that curve right there.
01:10:12.360 -- OK.
01:10:14.330 -- Good, I just want you to
01:10:16.430 -- appreciate. The difference here of the designs of what a
01:10:20.262 -- supersonic aircraft does and what a subsonic aircraft does.
01:10:23.034 -- Let's look at two of 'em here.
01:10:25.890 -- OK, the plane on the left is a P51 Mustang, a workhorse in
01:10:32.338 -- World War Two great great airplane fun fact. This went
01:10:37.298 -- from this. Went from initial design when pencil first went to
01:10:42.754 -- paper to prototype in 90 days.
01:10:48.400 -- Now, remember the United States was on war footing at the time,
01:10:52.684 -- so it had to get things out quickly, but this plane was
01:10:56.968 -- designed in that length of time there. Look at underneath here.
01:11:00.895 -- These are external fuel tanks.
01:11:03.960 -- See that right there? See one there there in there?
01:11:07.650 -- What's that shape?
01:11:10.050 -- It's like a teardrop design.
01:11:11.980 -- Right? Here is, I think this is an F15 right here. Same kind of
01:11:17.335 -- design. Look at the inlet right there. You know now why those
01:11:21.115 -- inlets are designed the way they are. Look at those external
01:11:24.580 -- tanks. What does that look like?
01:11:28.640 -- Kind of a bull. It's got a very sharp point there
01:11:31.896 -- and there. Why is that?
01:11:36.080 -- Minimize what? Drag, let's look at those. So the P51 Mustang has
01:11:42.820 -- a teardrop shape.
01:11:45.740 -- External fuel tanks.
01:11:48.260 -- Which turns out to be wonderful for subsonic flows.
01:11:52.980 -- OK, this minimizes both the pressure and the friction drag.
01:11:57.990 -- OK, if I had that fuel tank flying supersonically, what
01:12:01.190 -- would you see?
01:12:03.850 -- Normal shock bow shock right there in the front. Great sub.
01:12:09.295 -- Sonically awful supersonically.
01:12:11.550 -- OK, the F15 right there has a tank that looks like this.
01:12:19.070 -- Right there sharpoint. So now you get oblique shockwaves that
01:12:24.200 -- form minimizing the drag.
01:12:28.290 -- Appreciate now the difference between subsonic flows and
01:12:31.818 -- supersonic flows in aircraft design. Good any questions?
01:12:37.110 -- OK. Dismiss, good luck.
01:12:41.790 -- And let me know if you have any questions on the
01:12:44.375 -- homeworks at all.
01:12:46.510 -- What a beautiful day for gas dynamics. I can't believe
01:12:49.330 -- they pay me money to teach this class.
01:12:53.230 -- I should have to pay to teach it.
01:12:56.740 -- Have a great day.
00:00:27.640 -- Alright, for today we're going to start in Chapter 3. We're
00:00:31.336 -- going to go over. Some were basically kind of reviewing at
00:00:35.032 -- this point, so a couple of things to show everybody on the
00:00:39.064 -- website. Is if we go to I have to move this up here. Sorry I
00:00:45.806 -- forgot I have a preview in a program one so up here the
00:00:50.928 -- lectures we have. Intro class review so for this I have been
00:00:55.656 -- an I will continue to do so. Uploading my intro class of
00:01:00.384 -- lectures. Hopefully most of these links should work OK good
00:01:04.324 -- and they were working on the right files. That's even
00:01:08.264 -- important. Super important
00:01:09.446 -- actually. So there's the.
00:01:13.100 -- And.
00:01:14.470 -- Basic Intro class review lectures. Like I said, I'll
00:01:16.882 -- be putting up a whole bunch more, especially as we run
00:01:19.830 -- into more stuff that's more pertinent to the stuff we're
00:01:22.510 -- looking at and or reviewing.
00:01:25.260 -- And on today's we are going to be in Module 3 which just from
00:01:28.550 -- correspond to chapter three. I'm not quite sure why use the term
00:01:31.370 -- module, but I did and there it is, so we're using it.
00:01:35.560 -- So here my links are working. Yeah, I have a few more links
00:01:39.408 -- for other things to look at, so my central limit theorem we're
00:01:42.960 -- going to talk about that today review that I have two
00:01:46.216 -- different lectures for that. One of 'em actually shows a
00:01:49.176 -- simulation which will be. I don't know. I always enjoyed
00:01:52.136 -- this simulation. Once I finally thought so it was kind of nice
00:01:55.688 -- to see. And we're also going to go through this probability
00:01:58.944 -- distributions handout. I wasn't actually going to put this up
00:02:01.904 -- and I'm going to do most of it on the document camera, but.
00:02:07.070 -- I decided to at the last minute and it literally is a last
00:02:10.567 -- minute hand out, so don't expect anything gorgeous. No pretty
00:02:13.257 -- colors, sorry, no pretty colors here. Have a couple of nice
00:02:16.216 -- looking tables or just not sitting where I want them to,
00:02:19.175 -- but the handout itself will work just fine, and that's actually
00:02:22.134 -- primarily we're going to go through today and then we'll see
00:02:25.093 -- if we get a chance to look at at least one of these central limit
00:02:29.128 -- theorem. Handouts, so today we are going to look at this, but
00:02:33.546 -- we are going to walk through all this, so we want to do most of
00:02:37.836 -- this on the document camera, but I wanted to show something
00:02:40.982 -- first, because, well, this thing can graph so much nicer than I
00:02:44.414 -- can. So alright first.
00:02:49.140 -- We need to review some basic terms from
00:02:53.228 -- probability and we want to.
00:02:57.150 -- Zoom in just to hear will come back to the computer in just a
00:03:02.358 -- bit. So remember, we're going to be talking about is
00:03:06.078 -- probability distributions.
00:03:10.950 -- We'll start out with this simple case just to work through. Now
00:03:14.826 -- that we're going to get into a super highly complex one, but
00:03:18.702 -- will start out with a simple case. Alright, so we have this
00:03:22.578 -- hypothesize data, and that's what this worksheet that I made
00:03:25.808 -- is going to basically following through it. Oh, sorry
00:03:28.715 -- hypothesis. Can you tell that to normal term? I use hypothesized.
00:03:36.740 -- Population.
00:03:41.420 -- And it's an old example. It's probably not extremely current
00:03:45.010 -- in terms of its probabilities. Fitting it is OK. It will still
00:03:49.318 -- work. So here we have the number of TV sets that are owned.
00:03:55.960 -- Per household.
00:04:01.570 -- Nowadays it might be more more interesting to look at
00:04:05.960 -- phones or computers, but everybody's got something
00:04:09.033 -- alright. So in this population, well, we can
00:04:12.545 -- either have the TV's can take on values of 0123 or four. Do
00:04:18.252 -- I have for USF 4?
00:04:21.920 -- And then we have some probabilities associated
00:04:23.845 -- with those.
00:04:26.400 -- P of TV's.
00:04:29.250 -- So the probability of those.
00:04:31.860 -- So I'm just going to write
00:04:33.006 -- another wreath. We can make a nice pretty table here
00:04:35.404 -- when we're done.
00:04:40.190 -- Alright.
00:04:42.870 -- OK, so this example is at least.
00:04:46.220 -- Now might be over 10 years old, but it's at least 10 years old.
00:04:50.100 -- So the probability that probably not quite so accurate anymore,
00:04:53.140 -- but that's OK for what we want to do here. So this is the
00:04:57.396 -- number of TV sets owned per household. And if you want to
00:05:01.044 -- think about it this way for remember what this term is is
00:05:04.692 -- the number of TV's this is going to be a random variable.
00:05:10.630 -- Which remember is kind of like a function of valued function.
00:05:15.910 -- So a specific value of our distribution has a specific
00:05:19.610 -- probability associated with it.
00:05:23.440 -- Alright.
00:05:25.470 -- So actually, let's make a nice table. I should have done that
00:05:28.350 -- to begin with, but whatever.
00:05:30.770 -- I do things the hard way sometimes, so TV's
00:05:34.991 -- probability of TV's.
00:05:43.660 -- Your attentive and you want to redo it, go
00:05:45.928 -- for it, I understand.
00:05:48.120 -- Probably not necessary, as long as you got all your information,
00:05:51.442 -- but nice little table.
00:05:54.160 -- I could have done it vertically, whatever it however you want to
00:05:57.280 -- look at it. Either way, this will get us the basic idea.
00:06:02.790 -- And that's where my graph comes into play and I totally draw it.
00:06:06.287 -- But really, my my little my little handout can show so much
00:06:09.515 -- better than I could ever draw it. So if you want to look at
00:06:13.281 -- that real quick on the computer that is distribution TV's.
00:06:17.910 -- Thought about playing with colors, but I just left alone.
00:06:20.390 -- Figured you could get the gist of it. So we got our 20% here at
00:06:24.110 -- zero and two 40% at one and then our 10% at three and four.
00:06:28.490 -- Alright. So back to our examples here.
00:06:35.110 -- So one of the things, well, we have many things of interest
00:06:38.254 -- that we'd like to look at about. One of the major things
00:06:41.398 -- we want to look at is to look at some of our summary
00:06:44.804 -- statistics, and while looking at this, it would probably be
00:06:47.424 -- nice to know on average, how many TV's are owned per
00:06:50.306 -- household. So we want to find a mean.
00:06:53.970 -- And we also call this here in. With this we call this
00:06:58.134 -- an expected value.
00:07:02.070 -- Expected value alright, so an expected value is a mean, but
00:07:06.756 -- unlike our continuous distributions versus discrete
00:07:09.312 -- and this is more of one of those discrete answers 'cause you
00:07:14.424 -- can't own. 1 1/2 television sets. You could on average but
00:07:19.110 -- not. Literally.
00:07:22.380 -- You probably don't want broken ones. I think they think they're
00:07:26.208 -- counting functional TV's not nonfunctional TV's as well, so
00:07:29.340 -- this would be discreet.
00:07:33.200 -- So basically you want to think about it that your your variable
00:07:36.344 -- in our book uses why a lot versus X, but pick a letter. It
00:07:40.012 -- doesn't really matter. I'm going to use why just because their
00:07:42.894 -- book does, but what was I going with? This whole number values?
00:07:46.860 -- That's what these things can take on.
00:07:50.850 -- There's an S there. There we go.
00:07:54.640 -- So this mean here the way we're going to compute it is because
00:07:59.359 -- it's basically it's a weighted average, so not every value of
00:08:03.352 -- our random variable TV's.
00:08:05.560 -- Takes on equal probabilities, they don't have equal
00:08:08.240 -- probabilities, so we have a weighted average that we're
00:08:11.255 -- going to do.
00:08:13.440 -- And.
00:08:15.690 -- Since we're dealing with the population, this is what we
00:08:18.900 -- call deductive because we know exactly what's going to
00:08:21.789 -- be happening in the population versus a sample,
00:08:24.357 -- and most of our exploration there is going to be
00:08:27.567 -- inductive, but this ones deductive, because we can
00:08:30.135 -- actually see what's actually happening, so we're going to
00:08:33.024 -- call this thing mu the population mean of the
00:08:35.913 -- distribution, some other notation E of Y, like
00:08:38.481 -- function notation.
00:08:41.180 -- And to calculate this is the
00:08:44.180 -- sum. So Sigma sum at each Y times its
00:08:50.164 -- corresponding probability.
00:08:53.320 -- They just calculate about products and add them all up.
00:08:59.830 -- Alright, so why not? We're here. We should do this, for example.
00:09:05.870 -- So to calculate our expected value of Y or mean for this
00:09:11.258 -- we would take zero times its probability.
00:09:15.950 -- Plus one, so I was trying to parenthese ahead of time times
00:09:19.922 -- the probability of 1.
00:09:22.890 -- Two times its probability plus three times .1.
00:09:30.060 -- There's four times by 1.
00:09:33.490 -- There's also and then we get a lovely 1.5.
00:09:38.590 -- Oops, sorry, papers got broken.
00:09:42.410 -- So on average, we could expect a household to
00:09:45.830 -- have about 1 1/2 TV's.
00:09:49.290 -- It's like the 1 1/2 kids thing, though obviously we can't have a
00:09:51.994 -- half a TV or a half a kid, but it's an average, even if it's
00:09:55.114 -- not a part of the original
00:09:56.362 -- distribution. And that's OK.
00:09:59.780 -- So. This, unfortunately, you're going to have to torture with my
00:10:03.740 -- drawing anyways. If I drew out our little distribution.
00:10:09.860 -- So this will be our probability.
00:10:12.750 -- We have wide on the X axis.
00:10:16.310 -- Let's see here.
00:10:18.840 -- So I'm just kind of guesstimating I'm not an artist
00:10:22.290 -- by any stretch of the imagination. I can draw a decent
00:10:26.085 -- Bell curve. And occasionally decent rectangles.
00:10:31.040 -- Pretend those are both .1 and the other ones are point 2.4,
00:10:35.348 -- point 2.1. .1 there we go.
00:10:40.090 -- So if we imagine where we put the mean, that would be right
00:10:43.639 -- about here. Well, this is basically what we would consider
00:10:46.369 -- this center of mass. So if we actually try to balance this
00:10:49.645 -- thing on, that's exactly the point where it would balance the
00:10:52.648 -- center of mass right there. And that's where that mean is.
00:10:59.580 -- I think I just wanted to touch you with my drawing.
00:11:01.395 -- That's not what I think I needed to do here.
00:11:04.750 -- Right, and of course we love measures of location. That's
00:11:08.310 -- what the mean is. But we also love measures of spread so we
00:11:12.938 -- can see how much variation we actually have. So this is our
00:11:17.210 -- measure.
00:11:19.070 -- Of spread.
00:11:21.380 -- Variation.
00:11:27.190 -- It's one of 'em, but this one in particular.
00:11:31.040 -- The variance.
00:11:34.270 -- Is the average.
00:11:40.050 -- Important work here squared.
00:11:43.820 -- Distance.
00:11:46.410 -- Each point is from its mean.
00:11:53.280 -- Remove there.
00:11:59.540 -- So we can see how much variation we have in our data. Points were
00:12:03.670 -- in relation to the center.
00:12:05.750 -- Of the distribution.
00:12:11.170 -- At see here it's units.
00:12:14.290 -- R-squared units.
00:12:22.720 -- Measurement but yeah.
00:12:25.130 -- But it's not on the same scale as the mean, so not all the
00:12:29.568 -- time. Is this the one we want to directly deal with? But we still
00:12:34.006 -- have to calculate it. So to do that it's it's Greek symbol is a
00:12:38.444 -- Sigma squared. Yeah, my Sigma is mostly OK.
00:12:42.810 -- And one of my friends used to draw it and it looked like a
00:12:44.896 -- Theta and I was like Theta
00:12:45.790 -- squared shoes. Now it's a Sigma. OK, mine supposed to be a Sigma
00:12:50.030 -- at mostly kind of looks like one. You can also use via Y. Now
00:12:54.930 -- this V here is going to denote the actual true variance.
00:12:59.660 -- And of course, since we're dealing with the population,
00:13:01.694 -- that's OK. 'cause that's what we're going to be looking at.
00:13:04.180 -- But that reason I brought that up is that will come into play
00:13:07.118 -- here in just a bit, so.
00:13:09.070 -- Keep that in the back of your
00:13:10.183 -- head, all right. So what we're going to do
00:13:14.456 -- is look at Y minus mu.
00:13:18.550 -- Quantity squared times the probability of Y, so we'll take
00:13:23.300 -- each squared difference of each value between it and the mean.
00:13:29.170 -- Look at that squared distance and multiply it by the
00:13:32.280 -- probability of that data point and that gives us.
00:13:35.970 -- What we're looking for in terms of the variation.
00:13:42.290 -- Alright. So of course we're going to do that.
00:13:48.350 -- And I got a little carried away on my hand out, which is OK and
00:13:52.910 -- carried away in a good way, sort of. It might be a little
00:13:56.862 -- redundant for you, but I did actually expand some of these
00:14:00.206 -- formulas a little bit more, but I did show the actual work later
00:14:04.158 -- on, so we will actually do this. So Sigma squared is the variance
00:14:08.110 -- of Y. So we're going to take the first data point, which is a 0.
00:14:13.780 -- Minus the mean.
00:14:15.750 -- Squared and the probability of zero was a .2.
00:14:21.720 -- And we get to do this for all
00:14:23.704 -- five values. So next one 1 -- 1 1/2 ^2 * .4.
00:14:33.350 -- I have the right table. I'm just making sure I have the
00:14:35.318 -- right values OK.
00:14:37.450 -- And the next one 2 -- 1 1/2 ^2 * .2.
00:14:43.550 -- I have to move it down page or move it down the line.
00:14:47.400 -- 3 minus the mean squared times .1 and the last one 4 -- 1 1/2
00:14:53.715 -- squared times point. That's a two up there. Sorry times .1.
00:15:01.080 -- Well, you know this stuff and all these lovely little.
00:15:06.550 -- Squared differences in products all add up to 1.45.
00:15:14.620 -- Ann, if at anytime you're working through this on your own
00:15:17.887 -- and you get a different number than I do, don't hesitate to say
00:15:21.748 -- something. It happens, unfortunately, but it happens
00:15:23.827 -- and I won't be offended.
00:15:27.730 -- I used to wonder why I was like
00:15:29.498 -- why. How is it so easy to make mistakes? And I think it's
00:15:33.506 -- actually really easy on this end 'cause you get caught up in what
00:15:36.665 -- you're doing. You don't think about something that you're
00:15:38.852 -- dealing with right this moment when you're trying to talk about
00:15:41.525 -- something 5 minutes ahead of you know and think 5 minutes ahead.
00:15:44.441 -- Yeah, it's interesting, alright, but if I do make a mistake,
00:15:47.114 -- don't hesitate to let me know.
00:15:49.900 -- So. Of course, the variance leads us to the next one, which
00:15:55.212 -- is the standard deviation anisur standard measurement of spread.
00:16:02.230 -- And.
00:16:04.320 -- So it's a again a measure of spread.
00:16:09.480 -- Variation that's an R in there, sorry.
00:16:12.890 -- It is the average distance.
00:16:17.670 -- Without the squared.
00:16:20.910 -- Each point is from its mean.
00:16:23.690 -- Oh, there's an end in there.
00:16:32.600 -- So.
00:16:36.450 -- It's just the square root of the variance, and since it's
00:16:39.365 -- really what we end up wanting to do because its units of
00:16:42.545 -- measurement are the same as the mean, so they have single non
00:16:45.725 -- squared units of measurement.
00:16:47.830 -- It has seem.
00:16:55.120 -- Units of measurement azzameen
00:17:00.870 -- which is good? Want to keep things on the same scale?
00:17:06.180 -- And it literally is just the square root of the variance.
00:17:14.510 -- The positive square root, of course.
00:17:19.220 -- Remember, standard deviations invariances cannot be negative.
00:17:21.789 -- They can be 0.
00:17:23.810 -- Which is not very exciting, but they can't be negative.
00:17:27.280 -- 'cause if there is zero, you
00:17:28.678 -- have identical data points. Which I suppose is not
00:17:31.805 -- necessarily that it's not super exciting. There could be
00:17:34.550 -- a good case for it, but it might not be that exciting to
00:17:38.515 -- look at. So Sigma without the squared is our notation.
00:17:43.890 -- So SD of Y.
00:17:46.650 -- Probably, but as of why, but that might mean something else
00:17:50.005 -- in a different class, so I use SD and so we just take the
00:17:54.275 -- square root of our variance.
00:17:56.910 -- Or the square root of Sigma squared. Either way for us in
00:18:01.962 -- this example, is sqrt 1.45.
00:18:05.720 -- Which is fire Mario 1.2?
00:18:13.770 -- Or something close.
00:18:22.100 -- Alright, this would be great if we could always get population
00:18:25.070 -- values and we would never have to worry about doing. You know
00:18:28.310 -- we'd always be able to know everything about the population.
00:18:32.020 -- Not always the exact case in life. Unfortunately, there's a
00:18:36.010 -- lot of unknown.
00:18:38.650 -- And it's a two point 1.20 that some decimals off the end, but I
00:18:42.458 -- just found it to 1 decimal place as far as your work goes most of
00:18:46.538 -- the time using significant digits is not a horrible idea,
00:18:49.258 -- but I would say except in a rare case when we get to the last
00:18:53.338 -- chapters, you probably don't need to carry it more than two
00:18:56.330 -- to four decimal places are last chapters or there are some
00:18:59.322 -- concepts where we're going to have a small value, some sort of
00:19:02.586 -- density value which is very similar to like a growth or
00:19:05.578 -- decay rate, so you'd probably more like a decay rate, so you
00:19:08.842 -- probably want to make sure you might want to carry those out a
00:19:12.378 -- little bit further, but.
00:19:13.640 -- The most part significant digits or two to four decimal places
00:19:17.457 -- will be more than sufficient for what you need.
00:19:22.630 -- But unfortunately we don't have.
00:19:26.030 -- Population values all the time. He did. Life would be simple and
00:19:29.510 -- then we wouldn't probably need a whole discipline called
00:19:32.120 -- statistics for all this stuff because we wouldn't know the
00:19:35.020 -- entire population. But since we don't, we have to use
00:19:37.920 -- statistics. So what we're going to do is take samples and that's
00:19:41.400 -- really what you're doing. Here is looking at the samples from
00:19:44.590 -- surveys and what have you and trying to make estimations. Our
00:19:47.780 -- main estimations are going to be
00:19:49.520 -- a mean. A total which you may or may not have dealt with in your
00:19:55.123 -- intro class and a proportion. There are others, of course, but
00:19:58.764 -- those are our main.
00:20:00.180 -- Three statistics of interest while we're here in this course
00:20:02.840 -- and those would be the main three statistics of interest in
00:20:05.766 -- surveys as well. So.
00:20:08.360 -- And of course with that we always want to have a variance,
00:20:12.284 -- so we getting back to calculating this all right now.
00:20:16.590 -- You've probably seen that there are different.
00:20:20.540 -- Calculations formulas for population versus sample values.
00:20:25.880 -- So let's kind of take a peek at
00:20:28.336 -- the differences. Population.
00:20:33.650 -- Versus sample.
00:20:36.980 -- So population value for a mean.
00:20:40.470 -- Is mew. I'm going to write the word meaning here so we know
00:20:44.698 -- what this is at first case. It's been a little while since you've
00:20:47.402 -- seen some of these. So if we knew every single value in the
00:20:51.642 -- population, we would sum all of those up. So I'm going to use.
00:20:55.516 -- Not that you can tell the difference, but that's supposed
00:20:58.496 -- to be a capital Y versus a small way. Usually my capital wise are
00:21:02.668 -- straight just lines and my lower case. Why is usually kind of got
00:21:06.542 -- a curve to it?
00:21:08.830 -- I'll usually remind you as we get there, so we take every
00:21:12.514 -- value of the population.
00:21:14.700 -- And we divide it by. Now we have a new symbol, big End. Big N
00:21:19.875 -- represents population size.
00:21:22.480 -- I'm actually going to write that on my previous sheet of
00:21:25.351 -- paper that I'm going to bring down Tuesday here, so an is
00:21:28.483 -- always your sample size.
00:21:32.560 -- And Big N is going to be your population size.
00:21:37.320 -- Which is actually important. In this course we need to
00:21:39.380 -- know that for the surveys and stuff that we were analyzing.
00:21:45.350 -- Alright.
00:21:48.520 -- Before a sample.
00:21:51.400 -- Ala carte, why bar could be X bar. Yeah, Brooke. Uses why
00:21:54.856 -- we're going to stick with guys. We would take the sum of all of
00:21:58.888 -- our sample. Observations and divided by the number of
00:22:03.687 -- observations in our sample.
00:22:06.980 -- Depending on how we draw a sample, these could be
00:22:10.090 -- identical. But it's not going to happen terribly often, except in
00:22:14.340 -- my example. Today it was coincidence. I swear I actually
00:22:17.390 -- do a random sample, and the thing we're going to look at
00:22:21.050 -- today, and it turned out that the sample mean is going to end
00:22:25.015 -- up being exactly the population mean, but that doesn't always
00:22:28.065 -- happen, but it should be most of the time, pretty close.
00:22:33.250 -- Alright, variance.
00:22:36.060 -- So we call it Sigma squared.
00:22:40.420 -- I'm going to actually give you two different derivations
00:22:43.183 -- of the same formula.
00:22:46.410 -- There's one you've seen before.
00:22:47.860 -- Maybe sort of. So why that should be an eye for each
00:22:52.895 -- individual observation minus mu?
00:22:55.150 -- Quantity squared divided by big N.
00:22:58.680 -- Or in the discrete case, what you saw earlier?
00:23:04.730 -- That was the sum Y minus mu quantity squared times the
00:23:09.427 -- probability of Y.
00:23:15.730 -- Alright.
00:23:19.650 -- S squared following the same sort of formula over here.
00:23:24.040 -- It's going to be.
00:23:27.980 -- The sum why I -- Y bar quantity squared. We divide that by
00:23:33.661 -- little N -- 1.
00:23:36.370 -- Because it came from a sample and we're losing
00:23:38.773 -- some information. If you look there at the formula
00:23:41.176 -- I have. Why bar versus mu, since we don't know mu, we
00:23:44.380 -- lose. We lose our information. A degree of
00:23:46.516 -- freedom. So that's why we're dividing by N -- 1
00:23:49.186 -- little N -- 1.
00:23:51.980 -- But in the population case, we wouldn't actually lose any
00:23:55.010 -- information because we have it all. So, and we're using
00:23:58.040 -- the real mean.
00:24:01.050 -- This one here. Technically we still use via why, but it's more
00:24:05.154 -- ha with a hat on it, so anytime you see a hat on something
00:24:09.942 -- that's usually called an estimator and you're actually
00:24:12.678 -- going to see a hat on a V. More often than not. So this one
00:24:17.808 -- implies that we actually do
00:24:19.518 -- know. All the values and population. Here we are
00:24:23.030 -- estimating the variance, so it's the estimated variance of Y.
00:24:27.750 -- And it really isn't going to look hugely different.
00:24:36.970 -- As well, calculate the expected
00:24:38.450 -- value of Y. Or mew hat.
00:24:42.130 -- Yeah, this book likes to use hats on things, so if you
00:24:44.962 -- haven't seen that too much before, we're going to have
00:24:47.322 -- hats. Lots of hats.
00:24:51.340 -- Standard deviation, well, that's actually.
00:24:57.180 -- Not that exciting or different than what we were used to. So
00:25:01.212 -- Sigma is the square root of Sigma squared and over here S is
00:25:05.580 -- sqrt X ^2.
00:25:11.650 -- Alright.
00:25:15.800 -- Trying to keep my pages and pages in line here so in our
00:25:21.065 -- statistical studies we love to take samples and we make
00:25:25.115 -- inferences from those samples about the larger population. So
00:25:28.760 -- we want to make.
00:25:30.980 -- Well, it's an inference is an educated guess, but we're using
00:25:34.126 -- data and facts to back that up. So it is an educated guess.
00:25:37.844 -- Guess sounds so. I don't know Willy nilly versus.
00:25:42.350 -- An educated statement I don't know, but that's what we're
00:25:45.690 -- going to do. So a lot of times we want to make inferences about
00:25:50.366 -- unknown population parameters. So what do we do? We use our
00:25:54.040 -- sample statistics, so we're going to get back to our TV
00:25:57.714 -- example. 'cause it's completely exciting an in our TV example.
00:26:04.140 -- TV simple. Let's say I took a sample an it's not a very
00:26:09.301 -- big sample, it's only a sample size 4.
00:26:13.970 -- An out of this sample, we knew that we could have values that
00:26:18.169 -- were 0123 or four, but in this particular sample my values
00:26:21.722 -- were. Those are my sorry. These are supposed to be my curly
00:26:25.598 -- braces, but I suck at drawing them, so that's what it is.
00:26:30.900 -- These were my data points.
00:26:34.650 -- There we go.
00:26:37.580 -- 2013
00:26:42.340 -- now just for reference, our population had a sample or
00:26:44.780 -- had a size 4 as well, but we're going to take a sample
00:26:47.952 -- of size 4 and it could have been any values Now notice.
00:26:52.870 -- We actually had five different values that could happen. We
00:26:55.460 -- only chose for actually so big N is. I have a big I have a typo
00:26:59.604 -- on my thing. I gotta fix it big and is actually 5 here, alright?
00:27:04.520 -- So let's estimate mu. So mu hat. We usually just
00:27:07.920 -- call that Y bar X bar.
00:27:11.360 -- Pick a letter well, minus a few of 'em till pigsie.
00:27:17.060 -- But here it is. When we use the sum of our values
00:27:21.296 -- divided by your sample size. So we can do that.
00:27:27.960 -- Divided by 4, why are we doing it this way? Well in this case.
00:27:33.280 -- We're kind of assuming that they didn't have different
00:27:36.286 -- probabilities from our sample when we actually went to those
00:27:39.626 -- probabilities were different based on numbers in a
00:27:42.298 -- household, but from our sample, each of these had an
00:27:45.638 -- equal chance of being chosen.
00:27:48.740 -- So we do this and like I said before.
00:27:54.130 -- We actually get.
00:27:56.190 -- The same number, or pretty close to it, 6 force. I don't know.
00:27:59.986 -- Today is one of those days.
00:28:02.510 -- One of my favorite teachers in the math Department, so some
00:28:05.370 -- days are Calculator days, even for the most simple things like
00:28:08.230 -- 1 1/2. Which I already told you it was the same, but all of a
00:28:12.991 -- sudden my brain said no, you must test it again. So even
00:28:16.195 -- though I calculated it 2 hours ago, evidently I needed to do it
00:28:19.666 -- again. Alright now your sample mean is not always going to be
00:28:22.870 -- equal to your population mean. It should be relatively close
00:28:25.540 -- most of the time this just happened have been one of those
00:28:28.744 -- samples that I happened to draw and I did actually honestly draw
00:28:31.948 -- it randomly. And it just happened to be that this sample
00:28:35.507 -- mean was the same as a population mean, which is OK,
00:28:38.290 -- that's not a bad thing.
00:28:41.470 -- But now we're going to get into calculating our variance
00:28:46.070 -- and standard deviation.
00:28:49.020 -- So here's our variance. We can call it Sigma squared
00:28:52.170 -- hat or Sigma hat squared, probably Sigma hat squared.
00:28:56.380 -- Or you just call ask word that works too.
00:28:59.750 -- We're going to use the other formula, the second, well, the
00:29:02.335 -- first one I drew out, but not the first one we actually used.
00:29:06.540 -- Why I -- Y bar quantity squared divided by N -- 1?
00:29:12.260 -- That's the one we're going to use.
00:29:16.480 -- Amazon to all this lovely fun stuff.
00:29:22.350 -- And we got a zero. Remember using the values from the sample
00:29:25.878 -- and not the actual population.
00:29:28.710 -- 1 -- 1 1/2 squared and 3 -- 1 1/2 ^2.
00:29:38.620 -- Bye bye oh I was gonna say 4 -- 3. Now the answer is
00:29:42.904 -- three 4 -- 1.
00:29:47.900 -- We had five thirds or 1.67.
00:29:53.010 -- Versus what was it before 1.45?
00:29:58.000 -- So a little more variation in
00:29:59.842 -- this. Particular sample, then there wasn't a
00:30:01.996 -- population, that's OK.
00:30:05.390 -- And then for the standard deviation.
00:30:08.780 -- Sigma hat or S just take the square root of your S ^2.
00:30:15.100 -- Anne will get.
00:30:17.490 -- Our standard deviation 1.29.
00:30:22.170 -- As probably.
00:30:28.010 -- Alright.
00:30:30.940 -- Not very exciting, but I thought we do a nice little nice
00:30:34.816 -- overview. Just remind you so for random samples from infinite
00:30:38.046 -- populations, which is what we're kind of doing. The expected
00:30:41.276 -- value of the sample mean.
00:30:43.670 -- Is usually the true meaning that leads us toward what we're
00:30:47.850 -- looking at next, which is not just probability distributions,
00:30:51.270 -- but distributions of statistics.
00:31:02.590 -- Sample statistics so distributions of sample
00:31:05.278 -- statistics, or in other words, sampling
00:31:07.966 -- distributions. That's usually the more common
00:31:10.654 -- terminology.
00:31:16.340 -- So just a reminder, what a sampling distribution is is that
00:31:21.235 -- it looks it's the distribution.
00:31:28.200 -- Of all possible samples, Whoops, there's 2 S is there?
00:31:37.310 -- Of a sample statistic.
00:31:46.050 -- We like that we have a specific theorem that we really really
00:31:50.106 -- like. And I need to go find that real quick here now. We probably
00:31:56.269 -- going to look through one of these on the computer up here,
00:32:00.985 -- but it didn't want to go
00:32:03.343 -- through. Well, I wanted to show I didn't want to necessarily go
00:32:06.891 -- through both of 'em 'cause the other one really just kind of
00:32:09.663 -- summarizes this whole thing together. So that's something
00:32:11.511 -- you can look at the other link for. It's called CLT 2.
00:32:15.230 -- But we're going to do is we're going to look at the sampling
00:32:18.961 -- distribution an. I actually have a couple of examples to
00:32:21.831 -- show through simulation how this actually works and why
00:32:24.414 -- we're still able to actually use a normal model. Most of
00:32:27.571 -- the time for analysis, and we're going to do a normal
00:32:30.728 -- model in this classroom as well for this course.
00:32:34.600 -- Not all your surveys are going to have variables that follow
00:32:38.131 -- normal models. OK, not all of 'em, but provided we look at we
00:32:42.304 -- have large enough samples and what have you most of the time
00:32:46.156 -- we should be OK, but not every time. There are exceptions to
00:32:50.008 -- that rule always. So first thing you should always do graph your
00:32:53.860 -- data if you don't know what your data looks like visually, then
00:32:57.712 -- you're only getting probably about 1/3 to half of the
00:33:00.922 -- picture. So alright, so we're gonna look at the central Limit
00:33:04.453 -- theorem. And for that one, our sampling distribution of the
00:33:08.129 -- sample mean is approximately normal with a mean mu and
00:33:11.539 -- standard deviation of the sampling distribution of the
00:33:14.267 -- sample mean. Is Sigma divided by square root of N. So since
00:33:18.359 -- we're looking at the distribution of the sample
00:33:21.087 -- mean, we don't just use our variance, we take the variance
00:33:24.838 -- divided by N or the standard deviation divided by the
00:33:28.248 -- square root of N. We call that Sigma over square root of N.
00:33:32.681 -- We used to call that a standard error.
00:33:36.760 -- That is provided that N is sufficiently large. This theorem
00:33:39.570 -- can also apply to other statistics, which is really,
00:33:42.099 -- really handy because we're going to be using those other
00:33:44.909 -- statistics as well. The sample proportion an one of 'em I
00:33:48.000 -- didn't actually have on here. The sample total which could be
00:33:51.091 -- used in case I don't know if you guys have ever dealt with the
00:33:55.025 -- total before, but it could be nice, say for an airline we need
00:33:58.678 -- to know how many passengers are boarding the plane right? And
00:34:01.769 -- the other thing we do is we weigh how much your bags weigh.
00:34:05.810 -- We need to know the weight of your bags, how much junk
00:34:09.002 -- you're taking with you on the plane, in addition to the
00:34:11.928 -- weight of everything else on the plane, the humans on the
00:34:14.854 -- plane, everything.
00:34:16.640 -- So it might be nice to know what the average weight per person
00:34:20.085 -- should be. The maximum average weight per person, but that's
00:34:22.735 -- not the only thing of interest. It could actually be of interest
00:34:25.915 -- to look at the entire plane full of people's total weight. That's
00:34:29.095 -- just one example. It's not the only one, but it's one of the
00:34:32.540 -- few examples that you could use a total for, and so that's how
00:34:35.985 -- that's going to play in when we start getting to that.
00:34:39.770 -- Alright, so for the most part, the sample size should be
00:34:43.752 -- approximately at least 30.
00:34:46.100 -- If your distribution wasn't already normal to quote
00:34:48.956 -- unquote, guarantee the normality I say and kind of
00:34:52.169 -- using that term guarantee a little. Loosely, there's no
00:34:55.382 -- guarantees, but to get us the approximate normality,
00:34:58.238 -- we should have a sample size of at least 30. Now, if your
00:35:02.879 -- original distribution you already know is inherently
00:35:05.378 -- normal, that sample size stipulation is not required.
00:35:08.234 -- You could have a sample size is smallest 2.
00:35:12.850 -- But if you don't know anything about your original
00:35:15.613 -- distribution, always safer to take a sample size of at least
00:35:18.990 -- 30. That being said, in surveys we take, sample size is usually
00:35:22.674 -- of probably at least 10 or more times than that than 30, so.
00:35:27.480 -- And we're going to sample proportion. We usually want to
00:35:30.000 -- sample size of at least 60. Most of your information from sample
00:35:33.024 -- surveys, alot of time, not most or all. But a lot of times are
00:35:36.552 -- going to be percent, so that would be of interest.
00:35:39.800 -- And again, I said here, if you're just distribution is
00:35:42.520 -- already inherently normal, your sample size stipulation can be
00:35:44.968 -- ignored. It's not that you're ignoring it, but it's not. It's
00:35:47.960 -- not relevant to what you need to worry about it, alright?
00:35:52.040 -- This one sorry. The book I was using used pie instead of P for
00:35:56.184 -- the proportion. Now it should be like most. I'm an intro books,
00:35:59.736 -- they always use P, but as soon as you hit like our 431 class,
00:36:03.880 -- that book uses pie 'cause everything else uses a Greek
00:36:06.840 -- letter. Why not? So why not intro class? Well unfortunately
00:36:09.800 -- will never find that answer out but we still go back to P in
00:36:13.944 -- this book. This book uses P for that terminology just to kind of
00:36:17.792 -- let you know. But you can interchange it with pie. It is
00:36:21.344 -- the same basic thing.
00:36:23.990 -- Alright, so in shorthand notation. Our sample mean X bar
00:36:28.110 -- or why bar is distributed normally with a mean mu and the
00:36:33.054 -- Sigma sub X bar is another notation for that standard
00:36:37.174 -- error. Sigma over square then.
00:36:41.150 -- Same thing for the one for the proportion and the total would
00:36:44.042 -- work as well. I thought this was my updated file that showed.
00:36:47.630 -- Totals so all this is nice and interesting in review. You'd not
00:36:51.950 -- be calculating Z scores in here.
00:36:54.890 -- So if you're hoping to see Z&T scores in here, I'm actually
00:36:58.454 -- going to see those, but that's OK, Alright? This is the
00:37:01.721 -- important part, so we actually see how this distribution works
00:37:04.691 -- and how the central Limit Theorem helps us to look at
00:37:07.958 -- normality. So we're actually going to look at a distribution
00:37:10.928 -- that's already normal, so it's not going to be that exciting
00:37:14.195 -- when we take the look at the sampling distribution, it's
00:37:17.165 -- still going to be normal. There are going to be some
00:37:20.432 -- differences, but then we're going to look at an exponential
00:37:23.402 -- distribution, which is obviously
00:37:24.590 -- not. A normal Bell curve distribution and a binomial
00:37:27.672 -- distribution, just so you can see how the central Limit
00:37:31.062 -- theorem works on even the non normal distributions.
00:37:35.200 -- You don't ever have to reproduce this unless you want to, and
00:37:38.608 -- which case if you want to borrow my code, just ask me, But what
00:37:42.584 -- this does is I'm basically going to take this is in our command,
00:37:46.276 -- so our norm. And you plug in how many values you want into that.
00:37:50.969 -- That will give you random numbers generated from a normal
00:37:53.659 -- distribution. If you don't specify the mean and standard
00:37:56.080 -- deviation, it will assume the mean is 0 and the standard
00:37:59.039 -- deviation is 1, just like the Z
00:38:00.922 -- distribution. So we needed.
00:38:04.070 -- And in this case I actually gave it a different mean in a
00:38:08.256 -- different standard deviation than the Z distribution. So I
00:38:11.154 -- took a sample of 500.
00:38:13.810 -- Out of a normal distribution and I set the mean at 100 and the
00:38:18.612 -- standard deviation at 10 and I said, oh, let's look at the mean
00:38:23.071 -- so mean for that particular sample was 100.25.
00:38:26.970 -- So close.
00:38:29.170 -- And here's our histogram. So the spread on this one goes from
00:38:33.898 -- about 65 to 135, give or take.
00:38:39.630 -- And another random sample just to show the mean change
00:38:42.480 -- to her. But we're still right around that 100 mark.
00:38:46.440 -- And then. For some silly reason, I decided I need to put a curve
00:38:51.432 -- on it. I hardly ever put curves on my on my distributions like
00:38:54.890 -- this, but this one was like I'm going to put that curve on
00:38:58.348 -- there. So there it is. So it is a normal distribution still
00:39:01.540 -- spread out between 65 and 135 center right about 100.
00:39:05.510 -- Oh, rest of my code fell off, sorry.
00:39:09.260 -- Alright, so for this simulation process I'm setting the mean and
00:39:12.714 -- the standard deviation. I'm going to take samples of size
00:39:16.232 -- 5 and I'm going to do that 500 times. We're going to have 500
00:39:20.236 -- samples of size 5, so we can look at the means of all of
00:39:24.240 -- those, and that's what I'm calculating here.
00:39:28.350 -- And then we look at histogram and there is the distribution of
00:39:32.898 -- the sampling distribution of the sample mean. So the spread
00:39:36.688 -- changes 'cause we're dividing it by the square root of N. So it's
00:39:41.615 -- now spread from about 85 to maybe 115 versus 65135.
00:39:46.160 -- So the curve got skinnier and a little bit taller and that
00:39:50.324 -- happens. But it's still a normal distribution, but
00:39:53.151 -- this is now the distribution of X bar versus X.
00:39:56.910 -- And they are just kind of arbitrary values. I guess I
00:39:59.968 -- just. Grabbed grabbed a mean in a standard deviation and
00:40:03.884 -- just used it so.
00:40:05.900 -- Normal spread change though.
00:40:08.720 -- That's important to look at.
00:40:10.480 -- Exponential distribution. I don't know why I really like
00:40:12.919 -- this distribution. If you took 201 or 251 then chances are you
00:40:16.171 -- probably didn't see this. You may have heard about it, but you
00:40:19.423 -- probably didn't see this. If you take 301, they may have seen
00:40:22.675 -- this, but don't stress it if you
00:40:24.572 -- haven't seen it. I'm not going to test you on this formula,
00:40:29.394 -- but this just shows you the formula I'm using, so it's an
00:40:33.786 -- exponential distribution. Exponential is really great
00:40:35.982 -- for modeling the waiting time between events.
00:40:39.620 -- Other processes too, but that's one of its big big draws.
00:40:43.920 -- Now let's see. Here we are going to be looking at this with this
00:40:47.784 -- one. We're going to use a distribution with a rate of 1.
00:40:52.180 -- Alright, so random number again, a different
00:40:54.595 -- distribution R has told whole bunch of different
00:40:57.355 -- distributions. You can randomly generate numbers
00:40:59.425 -- out of which is great.
00:41:03.170 -- We need N.
00:41:05.160 -- Anna rate. So with this one we're going to sample size 500.
00:41:10.040 -- We will find the mean.
00:41:11.880 -- That's pretty close to 1.
00:41:14.470 -- That sample #1 sample #2.
00:41:18.450 -- Actually knows same sample. This sample number one.
00:41:21.362 -- Sorry, obviously not a normal distribution.
00:41:25.370 -- And do it again. This time the mean was to even just a hair
00:41:28.968 -- lower. But we're still pretty close to the one mark.
00:41:33.720 -- There we go. Being silly had to add that curve in again. So
00:41:37.984 -- there's our our curve or exponential curve and the
00:41:40.936 -- regular distribution of it.
00:41:43.260 -- So now we're going to do the same thing, except for I'm
00:41:46.776 -- going to be taking samples of size 30 and I'm going to
00:41:50.292 -- take 500 samples of size 30 to calculate. 500 means joy.
00:41:54.530 -- It's kind of fun to do. Well, this is the first time so.
00:41:58.840 -- This one, a sample size of 30 almost gives it the normality.
00:42:03.016 -- It's not perfect, but it's.
00:42:05.570 -- Close enough, that's the one thing that's hard to once you
00:42:08.309 -- get out. Intro class is looking at some of these
00:42:10.799 -- curves, and some of these they might not look normal to. You
00:42:13.787 -- might want to go. Some of these are going to be normal
00:42:16.775 -- enough. This one is actually good.
00:42:19.450 -- Obviously not exponential anymore and then binomial. So
00:42:23.266 -- remember binomial distribution is one of those discrete
00:42:27.082 -- distributions for absence or presence, so success or failure.
00:42:33.220 -- So this one is again 500 samples with a binomial
00:42:38.020 -- distribution. Its probability of success was .8 an. We did
00:42:42.820 -- sample sub size 10.
00:42:46.370 -- But this person will probably do 500, though again binomial. You
00:42:49.967 -- can randomly generate, so this first one is actually 500. Later
00:42:53.564 -- on when we do, the 500 samples were going to take 500 samples
00:42:57.815 -- of size 30. I think or is it 10, probably 10? I don't know. I'll
00:43:02.530 -- double check. I looked through it today and then I forgot.
00:43:06.200 -- So the eight the mean should be 8, so the mean for a
00:43:11.127 -- binomial is N * P, so 10 times .8 gives us 8 and this one's
00:43:16.812 -- pretty darn close 7.98.
00:43:20.370 -- Not even a continuous distribution.
00:43:25.820 -- There we go again, and this one that means just a hair over
00:43:29.668 -- eight. OK, so that was our second random sample and there's
00:43:32.924 -- our second. Histogram.
00:43:36.780 -- Same process we're doing samples of size 10, but
00:43:39.012 -- we're taking 500 of them.
00:43:42.220 -- And look at that all of a sudden. It's not the prettiest
00:43:46.084 -- thing I've ever seen, seen prettier distributions, but
00:43:48.660 -- it's still approximately normal.
00:43:51.700 -- Excuse me, centered right about 8:00, so that's what the central
00:43:54.989 -- limit Theorem does. Remember, when I took my intro course, I
00:43:58.278 -- was just like it was just kind of this concept. You had to just
00:44:02.464 -- think about it was like, OK, I'm sure I'll use it, but actually
00:44:06.351 -- saying it for me it made a huge
00:44:08.743 -- difference this other. Thing that you get Lord death right?
00:44:12.372 -- I zoomed in, sorry this other one that you can look at is
00:44:16.090 -- just moves a nice little handout that my 200 level class
00:44:19.236 -- professor had given to us. So I asked him if I could steal it.
00:44:23.240 -- Well, I said I asked him if I could borrow it so I said well
00:44:27.530 -- can I. Can I borrow it and give it to my class and you said OK,
00:44:32.106 -- that's fine so I stole it. There it is but I did I did put
00:44:36.396 -- his name down there so.
00:44:39.460 -- Alright. So we're looking at this thing. We're probably not
00:44:43.222 -- gonna be able to finish this up today, which is OK. We can
00:44:46.576 -- finish this up later, but we can kind of set ourselves up for the
00:44:50.188 -- end of this. So what we want to do?
00:44:54.510 -- Is we have our population at see here.
00:44:59.810 -- I left my pen open, sorry.
00:45:03.800 -- So this is our original population values.
00:45:08.530 -- And we're going to.
00:45:11.320 -- I think in this case just take samples.
00:45:19.530 -- Size 2
00:45:22.310 -- just keep simple.
00:45:25.620 -- Now.
00:45:28.390 -- In this case.
00:45:31.100 -- This is this is our population and this is the number the
00:45:35.132 -- sample size we're going to do. We want to actually look at all
00:45:39.500 -- possible samples for this so.
00:45:49.860 -- All possible samples.
00:45:54.600 -- From in this case, what we're doing is those were the number
00:45:58.344 -- of TV's in the House, but what we're going to do is we're going
00:46:02.712 -- to be looking at from 4 houses.
00:46:06.240 -- So let's say we have House 1-2, three and four.
00:46:13.590 -- So this has a population of four different.
00:46:19.210 -- Possibility so for houses small town. There we go
00:46:22.666 -- more than Moscow.
00:46:26.250 -- One of the things we got excited about when I was a kid
00:46:29.396 -- we were driving. I think we were driving to California and
00:46:32.058 -- we were driving through southern Idaho really late
00:46:33.994 -- tonight. My dad got all excited how to wake all of us up. It
00:46:37.382 -- was like 3:00 o'clock in the morning. 'cause one of the
00:46:40.044 -- towns we came from California so this was a pretty cool
00:46:42.706 -- concept to us. 'cause it was cute, neat. One of the towns
00:46:45.610 -- actually like listed on the animals and I can't remember
00:46:48.030 -- what town it is but listed all the animals, the cows, the
00:46:50.934 -- humans telling my dad had to wake us all up. Look, look at
00:46:54.080 -- this look at this.
00:46:57.060 -- Alright, so small town that was a small town, not as small as
00:47:01.038 -- this little town we're going to deal with, so we're going to
00:47:04.710 -- sample the houses. And then we're going to ask them.
00:47:11.280 -- How many?
00:47:13.980 -- TV's do you own?
00:47:20.060 -- All right, we're going to look at all possible samples.
00:47:25.100 -- So if we just line 'em up.
00:47:29.080 -- One and two can be one of the samples 'cause we're
00:47:31.621 -- taking samples of size 2.
00:47:34.540 -- Now we're obviously going to assume something here that's
00:47:37.528 -- going to be kind of important for us to talk about. Kind of
00:47:41.844 -- important. That's an understatement.
00:47:44.660 -- Is that we're doing this?
00:47:49.350 -- Without replacement.
00:47:52.210 -- So what I'm doing here is that when I choose a house,
00:47:56.086 -- it can no longer be chosen for the observation #2. So
00:47:59.639 -- if it's been chosen for observation number one, it
00:48:02.546 -- can't be chosen again for observation #2, so we
00:48:05.453 -- couldn't go to the House number one twice or House
00:48:08.683 -- number 2 twice, etc.
00:48:12.680 -- Now two and three can be chosen, and this is all
00:48:15.969 -- possible samples. It's not what we actually did, but
00:48:18.660 -- we're looking at the possibilities.
00:48:21.830 -- I don't know. I like Roman numerals. I always have it
00:48:24.888 -- thing from when I was a kid. I apologize, but I'm not
00:48:28.224 -- that sorry.
00:48:31.500 -- So these are all possible samples. We had six of them.
00:48:37.480 -- And that's where we get to pick
00:48:39.545 -- up next time. Figure out what to do with this thing.
00:48:46.780 -- And that's it, and we will finish this tomorrow or
00:48:49.910 -- finish this next class.