Demo a Course Session

Duration:"00:47:15.1170000"

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 -- answer 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.

Duration:"01:16:32.4080000"

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 handout 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 -- consequence 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 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%

00:42:53.060 --

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 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.

Duration:"00:40:29.6340000"



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 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

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.

Duration:"00:52:28.7600000"



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 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 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 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 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 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

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 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 me use, 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 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 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 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 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 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 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 because 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'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 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 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 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 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 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.

Duration:"00:56:27.8400000"

00:00:21.260 -- Alright, good morning.

00:00:22.202 -- Welcome to class city.

00:00:23.460 -- You got your homework turned in?

00:00:25.340 -- How was the assignment? Do OK with it.

00:00:29.170 -- How was the airfoil problem the long?

00:00:34.050 -- But solvable.

00:00:34.688 -- Just kind of have to be patient

00:00:36.921 -- as you work your way through it.

00:00:39.240 -- OK, so since you're here right now,

00:00:40.970 -- I will tell you next time we

00:00:42.503 -- have a midterm next Thursday.

00:00:43.930 -- We have mentored.

00:00:45.007 -- There will be an airfoil problem on the exam.

00:00:48.200 -- So study that up.

00:00:49.276 -- We'll review it a little bit later on today,

00:00:51.830 -- OK? Alright I have.

00:00:54.616 -- I have colleagues here at the university

00:00:57.292 -- and some friends in the law school.

00:00:59.620 -- That's probably a bad sign,

00:01:01.420 -- right that I have lawyer friends.

00:01:05.130 -- And I always ask, you know,

00:01:07.130 -- do you do you bring up current

00:01:09.265 -- events in your classes?

00:01:10.790 -- You know they're always the Supreme

00:01:12.854 -- Court litigation stuff going on?

00:01:14.450 -- I hear no all the time.

00:01:16.450 -- It takes too much work.

00:01:18.120 -- We do not do that in gas tax.

00:01:20.780 -- OK, just to be clear,

00:01:22.450 -- so.

00:01:25.150 -- This is. The seven minutes of terror.

00:01:29.460 -- OK, the perseverance Lander

00:01:30.888 -- that landed on Mars last week.

00:01:33.030 -- So I thought that I would go through

00:01:35.854 -- step by step to let you know what's

00:01:38.812 -- going on and then watch a video of it.

00:01:41.950 -- Just 'cause I'm the teacher, right?

00:01:44.096 -- And I say we could watch videos,

00:01:46.590 -- we're going to watch videos, all right.

00:01:49.092 -- Let's check this out.

00:01:50.520 -- So 10 minutes before landing,

00:01:52.300 -- the shell comes off and this

00:01:54.346 -- is what the Lander looks like,

00:01:56.590 -- and you can see that it maneuvers itself.

00:01:59.850 -- OK, now it's 8 minutes to entry.

00:02:01.870 -- Gets balance right here.

00:02:02.958 -- What kind of body would we call that right

00:02:05.550 -- there, that kind of rounded surface?

00:02:09.750 -- That's a blunt body. OK,

00:02:12.520 -- that's how that's how any kind of spaceship.

00:02:18.460 -- Enters the atmosphere with a blunt body and

00:02:21.100 -- we'll see that once it starts to enter.

00:02:23.630 -- Look at this, although they

00:02:25.170 -- don't draw it as a shock.

00:02:27.180 -- We know because we are experienced

00:02:29.352 -- gas dynamicist that is a

00:02:31.554 -- shockwave that occurs right there

00:02:33.274 -- and notice that's a peak heating.

00:02:35.290 -- OK, let's look at some of these numbers.

00:02:39.140 -- Here it is 78 miles in altitude

00:02:42.262 -- above the Martian surface.

00:02:44.430 -- Look at this velocity 13,000.

00:02:47.670 -- Mph. Let's move in.

00:02:52.573 -- That is about 5900 meters per second, OK?

00:03:00.730 -- And let's see here.

00:03:02.322 -- So I did a little bit of calculation here.

00:03:06.760 -- The Martian atmosphere temperature.

00:03:08.512 -- Obviously it ranges like

00:03:10.264 -- every other atmosphere,

00:03:11.870 -- but kind of a standard temperature

00:03:15.272 -- is 60 degrees Centigrade below 0.

00:03:18.410 -- Minus 60 pretty cold, so that is 213

00:03:22.210 -- Kelvin enters at 5900 meters per second.

00:03:25.730 -- It's Martian atmosphere is primarily

00:03:28.170 -- carbon dioxide is about 95% carbon dioxide,

00:03:31.548 -- and so it's gamma is about 1.3

00:03:34.831 -- and its molecular weight is 44.

00:03:37.930 -- So we can calculate RA 314 divided

00:03:41.038 -- by 44 and it works out 'cause you

00:03:44.993 -- know how to calculate Mach numbers.

00:03:48.430 -- This works out to be a Mach number of 25.6.

00:03:53.910 -- Entering the atmosphere.

00:03:58.570 -- That's moving. That's moving OK,

00:04:00.990 -- so one of the one of the one of the

00:04:03.975 -- primary designs for this blunt body is that

00:04:07.951 -- you're traveling on Mach number of 25.6.

00:04:10.940 -- Tell me do parachutes work

00:04:12.895 -- very well in a lot #25.6 no.

00:04:15.720 -- OK, we will talk actually a little

00:04:18.303 -- bit later on in the semester that

00:04:21.516 -- there are some supersonic parachutes.

00:04:24.050 -- OK, there are some,

00:04:25.506 -- but this is not designed to do that.

00:04:28.390 -- So what the blunt body does is

00:04:30.721 -- that it absorbs that kinetic

00:04:32.582 -- energy and slows it down.

00:04:34.550 -- OK, so it's moving that amount #25.

00:04:37.080 -- Here's where it heats up that peak

00:04:39.460 -- heating and then it starts to decelerate,

00:04:42.150 -- and then finally finally it starts

00:04:44.244 -- to slow down and what it can do

00:04:47.223 -- this hypersonic aero maneuvering

00:04:48.731 -- what it does is kind of vibrate.

00:04:51.200 -- So it goes like this and it

00:04:53.657 -- converts that kinetic energy into.

00:04:55.650 -- Thermal energy and slows down.

00:04:57.510 -- That's what this maneuvering is.

00:04:59.360 -- And then finally.

00:05:00.803 -- Begins to deploy a parachute.

00:05:03.210 -- Does that at 400 meters per second.

00:05:05.980 -- Now, if you think in air that's

00:05:08.696 -- a little over Mach one,

00:05:10.730 -- it turns out for the Martian atmosphere and

00:05:14.098 -- these times it's a little bit over Mach one,

00:05:17.470 -- so it turns out that this supersonic,

00:05:20.240 -- but not real high,

00:05:21.820 -- not like hypersonic.

00:05:23.010 -- A parachute begins to deploy,

00:05:24.990 -- then the heat shield that

00:05:26.970 -- protected the spacecraft.

00:05:28.160 -- Right here, drops off.

00:05:29.988 -- OK falls off and then over here.

00:05:33.290 -- This is kind of neat.

00:05:35.210 -- The shell drops off and then there's

00:05:37.975 -- a little radar system right there

00:05:40.454 -- in that beams down to the surface to

00:05:43.756 -- determine the best place to land.

00:05:46.150 -- So it's like an autopilot.

00:05:48.140 -- It's like it's like an auto,

00:05:50.530 -- an artificial intelligence system.

00:05:52.602 -- That search arounds figures

00:05:54.674 -- out the best place to land OK?

00:05:57.420 -- Starts to collect that data and

00:05:59.706 -- then this shell drops off right

00:06:02.030 -- here and the Lander starts to

00:06:04.190 -- descend to starts to descend.

00:06:06.280 -- Excuse me, comes down,

00:06:07.776 -- it's got 4 little rockets right here

00:06:10.480 -- and these are called cold gas thrusters.

00:06:13.210 -- That's pressurized gas there,

00:06:14.902 -- so it's not necessarily a combustion

00:06:17.507 -- process starts to descend right here and

00:06:20.020 -- then it deploys what's called a sky crane,

00:06:22.830 -- so this hovers right here with those rockets,

00:06:25.910 -- and what the sky crane does.

00:06:28.380 -- Is that it lowers the Lander CABI

00:06:30.606 -- wires so that the Lander lands,

00:06:32.790 -- and then it cuts those wires in the sky,

00:06:35.840 -- crane falls away and you have

00:06:37.730 -- soft landing on the surface?

00:06:41.890 -- First off, if you're an engineer,

00:06:43.400 -- you can't look at that and say

00:06:45.213 -- that is not absolutely awesome.

00:06:47.230 -- To see that in action OK Now is actually.

00:06:49.980 -- That's kind of a complicated process if you

00:06:52.188 -- think of everything that's going on there,

00:06:54.550 -- you know one of the few seniors here that

00:06:57.016 -- learn about the Kiss principle, right?

00:06:59.125 -- Keep your design simple so that they work.

00:07:01.570 -- This is actually a pretty

00:07:03.090 -- complicated process.

00:07:03.700 -- Past Mars Landers.

00:07:06.030 -- Past Mars Landers actually have a

00:07:08.022 -- kind of a balloon on the outside

00:07:10.452 -- number of balloons that cover this in,

00:07:12.870 -- so that when it lands,

00:07:14.580 -- it actually bounces on the

00:07:16.230 -- surface of Mars until it stops.

00:07:18.340 -- Then the balloons deflated, opens up.

00:07:20.390 -- Then you have a Lander,

00:07:22.100 -- so spirit and opportunity landed like that.

00:07:24.500 -- OK, so.

00:07:26.460 -- That the pretty complicated

00:07:27.820 -- sort of landing process.

00:07:29.180 -- Now let's watch the video.

00:07:30.880 -- They actually in fact I heard

00:07:32.896 -- this on the news is that is that

00:07:35.661 -- in part of the design process,

00:07:37.680 -- they told a couple of engineers say hey,

00:07:40.400 -- listen,

00:07:40.804 -- would it be kind of cool to take high def?

00:07:44.890 -- Pictures,

00:07:45.179 -- High resolution pictures of the

00:07:46.624 -- landing process so they actually went.

00:07:48.450 -- I want to see RadioShack that kind

00:07:50.585 -- of dates me so they went to off

00:07:53.100 -- the shelf cameras and were able to

00:07:55.232 -- put it on the Lander so that you

00:07:57.360 -- can see the landing in process.

00:08:00.400 -- Dumb question, would you like to see it?

00:08:03.570 -- I thought so, so here we go.

00:08:05.630 -- So now that you know.

00:08:07.100 -- So now that you know the landing process,

00:08:09.450 -- let's see what we got here and fly

00:08:11.482 -- right maneuver where the spacecraft

00:08:13.007 -- will jettison the entry balance

00:08:14.612 -- masses in preparation for parachute

00:08:16.198 -- deploy and to roll over to give the

00:08:18.563 -- radar a better look at the ground.

00:08:23.330 -- Public it indicates she's deployed.

00:08:26.610 -- The navigation has confirmed that

00:08:28.390 -- the parachute has deployed an.

00:08:30.170 -- We're seeing significant

00:08:31.235 -- deceleration in the velocity.

00:08:32.660 -- Our current velocity is 450 meters

00:08:34.658 -- per second at an altitude of about 12

00:08:37.545 -- kilometres from the surface of Mars.

00:08:42.310 -- He tilts up. Perseverance is now,

00:08:45.000 -- so she was dropping off these

00:08:47.010 -- and then literally on Mars.

00:08:48.750 -- Just allow both the radar and the

00:08:50.920 -- cameras to get their first look

00:08:52.991 -- at the surface current velocity

00:08:54.771 -- it 145 meters per second at an

00:08:57.263 -- altitude of about 10 Columbia 9

00:08:59.321 -- 1/2 kilometers above the surface.

00:09:03.160 -- Now that's pretty cool to see that close.

00:09:06.420 -- Picture in that sort of definition of

00:09:08.667 -- the Martian surface. That's pretty neat.

00:09:14.260 -- And if you look on the bottom,

00:09:16.470 -- you can kind of see it's.

00:09:18.370 -- It gives the step filter contract about

00:09:20.540 -- players entry .3 meters that there

00:09:22.546 -- should come out altitude 7.4 kilometers

00:09:24.574 -- now has radar lock on the ground.

00:09:26.580 -- Current city is about 100 meters per second,

00:09:29.110 -- 6.6 kilometres of the surface.

00:09:35.220 -- President is continuing to

00:09:36.888 -- descend on the parachute.

00:09:38.560 -- We're coming up on the initialization of

00:09:41.507 -- terrain relative navigation and subsequently

00:09:43.641 -- the priming of the landing engines.

00:09:46.060 -- Our current velocity is about 90 meters per

00:09:49.724 -- second at an altitude of 4.2 kilometers.

00:09:53.320 -- Now, whether or not your geologist,

00:09:55.090 -- you could look at that surface

00:09:56.692 -- and say that there's windblown.

00:09:58.340 -- You can see the guys mentioned

00:10:00.086 -- that the reservation system has

00:10:01.595 -- produced a valid solution here,

00:10:03.060 -- and part of strain out the navigation water

00:10:05.500 -- or liquid that was there be a nominal.

00:10:07.780 -- We have timing of the landing engine's.

00:10:15.160 -- Back Shell survival at sea is 83

00:10:17.589 -- meters per second at about 2.6

00:10:19.838 -- kilometers from the surface of Mars,

00:10:22.110 -- we have confirmation that the

00:10:23.940 -- back shell has separated.

00:10:25.410 -- We are currently performing

00:10:26.870 -- the divert maneuver.

00:10:27.970 -- Travelocity is about 75 meters per

00:10:30.256 -- second at an altitude of about a

00:10:32.874 -- kilometre off the surface of Mars.

00:10:34.920 -- Here, in safety Bravo.

00:10:38.090 -- We have completed our

00:10:39.770 -- terrain relative navigation.

00:10:41.030 -- Current speed is about 30

00:10:43.130 -- meters per second altitude,

00:10:44.810 -- about 300 meters off the surface of Mars.

00:10:50.680 -- We have started our constant

00:10:52.570 -- velocity accordion, which means

00:10:54.084 -- we are conducting the skycrane,

00:10:55.970 -- so this is where the sky Crane

00:10:58.616 -- maneuver deploys at it drops,

00:11:00.510 -- there's the Lander right there?

00:11:02.400 -- I mean, that's that is totally cool,

00:11:05.040 -- 20 meters off the surface.

00:11:13.220 -- And you see that in the top view,

00:11:15.220 -- that's the that's the sky cream.

00:11:16.720 -- Rocky go delta.

00:11:18.148 -- Captain confirmed persevered

00:11:19.576 -- safely on the surface of Mars,

00:11:22.130 -- ready to begin seeking

00:11:23.930 -- the sands of past life.

00:11:29.220 -- There you go.

00:11:32.380 -- How great is that?

00:11:33.828 -- And that is pretty cool.

00:11:35.640 -- And since you are,

00:11:37.088 -- since you were all gas dynamicist's,

00:11:39.260 -- you know you could figure out the

00:11:42.186 -- aerodynamics of a good portion

00:11:44.444 -- of everything that we saw there.

00:11:47.210 -- In a couple of weeks actually,

00:11:49.100 -- starting next week,

00:11:50.528 -- we'll learn about rocket nozzles.

00:11:52.910 -- OK, and how they work good.

00:11:56.780 -- Any questions at all for get going?

00:11:58.950 -- Is it not a beautiful day

00:12:00.600 -- for gas at the office today?

00:12:02.670 -- It is awesome, OK?

00:12:05.420 -- Thursday is an exam at the end of class.

00:12:08.480 -- Today we will have a little review

00:12:11.360 -- an what's going to be on the exam,

00:12:13.920 -- but I want to just rehash

00:12:15.780 -- expansion waves to make sure

00:12:17.462 -- you've got that down expansion,

00:12:19.360 -- which are very important concept.

00:12:21.400 -- OK so.

00:12:23.500 -- Recall that an expansion wave

00:12:25.810 -- occurs when you have a supersonic

00:12:28.231 -- flow that turns away from itself,

00:12:30.650 -- so it's coming down this way

00:12:32.828 -- known as the mod numbers.

00:12:35.010 -- One goes through goes through

00:12:36.995 -- a turn away from itself.

00:12:38.980 -- There are some questions

00:12:40.568 -- last time after class,

00:12:42.160 -- so I just want to clarify

00:12:44.542 -- this that this line,

00:12:46.130 -- this leading Mauchline in this

00:12:48.205 -- trailing Mauchline constitute

00:12:49.450 -- what's called an expansion fan.

00:12:51.290 -- OK, those are expansion waves that.

00:12:53.810 -- That occur when a supersonic

00:12:56.005 -- flow turns away from itself.

00:12:58.200 -- OK, this leading Mauchline,

00:12:59.992 -- then that's the front edge of that fan,

00:13:03.470 -- and it occurs at this Mach angle.

00:13:06.540 -- Musa one OK?

00:13:09.470 -- It's it's the air turns through this

00:13:11.990 -- fan once it leaves the fan then it

00:13:14.769 -- falls the wall as it comes down here OK.

00:13:17.890 -- In the turning angle Delta OK,

00:13:20.590 -- recall a number of things.

00:13:22.840 -- The flow is isentropic.

00:13:24.640 -- No losses.

00:13:25.540 -- That means the stagnation pressure

00:13:27.790 -- remains constant across that turn,

00:13:30.040 -- their model number goes up.

00:13:33.190 -- It accelerates,

00:13:33.940 -- in fact,

00:13:34.690 -- I mentioned nozzles rocket nozzles will learn

00:13:37.357 -- how expansion waves work in rocket nozzles,

00:13:39.870 -- and that's how they that's how a

00:13:42.201 -- nozzle can produce supersonic flow.

00:13:44.320 -- Since the model number goes up,

00:13:46.550 -- the pressure goes down,

00:13:48.030 -- static pressure goes down,

00:13:49.510 -- static temperature goes down,

00:13:51.350 -- so these temperature and pressure ratios

00:13:54.182 -- right here are both less than one.

00:13:56.420 -- OK, again,

00:13:57.056 -- as you're going through problems,

00:13:58.650 -- make sure that you've got that.

00:14:01.850 -- That when you calculate those numbers

00:14:03.602 -- that your pressure and temperatures are

00:14:05.476 -- all trending in the right directions.

00:14:07.450 -- OK, if you haven't seen it already

00:14:09.445 -- in your homework problems,

00:14:10.870 -- but I've seen it all the time in exams.

00:14:13.670 -- You're under the pressure.

00:14:14.878 -- You gotta get the problem solved and

00:14:17.060 -- you accidentally invert a number.

00:14:18.640 -- Or you read it wrong.

00:14:20.200 -- OK,

00:14:20.540 -- check and make sure that all those

00:14:22.920 -- pressure temperature Mach number

00:14:24.332 -- values are going in the right direction.

00:14:26.670 -- OK.

00:14:28.710 -- Good we showed then what the

00:14:31.032 -- Prandtl Meyer angle was and it is.

00:14:33.400 -- It's a fictitious angle.

00:14:34.828 -- OK so it's not like as an angle.

00:14:37.740 -- You can take your protractor

00:14:39.540 -- out and measure it in the flow.

00:14:42.070 -- It's a mathematical construct that occurs

00:14:44.590 -- for every Mach number greater than one.

00:14:47.480 -- OK,

00:14:47.824 -- and as you learn in your in your homework

00:14:51.004 -- that you could go through the book

00:14:53.573 -- or go through com prop and be able

00:14:56.444 -- to figure out what that value of

00:14:58.840 -- Theta is that parental Meyer angle.

00:15:00.970 -- And this is the big relationship right here.

00:15:04.580 -- That the turning angle.

00:15:06.292 -- Right there is related to the

00:15:08.946 -- difference in the parental Myer angles.

00:15:12.100 -- OK. Alright, and this is this is

00:15:14.564 -- the key relationship right here.

00:15:16.420 -- When you're solving expansion weight

00:15:17.885 -- problems, you know any two values.

00:15:19.650 -- If you know the downstream Mach

00:15:21.360 -- number and the turning angle,

00:15:22.870 -- you can get the upstream Mach number.

00:15:24.920 -- If you know the upstream Mach number

00:15:26.901 -- in the downstream Mach number,

00:15:28.440 -- you get the turning angle.

00:15:29.900 -- If you know the turning angle

00:15:31.562 -- in the upstream Mach number,

00:15:33.120 -- you can get announcement number.

00:15:34.590 -- OK, so you'll be able to deduce

00:15:36.522 -- two pieces of information and

00:15:38.168 -- from this relationship right here

00:15:40.043 -- you can get the third OK.

00:15:42.030 -- Straight forward there.

00:15:46.230 -- OK, shock expansion theory.

00:15:48.054 -- You use this when you solve problems, OK?

00:15:54.360 -- So again, I don't want you to use

00:15:57.120 -- the airfoil function on com prop.

00:15:59.500 -- I want you to be able to look at

00:16:01.831 -- an airfoil like this and determine

00:16:04.504 -- if you have oblique shockwaves.

00:16:06.840 -- If you have expansion ways all

00:16:09.420 -- based on the geometry of the turn

00:16:12.395 -- right here and the angle of attack.

00:16:15.120 -- OK, let's let's, let's talk about.

00:16:17.160 -- Let's talk about a problem here real

00:16:19.505 -- quickly to make sure you've got this down.

00:16:22.260 -- 'cause this is, this is an issue

00:16:24.514 -- that comes up many times here.

00:16:26.680 -- Let's say you have a triangular

00:16:28.720 -- shaped airfoil.

00:16:29.400 -- OK, and then we'll just put

00:16:31.494 -- this at an angle of attack of 0.

00:16:34.160 -- So looks like this.

00:16:35.520 -- It's a symmetric airfoil.

00:16:39.930 -- Looks like this here and you have some

00:16:42.314 -- Mach number that's greater than one.

00:16:44.400 -- And let's say that we have a

00:16:46.647 -- turning angle here of 10 degrees

00:16:48.601 -- just to pick a number, OK.

00:16:50.590 -- K alpha is equal to 0,

00:16:53.290 -- so zero angle of attack there.

00:16:55.290 -- What kind of waveform do

00:16:56.955 -- you see at the bottom?

00:17:01.400 -- Nothing. What's the pressure that

00:17:03.725 -- acts on the bottom right there?

00:17:06.620 -- It's whatever your that's greater than one.

00:17:08.730 -- It's whatever your piece

00:17:09.934 -- of one is right there.

00:17:11.440 -- Whatever the atmospheric pressure is.

00:17:13.770 -- OK. Right, what do you see on

00:17:17.368 -- this top surface right up here?

00:17:21.580 -- Well, big shockwave.

00:17:22.555 -- The flow is turning into itself.

00:17:24.510 -- Changes 10 degrees,

00:17:25.740 -- so in oblique shock.

00:17:27.380 -- Forms right there and what's

00:17:29.240 -- the direction of the flow?

00:17:31.100 -- Flow follows the wall.

00:17:34.730 -- So it goes up this way.

00:17:36.790 -- OK, so here's where the here's

00:17:38.680 -- where the messed up part happens.

00:17:40.830 -- What occurs around the turn here?

00:17:44.510 -- Expansion wave.

00:17:46.340 -- OK, here's the question.

00:17:47.572 -- What is the turning angle

00:17:49.112 -- across the top there?

00:17:53.090 -- For me. 20 degrees.

00:17:59.000 -- Everybody see why that is.

00:18:01.140 -- If not, let's talk about it.

00:18:03.400 -- OK, if this is where the mistake comes in.

00:18:06.970 -- If it were 10 degrees.

00:18:09.810 -- Then what would be the direction

00:18:11.904 -- of the flow coming out here?

00:18:13.990 -- It would be horizontal.

00:18:16.970 -- If that turning angle 10 degrees

00:18:18.938 -- 'cause it goes down this way goes up

00:18:21.458 -- 10 degrees and so it's going to turn

00:18:23.919 -- another 10 degrees just to go horizontal.

00:18:26.330 -- OK, however the flow is not

00:18:28.772 -- horizontal on that side.

00:18:30.400 -- It goes down this way another 10 degrees.

00:18:34.230 -- OK, so this turning angle.

00:18:37.910 -- Right there. 20 degrees.

00:18:42.960 -- OK. Alright.

00:18:46.690 -- OK, you can calculate the

00:18:48.475 -- pressure in this region.

00:18:49.910 -- You can calculate the pressure

00:18:51.700 -- on this plate right here?

00:18:53.490 -- Can you calculate the pressure

00:18:55.280 -- in this region right here?

00:18:57.070 -- Absolutely OK with those three pressures.

00:18:59.910 -- Determine what the force is.

00:19:02.860 -- And then some of the forces.

00:19:05.930 -- In the vertical.

00:19:08.090 -- And the horizontal directions

00:19:09.534 -- get the lift in the draft.

00:19:11.700 -- OK.

00:19:12.160 -- Good.

00:19:12.620 -- OK, this is not so critical in

00:19:15.840 -- your calculations of lift and drag,

00:19:19.470 -- but an actually an oblique

00:19:22.250 -- shockwave forms here.

00:19:23.920 -- And the reason that is is because

00:19:26.188 -- the flow now comes down this way

00:19:28.809 -- and it's going to turn horizontally.

00:19:31.180 -- Across here.

00:19:31.922 -- So the float turns into itself,

00:19:34.150 -- and so there's going to be

00:19:35.890 -- an oblique shockwave,

00:19:36.760 -- but you don't have to worry about that

00:19:38.864 -- for to calculate the lift and drag.

00:19:41.090 -- OK, so it's this turning angle right there.

00:19:44.010 -- That causes problems.

00:19:45.102 -- I wanna make sure you got that down OK?

00:19:48.280 -- Excellent.

00:19:50.460 -- OK.

00:19:52.370 -- Overhead,

00:19:52.728 -- alright,

00:19:53.086 -- so actually all this is just a

00:19:55.592 -- review of what we talked about here

00:19:57.736 -- that the lift is equal to the sum

00:20:00.230 -- of the vertical components of all

00:20:02.150 -- the forces that act on the plate.

00:20:06.740 -- Very good. The lift coefficient

00:20:08.890 -- often used in the in determining

00:20:11.560 -- what kind of airfoil you want to

00:20:14.570 -- use and over what flight regimes.

00:20:17.590 -- OK is just the lift lift force.

00:20:21.340 -- Divided by 1/2 rho V squared,

00:20:23.660 -- this is the dynamic pressure.

00:20:26.230 -- Multiplied by the.

00:20:28.420 -- Multiplied by S here,

00:20:31.340 -- that is the area of the wing.

00:20:35.170 -- That's what S is there.

00:20:36.980 -- OK, and we showed in class in

00:20:39.101 -- the past that 1/2 rho V squared

00:20:41.245 -- is the same as one half P Mach

00:20:43.857 -- number squared right there.

00:20:47.600 -- OK.

00:20:50.040 -- So let's look at some.

00:20:51.890 -- Let's look at some things.

00:20:53.730 -- Let's let's just solve for L for

00:20:56.999 -- this relationship right here.

00:20:58.910 -- L and let's see what we can

00:21:01.101 -- do to generate lift here. OK.

00:21:03.381 -- We could have a better lift coefficient.

00:21:06.960 -- OK, so that's going to depend.

00:21:08.720 -- That's going to depend on

00:21:10.190 -- the shape of the airfoil.

00:21:11.660 -- OK, we could get better lift if you

00:21:14.212 -- have a higher freestream pressure.

00:21:16.990 -- OK, are you higher in the atmosphere,

00:21:19.430 -- lower in atmosphere that governs that?

00:21:22.880 -- But look at this right here. The lift.

00:21:26.392 -- Goes like the Mach number squared.

00:21:30.870 -- So what does that say about how fast you fly?

00:21:33.580 -- What is that going to generate?

00:21:36.120 -- More. Lift.

00:21:39.910 -- OK, so that means you can have

00:21:42.570 -- some very poorly shaped airfoils.

00:21:45.090 -- OK, so this Isabelle.

00:21:46.890 -- This lift coefficient might be bad.

00:21:49.590 -- But if your Mach number is high enough.

00:21:52.050 -- You can generate a lot of lift.

00:21:54.650 -- My number squared.

00:21:56.045 -- Also says that the that the wing

00:21:59.398 -- area S the larger the wing area,

00:22:02.320 -- the greater the lift you have.

00:22:05.290 -- OK, now this actually works well.

00:22:07.580 -- Not this relationship right here,

00:22:09.480 -- but this relationship.

00:22:10.953 -- If we wrote this is 1/2 Rho

00:22:14.496 -- v ^2 s think about a glider.

00:22:17.250 -- Moves very slow, right?

00:22:19.790 -- V is pretty small.

00:22:21.042 -- How does a glider make up for the

00:22:23.660 -- lift that it has to generate?

00:22:25.640 -- Was it have?

00:22:28.740 -- Lots of wing area.

00:22:30.344 -- Wings are very, very long,

00:22:32.350 -- so as if we wrote this as rho V squared.

00:22:36.080 -- Just substituting this right here.

00:22:37.940 -- If V is small then you could have

00:22:40.428 -- a large wing area right here to

00:22:43.223 -- generate whatever list you have nice.

00:22:45.770 -- OK, so that's the balance.

00:22:47.640 -- In aerodynamic design there OK?

00:22:50.770 -- We we've seen this diagram a few times and

00:22:53.920 -- showing the difference between a subsonic

00:22:56.374 -- airfoil you seem kind of nice rounded,

00:22:59.390 -- symmetric airfoil right there.

00:23:00.958 -- This is subsonic.

00:23:02.140 -- Speeds is getting close to transonic

00:23:04.408 -- here at the transonic regime.

00:23:06.450 -- Remember we said that now all the shockwaves

00:23:10.146 -- start to form right up there in the top.

00:23:13.870 -- And then that subsonic airfoil

00:23:15.900 -- has a bow shock here when it's

00:23:19.414 -- traveling supersonically,

00:23:20.810 -- whereas whereas a supersonic airfoil

00:23:23.765 -- like we see right here very thin.

00:23:27.830 -- OK, here you can see the shaded area right

00:23:30.827 -- here is going to give you it's roughly

00:23:33.616 -- related to the amount of drag that you see.

00:23:36.700 -- OK, so this shaded area right here

00:23:39.129 -- is smaller compared to its companion

00:23:41.223 -- there at the top same Mach number,

00:23:43.520 -- but different airfoil has

00:23:45.388 -- a nice rounded shape.

00:23:47.260 -- And then Supersonically right here,

00:23:48.920 -- you see, an oblique shockwave that forms.

00:23:51.870 -- Across that very thin airfoil,

00:23:53.290 -- whereas whereas you would have a bow shock

00:23:55.714 -- if you have a subsonic airfoil there.

00:23:58.250 -- OK, so again subsonic airfoils are great.

00:24:01.640 -- Flying subsonic Lee, they're awful.

00:24:04.060 -- Supersonically supersonic airfoils

00:24:05.512 -- are awful subsonic Lee,

00:24:07.450 -- but very good supersonically OK.

00:24:09.870 -- And then we talked a little bit

00:24:12.894 -- last time about these airfoils.

00:24:15.670 -- Here you see,

00:24:17.764 -- it's very thin.

00:24:19.860 -- K very thin right across here.

00:24:21.620 -- It's got a little curve there at the

00:24:23.764 -- bottom gives a little camber little

00:24:25.599 -- curvature so that you can generate lift.

00:24:27.800 -- Subsonic Lee.

00:24:28.722 -- This is the Russian.

00:24:30.570 -- This is the Russian aircraft that didn't

00:24:33.125 -- have that crashed really bad.

00:24:35.550 -- That's our 71 and then we talked about this

00:24:39.042 -- plane and F15 that flew with only one wing.

00:24:42.450 -- OK, Why was able to fly with one wing?

00:24:45.890 -- Well,

00:24:46.273 -- if we went back to our CISA Bell right?

00:24:49.720 -- He already went over it.

00:24:53.180 -- Right here, so it loses a wing.

00:24:55.880 -- So that means this area S decreases.

00:24:58.240 -- But as you increase the

00:24:59.920 -- Mach number right there,

00:25:01.270 -- you can generate the lift that you

00:25:03.552 -- need in order to fly in order to

00:25:06.088 -- in order to stay straight and level

00:25:08.430 -- roughly straight and level OK.

00:25:13.130 -- So actually before we get to that, here is.

00:25:19.100 -- So Mr Castle, sorry it's a Navy plane.

00:25:21.560 -- You OK with that? OK good.

00:25:24.310 -- So this is an F18 Hornet right here.

00:25:28.120 -- I want you to look. At the wings.

00:25:34.040 -- Can you get an idea for what that

00:25:36.152 -- for those wings look like right?

00:25:38.100 -- There? You see how thin those are?

00:25:40.870 -- It's got a very sharp leading edge.

00:25:44.130 -- OK, very thin all the way across there.

00:25:47.850 -- OK, now this actually has a really,

00:25:51.100 -- really cool design to fly.

00:25:53.430 -- Subsonic Lee. We saw F-14.

00:25:55.750 -- Tomcat had the swing wings.

00:25:58.080 -- Variable variable, swept wings.

00:25:59.860 -- If you look closely on

00:26:02.085 -- this airplane right here.

00:26:06.620 -- There is a hinge there and

00:26:09.608 -- the hinge right there. OK.

00:26:11.979 -- So what this plane does this is this

00:26:15.011 -- is awesome engineering as well.

00:26:17.960 -- OK, what this plane does with

00:26:20.234 -- this hinge here and the ailerons

00:26:22.601 -- here in the back. What it does?

00:26:28.420 -- OK, so as you say,

00:26:30.290 -- unhinged is probably a little bit unhinged,

00:26:32.900 -- so OK, so there's a hinge right there.

00:26:36.530 -- Flat plate for the surface of the wing

00:26:39.210 -- and then the other runs here in the back.

00:26:42.240 -- OK, so there's a kind of a classic

00:26:44.456 -- real thin supersonic airfoil.

00:26:46.270 -- What this leading edge hinge

00:26:47.965 -- does when it flies subsonic Lee.

00:26:49.970 -- Is that this actually bends down a little

00:26:52.906 -- bit and I'm just going to exaggerate

00:26:55.447 -- it so you can see what's going on.

00:26:58.450 -- So look like this.

00:27:01.320 -- Then the wing comes across here.

00:27:04.430 -- And then either on comes

00:27:05.855 -- back down there a little bit.

00:27:07.580 -- You see what that does to the wing?

00:27:11.290 -- Provides a little bit of curvature

00:27:14.170 -- little camber to the wing there so

00:27:17.438 -- that subsonic Lee you can take off and

00:27:21.124 -- land relatively relatively safely.

00:27:23.780 -- OK, that's that's pretty cool

00:27:25.880 -- design right there.

00:27:27.140 -- OK, will show some pictures a little

00:27:29.975 -- bit later on in the semester,

00:27:32.600 -- but one of the one of the research areas

00:27:35.750 -- that Aerodynamicists are working on now.

00:27:38.900 -- So let's say you're in the class,

00:27:41.840 -- which you really like materials.

00:27:43.940 -- OK, your control systems,

00:27:45.620 -- and not necessarily interested in that.

00:27:48.140 -- Fluids in the aero part they're

00:27:50.660 -- working water called smartwings.

00:27:52.340 -- OK, so smart wing.

00:27:55.400 -- So it has a surface like this cake

00:27:57.480 -- kind of a classic subsonic airfoil,

00:27:59.740 -- but this airport is actually

00:28:01.770 -- made up of a bunch of sections.

00:28:04.930 -- Right here. OK, on the wing.

00:28:08.774 -- So kind of looks like this,

00:28:11.390 -- and so those sections and

00:28:13.010 -- I'm just putting a squares.

00:28:14.630 -- I think they're hexagons.

00:28:15.926 -- I don't recall right off hand,

00:28:17.870 -- but you get the idea.

00:28:19.490 -- OK,

00:28:19.804 -- each of these little sections and

00:28:21.688 -- what they what they do depending on

00:28:24.008 -- the flight regime that you're in,

00:28:25.970 -- flying really fast or really slow,

00:28:27.920 -- it actually changes the shape

00:28:29.535 -- of the wing or your fly.

00:28:33.950 -- OK, so there are little.

00:28:37.460 -- Their little devices, little little motors,

00:28:39.530 -- little servos here on the inside that can

00:28:42.322 -- move up and down and change the shape of

00:28:45.297 -- the wing to make it the most aerodynamic

00:28:48.091 -- and the most the most efficient wing

00:28:51.070 -- for that particular flight regime.

00:28:54.050 -- That school engineering. OK.

00:28:59.040 -- Good, here's a here's another

00:29:01.895 -- little bit of supersonic design.

00:29:04.750 -- OK, I don't mean to.

00:29:06.570 -- I don't mean to brag on

00:29:08.682 -- this little guy right here.

00:29:10.570 -- Here's the X one.

00:29:11.966 -- We mentioned that it did break.

00:29:14.580 -- The sound barrier didn't have swept wings.

00:29:17.130 -- Not not the best compressible flow design.

00:29:19.670 -- Here's another part that Aerodynamicists

00:29:21.884 -- didn't understand at the time,

00:29:23.680 -- but is not a good design here is

00:29:27.192 -- that if you look at the tail.

00:29:30.490 -- OK, if you look at the tail right there,

00:29:33.700 -- notice it's got cut.

00:29:35.256 -- Classic tail design tail here and you see

00:29:38.286 -- you see here these tabs here in the back.

00:29:41.200 -- In the back there so that when this

00:29:43.816 -- is flying this remains flat and

00:29:45.979 -- then those back parts go up and

00:29:48.565 -- down again so that can maneuver

00:29:50.791 -- the plane going like this.

00:29:52.696 -- Turns out that Supersonically that is

00:29:55.054 -- not a very good way to maneuver in aircraft.

00:29:58.680 -- OK, so let's see why.

00:30:00.390 -- If this is flying super fast or if it's

00:30:03.261 -- flying this in the in the supersonic regime,

00:30:06.220 -- there were going to form there at the front.

00:30:12.230 -- Oblique shockwaves OK,

00:30:13.565 -- so an oblique shockwave forms there and

00:30:16.755 -- then these fins go up and down like this,

00:30:19.700 -- and so another shockwave is

00:30:21.665 -- going to format that side.

00:30:23.630 -- So you get 2 shocks that would form

00:30:26.014 -- and what aerodynamics is found is that

00:30:28.404 -- you could get a shock interaction

00:30:30.724 -- between those two shockwaves that

00:30:33.329 -- form their supersonic aircraft.

00:30:35.420 -- Nowadays will go back to

00:30:37.385 -- our F18 Hornet right here,

00:30:39.350 -- and you look at the tail.

00:30:42.720 -- Right here, OK, notice that there's no.

00:30:45.740 -- There's no part in the back there

00:30:49.408 -- that the whole tail moves up and down.

00:30:53.600 -- Like this or like this OK?

00:30:56.020 -- And it turns out that aerodynamically

00:30:58.438 -- again when this, when this turns,

00:31:01.363 -- say down like that.

00:31:03.650 -- Shockwave's going to form there at the

00:31:06.037 -- top and then you get smooth flow all

00:31:08.703 -- the way across the rest of the elevator.

00:31:11.420 -- OK, so that's another sign that you

00:31:13.583 -- could tell of what a supersonic

00:31:15.632 -- aircraft looks like if that whole

00:31:17.750 -- tail moves as opposed to just the

00:31:20.057 -- just the back portion of the tail.

00:31:24.300 -- Alright.

00:31:26.960 -- Aren't you gonna glad

00:31:27.896 -- you came to class today?

00:31:29.070 -- I'm glad that you came in class. OK.

00:31:33.690 -- Alaskan, so one of the things that now

00:31:36.554 -- that we've got this here, so let's see.

00:31:39.450 -- Did you send this to me?

00:31:41.610 -- Yes, so I got an email from Samantha.

00:31:44.490 -- He ran. She said that you should check

00:31:46.826 -- out this particular design and it's

00:31:49.187 -- called a coleopter Anna Coleoptile.

00:31:51.330 -- If you notice there that it has a,

00:31:54.210 -- it has a rounded wing on it.

00:31:57.880 -- That's that's kind of cool

00:31:59.755 -- right across there.

00:32:00.880 -- So here's the rounded wing.

00:32:02.760 -- Now it turns out that when

00:32:05.268 -- you have a finite wing.

00:32:07.520 -- Like you have here,

00:32:08.820 -- the pressure on the bottom of the

00:32:11.148 -- wing is higher than the pressure

00:32:13.014 -- on the top of the wing does.

00:32:15.150 -- That's how you generate lift.

00:32:16.740 -- But what happens is you have

00:32:18.504 -- what are called end effects.

00:32:20.240 -- So in end effect occurs when

00:32:22.052 -- the air on the bottom of the

00:32:24.311 -- wing kind of leaks out here,

00:32:26.280 -- and so you have high pressure air on the

00:32:29.133 -- bottom and low pressure air on the top.

00:32:31.690 -- And what that does is generate

00:32:33.826 -- these vertical structures go like

00:32:35.582 -- this and that causes a lot of drag.

00:32:37.730 -- It's a parasitic drag problem.

00:32:39.590 -- End effect drag problem OK in in

00:32:43.545 -- aerodynamics class you'd learn about how

00:32:46.926 -- wings can be shaped to minimize that.

00:32:50.610 -- The Spitfire that flew in World War

00:32:53.263 -- Two has elliptical shaped wings.

00:32:55.390 -- Beautiful beautiful aircraft but those

00:32:57.545 -- elliptical shapes right there minimized

00:32:59.759 -- the formation of those tip foresees.

00:33:01.750 -- OK so menace and minimizes

00:33:03.740 -- the drag flights faster.

00:33:05.340 -- OK so one of the nice things

00:33:08.028 -- about this design over here of

00:33:10.389 -- the coleopter is that it's got a

00:33:13.105 -- rounded wing so that there are no

00:33:16.024 -- tip effects that go across there.

00:33:18.489 -- Alright so you might think well.

00:33:20.910 -- How the heck does that generate

00:33:23.046 -- any kind of lift?

00:33:24.470 -- OK,

00:33:24.817 -- well it turns out that the angle

00:33:27.246 -- of attack is also related to the

00:33:29.756 -- amount of lift that you have,

00:33:31.950 -- so the higher angle of attack that you get,

00:33:35.150 -- the more lift you can generate.

00:33:37.290 -- Now for those that are paper airplane.

00:33:40.370 -- Enthusia STS

00:33:44.170 -- is it cold after I hear? Ever seen this?

00:33:48.130 -- And you can fly it. We'll see,

00:33:50.858 -- maybe we can catch this in slo-mo

00:33:53.266 -- from our from our from the cameras

00:33:55.583 -- here so you can fly this like this.

00:34:00.800 -- Not very good. Not very good.

00:34:04.260 -- But the idea is that you can generate lift.

00:34:07.500 -- Now you might think how can you

00:34:09.803 -- generate lift from something like that.

00:34:12.180 -- And again notice that in the picture

00:34:14.812 -- here it's flying at an angle of attack.

00:34:17.580 -- OK, so an airplane like this right here.

00:34:21.240 -- OK, actually has a slight angle of attack

00:34:24.008 -- built into it so that when it takes off.

00:34:26.710 -- In fact, if you if you look at it

00:34:29.374 -- straight and level right here, that Wing

00:34:31.812 -- has a little bit of an angle of attack.

00:34:34.760 -- OK, so that angle of attack is going to

00:34:37.460 -- create a small shockwave on the bottom.

00:34:39.920 -- Slight expansion wave on the top

00:34:41.852 -- so it generates lift.

00:34:43.140 -- OK, now it also answers the age old question

00:34:46.632 -- of how can a plane like this fly inverted?

00:34:50.330 -- OK, again, if this plane and you'll

00:34:52.283 -- notice it notice any aircraft when it

00:34:54.374 -- flies upside down screen level like

00:34:56.233 -- the Blue Angels and the Thunderbirds

00:34:57.871 -- when they do a flight show like that,

00:35:00.422 -- there's always a little bit of an

00:35:02.957 -- angle of attack as it goes down so

00:35:05.253 -- that it generates lift. OK, good.

00:35:10.130 -- That's your area lesson for today.

00:35:12.980 -- Alright. Got any questions?

00:35:14.924 -- Any questions? Anything we've seen so far?

00:35:18.570 -- OK.

00:35:21.500 -- So let's see here what we'd like to do now is

00:35:24.903 -- to figure out what we know for turning angle,

00:35:27.980 -- you can accelerate the flow.

00:35:29.600 -- What is the maximum turning angle?

00:35:31.540 -- How much can you turn?

00:35:33.810 -- And then what is the associated

00:35:35.670 -- Mach number associated with that?

00:35:37.230 -- OK, so here's our parental Meyer angle.

00:35:39.410 -- Theta of M is equal to this relationship,

00:35:41.900 -- right here it's got an arctangent of

00:35:43.972 -- the Mach number there and arctangent

00:35:45.881 -- of the Mach number right over here, OK.

00:35:48.439 -- So this is this will harken

00:35:50.353 -- back to your calculus days.

00:35:52.400 -- I know that was three or four years ago,

00:35:54.870 -- right so?

00:35:57.110 -- We're going to take a limit.

00:35:59.800 -- And what we're going to do is we're going

00:36:02.500 -- to take the limit as this angle as Phi.

00:36:05.330 -- We want to know what that

00:36:07.988 -- maximum turning angle is.

00:36:09.760 -- Of of our flow here.

00:36:11.500 -- So notice that in these arc

00:36:13.480 -- tangents right here we have gammas.

00:36:15.660 -- Cagamas are going to be constant for the air.

00:36:18.780 -- The only variable that we

00:36:20.515 -- have is the Mach number.

00:36:22.250 -- So we want to say if this

00:36:25.029 -- Mach number goes to Infinity.

00:36:27.460 -- What's the turning angle going to be?

00:36:30.510 -- OK, so just from mathematics right here.

00:36:33.560 -- The arctangent of Infinity is π / 2.

00:36:40.720 -- So we figure that one out.

00:36:43.170 -- Arctangent of Infinity is π / 2.

00:36:45.917 -- Let's go to the overhead here.

00:36:49.080 -- So have a triangle.

00:36:52.060 -- OK, and this is fi right here.

00:36:56.910 -- What's the tangent of Phi?

00:37:01.070 -- Say X. Why? Z what's tangent feet?

00:37:10.090 -- Tangent of Phi. The rise over the run. Y / X.

00:37:16.840 -- OK, alright, so let's let's

00:37:19.200 -- expand fi a little bit here.

00:37:26.040 -- Here is why an X.

00:37:30.280 -- 10s if he's going to get bigger

00:37:32.555 -- there right? 'cause your eyes.

00:37:34.314 -- Why is large and X is small here?

00:37:37.390 -- What happens when you have

00:37:38.560 -- a triangle look like this?

00:37:46.200 -- Look at this rise divided by

00:37:48.858 -- that teeny tiny run right there.

00:37:51.840 -- And then eventually.

00:37:53.553 -- You just have a straight line.

00:37:56.980 -- This is 90 degrees right here.

00:37:59.530 -- Which is the same. Is π / 2?

00:38:05.520 -- So the arctangent.

00:38:07.371 -- A fee from our little trigonometry

00:38:11.073 -- bit right there is π / 2.

00:38:14.287 -- That's why the limit as

00:38:16.972 -- fee tends to Infinity. OK.

00:38:21.280 -- Net fee is just the rise over the run.

00:38:23.900 -- That's why are y / X here is equal to π / 2.

00:38:27.653 -- OK, so if we take that limit then

00:38:29.757 -- Theta taking the limit here as

00:38:31.596 -- N goes to Infinity here an as N

00:38:34.053 -- goes to Infinity here substituting

00:38:35.528 -- in π / 2 to both of those terms,

00:38:38.160 -- we get that Theta is equal to π /

00:38:40.780 -- 2 times the square root of gamma

00:38:42.803 -- plus one divided by gamma minus one.

00:38:44.850 -- All this minus one right here.

00:38:47.720 -- So you can actually get a turning

00:38:51.066 -- angle of 130 degrees.

00:38:53.440 -- If you could turn that flow 130 degrees,

00:38:55.910 -- you would accelerate it to Infinity.

00:38:59.920 -- Obviously that's not going to happen OK,

00:39:02.030 -- Anna 130 degrees.

00:39:02.930 -- I mean, if you think about it,

00:39:05.040 -- it's got not just 90 degrees,

00:39:06.840 -- but now it's coming back

00:39:09.105 -- in the opposite direction.

00:39:10.920 -- Not an actual physical limit,

00:39:12.320 -- but a theoretical limit on what

00:39:14.474 -- the term could be in your flow.

00:39:16.940 -- OK, little bit of mathematical

00:39:20.750 -- gymnastics there OK?

00:39:23.040 -- This is the problem that was

00:39:24.912 -- cancelled in your homework assignment,

00:39:26.850 -- but I want you to make sure that

00:39:29.258 -- we've got shock tubes down so that you

00:39:32.133 -- understand how a shock tube works.

00:39:34.460 -- OK, so again,

00:39:35.264 -- this is 1/2 of it as a rehash from

00:39:38.039 -- when we did normal shocks and moving

00:39:40.909 -- normal shocks in a shock tube.

00:39:43.110 -- You have a driver section.

00:39:44.840 -- OK, high pressure gas right?

00:39:46.570 -- Over here there's a membrane that

00:39:48.856 -- separates this high pressure region

00:39:50.786 -- from a low pressure region right over here.

00:39:53.520 -- And this is what the pressure

00:39:55.986 -- profile looks like.

00:39:57.220 -- So you have a high pressure

00:39:59.542 -- region in the driver region.

00:40:01.740 -- A low pressure area right

00:40:03.795 -- here in the DRIVIN region.

00:40:05.850 -- OK then we break the membrane

00:40:08.406 -- and what that membrane does is

00:40:10.967 -- that it creates a normal shock

00:40:13.277 -- and that normal shockwave then

00:40:15.692 -- propagates down from left to right.

00:40:18.810 -- OK,

00:40:19.214 -- and so this stuff that we talked

00:40:22.042 -- about in class when we did

00:40:24.901 -- normal shops right across here.

00:40:27.350 -- Right across here,

00:40:28.307 -- we figured out what the pressure

00:40:30.602 -- profiles the Mach number is.

00:40:32.200 -- The Mach number of the wave, and so on.

00:40:35.225 -- We figured all this stuff out and we

00:40:38.262 -- ignored the things on the left hand side OK.

00:40:41.780 -- Kate, now that we know about expansion waves.

00:40:44.930 -- What's going on here between 3:00 and

00:40:47.450 -- 1:00 is an expansion wave problem.

00:40:50.330 -- Just like we talked about before.

00:40:53.610 -- OK,

00:40:54.067 -- so what's happening here is that again

00:40:57.266 -- you have a high pressure region here.

00:41:00.780 -- There is a lower pressure region

00:41:02.898 -- right across here in Region 3.

00:41:04.980 -- In fact,

00:41:05.714 -- if you look at the pressure

00:41:07.916 -- profile here it is.

00:41:09.180 -- This is the driven section

00:41:10.930 -- has a low pressure,

00:41:12.330 -- goes across the shock.

00:41:13.610 -- That's what this region is right

00:41:15.597 -- across here and then that pressure

00:41:17.727 -- remains constant to what's called

00:41:19.582 -- the contact surface right here.

00:41:21.430 -- And that's where Region 3 contacts Region 4.

00:41:24.230 -- And then there is a gradual

00:41:26.606 -- increase in the pressure right up

00:41:29.075 -- here and then the driver section.

00:41:31.480 -- OK,

00:41:32.021 -- so this gas this high pressure gas

00:41:35.808 -- begins to expand. As it goes across.

00:41:40.872 -- OK.

00:41:41.640 -- Draw that out, drawn out process out here,

00:41:44.900 -- OK?

00:41:46.430 -- So.

00:41:52.870 -- So here's the membrane.

00:41:54.442 -- Here's the high pressure section.

00:41:56.410 -- Here is the low pressure section here.

00:41:59.160 -- OK, when the membrane breaks.

00:42:03.680 -- Creates a shockwave.

00:42:06.900 -- Propagates down this direction here,

00:42:09.680 -- and an expansion wave

00:42:12.464 -- forms as this gas expands.

00:42:15.950 -- Into this region right over

00:42:18.695 -- here so that wave propagates.

00:42:21.440 -- Starts out small here,

00:42:23.416 -- gets a little bit larger.

00:42:25.890 -- And then that distance increases

00:42:27.525 -- right across there as this high

00:42:29.536 -- pressure gas region expands

00:42:30.820 -- and goes across this way.

00:42:35.040 -- As a as my kids say, you know fun

00:42:39.299 -- fact fun fact about expansion waves.

00:42:42.140 -- OK, this behaves exactly like.

00:42:46.240 -- A traffic jam or traffic

00:42:49.620 -- flow at a red light. OK.

00:42:54.940 -- But how could that be OK?

00:42:57.100 -- Think about this.

00:42:58.180 -- You have a red light right here.

00:43:04.440 -- We want to make this authentic,

00:43:06.310 -- so there's a red light right there.

00:43:08.480 -- OK? And there are cars lined up.

00:43:11.840 -- And it's all bumper to bumper, right?

00:43:13.897 -- I'm sure all of you keep a safe

00:43:16.593 -- distance right when you break.

00:43:18.520 -- In traffic, especially,

00:43:19.669 -- the traffic jams here in Moscow ID OK,

00:43:22.620 -- so looks like this and you can have a line

00:43:26.893 -- of cars that go all the way back here.

00:43:30.670 -- OK, so the red light then turns green.

00:43:35.490 -- Green light right here.

00:43:40.130 -- There's a green light OK,

00:43:42.170 -- and now the cars go do all seven of

00:43:45.050 -- these cars move at the same speed and

00:43:48.453 -- propagate through their drive-thru light?

00:43:51.150 -- No, what happens?

00:43:52.809 -- First car goes it's here.

00:43:55.580 -- And there's a larger distance

00:43:57.390 -- between that one and the next one,

00:43:59.880 -- and then that distance gets a

00:44:01.890 -- little bit smaller and smaller,

00:44:03.810 -- and the way in the back here is that,

00:44:07.040 -- and I've had this happen to me before.

00:44:09.900 -- Is that you see the green light here,

00:44:12.760 -- but the distance when I'm parked

00:44:14.710 -- way back here the distance between

00:44:16.863 -- me and that car hasn't changed.

00:44:19.210 -- Heckuva lot for 1015 seconds,

00:44:21.000 -- however long it takes to

00:44:22.790 -- propagate that through these cars.

00:44:24.580 -- Driving through here.

00:44:26.062 -- It turns out have the same

00:44:29.026 -- mathematical modeling associated

00:44:30.809 -- with expansion ways right there.

00:44:33.820 -- Same modeling process here.

00:44:35.308 -- This membrane breaks the gases expand.

00:44:37.540 -- This wave goes faster than this one,

00:44:40.140 -- which goes faster than this one and so on.

00:44:43.490 -- So as this as this pressure wave

00:44:46.381 -- propagates there through the back

00:44:48.431 -- has the same properties as cars

00:44:50.645 -- parked not part but in a in a

00:44:53.369 -- traffic line right across there.

00:44:57.160 -- There you can tell your folks

00:44:58.984 -- over the break that you learned

00:45:01.030 -- something here in gas dynamics.

00:45:02.940 -- OK, like I said, fun fact.

00:45:05.720 -- OK, shock tubes in let's go and here is

00:45:09.374 -- what happens after the membrane breaks.

00:45:13.010 -- OK, so let's look at the

00:45:15.920 -- expansion wave process.

00:45:17.380 -- OK, so here again these waves

00:45:19.840 -- propagate back and as they propagate

00:45:22.606 -- back eventually this high pressure

00:45:25.216 -- region is going to completely expand.

00:45:28.560 -- OK, so the initial state of our

00:45:32.179 -- of our shock tube right here.

00:45:35.730 -- Of this pressure difference

00:45:36.918 -- across here from one to two,

00:45:38.700 -- we have a high pressure region

00:45:40.380 -- in a low pressure region,

00:45:41.970 -- right across here,

00:45:42.858 -- that's called the diaphragm pressure ratio.

00:45:46.930 -- OK, right across there after that

00:45:49.348 -- diaphragm or the membrane breaks,

00:45:51.510 -- then the pressure in Region 3 is the

00:45:55.190 -- same as the pressure in region 4.

00:45:58.840 -- And the speed in Region 3 is the

00:46:01.216 -- same as the speed in region 4.

00:46:03.520 -- So if we go back to our

00:46:05.627 -- little drawing right here,

00:46:06.950 -- these two pressures are the same.

00:46:08.820 -- Cross this contact surface and the speed

00:46:11.004 -- of the air or the gases between those

00:46:13.570 -- two regions are the same as well. What?

00:46:16.176 -- What do you think would be different?

00:46:19.580 -- Speeds the same pressures the same.

00:46:21.010 -- What do you think would be

00:46:23.380 -- different across there?

00:46:24.570 -- What happens to the temperature

00:46:27.760 -- across a shockwave?

00:46:29.680 -- It goes up what happens to the

00:46:32.214 -- pressure downstream of this expansion

00:46:34.112 -- wave as it starts to expand.

00:46:36.320 -- Goes down. OK, so there's a.

00:46:38.530 -- There's a temperature difference between

00:46:40.365 -- 3:00 and 4:00, right across there.

00:46:42.586 -- OK, T4 is going to be higher than T2,

00:46:45.890 -- and T3 is going to be higher than T1.

00:46:49.830 -- That's really what defines that difference

00:46:51.882 -- right across there is that temperature OK,

00:46:54.380 -- but the pressures in those

00:46:56.130 -- two regions are the same.

00:46:57.880 -- The speed in those two regions of the same.

00:47:01.030 -- OK, so again, across an expansion wave,

00:47:03.480 -- the flow is isentropic.

00:47:04.764 -- So we can use our isentropic relations here.

00:47:07.680 -- P3 or P1 is equal to T3 over T1 to

00:47:10.572 -- the gamma over gamma minus one power.

00:47:13.630 -- Or we could write that in terms

00:47:16.843 -- of the densities as well.

00:47:19.090 -- You know this is nice because

00:47:20.590 -- once we get it in this form,

00:47:22.390 -- we can write this in terms

00:47:24.412 -- of the speed of sound.

00:47:26.330 -- So now P3 over P11 minus gamma one.

00:47:30.120 -- Over 2 times V 2 /, 81 squared.

00:47:33.334 -- So this is the Mach number.

00:47:36.680 -- OK, in the expansion region right there to

00:47:39.272 -- the two gamma sub one over gamma minus one,

00:47:42.260 -- right across there.

00:47:43.769 -- OK if we solve for V then.

00:47:47.290 -- We can solve for V2.

00:47:48.760 -- We can solve for the speed and region 2.

00:47:52.260 -- Right across there and it is just

00:47:54.808 -- a function of the speed of sound.

00:47:57.450 -- He said one and this pressure ratio,

00:48:00.050 -- the diaphragm pressure ratio P2 over P1 OK.

00:48:04.450 -- What it gives us here is that since gamma

00:48:07.105 -- minus one this power right over here,

00:48:09.410 -- gamma minus 1 / 2 gamma is less

00:48:11.658 -- than one as PP1 tends to Infinity.

00:48:14.060 -- This term is going to go to zero,

00:48:16.540 -- and So what this does is you

00:48:18.990 -- can generate a maximum.

00:48:20.780 -- Speed in region 2.

00:48:22.528 -- From this term right there twice,

00:48:25.150 -- the speed of sound divided

00:48:26.930 -- by gamma minus one.

00:48:28.360 -- OK, enough gammas for air,

00:48:29.850 -- that's one point 4 -- 1 or that's two

00:48:32.588 -- times the speed of sound divided by .4.

00:48:34.920 -- What it tells you can do is you

00:48:37.608 -- can generate pretty high speeds.

00:48:39.820 -- With the shock tube.

00:48:42.040 -- OK, they don't last for very long.

00:48:44.230 -- OK,

00:48:44.522 -- but you can get hypersonic flows from a

00:48:46.858 -- shock tube that comes right across here,

00:48:49.240 -- OK?

00:48:51.270 -- Good,

00:48:51.614 -- so that was the problem

00:48:53.334 -- that was cancelled is

00:48:54.781 -- to figure out the expansion portions

00:48:56.983 -- of the flow. OK, any questions? Yes.

00:49:04.810 -- OK, I'm getting to that,

00:49:06.520 -- so I answer part one.

00:49:08.220 -- No shock tube questions,

00:49:09.572 -- no moving shock problems on the exam.

00:49:14.180 -- What's that? And an airfoil

00:49:18.041 -- is not a shock tube, so yes,

00:49:20.052 -- so there is an airfoil problem.

00:49:21.770 -- There is not a shock to problem.

00:49:23.780 -- OK, I just want to make sure that

00:49:26.068 -- you understood kind of the basic

00:49:27.825 -- workings of what a shock tube are.

00:49:29.810 -- OK, not answer your question here.

00:49:31.530 -- Exams are open book.

00:49:33.690 -- Open notes OK.

00:49:37.600 -- And it's also open laptops if you

00:49:40.414 -- choose that you've got a laptop to use

00:49:43.203 -- com prop that is OK to use as well,

00:49:45.950 -- but there are no communications

00:49:48.122 -- with the outside world or the

00:49:50.278 -- inside world on your laptop.

00:49:51.870 -- So if you have a phone 'cause you

00:49:54.558 -- want to use your app, that's OK too.

00:49:59.430 -- OK, so open book, open notes, open com,

00:50:02.910 -- prop, or if you've got some other,

00:50:05.960 -- you know if you use some other device

00:50:08.848 -- for your compressible flow tables,

00:50:11.610 -- that's OK as well, yes?

00:50:17.110 -- Open Book open notes.

00:50:24.070 -- There are five problems on the exam,

00:50:25.450 -- so if you spend all your time looking

00:50:26.946 -- at your notes, you're not going to

00:50:28.408 -- have time to solve the problems.

00:50:32.510 -- OK, so be sure that your

00:50:34.682 -- your laptops are charged.

00:50:36.130 -- We don't have a whole lot of.

00:50:38.660 -- Let's see. Do you have?

00:50:40.470 -- Do you have plugs underneath?

00:50:44.200 -- Not sure what you got there. OK.

00:50:48.770 -- And you will have the hour and 15

00:50:51.746 -- minutes to solve the exam as well.

00:50:54.520 -- OK, I will talk to the greater today and we

00:50:57.548 -- will try and get this graded by tomorrow.

00:51:00.370 -- So if you want to pick him up

00:51:02.818 -- tomorrow you could come by my

00:51:04.854 -- office and pick up the exams.

00:51:06.870 -- That's fine.

00:51:07.616 -- The solutions are also will

00:51:09.481 -- be available this afternoon.

00:51:11.310 -- So even though even if you don't

00:51:12.689 -- have your homework assignment,

00:51:13.750 -- if the greater can't get it back in time.

00:51:17.120 -- Can you can always look online on TV learn?

00:51:21.160 -- OK. Good, so let's see here.

00:51:24.980 -- I would say that the exam problems

00:51:26.807 -- are going to be very similar to what

00:51:28.959 -- you see in the homework problems.

00:51:30.830 -- Let's just let's just review.

00:51:33.610 -- Write reviews 'cause then you

00:51:35.720 -- can see how intelligent you have

00:51:38.277 -- become over the last one week.

00:51:40.500 -- Six now, let's see what we've discussed here,

00:51:43.740 -- OK?

00:51:45.680 -- Just going through just going through the

00:51:49.460 -- going through the table of contents here.

00:51:53.280 -- OK, let's see here.

00:51:54.708 -- You probably won't have any problem that

00:51:57.396 -- just discusses the continuity equation.

00:51:59.780 -- You probably won't have a problem

00:52:02.222 -- that just has the ideal gas law,

00:52:05.050 -- but you could certainly use it.

00:52:08.650 -- OK,

00:52:09.065 -- some fundamental aspects

00:52:10.310 -- of compressible flow.

00:52:11.560 -- You know how to calculate the speed of sound?

00:52:16.350 -- OK, and you know how to calculate

00:52:18.380 -- Mach numbers and Mach waves.

00:52:19.950 -- I would say that's going to

00:52:21.840 -- be something to study up on.

00:52:23.550 -- Make sure that you understand what

00:52:25.224 -- a Mach wave is and how different

00:52:27.348 -- it is from an oblique shockwave.

00:52:29.250 -- What is the difference?

00:52:32.300 -- What's the difference between an

00:52:33.910 -- oblique shockwave animac wave?

00:52:37.280 -- A Mach wave is an infinitesimally

00:52:40.880 -- weak oblique shockwave.

00:52:42.680 -- OK. Good. Uh, now?

00:52:49.560 -- You probably won't have a problem.

00:52:50.820 -- It says, just calculate the speed of sound,

00:52:52.500 -- but you will likely have a problem.

00:52:53.970 -- You'll have to calculate the speed of sound.

00:52:56.440 -- OK, so this is all fundamental

00:52:59.068 -- stuff right here.

00:53:00.390 -- OK, in chapter four we learned about

00:53:02.938 -- isentropic flows and the difference

00:53:05.251 -- between stagnation conditions,

00:53:06.980 -- static conditions and critical

00:53:08.732 -- conditions right there, OK?

00:53:12.020 -- You probably won't have a problem

00:53:14.138 -- that says what is the stagnation

00:53:16.500 -- temperature of such and such.

00:53:18.680 -- You might OK, but you'll be able to.

00:53:21.820 -- You'll need to calculate what stagnation

00:53:25.144 -- temperatures are and pressures.

00:53:27.360 -- Pretty straightforward using

00:53:28.542 -- the tables using comp.

00:53:30.120 -- However you want to do it OK.

00:53:32.880 -- We learned about shockwaves.

00:53:35.072 -- OK, and you'll probably see some problems

00:53:38.584 -- that have a normal shocks in him.

00:53:41.890 -- OK, that's very important in gas dynamics.

00:53:46.210 -- OK, pitot tubes. Are also important,

00:53:51.190 -- so make sure you got those down.

00:53:54.150 -- Don't worry bout moving normal shocks.

00:53:58.200 -- OK.

00:54:00.360 -- Are oblique shock waves are important?

00:54:02.620 -- Make sure that you've got those down

00:54:05.147 -- and expansion waves are also important.

00:54:07.520 -- Make sure that you've got that down.

00:54:12.350 -- And make sure that you've got

00:54:14.702 -- the reflection part of oblique

00:54:16.895 -- shockwaves down pretty well. OK.

00:54:21.310 -- Again, as you learn in your problems,

00:54:23.570 -- a lot of it's just the geometry.

00:54:25.830 -- Make sure you've got the

00:54:27.890 -- direction of the flow right. OK.

00:54:32.670 -- Just to re clarify. We're going to

00:54:36.557 -- go back to our normal shocks here.

00:54:42.040 -- Region 1. Region 2 flow goes

00:54:44.950 -- this way comes out this way.

00:54:47.930 -- This is the region that is upstream.

00:54:53.590 -- It is also ahead. Of the shock.

00:54:59.930 -- OK, this is the supersonic flow.

00:55:03.280 -- Regime this is the subsonic flow.

00:55:07.240 -- Regime this is. Downstream.

00:55:12.700 -- Of the shock, this is also behind.

00:55:17.250 -- The shock there means the same thing.

00:55:21.530 -- OK, sometimes a problem will ask

00:55:23.396 -- what's the temperature and

00:55:25.004 -- pressure just downstream of the shock?

00:55:26.870 -- Where does that where you where?

00:55:28.750 -- Are you looking to calculate that?

00:55:31.280 -- Right there.

00:55:34.070 -- Just downstream so you don't have to

00:55:35.876 -- worry about any of the flow anywhere else.

00:55:38.000 -- What's the temperature and pressure?

00:55:39.310 -- The Mach number just downstream of the

00:55:41.564 -- shock just in that region right there?

00:55:44.080 -- OK. Excellent. Any questions?

00:55:52.850 -- OK, have a wonderful day study

00:55:55.724 -- up for the exam and we'll see

00:55:59.529 -- Thursday bright and early.