WEBVTT 00:00:00.499 --> 00:00:02.954 The following content is provided under a Creative 00:00:02.954 --> 00:00:03.620 Commons license. 00:00:03.620 --> 00:00:05.770 Your support will help MIT OpenCourseWare 00:00:05.770 --> 00:00:10.440 continue to offer high quality educational resources for free. 00:00:10.440 --> 00:00:12.590 To make a donation, or to view additional materials 00:00:12.590 --> 00:00:15.620 from hundreds of MIT courses visit MIT OpenCourseWare 00:00:15.620 --> 00:00:19.360 at ocw.mit.edu. 00:00:19.360 --> 00:00:24.250 PROFESSOR STRANG: Okay, so this is I 00:00:24.250 --> 00:00:27.310 could say delta function day. 00:00:27.310 --> 00:00:30.560 Break from linear algebra mostly. 00:00:30.560 --> 00:00:34.250 So we're looking on another type of right-hand side. 00:00:34.250 --> 00:00:36.720 Before in the differential equation 00:00:36.720 --> 00:00:39.720 and in the difference equation. 00:00:39.720 --> 00:00:44.750 So the right-hand sides up to now, the one we looked at 00:00:44.750 --> 00:00:51.580 was a uniform constant load second derivative equal one. 00:00:51.580 --> 00:00:55.320 Now a point load. 00:00:55.320 --> 00:00:58.300 Well in a way, we're now solving a whole bunch of problems 00:00:58.300 --> 00:01:02.470 because the point load can be in different places. 00:01:02.470 --> 00:01:05.780 So instead of solving one problem with one 00:01:05.780 --> 00:01:10.790 on the right-hand side, we're solving with a delta function. 00:01:10.790 --> 00:01:15.010 Now a delta function is, you probably have seen and heard 00:01:15.010 --> 00:01:19.220 the words and seen the symbol, but maybe not done much 00:01:19.220 --> 00:01:22.090 with a delta function. 00:01:22.090 --> 00:01:25.620 It takes a little practice but it's really worth it. 00:01:25.620 --> 00:01:31.490 It's a great model of maybe what can't quite happen physically, 00:01:31.490 --> 00:01:36.470 to have a load acting exactly at a point and nowhere else. 00:01:36.470 --> 00:01:40.040 So the delta function is, I drew it's picture, 00:01:40.040 --> 00:01:46.880 the delta function is zero, this is delta of x is zero except 00:01:46.880 --> 00:01:50.600 at that one point, the origin, x=0, 00:01:50.600 --> 00:01:53.910 and then all along back to zero again. 00:01:53.910 --> 00:02:00.300 So nothing's happening, no load except at that one point. 00:02:00.300 --> 00:02:05.510 And let me just, so there's no hesitation 00:02:05.510 --> 00:02:13.840 in when I change from x to x-a, what does that do to a graph? 00:02:13.840 --> 00:02:21.440 If I have a function of x and I instead shift the function 00:02:21.440 --> 00:02:27.600 to f(x-a), I shift x to x-a, well in this case, 00:02:27.600 --> 00:02:30.700 and in all cases, it will just shift the graph. 00:02:30.700 --> 00:02:37.150 So if I drew a picture of delta, of x-a, 00:02:37.150 --> 00:02:41.360 the load now would happen when this is zero, 00:02:41.360 --> 00:02:45.660 because it's delta at zero is the impulse, 00:02:45.660 --> 00:02:48.490 and now this is zero at x=a. 00:02:48.490 --> 00:02:52.440 In other words, the load moved to the point a. 00:02:52.440 --> 00:02:59.080 So there is the shifting load, but the load 00:02:59.080 --> 00:03:05.030 could fall anywhere between zero and one. 00:03:05.030 --> 00:03:07.940 So delta of x, the load actually falls at zero. 00:03:07.940 --> 00:03:10.750 Well we don't quite want that load at the boundary. 00:03:10.750 --> 00:03:14.900 So let's think of the point a, the load point as somewhere 00:03:14.900 --> 00:03:17.410 between zero and one. 00:03:17.410 --> 00:03:22.610 Can I just take a little time to recall the main fact 00:03:22.610 --> 00:03:24.580 about delta functions? 00:03:24.580 --> 00:03:29.960 When I say recall, it could very well be new to you. 00:03:29.960 --> 00:03:32.990 So that's what the delta function-- 00:03:32.990 --> 00:03:36.110 that's my best graph of the delta function. 00:03:36.110 --> 00:03:39.120 But of course I'm, in using the word function, 00:03:39.120 --> 00:03:43.470 I'm kind of breaking the rules because no function-- I mean 00:03:43.470 --> 00:03:46.200 the function is, functions can be zero there, 00:03:46.200 --> 00:03:48.060 can be zero there, but they're not 00:03:48.060 --> 00:03:52.810 supposed to be infinite at a single point in between, 00:03:52.810 --> 00:03:54.300 but this one is. 00:03:54.300 --> 00:04:00.170 Let me go back to delta of x to match these figures. 00:04:00.170 --> 00:04:04.450 Of course, they would also just shift along by a. 00:04:04.450 --> 00:04:09.490 Maybe no harm in that. 00:04:09.490 --> 00:04:16.160 Sorry, I'll stay there and now I want to integrate. 00:04:16.160 --> 00:04:22.260 And that's when a delta function comes into its own. 00:04:22.260 --> 00:04:26.330 Its value of infinity is a little bit uncertain. 00:04:26.330 --> 00:04:27.670 What does that mean? 00:04:27.670 --> 00:04:31.270 But when we integrate it, what's the key fact 00:04:31.270 --> 00:04:32.790 about delta function? 00:04:32.790 --> 00:04:37.330 That the integral of a delta function from, 00:04:37.330 --> 00:04:39.470 let's say, let's integrate the whole thing, 00:04:39.470 --> 00:04:42.310 we can safely start way at the far left 00:04:42.310 --> 00:04:44.340 and go away to the far right because it's zero 00:04:44.340 --> 00:04:49.300 all the time there except at one point, and you know. 00:04:49.300 --> 00:04:53.030 So what's the area under that spike? 00:04:53.030 --> 00:04:54.939 It is one. 00:04:54.939 --> 00:04:55.480 That's right. 00:04:55.480 --> 00:04:58.380 So that's the fact, the sort of central fact 00:04:58.380 --> 00:05:00.190 about a delta function. 00:05:00.190 --> 00:05:02.030 That the area is one. 00:05:02.030 --> 00:05:04.710 Oh well, let me, while I'm really writing down 00:05:04.710 --> 00:05:14.010 the central fact, let me write it more specifically, more 00:05:14.010 --> 00:05:14.830 generally. 00:05:14.830 --> 00:05:20.040 Suppose I integrate, and this is delta functions now really 00:05:20.040 --> 00:05:24.640 showing up, if I integrate a delta function against some, 00:05:24.640 --> 00:05:28.200 times some nice function. 00:05:28.200 --> 00:05:31.630 Now have you ever thought about that? 00:05:31.630 --> 00:05:35.990 What would be the answer if I integrate the delta function 00:05:35.990 --> 00:05:38.540 against some nice function? 00:05:38.540 --> 00:05:43.980 So I'm still getting zero from this term all the way along 00:05:43.980 --> 00:05:47.510 until I hit the spike and then after it goes back 00:05:47.510 --> 00:05:48.680 to zero again. 00:05:48.680 --> 00:05:54.030 So, whatever, it's gotta be at the spike, at x=0, 00:05:54.030 --> 00:05:57.790 because I put the spike here at zero, the impulse. 00:05:57.790 --> 00:06:02.980 So what do you think's the answer for that one? 00:06:02.980 --> 00:06:04.790 Yeah, It's the function. 00:06:04.790 --> 00:06:10.000 So, yes, tell me again and I'll write it down. g, 00:06:10.000 --> 00:06:12.240 it's a value of this function g. 00:06:12.240 --> 00:06:16.260 We don't care what it is to the left and to the right at zero 00:06:16.260 --> 00:06:20.390 because it's really at zero that this thing turns on 00:06:20.390 --> 00:06:24.690 and its value at that point is just-- 00:06:24.690 --> 00:06:31.260 gives us the amplitude of the impulse, which is g(0). 00:06:31.260 --> 00:06:34.400 And of course if g is the constant function one, 00:06:34.400 --> 00:06:36.310 I'm back to that formula. 00:06:36.310 --> 00:06:39.850 But this is maybe the thing to watch for. 00:06:39.850 --> 00:06:43.100 Actually there's a lot built into that little thing. 00:06:43.100 --> 00:06:46.580 We'll come back to that. 00:06:46.580 --> 00:06:51.580 So that's delta functions integrated 00:06:51.580 --> 00:06:53.590 and now here are some pictures. 00:06:53.590 --> 00:06:57.640 These are the good pictures. 00:06:57.640 --> 00:07:03.540 So here's one integral of the delta function. 00:07:03.540 --> 00:07:06.220 It's a step function. 00:07:06.220 --> 00:07:08.760 And the step of course will occur at the point 00:07:08.760 --> 00:07:14.290 a if the integral of the delta function at a point a 00:07:14.290 --> 00:07:17.220 will be the step function. 00:07:17.220 --> 00:07:20.060 Where the action happens. 00:07:20.060 --> 00:07:23.620 The jump happens, I could call it a jump function. 00:07:23.620 --> 00:07:25.880 At that point a. 00:07:25.880 --> 00:07:28.840 Because, just for the reason we said. 00:07:28.840 --> 00:07:32.480 That if we integrate, the integral is zero. 00:07:32.480 --> 00:07:35.940 And then as soon as our integral passes this point, 00:07:35.940 --> 00:07:40.410 so this is integral of the, this is-- I integrated. 00:07:40.410 --> 00:07:44.600 I integrate to get to this picture. 00:07:44.600 --> 00:07:48.930 I start with that delta function and I integrate and it suddenly 00:07:48.930 --> 00:07:52.190 jumps to one as soon as the integral goes 00:07:52.190 --> 00:07:56.030 past the spike, the impulse. 00:07:56.030 --> 00:07:57.390 So a step function. 00:07:57.390 --> 00:07:59.710 Very handy function, step function. 00:07:59.710 --> 00:08:02.610 Sometimes called a Heaviside function 00:08:02.610 --> 00:08:06.910 named after the guy who-- the electrical engineer 00:08:06.910 --> 00:08:13.190 I think who first sort of work out the rules for using these. 00:08:13.190 --> 00:08:18.380 Let's integrate one more time because we have second order 00:08:18.380 --> 00:08:22.810 equations, second derivatives, so we better integrate twice 00:08:22.810 --> 00:08:26.160 to see what sort of answer we get. 00:08:26.160 --> 00:08:28.200 Now integrate the step function. 00:08:28.200 --> 00:08:31.100 So again, the integral is zero all the way to the left, 00:08:31.100 --> 00:08:34.870 so I'm still getting zero, but now beyond this point 00:08:34.870 --> 00:08:36.810 I'm integrating one. 00:08:36.810 --> 00:08:40.810 And the integral of one is x. 00:08:40.810 --> 00:08:44.410 So now that I would call a ramp function. 00:08:44.410 --> 00:08:48.760 That's a nice short word for this valuable function. 00:08:48.760 --> 00:08:56.060 A ramp function is the function that's zero and then x. 00:08:56.060 --> 00:09:00.600 So tell me about that ramp function. 00:09:00.600 --> 00:09:02.660 Just think about it. 00:09:02.660 --> 00:09:08.070 What happens to its derivative at the point a? 00:09:08.070 --> 00:09:11.700 As I run along and I hit this key point, 00:09:11.700 --> 00:09:18.230 what happens to the derivative of the ramp? 00:09:18.230 --> 00:09:20.870 What does the derivative do? 00:09:20.870 --> 00:09:22.820 Focus on that ramp now. 00:09:22.820 --> 00:09:27.300 What does the derivative do at that point? 00:09:27.300 --> 00:09:28.620 It jumps. 00:09:28.620 --> 00:09:29.800 The derivative jumps. 00:09:29.800 --> 00:09:33.250 The slope is the derivative, the slope jumps from zero 00:09:33.250 --> 00:09:34.590 and here the slope is one. 00:09:34.590 --> 00:09:36.620 And of course that's what that's telling us. 00:09:36.620 --> 00:09:38.280 Here's the picture of the derivative. 00:09:38.280 --> 00:09:41.660 What does the second derivative do? 00:09:41.660 --> 00:09:46.600 Well, since I integrated twice I guess 00:09:46.600 --> 00:09:49.100 going back two steps I'll find out 00:09:49.100 --> 00:09:51.200 what the second derivative is. 00:09:51.200 --> 00:09:55.960 So the first derivative takes a jump. 00:09:55.960 --> 00:09:59.990 The second derivative is the derivative of that jump, 00:09:59.990 --> 00:10:01.660 so it's got the impulse. 00:10:01.660 --> 00:10:04.720 So the second derivative, it's a straight line here, 00:10:04.720 --> 00:10:06.780 second derivative a straight line. 00:10:06.780 --> 00:10:09.380 This is straight line here, second derivative 00:10:09.380 --> 00:10:13.130 of a straight line is a straight line. 00:10:13.130 --> 00:10:17.610 But at that point the first derivative jumps, 00:10:17.610 --> 00:10:21.010 the second derivative has that delta function. 00:10:21.010 --> 00:10:24.530 In other words, that's that stuff. 00:10:24.530 --> 00:10:30.360 If I keep integrating -- and I don't need higher integrals 00:10:30.360 --> 00:10:35.250 in today's lecture -- another integral would be what? 00:10:35.250 --> 00:10:37.600 If I integrate this function, then it's 00:10:37.600 --> 00:10:38.880 running along the zero. 00:10:38.880 --> 00:10:41.380 What's the integral of this? 00:10:41.380 --> 00:10:44.680 Doesn't quite turn that steeply. 00:10:44.680 --> 00:10:48.070 What's that curve there? 00:10:48.070 --> 00:10:49.925 If I've integrated the ramp. 00:10:49.925 --> 00:10:50.800 Here is the integral. 00:10:50.800 --> 00:10:55.340 First, the next step up, the integral of the ramp would be? 00:10:55.340 --> 00:10:59.630 It'll be x squared, yeah, it'll be a parabola. 00:10:59.630 --> 00:11:02.480 x squared over two, the integral of that. 00:11:02.480 --> 00:11:05.470 And now what do I get when I integrate this one? 00:11:05.470 --> 00:11:07.950 I get something very important. 00:11:07.950 --> 00:11:13.200 Not important today, but important in a few weeks. 00:11:13.200 --> 00:11:15.920 And very useful in computing. 00:11:15.920 --> 00:11:19.100 These have turned out to be just the right thing. 00:11:19.100 --> 00:11:21.140 So again, I'm integrating that. 00:11:21.140 --> 00:11:24.700 Everybody can tell me, what is that? 00:11:24.700 --> 00:11:27.460 What's that curve now? 00:11:27.460 --> 00:11:29.410 It's the next integral of course. 00:11:29.410 --> 00:11:35.250 The area under that will be x cubed over six. 00:11:35.250 --> 00:11:37.060 So now that is a function. 00:11:37.060 --> 00:11:39.180 Yeah, it's worth maybe just for practice. 00:11:39.180 --> 00:11:42.110 What's the deal with that function? 00:11:42.110 --> 00:11:44.870 That's pretty smooth function. 00:11:44.870 --> 00:11:50.370 Because it certainly passes-- right, it meets at that point. 00:11:50.370 --> 00:11:54.560 The first derivative meets at that point. 00:11:54.560 --> 00:11:57.940 The second derivative meets at that point. 00:11:57.940 --> 00:12:00.480 The third derivative does what? 00:12:00.480 --> 00:12:02.150 Of this line. 00:12:02.150 --> 00:12:05.020 The third derivative, take three steps back down the line 00:12:05.020 --> 00:12:08.481 and you see that the third derivative jumps. 00:12:08.481 --> 00:12:08.980 Right? 00:12:08.980 --> 00:12:12.120 The third derivative of that is the third derivative, 00:12:12.120 --> 00:12:17.570 would be, shall I-- for C, for cubic spline or something, 00:12:17.570 --> 00:12:21.280 the third derivative will be zero there. 00:12:21.280 --> 00:12:24.700 And the third derivative of that is exactly 00:12:24.700 --> 00:12:29.860 like back to that, back to that, back to one is one. 00:12:29.860 --> 00:12:34.750 So the third derivative. so the cubic spline's so smooth 00:12:34.750 --> 00:12:38.640 your eye doesn't see that. 00:12:38.640 --> 00:12:42.390 They're very useful for drawing many, many purposes. 00:12:42.390 --> 00:12:45.550 CAD programs would use such things constantly 00:12:45.550 --> 00:12:50.460 because they're convenient, they have nice pieces 00:12:50.460 --> 00:12:53.070 that you can fit together and they fit together 00:12:53.070 --> 00:12:55.180 very smoothly. 00:12:55.180 --> 00:12:59.930 But they really are two separate functions. 00:12:59.930 --> 00:13:02.270 So that's up to cubic spline. 00:13:02.270 --> 00:13:04.770 But our focus is-- 00:13:04.770 --> 00:13:09.550 These would solve, what equations would those solve? 00:13:09.550 --> 00:13:13.290 Well, that takes how many derivatives to get to a delta? 00:13:13.290 --> 00:13:17.820 So what would be the equation? 00:13:17.820 --> 00:13:22.630 What would be the right-hand side? 00:13:22.630 --> 00:13:25.520 Let me take the fourth derivative. 00:13:25.520 --> 00:13:27.530 I'll just ask the question that way. 00:13:27.530 --> 00:13:29.270 What would be the fourth derivative 00:13:29.270 --> 00:13:31.820 of that cubic spline? 00:13:31.820 --> 00:13:32.800 A delta, right? 00:13:32.800 --> 00:13:34.740 Four steps back. 00:13:34.740 --> 00:13:40.620 So what is, physically, what are we seeing here? 00:13:40.620 --> 00:13:44.100 Do you recognize what kind-- If I ask now 00:13:44.100 --> 00:13:47.230 people in mechanics, When will we 00:13:47.230 --> 00:13:49.700 meet a fourth order equation? 00:13:49.700 --> 00:13:53.560 Fourth derivative equals a load. 00:13:53.560 --> 00:13:58.110 Anybody know the physical situation 00:13:58.110 --> 00:14:02.410 where fourth derivative? 00:14:02.410 --> 00:14:03.590 Beams, yeah. 00:14:03.590 --> 00:14:05.790 It's the equation for a beam. 00:14:05.790 --> 00:14:10.710 A beam has-- The bending of a beam. 00:14:10.710 --> 00:14:11.530 So it's a beam. 00:14:11.530 --> 00:14:15.260 This eraser isn't too very much like a beam, 00:14:15.260 --> 00:14:20.380 but anyway I put the chalk on it, well nothing happened. 00:14:20.380 --> 00:14:22.680 Sit on it, whatever. 00:14:22.680 --> 00:14:28.430 It'll bend and that bending will be given by a beam equation. 00:14:28.430 --> 00:14:31.050 So later we'll meet the beam equation. 00:14:31.050 --> 00:14:40.150 So most equations of physics, mechanics, biology, everything 00:14:40.150 --> 00:14:45.780 are second order, Newton's Laws often the reason. 00:14:45.780 --> 00:14:49.730 But we get up to fourth order sometimes. 00:14:49.730 --> 00:14:52.260 And very seldom get higher. 00:14:52.260 --> 00:14:53.380 Hopefully. 00:14:53.380 --> 00:14:58.500 Beams or plates, that table would be a plate 00:14:58.500 --> 00:15:05.530 and it would have a fourth order equation. 00:15:05.530 --> 00:15:07.850 Let's start solving this problem. 00:15:07.850 --> 00:15:11.620 What's the solution, what's the general solution 00:15:11.620 --> 00:15:13.200 to that equation? 00:15:13.200 --> 00:15:15.830 Minus the second derivative, so notice the minus 00:15:15.830 --> 00:15:19.840 that I like, and the load has now moved to the point a. 00:15:19.840 --> 00:15:25.920 So the solution u(x), let's write down all solutions. 00:15:25.920 --> 00:15:27.390 Tell me one solution, first. 00:15:27.390 --> 00:15:29.280 One particular solution. 00:15:29.280 --> 00:15:32.410 What is one function for which minus 00:15:32.410 --> 00:15:36.480 the second derivative would be the delta? 00:15:36.480 --> 00:15:38.030 That's what we've got over there. 00:15:38.030 --> 00:15:42.110 So just bring that blackboard over here. 00:15:42.110 --> 00:15:45.120 Change its sign because that minus, 00:15:45.120 --> 00:15:46.750 and what are you going to tell me? 00:15:46.750 --> 00:15:51.250 Minus a ramp. 00:15:51.250 --> 00:15:53.170 Minus a ramp. 00:15:53.170 --> 00:15:57.830 And the ramp, of course, will ramp up at the point 00:15:57.830 --> 00:16:04.820 a so that it's the second derivative of that, 00:16:04.820 --> 00:16:07.650 the second derivative of R will be delta. 00:16:07.650 --> 00:16:12.270 The minus is correct and the point is correct. 00:16:12.270 --> 00:16:17.590 Now does that solve our problem? 00:16:17.590 --> 00:16:20.610 No. 00:16:20.610 --> 00:16:22.240 The ramp is going upwards. 00:16:22.240 --> 00:16:23.570 It's not zero. 00:16:23.570 --> 00:16:25.650 What am I forgetting? 00:16:25.650 --> 00:16:27.180 What do I not yet have? 00:16:27.180 --> 00:16:30.520 There's more to this solution. 00:16:30.520 --> 00:16:34.360 Just as there was for a uniform load. 00:16:34.360 --> 00:16:36.850 What was the more? 00:16:36.850 --> 00:16:47.400 Constant and-- and I want two homogeneous solutions, 00:16:47.400 --> 00:16:52.270 null solutions, two solutions with second derivative equal 00:16:52.270 --> 00:16:53.030 zero. 00:16:53.030 --> 00:16:57.340 One of them is C and the other one is Dx. 00:16:57.340 --> 00:16:59.440 That's the whole solution. 00:16:59.440 --> 00:17:04.450 So what I want to-- I mean we need that C+Dx. 00:17:04.450 --> 00:17:06.030 We've got two boundary conditions 00:17:06.030 --> 00:17:08.340 to satisfy, just as before. 00:17:08.340 --> 00:17:10.820 So I need two constants, that'll do it perfectly 00:17:10.820 --> 00:17:13.430 and I'll get an exact answer. 00:17:13.430 --> 00:17:18.170 And so this is a ramp. 00:17:18.170 --> 00:17:19.350 Oh yeah. 00:17:19.350 --> 00:17:23.760 Before I go further, how would I think about this? 00:17:23.760 --> 00:17:27.160 This is a ramp that turns which way? 00:17:27.160 --> 00:17:28.270 Down. 00:17:28.270 --> 00:17:29.840 Right? 00:17:29.840 --> 00:17:35.330 With that minus sign, that ramp turns down at the point x=a. 00:17:35.330 --> 00:17:37.630 Right? 00:17:37.630 --> 00:17:43.840 It's derivative goes from zero to minus one. 00:17:43.840 --> 00:17:49.790 The slope of this guy drops by one because of the minus sign. 00:17:49.790 --> 00:18:00.340 Sorry the slope of the ramp function, of minus the ramp. 00:18:00.340 --> 00:18:03.630 It goes at zero, drops by one. 00:18:03.630 --> 00:18:07.070 And what this is going to do is take that ramp 00:18:07.070 --> 00:18:11.700 and adjust it to go through the fixed ends. 00:18:11.700 --> 00:18:13.000 Oh, let's just do it. 00:18:13.000 --> 00:18:13.780 Let's just do it. 00:18:13.780 --> 00:18:15.530 What are C and D? 00:18:15.530 --> 00:18:16.480 What are C and D? 00:18:16.480 --> 00:18:18.660 My point a-- let me draw a graph, 00:18:18.660 --> 00:18:21.110 that's always the best thing. 00:18:21.110 --> 00:18:26.150 Always draw a graph of these solutions. 00:18:26.150 --> 00:18:29.730 So let me put in the point a. 00:18:29.730 --> 00:18:34.830 So I'm drawing now a picture of the solution from zero to one. 00:18:34.830 --> 00:18:40.470 I'll graph it. 00:18:40.470 --> 00:18:42.260 What do I have here? 00:18:42.260 --> 00:18:47.160 Shall we just plug in the boundary conditions 00:18:47.160 --> 00:18:48.260 and find C and D? 00:18:48.260 --> 00:18:50.060 That's the direct way. 00:18:50.060 --> 00:18:52.880 What is C? 00:18:52.880 --> 00:18:55.030 C I'm going to plug in. 00:18:55.030 --> 00:18:57.760 Hopefully I might find it from just the first boundary 00:18:57.760 --> 00:18:59.090 condition. 00:18:59.090 --> 00:19:02.690 If I'm starting from zero, well this guy certainly 00:19:02.690 --> 00:19:04.770 starts at zero, right? 00:19:04.770 --> 00:19:09.010 The ramp hasn't done anything until it gets to a. 00:19:09.010 --> 00:19:10.820 And this guy is certainly zero. 00:19:10.820 --> 00:19:12.880 So what is C? 00:19:12.880 --> 00:19:15.760 Gone, right. 00:19:15.760 --> 00:19:19.200 Now what is D? 00:19:19.200 --> 00:19:22.810 Well alright, what's D? 00:19:22.810 --> 00:19:23.310 Let's see. 00:19:23.310 --> 00:19:30.020 Let me draw the-- So there's a Dx, and D won't be zero. 00:19:30.020 --> 00:19:34.480 I want that thing to be zero at point one. 00:19:34.480 --> 00:19:39.390 So I want to determine D. Let me determine D. 00:19:39.390 --> 00:19:45.180 So what is minus the ramp at x=1? 00:19:45.180 --> 00:19:46.637 I'm plugging in x=1. 00:19:46.637 --> 00:19:47.220 Is that right? 00:19:47.220 --> 00:19:49.360 I'm going straight forward here. 00:19:49.360 --> 00:19:52.660 Plugging in x=1 into this boundary condition, 00:19:52.660 --> 00:19:54.780 ready for this guy. 00:19:54.780 --> 00:19:56.070 What's the ramp? 00:19:56.070 --> 00:20:02.980 So it's minus and the ramp is, well 00:20:02.980 --> 00:20:07.310 if the ramp is shifted over then that's shifted over. 00:20:07.310 --> 00:20:12.220 So at x=1, what's the ramp? 00:20:12.220 --> 00:20:15.400 How high has that ramp gone? 00:20:15.400 --> 00:20:17.340 1-a. 00:20:17.340 --> 00:20:17.840 Right? 00:20:17.840 --> 00:20:19.030 The ramp is x-a. 00:20:21.930 --> 00:20:25.860 At the point x=1 it will be 1-a. 00:20:25.860 --> 00:20:29.890 So I think I get one, -(1-a) out of that. 00:20:29.890 --> 00:20:35.500 Minus the ramp plus D times what? 00:20:35.500 --> 00:20:37.880 One, I'm plugging in x=1. 00:20:37.880 --> 00:20:40.610 And that's supposed to equal? 00:20:40.610 --> 00:20:43.310 Zero, good. 00:20:43.310 --> 00:20:49.370 So I'm doing this sort of the systematic way of writing down 00:20:49.370 --> 00:20:51.230 the general solution. 00:20:51.230 --> 00:20:55.060 Discovering that D, what do I discover D is? 00:20:55.060 --> 00:20:56.680 Put it on the other side. 00:20:56.680 --> 00:21:00.520 D is 1-a. 00:21:00.520 --> 00:21:08.420 And of course, don't forget that it's multiplying the x. 00:21:08.420 --> 00:21:13.010 Let me just draw the picture. 00:21:13.010 --> 00:21:15.580 Here's how I think about it. 00:21:15.580 --> 00:21:20.690 The solution is, away from x=a, what does the solution look 00:21:20.690 --> 00:21:22.730 like? 00:21:22.730 --> 00:21:28.000 To the left of x=a what's my graph going to be? 00:21:28.000 --> 00:21:31.630 It's going to be? 00:21:31.630 --> 00:21:33.690 A straight line, right? 00:21:33.690 --> 00:21:38.400 To the left of here there is no load. 00:21:38.400 --> 00:21:43.370 The equation is second derivative equals zero. 00:21:43.370 --> 00:21:45.730 The solution to that is a straight line. 00:21:45.730 --> 00:21:49.130 In other words, until I get to a, this thing hasn't started. 00:21:49.130 --> 00:21:51.360 It's only this straight line. 00:21:51.360 --> 00:21:53.830 The solution does something like that. 00:21:53.830 --> 00:21:55.650 It's a straight line. 00:21:55.650 --> 00:21:59.180 And I guess, actually, that's what it is. 00:21:59.180 --> 00:22:02.710 Because the C isn't here and that's all we've got left. 00:22:02.710 --> 00:22:06.830 So that's that straight line. 00:22:06.830 --> 00:22:10.820 What is it for the second half? 00:22:10.820 --> 00:22:14.660 Tell me what the solution looks like in the second half. 00:22:14.660 --> 00:22:17.260 In between a and one. 00:22:17.260 --> 00:22:19.600 It's going downhill. 00:22:19.600 --> 00:22:22.030 Why? 00:22:22.030 --> 00:22:24.770 Because it gotta get back to zero. 00:22:24.770 --> 00:22:28.690 And how's it going downhill? 00:22:28.690 --> 00:22:31.030 It has to be linear. 00:22:31.030 --> 00:22:34.280 In this region, has to be linear. 00:22:34.280 --> 00:22:37.200 Why? 00:22:37.200 --> 00:22:38.800 How do I know it's linear here? 00:22:38.800 --> 00:22:43.040 Because one way is to say the equation in that region 00:22:43.040 --> 00:22:46.380 is second derivative equal zero. 00:22:46.380 --> 00:22:49.250 Second derivative equal zero, straight line. 00:22:49.250 --> 00:22:50.830 This is my solution. 00:22:50.830 --> 00:22:55.820 It's (1-a)x here and it's whatever it is to get back 00:22:55.820 --> 00:22:58.320 to zero. 00:22:58.320 --> 00:23:02.100 What will it take to get back to zero? 00:23:02.100 --> 00:23:03.540 Let's see. 00:23:03.540 --> 00:23:07.090 Well we could plug in, we've got one expression here. 00:23:07.090 --> 00:23:09.350 Or I could just look at that. 00:23:09.350 --> 00:23:12.870 I could say, okay what's the equation for the straight line 00:23:12.870 --> 00:23:17.510 that's at this point, what is the, yeah, it's (1-a)x. 00:23:23.230 --> 00:23:26.360 I want it to be linear. 00:23:26.360 --> 00:23:31.520 I want it to get to zero. 00:23:31.520 --> 00:23:32.390 Let's see. 00:23:32.390 --> 00:23:36.460 If I want that, it would be great to have 1-x times 00:23:36.460 --> 00:23:37.460 something. 00:23:37.460 --> 00:23:39.910 I have to figure out what. 00:23:39.910 --> 00:23:44.160 Because with the 1-x at x=1, that'll drop off. 00:23:44.160 --> 00:23:45.460 That's linear. 00:23:45.460 --> 00:23:49.960 What number, what's the key here? 00:23:49.960 --> 00:23:58.230 That slope, I want to match them up there. 00:23:58.230 --> 00:24:01.750 And that's the point x=a. 00:24:01.750 --> 00:24:04.640 This is supposed to match that at x=a. 00:24:04.640 --> 00:24:09.380 Do you have an idea for what I should take? 00:24:09.380 --> 00:24:15.740 What do I put right there? a. 00:24:15.740 --> 00:24:22.120 Look at the symmetry in those two sides. (1-a)x going up. 00:24:22.120 --> 00:24:24.700 (1-x)a going down. 00:24:24.700 --> 00:24:28.990 At x=a it hits that point, right. 00:24:28.990 --> 00:24:30.750 So we've solved it. 00:24:30.750 --> 00:24:34.130 We could think about this different ways. 00:24:34.130 --> 00:24:39.210 I could have got that 1-x, let's see, 00:24:39.210 --> 00:24:42.290 I could have got it from the formula. 00:24:42.290 --> 00:24:47.040 In a way I like to get it from the picture, I see it, sort of, 00:24:47.040 --> 00:24:49.000 I see the point. 00:24:49.000 --> 00:24:51.410 What happened at that point? 00:24:51.410 --> 00:24:54.160 What are the jump conditions? 00:24:54.160 --> 00:24:56.390 This is another way to ask, to see 00:24:56.390 --> 00:24:58.410 how the delta function works. 00:24:58.410 --> 00:25:00.740 What are they jump conditions? 00:25:00.740 --> 00:25:03.530 I want to know, when I ask about jump conditions, 00:25:03.530 --> 00:25:06.570 I want to know what are the conditions on u(x), 00:25:06.570 --> 00:25:08.030 the displacement? 00:25:08.030 --> 00:25:12.850 What are the conditions on the slope, u'(x)? 00:25:12.850 --> 00:25:19.790 That'll be the strain when we're speaking about elasticity. 00:25:19.790 --> 00:25:23.020 Just for u(x), what's the statement 00:25:23.020 --> 00:25:25.480 about u(x) from the left and from the right 00:25:25.480 --> 00:25:32.760 at that critical point, the point of the load. 00:25:32.760 --> 00:25:38.020 From the left and from the right u(x) is? 00:25:38.020 --> 00:25:42.000 The same. u(x) matches up. u(x) from the left 00:25:42.000 --> 00:25:44.500 is that height. u(x) from the right is that. 00:25:44.500 --> 00:25:48.860 I want to write down those jump conditions. 00:25:48.860 --> 00:25:53.050 Because that's another way to see this. u(x), 00:25:53.050 --> 00:26:02.200 u(a) from the left should equal u-- 00:26:02.200 --> 00:26:04.500 do you want me to say u is continuous? 00:26:04.500 --> 00:26:12.650 I'll just say it in words. u(x) is continuous, 00:26:12.650 --> 00:26:15.200 that just means it doesn't jump, at x=a. 00:26:18.050 --> 00:26:22.770 So that's, you could say that's a non-jump condition. 00:26:22.770 --> 00:26:24.390 The function itself doesn't jump. 00:26:24.390 --> 00:26:25.020 Why not? 00:26:25.020 --> 00:26:28.190 Because we're talking about some elastic bar on which 00:26:28.190 --> 00:26:30.160 we put a point load. 00:26:30.160 --> 00:26:32.600 The thing isn't going to break. 00:26:32.600 --> 00:26:36.510 The displacement is going to be continuous. 00:26:36.510 --> 00:26:42.150 But what's the condition on u'(x), the derivative, 00:26:42.150 --> 00:26:43.560 the slope? 00:26:43.560 --> 00:26:46.390 So that's the function and now tell me 00:26:46.390 --> 00:26:48.400 what's the deal on the slope? 00:26:48.400 --> 00:26:51.350 What's the comparison between the-- I 00:26:51.350 --> 00:26:54.420 have a slope of whatever it is going along here 00:26:54.420 --> 00:26:56.980 and I have a slope of-- a new slope. 00:26:56.980 --> 00:27:03.130 So u'(x), the slope jumps, right? 00:27:03.130 --> 00:27:06.710 And how much does it jump? 00:27:06.710 --> 00:27:08.120 Minus one. 00:27:08.120 --> 00:27:09.860 It drops by one. 00:27:09.860 --> 00:27:13.420 The slope, because of my minus. 00:27:13.420 --> 00:27:17.849 So this tells me that-- Yeah, let me write that down. 00:27:17.849 --> 00:27:18.640 u'(x) drops by one. 00:27:26.430 --> 00:27:32.550 This is another way to say what the equation is asking. 00:27:32.550 --> 00:27:36.540 The equation is looking for two pieces of straight lines 00:27:36.540 --> 00:27:41.260 that meet at a but their slope drops by one. 00:27:41.260 --> 00:27:42.800 By the way, what were the slopes? 00:27:42.800 --> 00:27:45.880 It's good to graph the slopes, too. 00:27:45.880 --> 00:27:52.610 Let me graph the slopes. 00:27:52.610 --> 00:27:58.590 The slope u', the derivative du/dx. 00:27:58.590 --> 00:28:00.790 What's the slope here? 00:28:00.790 --> 00:28:03.280 Slope is 1-a at this point, right? 00:28:03.280 --> 00:28:06.350 The derivative is 1-a along here. 00:28:06.350 --> 00:28:09.790 So slope is 1-a. 00:28:09.790 --> 00:28:14.680 And now at x=a the slope changes to this one. 00:28:14.680 --> 00:28:17.730 And what's the slope of that second part? 00:28:17.730 --> 00:28:18.770 Minus a. 00:28:18.770 --> 00:28:19.660 Look. 00:28:19.660 --> 00:28:21.190 It did it right. 00:28:21.190 --> 00:28:24.430 Minus a is the slope along here. 00:28:24.430 --> 00:28:28.330 Do you see 1-a? 00:28:28.330 --> 00:28:29.520 It dropped by one. 00:28:29.520 --> 00:28:36.610 The one disappeared to leave a slope of minus a. 00:28:36.610 --> 00:28:41.970 I guess if I just imagine a bar. 00:28:41.970 --> 00:28:45.110 I'm fixing it at both ends. 00:28:45.110 --> 00:28:50.870 There's a bar. 00:28:50.870 --> 00:28:53.450 I'm just thinking for people who like 00:28:53.450 --> 00:28:57.510 to see a physical picture of what's happening, 00:28:57.510 --> 00:29:03.490 that's what this is, we'll do it properly very, very soon. 00:29:03.490 --> 00:29:05.500 I've got a bar. 00:29:05.500 --> 00:29:08.700 It's a very light bar. 00:29:08.700 --> 00:29:12.320 Its weight is not a problem here. 00:29:12.320 --> 00:29:14.460 But it's got a load at the point. 00:29:14.460 --> 00:29:16.340 So I'll measure x going downwards. 00:29:16.340 --> 00:29:22.900 And at the point x=a I'm hanging a heavy load. 00:29:22.900 --> 00:29:24.640 A load. 00:29:24.640 --> 00:29:30.940 How do I draw a load? 00:29:30.940 --> 00:29:40.110 Maybe I'll make a big weight or something. 00:29:40.110 --> 00:29:49.180 What's going to happen to this dumb bar when I do that? 00:29:49.180 --> 00:29:50.472 Just tell me physically. 00:29:50.472 --> 00:29:51.430 What's going to happen? 00:29:51.430 --> 00:29:54.810 What's going to happen above the load? 00:29:54.810 --> 00:30:00.300 It's going to stretch, right, tension. 00:30:00.300 --> 00:30:02.740 The load is going to pull the bar down, 00:30:02.740 --> 00:30:05.350 it's going to stretch this part. 00:30:05.350 --> 00:30:07.340 And because nothing special is happening, 00:30:07.340 --> 00:30:09.780 it's going to stretch it linearly. 00:30:09.780 --> 00:30:13.180 And then what's going to happen below the load? 00:30:13.180 --> 00:30:15.810 Compression. 00:30:15.810 --> 00:30:19.390 So the slope will go negative. 00:30:19.390 --> 00:30:23.730 And nothing special happened so the slope will be negative 00:30:23.730 --> 00:30:25.240 but it'll be constant. 00:30:25.240 --> 00:30:27.600 The slope will drop from this to this. 00:30:27.600 --> 00:30:34.650 The displacement, that point will go down a little bit. 00:30:34.650 --> 00:30:38.687 That little bit it goes down is actually the height of this, 00:30:38.687 --> 00:30:40.020 because that's the displacement. 00:30:40.020 --> 00:30:41.940 It'll go down a little bit. 00:30:41.940 --> 00:30:44.990 It'll stretch above, it'll compress below, 00:30:44.990 --> 00:30:52.700 and we see that in that picture of the displacement. 00:30:52.700 --> 00:30:55.070 The displacement's all down. 00:30:55.070 --> 00:30:56.000 Right? 00:30:56.000 --> 00:31:01.320 Displacement-- You know, nature is still going to-- All the bar 00:31:01.320 --> 00:31:02.340 is going to move down. 00:31:02.340 --> 00:31:07.310 That's why this function doesn't, this function, 00:31:07.310 --> 00:31:09.340 the displacement function is positive. 00:31:09.340 --> 00:31:10.880 It goes all down. 00:31:10.880 --> 00:31:15.500 But the slope function is positive here, 00:31:15.500 --> 00:31:19.780 so tension is positive slope, stretch. 00:31:19.780 --> 00:31:24.740 And compression is negative. 00:31:24.740 --> 00:31:31.360 Well all that to solve this equation. 00:31:31.360 --> 00:31:38.450 Maybe while we're on a roll, let's solve the free-fixed guy. 00:31:38.450 --> 00:31:41.440 So this is our-- might as well be systematic. 00:31:41.440 --> 00:31:43.160 This is the fixed-fixed problem. 00:31:43.160 --> 00:31:46.770 Let me below it solve the free-fixed problem. 00:31:46.770 --> 00:31:50.490 So it'll be minus u'', that's the second derivative, 00:31:50.490 --> 00:31:53.840 equals delta at x-a. 00:31:53.840 --> 00:31:55.680 Same setup. 00:31:55.680 --> 00:32:01.730 But now the top end is, so it's free at the top. 00:32:01.730 --> 00:32:04.290 What does that mean? 00:32:04.290 --> 00:32:12.110 Slope is zero at the top but it's still fixed at the bottom. 00:32:12.110 --> 00:32:21.490 So this will be now free-fixed. 00:32:21.490 --> 00:32:24.040 Let me go straight to the picture. 00:32:24.040 --> 00:32:27.700 Let me go straight to the picture of u(x). 00:32:27.700 --> 00:32:32.770 So there is x=0, there's x=1, here's the load at a. 00:32:32.770 --> 00:32:37.050 What's up? 00:32:37.050 --> 00:32:38.940 And while you're thinking about that, 00:32:38.940 --> 00:32:43.300 let me draw a picture to match this picture. 00:32:43.300 --> 00:32:52.640 A bar fixed at the bottom but not at the top. 00:32:52.640 --> 00:32:59.120 And it's got its load here hanging down. 00:32:59.120 --> 00:33:08.240 But let's do it math first, and then check with the picture. 00:33:08.240 --> 00:33:09.890 What have we got, two or three ways now 00:33:09.890 --> 00:33:11.490 to try to get the answer? 00:33:11.490 --> 00:33:14.590 The systematic way would be to write down 00:33:14.590 --> 00:33:22.710 this solution and plug in the two boundary conditions. 00:33:22.710 --> 00:33:24.780 That'd be a straightforward way. 00:33:24.780 --> 00:33:27.350 Yeah, we could even start by that. 00:33:27.350 --> 00:33:33.170 So u(x) is the particular solution, 00:33:33.170 --> 00:33:46.350 the ramp plus any Cx+D. And just plug in x=0 that'll be easy. 00:33:46.350 --> 00:33:51.550 If I plug in x=0 in the free condition, 00:33:51.550 --> 00:33:53.240 what does that tell me? 00:33:53.240 --> 00:33:58.820 At x=0, this corner, this ramp hasn't started so the slope is 00:33:58.820 --> 00:34:00.040 zero. 00:34:00.040 --> 00:34:01.730 The slope of the constant is zero. 00:34:01.730 --> 00:34:05.280 What do I learn from this boundary condition? u'(0)=0. 00:34:08.650 --> 00:34:10.300 That C is zero. 00:34:10.300 --> 00:34:12.740 Before I learned that D was zero, but now 00:34:12.740 --> 00:34:18.540 from that condition I'm going to learn C is zero. 00:34:18.540 --> 00:34:21.160 Do the picture for me. 00:34:21.160 --> 00:34:24.090 Do the picture for me. 00:34:24.090 --> 00:34:27.770 What's the graph of-- this is a graph of u(x). 00:34:30.990 --> 00:34:36.000 Remember now it starts from zero slope 00:34:36.000 --> 00:34:38.330 because it's free at the top. 00:34:38.330 --> 00:34:44.880 What does the graph look like in the first part? 00:34:44.880 --> 00:34:49.370 It's a straight line, has to be a straight line because there's 00:34:49.370 --> 00:34:51.050 no force. 00:34:51.050 --> 00:34:54.260 And what kind of a line? 00:34:54.260 --> 00:34:57.920 It's going to be horizontal because it starts off 00:34:57.920 --> 00:34:59.290 horizontal. 00:34:59.290 --> 00:35:05.080 The slope has to be zero at zero and nothing changes until a. 00:35:05.080 --> 00:35:08.960 So it comes along there. 00:35:08.960 --> 00:35:11.220 Right? 00:35:11.220 --> 00:35:14.110 Now I've started out with the right, left, 00:35:14.110 --> 00:35:18.950 the correct boundary condition at zero, which was no slope. 00:35:18.950 --> 00:35:22.330 And now what's it going to do the other half? 00:35:22.330 --> 00:35:25.200 From a to one. 00:35:25.200 --> 00:35:31.090 It's going to be again, it'll be a straight line, right? 00:35:31.090 --> 00:35:33.380 Because there's no force there. 00:35:33.380 --> 00:35:37.680 And what happens at-- all the action of course 00:35:37.680 --> 00:35:41.900 is at this point a, and what action is it? 00:35:41.900 --> 00:35:45.000 Tell me what sort of a line. 00:35:45.000 --> 00:35:49.940 How do I finish the picture? 00:35:49.940 --> 00:35:52.370 What do I do? 00:35:52.370 --> 00:35:53.770 I start here, right? 00:35:53.770 --> 00:36:01.070 Because the bar's not falling apart. u is continuous. 00:36:01.070 --> 00:36:03.780 I don't get a gap suddenly. 00:36:03.780 --> 00:36:07.390 And now what do I do from there? 00:36:07.390 --> 00:36:11.130 Only thing I can possibly do, because I have to end up here 00:36:11.130 --> 00:36:16.942 and it has to be a straight line, that's it. 00:36:16.942 --> 00:36:18.900 That's what the picture will have to look like. 00:36:18.900 --> 00:36:30.110 What does that correspond to in the picture for the bar? 00:36:30.110 --> 00:36:33.300 Well what happens with this bar? 00:36:33.300 --> 00:36:38.740 Above the weight, what happens to this top part 00:36:38.740 --> 00:36:42.630 of the bar in that picture? 00:36:42.630 --> 00:36:45.020 And what happens to the lower part of the bar? 00:36:45.020 --> 00:36:51.300 So this was at the point x=a, this is x=0, this is x=1. 00:36:51.300 --> 00:36:57.890 What happens above the bar, above the weight? 00:36:57.890 --> 00:37:01.780 It just-- A rigid motion, just goes down. 00:37:01.780 --> 00:37:04.620 Because what happens below the weight? 00:37:04.620 --> 00:37:07.510 The same compression or compression still happening. 00:37:07.510 --> 00:37:08.950 This is still squeezed. 00:37:08.950 --> 00:37:12.140 Shall I try to draw it? 00:37:12.140 --> 00:37:14.830 So this is after the weight. 00:37:14.830 --> 00:37:21.300 This got squeezed but this part did not get squeezed. 00:37:21.300 --> 00:37:25.970 And that's what we're seeing here. 00:37:25.970 --> 00:37:28.120 A fixed displacement. 00:37:28.120 --> 00:37:30.970 So this means, that picture means 00:37:30.970 --> 00:37:36.600 that all the pieces of the bar here 00:37:36.600 --> 00:37:39.370 got moved down by the same amount, whatever this, 00:37:39.370 --> 00:37:41.860 we don't know that number yet. 00:37:41.860 --> 00:37:49.240 And then below it they got compressed. 00:37:49.240 --> 00:37:56.430 Well we're almost there but we don't yet have that solution. 00:37:56.430 --> 00:38:02.130 Come back to this picture. u(x) is continuous, got it. 00:38:02.130 --> 00:38:04.690 And what's the real condition that's 00:38:04.690 --> 00:38:09.980 going to determine where we are, what those heights 00:38:09.980 --> 00:38:12.390 are, the numbers in there. 00:38:12.390 --> 00:38:16.060 It's gotta look like that, but we get more than 00:38:16.060 --> 00:38:19.970 that, we gotta know what are the actual, what is that height. 00:38:19.970 --> 00:38:20.950 What is this? 00:38:20.950 --> 00:38:22.050 What's the slope? 00:38:22.050 --> 00:38:27.250 Here the slope is zero. 00:38:27.250 --> 00:38:34.230 Here the slope is what? 00:38:34.230 --> 00:38:36.170 What's the slope in the second part? 00:38:36.170 --> 00:38:37.080 That's the key. 00:38:37.080 --> 00:38:40.610 And you know what it has to be because what 00:38:40.610 --> 00:38:42.740 happens to the slope? 00:38:42.740 --> 00:38:46.910 If I have the second derivative as a delta function 00:38:46.910 --> 00:38:56.420 with that minus sign, the slope drops by one. 00:38:56.420 --> 00:39:02.690 And the slope here is zero, so the slope here is minus one. 00:39:02.690 --> 00:39:07.620 And now it has to get through there, so what is the function? 00:39:07.620 --> 00:39:11.990 What's the function that has a slope of minus one 00:39:11.990 --> 00:39:22.030 and comes down to zero? 00:39:22.030 --> 00:39:24.480 It's gotta have a minus x in it and what's 00:39:24.480 --> 00:39:30.800 the constant to make it come out right? 00:39:30.800 --> 00:39:34.510 What do I write now here for u(x)? 00:39:34.510 --> 00:39:35.010 1-x. 00:39:37.830 --> 00:39:41.230 That has a slope of minus one, the derivative is minus one, 00:39:41.230 --> 00:39:44.140 at x=1 it comes to zero, that's it. 00:39:44.140 --> 00:39:48.680 And what do I write, what's u(x) up here? 00:39:48.680 --> 00:39:54.550 And therefore, right there? 00:39:54.550 --> 00:39:59.070 What's the displacement there, of all this bit that 00:39:59.070 --> 00:40:03.200 moves down, how much does it move down? 00:40:03.200 --> 00:40:04.360 1-a. 00:40:04.360 --> 00:40:05.680 Why 1-a? 00:40:05.680 --> 00:40:06.780 That's the right answer. 00:40:06.780 --> 00:40:07.279 1-a. 00:40:09.750 --> 00:40:12.430 Why's that? 00:40:12.430 --> 00:40:15.600 Because it had to match up at x=a. 00:40:15.600 --> 00:40:19.230 At x=a, this and that match up. 00:40:19.230 --> 00:40:23.590 At x=a, that slope, that function and that function 00:40:23.590 --> 00:40:24.370 match up. 00:40:24.370 --> 00:40:31.570 So the slope picture is zero and-- Oh, I'm sorry, 00:40:31.570 --> 00:40:36.500 can't draw it because I'm at the bottom of the board. 00:40:36.500 --> 00:40:39.120 The slope picture, maybe I can draw it here, 00:40:39.120 --> 00:40:42.740 the slope picture is zero along here 00:40:42.740 --> 00:40:45.300 and then it drops by one to 1-a. 00:40:45.300 --> 00:40:47.600 So that's a picture of u'. 00:40:47.600 --> 00:40:51.470 Zero and minus one. 00:40:51.470 --> 00:40:59.930 This is the thing to look at. 00:40:59.930 --> 00:41:03.150 That's hard work, when you're seeing delta functions 00:41:03.150 --> 00:41:03.790 the first time. 00:41:03.790 --> 00:41:06.590 But of course the functions did not get complicated. 00:41:06.590 --> 00:41:11.220 We kept a clean example. 00:41:11.220 --> 00:41:18.240 And which we matched up with a figure and we've got the answer 00:41:18.240 --> 00:41:20.440 and we've got a couple of ways to do it. 00:41:20.440 --> 00:41:25.510 One is this standard, systematic, plug-in boundary 00:41:25.510 --> 00:41:26.650 condition way. 00:41:26.650 --> 00:41:31.700 The other way is this. u(x) does something here, 00:41:31.700 --> 00:41:34.460 then the slope has to drop by one. 00:41:34.460 --> 00:41:40.120 And that's the key to everything with a boundary condition. 00:41:40.120 --> 00:41:43.410 So in a way, we have a piece to the left and a piece 00:41:43.410 --> 00:41:44.660 to the right. 00:41:44.660 --> 00:41:47.140 Two constants here, two constants here, 00:41:47.140 --> 00:41:50.750 and somewhere there are four conditions that 00:41:50.750 --> 00:41:52.940 settle those four constants. 00:41:52.940 --> 00:41:55.450 You know, we could have a straight line here, 00:41:55.450 --> 00:41:57.990 a straight line here, that's two and two. 00:41:57.990 --> 00:42:00.280 But what are the four conditions that 00:42:00.280 --> 00:42:01.690 settle those four constants? 00:42:01.690 --> 00:42:05.460 Well we have a boundary condition here, that's one. 00:42:05.460 --> 00:42:07.890 Boundary condition here is two. 00:42:07.890 --> 00:42:11.970 We need two more conditions to settle 00:42:11.970 --> 00:42:19.410 the two pairs of constants, and there they are. 00:42:19.410 --> 00:42:27.540 Two conditions at the jump, at the discontinuity. 00:42:27.540 --> 00:42:36.270 Now I've got to do the discrete case. 00:42:36.270 --> 00:42:39.170 Are you up for the discrete case? 00:42:39.170 --> 00:42:47.320 The case where we're doing-- We have a difference equation, 00:42:47.320 --> 00:42:52.480 so we're doing Ku equal a column of the identity. 00:42:52.480 --> 00:43:03.460 Column of I. Let me take a specific column. 00:43:03.460 --> 00:43:06.420 Say [0, 1, 0, 0, 0]. 00:43:06.420 --> 00:43:12.550 Let's suppose we have five-- I'm going to draw a picture now. 00:43:12.550 --> 00:43:15.680 We have five, because I made it five by five. 00:43:15.680 --> 00:43:21.260 One, two, three, four, five, here is zero and here is six. 00:43:21.260 --> 00:43:25.720 So h is 1/(5+1), 1/6, that's the delta x. 00:43:25.720 --> 00:43:36.650 And my equation says-- So what does my equation say? 00:43:36.650 --> 00:43:46.130 Remember what K is. u is then u_1, u_2, u_3, u_4, and u_5, 00:43:46.130 --> 00:43:48.080 the unknowns. 00:43:48.080 --> 00:44:04.130 K is our old friend with twos and minus ones and minus ones. 00:44:04.130 --> 00:44:07.140 I'm going to find the solution. 00:44:07.140 --> 00:44:15.280 And this'll be the solution that has a load at this point. 00:44:15.280 --> 00:44:18.240 This is like my point a, right? 00:44:18.240 --> 00:44:21.400 Here in the continuous case, a could run anywhere 00:44:21.400 --> 00:44:23.050 between zero and one. 00:44:23.050 --> 00:44:27.420 In the discrete case, I've got five possible load points 00:44:27.420 --> 00:44:29.120 and I've picked the second one. 00:44:29.120 --> 00:44:31.830 Five columns of the identity matrix, five 00:44:31.830 --> 00:44:36.050 places to put that one, I put it there. 00:44:36.050 --> 00:44:42.660 Now can I draw the picture here? 00:44:42.660 --> 00:44:45.000 Which should we do first? 00:44:45.000 --> 00:44:46.950 Should we do free-fixed? 00:44:46.950 --> 00:44:52.280 Because that came out even easier than fixed-fixed. 00:44:52.280 --> 00:44:55.920 Notice the solution here had two parts. 00:44:55.920 --> 00:44:59.660 This is the way I would write that answer. 00:44:59.660 --> 00:45:01.550 Because you could draw a picture, 00:45:01.550 --> 00:45:06.560 but if you want to write the formula, what would I do? 00:45:06.560 --> 00:45:10.410 I would break it into two pieces. 00:45:10.410 --> 00:45:16.210 1-a up to the point a because that's 00:45:16.210 --> 00:45:18.120 what it was running along here. 00:45:18.120 --> 00:45:23.480 And then down here it was 1-x, x >=a. 00:45:31.580 --> 00:45:33.830 That's important to mention. 00:45:33.830 --> 00:45:38.820 You have to have some guidance on how to write the answer. 00:45:38.820 --> 00:45:42.210 And when the answer has two parts, 00:45:42.210 --> 00:45:44.490 this is a good way to write it, in two parts. 00:45:44.490 --> 00:45:46.770 It's a little too-- you're compressing 00:45:46.770 --> 00:45:50.580 it too much to write, to use that ramp function. 00:45:50.580 --> 00:45:56.310 Better to split it apart into before a and after a. 00:45:56.310 --> 00:45:59.660 What's going to happen over here? 00:45:59.660 --> 00:46:04.860 Oh yeah, can we take a shot at this problem? 00:46:04.860 --> 00:46:11.880 And let me mention again, in the review that'll 00:46:11.880 --> 00:46:17.130 be in here this afternoon and every Wednesday afternoon I'll 00:46:17.130 --> 00:46:18.880 just be ready for questions. 00:46:18.880 --> 00:46:22.040 Please bring questions. 00:46:22.040 --> 00:46:25.360 They can be questions on the homework. 00:46:25.360 --> 00:46:28.200 Even better if they're questions on other problems, 00:46:28.200 --> 00:46:33.060 questions on the lecture. 00:46:33.060 --> 00:46:42.000 Questions are essential to make that help session helpful. 00:46:42.000 --> 00:46:46.470 What do you think's cooking here? 00:46:46.470 --> 00:46:54.060 At a typical-- Somewhere in the middle here, 00:46:54.060 --> 00:46:57.770 I'm going to draw the u's. 00:46:57.770 --> 00:47:00.560 Shall I just draw them? 00:47:00.560 --> 00:47:02.640 And now what's my condition? 00:47:02.640 --> 00:47:04.710 I gotta put the boundary conditions on. 00:47:04.710 --> 00:47:07.800 Oh, I have put the boundary conditions on it. 00:47:07.800 --> 00:47:11.350 By putting that two there, I'm up to here. 00:47:11.350 --> 00:47:15.280 Okay, let's do that one. 00:47:15.280 --> 00:47:20.780 When I chose K and put a two in there 00:47:20.780 --> 00:47:24.620 I was picking the fixed-fixed boundary condition. 00:47:24.620 --> 00:47:29.010 So can I just say it's going to be beautiful. 00:47:29.010 --> 00:47:32.530 The solution over there is going to look like this. 00:47:32.530 --> 00:47:37.720 The solution over here is going to be up, up, up. 00:47:37.720 --> 00:47:42.960 It's going to be a straight line but only points in a line 00:47:42.960 --> 00:47:46.660 and it'll be straight line down. 00:47:46.660 --> 00:47:50.280 That value, that value, that value. 00:47:50.280 --> 00:47:53.760 Those will be u_1, u_2, u_3, u_4, and u_5. 00:47:58.470 --> 00:48:05.690 And once more, this is going to drop by one again. 00:48:05.690 --> 00:48:08.620 Actually I didn't have to redraw the picture. 00:48:08.620 --> 00:48:11.140 It falls right on. 00:48:11.140 --> 00:48:22.540 In case x is 2/6 so that it fits that picture, 00:48:22.540 --> 00:48:28.250 I'm claiming we have another extremely lucky case. 00:48:28.250 --> 00:48:34.380 If we can use the word lucky for math. 00:48:34.380 --> 00:48:37.560 I'm claiming that the way the-- You remember 00:48:37.560 --> 00:48:41.180 for the uniform load with a one, when 00:48:41.180 --> 00:48:44.840 we had second derivative equal one, the solution 00:48:44.840 --> 00:48:49.160 was a perfect parabola and the discrete solution, 00:48:49.160 --> 00:48:52.570 the difference equation was right on the parabola 00:48:52.570 --> 00:48:54.700 for this fixed-fixed case. 00:48:54.700 --> 00:48:57.540 It's going to happen again. 00:48:57.540 --> 00:48:59.580 It won't always happen. 00:48:59.580 --> 00:49:04.330 Those are the only two important right-hand sides I know. 00:49:04.330 --> 00:49:07.440 They're the two most important right-hand sides and those 00:49:07.440 --> 00:49:09.900 are the two lucky ones. 00:49:09.900 --> 00:49:13.570 If we have a constant that lies right on a parabola, 00:49:13.570 --> 00:49:21.660 if we have a delta function, it lies right on a ramp. 00:49:21.660 --> 00:49:23.300 And there it is. 00:49:23.300 --> 00:49:26.630 So that's what the solution looks like. 00:49:26.630 --> 00:49:31.720 Now, I have to figure out what these numbers are, I guess. 00:49:31.720 --> 00:49:33.870 Yes, what are those numbers? 00:49:33.870 --> 00:49:35.060 Oh, well. 00:49:35.060 --> 00:49:40.900 Actually, if it falls right on, I know the numbers. 00:49:40.900 --> 00:49:46.890 So a is 2/6. 00:49:46.890 --> 00:49:49.540 So let me keep 2/6. 00:49:49.540 --> 00:49:51.550 So a is 2/6. 00:49:51.550 --> 00:49:53.140 That's that value. 00:49:53.140 --> 00:49:58.870 So let me say what I think u is. 00:49:58.870 --> 00:50:00.590 So this was a picture of u. 00:50:00.590 --> 00:50:04.990 That's u_1, 2, 3, 4, and 5 and now 00:50:04.990 --> 00:50:06.780 I think it lies right on that. 00:50:06.780 --> 00:50:19.160 So it's going to be (1-2/6)x going up and (1-x)2/6 going 00:50:19.160 --> 00:50:21.380 down. 00:50:21.380 --> 00:50:28.780 My point is that I'll be able to figure out what that-- this 00:50:28.780 --> 00:50:33.420 is u, this is the u. 00:50:33.420 --> 00:50:38.140 You're going to say, why? 00:50:38.140 --> 00:50:40.530 Let me pause before putting in numbers 00:50:40.530 --> 00:50:46.300 and say why is it, how do I know that the solution is 00:50:46.300 --> 00:50:53.150 right on the function, the continuous solution. 00:50:53.150 --> 00:50:59.150 Well, can I draw a set of pictures 00:50:59.150 --> 00:51:03.420 just like those guys for discrete? 00:51:03.420 --> 00:51:07.700 Yeah, let me just draw those for discrete here. 00:51:07.700 --> 00:51:12.890 That shows you the magic. 00:51:12.890 --> 00:51:24.630 So there is a-- I'm going to draw a vector now. 00:51:24.630 --> 00:51:27.220 I'm going to have to lift the chalk, it won't be a function 00:51:27.220 --> 00:51:29.290 and it'll be the delta vector. 00:51:29.290 --> 00:51:32.600 So it'll be the delta vector, delta 00:51:32.600 --> 00:51:37.730 with-- So there is point one, zero, one, two, up to six. 00:51:37.730 --> 00:51:41.250 It'll be the delta vector. 00:51:41.250 --> 00:51:45.630 Well if I just draw the delta vector, 00:51:45.630 --> 00:51:49.000 the delta vector has a one there. 00:51:49.000 --> 00:51:51.360 So this is the delta vector. 00:51:51.360 --> 00:51:52.810 Do I need? 00:51:52.810 --> 00:51:57.610 Well you can see that the delta vector is now 00:51:57.610 --> 00:52:01.140 going to be the vector of all zeroes 00:52:01.140 --> 00:52:06.110 and it's got a one at the key-- at the impulse and then zero. 00:52:06.110 --> 00:52:07.940 So it's a discrete impulse. 00:52:07.940 --> 00:52:09.610 That would be a better word. 00:52:09.610 --> 00:52:10.870 Discrete impulse. 00:52:10.870 --> 00:52:13.520 Impulse at zero. 00:52:13.520 --> 00:52:16.700 So let's stay with an impulse at zero. 00:52:16.700 --> 00:52:20.960 Alright. 00:52:20.960 --> 00:52:25.710 What's my next picture? 00:52:25.710 --> 00:52:28.780 Again let me put in zero. 00:52:28.780 --> 00:52:31.390 One, two, three, onwards. 00:52:31.390 --> 00:52:35.450 Minus one, so on. 00:52:35.450 --> 00:52:36.570 What do I want to do now? 00:52:36.570 --> 00:52:38.460 What do I draw second? 00:52:38.460 --> 00:52:40.890 I always look over here. 00:52:40.890 --> 00:52:44.080 What did I draw second over here? 00:52:44.080 --> 00:52:46.450 The step. 00:52:46.450 --> 00:52:51.620 Now why did I draw a step function? 00:52:51.620 --> 00:52:54.510 How did I get from here to here? 00:52:54.510 --> 00:52:56.260 I integrate. 00:52:56.260 --> 00:52:57.890 I took the integral. 00:52:57.890 --> 00:53:02.530 So how will I get from here to this picture? 00:53:02.530 --> 00:53:07.300 I don't integrate, I add, sum. 00:53:07.300 --> 00:53:12.110 So coming along from the left, all these all along here, 00:53:12.110 --> 00:53:14.560 this sum is all zero because it was always zero. 00:53:14.560 --> 00:53:19.070 So it's zero, zero, zero, zero. 00:53:19.070 --> 00:53:21.660 And then, whoops, wait a minute. 00:53:21.660 --> 00:53:24.480 It says it a one there? 00:53:24.480 --> 00:53:26.630 Yeah, I think it must be. 00:53:26.630 --> 00:53:30.100 So here it wasn't a zero, wrong. 00:53:30.100 --> 00:53:31.880 Here it's a one. 00:53:31.880 --> 00:53:33.070 And what is it next? 00:53:33.070 --> 00:53:34.620 What's next to it? 00:53:34.620 --> 00:53:37.720 One, because I'm adding more and more zeroes 00:53:37.720 --> 00:53:39.880 but I have that one now, okay. 00:53:39.880 --> 00:53:43.060 A discrete step. 00:53:43.060 --> 00:53:46.380 It's a discrete step, zeroes and then ones. 00:53:46.380 --> 00:53:48.230 Now comes the second. 00:53:48.230 --> 00:53:52.860 So what am I going to call that? 00:53:52.860 --> 00:53:54.670 A step, right? 00:53:54.670 --> 00:54:01.620 It'll be a step function, step vector. 00:54:01.620 --> 00:54:07.410 If the sums of the delta vector gave me the step vector, 00:54:07.410 --> 00:54:11.530 how do I go the other way? 00:54:11.530 --> 00:54:13.990 What do I do to the step vector to get back 00:54:13.990 --> 00:54:18.730 to the delta vector? 00:54:18.730 --> 00:54:20.180 Differences, right? 00:54:20.180 --> 00:54:23.500 Sums in one direction, differences in the other. 00:54:23.500 --> 00:54:31.010 So the differences of the step vector are the delta vector. 00:54:31.010 --> 00:54:33.780 The step is the sum of the deltas 00:54:33.780 --> 00:54:37.600 and the delta is the differences of the step. 00:54:37.600 --> 00:54:39.870 Now for the crucial next guy. 00:54:39.870 --> 00:54:42.450 What's it going to be? 00:54:42.450 --> 00:54:44.850 I add. 00:54:44.850 --> 00:54:46.090 Wait a minute. 00:54:46.090 --> 00:54:49.680 What's up? 00:54:49.680 --> 00:54:55.680 I'm looking for that picture. 00:54:55.680 --> 00:54:58.550 Do I get it? 00:54:58.550 --> 00:55:02.450 Yeah, I hope so. 00:55:02.450 --> 00:55:03.780 Oh, look, we ran out of time. 00:55:03.780 --> 00:55:05.940 I don't have to do this, but I will. 00:55:05.940 --> 00:55:12.720 So as I add I get zeroes and then it's one, 00:55:12.720 --> 00:55:15.200 and then I add on one more one. 00:55:15.200 --> 00:55:16.880 Look. 00:55:16.880 --> 00:55:18.420 You see what's happening. 00:55:18.420 --> 00:55:21.090 I run along at zero but I'm going 00:55:21.090 --> 00:55:26.330 to look at the book to see whether that jump should 00:55:26.330 --> 00:55:28.190 come here or here. 00:55:28.190 --> 00:55:32.070 So I've got a little bit of this to finish next time 00:55:32.070 --> 00:55:35.970 and I'm open for any questions this afternoon. 00:55:35.970 --> 00:55:38.810 Okay, thanks and sorry to keep you late.