1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:02
The following content is
provided under a Creative
3
00:00:03 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:10
offer high-quality educational
resources for free.
6
00:00:10 --> 00:00:13
To make a donation, or to view
additional materials from
7
00:00:13 --> 00:00:15
hundreds of MIT courses visit
MIT OpenCourseWare
8
00:00:15 --> 00:00:19
at ocw.mit.edu.
9
00:00:19 --> 00:00:24
PROFESSOR STRANG: Ok,
so this is I could say
10
00:00:24 --> 00:00:27
delta function day.
11
00:00:27 --> 00:00:30
Break from linear
algebra mostly.
12
00:00:30 --> 00:00:34
So we're looking on another
type of right-hand side.
13
00:00:34 --> 00:00:38
Before in the differential
equation and in the
14
00:00:38 --> 00:00:39
difference equation.
15
00:00:39 --> 00:00:45
So the right-hand sides up to
now, the one we looked at was a
16
00:00:45 --> 00:00:51
uniform constant load second
derivative equal one.
17
00:00:51 --> 00:00:55
Now a point load.
18
00:00:55 --> 00:00:57
Well in a way, we're now
solving a whole bunch of
19
00:00:57 --> 00:01:02
problems because the point load
can be in different places.
20
00:01:02 --> 00:01:06
So instead of solving one
problem with one on the
21
00:01:06 --> 00:01:10
right-hand side, we're solving
with a delta function.
22
00:01:10 --> 00:01:15
Now a delta function is, you
probably have seen and heard
23
00:01:15 --> 00:01:19
the words and seen the symbol,
but maybe not done much
24
00:01:19 --> 00:01:22
with a delta function.
25
00:01:22 --> 00:01:25
It takes a little practice
but it's really worth it.
26
00:01:25 --> 00:01:30
It's a great model of maybe
what can't quite happen
27
00:01:30 --> 00:01:34
physically, to have a
load acting exactly at a
28
00:01:34 --> 00:01:36
point and nowhere else.
29
00:01:36 --> 00:01:40
So the delta function is, I
drew it's picture, the delta
30
00:01:40 --> 00:01:47
function is zero, this is delta
of x is zero except at that one
31
00:01:47 --> 00:01:52
point, the origin, x=0, and
then all along back
32
00:01:52 --> 00:01:53
to zero again.
33
00:01:53 --> 00:02:00
So nothing's happening, no load
except at that one point.
34
00:02:00 --> 00:02:08
And let me just, so there's no
hesitation in when I change
35
00:02:08 --> 00:02:13
from x to x-a, what does
that do to a graph?
36
00:02:13 --> 00:02:21
If I have a function of x and I
instead shift the function to
37
00:02:21 --> 00:02:28
f(x-a), I shift x to x-a, well
in this case, and in all cases,
38
00:02:28 --> 00:02:30
it will just shift the graph.
39
00:02:30 --> 00:02:38
So if I drew a picture of
delta, of x-a, the load now
40
00:02:38 --> 00:02:43
would happen when this is zero,
because it's delta at zero is
41
00:02:43 --> 00:02:47
the impulse, and now
this is zero at x=a.
42
00:02:48 --> 00:02:52
In other words, the load
moved to the point a.
43
00:02:52 --> 00:03:00
So there is the shifting load,
but the load could fall
44
00:03:00 --> 00:03:05
anywhere between zero and one.
45
00:03:05 --> 00:03:07
So delta of x, the load
actually falls at zero.
46
00:03:07 --> 00:03:10
Well we don't quite want
that load at the boundary.
47
00:03:10 --> 00:03:14
So let's think of the point a,
the load point as somewhere
48
00:03:14 --> 00:03:17
between zero and one.
49
00:03:17 --> 00:03:22
Can I just take a little time
to recall the main fact
50
00:03:22 --> 00:03:24
about delta functions?
51
00:03:24 --> 00:03:29
When I say recall, it could
very well be new to you.
52
00:03:29 --> 00:03:33
So that's what the delta
function, that's my best
53
00:03:33 --> 00:03:36
graph of the delta function.
54
00:03:36 --> 00:03:40
But of course I'm, in using the
word function, I'm kind of
55
00:03:40 --> 00:03:43
breaking the rules because no
function, I mean the function
56
00:03:43 --> 00:03:47
is, functions can be zero
there, can be zero there, but
57
00:03:47 --> 00:03:52
they're not supposed to be
infinite at a single point in
58
00:03:52 --> 00:03:54
between, but this one is.
59
00:03:54 --> 00:04:00
Let me go back to delta of
x to match these figures.
60
00:04:00 --> 00:04:04
Of course, they would also
just shift along by a.
61
00:04:04 --> 00:04:09
Maybe no harm in that.
62
00:04:09 --> 00:04:16
Sorry, I'll stay there and
now I want to integrate.
63
00:04:16 --> 00:04:22
And that's when a delta
function comes into its own.
64
00:04:22 --> 00:04:26
It's value of infinity is
a little bit uncertain.
65
00:04:26 --> 00:04:27
What does that mean?
66
00:04:27 --> 00:04:31
But when we integrate
it, what's the key fact
67
00:04:31 --> 00:04:32
about delta function?
68
00:04:32 --> 00:04:38
That the integral of a delta
function from, let's say, let's
69
00:04:38 --> 00:04:41
integrate the whole thing, we
can safely start way at the far
70
00:04:41 --> 00:04:44
left and go away to the far
right because it's zero all the
71
00:04:44 --> 00:04:49
time there except at one
point, and you know.
72
00:04:49 --> 00:04:53
So what's the area
under that spike?
73
00:04:53 --> 00:04:54
It is one.
74
00:04:54 --> 00:04:55
That's right.
75
00:04:55 --> 00:04:58
So that's the fact, the
sort of central fact
76
00:04:58 --> 00:05:00
about a delta function .
77
00:05:00 --> 00:05:02
That the area is one.
78
00:05:02 --> 00:05:05
Oh well, let me, while I'm
really writing down the central
79
00:05:05 --> 00:05:14
fact, let me write it more
specifically, more generally.
80
00:05:14 --> 00:05:20
Suppose I integrate, and this
is delta functions now really
81
00:05:20 --> 00:05:24
showing, if I integrate a delta
function against some, at
82
00:05:24 --> 00:05:28
times, some nice function.
83
00:05:28 --> 00:05:31
Now have you ever
thought about that?
84
00:05:31 --> 00:05:35
What would be the answer if I
integrate the delta function
85
00:05:35 --> 00:05:38
against some nice function?
86
00:05:38 --> 00:05:43
So I'm still getting zero from
this term all the way along
87
00:05:43 --> 00:05:47
until I hit the spike and then
after it goes back
88
00:05:47 --> 00:05:48
to zero again.
89
00:05:48 --> 00:05:54
So, whatever, it's gotta be at
the spike, at x=0, because I
90
00:05:54 --> 00:05:57
put the spike here at
zero, the impulse.
91
00:05:57 --> 00:06:02
So what do you think's
the answer for that one?
92
00:06:02 --> 00:06:04
Yeah, It's the function.
93
00:06:04 --> 00:06:10
So, yes, tell me again and
I'll write it down. g, it's
94
00:06:10 --> 00:06:12
a value of this function g.
95
00:06:12 --> 00:06:16
We don't care what it is to the
left and to the right at zero
96
00:06:16 --> 00:06:21
because it's really at zero
that this thing turns on and
97
00:06:21 --> 00:06:26
its value at that point is
just, gives us the amplitude of
98
00:06:26 --> 00:06:29
the impulse, which is g(0).
99
00:06:31 --> 00:06:34
And of course if g is the
constant function one, I'm
100
00:06:34 --> 00:06:36
back to that formula.
101
00:06:36 --> 00:06:39
But this is maybe the
thing to watch for.
102
00:06:39 --> 00:06:43
Actually there's a lot built
into that little thing.
103
00:06:43 --> 00:06:46
We'll come back to that.
104
00:06:46 --> 00:06:52
So that's delta functions
integrated and now here
105
00:06:52 --> 00:06:53
are some pictures.
106
00:06:53 --> 00:06:57
These are the good pictures.
107
00:06:57 --> 00:07:03
So here's one integral
of the delta function.
108
00:07:03 --> 00:07:06
It's a step function.
109
00:07:06 --> 00:07:11
And the step of course will
occur at the point a if the
110
00:07:11 --> 00:07:14
integral of the delta function
at a point a will be
111
00:07:14 --> 00:07:17
the step function.
112
00:07:17 --> 00:07:20
Where the action happens.
113
00:07:20 --> 00:07:23
The jump happens, I could
call it a jump function.
114
00:07:23 --> 00:07:25
At that point a.
115
00:07:25 --> 00:07:28
Because, just for
the reason we said.
116
00:07:28 --> 00:07:32
That if we integrate,
the integral is zero.
117
00:07:32 --> 00:07:35
And then as soon as our
integral passes this point,
118
00:07:35 --> 00:07:40
so this is integral of the,
this is, I integrated.
119
00:07:40 --> 00:07:44
I integrate to get
to this picture.
120
00:07:44 --> 00:07:48
I start with that delta
function and I integrate and it
121
00:07:48 --> 00:07:53
suddenly jumps to one as soon
as the integral goes past
122
00:07:53 --> 00:07:56
the spike, the impulse.
123
00:07:56 --> 00:07:57
So a step function.
124
00:07:57 --> 00:07:59
Very handy function,
step function.
125
00:07:59 --> 00:08:03
Sometimes called a heavy side
function named after the guy
126
00:08:03 --> 00:08:10
who, the electrical engineer I
think who first sort of work
127
00:08:10 --> 00:08:13
out the rules for using these.
128
00:08:13 --> 00:08:18
Let's integrate one more time
because we have second order
129
00:08:18 --> 00:08:23
equations, second derivatives,
so we better integrate twice to
130
00:08:23 --> 00:08:26
see what sort of answer we get.
131
00:08:26 --> 00:08:28
Now integrate the
step function.
132
00:08:28 --> 00:08:31
So again, the integral is zero
all the way to the left, so I'm
133
00:08:31 --> 00:08:35
still getting zero, but now
beyond this point I'm
134
00:08:35 --> 00:08:36
integrating one.
135
00:08:36 --> 00:08:40
And the integral of one is x.
136
00:08:40 --> 00:08:44
So now that I would
call a ramp function.
137
00:08:44 --> 00:08:48
That's a nice short word for
this valuable function.
138
00:08:48 --> 00:08:56
A ramp function is the function
that's zero and then x.
139
00:08:56 --> 00:09:00
So tell me about
that ramp function.
140
00:09:00 --> 00:09:05
Just think about it. what
happens to its derivative
141
00:09:05 --> 00:09:08
at the point a?
142
00:09:08 --> 00:09:12
As I run along and I hit this
key point, what happens to
143
00:09:12 --> 00:09:18
the derivative of the ramp?
144
00:09:18 --> 00:09:20
What does the derivative do?
145
00:09:20 --> 00:09:22
Focus on that ramp now.
146
00:09:22 --> 00:09:27
What does the derivative
do at that point?
147
00:09:27 --> 00:09:28
It jumps.
148
00:09:28 --> 00:09:30
The derivative jumps the slope.
149
00:09:30 --> 00:09:33
Is the derivative, the
slope jumps from zero and
150
00:09:33 --> 00:09:34
here the slope is one.
151
00:09:34 --> 00:09:36
And of course that's
what that's telling us.
152
00:09:36 --> 00:09:38
Here's the picture
of the derivative.
153
00:09:38 --> 00:09:41
What does the second
derivative do?
154
00:09:41 --> 00:09:48
Well, since I integrated twice
I guess going back two steps
155
00:09:48 --> 00:09:51
I'll find out what the
second derivative is.
156
00:09:51 --> 00:09:55
So the first derivative
takes a jump.
157
00:09:55 --> 00:09:59
The second derivative is the
derivative of that jump,
158
00:09:59 --> 00:10:01
so it's got the impulse.
159
00:10:01 --> 00:10:05
So the second derivative, it's
a straight line here, second
160
00:10:05 --> 00:10:06
derivative a straight line.
161
00:10:06 --> 00:10:09
This is straight line here,
second derivative of a straight
162
00:10:09 --> 00:10:13
line is a straight line.
163
00:10:13 --> 00:10:18
But at that point the first
derivative jumps, the second
164
00:10:18 --> 00:10:21
derivative has that
delta function.
165
00:10:21 --> 00:10:24
In other words,
that's that stuff.
166
00:10:24 --> 00:10:30
If I keep integrating and I
don't need higher integrals
167
00:10:30 --> 00:10:35
in today's lecture, another
integral would be what?
168
00:10:35 --> 00:10:37
If I integrate this
function, then it's
169
00:10:37 --> 00:10:38
running along the zero.
170
00:10:38 --> 00:10:41
What's the integral of this?
171
00:10:41 --> 00:10:44
Doesn't quite turn
that steeply.
172
00:10:44 --> 00:10:48
What's that curve there?
173
00:10:48 --> 00:10:49
If I've integrated the ramp.
174
00:10:49 --> 00:10:50
Here is the integral.
175
00:10:50 --> 00:10:55
First, the next step up, the
integral of the ramp would be?
176
00:10:55 --> 00:10:59
It'll be x squared, yeah, it'll
be a parabola. x squared over
177
00:10:59 --> 00:11:02
two, the integral of that.
178
00:11:02 --> 00:11:05
And now what do I get when
I integrate this one?
179
00:11:05 --> 00:11:07
I get something very important.
180
00:11:07 --> 00:11:13
Not important today, but
important in a few weeks.
181
00:11:13 --> 00:11:15
And very useful in computing.
182
00:11:15 --> 00:11:19
These have turned out to
be just the right thing.
183
00:11:19 --> 00:11:21
So again, I'm integrating that.
184
00:11:21 --> 00:11:24
Everybody can tell
me, what is that?
185
00:11:24 --> 00:11:27
What's that curve now?
186
00:11:27 --> 00:11:29
It's the next
integral of course.
187
00:11:29 --> 00:11:35
The area under that will
be x cubed over six.
188
00:11:35 --> 00:11:37
So now that is a function.
189
00:11:37 --> 00:11:39
Yeah, it's worth maybe
just for practice.
190
00:11:39 --> 00:11:42
What's the deal with
that function?
191
00:11:42 --> 00:11:44
That's pretty smooth function.
192
00:11:44 --> 00:11:50
Because it certainly passes
right, it meets at that point.
193
00:11:50 --> 00:11:54
The first derivative
meets at that point.
194
00:11:54 --> 00:11:57
The second derivative
meets at that point.
195
00:11:57 --> 00:12:00
The third derivative does what?
196
00:12:00 --> 00:12:02
Of this line.
197
00:12:02 --> 00:12:04
The third derivative takes
three steps back down the
198
00:12:04 --> 00:12:08
line and you see that the
third derivative jumps.
199
00:12:08 --> 00:12:08
Right?
200
00:12:08 --> 00:12:13
The third derivative of that is
the third derivative, would be,
201
00:12:13 --> 00:12:18
shall I for C, for cubic spline
or something, the third
202
00:12:18 --> 00:12:21
derivative will be zero there.
203
00:12:21 --> 00:12:25
And the third derivative of
that is exactly like back to
204
00:12:25 --> 00:12:29
that, back to that,
back to one is one.
205
00:12:29 --> 00:12:34
So the third derivative. so
the cubic spline's so smooth
206
00:12:34 --> 00:12:38
your eye doesn't see that.
207
00:12:38 --> 00:12:42
They're very useful for
drawing many, many purposes.
208
00:12:42 --> 00:12:46
CAD programs would use such
things constantly because
209
00:12:46 --> 00:12:50
they're convenient, they have
nice pieces that you can
210
00:12:50 --> 00:12:55
fit together and they fit
together very smoothly.
211
00:12:55 --> 00:12:59
But they really are two
separate functions.
212
00:12:59 --> 00:13:02
So that's up to cubic spline.
213
00:13:02 --> 00:13:04
But our focus is--
214
00:13:04 --> 00:13:09
These would solve, what
equations would those solve?
215
00:13:09 --> 00:13:13
Well, that takes how many
derivatives to get to a delta?
216
00:13:13 --> 00:13:17
So what would be the equation?
217
00:13:17 --> 00:13:22
What would be the
right-hand side?
218
00:13:22 --> 00:13:25
Let me take the
fourth derivative.
219
00:13:25 --> 00:13:27
I'll just ask the
question that way.
220
00:13:27 --> 00:13:29
What would be the fourth
derivative of that
221
00:13:29 --> 00:13:31
cubic spline?
222
00:13:31 --> 00:13:32
A delta, right?
223
00:13:32 --> 00:13:34
Four steps back.
224
00:13:34 --> 00:13:40
So what is, physically,
what are we seeing here?
225
00:13:40 --> 00:13:45
Do you recognize what kind? if
I ask now people in mechanics,
226
00:13:45 --> 00:13:49
When will we meet a
fourth order equation?
227
00:13:49 --> 00:13:53
Fourth derivative
equals a load.
228
00:13:53 --> 00:13:58
Anybody know the physical
situation where
229
00:13:58 --> 00:14:02
fourth derivative?
230
00:14:02 --> 00:14:03
Beams, yeah.
231
00:14:03 --> 00:14:05
It's the equation for a beam.
232
00:14:05 --> 00:14:10
A beam has, the
bending of a beam.
233
00:14:10 --> 00:14:11
So it's a beam.
234
00:14:11 --> 00:14:16
This eraser isn't too very much
like a beam, but anyway I put
235
00:14:16 --> 00:14:20
the chalk on it, well
nothing happened.
236
00:14:20 --> 00:14:22
Sit on it, whatever.
237
00:14:22 --> 00:14:26
It'll bend and that
bending will be given
238
00:14:26 --> 00:14:28
by a beam equation.
239
00:14:28 --> 00:14:31
So later we'll meet
the beam equation.
240
00:14:31 --> 00:14:40
So most equations of physics,
mechanics, biology, everything
241
00:14:40 --> 00:14:45
are second order, Newton's
Laws often the reason.
242
00:14:45 --> 00:14:49
But we get up to fourth
order sometimes.
243
00:14:49 --> 00:14:52
And very seldom get higher.
244
00:14:52 --> 00:14:53
Hopefully.
245
00:14:53 --> 00:14:59
Beams or plates, that table
would be a plate and it would
246
00:14:59 --> 00:15:05
have a fourth order equation.
247
00:15:05 --> 00:15:07
Let's start solving
this problem.
248
00:15:07 --> 00:15:11
What's the solution, what's
the general solution
249
00:15:11 --> 00:15:13
to that equation?
250
00:15:13 --> 00:15:15
Minus the second derivative,
so notice the minus that
251
00:15:15 --> 00:15:19
I like, and the load has
now moved to the point a.
252
00:15:19 --> 00:15:25
So the solution u(x), let's
write down all solutions.
253
00:15:25 --> 00:15:27
Tell me one solution, first.
254
00:15:27 --> 00:15:29
One particular solution.
255
00:15:29 --> 00:15:33
What is one function for which
minus the second derivative
256
00:15:33 --> 00:15:36
would be the delta.
257
00:15:36 --> 00:15:38
That's what we've
got over there.
258
00:15:38 --> 00:15:42
So just bring that
blackboard over here.
259
00:15:42 --> 00:15:45
Change its sign because
that minus, and what are
260
00:15:45 --> 00:15:46
you going to tell me?
261
00:15:46 --> 00:15:51
Minus a ramp.
262
00:15:51 --> 00:15:53
Minus a ramp.
263
00:15:53 --> 00:16:01
And the ramp, of course, will
ramp up at the point a so that
264
00:16:01 --> 00:16:05
it's the second derivative of
that, the second derivative
265
00:16:05 --> 00:16:07
of R will be delta.
266
00:16:07 --> 00:16:12
The minus is correct and
the point is correct.
267
00:16:12 --> 00:16:17
Now does that solve
our problem?
268
00:16:17 --> 00:16:20
No.
269
00:16:20 --> 00:16:22
The ramp is going upwards.
270
00:16:22 --> 00:16:23
It's not zero.
271
00:16:23 --> 00:16:25
What am I forgetting?
272
00:16:25 --> 00:16:27
What do I not yet have?
273
00:16:27 --> 00:16:30
There's more to this solution.
274
00:16:30 --> 00:16:34
Just as there was
for a uniform load.
275
00:16:34 --> 00:16:36
What was the more?
276
00:16:36 --> 00:16:47
Constant and I want two
homogeneous solutions, null
277
00:16:47 --> 00:16:53
solutions, two solutions with
second derivative equal zero.
278
00:16:53 --> 00:16:57
One of them is C and
the other one is Dx.
279
00:16:57 --> 00:16:59
That's the whole solution.
280
00:16:59 --> 00:17:02
So what I want to, I
mean we need that C+Dx.
281
00:17:04 --> 00:17:07
We've got two boundary
conditions to satisfy,
282
00:17:07 --> 00:17:08
just as before.
283
00:17:08 --> 00:17:10
So I need two constants,
that'll do it perfectly and
284
00:17:10 --> 00:17:13
I'll get an exact answer.
285
00:17:13 --> 00:17:18
And so this is a ramp.
286
00:17:18 --> 00:17:19
Oh yeah.
287
00:17:19 --> 00:17:23
Before I go further, how
would I think about this?
288
00:17:23 --> 00:17:27
This is a ramp that
turns which way?
289
00:17:27 --> 00:17:28
Down.
290
00:17:28 --> 00:17:29
Right?
291
00:17:29 --> 00:17:33
With that minus sign, that ramp
turns down at the point x=a.
292
00:17:35 --> 00:17:37
Right?
293
00:17:37 --> 00:17:43
It's derivative goes
from zero to minus one.
294
00:17:43 --> 00:17:49
The slope of this guy drops by
one because of the minus sign.
295
00:17:49 --> 00:18:00
Sorry the slope, of the ramp
function of minus the ramp.
296
00:18:00 --> 00:18:03
It goes at zero, drops by one.
297
00:18:03 --> 00:18:10
And what this is going to do is
take that ramp and adjust it
298
00:18:10 --> 00:18:11
to go through the fixed ends.
299
00:18:11 --> 00:18:13
Oh, let's just do it.
300
00:18:13 --> 00:18:13
Let's just do it.
301
00:18:13 --> 00:18:15
What are C and D?
302
00:18:15 --> 00:18:16
What are C and D?
303
00:18:16 --> 00:18:19
My point a-- let me draw
a graph, that's always
304
00:18:19 --> 00:18:21
the best thing.
305
00:18:21 --> 00:18:26
Always draw a graph
of these solutions.
306
00:18:26 --> 00:18:29
So let me put in the point a.
307
00:18:29 --> 00:18:34
So I'm drawing now a picture of
the solution from zero to one.
308
00:18:34 --> 00:18:40
I'll graph it.
309
00:18:40 --> 00:18:42
What do I have here?
310
00:18:42 --> 00:18:47
Shall we just plug in
the boundary conditions
311
00:18:47 --> 00:18:48
and find C and D?
312
00:18:48 --> 00:18:50
That's the direct way.
313
00:18:50 --> 00:18:52
What is C?
314
00:18:52 --> 00:18:55
C I'm going to plug in.
315
00:18:55 --> 00:18:57
Hopefully I might find
it from just the first
316
00:18:57 --> 00:18:59
boundary condition.
317
00:18:59 --> 00:19:02
If I'm starting from zero,
well this guy certainly
318
00:19:02 --> 00:19:04
starts at zero, right?
319
00:19:04 --> 00:19:09
The ramp hasn't done anything
until it gets to a.
320
00:19:09 --> 00:19:10
And this guy is certainly zero.
321
00:19:10 --> 00:19:12
So what is C?
322
00:19:12 --> 00:19:15
Gone, right.
323
00:19:15 --> 00:19:19
Now what is D?
324
00:19:19 --> 00:19:22
Well alright, what's D?
325
00:19:22 --> 00:19:23
Let's see.
326
00:19:23 --> 00:19:30
Let me draw the-- so there's
a Dx and D won't be zero.
327
00:19:30 --> 00:19:34
I want that thing to
be zero at point one.
328
00:19:34 --> 00:19:37
So I want to determine D.
329
00:19:37 --> 00:19:39
Let me determine D.
330
00:19:39 --> 00:19:44
So what is minus
the ramp at x=1?
331
00:19:45 --> 00:19:46
I'm plugging in x=1.
332
00:19:46 --> 00:19:47
Is that right?
333
00:19:47 --> 00:19:49
I'm going straight
forward here.
334
00:19:49 --> 00:19:52
Plugging in x=1 into this
boundary condition,
335
00:19:52 --> 00:19:54
ready for this guy.
336
00:19:54 --> 00:19:56
What's the ramp?
337
00:19:56 --> 00:20:04
So it's minus and the ramp is,
well if the ramp is shifted
338
00:20:04 --> 00:20:07
over then that's shifted over.
339
00:20:07 --> 00:20:12
So at x=1, what's the ramp?
340
00:20:12 --> 00:20:15
How high has that ramp gone?
341
00:20:15 --> 00:20:15
1-a.
342
00:20:17 --> 00:20:17
Right?
343
00:20:17 --> 00:20:19
The ramp is x-a.
344
00:20:19 --> 00:20:21
345
00:20:21 --> 00:20:25
At the point x=1
it will be 1-a.
346
00:20:25 --> 00:20:29
So I think I get 1,
-1 - a out of that.
347
00:20:29 --> 00:20:35
Minus the ramp plus
D times what?
348
00:20:35 --> 00:20:37
One, I'm plugging in x=1.
349
00:20:37 --> 00:20:40
And that's supposed to equal?
350
00:20:40 --> 00:20:43
Zero, good.
351
00:20:43 --> 00:20:48
So I'm doing this sort of the
systematic way of writing
352
00:20:48 --> 00:20:51
down the general solution.
353
00:20:51 --> 00:20:55
Discovering that D, what
do I discover D is?
354
00:20:55 --> 00:20:56
Put it on the other side.
355
00:20:56 --> 00:20:58
D is 1-a.
356
00:21:00 --> 00:21:08
And of course, don't forget
that it's multiplying the x.
357
00:21:08 --> 00:21:13
Let me just draw the picture.
358
00:21:13 --> 00:21:15
Here's how I think about it.
359
00:21:15 --> 00:21:20
The solution is, away
from x=a, what does the
360
00:21:20 --> 00:21:22
solution look like?
361
00:21:22 --> 00:21:28
To the left of x=a what's
my graph going to be?
362
00:21:28 --> 00:21:31
It's going to be?
363
00:21:31 --> 00:21:33
A straight line, right?
364
00:21:33 --> 00:21:38
To the left of here
there is no load.
365
00:21:38 --> 00:21:43
The equation is second
derivative equals zero.
366
00:21:43 --> 00:21:45
The solution to that
is a straight line.
367
00:21:45 --> 00:21:49
In other words, until I get to
a, this thing hasn't started.
368
00:21:49 --> 00:21:51
It's only this straight line.
369
00:21:51 --> 00:21:53
The solution does
something like that.
370
00:21:53 --> 00:21:55
It's a straight line.
371
00:21:55 --> 00:21:59
And I guess, actually,
that's what it is.
372
00:21:59 --> 00:22:02
Because the C isn't here and
that's all we've got left.
373
00:22:02 --> 00:22:06
So that's that straight line.
374
00:22:06 --> 00:22:10
What is it for the second half?
375
00:22:10 --> 00:22:14
Tell me what the solution looks
like in the second half.
376
00:22:14 --> 00:22:17
In between a and one.
377
00:22:17 --> 00:22:19
It's going downhill.
378
00:22:19 --> 00:22:22
Why?
379
00:22:22 --> 00:22:24
Because it gotta
get back to zero.
380
00:22:24 --> 00:22:28
And how's it going downhill?
381
00:22:28 --> 00:22:31
It has to be linear.
382
00:22:31 --> 00:22:34
In this region,
has to be linear.
383
00:22:34 --> 00:22:37
Why?
384
00:22:37 --> 00:22:38
How do I know it's linear here?
385
00:22:38 --> 00:22:44
Because one way is to say the
equation in that region is
386
00:22:44 --> 00:22:46
second derivative equal zero.
387
00:22:46 --> 00:22:49
Second derivative equal
zero, straight line.
388
00:22:49 --> 00:22:50
This is my solution.
389
00:22:50 --> 00:22:55
It's (1-a)x here and
it's whatever it is
390
00:22:55 --> 00:22:58
to get back to zero.
391
00:22:58 --> 00:23:02
What will it take to
get back to zero?
392
00:23:02 --> 00:23:03
Let's see.
393
00:23:03 --> 00:23:07
Well we could plug in, we've
got one expression here.
394
00:23:07 --> 00:23:09
Or I could just look at that.
395
00:23:09 --> 00:23:12
I could say, okay what's the
equation for the straight line
396
00:23:12 --> 00:23:17
that's at this point, what
is the, yeah, it's (1-a)x.
397
00:23:17 --> 00:23:23
398
00:23:23 --> 00:23:26
I want it to be linear.
399
00:23:26 --> 00:23:31
I want it to get to zero.
400
00:23:31 --> 00:23:32
Let's see.
401
00:23:32 --> 00:23:36
If I want that, it would
be great to have 1-x
402
00:23:36 --> 00:23:37
times something.
403
00:23:37 --> 00:23:39
I have to figure out what.
404
00:23:39 --> 00:23:44
Because with the 1-x that
x=1, that'll drop off.
405
00:23:44 --> 00:23:45
That's linear.
406
00:23:45 --> 00:23:49
What number, what's
the key here?
407
00:23:49 --> 00:23:58
That slope, I want to
match them up there.
408
00:23:58 --> 00:23:59
And that's the point x=a.
409
00:24:01 --> 00:24:04
This is supposed to
match that at x=a.
410
00:24:04 --> 00:24:09
Do you have an idea for
what I should take?
411
00:24:09 --> 00:24:15
What do I put right there? a.
412
00:24:15 --> 00:24:21
Look at the symmetry in those
two sides. (1-a)x going
413
00:24:21 --> 00:24:24
up. (1-x)a x going down.
414
00:24:24 --> 00:24:28
At x=a it hits that
point, right.
415
00:24:28 --> 00:24:30
So we've solved it.
416
00:24:30 --> 00:24:34
We could think about
this different ways.
417
00:24:34 --> 00:24:39
I could have got that 1-x,
let's see, I could have
418
00:24:39 --> 00:24:42
got it from the formula.
419
00:24:42 --> 00:24:46
In a way I like to get it
from the picture, I see it,
420
00:24:46 --> 00:24:49
sort of, I see the point.
421
00:24:49 --> 00:24:51
What happened at that point?
422
00:24:51 --> 00:24:54
What are the jump conditions?
423
00:24:54 --> 00:24:56
This is another way to
ask, to see how the
424
00:24:56 --> 00:24:58
delta function works.
425
00:24:58 --> 00:25:00
What are they jump conditions?
426
00:25:00 --> 00:25:03
I want to know, when I ask
about jump conditions, I want
427
00:25:03 --> 00:25:08
to know what are the conditions
on u(x), the displacement?
428
00:25:08 --> 00:25:10
What are the conditions
on the slope, u'(x).
429
00:25:12 --> 00:25:15
That'll be the strain
when we're speaking
430
00:25:15 --> 00:25:19
about elasticity.
431
00:25:19 --> 00:25:24
Just for u(x), what's the
statement about u(x) from
432
00:25:24 --> 00:25:30
the left and from the right
at that critical point,
433
00:25:30 --> 00:25:32
the point of the load.
434
00:25:32 --> 00:25:38
From the left and from
the right u(x) is?
435
00:25:38 --> 00:25:42
The same. u(x) matches up. u(x)
from the left is that height.
436
00:25:42 --> 00:25:44
u(x) from the right is that.
437
00:25:44 --> 00:25:48
I want to write down
those jump conditions.
438
00:25:48 --> 00:25:57
Because that's another way to
see this. u(x), u(a) from the
439
00:25:57 --> 00:26:04
left should equal u-- do you
want me to say u is continuous?
440
00:26:04 --> 00:26:13
I'll just say it in words. u(x)
is continuous, that just means
441
00:26:13 --> 00:26:15
it doesn't jump, at x=a.
442
00:26:15 --> 00:26:18
443
00:26:18 --> 00:26:22
So that's, you could say
that's a non-jump condition.
444
00:26:22 --> 00:26:24
The function itself
doesn't jump.
445
00:26:24 --> 00:26:25
Why not?
446
00:26:25 --> 00:26:28
Because we're talking about
some elastic bar on which
447
00:26:28 --> 00:26:30
we put a point load.
448
00:26:30 --> 00:26:32
The thing isn't going to break.
449
00:26:32 --> 00:26:36
The displacement is
going to be continuous.
450
00:26:36 --> 00:26:41
But what's the condition
on u'(x), the
451
00:26:41 --> 00:26:43
derivative, the slope?
452
00:26:43 --> 00:26:46
So that's the function and
now tell me what's the
453
00:26:46 --> 00:26:48
deal on the slope?
454
00:26:48 --> 00:26:52
What's the comparison between
the-- I have a slope of
455
00:26:52 --> 00:26:54
whatever it is going
along here and I have a
456
00:26:54 --> 00:26:56
slope of a new slope.
457
00:26:56 --> 00:27:03
So u'(x), the slope
jumps, right?
458
00:27:03 --> 00:27:06
And how much does it jump?
459
00:27:06 --> 00:27:08
Minus one.
460
00:27:08 --> 00:27:09
It drops by one.
461
00:27:09 --> 00:27:13
The slope, because of my minus.
462
00:27:13 --> 00:27:18
So this tells me that, yeah,
let me write that down.
463
00:27:18 --> 00:27:18
u'(x) drops by one.
464
00:27:18 --> 00:27:26
465
00:27:26 --> 00:27:32
This is another way to say
what the equation is asking.
466
00:27:32 --> 00:27:36
The equation is looking for two
pieces of straight lines that
467
00:27:36 --> 00:27:41
meet at a but their
slope drops by one.
468
00:27:41 --> 00:27:42
By the way, what
were the slopes?
469
00:27:42 --> 00:27:45
It's good to graph
the slopes, too.
470
00:27:45 --> 00:27:52
Let me graph the slopes.
471
00:27:52 --> 00:27:58
The slope u', the
derivative du/dx.
472
00:27:58 --> 00:28:00
What's the slope here?
473
00:28:00 --> 00:28:03
Slope is 1-a at
this point, right?
474
00:28:03 --> 00:28:06
The derivative is
1-a along here.
475
00:28:06 --> 00:28:07
So slope is 1-a.
476
00:28:09 --> 00:28:14
And now at x=a the slope
changes to this one.
477
00:28:14 --> 00:28:17
And what's the slope
of that second part?
478
00:28:17 --> 00:28:18
Minus a.
479
00:28:18 --> 00:28:19
Look.
480
00:28:19 --> 00:28:21
It did it right.
481
00:28:21 --> 00:28:24
Minus a is the
slope along here.
482
00:28:24 --> 00:28:27
Do you see 1-a?
483
00:28:28 --> 00:28:29
It dropped by one.
484
00:28:29 --> 00:28:36
The one disappeared to
leave a slope of minus a.
485
00:28:36 --> 00:28:41
I guess if I just
imagine a bar.
486
00:28:41 --> 00:28:45
I'm fixing it at both ends.
487
00:28:45 --> 00:28:50
There's a bar.
488
00:28:50 --> 00:28:55
I'm just thinking for people
who like to see a physical
489
00:28:55 --> 00:29:00
picture of what's happening,
that's what this is, we'll do
490
00:29:00 --> 00:29:03
it properly very, very soon.
491
00:29:03 --> 00:29:05
I've got a bar.
492
00:29:05 --> 00:29:08
It's a very light bar.
493
00:29:08 --> 00:29:12
It's weight is not
a problem here.
494
00:29:12 --> 00:29:14
But it's got a load
at the point.
495
00:29:14 --> 00:29:16
So I'll measure a
going downwards.
496
00:29:16 --> 00:29:22
And at the point x=a I'm
hanging a heavy load.
497
00:29:22 --> 00:29:24
A load.
498
00:29:24 --> 00:29:30
How do I draw a load?
499
00:29:30 --> 00:29:40
Maybe I'll make a big
weight or something.
500
00:29:40 --> 00:29:49
What's going to happen to this
dumb bar when I do that?
501
00:29:49 --> 00:29:50
Just tell me physically.
502
00:29:50 --> 00:29:51
What's going to happen?
503
00:29:51 --> 00:29:54
What's going to happen
above the load?
504
00:29:54 --> 00:30:00
It's going to stretch,
right, tension.
505
00:30:00 --> 00:30:03
The load is going to pull
the bar down, it's going
506
00:30:03 --> 00:30:05
to stretch this part.
507
00:30:05 --> 00:30:07
And because nothing special
is happening, it's going
508
00:30:07 --> 00:30:09
to stretch it linearly.
509
00:30:09 --> 00:30:13
And then what's going to
happen below the load?
510
00:30:13 --> 00:30:15
Compression.
511
00:30:15 --> 00:30:19
So the slope will go negative.
512
00:30:19 --> 00:30:23
And nothing special happened
so the slope will be negative
513
00:30:23 --> 00:30:25
but it'll be constant.
514
00:30:25 --> 00:30:27
The slope will drop
from this to this.
515
00:30:27 --> 00:30:34
The displacement, that point
will go down a little bit.
516
00:30:34 --> 00:30:38
That little bit it goes down is
actually the height of this,
517
00:30:38 --> 00:30:40
because that's the
displacement.
518
00:30:40 --> 00:30:41
It'll go down a little bit.
519
00:30:41 --> 00:30:45
It'll stretch above, it'll
compress below, and we see
520
00:30:45 --> 00:30:52
that in that picture
of the displacement.
521
00:30:52 --> 00:30:55
The displacement's all down.
522
00:30:55 --> 00:30:56
Right?
523
00:30:56 --> 00:31:01
Displacement, you know, nature
is still going to, all the
524
00:31:01 --> 00:31:02
bar is going to move down.
525
00:31:02 --> 00:31:07
That's why this function
doesn't, this function,
526
00:31:07 --> 00:31:09
the displacement
function is positive.
527
00:31:09 --> 00:31:10
It goes all down.
528
00:31:10 --> 00:31:16
But the slope function is
positive here, so tension
529
00:31:16 --> 00:31:18
is positive slope.
530
00:31:18 --> 00:31:24
Stretch and compression
is negative.
531
00:31:24 --> 00:31:31
Well all that to
solve this equation.
532
00:31:31 --> 00:31:38
Maybe while we're on a roll,
let's solve the free-fixed guy.
533
00:31:38 --> 00:31:41
So this is our-- might
as well be systematic.
534
00:31:41 --> 00:31:43
This with the
fixed-fixed problem.
535
00:31:43 --> 00:31:46
Let me below it solve
the free-fixed problem.
536
00:31:46 --> 00:31:48
So it'll be u''.
537
00:31:49 --> 00:31:52
That's the second derivative
equals delta at x-a.
538
00:31:53 --> 00:31:55
Same set up.
539
00:31:55 --> 00:32:01
But now the top end is,
so it's free at the top.
540
00:32:01 --> 00:32:04
What does that mean?
541
00:32:04 --> 00:32:12
Slope is zero at the top but
is still fixed at the bottom.
542
00:32:12 --> 00:32:21
So this will be now free-fixed.
543
00:32:21 --> 00:32:24
Let me go straight
to the picture.
544
00:32:24 --> 00:32:25
Let me go straight to
the picture of u(x).
545
00:32:27 --> 00:32:32
So there is x=0, there's
x=1, here's the load at a.
546
00:32:32 --> 00:32:37
What's up?
547
00:32:37 --> 00:32:41
And while you're thinking about
that, let me draw a picture
548
00:32:41 --> 00:32:43
to match this picture.
549
00:32:43 --> 00:32:52
A bar fixed at the bottom
but not at the top.
550
00:32:52 --> 00:32:59
And it's got its load
here hanging down.
551
00:32:59 --> 00:33:08
But let's do it math first, and
then check with the picture.
552
00:33:08 --> 00:33:10
What do we got, two or
three ways now to try
553
00:33:10 --> 00:33:11
to get the answer?
554
00:33:11 --> 00:33:17
The systematic way would be
to write down this solution
555
00:33:17 --> 00:33:22
and plug in the two
boundary conditions.
556
00:33:22 --> 00:33:24
That'd be a
straightforward way.
557
00:33:24 --> 00:33:27
Yeah, we could even
start by that.
558
00:33:27 --> 00:33:34
So u(x) is the particular
solution, the ramp
559
00:33:34 --> 00:33:35
plus any Cx+D.
560
00:33:37 --> 00:33:46
And just plug in x=0
that'll be easy.
561
00:33:46 --> 00:33:51
If I plug in x=0 in the
free condition, what
562
00:33:51 --> 00:33:53
does that tell me?
563
00:33:53 --> 00:33:57
At x=0, this corner, this
ramp hasn't started
564
00:33:57 --> 00:34:00
so the slope is zero.
565
00:34:00 --> 00:34:01
The slope of the
constant is zero.
566
00:34:01 --> 00:34:05
What do I learn from this
boundary condition? u'(0)=0.
567
00:34:05 --> 00:34:08
568
00:34:08 --> 00:34:10
That C is zero.
569
00:34:10 --> 00:34:13
Before I learned that D was
zero, but now from that
570
00:34:13 --> 00:34:18
condition I'm going
to learn C is zero.
571
00:34:18 --> 00:34:21
Do the picture for me.
572
00:34:21 --> 00:34:24
Do the picture for me.
573
00:34:24 --> 00:34:27
What's the graph of--
this is a graph of u(x).
574
00:34:27 --> 00:34:30
575
00:34:30 --> 00:34:36
Remember now it starts
from zero slope because
576
00:34:36 --> 00:34:38
it's free at the top.
577
00:34:38 --> 00:34:44
What does the graph look
like in the first part?
578
00:34:44 --> 00:34:48
It's a straight line, has
to be a straight line
579
00:34:48 --> 00:34:51
because there's no force.
580
00:34:51 --> 00:34:54
And what kind of a line?
581
00:34:54 --> 00:34:57
It's going to be horizontal
because it starts
582
00:34:57 --> 00:34:59
off horizontal.
583
00:34:59 --> 00:35:03
The slope has to be zero
at zero and nothing
584
00:35:03 --> 00:35:05
changes until a.
585
00:35:05 --> 00:35:08
So it comes along there.
586
00:35:08 --> 00:35:11
Right?
587
00:35:11 --> 00:35:15
Now I've started out with the
right, left, the correct
588
00:35:15 --> 00:35:18
boundary condition at
zero, which was no slope.
589
00:35:18 --> 00:35:22
And now what's it going
to do the other half?
590
00:35:22 --> 00:35:25
From a to one.
591
00:35:25 --> 00:35:31
It's going to be again, it'll
be a straight line, right?
592
00:35:31 --> 00:35:33
Because there's no force there.
593
00:35:33 --> 00:35:38
And what happens at-- all the
action of course is at this
594
00:35:38 --> 00:35:41
point a, and what action is it?
595
00:35:41 --> 00:35:45
Tell me what sort of a line.
596
00:35:45 --> 00:35:49
How do I finish the picture?
597
00:35:49 --> 00:35:52
What do I do?
598
00:35:52 --> 00:35:53
I start here, right?
599
00:35:53 --> 00:36:01
Because the bar's not falling
apart. u is continuous.
600
00:36:01 --> 00:36:03
I don't get a gap suddenly.
601
00:36:03 --> 00:36:07
And now what do I
do from there?
602
00:36:07 --> 00:36:10
Only thing I can possibly do,
because I have to end up
603
00:36:10 --> 00:36:16
here and it has to be a
straight line, that's it.
604
00:36:16 --> 00:36:18
That's what the picture
will have to look like.
605
00:36:18 --> 00:36:30
What does that correspond to
in the picture for the bar?
606
00:36:30 --> 00:36:33
Well what happens
with this bar?
607
00:36:33 --> 00:36:38
Above the weight what's what
happens to this top part of
608
00:36:38 --> 00:36:42
the bar in that picture?
609
00:36:42 --> 00:36:45
And what happens to the
lower part of the bar?
610
00:36:45 --> 00:36:50
So this was at the point x=a,
this is x=0, this is x=1.
611
00:36:51 --> 00:36:57
What happens above the
bar, above the weight?
612
00:36:57 --> 00:37:01
It just, a rigid motion,
just goes down.
613
00:37:01 --> 00:37:04
Because what happens
below the weight?
614
00:37:04 --> 00:37:07
The same compression or
compression still happening.
615
00:37:07 --> 00:37:08
This is still squeezed.
616
00:37:08 --> 00:37:12
Shall I try to draw it?
617
00:37:12 --> 00:37:14
So this is after the weight.
618
00:37:14 --> 00:37:21
This got squeezed but this
part did not get squeezed.
619
00:37:21 --> 00:37:25
And that's what
we're seeing here.
620
00:37:25 --> 00:37:28
A fixed displacement.
621
00:37:28 --> 00:37:34
So this means, that picture
means that all the pieces of
622
00:37:34 --> 00:37:39
the bar here got moved down by
the same amount, whatever this,
623
00:37:39 --> 00:37:41
we don't know that number yet.
624
00:37:41 --> 00:37:49
And then below it
they got compressed.
625
00:37:49 --> 00:37:56
Well we're almost there but we
don't yet have that solution.
626
00:37:56 --> 00:38:02
Come back to this picture.
u(x) is continuous, got it.
627
00:38:02 --> 00:38:08
And what's the real condition
that's going to determine where
628
00:38:08 --> 00:38:12
we are, what those heights
are, the numbers in there.
629
00:38:12 --> 00:38:16
It's gotta look like that, but
we get more than that, we gotta
630
00:38:16 --> 00:38:19
no what are the actual,
what is that height.
631
00:38:19 --> 00:38:20
What is this?
632
00:38:20 --> 00:38:22
What's the slope?
633
00:38:22 --> 00:38:27
Here the slope is zero.
634
00:38:27 --> 00:38:34
Here the slope is what?
635
00:38:34 --> 00:38:36
What's the slope in
the second part?
636
00:38:36 --> 00:38:37
That's the key.
637
00:38:37 --> 00:38:40
And you know what it
has to be because what
638
00:38:40 --> 00:38:42
happens to the slope?
639
00:38:42 --> 00:38:47
If I have the second derivative
as a delta function with
640
00:38:47 --> 00:38:56
that minus sign, the
slope drops by one.
641
00:38:56 --> 00:39:02
And the slope here is zero, so
the slope here is minus one.
642
00:39:02 --> 00:39:07
And now it has to get through
there, so what is the function?
643
00:39:07 --> 00:39:11
What's the function that
has a slope of minus one
644
00:39:11 --> 00:39:22
and comes down to zero?
645
00:39:22 --> 00:39:25
It's gotta have a minus x in
it and what's the constant
646
00:39:25 --> 00:39:30
to make it come out right?
647
00:39:30 --> 00:39:33
What do I write now
here for u(x).
648
00:39:34 --> 00:39:35
1-x.
649
00:39:35 --> 00:39:37
650
00:39:37 --> 00:39:41
That has a slope of minus one,
the derivative is minus one,
651
00:39:41 --> 00:39:44
at x=1 it comes to
zero, that's it.
652
00:39:44 --> 00:39:48
And what do I write,
what's u(x) up here?
653
00:39:48 --> 00:39:54
And therefore, right there?
654
00:39:54 --> 00:39:59
What's the displacement there,
of all this bit that moves
655
00:39:59 --> 00:40:03
down, how much does
it move down?
656
00:40:03 --> 00:40:03
1-a.
657
00:40:04 --> 00:40:04
Why 1-a?
658
00:40:05 --> 00:40:06
That's the right answer.
659
00:40:06 --> 00:40:07
1-a.
660
00:40:07 --> 00:40:09
661
00:40:09 --> 00:40:12
Why's that?
662
00:40:12 --> 00:40:13
Because it had to match
663
00:40:13 --> 00:40:15
up at x=a.
664
00:40:15 --> 00:40:19
At x=a, this and that match up.
665
00:40:19 --> 00:40:23
At x=a, that slope,
that function and that
666
00:40:23 --> 00:40:24
function match up.
667
00:40:24 --> 00:40:32
So the slope picture is zero
and, oh I'm sorry, can't draw
668
00:40:32 --> 00:40:36
it because I'm at the
bottom of the board.
669
00:40:36 --> 00:40:40
The slope picture, maybe I can
draw it here, the slope picture
670
00:40:40 --> 00:40:44
is zero along here and then
it drops by one to 1-a.
671
00:40:45 --> 00:40:46
So that's a picture of u'.
672
00:40:47 --> 00:40:51
Zero and minus one.
673
00:40:51 --> 00:40:59
This is the thing to look at.
674
00:40:59 --> 00:41:01
That's hard work.
675
00:41:01 --> 00:41:03
When you're seeing delta
functions the first time.
676
00:41:03 --> 00:41:06
But of course the functions
did not get complicated.
677
00:41:06 --> 00:41:11
We kept a clean example.
678
00:41:11 --> 00:41:18
And which we matched up with a
figure and we've got the answer
679
00:41:18 --> 00:41:20
and we've got a couple
of ways to do it.
680
00:41:20 --> 00:41:25
One is this standard,
systematic, plug-in
681
00:41:25 --> 00:41:26
boundary condition way.
682
00:41:26 --> 00:41:32
The other way is this. u(x)
does something here, then the
683
00:41:32 --> 00:41:34
slope has to drop by one.
684
00:41:34 --> 00:41:38
And that's the key to
everything with a
685
00:41:38 --> 00:41:40
boundary condition.
686
00:41:40 --> 00:41:42
So in a way, we have a
piece to the left and
687
00:41:42 --> 00:41:44
a piece to the right.
688
00:41:44 --> 00:41:47
Two constants here, two
constants here, and somewhere
689
00:41:47 --> 00:41:52
there are four conditions that
settle those four constants.
690
00:41:52 --> 00:41:55
You know, we could have a
straight line here, a straight
691
00:41:55 --> 00:41:57
line here, that's two and two.
692
00:41:57 --> 00:42:00
But what are the four
conditions that settle
693
00:42:00 --> 00:42:01
those four constants?
694
00:42:01 --> 00:42:05
Well we have a boundary
condition here, that's one.
695
00:42:05 --> 00:42:07
Boundary condition here is two.
696
00:42:07 --> 00:42:13
We need two more conditions
to settle the two pairs of
697
00:42:13 --> 00:42:19
constants, and there they are.
698
00:42:19 --> 00:42:27
Two conditions at the jump,
at the discontinuity.
699
00:42:27 --> 00:42:36
Now I've got to do
the discrete case.
700
00:42:36 --> 00:42:39
Are you up for the
discrete case?
701
00:42:39 --> 00:42:47
The case where we're doing, we
have a difference equation,
702
00:42:47 --> 00:42:52
so we're doing KU equal a
column of the identity.
703
00:42:52 --> 00:43:00
Column of I.
704
00:43:00 --> 00:43:03
Let me take a specific column.
705
00:43:03 --> 00:43:06
Say, .
706
00:43:06 --> 00:43:09
Let's suppose we have five.
707
00:43:09 --> 00:43:12
I'm going to draw
a picture now.
708
00:43:12 --> 00:43:15
We have five because I
made it five by five.
709
00:43:15 --> 00:43:21
One, two, three, four, five,
here is zero and here is six.
710
00:43:21 --> 00:43:34
So h is 1/(5+1), 1/6,
that's the delta x.
711
00:43:34 --> 00:43:36
So what does my equation say?
712
00:43:36 --> 00:43:41
Remember what K is.
713
00:43:41 --> 00:43:48
U is the n u_1, u_2, u_3,
u_4, and u_5, the unknowns.
714
00:43:48 --> 00:44:04
K is our old friend with twos
and minus ones and minus ones.
715
00:44:04 --> 00:44:07
I'm going to find the solution.
716
00:44:07 --> 00:44:15
And this'll be the solution
that has a load at this point.
717
00:44:15 --> 00:44:18
This is like my point a, right?
718
00:44:18 --> 00:44:21
Here in the continuous
case, a could run anywhere
719
00:44:21 --> 00:44:23
between zero and one.
720
00:44:23 --> 00:44:27
In the discrete case, I've got
five possible load points and
721
00:44:27 --> 00:44:29
I've picked the second one.
722
00:44:29 --> 00:44:32
Five columns of the identity
matrix, five places to put
723
00:44:32 --> 00:44:36
that one, I put it there.
724
00:44:36 --> 00:44:42
Now can I draw the
picture here?
725
00:44:42 --> 00:44:45
Which should we do first?
726
00:44:45 --> 00:44:46
Should we do free-fixed?
727
00:44:46 --> 00:44:52
Because that came out even
easier than fixed-fixed.
728
00:44:52 --> 00:44:55
Notice the solution
here had two parts.
729
00:44:55 --> 00:44:59
This is the way I would
write that answer.
730
00:44:59 --> 00:45:02
Because you could draw a
picture, but if you want
731
00:45:02 --> 00:45:06
to write the formula,
what would I do?
732
00:45:06 --> 00:45:10
I would break it
into two pieces.
733
00:45:10 --> 00:45:16
1-a up to the point a
because that's what it
734
00:45:16 --> 00:45:18
was running along here.
735
00:45:18 --> 00:45:23
And then down here
it was 1-x, x>=a.
736
00:45:23 --> 00:45:31
737
00:45:31 --> 00:45:33
That's important to mention.
738
00:45:33 --> 00:45:38
You have to have some guidance
on how to write the answer.
739
00:45:38 --> 00:45:42
And when the answer has two
parts, this is a good way
740
00:45:42 --> 00:45:44
to write it, in two parts.
741
00:45:44 --> 00:45:47
It's a little too, you're
compressing it too much to
742
00:45:47 --> 00:45:50
write, to use that
ramp function.
743
00:45:50 --> 00:45:56
Better to split it apart
into before a and after a.
744
00:45:56 --> 00:45:59
What's going to
happen over here?
745
00:45:59 --> 00:46:04
Oh yeah, can we take a
shot at this problem?
746
00:46:04 --> 00:46:12
And let me mention again in the
review that'll be in here this
747
00:46:12 --> 00:46:17
afternoon and every Wednesday
afternoon I'll just be
748
00:46:17 --> 00:46:18
ready for questions.
749
00:46:18 --> 00:46:22
Please bring questions.
750
00:46:22 --> 00:46:25
They can be questions
on the homework.
751
00:46:25 --> 00:46:28
Even better if they're
questions on other problems,
752
00:46:28 --> 00:46:33
questions on the lecture.
753
00:46:33 --> 00:46:42
Questions are essential to make
that help session helpful.
754
00:46:42 --> 00:46:46
What do you think's
cooking here?
755
00:46:46 --> 00:46:54
At a typical, somewhere
in the middle here, I'm
756
00:46:54 --> 00:46:57
going to draw the u's.
757
00:46:57 --> 00:47:00
Shall I just draw them?
758
00:47:00 --> 00:47:02
And now what's my condition?
759
00:47:02 --> 00:47:04
I gotta put the boundary
conditions on.
760
00:47:04 --> 00:47:07
Oh, I have put the boundary
conditions on it.
761
00:47:07 --> 00:47:11
By putting that two
there, I'm up to here.
762
00:47:11 --> 00:47:15
Ok, let's do that one.
763
00:47:15 --> 00:47:22
When I chose K and put a two
in there I was picking the
764
00:47:22 --> 00:47:24
fixed-fixed boundary condition.
765
00:47:24 --> 00:47:29
So can I just say it's
going to be beautiful.
766
00:47:29 --> 00:47:32
The solution over there is
going to look like this.
767
00:47:32 --> 00:47:37
The solution over here is
going to be up, up, up.
768
00:47:37 --> 00:47:42
It's going to be a straight
line but only points in
769
00:47:42 --> 00:47:46
a line and it'll be
straight line down.
770
00:47:46 --> 00:47:50
That value, that
value, that value.
771
00:47:50 --> 00:47:52
Those will be u_1, u_2,
u_3, u_4, and u_5.
772
00:47:52 --> 00:47:58
773
00:47:58 --> 00:48:05
And once more, this is going
to drop by one again.
774
00:48:05 --> 00:48:08
Actually I didn't have
to redraw the picture.
775
00:48:08 --> 00:48:11
It falls right on.
776
00:48:11 --> 00:48:22
In case x is 2/6 so that
it fits that picture, I'm
777
00:48:22 --> 00:48:28
claiming we have another
extremely lucky case.
778
00:48:28 --> 00:48:33
If we can use the
word lucky for math.
779
00:48:33 --> 00:48:38
That I'm claiming that the way,
you remember for the uniform
780
00:48:38 --> 00:48:43
load with a one, when we had
second derivative equal one,
781
00:48:43 --> 00:48:48
the solution was a perfect
parabola and the discrete
782
00:48:48 --> 00:48:51
solution, the difference
equation was right on the
783
00:48:51 --> 00:48:54
parabola for this
fixed-fixed case.
784
00:48:54 --> 00:48:57
It's going to happen again.
785
00:48:57 --> 00:48:59
It won't always happen.
786
00:48:59 --> 00:49:02
Those are the only two
important right-hand
787
00:49:02 --> 00:49:04
sides I know.
788
00:49:04 --> 00:49:07
They're the two most important
right-hand sides and those
789
00:49:07 --> 00:49:09
are the two lucky ones.
790
00:49:09 --> 00:49:13
If we have a constant that
lies right on a parabola, if
791
00:49:13 --> 00:49:21
we have a delta function,
it lies right on a ramp.
792
00:49:21 --> 00:49:23
And there it is.
793
00:49:23 --> 00:49:26
So that's what the
solution looks like.
794
00:49:26 --> 00:49:31
Now, I have to figure out what
these numbers are, I guess.
795
00:49:31 --> 00:49:33
Yes, what are those numbers?
796
00:49:33 --> 00:49:35
Oh, well.
797
00:49:35 --> 00:49:40
Actually, if it falls right
on, I know the numbers.
798
00:49:40 --> 00:49:46
So a is 2/6.
799
00:49:46 --> 00:49:49
So let me keep 2/6.
800
00:49:49 --> 00:49:51
So a is 2/6.
801
00:49:51 --> 00:49:53
That's that value.
802
00:49:53 --> 00:49:58
So let me say what
I think U is.
803
00:49:58 --> 00:50:00
So this was a picture of U.
804
00:50:00 --> 00:50:05
That's u_1, 2, 3, 4, and
5 and now I think it
805
00:50:05 --> 00:50:06
lies right on that.
806
00:50:06 --> 00:50:18
So it's going to be
(1-2/6)x going up and
807
00:50:18 --> 00:50:21
(1-x)2/6 going down.
808
00:50:21 --> 00:50:28
My point is that I'll be able
to figure out what that--
809
00:50:28 --> 00:50:33
this is u, this is the u.
810
00:50:33 --> 00:50:38
You're going to say, why?
811
00:50:38 --> 00:50:44
Let me pause before putting in
numbers and say why is it, how
812
00:50:44 --> 00:50:49
do I know that the solution
is right on the function,
813
00:50:49 --> 00:50:53
the continuous solution.
814
00:50:53 --> 00:50:59
Well, can I draw a set of
pictures just like those
815
00:50:59 --> 00:51:03
guys for discrete?
816
00:51:03 --> 00:51:07
Yeah, let me just draw
those for discrete here.
817
00:51:07 --> 00:51:12
That shows you the magic.
818
00:51:12 --> 00:51:24
So there is a, I'm going
to draw a vector now.
819
00:51:24 --> 00:51:27
I'm going to have to lift the
chalk, it won't be a function
820
00:51:27 --> 00:51:29
and it'll be the delta vector.
821
00:51:29 --> 00:51:33
So it'll be the delta
vector, delta with, so
822
00:51:33 --> 00:51:34
there is point one.
823
00:51:34 --> 00:51:37
Zero, one, two, up to six.
824
00:51:37 --> 00:51:41
It'll be the delta vector.
825
00:51:41 --> 00:51:46
Well if I just draw the
delta vector, the delta
826
00:51:46 --> 00:51:49
vector has a one there.
827
00:51:49 --> 00:51:51
So this is the delta vector.
828
00:51:51 --> 00:51:52
Do I need?
829
00:51:52 --> 00:51:59
Well you can see that the delta
vector is now going to be the
830
00:51:59 --> 00:52:03
vector of all zeroes and it's
got a one at the key, at
831
00:52:03 --> 00:52:06
the impulse and then zero.
832
00:52:06 --> 00:52:07
So it's a discrete impulse.
833
00:52:07 --> 00:52:09
That would be a better word.
834
00:52:09 --> 00:52:10
Discrete impulse.
835
00:52:10 --> 00:52:13
Impulse at zero.
836
00:52:13 --> 00:52:16
So let's stay with
an impulse at zero.
837
00:52:16 --> 00:52:20
Alright.
838
00:52:20 --> 00:52:25
What's my next picture?
839
00:52:25 --> 00:52:28
Again let me put in zero.
840
00:52:28 --> 00:52:31
One, two, three, onwards.
841
00:52:31 --> 00:52:35
Minus one, so on.
842
00:52:35 --> 00:52:36
What do I want to do now?
843
00:52:36 --> 00:52:38
What do I draw second?
844
00:52:38 --> 00:52:40
I always look over here.
845
00:52:40 --> 00:52:44
What did I draw
second over here?
846
00:52:44 --> 00:52:46
The step.
847
00:52:46 --> 00:52:51
Now why did I draw
a step function?
848
00:52:51 --> 00:52:54
How did I get from
here to here?
849
00:52:54 --> 00:52:56
I integrate.
850
00:52:56 --> 00:52:57
I took the integral.
851
00:52:57 --> 00:53:02
So how will I get from
here to this picture?
852
00:53:02 --> 00:53:07
I don't integrate, I add, sum.
853
00:53:07 --> 00:53:12
So coming along from the left,
all these all along here, this
854
00:53:12 --> 00:53:14
sum is all zero because
it was always zero.
855
00:53:14 --> 00:53:19
So it's zero, zero, zero, zero.
856
00:53:19 --> 00:53:21
And then, whoops,
wait a minute.
857
00:53:21 --> 00:53:24
It says it a one there?
858
00:53:24 --> 00:53:26
Yeah, I think it must be.
859
00:53:26 --> 00:53:30
So here it wasn't
a zero, wrong.
860
00:53:30 --> 00:53:31
Here it's a one.
861
00:53:31 --> 00:53:33
And what is it next?
862
00:53:33 --> 00:53:34
What's next to it?
863
00:53:34 --> 00:53:38
One, because I'm adding
more and more zeroes but I
864
00:53:38 --> 00:53:39
have that one now, okay.
865
00:53:39 --> 00:53:43
A discrete step.
866
00:53:43 --> 00:53:46
It's a discrete step,
zeroes and then ones.
867
00:53:46 --> 00:53:48
Now comes the second.
868
00:53:48 --> 00:53:52
So what am I going
to call that?
869
00:53:52 --> 00:53:54
A step, right?
870
00:53:54 --> 00:54:01
It'll be a step
function, step vector.
871
00:54:01 --> 00:54:07
If the sums of the delta vector
gave me the step vector,
872
00:54:07 --> 00:54:11
how do I go the other way?
873
00:54:11 --> 00:54:13
What do I do to the step
vector to get back
874
00:54:13 --> 00:54:18
to the delta vector?
875
00:54:18 --> 00:54:20
Differences, right?
876
00:54:20 --> 00:54:23
Sums in one direction,
differences in the other.
877
00:54:23 --> 00:54:31
So the differences of the step
vector are the delta vector.
878
00:54:31 --> 00:54:35
The step is the sum of the
deltas and the delta is the
879
00:54:35 --> 00:54:37
differences of the step.
880
00:54:37 --> 00:54:39
Now for the crucial next guy.
881
00:54:39 --> 00:54:42
What's it going to be?
882
00:54:42 --> 00:54:44
I add.
883
00:54:44 --> 00:54:46
Wait a minute.
884
00:54:46 --> 00:54:49
What's up?
885
00:54:49 --> 00:54:55
I'm looking for that picture.
886
00:54:55 --> 00:54:58
Do I get it?
887
00:54:58 --> 00:55:02
Yeah, I hope so.
888
00:55:02 --> 00:55:03
Oh, look, we ran out of time.
889
00:55:03 --> 00:55:05
I don't have to do
this, but I will.
890
00:55:05 --> 00:55:13
So as I add I get zeroes and
then it's one, and then
891
00:55:13 --> 00:55:15
I add on one more one.
892
00:55:15 --> 00:55:16
Look.
893
00:55:16 --> 00:55:18
You see what's happening.
894
00:55:18 --> 00:55:21
I right along at zero but I'm
going to look at the book
895
00:55:21 --> 00:55:28
to see whether that jump
should come here or here.
896
00:55:28 --> 00:55:32
So I've got a little bit of
this to finish next time
897
00:55:32 --> 00:55:35
and I'm open for any
questions this afternoon.
898
00:55:35 --> 00:55:38
Okay, thanks and sorry
to keep you late.