MAE107 Introduction to Modeling and Analysis of Dynamic Systems
Prof. R.T. MCloskey, Mechanical and Aerospace Engineering, UCLA
Subject outline:
1. Intro to rst order ODEs: ode, initial value problems, stability
2. Intro to second order ODEs: same subject

MAE107
Prof. MCloskey
Problem 1
The approximate Fourier transform of the normalized impulse that was used as the input in Lab #2
is shown below:
Magnitude of Fourier transform of unit impulse
1
Magnitude (Voltsec)
10
0
10
1
10
2
10
1
10
0
1
10
10
2
10
3
1

MAE107
Prof. MCloskey
Problem 1
1. The Fourier series coefficients for the periodic pulse train are computed from
Z
ck =
(0
=
1 jk0 t
e
dt
1
jk0
ejk0 1
1
k 6= 0
k=0
where 0 = 2/T = 4.
2. Using the power series definition of the exponential function,
ejk0

MAE107
Prof. MCloskey
Problem 1
Here is the Matlab code to perform the moving average smoothing of the impulse response data:
t = imp1.X.Data;
ts = 0.0001;
uin = imp1.Y(1).Data - mean(imp1.Y(1).Data(1:1/ts);
a1 = sum(uin)*ts;
lp = imp1.Y(3).Data - mean(im

MAE107
Prof. MCloskey
Problem 1
The circuit model for the first stage is simply the familiar low-pass filter with ODE: R1 C2 v a + va =
v1 , where v1 is the input voltage and va is the output voltage that appears at the input of the
first isolation amplif

MAE107
Prof. MCloskey
,
Problem 1
1. h1 = e1 t (t) and h2 = b1 (t) + (b2 + b1 2 )e2 t (t)
2. The systems impulse response is the convolution of h1 and h2 (order doesnt matter for
scalar-valued functions),
Z
h(t) =
h2 (t )h1 ( )d
Z
=
b1 (t ) + (b2 + b1 2

MAE107
Prof. MCloskey
Problem 1
1. The convolution expression for y:
Z
h(t )u( )d
Z
h(t )u( )d
= |y(t)| =
Z
|h(t )u( )| d
Z
=
|h(t )| |u( )| d
Z
|h(t )| md
Z
|h( )| d
=m
y(t) =
= mAh
2. The result from Part 1 shows that y is well-behaved as a func

MAE107
Prof. MCloskey
Problem 1
Without the feedback of y through the gain a, the value of y is b(t) for all t. Note that an
impulse appears in y but note that y(0+ ) = b(0+ ) = 0. Now, if the feedback path is included,
then the impulse remains in y, howe

MAE107
Prof. MCloskey
Problem 1
1. Let the current through C1 be denoted i1 and the current through 9R be denoted i2 . The
following relations are determined,
i1 = C1 (u y)
(u y) = 9Ri2
y = R(i1 + i2 )
Substituting the first two relations into the third y

nth Order Linear Systems
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126
Prof. R.T. MCloskey, UCLA
Introduction to nth order ODEs
Consider the nth order linear time-invariant ODE:
dn1
d
dn
(y(t)+a1 n1 (y(t) + + an1 (y(t) + an y(t) =
dtn
dt
dt
n1
d
d
b1 n1 (u(t) + + bn1

The Laplace Transform
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239
Prof. R.T. MCloskey, UCLA
The Bilateral Laplace Transform
The Laplace transform comes in two avors. The most common version is the unilateral Laplace transform,
denoted L , which operates on function

Revisiting the Frequency Response Function
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148
Prof. R.T. MCloskey, UCLA
General perspective of a systems frequency response
The idea behind the frequency response function for a linear system is simple: a sinusoidal input ev

The Fourier Transform
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209
Prof. R.T. MCloskey, UCLA
Background
The Fourier transform generalizes Fourier series to functions which are not periodic but still dened on the interval
(, ). The Fourier analysis of periodic functi

Fourier Series
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177
Prof. R.T. MCloskey, UCLA
Periodic functions and Fourier Series
Background
Fourier series are named after Joseph Fourier, a 18th century French mathematician who studied problems in
heat transfer and mechan

Convolution
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57
Prof. R.T. MCloskey, UCLA
Convolution derivation from linearity and time-invariance
Analysis of the rst order ODE y = ay + bu naturally led to the convolution derivation of the zero-state response. In
fact, we

Second Order Systems
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91
Prof. R.T. MCloskey, UCLA
Second Order Linear Differential Equations
The second order, linear, time-invariant ODE is given by:
y + a1 y + a2 y = b1 u + b2 u,
(31)
where
y=
t=
u=
a1 , a2 , b1 , b2 =
dep

Solution
Prof. MCloskey
Problem 1
1. u = et (t), R:
Z
Laplace transform:
u(t)est dt
u
(s) =
Z
=
et est dt
0
=
1
s
for Re(s) >
2. u = et (t), R:
Z
Laplace transform:
u
(s) =
0
Z
=
u(t)est dt
et est dt
=
1
for Re(s) <
s
3. u = et , R. The Laplace transfor