nth Order Linear Systems
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63
Prof. R.T. MCloskey, UCLA
Introduction to nth order ODEs
Consider the nth order linear timeinvariant ODE:
dn1
d
dn
(y (t)+a1 n1 (y (t) + + an
MAE107 Homework #1 Solution
Prof. MCloskey
Problem 1
The circuit to analyze is
+
Vin
R1
C2
V
i1
i2
+
R2
C1
Vout
Let i1 be the current through capacitor C1 and let i2 be the current through
DiscreteTime Fourier Series
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169
Prof. R.T. MCloskey, UCLA
Discretetime Signals
A continuoustime signal produces a discretetime signal through the process of sampling.
ME 352  Machine Design I
Name of Student_
Fall Semester 2013
Lab Section Number_
Homework No. 1 (30 points). Due at the beginning of lecture on Wednesday, August 28th.
Important notes for homework as
MAE107 Homework #2 Prof. M'Closkey Due Date
The homework is due by Friday, 5PM, January 23, 2009 to Mr. David Shatto in the 38137 foyer (3rd floor Engineering 4).
Mandatory Reading
Please review the
MAE107 Homework #1
Prof. MCloskey
Due Date
The homework is due at beginning of recitation on 4pm, Friday, January 16, 2016.
Mandatory Reading
Please read the following sections from the course text:
1
MAE107 Homework #6 Prof. MCloskey Due Date
The homework is due at 5PM on Thursday, June 3, 2010, to David Shatto (38138 foyer, Engineering 4).
Problem 1
Consider the following 2 2 matrices, A1 = A2 =
MAE107 Homework #3 Solution
Prof. M’Closkey
Problem 1
Consider,
∞
−∞
Set s = t − τ to get
−∞
∞
h(t − τ )u(τ )dτ.
h(s)u(t − s)(−ds) =
∞
−∞
h(s)u(t − s)ds.
Note that we can use any symbol for the variab
Unit Impulse, Unit Step, and Impulse Response Revisited
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83
Prof. R.T. MCloskey, UCLA
The Unit Impulse Function
The unit impulse function is a signal dened on (, ). It is
MAE107
Spring2014
Homework1
DueApril11/2014
Reading assignment:
Chapter1: Study particularly the dynamic analysis and block diagrams
Chapter2: Section 2.6, Material related to EE100
Problem 1: Find th
MAE107  Dynamic Systems Laboratory
Winter 2014
Lab #1
Lab Rules
No food or drinks in the lab
No ear plugs in or cell phones in sight
No USB memory sticks allowed in the lab, use your SEASnet accou
MAE107 Homework #1 Prof. M'Closkey Due Date
The homework is due by 5PM, Thursday, January 15, 2009 to Mr. David Shatto in the 38137 foyer (3rd floor Engineering 4).
Mandatory Reading
Please read the
9.16 A magnesiumlead alloy of mass 5.5 kg consists of a solid phase that has a composition that is just
slightly below the solubility limit at 200C (390F).
(a) What mass of lead is in the alloy?
(b)
MAE107 Homework #7 Solution
Prof. Luong
Problem 1
1. u = et (t), > 0:
Z
Z
et dt =
u(t)dt =
Absolutely integrable:
0
Z
1
u(t)ejt dt
Fourier transform: u
() =
Z
=
et ejt dt
0
=
1
+ j
2. u = et (t), >
MAE107 Homework #3 Solution
Prof. Luong
Problem 1
1 t < 0
y(t) = 2
0t<1
y(t) = 2
1t<2
y(t) = 2
2t<3
y(t) = 2
3t<4
y(t) = 2
Rt
1 e d
R0
1 e d
R0
1 e d
R0
t3 e d
R1
= 2(et e1 )
+2
+2
Rt
0
R1
+2
t3 e d
0
MAE107 Homework #3
Prof. Luong
Due Date
The homework is due by noon, Monday, October 26, 2015.
Problem 1
Objective: Practice performing a graphical convolution.
4
0.8
3
0.6
u(t)
1
h(t)
5
2
0.4
1
0.2
0
MAE107 Homework #6 Solution
Prof. MCloskey
Problem 1
1. Let A be the height of the pulse with duration . The energy in one period of the signal is
Z
Z 1
Z
2
2
kukE =
u(t) dt =
u(t) dt =
A2 dt = A
MAE 107
Homework4
Due May 2, 2014
Problem 1
For the following equations
1) Find the general expression for the impulse response function. Explain your procedure.
2) Obtain the general solution for u(t
MAE107 Homework #4 Solution
Prof. Luong
Problem 1
1. If u = , then y = h where
h(t) =
b1
b2
2 e2 t 1 e1 t (t) +
e2 t e1 t (t)
2 1
2 1
then y = h,
If u = ,
y(t) = h(t)
=
b1
b1
22 e2 t 21 e1 t (t) +
2
MAE107 Homework #7
Prof. Luong
Due Date
The homework is due 9AM on Friday, December 4, 2015.
Background
In Problem 1 you will build a library of functions and their Fourier transforms. Hopefully, you
MAE107 Homework #2 Solution
Prof. MCloskey
Problem 1
1. The block diagram ODEs and gains are,
vin (t)
K
M (t)
+ bJ = M
J
2. First, solve the IVP:
(t)
1
v1 (t)

RC v out + vout = v1
vout(t)
+ b =
MAE107 Homework #4
Prof. Luong
Due Date
The homework is due Noon, Monday, November 9, 2015, at the beginning of lecture.
Problem 1
Consider the second order differential equation
y + a1 y + a2 y = b1
MAE107 Homework #6
Prof. Luong
Due Date
The homework is due at 12PM on Monday, November 23, 2015, at the beginning of lecture.
Problem 0
Figure 1: taken from System Dynamics by K. Ogata
Simplify this
MAE107 Homework #6 Solution
Prof. Luong
Problem 0
1
Problem 1
1. The frequency response function is
Z
h(t)ejt dt
() =
1
Z
1 ejt dt
=
12
(3
=
2
1
j
ej
e
j 12
=0
6= 0
2. The Fourier coefficients as
MAE 107 Spring 2016 HW 1 Solution
April 8, 2016
Problem 6.16
(4 pts)
y = y + x
a = 1 and b = 1
x(t) = e2t u(t)
y(0) = 2
So
t
y(t) = 2e
t
Z
+
0
e(t ) e2 d = 3et e2t
for t > 0
Problem 6.17
(5 pts)
v3 =
Problem 1
1. Let i1 be the current flowing through the resistor between vref and ve . Let i2 be the current
flowing through the resistor between and ve . Using these definitions we have
ve = R(i1 + i2
MAE107
Prof. MCloskey
Problem 1
1. When drawing the free body diagrams we can assume y2 > y1 > 0 and y 2 > y 1 > 0. Thus,
the springs are stretched relative to their equilibrium lengths and the damper
MAE107
Prof. MCloskey
Problem 1
The circuit model for the first stage is simply the familiar lowpass filter with ODE: R1 C2 v a + va =
v1 , where v1 is the input voltage and va is the output voltage
Prof. MCloskey
Problem 1
R
1. A graph of f is shown below. Since f (t)dt = 1 and is only nonzero in a neighborhood
of t = 0, we can classify f as a unit impulse function assuming is sufficiently smal
MA107 Modeling and Analysis
of Dynamic Systems
Professor TsuChin Tsao
Fall 2017
Lecture 1 Outline
Syllabus & Class Objectives
Preview of past final exams (07, 16) & HW1
Introduction to Dynamic Inv
Lab Report #2
MAE 107L  Dynamic Systems Laboratory
Fall 2017
Objective
This lab session aimed to illustrate the frequency response of a system as a sinusoidal
input was sent to the system. Using simi
Lab Report #3
MAE 107L  Dynamic Systems Laboratory
Winter 2017
Objective
The objective of this lab is to obtain the input and output from the low pass and high pass circuit.
Using these values, the t
Lab Report #7
MAE 107L  Dynamic Systems Laboratory Fall
2017
Matlab Review and Lab Objectives
Recall that we defined the chirp for one period as
u(t) = cos(!(t),
with
!
and where T is the period of t