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 C2 and R2 (since no
current ows into the Vout measureme
nth Order Linear Systems
Do not distribute these notes
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) + + an1 (y (t) + an y (t) =
dtn
dt
dt
n 1
d
d
b1 n1 (u(t) + +
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 assignments (applicable to all homework this semester):
(
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 variable of integration.
Problem 2
˜
The deﬁnition of h is
˜
DiscreteTime Fourier Series
Do not distribute these notes
169
Prof. R.T. MCloskey, UCLA
Discretetime Signals
A continuoustime signal produces a discretetime signal through the process of sampling. The most common
form of sampling takes a snapshot of t
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 = Answer the following: 1. Compute the matrix exponentia
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. All of Chapter 1. This introductory chapter broadly d
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 following sections from the course text: 1. Sections 6.
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 of fundamental importance for linear systems. The
idea
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 the solution to the equations below.
a)
b)
Solve the equa
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 account to login to the computer
Groups should remain the s
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) If the alloy is heated to 350C (660F), how much more le
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 following sections from the course text: 1. All of Chap
MAE107
Fall2014
Homework2
DueApril18
Reading assignment:
Chapter6: Just as a complement to what was given in class
Problem 1: Calculate the particular solution for the equations shown.
a)
b)
You must use the transfer function approach. How does the partic
MAE107 Homework #8 Solution
Prof. MCloskey
Problem 1
1. Compute the Laplace transform of the autocorrelation function,
Z 1 Z 1
L(Ruu ) =
u(t + )u( )d e st dt
1
1
Z 1 Z 1
s(t+ )
=
u(t + )u( )e
dt d
1
1
Z 1 Z 1
s(t+ )
=
u(t + )e
dt u( )es d
1
 1
cfw_z
= u
MAE107 Homework #1 Solution
Prof. MCloskey
1. Let the current through R1 (from left to right) be denoted i, and let the currents through
R2 and C1 (from y to ground) be denoted i1 and i2 , respectively. The following relationships
can be derived by applyi
Pol 1
rb m
e
Problem 2
1.
Time Domain Approach
The impulse response of
y + 2y = 4u
is
h(t) = 4e2t (t),
so the solution to the initial value problem is
t
y(t) = e2t y(0) +
4e2(t ) u( )d,
t 0.
0
Using the periodicity of the solution, we dene a constant to b
clc; clear all; close all;
R = 10000;
L = 3.45e3;
C = 1.05e9;
V = 10;
%RC = 0.1;
%w = 10e3:.1:10e3;
w = 10e4:.5:10e5;
%H = 1./(RC*j*w+1);
H = V./(R+(L*j*w./(1(L*C*w.^2);
figure(1)
loglog(w,abs(H)
title('Amplitude of Frequency Response')
xlabel('Freque
MATLAB CODE:
clc; clear all; close all;
RC = 0.1;
w = 10e3:.1:10e3;
H = 1./(RC*j*w+1);
figure(1)
loglog(w,abs(H)
title('Magnitude of Frequency Response')
xlabel('Frequency, w [rad/s]')
ylabel('H(j*w)')
grid on
axis([10e3 10e3 10e3 10e2])
figure(2)
se
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
will get a sense of what kind of functions possess a Fo
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 = K vin
J
J
(0) = 0
vin (t) = vin,0 , t 0.
Since the non
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 = A2 .
T
0
0
Since we require kukE = 1, then A = 1/ .
2. T
MAE107 Homework#6 Solution
Spring 2011
Problem 1
For the square wave with height 1, onduration 0.5, and rest duration 0.5, the Fourier series
coefficients are
Z
Z 1
jk! 0 t
ck =
u(t)e
dt =
u(t)e jk!0 t dt
T
0
Z 0.5
=
e jk!0 t dt
(!0 = 2)
0
1
=
1 e jk ,
Problem 2
The ODE for the lefthand mass is
2m
x2 = f1 + k(x3 x2 )
2m
x2 + kx2 = f1 + kx3 .
The ODE for the righthand mass is
m
x3 = k (x3 x2 ) kx3
m
x3 + 2kx3 = kx2 .
Assuming solutions of the form x2 (t) = x
2 est , x3 (t) = x
3 est , f1 (t) = f1 est ,
MAE 107
Homework5
Due April 9, 2014
Reading assignment
Read Chapter 16 on Fourier series. Be careful with the notation used.
Problem 1
Consider the circuit shown below.
Assume R1=10000, R2=1000 and C1 =1E5 F. Draw the straight line frequency
response plo
MAE107  Dynamic Systems Laboratory
Spring 2017
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 account to login to the computer
Groups should remain the s
For each lab, you are expected to submit the requested figures and responses to all questions posed
in the report template below. A well composed lab report has (and is graded upon having):
1. All the requested figures, with appropriate axis labels, figur