ME 360
Modeling, Analysis and Control of Dynamic Systems
W 2012
Homework Set #2
Due on Thu. 1/19/2012 (Late homework will NOT be accepted)
Please read Chapters 1 and 2 in your textbook (Palm, System Dynamics, 2nd ed.), and please
show your work in solving
ME360 (winter 2015)
HW2 Solutions
Problem 1.
The small angle approximation is: sin . This approximation as the name suggests holds when is
small or close to zero. If we write the Taylor series expansion for sin around 0 we have that
sin = sin 0 + [
(sin )
part d.
The Simulink model:
Note that we cannot use transfer function block in Simulink, because there is no input in this problem.
Instead, we have initial conditions that we should specify in the integrator blocks. In integrator 1, we
insert 0 as initia
Problem 2.
Part 1.
Consider the equation in the form: =
The Simulink model:
The output:
+
.
Part 2.
The transfer function for this system can be found as follows:
Taking the Laplace transform of the both sides of the EOM:
2 () + () + () = () () =
()
1
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Amplitude
This step response
is for a system:
Step Response
2
1.8
1.6
1.4
1.2
a)
b)
c)
d)
1
First order
0
First order with a zero
Second order
Second order with a zero
2
4
6
Time (seconds)
8
$200
Energy is dissipated through:
a) inductors
b) dampers
b=[2 4];
a=[1 4 3 0];
0umerator
0enominator
[r,p,k]=residue(b,a)
r =
-0.3333
-1.0000
1.3333
p =
-3
-1
0
k =
[]
Note that in Matlab the syntax is like this:
() =
()
1
2
3
=
+
+
+ ()
()
1 2 3
So we have
() =
() 1/3
1
4/3
=
+
+
()
+3 +1
which is the same an
ME 360
Modeling, Analysis and Control of Dynamic Systems
WN 2015
Midterm #1 Review - Solutions
Problem 1: (P 4.54 in Palm III)
For the lever system shown in the figure, the position = 0 corresponds to the equilibrium
position when f = 0. The lever has ine
ME 360
Modeling, Analysis and Control of Dynamic Systems
WN 2015
Midterm #1 Review
Problem 1: (P 4.54 in Palm III)
For the lever system shown in the figure, the position = 0 corresponds to the equilibrium
position when f = 0. The lever has inertia I about
ME 360
Modeling, Analysis and Control of Dynamic Systems
WN 2015
Final Exam Review
Final Exam will be held Thursday, April 23rd, 2015 from 4:00-6:00 pm.
It will be held in the auditorium in Chrysler.
The exam is closed book and closed notes, no calculator
Problem 3.
% HW-1 problem 3
% part a
a=3+2j;
b=1-1j;
c1=a+b
c2=a*b
c3=a/b
figure(1)
hold on
plot(real(a),imag(a),'bo'); text(real(a),imag(a)+.5,'a')
plot(real(b),imag(b),'ro'); text(real(b),imag(b)+.5,'b')
plot(real(c1),imag(c1),'gx','MarkerSize',10);
tex
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#2
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
September 15 (Thursday), 2016
September 22 (Thursday), 2016, start of class
1. Solve the following differential
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#10
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
November 17 (Thursday), 2016
November 23 (Wednesday), 2016, midnight, box on front of G034 Autolab
1. The DArs
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#12
University of Michigan, Fall 2015
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
December 6 (Tuesday), 2015
December 13 (Tuesday), 2016, start of class
1. Find the Laplace transform of the fu
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#6
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
1.
October 13 (Thursday), 2016
October 20 (Thursday), 2016, start of class
Consider the function
sin
y(t ) 2
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#11
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
November 29 (Tuesday), 2016
December 6 (Tuesday), 2015, start of class
1. Consider the block diagram on the fi
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#7
University of Michigan, Fall 2015
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
October 20 (Thursday), 2016
October 27 (Thursday), 2016, start of class
1. Consider the system below such that
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#9
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaoche He
Assigned:
Due:
November 10 (Thursday), 2016
November 17 (Thursday), 2016, start of class
1. Consider the electric circuit in t
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#8
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
1.
November 3 (Thursday), 2016
November 10 (Thursday), 2016, start of class
Consider the differential equation
ME360
MODELING, ANALYSIS AND CONTROL OF DYNAMIC SYSTEMS
HW#5
University of Michigan, Fall 2016
Prof Gbor Orosz and Mr Chaozhe He
Assigned:
Due:
October 6 (Thursday), 2016
October 13 (Thursday), 2016, start of class
1. The cylinder on the figure rolls with
Problem 1
(a)
The function is given by
=
1
1
1
1
= (T 4 20 sin ) 3 = 10 sin 2 3 + T
2
10
10
2
(b)
Torque T = cos(3t)
Torque T = us (t)
1.2
1.5
Response
Input T
1
Response
Input T
1
0.8
0.6
0.5
0.4
0
0.2
0
0.5
0.2
0.4
0
2
4
6
8
10
1
0
Time [s]
2
4
6
T
ME 360
Modeling, Analysis and Control of Dynamic Systems
WN 2015
Midterm #2 Review
Problem 1:
For the step response, choose the transfer function that matches it (there should only be one
transfer function per response).
Step Response
Step Response
3
1
1
5
Problem (10 Points)
a) Kinematic diagram (5 points)
Must have links numbered, and letters (or
circled numbers, etc,) on the joints
b) DOF using Grueblers equation (5 points)
F = 3(n "1) " 2 f1 " f 2
n = 10
f1 = 13
f2 = 0
F = 3(10 "1) " 2(13) " 0
F = 27