1.
CFN (Textbook) Problem 8.10.
2.
For each wheel: M
3600
900lb
4
From the problem:
Mp=0.35
ts=3
Therefore:
M p 0.35 exp
1 2
ln(0.35)
1.05
1 2
1 2
1.1025(1 2 ) 2 2 10.97 2 1.1025
0.32
Now,
ts 3
n
4.6
4.6
n 0.32n
4.6
4.6
n
3 0.32
0.96
n
1.
CFN (Textbook) Problem 8.49.
ME372
HW#10
2.
Draw block diagrams for the following models:
a) 2 6 x 12 x f (t )
x
b) 5x 2 x f (t )
x
a)
0.5 f (t ) 6 x 12 x
x
f(t)
x
+
1
s
0.5


x
x
1
s
6
12
b)
f (t ) 5x 2 x
x
f(t)
+


1
x
s
x
1
s
x
1
s
x
5
2
3.
C
ME 372  Spring 2013
HW #7
03/07/13
For your quiz on Monday, March 11, you will need to draw the Bode plots
of one firstorder and one secondorder system, each selected from the
problems listed below.
A generic firstorder system with a timeconstant
has
ME 372 Spring 2013
HW #1
01/09/13
You will be quizzed on one of the following problems at the beginning of
class on Monday, 01/14/13. On the quiz, it is not sufficient to know
the answer. You must show ALL steps necessary to obtain the
answer.
For each o
ME 372  Spring 2013
HW #8
03/21/13
For your quiz on Friday, March 22, you will need to develop the equations
of motion for one of the systems shown below. You must also put the EOM
in statespace form, as indicated.
x1
K1
M1
no slip
r
J
inextensible cabl
ME 372  Spring 2013
HW #3
01/24/13
You will be quizzed on one of the following problems on Wednesday,
January 30, 2013.
1.
Given the system described by the following equation:
4 y 2 y 5 t 8,
where y 0 4 .
DERIVE the complete expression for y t . (That
ME 372 Spring 2013
HW #2
01/16/13
You will be quizzed on one of the following problems at the beginning of
class on Wednesday, 01/23/13. On the quiz, it is not sufficient to
know the answer. You must show ALL steps necessary to obtain the
answer.
For each
ME 372  Spring 2013
HW #4
02/2/13
You will be quizzed on one of the following problems on Friday, February 8,
2013.
In the following problems, you will be examining massspringdampers system acted on by a
force, as shown in the figure below. For each p
ME 372  Spring 2013
HW #5
02/20/13
You will be quizzed on two of the following problems on Monday, February
25, 2013; one where you must find X(s) for a given x(t) and one where
you must x(t) for a given X(s).
1.
Given
xt
sin
10 4t
sin
Given x t
Answer:
ME372 Dynamic Systems
Solutions to Homework # 3
Part a:
x2
x1
B1 x1
M1
K1 x1
B2 x2 x1
K 2 x2 x1
M2
M 1 1
x
For M1 :
For M2 :
f
f
0 M 2 2 B2 x2 x1 K 2 x2 x1 f a 0
x
M11 B1 B2 x1 K1 K 2 x1 B2 x2 K 2 x2
x
M 2 2 B2 x2 K 2 x2 f a B2 x1 K 2 x1
x
State space
ME372 Dynamic Systems
Solutions to Homework # 4
Problem 1:
CFN 5.6
a)
1 , 1
2 , 2
a t
K 22
B1 2
J2
K11
J1
J 2 2
J11
Figure 1. Free Body Diagram
a t B1 2 J11 K11 (1)
J 2 2 k 2 2 B1 2 (2)
Put (2) into (1) and rewrite all the equations:
J11 B1 K11 a t B
CHAPTER1 '
INTRODUCTION
In this chapter we present the rationale for the book, deﬁne several terms
that will be used throughout, and describe various types of systems. The
chapter concludes with a description of the particular types of systems to be
consi
Example 1
x1
K1
M1
B1
x2
K2
fa(t)
M2
x1
K1
M1
B2
Assume x1 and x2 measured from unstretched
spring position.
ME 372 Dynamic Systems
Replace the film damper with a discrete damper.
Example 1: Spring Forces
fK1
fa(t)
M2
fK1
K1
Stretch in K1 =
ME 372 Dynamic
System Order
Solving for the System Response
So far, we have focused on solving for the
equations of motion (EOM) for translational
systems.
Now we will use the EOM to help us _
for the system responses.
The order of a system is the minimum number of
_ r
Elements of Translational Systems
Mass: stores _ energy
 variable = velocity
Spring: stores _ energy
 variable = displacement
Damper: _ energy
 variable = velocity
Element Law: Mass
Three quantities of interest:
x=
v=
a=
x, v, a
M
f
K
B
K
M
Assuming
Element Law: Viscous Damping
The force required to
pull the dashpot is
proportional to its rate
of change of length
Element Law: Viscous Friction
friction always
opposes motion!
Dashpot/damper:
v1
B
fB
v2
L
fB
ME 372 Dynamic Systems
ME 372 Dynamic Syste
The University of Alabama
Department of Mechanical Engineering
75 Minutes
ME 372 test#1 Fall 2010 (9/30) S.N. Mahmoodi
Your Full Name:
Your Section:
MASTER
Exam Rules:
1. Closed book and notes (use only the provided formula sheet).
2. Do not use any paper
The University of Alabama
Department of Mechanical Engineering
50 Minutes
ME 372 test#1 Spring 2011 (2/25) S.N. Mahmoodi
Your Full Name:
Your Section:
MASTER
Exam Rules:
1. Closed book and notes (use only the provided formula sheet).
2. Do not use any pap
Revised Example 1
Example 1: Introduction to Relative Coordinates
x2
x1
M1
K
M2
B2
B1
fa(t)
x2
x1
B2
M2
K
M1
B0
fa(t)
B1
B0
x1 and x2 measured from equilibrium positions.
x2 is measured from fixed point on M1!
x1 and x2 measured from equilibrium positions
ME372 Modeling & Analysis of Dynamic Systems
Solutions to Homework # 5
Problem 1:
CFN (Textbook) Problem 6.7.
Sol.
The input/output equation can be obtained by applying formulae of opamp and using
KCL and ohms law.
For Node A :
Nodal equations for A are
ME 372  Spring 2013
HW #6
03/06/13
You will be quizzed on one of the following problems on Friday, March 8,
2013.
In the following problems, you will be examining massspringdampers system acted on by
multiple forces, as shown in the figure below. For e
ME 372  Spring 2013
HW #9
04/11/13
For your quiz on Friday, April 12, you will need to develop the equations of
motion for one of the systems shown below. You must also put the EOM in
statespace form, as indicated.
C1
L1
ein
1.
diL1
deC1
eout
R2
R1
R2
d
ME 372: Dynamic Systems
Independent Energy Elements
Typically, the number of states required to
define a system is equal to the number of
_ energy storage elements.
Topic 4: Modeling of Electrical
Systems
Section 2: General Procedure
Independence Example
1/8/2016
ME 372: Dynamic Systems
Topic 7: Transfer Function
Analysis
Frequency Response
The steadystate response of a system
to a sinusoidal input is called the
_.
Can be found by 3 ways:
1) Simulate several (all?) possible sinusoidal inputs
Section 4:
1/8/2016
Mass Spring  Damper
ME 372: Dynamic Systems
fa(t)
Topic 7: Transfer Function
Analysis
x
M
Section 1: Transfer Functions
K
MSD (cont.)
MSD (cont.)
Rewrite as a secondorder system:
Mx B x K x f a t
L"
Assumptions:
x=0 at static equilibrium
x(0)
1/8/2016
ME 372: Dynamic Systems
Matrix Form of SV Equations
For linear, time invariant systems, the state
equations can be written
qnx1 = Anxn qnx1 + Bnxm umx1
ynx1 = Cnxn qnx1 + Dnxm umx1
Topic 2: State Space Equations
where
Section 2: StateSpace Form
Using MATLAB Simulink to solve a second order differential equation
Let's consider a massspring damper system, the equation of motion for this system is:
Mx Bx Kx f (t )
(1)
Note that Simulink is a numerical software and cannot solve a symbolic equation.
Lever
ME 372: Dynamic Systems
q
Assume small motions
such that:
Topic 3: Modeling of Rotational
Mechanical Systems
a
sinq
q in _
cosq
Section 2: Linking Mechanisms
b
Lever: Kinematics
With small motion
assumption :
x2
x1
x1
Lever: Loads
q
q
x1
a
:
x
1/8/2016
ME 372: Dynamic Systems
Translational vs. Rotational Systems
Mass, M
Mass Moment of Inertia, J
Topic 3: Modeling of Rotational
Mechanical Systems
Spring, K
Torsional Spring, K
Section 1: Elements
Damper, B
Torsional Damper, B
stores _ ener
ME 372: Dynamic Systems
Topic 4: Modeling of Electrical
Systems
Section 1: Elements
Electrical Systems
Electrical variables:
voltage, e
current, i
Mechanical variables:
force
velocity & displacement
Electrical elements:
inductance, L
capacitance, C
1/8/2016
ME 372: Dynamic Systems
Equilibrium Point
Topic 1: Modeling of Translational
Mechanical Systems
Equilibrium Point: A state or position of a
system in which the system remain steady
in the absence of external effects for all
time.
Section 3: Syste