Preparation for Chapter
4 and Chapter 5
University of Maryland
B. Balachandran & E. Magrab
System Response
x t
k
m
f t
c
f t
m, k , c
x t
Input
System
Output
University of Maryland
B. Balachandran & E. Magrab
mx cx k x f (t)
Free Vibration (Chapter 4)

ENME 361 Fall 2016 Midterm 1 practice problems
Problem 1: A pendulum with a point mass m is attached to a rolling disc at its center of mass
(point O) only, see the figure below. The disc rolls without slipping.
i)
ii)
iii)
iv)
Determine the degrees of fr

3. A heavy machine, weighing 3000 N, is supported on a resilient foundation. The static
deection of the foundation due to the weight of the machine is found to be 7.5 cm. It is
observed that the machine vibrates with an amplitude of 1 cm when the base of

4. An important consideration in the construction of manufacturing facilities is the magnitude of
the vibratory forces transmitted between adjacent pieces of equipment. Oen times large
pieces of equipment transmit base
excitations to smaller pieces of equ

I. As shown in Fig. 1, a displacement 52(3) is applied to the end of the spring with stiffness k2.
Consider the acceleration 90) of the control tab as the output, the velocity t) as the input.
Assume all the initial conditions are zero and the direction o

2. Response to periodic excitations
A single degree-of-freedom system is driven by the periodic
saw-tooth wave excitation shown in the gure below. If
T = 2 s and PE, =10 N, nd the steady-state response of the
system by considering the rst two harmonics of

4. Determine the response of a vibratory system governed by the following equation.
560) + 0.23%(1) + 3.5x(t) : 1.5 sin(a)t)u(t) + 46(t 3)
Assume that m : 1 kg, the initial conditions are x(0) : 0.1 m and 5c(0) : 0 m/s, and the excitation
frequency a) = 1

3. For the system shown in Fig. 2, x and y denote, respectively, the absolute displacements of
the mass m and the end Q of the dashpot c;
(a) Derive the equation of motion of the mass m,
(5) nd the steady-state displacement of the mass m, and
(6) nd the f

2. The quarter car model shown in Fig. 2 weighs 4448 N. The spring constant of its
suspension is k = 35,024 N/m and the damping coefcient due to its shock absorbers is c
22,919 Ns/m. The road surface parameters are h = 5.1 mm and A: 2.44 m. The cars
horiz

1. For the SD01 system of Fig. 1, given mzlu kg, k : 4UUU N/m, and c 2200 IVs/m, calculate:
(a) The undamped natural frequency.
(b) The damped natural frequency.
(0) The peak frequency (where the amplitude of the response is maximum).
(1) The steady-state

ENME 350: HW19
Due November 9, 2016 by 8:00 AM in class
Fall, 2016
Please put your name, section number, and 3-digit ID number in the upper right corner.
1. An FET transistor has Vto=1V. Plot how the drain-source current iD changes between
0<VDS<4V for VG

280
CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
#
response of the mass m. Consider the base velocity y
$
as the input, the acceleration response x 1t2 as the output, and determine the frequency-response function of
this sy

Exercises
5.20 The area of the hysteresis loop of a cyclically
2
loaded system, which is the energy dissipated per
forcing cycle, is measured to be 10 N # m, and the
measured maximum response Xo of the deection is
2 cm. Calculate the equivalent viscous da

ENME 361 - Spring 2016: Vibrations, Controls, & Optimization I
ENME 361
th
Homework #1
(Due Thursday, Sep 8 2016)
1.
Figure 1
Figure1
Also determine the kinetic energy of the system. (20 pts.)
2. Consider the pendulum of mass m attached to a moving pivot

7
Nov. 10th 2016)
1. For the SDOF system of Fig. 1, given m=10 kg, k = 4000 N/m, and c =200 Ns/m, calculate:
(a) The undamped natural frequency.
(b) The damped natural frequency.
(c) The peak frequency (where the amplitude of the response is maximum).
(d)

ENME 361 Fall Spring 2016: Vibrations, Controls, & Optimization I
ENME361
th
Homework #2
(Due Thursday, September 15 2016)
1. Consider the crank-mechanism system shown in
Figure 1. Determine the rotary inertia of this
system about the point O and express

2/1/2016
Chapter 1 - Preliminaries from Dynamics
ENME361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 1
Preliminaries from Dynamics
Assignment 1 (due on 2/9/16)
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME 361, Spring 2

Midterm 2 Review
ENME 361
Vibrations, Control, and Optimization I
Spring 2016
Exam 2 Review
Prof. Miao Yu
Department of Mechanical Engineering
ENME361: Midterm II Review
We have covered the following sections in the textbook:
Single Degree-of-Freedom Sys

3/7/2016
Chapter 4 Free Response Characteristics
ENME 361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 4
Free Response Characteristics
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME 361 Spring 2016
1
Chapter 4 Free Respon

Chapter 5 Periodic Excitations
ENME 361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 5
Periodic Excitations
Reading assignment: Chapter 4 & 5 and Appendix A and D
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME 361 Spring

2/3/2016
Chapter 2 Modeling of Vibratory Systems
ENME361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 2
Modeling of Vibratory Systems
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME 361, Spring 2016
1
Chapter 2 Modeling of

ENME 361
VIBRATION, CONTROLS AND OPTIMIZATION I
(Spring 2016)
Acknowledgment: Professor Balachandran (UMD) and Professor Magrabs
Instructors and TA Contact Information
Instructor (0101&FR01):
Telephone:
E-Mail:
Office:
Office Hours:
TA:
E-Mail:
Office:
Of

Chapter 5 Periodic Excitations
ENME 361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 5
Periodic Excitations
Reading assignment: Chapter 4 & 5 and Appendix A and D
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME 361 Spring

4/19/2016
Chapter 7 Multi-degree-of-freedom Systems
ENME361
Vibrations, Control, and Optimization I
Spring 2016
Chapter 7
Multi-degree-of-freedom Systems
Reading assignment: Appendix E
Acknowledgement: Professors B. Balachandran and E. B. Magrab
ENME361 S

TABLE 2.2 Mass moments of inertia about axis z normal
to the x-y plane and passing through the center of mass.
L
Slender bar
L/2
z
JG
1
mL2
12
JG
1
mR 2
2
JG
2
mR 2
5
R
Circular disk
z
R
Sphere
z
TABLE 2.3 Spring constants for some common elastic eleme

ENME 361 Fall 2016 Midterm 1 practice problems
Problem 1: A pendulum with a point mass m is attached to a rolling disc at its center of mass
(point O) only, see the figure below. The disc rolls without slipping.
i)
ii)
iii)
iv)
Determine the degrees of fr

CHAPTER 5 Single Degree-of-Freedom Systems Subjected to Periodic Excitations
282
a) Expand Eq. (a) in a Fourier series and show that
1.5
1
tp 2p/v b
a0 0
f(t)/Fo
0.5
ai
0
1
T Mtp 2p/vo
tb Ntp
0
t
(a)
Amplitude
1
0.5
0
0.5
1
f(t)/F
x(t)/(F/k)
0
50
100
t

ENME361
Homework #5
th
(Due Thursday, October 20 2016)
1. For a uniform bar with a mass of m and a length of L. Considering small angular oscillations,
for what torsion stiffness is the system shown below unstable ?
g
O
2. Consider an electric motor driv

ENME361
th
Homework #6
(Due Tuesday, Nov 1 2016)
1.
As shown in Fig. 1, a displacement y ( t ) is applied to the end of the spring with stiffness k2 .
Consider the acceleration !(t) of the control tab as the output, the velocity y! (t) as the input.
Assum

ENME 361 Fall 2016 Midterm 2 practice problems
Problem 1: A 150 kg mill is suspended by a spring-damper combination with a stiffness of 30
103 N/m and a viscous-damping constant of 1500 Ns/m. The mill is initially at rest.
i)
If the mill is excited by a

ENME 361 Fall 2016: Vibrations, Controls, & Optimization I
ENME361
Homework #3
th
(Due Tuesday, Sept 27 2016)
1. Determine the equation governing the system studied in Example 3.15 (see Fig. 1) by carrying
out a force/moment balance.
x2
c
Jo, m2 o L1
kt
L