1. Free vibration with damping
Natural frequency =
Damped frequency = 1 2
Damping ratio =
=
2
2
Determine amplitude and phase angle from initial conditions
=
0 + 0
0 2 +
2
= 1
The response is = cos +
2. Forced vibration single harmonic driving

MEEM 3750. System Dynamics
ELECTRICAL DYNAMIC SYSTEMS
A. Gauchia (Sources: Palm System Dynamics; G. Parker slides)
Page 1 of 13
MEEM 3750. System Dynamics
A. Gauchia (Sources: Palm System Dynamics; G. Parker slides)
Page 2 of 13
MEEM 3750. System Dynamics

MEEM 3750. System Dynamics
DC GAIN, FIRST AND SECOND ORDER SYSTEMS
A. Gauchia (Partially based on G. Parker slides)
Page 1 of 12
MEEM 3750. System Dynamics
A. Gauchia (Partially based on G. Parker slides)
Page 2 of 12
MEEM 3750. System Dynamics
SECOND ORD

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u, c

MEEM 3750 Lecture 2
Linearization and
Function Estimation
Jim De Clerck
906 MEEM
jdeclerck@mtu.edu
Mechanical Engineering-Engineering Mechanics
Michigan Technological University
Copyright 2016
Lecture 2 Linearization and Function Estimation
MEEM3750
1
Lea

MEEM 3750. System Dynamics
NUMERICAL INTEGRATION
A. Gauchia (Partially based on G. Parker slides)
Page 1 of 3
MEEM 3750. System Dynamics
A. Gauchia (Partially based on G. Parker slides)
Page 2 of 3
MEEM 3750. System Dynamics
A. Gauchia (Partially based on

MEEM 3700
Mechanical Vibrations
Mohan D. Rao
Chuck Van Karsen
Mechanical Engineering-Engineering Mechanics
Michigan Technological University
Copyright 2003
MEEM 3700
1
Lecture 6-SDOF Damped
Vibration
Single Degree of Freedom Damped Free Vibration
M
K
C
K

MEEM3700
Summer 2012
25 points
HW Assignment Number 2 Due Tuesday, May 28, 2013.
1) A bungee cord of length 50 meters and stiffness 1800 N/m
is used by a jumper whose mass id 65 kg. Determine the
equation of motion (and initial conditions) for the person

MEEM3700
Summer 2013
25 points
HW Assignment Number 6 Due Thursday, June 20, 2013.
1. For the system below, the disk is held in the equilibrium position while the mass is
displaced 2 mm. The system is then released from rest.
a) Determine the natural freq

MEEM 3750. System Dynamics
VIBRATIONS AND SPRINGS
A. Gauchia (Sources: Palm System Dynamics; G. Parker slides)
Page 1 of 18
MEEM 3750. System Dynamics
A. Gauchia (Sources: Palm System Dynamics; G. Parker slides)
Page 2 of 18
MEEM 3750. System Dynamics
A.

MEEM 3750. System Dynamics
FLUID DYNAMIC SYSTEMS
A. Gauchia (Sources: Palm System Dynamics; G. Parker slides)
Page 1 of 14
MEEM 3750. System Dynamics
This equation is extremely useful when doing equilibrium. Think of the mass flow
rate as current in elect

Problem 1. Derive the EOM of SDOF system with both FBD and energy method.
Use the generalized coordinate shown. ( is measured from equilibrium position)
FBD:
= 2
Kinematics:
2 =
(1)
=
EOM:
4 + 2 + = 4
2016 Spring F.Long
= (2)
Energy method:
1. Tota

SDOF free vibration
Learning objectives:
1. Number of degrees of freedom of a system;
2. Single degree of freedom (SDOF) system modeling; Derive the equation of
motion using FBD method and energy method.
3. Free vibration of SDOF system without damping;

SDOF free vibration
Learning objectives:
1. Number of degrees of freedom of a system;
2. Single degree of freedom (SDOF) system modeling; Derive the equation of
motion using FBD method and energy method.
3. Free vibration of SDOF system without damping;

ME-EM 3700-Mechanical Vibrations (Spring 2016)
Exam 2
March 18, 2016
Note:
Your work must be organized and neat. To receive full credit, you must show all
pertinent reasoning, FBDs, equations and calculations.
By signing below, I certify this is my own w

Solution to Take Home Quiz Number 2
Consider the following three degree-of-freedom system:
x1
x2
f1(t)
f2(t)
m2=m
m1=2m
k1=3kk
1.
2.
3.
4.
7.
k2=k
x3
m3=2m
k3=k
k4=3k
Draw free-body diagrams.
Write the differential equations of motion.
Put the equations o

Solution to Homework Problem Number 17
Consider the following three degree-of-freedom system:
x1
x2
f1(t)
f2(t)
m2=m
m1=2m
k1=3kk
1.
2.
3.
4.
5.
6.
7.
8.
x3
m3=2m
k2=k
k4=3k
k3=k
Draw free-body diagrams.
Write the differential equations of motion.
Put the

MEEM 3700
Fall 2008
PRATICE Exam
Shapton
This is a closed book, and closed notes exam. In order to earn full credit, you
must:
DRAW FREE BODY DIAGRAMS and other figures as appropriate
Show all steps of your work
Write neatly and legibly
Circle your fi

Fall 08
MEEM 3700
Shapton
Exam 3
Name_SOLUTION_
This is a closed book, and closed notes exam. In order to earn full credit, you
must:
DRAW FREE BODY DIAGRAMS and other figures as appropriate
Show all steps of your work
Write neatly and legibly
Circle

MEEM 3700
Shapton
Fall 2010
Exam 3
Name_
This is a closed book, and closed notes exam. In order to earn full credit, you
must:
DRAW FREE BODY DIAGRAMS and other figures as appropriate
Show all steps of your work
Write neatly and legibly
Circle your fi

MEEM 3700
Practice Exam 2
Name_
This is a closed book, and closed notes exam. In order to earn full credit, you
must:
DRAW FREE BODY DIAGRAMS and other figures as appropriate
Show all steps of your work
Write neatly and legibly
Circle your final answe

MEEM 3700
Shapton
Fall 2010
Test 2
This is a closed book, and closed notes exam. In order to earn full credit, you must:
DRAW FREE BODY DIAGRAMS and other figures as appropriate
Show all steps of your work
Write neatly and legibly
Circle your final an

Fall 2010
MEEM 3700
Exam 1
Problem 1 (33%). Determine the differential equation of motion for the system shown
in terms of the coordinate . Assume the mass is a point mass and assume small
angles. Use Newtons Law.
L
b
a
m
k
c
Show your free body diagram
k

MEEM3700
Summer 2013
25 points
HW Assignment Number 5 Due Monday, June 17, 2013.
1.
A single degree of freedom system with mass m=10 kg, stiffness k=1000
N/m and damping coefficient c=10 N/m/s, is subjected to a periodic series
of force pulses. The force

MEEM3700
Summer 2013
25 points
HW Assignment Number 4 Due Monday, June 10, 2013.
A single degree of freedom system with mass m=10 kg, stiffness k=1000
N/m and damping coefficient c=10 N-sec/m, is subjected to a periodic
series of force pulses. The force p

MEEM3700
Summer 2013
25 points
HW Assignment Number 3 Due Thursday, May 30, 2013.
For story problems, clearly indicate:
Knowns
Unknowns
Assumptions
Analysis (must include free body diagrams)
Results (must include units)
Interpretation or verification of r