This document outlines the key topics in Chapter 5 of the textbook for the MCE 440 - Vibrations course taught by Eng. Omar Al-Ostah. The chapter discusses 2 degree of freedom (DOF) systems, including: 1) Developing the equations of motion for a 2 DOF spring-mass-damper system using Newton's second law applied to each mass. 2) Expressing the equations of motion in matrix form using the mass, damping, and stiffness matrices. 3) Analyzing the frequency response and complex frequency response of a damped 2 DOF system under harmonic forcing. 4) Working through an example problem of a plate supporting a pump.

1. MCE 440 – Vibrations
Teacher/Examiner: Eng. Omar Al-Ostah
Email: o.alostah@aiu.edu.kw
Office: A-S-332, Building A, South Wing, Third Floor

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Ch5: 2 DOF systems
Lecture No.X
School of Engineering
and Computing
Chapter Outline
5.1 Introduction
5.2 Equation of Motion for Forced Vibration
5.3 Free-Vibration Analysis of an Undamped System
5.4 Torsional System
5.5 Coordinate Coupling and Principal Coordinates
5.6 Forced-Vibration Analysis
5.7 Semidefinite Systems
5.8 Self-Excitation and Stability Analysis

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Lecture No.X
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and Computing Ch5: 2 DOF systems
5.1 Introduction
 The general rule for the computation of the number of degrees of
freedom can be stated as follows:
 2 DOF System require two independent coordinates to describe
their motion;
Two masses in the system X two possible types of motion of each mass

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.1 Introduction
 There are two equations of motion for a 2DOF system, one for each
mass (more precisely, for each DOF).
 They are generally in the form of couple differential equation that is,
each equation involves all the coordinates.

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.2 Equation of Motion for Forced Vibration
• Consider a viscously damped two degree of freedom spring-mass
system, shown in Fig.5.5.
Figure 5.5: A two degree of freedom spring-mass-damper system

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.2 Equation of Motion for Forced Vibration
 The application of Newton’s second law of motion to each of the
masses gives the equations of motion:
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2
.
5
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1
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5
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2
2
3
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2
1
2
2
1
2
1
2
2
1
2
1
1
1
F
x
k
k
x
k
x
c
c
x
c
x
m
F
x
k
x
k
k
x
c
x
c
c
x
m











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3
.
5
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[ t
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x
k
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c
t
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m


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

 Both equations can be written in matrix form as
where [m], [c], and [k] are called the mass, damping, and stiffness
matrices, respectively, and are given by

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.2 Equation of Motion for Forced Vibration

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.2 Equation of Motion for Forced Vibration

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
5.2 Equation of Motion for Forced Vibration

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Lecture No.X
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and Computing Ch5: 2 DOF systems

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and Computing Ch5: 2 DOF systems

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and Computing Ch5: 2 DOF systems
3.5 Response of a Damped System Under
• Frequency Response
The complex frequency response is given by:

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
Example 3.1: Plate Supporting A Pump

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Lecture No.X
School of Engineering
and Computing Ch5: 2 DOF systems
Example 3.1: Solution

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School of Engineering
and Computing Ch5: 2 DOF systems
Problems on section 3.3 response of an undamped system under harmonic force

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and Computing
END
Ch5: 2 DOF systems