Chapter 7. Free and Forced Response of Single-Degree-of-Freedom Linear Systems

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Chapter 7. Free and Forced Response of
Single-Degree-of-Freedom Linear Systems
7.1 Introduction

• Vibration System oscillates about a certain
  equilibrium position.
• Mathematical models (1) Discrete-parameter
  systems, or lumped systems.
• (2)
  Distributed-parameter systems, or continuous
  systems.
• Usually a discrete system is a
  simplification of a continuous system through a
  suitable lumping modelling.
• Importance performance, strength, resonance,
  risk analysis, wide engineering applications

Single-Degree-of-Freedom (Single DOF) linear
system

• Degree of freedom the number of independent
  coordinates required to describe a system
  completely.
• Single DOF linear system
• Two DOF linear system

System response

• Defined as the behaviour of a system
  characterized by the motion caused by excitation.
• Free Response The response of the system to the
  initial displacements and velocities.
• Forced Response The response of the system to
  the externally applied forces.

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7.2 Characteristics of Discrete System Components
The elements constituting a discrete mechanical
system are of three types The elements relating
forces to displacements, velocities and
accelerations.

• Spring relates forces to displacements

X1
X2
Fs
Fs
Fs
Slope K is the spring stiffness, its unit is N/m.
X2-X1
0
Fs is an elastic force known as restoring force.
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• Damper relates force to velocity

The damper is a viscous damper or a dashpot
Fd
Fd
c
Fd
Slope C is the viscous damping coefficient, its
unit is Ns/m
0
Fd is a damping force that resists an increase in
the relative velocity
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• Discrete Mass relates force to acceleration

m
Fm
Fm
Slope m, its unit is Kg
0
Note 1. Springs and dampers possess no mass
unless otherwise stated 2. Masses are
assumed to behave like rigid bodies
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• Spring Connected in Parallel

k1
x1
x2
Fs
Fs
k2

• Spring Connected in Series

x2
x0
x1
Fs
Fs
k1
k2
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7.3 Differential Equations of Motion for First
Order and Second Order Linear Systems

• A First Order System Spring-damper system

k
x(t)
Free body diagram
Fs(t)
F(t)
F(t)
Fd(t)
c

• A Second Order System Spring-damper-mass system

k
x(t)
Free body diagram
Fs(t)
m
m
F(t)
F(t)
Fd(t)
c
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7.4 Harmonic Oscillator
k
x(t)
Second order system
m
F(t)
Undamped case, c0
(1)
Solution
is called Phase angle
With initial conditions
and
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Period(second)
Natural frequency Hertz(Hz)
Example
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7.5 Free Vibration of Damped Second Order
Systems

• A Second Order System Spring-damper-mass system

k
x(t)
Free body diagram
Fs(t)
m
m
F(t)
F(t)
Fd(t)
c

• Express it in terms of nondimensional
  parameters

(1.7.1)
Viscous damping factor

• The solution of (1.7.1) can be assumed to have
  the form,

We can obtain the characteristic equation
With solution
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(1)
Undamped case, the motion is pure oscillation

s1 ,s2 are complex conjugates.
(2)
Underdamped Case
(3)
Critical damping
(4)
Overdamped case, the motion is aperiodic and
decay exponentially in terms of
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Critical damping
Overdamped case
12
,
where
the frequency of the damped free vibration
Figure 1.7.3

Underdamped Case
as
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7.6 Logarithmic Decrement

• Experimentally determine the damping of a system
  from the decay of the vibration amplitude during
  ONE complete cycle of vibration

Let
, We obtain
Introduce logarithmic decrement
for small damping,

• For any number of complete cycles

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7.7 Energy Method
Total energy of a spring-mass system on a
horizontal plane