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Mechanical
Vibrations
Singiresu S. Rao
SI Edition
Free Vibration of
SDOF
© 2005 Pearson Education South Asia Pte Ltd.2
2.1 Introduction
2.2 Free Vibration of an Undamped
Translational System
2.3 Free Vibration of an Undamped
Torsional System
2.4 Stability Conditions
2.5 Free Vibration with Viscous Damping
Chapter Outline
© 2005 Pearson Education South Asia Pte Ltd.3
2.1 Introduction
• Free Vibration occurs when a system oscillates
only under an initial disturbance with no external
forces acting after the initial disturbance
• Undamped vibrations result when amplitude of
motion remains constant with time (e.g. in a
vacuum)
• Damped vibrations occur when the amplitude of
free vibration diminishes gradually overtime, due
to resistance offered by the surrounding medium
(e.g. air)
© 2005 Pearson Education South Asia Pte Ltd.4
2.1 Introduction
• Several mechanical and structural
systems can be idealized as single
degree of freedom systems, for example,
the mass and stiffness of a system
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Equation of motion
• Newton’s second law
• Principle of Conservation of
Energy
• Rayleigh method
• Lagrange equation
• D’Alembert principle
5
© 2005 Pearson Education South Asia Pte Ltd.6
If mass m is displaced a distance when acted
upon by a resultant force in the same direction,
Free Vibration of an Undamped Translational
System
)(tx
)(tF
dt
txdm
dt
dtF
)()(
If mass m is constant, this equation reduces to
2
2
( )( )
d x tF t m mx
dt
where is the acceleration of the mass2
2 )(
dt
txdx
© 2005 Pearson Education South Asia Pte Ltd.7
Free Vibration of an Undamped Rotational
System
For a rigid body undergoing rotational motion,
Newton’s Law gives
( )M t J
where is the resultant moment acting on the
body and and are the resulting
angular displacement and angular acceleration,
respectively.
M
22 /)( dttd
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Newton’s second law: SDOF
8
( )F t mxUndamped Translational System
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Newton’s second law: SDOF
9
( )M t JUndamped Rotational System
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Newton’s second law: SDOF
10
( )F t mx
kx mx
0mx kx
Equation of motion(free vibration of SDOF)
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Undamped free vibration of SDOF
11
Laplace transform
Natural frequency of n k m
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Undamped free vibration of SDOF
12
where
Inverse Laplace, Solution of the governing equation
or
(1)
(2)
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Undamped free vibration of SDOF
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A system is said to be conservative if no energy is
lost due to friction or energy-dissipating nonelastic
members.
Principle of Conservation of Energy
If no work is done on the conservative system by
external forces, the total energy of the system
remains constant. Thus the principle of
conservation of energy can be expressed as:
constant
( ) 0
T U
dT U
dt
or
© 2005 Pearson Education South Asia Pte Ltd.15
Principle of Conservation of Energy:
SDOF
The kinetic and potential energies are given by:
21
2T mx and 21
2U kx
2 2
constant
1 1 constant
2 2
T U
mx kx
( ) 0
1 12 2 0
2 2
dT U
dt
m x x k x x
0mx kx
Equation of motion(free vibration of SDOF)
© 2005 Pearson Education South Asia Pte Ltd.16
Free Vibration of an Undamped Torsinal
System
Torsional Spring Constant:4
32t
Gdk
l
( )M t J
For a rigid body undergoing
rotational motion, Newton’s Law gives
tk J
0tJ k
Equation of motion(free vibration of SDOF)
© 2005 Pearson Education South Asia Pte Ltd.17
Example 2.8
Effect of Mass on ωn of a Spring
Determine the effect of the mass of the spring on
the natural frequency of the spring-mass system
shown in the figure below.
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Example 2.8 Solution
(E.1)2
12
l
xydy
l
mdT s
s
The kinetic energy of the spring element of length
dy is
where ms is the mass of the spring. The total
kinetic energy of the system can be expressed as
2 22
20
2 2
kinetic energy of mass ( ) kinetic energy of spring ( )
1 1
2 2
1 1(E.2)
2 2 3
m s
ls
y
s
T T T
m y xmx dy
l l
mmx x
© 2005 Pearson Education South Asia Pte Ltd.19
Example 2.8 Solution
(E.3)2
1 2kxU
The total potential energy of the system is given by
2 2 2
constant
1 1 1 constant
2 2 3 2
s
T U
mmx x kx
( ) 0
03
03
s
s
dT U
dt
mmx x kx
mm x kx
© 2005 Pearson Education South Asia Pte Ltd.20
Example 2.8 Solution
we obtain the expression for the natural
frequency:
(E.7)
3
2/1
s
n mm
k
Thus the effect of the mass of spring can be
accounted for by adding one-third of its mass to
the main mass.
© 2005 Pearson Education South Asia Pte Ltd.21
where c = damping constant
Free Vibration with Viscous Damping: SDOF
• Damping force: F cx
Newton’s law yields the equation of motion:
0
mx cx kx
mx cx kx
or
2
2
0
4
2
ms cs k
c c mks
m
Characteristic equation
© 2005 Pearson Education South Asia Pte Ltd.22
Free Vibration with Viscous Damping: SDOF
• Critical Damping Constant and Damping Ratio:
2 4 0
2 2 2c n
c mk
kc mk m m
m
The critical damping cc is defined as the value of
the damping constant c for which the radical in
becomes zero:
or
The damping ratio ζ is defined as:
c
c
c
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Case1.
Free Vibration with Viscous Damping: SDOF
(0 1 or )cc c
Assuming that ζ ≠ 0, consider the following 3 cases:
For this condition, Poles is negative and the roots
are:2
1
2
2
1
1
n n n d
n n n d
s i i
s i i
Damped natural frequency
21d n
1i
© 2005 Pearson Education South Asia Pte Ltd.24
Free Vibration with Viscous Damping: SDOF
where (C’1,C’2), (X,Φ), and (X0, Φ0) are arbitrary
constants to be determined from initial conditions.
1 2
1 2
2
0 0
( )
cos sin
sin
cos 1 (2.70)
n d n d
n
n
n
i t i t
t
d d
t
d
t
n
x t C e C e
e C t C t
Xe t
X e t
and the solution can be written in different forms:
© 2005 Pearson Education South Asia Pte Ltd.25
Free Vibration with Viscous Damping: SDOF
)71.2(1
and 2
00201
n
nxxCxC
0 00
2( ) cos sin
1
nt nd d
n
x xx t e x t t
and hence the solution becomes
For the initial conditions at t = 0,
Eq. describes a damped harmonic motion. Its
amplitude decreases exponentially with time, as
shown in the figure below.
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Free Vibration with Viscous Damping: SDOF
Underdamped Solution
0 00
2( ) cos sin
1
nt nd d
n
x xx t e x t t
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Due to repeated roots, the solution of Eq.(2.59) is
given by
Free Vibration with Viscous Damping: SDOF
1 22
cn
cs s
m
)78.2()()( 21
tnetCCtx
)79.2( and 00201 xxCxC n
In this case, the two roots(Poles) are:
Case2. )/2or or 1( mkmc/cc c
Application of initial conditions gives:
Thus the solution becomes:
)80.2()( 000
t
nnetxxxtx
© 2005 Pearson Education South Asia Pte Ltd.28
Due to repeated roots, the solution of critical
damped system is given by
Free Vibration with Viscous Damping: SDOF
1 22
cn
cs s
m
1 2( ) ( ) ntx t C C t e
1 0 2 0 0and nC x C x x
In this case, the two roots are:
Case2.
( 1 or )cc c
Application of initial conditions gives:
Thus the solution becomes:
0 0 0( ) nt
nx t x x x t e
© 2005 Pearson Education South Asia Pte Ltd.29
In this case, the solution Eq.(2.69) is given by:
The roots are real and distinct and are given by:
Free Vibration with Viscous Damping: SDOF
2
1
2
2
1 0
1 0
n n
n n
s
s
2 21 1
1 2( )n n n nt t
x t C e C e
For the initial conditions at t = 0,
Case3. )/2or or 1( mkmc/cc c
2
0 0
12
2
0 0
12
1
2 1
1
2 1
n
n
n
n
x xC
x xC
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Since , the motion will eventually
diminish to zero, as indicated in the figure below.
Free Vibration with Viscous Damping: SDOF
tetn as 0
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1
2
1
1
0 1 01
2 0 2 0
cos( )
cos( )
n
n
n
n d
n d
t
d
t
d
t
t
X e tx
x X e t
ee
e
Solution of Underdamped free vibration:
© 2005 Pearson Education South Asia Pte Ltd.
32
The logarithmic decrement can be obtained
1
22
2 2ln
21n d n
d
x c
x m
2 22
Hence,
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Home work - 1
33
Determine the differential equation of motion and
the equivalent stiffness for the system (N/m) of Figure
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Home work - 2
34
For free vibration of underdamped system( = 0.8) as shown in Figure.
Determine stiffness of system
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Home work - 3
35
The free response of the undamped system of Figure.
Determine
- Natural frequency, Initial velocity and Initial displacement
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Home work - 4
36
The free response of the underdamped system in Figure.
Calculate the damping ratio, logarithm decrement
© 2005 Pearson Education South Asia Pte Ltd.
Chapter summaries
• Equation of motion for SDOF undamped
system
• Equation of motion for SDOF damped system
37
© 2005 Pearson Education South Asia Pte Ltd.
Chapter summaries
• Undamped free vibration response
• Overdamped free vibration response
• Critically damped free vibration response
38
© 2005 Pearson Education South Asia Pte Ltd.
Chapter summaries
• Underdamped free vibration response
• Logarithm decrement
39