PPT: Vibrations Mechanical Engineering Notes | EduRev

Theory of Machines (TOM)

Mechanical Engineering : PPT: Vibrations Mechanical Engineering Notes | EduRev

 Page 1


© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Contents
19 - 2
Introduction
Free Vibrations of Particles.  Simple Harmonic Motion
Simple Pendulum (Approximate Solution)
Simple Pendulum (Exact Solution)
Sample Problem 19.1
Free Vibrations of Rigid Bodies
Sample Problem 19.2
Sample Problem 19.3
Principle of Conservation of Energy
Sample Problem 19.4
Forced Vibrations
Sample Problem 19.5
Damped Free Vibrations
Damped Forced Vibrations
Electrical Analogues
Page 2


© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Contents
19 - 2
Introduction
Free Vibrations of Particles.  Simple Harmonic Motion
Simple Pendulum (Approximate Solution)
Simple Pendulum (Exact Solution)
Sample Problem 19.1
Free Vibrations of Rigid Bodies
Sample Problem 19.2
Sample Problem 19.3
Principle of Conservation of Energy
Sample Problem 19.4
Forced Vibrations
Sample Problem 19.5
Damped Free Vibrations
Damped Forced Vibrations
Electrical Analogues
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Introduction
19 - 3
•
Mechanical vibration is the motion of a particle or body which 
oscillates about a position of equilibrium.  Most vibrations in 
machines and structures are undesirable due to increased stresses 
and energy losses.
•
Time interval required for a system to complete a full cycle of the 
motion is the period of the vibration.
•
Number of cycles per unit time defines the frequency of the vibrations.
•
Maximum displacement of the system from the equilibrium position is 
the amplitude of the vibration.
•
When the motion is maintained by the restoring forces only, the 
vibration is described as free vibration.  When a periodic force is 
applied to the system, the motion is described as forced vibration.
•
When the frictional dissipation of energy is neglected, the motion 
is said to be undamped.  Actually, all vibrations are damped to 
some degree.
Page 3


© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Contents
19 - 2
Introduction
Free Vibrations of Particles.  Simple Harmonic Motion
Simple Pendulum (Approximate Solution)
Simple Pendulum (Exact Solution)
Sample Problem 19.1
Free Vibrations of Rigid Bodies
Sample Problem 19.2
Sample Problem 19.3
Principle of Conservation of Energy
Sample Problem 19.4
Forced Vibrations
Sample Problem 19.5
Damped Free Vibrations
Damped Forced Vibrations
Electrical Analogues
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Introduction
19 - 3
•
Mechanical vibration is the motion of a particle or body which 
oscillates about a position of equilibrium.  Most vibrations in 
machines and structures are undesirable due to increased stresses 
and energy losses.
•
Time interval required for a system to complete a full cycle of the 
motion is the period of the vibration.
•
Number of cycles per unit time defines the frequency of the vibrations.
•
Maximum displacement of the system from the equilibrium position is 
the amplitude of the vibration.
•
When the motion is maintained by the restoring forces only, the 
vibration is described as free vibration.  When a periodic force is 
applied to the system, the motion is described as forced vibration.
•
When the frictional dissipation of energy is neglected, the motion 
is said to be undamped.  Actually, all vibrations are damped to 
some degree.
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 4
• If a particle is displaced through a distance x
m
 from its 
equilibrium position and released with no velocity, the 
particle will undergo simple harmonic motion,
( )
0 = +
- = + - = =
kx x m
kx x k W F ma
st
? ?
d
•
General solution is the sum of two particular solutions,
( ) ( ) t C t C
t
m
k
C t
m
k
C x
n n
? ? cos sin
cos sin
2 1
2 1
+ =
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
=
• x is a periodic function and ?
n
 is the natural circular 
frequency of the motion.
• C
1
 and C
2
 are determined by the initial conditions:
( ) ( ) t C t C x
n n
? ? cos sin
2 1
+ =
0 2
x C =
n
v C ?
0 1
= ( ) ( ) t C t C x v
n n n n
? ? ? ? sin cos
2 1
- = = ?
Page 4


© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Contents
19 - 2
Introduction
Free Vibrations of Particles.  Simple Harmonic Motion
Simple Pendulum (Approximate Solution)
Simple Pendulum (Exact Solution)
Sample Problem 19.1
Free Vibrations of Rigid Bodies
Sample Problem 19.2
Sample Problem 19.3
Principle of Conservation of Energy
Sample Problem 19.4
Forced Vibrations
Sample Problem 19.5
Damped Free Vibrations
Damped Forced Vibrations
Electrical Analogues
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Introduction
19 - 3
•
Mechanical vibration is the motion of a particle or body which 
oscillates about a position of equilibrium.  Most vibrations in 
machines and structures are undesirable due to increased stresses 
and energy losses.
•
Time interval required for a system to complete a full cycle of the 
motion is the period of the vibration.
•
Number of cycles per unit time defines the frequency of the vibrations.
•
Maximum displacement of the system from the equilibrium position is 
the amplitude of the vibration.
•
When the motion is maintained by the restoring forces only, the 
vibration is described as free vibration.  When a periodic force is 
applied to the system, the motion is described as forced vibration.
•
When the frictional dissipation of energy is neglected, the motion 
is said to be undamped.  Actually, all vibrations are damped to 
some degree.
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 4
• If a particle is displaced through a distance x
m
 from its 
equilibrium position and released with no velocity, the 
particle will undergo simple harmonic motion,
( )
0 = +
- = + - = =
kx x m
kx x k W F ma
st
? ?
d
•
General solution is the sum of two particular solutions,
( ) ( ) t C t C
t
m
k
C t
m
k
C x
n n
? ? cos sin
cos sin
2 1
2 1
+ =
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
=
• x is a periodic function and ?
n
 is the natural circular 
frequency of the motion.
• C
1
 and C
2
 are determined by the initial conditions:
( ) ( ) t C t C x
n n
? ? cos sin
2 1
+ =
0 2
x C =
n
v C ?
0 1
= ( ) ( ) t C t C x v
n n n n
? ? ? ? sin cos
2 1
- = = ?
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 5
( ) ? ? + = t x x
n m
sin
= =
n
n
?
p
t
2
period
= = =
p
?
t 2
1
n
n
n
f natural frequency
( ) = + =
2
0
2
0
x v x
n m
?
amplitude
( ) = =
- n
x v ? ?
0 0
1
tan phase angle
•
Displacement is equivalent to the x component of the sum of two vectors
which rotate with constant angular velocity 
2 1
C C
? ?
+
.
n
?
0 2
0
1
x C
v
C
n
=
=
?
Page 5


© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Contents
19 - 2
Introduction
Free Vibrations of Particles.  Simple Harmonic Motion
Simple Pendulum (Approximate Solution)
Simple Pendulum (Exact Solution)
Sample Problem 19.1
Free Vibrations of Rigid Bodies
Sample Problem 19.2
Sample Problem 19.3
Principle of Conservation of Energy
Sample Problem 19.4
Forced Vibrations
Sample Problem 19.5
Damped Free Vibrations
Damped Forced Vibrations
Electrical Analogues
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Introduction
19 - 3
•
Mechanical vibration is the motion of a particle or body which 
oscillates about a position of equilibrium.  Most vibrations in 
machines and structures are undesirable due to increased stresses 
and energy losses.
•
Time interval required for a system to complete a full cycle of the 
motion is the period of the vibration.
•
Number of cycles per unit time defines the frequency of the vibrations.
•
Maximum displacement of the system from the equilibrium position is 
the amplitude of the vibration.
•
When the motion is maintained by the restoring forces only, the 
vibration is described as free vibration.  When a periodic force is 
applied to the system, the motion is described as forced vibration.
•
When the frictional dissipation of energy is neglected, the motion 
is said to be undamped.  Actually, all vibrations are damped to 
some degree.
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 4
• If a particle is displaced through a distance x
m
 from its 
equilibrium position and released with no velocity, the 
particle will undergo simple harmonic motion,
( )
0 = +
- = + - = =
kx x m
kx x k W F ma
st
? ?
d
•
General solution is the sum of two particular solutions,
( ) ( ) t C t C
t
m
k
C t
m
k
C x
n n
? ? cos sin
cos sin
2 1
2 1
+ =
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
=
• x is a periodic function and ?
n
 is the natural circular 
frequency of the motion.
• C
1
 and C
2
 are determined by the initial conditions:
( ) ( ) t C t C x
n n
? ? cos sin
2 1
+ =
0 2
x C =
n
v C ?
0 1
= ( ) ( ) t C t C x v
n n n n
? ? ? ? sin cos
2 1
- = = ?
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 5
( ) ? ? + = t x x
n m
sin
= =
n
n
?
p
t
2
period
= = =
p
?
t 2
1
n
n
n
f natural frequency
( ) = + =
2
0
2
0
x v x
n m
?
amplitude
( ) = =
- n
x v ? ?
0 0
1
tan phase angle
•
Displacement is equivalent to the x component of the sum of two vectors
which rotate with constant angular velocity 
2 1
C C
? ?
+
.
n
?
0 2
0
1
x C
v
C
n
=
=
?
© 2010 The McGraw-Hill Companies, Inc. All rights reserved. 
 
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
Ninth
Edition
Free Vibrations of Particles.  Simple Harmonic Motion
19 - 6
( ) ? ? + = t x x
n m
sin
•
Velocity-time and acceleration-time curves can be 
represented by sine curves of the same period as the 
displacement-time curve but different phase angles.
( )
( ) 2 sin
cos
p ? ? ?
? ? ?
+ + =
+ =
=
t x
t x
x v
n n m
n n m
?
( )
( ) p ? ? ?
? ? ?
+ + =
+ - =
=
t x
t x
x a
n n m
n n m
sin
sin
2
2
? ?
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