FUNDAMENTALS OF

VIBRATIONS
BASIC CONCEPTS
MEMB443 Mechanical Vibrations
LEARNING OBJECTIVES
Upon completion of this lecture, you should be able to:
Understand the concepts of degree-of-freedom, and discrete
and continuous systems.
Compute the stiffness of some simple spring elements.
Determine the equivalent mass or inertia, and equivalent
spring and damping constants of vibrating systems.
Understand the definitions of free and forced vibration,
undamped and damped vibration, linear and non-linear
vibration, and deterministic and random vibration.
MEMB443 Mechanical Vibrations
VIBRATION IN ENGINEERING
PRACTICE
Good
Machine condition monitoring.
Vibrating sieves, mixers and tools.
Electric massaging units, dentist drills, electric
toothbrushes.
Bad
Noise, vibration and harshness (NVH).
Machinery and structural failures.
Motion sickness, white finger syndrome, etc.
MEMB443 Mechanical Vibrations
TACOMA NARROWS BRIDGE
This photograph shows the twisting motion of
the center span just prior to failure.
The nature and severity of the torsional
movement is revealed in this picture taken
from the Tacoma end of the suspension span.
When the twisting motion was at the
maximum, elevation of the sidewalk at the
right was 28 feet (8.5m) higher than the
sidewalk at the left.
MEMB443 Mechanical Vibrations
TACOMA NARROWS BRIDGE (cont.)
MEMB443 Mechanical Vibrations
POSITIVE DISPLACEMENT BLOWER
Shearing of shaft attributed to high torsional vibrations. Rotor operating
speed was within the vicinity of a torsional natural frequency.
MEMB443 Mechanical Vibrations
POSITIVE DISPLACEMENT BLOWER
(cont.)
MEMB443 Mechanical Vibrations
EXAMPLES OF VIBRATING
SYSTEM
Masses attached to springs.
Flexible Rods.
Pendulums.
MEMB443 Mechanical Vibrations
BASIC CONCEPTS OF VIBRATION
When a particle or a rigid body in stable equilibrium is
displaced by the application of an additional force,
mechanical vibration will result.
Some important concepts in mechanical vibration theory
can be categorized into the followings:
Elementary parts of vibrating systems.
Degree of freedom.
Discrete and continuous system.
MEMB443 Mechanical Vibrations
ELEMENTARY PARTS OF VIBRATING
SYSTEMS
To be subject to vibration, a system must be able to store
energy in two different forms and allow energy to be
transferred from one to the other.
In particular, for vibration to exist, there must be a transfer
of energy from potential to kinetic and vice-versa.
Potential energy is due to either gravity or the elasticity of
the system, whilst the kinetic energy is due to the motion
of the mass.
The simplest mechanical oscillators are the pendulum and
the spring-mass system. The corresponding simplest
electrical oscillator is the capacitor-inductor system.
MEMB443 Mechanical Vibrations
SPRING ELEMENTS
A spring is a mechanical link that is generally assumed to
have negligible mass and damping.
A force is developed in a spring whenever there is a
relative motion between two ends of the spring.
Work done in deforming a spring is stored as potential
energy in the spring.
kx F
S
=
u
T S
k M =
Linear Spring
Torsional Spring
MEMB443 Mechanical Vibrations
SPRING ELEMENTS (cont.)
A spring element is generally made of an elastic material.
The stiffness in a spring element can be related more
directly to its material (elastic modulus) and geometric
properties.
A spring-like behavior results from a variety of motion
configurations, including:
Longitudinal motion (vibration in the direction of the
length).
Transverse motion (vibration perpendicular to the length).
Torsional motion (vibration rotating around the length).
MEMB443 Mechanical Vibrations
STIFFNESS OF SPRING ELEMENTS
Stiffness associated with the longitudinal vibration of a
slender prismatic bar.
MEMB443 Mechanical Vibrations
STIFFNESS OF SPRING ELEMENTS
(cont.)
Stiffness associated with the torsional vibration of a shaft.
MEMB443 Mechanical Vibrations
STIFFNESS OF SPRING ELEMENTS
(cont.)
Stiffness associated with a helical spring.
MEMB443 Mechanical Vibrations
STIFFNESS OF SPRING ELEMENTS
(cont.)
Beam stiffness associated with the transverse vibration of
the tip of a beam.
MEMB443 Mechanical Vibrations
EQUIVALENT SPRING CONSTANT
Springs in Parallel Springs in Series

n eq
k k k k + + + = ....
2 1
n eq
k k k k
1
....
1 1 1
2 1
+ + + =

MEMB443 Mechanical Vibrations
MASS OR INERTIA ELEMENTS
The mass or inertia element is assumed to be a rigid body.
A rigid bodys inertia is responsible for the resistance to
acceleration of a system.
Work done on a mass is stored in the form of kinetic
energy of the mass.
Linear Motion
Rotational Motion
x m F

=
u

I M

=
MEMB443 Mechanical Vibrations
EQUIVALENT MASS OF A SYSTEM
Translational Masses Connected by a Rigid Bar.
Pivot point
1
l
2
l
3
l
3
x

2
x

1
x

2
m
3
m 1
m
Original system

eq
m
eq
x

1
l
Equivalent system

From trigonometric
relationship



Assume

eq
x
l
l
x
1
2
2
=
eq
x
l
l
x
1
3
3
=
1
x x
eq

=
MEMB443 Mechanical Vibrations
EQUIVALENT MASS OF A SYSTEM
(cont.)
Translational Masses Connected by a Rigid Bar.
2 2
3 3
2
2 2
2
1 1
2
1
2
1
2
1
2
1
eq eq
x m x m x m x m

= + +
Equate the kinetic energy of the three masses to that of the
equivalent systems mass
The equivalent systems mass is therefore obtained
3
2
1
3
2
2
1
2
1
m
l
l
m
l
l
m m
eq
|
|
.
|

\
|
+
|
|
.
|

\
|
+ =
MEMB443 Mechanical Vibrations
EQUIVALENT MASS OF A SYSTEM
(cont.)
Coupled Translational and Rotational Masses.
m
0
J
r
x
From kinematics, the relationship
between the linear and the angular
velocity is

To obtain the equivalent translational
mass

To obtain the equivalent rotational
mass
r
x

= u
x x
eq

=
u u

=
eq
MEMB443 Mechanical Vibrations
EQUIVALENT MASS OF A SYSTEM
(cont.)
Equivalent Translational
Mass

Equivalent Rotational Mass
2
2
0
2
2
1
2
1
2
1
eq eq
x m
r
x
J x m

=
|
.
|

\
|
+
2
0
r
J
m m
eq
+ =
( )
2 2
0
2
2
1
2
1
2
1
eq eq
J J r m u u u

= +
0
2
J mr J
eq
+ =
Coupled Translational and Rotational Masses.
MEMB443 Mechanical Vibrations
DAMPING ELEMENTS
A damper is generally assumed to have negligible mass
and stiffness.
A force is developed in a damper whenever there is a
relative velocity between two ends of the damper.
The damper dissipates energy from a system in the form of
heat or sound.
x c F
D

=
u

T D
c M =
Linear Damper
Torsional Damper
MEMB443 Mechanical Vibrations
DAMPING ELEMENTS MODELS
Viscous Damping
The damping force is proportional to the velocity of the
vibrating body.
Coulomb Damping
The damping force is constant in amplitude but opposite
the direction to that of the motion of the vibrating body.
Hysteretic Damping
The energy dissipated per cycle is proportional to the
square of the vibration amplitude.
MEMB443 Mechanical Vibrations
EQUIVALENT DAMPING CONSTANT
Dampers in Parallel Dampers in Series
n eq
c c c c + + + = ....
2 1
n eq
c c c c
1
....
1 1 1
2 1
+ + + =

MEMB443 Mechanical Vibrations
DEGREE OF FREEDOM
The minimum number of independent coordinates required
to determine completely the positions of all part of a
system at any instant of times.
x
y
z
u
x
u
y
u
z
Unconstrained rigid body with 6 d.o.f.
MEMB443 Mechanical Vibrations
DISCRETE AND CONTINUOUS SYSTEM
Systems with a finite number of degrees of freedom are
called discrete or lumped parameter systems, and those
with an infinite number of degrees of freedom are called
continuous or distributed systems.
Discrete System Continuous System
Solution: 2
nd
Order Ordinary
Differential Equation
Solution: Partial Differential
Equation
MEMB443 Mechanical Vibrations
CLASSIFICATION OF VIBRATION
Free and Forced Vibration.
Undamped and Damped Vibration.
Linear and Nonlinear Vibration.
Deterministic and Random
Vibration.
MEMB443 Mechanical Vibrations
FREE AND FORCED VIBRATION
Free Vibration
Oscillation occurring at a natural frequency, after an initial
force input.


Forced Vibration
Oscillation occurring at the frequency of a driving force
input.
0
0
= + +
= +
kx x c x m
kx x m


) (
) (
t F kx x c x m
t F kx x m
= + +
= +


MEMB443 Mechanical Vibrations
UNDAMPED AND DAMPED VIBRATION
Undamped Vibration
No energy is lost or dissipated in friction or other
resistance during oscillation.


Damped Vibration
Energy is lost or dissipated during oscillation.
) (
0
t F kx x m
kx x m
= +
= +


) (
0
t F kx x c x m
kx x c x m
= + +
= + +


MEMB443 Mechanical Vibrations
LINEAR AND NONLINEAR VIBRATION
Linear Vibration
The cause (force) and effect (response) are proportionally
related. Principle of superposition holds.

Nonlinear Vibration
Relationship between cause and effect is no longer
proportional.
MEMB443 Mechanical Vibrations
DETERMINISTIC AND RANDOM
VIBRATION
Deterministic Vibration
The instantaneous values of the vibration amplitude at any
time (t) can be determined from mathematical expressions.
Random Vibration
Future instantaneous values of the vibration amplitude
cannot be predicted in a deterministic sense.
MEMB443 Mechanical Vibrations
EXAMPLE 1
A composite propeller shaft, which is made of steel and
aluminium, is shown below. Determine the torsional
spring constant of the shaft The shear modulus G of steel
is 80 GPa and for aluminium 26 GPa.
MEMB443 Mechanical Vibrations
EXAMPLE 1 (cont.)
The polar area moment of inertia for a hollow shaft is
given by the following equation where D and d are the
outer and inner diameters, respectively.



Torsional stiffness for the shaft is:
( )
32
4 4
d D
I
P

=
t
( )
l
d D G
l
GI
k
P
t
32
4 4

= =
t
MEMB443 Mechanical Vibrations
EXAMPLE 1 (cont.)
For the steel shaft, torsional stiffness is:



For the aluminium shaft, torsional stiffness is:



Shafts are in parallel, therefore:
( )
Nm/rad 10 34 . 5
5 32
15 . 0 25 . 0 10 80
6
4 4 9
=

=
t
tS
k
( )
Nm/rad 10 207 . 0
5 32
1 . 0 15 . 0 10 26
6
4 4 9
=

=
t
tA
k
Nm/rad 10 547 . 5
6
= + =
tA tS eq
k k k
MEMB443 Mechanical Vibrations