Vibration
MENG 384
Chapter 1
Introduction to Vibration
By: Dr. Muhammad AL-Qassab
Vibration: Any motion that repeats itself after an interval
of time is called vibration or oscillation.
All bodies possessing mass and elasticity are capable of
vibration. Thus, most engineering machines and structures
experience vibration to some degree, and their design
generally requires consideration of their oscillatory
behavior.
Examples:
Swinging of a Pendulum
Motion of Strings (Cables)
Spring Mass System
Torsional Vibration of Shafts
Elementary Parts of Vibrating Systems:
Mass (m, kg) or Inertia (J, kg.m²)
Spring or Elasticity (k, N/m)
May and may not include a Damper (c, N.s/m)
Spring Elements:
Spring is
characterized by
its stiffness
(Spring Constant,
k)
k depends on
material
properties (E, G)
and geometries (l,
A,I)
Spring Elements: Beams
Fixed-Fixed Beam Carrying
Electric Motor
Actual System
Cantilever Beam Carrying Electric
Motor
Equivalent System
Degree of Freedom
The number of independent coordinates required to
describe the motion of a system is called degrees of
freedom of the system.
Examples:
Single degree of freedom systems
Two-degree of freedom system
A cantilever beam (an infinite-number-of-degree of freedom system)
Classification of Vibration
Free Vibration:
Free vibration takes place when a system oscillates under its own after initial
disturbance. No external forces act on the system. The system under free
vibration will vibrate at one or more of its natural frequencies which are
properties of the dynamical system.
Forced Vibration:
Forced vibration takes place under the excitation of external forces. In this case
the system is forced to vibrates at the excitation frequency. If the excitation
frequency coincides with the natural frequency, resonance is encountered and a
dangerously large oscillations may results
Undamped and Damped Vibration:
If no energy is lost or dissipated in friction or other resistance oscillation, the
vibration is known as undamped vibration. Otherwise, it is called damped
vibration.
Linear Vibration
Nonlinear Vibration
Harmonic Motion
Harmonic motion is often represented as
the projection on a straight line of a point
that is moving on a circle at constant
speed (crank tip point of the Scotch yoke
mechanism). It is the simplest type of
periodic motion. With the angular speed
of the crank O-P designated by , the
displacement x can be written as
One
cycle of
motion
x = A sin t
where A is the crank length
and is generally measured in rad/s and referred to as the circular frequency.
Because motion repeats itself in 2 radians, we have the relationship
=2 / =2 f
Where is the period of oscillation. It is the necessary time for the motion to
repeat itself. Measured in seconds. And f is the frequency given by f = 1/ ,
measured in cycles per seconds, or more commonly Hertz (Hz).
The velocity and acceleration are determined by differentiation:
x = A sin t
Relationship of Displacement, Velocity and
Acceleration
Displacement:
Peak Displacement = A
Velocity: It is the rate of
change of displacement
Peak Velocity = A
Acceleration: It is the rate
of change of velocity
Peak Acceleration = ² A
Phase: Velocity leads
Displacement by 90º.
Acceleration leads Velocity by
90º and Displacement by 180º.
Harmonic Analysis
The French mathematician J. Fourier (1768-1830) showed
that any periodic motion can be represented by a series of
sines and cosines that are harmonically related.
where
To determine the coefficients
we multiply both sides of the series by
or
and integrate each term over the period .
Sine and cosine functions have the following properties:
Therefore,
For even and odd functions, we have
Gibbs phenomenon
Vibration Terminology
Peak Value:
Average Value: It indicates steady or static
value. Example, average value of sine
wave is zero. For rectified sine wave
Mean square value:
Example, for sine wave
1.5
Sin(wt)
sin(wt)^2
1
0.5
0
-0.5
0
2
-1
-1.5
Root mean square value, rms:
For sine wave, rms=0.707 A
Decible: Power ratio
4
6
8