Professor Mike Brennan
Introduction to Vibration
Introduction to Vibration
• Nature of vibration of mechanical systems
• Free and forced vibrations
• Frequency response functions
• For free vibration to occur we need
– mass
– stiffness
Fundamentals
m
k
c
• The other vibration quantity is damping
Fundamentals - damping
Fundamental definitions
sin( )x A t
t
( )x t
A
T
T
Period 2T
Frequency 1f T
(seconds)
(cycles/second) (Hz)
2 f (radians/second)
Phase
sin( )x A t
t
( )x t
A
sin( )x A t
Green curve lags the blue curve by radians 2
Harmonic motion
t
( )x t
angular
displacement
One cycle of motion
2π radians
A
t
Complex number representation
of harmonic motion
a
+ imaginary
+ imaginary
+ real - real
b
A
a jb x
cos sinA jA x
cos sinA j x
Euler’s Equation
cos sinje j
So jAe x
magnitude
phase
magnitude 2 2A a b x phase 1tan b a
Relationship between circular motion in the
complex plane with harmonic motion
Imaginary part – sine wave
Real part – cosine wave
Free Vibration
• System vibrates at its natural frequency
t
( )x t
sin( )nx A t
Natural frequency
Forced Vibration
• System vibrates at the forcing frequency
t
( )x t
sin( )fx A t
Forcing frequency
( )f t( )x t
Mechanical Systems
• Systems maybe linear or nonlinear
system
• Linear Systems
1. Output frequency = Input frequency
2. If the magnitude of the excitation is changed, the
response will change by the same amount
3. Superposition applies
input excitation output response
• Same frequency as input
• Magnitude change
• Phase change
• Output proportional to input
Mechanical Systems
• Linear system
Linear
system
Mechanical Systems
• Linear system
M
system
input excitation
output response, y a
b
( )by aM baM M
• output comprises frequencies
other than the input frequency
• output not proportional to input
Mechanical Systems
• Nonlinear system
Nonlinear
system
Mechanical Systems
• Nonlinear systems
• Generally system dynamics are a function of frequency
and displacement
• Contain nonlinear springs and dampers
• Do not follow the principle of superposition
linear
hardening
spring
softening
spring
Mechanical Systems
• Nonlinear systems – example: nonlinear spring
displacement
x
force
f
k f
x
For a linear system
f kx
Mechanical Systems
• Nonlinear systems – example: nonlinear spring
displacement
x
force
f
stiffnessf
x
Static displacement
Peak-to-peak vibration
(nonlinear)
Peak-to-peak vibration
(approximately linear)
Degrees of Freedom
• The number of independent coordinates required to
describe the motion is called the degrees-of-freedom
(dof) of the system
Independent
coordinate
• Single-degree-of-freedom systems
Degrees of Freedom
• Single-degree-of-freedom systems
Independent
coordinate
m
k
x
Idealised Elements
• Spring
k 1f 2f
1x2x
• no mass
• k is the spring constant
with units N/m
1 1 2f k x x
2 2 1f k x x
1 2f f
Idealised Elements
• Addition of Spring Elements
k1
1 2
1
1 1total
k k
k
k2
ktotal is smaller than the smallest stiffness
Series
ktotal is larger than the largest stiffness
k1
k2 1 2total kk k Parallel
Idealised Elements
• Addition of Spring Elements - example
kT
kR
f
x
stiffnessf
x
• Is kT in parallel or series with kR ? Series!!
Idealised Elements
• Viscous damper
c 1f 2f
1x2x
• no mass
• no elasticity
1 1 2f c x x
2 2 1f c x x
1 2f f
• c is the damping constant
with units Ns/m Rules for addition of
dampers is as for springs
Idealised Elements
• Viscous damper
1f 2f
x
• rigid
• m is mass with
units of kg
1 2f f mx
2 1f mx f
Forces do not pass unattenuated
through a mass
m
Free vibration of an undamped
SDOF system
k
m
System equilibrium
position
System vibrates about its equilibrium position
k
Undeformed
spring
Free vibration of an undamped
SDOF system
k
m
System at
equilibrium
position
k
Extended position
m m mx
kx
0mx kx
inertia force stiffness force
Simple harmonic motion
The equation of motion is:
0mx kx
0k
x xm
2 0nx x
where n
k
mis the natural frequency of the system
The motion of the mass is given by sino nx X t
k
m x
Simple harmonic motion
k
m x
Real Notation Complex Notation
Displacement
Velocity
Acceleration
sino nx X t nj tx Xe
cosn o nx X t nj t
nx j Xe
2 sinn o nx X t 2 nj t
nx Xe
x
x
x
Simple harmonic motion
tReal
Imag
Free vibration effect of damping
k
m x
c
The equation of motion is
0cx kxm x
inertia
force stiffness
force
damping
force
ntx Xe
Free vibration effect of damping
time
2d
d
T
d
sinnt
dx Xe t
Damping ratio
Damping perioddT
Phase angle
Free vibration effect of damping
The underdamped displacement of the mass is given by
sinnt
dx Xe t
= Damping ratio = 2 0 1nc m
n = Undamped natural frequency = k m
d = Damped natural frequency = 21n
= Phase angle
Exponential decay term Oscillatory term
Underdamped ζ<1
Critically damped ζ=1
Overdamped ζ>1
Free vibration effect of damping
t
x t
Undamped ζ=0
Degrees-of-freedom
k m
Single-degree-of-freedom system
Multi-degree-of-freedom (lumped parameter systems)
N modes, N natural frequencies
k m
1x
1x
k m
2x
k m
3x
k m
4x
Degrees-of-freedom
Infinite number of degrees-of-freedom (Systems having
distributed mass and stiffness) – beams, plates etc.
Example - beam
Mode 1 Mode 2 Mode 3
Free response of
multi-degree-of-freedom systems
Example - Cantilever
X
x t
t
+
+
+
1
2
3
4
Response of a SDOF system to
harmonic excitation
k
m x
c
sinF t
t
( )fx t
t
( )px t
t
( ) ( )p fx t x t
Steady-state
Forced vibration
k
m x
c
Steady-state response of a SDOF
system to harmonic excitation
sinF t The equation of motion is
sinmx cx k F tx
The displacement is given by
sinox X t
where
X is the amplitude
is the phase angle between the response and the force
Frequency response of a SDOF system
k
m x
c
sinF t
The amplitude of the
response is given by
2 22
o
FX
k m c
The phase angle is given by
1
2tan
c
k m
Stiffness force okX
Damping
force
ocX
Inertia force 2
omX
Applied force
F
Frequency response of a SDOF system
k
m x
c
j tFe
The equation of motion is
j tFmx cx x ek
The displacement is given by
j tx Xe
This leads to the complex amplitude given by
2
1X
F k m j c
or
2
1 1
1 2n n
X
F k j
Complex notation allows the amplitude and phase information
to be combined into one equation
Where 2
n k m and 2c mk
Frequency response functions
Receptance 2
1X
F k m j c
Other frequency response functions (FRFs) are
AccelerationAccelerance =
Force
VelocityMobility =
Force
ForceApparent Mass =
Acceleration
ForceImpedance =
Velocity
ForceDynamic Stiffness =
Displacement
Increasing damping
Representation of frequency response data
Log frequency
Log receptance
1
k
Increasing damping phase
n
-90°
Vibration control of a SDOF system
Log frequency
Log
1
k
oX
F
k
m x
c
j tFe
2 22
1oX
Fk m c
Frequency Regions
Low frequency 0 1oX F k Stiffness controlled
Stiffness
controlled
Resonance 2 k m 1oX F c Damping controlled
Damping
controlled
High frequency 2
n 21oX F m Mass controlled
Mass
controlled
Representation of frequency response data
Recall
2
1 1
1 2n n
X
F k j
This includes amplitude and phase information. It
is possible to write this in terms of real and imaginary
components.
2
2 22 2 2 2
1 21 1
1 2 1 2
n n
n n n n
Xj
F k k
real part imaginary part
Real and Imaginary parts of FRF
frequency
ReX
F
ImX
F
n
Real and Imaginary parts of FRF
Increasing
frequency
ReX
F
ImX
F
n
1 k
Real and Imaginary components can be plotted on one
diagram. This is called an Argand diagram or Nyquist plot
3D Plot of Real and Imaginary parts of FRF
frequency
ReX
F
Im
X
F
0
0.1
Summary
• Basic concepts
– Mass, stiffness and damping
• Introduction to free and forced vibrations
– Role of damping
– Frequency response functions
– Stiffness, damping and mass controlled frequency
regions