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Introduction to Vibration
Mike BrennanUNESP, Ilha Solteira
São PauloBrazil
Vibration• Most vibrations are undesirable, but there are many
instances where vibrations are useful
– Ultrasonic (very high frequency) vibrations
• Tooth cleaning
• Imaging of internal organs• Imaging of internal organs
• Welding
• Structural Health MonitoringStructural Health MonitoringStructural Health MonitoringStructural Health Monitoring
– Vibration conveyers
– Time-keeping instruments
– Impactors
– Music
– Heartbeat
Introduction to Vibration
• Nature of vibration of mechanical systems
• Free and forced vibrations• Free and forced vibrations
• Frequency response functions
• For free vibration to occur we need
– mass
– stiffness
Fundamentals
m
– stiffness k
c
• The other vibration quantity is damping
Fundamentals -potential and kinetic energy
energy.mov
Fundamentals - damping
Fundamental definitions
t
( )x t
A
sin( )x A tω=
T
T
Period 2T π ω=
Frequency 1f T=
(seconds)
(cycles/second) (Hz)
2 fω π= (radians/second)
Phase
sin( )x A tω=
t
( )x t
A
sin( )x A tω=
sin( )x A tω φ= +
φω
Green curve lags the blue curve by radians2π
Harmonic motion
( )x tω
angulardisplacement
A
ω
tφ ω=
displacement
One cycle of motion2π radians
tφ ω=
Complex number representation of harmonic motion
a
+ imaginary
+ real- realφ
b
A
a jb= +x
cos sinA jAφ φ= +x
( )cos sinA jφ φ= +x
+ imaginary
Euler’s Equation
cos sinje jφ φ φ± = ±
So jAe φ=x
magnitude
phase
magnitude 2 2A a b= = +x phase ( )1tan b aφ −=
Relationship between circular motion in the complex plane with harmonic motion
Imaginary part – sine wave
Real part – cosine wave
Sinusoidal signals – other descriptions
( )x t
0
1sin dt
T
avx A tT
ω= ∫
For a sine wave
• Average value
t
TFor a sine wave
0avx =
For a rectified sine wave
0.637avx A=
Sinusoidal signals – other descriptions
( )x t
• Average value
DC
t
Average value of a signal = DC component of signal
Sinusoidal signals – other descriptions
( )x t
For a sine wave
• Mean square value
( ) 22
0
1sin dt
T
meanx A tT
ω= ∫
tT
For a sine wave2 20.5meanx A=
• Root Mean Square (rms)
2 2rms meanx x A= =Many measuring devices, for example a digital voltmeter, record the rms value
Sinusoidal signals – Example
• A vibration signal is described by:
0.15sin200x t=• Amplitude (or peak value) = 0.15 m• Average value = 0• Mean square value = 0.01125 m2
• Root mean square value = 0.10607 m• Peak-to-peak value = 0.3 m• Frequency = 31.83 Hz
Vibration signals
( )x t
• Periodic or deterministic (not sinusoidal)
• Heartbeat• IC Engine
t
T T
T is the fundamental period
Fourier Analysis(Jean Baptiste Fourier 1830)
+( )x t
• Representation of a signal by sines and cosine waves
+
+:
t
Fourier Composition of a Square wave
frequency
Vibration signals
( )x t
• Transient
• Gunshot• Earthquake• Impact
t
• Impact
Vibration signals
( )x t
• Random
• Uneven Road• Wind• Turbulence
t
• Turbulence
Free Vibration
• System vibrates at its natural frequency( )x t
t
sin( )nx A tω=Natural frequency
Forced Vibration
• System vibrates at the forcing frequency( )x t
( )f t( )x t
t
sin( )fx A tω=Forcing frequency
Mechanical Systems
• Systems maybe linear or nonlinear
systeminput excitation output response
• 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
Mechanical Systems
• Linear system
Linearsystem
• Same frequency as input• Magnitude change• Phase change• Output proportional to input
system
Mechanical Systems
• Linear system
M
input excitation
output response, ya
Msystem
output response, y
b
( )by aM baM M= + = +
Mechanical Systems
• Nonlinear system
Nonlinearsystem
• output comprises frequenciesother than the input frequency
• output not proportional to input
system
Mechanical Systems
• Nonlinear systems
• Generally system dynamics are a function of frequencyand displacement
• Contain nonlinear springs and dampers
• Do not follow the principle of superposition
linear
hardeningspring
Mechanical Systems
• Nonlinear systems – example: nonlinear spring
kf
linear
softeningspring
displacement
x
force
f
x
For a linear system
f kx=
Mechanical Systems
• Nonlinear systems – example: nonlinear spring
force
f
Peak-to-peak vibration(approximately linear)
displacement
x
f
stiffnessfx
∆=∆
Static displacement
Peak-to-peak vibration(nonlinear)
Degrees of Freedom • The number of independent coordinates required to describe the motion is called the degrees-of-freedom(dof) of the system
• Single-degree-of-freedom systems
θ
Independentcoordinate
Degrees of Freedom
• Single-degree-of-freedom systems
x
Independentcoordinate
m
k
x
Idealised Elements
• Spring
k1f 2f
x x( )1 1 2f k x x= −
( )= −1x 2x
• no mass• k is the spring constantwith units N/m
( )2 2 1f k x x= −
1 2f f= −
Idealised Elements
• Addition of Spring Elements
k1
1 2
11 1total
k k
k =+
k2
k is smaller than the smallest stiffness
Series
ktotal is smaller than the smallest stiffness
ktotal is larger than the largest stiffness
k1
k2 1 2total kk k= +Parallel
Idealised Elements
• Addition of Spring Elements - example
kR
f
x
kT
stiffnessfx
=
• Is kT in parallel or series with kR ? Series!!
Idealised Elements
• Viscous damperc
1f 2f
xɺ xɺ( )1 1 2f c x x= −ɺ ɺ
( )= −ɺ ɺ1xɺ 2xɺ
• no mass• no elasticity
( )2 2 1f c x x= −ɺ ɺ
1 2f f= −
• c is the damping constantwith units Ns/m
Rules for addition ofdampers is as for springs
Idealised Elements
• Viscous damper
1f 2f
1 2f f mx+ = ɺɺ
f mx f= −ɺɺ
m
xɺɺ
• rigid• m is mass with units of kg
2 1f mx f= −ɺɺ
Forces do not pass unattenuatedthrough a mass
Free vibration of an undamped SDOF system
System equilibriumposition
Undeformed spring
k
m
System vibrates about its equilibrium position
k
Free vibration of an undamped SDOF system
System at equilibrium
position
Extended position
m m mxɺɺ
k
mk
kx−
0mx kx+ =ɺɺ
inertia force stiffness force
Simple harmonic motion
The equation of motion is:
0mx kx+ =ɺɺ
0k
x x⇒ + =ɺɺk
m x
0k
x xm
⇒ + =ɺɺ
2 0nx xω⇒ + =ɺɺ
where 2n
km
ω = is the natural frequency of the system
The motion of the mass is given by ( )sino nx X tω=
k
Simple harmonic motion
k
m x
Real Notation Complex Notation
Displacement( )sino nx X tω= nj tx Xe ω=
kVelocity
Acceleration
( )o n
( )cosn o nx X tω ω=ɺ nj tnx j Xe ωω=ɺ
( )2 sinn o nx X tω ω= −ɺɺ 2 nj tnx Xe ωω= −ɺ
x
xɺɺ
Simple harmonic motion
Imag
ω
xɺtω
Real
Free vibration effect of damping
k
m x
c
The equation of motion is
0cx kxm x+ + =ɺɺɺ
inertia force
stiffness force
dampingforce
ntx Xe ζω−=
Free vibration effect of damping
timetime
2d
d
Tπ
ω=
d
φω ( )sinnt
dx Xe tζω ω φ−= +
Damping ratioζ =Damping perioddT =Phase angleφ =
Free vibration - effect of damping
The underdamped displacement of the mass is given by
( )sinntdx Xe tζω ω φ−= +
Exponential decay term Oscillatory term
ζ = Damping ratio = ( ) ( )2 0 1nc mω ζ< <
nω = Undamped natural frequency = k m
dω = Damped natural frequency = 21nω ζ= −φ = Phase angle
Exponential decay term Oscillatory term
Free vibration - effect of damping
Free vibration - effect of damping
t
( )x t
Underdamped ζ<1
Critically damped ζ=1
Overdamped ζ>1
Undamped ζ=0
Variation of natural frequency with damping
d
n
ωω
1
ζ10
Degrees -of-freedom
km
Single-degree-of-freedom system
1x
Multi-degree-of-freedom (lumped parameter systems)N modes, N natural frequencies
km
1x
km
2x
km
3x
km
4x
Degrees -of-freedomInfinite number of degrees-of-freedom (Systems having distributed mass and stiffness) – beams, plates etc.
Example - beam
Mode 1
Degrees -of-freedomInfinite number of degrees-of-freedom (Systems having distributed mass and stiffness) – beams, plates etc.
Example - beam
Mode 1 Mode 2
Degrees -of-freedomInfinite 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 +
1ω
2ω
( )x t
t
+
+3ω
4ω
Response of a SDOF system to harmonic excitation
m x
( )sinF tω
t
( )fx t
( )x t
Steady-stateForced vibration
k c
t
( )px t
t
( ) ( )p fx t x t+
k
m x
c
Steady -state response of a SDOF system to harmonic excitation
( )sinF tω The equation of motion is
( )sinmx cx k F tx ω+ + =ɺɺ ɺ
The displacement is given byc
( )sinox X tω φ= +
where
X is the amplitude
φ is the phase angle between the response and the force
Frequency response of a SDOF system
k
m x
c
( )sinF tωThe amplitude of the response is given by
( ) ( )2 22o
FX
k m cω ω=
− +
The phase angle is given by
12tan
ck m
ωφω
− = −
Stiffness force okX
Damping force
ocXω
Inertia force 2omXω
Applied force
F
φ
Frequency response of a SDOF system
k
m x
c
j tFe ω
The equation of motion is
j tFmx cx x ek ω+ + =ɺɺ ɺ
The displacement is given by
j tx Xe ω=x Xe=This leads to the complex amplitude given by
2
1XF k m j cω ω
=− +
or( )2
1 1
1 2n n
XF k jω ω ζ ω ω
= − +
Complex notation allows the amplitude and phase information to be combined into one equation
Where 2n k mω = and ( )2c mkζ =
Frequency response functions
Receptance2
1XF k m j cω ω
=− +
Other frequency response functions (FRFs) are
AccelerationAccelerance =
ForceForce
Apparent Mass = Acceleration
Accelerance = Force
VelocityMobility =
Force
Apparent Mass = Acceleration
ForceImpedance =
Velocity
ForceDynamic Stiffness =
Displacement
Increasing damping
Representation of frequency response data
Log receptance
1k
Log frequency
Increasing dampingphase
nω
-90°
Vibration control of a SDOF system
k
m xc
j tFe ω
( ) ( )2 22
1oXF k m cω ω
=− +
Frequency Regions
Low frequency 0ω → 1oX F k⇒ = Stiffness controlled
Resonance 2 k mω = 1oX F cω⇒ = Damping controlled
Log frequency
Log
1k
oXF
Stiffnesscontrolled
Dampingcontrolled
High frequency 2nω ω>> 21oX F mω⇒ = Mass controlled
Masscontrolled
Representation of frequency response data
Recall( )2
1 1
1 2n n
XF k jω ω ζ ω ω
= − +
This includes amplitude and phase information. Itis possible to write this in terms of real and imaginary is possible to write this in terms of real and imaginary components.
( )( )( ) ( ) ( )( ) ( )
2
2 22 2 2 2
11 1 2
1 2 1 2
n n
n n n n
Xj
F k k
ω ω ζ ω ω
ω ω ζ ω ω ω ω ζ ω ω
− = +
− + − +
real part imaginary part
Real and Imaginary parts of FRF
frequency
ReXF
frequency
ImXF
nω
Real and Imaginary parts of FRF
ReXF
φ1 k
Real and Imaginary components can be plotted on one diagram. This is called an Argand diagram or Nyquist plot
Increasingfrequency
ImXF
nω
3D Plot of Real and Imaginary parts of FRF
ReXF
Im
XF
0ζ =
frequency0.1ζ =
Summary
• Basic concepts
– Mass, stiffness and damping
• Introduction to free and forced vibrations• Introduction to free and forced vibrations
– Role of damping
– Frequency response functions
– Stiffness, damping and mass controlled frequency
regions