Mike Brennan UNESP, Ilha Solteira São Paulo Brazil · UNESP, Ilha Solteira São Paulo Brazil....
Transcript of Mike Brennan UNESP, Ilha Solteira São Paulo Brazil · UNESP, Ilha Solteira São Paulo Brazil....
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Introduction to Vibration
Mike BrennanUNESP, Ilha Solteira
São PauloBrazil
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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
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Introduction to Vibration
• Nature of vibration of mechanical systems
• Free and forced vibrations• Free and forced vibrations
• Frequency response functions
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• For free vibration to occur we need
– mass
– stiffness
Fundamentals
m
– stiffness k
c
• The other vibration quantity is damping
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Fundamentals -potential and kinetic energy
energy.mov
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Fundamentals - damping
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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)
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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
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Harmonic motion
( )x tω
angulardisplacement
A
ω
tφ ω=
displacement
One cycle of motion2π radians
tφ ω=
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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 −=
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Relationship between circular motion in the complex plane with harmonic motion
Imaginary part – sine wave
Real part – cosine wave
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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=
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Sinusoidal signals – other descriptions
( )x t
• Average value
DC
t
Average value of a signal = DC component of signal
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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
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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
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Vibration signals
( )x t
• Periodic or deterministic (not sinusoidal)
• Heartbeat• IC Engine
t
T T
T is the fundamental period
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Fourier Analysis(Jean Baptiste Fourier 1830)
+( )x t
• Representation of a signal by sines and cosine waves
+
+:
t
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Fourier Composition of a Square wave
frequency
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Vibration signals
( )x t
• Transient
• Gunshot• Earthquake• Impact
t
• Impact
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Vibration signals
( )x t
• Random
• Uneven Road• Wind• Turbulence
t
• Turbulence
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Free Vibration
• System vibrates at its natural frequency( )x t
t
sin( )nx A tω=Natural frequency
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Forced Vibration
• System vibrates at the forcing frequency( )x t
( )f t( )x t
t
sin( )fx A tω=Forcing frequency
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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
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Mechanical Systems
• Linear system
Linearsystem
• Same frequency as input• Magnitude change• Phase change• Output proportional to input
system
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Mechanical Systems
• Linear system
M
input excitation
output response, ya
Msystem
output response, y
b
( )by aM baM M= + = +
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Mechanical Systems
• Nonlinear system
Nonlinearsystem
• output comprises frequenciesother than the input frequency
• output not proportional to input
system
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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
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linear
hardeningspring
Mechanical Systems
• Nonlinear systems – example: nonlinear spring
kf
linear
softeningspring
displacement
x
force
f
x
For a linear system
f kx=
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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)
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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
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Degrees of Freedom
• Single-degree-of-freedom systems
x
Independentcoordinate
m
k
x
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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= −
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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
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Idealised Elements
• Addition of Spring Elements - example
kR
f
x
kT
stiffnessfx
=
• Is kT in parallel or series with kR ? Series!!
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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
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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
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Free vibration of an undamped SDOF system
System equilibriumposition
Undeformed spring
k
m
System vibrates about its equilibrium position
k
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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
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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
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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
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x
xɺɺ
Simple harmonic motion
Imag
ω
xɺtω
Real
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Free vibration effect of damping
k
m x
c
The equation of motion is
0cx kxm x+ + =ɺɺɺ
inertia force
stiffness force
dampingforce
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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 =
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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
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Free vibration - effect of damping
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Free vibration - effect of damping
t
( )x t
Underdamped ζ<1
Critically damped ζ=1
Overdamped ζ>1
Undamped ζ=0
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Variation of natural frequency with damping
d
n
ωω
1
ζ10
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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
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Degrees -of-freedomInfinite number of degrees-of-freedom (Systems having distributed mass and stiffness) – beams, plates etc.
Example - beam
Mode 1
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Degrees -of-freedomInfinite number of degrees-of-freedom (Systems having distributed mass and stiffness) – beams, plates etc.
Example - beam
Mode 1 Mode 2
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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
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Free response of multi-degree -of-freedom systems
Example - Cantilever
X +
1ω
2ω
( )x t
t
+
+3ω
4ω
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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+
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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
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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
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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 =
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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
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Increasing damping
Representation of frequency response data
Log receptance
1k
Log frequency
Increasing dampingphase
nω
-90°
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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
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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
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Real and Imaginary parts of FRF
frequency
ReXF
frequency
ImXF
nω
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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
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3D Plot of Real and Imaginary parts of FRF
ReXF
Im
XF
0ζ =
frequency0.1ζ =
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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