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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