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 -

potential and kinetic energy

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