professor mike brennan - unesp · • the flow induced vibration of pipes • aerodynamically...

Sources of Vibration

Professor Mike Brennan

Sources of Vibration

impacts rotating machines

gears

acoustic

excitation

vortex shedding

flutter

Self-excited vibration

Impact as a source of vibration

Impact force

Piston slap

pile driver

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• Impacts occur in many engineering situations. As they take place over

a very short time-period, the frequency content of the impacting force

tends to be rich. Some examples are illustrated below


ovm

local strain

m

m model

transmitted force, f

k

• Consider an elastic sphere that strikes a rigid surface as shown below

Example

• Once the mass-spring system is in contact with the surface it will begin

to oscillate with a period

2m

Tk

• The transmitted force is the product of the displacement and the

contact stiffness. Thus there is a a half-sine force-time curve.

• The displacement will be sinusoidal during the half-period the body

stays in contact with the surface.


Time-history

Energ

y s

pectr

um

of

the f

orc

e

frequency 1.5

cT

2.5

cT

4

1

(-12dB/octave)

Energy spectrum

• To increase the contact time the stiffness of the impacting body should be

decreased. The high frequency content is then reduced.

Contact time = 2T m k

time

( )f t ov km

cT

Vibrations generated by rotating machinery

Gears

• The driving wheel is assumed to rotate at constant angular velocity Ω1

and the driven wheel has an average angular velocity Ω2, and has

angular displacement

2 2t

where 12 1

2

r

r and is known as th e transmission error

Transmission error is defined as the difference between the actual

position of the output gear and the position it would take if it were perfect


Gears

• transmission error occurs due to

• the fundamental meshing frequency of the meshing force is given by

fundamental meshing frequency = number of teeth

deflection of gear teeth under load manufacturing defects


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local strain model

o

Gears Typical spectrum

frequency (Hz)

am

plit

ud

e (

dB

)


Rotating unbalance

em

xtm

2k 2kc

In a rotating machine, unbalance exists if the mass centre of the rotor does

not coincide with the axis of rotation

The unbalance me is measured in terms of an equivalent point mass m with

an eccentricity e.

The equation of motion is given by

2 sintm x cx kx me t

Assuming a harmonic response

results in sinx X t

2

2

1

1 2n n

meX

k j

where and 2n t tk m c km


Rotating unbalance

em

xtm

2k 2kc

2

2

1

1 2

t

n n n

Xm

me j

10-1

100

101

10-2

10-1

100

101

102

n

tm X

me

increasing

damping

• At low speeds is small and hence the vibration amplitude is also small n

• At resonance when damping controls the vibration amplitude 1n

• At high speeds when , which means that the

vibration is constant and is independent of speed

1, Xn tme m


Reciprocating unbalance

m

xtm

2k 2kc

e

• The excitation force is equal to the inertia force

of the reciprocating mass, which is approximately

equal to

2dynamic force me si sin2n te

tL

Note the second harmonic

L


Critical speed of a rotating shaft

The critical speed occurs when the speed of rotation of the shaft is

equal to the frequency of lateral vibration of the shaft

•Mass of the shaft is negligible

•Lateral stiffness of the shaft is k

•M is the mass of the rotating disk

•G is the mass centre of the disk

•S is the geometric centre of the disk

•The bearings are much stiffer than

the shaft


Critical speed of a rotating shaft

em

tm

2k c

2k

c

x

y The equations of motion in the x and y directions are

2 sintm x cx kx me t

2 costm y cy ky me t

Assuming a harmonic response gives

2

2

1

1 2

t

n n n

Xm

me j

2j

Y Xe

and

Since the two harmonic motions x(t) and y(t) have the same amplitude, and are in

quadrature, their sum a circle. The radius r = X or Y

O

G

S

n O

G

S

n

O

GS

n


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Critical speed of a rotating shaft – effects of bearing stiffness

• The stiffness of the bearings acts in series with the stiffness of the shaft. Thus

the system becomes more flexible lowering the critical speed


Critical speed of a rotating shaft – effects of bearing stiffness

The equations of motion in the x and y directions are

2 sinxtm x cx x me tk

2 cosytm y cy y me tk

If the system has two natural frequencies and therefore two critical speeds. yxk k

Since the two harmonic motions x(t) and y(t) have different amplitudes, and are in

quadrature, their sum an ellipse.

x

ymass centre

x

y

x

y

x y

Acoustic Excitation

txl

is the wavelength of the impinging sound field

is the trace wavelength and is given by t sint

The trace velocity is the speed at which the wavefront moves along the beam

and is given by where c is the speed of sound in air sintc c

The pressure distribution on the beam is given by

( , ) tj t k xp x t Pe

2where is the trace wavenumber given by t t

t t

k kc

Acoustic Excitation

The maximum response of the beam occurs at

A RESONANT FREQUENCY, when the length of the beam is equal to

an integer number of half-wavelengths

A COINCIDENT FREQUENCY, when the trace wavelength is equal to a

flexural wavelength

Acoustic Excitation

wavespeeds

frequency

Sound waves

Bending waves

Speed of sound

Critical

frequency

Acoustic Excitation

Critical frequency

The critical frequency fc is the

lowest frequency at which

coincidence can occur.

It is a frequency when the speed

of the impinging sound wave c,

equals the speed of the flexural

bending wave cb.

The critical frequency for a plate can

be determined using the equation

Product of plate thickness and

critical frequency in air (20°C)*

Material hfc (ms-1)

Steel

Aluminium

Brass

Copper

Glass

Perspex

Chipboard

Plywood

Light concrete

12.4

12.0

17.8

16.3

12.7

27.7

23

20

34

* For water multiply by 18.9 2

1.8c

l

cf

hc

where cl is the longitudinal wavespeed (≈5000 ms-1 for steel and aluminium)

and h is the plate thickness

Acoustic Excitation

Coincidence angle

The coincidence angle θco, is given by 1

12sinco c

c 1

co

incid

en

ce

an

gle

θco

2

Cri

tica

l fr

eq

ue

ncy

no coincidence

Self-excited vibrations

The force acting on a vibrating system is usually independent of the

motion of the system.

In a self-excited system the excitation force is a function of the

motion of the system.

Examples of such systems where this phenomenon occurs are

• Instabilities of rotating shafts

• The flutter of turbine blades

• The flow induced vibration of pipes

• Aerodynamically induced motion of bridges

• Brake squeal


Stable system

Vibrating system

Unstable system

Vibrating system +

feedback system


To see the circumstances that lead to instability, consider a SDOF system

k

m x

c

The equation of motion is

0mx cx kx

Substituting for 2

n k m and 2c mk

gives 22 0n nx x x

Assuming a solution of the form gives the characteristic equation stx Xe

2 22 0n ns s

which has a solution 2

1,2 1n ns

For oscillatory motion are complex. The system will be dynamically

Stable provided 1,2s

0

unstable stable


Locus of

real axis

imaginary axis

1 1

0

0

1

0 1

0 1

0

0

1 2 and S S


Simple example

The vibration of an aircraft wing can be crudely modelled as

mx cx kx Ax

where m, c and k are the mass, damping and stiffness values of the wing

modelled as a SDOF system and where is an approximate model

of the aerodynamic force on the wing.

Ax

(1)

Rearranging (1) gives

0mx c A x kx

If C > A the system is stable If C < A the system is unstable

This is an example of flutter instability

Flow induced vibrations

There are three mechanisms for inducing vibration by

fluid flow

• Turbulence

• Vortex shedding

• Instabilities

Vibrations induced by turbulence

• Most common cause of flow induced vibration

• swaying of trees by gusts of wind

• vibration of a car aerial due to turbulent flow

• anchored ship bobbing in a tubulent sea

• heat exchanger vibration due to turbulent flow

• vibration of aircraft fuselage due to turbulent boundary layer

• random vibration problem

Vibrating system

2

gg ffS H j S

ffS ggS

Vortex induced vibration

• Note vortices being shed alternatively from each side of

the structure

Dynamic

forces

Vortex induced vibration

• Any structure with sufficiently bluff trailing edge sheds vortices in a

subsonic flow

Dynamic

forces

D U

• The separation of the flow is a function of the Reynolds number (Re)

inertial forceRe

viscous force

uD

v

u is the flow velocity

D is width normal to the free stream

v is the kinematic viscocity of the fluid which is the viscocity divided by the fluid

density

Vortex induced vibration regimes of flow

across a circular cylinder [after Blevins]

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local strain model

transmitted force ÃÃFÄÄ

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o

Frequency of vortex shedding

• The frequency of vortex shedding fs is given by

s

Suf

D

u is the flow velocity

D is width normal to the free stream

Vortex induced vibration – other effects

Self-Limiting Effect

If the amplitude of vibration increases beyond approximately 0.5D, the

symmetric pattern of alternatively spaced vortices begins to break up.

This means that the vortex induced forces are self-limiting at vibrations

of the order of D

Lock-In Effect

When the vortex shedding frequency

is close to the cylinder natural

frequency, the frequency of vortex

shedding can shift to the cylinder

natural frequency. This is known as

the lock-in effect

Vortex induced vibration - solutions

• Increase internal damping

• Avoid resonance nf Du

S

where fn is the natural frequency of the structure

• Take measures to eliminate vortex shedding

Galloping vibrations and stall flutter

If a structure vibrates in a steady flow, the flow in turn oscillates relative to the

moving structure, and acts as an oscillating aerodynamic force on the structure.

If this force tends to diminish the vibrations of the structure, then the structure is

said to be aerodynamically stable.

If this force tends to increase the vibrations of the structure, then the structure is

said to be aerodynamically unstable.

k c

m u

x

Wing stiffness

and damping

Flutter – aircraft wings

Examples

Galloping – ice coated power lines

k c

m u

x

Cable stiffness

and damping

cable

ice

Example - flutter


k c

m u

x

Wing stiffness

and damping

Example

x

u

relu

Causes a static

displacement

Causes instability

The vertical force acting on the aerofoil due to the fluid flow is given by

[Blevins]

21 1

2 2L

L D

cf u Dc uD c x

ρ is the density of the fluid

cL is the lift coefficient

cD is the drag coefficient

u is the free stream velocity


The equation of motion is

mx cx kx f

Substituting for the force component related to velocity gives

1

2L

D

cuDmx cx k c xx

or

Unstable when this < 0

10

2L

D

cmx c x kD c xu

2

LD

cu

cD c

Thus the minimum flow velocity to cause instability is

The system is unstable if the lift force increases when the model is slightly

rotated in a steady wind such that 0L Du c c

section xc Re


xc

note negative gradient

x LD

c cc

vertical force coefficientxc

Example – Tacoma Narrows Bridge

Example – bridge vibrations

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local strain model

transmitted force ÃÃFÄÄ

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o

stable

unstable

ae

rod

yn

am

ic d

am

pin

g

structural damping

Noise from Vortices

U

Vortices impinge on

structure generating sound

Vortices propagate at

speed of the flow

Sound propagates at

speed of sound

Noise from Vortices

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Incre

asin

g f

low

sp

eed

Summary

• Impacts

• Rotating machinery

– Gears

– Mass unbalance

• Acoustic excitation

• Self-excited vibration

• Flow-induced vibration

The following sources of vibration have been discussed

References

• R.H. Lyon 1987. Machinery Noise and Diagnostics, Butterworth

Publishers.

• F.S. Tse, I.E. Morse and R.T. Hinkle 1978. Mechanical

Vibrations, Theory and Applications, Second Edition, Allyn and

Bacon, Inc.

• F.J. Fahy 1985. Sound and Structural Vibration; Radiation,

Transmission and Response, Academic Press.

• R.D. Blevins 1977. Flow Induced Vibration. Van Nostrand

Reinhold Company.

professor mike brennan - unesp · • the flow induced vibration of pipes • aerodynamically...

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