Vibration Isolation - UNESP · Vibration isolation – The mobility approach Source Y s A F Equipment to be mounted here Requirement • An item of flexible equipment is required

Vibration Isolation

Professor Mike Brennan

Vibration Isolation

• In vibration Isolation we modify the transmission path

Vibration

source Receiver Transmission path

• Simple approach

• Mobility approach – concept of isolator effectiveness

• Wave effects in the isolator

• Power flow approach

• Examples

Vibration Isolation by Resilient Connections

machines pipework

Low frequency models of vibration isolation

• Two problems

j t

tF e

Isolator

Receiver

Source

j t

eF e

j tXe

k c

m

(1)

Isolator

Host structure

equipment

k c

m

j t

tX e

j t

eX e

(2)

Low frequency models of vibration isolation

• The simplest model for vibration isolation is:

j t

tF e

Isolator

Receiver

Source

j t

eF e

j tXe

k c

m

The transmitted force, Fr is given by:

The equation of motion is:

( )mx cx kx f t For harmonic excitation:

2- em j c k X F

tF k j c X

(1)

(2)

Combining (1) and (2) gives the

force transmissibility

2

tF

e

F k j cT

F k m j c

Force transmissibility from SDOF model

(viscous damping)

• Transmissibility

can be written as:

2

1 2

1 2

nF

n n

j

T

j

where

,n

k

m

2 n

c

m

=0.01 =0.03 =0.1 =0.3

1 10 10

-3

10 -2

10 -1

10 0

10 1

10 2

Non-dimensional frequency n

Forc

e t

ran

sm

issib

ility

, T

F

2n

amplification isolation

Displacement transmissibility

Displacement Transmissibility

=

Force Transmissibility

t t

e e

X FT

X F

Isolator

Host structure

equipment

k c

m

j t

tX e

j t

eX e

Transmissibility- notes

•At low frequencies when n then 1T

•At resonance when n then 1

12

T

•At high frequencies when n then 2

1 2n

n

j

T

High damping T reduces at 20 dB/decade

Low damping T reduces at 40 dB/decade

Transmissibility – Low frequency

Below resonance At resonance Well-above resonance

Random excitation

Discussion of the SDOF model

• The simple model has a number of limitations for practical applications:

• Isolators often behave as hysteretically damped springs at

low frequencies. Replace (k+jc) by k(1+j) .

• The foundation is not rigid.

• At high frequencies internal resonances occur in the isolators.

• At high frequencies the source and receiver have modal behaviour

and can no longer be considered to be rigid.

• There are usually multiple mounts / mounting points.

• Usually more than one direction is important, including rotations.

Calculation of the fundamental natural frequency

from the static deflection

k

Undeformed

spring

k

m

System equilibrium

position

n

k

m

n

g

mg

k Static deflection

Isolator configurations- examples

Compression

l

A

stiffness, EA

kl

Shear

A

h

stiffness, GA

kh

2 1E G

Poisson's ratio

Young's modulusE

shear modulusG

Damping mechanism: complex Young’s modulus, * 1E E j

Loss factor

Isolator configurations- examples

Vibration isolation – The mobility approach

Source

sYA

Equipment to be mounted here F

Requirement • An item of flexible equipment is required to be

mounted on a flexible structure (source) that is

excited by unknown forces

• minimum vibration transmission

f

AVfree

• Before attaching the equipment the free velocity at A due to F

is

• The source mobility at point A is Ys. Now attach the equipment

(receiver) with mobility Yr to A through an isolator with mobility Yi


Source Isolator Receiver

sYiY rY

fVrV

F A

• The problem is to find Vr in terms of and the system mobilities,

i.e., given , Ys,Yi and Yr, what is Vr?

f

AVf

AV

• Assumption: The isolator is massless, i.e., the inertia forces

in the isolator are negligible


( ) (2)

( ) (

without isolator

with isolator 1)r

r

VE

VIsolator effectiveness

So

1 i

s r

YE

Y Y

Source Isolator Receiver

sYiY rY

fV rVFree velocity

. rr f

r s i

YV V

Y Y Y

(1)

• If no isolator is fitted, i.e., the receiver is rigidly connected directly

to the source, then Yi=0, and

. rr f

r s

YV V

Y Y

(2)

Vibration isolation - notes

1 i

s r

YE

Y Y

• E should be as large as possible for good isolation

• E depends on the source, isolator and receiver mobilities

• For good isolation i s rY Y Y

• We require high isolator mobility

• At high frequencies:

Ys and Yr can be large ~ resonance frequencies

Yi can be small ~ wave effects in the isolator

(the above theory is not strictly applicable because of inertia forces in

the isolator)

Some Measurements on a Container Ship





0 5 10 15 20 25 30 35 40

0

100

200

300

400

500

600

700

800

Engin

e s

peed(r

pm

)

Time(sec)


Horizontal transverse motion


Vertical motion


movie

MOV00025.MPG


Vertical motion; 75% engine load and synchronized speed of 720 rpm


Transverse motion; 75% engine load and synchronized speed of 720 rpm


Vertical acceleration either side of an engine mount


Transverse acceleration either side of an engine mount

Vibration isolation at high frequencies

source

receiver

isolator

Wave effects in the isolator

• At high frequencies there are standing waves in

the isolator

• Simple theory based upon the massless element

is not appropriate

• Suitable model is a distributed parameter

element for the isolator (including mass and

stiffness)

Wave effects in isolators

j tVe

j tFe

Example: Cylindrical distributed parameter element

l

A Appropriate quantity is the transfer impedance

sin

F jkEA

V kl (No damping) from tables

kE

(longitudinal wavenumber)

Young's modulusE

density

F

V

10 - 1

10 0

10 1

10 - 1

10 0

10 1

Non-dimensional frequency



j tVe

j tFe

l

sin

F jkEA

V kl

As 0, then

skF jEA

V l j

where s

AEk

l static stiffness

(1)

Substituting for ks in (1) gives

sin

sj k mF

V kl

where mass of isolatorm Al

(2)

The modulus of equation (2) has a

minimum when sin 1, which iskl

min

s

Fk m

V

Wave effects in isolators - damping

We can include damping by using the complex Young’s modulus

* 1E E j

This results in a complex wavenumber * 1 2k k j

Substituting into (1) gives

* *

*

1 1 2

sinsin cos cos sin

2 2

jkEA j jF jk E A

jkl jklV k lkl kl

max

2 skF

V so

Assuming sin 0

and <<1,gives

kl

0 1 12

jkl

0 0

1 sk m To find the first natural frequency set , to give sin 0kl


isolator

mass m

stiffness ks

j tVe

j tFe

10 -1

10 0

10 1

10 -1

10 0

10 1

ks/

2ks/

(ksm)

F

V

1

• At low frequencies the transmitted force is controlled by the static stiffness

• At high frequencies, above the first isolator resonance, the isolator

impedance . Thus decreasing the isolator mass will improve the

situation.

• Decreasing the mass will also increase

• At isolator resonances, the transmission of vibration is controlled by the

isolator damping

sk m

1 sk m

Isolator effectiveness

Frequency (arbitrary units)

10 -1

10 0

10 1

10 2

-40

-20

0

20

40

60

80

100

Iso

lato

r e

ffe

ctive

ne

ss (

dB

)

Effectiveness of isolator with rigid

source and receiver and massless

isolator

Isolator resonance

frequencies

Fundamental resonance

frequency

Source and receiver

resonance frequencies

10 -1

10 0

10 1

10 2 10

-3

10 -2

10 -1

10 0

10 1

10 2

10 3

Wave Effects in Isolators

10 -1

10 0

10 1

10 2 10

-3

10 -2

10 -1

10 0

10 1

10 2

10 3

No mass in isolator

0.01

0.1T

ran

sm

issib

ility

, T

Non-dimensional frequency, n

j t

eF e

j t

tFe

mass of isolator

0.05mass of equipment

Transmissibility of a simple mounting system on the assumption that

the mount behaves as a rod with internal damping

Further considerations

1. Isolators should be placed symmetrically with respect to

the centre of gravity

2. The centre of gravity should be located as low as possible

to avoid rocking effects

machine

mass

rubber or cork etc

isolation

foundation

centre of gravity

Example “floating floor”

Further considerations

3. Greater isolation can be achieved if a compound system is

used

machine

added mass

j t

eF e

j t

tFe 10

-110

010

110

2-80

-60

-40

-20

0

20

40

compound system

Simple system

1

Tra

nsm

issib

ility

(dB

)

Miscellaneous - isolation from

earthquake motion


earthquake motion


earthquake motion

Miscellaneous – rotating shafts

Miscellaneous – Isolation of buildings

and railways

Building isolator

Power flow approach

machinery

(source)

pipework

pipe hangers

isolators

substructure (receiver) 6 dof – displacements and rotations

Consider the machinery installation

• Can measure pressures, accelerations and strain

• Cannot compare measurements, thus cannot determine which paths

are important

• Isolators, pipes, shafts, electrical connections, acoustics, etc

• Unifying concept is power transmission or “power flow”

Vibration power input to a structure

structure

force, f

velocity, v

Power input to the structure is given by

0

T

in

1 = f(t) v(t) dt

T (1)

For harmonic inputs ( ) Re j tf t Fe and

( ) Re j tv t Ve equation (1) simplifies to

*1 1

cos Re2 2

in F V FV

Phase angle between

force and velocity

Complex

conjugate

Force source 21Re

2in F Y Velocity source

21Re

2in V Z

mobility of structure impedance of structure

1

2

beams

Power input to infinite beams and plates is constant

1

2

• Moments are better at inputting

power at high frequencies

machine

plates

is constant

pow

er

frequency

Plate

force

Plate

moment Beam

force Beam

moment

Summary

• Vibration isolation – simple approach

• Vibration isolation – mobility approach

• Wave effects in isolators

• Vibration power

References

• C.M. Harris, 1987, Shock and Vibration Handbook, Third

Edition, McGraw Hill.

• R.G. White and J.G. Walker, 1982, Noise and Vibration, Ellis

Horwood Publishers.

• L.L. Beranek and I.L. Ver, 1992. Noise and Vibration Control

Engineering, John Wiley and Sons.

• S.S. Rao, 1990, Mechanical Vibrations, Second Edition, Adison

Wesley.

Documents

Vibration Isolation - UNESP · Vibration isolation – The mobility approach Source Y s A F Equipment to be mounted here Requirement • An item of flexible equipment is required