3 Expanded Understanding

7/31/2019 3 Expanded Understanding

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Noise and Vibration Control

3. Expanded understanding of Vibration isolation

Two types of passive vibration control:

(i) vibration isolation and (ii) vibration absorption.

Vibration isolation requires tuning the natural frequencyand damping ratio of a single-DOF system to reduce the"transmissibility ratio" between input and output.

Vibration absorption is a method of adding a tuned mass-spring absorber to a system to create an anti-resonance at a

resonance of the original system.

Indian Institute of Technology Roorkee


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Strategy to reduce noise and vibrations:

Interrupt the propagation path between the source andthe receiver.

Elastic mounting-effective and an inexpensive approach

Examples-

Strongly vibrating machines in factories, dwellings, andoffice buildings can be placed on elastic elements.

The passenger compartments in vehicles are isolated from

wheel-generated vibrations by incorporating springs betweenthe wheel axles and the chassis.

Indian Institute of Technology Roorkee 2/53


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3.1 Where transmissibility is and is not useful

Undesirable effects of vibration are reduced by inserting a

resilient member (isolator) between the vibrating mass and

the source of vibration so that a reduction in the dynamicresponse of the system is achieved

Isolation Systems Active or Passive

Whether or not external power is required for isolator to

perform its function.

Passive Isolator Consists of a resilient member (stiffness

and an energy dissipater (damping)) .

Eg. Metal springs, corle, felt, pneumaticActive Isolator Consists of a servo mechanism with a

sensor , signal processor and an actuator.



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Two types of situations:

1. The foundation of a machine is protected against largeunbalanced forces or impulsive forces.

Modeling the system as a single d.o.f. system

the force is transmitted to the foundation through spring

and damper .The force transmitted to the base (Ft) is given by

Ft(t)=kx(t)+cx(t)

dv

m

x(t

Figure 1. Single d.o.f. system

To achieve isolation, the force transmitted to the foundation

should be less than excitation force.

This can be possible if the forcing frequency is greater than

(21/2) times the natural frequency of the system.



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2. In the second type, the system is protected against the

motion of its foundation (as in the case of protection of adelicate instrument from the motion of its container).

Modeling the delicate instrument as a single d.o.f. system

, the force transmitted to the instrument (mass m) is given

by:

dv

m

x(t)

y(t

m

dv

x(t)

y(tFt(t)=m x(t)=k[x(t)-y(t)]+c[x(t)-y(t)]

where (x-y) and (x-y) denote the relativedisplacement and relative velocity of the spring and

damper respectively.



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Vibrationsource Receiver

Propagation path

Vibration

isolation

Wi

Wr

Wt

L

3.2 Some common misconceptions regarding

inertia bases, damping, and machine speed

Figure 1. A situation in which the vibrations emanatingfrom a machine are reduced by isolation.

- powerWiimpinges on the isolators,

- powerWris reflected back towards the source,

- and powerWtis transmitted to the floor.



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source isolation& receiver isolation

a) b

Figure 2 Two different strategies for vibration isolation: a) source

isolation of machines; and, b) shielding isolation of sensitive

equipment.



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Principle of vibration isolation

When a wave propagating in an elastic medium fallsupon an abrupt change (discontinuity) in the properties ofthe medium, only a portion of the wave passes throughthat discontinuity.

The remaining portion of the wave is reflected backtowards the direction from which the incident wave arrives.

The magnitude of the reflected portion of the wavedepends on the magnitude of the change in properties.



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vibration isolator & blocking masses

Wi

Wr

Wt

a)b) Wi

Wr

Wt

m

Figure 3 Two different vibration isolation methods.

a) Reflection against a soft element.

b) Reflection against a mass.

Wi = incident power,Wr= reflected power and Wt = transmitted power.



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Measures Of Transmission Isolation

insertion loss DIL :-

[dB] (1a)

[dB] (1b)

afterv

beforev

vIL LLD =

Fbefore

Without isolator With isolator

Fafter

after

F

before

F

F

IL LLD=

2

2~log10

ref

v

v

vL =

2

2~

log10

ref

F

F

FL =

Figure 4. The insertion loss can be defined as the difference in the

force level acting on the foundation before and after the

implementation of isolation.

Where velocity and force levels Lv

and LF

are defined as:

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Conclusions:

The vibration isolators must be designed such that themachines mounting frequency does not coincide with anyimportant excitation frequency.

A positive effect is obtained from the isolators atfrequencies above the mounting frequency.

The mounting resonance frequency should be as low aspossible. In practice, machine mounting is often designedso that the mounting resonance frequencyfalls in the 2-10Hz band.


C


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3.3 Design of vibration isolators

rule of thumb:

(i) The isolators static stiffness must be chosen so low thatthe highest mounting resonance falls far below the lowestinteresting excitation frequency.

(ii) The mounting positions on the foundation should be as stiffas possible.

(iii) The points at which the machine is coupled to the isolators

should also be as stiff as possible.

(iv) The isolator should, if possible, be designed so that its firstinternal anti-resonance falls well above the highest excitation

frequency of interest.


N i d Vib ti C t l


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If that rule cannot be followed due to practical reasons, thenone should ascertain by measurements or computations thatat least the following alternative rules are fulfilled:

(v) The isolator must be designed so that its internalresonances do not coincide with strong components of theexcitation spectrum.

(vi) The isolator must, furthermore, be designed so that itsantiresonance frequencies do not coincide with theresonance frequencies of the foundation.


N i d Vib ti C t l


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methods to improve vibration isolation

double layered isolation:A combination ofelastic elements and a blocking mass

(figure 3-16).

In practice, a double layered isolation is realized by

interposing a large mass between the machine and thefoundation.

The blocking mass should behave as a rigid body up to

frequencies that are as high as possible.

Wi

Wr

Wt

dyn

m

dyn

Figure 5. Schematic illustration of double layered isolation with

two compliant elements and one stiff element.




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Example - Passenger railway wagons.

The vibration source, i.e., the wheel-rail contact zone, is

isolated first by a primary suspension between the

bearings and the frame of the bogies.

To further improve passenger comfort and obtain

smooth ride characteristics,

a secondary suspension, or comfort suspension, isinterposed between the bogies and the body of the wagon




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Figure 6.An example of double elastic mounting : Attachment of a

railway wagon chassis to a bogie.

The primary suspension between the bearings and the frame is

commonly built up of stiff chevron elements, (rubber).

The secondary suspension, which connects the bogie to the chassis

of the railcar consists of very compliant air springs or spiral springs.

Primary

suspension

Secondary

suspension




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3.4 Accounting for support and machine frame flexibility,

isolator mass and wave effects, source reactionSome Vibration Isolation Computational Models

In order for computational models to serve as practical tools for thecomparison of different vibration isolation alternatives, simplified

models must inevitably be used.1. Rigid body ideal spring rigid foundation

At much lower excitation frequencies, considerably simplifiedmodels of the components are used.

Example: a machine mounted at four points on a system of concretejoists.

>The machine has an axle that generates sinusoidal bearing forcesat the rotational frequency.

>At very low disturbance frequencies (i.e., low rotational speeds),the deformations of the machine itself are negligible, i.e., themachine acts as a rigid body.

Mathematically, the machines movements can be described bymeans of equations from rigid body mechanics.




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Figure 5 a) Electric motor elastically mounted to a large steel plate

via four vibration isolators. b) Simplified model of the system in a.

c) The system in b represented by its separated subsystems.

a)m

4

m

x

4F

1

F1

x

F1

4F1

b) c)

x

Fstr

Fstr

Single

isolator

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As the rotational speed of the axle increases

force changes so rapidly that not all parts of the machine

have time to react

wave propagation in the machine.

With further increase in rotational speed - amplitude of the

machine deformations has a strong peak resonance occurs.

machine no longer regarded as a rigid body.

Thumb rule: the rigid body assumption valid up to

frequencies of 1/3 of the first resonance frequency, i.e., for

low Helmholtz numbers.

At very low excitation frequencies - the joists can be

regarded as a rigid foundation

only at low frequencies, say up to 1/3 of the first resonance

frequency, i.e., once again at low Helmholtz numbers.



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To reduce the vibrations transmitted from the machine into

the system of joists

incorporating soft vibration isolators at the mountingpositions between the machine and the joists.

Under the influence of forces from the machine, the springs

are deformed.

At low excitation frequencies, no considerable wave

propagation.

Yet another consequence - isolator can be considered

massless.

In contrast to the joists, the isolator is compliant.

The isolator can be regarded as an ideal massless spring.

As the frequency increases, the motion in the spring takes

on the character of wave propagation more and more.

Once again, at a certain point, the situation becomes

resonant.

Same thumb rule: spring idealization applies up to about 1/3

of the first resonance frequency.

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Figure 7 The insertion loss for a rigid body mounted elastically to arigid foundation.

The mounting resonance is tuned to f0 = 3.18 Hz.

Note the deep trough in the insertion loss at the mounting resonance,and its negative values elsewhere at low frequencies.The vibration isolation system is therefore counterproductive at lowfrequencies;

it is essential that the excitation frequency not fall in the vicinity of themachines mounting resonance frequency.

1-40

-20

0

100

80

40

20

60

f0 10 100 1000Frequency [Hz]

DIL[dB]




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conclusion

the vibration isolators must be designed to prevent thecoincidence of the machines mounting frequency with anyimportant excitation frequency.

a positive effect is obtained from the isolators at frequenciesabove the mounting frequency.

a mounting resonance frequency must be as low as possible

In practice, machine mounting is often designed so that themounting resonance frequencyfalls in the 2-10 Hz band.




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2. Flexible foundation

As the excitation frequency increases, the deformation of thefoundation due to the excitation force soon becomes too large

to ignore.

Use a model with flexible foundation

A number of different models available

infinite plate model - if the foundation is a system of joists with

considerable dimensions,

mass-damper system - If the foundation exhibits a resonance,




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m

4

m

4F1

F1

F14F1

b)

Fr

Fs

1

1 1

2

22

a)

Single isolator

Figure 8 Simple model of a machine mounted to a flexible foundation.

12

1

2

4Fx

= excFdt

dm

The equation of motion, Hookes law, and the mobility of a plate yield

the following system of equations:

)( 211 xxF =




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Without isolators, the force on the foundation can be determined

by excluding the second of the equations from the system givenabove, and setting x1 equal to x2. The system then has thesolution

1

1

2 4)( FYx platei=

=+

=

plateexc

with

im

m

F Y

F

)(441

442

2

1

Eliminate x1 and x2 ,

plateIm

m

I

m

i

mi

YYY

Y

Y

Y

++=

===

4

1

platem

m

plateexc

without

mi

mi

F YY

Y

Y

F

+=

+=

1

14 1

The insertion loss is therefore

platem

plateIm

ILDYY

YYY

+

++= log20

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Figure 9 Insertion loss for a rigid body elastically mounted to aninfinite steel plate. Compared to an ideal, rigid foundation, the

amplification peak at the mounting resonance frequency is reduced

and the rate of increase of the insertion loss falls off.

1

-40

-20

0

100

80

40

20

60


DIL[dB]

Rigid foundation

Compliant foundation

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3. Wave propagation in the isolator

When the excitation frequency has increased so much that the

deformation field in the isolator is a wave motion, the ideal spring

model becomes less and less tenable.

Depending on the isolator design, different models forwave

propagation in the isolatormay be appropriate.

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Figure 10 Insertion loss of isolators.

Rigid foundation

Compliant foundation

Wave propagation

in the isolator

Frequency [Hz]1-4 0

-2 0

0

1 0 0

8 0

4 0

2 0

6 0

f0 1 0 1 0 0 1 0 00Fr ek v en s [ H z]

D IL [ d B ]

O ef terg iv l ig t under lag

Ef terg iv l ig t u nder lag

V g u tb r ed n in g i

isolatorn6 4 H z

1 3 0 H z

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Deformable machine

So far, the machine is assumed as a rigid point mass.

The insertion loss is then very low at excitation frequencies near the

mounting frequency.

Considering model with several of the machines six rigid body d.o.f.

into account, several mounting resonance frequencies will be exhibited.

The most general case will therefore have critical frequencies at six

different mounting resonances.

Every real machine also exhibits internal resonances at certain

frequencies.

Typically, the first resonance frequency of a compact machine with a

100-kg mass, e.g., a small internal combustion engine, falls in the 100Hz - 500 Hz range.

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The possibility of wave propagation in the machine also

effects the isolation insertion loss obtained from elasticmounting.

If the machine stiffness varies significantly, due to

resonances and antiresonances, then even the insertion loss

will vary.the isolation performance is degraded above the machines

first resonance frequency.

Figure 11 shows the insertion loss of a simple system

consisting of a machine with internal resonances.The foundation is rigid and the isolator is the same.the machine has resonances at 185, 345 and 535 Hz, and

antiresonances at 160, 205 and 495 Hz.

at the resonance frequencies, at which the machine is compliant,there are insertion loss minima, i.e., frequencies at which the isolation

is poor.

extra isolation is obtained at the antiresonances, at which the

machine is very stiff.

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1

-40

-20

0

100

80

40

20

60


DIL[dB]

Machine structure with

internal resonances

Rigid machine structure 50 kg

Fext

50 (1 + 0,1i) MN/m

25 kg

15 kg

10 kg

Machine with Internal

Resonances

Figure 11 Insertion loss of a simple system consisting of a machine

with internal resonances. The right side of the figure shows a

mechanical model of the machine, and the input data used.

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General formula for insertion loss

FM

FIMILD

YY

YYY

+

++= log20

Machine

YM

Vibration

isolator(s)

YI

Foundation

YF

Figure 12 General vibration isolation problem. Every element of the

system is dynamically and acoustically characterized by its mobility.

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3.5 Commercial vibration isolators and dynamic stiffness

Various types of Vibration isolators:-

steel coil springs, rubber isolators, and gas springs.

The two fundamental properties of an isolator are itsdynamic stiffness and loss factor.

The stiffness is the property that largely determines thesuitability of an isolator.

The loss factor is significant as an amplitude-limitingparameter at resonances.




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Figure 13 Examples

of commercially-

available vibrationisolators




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Isolator dynamic stiffness

The performance of an isolator is largely determined by

the transfer stiffness.

That stiffness is the inverse ratio of a fixed deformationapplied to one end, and the resulting force obtained at the

other, blocked, end.

Reliable dynamic stiffness data is obtained by separatemeasurements for every individual isolator.




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The dynamic stiffnesses given in manufacturers tablesare usually only corrected static stiffnesses.

At high frequencies, the deviations between the true

dynamic stiffnesses, and the corrected static stiffnesses,are very large.

That is illustrated in figure 9, in which the measured

dynamic stiffness of a common circular cylindrical rubberisolator is shown.

The relative deviation between the corrected static

stiffness and the true dynamic stiffness can reach severalhundred percent.




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Static stiffness

Frequency[Hz]

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0 100 200 300 400 500 600 700

Dynamic stffness

[MN/m]

Corrected static stiffness

Figure 14. Measured dynamic stiffness of a circular cylindricalrubber isolator, compared to the static stiffness from themanufacturers catalog data, and the corrected static stiffness.

Apparently, the corrected static stiffness is only in agreement withthe true, measured dynamic stiffness at relatively low frequencies.



3 6 R l f A i I l i S d h d


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3.6 Role of Active Isolation Systems and methods toimprove vibration isolation

22

2

2

2

nn

nn

wsws

wsw

z

x

++

+=

&

&

damper

primary

mass

z

x

passive mount

disturbance

MPassive Vibration Control

System: fundamental system

examples

buildings

low end automobiles

Fig.15




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Application of active techniques to problems in vibration

isolation.

Two classes of problem:

(i) instances where we wish to isolate a vibrating body (such

as a machine of some kind) from a 'receiving structure'

(such as a car body, ship hull, aircraft fuselage or building)

and(ii) instances where we wish to isolate a body (such as

sensitive equipment or a railway car) from vibrations

imposed by another source (such as ground vibrations or

railway track unevenness).




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In both classes of problem, the source of the vibrations may

be either deterministic (i.e. having a perfectly predictablewaveform) or random (i.e. having a waveform that is not

perfectly predictable).

Most problems of the first kind, have a deterministic sourceof vibrations.

E.g.: whenever the source of vibrations is a rotating or

reciprocating machine.

In these cases we can adopt a feedforward control

approach to the problem.

Knowing the frequency of the vibration source , thenecessary control forces can be synthesised using the

adaptive feedforward techniques




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The second class of problem, is mostly dealt with using

feedback techniques.

e.g., design of active vehicle suspension systems.

Thus the body to be isolated is the passenger cabin of

the vehicle and the source of vibrations is the variable

height of the road surface, the latter being a random

process.




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Isolation of periodic vibrations of an SDOF system

Consider the isolation of an SDOF system from a

flexible substructure.

Assume that the single d.o.f. system is harmonicallyexcited, such that in practice, an adaptive feedforward

control system could be used.

This in turn assumes that the primary excitation forces inthe machine are deterministic (perfectly predictable)

e.g., the isolation of machines such as engines, pumps

and compressors from flexible structures such as the

hulls of ships and submarines or the bodies ofautomobiles.




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Ref. Fig.16,a secondary force can be applied at three possible

locations associated with the mass-spring-damper

system.

Assumption:

the primary complex excitation force fe is applied

to the mass M of the system.

(i) secondary force is also applied to the mass

and that to achieve zero response of the system

we require simply that

fs = -fp.




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Fig. 16. Active isolation of a harmonically excited SDOF system from

a receiving structure.

Three arrangements are shown for the application of a secondary

force (a) directly to the mass of the system,

(b) directly to the receiver




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(ii) Fig. 16 (b). :

Here the secondary force is applied directly to thereceiving structure with the objective of reducing the

response of the receiving structure to zero.

Assume that the receiver can be characterised bycomplex input receptance R (j),

such that its complex displacement wR is related to the

applied force f by wR= R(j)f.

The force applied to the receiver is the sum of the

secondary force and the forces applied via the spring

and viscous damper.

Thus we can write

WR= R(j)[fs + K(ws- wR) + j C(ws- wR)],




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where

ws - the complex displacement of the mass M,subscript s - the displacement of the 'source'.

For wR = 0

fs = - (K + jC)ws

For wR = 0




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or

where

n =(K/M)1/2 and damping ratio =C/2Mn

where

Non- dimensional frequency




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Fig.12. The magnitude of the secondary force required relative to that

of the primary force for the three arrangements shown in Fig. 11:(a) cancellation of the force applied to the mass;

(b) cancellation at the receiver;

(c) cancellation at the receiver with reaction against the mass.




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Fig. 16. Active isolation of a harmonically excited SDOF systemfrom a receiving structure.

(c) directly to the receiver with reaction against the mass of the

system



(iii) Fi 11


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(iii) Fig.11 c.:

Here the secondary force is applied in parallel with

the spring and damper such that it acts on the

receiver with a reaction against the source.

Example, an electrodynamic exciter were used with

the body of the exciter rigidly fixed to the source and

the excitation applied to the receiver;

Such an arrangement generates equal andopposite forces applied to both source and receiver.

For wR = 0




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shows that the displacement of the mass is exactly as if it

were freely suspended;the dynamic displacement of the mass is determined only

by its inertia.

Non- dimensionally



Active Vibration Control System:


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Active Vibration Control System:

controller

primarymass

Base z

x

passive mount

disturbance

M

u

advantages: performance

disadvantages: cost, complexity

examples: luxury cars




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TH ANK YOU


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