Chapt.7 (Vibration Suppresion_Control)

8/7/2019 Chapt.7 (Vibration Suppresion_Control)

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Pukyong National University Intelligent Mechanics Lab.

William J. Palm III : Mechanical Vibration

7. Vibration Suppression and Control

Outline

This chapter considers how to design systems to eliminate or at least reduce

the effects of unwanted vibration.

h Acceptable vibration levels

h Sources of vibration

h Isolator design for fixed-based systems

h Isolation with base motion

h Dynamic vibration absorbers

h Active vibration control

h Chapter review


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Introduction

h Elimination orreduction of source causing the vibration :

y Balancing translating or rotating massesy Minimizing clearances such as bearings and pin joints

y Streamlining objects exposed to wind or currents

h Redesign of the system :

y Changing the natural frequency

y Dissipating the energy of vibration by adding damping Oil and friction dampers

Damping treatments which is coatings of damping materials

y Isolating the source by using isolator

y Using a vibration absorber

y Using an active control system

Wheel Weight


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Acceptable Vibration Levels

hWhat levels of vibration are harmful?

h For harmonic motion,

hWhat levels affect health and comfort?

Vibration Nomographs

2

( ) sin( )( ) cos( )

( ) sin( )

x t t x t t

x t

AA

A t

[

[

[ J[ J

[ J

! !

!

&

&&

h Amplitude :

2 2

, ,x x

x

x

x x

[ [

[ [ [

! ! !

! ! !

&

&& &

log log

log log

l

log

og

log

l l gog o

x

x

x

x

x

x

[

[

[

!

!

!

&

&&

&

&&

&

Log-Log scale

Velocity amplitude as a function of frequency [ for givendisplacement amplitude. All having a slope of+1

Velocity amplitude as a function of frequency [ for givenacceleration amplitude. Family of lines has a slope of-1

Vibration Nomograph


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Vibration Nomograph


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hMaximum allowable amplitudes of displacement, velocity and acceleration

Vibration Nomographs : Example

h Boundary formed by lines corresponding to these maximum values defines

the allowable operating region for the system

h Acceleration values are often

quoted as root mean square

(rms) values

For harmonic acceleration,

2rms

aa !

y Log-Log scale

Specification of vibration levels on nomograph


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h Difficult to quantify effects precisely, partly because of individual variability and

subjective responses in some cases

Effects of Vibration on People

y Immediate mechanical damage to the bodyy Longer-term health effects y Discomfort

hMaximum acceleration amplitude : limit most often specified forcomfort & health

often specified by gravitational acceleration constant g

hMaximum displacement amplitude :

often a function of available space,not usually related to discomfort

h Above 9kHz : beyond threshold of

perception by humans

h Tolerance of vibration : depend on

frequency and acceleration

Acceleration Comfort Level

0.03 g

0.03 ~ 0.08 g

0.08 ~ 0.13 g

0.13 ~ 0.2 g

> 0.2 g

Not uncomfortable

Somewhat uncomfortable

Uncomfortable

Very uncomfortable

Extremely uncomfortable

Duration

4 8

People fatigue

most quickly


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h Safe Exposure Limit

Effects of Vibration on People

y Recommended vertical acceleration limit to avoid health problemswhen exposure time is 8 hours

y Vibration below the reduced comfort level allows activitiessuch as reading, writing and eating to take place comfortably


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What are the Exposure Action and Limit Values (EAV/ELV)?

y The Control of Vibration at Work Regulations 2005 require you to take specific action

when the daily vibration exposure reaches a certain action value.y Exposure Action Value (EAV) is a daily amount of vibration exposure above which

employers are required to take action to control exposure. The greater the exposure

level, the greater the risk and the more action employers will need to take to reduce

the risk. Forhand-arm vibration the EAV is a daily exposure of2.5 m/s2 A(8).

y Level of vibration exposure that must not be exceeded: Exposure Limit Value (ELV)ELV is the maximum amount of vibration an employee may be exposed to on any

single day. Forhand-arm vibration the ELV is a daily exposure of5 m/s2 A(8).

y The Regulations allow a transitional period for the limit value until July 2010.

y Taken all reasonably practicable actions to reduce exposure as much as you can.

How vibration level and duration affect exposure (www.hse.gov.uk)


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Sources of Vibration

y Vibration can be caused by many types of excitation:

Hydraulic and aerodynamic forces due to fluid flow Reciprocating machinery

Rotating unbalanced machinery

Motion induced in vehicles traveling surfaces

Ground motion caused by earthquakes

Machine Primary Motion Vibration Type

Fans, Blowers

Centrifugal Pumps

Compressors

Generators, Motors

Turbines, Lathes

Washing machines

Rotation Sinusoidal

Piston EnginesReciprocating Pumps

Screening Machines

Weaving machines

Reciprocation Sinusoidal

Forging Hammers

Molding Presses

Punching Machines

Impact Transient


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Vibration Induced by Fluid Flow

y Generated by forces exerted on object by fluid motion

y Motion of vibrating object can alter fluid flow conditions, thus changing fluid forces

y Examples of vibration caused by fluid motion :

Wave action on structures

Vortex-induced vibration

Vibration caused by internal flows such as flow through pipes, horses with bends

Structural vibration caused by fluctuating aerodynamic forces such as turbulence


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Vortex-Induced Vibration

y Fluid flowing over object can sometimes separate from downstream side of object

y Vortices shed alternately from top and bottom of object, produces oscillating lift

y Resulting vibration of cylinder :

Increase lift force generated by shedding vortices

Cause shedding frequency to shift from that occurring with stationary cylinder

to natural frequency of cylinder

Increase drag force on cylinder

Change vortex pattern

Drag

Lift

Upstream Downstream


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Vibration from Reciprocating Engines

y Vibration of reciprocating engines caused by

Unbalanced motions of piston, connecting rod, crank Fluctuating steam or gas pressure in cylinder

y Vibration is transmitted to chassis or foundation by enginesand crankshaft undergoes torsional vibration

y Designers try to balance engines as much as possible

but its not possible to eliminate all vibration Counterweight

y Piston displacement & acceleration for single cylinder:

2

2

( ) cos cos4 4

( ) cos cos4

2

2

p

p

R Rx t R R t t

L L

Rx t R t tL

[ [

[ [[

}

} &&

Excitation frequency : [, 2[

This analysis is for single-cylinder, ignores dynamics

of crank & connecting rod, lateral force and effects of

pressure variation


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Isolator Design for Fixed-base Systems

y Excitation acts on mass whose resulting motion will produce vibration

in adjacent objects unless it is isolated During forging hammer strikes object to be formed

Impact can damage supporting structure and floor if system is not properly designed

y To isolate the vibration transmitted, insert isolatorbetweensupporting structure and mass being excited

Consider design of isolator Excitation : force transmitted to mass by vibration, Force isolation

Excitation : motion transmitted to mass by vibration, Displacement isolation

Force transmitted Displacement transmitted


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y Isolation system model which includes the mass of base : m2 : base or foundation mass

k2 : stiffness of floor or other supporting structure

Force f(t) is transmitted completely to base mass

Fixed-Base Model

y Isolation system model which treats the base as fixed :

m2 : very large (p g)

k2 : very stiff(p g)

Example : concrete foundation

Pumping Station


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y Equation of motion :

Isolation for Harmonic Excitation

( )m x cx kx f t !&& &

y Force transmitted to base :tf cx kx! &

y Transfer function :

2

( ) ( ) ( )

( ) ( ) ( )

t tF s F s X s cs k

F s X s F s ms cs k

! !

2( ) 1

( )

X s

F s ms cs k!

y Frequency transfer function :

2

( ) 1

( )

X i

F i k m c i

[[ [ [

! 2

( )

( )

tF i k c i

F i k m c i

[ [[ [ [

!

y Amplitude ratio :

2 2 2 2 2 2

1 1 1

( ) ( ) (1 ) (2 )

X

F kk m c r r[ [ :

! !

Displacement transmissibility

2 2 2

2 2 22 2 2

( ) 1 (2 )

(1 ) (2 )( ) ( )

tk c r

F r r k m c

[ ::[ [

! !

Force transmissibility

,2 /n

cr

mk k m

[ [:

[! ! !


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y Static deflection caused by constant forceF:

Displacement Isolation for Harmonic Excitation

y Force transmitted to base : /st F kH !

y Nondimensional amplitude ratio :

2 2 2

1

(1 ) (2 )st

kX

F r r

X

:H! !

y At high forcing frequency, responseamplitude approaches zero

Systems inertia prevents it from

following a rapidly varying forcing

function

This effect is due primarily to systems

inertia, not its damping

y Ifm is given, must choose kso that not close to r = 1 to obtain good isolation

y In some cases, also increase or decrease mass m to improve isolation

y However, cannot decrease mass because minimized already for well-designed machine

y Whether or not increase mass depends on applications

Amplification

Isolation

Displacement

Transmissibility


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y All curves pass through

Force Isolation for Harmonic Excitation

y To obtain good force isolation,to decrease transmitted force to

foundation, need to make

2 1.414r ! !

y Near resonance (r } 1),Ft/Fis highly

dependent on value of^

y Ifr > 1, ^ is small, then approximation formula

Force

Transmissibility

Isolation

/ MinimizetF Fp

y When r < 1.414, increasing ^ willdecreaseFt/F improve isolation

y When r > 1.414,Ft/F is not so highlydependent on ^,

decreases as ^decreases

2

2

11,

1

t r

r

F Tr

F r T

! !


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Formulas for fixed-base harmonic excitation


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Example : Undamped Isolator Design

Design an umdamped isolator for a 20 kg mass subjected to a harmonic forcing

function whose amplitude and frequency are 600 N and 17 Hz.The isolator should transmit to the base no more than 10% of the applied force.

Determine the resulting displacement amplitude.

y Displacement amplitude :

0.1,r

!

y Isolator stiffness :

2 1 1 0 11.1

0.1

r

r

r

! ! !

2 2

22

n

m

rk

[ [

[! !

2 2

2

420(2 17)

12.0744 10 /m

1N

mk

r

[ T v! v! !

2 4

600 1 600 1

1 2.0744 10.0029m 2.9m

0 11m

1X

k r! ! !!

v


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y Equation of motion : ( )M x cx kx f t !&& &

y Vertical component of unbalance force : 2 sinu R Rf mR t[ [!

y Amplitude ratio :

2 2 2 2 2 2 2

1 1 1

( ) ( ) (1 ) (2 )R R R

X X

F mR kk c r r[ [ [ :! ! !

,2

/

R R

n

c

Mk

r

k M

:

[ [

[

!

! !

Isolation from Rotating Unbalance

2 2 2

2 2 2 22 2 2

( ) 1 (2 )

(1 ) (2 )( ) ( )

Rt t

r

R R R

k cF F r

F mR r rk c

[ :[ :[ [

! ! ! !

2, ,R Rm F mR[ [ [p p p

System with rotating unbalance


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Formulas for rotating unbalance


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Example : Support Vibration due to Rotating Unbalance

y System mass included 23% of beam mass :

y Spring constant :3 11 3

3 3

62 10 0.1 0.01

4 4(0.15)1.4815 10 N/m

Ewhk

Lv

v v v! ! !

AC motor runs at a constant speed, 1750 rpm. Amotor mass of 8 kg is mounted

on a steel cantilever beam with 15 cm long, 10 cm wide, and 1 cm thick.The rotating part of the motor has a mass of 4 kg and an eccentricity of 0.3 mm.

The damping ratio for beam is difficult to determine but ^ = 0.1.Estimate the amplitude of vibration of the beam at steady-state.

3

motor beam0.23 8 0.23(7.8 10 )(0.1)(0.01) 8.269kg M M M ! ! v !

y Unbalanced mass : 4 kgm !

2 2

/ 423 rad/s, 1750 rpm 183 rad/s, / 183 / 423 0.433

4(0.0003)(183) 40.4 , 0.1

n R R n

R

k M r

F mR

[ [ [ [

[ :

! ! ! ! ! ! !

! ! ! !

2

2 2

5

2

1

(1 ) (2 )3.3 10 mR

mRX

k r r

[

:

! !

v

y Steady-state amplitude :

y Total amplitude included static displacement :55 6

3.3 10 8(9.8)/1.4 8.8 515 9 1 m10 0 vv v !


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Example : Support Vibration due to Rotating Unbalance

y In this region, assumed value of^ would be critical in amplitude calculation.In practice, such a design must be avoided

y In vibration analysis, the most important

quantity to know is the natural frequency

y For motor speedforcing frequency very close to natural frequency

3500 rpm 366rad/s,R[ ! !/ 366/423 0.865R nr [ [! ! !

Amplification

Isolation

Displacement

Transmissibility

Resonance

Region (s 20%)

y If natural frequency is not close to forcingfrequency, not usually necessary to know

precise amount of damping


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Example : Isolation of a Motor

Motors are mounted to a base with an isolator consisting of an elastic pad.

Pad serves to reduce motors rotating unbalance force transmitted to the base.A motor has a mass of 2 kg and runs at 5000 rpm.

Neglect damping in pad and calculate pad stiffness required to provide a 95%

reduction in force transmitted from motor to base.

y If pads damping is slight (^=0.1), exact expression gives Tr

=0.07,

a 93% reduction that is close to desired value of 95%

0.05r

T !

y Isolator stiffness :

22

1 11

0.05

0.05

r

r

Tr

T

! ! !

24

2

2

(5000 2 / 60)2

212.61 10 N/mRk M

r

[ Tv

v! ! !

y 95% force reduction :2

2 2

2

R

R

n

Mr

k

[ [[

! !


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Example : Transmitted Force

When machine is rotating at 3600 rpm, the effect of rotating unbalance is to exert

a force of 350 N on machine, whose total mass is 150 kg.The isolator value are

(a) Compute steady-state amplitude of displacement

(b) Compute magnitude of force transmitted to foundation at steady-state

y Displacement :

2

3

2.46

350

( 3 10 k377 m) gmR

v! !

2

2 2

5

2 2.8 10 m

1

(1 ) (2 )

RmR

X k r r

[

:

! ! ! v

22 2 2

2 7

150(377)

1.6 11.333

0

R

R

n

Mr

k

[[

[! ! ! !

v

71.6 10 N/m, 0.3.k :! v !

2 , 3600(350N 377ra2 ) / 6 d0 /sR RmR[ [ T! ! !

22

2 2 2

1 (2 )

(1 )554N

(2 )t R

rF mR

r r

:[

:

! !

y Transmitted force :


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Practical Isolator Design

y Vibration isolator design is to compute required values for materials damping & stiffness

y Designers then must look at vendor catalogs for existing mounts and materialsthat have values of damping & stiffness near required values

y Commercially available isolator : mount, elastic material

y Also, must take into account any requirements or constraintson size, shape, weight imposed on mount by particular application


y If none can be found, there is often enough latitude to recompute another set ofdamping & stiffness values General case

y Other factors must also be considered such as cost, ease of installation, reliability,and availability

y In many applications, inputs is not constant frequency, not harmonic

Example : reciprocating engines, vehicle suspension, earthquakes


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Minimum Vibration Isolation Efficiency


Type of Equipment CriticalArea:

Church, Restaurants,Stores,Office Bldgs,Schools, Hospitals,Broadcasting Studios

Non-CriticalAreas:

Laundries, Factories,Subbasements, Garages,Warehouses

Air-conditioners (self-

contained)

90% 70%

Air handling units 80% 70%

Compressors (centrifugal) 99% 80%

Compressors Up to 10 HP

(reciprocating) 15 50 HP

60 150 HP

85%

90%

95%

70%

75%

80%

Heating & Ventilating 80% 70%

Cooling Towers 80% 70%

Condensers Air cooled

Evaporative

80% 70%

Piping 90% 70%

Pumps Up to 3 HP

5 HP or over

80%

95%

70%

80%

Steam Generators (packaged) See selection guide See selection guide


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Commercially Available Isolators

y Common type of isolator is made ofrubberor anotherelastomer

y Many companies offer vibration isolators which is a wide range of designs available


Rubber isolators Belleville springs

Nonlinear springs

Hardening

springs

Nonlinear springs

with mechanical stops


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y Equation of motion : ) or( ( )tmx f c y x k y x m x cx k x cy ky! ! ! && & & && & &y Force transmitted to mass : ( ) ( )tf c y x k y x! & &


2

2

( )( )

( )

tF s ms

cs kY s ms cs k

!

2

2 2 2

2( )

( ) 2

n n

n n

sX s cs k

Y s ms cs k s s

:[ [

:[ [

! !

y Dimensionless amplitude ratio :2

2 2 2

1 (2 )

(1 ) (2 )

X r

Y r r

:

:

! Displacement transmissibility

22

2 2 2

1 (2 )

(1 ) (2 )

tF r

rkY r r

::

!

Force transmissibility

,2 /n

cr

mk k m

[ [:

[! ! !

Isolation with Base Motion

y Common input : motion of a base support (base excitation)

Ratio of mass motion to base motion

Ratio of transmitted force to base motion

2 2,tt

F Xr F r kX

kY Y! !


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y Displacement transmissibility X/Y: r } 1(resonance region), curve is at maximum This means that maximum base motion is transferred to mass (amplification)

If

If displacement transmissibility decreases as r, ^are increased

Amplification

Isolation

2, / 1; 2, / 1r X Y r X Y " u

2,r u

y Force transmissibilityFt/ kY: r } 1(resonance region), curve is at maximum If force transmissibility does not necessarily decreases as r is increased

For example, if^= 1, this increases with r

2,ru

^

y Formulas and plots can be used to design isolators to protect objects fromunwanted vibration


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Formulas for base excitation


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Example : Transmissibility of PCB

y Equivalent mass (= 1/2 of beam mass) :

y Board stiffness :3 10 3

3

4

3

16 16 1.38 10 0.178 0.0016

0

2.012 10 N m.2

/Ewh

kL

v v v v! ! ! v

PCB (printed circuit board) supported by a chassis that is attached to vibrating

motor. The board is 1.6 mm thick, 178 mm wide, 200 mm long, mass of 0.45 kg.

Neglect damping, model the board as a fixed-fixed beam.

Determine its stiffness & natural frequency, and compute displacement

transmissibility if chassis vibrates at 60 Hz due to motor unbalance.

Youngs modulus of epoxy fiberglass board is

0.45 / 2 0.225kge

m ! !

y Frequency ratio :

42.012 10

0.299rad/s

225n[

v! !

60(21.261

)

299n

r[ T[

! ! !

21.6

19 169or

1%

rT

r

! !

y Displacement transmissibility :


10 21.38 10 N/mE! v

y Natural frequency :

PCB may be sensitive to vibration because vibration can loosen the soldered joints

attaching the components (resisters, capacitors)


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Dynamic Vibration Absorbers

yVibration Absorber :

Useful for situation with constant forcing frequency

Device consisting ofanother mass and a stiffness element that are attachedto main mass to be protected from vibration

2 DOF system consisting ofmain mass and absorber mass

Two natural frequencies

If forcing and natural frequency are known, we can select values for absorbers

mass and stiffness

Vibration energy of main mass is transferred to absorber systemThen resulting absorber motion will be large

Another term : dynamic vibration absorberorturned mass damper

yVibration Isolator :

Device consisting of a stiffness and damping elements

Intended to isolate one part of structure from an excitation or from another part


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Example of Dynamic Absorbers


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y Devices that run at constant speed such as

saws, sanders, shavers, passenger cars, power lines,buildings, bridges, devices powered by AC motors

Example of Dynamic Absorbers

Vibration absorber for exhaust pipe Stockbridge damper for power line

Vibration absorber for tall building


36/47


y Equation of motion : 1 1 1 1 2 1 2

2 2 2 1 2

( )

( )

m x k x k x x f

m x k x x

!

!

&&

&&


2

1 2 21 2 2 2

1 1 2 2 2 2

( )( )

( ) ( )( )

X s m s kT s

F s m s k k m s k k

! !


Analysis of Vibration Absorber


Main

mass

Absorbermass

2

1 1 2 1 2 2

2

2 1 2 2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) 0

m s k k X s k X s F s

k X s m s k X s

!

!

2 22 2 2 2

1 1 2 2 2 2

( )( )

( ) ( )( )

X s kT s

F s m s k k m s k k! !

22

2

2 2 2

1 2 2 2 21 2 1 2 2 2 1 2 1 2 2

1 1 2

2

2

2 21 2 21

1

2

1 1

11

( )( )( ) 1 1

11

1 1

m

k m kT i

k k m k m k k k m m k

r

k k kr r

k

k k

k

k k

[[

[[ [

[

! !

!

1 2

1 21 1 2 2

,/ /n n

r r

k m k m

[ [ [ [[ [

! ! ! !


37/47


y Define some parameters :2 2 2

2 2 2 22 2 22 2 2 1 21 2

1 1 1 1 1 2 1 1 1 2

, , ,n n n

n n n n n

m k m m kb b r b r

m k m k m

[ [ [[ [ Q Q Q

[ [ [ [ [

! ! ! ! ! ! ! !



2

2 2 2 22 21 2 1 2 2 2 1 2 2

1 2

1 1

1 1

( ) ( )( )1 1

kT i

k k m k m k k k kr r

k k

[ [ [! !


2

1 21

2

2

22 2 2 4 2 2

1 22 2

1 2 22

( ) 11( )

( ) 1

11

[1 (1 ) ]1 1

X i r T i

F i k b b r

r

k bb b rr r

[[

[ Q QQ

! ! !

2 4 2 2

1 2 2

22

1 1

[1 (1 ) ] 1

( )( )

( ) k b r

X iT i

F i b r[ Q[

[

!

!

Both frequency transfer functions are real numbers

y Because , we have1 1( ) ( ) ( )X s T s F s! 1 1( ) ( ) ( )X i T i F i[ [ [!

y If steady-state motion ofm1( ) sin ,f t F t[!

1 1 1 1 1( ) sin( ), ( )x t X t X T i F[ J [! ! 1 1( ) 0,If 0T i X[ ! !


38/47


y Because r2 cannot be negative by definition, absorber design equation is givenby r2 = 1

22 2

2 2

or1 nn

kr

m

[[ [

[! ! ! !



y Thus mass m1 will be motionless if we select an absorber having same naturalfrequency [n2 as frequency [ of applied force

If this is done, absorber is said to be tuned to input frequency

2

2 1 2

1 0, (I ) 0 1r T i r [ ! ! @ ! s

y Ifr2 = 1, expression forT2 (i[) becomes

2 2

1

2

2

2

1 1 1

1 (1

( )( )

( ) ) 1

X iT i

F i k b b k

[

[ Q

[ !

! !

y Thus, if absorber is designed so that r2 = 1, then

2 2

2

( ) ( ) ( )1

( )X i T i F i F ik

[[ [ [! !

Amplitude of absorbers motion : 2 22

(1

)Xk

X i F[! !


39/47


y Because transfer function T2 (i[) is real and negative, absorbers spring force

acting on main mass is

2 2 1 2 2 2 2 2 2( ) sin( ) sink x x k x k X t k X t[ T [ ! ! !



Thus, if absorber is tuned to input frequency and its motion has reached

steady-state, force acting on absorbers mass has same magnitude Fas

applied force but is in opposite direction

Net force acting on main mass to be zero, therefore, it does not move

y In practice, allowable clearance for absorbers motion X2 puts a limit on allowablerange of absorbers stiffness k2

y Absorbers stiffness k2 must be able to support forceFand resulting compression

or extension X2

By setting

1 1 / 1k X F

y Because X2 =F/k2, 2 2 2 2( / ) sin sink x k F k t F t[ [! !

y Frequency range over which absorber is effective

1 1( ) 1,kT i[ ! s 2

21 1 2 2 2 2 2

2 2

1(

1 11)

rkT i

b b r r b[

Q Q

! !

s

2 4 2 2 2

2 2 21 , (2 ) 2 0r b r b b rQ Q! !


40/47


Example : Sensitivity Analysis in Absorber Design

y Absorbers design requires that

y Natural frequency of machine :

A machine with supports has a measured natural frequency of 3.43Hz.

Machine will be subjected to a rotating unbalance force of 13 N andfrequency of 3 Hz. Design a dynamic vibration absorber for this machine.

Available clearance foe absorbers motion is 25 mm.

y Maximum allowable clearance : 25 mm

2 2

2 520N/m13

0.025F

k k

k !! ! p

y Thus absorbers mass :


1 2 (3.43) 6.86 rad/sn T[ T! !

y If absorbers natural frequency is not exactly equal to input frequency, main mass will vibratey Vibration amplitude depends on difference between input and absorbers frequency

y Frequency of applied force : 2 (3 r d/) 6 a s[ TT! !

2 6n[ [ T! !

22

2

6nk

m[ T! ! 2

2 236

km

Tp !

22 2 2

520

36 361.46kg

km

T T! ! !


41/47


42/47


Assignment :Vibration Absorber


Rotating unbalance of motor is at 3.15 rad/s.

Is vibration of M1 significantly absorbed?

1 1 2 2M 50 kg, k 450 N/m, M 10 kg, k 90N/m! ! ! !

M1 : Total mass of machine, motor, motor unbalance

K1 : Mount stiffnessM2 : Absorber mass

K2 : Absorber stiffness

I l i i h B M i


43/47


Assignment : Design of Vibration Absorber


Motor-generator set is designed to operate between 2,000~4,000 rpm. However,

this set vibrates violently at 3,000 rpm due to unbalance in rotor.W

hen a cantileverwith a 2 kg trial mass tuned to 3,000 rpm is attached to the set, the resultant natural

frequency are 2,500 and 3,500 rpm.

Design the vibration absorber (mass and stiffness) so that the natural frequencies of

the total system fall outside the operating speed range of the motor-generator set.

1 22,500 rpm 261.80 rad/s, 3,500 rpm 366.52rad/sn n; ! ! ; ! !

3, 000rpm 314.16 rad/s[ ! !

I l ti ith B M ti


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Damping in Vibration Dampers

y Because absorber model has no damping, its not stable but neutral stable

y Equation for a model with no damping are strictly true only if system is stableand at steady-state

y Nevertheless, these equations are widely used because inclusion of dampingcomplicates mathematics

y Existence of damping is a very complicated topic,the results are not easily presented in a concise form


y Transient response could affect acceptability of absorber design

1 1 2 2( ) , ( )X T i F X T i F[ [! !

y In practice, design of vibration absorber is based on steady-state response

I l ti ith B M ti


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Torsional Dampers


Fluidampr


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Changing natural frequency either by increasing or decreasing the mass

or by increasing the stiffness

We have studied several ways to reduce unwanted vibration includingredesign of system by

Chapter Review

Dissipating the energy of vibration by adding damping

Isolating the source by using an isolator consisting of a stiffness element

and a damping element placed between source and surrounding environment

Using a vibration absorber


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Pukyong National University Intelligent Mechanics Lab

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Chapt.7 (Vibration Suppresion_Control)

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