chapt.7 (vibration suppresion_control)
TRANSCRIPT
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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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Acceptable Vibration Levels
Vibration Nomograph
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Acceptable Vibration Levels
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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Acceptable Vibration Levels
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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Acceptable Vibration Levels
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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Acceptable Vibration Levels
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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Sources of Vibration
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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Sources of Vibration
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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Sources of Vibration
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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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Isolator Design for Fixed-base Systems
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
Isolator Design for Fixed-base Systems
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
Isolator Design for Fixed-base Systems
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
Isolator Design for Fixed-base Systems
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! & &
y Transfer function :
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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Isolation with Base Motion
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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Isolation with Base Motion
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 :
Isolation with Base Motion
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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Dynamic Vibration Absorbers
Example of Dynamic Absorbers
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Dynamic Vibration Absorbers
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
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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
!
!
&&
&&
y Transfer function :
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
! !
y Frequency transfer function :
Analysis of Vibration Absorber
Dynamic Vibration Absorbers
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
[ [ [ [[ [
! ! ! !
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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
[ [ [ [ [
! ! ! ! ! ! ! !
Analysis of Vibration Absorber
Dynamic Vibration Absorbers
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
[ [ [! !
y Frequency transfer function :
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[ ! !
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Pukyong National University Intelligent Mechanics Lab.
y Because r2 cannot be negative by definition, absorber design equation is givenby r2 = 1
22 2
2 2
or1 nn
kr
m
[[ [
[! ! ! !
Analysis of Vibration Absorber
Dynamic Vibration Absorbers
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[! !
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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 [ ! ! !
Analysis of Vibration Absorber
Dynamic Vibration Absorbers
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! !
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Pukyong National University Intelligent Mechanics Lab.
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 :
Isolation with Base Motion
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! ! !
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Pukyong National University Intelligent Mechanics Lab.
Assignment :Vibration Absorber
Isolation with Base Motion
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
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Assignment : Design of Vibration Absorber
Isolation with Base Motion
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
Isolation with Base Motion
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
Isolation with Base Motion
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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