Vibrations & Shock

8/18/2019 Vibrations & Shock

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MACHINE DESIGN EXCEL SPREAD SHEETS

Damped Vibrations With or!in" #n!tionThe inertia forces of rotating and oscillating machinery cause elastic supports to vibrate.

Vibration amplitudes can be reduced by installing vibration damping mounting pads or springs.

Simp$e Vibratin" S%stemsExternal forcing function F(t) varies with time and is externally applied to the mass M.

Fm is the maximum applied force.

M is the mass of the vibration obect that is e!ual to "#g.

g is the gravitational constant$ %&.& ft#sec'&.

is the displacement from the e!uilibrium position.

is the damping constant force per second velocity

and is proportional to velocity.

* is the spring stiffness force per inch.

&ndamped Vibrations+f the mass M shown above is displaced through distance x and released it will vibrate freely.

,ndamped vibrations are called free vibrations. -oth x and g are measured in inch units.

Inp#t

"eight$ " & lb

/pring stiffness$ 0 12 lb#in

Ca$!#$ation

Gra'itationa$ Content( " ) *+,+ -t.se!/+

0 ) *,12+Stati! De-$e!tion( 3 ) W . 4

) 5,+5 in

Mass( M ) W . 6"71+8

) 5,559 $bm:se!/+.in

Nat#ra$ re;#en!%( -n ) 61.+7087647.M8/,9 H<

) =>,59 H<

An"#$ar -re;#en!%( ? ) +707-n

) 2*2 radn.se!

opy write$ 3 Machine 4esign /preadsheet alculations by 5ohn 6 7ndrew$ 8 5uly &228

"e will assume$ F(t) Fm9/in(:t)

;mega$ : is the angular fre!uency as defined below.

4isplacement vsTime


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or!ed &ndamped Vibrations Inp#tMotor weight$ " =2 lb

Motor speed$ > 11=2 rpm

1B=2 rpm

,2 H


3/34

Dist#rbin" -or!e -re;#en!%( - ) N . =5 H<

) +>,1 H<

Dist#rbin" -or!e an"#$ar -re;#en!%( -a ) +707- rad.se!) 1*,* rad.se!

#t o- ba$an!e -or!e d#e to rotatin" mass

) Wi7-a/+7e . "

) 9+1= $b-

or!in" -re;#en!% . Nat#ra$ -re;#en!% ) r ) - . -n

) 1,22

Amp$it#de ma"ni-i!ation -a!tor( M ) 1.6 61 :r/+8 6+7Cr8/+8

) 5,=1

Vibration amp$it#de( 3 ) 6M876 . 48 in

) 5,1>= in

Transmissibi$it%( TR ) 6M8761 6+7r7C8/+8/,9 ) 5,2

Transmissibi$it% or!e( tr ) 6TR87

) 2=11 $b-

Criti!a$ Dampin"Criti!a$ dampin" o!!#rs Fhen the 'ibration amp$it#de is stab$e

C ) Dampin" Coe--i!ient

C!rit ) Criti!a$ Dampin" Coe--,

C!rit ) +767M8/,9

) S%stem sti--ness

M ) Vibratin" Mass


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Transmissibi$it% 6TR8Transmissibility is the ratio of the force

transmitted to a machineDs supports

due to a periodic imbalance in an engine$

pump$ compressor$ pulverier$ motor$ etc.

The amplitude of vibrations in machinery

mountings can be reduced with resilient

pads or springs called isolators.

The isolated system must have a natural

fre!uency less than 2.B2B x the disturbing

periodic imbalance force.

The vibration amplitude will increase if the

isolated system has a natural fre!uency

higher than 2.B2B x the disturbing fre!uency.

Transmissibility ratio is e!ual to the$ mass displacement amplitude # base displacement amplitude.

TR ) X+ . X1

The transmissibility ratio T6$ is the vibration amplitude reduction.

Inp#t

4isturbing force fre!uency$ fd 18.2 G

,ndamped natural fre!uency$ fn 1&.2 G

Ca$!#$ation

Transmissibi$it%( TR ) 1.61:6-d.-n8/+8

TR ) :1,+= C

I- mo#ntin" damper pad nat#ra$ -re;#en!% is 4noFn

Inp#t

Transmissibility$ T6 2.= C

4isturbing force fre!uency$ fd 1@ G

Ca$!#$ations

S%stem nat#ra$ -re;#en!%( -n ) -d . 6161.TR88/5,9

AnsFer -n ) ,1 G

/prings are employed as vibration isolators.

Series Sprin"s Combined Sti--ness Inp#t01 12 lbf#in

0& 1= lbf#in

Ca$!#$ation

1 . 4 ) 1 . 41 1 . 4+

4 ) 64174+8 . 641 4+8


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AnsFer 4 ) = lbf#in

Para$$e$ Sprin"s Combined Sti--nessInp#t

01 1& lbf# in

0& &@ lbf# in

Ca$!#$ation

AnsFer 4 ) 41 4+

4 ) *= $b-. in

Criti!a$ Speed o- Rotatin" Sha-tThe critical speed of a shaft is itsnatural fre!uency. The amplitude of

any vibrating system will increase

if an applied periodic force has the

same or nearly same fre!uency.

6esonance occurs at the critical

speed.

Inp#t

Flywheel mass$ " =2 lbm/haft diameter$ 4 1.222 in

/teel /haft$ E &H222222 lb#s! in

-earing center distance$ I& &2 in

Flywheel overhang$ I1 ? in

1 in/2

The ba$$ bearin"s a!t as pi'otin" s#pports

$%Fhee$ stati! de-$e!tion is

3 ) W7L1/+76L1L+8 .*7E7I in

) 5,5+1 in

Nat#ra$ -re;#en!%( - ) 61 . +70876" . 38/,9 H<

) +1,= H


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Beam Sti--ness 648( De-$e!tion 638( and Nat#ra$ re;#en!% 6 - 8Canti$e'er( $oad W at ree End Inp#t

Ioad at Free End$ " 822 lbf

Iength$ I %2 in

JoungDs Modulus$ E &H222222 lb#s! in

Moment of +nertia$ + @.222 in'@

Ca$!#$ation

De-$e!tion( 3 ) W7L/* . 6*7E7I8 in

AnsFer 3 ) 5,52 in

Sti--ness( 4 ) *7E7I.L/* $b-.in

AnsFer 4 ) 1+> $b-.in

Nat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9

- ) 1*+1 H<

Canti$e'er( &ni-orm Load F Inp#t

,niform Ioad$ w @=2 lbf#in

Iength$ I @ in


Moment of +nertia$ + &.222 in'@

Ca$!#$ation

De-$e!tion( 3 ) F7L/2 . 67E7I8 in

AnsFer 3 ) 5,551 in

Sti--ness( 4 ) 7E7I.L/* $b-.in

Nat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9

- ) >+ H<

Beam( Pinned ends( W at Mid Span Inp#t

Ioad at Mid /pan$ " @22 lbf

Iength$ I 82 in


Moment of +nertia$ + %.222 in'@

Ca$!#$ation

De-$e!tion( 3 ) W7L/* . 627E7I8 in

AnsFer 3 ) 5,5+1 in

Sti--ness( 4 ) 27E7I.L/* $b-.in

AnsFer 4 ) 1>***,********** $b-.inNat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9

- ) +>+ H<

Beam( Pinned ends( &ni-orm Load F Inp#t

,niform Ioad$ w =22 lbf#in

Iength$ I @2 in



Ca$!#$ation


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De-$e!tion( 3 ) 97F7L/2 . 6*27E7I8 in

AnsFer 3 ) 5,+ in

Sti--ness( 4 ) *27E7I.697L/*8 $b-.in

AnsFer 4 ) =>=55 $b-.in

Nat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9

- ) +12 H<

Beam( i3ed Ends( Load W at Mid Span Inp#t

Ioad at Mid /pan$ " B22 lbf

Iength$ I ?2 in



Ca$!#$ation

De-$e!tion( 3 ) W7L/* . 61>+7E7I8 in

AnsFer 3 ) 5,5*+ in

Sti--ness( 4 ) 1>+7E7I.L/* $b-.in

AnsFer 4 ) +195 $b-.in

Nat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9- ) 1>11 H<

Beam( i3ed ends( &ni-orm Load F Inp#t

,niform Ioad$ w 822 lbf#in

Iength$ I =2 in



Ca$!#$ation

De-$e!tion( 3 ) F7L/2 . 6*27E7I8 in

AnsFer 3 ) 5,1= in

Sti--ness( 4 ) *27E7I.6L/*8 $b-.in

AnsFer 4 ) 11= $b-.in

Nat#ra$ -re;#en!%( - ) 61.+0876" . 38/5,9

- ) *=9 H<

P$ate Nat#ra$ re;#en!% 6-8Re!tan"#$ar p$ate nat#ra$ -re;#en!%( - ) 6 . +708766D7"8.6F7a/288

6ectangular Alate$ simply supported edges ( ss

6ectangular Alate$ fixed edges ( -i3ed

Vibration Coe--i!ients

a . b ( ss ( -i3ed

1,5 1>, *=,5

5, 1=,+ +>,>

5,= 1*,2 +9,>5,2 11,9 +*,=

5,+ 15,* ++,=

5,5 >, ++,2

Re!tan"#$ar P$ate Nat#ra$ re;#en!% 6-8Inp#t

Modulus of elasticity$ E &.H2EK2B lbf#in'&

Alate thic0ness$ t 2.= in

Cir!#$ar Sti--ness a!torsircular Alate$ simply supportededges$ * @.HH.

ircular Alate$ fixed supported edges$* 12.&.


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AoissonDs ratio$ v 2.%

Alate short side$ a %8 in

Alate long side$ b @=.2 in

From the table above$ *$ss or *fixed 18.&

Ioad per unit area$ w =2 lb#in'&

Ca$!#$ation

AnsFer a . b ) 5,5

D ) E7t/* . 61+761 : J/+88

AnsFer D ) **1>=5

0 ) *,12+

Gra'itationa$ a!!e$eration( " ) *=,2 in.se!/+

Re!tan"#$ar P$ates( - ) 6 . +708766D7"8.6F7a/288

AnsFer - ) *,>* H<

Cir!#$ar P$ate Nat#ra$ re;#en!% 6-8 Inp#tIoad per unit area$ w =2 lb#in'&

Modulus of elasticity$ E &.H2EK2B lb#in'&

Alate thic0ness$ t 2.=

AoissonDs ratio$ v 2.%

Alate radius$ r %8 in

From the table above$ *$ss @.HH

*fixed 12.&

Ca$!#$ation

0 ) *,12+

" ) *=,2 in.se!/+

D ) E7t/* . 61+761 : J/+88

AnsFer D ) **1>=5

Simp$% s#pported ed"es( - ) 6 . +708766D7"8.6F7r/288

AnsFer - ) 1,+1* H<

i3ed ed"es( - ) 6 . +708766D7"8.6F7r/288

AnsFer - ) +,2> H<

Ba$an!in" Rotatin" Sha-ts

Masses in the Same P$aneFor static balanceL

Two masses$ M1 and M& must be in the

same plane and 1?2 degrees out of

phase and moments must balanceL

Kmi7Ri ) 5


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M17R1 M+7R+ ) 5

Masses in Di--erent P$anesFor static and dynamic balance there must

be no unbalanced moments and couples.

"hen the masses are in the same plane

static and dynamic balance occurs whenL

Kmi7Ri7Xi ) 5

M+7R+7X+ M*7R*7X* M27R27X2 ) 5

The cran0 (Mc) is statically and dynamically

balanced by two counter weights$ M1 M&$

all three masses are in the same plane.

Find the masses of the two counterweights.

Inp#t Example only

Mass 1 .


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M!7E7X1 ) M+7R+76X1X+8

M+ ) M!7E7X1 . R+76X1X+8

AnsFer M+ ) +,>95*1599> $bm

Condition -or stati! ba$an!e

Kmi7Ri ) 5

5 ) M17R1M+7R+:M!7E

Mass re;#ired to ba$an!e M!( M1 ) 6:M+7R+M!7E8 . R1AnsFer M1 ) *,*=>9=9+12 $bm

or!ed( Stead% State Vibration E3amp$e

alculate the two spring support stiffness

(0) if the horiontal vibration amplitude is to

be no more than 2.&= inches.

Estimated friction is =N of the critical

damping factor (c).

Inp#t

Motor speed$ > %82 rpm

MotorKompressorKTable Mass$ " ?2 lbm

ritical damping coefficient c

Friction damping coefficient f

(Friction# ritical) damping factor ratio$ 46 f # c

2.2=

7llowable vibration amplitude$ J 2.&= in

Ca$!#$ation

Motor speed( ? ) +707N . =5

AnsFer ? ) *,52 rad . se!

" ) *=,2 in.se!/+

M ) W . "

AnsFer M ) 5,+55 m:se!/+.in

Tota$ sprin" s#pport sti--ness( t ) +7

t ) M7?/+

AnsFer t ) +>2,* $b- . in

) t . +

AnsFer ) 12,+ $b- . in

Criti!a$ 'a$#e o- dampin" -a!tor( C! ) +76t7M8/,9

AnsFer C! ) 19,=1

ri!tion dampin" -a!tor( C- ) C!7DR


11/34

AnsFer C- ) 5,1

The motor periodi! imba$an!e -or!e( ) o7Sin6?7t8 $b-

The motor pea4 imba$an!e -or!e( o ) C-7?7 $b-

At resonan!e( ) o . C!7? in

o ) C-7?7

AnsFer o ) ,*= $b-

Verti!a$ Vibration Damper Se$e!tion 7 metal tumbling drum driven by an electric

motorCgear$ right$ rotates at 12?2 rpm causing

a disturbing vibration to the floor on which it is

mounted.

The loaded drum$ motor$ and support base .

weigh @22 lbm.

Vibration Iso$ator Se$e!tionSe$e!t 2 'ibration iso$ators that Fi$$ pro'ide

5 'ibration red#!tion app$ied to the -$oor, Inp#t

/ystem weight$ " &22 lbm

>umber of isolators$ > @Vibration reduction$ V6 2.?2

4isturbing fre!uency$ Fd 12?2 rpm

Ca$!#$ation

Wei"ht per iso$ator( F ) W . N $bm

AnsFer F ) 95

Transmissibi$it%( T ) 1 : VR

AnsFer T ) 5,+5

AnsFer d ) 1 rps

Transmissibi$it%( T ) 61 . 61:6d . n8/,98S%stem nat#ra$ -re;#en!%( n ) d . 61 61.T88/,9

AnsFer n ) ,*9 H<

" ) *=,2 -t . se!/+

Sti--ness( ) W . 3

De-$e!tion( 3 ) W .

&ndamped nat#ra$ -re;#en!%( n ) 61 . +08767" . W8/,9 H<

n ) 61 . +0876" . 38/,9


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n ) *,1+761 . 38/,9

So$'in" -or de-$e!tion in the abo'e( 3 ) 6*,1+8/+ . 6n8/+

AnsFer 3 ) 5,11 in

S#""ested ma3 transmissibi$it%( Tma3 ) 15

Re-, @En"ineered So$#tions@ a Barr% Contro$s p#b$i!ation,

At resonan!e transmissibi$it%( T ) 1. 6+7C . C!rit8

C . C!rit ) 1. 6+7T8

AnsFer C . C!rit ) 5,59

Iso$ator Se$e!ted


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The O-arry ontrolsO information presented here may be found on the web atL

O-arry 8%%7 /eries Mounts are medium weight mounts normally

used for vertically applied loads to prevent transmission of noise

and vibration caused by rotation of imbalanced e!uipment

(i.e. generators$ blowers$ pumps$ etc...)

IowCprofile$ low fre!uency elastomeric noise and vibration

www.barrycontrols.com

http://www.barrycontrols.com/http://www.barrycontrols.com/


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isolators for medium weight industrial e!uipment.O

The above graph shows a static load of 122 lbs produces a deflection of 2.&B= inches.


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This is the end o- this spread sheet,


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8H.2=&==


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MACHINE DESIGN EXCEL SPREAD SHEETS

Sho!4 Loads 7 shoc0 load is caused by a nearly instantaneous

rise and fall of acceleration.

/hoc0 input pulse is normally

expressed in gDs.

ree a$$ Impa!t Sho!4

7 typical free fall shoc0 test is an 11

millisecond second half sine waveform

with a pea0 acceleration of 1= g.

The above graph shows a static load of 122 lbs produces a natural fre!uency of B.& G.

Sho!4 Imp#$se De-$e!tion

7n electronic device is to be subected to a

1=g half sine shoc0 lasting 11 milliseconds.

The unit is mounted on a 12 G naturalfre!uency isolation system.

4etermine the maximum shoc0 transmission

Inp#t

Galf sine shoc0 acceleration$ a 1& g

/hoc0 pulse time$ t 2.21? sec

g %?8.@ in# sec'&

+solator natural fre!uency$ Fn &2 G

opy write$ 3 Machine 4esign /preadsheet alculations by 5ohn 6 7ndrew$ 8 5uly &228


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information shown here may be found atL

*orfund division of -aldor Motor corp.

and at the direct lin0 above.

OEffective vibration control for loads up to

. /tatic deflections up to 1.%8O. 7vailablewith$ or without adustable snubbing.O

O7pplications includeL /tationary e!uipment$

GV7$ ompressors$ Aumps$ Motor

, H<

Ha$- Sine Sho!4 P#$se re;#en!%( p ) 1. 6+ 7 t8AnsFer p ) 1==, H<

Sho!4 Absorber Se$e!tion

Ma3 Verti!a$ Sho!4 Transmitted( G' ) Wi 76+707n8. "

AnsFer G' ) >,5 "

Re;#ired A'era"e Sprin" Rate( s ) 6+707n8/+76W."8

AnsFer s ) 1** $b.in

Combined Iso$ator Verti!a$ re;#en!%( ! ) *,1*76s . Wi8

AnsFer ! ) 1>, H<

Ma3im#m D%nami! Tra'e$( Dt ) G'7" . 6+707s8/+

AnsFer Dt ) 5,++ in

Ma3 Ha$- Sine P#$se Ve$o!it%( V' ) +7"7G'7t . 0

AnsFer V' ) *=,> in.se!

httpL##www.baldor.com

http://www.baldor.com/http://www.baldor.com/


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7boveL *orfund division of -aldor Motor corp.

This is the end o- this spread sheet,

Vibrations & Shock

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