explanation

ReprintedfromVMEbusSystems/December2005Copyright2005

WhatisthemeaningofPSD

ing2/Hzunits?

ByWayneTustin

2

TESTING,1,2,3...Special

Tom’sassignment

Tomisaskedtophysicallymeasurethevibrationexistingatsome

hersvibration-measuring

arily,however,

Tombypassthefilter.

Beforetakingmeasurements,Tomcalibrateshismeasurement

orarilyattacheshisaccelerometertothetable

ofasmallelectrodynamiccalibrationshakerthatdevelops1g

RootMeanSquare(RMS);thatis,1.414gpeak,atthearbitrarily

ustshis“tgain”

controlsothathistrueRMSreadoutdisplays1Vrms.

Whatisthensitivityofhismeasurementsystem?Thatis,what

isTom’stransfernsitivityfromaccelerationingtovolts?1:1,

right?WheneverTomes1Vonthemeter,heknowsthathis

erTomes2Von

themeter,heknowsthathisaccelerometerixperiencing2g

andsoon.

s

eps

thefrequencyofhiscalibrationshaker(maintaining1-gRMS

=1.414gpeak)ts

hisvoltmeterreading(whichsurprisinglyremainsconstantat

1Vrms)2showsthecalibrationresults.

Intherealworld,ofcour,theresponwillneverbethisflat.

Thegraphwasdrawnwitharuler.

SomethingmotivatesTomtointroducethebandpassfiltershown

ststhefiltercenter

repeatstheprevioussweepandgraph.

ncydepicted

’snsitivityisstill1V/gbutonlyfrom920Hz

transitionsatthe“cutoff”,

too,yreasonforincludingFig-

ure3istheremotepossibilitythatsomereadersmightnotbe

familiarwithbandpassfilters.

Finally,oves

hisaccelerometerfromthecalibrationshakerandattachesitatthe

Editor’snote:WayneTustinisonetheelectronicsindustry’firstofaries,Wayneusa

“Tom,Dick,andHarry”fabletohighlighttheimportanceofcontextinconductingvibrationtesting.

vibration

input

motion

bypass

t

gain

t

center

frequency

t

bandwidth

“B”

crystal

accelerometer

accelerometer

amplifier

variable

bandpass

filter

TRMS

electronic

voltmeter

Figure1

Copyright2005ReprintedfromVMEbusSystems/December2005

the4Vrmsmeterreading,Tominfersthathisaccelerometerwas

nsing4gRMS.

Dick’sassignment

pendentlyrepeatachof

Tom’ssteps,withoneexception:Helectsa40Hzbandwidth.

Atthispointsomekeyinformationshouldbenotedthatneither

TomnorDickwouldhavewithouttakingfurthermeasurements:

Thevibrationthatintereststhemis(1)randomanditis(2)white

eof(1)and(2),theirvoltmeter

readoutswillbeproportionaltothesquarerootofbandwidth.

Dick’sbandwidthisone-fourthofTom’s,sohisvoltmeterread-

ingwillbeone-halfofTom’,Dick’svoltmeterindicates

e2Vrmsreadingofhismeter,Dickinfersthat

hisaccelerometerisnsing2-gRMS.

Harry’sassignment

TomandDickcomparereadingsandarepuzzledatthediffer-

ndependentlyrepeats

eachofTom’sandDick’sstepswithoneexception:Forsomerea-

son,Harrylectsa10Hzbandwidthfilter,one-fourthofDick’s.

Hegets1V(one-halfofDick’s)andreports1-gRMStoTomand

threearepuzzled–threevalues.

Whowascorrect?

WasTom’sreadingcorrect?WasDick’s?WasHarry’s?Inan,

allwerecorrectbutallerredinfailingtostatethefrequencyrange

overwhicheachhadmeasured:920-1080,980-1020,995-1005Hz,

eymighthaveagreedthateveryonemustu

thesamebandwidth.

Historically,thereportsofsomeearlyrandomvibrationinves-

tigatorsfailedtologthebandwidthstheinvestigatorshadud.

Theirdatawasntiallyuless.

Manipulatedata

Table1indicatesthathadTom,Dick,andHarryeachsquaredhis

RMSreadingsanddividedbythebandwidtheachud;theneach

wouldhavereported0.1g2/Hz.

Variouslaboratories,firms,andagenciessimilarlycommunicate

ectofbandwidth

iscanceledout(normalized).Nearlyeveryoneusthelf-

explanatorytermg2/wworkersuametricversion

ofPSD/ASD:m2/s3.

ThefinalcolumnofTable1,withtheunitsg2/Hz,iscommonly

rkerscallitAccelerationSpectralDensity

(ASD)nesntially

thesamething.

Power?Examinethespectrum

Howcanweuthetermpowerinthiscontext?Notethatvolt-

agee2reprentsaccelerationingravitationalmultiplesand

2/Hzisthevibrationpowerper

unitbandwidth.

Butwait,allthatTom,Dick,andHarryhavelearnedisthePSD/

occurringatotherfrequencies?

Tofindout,eachvarieshisfilter’scenterfrequencyfrom,say,

0to5000Hz.

Forexample,suppothateachfindsthatPSD/ASDisconstant

at0.1g2/Hzfrom20to2000Hzbutiszerooutsidethatband.

Rectangularspectramakeiteasytocalculatetheunder-spectrum

=0.1g2/Hz_1980Hz=thesquareroot,

eacharrivesat14gRMS.

Spectrumofshakerrandomvibration

Whereisthislocationofinterest?Itcouldnothavebeenon

anyvehicleormachine,wherespectraarealwaysveryjagged.

Thisspectrumcouldonlybefoundonashakerthathadbeen

adjustedtodeliveraflat,whitespectruminaccordancewithan

earlycirca1955randomvibrationtestspecification,asshownin

Figure4.

s

e

n

si

t

i

vi

t

y

frequency

Figure2

s

e

n

si

t

i

vi

t

y

frequency,Hz9201080

Figure3

Frequency,Hz

202000

0.1g2/Hz

PSD

Figure4

BW(Hz)volts(RMS)g(RMS)g2g2/Hz

16044160.1

402240.1

101110.1

“Allerredinfailingtostatethe

frequencyrangeoverwhich

eachhadmeasured.”

Table1

ReprintedfromVMEbusSystems/December2005Copyright2005

TESTING,1,2,3...Special

Shakerforce

Thenumberwejustcalculated,14gRMS,sus

withsinusoidalvibra-

tiontesting,wecalculateF=pall

themass(weights)involved,includingtestarticles,fixtureand

shakerarmature,ovidesthe

RMSforcetheshakermustdeliver.

l

testandscreeningspectracontainPSD/

extrafeatureofdigitalspectrumcontrolistheautomaticcalcula-

ime,testengineersdidthocalcula-

tionswithhandcalculators.

dnever

ceptofanalysisintermsofvolts2/Hzwas

alreadywellestablishedintelephonylongbeforesay1950,when

sspopular

normalizingalternativeisvolts/rootHz.Ω

WayneTustinispresidentandfounderofEquipmentReliability

Institute,aspecializedSantaBarbara,Calif.,engineering

schoolthatfocusonthereliabilityofhardwareforspace,

air,a,’s

courxplorethebasicsofvibrationandshocktesting,

measurement,analysis,andcalibration.

Foradditionalinformation,contactWayneat:

EquipmentReliabilityInstitute

1520SantaRosaAvenue

SantaBarbara,CA93109

Tel:805-564-1260

Fax:805-966-7875

E-mail:tustin@

Website:

23

2

23

2

23