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CHAPTER

ONEINTRODUCTION TO CONTROL SYSTEMS

1.1INTRODUCTION

The automaticcontrolof machinesand processesis fundamentalto the successfuloperationof modernindustry.Modern manufacturing,processingand transportationsystemsareheavilydependenton automaticcontrolsystems.The benefitsof automaticcontrolincludemoreconsistentoperation,greatersafetyfor the processor machineandoperatingpersonnelandreducedoperatingcostsdueto improvedutilizationandreductionin manpowerrequirements.

The needfor automaticcontrol continuesto grow in terms of the range ofapplicationsandperformancerequirements.Thedevelopmentof highperformancecivilandmilitaryaircraft,missiletechnologyandspacevehicleshasplacedgreatdemandsonthespeedandaccuracyof attitudecontrolsystems.In manufacturing,thedevelopinguseof robotsand automatedproductionhasfurtherincreasedtheneed for reliable,high performancecontrolsystems.In the processindustry,stricterrequirementsforproduetquality,energyefficiencyand pollutionleve]sp]acetighterlimits on processcontrolsyste.ms.

The reducingcostand increasedperformanceof digitaJ coniputershas had asignificantimpacton thewaycontrolsystemsaredesignedandimplemented.Powerfu]mini- and mieroeomputerswith superbgraphicscapabilitiesare readilyavailablefordesigningand simulatingcontrol systems.Computersare also widely used forimplementingautomaticcontrol in an increasingvarietyof industrialand domesticapplieations. Even masS-producedconsumer products can indude powerfuImicroprocessorsystemsto monitorandcontrolthesystem.For example,theaveragemusiccentreor videocameracontainsmorecomputingpowerandautomaticcontrolthana typicalengineering]aboratoryof themid ]960s.

1.2BASIC CONTROL SYSTEM HUMINOLOGV

A cQi1lrol sysiemconsIstsof a comroller and apiani. We usethegeneraltermplant todeseribethemachine,veliicleor processwhichis beingcontrolled.The controllercan

~2 CONTROLSYSTEM DESfGNANDSfMULAnON

Control

elements

Controller

Plani

Measurement

elements

Figure LI A general control system

beaperson,in whichcasewehaveamanualcontrolsystem.Alternatively,iRanautomaticcontrolsystemthe controlleris a device,electroniccircuit, computer,or mechanicallinkage,etc. Figure 1.1showsthegeneralarrangement.

The interface betweenthe plant and the controller requires,actuators(controlelements)to provide the control action. In addition instrumentation,detectorsandsensors(measurementelements)areneededtoprovideinformationabouttheplantstatusto thecontroller. The informationpassingbetweenthecontroller and theplant is intheform of signals.Thesesignalscanbe verydiverse,for exampleelectrical,pneumaticor mechanical,etc.The term'transmitter'iscommonlyusedto describethemeasurementelementin a processcontrol systembecausethe transmittersendsan electricalorpneumaticsignal representingthemeasuredvalueto thecontroller.

Controllers are usual1yimplementedelectronically,eitherusing analoguecircuitsor a digital computer (microprocessor).Pneumaticand hydrauliccontrollersare alsato befound. Actuators arecommonlypneumatic,electricor hydraulicdependingonthe applicationand powerlevelrequired.

The behaviourand performanceof a control systemdependson theinteractionofall theelements.The individualcomponentscannotgeneralJybeconsideredin isolation.The plant itself is probablythemostimportantelementin any control system;thebestcontrolJerin the world cannotmakean inadequateplant operatewelL.

Feedback

In everydaylife, feedbackoccurswhenwe are madeawareof theconsequencesof ouractions.Feedbackis sonaturalthatwetakeit for granted.Imaginetryingto accomplishthe simpJestof tasks without feedback,for example,trying to waJk without visualfeedback.Feedbacknot only givesverilicationof our actions:it allowsus to copewith:ci changingenvironmentby adjustingour actions in thepresenceof unforeseenevents

-and changingconditIons,Feedback has similar advantageswhen applied to automaticcontrol. Feedback

occursin automaticcontrolsystemswhenthecontrol actiondependsuponthemeasuredstateof themadiine or processbeingcontrolled.Feedbackgivesan automaticcontrolsystemthe abIlity to deal wIth unexpecteddisturbancesand changesin the plantbehaviour.

-

fNTRoDucnoNTOCONTROLSYSTEMS3

SeqiieiitialandQuantitativeControlSystems

A sequentialcontrolsysteminvolves logiccontrolfunctions.The sensorsmonitoring theplant provide switehedoutputs which produceonly on/off signals. For exampleanautomaticdoor may belitted with limit switchesto detecttheposition of thedoor andaninfrareddetectorwithaswitchedoutputtosenseanapproachingperson.The controlfunetioninvolvesthe useof logical rulesso that the actuatarsoperate in the coneet

sequenceand at the conect time. Sequentialcontrol systemsare comman in factoryautomation,automaticwarehousesand the controlof batchoperations.The designofsequentialcontrol systemsinvolvesproblems in logic and is not coveredin this book.

The objectivesof a quantitativecontrolsystemare dilferent.This type of controlsystemis concemed with controlling the actual value of some plant quantity.Measurementelementsprovide quantitativeinformationto the controiier rather thanjuston/olf signals.The divisionbetweensequentialcontrolandquantitativecontrol canbevagueand samesystemscan be consideredin bothways.

As an example,a modernautomaticwashingmachineclearly involvessequentialcontrol to switch the various solenoid valvesand pumps on and oIT in the requiredprogressionfor the selectedwash program.Quantitativecontrol is usedfor the washdrum rotational speed.Here the actual drum speedis measuredand controlled byalteringthe power deliveredto the motoL A lessclear exampleis the thennostaticcontrol of wash temperature.The thermostatswitchesthe heaterelementon and offdependingon whetherthewashtemperatureis too law or too high. The control signalis clearlyon/olT but sinceit is the actualvalueof thetemperaturewhich is importantthesystemcan alsa be consideredas quantitative.

The behaviourof a quantitativecontrol systemdependsfundamentaiiyon the rateandextentto which theplant respondsto thecontrolaction. Such dynamicbehaviourisdIfficultto predictandthedesignof quantitativecontrolsystemsto achieveacceptableresponseis no trivial matter.This book is concemedwith the behaviourand designofquantitatIvecontrol systems.

1.3OBJECTIVES OF AUTOMATIC CONTROL SYSTEMS

Regulatioii

A control systemfor maintaining the plant output constant at the desiredvalue inthepresenceof externaldisturbancesis calleda regulator.Disturbanceswill causetheplantoutput to deviateand theregu]atormustapplycontrol action or controlefforttoattemptto maintain theplant output at thereferencevaluewith theminimumof error.Feedbackis fundamentalto regulationbecaus~only feedbackcan provide information

abouttheactuaJplant output.A good regulatorwill minimize,theeffectsof disturba~son theplant output.

Tr:iijectory F ollowing

Quiteoftena control systemis requiredto maketheplantoutput [o]]owa certainprofileor trajectory.A servasystemisa control systemspecificaliydesigned10 foiiow a changing

4 CONTROL SYSTEM DESIGN AND SIMULATION

referencevalue.The servoproblem,as it is called,is of majorconcemin transportation,defenceand manufacturingsystems.The servomustapply control effortto make theplant output follow thedesiredpath with theminimumof erroL

it is clearthat theregulationandservoproblems areverysimilarand indeedmanycontrol systemsgive good regulationagainst disturbancesand dose following of achangingreference,

1.4CONTROL STRATEGIES

In order to examinesomedifferentcontrol strategieslet us consider a simplelevelcontrolproblem.Figure 1.2showsa tank holding liquid for feedingsomeprocess.Theprocessbeingsuppliedrequiresa constantheadfeedandso a control systemis requiredto keepthe tank leve!constantat somereferencelevel.A valveis locatedin thetankinIetto alter or modulatetheflow rate.

Open-IoopControl

The simpleststrategyis to havea dial on the inlet valve,By experimentthevalvecanbe moved to different positions and a note made of the dial position and thecorrespondinglevel in the vessel.The dial can be calibratedin 'metres'.Thus if it is

decidedto operateat a different!evel,the valve can be movedto the correspondingposition on the diaL.This strategyis termedopen-Ioopcontrol.Open-loop control issimp!eand will workwell providedthereis no changein theflow of liquid from thevesseland all other parametersaffectingthelevel in thevesselremainconstanL

FeedforwardControl

r.~g:

;8

J

-i

~

INTRODUCTiON TO CONTROL SYSTEMS 5

bedeterminedby examiningthe calibrationcurvefot the new Dowand then openingor clasingthe inIet valveaccordingly,This strategyis termedfeedforwardcontrol.

feedforwardcontrolrequiresasetofcalibrationcurvesor amodel of therelatianshipbetweenthe valve position, outDowand level. The relationship can be obtained

experimentallyby measuringtheleve!for variousoutflowratesand inletvalvepositions.Alternative!ythe model can be formulatedfrom a theoretical analysis of the tank,Anotherconsequenceof this strategyis that a measurementof the outlet Dow rate is

requiredto calculatethenecessarychangein the position of the inlet valve.Feedforwardcontrol is an improvementoversimpleopen-Ioop control. But it only

catersfor theone variablewhich is beingmonitored(in this easethe outflow) and relieson a goodmodelof theplanL if themodelis inaccurateor thebehaviour of thesysternvarieswith time then the feedforwardstrategymay not work too weii. Disturbancescanoriginatefrom many causeswhich may not be included in the model, or are notmonitored.For example,the supplypressureupstreamof the inle! valve may changeor the densityof the liquid could alter. THesevariationswiii cause the relationshipbetweeninlet valve position and tank inDowto changeand so the tank level will beincorrect,The feedforwardstrategywiii not correctfor thesefactorso

FeedbackControl

Ratherthan adding more feedforwardmeasurementsto compensatefor theseotherfactors,the obvious solution tomaintain the levelin the vesselis to monitor the levelitselfand :idjust the inlet valve if the level deviatesfrom .the desired value, Such afeedback strategyis error driven in that thecontroleffortis a function of thedifferencebetweentherequiredlevelandactuallevel.The relationshipbetweentheerror and thecontroleffort is calledthe controllaw. Feedbackcontrol, unlike feedforward,can giveregulationagainstunmeasuredor unmodelleddisturbances,

The major causeof disturbancesaffectingthe tank levelis likely to be changesin thetankoutDowrate.An increasedoutDowwill causethelevelto drop.A maresophisticatedstrategyis to usea set of calibrationsover a numberof outflow rates.By monitoringtheoutDowrate whentheplant is in operation, thecorrectposition of the valvecan

rJ

---t--i InflowTi

",""'"""'i r_jJ_ -10c"". Figuce 1.2 Example plani for levc1control

011/0ff FeedbackControl

The simplestmethodof monitoringthelevelis by meansof a levelswitch(Doatswitch).The levelswitch is mountedin the tank at the desiredlevel. The switch producesabinary(on/off) signal that indicateswhetherthe !evelis above or below the requiredvalue.The signalcanbeusedto operatetheinletvalvedirectly,When thelevelis abovethereferencethe inlet valveis closedand whenbelo,wit is opened,The control law inon/offcontrol thereforeswitchesthe control effortbetweenextremesdependingon thesignof theerrOL

On/off control certainlyovercomesthecriticismsof theopen-Ioop and feedforwardstrategies,Whateverthecauseof thechangein level,if thedeviationin thelevelis largeenoughto activatetheswitchthencontrol actionwill beapplied to correctthesituation,The requiredlevel (refereneie)in thissimpleschemeis determinedby the positionof thelevelswitchon thetank,On/off controlrequiresonly verysimpleequipmentin theformof levelswitchesand a simplesolenoidtypeactuatorto open or shut the valve,

Tliere are severalproblemswith on/of[ control.One problem concemsthe violentDuctuationsin inlet Dowas thevalveswitchesbetweenfuJly open and fuJly shut.Thesenowchangesmay appearas a significantdisturbanceto any processfeedingthe tank,

6 CONTROL SYSTEM DESIGN AND SIMULATION

'f'i

~'~

INTRODUCTION TO CONTROL SYSTEMS 7

One overridingconsideratianin control systemdesignand imp]ementatIonis safety.This mayrelateto thereliabilityand robustnessof thecontrolstrategy,but moreoftendealswiththemonitoringof exceptiona!conditionsandthesubsequentalarm andsafetysystemsassociatedwith thedetectionof a malfunctionor of a variablethathasreached

Another problemconcernsplants which do not respondimmediatelyto the controleffort appIied,Such delaysare commanin more complexplantsand particularlyintemperaturecontrol systems,Any delayin the plant responsemeansthat theprocessoutputwill continueto riseevenafterthe upperswitchinglimit is reached,EventualIytheoutputwill respondto thecontrol effortand startto fall, but of coursetheprocessoutputwill onceagaincontinueta droppastthelowerswitchinglimiL Thus theprecisionof on/off controldependsheavilyon'any plant delay,

Evenwithnegligibledelay,thetanklevelwill beconstantlyfluctuating,if theswitchis very sensitiyeit only requiresa smail changein level to changestate,A sensitiyeswitch wiIl causethe inlet valve to switch betweenfully open and fully closed veryfrequentlyas thelevelcycIesaroundtherequiredsetting,Alsa anywavesor ripplesontheIiquid surfacecould causea verysensitiyeswitchto be activatedvery rapidly, Thefrequencyof switchingcanbe tradedoff againstalass of precisionin the levelcontrolby usinga lesssensitiyeswitchor evenby usingtwo switches,oneat high levelto dosethe inlet valveand anatherat low levelto open il.

Conlrolledvariable

Plant

Oisturbances

Measurement

Totalcontrol

+ 0eiio;ower( . ,i amplificationandactuator

Modelandcalculalion

offeedforwardconlribution

Measurement

>

'" Feedbackloop t

Feedforward

contribulion _Feedback

contributiOn\Feedback i +controller

Measured

value

dangerouslevels.Safetysystemsareusuallyseparatefromthenormalcontrolsystemandcan be treatedindependentlyin termsof strategyand implementation,

Safetysystemsusually involvedetectionof unsafeor potentiallyunsafeconditionsusingon/off or switchingsensors.Standbyequij:imentmay needto be startedif thereis a failureor the plant may needto be shut down safely.Safetysystemsinvolve thesameproblemsof binary logic as sequentiaIcontrol systems. '

Quite oftencontrol systemscan be designedto fail safe. With the correctdesign,failuresof instrumentationand control equipmentcan resultin saferather than unsafesituations.For exampleconsiderthe levelcontrol problemexaminedabove,It may bethat theliquid heldin thetank is corrosiveor otherwisedangerous.lt would thereforebe unsafeif the tank were to overflowdue to a control systemfailure. The fail safe

philosophyis to designthe control systemand associatedinstrumentationsuch that afailure in any one elementcausesthe control errortto act in a safeway (i.e. the tankinlet valveshoulddose)~

For a fail safemeasurement,a failurein themeasuringelementshouldproducethe

samesignalasexistsin thedangerouscondition,In thecaseof the levelcontrol systema leveltransmitterfailureshouldlook like a high levelin thetank.This can be achieved

by using a law signal to representa high level (a reverseactingtransmitter).Loss ofthemeasurementsignalwould thuslook ]ikea highlevelin thetankandso thecontrollerwould shut thevalvepreventingtank overflow.

The fail safephilosophy can alsa be applied to the control systemactumor.Theargumenthereis simpler, theactuatorshouldmoveto thesafepasition in the eventofan actuatarpowerfailure. For thetank inlet control valvethis meansthat the vaIveshould dose if the actuator signal is lost. This can be,accomplishedby making theactuatarpush againsta spring to open the valve (i.e.a spring return actuator),Thespring would thenclose thevalvein the absenceof actuatorpower,

Figure1.3 Generalblackdiagramof a controlsystem

~Th

:il

.~.~t,~~.~~~

i~{}

--

ModulatingFeedbackControl

Rather than switchthe inlet valveopen and shut, a more subtIeappro'achis to inchthe valveby an amountwhich dependson the differencebetweenthe actualIevelandthe desiredlevel.This strategycan be termedmoduZatingfeedback control.

Modulatingcontrol impliesa moreelaboratelevelmeasurementandvalveactuator.In thefirst placeit requiresa signalrelatedto the actuaIlevel(Le,a level transmitter),Secondlythevalveactuatarmustbe able to open and dose the inIet valvegradually(moduIate the valve opening). Furthermore the valve itself must have a smoothcharacteristicso that its resistanceto Dow is infinitelyvariable.

The control law can profoundlyeffecttheway in which a feedbackcontrol systembehaves.SimpIe on/off control can sametimesgive acceptableperformance;mostdomesticheatingsystemsare controlled this way. The important characteristicofmodulatingcontrol is that it is capableof providing a rangeof control effort and canproducesmail,aswell as large,conections.With a welldesignedcontrol law, feedbackcontrol can provide goad regulationand trajectoryfollowing. One of the primaryfunctionsof a controlengineeris to designor selectan appropriatecontrol law for theplant whichgivesacceptableperformance.

Figure 1.3showsa generalblackdiagramof acontrol systemwith both feedforwardand feedbackcontrol. The box labelled 'feedbackcontroller' is where the feedback

contribution to thecontrol effort is produced.if thereis a feedforwardcomponent,itis addedto thefeedbackcontributionto makeup thetotal controleffartsignal.

1.5 SAFETY--

\x)

8 CONTROL SYSTEM DESIGN AND SIMULATION INTRODUCTION TO CONTROL SYSTEMS 9

Temperaturetransmitter

The aboveargumentsarebasedon thepremisethatthetankshouldnot beallawedto overllow. if theIiquid held in the tank is essentialto thedownstreamprocess(i.e.lubricatingoil or coolant)thenthe tank should not be allawedto mn dry in the eventof a faiIure.The situationis now reversedand the fail safeargumentimplies that adirect acting level transmittershould be used. Similarly the control valve must bedesignedto open,shouldtheactuatorpower fai!.

Safety considerationsare very dependent on the process or system, andnotwithstandingtheir importance,can only be touchedupon ina generaltext of thisnature. Experienceand a thorough knowledgeof the systemoperationare requiredbeforesensibledecisionson safetyaspectscan be made.it mustbe emphasizedthatmostcontrolengineersspendfar rnoretimeconsideringsafetyaspectsthan in designingthe control systemfor normaloperation.

Disturbances

Temperature controller

i---------~-------i

! i i Control

i+Di1i'__\_:~~~~]\ 3!imit Controller

Relerence Errar outputtemperature

Measured

temperature'\

Disturbances

FuelIlow

\ . i Processlurnace

Processoutlet

temperature

\

1.6EXAMPLES OF CONTROL SYSTEMSFigure 1.5 Biock diagram for temperaturecontrol of a process (urnace

FuruaceControl

Controlof a RoboticArm

Robatsareincreasinglyusedformaterialshandling,automaticassemblyandfabrication.Robotic armsnormallyhaveseveraljoints or axeseachfittedwith an actuatorenabljngthearmto movein a varietyofwaysto pasitionandorientatethegripper.The actuators

are controlled by a dedicatedcomputerso that the conect sequenceo[ motions iscarriedout.The computernormallydoesthisby replayinga storedsequenceof desiredmotions Sametimesmoresophisticatedsystemsusetelevisiancamerasor touehsensors

to heIpdecideon the requiredgrippermotion,

valueof the processoutlet ternperature(the referenceternperature)can be set. Thecontrollercomparesthe measuredtemperaturewith thereferencetemperatureand thefuelflow is increasedor decreasedaccordingly.The exactrelatianshipbetweentheerrorin temperatureand the fuel valvemavementis determinedby thecontrol law.

A black diagramof the control systemis shown in Fig. 1.5.The main sourceofdisturbancesare changesin the processfluid flowrate. Changes in throughput areinevitablefor nuqierousreasons.For example,if thereis unsufficientfeedstockor ifthereis a suddenincreasein demandbecauseof a weatherchangeor evenfluctuationsin thestock market.Feedforwardcontrol would be relativelyeasy to implementby

monitoringtheflow of processfluid andaddingin a componentto thefeedbackcontrolef[ort.

Turning to the safetyof this [urnacecontrol system,[ail safeprecautionscan betakento ensurethat a control equipmentfailure resultsin the burner shuttingdown.Howevertherearemanyothersafetyconsiderationsto beconsideredin firedequipmentsuchas this. For example,during normaloperation disturbancesin the processfluidcould causethe furnaceoutlet temperatureto rise. If the correctingcontrol effort isexcessive,thefuel valvemay be shut of[ causinga bumer flarnefailure. When thefuelflow is restoredby the controlleropeningthe valve,thepotentialfor an explosionwillexistbecauseo[ the unignitedfuel enteringthe not fumace.Solutions to this problemcan Includeplacinglimits on the [uel control valve travel, useof name detectorsandautomaticre-ignitionof the extinguishedflame.

Heatedprocess-Iluidtotreatmentplant

Temperaturetransmitter

To otherconsumers

ii.---Measured temperaturei

___+-----__Desired/referenceControJ,.. V temperaturevalve Temperature

contrailerFueloil

\

Pump

Furnaces are used in the process industry for heating feedstocksprior to furthertreatment.For example,in an oII refinery,cmde oil is heatedbeforeit entersthecmdedistillationcolumnwhereit issplitupintofractionswhicheventuallyproducemarketabIeproductssuchasaviatIon[uelor roadbItumen.The initial heatingis carefullycontrolledas thesubsequentfractionationis quitedependenton thedegreeof vaporizationof thecrude oil [eed.

Figure lA showsa simplifieddiagramof a processfurnace.The temperatureof theprocessfluid is measuredat theoutletof thefumaceby a temperaturetransmitterwhichsends asignalto thetemperaturecontroller.The controllerhasadial sothat thedesired

Figure 1.4 Process furnace

ls>

W-8

LO CONTROL SYSTEM DESIGN AND SIMULATIONINTRODUCTION TO CONTROL SYSTEMS ]]

Figure1.7 Blackdiagrainof roboticarincontrolsystem(oneaxisonly)

Digrtallyencoded

armposition

r

r

r

r

r

riControl computer i

______________________ 1

The responseof anynatural systemto stimuliis not instantaneous,it takestimefor thesystemto respond.We are all awarethatbeforewe canmake a cup of instantcoffee,it takestimefor thekettleto boil after wehaveswitchedit on. When acar hitsa bumpin theroad thesuspensianbouncesor oscillatesfor a while beforeit settlesback to anequilibriumposition.When wehit the acceleratorpedalof our car, it takestimefor itto accelerateto a new speed,evep.if we happento own the latestLamborghini! The

way the responseof a systemevolvesas functionof time to a particular stimulus istermedthedynamicresponse.

The dynamicbehaviourol systemscanbemodelledbydifferentialequations.Modelscanbeobtainedbyapplyingthebasicequationsofphysicsandmechanicsto thesystem.This approachcanbeapplied to systemswheretheunderlyingprinciplesaredear andwherethesystemis sufficientlysimple,or canbebrokendown into simplesubsystems,to usea 'first principles'approach.Anather approachis to observethe behaviourofthesystemin its normal working enyironmentor introducetestsignals.A model canthenbeproposedfrom the observationsof the input/oiitputbehaviour.

The solution of differentiaJequationsby hand methodsis difficult and so a rangeof numerical methodshave been developedwhich can be implementedon digita!

computers.Many computer techniquesand packagesexist for solvlng differentia]equationsand simulatingthe behaviourof dynamicsystems.

Control systemdesign involves simulating the dynamic behaviour-of feedback

systems,but it alsa involves other techniqueswhich require the use of computers.Dynamic simulationa!oneis not enough.Design methods for control systemshaveevolyedsincethe 1930s.At first methodsweredevlolopedwhichwerebasedon short-cuthandcalculations.Analoguecomputerswereusedto do the final simulationandverifythedesign.This stateof affairsdid not changesignificantlyunli! theearly 1950swhenthefirstdigital computerscameinto commercia]andscientificuse.Computerprograms

In practicehowever,the control law is usuallychosento give adequateperformancewitha rangeof loads.

1.7CONTROL SYSTEM i;>YNAMICS,MODELLING,SIMULATION ANDDESIGN

Reference

pasitlon

ij~.:~-j:!

.~iii.~i~]i,j

Gripper

Servomotor

andgearbox

Figure1.6 Roboticann

The control computermustcontrol all therobotjoints simuItaneously.Sametimeseachaxishasits own separatemicroprocessorwhichreceivescommanclsfrom themaincontrol computer.Howeverimplemented,theschemefor thePQsitioningof eachaxisis similar, so thecontrolaf just onearmjoint will be examinedasshownin Fig. 1.6.

In this example,thejoint is driven by a de electricmotor (servomotor)through agear box. A de motor is a flexibleactuatorwhich can be drivenin eitherdirectian atvariousspeedsbyalteringthedirectianandmagnitudeof themotorcurrent.The currentis suppliedto themotorby a poweramplifierwhich in turn receivesits input from thecomputeror microprocessor.The computerdealswith digital dataandso theinterfaceto the power amplifierrequiresa digital to analogueconverter.The angleof the armjoint is monitoredby an encoder.An encader is a transducerwhichppoducesa digitaloutputrepresentingthemeasiiredanglewhichcanbedirectlyinterfacedtb thecomputer.

As the computerretrievesthe sequenceof desired robot motions, the requiredpositionsfor theaxisarepassedto a partof theprogramwhichimplementsthecontrollaw. The control cakiilation involvessubtractingthe measuredarm angle from therequiredarmangleto find theangularpasitionerror.The computerthenusesthiserrorto detenninethemagnitudeanddirectianof thecontrol effort.This calculatedcontrol

errortis convertedto an analoguesignal which Is applied to the inpcitof the poweramplifier to diive the servomotorand hencerediice the error. Figure 1.7shows thearrangementin black diagramform.

The main requirementin thiscontrol systemis that of fastand acciiratetrajectoryfollowing. The controllercan usethe rateof changeof measuredpasitionto calculatetheyelocityof thejoint andso reducethemotor currentif thearm is mavingtf1tifast.The control law in a high-performancerobotmaybequitecompiexandwill betailoredto the way in which thearm is expectedto respond.

A difficulty arisesbecausethearm may responddifferentlywhencarryinga loadthan when empty-handed.The motor will require more current to accelerateanddeceleratewith the load thanwlthout. The control computermay know whetherthegripperis holding a load and in theorycould thencompensateby usingmorecurrent.

..:::....

CJ

-

12 CONTROL SYSTEM DESIGN AND SIMULATION INTRODUCTION TO CONTROL SYSTEMS 13

FigureP1.3 Oil cooler temperaturecontrol system

'"'""""

werewrjttento solvesetsof differentialequationsand resultsproducedon listingpaper.Graphic displayswereyirtuaiiy nonexistentand the interactionbetweenthe designerand thecomputerwasyerypoor. Even with thedevelopment.ofminicomputersin themid 1960sthesituationwasnot significantlybetter.

Graphics displayterminalsfirst becamewidelyavailablein theearly 1970s.Theirusefor controlsystemdesignwasgenerallyrestrictedto largeruniversitiesandindustria!companies.User interactionwas still poor, and often the designsuiteswerenot wellintegrated.The 1980ssawthearrivalof the'personalcomputer'(PC) andtheavailabilityof graphicsterminalsof highquality.Many softwarehousesadaptedexistingprogramsto the PC, but really did not take advantageof its potential as a designtool. Theancestryof theprogramswas dearly vIsible.

In addition to theimproyementsandcost reduction of the hardware,the secondhalf of the 1980ssawan increasingawarenessof well designed (userfriendly)software.Businesssodftwarewith programssuchaswordprocessors,spreadsheetsanddatabaseswas the first areain which this developmentoccmred. Later, as graphicscapabilitiesgrew, computer-aideddesignpaekagesfor draughting, three-dimensionalmodelling,printedeireuitdesign,ete.,appeared.Computer-AidedControl SystemDesign(CACSD)softwarewasslowto respandto this trendsineea fairly largeinvestmentwasrequiredwith a morerestrictedmarket.Taday thereareseveralgood quality,low costpackagesavailable.

PROBLEMS

1.1 Classify the following into sequential or quanlitative control systems and identify whether feedback ispresenI:

(a) The thermosta!;ctemperalure control in a central heating boiler.

(b) The programmer controlling a typical central heatingsystem.

(c) A cruise control system,as Iltted to a modern car.(d) The over-speedcutout Iltted to acar engine.

1.2 Examine lhe fail safe requirements for the inslrumentaiion and actuators in the furnace temperature

control syslem shown in Fig. 1.4. You may assumethat it is unsafefor the Processfluid to overheat.

1.3 Consider the oil cooling systemin Fig. P1.3. The oil temperatureis controlled by throttling the cooler

bypassvalve.Safetyconsiderationsindicate thai control equipmentfailure should not causelass of thecoolingfunction. Determine the required aclion for the temperaturetransducer and the control valve. What actionwill the canIraller take if the signal from the temperaturetransducer reduces in value?

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CLASSIFICA TION OF VIBRA TiON

Vibration can he classified in several ways. Some of the imfortantclassificationsareasfollows.

FreeandForcedVibration

rS:t'bt.~JFree VIbration: if a system,afteraninitialdisturbanee,is left to vibrateon its,,-tI~~:Sii"\ own,the ensuingvibrationis knownas free vibration.No repeatedextemal

forceactsonthesystem.Theoscillationof a simplependulumis anexampleof

freevibration.(ensue:happenlaterorasaresult;follow)

Forced VIliration, if a systemis subjectedto an extemalforce (often,a(c.of'lofll"l!\'.s' i\+rt'~imrepeatingtypeof force),theresultingvibrationis knownas forcedvibration.

Theoscillationthatarisesin machinessuchasdieselenginesis anexampleof

forcedvibration.

if the frequencyof the extemalforce coincideswith one of the natural

frequenciesof the system,a conditionknown as resonanceoccurs,and the

systemundergoesdangerouslylargeoscillations.Failuresof suehstrueturesas~bui1dings,bridges,andairplanewingshavebeenassociatedwiththe occurrence

ofresonance.

UndampedandDampedVIbratIon

if noenergyis lostordissipatedin frictionor otherresistanceduringoscillation,

thevibrationis knownasundampedvibration.If anyenergyis lostin thisway,

ontheotherhand,it is calleddampedvibration.In manyphysicalsystems,the

amountof dampingis so smallthatit canhe disregardedfor mostengineering

purposes.However,considerationof dampingbecomesextremelyimportantin

analyzingvibratorysystemsnearresonance.

1

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DeterministicandRandmVibratIn (To(iirf/Ic Ile i

Examplesof ranclamexcitationsarewindvelocity,roadroughness,andground

motionduringearthquakes.If theexcitationis random,theresultingvibrationis

calledrandomvibration.In thecaseof raridomvibration,thevibratoryresponse

of thesystemis alsorandom;it canbe describedonly in termsof statistical

quantitiesFigure1.10showsexamplesof deterministicandrandamexc-itations.

1.6 Vibr.ilonAnaly.l.~OOJu",

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whereF is thespringforce.x is thedeformation(displacementof oneendwith

respecttotheother),andk is thecoefticientofproportionality,calledthespring

stiffnessorspringconstant.

If wepIota graphbetweenF andx. theresultis a stniightlineaccordingto Eq.

(i. i). Theworkdonein aeformingaspringisstoredasstrainorpotentialenergy

in thespring.Actualspringsarei:ionlinearandfgllow Eq. (Ll) only within a

certainrangeof deformation.

MASS OR INERTIA ELEMENTS

Themassor inertiaelementis assumedto be a rigid body;it cangainor lose

kineticenergywheneverthe velocityof the body changes.From Newton's

secondlawofmotion,theproductof themassanditsaccelerationis eqmtito the

force appliedto the mass.Work is equalto the force multipliedby the

displacementin thedirectianof theforceandtheworkdoneonamassis stored

intheformofkineticenergyofthemass.

In most cases,we must use a mathematicalmodel to representthe actual

vibratingsystem,andthereareoftenseveralpossiblemodels.As statedearlier,

thepurposeof theanalysisdetermineswhichmathematicalmodelis appropriate.

Oncethemodelis chosen,themassor inertiaelementsof thesystemcanbe

easilyidentified.For example,considerthe cantileverbeamwith a tip mass

shownin Fig. 1.14(a).For a quickandreasonablyaccurateanalysis,themass

anddampingof thebeamcanbe disregarded;thesystemcanbemodeledasa

spring-masssystem,as shownin Fig. 1.i4(b).The tip massm representsthemasselement,andtheelasticityof thebeamdenotesthestiffnessof thespring.

Next,consideramultistorybuildingsubjectedto anearthquake.Assumingthat

themassof theframeis negligiblecomparedto themassesof theflooirs,the

buildingcanbemodeledasa multidegreeof freedamsystem,asshownin Fig.

i.18.Themassesatthevariousfloorlevelsrepresentthemasselements,andthe

elasticitiesoftheverticalmembersdenotethespringelements.4

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DAMPING ELEMENTS

A damperis assumedto haveneithermassnor elasticity,and dampingforce

existonlyif thereis relativevelocitybetweenthetwo endsof thedamper.The

energyor work iI1,putto a damperis eonvertedinto heator sound:hencethe

dampingelementis noneonservative.The d~ping maybe oneor moreof the

followingtypes

ViscousDamping. Viseousdampingis the mosteommonlyuseddamping(\Ji'-z,kO"2- S~f\tlM-) . imeehanismin vibrationanalysis.This typeof dampingis presentwhenevera

viscousfluid flows throughasIot, arounda pistonin a eylinder,or aroundthe

journalin a bearing.In viseousdamping,thedampingforceis proportionalto

theveloeityof thevibratingbody.

Coulombor Dry Friction Damping.Herethedampingforce is eonstantin{J

1.0 FUNDAMENTALS OF VIBRATION1.1 What is Vibration?Mechanlcalvlbratlon is a form of motlon oscillation. It occursIn all forms of machlnery and equlpment. it Is what you feel whenyou put your hand on the hood of acar, the engine of whlch Is runnlng, or on the base of an electrlc motor when the motor isrunnlng. Perhapsthe simplest illustratlon of a mechanlcalvibrationis a vertical sprlng wlth welght, as shown in FIgure 1. In thisposltion, the defiectlonof the spring from its free state Is just sufficlentto counterbalancethe welght W. This defiectlon ls calledthe static deflectionof the sprlng. The pasition in which the spring is at rest is #1. The spring Is then slowly extendedto posltion#2, and released.The subsequent motlon of the welght as a functionof time, when there is negllgible reslstance to the motlon, iswavy and repetltlveas shown in the graph. it exhlblts manyof the basic characteristlcsof mechanlcalvlbratlons. The maximumdlsplacementfrom the rest or mean posltlon Is calied the AMPLITUDE of the vlbratlon. The vibratory motion repeats itself atregular Intervals(Al, A2, A3). The Interval of time wlthin whlch the motion sequencerepeats Itself is called a CYCLE or PERIOD.The number of cycles executed in a unit time (for example,during one second or during one mlnute), is known as theFREQUENCY. In a high-speed osclllatlon the frequency is high and conversely.When, as In Figure i, the-spring-weight system isnot drlven by an outside source,'the vlbration Is a FREE VIBRATION and the frequency Is called the NATURAL FREQUENCY of thesystem.

In general,vlbratory motion mav or mav not be repetltiveand its shapeas a funetlon of time mav be simple or complex.Typical vlbratlons, whlch are repetltive and continuous, are those of the base or houslng of an electric motor, household fans,

vacuum deaners and sewing rrrachlnes,for example. Vlbrations of short duration and variable intensity are frequently Initiated bya sudden impactor shock load; for example, rocket equlpmentupon takeoff, equipmentsubjeet to Impaet and drop tests, apackagefallingfrom a height, or a Iading in a freight car.

In many machines,the vibration Is not part of its regular or Intendedoperation and funetion, but rather it cannot be avoided.The task of vlbratlon isolation is to controi this unwantedvibrationso that Its adverse effects are kept withln acceptablelimlts.

:Fi~iite1T195

1.2 What Causes Vibration?The basic cause Is aiready evldent In the slmple mass-springsystemof flgure i. It is an UNBALANCED fORCE, or system offorces (In Figure ilt Is the spring force aeting on the weight) aetingon or through an ELASTIC OR RESILlENT MATERIAL (InFigure 1, this is the spring). The unbalanced force mav be due to mass unbalance, such as In an eccentrlcaily mountedrotor, or ltmav be due to the variable inertla forces In machinery, which does not move unlformly, e.g. crank-and-conneetlng-rod motion,I.inkages,cam-followersystems. In the latter, the speedsand dlreetlonsof motian of machine parts are contlnuously changing,e.g. the neediemotion in a household sewing machine, bucketmotions In earth-movlng machlnery, ete. Force unbalancecanarlse also from electrle,hydraulic and acoustlc sources, e.g. transformer hum, water hammer, a loudspeaker, ete.

1.3 Adverse Effects of UncontrolledVibrationsThe objectlonableresults of machlne vlbrations, If left uncontrolied,can be several:

High stresses and foree levels mav be set up as a result of vibratlons and in extreme cases mav lead to part fallure. Suchfallure can be sudden or gradualras~infatlgue.More frequently,therels Increasedw.earof parts anduns~tisfaetory equlpmentperformance.This requlres increased malntenance and mayaiso Involvedowntlme of equipmenL For example, In a machinetoolwith excessivevibrations,_parts mav be inaccurately machined and subsequently rejected. In other casesi an inadequatelycushionedmachlnemav walk away on Its foundation. And finaliy, noise may becomeexcesslve, Independent of stress levels,consumer produetacceptancemaybe jeopardized, and w.orklngconditionsmav becomeunacceptable, Usually, the objectionableresults are a combination of these circumstances.

1.4 Principles of Vibration and Shock IsolationIn dlscussingvlbratlon Isolation, it is useful to identlfy the three basicelements of all vlbratlng systems: the equipment(component,machine,motor, instrument or part); the vlbratlon mountor Isolator (resillent member); and the base (fioor, baseplate, concretefoundation, ete.); the vibratlon mount Is a resillentmember(rubber pad, sprlng or the like), whlch is interposedbetweenthe equlpmentand the base. It is usualiy qulte smaiL.

lf the equipmentIs the souree of the vlbratlon, the purposeof the vlbratlon mount is to reduee the force transmitted fromequlpmentto base. The directlon of force transmission is from equipmentto base. This is probably the most common case.

lf the base is the soureeof the vibration, the purposeof the vibratlonmount Is to reduce the vlbratlng motlon transmitted fromthe base to the equipmenL The direction of motion transmission Is from base to equipmenL This case arises, for Instance, InproteetingdelicatemeasuringInstruments from vlbrating fioors, ete.

In either case, the prlnclpie of the cushioning aetlonof the vibration Isolator Is the same. The Isolator is a reslllent member. Itaets both as a time delay and a source of temporary energystorage, whlch evens out the force or motlon dlsturbance on one sldeof the vlbratlonmount and transmits or meters out a lesser, controlleddlsturbance, at the other end of the mount.

A good vibration mount, thereforei slows equipment response to a force- or motian disturbance. In engineering termsi thecharaeterlsticof a good vibratlon mount is that the natural frequencyof the equlpmentwlth the mount Is substantially lower thanthe frequencyof the vibratlon source (forcing frequency).The designof a suitable vlbratlon mount insures that this Is the case.Conversely a poorly designed mount, having an undesirable frequency characteristic, can be WOrse than no mount at ali.

In additionto its funetion as a time delay and source of temporaryenergy storage, vibration mounts can also function asenergy dlssipatorsor absorbers. This effeet Is usualiy producedby the damping charaeteristicsof materials, viscous fiulds, slidlngfrietion, and dashpots,

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21

a7f!ioughin generalthesemavor mavnot bepartof the isolatar.The damping,or energy-dissipatingeffectof an isolatarmavbenegligible or substantlal depending on the applkation. The main purpese of isolatar dampIng \s to re.duce or to attenuate tne.vlbratlons as rapicny as posslble. Damping \s partic"ularly important at certaln speeds, which cause a candition known asRESONANCE. Th'is occurs when the natural frequency of the equipment with isolatar (oincides with the frequency of the source ofthe vibration. For examp!e, jf an electric motor runs at 3600 RPM, then an isoJator-equipment natural frequency of 3600 cyclesper minute corresponds to a conditlan of resanance, if a machine operates near resonancei or has to pass through aresonantspeed in order to attain operat1ng speed, damping is important in preventing the buildup of vibration to an unsatisfactory level.

In summary, then, a good vibration mount functions as_a time delay, temporary energy absorber and possibly to same extentas an energy dlsslpator, or damper. The engineering design of a vibration mount consists in identifying the characteristics of thesouree of the vibration, the mechan"icalcharacteristics of the equipment and the determtnation of the rnount characteristics, inorder to achieve a speclfied degree of vibration reduction.

1.5 Principles of Nois_e ReductionA good vlbration mount can be effective in reducJng noise as we1Jas in reducing the transmission of forces and motions.(a) What is Noise?Sound is a vlbratlon of air. The air in this case is an elastic member. The vlbrations of the air have a frequencyand an intensity(loudness). The frequency can be expressed in eyeles per se~ond or eycfes per mlnute. The audible frequencles range from about100cycfes/see. to about 18,000cyclesisec.,although sens'itive human ears mav have a somewhat larger range. Intenslty orloudness, is measured in decibels. Asound intensity of 15decibels would usually be regarded as quie~,whlle a decibel level of 60and up iS usual1y regarded as loud and objectionable. Noise mav be regarded as objectionable sound.

More speelfically, the decibells essentlally a comparison of the pressure of the sound to that of a standard or reference sound(.0002 micrabars, usuaiiy). In arder to arrive at a reasonable seale of values, the logarithm of thIS ratlo to the base 10 is usedandmultipledbytwenty.

Typicalvaloesof levelsof soundintensltyandnoiseintensityareshownin thefollowingTabiesla and1b:

TABLE la VALUES OF SOUND Various industria! operations and related noise levels recorded atAND NOISE INTENSITY distances of from one to three feet from machine. * *

TABLE lb VALUES OF SOUNOAND NOISE INTENSITY

**From: Acoustical Enclosures Muffle Plant Nolse" byS.Wasserman and A.Oppernheim, P!ant Engineering, January1965.

Mark's Mechanlcal Engineers' HandBoek, Sixth Editfon,Hill Book Co. ine., New York, 1958, Sedian 12,p.153N to Specify Audible Naise" by E.A. Harris and W.E.MachineDesignNov.9, 1961,p.168.

T197

(b) What Causes Ndise?A comman cause is the impad or vibration of a solid material, which sets alr In motion; for example, a hammer striking a nail, ora vibrating equipment pane!.In machlnery, in particular, there are many commoniy found sources of noise. These are usuallyassociatedwith the operating frequencyof machine motions, e.g. the RPM of an eiectric motor or of gears, the rate of toothengagementin gear teeth, the frequenciesassociated with reciprocatlngmachinery, etc. lt is possible alsa that vibrations mav begeneratedin one part of the equipment, but mav set up noise and vlbratlon In anather part of the equipment, such as doors,panels, chassis, fiexible lines, printed-circuitboards ete.

(c) The Adverse Effects of UncontrolJed NoiseThere are severaL.First,nolse may be an indication of faulty equipmentoperation, e.g. cracked parts, faulty bearings, excessiyerotor unbalance, Improper lubrication, loose parts, etc. It is possiblealso, how~ever,for a machineto fundian satisfadorilymechanlcally,but to be rejected by the Customer, If it is too nolsy, e.g. nOisyhouseholdappllances,alr condltloners, etc. Second,human efficiency and fatigue mav be adversely affected, for examplein production Iines in a nolsy fadory, or in the office. Shortof expenSive--ancJ"diffidJitiiiv~estigatibhs,agooirVibration mountcanoften be-an effectiveway to reducenoise !evelsto withinacceptablelimits.

(d) How Can Noise be Reduced?There are many ways. One of the most practical and effectlvemav be the use of vibration mounts. As a general rule, a welldesigned vibration Isolator will also help reduce ndise. In the case of panel fiutter, for example, a well designed vibratlon mountcould reduce or eliminatethe noise. This can be achieved by elimlnatingthe fiutter of the panel Itself, or by preventing itstransmission to ground, or by a combinationof the two.The range of audible frequenciesis so high that the natural frequenciesof a vibration mount can usually be designed to be well below the nolse-produeingfrequency.

In order to reduce nolse, try to Identifythe source of the nolse, e.g. transformer hum, panel Rutter, gear tooth engagement,rotor unbalance, etc. Next identify the noise frequency. A vibration mount designed in accordancewiththe guidelines forvibration and shock control can then ad as a barrier either in not canduding the sound, or in attenuating the vlbration, whicli isthe source of the noise.

2.0 BASIC DEFINITIONS AND CONCEPTS IN VIBRATION ANALYSIS2.1 Kinematlc CharacteristicsCOORDINATE - A quantity, such as a length or an angle, which helps define the positionof a moving part. In Figure 1, x is acoordinate, which defines the positionof the weight, W.DlSPLACEMENT - A change in pasition. lt is a vedor measuredrelative to a specifiedposition, or frame of reference. The changein x (Figure 1) measured upward, say, from the bottom pasition, iS a dlsplacement.A displacementcan be positive or negative,translational or rotatlona!.For example, an upward dlsplacementmav be positive; and a downward displacement negatiye.Similarly, a clockwlse rotation mav be positive and a counterclockwiserotatian negative. Units: inches, feet, or, in the case ofrotations: degrees, radians, etc.VELOCm - The rate of changeof dlsplacement.Units: in/sec, M.P.H., etc. Velocity has a diredion. lt is a vedor. lts magnitude isthe speed. Angular velocity mlght be measured In radians/sec or deg/sec, elockwiseor counterclockwise.ACCELERA110N- The rate of changeof velocity. Units: in/see' etc. It is a veetor and has magnitudeand direct"ion,e.g. 5 in/see'North. Angular acceleratlonmlght be measured in radians/see' or deg/sec', clockwlseor eounterclockwlse.

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2'2-

VIBRATORY MOTION - An oScillating motior,; for example, that of the weight W, in figure 1.SIMPLE HARMONIC MOTION - A form of vibratory motlan. The motianas a fundian of the time iS of the form X =a sin wt, wherea, w are constants.The maximum dlsplacement, a, from the mean pasition(X =O) is the amplitude; the frequency (rate at whichmotlan repeats Itself) Is w/2it eycles/sec, where has the dlmensionsof reciprocaltime, e.g. reclprocal seconds. The motlan is alsacal/edharmonic or sinusoidal motion.AMPLITUDE - flgure 2 shows a vibrating motian, which repeatsitseif everyT seconds. The maximum values of the displacement,x, from the referencepasition (x =O) are called amplitudes.These are (al, a2... ). The largest of these is ealledthe pea kamplitude..

FREQUENCY - Rate at whlch motian repeats itself per unit time. if the motian repeats itself every Tseconds, the frequencyis l/Teyclesper second.PERIOD, CYCLE The intervai of time within which the motian repeatsitself. in figure 2, this is Tsecongs. The term cycle tends torefer alsa to the sequenceof events wlthln one period.STEADY-STATE MOTION - A periodic matlan of a mechanicalsystem, e.g. a eontinuouslyvibrating penduiumof constantamplitude.TRANSIENT MOTION - Amotion whieh changes with time in a non-periodicmanner; orten the motian declines (attenuates) to anegligiblevalue after a finite period oftime (e.g. impaeteffectswhiehdecaywith time, ete.).PERIODIC AND NON-PERlOoiC MOTIONS - Amotion, whieh repeatsItself is periodie; amotion, whleh does not repeat itself, isnon-periodie.HARMONICS - Any motian can be considered as made up of a seriesof simple harmonic motions of different frequencies andamplitudes.The lowest-frequency component Is usually calledthe fundamentalfrequency; higher frequencycomponents areealled harmonicsor super-harmonlcs. Thelr frequeneiesare exact multlplesof the fundamental frequency. Sametimes,components of the frequencies of which are a fraetion of driving frequencyare signlficant (e,g. the "half-frequency" whirl ofrotating sharts, ete). Such components are calied subharmonics.PULSE - Usually a displacement-time or force-time funetiondeseribingan input into a dynamical system.PULSE SHAPE - The shape of the tlme-dlsplacement or force-displaeementcurve of a pulse. Typicaliy, this might be a squarewave, a rectangularpulsei or a half sine-wave pulse. In general, however,the shape can be an arbitrary functian of the time.

Tl99

SHOCK MOTION - A motlon in which there iS a sharp, nearly suddenchangein veioclty, e.g. a hammer blow on anaii, a packagefailing to ground from a height, ete. its mathematical ideallzatlonIs that of amotion in which the veioeitychanges suddenly.Themathematicalidealizatian of a sudden velocity change often representsa dose approxirnation to the real dynamic behavior of thesystem.

2.2 Rigid-Body CharaeteristicsMASS - Weight in Ibs. divided by,the gravitational constant (g = 32.2 ft/see', or 386 in/see2).CENTER OF GRAVin - Point of support at which a body would be in balance.MOMENT OF INERTIA - The moment of inertla of a rigid body about a given axis in the body Is the sum of the produetof the massof eachvolume eiementand the square of its distancefrom the axls. Units are in-ib-sec2, for example. Moments of inertia of thestandardshapes are ,blated in h-andb06ks(see par.4.2, MechanicalSystem Characterlstics). If insteadof the mass of theelement, the volume ls used, the result is also called a moment of inertia. Dependingon the applieation,mass-, vOlume-, or areamomentsof Inertia can be used.PRODUCTUFINERTIA ' The 'produetofinertia ofa rigid body abouttwo interseding, perpendicular axes in the body lsthe sum ofthe produetof the mass (volumes,'areas) of a constltuentelementandthe produetof the distances of the element from the twoperpendlcularaxes. Units same as moment of inertia. Tabulationsare availablein handbooks.PRINCIPAL AXES OF INERTIA - At any point of a rigid bOdy,mutuallyperpendicular(orthogonal) axes can be chosen so that theproduetsof inertla about these axes vanish. The orthogonalset ofaxes are called principal axes of inertia. These can often beidentified by the symmetry of the bOdy,beeausethe principal axescoincidewith the axes of symmetry. (An axis of symmetry is aline in the body, such that the body can be rotated a fraetionof a turn about the line without changing its outline in space).

2.3 Spring and Compliance CharacteristicsTENSION - When a body Is stretehed from its free configuration,its partielesare said to be in tension (e.g. a stretched bar), Thetensile foree per unit area is called the tensile stress (Units: Ibs/in2),COMPRESSION - When a body is compressed from its free connguration,its partieies are said to be in compressian (e,g, a eolumnin axial loading). The compressive force per unit area is calledthe eompressivestress (Units: Ibs/in2).SHEAR - When a body is subjeet to equai and opposite forees,whiehare not collinear, the forees tend to shear the body in two.The body is said to be in a state of shear, e.g. a rubber pad under foreesparallel to its upper and lower faees.The shear foree perunitarea is calledthe shear stress (Units: Ibs/in'). Most bodies in a state of stress are in tension, compresslan and shearsimultaneously, e,g. a beam in bendingoSPRING CONSTANT - When a helical spring is stretched or compressed(amount x, say), the applied force, F, is proportionaltothe dispiacement,x (Hooke's law). The constant of proportlonality(k) is calledthe spring constant or gradient: F = kx. Moregenerally, k is the ratio of a force inerement to the correspondingdisplacementinerement of the spring. In the usual helica!spring this ratio is a eonstant(independent of displaeement)within the elastic limit of the spring materia!.However, k may bevariable, e.g. in a nubberpad, since the rubber is nearly incompressible.Units: Ibs/in. if the spring defieetin torsion, the units ofk mav be in-Ib/radian or in-Ib/degree, ete.FORCE-DEfLECTION CHARACTERISTIC Of A SPRING - This refers to the shape of the force-defiection curve. The most familiarcase is a straight line through the origin of coordinates (eonstantk). it is possibie, however, for the spring "constant" to vary. if itincreases

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with displacement(e.g. as in a rubber pad), the spring is called a h~rdspring.lfit decreaseswith displacement (e.g. as in aBelleville spring), the sprlng Is called a soft sprlng.ENERGY STORAGE - This Is the area under the force-defieetloncurve of the spring. It represents the strain energy stored iri thespring (Unlts: in-Ibs, or ft-Ibs, etc.).PRElOAD - A spring used in equipmentmay or may not have a rest (on the shelf) posltion in which it has its natural, free, orunstretched length. If its rest-position length is not its free length, the spring is in tension or compression. The amount of thistension or compressionis called the preload. When measured in force units, it is a preload force; when measured in defiectionfrom free posltion, it is a preload defiection.ELASTlC MODULUS (E) AND SHEAR MODULUS (G) - These are materlaiproperties,whlch charaeterize material compliance intension or in compressian(E) and in shear (G). Theyare defined as the ratio of stress to strain, where strain refers to the changein length (or deformation) per unit length. E involves tensile or compressiyestress and G involves shear stress. Units: Ibs/in'. Inmany practical applications,E and G can be regarded as constants, withln alimit of material stress known as the proportionalllmit. Metals loaded below the proportional limit are exampies. Rubber and plasties,however, usually have no well definedproportionai Iimit.

2.4 Damping, Friction and Energy-Dissipation CharacteristicsSTATIC FRICTlON, SLIDING FRICTlDN, COULOMB DAMPING - These are all terms used for the frictional resistance eneounteredwhen one body slides relativeto another, e.g. a weight dragged on the ground. The frietional force is approximately proportionalto the contaet force betweenthe two bodies and opposed to the directianof relativemotion. The constant of proportionality, m, isknown as the coeffieientof frietion. It a 10 Ib weight is dragged along a horlzontalfioor with a coefficient of friction, ~ 0.2, thefrictional resistanceis 0.2 x 10 ~ 2 Ibs. Sometlmes a distlnetion is madebetweenthe value of the coefficient of friction whenmotian is just impending(starting frietion) and the value during motion (kinetic friction).-The coefficient of friction in the lattercase is generally somewhat lower. Table 2 shows typical values of the coefficientof friction for various materials and operatingconditions.VISCOUS DAMPING - if a body moves relative to a second bOdy,viscous dampingrefers to a resisting force, which is proportionalto the relative velocity betweenthe two bodies and opposes the directlanof relativeveioclty between them. The constant ofproportionality is known as the coeffieientof viscous damping, c. Units: ibs per unit velocity, Le. ibs/(in/sec). Such damping isencountered, for exampie, in hydraulic dashpots and devlces, whlch metera liquid through an orifice. The more viscous the nuid,the greater the damping. if c ~ 0.5 Ibs/(in/sec) and the body moves through a viseousfiuid at 10 in/see, the viscous dampingforce Is 0.5 x 10 ~ 5 Ibs. Typical example: hydraullc door c1osers.CRITICAL DAMPING - Vaiue of damping constant just sufficientiy high in a mass-sprlng-damplngsystem so as to preventvibration.DAMPING RATlO - The ratio,of the damping constant to the criticai dampingconstantfor that system.

2.5 Vibration Characteristics of Mechanical SystemsMATHEMATlCAL MODEL - An idealized representation of the real mechanicalsystem,slmplined so that it can be analyzed.Therepresentationoften consistsof rigid masses and dashpots. Hopefully, the representationis sufficiently realistic so that theresults of the analysiscorrespondreasonably c10selyto the behavior of the physicalsystem from which it was derived.

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LUMPED-AND DISTRIBUTED-PARAMETER SYSTEMS - In a lumped-paranietersystem, the mass-, elastic- and damping propertiesare separatedor lumped into distinet components each of which has only mass or oniy elasticity or only damping, but not morethan one of these per component.In a distributed-parameter system, a componentmay have combined mass and elasticity anddamping, distrlbutedcontinuouslythrough the component. The latter models tend to be more realistic, but more diffieulttoanalyze.DEGREE OF FREEDOM- This Is the number of independentquantities(dimensions),which must be known in order to be able todraw the mechanlcalsystem in any one position, the fixed dimenslonsof the systembeing known. The slmple mass-springsystem of Figure 1 has one degreeof freedom; a mechanicaldifferential, for example,has two degrees of freedam; a rigid bodymoving freely in space has 6 degreesof freedom.FORCE AND MOTlON EXCITATION - If a force is applied to a dynamicalsystem, it usually is a source of vibration (e.g. centrifugalforce due toai1 unba"Wite-rbtbr).Thevibrations-are then said to 'bedue to forceexeitation.If, on the other hand, the foundation(or other part) of a machineis subject to a forced motion (vibration dr shock), the resulting machine vibration is said to be due tomotian excitation,e.g.anearthquake actuating a seismograph.FREE VIBRATlDN--If theweight in-Figure 1 is'movedou\' of-i\'sequilibriumposition,and released, the system wiil vibrate withoutthe actian of any external forees. Such an asciiiation is called a free vibration.FORCED VIBRATION - If an external force is applied to the weight in Figure l, whlehcauses it to vibrate (e.g. a force varyingharmonicallywith time, say), the resulting motion of the spring-mass system is ealieda foreed vibration. if the base. which isupports the spring, undergoesa forced motion, which in turn causesthe weightto vibrate, the vibration is also forced.RANDOMVlBRATION - Equipmentmay be caused to vibrate by applledforcesor motlons, the frequency (or frequencycomponents) of which vary in arandam manner with time (e.g. wind gusts on a missiie). The resulting vlbration is called random.NATURAL FREQUENCY- When mechanicalequipment vibrates freely, the resultingnumber of oscillations per unit time is calledthe frequency (cycles/see).According to whether the system is free without damping,or free with damping, the frequency iscailed the free-undampednatural frequency or the free-damped natural frequency.The natural frequency is a function of themass distributionand complianceof the system. For a simple mass-springsystem(Figure 1), which represents a reasonable

approximationto many reai mechanlcal systems, the natural frequencyis equalt0.a.-",,\i''-'~. ' radians per second,IN ",il,,!

where k is the spring constant, Ibs/in; W is the weight, ibs; 9 is the gravitationalconstant, 386 In/sec'; and Xst is the statlcdefiectionof the sprlng, In. Thus, fiexlble systems tend to have low natural frequenciesand rlgid systems tend to have highnaturai frequencies.At the same time, the natural frequency can be changedby alteringthe compliance and mass distribution ofthe system. The simple expressionsfor natural frequencyjust giyen, yield the natural-frequencycurve of the basic vibration chartgiven in Par. 4; Case A. In the ehart theyare plotted on a logarithmicscale and the frequency ls given in cycles per minute,rather than in radians per second. A system may have more than one naturalfrequency, in which case the lowest of these isoften the most significant.In general, the number of natural frequeneiesis equalto the degree of freedom of the system.FOReING FREQUENCY- The number of oscillations per unit time of an externalforce or displacernent,applied to a vibratlngsystem.

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