Project Two Report

8/8/2019 Project Two Report

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ProjectTwoSingle degree of freedom analysisAaronCanning

ScottHarrison

StephenShew

DavidSteele

JacksonSupit

NathanMcCosker

8/3/2009


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Projectoverview

Thisreportisananalysisoftheperformanceofasimpleboxtrailerssuspensionsystem.Theanalysis

willbebasedonthetrailerbeingviewedasamechanicalsystemwithasingledegreeoffreedomfor

anymovement. Itwillbeassumed that the towbarof the trailer isconnected toa singlevertical

slider,henceeliminatinganyrollorpitch.Thustherewillonlybeverticalmotioninthesystem.

The systemwillbeanalysed inmanydifferentsituations including freevibration, forcedand road

surfaceforcedscenarios.Allofwhich,willbeanalysedregardingresonance,transmissibilityfactoras

wellasbeingmodelledinsimulink.


4/43

Listofassumptions

Forthewholesystem

The Maximum displacement of the trailer in the vertical direction is 0.2m. Hence theamplitude,X,is0.2

Critical dampening is designed to occur when the trailer is fully loaded (2000kg). This isassumedbecausegenerallymoreloadmeansmoreobjectswillbecarriedandhencethere

willbemoreprotectionneeded.

Thetrailerisstationaryonlevelgrounduntiltheroadsurfaceanalysis. Thespringsactasoneinthesystemandareviewedasasinglespringanddampener.

Engineforcedvibration

The eccentric mass of the skid mounted compressor it 100grams with an eccentricity of10cm. It isassumed the imbalance isdue to the rotationof thecrankshaftaswellas the

movementofpistons.Theengineisverybadlymaintained.

Thecompressorhasbeenmounteddirectlyovertheaxleofthetrailertodirectthemotionoftheeccentricmassthroughthesystemandsimplifythecalculation.

Theengineiscompletelyfixedtothebackofthetrailerwithnoloosemovement. Themassesofthetrailerandengineareusedtogetherasonemass.

RoadInductedVibrations

Roadsurfaceismodelledbyasinewavetosimulatethetrailerbeingpulledacrosscorrugations.

Corrugationshave:o Amplitudeof0.1mo Wavelengthof2m

Thevelocityofthetrailerisfrom0120km/hr Thetireonthetrailerremainsincontactwiththeroadatalltimes


5/43

DerivationofFreevibrationresponse

Asthereisalimitontheamountofinformationwehaveonthetraileranditsload,furtherresearch

and analysis was conducted to gain all the information that was needed to describe the free

vibrationresponse.ThisinformationcanbeseenonthefreebodydiagraminappendixB.Itconsists

ofthecombinedspringconstants,Kofthetwoleafsprings,thedampeningsuppliedbythesystem

andthemassoftheloadtobecarried(theskidmountedenginepoweredaircompressor).

Equationoffreevibrationresponse;

(t)=0The weight of the trailer is known skid mounted engine, research was conducted to establish a

suitable device to use with the trailer. It was found that a___________ weighing approximately

500kgwasaviableoptionconsideringitssizeandmass.Thismeansthatthemassofthetrailerwill

varyfrom400kg(empty)to900kg(compressormounted).

Forthespringconstant,Kitwasagreedthatitshouldbedesignedforatraileratmaximumloadto

withstand the stressesapplied to the system.Hence,when calculating the spring constant itwas

assumedthattheMwasthegrossweightofthetrailer,M=2000kg.Usingequation1.4(Hookslaw)

inappendixBandamaximumdisplacement/amplitudeof0.2mitwasfoundthatK=98000N/m.

Tofindthevalueofthedampeningconstant itwasassumedthatthetrailerwasatfullcapacityas

that iswhen critical dampeningwillbemostdesirable.Therefore thedampening constantof the

systemwasfound(usingequations1.1,1.2and1.3)tobe28000N.sec/m.

With all the values of the free body diagram found they can be formed into an equation and

modelledinsimulinktobegraphedaccordingly.Thefreevibrationequationsandthegraphscanbe

foundinappendixBaswellasthesimulinkmodelinappendixF.

It is also is noted that from equation 1.1 the resonant frequencies of the unloaded and loaded

trailersare15.65rad/secand10.43rad/sec.Therefore,withoutsufficientdampeninganyexcitation

forcewithsimilarfrequenciescouldpotentiallycauseresonancetooccur.


6/43

Forcedvibrationadditionandaffectonsystem

To analyse forced vibration on the system, a 500Kg internal combustion compressor has been

mountedonto thebackof the trailer.Thecompressorconsistsof four straightcylinders.The isa

mild imbalancewithintheenginewhich isrepresentedbyaneccentricmasswithavalueof0.1Kg

andaneccentricityof0.1m.

Equationofthenewsystem

Where

sinTherefore

sinWiththevaluesforallthevariablesenteredintotheequation;

900 28000 98000 0.1 sinWhereisproportionaltotheenginespeed.Thisfunctionwasmodelledinsimulinkandgraphedoverasetperiodtoanalysetheresponseofthesystemtotheengine.ThesimulinkmodelcanbefoundinappendixFandthegraphsinappendixC.

Itcanbe seenby thegraphs inappendixC thatundercriticaldampening theexcitation from the

engine causes little or no super positioning of the system at all. However when the damping

constantisreducedtolessthancriticaltheeffectsbecomesmoreobservable.Thetransfersystemis

very apparent initially and with less damping can be observed longer before a steady state is

achieved.

Itcanalsobeseenbythegraphsthatvaryingenginespeedsdoaffectthesystemduehoweverthey

onlybecome considerablearoundnof the system.Byuseofequation2.1, thenondimensional

steady state canbe foundandmodelledagainst the ratioofengine frequency, to the systemsnaturalfrequency,.HencetheresonanceofthewholesystemcanbeanalysedasinappendixCtorevealwhatenginespeedcauseresonanceinthesuspensiontooccur.ThisallowstheRPMvalue

oftheenginewhichcausesresonancetobefound.


7/43

From the results found inappendixC, theengineonlybegins to resonatewith the suspensionat

110RPM. The engine itself like any typical four cylinder internal combustion system operates

between3006000RPMandhasan idlespeedof300 700RPMapproximately.Theonlytimewhen

theenginewillbeoperatingaround110RPMisduringtheignitionphasewhenitisatthatspeedfor

onlyaninstant.Thereforeevenwithoutthedampeningofthesuspensiontheenginewouldnotbe

abletoachieveresonanceintimetocauseanyseriousissues.


8/43

Roadforcedvibrationadditionandaffectonsystem

Tomodela trailerbeing towedacrossacorrugated roadbetween the speedsof0120km/hr, the

abovesystemwillbeusedtoanalysethedisplacementofthewheelhubwithtime. Thecorrugated

roadwillberepresentedbyasinewavewithawavelengthof5mandanamplitudeof0.1m. The

vibrationsofthesystemisexcitedbythemotionofthesystemoverthecorrugations. Thusknowing

thespringconstant,dampingcoefficientandfrequency(duetothetrailer'svelocity),agraphcanbe

plottedtodetermineresonancecharacteristics. Thiscanbeachievedbyplottingthetransmissibility

factorVSFrequencyratio(ascanbeseeninAppendixD).

M

kc

V


9/43

Equationofmotion:

TheNonDimensionalsteadystatesolutionofthesystem

And

tan It can be seen from the graphs in appendix D, that when the system is critically damped the

suspension response is very minimal. When critically damped the trailer simply follows the

undulationsofthecorrugatedroad. Howeverwhenthedampingconstantisloweredtheeffectsof

thecorrugatedroadbecomemuchmorenoticeable. Withnodampingthemaximumdisplacement

spicksover0.15m(anadditional30%ofmovementcomparedtocritical).


10/43

Additionalnon-linearities

Whataresomepossiblenon-linearities ofthetrailer?

Nonlinearities can occur if the force exerted by the leaf spring is a nonlinear function of thedisplacement.Areal lifeexampleofthis is ifthewheelbecomesairbornor ifthespringbecomes

fullycompressed,whenreferringtoasuspensionsystemsuchasonthetrailer itcanbesaidthat

thesuspensionhasbeenbottomedout.Oncethespringcannotcompressanymorenonlinearities

occurs.Asforifthetyrebecomesairbornthedisplacementofthespringovertimewouldnotbeina

linearform,duetothecharacteristicsandnatureofthekinematicequation.Inadditiontothisthe

spring could be stretched to yield by the means of the mass of axel and wheel rims tyres etc.

Howeverinthisparticularmodelthiswasnotexamined.

Inamorecomplexmodel rather thanthespringhavingaconstantkvalue itchangesthroughout

eachleafspring.ThiscanbeseeninthebelowFEAsimulations1

1Source:2002ABAQUSUsersConference


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Simulinkmodelincludingnon -linearities

ThesimulinkmodelandinputvaluescanbefoundonAppendixF.Asitcanbeseenfromthebelowgraphthatinitialythemodelisoutofanacceptablerangewitha

peakat1.5mobviouslytheleafspringsshouldnottravelthesetypesofdistances.Howeveraftera

periodof time it settlesdown toabout100125mmwhichwould thenbe inanacceptable range

beforebeingcompressed.Aftermuchtweakingofthemodelithasbecomeapparentthattherange

valueswhichareentered into themodelmustbe furtherexamined.Although, itappearsthat the

modelisactuallysimulatingcorrectlyresultinginatrueresponseandoutput.Errorscanoccursuch

as identifying that the response is in metres and not in centre metres and the range values are

correct.


12/43

Accelerometer

Anaccelerometer isan instrument thatmeasures theaccelerationofavibratingbody.When the

natural frequency of the device is high compared to that of the vibration to be measured, the

instrumentindicatesacceleration.

An accelerometer based on spring mass system has been designed to measure the measure the

verticalmotionofthetrailer.Theconfigurationhasamaximumerrorof0.01%withinthefrequency

rangeof0to20Hz.

Model:

Specifications:

Mass,m=5grams

Springstiffness,k=4500N/m

Dampingcoefficient,c=6.64Nm/s

Dampingfactor=0.7

Assumptions:

Accelerometerisfirmlymountedonthechassisofthetrailer

Themaximumfrequencyoftheroadsurfacetobedrivenonis18Hz

m

k

x

a

m

k

x

a


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Theaccelerometerresponseisameasureofthemovementorvibrationofthedevice.Tocalculate

theaccelerometerresponse,thefollowingequationisused

Thegraphofresponseshowsthattheinstrumentisfunctionalatlowfrequencyandthusitisan

accelerometer.Infact,ifthemeasuredfrequenciesarehigherthatthenaturalfrequencyofthe

accelerometer,theamplituderesponsebecomesflat.

Theusefulrangeoftheaccelerometer,theaccelerometererror,canbecalculatedusingthe

equation

11 2

Theerror in theaccelerometer isnegligiblewithin the frequency rangeof0 to20Hz.The reading

accuracydecreasesifthedeviceisusedoutsideofthisrange.Thegraphbelow,accelerationerrorvs

frequency with as a parameter, shows that 0.7 is an ideal damping factor. In addition, = 0.7

extendstheusefulfrequencyrangeandalsopermitstogettheleastamplitudedistortion.

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

IZ/YI

/n

AccelerometerResponse

0

0.7

Damping

Factor


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15/43

Reflection

-Realandidealisedmodels

TheImportance ofconsidering non-linearities

Nonlinearitiesofthesystemshouldbeconsideredtoincreasetheaccuracyofthesimulinkmodelso

thatitreasonblesthatofareallifesituation.

Byconsideringthenonlineraritesofthesystemcanalsopreventanddetectdesign flaws for the

operation of the component or object. When modelling in simulink the input and output is

determinedonthestructuringofthemodelasaresult ifthemodeldoesntaccommodateforany

nonlineraties(onlyconsiderslinear)thesystemmaybecomeunstableinreallifewhichmaycausea

faultundercertainoperationalconditions,however thecomputermodelmay reasonablea stable

outputresult.

Thekeypointwhencomparingrealandidealisedmodelsisyouonlygetwhatyouputin,forother

wordsifyouputincorrectvaluesintothesystemorthesimulinksystemisnotdesignforthoseinput

valuesobvisalyonlyincorrectsolutionswillbereturnedtotheoutput.

Sometimesitisnearimpossibletosimulatethereallifemodel,howevertheidealisedmodelmay

becloseenoughandbeintherangethatitdoesntmatterthateverysinglevariablehasbeentaken

into account. Such as for the trailer the leaf springs has been modelled in ABAQUS and it was

discovered that for the leaf arrangement which would be very simulink to the trailer that force

displacementissoclosetolinearitcanbeassumedtobelinear2,howeverthesystemwouldbeina

nonlinear formdependingon thedesired trend line.So this leaves twooptions;make themodel

morecomplexandincludenonlinearitiesresultinginmorechancestomakeaprogrammingmistake

ortakethesecondoptionofjustusinglineararrangementwhichwouldbelessproblematicandgive

suchthesameoutputresultoftheotheroption.

Howeverthetruthis,ifyouhavetomakethemodelasrealisticaspossibletoreflectthebehaviour

ofthecomponentorobjectnonlinearitesmustbeconsidered.

-Differences/advantagesoffrequencydomainanalysis

Thedifferencebetween frequencydomainanalysisandtimedomainanalysis includesthemethod

forgraphing thedata.Frequencydomainanalysisplotsthe frequency ratioofagainstthenon

2RefertoAppendixIoutsourced2002ABAQUSUsersConference


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dimensionalamplitudeMX,amtheresultinggraphwillshowthe largeinfluencethatthedamping

hasoverthesystemwhennearresonance.Through inspectionoffrequencydomainanalysisplots,

the inertia and damping forces can be observed to be large, small, or balanced by other forces,

dependingonvaluesof .Whenthe

valueissmall,boththeinertiaanddampingforcesaresmall.Whenthevalueis1,

the larger inertia force is balanced by the spring force, and the damping force is overcome by

impressedforces.Whenthevalueofislargertheimpressedforceisexpendedalmostentirelyin

overcomingthelargeinertiaforce.

There are some advantages to using time domain over frequency domain analysis. It is easier to

identifyandfixproblemsorinconsistenciesinthedesign,andthroughtheuseofSimulinkitisalso

easiertocreateandefficientsystemandmakechangesinthedesignsoitsuitstheconditionsitwill

beappliedto.

The disadvantages include not being able to accurately reflect on the physical properties of the

system, and relying on the skills and knowledge of the operator using the Simulink program to

effectivelyapplythephysicalmodeltotheprogram.

-Briefinvestigationofthealternativemodellingapproachie.Rotational

movement

Thespringsoftheboxtrailerare inthesimplestformofasingledegreefreedommodel.Theycan

eitherbemodelledmovingtranslationalongonedirection(vertically)orcanrotateaboutoneaxis

(rotationalmotion)whichwillbediscussedbrieflyshowingthedifference inequationsandhow it

wouldaffecttheresonanceofthesprings.Theequationsbelowarethemost importantequations

thatwereusedinthecompletionofourspringmodellingthereforetheycanbeviewedagainstone

another.


17/43

Vertical Motion Rotational Motion

Wn = 2 (Sqre root k/m)

(O) + c/m (O) + 4k/m (O)

Cc=c/4(sqrtrootk.m)

X=M(F0/K) (;)=M(M0/K0)

Resonanceofamechanicalsystemisthestateofthesysteminwhichanabnormallylargevibration

isproducedinresponsetoanexternalstimulus,occurringwhenthefrequencyofthestimulusisthe

same, or nearly the same, as the natural vibration frequency of the system. The comparison of

resonance from theverticalmotion to thatof the rotationalmotion canbeviewedbelowas the

responsetimesadifferent.

Verticalresonance

400 28000 98000 0 (unloaded)900 28000 98000 0 (loaded)Resonantfrequenciesoftheunloadedandloadedtrailersare15.65rad/secand

10.43rad/sec.


18/43

Rotationalresonance

400 28000 98000 0 (unloaded)

900 28000 98000 0 (loaded)

Resonantfrequenciesoftheunloadedandloadedtrailersare31.30rad/secand

20.85rad/sec.There isa larger time frameof resonancebetween theunloadedand loaded trailer

when looking at the spring of thebox trailer from a rotational motion view. The response times

differconsiderably,theloadedtrailerbeing10.4rad/secandtheunloadedtrailer15.65rad/sec.The

responsetimesaredoublethatoftheverticalmotiontimes


19/43

Appendix -A - Listofgiveninformation

Specifications

Type: ElCheapo

Mass:(Tare) 400kg

Mass:(Gross) 2000kg

ChassisDimensions:

Overalllength: 3600mm

OverallWidth: 1700mm

OverallHeight: 800mm

BoxLength: 2100mm

Boxwidth: 1200mm

Boxheight: 400mm

Axleposition: 1100mmfromthefrontofthetrailer

Tyres: 185mmx355mm(Rimdiameter)


20/43

Appendix -BFreevibrationmodellingandfreebodydiagrams

Layoutanddimensionsofthetrailer

Freebodydiagram Functionaldiagram


21/43

Equationsusedtoderivethemodellingequation

Equation1.1 Equation1.2 2Equation1.3 = 1Critical)Equation1.4 Hookslaw,

Fromdiagram,thefollowingequationcanbeformed;

0Usingtheinformationfound,theequationfortheunloadedtraileris;

400 28000 98000 0andfortheloadedtraileris;

900 28000 98000 0Thegraphson the followingpagehavebeenmade toshowwhat theequation representsand to

display it in different situations primarily focusing on different values of C. They show the

displacement of the trailers axle in metres (centre point of the wheel) with respect to time in

seconds.

Note:Theempty (M=400kg)and loaded (M=900kg)systemswere inputted intosimulinkhowever

theonlynotable changedetectedwas inamplitudewhere the loaded system rangedhigherbya

verysmallamount.Withthissmalldifferencedetectedallfurtheranalysiswillbeconductedwiththe

loadedtrailerduetothevariationbeingfoundinthisinitialtest.


22/43

Unloaded Loaded

Underdampened Criticaldampening

NoDampening OverDamped

Theeffectsofdifferentdampingvaluesonthesystemcanbeobserved.


23/43

Appendix -CForcedvibration,unbalancedengine

Diagramofengineaffectingthetrailer

Equationsusedintheanalysis

Equation2.1,frompage54ofTheoryofvibrationwithapplications5thedition,

1 2

Wherethedampingratioisrepresentedby . Equationofsystemwhenloaded;

sin sin900 28000 98000 0.1 sin sin FortheLaplacetransform;

LetX(s)=x,Therefore;

Forthetransferfunction;


24/43

Simulinkgraphsdisplayingdampeningandenginespeedeffectsonthesystem

Nodampingwithforceatresonancefrequency Forcedvibrationwithcriticaldampingresonancefrequency

Forcedvibrationwithverylittledampingatresonancefrequency Littledampingwithengineat3000RPM


25/43

Littleafterresonantfrequencywithlittledamping Littlebeforeresonantfrequencywithlittledamping

Plotsofforcedvibrationwithrotatingunbalance(dampingconstant=(1/2.8)

Fromabove itcanbe seen thatat resonance the frequency ratio isabout1.1.Considering thata

systemwithnodampingwouldhavearatioof1,theeffectofthedamping issomewhatevident.

Withoutsuchafunctionthetrailerssuspensionwouldreachresonanceataratioof1causinglarge

excitationofthesystempotentiallydamagingthetrailer.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12

MX/me

w/wn

MX/meVsw/wn


26/43

Dampeningfactorof1,hencecriticaldampening

Inthisgraphthesystemisatcriticaldampeningwhichmakesdetermininganapproximatepointof

resonancefromitimpossible.Thisistheperfectsettingforthetrailerasthedampeningnullifiesa

largeamountofimpactandstopedthesystemfromresonating.

FromThefirstgraphofMX/meVs thefrequencyratiowasdeterminedtobeatapproximately1.1whenthesystemtentedtowardsresonance.Fromthistheenginespeedcanbedeterminedand

accommodatedfor.

=1.1,Were

10.43rad/sec

from equation 1.1

WhereM=massofentiresystem=900Kg(notethemassoftheentiresystemandnottheeccentric

masswastaken)

K=98000N/m

and C=10000

Therefore=11.48rad/secand109.61RPM


27/43

Fromthisitcanbesaidthatforasystemdampedpartially,theenginewillstarttoresonatewiththe

suspensionat100 110RPM.Tostopthisfromhappening,thetrailerhasbeendampedsufficiently

asinthesecondgraphwhereitcanbeseenastheengineapproachesapproximately110RPM(value

changeswithadifferentC)resonanceisnotexistent.

A smallobservation thatwasmadewas that the resonance frequency increaseswithdampening.

Withnodampeningthesystemreachesresonanceat100RPM,withC=10000thesystemresonates

at110RPMasabove.


28/43

Appendix -DRoadsurfacevibration

NoDampening

Underdamped


29/43

Critical


30/43

SimulinkModel:RoadSurfaceinducedVibration


31/43

ResonanceInductedbyRoad

Knowing the spring constant, damping coefficient and frequency (due to the trailer's velocity), a

graphcanbeplottedtodetermineresonancecharacteristics. Thiscanbeachievedbyplottingthe

transmissibilityfactorVSFrequencyratioascanbeseenbelow.

SpringConstantof28000Ns/m(criticaldamping)

At 28 000Ns/m the system is critically damped and doesn't allow the system to resonate at any

speed. Thisiswhythisdampeningconstantwaschosenforthedampenerinourtrailerbecause it

wouldpreventanyunexpectedresonancethatcouldmakethetrailerbounceviolentlyovertheroad

under the right conditions. Yet thedampener is still softenough toabsorbany impactand then

returntothesystemstoitsnaturalposition.

ResonanceInducedbyRoad


32/43

SpringConstantof15000Ns/m(underdamped)

Forillustrativepurposes,itcanseenthatwithareduceddampingconstantof15000Ns/mresonance

occurswhenthefrequencyratio isapproximately0.8. Asthedampingconstantapproaches0,the

frequency ratio at which resonance will occur will approach 1. (with no damping resonance will

occurat1.)Thespeedatwhichresonancewilloccurcanthenbecalculated.

0.8 7rad/sfromequaton1.1

0.8x7

5.6/FindFrequency

2 2

0.89

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0.

00

0.

10

0.

20

0.

30

0.

40

0.50

0.

60

0.70

0.

80

0.

90

1.

00

1.

10

1.

20

1.

30

1.

40

1.50

1.

60

1.70

1.

80

1.

89

1.

99

2.

09

2.

19

2.

29

2.

39

2.

49

TransmissibilityX/y

TransmissibilityVSFrequencyRatioResonanceInducedbyRoad


33/43

FindVelocity

0.89x5 4.45/ 16/Thereforeat16km/hrandadampingconstantof15000Ns/mthetrailerwillbeatresonance.

MagnificationFactor

Themagnificationfactorisanondimensionalexpressionfortheamplitudeofoscillation. Thisis

determinedfrom:

11 2

0

0.2

0.4

0.6

0.8

1

1.2

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 1.99 2.19 2.39

MagnificationFactor

MagnificationFactorVsFrequencyRatio

15000Ns/m

28000Ns/m


34/43

Itcanbeseenfromthegraphabovethatwithadampingconstantsof15000Ns/mtheamplitudeof

oscillationisalwaysgreaterthanthatofthechosendamperof28000Ns/m. Onceagainthisiswhya

dampingconstantof28000Ns/mwasusedtodampenthetrailersdisplacement.

DerivingtheTransferFunction

Equationofmotion

LaplaceIntegralTransform

ThereforeTransferFunction

Matlabcannowusedtographabodeplotforthesystem.


35/43

BodePlot:RoadInductedVibrations


36/43

The above bode plot shows that at very low frequencies and frequencies just below 10 the

suspensionresponseofthetrailerisminimal.Aroundafrequencyratioof10thesuspensionsystem

isshowntobe inaccurate.Afterthispointthefrequencyoftheroadsurfacebecomessohighthat

the trailer feels little impact from the roads surface. This isbecause the trailer wheel no longer

followstheprofileoftheroad'ssurfaceandliftoffoccurs.

MatlabCode

num=[2800098000]

den=[20002800098000]

sys=tf(num,den)

bode(sys)

margin(sys)


37/43

Appendix -FSimulinkdiagramsusedinproject

Freevibrationresponse

Engineforcedvibration


38/43

Non-linearities


39/43


40/43


41/43

Appendix -HExtraResonancecalculation

EXTRA Eccentricmassusedforengine

UsingthesameworkingasappendixCandthecriticaldampeningconstantof28000,

n=989.95rad/sec

Fromthegraph,resonanceisapproachedatapprox1.1.

Thereforew=1088.94rad/sec=10398.65RPM.

The RPM value for the eccentric mass to reach resonance with the dampener and spring of the

trailerwasfoundtobe10400RPM.WhichliketheappendixCresultsisalsooutsideoftheoperating

rangeoftheengine.

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

MX/me

w/wn

MX/meVsw/wn


42/43

Appendix -IForceVsDisplacement:LeafSpring


43/43

References

WilliamT.Thomson&MarieDillonDahleh,Theoryofvibrationwithapplications5thEdition,PrenticeHall,UppersaddleriverNJ,published1998

CharlesM.Close,DeanKFrederick,JonathanC.Newell,ModellingandAnalysisoddynamicsystems3

rdEdition,JohnWiley&SonsInc.published2002

WelcometoVibrationDataLaplaceTransformTable,TomIrvine,VibrationData(searchengineandtutorialsite),viewed31/07/09

http://www.vibrationdata.com/Laplace.htm Gillespie,T.D.,FundamentalsofVehicleDynamics,SocietyofAutomotiveEngineers,Inc,

1992.

Liu,W.,NonlinearAnalysisTheoryofSingleLeafSteelSprings,SAEPaper881744,1988. SAE,ManualonDesignandApplicationofLeafSpring,SAEHS788,APR80. SAE,SpringDesignManual,SAEAE21,1996 SAEStandard,LeafSpringsforMotorVehicleSuspensionMadetoCustomaryU.S.Units

SAE

J510NOV92,SAEHandbook,Vol.2,p20.09,1998.

Tavakkoli,S,Aslani,F.,andRohweder,D,AnalyticalPredictionofLeafSpringBushingLoads

UsingMSC/NASTRANandMDI/ADAMS,MSCWorldUsersConference,1996. Wachtel,D.W.,Adkins,D.E.,May,J.M.,andHohnstadt,W.E.,AdvancesintheDesign,

Analysis,andManufacturingofSteelLeafSprings,SAEPaper872256,1987.

Documents

Project Two Report