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Transcript of 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.
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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
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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.
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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.
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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.
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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
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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).
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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.
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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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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.
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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.
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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
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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)
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Appendix -BFreevibrationmodellingandfreebodydiagrams
Layoutanddimensionsofthetrailer
Freebodydiagram Functionaldiagram
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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.
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Unloaded Loaded
Underdampened Criticaldampening
NoDampening OverDamped
Theeffectsofdifferentdampingvaluesonthesystemcanbeobserved.
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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;
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Simulinkgraphsdisplayingdampeningandenginespeedeffectsonthesystem
Nodampingwithforceatresonancefrequency Forcedvibrationwithcriticaldampingresonancefrequency
Forcedvibrationwithverylittledampingatresonancefrequency Littledampingwithengineat3000RPM
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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
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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
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Fromthisitcanbesaidthatforasystemdampedpartially,theenginewillstarttoresonatewiththe
suspensionat100 110RPM.Tostopthisfromhappening,thetrailerhasbeendampedsufficiently
asinthesecondgraphwhereitcanbeseenastheengineapproachesapproximately110RPM(value
changeswithadifferentC)resonanceisnotexistent.
A smallobservation thatwasmadewas that the resonance frequency increaseswithdampening.
Withnodampeningthesystemreachesresonanceat100RPM,withC=10000thesystemresonates
at110RPMasabove.
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Appendix -DRoadsurfacevibration
NoDampening
Underdamped
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Critical
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SimulinkModel:RoadSurfaceinducedVibration
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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
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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
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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
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Itcanbeseenfromthegraphabovethatwithadampingconstantsof15000Ns/mtheamplitudeof
oscillationisalwaysgreaterthanthatofthechosendamperof28000Ns/m. Onceagainthisiswhya
dampingconstantof28000Ns/mwasusedtodampenthetrailersdisplacement.
DerivingtheTransferFunction
Equationofmotion
LaplaceIntegralTransform
ThereforeTransferFunction
Matlabcannowusedtographabodeplotforthesystem.
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BodePlot:RoadInductedVibrations
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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)
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Appendix -FSimulinkdiagramsusedinproject
Freevibrationresponse
Engineforcedvibration
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Non-linearities
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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
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Appendix -IForceVsDisplacement:LeafSpring
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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.