Matlab question | Homework Writings

MinorProject II
DYNAMIC EQUATIONS
Solve oneof the following problemsor one of yourowndesign(subject toapproval bymodule coordinator.Note,themoduleissubstantially concerned withmethodsfor solvingequations numerically,accordingly aboutone third ofthegrade stepsare earned by writingand reporting good MATLABcode tosolve the equationsarising bymeansof one (or more) of themethodsoutlinedin the module notes. N.B. Grade stepsare notearnedif a built-inMATLABfunctionorsome other packageisusedto solve theequations. To clarify,thismeansa functionlike inv,solve, fzero,roots, ode45, etc.whichliterally solvesproblem ina single command.Obviously thisdoes not preclude the use of such functionsasaidsto helpyoutosolve the problem.It onlyprecludes them from being part of the final reported solution.AsalwayskeepcodeDRY andmodular. Where code isused repeatedlyit shouldtake the form of a MATLAB functionm-file andit should be
presentedinfullin yourreportafter thefashion ofthepartialsamplereport.
Recall previouscommentsconcerningthedevelopment ofMATLAB function M-files. A large component of your results, regardlessof what project youelect to do,iscomprised ofMATLAB function M-files. If you wish toobtainall ofthegrade stepsassigned to thisportion of your work then you must adheretothe requirementsthat I havepreviously imposed: (i) help filesmust be written, (ii) version information must be present, (iii)error checkingcode should beincludedto ensure greater robustnessandUHOLDELOLWLYWKHUHVKRXOGEHQR³P DJLFQXPEHUV´HPEHGGHGLQWKH
code,(v) namesof filesand variablesshould be wellchosen,being descriptive of the function or meaning,(vi) commentsshould beextensiveand M-filesshould have greater ratherthanlesser functionality,meaning for example that rather thanmakingexplicitassumptionsconcerning parameter valueswithinthe codethese might insteadbe input arguments, (vii) code shouldnot be
excessivelyinefficient,meaning in practicethat youshouldpay close attentiontowhatisdone inside loops. A certain number of grade steps willbeset aside for the writing ofMATLAB function m-files. You mayexpecttolose a significantportionof these assignedgrade stepsshould you ignore entirelythe seven requirementslisted above.Anotherrequirement ona MATLAB function
m-fileisthatitshould notdotoomanythings.Multiplethings areachieved by multiplem-files.
Codewhereindividualm-filesdo toomanythingsisnotmodular.
Thisprojectisundertaken aspart ofa team.Ingeneral teamscomprise3 members. Some exceptionalteamsmay comprise only 2 members.Eachproject involvesthree broad components: (i) the translation oftheprobleminto a mathematicalequation or system ofequationsamenable to solution by themethodsofthismodule (i.e. modelling),(ii) the solution ofthisequation or system
of equations bymeansof some numerical method ormethodsdiscussedinthemodule and(iii) the visualisation of the resulting solution. The reportingof the work involveseach member of the team writinga report onthe work asa whole but emphasisingtheir particular contribution anda brief teampresentationin Powerpoint or similar. Teamswill need to organise themselves.This componentisthe onlycomponent where deadlineisnot set.Under the unusual circumstancesin
whichwe findourselves the normalmethod ofassessment where teams makelive presentationto module coordinator will nottake place,insteada recording of presentation,withallteammembers contributingtothe audio,must be sent byemailalong with reports.

Problem2.2:VehicleSuspension
A vehicleisassumedto becompletely symmetricaboutitscentralaxis,i.e.theleft handsideisthe exactmirror image of the right handside. A modelfor the suspension system of half ofthe vehicle is showninFigure 2.3. The followingparameter values have beendeemedappropriate:
2mS1794kg sprungmass
2muf87 kg , 2mur140 kg frontandrear unsprungmass
a 1.271m , b 1.716 m distancefromcentreofsprungmasstoforwardandrearaxles
, kr18615N/m springconstantforfrontandrearshock
absorbers
C f 1190Nsec/m, Cr1000
Nsec/m
damping
constantforfrontandrearshock
absorbers
IS 3443kgm2 momentofinertiaofsprungmass
kt101115N/m tyrespring
constant
The wheelbasea b and weight are approximately the valuesfor a sixth generation BMW5-series. Note thatinthemodel half of the sprung mass,i.e. half of 1794 kg, issupported by thesuspension
systemon onesideofthevehicle.
Developa system of ordinary differential equationstodescribe the performance of thehalfvehicle model as the vehicletravelsat a constant speed of60km/hr over a certainroad surface. Choose yourownmodel for the roadsurface but besuretoinclude a seam, i.e. a sharpstep,some undulationand a stochastic (i.e. random)component.Inthisregard you mayfindthe MATLAB commandrand to be useful. You will finda considerableamount of information concerningroad
surfacesonline. Animatethemovementof thevehicleasittravelsalong theroad.

mS
T

uf
a b
kfCf

krCr
m

murkt

kt

Figure2.3:simplifiedmodelofsuspension systemof a half car
A possibledivision ofresponsibility:
Teammember #1focuseson development ofand presentation ofanalysisand selection of road surface model. Themost commonerrormadeinthepast byteam member responsible formodelling thissystem wastoproducea linear model. You mayvery well haveseenananalysisin other modulesof thismechanism whichresultedina linear model. Thisanalysiswasapproximate, although thatmight not havebeen madeclear to you.Most analysesof thismechanism onthe internet arealsoapproximateandagaindo notmakethisclear.Theactual modelisnot linearand since weproposetosolve numerically,not analytically, you must model moreaccurately.The
assumptionswhichleadtoa linear modeldo notneed to bemadeandshould notbemade.
Teammember #2focusesonnumericalsolution oftheresulting differentialequationsby some
methodpresentedinthemodule.
Teammember #3focuseson theanimatedpresentation of results.
Thisdivision of responsibilityisa suggestiononly.Eachteam may organisethemselvesasthey please, butpart of the presentationandreportingwillbetoexplain thisorganisation.