Mass-Spring-System(弹簧系统)

Mass-SpringSystemsABasicToolforModelingDeformableObjectsPart1ResearchResearchNumericalSolutionModelingPhysicalSimulation:Howto…?2PhysicalSystemParticleSystem(DiscreteModel)PDE(ContinuousModel)FiniteElementsOrdinaryDifferentialEquationsFiniteDifferencesResearchNumericalSolutionModelingPhysicalSimulation:Howto…?3ParticleSystem(DiscreteModel)PDE(ContinuousModel)FiniteElementsOrdinaryDifferentialEquationsFiniteDifferencesDeformableObjectsResearchDeformableObjects•Deformableobjects–changesizeandshapeduetoappliedforces–canbedeformedbutresistdeformation•Commonmaterialproperties–Elastic:deformationsarereversible–Viscous:amplitudeofoscillationsisreduced–Plastic:irreversibledeformations–Anycombinationthereof4ResearchNumericalSolutionModelingPhysicalSimulation:Howto…?5ParticleSystem(DiscreteModel)PDE(ContinuousModel)FiniteElementsOrdinaryDifferentialEquationsFiniteDifferencesDeformableObjectsResearchNumericalSolutionModelingPhysicalSimulation:Howto…?6Mass-SpringSystems(DiscreteModel)OrdinaryDifferentialEquationsDeformableObjectsResearchMass-SpringSystemsMass-springsystemsareparticlesystemswithspecialinteractionforcesStepstowardssimulation1.Spatialdiscretization:sampleobjectwithmasspoints2.Forces:defineinternal(springs!)andexternalforces3.Dynamics:setupequationsofmotion4.Temporaldiscretization:solveequationsofmotion7ResearchClothSimulationApplications8Bridsonetal.,2002Choi&Ko,2002ResearchApplications9MedicalsimulationKuehnapfeletal.,1993HairanimationSelleetal.,2008FacialanimationLeeetal.,1995ResearchOutlineStepstowardssimulation1.Spatialdiscretization:sampleobjectwithmasspoints2.Forces:defineinternal(springs!)andexternalforces3.Dynamics:setupequationsofmotion4.Temporaldiscretization:solveequationsofmotion10Mass-SpringSystemsResearchSpatialDiscretizationSampleobjectwithmasspoints•Totalmassofobject:M•Numberofmasspoints:n•Massofeachpoint:m=M/n(uniformdistribution)Eachpointholdsproperties•Mass•Position•Velocity11im)(tix)(tivResearchOutlineStepstowardssimulation1.Spatialdiscretization:sampleobjectwithmasspoints2.Forces:defineinternal(springs!)andexternalforces3.Dynamics:setupequationsofmotion4.Temporaldiscretization:solveequationsofmotion12Mass-SpringSystemsResearchForcesWhataretheforcesthatactonparticlei?13Externalforces–GravityInternalforces–Elasticspringforces–Viscousdampingforces281.900smigimFTotalforceextintiiiFFF(F)i=miNote:forcesare3D,3RFiResearchElasticity:Abilityofaspringtoreturntoitsinitialformwhenthedeformingforceisremoved.LlkFHooke’sLawInternalForces:ElasticSpringsSpringForce:•Ceiiinosssttuv.(Hooke,1676)•Uttensio,sicvis.(Hooke,1678)Forceislinearw.r.t.extension!FlLFInitialspringlengthLCurrentspringlengthlSpringstiffnesskResearchInternalForces:ElasticSpringsInitialspringlengthLCurrentspringlengthlSpringstiffnesskFlLFixjxLlkFForcein1DjijijiiLkxxxxxxFForcein3DResearchElasticEnergyFlLFixjxForpurelyelasticsprings(materials)•Forcedependsonlyonposition•NoenergylostduringdeformationWorkdonebyforcesdxLxkWlLElasticspringenergy2)(21LlkWEiiExFForce=–gradientofenergyResearchInternalforcesintFExternalforcesextF•Gravity•Contactforces•AllforcesthatarenotcausedbyspringsTotalspringforceextiintiiFFF0x3x2x1xF0intkii|i1,2,3liLixix0li11,kL22,kL33,kLForcesatMassPointResultingforceatpointiResearchDissipativeForces•Real-worldmechanicalsystemsdissipateenergyovertimeInternalfrictionThermalenergy(irreversibleprocess)•ControllabledissipationusefulforphysicssimulationsDowewantthingstomoveindefinitely?•Dissipationformass-springsystems)(ttpdvFPointdampingisdampingcoefficient+Simpleandefficient–Dampsallmotion(translationsandrotations)ResearchOutlineStepstowardssimulation1.Spatialdiscretization:sampleobjectwithmasspoints2.Forces:defineinternal(springs!)andexternalforces3.Dynamics:setupequationsofmotion4.Temporaldiscretization:solveequationsofmotion20Mass-SpringSystemsResearchDynamicsForceisknownforeveryparticle.Howdowedeterminemotion?•Kinematicrelationsdttdtii)()(xvVelocity22)()()(dttddttdtiiixvaAcceleration•MotionfollowsfromiiimaFNewton’s2ndLawtixResearchEquationsofMotioniiimaFNewton’s2ndLaw22)()(dttdtiixaAccelerationEquationsofmotionforonemasspoint(3equations)ttdttdmiiiiextintFFx22Equationsofmotionforsystemofmasspoints(3nequations)ttdttdextintFFxM22isadiagonalmatrixnn33RMResearchEquationsofMotion)(ttpdvFSpecialcase:pointdampingEquationsofmotion(3nequations)Motionisdeterminedbyasystemof3n2ndorderOrdinaryDifferentialEquations)(extint22ttdttddttdFFxDxMisdiagonalwithentriesnn33RDResearchOutlineStepstowardssimulation1.Spatialdiscretization:sampleobjectwithmasspoints2.Forces:defineinternal(springs!)andexternalforces3.Dynamics:setupequationsofmotion4.Temporaldiscretization:solveequationsofmotion24Mass-SpringSystemsResearch•Adifferentialequationdescribesanunknownfunctionthroughitsderivatives•Anordinarydifferentialequationcontainsonlyderivativeswithrespecttoasinglevariable•ExpressedasfunctionofhighestderivativeDifferentialEquations25Example)