Sensitivity Calculation

The sensitivity of an output y (u) to an input u is defined as the change in y due to a unit change in u.

Around an operating point ( y 0 , u 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqefqvATv2CG4uz3bIuV1wyUbqeduuDJXwAKbYu51MyVXgaruWq VvNCPvMCG4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC 0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yq aqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaGaaiGadmWaamaaci GaaqqaceqbcaGcbaGaaiikaiaadMhadaWgaaWcbaGaaGimaaqabaGc caGGSaGaamyDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@42DF@ one can write the Taylor’s series as:(1)
y = y 0 + ( y u ) ( u u 0 )   +   O ( b ) = y 0 + ( y u ) Δ u   +   O ( b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGaamyEaiabg2da9iaadMhadaWgaaWcbaGaaGimaaqabaGc cqGHRaWkdaqadaqaamaalaaabaGaeyOaIyRaamyEaaqaaiabgkGi2k aadwhaaaaacaGLOaGaayzkaaWaaeWaaeaacaWG1bGaeyOeI0IaamyD amaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiaabccaqqqFc0 Na9jqBHDMBLnxA11gapeGaey4kaSIaaeiiaiaad+eacaGGOaGaamOy aiaacMcapaGaeyypa0JaamyEamaaBaaaleaacaaIWaaabeaakiabgU caRmaabmaabaWaaSaaaeaacqGHciITcaWG5baabaGaeyOaIyRaamyD aaaaaiaawIcacaGLPaaacqqHuoarcaWG1bGaaeiiaabb8jaFcWNaTf 2zUv2CPvxBa8GacqGHRaWkcaqGGaGaam4taiaacIcacaWGIbGaaiyk aaaa@6D86@

The quantity ( y u ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaeWaaeaadaWcaaqaaiabgkGi2kaadMhaaeaacqGHciIT caWG1baaaaGaayjkaiaawMcaaaaa@4172@ is called the first order sensitivity of the quantity y to the input u.

Sensitivity analysis is the study of how the change or uncertainty in the output of a mathematical model or system (y) can be apportioned to different sources of change or uncertainty in its inputs (u).

For Design Sensitivity Analysis, the quantity (u) is the design b. Here we are asking the question: How does the response y change for a given change in the design b.

In a multibody simulation, the response y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGaamyEaaaa@3C13@ is typically a function of the system states x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGaamiEaaaa@3C12@ and x ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGabmiEayaacaaaaa@3C1B@ , and, perhaps, explicitly on the design b. The system states x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGaamiEaaaa@3C12@ consist of (a) Displacements, (b) Velocities, (c) Lagrange Multipliers (or constraint reaction forces), (d) User defined differential equations, (e) User defined algebraic equations originating from Variables and LSE/GSE/TFSISO outputs, and, (f) Internally created intermediate states that simplify computation.

In mathematical terms:(2)
y = y ( x , x ˙ , b ) Δ y Δ b = ( y x ) x b + ( y x ˙ ) x ˙ b + y b  + higher order terms MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGceaqabeaacaWG5bGaeyypa0JaamyEamaabmaabaGaamiEaiaa cYcaceWG4bGbaiaacaGGSaGaamOyaaGaayjkaiaawMcaaaqaaiabgs JiCnaalaaabaGaeuiLdqKaamyEaaqaaiabfs5aejaadkgaaaGaeyyp a0ZaaeWaaeaadaWcaaqaaiabgkGi2kaadMhaaeaacqGHciITcaWG4b aaaaGaayjkaiaawMcaamaalaaabaGaeyOaIyRaamiEaaqaaiabgkGi 2kaadkgaaaGaey4kaSYaaeWaaeaadaWcaaqaaiabgkGi2kaadMhaae aacqGHciITceWG4bGbaiaaaaaacaGLOaGaayzkaaWaaSaaaeaacqGH ciITceWG4bGbaiaaaeaacqGHciITcaWGIbaaaiabgUcaRmaalaaaba GaeyOaIyRaamyEaaqaaiabgkGi2kaadkgaaaGaaeiiaabb8jaFcWNa Tf2zUv2CPvxBa8qacaqGRaGaaeiiaiaabIgacaqGPbGaae4zaiaabI gacaqGLbGaaeOCaiaabccacaqGVbGaaeOCaiaabsgacaqGLbGaaeOC aiaabccacaqG0bGaaeyzaiaabkhacaqGTbGaae4Caaaaaa@7F59@

The equations of motion provide an implicit relationship between (x and b) and ( x ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaGabmiEayaacaaaaa@3C1B@ and (x, b)). The quantities x b x ˙ b  and  y b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaSaaaeaacqGHciITcaWG4baabaGaeyOaIyRaamOyaaaa caqGSaGaaeiiamaalaaabaGaeyOaIyRabmiEayaacaaabaGaeyOaIy RaamOyaaaacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiamaalaaabaGa eyOaIyRaamyEaaqaaiabgkGi2kaadkgaaaaaaa@4EB3@ need to be computed first. Once these are known, the design sensitivity, Δ y Δ b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaSaaaeaacqqHuoarcaWG5baabaGaeuiLdqKaamOyaaaa aaa@3FD6@ , can be computed.

The calculation of Δ y Δ b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaSaaaeaacqqHuoarcaWG5baabaGaeuiLdqKaamOyaaaa aaa@3FD6@ is called Design Sensitivity Analysis (DSA) in MotionSolve. This is a new analysis method in MotionSolve. It always accompanies a regular analysis, such as static analysis, quasi-static analysis, kinematic analysis or dynamic analysis.

The job of the regular analysis is to compute the states x, ẋ and the outputs y for a given design b.

Once these are known, the DSA analysis will compute the sensitivity, Δ y Δ b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaaSaaaeaacqqHuoarcaWG5baabaGaeuiLdqKaamOyaaaa aaa@3FD6@ . When there are Ny responses and Nb design variables, [ Δ y Δ b ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbd9MBZ9 gBHnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMC G4uz3bqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaGaciGadmWaamaaciGaaqqace qbcaGcbaWaamWaaeaadaWcaaqaaiabfs5aejaadMhaaeaacqqHuoar caWGIbaaaaGaay5waiaaw2faaaaa@41C8@ is a matrix of dimension Ny x Nb.

There are three well-known methods for computing design sensitivity: Finite Differencing, Direct Differentiation, and Adjoint Approach.