Coupling

Coupling refers to the forces and moments generated in a bushing to oppose the overall deformation of the bushing. These forces and moments are independent of any coordinate system that might be used to measure the deformation or deformation velocity. Coupling is an important factor when the bushing characteristics are non-linear.

Below is information on the formulation for coupling followed by three examples that show how coupling affects bushing force and torque output:

Coupling Formulation

The Altair Bushing Model supports three options for coupling:

  • Cylindrical coupling (2-dimensional)
  • Spherical coupling (3-dimensional)
  • No coupling

Two-dimensional coupling and three-dimensional coupling are analogous, and therefore, this guide explains coupling in terms of the two-dimensional concept.

For cylindrical coupling, assume:

  • A bushing has been fitted in two radial directions: x and y.
  • The internal states for the bushing are d, the bushing deformation, v, the bushing velocity, and q. These can change with direction.
  • G x ( d , v , q ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWG4baabeaakiaacIcacaWGKbGaaiilaiaadAhacaGGSaGa amyCaiaacMcaaaa@3D88@ defines the force function in the x-direction, as obtained by the fitting process.
  • G y ( d , v , q ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacaWG5baabeaakiaacIcacaWGKbGaaiilaiaadAhacaGGSaGa amyCaiaacMcaaaa@3D89@ defines the force function in the y-direction, as obtained by the fitting process.
  • During simulation, at any time t, the bushing undergoes deformations of (x, y) and deformation velocities of ( x ˙ , y ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqadI hagaGaaiaacYcacaaMe8UabmyEayaacaGaaiykaaaa@3B99@ .

The diagram below shows the deformations and force vectors in the bushing. The J Marker is used as the coordinate system for all calculations.



Figure 1.
e ^ x j a n d e ^ y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadIhacaWGQbaabeaakiaaysW7caWGHbGaamOBaiaa dsgacaaMe8UabmyzayaajaWaaSbaaSqaaiaadMhacaWGQbaabeaaaa a@4201@
are the unit vectors along the x- and y-axes of the J marker. For cylindrical coupling, axial deformation is uncoupled. Only the x- and y-forces are coupled.
e ^ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadkhaaeqaaaaa@3813@
is a unit vector along the deformation vector.
r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36ED@
is the magnitude of the deformation. Its components along e ^ x j a n d e ^ y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadIhacaWGQbaabeaakiaaysW7caWGHbGaamOBaiaa dsgacaaMe8UabmyzayaajaWaaSbaaSqaaiaadMhacaWGQbaabeaaaa a@4201@ are denoted as x and y respectively.
e ^ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaad6gaaeqaaOGaaGjbVdaa@39A6@
is a unit vector orthogonal to the deformation vector. It points to the tangential velocity that may exist in the bushing.

The table below shows the various quantities of interest and how they are calculated:

Quantity Formula
Deforming vector r =r e ^ r =x e ^ xj +y e ^ yj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaala Gaeyypa0JaamOCaiqadwgagaqcamaaBaaaleaacaWGYbaabeaakiab g2da9iaadIhaceWGLbGbaKaadaWgaaWcbaGaamiEaiaadQgaaeqaaO Gaey4kaSIaamyEaiqadwgagaqcamaaBaaaleaacaWG5bGaamOAaaqa baaaaa@4535@
Unit vector along radial deformation e ^ r =( x r ) e ^ xj +( x r ) e ^ yj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaadkhaaeqaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaa dIhaaeaacaWGYbaaaaGaayjkaiaawMcaaiqadwgagaqcamaaBaaale aacaWG4bGaamOAaaqabaGccqGHRaWkdaqadaqaamaalaaabaGaamiE aaqaaiaadkhaaaaacaGLOaGaayzkaaGabmyzayaajaWaaSbaaSqaai aadMhacaWGQbaabeaaaaa@474E@
Unit vector perpendicular to deformation e ^ n =( y r ) e ^ xj +( x r ) e ^ yj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyzayaaja WaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiab gkHiTiaadMhaaeaacaWGYbaaaaGaayjkaiaawMcaaiqadwgagaqcam aaBaaaleaacaWG4bGaamOAaaqabaGccqGHRaWkdaqadaqaamaalaaa baGaamiEaaqaaiaadkhaaaaacaGLOaGaayzkaaGabmyzayaajaWaaS baaSqaaiaadMhacaWGQbaabeaaaaa@4838@
Bushing radial deformation r= r · r = ( x 2 + y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9maakaaabaGabmOCayaalaGaeS4JPFMabmOCayaalaaaleqaaOGa eyypa0ZaaOaaaeaadaqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaa GccqGHRaWkcaWG5bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzk aaaaleqaaaaa@4407@
Bushing tangential deformation r= r · e ^ n =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iqadkhagaWcaiabl+y6NjqadwgagaqcamaaBaaaleaacaWGUbaa beaakiabg2da9iaaicdaaaa@3F4F@
Deformation velocity vector r ˙ = x ˙ e ^ xj + y ˙ e ^ yj = v r e ^ r + v n e ^ n = ( x· x ˙ +y· y ˙ ) r e ^ r + ( x· y ˙ y· x ˙ ) r e ^ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaacy aalaGaeyypa0JabmiEayaacaGabmyzayaajaWaaSbaaSqaaiaadIha caWGQbaabeaakiabgUcaRiqadMhagaGaaiqadwgagaqcamaaBaaale aacaWG5bGaamOAaaqabaGccqGH9aqpcaWG2bWaaSbaaSqaaiaadkha aeqaaOGabmyzayaajaWaaSbaaSqaaiaadkhaaeqaaOGaey4kaSIaam ODamaaBaaaleaacaWGUbaabeaakiqadwgagaqcamaaBaaaleaacaWG Ubaabeaakiabg2da9maalaaabaWaaeWaaeaacaWG4bGaeS4JPFMabm iEayaacaGaey4kaSIaamyEaiabl+y6NjqadMhagaGaaaGaayjkaiaa wMcaaaqaaiaadkhaaaGabmyzayaajaWaaSbaaSqaaiaadkhaaeqaaO Gaey4kaSYaaSaaaeaadaqadaqaaiaadIhacqWIpM+zceWG5bGbaiaa cqGHsislcaWG5bGaeS4JPFMabmiEayaacaaacaGLOaGaayzkaaaaba GaamOCaaaaceWGLbGbaKaadaWgaaWcbaGaamOBaaqabaaaaa@6A9A@
Bushing radial deformation velocity v r = ( x· x ˙ +y· y ˙ ) r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGYbaabeaakiabg2da9maalaaabaWaaeWaaeaacaWG4bGa eS4JPFMabmiEayaacaGaey4kaSIaamyEaiabl+y6NjqadMhagaGaaa GaayjkaiaawMcaaaqaaiaadkhaaaaaaa@457E@
Bushing tangential deformation velocity v n = ( x· y ˙ y· x ˙ ) r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGUbaabeaakiabg2da9maalaaabaWaaeWaaeaacaWG4bGa eS4JPFMabmyEayaacaGaeyOeI0IaamyEaiabl+y6NjqadIhagaGaaa GaayjkaiaawMcaaaqaaiaadkhaaaaaaa@4585@
Bushing force vector F = F x e ^ xj + F y e ^ yj = F r e ^ r + F n e ^ n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaala Gaeyypa0JaamOramaaBaaaleaacaWG4baabeaakiqadwgagaqcamaa BaaaleaacaWG4bGaamOAaaqabaGccqGHRaWkcaWGgbWaaSbaaSqaai aadMhaaeqaaOGabmyzayaajaWaaSbaaSqaaiaadMhacaWGQbaabeaa kiabg2da9iaadAeadaWgaaWcbaGaamOCaaqabaGcceWGLbGbaKaada WgaaWcbaGaamOCaaqabaGccqGHRaWkcaWGgbWaaSbaaSqaaiaad6ga aeqaaOGabmyzayaajaWaaSbaaSqaaiaad6gaaeqaaaaa@4D05@
Radial force F r x r | x r | G x ( sign( x )·r, v r , q rx )+ y r | y r | G y ( sign( y )·r, v r , q ry ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGYbaabeaakiablYLianaalaaabaGaamiEaaqaaiaadkha aaWaaqWaaeaadaWcaaqaaiaadIhaaeaacaWGYbaaaaGaay5bSlaawI a7aiaadEeadaWgaaWcbaGaamiEaaqabaGcdaqadaqaaiaadohacaWG PbGaam4zaiaad6gadaqadaqaaiaadIhaaiaawIcacaGLPaaacqWIpM +zcaWGYbGaaiilaiaadAhadaWgaaWcbaGaamOCaaqabaGccaGGSaGa amyCamaaBaaaleaacaWGYbGaamiEaaqabaaakiaawIcacaGLPaaacq GHRaWkdaWcaaqaaiaadMhaaeaacaWGYbaaamaaemaabaWaaSaaaeaa caWG5baabaGaamOCaaaaaiaawEa7caGLiWoacaWGhbWaaSbaaSqaai aadMhaaeqaaOWaaeWaaeaacaWGZbGaamyAaiaadEgacaWGUbWaaeWa aeaacaWG5baacaGLOaGaayzkaaGaeS4JPFMaamOCaiaacYcacaWG2b WaaSbaaSqaaiaadkhaaeqaaOGaaiilaiaadghadaWgaaWcbaGaamOC aiaadMhaaeqaaaGccaGLOaGaayzkaaaaaa@703C@
Tangential force F n y r | y r | G x ( n, v n , q nx )+ x r | x r | G y ( r, v n , q ny ) = y r | y r | G x ( 0, v n , q nx )+ x r | x r | G y ( 0, v n , q ny ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGgb WaaSbaaSqaaiaad6gaaeqaaOGaeSixIa0aaSaaaeaacaWG5baabaGa amOCaaaadaabdaqaamaalaaabaGaamyEaaqaaiaadkhaaaaacaGLhW UaayjcSdGaam4ramaaBaaaleaacaWG4baabeaakmaabmaabaGaamOB aiaacYcacaWG2bWaaSbaaSqaaiaad6gaaeqaaOGaaiilaiaadghada WgaaWcbaGaamOBaiaadIhaaeqaaaGccaGLOaGaayzkaaGaey4kaSYa aSaaaeaacaWG4baabaGaamOCaaaadaabdaqaamaalaaabaGaamiEaa qaaiaadkhaaaaacaGLhWUaayjcSdGaam4ramaaBaaaleaacaWG5baa beaakmaabmaabaGaamOCaiaacYcacaWG2bWaaSbaaSqaaiaad6gaae qaaOGaaiilaiaadghadaWgaaWcbaGaamOBaiaadMhaaeqaaaGccaGL OaGaayzkaaaabaGaaGjbVlaaysW7caaMe8UaaGjbVlabg2da9maala aabaGaamyEaaqaaiaadkhaaaWaaqWaaeaadaWcaaqaaiaadMhaaeaa caWGYbaaaaGaay5bSlaawIa7aiaadEeadaWgaaWcbaGaamiEaaqaba GcdaqadaqaaiaaicdacaGGSaGaamODamaaBaaaleaacaWGUbaabeaa kiaacYcacaWGXbWaaSbaaSqaaiaad6gacaWG4baabeaaaOGaayjkai aawMcaaiabgUcaRmaalaaabaGaamiEaaqaaiaadkhaaaWaaqWaaeaa daWcaaqaaiaadIhaaeaacaWGYbaaaaGaay5bSlaawIa7aiaadEeada WgaaWcbaGaamyEaaqabaGcdaqadaqaaiaaicdacaGGSaGaamODamaa BaaaleaacaWGUbaabeaakiaacYcacaWGXbWaaSbaaSqaaiaad6gaca WG5baabeaaaOGaayjkaiaawMcaaaaaaa@8AEC@
Force in the x-direction F x = F · e ^ xj = x r F r y r F n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG4baabeaakiabg2da9iqadAeagaWcaiabl+y6Njqadwga gaqcamaaBaaaleaacaWG4bGaamOAaaqabaGccqGH9aqpdaWcaaqaai aadIhaaeaacaWGYbaaaiaadAeadaWgaaWcbaGaamOCaaqabaGccqGH sisldaWcaaqaaiaadMhaaeaacaWGYbaaaiaadAeadaWgaaWcbaGaam OBaaqabaaaaa@4941@
Force in the y-direction F y = F · e ^ yj = y r F r + x r F n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG5baabeaakiabg2da9iqadAeagaWcaiabl+y6Njqadwga gaqcamaaBaaaleaacaWG5bGaamOAaaqabaGccqGH9aqpdaWcaaqaai aadMhaaeaacaWGYbaaaiaadAeadaWgaaWcbaGaamOCaaqabaGccqGH RaWkdaWcaaqaaiaadIhaaeaacaWGYbaaaiaadAeadaWgaaWcbaGaam OBaaqabaaaaa@4938@

Coupling Examples

Example 1: Isotropic Bushing with No Damping and Constant Rotating Deflection
A constant deflection of 5 units stretching the bushing and rotating at 2*π radians/sec is imposed on the bushing. Rotation occurs in the X-Y plane of the J-Marker.

The following equations show the forces F x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG4baabeaaaaa@37EA@ and F y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG5baabeaaaaa@37EB@ computed by the coupling formulation. The plot shows that F y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG5baabeaaaaa@37EB@ vs. F x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWG4baabeaaaaa@37EA@ is a circle as expected.



Figure 2.
Example 2: Isotropic Bushing with No Damping and Constant Rotating Force
A constant tensile force of 125 units, rotating at 2*π radians/sec is imposed on the bushing. The force rotates in the X-Y plane of the J-Marker.

The following equations show the deformations x and y as computed by the coupling formulation. The plot shows y vs. x is a circle as expected.



Figure 3.
Example 3: Anisotropic Bushing with No Damping and Constant Rotating Force
A constant tensile force of 125 units, rotating at 2*π radians/sec, is imposed on the bushing. Rotation occurs in the X-Y plane of the J-Marker.

The following equations show the deformations of x and y as computed by the coupling formulation. The plot shows that since the bushing is non-isotropic, the y vs. x plot is not a circle, but a smooth, elliptical, closed-curve as expected.



Figure 4.