Mount Stiffness

The bodies connected by the bushing are flexible and may deflect under the load being transmitted. This phenomenon is modeled with the Mount Stiffness feature. Mount stiffness models the structural stiffness of the bodies, thus mounting the bushing as a linear spring and damper in series with the bushing in each direction.

Modeling Mount Stiffness

The mount or structure surrounding a bushing deflects under the force transmitted by the bushing. The local stiffness models this deflection. If the I-Body and J-Body connected by the bushing are modeled as rigid-bodies, then you must estimate structural deflection separately and add that value to the bushing-rubber deflection. Mount stiffness is NOT required if the two bodies are modeled as flexible bodies.

To include local structural stiffness in the bushing model, you enter the combined local stiffness of the I-Body and J-Body, and then set the local stiffness flag to active.

When one or both bodies is flexible, you specify that the mount stiffness is conditionally active only when the I-Body and J-Body are both rigid.

Let:

d k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@37FB@
be the bushing deformation in the kth direction.
s k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGRbaabeaaaaa@380A@
be the net local structural deflection in the kth direction.
s i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbGaam4Aaaqabaaaaa@38F8@
be the local structural deflection of the I-body in the kth direction.
s j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGQbGaam4Aaaqabaaaaa@38F9@
be the local structural deflection of the J-body in the kth direction.
K s k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGZbGaam4Aaaqabaaaaa@38DA@

be the local structural stiffness in the kth direction.

K i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGPbGaam4Aaaqabaaaaa@38D0@
be the local structural stiffness of the I-Body in the kth direction.
K j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGQbGaam4Aaaqabaaaaa@38D1@
be the local structural stiffness of the J-Body in the kth direction.
C s k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaam4Aaaqabaaaaa@38D2@
be the local structural damping in the kth direction.
C i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGPbGaam4Aaaqabaaaaa@38C8@
be the local structural damping I-Body in the kth direction.
C j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGQbGaam4Aaaqabaaaaa@38C9@
be the local structural damping of the J-Body in the kth direction.
F k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGRbaabeaaaaa@37DD@
be the bushing force in the kth direction.

Then the net structural deflection for the kth direction is:

s k = s ik + s jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGRbaabeaakiabg2da9iaadohadaWgaaWcbaGaamyAaiaa dUgaaeqaaOGaey4kaSIaam4CamaaBaaaleaacaWGQbGaam4Aaaqaba aaaa@400B@

Since the I-body structural stiffness, the J-body structural stiffness and the bushing act in series, the forces in kth direction in the bushing, the I-body and J-body are all equal.

Therefore:
F k = K i k s i k = K j k s j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGRbaabeaakiabg2da9iaadUeadaWgaaWcbaGaamyAaiaa dUgaaeqaaOGaam4CamaaBaaaleaacaWGPbGaam4AaaqabaGccqGH9a qpcaWGlbWaaSbaaSqaaiaadQgacaWGRbaabeaakiaadohadaWgaaWc baGaamOAaiaadUgaaeqaaaaa@45CB@
s k = s i k + s j k = F k K i k + F k K j k = F K i k + K j k K i k K j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGRbaabeaakiabg2da9iaadohadaWgaaWcbaGaamyAaiaa dUgaaeqaaOGaey4kaSIaam4CamaaBaaaleaacaWGQbGaam4Aaaqaba GccqGH9aqpdaWcaaqaaiaadAeadaWgaaWcbaGaam4Aaaqabaaakeaa cGaJao4samacmc4gaaWcbGaJakacmc4GPbGaiWiGdUgaaeqcmciaaa GccqGHRaWkdaWcaaqaaiaadAeadaWgaaWcbaGaam4Aaaqabaaakeaa cGaJao4samacmc4gaaWcbGaJakacmc4GQbGaiWiGdUgaaeqcmciaaa GccqGH9aqpcaWGgbWaaSaaaeaacaWGlbWaaSbaaSqaaiaadMgacaWG RbaabeaakiabgUcaRiaadUeadaWgaaWcbaGaamOAaiaadUgaaeqaaa GcbaGaam4samaaBaaaleaacaWGPbGaam4AaaqabaGccaWGlbWaaSba aSqaaiaadQgacaWGRbaabeaaaaaaaa@6703@
K s k = K i k K j k K i k + K j k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaam 4samaaBaaaleaacaWGZbGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaa dUeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaam4samaaBaaaleaaca WGQbGaam4AaaqabaaakeaacaWGlbWaaSbaaSqaaiaadMgacaWGRbaa beaakiabgUcaRiaadUeadaWgaaWcbaGaamOAaiaadUgaaeqaaaaaaa a@47A2@

Through similar logic, one can conclude:

C sk = C ik C jk C ik + C jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaWGZbGaam4AaaqabaGccqGH9aqpdaWcaaqaaiaadoeadaWg aaWcbaGaamyAaiaadUgaaeqaaOGaam4qamaaBaaaleaacaWGQbGaam 4AaaqabaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGRbaabeaakiab gUcaRiaadoeadaWgaaWcbaGaamOAaiaadUgaaeqaaaaaaaa@463C@

The structure and the bushing act in series. Therefore they must generate the same force:

F k = K sk s k C sk s ˙ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaam OramaaBaaaleaacaWGRbaabeaakiabg2da9iabgkHiTiaadUeadaWg aaWcbaGaam4CaiaadUgaaeqaaOGaam4CamaaBaaaleaacaWGRbaabe aakiabgkHiTiaadoeadaWgaaWcbaGaam4CaiaadUgaaeqaaOGabm4C ayaacaWaaSbaaSqaaiaadUgaaeqaaaaa@4614@
s ˙ k = ( F k + K sk s k ) C sk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Cayaaca WaaSbaaSqaaiaadUgaaeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaa dAeadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcaWGlbWaaSbaaSqaai aadohacaWGRbaabeaakiaadohadaWgaaWcbaGaam4Aaaqabaaakiaa wIcacaGLPaaaaeaacaWGdbWaaSbaaSqaaiaadohacaWGRbaabeaaaa aaaa@4577@

The above equation is integrated to yield s k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGRbaabeaaaaa@380A@ , where the structural deformation is known and accounted for.