Question 4.7: Calculate the torsional stiffness kt of the rubber bushing s...
Calculate the torsional stiffness k_t of the rubber bushing shown in Figure 4.40. Assume that the rubber is bonded both to the steel shaft and to the outer steel tube, which is in turn attached to a machine housing. Assume that the metal parts do not deform, and that the shear modulus of rubber is G.
Given: Rubber bushing of known dimension and shear modulus.
Find: Torsional stiffness k_t .
Assume: Hooke’s law applies.
Axisymmetric torque is resisted by constant shear stresses, T = \tau (2 \pi rL)r. From Hooke’s law,
\gamma =\frac{T}{G}=\frac{T}{(2\pi r^2L)G} .
The incremental shaft rotation is rd \phi ≅ \gamma dr , the total shaft rotation is
\phi =\int{d\phi =\frac{T}{2\pi LG}}\int\limits_{d/2}^{D/2}{\frac{dr}{r^3}=\frac{T}{\pi LG} \left\lgroup\frac{1}{d^2} -\frac{1}{D^2} \right\rgroup } ,
and the stiffness is
k_t=\frac{T}{\phi }=\frac{\pi LG}{1/d^2-1/D^2}