Derive the equation of torsional oscillations of a…
Derive the equation of torsional oscillations of a homogeneous cylindrical shaft.
Explanation of Method
Apply Saint-Venant's torque–twist relation to an infinitesimal shaft element, apply Newton's second law for rotation, and take the limit as the element length shrinks to zero.
Step 1 · Torque–twist relation
For an element of a shaft with shear modulus $G$ and polar area moment $I_p$, the internal torque is \[ M_t(x,t) = GI_p\,\theta_x. \]
Step 2 · Mass moment of inertia and Newton's second law
An element of length $dx$ has mass moment of inertia $dI=\rho I_p\,dx$. The net torque $dM_t=(\partial M_t/\partial x)\,dx$ produces angular acceleration: \[ \frac{\partial M_t}{\partial x}\,dx = \rho I_p\,dx\,\theta_{tt}. \]
Step 3 · Derive the PDE
Substituting $M_t=GI_p\theta_x$ (with $G,\rho,I_p$ constant for a homogeneous shaft) and dividing by $\rho I_p$: \[ \theta_{tt} = \frac{G}{\rho}\,\theta_{xx}. \]
Remark
The general solution $\theta=F(x+ct)+G(x-ct)$ represents torsional waves propagating in both directions with speed $c=\sqrt{G/\rho}$, independent of the cross-sectional geometry.
Result
The torsional wave equation is: