Question 5.8: A Single-Degree-of-Freedom Rotational Mass–Spring–Damper Sys...
A Single-Degree-of-Freedom Rotational Mass-Spring-Damper System
Consider a simple disk-shaft system shown in Figure 5.54a, in which the disk rotates about a fixed axis through point \mathrm{O}. A single-degree-of-freedom torsional mass-spring-damper system in Figure 5.54b can be used to approximate the dynamic behavior of the disk-shaft system. I_{\mathrm{O}} is the mass moment of inertia of the disk about point \mathrm{O}, \mathrm{K} represents the elasticity of the shaft, and B represents torsional viscous damping. Derive the differential equation of motion.
The free-body diagram of the disk is shown in Figure 5.54c. Because the disk rotates about a fixed axis, we can apply Equation 5.32
\sum M_{O}=I_{O}\alpha. (5.32)
about the fixed point \mathrm{O}. Assuming that counterclockwise is the positive direction, we have
\begin{gathered} +\curvearrowleft: \sum M_{\mathrm{O}}=I_{\mathrm{O}} \alpha, \\ \tau-K \theta-B \dot{\theta}=I_{\mathrm{O}} \ddot{\theta} . \end{gathered}
Thus,
I_{\mathrm{O}} \ddot{\theta}+B \dot{\theta}+K \theta=\tau.