Question 3.3.4: A rack-and-pinion, shown in Figure 3.3.3, is used to convert......
A rack-and-pinion, shown in Figure 3.3.3, is used to convert rotation into translation. The input shaft rotates through the angle θ as a result of the torque T produced by a motor. The pinion rotates and causes the rack to translate. Derive the expression for the equivalent inertia I_{{e}} felt on the input shaft. The mass of the rack is m, the inertia of the pinion is I, and its mean radius is R.
The kinetic energy of the system is (neglecting the inertia of the shaft)
\mathrm{KE}={\frac{1}{2}}m\dot{x}^{2}+\frac{1}{2}{ I}\dot{\theta}^{2}where \dot{x} is the velocity of the rack and \dot{\theta} is the angular velocity of the pinion and shaft. From geometry, x = Rθ, and thus \dot{x}=R\dot{\theta}. Substituting for \dot{x} in the expression for KE, we obtain
{\mathrm{KE}}={\frac{1}{2}}m\left(R{\dot{\theta}}\right)^{2}+{\frac{1}{2}}I{\dot{\theta}}^{2}={\frac{1}{2}}\left(m R^{2}+I\right){\dot{\theta}}^{2}Thus the equivalent inertia felt on the shaft is
I_{e}=m R^{2}+I (1)
and the model of the system’s dynamics is I_{e}{\ddot{\theta}}=T, which can be expressed in terms of x as I_{e}{\ddot{x}}=R T.