Chapter 3, Problem 3.40P

A single link of a robot arm is shown in Figure P3.40. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque $T_m$ through two pairs of spur gears. We model the arm as a pendulum with a concentrated mass m. Thus we take the arm’s moment of inertia $I_G$ to be zero. The gear ratios are $N_1=2$ (the motor shaft has the greater speed) and $N_2=1.5$ (the shaft connected to the link has the slower speed). Obtain the equation of motion in terms of the angle O $\theta$ , With $T_m$ as the input. Neglect the shaft inertias relative to the other inertias. The given values for the motor and gear inertias are

$\begin{array}{l}{I}_m=0.05\text{kg}\cdot {\text{m}}^2{I}_{G_1}=0.025\text{kg}\cdot {\text{m}}^2{I}_{G_2}=0.1\text{kg}\cdot {\text{m}}^2\\ I_{G_3}=0.025\text{kg}{\text{.m}}^2{I}_{G_4}=0.08\text{kg}\cdot {\text{m}}^2\end{array}$

The values for the link are

$m=10\text{kg}L=0.3\text{m}$