A single link of a robot arm is shown in Figure P3.40. The

Chapter , Problem 3.40

(choose chapter or problem)

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 \(\theta\), 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

\(\begin{array}{c} I_{m}=0.05 \mathrm{~kg} \cdot \mathrm{m}^{2} \quad I_{G_{1}}=0.025 \mathrm{~kg} \cdot \mathrm{m}^{2} \quad I_{G_{2}}=0.1 \mathrm{~kg} \cdot \mathrm{m}^{2} \\ I_{G_{3}}=0.025 \mathrm{~kg} \cdot \mathrm{m}^{2} \quad I_{G_{4}}=0.08 \mathrm{~kg} \cdot \mathrm{m}^{2} \end{array}\)

The values for the link are

m = 10 kg L = 0.3 m