Chapter 6, Problem 6.43P

A single link of a robot arm is shown in Figure P6.43. 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 spur gears. Suppose that the equivalent inertia felt at the motor shaft is $\text{0}\text{.215 kg}\cdot {\text{m}}^2$ . As the arm rotates, the effect of the arm weight generates an opposing torque that depends on the arm angle, and is therefore nonlinear. For this problem, however, assume that the effect of the opposing torque is a constant $\text{4}\text{.2 N}\cdot \text{m}$ at the motor shaft. Neglect damping in the system. It is desired to have the motor shaft rotate through $3\pi /4$ rad in a total time of 2 s, using a trapezoidal speed profile with $t_1=0.3\text{s}$ and $t_2=1.7\text{s}$ .

The given motor parameters are $R_a=4\Omega$ , $L_a=3\times {10}^{-3}\text{H}$ , and $K_T=0.3\text{N}\cdot \text{m/A}$ . Compute the energy consumption per cycle; the maximum required torque, current, and voltage; the rms torque; and the rms current.