Question 5.24: A Single-Link Robot Arm with a Gearbox Consider the single-l...
A Single-Link Robot Arm with a Gearbox
Consider the single-link robot arm in Example 5.20. It is driven by a DC motor through a gearbox with a gear ratio N = 1/70. The mass moments of inertia of the motor and the load are I_m and I, respectively. The coefficients of torsional viscous damping of the motor and the load are B_m and B, respectively. τ_m is the torque generated by the motor. The parameter values are I_m = 3.87 × 10^{–7} \text{kg ⋅ m}^2, I = 0.0016 \text{kg ⋅ m}^2, B_m = 0 \text{N ⋅ m ⋅ s/rad}, and B = 0.004 \text{N ⋅ m ⋅ s/rad}. Assume that the robot arm is initially at rest and the motor torque is τ_m(t) = 0.003u(t), where u(t) is the unit-step function with a step time of 0 seconds. Build a Simscape model of the physical system and find the angular velocity output \dot{θ}(t).
The Simscape model of the system is shown in Figure 5.115. To model the robot arm, two Inertia blocks are included, one for the motor and the other for the load. Since the damping of the motor can be ignored, one Rotational Damper block is included to represent the damping of the load. The shaft of the motor and that of the load are coupled through a gearbox. Double-click the Gear Box block and type 70 (not 1/70) for the Gear ratio, which in Simscape is defined as the ratio of the input shaft angular velocity to that of the output shaft. This is the reciprocal of the gear ratio defined in Equation 5.55. The output of port W of the Ideal Rotation Motion Sensor is the corresponding angular velocity \dot{θ}(t), which is shown in Figure 5.116.
\frac{ω_2}{ω_1} = N (5.55)
Note that only the Simscape block diagrams are given in the previous two examples (5.22+5.23). The corresponding Simulink modeling is left to the reader as an exercise.
