An inverted pendulum mounted on a motor-driven cart was

Chapter 4, Problem 71

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A drive system with elastically coupled load (Figure P4.18) has a motor that is connected to the load via a gearbox and a long shaft.

The system parameters are: \(J_{\mathrm{M}}=\) drive-side inertia = \(0.0338 \mathrm{~kg}-\mathrm{m}^2\), \(J_L=\) load-side inertia \(=0.1287 \mathrm{~kg}-\mathrm{m}^2\), \(K=C_T=\) torsional spring constant \(=1700 \mathrm{~N}-\mathrm{m} / \mathrm{rad}\), and D= damping coefficient \(=0.15 \mathrm{~N}-\mathrm{m}-\mathrm{s} / \mathrm{rad}\).

This system can be reduced to a simple two inertia model that may be represented by the following transfer function, relating motor shaft speed output, \(\Omega(s)\), to electromagnetic torque input (Thomsen, 2011):

\(G(s)=\frac{\Omega(s)}{T_{e m}(s)}=\frac{1}{s\left(J_M+J_L\right)} \cdot \frac{\frac{J_L}{C_T} s^2+\frac{D}{C_T} s+1}{\frac{J_M J_L}{C_T\left(J_M+J_L\right)} s^2+\frac{D}{C_T} s+1}\)

Assume the system is at standstill at t=0, when the electromagnetic torque, \(T_{e m}\), developed by the motor changes from zero to \(50 \mathrm{~N}-\mathrm{m}\). Find and plot on one graph, using MATLAB or any other program, the motor shaft speed, \(\omega(t)\), \(\mathrm{rad} / \mathrm{sec}\), for the following two cases:

(a) No load torque is applied and, thus, \(\omega=\omega_{n l}\).

(b) A load torque, \(T_L=0.2 \omega(t) \quad \mathrm{N}-\mathrm{m}\) is applied and \(\omega=\omega_L\).