Chapter 2, Problem 59P

Figure P2.36 shows a crane hoisting a load. Although the actual system's model is highly nonlinear, if the rope is considered to be stiff with a fixed length L, the system can be modeled using the following equations:

$\begin{array}{l}{m}_L{\ddot{x}}_{La}=m_{L}g\varphi \\ m_T{\ddot{x}}_T=f_T-m_{L}g\varphi \\ x_{La}=x_T-x_L\\ x_L=L\varphi \end{array}$

where m_Lis the mass of the load, my is the mass of the cart, x_T, and x_Lare displacements as defined in the figure, $\varphi$ is the rope angle with respect to the vertical, and f_Tis the force applied to the cart (Marttinen, 1990).

1. Obtain the transfer function from cart velocity to rope angle $\frac{\Phi \left(s\right)}{V_T\left(s\right)}$ .

Assume that the cart is driven at a constant velocity V₀ and obtain an expression for the resulting $\varphi (t)$ . Show that under this condition, the load will sway with a frequency ${\omega }_0=\sqrt{\frac{g}{L}}$ .

Find the transfer function from the applied force to the cart’s position, $\frac{X_T\left(s\right)}{F_T\left(s\right)}$ .

Show that if a constant force is applied to the cart, its velocity will increase without bound as $t\to \infty$ .