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:
where mL is the mass of the load, mT is the mass of the cart, xT and xL are displacements as defined in thefigure,ϕ is the rope angle with respect to the vertical, and f T is the force applied to the cart (Marttinen, 1990).
a. Obtain the transfer function from cart velocity to rope angle Φ(s)/VT(s).
b. Assume that the cart is driven at a constant velocityV0 and obtain an expression for the resulting ϕ(t). Show that under this condition, the load will sway with a frequency ω0 = √g/L.
c. Find the transfer function from the applied force to the cart’s position, XT (s)/FT (s).
d. Show that if a constant force is applied to the cart, its velocity will increase without bound as t→∞.
