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, m_T is the mass of the cart, x_T and x_L 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)/V_T(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, X_T (s)/F_T (s).

d. Show that if a constant force is applied to the cart, its velocity will increase without bound as t→∞.

m, = migp mpx, = fr - migp XLa = XT - XL XL = Lp