Question: The cart shown in Figure is attached to the wall by a spring of stiffness k. As shown, the spring is undeflected. The pendulum is attached to the cart by a frictionless pin and a torsional spring of stiffness kθ. The following data set applies: mc = 5 kg, mp = 1 kg, L = 0.5 m, kθ = 3.5 N m/rad, k = 200 N/m. Tasks:
(a) Select coordinates and draw the free-body diagram for each body.
(b) Obtain the EOMs for the system using Newtonian principles. Present in matrix format. Apply small-angle assumption.
(c) Determine the required conditions for stable equilibrium.
(d) Obtain the eigenvalues and normalized eigenvectors for the system.
(e) State the uncoupled modal equations of motion and the transformation from modal to physical coordinates.