Homework H.6.N Given: Cart A (of mass m) is supported by two rollers C and D each of mass m, and having outer radii of Rand 2R, respectively). The rollers are known to roll without slipping with respect to both the ground and cart A. A spring of stiffness k is attached between the left side of the cart and ground. Two additional springs, of stiffnesses k and 2k, are attached between the right side of cart A and a moveable base B. The base B is given a prescribed motion of r(t)=b cost. Let represent the motion of cart A, where x = x =0 when all three of the springs are unstretched. Find: For this problem: (a) Derive the differential equation of motion (EOM) for the structure in terms of the coordinate (b) Identify the natural frequency wn of oscillations for the system; (c) Derive the particular solution for the EOM, xp(t); and, (d) What is the amplitude of the motion for the cart when 2 = 2wn? Is the cart moving in-phase or out-of-phase with the base B at this frequency of excitation? XBO 2k w w w D 2R 1 cm no slip m )