Homework 6.G
Given: A homogeneous wheel of mass m and outer radius R rolls without slipping on a horizontal surface. Block A, also having a mass of m, is pinned to the center O of the wheel and is able to slide without friction on the same horizontal surface. A spring of stiffness k and a dashpot of damping constant c are connected between block A and the ground. Let x represent the motion of block A measured positively to the right. When x = 0 m, the spring is unstretched.
Find: For this problem:
a) Draw individual free body diagrams for block A and the wheel.
b) Derive the single differential equation of motion for the system in terms of the coordinate x, its time derivatives, and the following parameters: m, R, c, and k.
c) Determine numerical values for the undamped natural frequency wn, the damping ratio ζ, and the damped natural frequency wd.
d) Determine the response of the system x(t) for t > 0, assuming the system is released when the springs are unstretched with xâ‚€ = vo.
Use the following parameters in your analysis: m = 4 kg, k = 2250 N/m, R = 0.1 m, c = 60 kg/s, and vo = 8 m/s.