Answer Happy

Problem 2. (20 points) Consider the spring-mass-dashpot system mounted on a massless cart as shown in the figure below. Assume that the cart is standing still for t < 0 and the spring-mass- dashpot system on the cart is also standing still for t < 0. In this system, u(t) is the displacement of the cart and is the input to the system. At t = 0, the cart is moved at a constant speed, or constant. The displacement y(t) of the mass is the output. (The displacement is relative to the ground.) In this system, m denotes the mass, b denotes the viscous-friction coefficient, and k denotes the spring constant. We assume that the damping force of the dashpot is proportional to ý – ů and that the spring is a linear spring; that is, the spring force is proportional to y - u. Massless cart = (a) Find the equations of motion. (b) Write the state equations and state space model (* = Ax + Bu) of the system. Hint: Set xı = y and xy = ý-u. m