Consider the simplified quarter-car model of an automobile suspension given belo

Question

Consider the simplified quarter-car model of an automobile suspension given below where m is one quarter of the total mass of the vehicle, b is the damping in the suspension, and k is the stiffness of the suspension. The differential equation describing this system is mx + bx + kx = u where x(t) is the displacement of the car body and u(t) is the displacement of the tire, that is. the height of the road surface. Consider the response of our suspension subject to a sharp bump in the road. u(t) = delta (t). It turns out that the impulse response of a system is equivalent to the free response of a system to an initial condition on its velocity. In other words, our situation is described by the following differential equation where for simplicity m = 1, b = 2 and k = 4. x + 2x + 4a = 0, where x(0) = 0, x(0) = 1 (a) From the above differential equation, find X(s). Determine the poles of X(s). What does this tell us about the response of our car body x(t)? Can you apply the final value theorem to find x(t) as t rightarrow infinity ? Explain. Finish solving the differential equation, that is. find x(t). Does the function x(t) you found agree with your answer to Part (a)? Explain.

Explanation / Answer

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