Your Problem

The system shown above is modeled by the following differential equation. It is obtained by applying Newton's law of motion, F = m a, to the mass. In the following expression prime (') denotes differentiation with respect to time. Therefore, velocity, v=x'(t), and acceleration a = x''(t).

x''(t) + 0.2 x'(t) + 4.4 x(t) = sin(wt)

The steady state solution to the differential equation is the one that remains in effect long after the system has started and the transient terms have died out. (Transient refers to initial start-up behavior which dies out after a time) It can be shown that a steady state solution to this differential equation is


x(t) = A(w) sin(w t) - B(w) cos(w t)

44000 - 10000 w2
where A(w) = -----------------------------------
193600 - 87600 w2 + 10000 w4

2000 w
and B(w) = -----------------------------------
193600 - 87600 w2 + 10000 w4

Your problem consists in three parts:

(1) Verify that the solution given above satisfies the differential equation which models
the system.
(2) Find the amplitude of the response as a function of w. This is called the frequency
response. Plot this response.
(3) Find the frequency, w, of the driving force which produces the largest amplitude in
the response, x(t). This is called the resonant frequency.