Part 2: Basic Computational Skills Component
Forced oscillations Problem 16. The governing equation for a weakly damped harmonic oscillator with forcing is modeled by the equation:
x'' + 2x' + x = f(t), x(0) = 0, x'(0) = 0
(governing equation) (initial conditions)
where ζ is the damping term, ω^2 the frequency of the undamped oscillator, and f(t) is the forcing term. Recall from class that the steady-state Fourier-series solution (neglecting friction) of the system's response to the forcing term f(t) was given by the particular solution to the ODE:
x(t) = ∑[An cos(nωt) + Bn sin(nωt)]
where the Fourier series of the forcing term f(t) is given by the expression:
f(t) = A0 + ∑[An cos(nωt) + Bn sin(nωt)] (n≥1)
Suppose the Fourier series of the forcing term is known to be the 2t-periodic function triangular forcing f(t) on (-t, t) (shown in the figure below), which has the Fourier series:
f(t) = ∑[4/(n^2π^2) cos(nπt/t)] for n≥1
Figure 1: Triangular forcing term f(t).
Which of the following values for ζ will give the largest (amplitude) response in the steady-state solution? (a) 0.3 (b) 4.01 (c) 5.01 (d) 11.01 (e) None of these
