Question

A damped oscillator with natural frequency wo and damping K is driven by a period square wave force with amplitude A such that F(t)= A Find the Fourier series for F(t), and solve for the amplitude of the motion of the oscillator. For which frequency wn is the resonance condition the most closely satisfied? Plot the maximum amplitude (in units of A) as a function of wn for the conditions with the spring constant k 1, m 2, K 0.1, and T 20.

Answer 1

According to question, damped oscillator - w_0 frequency periodic force f(t) = A 0 {0 less than or equal to t < T T less than or equal to t < 2 T fourier series of f(t) = 2 fourier of f(t) = a_0/2 + sigma^infinity_n = 1 a_n cos n phi t/T + sigma^infinity_n = 1 bn sin n phi t/T where, a_0 = 1/T integral^2 T_0 f(t) dt a_n = 2/T integral f(t) cos n phi t/T dt bn = 1/T integral f(t) sin n phi t/T dt a_0 = 1/T [integral^T_0 A dt + integral^2 T_T 0 dt] a_0 = 1/T [A (t)^T_0] = 1/ [A] = A = a_n = 1/T [integral^T_0 A cos n phi t/T dt] = 1/T A integral^T_0 cos n phi t/T dt = A/T (sin n phi t/T)^T_0 T/n phi