Question

2. Resonant Frequency. Show that the resonant frequency of a driven (forced) DHO is: Note that this expression strongly resembles the "natural frequency" of the DHO, but it is not the same. The "2" is different. (Hint: take the first derivative of the amplitude A(o) of the particular solution, F/G(o), with respect to ω, and set this equal to zero. Pay special attention to the inner derivative. Simple algebra will then yield the result.) 1S

Answer 1

Force oscillation forces on body (1) resistance force = -k x Damping = -b d x/d t external periodic force = f_0 sin p t net force = f_0 sin p t - k x - b d x/d t m d x/dt^2 = f_0 sin p t - k x - b d x/d t d x/dt^2 + b/m d x/d t + k/m x = f_0/m sin p t let b/m = 2 r: k/m = w^2_0 = f_0/m = f_0 d x/dt^2 + 2 r d x/d t + w^2_0 x = f_0/m sin p t (D^2 + 2 D + w^2_0) x = f_0 sin p t will have c f & as right not equal to 0 c f = a e^-r t sin (w t + delta) p I = 1/(D^2 + 2 r D + w^2_0) f_0 sin p t

I have derived the complete expression for your convenience to go through all the steps. At the end, the expression is derived for resonant frequency of forced DHO and compared with natural frequency of DHO