As indicated by the cart-with-flywheel example discussed in this section, an unbalanced rotating machine part typically results in a force having amplitude proportional to the square of the frequency ω.

(a) Show that the amplitude of the steady periodic solution of the differential equation

(with a forcing term similar to that in Eq. (17)) is given by

(b) Suppose that c² m given by

Thus the resonance frequency in this case is larger (in contrast with the result of Problem 27) than the natural frequency ω₀ = √k/m.

Problem 27

According to Eq. (21), the amplitude of forced steady periodic oscillations for the system mx'' + cx' + kx = F₀ cos ωt is given by

mx" + cx' + kx = = m Aw cos wt