Oscillations and Waves > Forced and damped oscillations > Problem 3.5.36

Statement

a. What is the necessary forcing force for an oscillator of mass with damping coefficient to begin to perform harmonic oscillations with natural frequency according to the law ?

b. The amplitude of the forcing force is , its frequency . Determine the amplitude of the forced oscillations. How many times is it the deflection of the oscillator when a constant force is applied?

Solution

Substituting the particular (i.e. steady-state) solution into the expression , we obtain . (Note that the first and third terms cancel out, because the particular solution happens to be the solution to the homogeneous equation .) Thus, the required forcing is .

Conversely, if the forcing has amplitude and , then , and the amplitude of the forced oscillation is . When a constant force is applied instead, the deflection of the oscillator is . The amplitude of the forced oscillation is thus times this deflection.

Answer

a.

b. , which is times the deflection under constant force