Question

A point performs damped oscillations with frequency ω and damping coefficient β. Find the velocity amplitude of the point as a function of time t if at the moment t = 0

(a) its displacement amplitude is equal to a0;

(b) the displacement of the point x (0) = 0 and its velocity projection vx (0) =

Answer 1

Velocity amplitude as a function of time is defined in the following manner. Put then

for This means that the displacement amplitude around the time and we can say that the displacement amplitude at time t is Similarly for the velocity amplitude.

Clearly

(a) Velocity amplitude at time

where γ is another constant.

where a0 is real and positive.

Thus and we take - ( + ) sign if x0 is negative (positive). Finally the velocity amplitude is obtained as