Forced Vibration (damped, undamped)

Consider the spring-mass-dashpot system. The mass is displaced by a distance x and its free body diagram is shown. From this figure, we see that the equation of motion is

The solution to this equation consists of two parts, the complementary function, which is the solution of the homogeneous equation and the particular integral.

The particular solution in the equation is a steady-state oscillation of the same frequency as that of the excitation. We can assume the particular solution to be of the form

where X is the amplitude of oscillation and is the phase of the displacement with respect to the exciting force.

Substituting this in the equation of motion

or,