However, it is common for a damped system to have an external harmonic force acting on the system. For example, unbalanced rotating motors will cause sever forces on the mounting brackets. This system is called forced vibrations.
A forced vibration is one in which the system is excited by an external, time-varying force P, called a forcing function. In this case
In general, forcing functions are periodic, and since any periodic function can be expressed as a Fourier series, it is convenient to look at a forcing function of the type
P(t) = Posinωt
or P(t) = Pocosωt
Considering the sin term, the differential equation is
The solution of this differential equations consists of two parts, the complementary or homogenous solution,
x(t) = xc(t) + xP(t)