Consider the harmonic shaking force P_{0}cosΩt acting on a damped sping-mass system. The differential equation of motion with an external force exciting the system is
mΫ + cϓ + KY = P_{0}cosΩt
The solution of this equation consists of a complementary function plus a particular function. The complementary solution is the free vibrations. Thesewill die out because of tthe damping. The particular solution can be taken in the form
Y = Y_{0} cos(Ωt - Θ)
The maximum displacement Y0 can be expressed in terms of the maximum impressed force, P0 as follows:
Let Y_{st} = P_{0}/K, where Y_{st} is the deflection of the system due to the maximum dunamic input load acting as a static load. For additional simplification, let
Y_{Ω} = (Ω / Ω_{n}) and R_{c} = (c/c_{c})
This leads to the general amplification (not transmissibility) equation: