Presented here is the solution to the second order differential equation that governs the behavior of the forced vibration mode of our apparatus.
If there is a periodic forcing function [F(t) = A sin t], the beam will experience a forced vibration response, or periodic response.
This is modelled by:
(1)
It can be shown that the steady state solution for an underdamped system is given by:
(2)
The amplitude response, D, can be written as:
(3)
The phase response (shift), , is given by:
(5)
This ratio is defined as:
(6)
The response of the spring-mass-dashpot system to an instantaneously applied forcing function includes both a transient adjustment to the forcing function plus a steady-state response. When taking data, be sure to sample for a long enough time to allow the transient response to approach zero.