Forced Vibration Equation Solution

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)     

where:

D is the amplitude of the response and

is the phase response (shift).

The amplitude response, D, can be written as:

(3)     

The phase response (shift), , is given by:

(5)     

where

is the ratio of the forcing frequency to the phase shift.

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.