Free Vibration Equation Solution

Presented here is the solution to the second order differential equation that governs the behavior of the free vibration mode of our apparatus.

If there is no forcing function [F(t) = 0] and the beam is displaced and then released, it experiences a free vibration response. Free vibration is modeled by the Homogeneous Solution to the 2nd Order Differential Equation, which is:

(1)     .

The boundary conditions to the homogeneous solution are given by the physical system and are:

(2)     .

It can be shown that the solution to the differential equation with the specified boundary conditions is:

(3)     

where:

is the damped natural frequency and

A and B are given in Equation 4:

(4)     .