Governing Equations

By examining the forces acting upon the mass, it can be shown that the standard form of the equations governing our model is given by:

(1)     

where:

M is the effective mass,

C is the damping constant, and

k is the spring constant.

By introducing the natural frequency and the damping ratio, it can also be shown that Equation 1 can be written as:

(2)     

where:

is the natural frequency,

is the damping ratio,

K is the compliance (=1/k), and

F(t) is the forcing function.

The natural frequency can be written as:

(3)     

And, the damping ratio is given by:

(4)     

It is reasonable to estimate k and M for our system. But because the damping constant, C, is less well understood, the damping ratio, z, cannot be estimated directly.

If there is no forcing function [F(t) = 0] and the beam is displaced and then released, it experiences a free vibration response. The free response naturally gives rise to an additional parameter, damped natural frequency (ringing frequency).

If there is a periodic forcing function [F(t) = A sin t], the beam will experience a forced vibration response, or periodic response. The forced response gives the following parameters: resonant frequency, natural frequency, and damping ratio.