It Can Be Shown That . . .

Recall that the spring-mass-dashpot system is governed by the following 2nd Order Ordinary Differential Equation:

(1")     

The forced vibration response is based upon a sinusoidal forcing function. Substituting F(t)=Asint into Equation 1" yields:

(2")     

The solution can be broken down into a homogeneous solution and a particular solution:

(3")     

The homogeneous solution corresponds to the transient response of the system. Because we are interested in the steady state response of the dyanmic system in the lab, we are only interested in the particular solution.

It can be shown that the steady state solution to the 2nd Order Ordinary Differential equation is:

(2)      

where:

D is the amplitude of the response and

is the phase shift.

The amplitude, D, can be written as:

(3)     .

The phase shift, , is given by:

(5)     

where u is the ratio of the forcing frequency to the phase shift. This ratio is defined as:

(6)