It can be shown that . . .

Dividing Equation 1 by M yields:

(1")     .

Define the natural frequency, , as:

(3)     

and the damping ratio, ,as:

(4)     .

Through algebraic manipulation it is clear that:

(2")     

and

(3")    .

Thus Equation 1 can be rewritten as:

(4")     .

Equation 4" can be rewritten as:

(5")     .

where:

K is the compliance (=1/k).

By algebraic manipulation, Equation 5" can be rewritten as:

(2)     .