It Can Also Be Shown That . . .

Adding additional mass to the system will reduce the natural frequency.

This is best seen from the second order differential equation where the natural frequency is given by:

(1")     

where:

k is the spring constant and

m is the mass.

If additional mass is added, the equation can be written as:

(2")     

where:

M is the additional mass.

Dividing Equation 1" by Equation 2" yields:

(3")     .

After algebraic manipulation, Equation 3" becomes:

(4")     .

After further manipulation of Equation 4", the mass as a function of, , , and M is given by:

(5")     .

Inverting Equation 1" and Equation 2" and rewriting them as:

(6")     

and

(7")     .

Substituting Equation 4" into Equation 5" yields:

(8")     .

Solving Equation 8" for k, gives the spring constant as a function of , , and M is given by:

(9")     .