Recalling Equation 3 and Equation 2 of pages Free Vib. Sys. Parameters and Ringing Frequency respectively:
(1")
and
(2").
Rearranging Equation 1" for z yields:
(3").
Squaring both sides of Equation 2" and substituting Equation 3" yields:
(4").
After algebraic manipulation, the natural frequency is given as a function of the time constant and the ringing frequency by:
(1).
Substituting Equation 1 into Equation 3" yields:
(4").
Manipulating Equation 4" it is evident that:
(2).
