ABSTRACT
Equation (38) is the general solution to the single-degree-of-freedom, undamped, free-vibration response. Equation (38) shows that the response is harmonic in the natural frequency w,, where w, is expressed in units of radians per second. The natural frequency w, (in rad/s) is related to the natural frequency f, in cycles per second or hertz by
If x(O) = 3 in., x(O) = 0, and f, = 2 Hz, then the free-vibration response of the system is as shown in Fig. 2 and can be written as
x(t) = Ao sin(wnt + 1/>o) where
Ao = ~+ (~:) and
«Po = tan-• e::n) (44) Natural frequency is the most important dynamic characteristic of a single-degree-of-freedom system in most applications. Equation (30) can be rewritten as
.. c . k 0 x+-x+-x=
m m
which is of the standard form
c '= 2../fm (50)
The damping ratio is the ratio of actual damping of a system to the critical damping of that system. Critical damping exists when ( = 1, which corresponds to Cc = 2../fm.
