Page 42 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 2.6 Viscously Damped Free Vibration 29

F[t) Figure 2.6-1.
Hence, the general solution is given by the equation
(2.6-7)
where A and B are constants to be evaluated from the initial conditions jc(0)
and i(0).
Equation (2.6-6) substituted into (2.6-7) gives

^ ^ { c / I m f - k n ^ + Be U ( c / 2 m ) - k / m ) /) (2.6-8)
/
The first term, simply an exponentially decaying function of time. The
behavior of the terms in the parentheses, however, depends on whether the
numerical value within the radical is positive, zero, or negative.
When the damping term {c/2mY is larger than k/m, the exponents in the
previous equation are real numbers and no oscillations are possible. We refer to
this case as overdamped.
When the damping term {c/lm Y is less than k/m, the exponent becomes
an imaginary number, ±i^Jk/m - (c/2m)^ t. Because

± - (c/2my
\ = cosy —— i + /siny-^— ^

Vm V2m I ~ Vm \ 2m J

the terms of Eq. (2.6-8) within the parentheses are oscillatory. We refer to this case
as underdamped.
In the limiting case between the oscillatory and nonoscillatory motion,
{c/2mY = k/m, and the radical is zero. The damping corresponding to this case
is called critical damping, c .
= 2m\ — = 2mo)^ 2^¡km (2.6-9)
m ^
Any damping can then be expressed in terms of the critical damping by a
nondimensional number called the damping ratio:
(2.6-10)