Page 22 - Mechanical Engineers Reference Book

Vibrations 1/11

X

c > Zrnw,

Figure 1.13

c = Zmw,

X

Figure 1.14

tion. The time taken to return to this position depends on the
degree of damping (see Figure 1.15(c)). If c = 2mw, the
system is said to be critically damped. In this case it will
respond to a disturbance by returning to its equilibrium
position in the shortest possible time. In this case (see Figure
1.15(b))
= e-(c/2m)r(A+Br)
where A and B are constants. If c < 2mw, the system has a Figure 1.15
transient oscillatory motion given by
= e-(</2m)r [C sin(w; - c2i4m2)’”t + cos w: - ~~/4m~)”~t]
where C and D are constants. The period
2.ir where x is given by
‘T = -
(wf - c2/4.m2)112 x = ecnm sin [(J 112
(see Figure 1.15(a)).
Therefore
1.4.1.4 Logarithmic decrement
A way to determine the amount of damping in a system is to = cr12rn
measure the rate of decay of successive oscillations. This is
expressed by a term called the logarithmic decrement (6), where T is the period of damped oscillation.
which is defined as the natural logarithm of the ratio of any If the amount of damping present is small compared to the
two successive amplitudes (see Figure 1.16): critical damping, T approximates to 27r/w, and then
6 = log&1/x2) 8 = cdmw,