Page 49 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

36 Free Vibration Chap. 2

Figure 2.8-1. Free vibration with
Coulomb damping.
To determine the decay of amplitude, we resort to the work-energy principle
of equating the work done to the change in kinetic energy. By choosing a half-cycle
starting at the extreme position with velocity equal to zero and the amplitude equal
to the change in the kinetic energy is zero and the work done on m is also
zero.
\ k { X ^ - X l , ) - F , { X , + X _ , ) = Q
or
' ^ k { X , - X _ , ) = F ,

where is the amplitude after the half-cycle, as shown in Fig. 2.8-1.
By repeating this procedure for the next half-cycle, a further decrease in
amplitude of 2F^/k will be found, so that the decay in amplitude per cycle is a
constant and equal to
x , - x , = ^ (2.8-1)

The motion will cease, however, when the amplitude becomes less than A, at
which position the spring force is insufficient to overcome the static friction force,
which is generally greater than the kinetic friction force. It can also be shown that
the frequency of oscillation is = ^Jk/m , which is the same as that of the
undamped system.
Figure 2.8-1 shows the free vibration of a system with Coulomb damping. It
should be noted that the amplitudes decay linearly with time.

TABLE OF SPRING STIFFNESS
k, k2 1
o— VWV---- •-----W A ---- o k = \/k, + \/k2

k = k^ + kj