Page 41 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

28 Free Vibration Chap. 2

Because 86 is arbitrary, the quantity within the brackets must be zero. Thus, the
equation of motion becomes
0+11 + 0 = 0
mj j I
where sin 6 = 6 has been substituted. The natural frequency from the preceding
equation is
2^2 \ g
1 +
mi ) I

2.6 VISCOUSLY DAMPED FREE VIBRATION

Viscous damping force is expressed by the equation
Fd = cx ( 2.6- 1)
where c is a constant of proportionality. Symbolically, it is designated by a
dashpot, as shown in Fig. 2.6-1. From the free-body diagram, the equation of
motion is seen to be
mx cx kx = F{t) (2.6-2)

The solution of this equation has two parts. If F (t) = 0, we have the homogeneous
differential equation whose solution corresponds physically to that of free-damped
vibration. With F{t) ^ 0, we obtain the particular solution that is due to the
excitation irrespective of the homogeneous solution. We will first examine the
homogeneous equation that will give us some understanding of the role of
damping.
With the homogeneous equation
mx + ci + fcc = 0 (2.6-3)
the traditional approach is to assume a solution of the form
jc = (2.6-4)
where 5 is a constant. Upon substitution into the differential equation, we obtain
(m5^ + + k^e^^ = 0
which is satisfied for all values of t when

-
H-----5 H----= 0 (2.6-5)
m m
Equation (2.6-5), which is known as the characteristic equation, has two roots:

“^1,2 — 2m (2.6-6)