Page 45 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

32 Free Vibration Chap. 2

Figure 2.6-4. Aperiodic motion >
1.0.

and

-i(0 ) - ( l - -\)co,x{0)
B =

The motion is an exponentially decreasing function of time, as shown in Fig. 2.6-4,
and is referred to as aperiodic.

Critically damped motion. = 1.0.] For = 1, we obtain a double root,
= Sj= and the two terms of Eq. (2.6-7) combine to form a single term,
which is lacking in the number of constants required to satisfy the two initial
conditions.
The correct general solution is
X =={A (2.6-19)

which for the initial conditions jc(0) and i(0) becomes

X = {x(0) 4- [i(0) + w^a:(0)]
This can also be found from Eq. (2.6-16) by letting ^ 1. Figure 2.6-5 shows three
types of response with initial displacement jr(0).

Figure 2.6-5. Critically damped
motion i = 1.0.