Page 44 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 2.6 Viscously Damped Free Vibration 31

Oscillatory motion. [¿' < 1.0 (Underdamped Case).] By substituting Eq.
(2.6-11) into (2.6-7), the general solution becomes

(2.6-13)
This equation can also be written in either of the following two forms:

X = sin ^\/l — + (/>j (2.6-14)

= sin y^l - i^co^t + C2 cos \/l — (o^t) (2.6-15)
where the arbitrary constants X, 4> or Cj,C2 are determined from initial condi­
,
tions. With initial conditions jc(0) and i(0), Eq. (2.6-15) can be shown to reduce to

X = I ^ ^ ^ ^ ( ) Y^l — (x)^t + x(0) cos (2.6-16)

The equation indicates that the frequency of damped oscillation is equal to

(2.6-17)

Figure 2.6-3 shows the general nature of the oscillatory motion.

Figure 2.6-3. Damped oscillation
^<1.0.

Nonosclllatory motion. > 1.0 (Overdamped Case).] As exceeds unity,
the two roots remain on the real axis of Fig. 2.6-2 and separate, one increasing and
the other decreasing. The general solution then becomes

(2.6-18)
where
•^(0) +
A =