Page 158 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 158
Sec. 5.5 Digital Computation 145
where Wj = k/m and oj\ = 3k/m are obtained from the determinant of the matrix
equation. When plotted, these results appear as in Fig. 5.4-2.
Example 5.4-2 Forced Vibration in Terms of Normal Mode Summation
Express the equations for and X2 in Example 5.4-1 as the sum of the normal
modes.
Solution: Consider X^ and expand the equation in terms of partial fractions.
{2k - C, ^
- 0)^) 0)^ —uj UJ2
To solve for Cj, multiply by {(o\ —o)^) and let w =
^ {2k - mw])F^ ^
' m^{a>2 —(o]{
Similarly, C2 is evaluated by multiplying by {col ~ letting co = co2’ .
^ {2k - nuo\)F^ ^ Fj_
^ m^{a>] —0)2)
An alternative form of X^ is then
=
2
’ 2m 2 — 0> 2 it) 2 — it) 2
2k 1 - (ü)/ù)^Ÿ 3 - {(o/(O^Ÿ
Treating X2 in the same manner, its equation is
2k
1 - {u^/üJ^Ÿ 3 - (ü)/(o^Ÿ
Amplitudes A", and A"2 are now expressed as the sum of normal modes, their time
solution being
jc, = X, sin cot
= X^ sin cot
5.5 DIGITAL COMPUTATION
The finite difference method of Sec. 4.6 can easily be extended to the solution of
systems with two DOF. The procedure is illustrated by the following problem,
which is programmed and solved by the digital computer.
