Page 203 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

190 Properties of Vibrating Systems Chap. 6

Writing out the terms of Bq. (6.8-2), we have
0.5 r 2 0 [ 0 . 5 - 1
. - 1 .0 1 . 1 1 >^2
'
0 . 5 1 3 - 1 0 . 5 - 1 0.5 1
+ k -1
- 1 1 - 1 1 1 1 1
F,
1 . 5 0 / + k 0 . 7 5 0
0 >^2 0 6
which are uncoupled.
The solutions for and y2 are in the form
yTO) sm (ot
T "" T ( 0 ) cos ojjt + ------ sin (Ojt -t-
1 - (a>/o>,Ÿ
which can be expressed in terms of the original coordinates by the P matrix as
0.5 -1
1 1

Example 6.8-2
For Example 6.8-1, determine the generalized mass and the P matrix. Numerically,
verify Eqs. (6.7-5) and (6.7-6).
Solution: The calculations for the generalized mass are
2 0 0.5
M, = (0.5 1) = 1.5
0 1 1
2 0 -1
M2 = ( - \ 1) 0 1 1 = 3.0
By dividing the first column of P by and the second column by the P
matrix becomes
'0.4083 -0.5773
P =
0.8165 0.5773
Equations (6.7-5) and (6.7-6) are simply verified by substitution.

6.9 MODAL DAMPING IN FORCED VIBRATION

The equation of motion of an A^-DOF system with viscous damping and arbitrary
excitation F{t) can be presented in matrix form:
MX + CX + KX = F (6.9-1)
It is generally a set of N coupled equations.
We have found that the solution of the homogeneous undamped equation

M X ^ K X = 0 (6.9-2)