216
MULT1-DEGREE OF FREEDOM LINEAR STRUCTURES
For K: when équations (8.66) and (8.62) are combined, then
Kx„=w„Mxn
(8.71)
When this last resuit is premultiplied by x* and équation (8.70) is used, then
xjKx» = u’xJ'Mxn=w*
(8.72)
Consider the two cases for r L n. For M: let n = r in équation (8.71) and
take the transpose of both sides to obtain
x^ Kr = lj2 X? Mt
(8.73)
Since both K and M are symmetric, then
xJrK=^xï'M
(8.74)
Postmultiply this last resuit by xn:
xjf Kxn = u>2 X? M xn
(8.75)
Premultiply équation (8.71) by xJ", which gives
x;T K x„ = x^ M xn
(8.76)
Subtract équation (8.75) from the last resuit to give
(«„ - üj2)x^ M xn = 0
(8.77)
Since the frequencies are distinct (o?n wr for n / r), then
x)T M xn = 0,
for n r
(8.78)
Thus, this last resuit, together with équation (8.70), complétés the proof of the
orthogonality statement (8.68). For K: the right side of équation (8.76) is
for n — r by équation (8.70), but is equal to zéro for n / r by équation (8.78).
This complétés the proof of the orthogonality statement (8.69).
8.5 FORCED, DAMPED MOTION
Derixed in this section are steady State solutions to the structural équations of
motion (8.1). which include both forcing vector p and System damping. These
solutions utilize the free, undamped modal vectors xn and their associated undarnped frequencies wn, both derived in the last section. These solutions dépend
on two important matrix forms: the modal shape matrix X and the modal
damping matrix C. Following the définition of these forms, a transformation
is introduced that uncouples the équations of motion, allowing for the normal
mode solutions to be displayed in closed form. The method of solution is then
MULT1-DEGREE OF FREEDOM LINEAR STRUCTURES
For K: when équations (8.66) and (8.62) are combined, then
Kx„=w„Mxn
(8.71)
When this last resuit is premultiplied by x* and équation (8.70) is used, then
xjKx» = u’xJ'Mxn=w*
(8.72)
Consider the two cases for r L n. For M: let n = r in équation (8.71) and
take the transpose of both sides to obtain
x^ Kr = lj2 X? Mt
(8.73)
Since both K and M are symmetric, then
xJrK=^xï'M
(8.74)
Postmultiply this last resuit by xn:
xjf Kxn = u>2 X? M xn
(8.75)
Premultiply équation (8.71) by xJ", which gives
x;T K x„ = x^ M xn
(8.76)
Subtract équation (8.75) from the last resuit to give
(«„ - üj2)x^ M xn = 0
(8.77)
Since the frequencies are distinct (o?n wr for n / r), then
x)T M xn = 0,
for n r
(8.78)
Thus, this last resuit, together with équation (8.70), complétés the proof of the
orthogonality statement (8.68). For K: the right side of équation (8.76) is
for n — r by équation (8.70), but is equal to zéro for n / r by équation (8.78).
This complétés the proof of the orthogonality statement (8.69).
8.5 FORCED, DAMPED MOTION
Derixed in this section are steady State solutions to the structural équations of
motion (8.1). which include both forcing vector p and System damping. These
solutions utilize the free, undamped modal vectors xn and their associated undarnped frequencies wn, both derived in the last section. These solutions dépend
on two important matrix forms: the modal shape matrix X and the modal
damping matrix C. Following the définition of these forms, a transformation
is introduced that uncouples the équations of motion, allowing for the normal
mode solutions to be displayed in closed form. The method of solution is then