The steps are as follows: for a one-dimensional structure, let u(x, t) denote the
dependent variable of the PDE. Assume the expansion (1.60), i.e.
uðx; tÞ ¼
X N
j¼1
q j ðtÞu j ðxÞ;
ð1:84Þ
with u j (x) being the free mode shapes (eigenfunctions) associated with the PDE,
and q j the unknown time functions (called normal coordinates, mode participation
factors, or modal coefficients). Insert the expansion into the PDE, multiply by u i ,
integrate over the length of the structure, and obtain a set of ODEs in the variables
q j . The mode shapes u i typically constitute an orthogonal set of functions, and the
expansion (1.84) yields exact results as N ! ∞.
Consider as an example the PDE (1.83) for the continuous beam of Fig. 1.6,
with m = 0 and including a viscous damping-term:
qA € u þ cqA _
u þ EIu
0000
þ PðtÞ ~ dðx À x 0 Þ ¼ 0:
ð1:85Þ
The free mode shapes u j and natural frequencies x j of a simply supported beam
are given by (cf. (1.46), (1.47)):
u j ðxÞ ¼ sin jpx=l
ð
Þ; x
2
j ¼
jp
l
4 EI
qA
; j ¼ 1; N:
ð1:86Þ
One can easily show that the mode shapes u i (x) in (1.86) satisfy the following
relations of orthogonality:
Z l
0
u i u j dx ¼
1
2
ld ij ;
Z l
0
u i u
0000
j dx ¼
Z l
0
u
00
i u
00
j dx ¼
1
2
lðjp=lÞ
4 d ij ;
ð1:87Þ
where d ij denote the Kronecker delta and i, j = 1, N. Now, insert the expansion
(1.84) into (1.85), multiply by u i and integrate over x 2 [0, l] to obtain:
qA
Z l
0
u i
X
j
€ q j u j dx þ cqA
Z l
0
u i
X
j
_
q j u j dx
þ EI
Z l
0
u i
X
j
q j u
0000
j dx þ PðtÞ
Z l
0
u i
~ dðx À x 0 Þ dx ¼ 0;
ð1:88Þ
or, by interchanging integration and summation and using (1.86)–(1.87):
26
1 Vibration Basics