338
8 Free Vibration Analysis of Continuous Systems
Multiplying both sides by sin
nπ x
L
and integrating with respect to x throughout the
length of the beam, we get
∞
m=1
A m
L
0
sin
mπ x
L
sin
nπ x
L
dx = 0
( c )
Using orthogonality relationship given by Eq. (8.137), we get
A n
L
0
sin
nπ x
L
2
dx = 0
or
A n ·
L
2
= 0
or
A n = 0
( d )
Substituting second condition from Eq. (b) into Eq. (a), we get
v =
p m (B m ) sin
mπ x
L
(e)
Multiplying both sides of Eq. (e) by sin
nπ x
L
and integrating with respect to x
throughout the length of the beam, we get
L
0
v sin
nπ x
L
dx =
m
p m B m
L
0
sin
mπ x
L
sin
nπ x
L
dx
Using the orthogonality relationship given by Eq. (8.137), we get
2
nπ
v = p n B n
L
2
for n = 1, 2, 3, . . . .
or
B n =
4
nπ
v
p n
for n = 1, 2, 3, . . . .
(f)