332
8 Free Vibration Analysis of Continuous Systems
or
p 1 = (4.462)
2
E I
ρ AL 4
The natural frequencies and mode shapes for various end conditions for a beam
are given below [note that μ m = λ m L]:
(1) Left-hand end simply supported and right-hand end clamped
Y m = sin
μ m x
L
− α m sinh
μ m x
L
(8.113)
where
μ m =
4m + 1
4
π
m = 1, 2, 3, . . . ..
(8.114)
α m =
sin μ m
sinh μ m
(8.115)
(2) Both ends clamped
Y m = sin
μ m x
L
− sinh
μ m x
L
− α m
cos
μ m x
L
− cosh
μ m x
L
(8.116)
μ m =
2m + 1
2
π
m = 1, 2, 3, . . . ..
(8.117)
where
α m =
sin μ m − sinh μ m
cos μ m − cosh μ m
(8.118)
(3) Both ends free
Y m = sin
μ m x
L
+ sinh
μ m x
L
− α m
cos
μ m x
L
+ cosh
μ m x
L
(8.119)
μ m =
2m + 1
2
π
m = 1, 2, 3, . . . ..
(8.120)
α m =
sin μ m − sinh μ m
cos μ m − cosh μ m
(8.121)