8.8 Free Flexural Vibration of Beams with General End Conditions
333
(4) Left-hand end clamped and right-hand end 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
(8.122)
μ 1 = 1.875, μ 2 = 4.694, μ 3 = 7.855
m ≥ 4, μ m =
2m − 1
2
π
(8.123)
α m =
sin μ m + sinh μ m
cos μ m + cosh μ m
(8.124)
(5) Left-hand end simply supported and right-hand end free
Y m = sin
μ m x
L
+ α m sinh
μ m x
L
(8.125)
where
μ m =
4m − 3
4
π,
m = 2, 3, 4, . . .
(8.126)
α m =
sin μ m − sinh μ m
cos μ m − cosh μ m
(8.127)
(6) Left-hand end clamped and the right-hand end elastically restrained in rotation
Y m = sin
μ m x
L
− sinh
μ m x
L
+ α m
cos
μ m x
L
− cosh
μ m x
L
(8.128)
where
α m =
sin μ m − sinh μ m
cos μ m − cosh μ m
Different values of μ m for various values of φ = E I /RL, where R is the coefficient of elastic restraint and EI is the flexural rigidity of the beam, are given in
Table 8.2.
(7) Left-hand end simply supported and right-hand end elastically restrained
against rotation