8.7 Free Flexural Vibration of Beams with Other End Conditions
327
Fig. 8.10 Mode shapes of a
uniform simply supported
beam
At x = L ,
E I
d
2 Y
dx 2 = 0 and
d
dx
E I
d
2 Y
dx 2
= 0
(8.100)
The derivatives which are required from the end conditions are obtained from
Eq. (8.85) as
d
2 Y
dx 2 = λ
2
(−C 1 sin λx − C 2 cos λx + C 3 sinh λx + C 4 cosh λx)
(8.101)
d
3 Y
dx 3 = λ
3
(−C 1 cos λx + C 2 sin λx + C 3 cosh λx + C 4 sinh λx)
(8.102)
Putting the boundary conditions given by Eq. (8.99) into Eq. (8.101) after
considering the beam to be uniform, we get
−C 2 + C 4 = 0
−C 1 + C 3 = 0
(8.103)
or
C 2 = C 4
C 1 = C 3
(8.104)
Substituting the end conditions given by Eq. (8.100) into Eq. (8.102), and noting
the relation given by Eq. (8.104), we get