8.13 Free Vibration of Beams Including Shear Deformation …
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8.13 Free Vibration of Beams Including Shear Deformation
and Rotary Inertia Effects
Let y b , y s and y be the bending, shear and total deflection of beams, then
y = y b + y s
(8.169)
and
∂ y
∂ x
=
∂ y b
∂ x
+
∂ y s
∂ x
(8.170)
The bending and shear deformation effects have been considered separately, and
the total effect is considered to be the summation of the two.
Some of the equations derived earlier are written as
V = μAG
∂ y s
∂ x
(8.154)
∂ V
∂ x
= ρ A
∂
2 y
∂ t 2
(8.155)
M = −E I
∂
2 y b
∂ x 2
(8.77)
Negative sign has been put in Eq. (8.77), because positive direction of y is
downwards. Again
∂ M
∂ x
= V + ρ I
∂
3 y b
∂ x∂ t 2
(8.144)
Combining Eqs. (8.155), (8.169) and (8.170) for a uniform beam, we get
μAG
∂
2 y s
∂ x 2 = ρ A
∂
2 y b
∂ t 2 +
∂
2 y s
∂ t 2
(8.171)
or
ρ A
∂
2 y b
∂ t 2 + ρ A
∂
2 y s
∂ t 2 = μAG
∂
2 y s
∂ x 2
(8.172)
Combining Eqs. (8.169), (8.154), (8.77) and (8.144), we get
E I
∂
3 y b
∂ x 3 + μAG
∂ y s
∂ x
− ρ I
∂
3 y b
∂ x∂t 2 = 0
(8.173)