8.13 Free Vibration of Beams Including Shear Deformation …
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λ
4
=
mp
2
E I
and r
2
=
I
A
.
The simply supported beam mode shape is assumed as follows:
Y n (x) = A sin
nπ x
L
(b)
Equation (b) satisfies all the boundary conditions of the problem. Therefore, the
shear deformation and the rotary inertia do not have any effect on the mode shape
of the uniform simply supported beam. Substituting Y n from Eq. (b) into Eq. (a), we
get
nπ
L
4 − λ
4
− λ
4 n
2
nπ
L
2
1 +
E
μG
+ λ
8 r
4
E
μG
= 0
( c )
Substituting the values of λ and r in Eq. (c), we get
p
4
−
1
ρ
(E + μG)n
2
π
2
L 2
+
μG A
I
p
2
+
μG En
4
π
4
ρ 2 L 4
= 0
( d )
Depending on the geometrical and material properties of the beam, n frequencies
can be calculated from Eq. (d). The natural frequencies have been plotted for three
values of E/μG in Fig. 8.16. It is seen in the figure that the correction due to shear
deformation and rotary inertia increases with mode number and decreases with the
increase of slenderness ratio.
Fig. 8.16 Effect of
geometric and material
properties on natural
frequencies