8.8 Free Flexural Vibration of Beams with General End Conditions
331
Table 8.1 Values of μ m for a beam having the same degree of elastic restraint against rotation at
both ends
φ
m
1
2
3
4
5
6
0.01
4.642
7.711
10.802
13.895
16.991
20.089
0.05
4.374
7.329
10.339
13.375
16.431
19.592
0.10
4.156
7.069
10.066
13.106
16.172
19.257
0.50
3.577
6.547
9.613
12.712
15.827
18.950
1.00
3.399
6.428
9.525
12.643
15.770
18.902
10.00
3.173
6.399
9.436
12.575
15.715
18.855
Combining Eqs. (c), (d) and (g), we get
C 1
C 2
=
[(sin μ + sinh μ) + μφ(cos μ + cosh μ) + 2μφ × (cosh μ + μφ sinh μ)]
(cos μ − cosh μ) − μφ(sin μ + sinh μ)
(j)
From Eqs. (h) and (j), one obtains the following transcendental equation
cosh μ − cos μ + 2φμ sinh μ
sin μ − sinh μ
=
(sin μ + sinh μ) + μφ(cos μ + cosh μ) + 2μφ(cosh μ + φ sinh μ)
(cos μ − cosh μ) − μφ(sin μ + sinh μ)
(k)
From Eq. (k), depending on the magnitude of coefficient of restraint, different
values of μ can be obtained.
Values of μ m for a beam for various values of the degrees of elastic rest-raint
against rotation, which are same at both ends, have been shown in Table 8.1.
The mode shape equation for this case is
Y m =
sin
μ m x
L
− sinh
μ m x
L
+ α m
cos
μ m x
L
− 2μ m ϕ sinh
μ m x
L
− cosh
μ m x
L
(l)
where
α m =
sin μ m − sinh μ m
cosh μ m − cos μ m + 2φμ m sinh μ m
(m)
From Table 8.1, for φ = 0.1, μ 1 = 4.462
Therefore,
λ 1 L = 4.462