334
8 Free Vibration Analysis of Continuous Systems
Table 8.2 Values of μ m for a beam having one edge clamped and the other end elastically restrained
against rotation
φ
m
1
2
3
4
5
6
0.01
4.686
7.782
10.898
14.015
17.134
20.254
0.05
4.547
7.585
10.661
13.751
16.850
19.958
0.10
4.431
7.450
10.552
13.615
16.720
19.834
1.00
4.042
7.134
10.257
13.388
16.523
19.660
1000.00
3.927
7.069
10.211
13.352
16.494
19.635
Y m = sin
μ m x
L
− α m sinh
μ m x
L
(8.130)
where
α m =
sin μ m
sinh μ m
(8.131)
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 8.3.
Natural frequencies of uniform beams having various boundary conditions are
given in Table 8.4 where
p n = c n
E I
m L 4
p n is given in cycles/s.
Table 8.3 Values of μ m for one edge simply supported and the other edge elastically restrained
against rotation
φ
m
1
2
3
4
5
6
0.01
3.890
7.004
10.119
13.256
16.354
19.474
0.05
3.770
6.820
9.891
12.977
16.075
19.180
0.10
3.665
6.688
9.752
12.840
15.943
19.055
0.50
3.367
6.418
9.520
12.640
15.768
18.900
1.00
3.274
6.356
9.475
12.605
15.739
18.876
1000.00
3.142
6.284
9.425
12.567
15.708
18.850
8 Free Vibration Analysis of Continuous Systems
Table 8.2 Values of μ m for a beam having one edge clamped and the other end elastically restrained
against rotation
φ
m
1
2
3
4
5
6
0.01
4.686
7.782
10.898
14.015
17.134
20.254
0.05
4.547
7.585
10.661
13.751
16.850
19.958
0.10
4.431
7.450
10.552
13.615
16.720
19.834
1.00
4.042
7.134
10.257
13.388
16.523
19.660
1000.00
3.927
7.069
10.211
13.352
16.494
19.635
Y m = sin
μ m x
L
− α m sinh
μ m x
L
(8.130)
where
α m =
sin μ m
sinh μ m
(8.131)
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 8.3.
Natural frequencies of uniform beams having various boundary conditions are
given in Table 8.4 where
p n = c n
E I
m L 4
p n is given in cycles/s.
Table 8.3 Values of μ m for one edge simply supported and the other edge elastically restrained
against rotation
φ
m
1
2
3
4
5
6
0.01
3.890
7.004
10.119
13.256
16.354
19.474
0.05
3.770
6.820
9.891
12.977
16.075
19.180
0.10
3.665
6.688
9.752
12.840
15.943
19.055
0.50
3.367
6.418
9.520
12.640
15.768
18.900
1.00
3.274
6.356
9.475
12.605
15.739
18.876
1000.00
3.142
6.284
9.425
12.567
15.708
18.850