x nj ¼
a nj ^ c 0
R
; j ¼ 1; 2; Á Á Á ;
ðD:23Þ
where numerical values for the constant a nj are listed in Table D.5. The corresponding mode shapes are:
u nj ¼ J n x nj r=^ c 0
À
Á
cosðnhÞ:
ðD:24Þ
D.6 Plates
The equation governing free transverse motions w of a thin plate is:
qh
K
€
w þ r
2
r
2 w ¼ 0;
ðD:25Þ
where K is the plate bending stiffness:
K ¼
Eh
3
12ð1 À m 2 Þ
:
ðD:26Þ
Using ð n; tÞ to denote coordinates normal and tangential, respectively, to the
boundary, the common types of boundary conditions are: clamped ðw ¼ w
0
n ¼ 0Þ,
hinged ðw ¼ w
00
n n þ mw
00
t t ¼ 0Þ, and free ðw
00
n n þ mw
00
t t ¼ ½w
00
n n þ ð2 À mÞw
00
t t Š
0
n ¼ 0Þ.
D.6.1 Rectangular Plates
With a rectangular plate having side lengths a and b, we assume it to be supported
along the lines x = 0, x = a, y = 0, and y = b.
When the plate is hinged, i.e. simply supported, at all four edges, the natural
frequencies and mode shapes are, respectively:
Table D.5 Values of a nj to be used with Eq. (D.23)
a
n = 0
1
2
3
j = 1
2.404
3.832
5.135
6.379
2
5.520
7.016
8.417
9.760
3
8.654
10.173
11.620
13.017
4
11.792
13.323
14.796
16.224
a Adapted from Flügge (1962) with permission from the publisher
Appendix D: Natural Frequencies and Mode Shapes for Structural Elements
493