D.1 Strings
For a uniform elastic string or cable with negligible bending stiffness, and subjected
to constant tension N at the fixed ends, the free and undamped transverse motions w
(x, t) are governed by:
€
w ¼ c
2
0 w
00
;
ðD:1Þ
where c 0 ¼
ffiffiffiffiffiffiffiffiffiffiffi ffi
N=qA
p
is the transverse wave speed. The jth natural frequency is:
x j ¼ jp
c 0
l
; j ¼ 1; 2; Á Á Á ;
ðD:2Þ
with corresponding mode shapes
u j ðxÞ ¼ sin jp
x
l
:
ðD:3Þ
D.2 Rods
D.2.1 Longitudinal Vibrations
For a uniform rod the free and undamped longitudinal motions u(x, t) are governed
by:
€ u ¼ c
2
0 u
00
:
ðD:4Þ
Table D.2. lists natural frequencies and mode shapes for some different
boundary conditions.
486
Appendix D: Natural Frequencies and Mode Shapes for Structural Elements