The investigation of the free vibrations of inclined taut cables has been a significant subject due to their wide applications in various engineering fields. For this subject, accurate analytical expression for the natural modes and the natural frequencies is of great importance. In this paper, the free vibration of an inclined taut cable is further investigated by accounting for the factor of the weight component parallel to the cable chord. Two coupled linear differential equations describing two-dimensional in-plane motion of the cable are derived based on Newton’s law. By variable substitution, the equation of the transverse motion becomes a Bessel equation of zero order when the equation of longitudinal motion is ignored. Solving the Bessel equation with the given boundary conditions, a set of explicit formulae is presented, which is more accurate for determining the natural frequencies and the modal shapes of an inclined taut cable. The accuracy of the proposed formulae is validated by numerical results obtained by the Galerkin method. The influences of two characteristic parameters $λ$ and $ε$ on the natural frequencies and modal shapes of an inclined taut cable are studied. The results are discussed and compared with those of other literatures. It appears that the present theory has an advantage over others in the aspect of accuracy, and may be used as a base for the correct analysis of linear and nonlinear dynamics of cable structures.

1.
Irvine
,
H. M.
, and
Caughey
,
T. K.
, 1974, “
The Linear Theory of Free Vibrations of a Suspended Cable
,”
Proc. R. Soc. London, Ser. A
0950-1207,
341
, pp.
299
315
.
2.
Irvine
,
H. M.
, 1978, “
Free Vibration of Inclined Cables
,”
J. Struct. Div.
0044-8001,
104
(
2
), pp.
343
347
.
3.
Henghold
,
W. M.
,
Russell
,
J. J.
, and
Morgan
,
J. D.
, 1977, “
Free Vibrations of Cable in Three Dimensions
,”
J. Struct. Div.
0044-8001,
103
(
5
), pp.
1127
1136
.
4.
Perkins
,
N. C.
, and
Mote
,
C. D.
, 1987, “
Three-Dimensional Vibration of Travelling Elastic Cables
,”
J. Sound Vib.
0022-460X,
114
(
2
), pp.
325
340
.
5.
Srinil
,
N.
,
Rega
,
G.
, and
Chuchepsakul
,
S.
, 2007, “
Two-to-One Resonant Multi-Modal Dynamics of Horizontal/Inclined Cables. Part I: Theoretical Formulation and Model Validation
,”
Nonlinear Dyn.
0924-090X,
48
(
3
), pp.
231
252
.
6.
Triantafyllou
,
M. S.
, 1984, “
The Dynamic of Taut Inclined Cables
,”
Q. J. Mech. Appl. Math.
0033-5614,
37
(
3
), pp.
421
440
.
7.
Triantafyllou
,
M. S.
, and
Grinfogel
,
L.
, 1986, “
Natural Frequencies and Modes of Inclined Cables
,”
J. Struct. Eng.
0733-9445,
112
(
1
), pp.
139
148
.
8.
Shih
,
B.
, and
,
I. G.
, 1984, “
Small-Amplitude Vibrations of Extensible Cables
,”
J. Eng. Mech.
0733-9399,
110
(
4
), pp.
569
576
.
9.
Al-Qassab
,
M.
, and
Nair
,
S.
, 2004, “
Wavelet-Galerkin Method for the Free Vibrations of an Elastic Cable Carrying an Attached Mass
,”
J. Sound Vib.
0022-460X,
270
(
1–2
), pp.
191
206
.
10.
Wang
,
L.
,
Zhao
,
Y.
, and
Rega
,
G.
, 2009, “
Multimode Dynamics and Out-of-Plane Drift in Suspended Cable Using the Kinematically Condensed Model
,”
ASME J. Vibr. Acoust.
0739-3717,
131
, p.
061008
.
11.
Rega
,
G.
, and
Srinil
,
N.
, 2007, “
Nonlinear Hybrid-Mode Resonant Forced Oscillations of Sagged Inclined Cables at Avoidances
,”
ASME J. Comput. Nonlinear Dyn.
1555-1423,
2
, pp.
324
336
.
12.
Wu
,
Q.
,
Takahashi
,
K.
, and
Nakamura
,
S.
, 2005, “
Formulae for Frequencies and Modes of In-Plane Vibrations of Small-Sag Inclined Cables
,”
J. Sound Vib.
0022-460X,
279
(
3–5
), pp.
1155
1169
.
13.
Sun
,
Y.
, and
Leonard
,
J. W.
, 1998, “
Dynamics of Ocean Cables With Local Low-Tension Regions
,”
Ocean Eng.
0029-8018,
25
(
6
), pp.
443
463
.
14.
Gobat
,
J. I.
,
Grosenbaugh
,
M. A.
, and
Triantafyllou
,
M. S.
, 2002, “
Generalized-α Time Integration Solutions for Hanging Chain Dynamics
,”
J. Eng. Mech.
0733-9399,
128
(
6
), pp.
677
687
.
15.
Gobat
,
J. I.
, and
Grosenbaugh
,
M. A.
, 2006, “
Time-Domain Numerical Simulation of Ocean Cable Structures
,”
Ocean Eng.
0029-8018,
33
(
10
), pp.
1373
1400
.
16.
Goyal
,
S.
,
Perkins
,
N. C.
, and
Lee
,
C. L.
, 2005, “
Nonlinear Dynamics and Loop Formation in Kirchhoff Rods With Implications to the Mechanics of DNA and Cables
,”
J. Comput. Phys.
0021-9991,
209
(
1
), pp.
371
389
.
17.
Goyal
,
S.
,
Perkins
,
N. C.
, and
Lee
,
C. L.
, 2008, “
Non-Linear Dynamic Intertwining of Rods With Self-Contact
,”
Int. J. Non-Linear Mech.
0020-7462,
43
(
1
), pp.
65
73
.
18.
Goyal
,
S.
, and
Perkins
,
N. C.
, 2008, “
Looping Mechanics of Rods and DNA With Non-Homogeneous and Discontinuous Stiffness
,”
Int. J. Non-Linear Mech.
0020-7462,
43
(
10
), pp.
1121
1129
.
19.
Stump
,
D. M.
,
Fraser
,
W. B.
, and
Gates
,
K. E.
, 1998, “
The Writhing of Circular Cross-Section Rods: Undersea Cables to DNA Supercoils
,”
Proc. R. Soc. London, Ser. A
0950-1207,
454
, pp.
2123
2156
.
20.
Zhu
,
W. D.
, and
Chen
,
Y.
, 2006, “
Theoretical and Experimental Investigation of Elevator Cable Dynamics and Control
,”
ASME J. Vibr. Acoust.
0739-3717,
128
, pp.
66
78
.
21.
Zhu
,
W. D.
, and
Zheng
,
N. A.
, 2008, “
Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation
,”
ASME J. Appl. Mech.
0021-8936,
75
(
3
), p.
031003
.
22.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, 1972,
Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables
,
Dover
,
New York
.