In this paper, the coupling effects among three elastic wave modes, flexural, tangential, and radial shear, on the dynamics of a planar curved beam are assessed. Two sets of equations of motion governing the in-plane motion of a curved beam are derived, in a consistent manner, based on the theory of elasticity and Hamilton’s principle. The first set of equations retains all resulting linear coupling terms that includes both static and dynamic coupling among the three wave modes. In the derivation of the second set of equations, the effects of Coriolis acceleration and high-order elastic coupling terms are neglected, which leads to a set of equations without dynamic coupling terms between the tangential and shear wave modes. This second set of equations of motion is the one most commonly used in studies on thick curved beams that include the effects of centerline extensibility, rotary inertia, and shear deformation. The assessment is carried out by comparing the dynamic behavior predicted by each curved beam model in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and mode shapes, and frequency responses. In order to ensure the comparison is based on accurate results, the wave propagation technique is applied to obtain exact wave solutions. The results suggest that the contributions of the dynamic and high-order elastic coupling terms to the response of a thick curved beam are quite significant and that these coupling terms should not be neglected for an accurate analysis of a thick curved beam with a large curvature parameter.

References

References
1.
Chidamparam
,
P.
and
Leissa
,
A. W.
, 1993, “
Vibration of Planar Curved Beams, Rings, and Arches
,”
ASME Appl. Mech. Rev.
,
46
, pp.
467
483
.
2.
Laura
,
P. A. A.
and
Maurizi
,
M. J.
, 1987, “
Recent Research on Vibrations of Arch-type Structures
,”
Shock Vibration Digest
,
19
, pp.
6
9
.
3.
Markus
,
S.
and
Nanasi
,
T.
, 1981, “
Vibration of Curved Beams
,”
Shock Vibration Digest
,
13
, pp.
3
14
.
4.
Love
,
A. E. H.
, 1944,
A Treatise on the Mathematical Theory of Elasticity
,
Dover
,
New York
.
5.
Kang
,
B.
and
Riedel
,
C. H.
, 2008, “
Wave Motion in a Planar Curved Beam
,”
Proceedings of IIAV 15th International Congress on Sound and Vibration
, Daejeon, Korea, July 6–10 (CD-ROM).
6.
Chidamparam
,
P.
and
Leissa
,
A. W.
, 1995, “
Influence of Centerline Extensibility on the In-Plane Free Vibrations of Loaded Circular Arches
,”
J. Sound Vib.
,
183
, pp.
779
795
.
7.
Graff
,
K. F.
, 1970, “
Elastic Wave Propagation in a Curved Sonic Transmission Line
,”
IEEE Trans. Sonics Ultrasonics
,
17
, pp.
1
6
.
8.
Lee
,
J.
, 2003, “
In-Plane Free Vibration Analysis of Curved Timoshenko Beams by the Pseudospectral Method
,”
KSME Int. J.
,
17
, pp.
1156
1163
.
9.
Issa
,
M. S.
,
Wang
,
T. M.
, and
Hsiao
,
B. T.
, 1987, “
Extensional Vibrations of Continuous Circular Curved Beams with Rotary Inertia and Shear Deformation: Free Vibration
,”
J. Sound Vib.
,
114
, pp.
297
308
.
10.
Yildrim
,
V.
, 1997, “
A Computer Program for the Free Vibration Analysis of Elastic Arcs
,”
Comput. Struct.
,
62
, pp.
475
485
.
11.
Yang
,
S. Y.
and
Sin
,
H. C.
, 1995, “
Curvature-Based Beam Elements for the Analysis of Timoshenko and Shear-Deformable Curved Beams
,”
J. Sound Vib.
,
187
, pp.
569
584
.
12.
Yang
,
F.
,
Sedaghati
,
R.
, and
Esmailzadeh
,
E.
, 2008, “
Free In-Plane Vibration of General Curved Beams Using Finite Element Method
,”
J. Sound Vib.
,
318
, pp.
850
867
.
13.
Mau
,
S. T.
and
Williams
,
A. N.
, 1988, “
Green’s Function Solution for Arch Vibration
,”
J. Eng. Mech.
,
114
, pp.
1259
1264
.
14.
Kang
,
B.
, 2007, “
Transfer Functions of One-Dimensional Distributed Parameter Systems by Wave Approach
,”
ASME J. Vibr. Acoust.
,
129
, pp.
193
201
.
15.
Mei
,
C.
, 2008, “
Wave Analysis of In-Plane Vibrations of H- and T-Shaped Planar Frame Structures
,”
ASME J. Vibr. Acoust.
,
130
, pp.
1004
1013
.
16.
Cook
,
R. D.
and
Young
,
W. C.
, 1985,
Advanced Mechanics of Materials
,
Macmillan
,
New York
.
17.
Boresi
,
A. P.
and
Schmidt
,
R. J.
, 2003,
Advanced Mechanics of Materials
,
John Wiley & Sons
,
Hoboken
.
18.
Cremer
,
L.
,
Heckl
,
M.
, and
Ungar
,
E. E.
, 1973,
Structure-Borne Sound
,
Springer-Verlag
,
Berlin
.
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