Wavelet-Galerkin Method for Free Vibrations of Elastic Cable

The wavelet-Galerkin method is applied to study the free vibration of a horizontally suspended catenary cable. Antiderivatives of the Daubechies compactly supported wavelets have been used with multilevel representations. Comparison between wavelet and Fourier methods is presented for natural frequencies, mode shapes, and dynamic tension of the cable. Both methods showed that they are converging fast in obtaining the natural frequencies and mode shapes. However, as the dynamic tension is obtained, the Fourier solution shows many oscillations and the existence of the Gibbs phenomenon at the cable supports, whereas, these oscillations do not appear in the wavelet solutions. Comparison is also made with the linear theory of cable vibration. Due to the inclusion of the inertia term of the longitudinal component in our solution, new modes have been found. Those modes are reverting but swapping modes in which the longitudinal displacement component is larger than the transverse displacement component.

[1]  David S. Saxon,et al.  MODES OF VIBRATION OF A SUSPENDED CHAIN , 1953 .

[2]  H. M. Irvine,et al.  The linear theory of free vibrations of a suspended cable , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[4]  Michael S. Triantafyllou.  The dynamics of translating cables , 1985 .

[5]  C. E. Smith,et al.  The Small Oscillations of a Suspended Flexible Line , 1973 .

[6]  William M. Henghold,et al.  Free Vibrations of Cable in Three Dimensions , 1977 .

[7]  Donald E. Bently,et al.  Modes of Vibration , 2002 .

[8]  W. J. Goodey,et al.  ON THE NATURAL MODES AND FREQUENCIES OF A SUSPENDED CHAIN , 1961 .

[9]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[10]  Jinchao Xu,et al.  Galerkin-wavelet methods for two-point boundary value problems , 1992 .

[11]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[12]  Fabrizio Vestroni,et al.  Planar non-linear free vibrations of an elastic cable , 1984 .

[13]  Iradj G. Tadjbakhsh,et al.  Small‐Amplitude Vibrations of Extensible Cables , 1984 .

[14]  Michael S. Triantafyllou,et al.  THE DYNAMICS OF TAUT INCLINED CABLES , 1984 .

[15]  A. Simpson.  On the oscillatory motions of translating elastic cables , 1972 .

[16]  Peter Hagedorn,et al.  On non-linear free vibrations of an elastic cable , 1980 .

[17]  C. Hwang,et al.  THE COMPUTATION OF WAVELET‐GALERKIN APPROXIMATION ON A BOUNDED INTERVAL , 1996 .