Wavelet-Galerkin method for the free vibrations of an elastic cable carrying an attached mass

A multilevel representation of Daubechies compactly supported wavelet has been used to study the free vibrations of elastic catenary cables carrying an attached mass. Anti-derivatives of wavelets are used to guarantee satisfaction of boundary conditions. Natural frequencies, mode shapes and dynamic tensions are obtained and compared with the classical Fourier series representation. The localization feature of wavelets has been implemented to enter the singularity region that is produced by the attached mass. More wavelets are used near the mass location and the spurious oscillations in the solution are minimized with few number of terms in the series. However, the Fourier solution shows many oscillations along the cable length and Gibbs phenomenon at the mass location. In both methods, reverting and swapping modes are discovered in which higher modes revert to lower modes and that the horizontal displacement components become greater than the vertical ones even for cables with small sag to span ratios.