Free Vibration Analysis of Frame Structures Using BSWI Method

Vibration analysis for complicated structures, or for problems requiring large numbers of modes, always requires fine meshing or using higher order polynomials as shape functions in conventional finite element analysis. Since it is hard to predict the vibration mode a priori for a complex structure, a uniform fine mesh is generally used which wastes a lot of degrees of freedom to explore some local modes. By the present wavelets element approach, the structural vibration can be analyzed by coarse mesh first and the results can be improved adaptively by multi-level refining the required parts of the model. This will provide accurate data with less degrees of freedom and computation. The scaling functions of B-spline wavelet on the interval (BSWI) as trial functions that combines the versatility of the finite element method with the accuracy of B-spline functions approximation and the multiresolution strategy of wavelets is used for frame structures vibration analysis. Instead of traditional polynomial interpolation, scaling functions at the certain scale have been adopted to form the shape functions and construct wavelet-based elements. Unlike the process of wavelets added directly in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. To verify the proposed method, the vibrations of a cantilever beam and a plane structures are studied in the present paper. The analyses and results of these problems display the multi-level procedure and wavelet local improvement. The formulation process is as simple as the conventional finite element method except including transfer matrices to compute the coupled effect between different resolution levels. This advantage makes the method more competitive for adaptive finite element analysis. The results also show good agreement with those obtained from the classical finite element method and analytical solutions.