Adaptive Frequency-Dependent Shape Functions for Accurate Estimation of Modal Frequencies

An adaptive linear FEM for modal analysis of structures is presented in this paper. The method uses frequency-dependent shape functions in addition to conventional low-order polynomial interpolation functions to improve accuracy in natural-frequency and mode-shape estimation. The proposed technique requires the iterative computation of the mass and stiffness matrices because shape functions are adjusted recursively as functions of estimated natural frequencies without mesh adaptation or refinement. Applied to frame structures, the method uses conventional polynomial interpolating functions for axial, bending, and torsional displacements and additional generalized coordinates multiplied by frequency-dependent functions derived from modal analysis of continuous beam models. Because these additional functions or their derivatives do not vanish at nodes located at the element boundaries, linear kinematic constraints are imposed to the augmented set of displacement coordinates to ensure displacement field continuity and compatibility conditions at element boundaries. Applications of the methodology to straight rod elements in longitudinal vibration and beams in flexural vibration are presented to compare accuracy obtained using conventional finite-element (FE) meshes with the proposed method.