A Wittrick-Williams-algorithm-compatible asymptotic dynamic stiffness formulation applied to non-uniform beam structures

This paper proposes an asymptotic dynamic stiffness formulation for analyzing the vibration of non-uniform beams with power-law varying cross-sections. Exact solutions of the governing differential equations of non-uniform beams, which employ special functions, inevitably lead to implicit dynamic stiffness formulations and mode count, thereby compromising computational efficiency and accuracy and rendering them incompatible with the Wittrick-Williams (WW) algorithm, which is a proven and efficient tool for free vibration analysis. To tackle these problems, this work derives exact solutions of governing differential equations for axial and flexural vibrations of non-uniform beams in terms of Bessel functions and hypergeometric series, then uses their asymptotic behavior to derive the asymptotic dynamic stiffness matrix with explicit analytical expressions, which is further leveraged to yield the explicit mode count. The proposed solutions are shown to greatly mitigate numerical problems in special functions and make the proposed formulation compatible with the WW algorithm. Numerical examples, covering beams and frames, showcase the accuracy and efficiency of the proposed asymptotic dynamic stiffness formulation through comparisons with finite element simulations. This method offers a promising tool for the design and vibration analysis of complex structures composed of beam elements, harnessing minimum number of degrees of freedom.