Gaussian expansion element method of the new dynamic modeling technique in non-uniform and variable cross-section structures

Abstract

The non-uniform and variable cross-section structures are widely used in engineering. However, its non-uniform characteristic has also caused some troubles in existing dynamic modeling methods, such as poor convergence, accuracy, efficiency, and difficulty in predicting high-order derivatives of mode shape. To solve these problems, this paper provides a new dynamic modeling technology for non-uniform structures based on element division and the Gaussian Expansion Method (GEM), named Gaussian Expansion Element Method (GEEM). In the GEEM, the non-uniform structure is divided into several elements, and their displacement fields are expanded to a set of Gaussian functions. For ensuring every element meets boundary and continuity conditions, these Gaussian basis functions are divided into modified (used to modify displacement fields near end supports) and main basis functions (describing internal displacement fields of element). Then, the dynamic model of element can be derived by the Rayleigh-Ritz method, and its state variables include the generalized coordinates of modified and main basis functions. Finally, the dynamic equation of the non-uniform structure can be obtained by assembling these equations of elements and eliminating the coordinates of modified basis functions by using boundary and continuity conditions. Numerical results show that the new method can predict the dynamic behaviors of the non-uniform beam, periodic structure, and frame structure, calculate their high-order derivatives of mode shape, and has higher accuracy, efficiency, and numerical stability than GEM. Notably, an adaptive selection scheme of Gaussian basis functions and element division rules are given to further enhance the new method's accuracy and efficiency.