In-plane free vibration analysis of multi-folded beam structures

Abstract

Folded beam structures have been widely used in mechanical engineering, aerospace engineering, and other fields due to their flexibility and extensibility in applications. Although the components of a folded beam are simple beam structures, building an accurate dynamic model is complex due to its multi-degree-of-freedom, multi-mode, and close-mode features. This study presents a novel method to analyze in-plane free vibration of a multi-folded beam structure with arbitrary numbers of beam segments with different beam lengths and folding angles. The governing equations of the system can be easily obtained based on the beam theory and the work-energy principle. However, the key issue is how to solve the eigenvalue problem of a multi degree of freedom system defined by a 6 n × 6 n matrix. As the coefficient solution vector of the modal function is unknown, solving equations defined by high-order determinants involving variables has considerable complexity. In this work, the intricate matrix is initially turned into a single matrix of block Hessenberg form using numerous "6 × 6" block square matrices. Furthermore, by combining the characteristics of the block Hessenberg matrix with the approach to solve block determinants, a considerable reduction in the determinant's size can be attained, diminishing it from a dimension of 6 n × 6 n to 6 × 6. This procedure markedly simplifies the computational intricacies involved in the analysis. Finally, four kinds of multi-folded beams, namely L-shaped beams, Z-shaped beams, C-shaped beams, and four-segment folded beams with arbitrary lengths, folding angles, and numbers of beam segments are considered and studied, and the mode shapes and the influence of geometric parameters on the first three natural frequencies are discussed in detail. The results are also verified by the finite element method (FEM), and good agreement is found between the two methods. The results show that the proposed method can offer a unified framework for the in-plane vibration analysis of multi-folded beam structures.