Dynamical modeling and response analysis of multi-folded beam structures with nonlinear joints

Multi-folded beams, composed of a series of simple beam structures interconnected by flexible joints, are extensively utilized in aerospace engineering due to their inherent flexibility and adaptability. While the characterization of the individual beam is straightforward, constructing an accurate dynamical model for multi-folded beam structures is challenging because of their multiple degrees of freedom, mode coupling, and the flexibility and nonlinearity of the joints. This paper presents an innovative approach to the development of an analytical nonlinear dynamical model for a multi-folded beam with any number of segments. Each segment can have unique geometric dimensions and folding angles. Initially, the flexible joints in the beams are modeled as artificial linear springs, and the coefficient matrix of the linear model for a multi-folded beam is rapidly developed by compiling the assumed mode matrices of each individual beam, organized by beam number. According to the eigenvectors of the linear model, the assumed modes are weighted to derive the Weighted Modes (WMs). Based on these WMs, the model of the entire structure is constructed, incorporating the nonlinearity of joints. The WM model was validated through comparisons with natural characteristics and dynamical responses present in the published literature and those obtained using the Finite Element Method (FEM). The advantage of the Mode Weighting Method (MWM), which achieves convergence with only a limited number of degrees of freedom, is demonstrated. Utilizing the low-dimensional WM model, this research further reveals the phenomenon of amplitude amplification induced by 3:1 internal resonance due to the nonlinearity of joints in multi-folded beam structures. These findings indicate that neglecting nonlinear factors in the structure can significantly impact structural safety, with internal resonance potentially leading to structural damage.