Mode Weighting Method for modeling and internal resonance of spatial folded beams

Spatial folded beams, exemplified by the locked configuration of spatial manipulators, enable more complex spatial operations than planar structures but exhibit strong couplings and high-dimensional characteristics that complicate modeling and control. In this study, the Mode Weighting Method (MWM) is developed for spatial assembled structures, enabling efficient extraction of spatially coupled modes and the construction of low-dimensional nonlinear dynamic models. The MWM defines flexible component orientations using Euler angles and models joints as artificial springs. It constructs the assumed mode (AM) model from individual component modes, then extracts spatially coupled global modes via mode weighting to establish a global mode (GM) model with joint nonlinearity. This reduced GM model accurately captures the complex structural coupling, as validated through finite element (FE) analysis and physical experiments. Compared with the FEM, the MWM achieves comparable accuracy with 75% fewer degrees of freedom and a 99.36% reduction in computation time for the computation of the first ten natural frequencies of the standard spatial four-folded beam under the same computational settings. Variations in Euler angles lead spatial structures to display richer internal resonances, including those between adjacent and separated frequencies, as well as more mode transitions. In some cases, internal resonance amplifies local vibrations to 1.8 times the linear level, threatening structural safety. By leveraging Euler angles, the GM model is well-suited for attitude-adjustable aerospace structures, while the MWM provides a concise analytical framework readily extendable to arbitrary spatial assembled structures.