The dynamics in a primary Spectral Submanifold (SSM) constructed over the slowest modes of a dynamical system provide an ideal reduced-order model for nearby trajectories. Modeling the dynamics of trajectories further away from the primary SSM, however, is difficult if the linear part of the system exhibits strong non-normal behavior. Such non-normality implies that simply projecting trajectories onto SSMs along directions normal to the slow linear modes will not pair those trajectories correctly with their reduced counterparts on the SSMs. In principle, a well-defined nonlinear projection along a stable invariant foliation exists and would exactly match the full dynamics to the SSM-reduced dynamics. This foliation, however, cannot realistically be constructed from practically feasible amounts and distributions of experimental data. Here we develop an oblique projection technique that is able to approximate this foliation efficiently, even from a single experimental trajectory of a significantly non-normal and nonlinear beam.

翻译：在动力系统最慢模态上构建的主谱子流形（SSM）动力学为邻近轨迹提供了理想的降阶模型。然而，若系统的线性部分呈现强非正规特性，则远离主SSM轨迹的动力学建模将变得困难。此类非正规性意味着，仅沿慢速线性模态的法向将轨迹投影至SSM，无法使这些轨迹与SSM上的降阶对应轨迹正确匹配。理论上，沿稳定不变叶理存在明确定义的非线性投影，可将完整动力学精确匹配至SSM降阶动力学。然而，从实际可行的实验数据量及其分布出发，该叶理构造难以实现。本文提出一种斜投影技术，即使对于显著非正规且非线性的梁结构单条实验轨迹，该技术仍能有效逼近此类叶理。