Szalai Robert, Ehrhardt David, Haller George
Department of Engineering Mathematics, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK.
Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Clifton BS8 1TR, UK.
Proc Math Phys Eng Sci. 2017 Jun;473(2202):20160759. doi: 10.1098/rspa.2016.0759. Epub 2017 Jun 14.
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken's delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.
在非线性振荡系统中，谱子流形（SSMs）是与平衡点的线性模态子空间相切的最平滑不变流形。谱子流形上动力学的幅频图提供了实验非线性模型识别中所寻求的经典主干曲线。在此，我们开发了一种方法，可从拟合实验振动信号的数据同化模型中解析计算谱子流形的形状及其相应的主干曲线。这种模型识别利用了塔肯斯延迟嵌入定理，以及对与该嵌入相关的采样映射的泰勒展开式进行最小二乘拟合。然后使用不变流形的参数化方法为采样映射构造谱子流形，该方法假设流形是谱子空间的嵌入，而非在谱子空间上的图。通过合成数据和实际实验数据的示例，我们证明了这种方法能够高精度地重现主干曲线。
