Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
复制标题
多自由度机械振动的非线性模型识别和谱子流形
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
G. Haller
中科院分区：
文献类型：
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作者：
R. Szalai;David A. Ehrhardt;G. Haller
在非线性振荡系统中，谱子流形（ssm）是与平衡点线性模态子空间相切的最光滑不变流形。ssm的动态幅频图提供了在实验非线性模型识别中寻求的经典骨干曲线。我们在这里发展了一种方法，可以从拟合实验振动信号的数据同化模型中解析计算ssm的形状及其相应的骨干曲线。该模型识别利用了Taken的延迟嵌入定理，以及与该嵌入相关的采样映射的泰勒展开的最小二乘拟合。然后使用不变流形的参数化方法为采样映射构造ssm，该方法假设流形是谱子空间的嵌入，而不是谱子空间上的图。通过合成和实际实验数据的实例，我们证明了该方法具有较高的主曲线再现精度。
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.