AI 中文总结
研究描述非线性连续体振动的偏微分方程，证明谱子流形存在，可据此从方程中严格提取骨干曲线和强迫响应曲线，无需先验离散化，通过弹性梁和薄板例子展示可直接从方程手动计算相关曲线。
AI 中文摘要
我们证明了在描述受迫阻尼连续体振动的非线性偏微分方程（PDEs）中谱子流形的存在性。我们的结果涵盖了受广义结构阻尼影响的结构振动。基于这些结果，无需先验离散化，就可以从PDEs中严格提取骨干曲线和强迫响应曲线。在涉及弹性梁和薄板的两个例子中，我们表明骨干曲线和强迫响应曲线甚至可以直接从PDEs手动计算得出。
英文摘要
We prove the existence of spectral submanifolds in nonlinear partial differential equations (PDEs) describing forced-damped continuum vibrations. Our results cover structural vibrations subject to generalized structural damping. Based on these results, backbone curves and forced response curves can be rigorously extracted from PDEs, without a priori discretization. On two examples involving an elastic beam and a thin plate, we show that backbone and forced response curves can even be calculated by hand directly from the PDEs.
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