AI 中文总结
研究波导动力学等场景中临界速度的计算方法，从多项式参数依赖特征值问题出发，结合等相速和群速条件得到奇异多项式多参数特征值问题，用既定算法求解，能同时计算所有临界点，经基准问题验证性能良好。
AI 中文摘要
在波导动力学和移动负载问题（如高速列车）中，临界速度表明强烈振动放大的开始。在运动方向不变的系统中，这些速度可从色散关系中确定为传播模式的相速度和群速度重合的点。对于具有复杂分支相互作用的多模态系统，通过追踪色散曲线间接找到这些点可能很麻烦且潜在不可靠。我们提出了一种直接方法来计算此类场景中的临界速度，特别是在半解析方法的背景下。从波数 - 频率关系的多项式参数依赖特征值问题出发，结合等相速度和群速度的附加条件，得到一个奇异多项式多参数特征值问题，可使用既定算法进行线性化和求解。所提出的方法能够同时计算所有临界点，而无需追踪色散曲线。通过几个基准问题证明了其性能，证实了对临界速度的准确和稳健识别。
英文摘要
In waveguide dynamics and moving-load problems (e.g., high-speed trains), critical velocities indicate the onset of strong vibration amplification. In systems that are invariant in the direction of motion, these velocities can be identified from dispersion relations as points where the phase and group velocities of a propagating mode coincide. Finding such points indirectly by tracing dispersion curves can be cumbersome and potentially unreliable for multimodal systems with complex branch interactions. We present a direct method for computing critical velocities in such scenarios, specifically in the context of semi-analytical methods. Starting from a polynomial parameter-dependent eigenvalue problem for the wavenumber-frequency relation, incorporating the additional condition of equal phase and group velocities yields a singular polynomial multiparameter eigenvalue problem that can be linearized and solved using established algorithms. The proposed approach enables the simultaneous computation of all critical points without requiring the tracing of dispersion curves. Its performance is demonstrated by several benchmark problems, confirming the accurate and robust identification of critical velocities.
