Intra-wave modulation effects with oscillating time-varying frequencies have been observed in physical systems including mechanical, ocean, power, and even biological ones. Such phenomena correspond to actual physical processes with which the underlying dynamics of these systems can be understood. In this work, we study intra-wave modulation effects in milling processes with the aid of iterative nonlinear chirp mode decomposition (INCMD), a recently proposed technique for analyzing complex dynamic responses. A nonlinear non-stationary template signal is provided to model milling vibration responses, and an INCMD-based strategy is developed to extract embedded modulation features. Through dynamic simulations and experimental verification, it has been demonstrated that distinct intra-wave modulation frequencies that are exactly equal to theoretical chatter frequencies calculated based on the Floquet theory exist to indicate different cutting states, and milling instability is accompanied by the variation of such a characteristic quantity. Bessel functions can mathematically relate explicit modulation patterns with intricate spectral distributions of milling responses. Moreover, the switch of the modulation pattern, which remains noticeable in the presence of noise, emerges far earlier than visible chatter marks do, indicating the superiority of the chatter detection and even prediction utilizing intra-wave modulation features.

在包括机械，海洋，动力甚至生物系统在内的物理系统中，都观察到了具有随时间变化的振荡频率的波内调制效应。这种现象对应于实际的物理过程，通过这些物理过程可以理解这些系统的基本动态。在这项工作中，我们借助迭代非线性线性调频模式分解（INCMD）研究铣削过程中的波内调制效应，这是最近提出的用于分析复杂动态响应的技术。提供非线性非平稳模板信号来模拟铣削振动响应，并开发了基于INCMD的策略来提取嵌入式调制特征。通过动态模拟和实验验证，已经证明存在与基于Floquet理论计算的理论颤动频率完全相等的不同的波内调制频率，以表示不同的切削状态，并且铣削不稳定性伴随着这种特征量的变化。贝塞尔函数可以在数学上将显式调制模式与铣削响应的复杂频谱分布相关联。而且，在有噪声的情况下仍可明显注意到的调制模式的切换出现的时间比可见的颤动标记要早得多，这表明颤动检测和甚至利用波内调制特性进行预测的优越性。铣削的不稳定性伴随着这种特征量的变化。贝塞尔函数可以在数学上将显式调制模式与铣削响应的复杂频谱分布相关联。而且，在有噪声的情况下仍可明显注意到的调制模式的切换出现的时间比可见的颤动标记要早得多，这表明颤动检测和甚至利用波内调制特性进行预测的优越性。铣削的不稳定性伴随着这种特征量的变化。贝塞尔函数可以在数学上将显式调制模式与铣削响应的复杂频谱分布相关联。而且，在有噪声的情况下仍可明显注意到的调制模式的切换出现的时间比可见的颤动标记要早得多，这表明颤动检测和甚至利用波内调制特性进行预测的优越性。