三自由度受约束机械系统的笛卡尔方法

约束力学系统的笛卡尔方法

Cartesian approach for constrained mechanical systems

在力学史上，研究力学系统的观点主要有两种：牛顿力学和笛卡尔力学。根据笛卡尔的观点，机械系统的运动可用N维位形空间中的一阶微分方程来描述。在本文中，我们开发的笛卡尔方法的机械系统的约束是线性的速度。

When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?

While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of …

Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

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. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone…

二自由度机械系统的横切流形

Transverse foliations for two-degree-of-freedom mechanical systems

研究二自由度机械系统在临界值以上微小能量下的动力学，通过凸性假设证明了横切流形的存在，并应用于Hénon-Heiles势能等系统。

高自由度系统非线性振动稳态响应快速计算

Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems

本文提出了一种积分方程方法，用于快速计算周期性和准周期性外部激励下高自由度非线性机械系统的响应，通过显式计算格林函数并推导出简单快速的皮卡迭代求解条件，即使对于非光滑系统也能保证收敛，同时结合标准连续方案提升性能。

Exact Model Reduction by a Slow-Fast Decomposition of Nonlinear Mechanical Systems

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the most flexible degrees of freedom. This Slow-Fast Decomposition (SFD) enslaves exponentially fast the stiff degrees of freedom to the flexible ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justifi…