隔振装置在机械设备中广泛应用，其非光滑特性可能导致整个系统具有多稳态响应共存、不稳定不变集以及复杂边界结构等非线性动力学特征。同时，作用在系统上的不确定扰动可能使机组产生一系列的动态效应，进而诱导系统在不同响应间切换和跃迁，甚至改变整个系统的动力学行为，并由此产生振动、噪声和故障等问题。因此，研究非光滑隔振系统的动力学特征以及噪声激励下系统的随机响应行为对系统响应预测，寿命预估等有着重要的作用。

为了解决随机系统大范围响应跃迁求解时的巨大计算消耗问题，本文首先在非线性随机动力学相关理论方法的框架下，结合主成分分析(Principal Component Analysis, PCA)技术和演化概率向量(Evolving Probabilistic Vector, EPV)技术，从数据驱动的角度提出了一种基于少量响应样本PCA的短时概率分布近似方法。该方法能够很好的实现Lévy噪声激励下系统响应跃迁的高效求解。另外，本文发展的数值计算方法具有进一步应用于研究其它非Gauss激励下系统瞬态概率密度演化行为的潜力。

其次，考虑到系统的基础牵连运动和限位器间歇接触的特点，在确定性框架下建立了包含间歇接触特征的系统确定性动力学模型，利用广义胞映射方法从全局动力学角度研究了系统的动力学特征。结果表明：不同工况下单自由度非光滑隔振系统普遍存在多稳态响应共存现象，支承刚度系数和限位器间隙的改变会导致系统响应在双稳态和多稳态之间切换，并且支承刚度系数的增大会加速无碰撞周期响应的湮灭。与之相反，限位器间隙的增加则会扩大无碰撞周期响应的吸引域范围。

再次，针对Gauss白噪声激励的随机动力学模型，基于短时Gauss近似的路径积分方法定性分析了系统响应的瞬态概率密度分布及其动力学演化行为，揭示了噪声诱导的不同随机分岔现象。分析显示：噪声改变了系统全局结构的表现形式，形成了随机吸引子，并且模糊了相邻吸引域的边界。当随机强度超过阈值后，甚至诱发出边界激变(Boundary Crisis)等危险分岔现象，此时响应概率密度会沿不稳定流形演化至其它共存的吸引域内部。

最后，通过和其它数值方法对比，验证了基于少量响应样本PCA的短时概率分布近似方法的有效性，并采用该方法研究了Lévy噪声激励下系统的随机响应行为，定性观察了噪声诱导的瞬态概率密度转移现象，定量讨论了噪声参数对响应逃逸时间的影响。具体结果为：稳定不变集和不稳定不变集等确定性全局结构奠定了随机响应的基本框架，通过对比不同类型噪声下响应概率密度的跃迁过程，发现扰动形式会造成瞬态演化存在一定差异，但演化的最终状态却保持一致，而且偏斜参数和噪声强度会明显改变响应的逃逸时间。

以上研究将有助于深入了解确定性全局结构和噪声的相互作用关系，以及它们在单自由度非光滑隔振系统随机稳定性中的作用。

Vibration isolation system, whose non-smooth features could induce some nonlinear dynamic characteristics such as coexistence of multiple steady-state responses, unstable invariant sets and complex boundary, is widely applied in some mechanical equipments. Furthermore, uncertain disturbances subjected to system could bring about some undesirable vibrations and boresome noises as well as catastrophe by the significant changes of system in different responses and even dynamic structures. Therefore, the investigations on dynamic characteristics and stochastic responses of non-smooth vibration isolation system under noises can gain insight into life estimation and response prediction of mechanical systems.

To release the huge computing cost on solving noise-induced large transition of responses, a short-time probability distribution approximation method based on a small number of response samples PCA is firstly proposed in this thesis, from a data-driven perspective by combining PCA and EPV techniques. The proposed method can work nicely to capture probability transition of system with input of Lévy excitation. Note that the numerical method developed in this paper has the potential to be further applied to investigation of the transient probability evolution of systems under other non-Gaussian excitations.

Secondly, a deterministic dynamic model with discontinuous impact characteristics is established by considering the basic implicated motion and intermittent impact of the vibration isolation system, and the dynamic characteristics of the system are studied from the perspective of global dynamics using the generalized cell mapping method. The results show that the coexistence of multiple steady-state responses is common in the non-smooth vibration isolation system under different working conditions. The change of linear stiffness coefficient and limiter clearance will cause the system response to vary between bistability and multi-stability, and the increase of linear stiffness coefficient will lead to the annihilation of non-collision periodic response, while on the contrary, the range of attraction domain corresponding to non-collision periodic response will be expanded when the limiter clearance increases.

Thirdly, the dangerous bifurcation phenomenon and probability density evolution of the system excited by Gauss white noise are analyzed qualitatively by a path integral method based on the short-time Gaussian approximation. The analysis shows that the noise changes the form of the global structure, creating stochastic attractors and blurring the boundaries of adjacent attractive domains. When the stochastic intensity exceeds a threshold, dangerous bifurcations such as boundary crisis may even be induced, and the response probability density will evolve to other coexisting basins of attraction along the unstable manifold.

Finally, the validity of the short-time probability distribution approximation method based on a small number of response samples PCA is verified by compared with other numerical methods. At the same time, the probability density transition of the response under Lévy noise excitation and the effect of noise parameters on the response escape time is studied by this method. The results are as follows: the deterministic global structure shows the base frame of dynamical behaviors, and different disturbance forms will cause certain differences in the transient evolution, but the final state of evolution remains consistent. Moreover, the skewness parameter and noise intensity will significantly change the escape time of response.

The above research will help to understand the interaction between deterministic global structure and noise, and their roles in the stochastic stability of single-degree-of-freedom non-smooth vibration isolation systems.

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