In recent decades, quasi-zero stiffness (QZS) vibration isolation systems with nonlinear characteristics have aroused widespread attention and strong research interest due to their enormous potential in low-frequency vibration isolation. This work comprehensively reviews recent research on QZS vibration isolators with a focus on the principle, structural design, and vibration isolation performance of various types of QZS vibration isolators. The negative-stiffness mechanism falls into two categories by different realization methods: passive and active/semi-active negative-stiffness mechanisms. Representative design, performance analysis, and practical application are elaborated for each category. The results show that passive vibration isolation systems have excellent low-frequency vibration isolation performance under specific payload and design parameters, whereas active/semi-active vibration isolation systems can better adapt to different environmental conditions. Finally, the development trends and challenges of QZS vibration isolators are summarized, and the solved and unsolved problems are highlighted. This review aims to give a comprehensive understanding of the QZS vibration isolation mechanism. It also provides guidance on designing new QZS vibration isolators for improving their vibration isolation performance and engineering applicability.

In order to effectively control the low-frequency vibration of ship machinery, based on the improved multi-line spectrum adaptive control algorithm, a two-degree-of-freedom (two-DOF) active-passive hybrid vibration isolator composed of an electromagnetic actuator, rubber spring, and the hydraulic device is proposed. The dynamic model of the two-DOF vibration isolation system is established and the main control force demand of the vibration isolation system at different damping forces is analyzed. By introducing the improved wavelet packet decomposition algorithm with the Hartley block least mean square algorithm to the filter-x least mean square (FxLMS) algorithm, an improved wavelet packet Hartley block filter-x least mean square (IWPHB-FxLMS) algorithm is established. The experimental results show that the IWPHB-FxLMS algorithm has better control performance. Compared with the traditional FxLMS algorithm, the IWPHB-FxLMS control algorithm improves the convergence speed by 91.7% and the line spectrum power spectrum attenuation by 58.1%. The active-passive hybrid vibration isolator is based on multi-line spectrum adaptive control and can achieve good control effects under the excitation of multi-frequency line spectrum and constant frequency line spectrum.

The nonlinear energy sink (NES) is an effective method to weaken or eliminate the line spectra feature of ship machinery equipment. However, due to the strong nonlinearity of the NES, there are multistable attractors in the NES vibration absorption system. Moreover, when the excitation characteristics or working conditions of the mechanical equipment change, the NES may migrate from the small-amplitude motion state to the large-amplitude motion state, resulting in a high-amplitude solitary branch response and thereby lowering its vibration absorption efficiency. In this study, the small-amplitude control of the NES nonlinear vibration absorption system has been studied by using the open-loop-plus-nonlinear-closed-loop (OLPNCL) method. First, the dynamic equation of the NES nonlinear vibration-absorbing system was formulated, and its global behavior was analyzed. The regulations of global features and coexisting attractors were found out. Second, the migration between different attractors was carried out under the OLPNCL control, which can ensure that the system always works at the lowest line spectrum intensity and the best overall vibration absorption performance. Finally, the simulation results verify the feasibility and effectiveness of the OLPNCL method and achieve effective vibration absorption in the low-frequency range and line spectrum control under variable operating conditions.

Lyapunov指数是识别系统非线性动力学特征的重要标志，但是目前的算法通用性不足且计算流程复杂。本文在经典的Lyapunov指数算法的基础上，基于摄动理论提出了一种适用于不连续系统的Lyapunov指数计算方法。首先，以系统状态参数初始值和沿相空间每个基本矢量的扰动量为初始条件，确定相轨迹。其次，采取差商近似导数法，获得Jacobi矩阵的近似矩阵。然后，对Jacobi矩阵进行特征值提取，得到系统的Lyapunov指数谱。最后，将新算法应用到二自由度干摩擦冲击振荡器系统实例中，并将计算结果与同步方法的计算结果进行对比，对新算法的有效性进行验证。该算法不仅适用于离散系统和连续时间系统，而且能够快速计算复杂不连续系统的Lyapunov指数，为确定复杂不连续系统的动力学行为提供了新思路。

Nonlinear energy sink (NES) systems, when applied to a physical system with multiple interference sources, exhibit abundant nonlinear dynamic behaviors. However, current research in this respect is limited within the theoretical scope of deterministic systems. According to the theory of cell mapping, this paper introduces a parallel restructured algorithm to improve the performance of cell mapping and cell processing, and a parallelized multidegrees-of-freedom (DOF) cell mapping (PMDCM) method is given. With the method, the global behavior of NES systems is analyzed so that the dynamical behavior of multiple stable attractors within typical parameter intervals can be captured. The research results show that for NES systems, there is the phenomenon of multiple stable attractors coexisting in multiple typical parameter intervals, which occurs between periodic and periodic attractor, periodic and quasi-periodic, periodic and chaotic attractor. While revealing the corresponding relationship between different types of attractors and their basin of attraction, these findings verify that the new cell mapping method has high computational efficiency and accuracy and can provide a theoretical basis for the study of high-dimensional nonlinear systems' global behavior and optimal control.