# Design Methodology and Application Dynamics of Compact Quasi-Zero Stiffness Isolators
## 1 Introduction  
With accelerating industrialization, vibration-related challenges have garnered significant attention across engineering applications. In the fields of precision instruments, aerospace, automotive, construction, etc., the impact of vibration is particularly significant. Vibration compromises the functional stability of precision instruments [1, 2, 3, 4], induces instability and deformation in ship mechanical structures [5], degrades vehicle stability and comfort [6, 7], impairs aerospace equipment fatigue life and attitude control accuracy [8, 9], and causes catastrophic destruction of buildings [10, 11, 12]. Therefore, effective isolation of vibration has become an important research direction in engineering technology. Vibration isolation is mainly divided into passive vibration isolation, active vibration isolation, and hybrid vibration isolation. Passive vibration isolation has a simple structure and high reliability, but there are problems such as a narrow frequency band of vibration isolation and poor vibration isolation effect under large amplitude. Active vibration isolation measures the state signals of the vibration isolation system through sensors, feeds the signals back to the control system, and then the actuator controls the vibration of thevibration isolation object, which realizes the results of widening the vibration isolation frequency band and improving the vibration isolation efficiency [13], but the high-precision sensors and actuators lead to the system's complexity and high cost. In order to combine the advantages of passive and active vibration isolation techniques, hybrid systems have emerged. These systems combine passive damping materials, mechanical elements, and active control mechanisms to achieve better vibration isolation. Hybrid systems are able to provide greater adaptability and versatility, especially when dealing with complex vibration environments [14; 15]. However, in the low-frequency range, especially when the vibration is close to zero frequency, i.e., static offset, it is often difficult for the above vibration isolators to effectively reduce the transmission of low-frequency vibration. Based on the classical linear vibration isolation theory, in order to obtain effective vibration isolation, it is necessary to ensure that the ratio of the excitation frequency to the intrinsic frequency of the system is greater than \(\sqrt{2}\)[16], as shown in Figure 1. In order to achieve effective suppression of low-frequency vibration, it is necessary to reduce the intrinsic frequency of the system. In the engineering application of the load mass, it is certain that we need to reduce the stiffness of the system to reduce the intrinsic frequency of the system, and the reduction of stiffness will lead to system stability deterioration. This contradiction has been restricting the practical application of low-frequency vibration isolation technology. Researchers have proposed a vibration isolator with high static stiffness and low dynamic stiffness, which can exhibit very low stiffness in the low-frequency range while providing excellent load stability through the parallel connection of positive and negative stiffness. This type of isolation is called quasi-zero stiffness (QZS). QZS isolation has both high static stiffness and low dynamic stiffness characteristics. High static stiffness means that the vibration isolator can provide high stiffness under the self-weight of the mechanical equipment and vibration isolation source. This means that under normal working conditions, the deformation of the vibration isolator is small, and it can effectively support heavy objects and maintain stability; low dynamic stiffness means that under the action of vibration impact, external excitation, etc., the vibration isolation device can provide low stiffness, and it can effectively absorb and reduce vibration transmitted through the equipment.