## 4 Current Progress of Magnetic Negative Stiffness in QZS Isolators
### QZS-Based Permanent Magnet Negative Stiffness Elements
Initially, the nonlinear behavior of a mass-spring model comprising series-connected positive and negative springs is examined, exhibiting free vibration and frequency responses consistent with theoretical analyses of Duffing's equation using cubic approximation. Additionally, the paper explores a model simulating a high-static, low-dynamic-stiffness isolator in a nonlinear state. This endeavor furnishes a flexible and utilitarian proof-of-concept framework for related systems, including vibration isolators, energy sinks, and other systems exhibiting nonlinearities. The experimental setup and parameter identification, theoretical and experimental examinations of the coupled vibrator models, and validation of a mass-spring model simulating a nonlinear HSLDS isolator yield fresh insights into comprehending and harnessing negative stiffness mechanisms. The facilities outlined in this study are instrumental in validating concepts related to systems  
Figure 24: Model of the MRA: (**a**) semi-sectional view; (**b**) front view; (**c**) outer magnetic ring; and (**d**) stiffness at different geometric parameters. like vibrational isolators, energy flow devices, and energy deposition devices incorporating negative stiffness elements. Wedge-shaped permanent magnets find extensive applications in various QZS vibration isolators owing to their versatile dimensions. Recently, [PERSON] [72] introduced a novel magnetic negative stiffness vibration isolator grounded in [PERSON]'s positive stresses to enhance the low-frequency vibration isolation efficacy of high-static-support stiffness systems. In Figure 26, this vibration isolator can furnish substantial negative stiffness while preserving a compact and lightweight form factor. In this study, the negative stiffness attributes of the vibration isolator are initially delineated via magnetic circuit analysis, followed by experimental validation of its passive vibration isolation capabilities. Subsequently, active control is integrated into the vibration isolation system to enhance its capacity for mitigating external disturbances and dampening system resonant response. The paper formulates the control equations for the hybrid active-passive vibration isolation system and demonstrates through simulation and experimental findings that a magnetic negative stiffness vibration isolator with hybrid active-passive control can notably lower  
Figure 25: System with a NNMS: (**a**) its built-up, NNMS containing 4r magnets; 2 magnets connected to mass 2 and the other 2 magnets are attached to fixed end; (**b**) its mass, spring and damper idealization; (**c**) photograph of the experimental setup with free vibration; (**d**) photograph of the experimental setup with dynamic vibration (without the sensors); (**e**) magnetic force of magnetic springs; and (**f**) stiffness of magnetic springs. the initial isolation frequency and effectively dampen the system resonant response relative to the passive-only approach. This work offers a promising avenue for achieving low-frequency broadband vibration isolation in systems characterized by high-static-support stiffness. The Designed Magnetic Negative Stiffness Isolator (MNSI), based on Maxwell's positive stress, markedly mitigates the system resonant response and expands the isolation bandwidth, rendering it particularly advantageous for low-frequency vibration isolation in systems featuring high-static-support stiffness. A direct method to enhance the isolation bandwidth entails augmenting the negative magnetic stiffness of the vibration isolator, which is achievable by augmenting the thickness of the permanent magnets and the pole regions of the moving components and/or diminishing the air gap thickness. To realize low-frequency vibration isolation, these vibration isolators necessitate design optimization for high negative magnetic stiffness, albeit at the expense of increased size and weight, constraining their applicability in aerospace and submarine settings where spatial and weight constraints are paramount in vibration isolation system design.