# Design Methodology and Application Dynamics of Compact Quasi-Zero Stiffness Isolators
## 2 Fundamentals of QZS
### Dynamic Characteristics of QZS Isolation  
QZS isolators exhibit unique advantages for low-frequency vibration isolation; however, their strong nonlinearity may lead to oversensitivity and reduced stability. They may also produce complex behaviors such as chaos, oscillations, and bifurcations, which are highly time-varying and spatially complex phenomena. Therefore, studying the nonlinear dynamics of QZS isolation helps us to design, control, and optimize the isolation system more accurately. Scholars have done much research in this area. Figure 3: Flowchart for the design of QZS isolators. Figure 2: Realization principle of QZS isolator: (**a**) Schematic diagram of parallel connection of springs; (**b**) Schematic diagram of force and displacement principle. [PERSON] et al. [19; 20; 21; 22] carried out a deep theoretical analysis of the system structure and mechanical properties of a three-spring quasi-zero-stiffness vibration isolation device, investigated the optimal relationship between the geometry, inclination, and relative stiffness of the springs, deduced the expressions of the system's force transmittance and displacement transmittance, and provided an explanation for the phenomenon of the damping threshold that exists for the displacement transmittance. [PERSON] et al. [23; 24] have investigated the SD oscillator under nonlinear damping perturbations and explored the residual dimensional bifurcation, periodic solution, and Hopf bifurcation phenomena of the SD oscillator. [PERSON] et al. [25], based on the classical triple-spring structure, introduced the softened springs with pre-compression and found that the system may exhibit the doubled-periodic bifurcation under asymmetric excitation. In a subsequent study [26], the harmonic equilibrium method was used to solve the principal resonance response of the asymmetric Duff oscillator to quantize the nonlinear error and to solve for the frequency of bifurcation of the system as well as the possibility of developing chaotic phenomena. [PERSON] et al. [27] investigated the periodic motions of SD oscillators under simple harmonic excitation and revealed the bifurcation properties of the system, the symmetry breaking around the saddle point, and the doubly periodic bifurcation leading to the chaotic. [PERSON] et al. [28] studied a two-degree-of-freedom forced nonlinear system, simulated a QZS single-degree-of-freedom system excited by a linear oscillator, applied the averaging method to derive the amplitude-frequency equations for the relative motion of a small mass at the excitation frequency, and determined the stability conditions for both the steady-state solution and the unsteady-state solution. [PERSON] et al. [29] found that the nonlinear damping can reduce the critical value of the system for the first time entering the critical value of the chaotic state and expand the parameter space in which chaos occurs. [PERSON] et al. [30] analyzed the role of nonlinear stiffness and nonlinear damping on the dynamic behavior of the Duffing system based on the output frequency response function method, and the conclusions show that nonlinear damping is beneficial in suppressing the resonance peaks of the vibration isolation system and in improving the stability of the system. [PERSON] et al. [31] based on the harmonic equilibrium method and numerical extension method to analyze the displacement, force transfer rate, and system stability of the QZS isolator under large amplitude excitation and derived the corresponding analytical expressions, and it was found that nonlinear damping helps to remove or reduce the unwanted isolated subharmonic branches. It cannot be ignored that all numerical models usually have some degree of uncertainty [32; 33], especially in the analysis of nonlinear dynamical systems, where complex dynamic behaviors can further amplify model errors. In this context, machine learning [34; 35] provides a new path to solve this problem through data-driven multi-parameter optimization and accurate modeling capabilities.