# An Air Spring Vibration Isolator Based on a Negative-Stiffness Structure for Vehicle Seat
## 1 Introduction  
Usually, whole-body vibration is associated with vehicle movement, which can affect the comfort, performance, and health of the driver [1; 2]. The human body responds strongly to vertical whole-body vibration at frequencies ranging from 0.5 to 5 Hz, which are below 2 Hz in horizontal axes [3; 4]. In order to increase the driver's comfort, safety, and health by reducing weariness caused by extended hours of driving or exposure to adverse road conditions, a low-frequency vibration isolator is developed by combining a positive stiffness element with a negative stiffness element. The traditional linear isolator model, which is frequently used to demonstrate vibration isolation, consists of a linear stiffness spring combined with a damper (spring-mass-damper system) [5; 6; 7]. Only when the excitation frequency exceeds \(\sqrt{2}\) times the natural frequency of the system can these structures achieve efficient vibration isolation. On the other hand, when the excitation frequency is less than \(\sqrt{2}\) times the natural frequency, the vibration of the driver's seat is increased in comparison with that of the base excitation [8]. This proves that the traditional vibration isolation systems can have a satisfying performance only at high-frequency excitation but not at low-frequency excitation. To solve this problem, vibration isolation systems with NSS have been recommended to overcome the disadvantages of classical linear isolation, which can effectively isolate vibrations in the low-frequency region while maintaining load capacity and low dynamic stiffness. In an early study by [PERSON] et al. [9], the NSS vibration isolator comprised two horizontal springs connected to the load via an actuator joint to produce negative stiffness. [PERSON] et al. [10] suggested a negative stiffness mechanism based on two rigid barshinged in the middle, supported on pivots at their outer ends and loaded in compression by opposing forces. [PERSON] et al. [11] continued to study this mechanism and developed a vibration isolator with NSS that employed two horizontal springs connected to the load via sliders and inclined links. The mechanical characteristics and experimental prototype of the proposed system were also analyzed. Later, these researchers applied an adaptive backstepping controller to enhance the isolation effectiveness of the vibration isolator system with NSS [12]. A nonlinear vibration isolation system with an NSS was investigated in terms of dynamics and power flow by [PERSON] et al. [13]. [PERSON] et al. [14] suggested to use a novel spring bar mechanism to improve the vibration isolation performance of the NSS. Moreover, other researchers designed vibration isolation systems that used a cam-roller mechanism combined with negative-stiffness elements [15]. [PERSON] et al. [16] proposed a negative stiffness vibration isolator based on a cam-roller-spring mechanism with a custom-designed cam profile. Other designs that produced negative stiffness characteristics include a magnet spring structure by [PERSON] et al. [17] and a pneumatic negative-stiffness mechanism introduced by [PERSON] et al. [18]. Since the negative stiffness mechanism has very little load capacity, it can only operate well when connected in parallel with a positive-stiffness element. By combining a negative-stiffness element with a positive-stiffness element in parallel, a quasi-zero-stiffness (QZS) vibration isolation system achieved both high static stiffness and low dynamic stiffness around the equilibrium position. [PERSON] et al. [19] proposed a QZS vibration isolator comprising a vertical spring operating in parallel with two oblique springs. [PERSON] et al. [20] presented the nonlinear-stiffness theoretical analysis of a QZS system with damping at the dynamic equilibrium position. The results of numerical simulations of differential equations for the QZS vibration isolator under different excitations, such as harmonic excitation, random excitation, and impulse excitation, were presented by [PERSON] et al. [21].