# An Air Spring Vibration Isolator Based on a Negative-Stiffness Structure for Vehicle Seat
## 1 Introduction
The vibration isolation efficiency of the QZS system was confirmed in [22], which indicated that the response of the QZS system under harmonic force was nearly almost two times lower than that of the classical linear vibration isolator. A nonlinear inertance mechanism (NIM) was designed using a pair of oblique inerters with one common hinged terminal and the other terminals fixed [23]. In addition, [PERSON] et al. [24] recommended a new design of the QZS system that used a cam-roller-spring mechanism. Based on this mechanism, [PERSON] et al. [25] further developed a QZS system with multiple frequency regions that could respond to a variety of isolated masses or applied forces. In addition, QZS vibration isolators based on linear magnetic springs [26] also attracted a attention. However, one disadvantage of the QZS vibration isolator is its limited displacement range when the stiffness is reduced to zero. In other studies, some researchers introduced a nonlinear vibration isolator that used a scissor-like structure (SLS) to enhance the vibration isolation effectiveness. The SLS vibration isolator described by [PERSON] et al. was composed of connecting horizontal springs, rods of the same length, and joints [27]. The results demonstrated that vibration isolators with an SLS can achieve good nonlinear vibration isolation performance while simultaneously overcoming the disadvantages of an existing QZS system. Inspired by the SLS presented in [27], an innovative QZS vibration isolator was constructed, adopting a single-layer SLS and parabolic cam-roller structures [28]. The proposed vibration system showed a small improvement in offset when compared to the system with only an SLS, but significantly reduced the magnitude of the vibration. In addition, a vibration isolator with SLS was developed by [PERSON] et al. by combining an n-layer scissor-like mechanism with electromagnetic generators [29]. The results showed that the structural parameters of the suggested system may be adjusted to obtain better energy-harvesting effectiveness. According to the above-mentioned research, the SLS can provide nonlinear stiffness, which is also effective for isolating low-frequency vibration. On the other hand, air springs are well known for their ability to easily provide variable force and load capacities by simply adjusting the gas pressure within the springs [30]. To further enhance the isolation performance of the vehicle seat, an air spring with negative stiffness was adopted as a vibration isolator of the driver's seat by [PERSON] et al. [31]. [PERSON] et al. [12] constructed a pneumatic vibration isolator employing an NSS to increase the isolation performance in low-excitation frequency ranges. Moreover, [PERSON] et al. [18] presented an NSS vibration isolator based on a set of two pneumatic actuators originally in a horizontal position and perpendicular to a reversible sleeve air spring. According to the analytical and experimental results, the resonance frequency of this structure was reduced by up to 58%, and the transmissibility modulus was increased by up to 78%. Later, when this structure was used as a vibration isolator in vehicle seats, comfort improvements ranging from 10% to 35% were observed [32]. Hence, the pneumatic vibration isolation system based on NSS achieved vibration isolation with different masses of the isolated objects. The important contribution of the present paper is the replacement of the traditional vibration isolator without NSS in a vehicle seat with a new vibration isolator with NSS, which consists of two air bellows combined with a crisscross mechanism to overcome the disadvantages of the classical isolator. Furthermore, a sleeve air spring was adopted to replace a positive-stiffness spring, which provided a stable support force when the load changed. Next, a mathematical model and the relationship between the nonlinear stiffness and the displacement of the ASVIS were derived.