# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 1 Introduction  
Vibrations are harmful in most engineering applications. Restricting the transmission of vibrations via the installation of vibration isolators is an effective method to reduce the harm induced by vibrations. Linear isolators generally have an extremely small stiffness for reducing the lowest effective isolation frequency and have relatively large damping for suppressing the resonance peak. Therefore, the loading capacity of the linear isolators is generally low and the displacement transmissibility in the high-frequency band is high [1]. The application of nonlinearity permits a wide range of novel design concepts in engineering; nonlinear design underpins the concept of quasi-zero stiffness (QZS) within the field of vibration isolation, which expands the effective isolation frequency range and ensures the isolator has a sufficient load capacity [2]. Although the QZS isolator was first proposed over decades ago, there is still considerable room for improvements in its design, and its potential for widespread use thus attracts considerable research attention. Smooth and discontinuous oscillators are widely used in the design of QZS elements and have become a basic component in the field [3]. [PERSON] et al. [4] introduced a permanent magnet spring in conjunction with a vertical linear spring and then formed an isolator. [PERSON] et al. [5] proposed a compact QZS isolator usingrepulsive magnet rings and a space-saving wave spring. [PERSON] et al. [6] designed a QZS isolator utilizing cam-roller-spring mechanisms. [PERSON] et al. [7] used buckled, parallel Euler beams and linear springs to construct a QZS vibration isolator. [PERSON] et al. [8] proposed a QZS isolator using a torsion bar spring and negative stiffness structure. [PERSON] et al. [9] proposed a QZS isolator composed of a cantilever beam and parallel magnets. [PERSON] et al. [10] designed a variable constant force configuration that can be used to provide the QZS characteristic in an isolator. [PERSON] et al. [11] studied an isolator system comprising an isolator with high-static low-dynamic stiffness (HSLDS) characteristics and a flexible plate with an arbitrary boundary. [PERSON] et al. [12] analyzed the force and displacement transmissibility of an HSLDS isolator. [PERSON] et al. [13] designed a nonlinear mechanical oscillator using the high-order QZS mechanism; the proposed oscillator has potential applications in vibration isolation as well as energy harvesting. [PERSON] et al. [14] studied the vibration isolation performance of parallel QZS isolators attached to fluid-conveying pipes. [PERSON] et al. [15] designed a negative stiffness system to isolate vibrations and then improve the comfort of vehicle seats. In addition, electromagnetic mechanisms have been widely used in the design of active vibration isolation devices [16]. The X-shaped structure is a novel component in the design of vibration isolators. [PERSON] et al. [17] designed a QZS isolator by connecting two X-shaped structures to a linear vibration isolation platform. [PERSON] et al. [18] studied the vibration isolation of a platform using an \(n\)-layer, scissor-like truss structure. [PERSON] et al. [19] proposed a novel X-shape QZS isolator, in which oblique springs and a cam-roller-spring mechanism were used to enhance its isolation performance. [PERSON] et al. [20] proposed an \(n\)-layer, vertically asymmetric, X-shaped structure for passive vibration isolation. Bioinspired designs provide an effective route to the realization of QZS mechanisms, and X-shaped structures belong to this category of structures [21]. Metamaterials such as origami and auxetic structures provide a large degree of tailorability and can be used to create structures or materials with extraordinary mechanical properties; such designs can be introduced to the field of vibration isolation [22; 23; 24; 25]. Cellular structures with negative stiffness components and origami-based structures with bi- and multi-stable states can be used to create QZS isolators [26; 27; 28].