## 1 Introduction  
Vibration isolation is an essential part of most engineering designs. Harmful vibration will affect the mechanical properties of materials and cause fatigue and wear, leading to the destruction of structures [1; 2]. The traditional linear vibration isolation system requires a lower natural frequency to achieve higher efficiency of vibration attenuation, which may degrade the isolation system's capability of supporting the payload [3; 4; 5]. Thus, there is a conflict between the wider effective frequency band and larger loading capacity. In nature, many animals have evolved simple limb trunk structures to isolate low-frequency vibrations, which can not only carry their large weight but still maintain good stability in high-speed motion. In recent years, the design of bio-inspired vibration isolators based on animal limb structures has been widely studied [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. [PERSON] and his coworkers presented a series of bio-inspired isolators composed of X-shaped structures and studied the dynamic characteristics of various vibration isolation systems [6; 7; 8; 9; 10; 11; 12; 13; 14]. For instance, inspired by the kangaroo's limbs, [PERSON] et al. [6] proposed a bio-inspired quadrilateral shape (BIQS) structure to suppress the vibration from the acquisition target to the satellite platform. They used the harmonic balance method to analyze the dynamic response of the vibration isolation system and studied the influence of BIQS' structural parameters on the vibration isolation performance. The results show that the vibration isolator has better performance than the ordinary spring-mass-damper vibration isolator. Considering leg posture adjustment, [PERSON] et al. [15] designed a bio-inspired polygon skeleton structure, in which the load capacity and working bandwidth can be adjusted by the initial parameters of the structure. The proposed structure can effectively suppress the vibration under various excitation conditions. According to the above studies, bio-inspired isolators composed of X-shaped structures present excellent vibration isolation performance. However, the studies of bio-inspiredisolators are currently limited to simple single-stage vibration isolation systems and the corresponding structural parameters' analysis. In power units (gas turbines, gear pumps, etc.) or precision instruments (optical interferometers, space telescopes, etc.) with complex vibration isolation systems [19; 20; 21; 22; 23; 24; 25], a single-stage vibration isolation system often leads to resonant peaks and standing wave effects at high frequencies [26; 27; 28; 29], which cannot meet the vibration isolation requirements. A dual-stage vibration isolation system draws great interest owing to its better high-frequency effect and higher stability [30; 31; 32; 33; 34; 35; 36]. In the dual-stage vibration isolation system, two layers of vibration isolators are installed between the control object and the base, and an intermediate mass is inserted between the two layers of vibration isolators. The inertia of the intermediate mass is used to offset part of the base excitation, to attenuate the vibration transmission rate from the base to the control object [30]. Linear isolators are usually used in the ordinary dual-stage vibration isolation system. Due to the large intermediate mass required in the dual-stage isolation system, the linear isolators will undergo excessive static deformation, resulting in instability. The bio-inspired isolators may solve this problem and further improve the effect of dual-stage vibration isolation. However, limited studies have been conducted on dual-stage vibration isolation systems using bio-inspired isolators. The influence mechanism of bio-inspired isolators on the dual-stage vibration isolation performance is still unclear. Moreover, in the real application environment of dual-stage vibration isolation systems, there are various complex excitation types, including translation and rotations.