## 1 Introduction
In addressing this inherent contradiction, the inerter, characterized by a two-terminal structure, emerges as a valuable solution, and it exerts a matching force on both ends that is directly proportional to the relative acceleration between these ends; this ratio is quantified by a parameter known as inertance, measured in kilograms [21]. The inerter has an inertial structure and has mass magnification properties; it provides an inertance far exceeding its mass without increasing the additional weight, which is in accordance with the vehicle design criteria. Based on the above mechanical properties, inerters have been utilized in vehicle suspension [22; 23], train suspension [24; 25], vibration isolation systems [26; 27; 28; 29], and DVAs [30; 31] because of their mechanical properties. Some scholars have combined NESs and inerters to construct a new type of NES, the nonlinear energy sink inerter (NESI), and carried out some studies. [PERSON] et al. [32] devised a parallel configuration for the NESI; its cubic spring, damper, and inerter are interconnected in parallel, and this design choice has been demonstrated to be more effective in attenuating vibrations within the primary system compared with traditional NES configurations. [PERSON] et al. [33] constructed a NESI with a damper and a nonlinear cubic spring that are first interconnected in parallel and then in series with the inerter, which decreases vibration amplitude more effectively. [PERSON] et al. [34] designed an asymmetric NESI to build structural vibration mitigation under harmonic and seismic excitations that has better vibration suppression effects and strong robustness for excitation changes. [PERSON] et al. [35] proposed a two-stage NESI for further reduced vibration transmissibility with tunable properties. In vehicle engineering, [PERSON] et al. [36] applied a NESI and quasi-zero stiffness to a quarter-vehicle-suspension model, which improved vehicle vibration-damping characteristics. Furthermore, [PERSON] et al. [37] assessed the dynamic behavior of a half-vehicle system incorporating a NESI when exposed to both pavement shock and harmonic excitations. In general, a NESI uses more inertia with an inerter with less mass and reduces the mass of the NES, which further reduces the main system's displacement and meets the lightweight design criteria. In this paper, a NESI is strategically employed in both the bodywork and the vehicle seat components, which are positioned between the bodywork and two tires and between the vehicle seat and the bodywork, respectively. A half-vehicle seat system with five degrees of freedom (DOFs) is adopted for analysis; it considers the vertical motion of the vehicle seat, the bodywork, front and rear tires, and the pitching motion of the bodywork. For coupling analysis of vertical and pitching motion, [PERSON] and [PERSON] [38] designed a double link sensor that is sensitive to translation and negligibly sensitive to tilt; therefore, it can mitigate the problem of tilt horizontal coupling when exposed to translation or tilt. [PERSON] and [PERSON] [39] studied the dynamics of a multiple-pendulum system under a translating and tilting pivot, which was applied to the inertial sensing area for high-precision instrumentation. Here, a NESI is adopted between the vehicle seat and the bodywork, which reduces seat vibration in the vertical direction. NESIs are adopted between the bodywork and two tires, which decreases vehicle vibration both in the pitching and vertical directions. Given the introduction of the nonlinear stiffness of the NESI, the half-vehicle seat system integrated with the NESI is a complicated system. The dynamic response of a system subjected to pavement harmonic and random excitations can be obtained using the pseudo-arc length method (PALM) and the harmonic balance method (HBM) [40] and the numerical method, respectively.