# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 1 Introduction  
With the rapid advancement of science and technology, the proliferation of high-speed and high-power equipment has significantly intensified mechanical vibrations and impacted forces in various engineering applications. These unwanted vibrations can severely affect the operational stability, precision, and structural integrity of critical systems. To mitigate these adverse effects, vibration isolation technologies play a crucial role by decoupling mechanical equipment from its supporting structures, thereby reducing the transmission of external disturbances. As a result, vibration isolators have been extensively implemented in precision instruments, navigation systems, electronic devices, and high-tech platforms such as satellites, missiles, ships, aircraft, and ground vehicles. In modern industrial environments, mechanical systems frequently experience complex three-directional dynamic loads, which necessitate the development of highly adaptable vibration isolation systems [1; 2; 3]. As a result, the demand for triaxial vibration isolators has increased substantially, drawing significant research interest toward their development and optimization. Three-directional vibration isolators are particularly important in applications involving aerospace, naval vessels, automotive engineering, and industrial machinery, where external excitations often induce vibrations in multiple spatial directions. With the growing demand for vibration control, researchers have devoted significant efforts to developing advanced three-directional vibration isolation systems leading to their application in sensitive instruments [4], floating raft vibration isolation systems [5], power unit vibration isolation systems [6], and optoelectronic devices [7]. [PERSON] et al. [8] focused on the dynamic characteristics of a three-dimensional vibration isolation system, performing theoretical analysis and experimental verification to evaluate the impact of various design parameters on vibration isolation performance. [PERSON] [9] proposed a novel three-dimensional vibration isolation bearing (3 DSL), which is composed of spring bearings (SBs) and laminated natural rubber bearings (LNR) connected in vertical series. Through design principle analysis and experimental testing of specimens, the vertical stiffness and vibration filtering performance of the 3 DSL were systematically investigated, and its effectiveness in subway vibration isolation was evaluated. The results demonstrate that the 3 DSL exhibits high vertical load-bearing capacity and low vertical stiffness, along with significant vibration filtering capabilities. Specifically, the vibration amplitude is slightly amplified in the low-frequency range (1-50 Hz) but significantly attenuated in the high-frequency range (50-200 Hz). This study offers a promising solution for vibration isolation applications in environments such as subways. [PERSON] et al. [10] examined the nonlinear dynamic behavior of a three-degree-of-freedom vibration isolation system and analyzed how different structural parameters influence its dynamic response. Additional studies have investigated seismic and industrial applications of three-dimensional vibration isolation systems. [PERSON] et al. [11] analyzed the application of a three-dimensional base seismic isolation system in high-rise buildings, focusing on the design and theoretical modeling of three-dimensional combined isolation bearings (3D-CIBs). [PERSON] et al. [12] studied the geometric nonlinear tuned inerter damper (GNTID) configuration and demonstrated that its incorporation into base-isolated structures significantly reduces the peak seismic response while enabling multi-directional vibration isolation. [PERSON] et al. [13] designed a horizontal vibration isolation system with adjustable quasi-zero stiffness along three axes, including horizontal translation stiffness in the x- and y-axes and vertical stiffness in the z-axis. Their results demonstrated that the system achieves a significantly large isolation bandwidth at low frequencies.