# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion
### Dynamic Response Analysis of the Three-Directional Vibration Isolator  
#### 3.3.1 Dynamic Modeling and Governing Equations  
Assuming linear viscous damping within the system, characterized by a damping coefficient \(c\), the energy dissipated by this damping force over one vibration cycle is set equal to that dissipated by an equivalent nonlinear damping force [20]. When subjected to a harmonic excitation force \(F_{n}=F_{0}\cos(\omega t)\), the equations of motion governing the three-directional vibration isolator under vertical and horizontal harmonic excitation forces [21] can be expressed as follows:  
\[\left\{ \begin{array}{l}mL\frac{d^{2}\widetilde{z}}{dt^{2}}+cl\frac{d \widetilde{z}}{dt}+F_{z}-mg=F_{0}\cos(\omega t)\\ mL\frac{d^{2}\widetilde{z}}{dt^{2}}+cl\frac{d\widetilde{z}}{dt}+F_{x}=F_{0} \cos(\omega t)\end{array} \right. \tag{15}\]  
Applying nondimensionalization to Equation (15) yields:  
\[\left\{ \begin{array}{l}\frac{d^{2}\widetilde{z}}{dt^{2}}+2\xi\frac{d \widetilde{z}}{dt}+f_{z}=f_{0}\cos(\Omega\tau)\\ \frac{d^{2}\widetilde{z}}{dt^{2}}+2\xi\frac{d\widetilde{z}}{dt}+f_{x}=f_{0} \cos(\Omega\tau)\end{array} \right. \tag{16}\]  
where \(\tau=\omega_{n}t\), \(\omega_{n}=\sqrt{K_{n}/m}\) represents equivalent undamped natural frequency, \(\Omega=\omega/\omega_{n}\) denotes the excitation frequency ratio, and \(\xi=c/2m\omega_{n}\) is the viscous damping factor, additionally, \(f_{0}=F_{0}/K_{z}\cdot L\) refers to the dimensionless excitation force amplitude, \(f_{z}=F_{z}-mg/K_{z}\cdot L\) indicates the dimensionless elastic force vertical of the vibration isolator under applied mass \(m\), and \(f_{x}=F_{x}/K_{z}\cdot L\) is the dimensionless elastic force horizontal of the vibration isolator under applied mass \(m\). It is noted that \(F_{z}=mg\), \(F_{x}=0\) at the static equilibrium position. As discussed in the previous section, the vertical and horizontal forces follow nonlinear variations with respect to displacement and initial inclination angle. Assuming small system displacements (\(z\) and \(x\)) and setting \(\delta=0.73\), \(\theta_{0}=37.6^{\circ}\), \(\lambda=0.005\), Taylor series expansions of \(f_{z}\) and \(f_{x}\) around the static equilibrium point (\(\widetilde{z}=0\), \(\widetilde{x}=0\)) are employed for approximate analysis. A comparison between the exact and approximate expressions of \(f_{z}\) and \(f_{x}\) is illustrated in Figure 7. It can be observed that the error between the exact and approximate expressions remains negligible when the system displacements \(z\) and \(x\) are relatively small. This confirms that the Taylor series expansion serves as a valid approximation technique for analyzing the system's dynamic response. Obviously, employing higher-order Taylor series expansions can better approximate the exact expressions. However, as the system displacement increases, the error between the exact and approximate expressions gradually becomes more pronounced, indicating the limitations of lower-order approximations. Considering small system displacements, \(f_{z}\) and \(f_{x}\) are approximated using a fifth-order Taylor series expansion. Consequently, Equation (16) can be expressed as follows:  
\[\left\{ \begin{array}{l}\frac{d^{2}\widetilde{z}}{dt^{2}}+2\xi\frac{d \widetilde{z}}{dt}+\alpha_{1}\widetilde{z}+\alpha_{2}\widetilde{z}^{2}+ \alpha_{3}\widetilde{z}^{3}+\alpha_{4}\widetilde{z}^{4}+\alpha_{5}\widetilde{ z}^{5}=f_{0}\cos(\Omega\tau)\\ \frac{d^{2}\widetilde{z}}{dt^{2}}+2\xi\frac{d\widetilde{z}}{dt}+\beta_{1} \widetilde{x}+\beta_{2}\widetilde{x}^{2}+\beta_{3}\widetilde{x}^{3}+\beta_ {4}\widetilde{x}^{4}+\beta_{5}\widetilde{x}^{5}=f_{0}\cos(\Omega\tau)\end{array} \right. \tag{17}\]The harmonic balance method [22] is employed to solve Equation (17). Assuming a solution of the form \(A=a\cos(\Omega r+\phi)\), and neglecting higher-order harmonics, the amplitude-frequency and phase-frequency relationships for the three-directional vibration isolator in both the vertical and horizontal directions can be derived as follows:  
\[\left\{