## 4 Finite Element Analysis
### Accuracy of Formulas
The response amplitudes of the base frame and peripheral frame are solved using modal superposition. Finally, the transmissibility is obtained by dividing the response amplitude of the peripheral frame by that of the base frame. Figure 14 presents the comparison between partial finite element simulation results and theoretical values. Figure 14a,c shows the results of negative stiffness and transmissibility calculated for LSs of \(h=0.3\) and \(u_{x}=0.3\), with different lengths \(l_{0}\) and widths \(b\). Figure 14b,d shows the results of negative stiffness and transmissibility calculated for LSs of \(l_{0}=80\) and \(b=80\), with different thicknesses \(h\) and compression amounts \(u_{x}\). In theoretical calculations and finite element simulations of negative stiffness values, \(\delta_{1}\) is taken as 0.2 mm. Considering the finite element analysis and subsequent experiments, using a larger value of \(\delta_{1}\) gives more intuitive results, and the millimeter-scale is also convenient to measure during the experiments, so the millimeter-scale \(\delta_{1}\) is used in the subsequent hydrostatic part of this study. As mentioned in the previous study on Equation (29), the numerical magnitude of \(\delta_{1}\) has basically no effect on the accuracy of the calculation of the negative stiffness, and, thus, the negative stiffness calculated using the millimeter-scale \(\delta_{1}\) can still be applied to the DCM. Figure 14e displays the numerical calculation of force \(U_{a}\) for LSs with \(l_{0}=80\) and \(b=80\). There is a significant discrepancy between theoretical values and finite element results, which increases with increasing LS thickness. The numerical calculation of \(U_{a}\) in the finite element simulation uses Equation (29), where the sum of terms \(U_{c}\) and \(U_{d}\) equals the force transmission value due to contact between the LS and the slider in the simulation steps mentioned above. Given the values of \(k_{1}\), \(\delta_{1}\), \(l_{0}\), and \(u_{x}\), the \(U_{a}\) term is deduced. From the comparison of finite element simulation results, it is observed that the error in theoretical calculations of negative stiffness values increases with thickness. As mentioned earlier in the derivation of the \(U_{a}\) term, there will be some error in using the superposition method for large deflection behavior; Figure 14a,b shows that the negative stiffness values calculated after neglecting \(U_{a}\) are closer to the finite element simulation results; the reason for this may be that there is a stiffness softening effect along the normal deformation of the beam after compression buckling as well, and the computation by the superposition method does not take into account the stiffness softening very well. [PERSON] (2018), for other structures with negative stiffness elements, used the elliptic integration method to solve this problem; however, because of the limitation of the boundary conditions, the study in this paper is unable to establish the differential equations required for elliptic integration. Although the inclusion of \(U_{a}\) introduces errors, neglecting \(U_{a}\) in theoretical calculations results in a slightly weaker negative stiffness effect than in reality. Designing based on formulas that neglect \(U_{a}\) may result in an overall negative stiffness even when the LS is connected in parallel with positive stiffness springs. To avoid design errors, the influence of \(U_{a}\) must be considered during calculations. Figure 14: **(a,c)** LS with an \(h\) of 0.3 mm and \(u_{x}\) of 0.3 mm; calculating the negative stiffness and transmissibility with different \(l_{0}\) and \(b\); (**b,d**) LS with a \(b\) of 80 mm and \(l_{0}\) of 80 mm; calculating the negative stiffness and transmissibility with different \(h\) and \(u_{x}\); (**e**) LS with a \(b\) of 80 mm and \(l_{0}\) of 80 mm; separately calculating the force value of the \(U_{a}\) term with different \(h\) and \(u_{x}\).