## 5 Experimental Verification  
In the previous chapters, we have completed the derivation and finite element simulation verification of the formulas, in which the core theoretical results include the numerical calculation formula of the negative stiffness, the analytical solution of the compression load, and compression after compression buckling of the fixed beams at both ends, as well as the formula of the stress distribution on the surface of the negatively stiffened element. Due to the limitation of experimental conditions, we completed the verification of the negative stiffness formula. In addition, the analytical solution of the compression load and compression amount, as the most weighted item in the negative stiffness formula, can be indirectly verified by the experimental results to be accurate. The experimental model is shown in Figure 18. In order to facilitate the processing and assembly, the experimental model is slightly different from the model in the previous chapters, but there is no change in the key components, such as coil springs and leaf springs. The experimental model used four sizes of leaf springs, respectively, for the static experiment, the experimental variables include thickness \(h\), width \(b\), length \(l_{0}\), and compression \(u_{x}\). The purpose of the static experiment is to measure the value of the negative stiffness; the experimental process is performed through the static module to drive the periphery frame along the vertical displacement; the purpose of the use of digital calipers and force transducers is to measure displacement and driving force, respectively, which will be used to obtain the overall stiffness coefficient. The negative stiffness is calculated by subtracting the stiffness coefficient of the coil spring from the overall stiffness  
Figure 17: (**a**) Displacement PSD; (**b**) response 3\(\sigma\) displacement. coefficient, and the stiffness coefficient of the coil spring in this experimental model is about 16.5 N/mm. While conducting the experiments, we have also completed the finite element simulation of this model, which is shown in Figure 18c, and the simulation process is consistent with the previous sections. The simulation results are also the stiffness coefficients of the negative stiffness element, which are compared with the theoretical calculations (the \(U_{d}\) term is omitted from the theoretical calculations; as mentioned in the previous sections, omitting the \(U_{d}\) term can save a lot of calculations, and, at the same time, it will not have a great influence on the accuracy), the experiments, and the finite element simulations. As shown in Figure 19, the negative stiffness of the experimental results shows a more obvious change with displacement, which is mainly due to the fact that the static experiment used ordinary digital calipers for the measurement of displacement, and there is a certain deviation in their measurement accuracy. As a whole, most of the errors between experiments and theoretical calculations are less than 15%, while the errors between finite element simulation results and theoretical results are within 10%. Therefore, the calculation accuracy of Equation (30) for the negative stiffness can basically meet the actual needs of engineering, and the correction factor for the \(\mathrm{U}\mathrm{a}\) term is also more accurate. It can be seen, in the figure, that the negative stiffness of the FEM is always lower than that of the analytical; this is because the value of the correction factor of (30) is conservative, which makes the negative stiffness effect of the analytical smaller compared to the FEM in order to prevent the negative stiffness effect from being stronger than the positive stiffness in the practical application. The experimental results are higher than the analytical and FEM results; this is probably because there are errors in both the FEM and analytical results, but the amount of error is small and within the acceptable range.