## 5 Conclusions  
This paper discusses how a dynamic vibration absorber can achieve lower frequencies by incorporating a negative stiffness component. This paper presents a theoretical analysis, FEM simulations, and experimental measurements. Axial loading is introduced on the springs to evaluate its influence on the negative stiffness effect. The efficiency of the negative stiffness resonator on vibration isolation is evaluated for different conditions of axial force. The novelty of this work is the derivation of a moderately complex beam bending analytical model instead of a spring-mass system. This is relevant because solving the beam problem encompasses more realistic phenomena compared to introducing the equivalent stiffness in the spring-mass equation of motion. The investigation is performed in three steps.  
This paper first recalls theoretical formulations describing the relation between restoring force and initial deflection, and shows how the standard linear force-displacement is altered with respect to the geometrical parameters of the spring-like components. This is supplemented by qualitative analysis and design rules based on static bending considerations.  
Along with the static aspect, the dynamic aspect is also explored. A theoretical model describing the flexural behavior of a beam supplemented by point mass, positive  
Figure 12: Structural inertance (ref: 1 m/s\({}^{2}\)/N) from experimental setup \(\delta=1.5\) mm (\(-\)), \(\delta=3.5\) mm (\(-\)), and \(\delta=5\) mm (\(-\)) and frequency shift evolving respectively to 30 Hz, 26 Hz, 20 Hz.  
Figure 13: **(a)** Structural inertance (ref: 1 m/s\({}^{2}\)/N) from analytical bending model with frequency shift from 30 Hz, 24 Hz, 16 Hz for \(\delta=1.5\) mm (\(-\)), \(\delta=3.5\) mm (\(-\)), and \(\delta=5\) mm (\(-\)), **(b)** analytical spring-mass model 5 with frequency shift from 29 Hz, 23 Hz, 15 Hz.  
stiffness, and axial force associated with negative stiffness components is explained. In this way, the key parameters are correlated to the negative stiffness mechanism as well as the overall bending kinematics governing the motion of the resonator. Finite element simulations confirmed static assumptions, and dynamic analysis confirmed the relevancy of the proposed model.  
Finally, according to the qualitative analysis suggested and the design procedure presented, a variable passive negative stiffness resonator is manufactured and tested. Although setting the overall stiffness to zero and overcoming the positive stiffness at equilibrium point is challenging due to the uncertainties when manufacturing the frame and junctions between the components, the decrease in frequency as axial force increases is evidenced experimentally. Comparisons with the analytical model demonstrate that the proposed bending model captures the behavior in terms of frequency and amplitude.  
Further improvements could focus on a more accurate bending behavior of the two pairs of curved beams, which is the key component responsible for the bending kinematic of the resonator. Qualitative analysis may be pushed forward to determine whether the axial or bending stiffness governs the performance of the structure. This would be related to the nonlinear response of the structure and would require the evaluation of a force-displacement relation in the scope of large displacements.  
Conceptualization, P.F.; methodology, P.F.; software, M.K.; validation, P.F. and M.K.; formal analysis, P.F.; writing--original draft preparation, P.F.; writing--review and editing, P.F. and M.I.; supervision, M.I. and O.B.; project administration, M.I. and O.B.; funding acquisition, M.I. and O.B. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the LabEx CeLyA (Centre Lyonnais d'Acoustique, ANR-10-LABX-0060) of Universite de Lyon, within the program \"Investissements d'Avenir\" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).  
Not applicable.  
The authors declare no conflict of interest.