## 1 Introduction  
A tuned mass damper (TMD) is a mechanical device designed to reduce the effects of vibrations on a structure. It consists of a mass, a spring, and a damper, which are tuned to a specific frequency close to the frequency of the primary system. It operates on the concept of anti-resonance in such a way that, when the primary system vibrates at its resonance frequency, the damper oscillates in the opposite direction, thereby countering the oscillations and decreasing its amplitude. When properly calibrated, these devices are effective even with a small additional mass (10% of the primary system). This concept was formalized in terms of a two degree of freedom system, together with associated design rules and optimized tuning parameters by [1]. For a more recent state of the art, one may refer to [2], which compiles chronological progress on the design and modeling and challenges in the field of passive vibration control. However, one limitation of the TMD is that stiffness should be reduced, or mass should be increased, as a lower frequency range of performance is expected. This would result in a large static displacement that would be unsuitable for some applications. To overcome this limitation, negative stiffness can be introduced to reach low-frequency ranges of efficiency. The negative stiffness system itself does not provide a stable operating regime and should be supplemented by a positive spring, which is expected to compensate for the negative stiffness and become stable. The overall stiffness will then be quasi-zero but still positive. The combination of these two stiffnesses of opposite signs gives the resonator great flexibility over the frequency range to be isolated, especially the low frequencies. The concept of negative stiffness was first proposed by [3], who used three springs to achieve the negative stiffness. The negative stiffness assembly is usually presented as two horizontal springs in series, whose connecting point is also attached to a vertical spring. In this way, the two horizontal springs provide the negative stiffness, while the vertical spring provides the positive stiffness. This assembly is already documented to identify its high static and low dynamic features, as well as the conditions under which the negative stiffness compensates for or overcome the positive one. A summary of various prototypes with quasi-zero stiffness characteristics is provided in [4] and numerous ways to produce quasi-zero stiffness are documented in [5]. Negative stiffness springs take the form of elastic rubber rods, slender beams, or bar systems, or even magnets repulsing each other. It is specifically introduced in the form of curved beams by [6] and it was shown that such nonlinear springs provide low-frequency performance while they support the static load of the equipment. Prestressed negative stiffness springs were investigated in [7], as an industrial vibration isolator for sensitive equipment such as microscopes. The substitution of the linear spring by a buckled beam to form Euler springs was described by [8] to achieve a vibration isolator with high-static low-dynamic stiffness properties. The buckled beam setup takes advantage of the snap-through behavior of beams to generate the negative stiffness mechanism. The aspect of symmetric buckling of Euler beams is reported in [9], which also constrains the connecting point to realize negative stiffness in the tangential direction. Very interesting is the derivation of static criteria describing the performance of compressed beams from a fine analysis of displacement and stresses. This mechanism was recently addressed by [10] from an analytical point of view. The specific nonlinear behaviors relying on the buckling instability motivated numerous fundamentals studies, as [11] followed by [12; 13].