# Study on Dynamic Characteristics of the Bistable Nonlinear Damper
## 1 Introduction  
A traditional passive vibration attenuation system mainly utilizes certain materials or damping elements, such as a rubber shock absorber, metal spring or air spring, for counteracting the vibration force; however, the natural frequency can hardly be lowered, on account of the contradiction between low rigidity and the static bearing capacity [1; 2; 3]. Recently, for low-frequency vibration control, scholars mainly adopt an air spring, variable-stiffness spring or multi-level vibration reduction system [4; 5; 6]. Despite effective reduction of the natural frequency of the vibration reduction system, these vibration reduction measures can hardly be extensively applied in actual projects because of the complex structure, the high cost, the great energy consumption and the addition of a control system [7]. In order to address the issue of a traditional vibration absorber not being applicable to a low-frequency environment, some scholars have introduced nonlinear elastic elements into vibration absorbing systems and designed a kind of nonlinear absorber. By setting reasonable parameters, the dynamic characteristics of the nonlinear vibration absorber are subjected to static load, so it can keep low stiffness while maintaining large bearing capacity, thereby achieving vibration absorption at low or ultra-low frequencies. [PERSON] et al. designed a nonlinear vibration absorber, which can achieve the preset dynamic stiffness, under a given static loading condition, while the constraint of deformation is based on the geometric nonlinearity of large-deformation elastic elements [8]. [PERSON] focused on thin cylindrical shell structures which can show interesting bistable behavior [9]. [PERSON] et al. investigated the coupled modeling method and free vibration characteristics of a graphene nanoplatelet (GPL)-reinforced blade-disk rotor system in which the blade has a pre-twist angle and a setting angle [10; 11]. Moreover, they presented an investigation on the coupled modeling and vibration behaviors of a spinning assembled cylindrical shell-plate structure. The method of hypothesis modes was adopted to obtain the free vibration results of theassembled cylindrical shell-plate structure [12]. Using [PERSON]'s and [PERSON]'s approaches, [PERSON] developed an interpretation of the bistable shapes, in terms of principal and anticlastic curvatures [13]. [PERSON] and [PERSON] employed a nonlinear absorber for energy collection and examined the vibration-absorbing and energy-collecting performance of the absorber, under instantaneous response conditions via computer simulation, and they experimentally validated the simulation results [14; 15]. By improving the single-pendulum nonlinear vibration damping system, scholars successively put forward an inverted-pendulum, transverse-pendulum, X-pendulum and conical-pendulum ultra-low-frequency horizontal vibration absorption system that can achieve a resonance period, from several seconds to dozens of seconds [16; 17; 18; 19; 20; 21]. Further, [PERSON] et al. used a symmetrical-torsion-bar spring, multi-level vibration-isolation absorber, a yield cylinder (Euler spring) and a negative-rigidness spring to realize passive absorption of low-frequency vibration, in a vertical direction [22; 23; 24]. [PERSON] et al. investigated the free vibration of a rotating functionally graded (FG), pre-twisted blade-shaft assembly reinforced with graphene nanoplatelets (GPLs) based on the coupled model proposed in this paper [25]. Moreover, they developed a super-parametric shell element to establish the rotating cylindrical shell model by employing the finite element method. Considering parameter uncertainties, the dynamic response of the rotating cylindrical shell is carried out [26]. [PERSON] et al. presented a method to easily and rapidly design bistable buckled beams subjected to a transverse point force [27].