# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 1 Introduction
developed an adjustable permanent-magnet bistable nonlinear energy sink to reduce vibration of a cantilever beam, and its broadband vibration suppression ability in the steady state frequency domain was validated [33]. [PERSON] et al. studied the targeted energy transfer of a cantilever beam attached with a bistable nonlinear energy sink, and the transient responses were verified with displacement responses, velocity responses, and phase portraits [34]. [PERSON] et al. verified the effectiveness of the ECD-nonlinear energy sink and the target energy transfer of a cantilever beam-ECD-nonlinear energy sink system with transient responses and wavelet spectra, respectively [14]. [PERSON] et al. validated the absorption performance with the transient response of the system, including displacement and voltage responses [13]. The numerical supports may be obtained by direct numerical approaches, such as finite element methods, finite difference methods, and quadrature methods. However, the available works [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34] did not examine the findings via direct numerical approaches. Generally, weights are balanced by the force of static deflection in linear vibrations. Hence, weights are usually neglected in the analysis of transverse beam vibrations in the vertical direction [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30], although it may be understood that static deflections shift steady-state responses. However, there are some works that account for weight when analyzing nonlinear vibrations. [PERSON] et al. experimentally investigated the influence of weight on the vibration of a two-degrees-of-freedom system coupled with a nonlinear energy sink [35]. [PERSON] et al. demonstrated the stochastic responses of a single-degree-of-freedom system attached to a nonlinear energy sink, considering weight [36]. [PERSON] et al. investigated the influence of weight on the stability of a conveying pipe at different angles to the vertical direction [37]. [PERSON] et al. discussed the influence of weight on the natural frequency of vertically placed-cantilever beams under lateral and compressive forces [38]. [PERSON] et al. and [PERSON] et al. investigated the natural frequencies of standing and hanging beams under vertical weight load [39; 40]. [PERSON] et al. addressed the effects of weight and sliding friction on the energy transfer dynamics of inclined vibro-impact nonlinear energy sinks [41]. [PERSON] et al. investigated the influence of weight on the natural frequency and modal function of a beam [42]. [PERSON] et al. explored the influence of equilibrium configuration for a cantilever flow pipe, considering the effect of weight on flutter [43]. [PERSON] et al. revealed the impact of weight on the sensitiveness of the modal response and the stability of the dynamic responses of a beam [44]. [PERSON] et al. verified the superiority of higher-order beam elements through dynamic simulation of a silicone rubber cantilever beam under the influence of weight [45]. [PERSON] et al. studied the influence of weight on a horizontally placed pendulum as a nonlinear energy sink attached to a discrete system [46]. [PERSON] et al. investigated the stability of a nonlinear energy sink at an angle to the horizontal, considering weight [47]. [PERSON] et al. experimentally discovered the influence of weight on the stability of cantilever flow pipelines with different flow velocities [48]. [PERSON] et al. investigated the hardening-type nonlinearity of a plate, considering the effect of weight [49]. [PERSON] et al. proposed a magnetic levitation system using the weight of a mass to control the vibrations of a plate [50]. [PERSON] et al. investigated the natural frequencies and the dynamic responses of a circular plate, considering the effect of weight [51]. However, some works consider the self-weight of beams when studying dynamic responses.