# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 1 Introduction  
Nonlinear energy sinks, as a promising vibration reduction approach, have been widely applied to various beams [1, 2]. The vibration of a continuum exhibits multiple vibration peaks over a wide frequency range. A nonlinear energy sink, a nonlinear absorber without linear stiffness, can adaptively reduce the resonant response of the continuum at each frequency within a wide frequency band, without introducing new resonant frequencies. When beams are modeled as continuous systems, the Galerkin method projects infinite-dimensional equations, arising from the continuous system, into a finite-dimensional subspace; therefore, the Galerkin method is usually used to truncate continuous systems into discrete systems. Therefore, response time histories and amplitude-frequency response curves can be numerically calculated. There have been some interesting numerical investigations into the vibrations of beams attached to nonlinear energy sinks. [PERSON] et al. optimized the parameters of nonlinear energy sinks placed on linear beams with different supports by computing the portion of the input energy dissipated by the nonlinear energy sinks [3]. [PERSON] et al. investigated the vibration reduction effectiveness of a nonlinear energy sink attached to an axially moving beam [4]. [PERSON] et al. also considered the vibration reduction effectiveness of nonlinear energy sinks for an axially moving beam under thermal shock [5]. [PERSON] et al. numerically investigated the impact of variable stiffness and the thermal effect on the vibration absorption efficiency of nonlinear energy sinks attached to a simply supported beam [6]. [PERSON] et al. investigated the vibration control of a cantilever beam under various parameters of a nonlinear energy sink [7]. [PERSON] et al. examined the effectiveness of parallel nonlinear energy sinks in suppressing the vibration of an axially moving beam induced by an impulse for different locations of the nonlinear energy sinks [8]. [PERSON] et al. conceptually proposed a symmetric single-sided vibro-impact nonlinear energy sink to suppress the transient vibration of a cantilever beam [9]. [PERSON] et al. conceptually proposed a tuned bistable nonlinear energy sink for a hinged-hinged beam to reduce its vibration [10]. [PERSON] et al. conceptually combined an acoustic black hole with a nonlinear energy sink to reduce the vibration of a nonlinear cantilevered beam, and the reduction effectiveness was numerically demonstrated by the portion of the input energy dissipated by the nonlinear energy sink [11]. [PERSON] et al. investigated the effectiveness of a parallel nonlinear energy sink attached to an axially moving beam under different axial speeds [12]. [PERSON] et al. investigated a nonlinear energy sink with an electromagnetic energy harvesting mechanism and installed it on a cantilever beam for vibration reduction [13]. In the above-mentioned applications of the Galerkin truncation, the retained truncation term numbers were 1 [13], 2 [4; 5; 6; 8; 9; 12], 4 [7], 6 [3], and 10 [11], respectively. Based on the applications of the Galerkin truncation, the response time histories [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13] and the amplitude-frequency response curves [4; 6; 7; 9; 10] were yielded numerically. In addition to the above-mentioned numerical investigations [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13], analytical approaches can be applied to approximately determine the steady-state periodical responses of harmonically excited beams with nonlinear energy sinks [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. Based on a complexification averaging method, [PERSON] et al. proposed an eddy-current-damping-enhanced nonlinear energy sink that could generate damping force through the interaction between the eddy current magnetic field passing through the magnetic field and the original magnetic field, in order to control the vibration of a cantilever beam [14]. [PERSON] et al.