# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 1 Introduction
optimized the parameters of a nonlinear energy sink to reduce the vibrations of a Euler-Bernoulli beam and a Timoshenko beam [15], and [PERSON] et al. investigated single and parallel nonlinear energy sinks to suppress the vibration of a simply supported beam [16]. In particular, the method of harmonic balance has been widely applied in the design and analysis of nonlinear energy sinks [17; 18; 19; 20; 21; 22; 23; 24; 25], detailed as follows. [PERSON] et al. revealed the effects of the moisture and temperature environment on the vibration of a laminated composite beam with a nonlinear energy sink [17]. [PERSON] et al. investigated the stability of an axially moving beam with a nonlinear energy sink [18]. [PERSON] et al. proposed a lever-like design to enhance the reduction in the vibration of nonlinear energy sinks attached to a viscoelastic beam [19]. [PERSON] et al. replaced the mass in a nonlinear energy sink with an inerter to reduce its weight with the same inertia, to propose an inertial nonlinear energy sink for an elastic beam [20]. [PERSON] et al. further investigated the vibration of a geometrically nonlinear beam with a boundary inertial nonlinear energy sink [21]. [PERSON] et al. designed multiple parallel nonlinear energy sinks for the vibration suppression of a Timoshenko beam [22]. [PERSON] et al. applied an inertial nonlinear energy sink to suppress the vibration of an asymmetric elastically supported beam [23]. [PERSON] et al. analyzed the vibration of a generally restrained pre-pressure beam with multiple nonlinear energy sinks [24]. [PERSON] al. analyzed the vibration of a slightly curved beam with a nonlinear energy sink at the end of the beam [25]. As a variation of the method of harmonic balance, the method of incremental harmonic balance has also been applied. Based on this method, [PERSON] et al. proposed a criterion for the parametric design of bistable nonlinear energy sinks attached to beams and plates [26]. [PERSON] et al. developed a semi-numerical analysis for an axially moving beam with a nonlinear energy sink and a piezoelectric attachment [27]. [PERSON] et al. investigated the influence of the natural frequency ratio of the beam and cable on the effect of a nonlinear energy sink [28]. [PERSON] et al. discussed the effectiveness of a nonlinear energy sink with various parameters attached to a cable-stayed beam [29]. [PERSON] et al. optimized the parameters of a nonlinear energy sink for reducing the vibration of a beam on a modified Winkler foundation [30]. In harmonic balance analysis, the harmonics order is an essential aspect. Actually, the above-mentioned works involving the harmonic balance method retained harmonics of 1 [22], 1, 3 [17; 18; 19; 20; 21; 22], and 0, 1, 2, 3 [25]. Some of the approximate analyses were supported by numerical simulations of response time histories [15; 20], amplitude-frequency response curves [16; 17; 18; 19; 20; 21; 23; 24; 25; 28; 29; 30], and periodic solutions in the phase plane [27]. The validation of Galerkin truncation-based works is an open issue. In most circumstances, only a few truncation terms are practically feasible, especially when approximate analysis is involved. In the above-mentioned applications of the Galerkin truncation, the retained truncation term numbers were 1 [13], 2 [4; 5; 6; 7; 8; 9; 12; 16; 17; 18; 27], 3 [30], 4 [7; 19; 20; 21; 23; 25], 5 [15; 26], 6 [3; 24], and 10 [11], respectively. Therefore, simulations and analysis based on the Galerkin truncation should be experimentally or numerically supported. There are some experimental works on the vibrations of beams with nonlinear energy sinks. [PERSON] et al. proposed a nonlinear energy sink based on a bistable cosine-shaped beam to reduce the vibration of a cantilever beam, and the reduction effectiveness was validated with amplitude-response curves [31]. [PERSON] et al. compared the surface plot of a beam-bistable nonlinear energy sink system's displacement frequency response, obtained by measurements and modals [32]. [PERSON] et al.