# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 7 Conclusions  
A fundamental issue was investigated to understand the effects of nonlinear energy sink weight on the vibration of transverse beams in the vertical direction. The governing equations considering and neglecting the nonlinear energy sink weight were derived. The Galerkin truncation was applied to discretize the governing equations. The harmonic balance method was applied to seek the approximate analytical expressions of the amplitude-frequency responses. Finite element simulations, independent of the governing equations, were implemented via ANSYS software (version number:2.2.1). This investigation yielded the following conclusions:  
1. Accounting for the nonlinear energy sink weight results in an additional linear coupling term and an additional quadratic stiffness term in the governing equation of the vertical transverse vibration beam with a nonlinear energy sink.
2. For the free vibration, the numerical amplitude spectra reveal that accounting for the nonlinear energy sink weight introduces a half super-harmonic response in addition to quantitative differences in the first primary harmonic response.
3. For the forced vibration, the amplitude-frequency response curves indicate that accounting for the nonlinear energy sink weight introduces half super-harmonic resonance in addition to quantitative differences in the first primary resonance.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Frequency** & **Amplitude** & **Amplitude Error** \\ \hline One-fourth point, neglecting & 59.12 Hz & 16.64 \(\times 10^{-3}\) m & 50.18\% \\ weight (HBM) & 58.71 Hz & 12.59 \(\times 10^{-3}\) m & 13.63\% \\ One-fourth point (ANSYS) & 58.88 Hz & 11.08 \(\times 10^{-3}\) m & / \\ One-eight point, neglecting weight & 59.12 Hz & 7.48 \(\times 10^{-3}\) m & 50.50\% \\ One-eight point, considering & & & \\ One-eight point (ANSYS) & 58.67 Hz & 5.42 \(\times 10^{-3}\) m & 9.05\% \\ One-eight point (ANSYS) & 58.88 Hz & 4.97 \(\times 10^{-3}\) m & / \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison of forced vibration with 2 nd-order excitation frequency, calculated by ANSYS software (version number:2.2.1) and harmonic balance method, when nonlinear energy sink is at 1/2 point of beam.  
Figure 18: Results of ANSYS software (version number: 2.2.1) simulation and harmonic balance method when nonlinear energy sink is placed at 1/2 point of beam with second-order primary frequency excitation: (**a**) frequency response of 1/8 point of beam; (**b**) frequency response of 1/4 point of beam.  
* The finite element solutions are closer to the model-based results when considering the nonlinear energy sink weight.
* The present investigation demonstrates that the nonlinear energy sink weight effects are significant for transverse beam vibrations in the vertical direction.  
X.F.: conceptualization, methodology, software, writing--original draft, preparation. S.W.: resources, writing--review and editing, supervision, funding acquisition. H.D.: resources, funding acquisition. L.-Q.C.: conceptualization, methodology, writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Science Fund for Distinguished Young Scholars, grant number 12025204, and the Program of Shanghai Municipal Education Commission, grant number 2019-01-07-00-09-E00018.  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors would like to thank all the peer reviewers and editors for their valuable contributions to this work.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.