# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 6 Finite Element Simulations and Verification
### The Free Vibration, Simulated by ANSYS Software  
For simulations of the free vertical transverse vibrations of a beam with a nonlinear energy sink using ANSYS software (version number:2.2.1), the displacement was set as 0 m and the initial velocity of the beam was set as 0.2 m/s in the software. The external excitation amplitude was set as 0. After calculation by the ANSYS software (version number:2.2.1), time history curves could be obtained. The fast Fourier transform was applied to obtain amplitude spectrum curves. Compare the results calculated by the Runge-Kutta method and ANSYS software (version number:2.2.1), as shown in Figure 14. The details of amplitude spectra are shown in Table 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Frequency** & **Amplitude** & **Amplitude Error** \\ \hline One-fourth point, neglecting & & & \\ weight (RKM) & 21.42 Hz & 5.87 \(\times\) 10\({}^{-6}\) m & / 37.1\% \\ One-fourth point, considering & & & \\ weight (RKM) & 10.15 Hz, 23.2 Hz & 2.32 \(\times\) 10\({}^{-6}\) m, 4.77 \(\times\) 10\({}^{-6}\) m & 9.9\%, 11.4\% \\ One-fourth point (ANSYS) & 10.16 Hz, 22.84 Hz & 2.11 \(\times\) 10\({}^{-6}\) m, 4.28 \(\times\) 10\({}^{-6}\) m & / \\ One-eight point, neglecting & & & \\ weight (RKM) & 21.42 Hz & 1.92 \(\times\) 10\({}^{-6}\) m & / 36.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of free vibration results calculated by ANSYS software (version number:2.2.1) and Runge–Kutta method when nonlinear energy sink is at 1/2 point of beam. For the assumption that weights are offset with the static deformation, the amplitude spectra determined by the Runge-Kutta method agree well with those calculated by ANSYS software (version number:2.2.1). When the nonlinear energy sink is placed at the half point of the beam, the amplitude spectra of the one-fourth point of the beam, corresponding with two peaks calculated by the Runge-Kutta method, are 10.15 Hz and 23.21 Hz, which correspond to amplitudes of 2.32 \(\times\) 10\({}^{-6}\) m and 4.77 \(\times\) 10\({}^{-6}\) m. The results calculated by the ANSYS software (version number:2.2.1) are 10.16 Hz and 22.84 Hz, whose corresponding amplitudes are 2.11 \(\times\) 10\({}^{-6}\) m and 4.28 \(\times\) 10\({}^{-6}\) m. When the weight is neglected, the results calculated by the Runge-Kutta method are 21.42 Hz and 5.87 \(\times\) 10\({}^{-6}\) m. The detailed data of the amplitude spectra of the one-eighth point of the beam yield the same conclusion. When weight is considered, the results calculated by ANSYS and the Runge-Kutta method agree well. When the nonlinear energy sink is placed at the half point of beam, the amplitude spectra and the detailed data are shown in Figure 14 and Table 6. All in all, the model-based numerical results of the free vibration considering the weight agree well with the results calculated by the ANSYS software (version number:2.2.1). beam are shown in Figure 15 and Table 7. It can be seen that the frequency results calculated by the Runge-Kutta method when the weight is considered are 10.69 Hz and 21.45 Hz, and when the weight is neglected, the frequency is 21.45 Hz. The frequency results calculated by ANSYS software (version number:2.2.1) are 9.29 Hz and 25.21 Hz. Using the Runge-Kutta method, the amplitudes of the one-fourth point of the beam, corresponding with the frequency referred to above, are 0.98 \(\times\) 10\({}^{-3}\) m and 1.89 \(\times\) 10\({}^{-3}\) m when the weight is considered, and when the weight is neglected, the amplitude is 3.61 \(\times\) 10\({}^{-3}\) m. The amplitudes obtained by ANSYS software (version number:2.2.1) are 1.06 \(\times\) 10\({}^{-3}\) m and 1.72 \(\times\) 10\({}^{-3}\) m. The errors of the amplitudes obtained by the Runge-Kutta method and ANSYS software (version number:2.2.1) were calculated. It can be seen that the errors are \(-\)7.55% and 9.88% when the weight is considered, and the error is 77.33% when the weight is neglected.