# 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
From Table 7, the frequency component of the one-eighth point of the beam is the same as that of the one-fourth point of the beam. Using the Runge-Kutta method, the amplitudes of the one-eighth point of the beam, corresponding with the frequency referred to above, are 0.31 \(\times\) 10\({}^{-3}\) m and 0.67 \(\times\) 10\({}^{-3}\) m when the weight is considered, and when the weight is neglected, the amplitude is 1.3 \(\times\) 10\({}^{-3}\) m. The amplitudes obtained by ANSYS software (version number:2.2.1) are 0.35 \(\times\) 10\({}^{-3}\) m and 0.77 \(\times\) 10\({}^{-3}\) m. It can be seen that \begin{table}
\begin{tabular}{c c c c} \hline \hline Item & **Frequency** & **Amplitude** & **Amplitude Error** \\ \hline One-fourth point, neglecting & & & \\ weight (RKM) & 21.45 Hz & 3.61 \(\times\) 10\({}^{-3}\) m & /.77.33\% \\ One-fourth point, considering & & & \\ weight (RKM) & 10.69 Hz, 21.45 Hz & 0.98 \(\times\) 10\({}^{-3}\) m, 1.89 \(\times\) 10\({}^{-3}\) m & \(-\)7.55\%, 9.88\% \\ One-fourth point (ANSYS) & 9.29 Hz, 25.21 Hz & 1.06 \(\times\) 10\({}^{-3}\) m, 1.72 \(\times\) 10\({}^{-3}\) m & / \\ One-eighth point, neglecting & & & \\ weight (RKM) & & & \\ One-eighth point, considering & & & \\ weight (RKM) & & & \\ One-eighth point (ANSYS) & 9.29 Hz, 25.21 Hz & 0.35 \(\times\) 10\({}^{-3}\) m, 0.77 \(\times\) 10\({}^{-3}\) m & / \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of forced vibration with 1 st-order excitation frequency, calculated by ANSYS software (version number:2.2.1) and Runge–Kutta method, when nonlinear energy sink is placed at 1/2 point of beam. Figure 15: Results of ANSYS software (version number:2.2.1) simulation and Runge–Kutta method when nonlinear energy sink is placed at 1/2 point of beam: (**a**) frequency response of 1/8 point of beam; (**b**) frequency response of 1/4 point of beam. the errors are \(-11.43\%\) and \(-12.98\%\) when the weight is considered, and the error is \(68.83\%\) when the weight is neglected. In both the one-fourth point and one-eighth point of the beam, the resonant frequency and amplitude calculated by the Runge-Kutta method when considering weight agree well with the result calculated by the ANSYS software (version number:2.2.1). For the forced vibration, the results calculated by harmonic balance analysis were compared with that calculated by the ANSYS software (version number:2.2.1), as shown in Figure 16. The details of the amplitude-frequency response curves are shown in Table 8.  
Firstly, the situation in which the nonlinear energy sink placed at the half point of the beam is discussed. From Table 8, the frequency components of the one-fourth point of the beam calculated by the harmonic balance method when the weight is considered are \(11.83\) Hz and \(23.67\) Hz, and when the weight is neglected, the frequency is \(21.43\) Hz. The frequency results calculated by ANSYS are \(9.29\) Hz and \(25.21\) Hz. Using the harmonic balance method, the amplitudes of the one-fourth point of the beam, corresponding to the frequency referred to above, are \(1.18\times 10^{-3}\) m and \(1.91\times 10^{-3}\) m when the weight is considered, and when the weight is neglected, the amplitude is \(4.82\times 10^{-3}\) m. The amplitudes obtained by the ANSYS software (version number:2.2.1) are \(1.06\times 10^{-3}\) m and \(1.72\times 10^{-3}\) m. The errors of amplitude obtained by the harmonic balance method and ANSYS software (version number:2.2.1) were calculated. It can be seen that the errors are \(-11.32\%\) and \(-11.05\%\) when the weight is considered, and the error is \(180.23\%\) when the weight is neglected. Figure 16: 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: **(a)** frequency response of \(1/8\) point of beam; **(b)** frequency response of \(1/4\) point of beam.