# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 6 Finite Element Simulations and Verification  
The finite element method was applied to simulate the beam vibrations. In particular, ANSYS software (version number:2.2.1) was used to calculate the responses of the beam with a nonlinear energy sink under an external excitation. Firstly, the element type BEAM188 is adopted. The material parameters of the beam are also set, namely the density,  
Figure 13: Amplitude\(-\)frequency response of forced vibration with second\(-\)order excitation frequency: (**a**) response of 1/8 point of beam when nonlinear energy sink is placed at 1/2 point of beam; (**b**) response of 1/4 point of beam when nonlinear energy sink is placed at 1/2 point of beam.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Frequency** & **Amplitude** \\ \hline One-fourth point, & 59.08 Hz & 11.08 \(\times\) 10\({}^{-3}\) m \\ neglecting weight & 59.09 Hz & 11.08 \(\times\) 10\({}^{-3}\) m \\ One-eighth point, & 59.08 Hz & 4.99 \(\times\) 10\({}^{-3}\) m \\ One-eighth point, & 59.09 Hz & 4.99 \(\times\) 10\({}^{-3}\) m \\ considering weight & 59.09 Hz & 4.99 \(\times\) 10\({}^{-3}\) m \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of forced vibration with 2 nd-order excitation frequency when nonlinear energy sink is at 1/2 point of beam.  
the Poisson's ratio, and the Young's modulus. Then, the beam model can be established using node coordinates. Secondly, the beam is meshed into 24 units and fixed support constraints are applied at both beam ends. The cross-sectional size parameters of the beam are entered. For the realization of a nonlinear energy sink, the MASS21 element, the COMBINE39 element, and the Spring14 unit are set to represent the mass, the nonlinear stiffness, and the linear damping of the nonlinear energy sink, respectively. The nonlinear stiffness is achieved by inputting loads and corresponding deformation values in the COMBINE39 element. The external excitation function is set and applied to the quarter node of the beam. Finally, the solver is set to a transient structure to obtain a time history duration of 80s and a time step of 0.001 s. The results obtained from ANSYS Software (version number:2.2.1) were compared with the model-based solutions calculated by the Runge-Kutta method and the harmonic balance method. It was shown that the finite element solutions were closer to the model-based results considering the effect of weights.  
The finite element simulation results were compared with the model-based numerical and analytical responses obtained in Section 5. These comparisons require defining the errors between the simulation and the model-based numerical results. The error between the results calculated by the Runge-Kutta method and ANSYS software (version number:2.2.1) is defined as follows:  
\[R_{1}=\frac{A_{RK}-A_{ANSYS}}{A_{ANSYS}}\times 100\% \tag{26}\]  
where \(A_{RK}\) is the amplitude calculated by the Runge-Kutta method and \(A_{ANSYS}\) is the amplitude calculated by the ANSYS software (version number:2.2.1).  
The error between the result calculated by the harmonic balance method and ANSYS software (version number:2.2.1) is defined as follows:  
\[R_{2}=\frac{A_{HBM}-A_{ANSYS}}{A_{ANSYS}}\times 100\% \tag{27}\]  
where \(A_{HBM}\) is the amplitude calculated by the harmonic balance method and \(A_{ANSYS}\) is the amplitude calculated by the ANSYS software (version number:2.2.1).