# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 5 The Effects of Weight on the Beam's Vibration
### The Forced Vibration of First-Order Primary Frequency, Simulated by the Runge-Kutta Method  
The influence of weight on the forced vibration with first-order external primary frequency is discussed in this section. The excitation amplitude is set as \(F=30\) N/m and  
Figure 8: Time history curve of free vibration: (**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
**Item** & **Frequency** & **Amplitude** \\ \hline One-fourth point, & 21.42 Hz & 5.87 \(\times\) 10\({}^{-6}\) m \\ neglecting weight & 10.15 Hz, 23.2 Hz & 2.32 \(\times\) 10\({}^{-6}\) m, 4.77 \(\times\) 10\({}^{-6}\) m \\ One-eighth point, & 21.42 Hz & 1.92 \(\times\) 10\({}^{-6}\) m \\ neglecting weight & 10.14 Hz, 23.21 Hz & 0.72 \(\times\) 10\({}^{-6}\) m, 1.58 \(\times\) 10\({}^{-6}\) m \\ considering weight & \\ \hline \end{tabular}
\end{table}
Table 3: Comparison of free vibration when nonlinear energy sink is placed at 1/2 point of beam.  
Figure 9: Amplitude Spectrum of free vibration: (**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.  
the excitation frequency as \(\Omega=21.43\) rad/s. All the initial values of the beam and nonlinear energy sink are set as 0.  
The Runge-Kutta method was utilized in Equation (23); then, time history curves could be obtained. The amplitude-frequency diagrams under first-order excitation frequency were obtained by the fast Fourier transform.  
In the case that the weights were assumed to be offset by the static deflection, the Runge-Kutta method and the fast Fourier transform were used to obtain the numerical responses based on Equation (24).  
As shown in Figure 10, the time histories of forced vibration with first-order excitation frequency were obtained. The red lines represent the case considering weight, and the blue lines represent the differences between the displacements when considering and neglecting weight. For the definition of the error in Figure 10, one can refer to Section 5.1. Figure 11 shows the amplitude-frequency responses under first-order excitation frequency.  
Figure 11: Amplitude–frequency response of forced vibration with first–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.  
Figure 10: Time history curve of forced vibration with first–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.  
The differences between the numerical and the analytical solutions in the both cases were analyzed. The conclusion can be reached that the displacements of the forced vibration when neglecting weight are larger than that when considering weight. The details of the amplitudes and frequencies when considering and neglecting weight are shown in Table 4. The amplitude-frequency response curves indicate half super-harmonic resonance, in addition to quantitative differences in the first primary resonance.