# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 3 Discretization via the Galerkin Method
\begin{array}{l}\phi_{1}(x)=\cos(4.7300x)-0.9825\sin(4.7300x)-\cosh(4.7300x)+0.9825\mathrm{sinh}(4.7300x)\\ \phi_{2}(x)=\cos(7.8532x)-1.0007\sin(7.8532x)-\cosh(7.8532x)+1.0007\mathrm{sinh }(7.8532x)\\ \phi_{2}(x)=\cos(10.9956x)-0.9999\sin(10.9956x)-\cosh(10.9956x)+0.9999\mathrm{ sinh}(10.9956x)\\ \phi_{4}(x)=\cos(14.1372x)-1.0000\sin(14.1372x)-\cosh(14.1372x)+1.0000\mathrm{ sinh}(14.1372x)\\ \phi_{5}(x)=\cos(17.2788x)-1.0000\sin(17.2788x)-\cosh(17.2788x)+1.0000\mathrm{ sinh}(17.2788x)\\ \phi_{6}(x)=\cos(20.4204x)-1.0000\sin(20.4204x)-\cosh(20.4204x)+1.0000\mathrm{ sinh}(20.4204x)\\ \end{array}  \tag{19}\]  
The first six mode functions can be used for convergence analysis via the Runge-Kutta method and dynamic response analysis via the harmonic balance method, respectively. To determine the truncation term number \(N\), the free and the forced vibrations are computed by up to two-, four-, and six-term Galerkin truncation. The parameters of the beam and nonlinear energy sink shown in Tables 1 and 2 are used in the following analysis. The parameters of the nonlinear energy sink may not be optimal, because the present work focuses on the effect of nonlinear energy sink weight on the responses, rather than the effect of the vibration reduction design. As for free vibration, the Runge-Kutta method is utilized based on Equation (15), with the first \(N\) terms. The excitation amplitude is set as \(\Gamma=0\) N/m. The dimensionless initial values of the beam and nonlinear energy sink are set as follows:  
\[ \begin{array}{l}q_{1}=0.01,\;\dot{q}_{1}=0,\;q_{n}=0,\;\dot{q}_{n}=0,\;n=2, 3,\ldots,N,\\ v=0,\;\dot{v}=0.\end{array}  \tag{20}\]  
The truncation coefficients and the generalized coordinates of the beam can be described as \(\phi_{l}(x_{c})(i=1,\ldots,N)\) and \(q_{l}(t)(i=1,\ldots,N)\) in Equation (13). Based on Equation (15), utilizing the Runge-Kutta method, Figure 2 can be obtained to illustrate the time history responses of the free vibration of the one-eighth and one-fourth points of the beam when the effect of weight is considered. \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Notation** & **Value** \\ \hline Young’s modulus & \(E\) & 210 GPa \\ Density & \(\rho\) & 7850 kg\(\cdot\)m\({}^{-3}\) \\ Length & \(L\) & 1.22 m \\ Cross-sectional area & \(A\) & 2.4 \(\times\) 10\({}^{-4}\) m\({}^{2}\) \\ Cross-sectional moment of inertia & \(I\) & 7.2 \(\times\) 10\({}^{-10}\) m\({}^{4}\) \\ Damping & \(C\) & 10 N\(\cdot\)s/m \\ \hline \end{tabular}
\end{table} 
Table 1: Physical and geometric parameters of beam. \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Notation** & **Value** \\ \hline Nonlinear stiffness & \(k\) & 2 \(\times\) 10\({}^{7}\) N/m\({}^{3}\) \\ Damping & \(\alpha\) & 1 N\(\cdot\)s/m \\ Mass ratio & \(\mu\) & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of nonlinear energy sink. As regards the weight being offset by the static deformation change, the numerical solutions via the Runge-Kutta method are calculated based on Equation (17). Figure 3 illustrates the time history responses of free vibration when the effect of weight is neglected. For forced vibrations, the amplitude-frequency responses with different Galerkin truncations (\(N=2\), 4, 6) are shown to determine the Galerkin truncation order. The excitation amplitude is set as \(F=30\) N/m. The initial displacements and velocities of the beam and nonlinear sink are all set as 0. By extracting the average value of amplitude during the steady-state response time period at different excitation frequencies, based on Galerkin truncation, the amplitude-frequency response curves of the forced vibration can then be obtained when the weights are considered or not, as shown in Figures 4 and 5.