# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 3 Discretization via the Galerkin Method  
Based on the weight functions \(\phi_{i}(x)(i=1,2,\ldots,N)\), the dimensionless beam transverse displacements are assumed to take the following form:  
\[w_{1}=w_{2}=\sum\limits_{i=1}^{\infty}\phi_{i}(x)q_{i}(t) \tag{13}\]  
where \(q_{i}(t)\) is the corresponding generalized coordinate. Consequently, Equation (7) can be discretized as follows:  
\[ \begin{array}{l}\sum\limits_{i=1}^{\infty}\phi_{i}(x)\ddot{q}_{i}(t)+c\sum \limits_{i=1}^{\infty}\phi_{i}(x)\dot{q}_{i}(t)+x^{2}\sum\limits_{i=1}^{\infty }\frac{\partial^{4}\phi_{i}(x)}{\partial x^{4}}q_{i}(t)\\ +\Bigg{\{}k\Bigg{[}\sum\limits_{j=1}^{\infty}\phi_{j}(x_{c})q_{j}(t)-v(t)- \sqrt[3]{\frac{-\mu\rho\text{gL}}{Ek}}\Bigg{]}^{3}+\lambda\Bigg{[}\sum\limits _{j=1}^{\infty}\phi_{j}(x_{c})\dot{q}_{j}(t)-\dot{v}(t)\Bigg{]}\Bigg{\}}\delta( x-x_{c})\\ =-f\cos(\omega t)\delta(x-s)\\ \mu\ddot{v}+k\Bigg{[}v(t)-\sum\limits_{j=1}^{\infty}\phi_{j}(x_{c})q_{j}(t)+ \sqrt[3]{\frac{-\mu\rho\text{gL}}{Ek}}\Bigg{]}^{3}+\lambda\Bigg{[}\dot{v}(t)- \sum\limits_{j=1}^{\infty}\phi_{j}(x_{c})\dot{q}_{j}(t)\Bigg{]}=-\frac{\mu \rho\text{gL}}{E}\end{array}  \tag{14}\]  
When both the weight functions and the trial functions are the mode functions of the beam without the damping, the external excitation, and the nonlinear energy sink, the [PERSON] method transforms Equation (14) into the truncated form, with the first \(N\) terms in Equation (13), as follows:  
\[ \begin{array}{l}M_{i}\ddot{q}_{i}(t)+cM_{i}\dot{q}_{i}(t)+\kappa^{2}K_{i}q_ {i}(t)\\ +\Bigg{\{}k\Bigg{[}\sum\limits_{j=1}^{N}\phi_{j}(x_{c})q_{j}(t)-v(t)- \sqrt[3]{\frac{-\mu\rho\text{gL}}{Ek}}\Bigg{]}^{3}+\lambda\Bigg{[}\sum\limits _{j=1}^{N}\phi_{j}(x_{c})\dot{q}_{j}(t)-\dot{v}(t)\Bigg{]}\Bigg{\}}\phi_{i}(x_{ c})=-f\cos(\omega t)\phi_{i}(s)\\ \mu\ddot{v}+k\Bigg{[}v(t)-\sum\limits_{j=1}^{N}\phi_{j}(x_{c})q_{j}(t)+\sqrt[3] {\frac{-\mu\rho\text{gL}}{Ek}}\Bigg{]}^{3}+\lambda\Bigg{[}\dot{v}(t)-\sum \limits_{j=1}^{N}\phi_{j}(x_{c})\dot{q}_{j}(t)\Bigg{]}=-\frac{\mu\rho gL}{E} \\ \text{where}\\ M_{i}=\int_{0}^{1}\phi_{i}(x)\phi_{i}(x)\text{d}x,K_{i}=\int_{0}^{1}\phi_{i}(x )\phi_{i}{}^{(4)}(x)\text{d}x\end{array}  \tag{16}\]  
In the case that the weights are balanced by the static deflection, Equation (11) is similarly truncated as follows:  
\[ \begin{array}{l}M_{i}\ddot{q}_{i}(t)+cM_{i}\dot{q}_{i}(t)+\kappa^{2}K_{i}q_ {i}(t)\\ +\Bigg{\{}k\Bigg{[}\sum\limits_{j=1}^{N}\phi_{j}(x_{c})q_{j}(t)-v(t)\Bigg{]}^ {3}+\lambda\Bigg{[}\sum\limits_{j=1}^{N}\phi_{j}(x_{c})\dot{q}_{j}(t)-\dot{v}( t)\Bigg{]}\Bigg{\}}\phi_{i}(x_{c})=-f\cos(\omega t)\phi_{i}(s)\\ \mu\ddot{v}+k\Bigg{[}v(t)-\sum\limits_{j=1}^{N}\phi_{j}(x_{c})q_{j}(t)\Bigg{]} ^{3}+\lambda\Bigg{[}\dot{v}(t)-\sum\limits_{j=1}^{N}\phi_{j}(x_{c})\dot{q}_{j }(t)\Bigg{]}=0\end{array}  \tag{17}\]  
The restoring forces for upward and downward motion are not symmetrical, due to the quadratic nonlinearity in Equation (17). In Equation (15), only cubic nonlinearity exists, and the restoring force is symmetrical. The modal function of the beam can be written as follows, without the damping, the external excitation, and the nonlinear energy sink:  
\[\phi(x)=H_{1}\cos(\beta x)+H_{2}\sin(\beta x)+H_{3}\cosh(\beta x)+H_{4}\text{ sinh}(\beta x) \tag{18}\]where the undetermined coefficients \(H_{j}(j=1,2,3,4)\) and the eigenvalue \(\beta\) can be obtained through the boundary conditions of the beam. Then, the first six natural frequencies of the beam can be obtained by using the parameters in Table 1, which are \(f_{1}=21.43\) Hz, \(f_{2}=59.07\) Hz,\(f_{3}=115.81\) Hz,\(f_{4}=191.45\) Hz,\(f_{5}=285.99\) Hz, and \(f_{6}=399.45\) Hz. The first six mode functions are as follows:  
\[