# The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method
## 3 Equation of Flexure of an Elastic Beam and Its Solution
\begin{array}{c}\int_{0}^{1}\biggl{[}-\sum\limits_{n=0}^{N}\left(\frac{2n+1 }{2}\pi\right)^{2}A_{n}\sin\frac{2n+1}{2}\pi x\\ +\alpha\;\cos\sum\limits_{n=0}^{N}A_{n}\sin\frac{2n+1}{2}\pi x\biggr{]}\sin \frac{2n+1}{2}\pi x\;\mathrm{d}x=0\\ n=1,2,3,\cdots,\;M,\end{array}  \tag{15}\]  
for the coupled amplitudes with the choice of order \(M\). The procedure outlined above is followed to obtain the solutions of amplitudes. To obtain the amplitude \(A_{0}\), the approximate solution is taken as  
\[\theta=A_{0}\sin\frac{\pi}{2}x, \tag{16}\]  
and the integration of Equation (7) yields  
\[575-\frac{1575\pi^{3}A_{0}}{16\alpha}-525A_{0}^{2}+35A_{0}^{4}-A_{0}^{6}=0. \tag{17}\]  
To find appropriate solution of \(A_{0}(\alpha)\), the curves from Equation (17) are plotted in Figure 1 to help the solution procedure of this equation. As shown in Figure 1, there are two real solutions of \(A_{0}\) with \(\alpha\) from the nonlinear algebraic equation in Equation (17). The amplitude \(A_{0}\) can be given with \(\alpha\), but the standard Maclaurin expansion only offers expression with a relatively small convergent radius. To obtain a power series solution with relatively larger convergent radius, an auxiliary expansion scheme with Chebyshev polynomial can be utilized. The procedure starts with the expression of \(A_{0}(\alpha)\), and the expansion is  
\[ \begin{array}{c}A_{0}(\alpha)=\sum\limits_{n=0}^{\infty}C_{n}T_{n}\left( \frac{\pi}{5}-1\right),\\ C_{n}=\frac{2}{5\pi}\int_{0}^{10}A_{0}(\alpha)\frac{T_{n}\left(\frac{\pi}{5}- 1\right)}{\sqrt{1-\left(\frac{\pi}{5}-1\right)^{2}}}\mathrm{d}\alpha,\end{array}  \tag{18}\]  
Figure 1: The solutions of \(A_{0}\) vs. parameter \(\alpha\). where \(T_{n}\) is the \(n\)th-order Chebyshev polynomial, and \(C_{n}\) is the coefficient of expansion. In case an explicit expression of \(A_{0}(\alpha)\) is not available, the symbolic expression from mathematical tools such as Mathematica(r) and MATLAB(r) can be utilized for the evaluation in Equation (18). Previous studies have shown that Equations (17) and (18) here have good stability and convergence [15; 32; 33; 34]. In this study, from Equations (17) and (18), the explicit expression of \(A_{0}(\alpha)\) with approximation is  
\[ \begin{array}{rl}A_{0}=&-1.1211313903689\times 10^{-7}+0.5160330121688547 \alpha\\ &-0.00009479584375745703a^{2}-0.045486335180154686\alpha^{3}\\ &+0.000012757993686000196\alpha^{4}+0.006466636289057949\alpha^{5}\\ &+0.006958737479441875\alpha^{6}-0.012229157808245239\alpha^{7}\\ &+0.008664382051570434\alpha^{8}-0.0039340381580180665\alpha^{9}\\ &+0.0012851620189843023\alpha^{10}-0.00031666548778055986\alpha^{11}\\ &+0.00006014045641459205\alpha^{12}\\ &-0.000008877160503395703\alpha^{13}\\ &+0.000001017529293530474\alpha^{14}\\ &-8.973639340531093\times 10^{-8}\alpha^{15}\\ &+5.973823554191147\times 10^{-9}\alpha^{16}\\ &-2.903563905834224\times 10^{-10}\alpha^{17}\\ &+9.721628341667188\times 10^{-12}\alpha^{18}\\ &-2.004035813851492\times 10^{-13}\alpha^{19}\\ &+1.916801557948531\times 10^{-15}\alpha^{20}\\ \end{array}  \tag{19}\]  
Then, from Equation (10) with the known value of \(A_{0}\), the approximate solution is modified to  
\[\theta=A_{0}\sin\frac{\pi}{2}x+A_{1}\sin\frac{3\pi}{2}x, \tag{20}\]  
where \(A_{1}\) is solved. Following the same procedure as demonstrated for \(A_{0}\), the nonlinear algebraic equation for amplitude \(A_{1}\) is  
\[ \begin{array}{rl}\frac{2\alpha}{3\pi}-\frac{9\pi^{2}A_{1}}{8}+\frac{1}{6542 9075\pi^{2}}2\alpha[230945A_{0}^{2}(189-27A_{0}^{2}+A_{0}^{4})\\ &-25194A_{0}(13365-1430A_{0}^{2}+59A_{0}^{4})A_{1}\\ &+14535\big{(}-5005-1118A_{0}^{2}+161A_{0}^{4}\big{)}A_{1}^{2}\\ &-6460A_{0}\big{(}-6561+695A_{0}^{2}\big{)}A_{1}^{3}+285(17017\\ &+3947A_{0}^{2}\big{)}A_{1}^{4}-1771470A_{0}A_{1}^{5}-138567A_{1}^{6}\big{]}= 0.\end{array}