## 3 Solution of the Nonlinear Problem
### Approximate Analytical Solution for Nonlinear Oscillation  
Consider the general form of the nonlinear differential equation governing the nonlinear free oscillation as below  
\[\ddot{q}+f(q)=0 \tag{51}\]  
According to the He's Method, the variational principle of Equation (51) can be defined by using the semi-inverse method as follows [41]  
\[J(q)=\int_{0}^{\frac{T}{4}}\bigg{\{}-\frac{1}{2}\dot{q}^{2}+F(q)\bigg{\}}dt \tag{52}\]  
where \(\dot{q}\) is the first derivative of \(q\) with respect to time, \(T\) is the period of the nonlinear oscillation and \(F(q)\) is a function of \(q\). The relation between \(F(q)\) and \(f(q)\) can be defined as below [41]  
\[\frac{\partial F}{\partial q}=f(q) \tag{53}\]  
It is assumed that the approximate solution of Equation (51) can be expressed as follows  
\[q(t)=A_{0}cos(\omega_{nl}t) \tag{54}\]  
where \(A_{0}\) and \(\omega_{nl}\) are the amplitude and the nonlinear natural frequency of the oscillation, respectively. Substituting Equation (54) into Equation (52) and keeping in mind the transformation \(\omega_{nl}t=\tau\) yields  
\[J(A_{0},\omega_{nl})=\frac{1}{\omega_{nl}}\int_{0}^{\frac{\eta}{2}}\bigg{\{}- \frac{1}{2}A_{0}{}^{2}\omega_{nl}^{2}sin^{2}\tau+F(A_{0}cos\tau)\bigg{\}}d\tau \tag{55}\]  
In order to determine \(\omega_{nl}\), the following stationary conditions based on the Ritz method should be satisfied [41]  
\[\frac{\partial J}{\partial\omega_{nl}}=0 \tag{56}\]  
\[\frac{\partial J}{\partial A_{0}}=0 \tag{57}\]  
It should be noted that for nonlinear oscillation, this approach will give inaccurate results. Hence, by modifying the conditions (56) and (57) into a simpler form as below, we will refer to the only condition (57).  
In order to simplify the analysis, the governing Equation (48) was rewritten into the general form mentioned in Equation (51), in which, \(f(q)\) is  
\[f(q)=\omega_{0}^{2}q+\eta_{2}q^{2}+\eta_{3}q^{3} \tag{58}\]  
Using of Equations (53), (54) and (58) together with the transformation \(\omega_{nl}t=\tau\) leads to  
\[F(q)=\frac{\omega_{0}^{2}}{2}A_{0}{}^{2}cos^{2}\tau+\frac{\eta_{2}}{3}A_{0}{} ^{3}cos^{3}\tau+\frac{\eta_{3}}{4}A_{0}{}^{4}cos^{4}\tau \tag{59}\]  
Substituting Equation (59) into Equation (55) leads to the variational form of Equation (47) as follows\[J(A_{0},\omega_{nl})=\frac{1}{\omega_{nl}}\int_{0}^{\frac{\pi}{2}}\left\{-\frac{1} {2}{A_{0}}^{2}\omega_{nl}^{2}sin^{2}\tau+\frac{\omega_{0}^{2}}{2}{A_{0}}^{2}cos^ {2}\tau+\frac{\eta_{2}}{3}{A_{0}}^{3}cos^{3}\tau+\frac{\eta_{3}}{4}{A_{0}}^{4} cos^{4}\tau\right\}d\tau \tag{60}\]  
Using Equations (57) and (60) leads to the stationary condition of Equation (60) with respect to \(A_{0}\) as  
\[\frac{\partial J}{\partial A_{0}}=\frac{1}{\omega_{nl}}\int_{0}^{\frac{\pi}{2} }\left\{-A_{0}\omega_{nl}^{2}sin^{2}\tau+\omega_{0}^{2}A_{0}cos^{2}\tau+\eta_ {2}{A_{0}}^{2}cos^{3}\tau+\eta_{3}{A_{0}}^{3}cos^{4}\tau\right\}d\tau=0 \tag{61}\]  
The nonlinear natural frequency of the system can be obtained from Equation (61) as follows  
\[\omega_{nl}=\sqrt{\omega_{0}^{2}+\frac{8}{3\pi}A_{0}\eta_{2}+\frac{3}{4}{A_{0 }}^{2}\eta_{3}} \tag{62}\]  
Using Equations (36), (54) and (62) leads to the following approximate solution  
\[w(x,t)=\psi(x)\cdot A_{0}cos\left(\left(\sqrt{\omega_{0}^{2}+\frac{8}{3\pi}A_ {0}\eta_{2}+\frac{3}{4}{A_{0}}^{2}\eta_{3}}\right)t\right) \tag{63}\]