# Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam
## 4 Application of the Homotopy Analysis Method  
Under the transformation \(\tau=\Omega t\), Equation (32) can be written as  
\[\Omega^{2}\eta^{\prime\prime}+\omega_{1}^{2}\eta+k_{f}^{2}(ca_{2}\eta^{2}+a_{3} \eta^{3})=0 \tag{46}\]  
with the following initial conditions:  
\[\eta(0)=A,\ \eta^{\prime}(0)=0 \tag{47}\]  
Thus, with these initial conditions, the initial guess of \(\eta(\tau)\) for a zero-order deformation equation is chosen as follows.  
\[\eta_{0}=A\cos\tau \tag{48}\]  
According to (46), we should choose the auxiliary linear operator  
\[L[\eta(\tau;q)]=\Omega^{2}\bigg{(}\frac{\partial^{2}\eta(\tau;q)}{\partial \tau^{2}}+\eta(\tau;q)\bigg{)} \tag{49}\]  
which has the property  
\[L(C_{1}\sin\tau+C_{2}\tau)=0 \tag{50}\]  
for any integration constants \(C_{1}\) and \(C_{2}\). Define the following nonlinear operator  
\[N[\eta(\tau;q),\Omega]=\Omega^{2}\frac{\partial^{2}\eta(\tau;q)}{\partial\tau ^{2}}+\omega_{1}^{2}\eta(\tau;q)+k_{f}^{2}\bigg{[}ca_{2}\eta^{2}(\tau;q)+a_{3} \eta^{3}(\tau;q)\bigg{]} \tag{51}\]  
The zero-order deformation is constructed as  
\[(1-q)L[\Phi(\tau;q)-\eta_{0}(\tau;q)]=q\hbar N[\Phi(\tau;q)] \tag{52}\]  
with the following initial conditions:  
\[\Phi(\tau;0)=\eta_{0}(t),\ \Phi(\tau;1)=\eta(\tau;q) \tag{53}\]  
The functions \(\Phi(\tau;q)\) and \(\Omega(q)\) can be expanded as power series of the \(q\) using [PERSON]'s theorem as  
\[\Phi(\tau;q)=\eta_{0}(\tau,q)+\sum_{m=1}^{+\infty}\eta_{m}(\tau,q)q^{m} \tag{54}\]  
\[\Omega(q)=\Omega_{0}+\sum_{m=1}^{+\infty}\Omega_{m}(\tau,q)q^{m} \tag{55}\]  
where \(\eta_{m}\) and \(\Omega_{m}\) are the m-order deformation derivatives.  
\[N_{m}(\eta_{m-1},\omega_{m-1})=\left.\frac{1}{(m-1)!}\frac{d^{m-1}N[\Phi, \omega]}{dq^{m-1}}\right|_{q=0} \tag{56}\]  
For free vibration without damping, periodic motion can be expressed by the following base functions  
\[\{\cos(m\tau),m=1,2,3,...\} \tag{57}\]  
For the first-order approximation of HAM, \(N_{1}\) is as follows:  
\[N_{1}=(\omega_{1}^{2}A-\Omega^{2}A+\frac{3}{4}k_{f}^{2}a_{3}A^{3})\cos\tau+ \frac{1}{2}k_{f}^{2}ca_{2}A^{2}\cos 2\tau+\frac{1}{4}k_{f}^{2}a_{3}A^{3}\cos 3 \tau+\frac{1}{2}k_{f}^{2}ca_{2}A^{2} \tag{58}\]  
Thus, \(\Omega_{0}\) can be written as  
\[\Omega_{0}=\sqrt{\omega_{1}^{2}+\frac{3}{4}k_{f}^{2}a_{3}A^{2}} \tag{59}\]Equation (59), does not contain the coefficient of quadratic term \(k_{f}^{2}ca_{2}\), which means that the nonlinear frequency is independent of a quadratic term. The cubic term of Equation (32) plays the dominant role in the nonlinear vibration process.  
Solving Equation (52) for the first order solution, \(\eta_{1}\), is obtained as  
\[\begin{split}\eta_{1}&=\frac{1}{96\Omega_{0}^{2}}[(48 \Omega_{0}^{2}A-48\omega_{1}^{2}A-32k_{f}^{2}ca_{2}A^{2}-33k_{f}^{2}a_{3}A^{3}) \cos\tau\\ &\qquad\qquad-16k_{f}^{2}ca_{2}A^{2}\cos 2\tau-3k_{f}^{2}a_{3}A^{3} \cos 3\tau+48k_{f}^{2}ca_{2}A^{2}]\end{split} \tag{60}\]  
Finally, the approximate solution of Equation (32) is  
\[\begin{split}\eta=\eta_{0}+\eta_{1}&=A\cos\Omega t +\frac{\hbar}{96\Omega_{0}^{2}}[(48\Omega_{0}^{2}A-48\omega_{1}^{2}A-32k_{f}^{ 2}ca_{2}A^{2}-33k_{f}^{2}a_{3}A^{3})\cos\tau\\ &\qquad-16k_{f}^{2}ca_{2}A^{2}\cos 2\tau-3k_{f}^{2}a_{3}A^{3}\cos 3 \tau+48k_{f}^{2}ca_{2}A^{2}]\end{split} \tag{61}\]