## 3 Explicit Solution to Rotation Angle for Large Deformation of Cantilever Beams by IHAM
### Zero Order Deformation Equation  
The nonlinear boundary value Equations (47) and (48) are then solved by IHAM to derive a explicit formula for \(\theta_{b}\). To start with, \(\theta(\xi)\) is expanded into the power series of \(\xi\):  
\[\theta(\xi)=\sum_{k=1}^{+\infty}a_{k}\xi^{k} \tag{54}\]  
Equation (52) is then identified as an initial guess. Accordingly, the initial guess of \(\theta(\xi)\) is expressed as:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]  
For simplicity, the below simplest auxiliary nonlinear differential equation is chosen  
\[\theta^{\prime\prime}+\alpha\Big{(}1-\frac{\varepsilon}{2!}\theta^{2}\Big{)}=0 \tag{56}\]  
which is subjected to the boundary conditions  
\[\theta(0)=0,\;\theta^{\prime}(1)=0, \tag{57}\]  
where \(\varepsilon\) is the correction coefficient of the convergent region, and \(\varepsilon\in[0,1]\). When \(\varepsilon=0\), Equation (56) becomes the linear Equation (50). When \(\varepsilon=1\), Equation (56) becomes:  
\[\theta^{\prime\prime}+\alpha\Big{(}1-\frac{1}{2!}\theta^{2}\Big{)}=0 \tag{58}\]  
In fact, the approximating second-order Taylor expansion for the cosine function in Equation (47) can be written as:  
\[\cos\theta\approx 1-\frac{1}{2!}\theta^{2}\]  
According to the original Equation (47) and the equivalent Equation (56), we define one nonlinear operator as:  
\[[\psi(\xi,q),q,\varepsilon]=\frac{\mathrm{d}^{2}\psi(\xi,q)}{\mathrm{d}\xi^{2 }}+\alpha\cos[q\cdot\psi(\xi,q)]-\frac{\alpha\varepsilon}{2!}\Big{(}1-q^{2} \Big{)}[\psi(\xi,q)]^{2} \tag{59}\]  
where \(q\in[0,1]\) is an embedded parameter, \(\psi(\xi,q)\) is a function dependent on \(\xi\) and \(q\), \(\varepsilon\) is a control parameter for the convergent region, and \(\varepsilon\in[0,1]\). When \(q=0\), there is:  
\[\mathrm{N}[\psi(\xi,0),0,\varepsilon]=\frac{\mathrm{d}^{2}\psi(\xi,0)}{ \mathrm{d}\xi^{2}}+\alpha-\frac{\alpha\varepsilon}{2!}[\psi(\xi,0)]^{2}\]  
If \(q=1\), we have  
\[\mathrm{N}[\psi(\xi,1),1,\varepsilon]=\frac{\mathrm{d}^{2}\psi(\xi,1)}{ \mathrm{d}\xi^{2}}+\alpha\cos[\psi(\xi,1)]\]  
Letting  
\[\mathrm{L}[\psi(\xi,q)]=\frac{\partial^{2}\psi(\xi,q)}{\partial\xi^{2}} \tag{60}\]  
Being an auxiliary linear operator, and \(\mathrm{h}\in[-1,0)\) denotes a nonzero auxiliary parameter (convergence-control parameter), a homotopy is then constructed as:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]0When \(q=0\), we have  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]1  
If \(q=1\), Equation (61) becomes:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]2  
Thus, by enforcing  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]3  
We have a family of equations:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]4  
which are subjected to the boundary conditions of:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]5  
The prime in Equation (63) denotes differentiation with respect to \(\xi\). When \(q=0\), Equation (62) incurs:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]6  
When \(q=1\), Equations (62) and (63) are equivalent to the original Equations (47) and (48), provided that:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]7  
Thus, as \(q\) increases from \(0\) to \(1\), \(\psi(\xi,q)\) evolves from the known initial guess \(\theta_{0}(\xi)\) to the solution \(\theta(\xi)\) in Equations (47) and (48). By expanding \(\psi(\xi;q)\) into a Taylor series of the embedded parameter \(q\) and using Equation (64), we have:  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]8  
where  
\[\theta_{0}(\xi)=\frac{\alpha}{2}(2-\xi)\xi \tag{55}\]9  
Assuming that \(\mathrm{h}\) and \(\varepsilon\) are properly chosen so that the series Equation (66) are convergent at \(q=1\), according to Equation (65), we have the series:  
\[\theta^{\prime\prime}+\alpha\Big{(}1-\frac{\varepsilon}{2!}\theta^{2}\Big{)}=0 \tag{56}\]0  
And the governing equations of \(\theta_{n}(\xi)\) can then be deduced from the zero-order deformation Equations (62) and (63).