## 3 Vertical Displacement of Cantilever Beam by IHAM  
### Zero Order Deformation Equation  
The nonlinear boundary value Equations (2.3b) and (2.4b) are then solved by IHAM to derive explicit form for \(V_{b}\). To start with, \(V(\xi)\) is expressed by power series of \(\xi\), i.e.,  
\[V(\xi)=\sum_{k=1}^{+\infty}b_{k}\xi^{k} \tag{3.1}\]  
where \(b_{k}\) is a constant coefficient. Since Equation (2.7b) from linear equation is an appropriate initial guess, we choose  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]  
The equivalent nonlinear Equation (2.3b) becomes  
\[V^{\prime}-\theta+\frac{e_{1}}{3!}\theta^{3}=0 \tag{3.3}\]  
subject to boundary condition  
\[V(0)=0 \tag{3.4}\]  
where \(e_{1}\) is the coefficient, and \(e_{1}\in[0,1]\). When \(e_{1}=0\), Equation (3.3) becomes linear Equation (2.5b), so that  
\[V^{\prime}-\theta=0 \tag{3.5}\]When \(\varepsilon_{1}{=}\) 1, Equation (3.3) becomes  
\[V^{\prime}-\theta+\frac{1}{3!}\theta^{3}=0 \tag{3.6}\]  
Equation (3.6) is an approximate expression of the second-order Taylor expansion of sine function in view of that  
\[\sin\theta\approx\theta-\frac{1}{3!}\theta^{3} \tag{3.7}\]  
According to the original Equation (2.3b) and the equivalent Equation (3.3), we define a nonlinear operator as  
\[\mathrm{N}_{v}[\
u(\xi;q),\psi(\xi,q),q,\varepsilon_{1}]=\frac{\mathrm{d}\
u( \xi;q)}{\mathrm{d}\xi}-\sin[q\cdot\psi(\xi,q)]\]  
\[-(1-q)\psi(\xi,q)+\frac{\varepsilon_{1}}{3!}\Big{(}1-q^{3}\Big{)}[\psi(\xi,q) ]^{3} \tag{3.8}\]  
where \(q\in[0,1]\) is an embedding parameter; \(\
u(\xi;q)\), \(\psi(\xi;q)\) are dependent on \(\xi\) and \(q\); \(\varepsilon_{1}\) is a control parameter to represent the convergent region, and \(\varepsilon_{1}\in[0,\ 1]\). When \(q=0\), there is  
\[\mathrm{N}_{v}[\
u(\xi;0),\psi(\xi,0),0,\varepsilon_{1}]=\frac{\mathrm{d}\
u( \xi;0)}{\mathrm{d}\xi}-\psi(\xi,0)+\frac{\varepsilon_{1}}{3!}[\psi(\xi,0)]^{3} \tag{3.9}\]  
If \(q=1\), it yields  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]0  
Letting  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]1  
be an auxiliary linear operator, and \(h\in[-1,0)\) denotes a nonzero auxiliary parameter (called convergence-control parameter), a homotopy is then constructed as  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]2  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]3  
When \(q=0\), we have  
\[V_{0}(\xi)=\frac{\alpha}{\xi}\xi^{2}(3-\xi) \tag{3.2}\]4  
If \(q=1\), there is  
\[ \begin{array}{c}\mathcal{H}_{v}[\
u(\xi;1),1]:=-h_{1}\mathrm{N}_{v}[\
u(\xi ;1),1,\varepsilon_{1}]\\ =-h_{1}\Big{\{}\frac{\mathrm{d}\
u(\xi;1)}{\mathrm{d}\xi}-\sin[\psi(\xi,1)] \Big{\}}\end{array}  \tag{3.14}\]  
Thus, by enforcing  
\[\mathcal{H}_{v}[\
u(\xi;q),q]=0\]  
we have a family of equations  
\[(1-q)\mathrm{L}_{v}[\
u(\xi;q)-V_{0}(\xi)]=qh_{1}\mathrm{N}_{v}[\
u(\xi;q), \psi(\xi,q),q,\varepsilon_{1}] \tag{3.15}\]  
subject to the boundary condition of  
\[\
u(0;q){=}0 \tag{3.16}\]When \(q=0\), we have  
\[\
u(\xi;0)=V_{0}(\xi),\psi(\xi;0)=\theta_{0}(\xi) \tag{3.17}\]  
When \(q=1\), Equations (3.15) and (3.16) are equivalent to the original Equations (2.3b) and (2.4b), provided that  
\[\
u(\xi;1)=V(\xi),\psi(\xi;1)=\theta(\xi) \tag{3.18}\]  
Thus, as \(q\) increases from \(0\) to \(1\), \(\
u(\xi;q)\) evolves smoothly from the specified initial guess \(V_{0}(\xi)\) to the solution \(V(\xi)\) of Equations (2.3b) and (2.4b). With reference to Equation (3.17), by expanding \(\
u(\xi;q)\) into Taylor series of the embedding parameter \(q\), we have  
\[\
u(\xi;q)=V_{0}(\xi)+\sum_{n=1}^{+\infty}V_{n}(\xi)q^{n} \tag{3.19a}\] \[\psi(\xi;q)=\theta_{0}(\xi)+\sum_{n=1}^{+\infty}\theta_{n}(\xi)q^{n} \tag{3.19b}\]  
where  
\[V_{n}=\left.\frac{1}{n!}\frac{\partial^{n}\
u(\xi;q)}{\partial q^{n}}\right| _{q=0},\;\theta_{n}=\left.\frac{1}{n!}\frac{\partial^{n}\psi(\xi;q)}{\partial q ^{n}}\right|_{q=0}\]  
It is worth noting that the zero order deformation Equation (3.15) contains auxiliary parameters \(\varepsilon_{1},h_{1}\).