## 3 Vertical Displacement of Cantilever Beam by IHAM
### Vertical Displacement at Free end of Cantilever Beam
#### 3.3.3 Control Parameters \(h_{1}\in[-1,0)\) for Convergence Rate  
By IHAM, when \(\varepsilon=8/\pi^{2}\), \(h=-0.5\), \(\varepsilon_{1}=24(\pi-2)/\pi^{3}\), \(h_{1}=-0.5\), the 10 th-order approximation of \(V_{b}\) is  
\[ \begin{array}{l}V_{b}=\frac{1}{3}\alpha-3.7639\times 10^{-2}\alpha^{3}+7.5564 \times 10^{-3}\alpha^{5}\\ -1.6013\times 10^{-3}\alpha^{7}+2.9034\times 10^{-4}\alpha^{9}-4.1074\times 10^{- 5}\alpha^{11}\\ +4.3984\times 10^{-6}\alpha^{13}-3.5126\times 10^{-7}\alpha^{15}+2.0566\times 10^{- 8}\alpha^{17}\\ -8.2692\times 10^{-10}\alpha^{19}+1.7210\times 10^{-11}\alpha^{21}\end{array}  \tag{3.39}\]  
When the relative error \(\Delta\leq 1\%\), the convergent regions of the 10 th-order approximation by IHAM are  
\[\alpha\in[0,3.3],~{}V_{b}\in[0,0.63],~{}0\leq\delta_{v}\leq 0.63L \tag{3.40}\]  
Table 3 compares the 10 th-order numerical solutions with the exact solutions. The relative error in Table 3 is calculated by  
\[\Delta=\frac{\left|V_{b}^{[\pi=10]}-V_{b}\right|}{V_{b}}\times 100\% \tag{3.41}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(a\)} & \multirow{2}{*}{**Exact Solution**} & \multicolumn{2}{c}{\(\varepsilon_{1}=0\)} & \multicolumn{2}{c}{\(\varepsilon_{1}=24(\pi-2)/\pi^{3}\)} & \multicolumn{2}{c}{\(\varepsilon_{1}=1\)} \\  & & **(HAM)** & & **(HAM)** & & **(HAM)** \\ \cline{2-7}  & \(V_{b}\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) \\ \hline
**1.8** & 0.46326 & 0.46494 & 0.361\% & 0.44997 & 2.87\% & **0.44800** & **3.30\%** \\ \hline
**1.9** & 0.47879 & 0.47448 & 0.899\% & 0.46146 & 3.62\% & **0.45975** & **3.98\%** \\ \hline
**2** & 0.49346 & 0.48139 & 2.44\% & **0.47184** & 4.38\% & **0.47058** & **4.64\%** \\ \hline \(\infty\) & **1** & **1** & **0** & **1** & **0** & **1** & **0** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\alpha\)} & \multicolumn{2}{c}{**Exact Solution**} & \multicolumn{2}{c}{\(\mathbf{h_{1}=1}\)} & \multicolumn{2}{c}{\(\mathbf{h_{1}=-0.5}\)} & \multicolumn{2}{c}{\(\mathbf{h_{1}=-0.3}\)} \\ \cline{2-7}  & \(V_{b}\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0.2 & 0.066364 & 0.066365 & **0.000951\%** & 0.06368 & **0.00522\%** & 0.066385 & **0.0316\%** \\
0.4 & 0.13098 & 0.13098 & **0.00286\%** & 0.13100 & **0.0183**\% & 0.13113 & **0.120\%** \\
0.6 & 0.19235 & 0.19236 & **0.00418**\% & 0.19241 & **0.0339\%** & 0.19283 & **0.252\%** \\
0.8 & 0.24945 & 0.24946 & **0.00464** & 0.24957 & **0.0480**\% & 0.25047 & **0.410\%** \\
1.0 & 0.30172 & 0.30174 & **0.00486**\% & 0.30190 & **0.0602**\% & 0.30348 & **0.583\%** \\
1.2 & 0.34901 & 0.34902 & **0.00369**\% & 0.34926 & **0.0713**\% & 0.35168 & **0.765\%** \\
1.4 & 0.39147 & 0.39145 & **0.00312**\% & 0.39178 & **0.0803**\% & 0.39519 & **0.950\%** \\
1.6 & 0.42941 & 0.42933 & **0.02017**\% & 0.42977 & **0.0829**\% & 0.43429 & 1.14\% \\
1.8 & 0.46326 & 0.4636 & **0.04675**\% & 0.46361 & **0.0750**\% & 0.46936 & 1.32\% \\
2.0 & 0.49346 & 0.49313 & **0.0665**\% & 0.49374 & **0.0572**\% & 0.50082 & 1.49\% \\
2.2 & 0.52042 & 0.52014 & **0.05339**\% & 0.52061 & **0.0371**\% & 0.52908 & 1.66\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: 10 th-order numerical solutions \(V_{b}^{[\pi=10]}\) by IHAM with different values of \(h_{1}\) (\(\varepsilon_{1}=24(\pi-2)/\pi^{3}\)) compared with exact solutions \(V_{b}\). After \(n=10\) iterative calculations, when \(h_{1}=-1\), the convergence region of \(V_{b}\) is \(0~{}<\alpha\leq 3\). When \(h_{1}=-0.5\), the convergence region of \(V_{b}\) is \(0~{}<\alpha\leq 3.2\). When \(h_{1}=-0.3\), the convergence region of \(V_{b}\) is \(0~{}<\alpha\leq 1.4\). It is seen that when \(h_{1}=-0.5\), the convergence region is \(128\%\) larger than that with \(h_{1}=-0.3\), while it is \(6.7\%\) larger than that with \(h_{1}=-1\) (See Table 3).