## 3 Vertical Displacement of Cantilever Beam by IHAM
### Vertical Displacement at Free end of Cantilever Beam
#### 3.3.4 30 th-Order Approximation Solution of Vertical Displacement  
When \(\varepsilon=8/\pi^{2}\), \(h=-0.1\), the 30 th-order approximation of \(\theta_{b}\) by IHAM is  
\[ \begin{array}{l}\theta_{b}=\frac{\alpha}{2}-4.3804\times 10^{-2}\alpha^{3}+ 7.2872\times 10^{-3}\alpha^{5}-1.3299\times 10^{-3}\alpha^{7}\\ +2.2991\times 10^{-4}\alpha^{9}-3.5481\times 10^{-5}\alpha^{11}+4.7492\times 10^{- 6}\alpha^{13}\\ -5.4234\times 10^{-7}\alpha^{15}+5.2245\times 10^{-8}\alpha^{17}-4.2081\times 10^{- 9}\alpha^{19}\\ +2.8138\times 10^{-10}\alpha^{21}-1.5540\times 10^{-11}\alpha^{23}+7.0725\times 1 0^{-13}\alpha^{25}\\ -2.6561\times 10^{-14}\alpha^{27}+8.2720\times 10^{-16}\alpha^{29}-2.1528\times 1 0^{-17}\alpha^{31}\\ +4.7252\times 10^{-19}\alpha^{33}-8.8286\times 10^{-21}\alpha^{35}+1.4162\times 1 0^{-22}\alpha^{37}\\ -1.9643\times 10^{-24}\alpha^{39}+2.3681\times 10^{-26}\alpha^{41}-2.4891 \times 10^{-28}\alpha^{43}\\ +2.2824\times 10^{-30}\alpha^{45}-1.8213\times 10^{-32}\alpha^{47}+1.2569 \times 10^{-34}\alpha^{49}\\ -7.4143\times 10^{-37}\alpha^{51}+3.6656\times 10^{-39}\alpha^{53}-1.4683 \times 10^{-41}\alpha^{55}\\ +4.4741\times 10^{-44}\alpha^{57}-9.1359\times 10^{-47}\alpha^{59}+9.2009 \times 10^{-50}\alpha^{61}\\ \end{array}  \tag{3.42}\]  
When the relative error \(\Delta\leq 1\%\), the convergent region of the 30 th-order approximation by IHAM is  
\[\alpha\in[0,5.1],~{}\Theta_{b}\in[0,0.79],~{}0\leq\theta_{b}\leq 70.6^{\circ} \tag{3.43}\]  
When \(\varepsilon_{1}=0\), \(h_{1}=-0.3\), the 30 th-order approximation of \(V_{b}\) by HAM is  
\[ \begin{array}{l}V_{b}=\frac{1}{3}\alpha-3.6198\times 10^{-2}\alpha^{3}+6.5568 \times 10^{-3}\alpha^{5}\\ -1.2305\times 10^{-3}\alpha^{7}+2.1160\times 10^{-4}\alpha^{9}-3.1714\times 1 0^{-5}\alpha^{11}\\ +4.0427\times 10^{-6}\alpha^{13}-4.3197\times 10^{-7}\alpha^{15}+3.8284 \times 10^{-8}\alpha^{17}\\ -2.7902\times 10^{-9}\alpha^{19}+1.6610\times 10^{-10}\alpha^{21}-8.0436 \times 10^{-12}\alpha^{23}\\ +3.1678\times 10^{-13}\alpha^{25}-1.0188\times 10^{-14}\alpha^{27}+2.6964 \times 10^{-16}\alpha^{29}\\ -5.9333\times 10^{-18}\alpha^{31}+1.0971\times 10^{-19}\alpha^{33}-1.7218 \times 10^{-21}\alpha^{35}\\ +2.3130\times 10^{-22}\alpha^{37}-2.6772\times 10^{-25}\alpha^{39}+2.6808 \times 10^{-27}\alpha^{41}\\ -2.3258\times 10^{-29}\alpha^{43}+1.7456\times 10^{-31}\alpha^{45}-1.1275 \times 10^{-33}\alpha^{47}\\ +6.2040\times 10^{-36}\alpha^{49}-2.8576\times 10^{-38}\alpha^{51}+1.0708 \times 10^{-40}\alpha^{53}\\ -3.0973\times 10^{-43}\alpha^{55}+6.1662\times 10^{-46}\alpha^{57}-6.2456 \times 10^{-49}\alpha^{59}\\ \end{array}  \tag{3.44}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\boldsymbol{\alpha}\)} & \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{2-9}  & \(V_{b}\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) & \(V_{b}\) & \(\Delta\) \\ \hline
2.4 & 0.54455 & 0.54472 & _0.0319\%_ & 0.54471 & _0.0296\%_ & 0.55457 & 1.84\% \\
2.6 & 0.56619 & 0.56745 & _0.222\%_ & 0.56646 & _0.0475\%_ & 0.57767 & 2.03\% \\
2.8 & 0.58567 & 0.58846 & _0.476\%_ & 0.58610 & _0.0474\%_ & 0.59864 & 2.21\% \\
3 & 0.60325 & 0.60709 & _0.636\%_ & 0.60340 & _0.0243\%_ & 0.61745 & 2.35\% \\
3.2 & 0.61918 & 0.62601 & 1.10\% & 0.61870 & _0.0777\%_ & 0.63428 & 2.44\% \\
3.4 & 0.63365 & 0.68393 & 7.94\% & 0.64405 & 1.64\% & 0.65542 & 3.44\% \\
3.6 & 0.64684 & — & — & 0.76181 & 17.8\% & 0.72504 & 12.1\% \\
3.8 & 0.65890 & — & — & — & — & — \\
4.0 & 0.66996 & — & — & — & — & — \\ \(\infty\) & 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 3: \(Cont.\)Table 4 compares the above numerical solution with the exact solution.