## 4 Horizontal Displacement of Cantilever Beam by IHAM
### Horizontal Displacement at Free End of Cantilever Beam
#### 4.3.2 Explicit Solution to Horizontal Displacement by IHAM  
The solutions by IHAM contains two auxiliary parameters \(\varepsilon_{2}\) and \(h_{2}\), which control and adjust the convergence region and rate of IHAM solution series. For example, the 10 th-order approximation of \(U_{b}\) with \(n=10\), \(\varepsilon=8/\pi^{2}\), \(h=-0.5\), \(\varepsilon_{2}=8/\pi^{2}\), \(h_{2}=-0.4\) is derived as  
\[ \begin{array}{l}U_{b}=6.6128\times 10^{-2}\alpha^{2}-1.2132\times 10^{-2} \alpha^{4}+2.6029\times 10^{-3}\alpha^{6}\\ -5.0108\times 10^{-4}\alpha^{8}+7.4259\times 10^{-5}\alpha^{10}-8.1166\times 10^{-6} \alpha^{12}\\ +6.5090\times 10^{-7}\alpha^{14}-3.7900\times 10^{-8}\alpha^{16}+1.5743\times 10^{-9 }\alpha^{18}\\ -3.4811\times 10^{-11}\alpha^{20}\end{array}  \tag{4.31}\]  
When the relative error \(\Delta\leq 1\%\), the convergent regions of the 10 th-order approximation by IHAM are  
\[\alpha\in[0,2.8],\;U_{b}\in[0,0.24],\;0\leq\delta_{h}\leq 0.24L \tag{4.32}\]  
The 20 th-order approximation of \(U_{b}\) with \(n=20\), \(\varepsilon=8/\pi^{2}\), \(h=-0.3\), \(\varepsilon_{2}=8/\pi^{2}\), \(h_{2}=-0.4\) is expressed as  
\[ \begin{array}{l}U_{b}=6.6663\times 10^{-2}\alpha^{2}-1.2647\times 10^{-2} \alpha^{4}+3.0553\times 10^{-3}\alpha^{6}\\ -7.8913\times 10^{-4}\alpha^{8}+1.9565\times 10^{-4}\alpha^{10}-4.2939\times 10^{-5 }\alpha^{12}\\ +7.8795\times 10^{-6}\alpha^{14}-1.1689\times 10^{-6}\alpha^{16}+1.3828\times 10^{-7 }\alpha^{18}\\ -1.3058\times 10^{-8}\alpha^{20}+9.9286\times 10^{-10}\alpha^{22}-6.1404\times 10^{- 11}\alpha^{24}\\ +3.1142\times 10^{-12}\alpha^{26}-1.3002\times 10^{-13}\alpha^{28}+4.4587\times 10^{- 15}\alpha^{30}\\ -1.2419\times 10^{-16}\alpha^{32}+2.7429\times 10^{-18}\alpha^{34}-4.5778\times 10^{- 20}\alpha^{36}\\ +5.1643\times 10^{-22}\alpha^{38}-2.9055\times 10^{-24}\alpha^{40}\end{array}  \tag{4.33}\]  
When the relative error \(\Delta\leq 1\%\), the convergent region of the 20 th-order approximation by IHAM is  
\[\alpha\in[0,3.4],\;U_{b}\in[0,0.29],\;0\leq\delta_{h}\leq 0.29L \tag{4.34}\]  
The 30 th-order approximation of \(U_{b}\) with \(n=30\), \(\varepsilon=8/\pi^{2}\), \(h=-0.1\), \(\varepsilon_{2}=8/\pi^{2}\), \(h_{2}=-0.3\) is \[U_{b}=6.6665\times 10^{-2}\alpha^{2}-1.1891\times 10^{-2}\alpha^{4}+2.4037 \times 10^{-3}\alpha^{6} \tag{4.35}\] \[-4.6144\times 10^{-4}\alpha^{8}+7.8466\times 10^{-5}\alpha^{10}-1.1452 \times 10^{-5}\alpha^{12}\] \[+1.4110\times 10^{-6}\alpha^{14}-1.4525\times 10^{-7}\alpha^{16}+1. 2395\times 10^{-8}\alpha^{18}\] \[-8.7140\times 10^{-10}\alpha^{20}+5.0250\times 10^{-11}\alpha^{22}-2. 3731\times 10^{-12}\alpha^{24}\] \[+9.1975\times 10^{-14}\alpha^{26}-2.9421\times 10^{-15}\alpha^{28}+7. 8331\times 10^{-17}\alpha^{30}\] \[-1.7531\times 10^{-18}\alpha^{32}+3.3309\times 10^{-20}\alpha^{34}-5. 4217\times 10^{-22}\alpha^{36}\] \[+7.6171\times 10^{-24}\alpha^{38}-9.2885\times 10^{-26}\alpha^{40}+9. 8639\times 10^{-28}\alpha^{42}\] \[-9.1295\times 10^{-30}\alpha^{44}+7.3481\times 10^{-32}\alpha^{46}-5. 1117\times 10^{-34}\alpha^{48}\] \[+3.0383\times 10^{-36}\alpha^{50}-1.5131\times 10^{-38}\alpha^{52}+6. 1056\times 10^{-41}\alpha^{54}\] \[U_{b}=6.6665\times 10^{-2}\alpha^{2}-1.1891\times 10^{-2}\alpha^{4}+2.4037 \times 10^{-3}\alpha^{6} \tag{4.35}\]0  
When the relative error \(\Delta\leq 1\%\), the convergent regions of the 30 th-order approximation by IHAM are  
\[U_{b}=6.6665\times 10^{-2}\alpha^{2}-1.1891\times 10^{-2}\alpha^{4}+2.4037 \times 10^{-3}\alpha^{6} \tag{4.35}\]1  
Table 6 compares the above numerical solutions with the exact solutions.