## 10 Conclusions  
The present research deals with a comprehensive analysis of the nonlinear vibration of CNTRC nanobeams subjected to an electromagnetic actuator, mechanical impact, and nonlinear Winkler-Pasternak foundation. A generalized nonlocal beam theory according to the Eringen condition was used, and a higher-order shear deformation beam theory was applied to some types of beams, including Euler-Bernoulli and Timoshenko beams, as a special case in the present formulation, accounting for NSGT. The mechanical impact was especially considered for small load lengths. To the best of the authors' knowledge, no study has been conducted to account for the extension of normal and shear strain components with a generalized curvature in the presence of the nonlinear forces mentioned above. The governing equations were obtained based on the von [PERSON] theory and then on the Hamilton principle. Any system with infinite degrees of freedom was discretized by the Galerkin-Bubnov technique, and then the ordinary differential equations were solved by the Optimal Auxiliary Functions Method, even if the small parameter did not appear in the governing equations or in the initial/boundary conditions. It should be emphasized that the presence of a so-called auxiliary function, according to some theories about differential equations (operator method, Cauchy method, the method of influence functions), implies a rapid convergence of the approximate solutions using only the first iteration. The solutions obtained by the Optimal Auxiliary Functions Method were validated by comparisons with the numerical results gained using a fourth-order Runge-Kutta procedure. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{2}{c}{**UD-Beam**} & \multicolumn{2}{c}{**X-Beam**} & \multicolumn{2}{c}{**O-Beam**} \\ \hline \(V_{CNT}\) & _ea_ & **Error** & \(p\) & **Error** & \(p\) & **Error** & \(p\) \\ \hline \multirow{3}{*}{0.12} & 0 & 0.1984 & 1.85 & 0.0237 & 0.116 & 0.1165 & 0.96 \\  & \(\sqrt{2}\) & 0.1123 & 0.71 & 0.1107 & 0.582 & 0.0287 & 0.265 \\  & 2 & 0.0293 & 0.2 & 0.0612 & 0.343 & 0.0128 & 0.131 \\ \hline \multirow{3}{*}{0.17} & 0 & 0.0543 & 0.2 & 0.125 & 0.382 & 0.0244 & 0.131 \\  & \(\sqrt{2}\) & 0.174 & 0.7 & 0.0957 & 0.315 & 0.061 & 0.413 \\  & 2 & 0.0745 & 0.33 & 0.0674 & 0.238 & 0.075 & 0.507 \\ \hline \multirow{3}{*}{0.28} & 0 & 0.2055 & 0.57 & 0.0928 & 0.216 & 0.1114 & 0.42 \\  & \(\sqrt{2}\) & 0.0241 & 0.072 & 0.1106 & 0.273 & 0.0672 & 0.283 \\ \cline{1-1}  & 2 & 0.0324 & 0.104 & 0.2213 & 0.578 & 0.0571 & 0.265 \\ \hline \end{tabular}
\end{table} 
Table 7: The errors and percentages for the results obtained in Table 1.  
Analytical solutions for critical buckling and the maximum deflection problems were studied by applying HPM, the Galerkin method in combination with the weighted residual method, and finally, by the optimization of results for simply supported composite nanobeams. Effects of nonlocality and length were investigated for some particular cases, as well as the influence of parameters, such as the volume fraction of CNTRC or size dependency for inhomogeneous or homogeneous nanobeam models. Tables 1-6 present some quantitative results obtained using OAFM, HPM, and Galerkin methods in comparison with the weighted residual method, and we compared them with those known in literature. Also, in Table 7, the errors and percentages were given corresponding to the presented results. The main novelties in this work were represented by:  
- Extension of normal and shear strain;  
- A generalized curvature;  
- Introduction of the functions \(f(x,t)\), \(g(x,t)\), and \(h(x,t)\);  
- Combination between the electromagnetic actuator, nonlinear elastic Winkler-Pasternak foundation, and mechanical impact;  
- The Optimal Auxiliary Functions Method;  
- Study of buckling and bending using Homotopy Perturbation Method in combination with the Galerkin method and then the optimization method. The obtained results were discussed and compared with the available results from literature.