Nonlinear Vibrations of a Nanobeams Rested on Nonlinear Elastic Foundation Under Primary Resonance Excitation

Abstract

In this paper, a comprehensive analysis of the nonlinear vibrations of nanobeams on nonlinear foundations under primary resonance excitation is presented. By utilizing advanced theories and highlighting the distinctions from previous work, we provide valuable insights into the behavior of these structures and their interaction with the supporting foundation. The results contribute to advancing the understanding and design of micro/nanoscale systems in a wide range of applications. The nanobeam is modeled in this paper as a Euler–Bernoulli beam with size-dependent properties. The material length scale parameter in this non-classical nanobeam model accounts for size effects at the nanoscale. For the nanobeam, two boundary conditions are taken into account: simply supported and clamped–clamped. The system's governing equation of motion is derived using the modified couple stress theory, and the accompanying boundary conditions are obtained by applying Hamilton's principle. This hypothesis enhances the analysis's precision by accounting for size effects. To arrive at an approximative analytical solution, the study employs an analytical method called the multiple-scale method. To manage primary resonance excitation in nonlinear systems, this technique is frequently used. The analysis takes into account a number of parameters, including the nonlinear foundation parameter (K_NL), Winkler parameter (K_L), Pasternak parameter (K_P), and material length scale parameter (l/h). These variables have a significant impact on how the nanobeam behaves on the nonlinear foundation. The study includes numerical results in graphical and tabular formats that show how the linear fundamental frequency, nonlinear frequency ratio, and vibration amplitude are affected by the material length scale parameter and stiffness coefficients of the nonlinear foundation. The research includes a comparison study with prior literature on related issues to verify the accuracy of the results acquired.

Appendix

$$\begin{gathered} \beta_{2} = + \sqrt {\frac{{ - K_{P} + \sqrt {\left( {K_{P} } \right)^{2} - 4(1 + \eta )(K_{L} - \omega^{2} )} }}{2(1 + \eta )}} \hfill \\ \beta_{3} = - \sqrt {\frac{{ - K_{P} - \sqrt {\left( {K_{P} } \right)^{2} - 4(1 + \eta )(K_{L} - \omega^{2} )} }}{2(1 + \eta )}} \hfill \\ \beta_{4} = + \sqrt {\frac{{ - K_{P} - \sqrt {\left( {K_{P} } \right)^{2} - 4(1 + \eta )(K_{L} - \omega^{2} )} }}{2(1 + \eta )}} \hfill \\ \end{gathered}$$

Simple–Simple Support

Support condition:

$$\begin{gathered} \beta_{1}^{2} \left( {\left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)} \right.\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2}^{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{2} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\left. {\beta_{4}^{2} } \right) \hfill \\ + \beta_{2}^{2} \left( {\left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)} \right.\left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\left. {\left. {\beta_{4}^{2} } \right)} \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\left. {\left. {\beta_{4}^{2} } \right)} \right) \hfill \\ + \left( {\left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)} \right.\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{2} \beta_{4}^{2} = 0 \hfill \\ \end{gathered}$$

The coefficients in the mode shapes

$$\begin{gathered} c_{2} = \frac{{\left( { - e^{{i\beta_{3} }} + e^{{i\beta_{4} }} } \right)\beta_{1}^{2} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\beta_{4}^{2} }}{{\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2}^{2} + \left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{3}^{2} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4}^{2} }} \hfill \\ c_{3} = \frac{{\left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{1}^{2} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{2}^{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)\beta_{4}^{2} }}{{\left( { - e^{{i\beta_{3} }} + e^{{i\beta_{4} }} } \right)\beta_{2}^{2} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{2} - \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4}^{2} }} \hfill \\ c_{4} = \frac{{\left( { - e^{{i\beta_{2} }} + e^{{i\beta_{3} }} } \right)\beta_{1}^{2} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\beta_{2}^{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)\beta_{3}^{2} }}{{\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2}^{2} + \left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{3}^{2} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4}^{2} }} \hfill \\ \end{gathered}$$

Clamped–Clamped Support

Support condition:

$$\begin{gathered} \beta_{1}^{{}} \left( {\left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)} \right.\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\beta_{3}^{{}} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\left. {\beta_{4} } \right) \hfill \\ + \beta_{2}^{{}} \left( {\left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)} \right.\left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{3} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\left. {\left. {\beta_{4} } \right)} \right)\left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\left. {\left. {\beta_{4} } \right)} \right) \hfill \\ + \left( {\left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)} \right.\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{3} \beta_{4} = 0 \hfill \\ \end{gathered}$$

The coefficients in the mode shapes

$$\begin{gathered} c_{2} = \frac{{\left( { - e^{{i\beta_{3} }} + e^{{i\beta_{4} }} } \right)\beta_{1} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{3} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\beta_{4} }}{{\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2} + \left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{3} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4} }} \hfill \\ c_{3} = \frac{{\left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{1} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{4} }} } \right)\beta_{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)\beta_{4} }}{{\left( { - e^{{i\beta_{3} }} + e^{{i\beta_{4} }} } \right)\beta_{2} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{4} }} } \right)\beta_{3} - \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4} }} \hfill \\ c_{4} = \frac{{\left( { - e^{{i\beta_{2} }} + e^{{i\beta_{3} }} } \right)\beta_{1} + \left( {e^{{i\beta_{1} }} - e^{{i\beta_{3} }} } \right)\beta_{2} - \left( {e^{{i\beta_{1} }} - e^{{i\beta_{2} }} } \right)\beta_{3} }}{{\left( {e^{{i\beta_{3} }} - e^{{i\beta_{4} }} } \right)\beta_{2} + \left( { - e^{{i\beta_{2} }} + e^{{i\beta_{4} }} } \right)\beta_{3} + \left( {e^{{i\beta_{2} }} - e^{{i\beta_{3} }} } \right)\beta_{4} }} \hfill \\ \end{gathered}$$