Dynamic pull-in of thermal cantilever nanoswitches subjected to dispersion and axial forces using nonlocal elasticity theory

Abstract

Precise analysis of nanoelectromechanical systems has an outstanding contribution in performance improvement of such systems. In this research, the dynamic instability of a cantilever nanobeam connected to a horizontal spring is analyzed. The system is subjected to thermal, electrostatic and molecular (Casimir and van der Waals) forces. By applying the Eringen’s nonlocal elasticity theory, the equilibrium equations are derived. The nonlinear dynamics governing equations of the actuated thermal switch are solved by reduced order method. Finally, the effects of several system parameters on the dynamic behavior of the nanocantilever are examined in detail. It is concluded that considering the nonlocal theory results in increasing the rigidity of cantilever nanobeams, unlike fixed-fixed nanobeams. Furthermore, the nonlocality affects more significantly by increasing the temperature of cantilevers; however, it is completely the opposite for double-clamped beams. The obtained results can be considered for modeling and analysis of several thermal micro and nanosystems.

Appendix 1

1.1 The governing equation of electrostatic and van der Waals attractions

$$\begin{aligned} &{\text{u}}^{\prime\prime}_{\text{n}} + \upomega _{\text{n}}^{2} {\text{u}}_{\text{n}} - 5(1 + \upmu {\text{N}}^{*} )\,\upomega _{1}^{2} {\text{u}}_{1}^{2} \int_{0}^{1} {\text{q}}_{1}^{3} {\text{dx}} + 10(1 + \upmu {\text{N}}^{*} )\upomega _{1}^{2} {\text{u}}_{1}^{3} \int_{0}^{1} {\text{q}}_{1}^{4} {\text{dx}} - 10(1 + \upmu {\text{N}}^{*} )\upomega _{1}^{2} {\text{u}}_{1}^{4} \int_{0}^{1} {\text{q}}_{1}^{5} {\text{dx}} \hfill \\ &\quad + 5(1 + \upmu {\text{N}}^{*} )\upomega _{1}^{2} {\text{u}}_{1}^{5} \int_{0}^{1} {\text{q}}_{1}^{6} {\text{dx}} - (1 + \upmu {\text{N}}^{*} )\upomega _{1}^{2} {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} - 5{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 10{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {\text{q}}_{1}^{4} {\text{dx}} \hfill \\ &\quad - 10{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 5{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {\text{q}}_{1}^{6} {\text{dx}} - {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {\text{q}}_{1}^{7} {\text{dx}} - \upmu {\text{u}}^{\prime\prime}_{1} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1} } {\text{dx}} + 5\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ &\quad - 10\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} + 10\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} - 5\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + \upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ &\quad + \upmu {\text{V}}^{2} \left(6 + 2\frac{{0.65{\text{d}}}}{\text{b}}\right){\text{u}}_{1}^{2} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1} } {\text{dx}} - \upmu {\text{V}}^{2} \left(6 + 4\frac{{0.65{\text{d}}}}{\text{b}}\right){\text{u}}_{1}^{3} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{2} } {\text{dx}} + \left[ {\upmu {\text{V}}^{2} \left(2 + \frac{{0.65{\text{d}}}}{\rm{b}}\right) - {\text{N}}^{*} } \right]{\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\ &\quad + \left[ {\upmu {\text{V}}^{2} \left(4 + 3\frac{{0.65{\text{d}}}}{\rm{b}}\right) - 5{\text{N}}^{*} } \right]{\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} + \left[ {\upmu {\text{V}}^{2} \left(2 + 3\frac{{0.65{\text{d}}}}{\rm{b}}\right) - 10{\text{N}}^{*} } \right]{\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} \hfill \\ &\quad + 2\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\rm{b}}{\text{u}}_{1}^{4} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{3} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\rm{b}} - 10{\text{N}}^{*} } \right]{\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} + 5{\text{N}}^{*} {\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + {\text{N}}^{*} {\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} \hfill \\ &\quad + 12\upmu \uplambda _{3} {\text{u}}_{1}^{2} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{{}} } {\text{dx}} + 3\upmu \uplambda _{3} {\text{u}}_{1}^{{}} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{{}} } {\text{dx}} - 3\upmu \uplambda _{3} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - {\text{V}}^{2} \left(1 + \frac{{0.65{\text{d}}}}{\rm{b}}\right)\int_{0}^{1} {{\text{q}}_{1}^{{}} } {\text{dx}} \hfill \\ &\quad + {\text{V}}^{2} \left(3 + 4\frac{{0.65{\text{d}}}}{\rm{b}}\right){\text{u}}_{1}^{{}} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - {\text{V}}^{2} \left(3 + 6\frac{{0.65{\text{d}}}}{\rm{b}}\right){\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + {\text{V}}^{2} \left(1 + 4\frac{{0.65{\text{d}}}}{\rm{b}}\right){\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} \hfill \\ &\quad - {\text{V}}^{2} \frac{{0.65{\text{d}}}}{\rm{b}}{\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} - \uplambda _{3} \int_{0}^{1} {{\text{q}}_{1}^{{}} } {\text{dx}} + 2\uplambda _{3} {\text{u}}_{1}^{{}} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - \uplambda _{3} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} = 0. \hfill \\ \end{aligned}$$

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1.2 The governing equation of electrostatic and Casimir attractions

$$\begin{aligned} &{\text{u}}^{\prime\prime}_{\text{n}} + \upomega_{\text{n}}^{2} {\text{u}}_{\text{n}} - 6(1 + \upmu {\text{N}}^{*} )\,\upomega_{1}^{2} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 15(1 + \upmu {\text{N}}^{*} )\upomega_{1}^{2} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - 20(1 + \upmu {\text{N}}^{*} )\upomega_{1}^{2} {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} \hfill \\ &\quad + 15(1 + \upmu {\text{N}}^{*} )\upomega_{1}^{2} {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} - 6(1 + \upmu {\text{N}}^{*} )\upomega_{1}^{2} {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} + (1 + \upmu {\text{N}}^{*} \upomega_{1}^{2} {\text{u}}_{1}^{7} \int_{0}^{1} {{\text{q}}_{1}^{8} } {\text{dx}} \hfill \\ &\quad- 6{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} + 15{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - 20{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + 15{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} - 6{\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{7} } {\text{dx}} \hfill \\&\quad + {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{6} \int_{0}^{1} {{\text{q}}_{1}^{8} } {\text{dx}} - \upmu {\text{u}}^{\prime\prime}_{1} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1} } {\text{dx}} + 6\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - 15\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} + 20\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} \hfill \\&\quad - 15\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + 6\upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} - \upmu {\text{u}}^{\prime\prime}_{1} {\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{7} } {\text{dx}} + \upmu {\text{V}}^{2} \left(6 + 2\frac{0.65{\rm{d}}}{\text{b}}\right){\rm{u}}_{1}^{2} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1} } {\text{dx}} \hfill \\&\quad - \upmu {\text{V}}^{2} \left(12 + 6\frac{0.65{\text{d}}}{\rm{b}}\right){\text{u}}_{1}^{3} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{2} } {\text{dx}} + \upmu {\text{V}}^{2} \left(6 + 6\frac{0.65{\text{d}}}{\text{b}}\right){\text{u}}_{1}^{4} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{3} } {\text{dx}} \hfill \\&\quad + \left[ {\upmu {\text{V}}^{2} \left(12 + \frac{0.65{\text{d}}}{\text{b}}\right) - {\text{N}}^{*} } \right]{\text{u}}_{1} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} \left(36 + 4\frac{0.65{\text{d}}}{\text{b}}\right) - 6{\text{N}}^{*} } \right]{\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\&\quad + \left[ {\upmu {\text{V}}^{2} \left(6 + 6\frac{0.65{\text{d}}}{\text{b}}\right) - 15{\text{N}}^{*} } \right]{\text{u}}_{1}^{3} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{3} } {\text{dx}} - \left[ {\upmu {\text{V}}^{2} \left(12 + 4\frac{0.65{\text{d}}}{\text{b}}\right) - 20{\text{N}}^{*} } \right]{\text{u}}_{1}^{4} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{4} } {\text{dx}} \hfill \\&\quad - 2\upmu {\text{V}}^{2} \frac{0.65{\text{d}}}{{\text{b}}}{\text{u}}_{1}^{5} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1}^{4} } {\text{dx}} + \left(\upmu {\text{V}}^{2} \frac{0.65{\text{d}}}{\text{b}} - 15{\text{N}}^{*} \right){\text{u}}_{1}^{5} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{5} } {\text{dx}} + 6{\text{N}}^{*} {\text{u}}_{1}^{6} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{6} } {\text{dx}} \hfill \\&\quad - 6{\text{N}}_{\rm {t}}{\text{u}}_{1}^{7} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{7} } {\text{dx}} + 20\upmu \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {\dot{{\text{q}}}_{1}^{2} {\text{q}}_{1} } {\text{dx}} + 4\upmu \uplambda_{4} {\text{u}}_{1} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1} } {\text{dx}} - 4\upmu \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {\ddot{{\text{q}}}_{1} {\text{q}}_{1}^{2} } {\text{dx}} \hfill \\&\quad - {\text{V}}^{2} \left(1 + \frac{0.65{\rm{d}}}{\text{b}}\right)\int_{0}^{1} {{\text{q}}_{1} } {\text{dx}} + {\text{V}}^{2} \left(4 + 5\frac{0.65{\text{d}}}{\text{b}}\right){\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - {\text{V}}^{2} \left(6 + 10\frac{0.65{\rm{d}}}{\text{b}}\right){\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} \hfill \\&\quad + {\text{V}}^{2} \left(4 + 10\frac{0.65{\rm{d}}}{\text{b}}\right){\text{u}}_{1}^{3} \int_{0}^{1} {{\text{q}}_{1}^{4} } {\text{dx}} - {\text{V}}^{2} \left(1 + 5\frac{0.65{\rm{d}}}{\text{b}}\right){\text{u}}_{1}^{4} \int_{0}^{1} {{\text{q}}_{1}^{5} } {\text{dx}} + {\text{V}}^{2} \frac{0.65{\rm{d}}}{\text{b}}{\rm{u}}_{1}^{5} \int_{0}^{1} {{\text{q}}_{1}^{6} } {\text{dx}} \hfill \\&\quad - \uplambda_{4} \int_{0}^{1} {{\text{q}}_{1} } {\text{dx}} + 2\uplambda_{4} {\text{u}}_{1} \int_{0}^{1} {{\text{q}}_{1}^{2} } {\text{dx}} - \uplambda_{4} {\text{u}}_{1}^{2} \int_{0}^{1} {{\text{q}}_{1}^{3} } {\text{dx}} = 0. \hfill \\ \end{aligned}$$

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