Large-Amplitude Vibration Analysis of an Electrostatically Actuated Nanobeam with Weak Interacting Forces

Abstract

Previous studies on the vibration of a nanobeam in the presence of weak interacting forces have been limited to small- to moderate-amplitude vibrations where the effects of weak forces are not significant. In this study, the large-amplitude vibration of a nonlinear nanobeam in the presence of the van der Waals, Casimir, electrostatic and fringing forces was investigated. The Bubnov–Galerkin approach was employed to transform the governing partial differential equation of the nanobeam to a nonlinear ordinary differential equation (NODE) characterized by strong inertia and static nonlinearities. The NODE for the nanobeam vibration was solved using the continuous piecewise linearization method (CPLM). The ability of the CPLM algorithm to accurately predict the large-amplitude vibration of the nanobeam was verified by comparing its results with highly accurate numerical solutions and other approximate analytical solutions found in the literature. It was observed that the results of the CPLM algorithm agreed well with corresponding results of highly accurate numerical solution. Also, the CPLM produced more accurate results than the published approximate analytical solutions. Furthermore, the effects of the van der Waals force, Casimir force, electrostatic force and fringing force on the frequency response of the large-amplitude nanobeam were studied and it was found that an increase in any of the weak forces causes a significant reduction in the vibration frequency of the nanobeam. The study shows that the CPLM can be applied for accurate nonlinear analysis of the large-amplitude vibration of conservative systems with complex mixed-parity nonlinearities.

Appendices

Appendix A

See Table 4.

Table 4 Value of coefficients in Eq. (13)

Appendix B

Energy Balance Method

Applying the basic principles of the energy balance method (EBM), the displacement solution for Eq. (14) can be expressed as [15]

$$u\left(\tau \right)=Acos\left(\omega \tau \right)$$

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where $\omega$ is the natural frequency, $\tau$ is the dimensionless time and $A$ is the amplitude. The natural frequency solution based on the EBM was derived as (Ghalambaz et al. 2016)

$$\omega =\frac{1}{30}\sqrt{\frac{4{725E}_{6}{A}^{10}+5580{E}_{5}{A}^{8}+6750{E}_{4}{A}^{6}+8400{E}_{3}{A}^{4}+10800{E}_{2}{A}^{2}+14400{E}_{1}}{\left({16a}_{2}-32{a}_{4}{A}^{2}+24{a}_{6}{A}^{4}-8{a}_{8}{A}^{6}+{a}_{10}{A}^{8}\right)}}$$

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Global Residue Harmonic Balance Method

The first-order approximation of the solution to Eq. (14) based on the global residue harmonic balance method was derived as [ 22]

$${u}_{1}\left(\tau \right)=\left(A+{a}_{31}\right)cos\left(\omega \tau \right)-{a}_{31}cos\left(3\omega \tau \right)$$

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where $\tau$ is the dimensionless time and $A$ is the amplitude. The natural frequency $\omega$ and the constant ${a}_{31}$ were derived as [22]

$$\omega =\sqrt{\frac{231{E}_{6}{A}^{10}+252{E}_{5}{A}^{8}+280{E}_{4}{A}^{6}+320{E}_{3}{A}^{4}+384{E}_{2}{A}^{2}+512{E}_{1}}{\left({512a}_{2}-1536{a}_{4}{A}^{2}+1920{a}_{6}{A}^{4}-1120{a}_{8}{A}^{6}+252{a}_{10}{A}^{8}\right)}+{\omega }_{1}}$$

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$${a}_{31}=\frac{{\Phi }_{1}{\Gamma }_{2}}{{\Phi }_{2}{\Gamma }_{1}+{\Phi }_{1}{\Gamma }_{3}}$$

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where

$${\omega }_{1}=\frac{{\Gamma }_{1}{\Gamma }_{2}}{{\Phi }_{2}{\Gamma }_{1}+{\Phi }_{1}{\Gamma }_{3}}$$

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$${\Gamma }_{1}={\frac{147}{128}a}_{10}{A}^{8}{{\omega }_{0}}^{2}-{\frac{35}{8}a}_{8}{A}^{6}{{\omega }_{0}}^{2}+{\frac{45}{8}a}_{6}{A}^{4}{{\omega }_{0}}^{2}-2{a}_{4}{A}^{2}{{\omega }_{0}}^{2}-{a}_{2}{{\omega }_{0}}^{2}+\frac{363}{256}{A}^{10}{E}_{6}+\frac{189}{128}{A}^{8}{E}_{5}+\frac{49}{32}{A}^{6}{E}_{4}+\frac{25}{16}{A}^{4}{E}_{3}+\frac{3}{2}{A}^{2}{E}_{2}+{E}_{1}$$

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$${\Gamma }_{2}={-\frac{21}{64}a}_{10}{A}^{9}{{\omega }_{0}}^{2}+{\frac{21}{16}a}_{8}{A}^{7}{{\omega }_{0}}^{2}-{\frac{15}{8}a}_{6}{A}^{5}{{\omega }_{0}}^{2}+{a}_{4}{A}^{3}{{\omega }_{0}}^{2}+\frac{165}{512}{A}^{11}{E}_{6}+\frac{21}{64}{A}^{9}{E}_{5}+\frac{21}{64}{A}^{7}{E}_{4}+\frac{5}{16}{A}^{5}{E}_{3}+\frac{{A}^{3}}{4}{E}_{2}$$

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$${\Gamma }_{3}={\frac{285}{128}a}_{10}{A}^{8}{{\omega }_{0}}^{2}-{\frac{21}{2}a}_{8}{A}^{6}{{\omega }_{0}}^{2}+{\frac{159}{8}a}_{6}{A}^{4}{{\omega }_{0}}^{2}-19{a}_{4}{A}^{2}{{\omega }_{0}}^{2}+9{a}_{2}{{\omega }_{0}}^{2}+\frac{363}{1024}{A}^{10}{E}_{6}+\frac{27}{128}{A}^{8}{E}_{5}-\frac{5}{16}{A}^{4}{E}_{3}-\frac{3}{2}{A}^{2}{E}_{2}-{E}_{1}$$

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$${\Phi }_{1}=\left(-{\frac{63}{128}a}_{10}{A}^{9}+\frac{35}{16}{a}_{8}{A}^{7}-\frac{15}{4}{a}_{6}{A}^{5}+3{a}_{4}{A}^{3}-{a}_{2}A\right)$$

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$${\Phi }_{2}=\left({\frac{21}{64}a}_{10}{A}^{9}-\frac{21}{64}{a}_{8}{A}^{7}+\frac{15}{8}{a}_{6}{A}^{5}+{a}_{4}{A}^{3}\right)$$

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$${\omega }_{0}=\sqrt{\frac{231{E}_{6}{A}^{10}+252{E}_{5}{A}^{8}+280{E}_{4}{A}^{6}+320{E}_{3}{A}^{4}+384{E}_{2}{A}^{2}+512{E}_{1}}{\left({512a}_{2}-1536{a}_{4}{A}^{2}+1920{a}_{6}{A}^{4}-1120{a}_{8}{A}^{6}+252{a}_{10}{A}^{8}\right)}}$$

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