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Analytical investigation and numerical verification of Casimir effect on electrostatic nano-cantilevers

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Abstract

In this paper, the two-point boundary value problem (BVP) of the nano-cantilever deflection subjected to Casimir and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the nano-beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. The pull-in parameters of the beam are computed under the combined effects of electrostatic and Casimir forces. Electrostatic microactuators and freestanding nanoactuators are considered as special cases of our study. The detachment length and the minimum initial gap of freestanding nanocantilevers, which are the basic design parameters for NEMS switches, are determined. The results of the analytical study are verified by numerical solution of the BVP. The centerline of the beam under the effect of electrostatic and Casimir forces at small deflections and at the point of instability is obtained numerically to test the validity of the shape function assumed for the beam deflection in the analytical investigation. Finally, the large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of nano-canilevers.

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Acknowledgments

The authors acknowledge the valuable comments of the reviewers. The first author (A. Ramezani) wishes to acknowledge the assistance of Dr. Panahi in preparation of this paper.

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Authors and Affiliations

  1. School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Asghar Ramezani

  2. Center of Excellence in Design, Robotics, and Automation (CEDRA), School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Aria Alasty & Javad Akbari

Authors
  1. Asghar Ramezani
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  2. Aria Alasty
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  3. Javad Akbari
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Corresponding author

Correspondence to Asghar Ramezani.

Appendix

Appendix

Solutions of the integrals appearing in Eq. 24 are

$${{\int {\frac{{z^{2}}}{{(a^{2} - z^{2})^{2}}}}}dz = - \frac{1}{{2a}}\tanh^{{- 1}} (\frac{z}{a}) + \frac{z}{{2(a^{2} - z^{2})}}}$$
(A.1)
$${{\int {\frac{{z^{3}}}{{(a^{2} - z^{2})^{2}}}}}dz = \frac{1}{2}\ln{\left| {z^{2} - a^{2}} \right|} + \frac{{a^{2}}}{{2(a^{2} - z^{2})}}}$$
(A.2)
$${{\int {\frac{{z^{2}}}{{(a^{2} - z^{2})^{4}}}}}dz = -\frac{1}{{16a^{5}}}\tanh^{{- 1}} (\frac{z}{a}) + \frac{{z(- 3z^{4} +8a^{2} z^{2} + 3a^{4})}}{{48a^{4} (a^{2} - z^{2})^{3}}}}$$
(A.3)
$${{\int {\frac{{z^{3}}}{{(a^{2} - z^{2})^{4}}}}}dz = \frac{{3z^{2} -a^{2}}}{{12(a^{2} - z^{2})^{3}}}}$$
(A.4)
$${{\int {\frac{{z^{2}}}{{a^{2} - z^{2}}}}}dz = - z + a \cdot \tanh^{{-1}} (\frac{z}{a})}$$
(A.5)
$${{\int {\frac{{z^{3}}}{{a^{2} - z^{2}}}}}dz = - \frac{1}{2}z^{2} -\frac{1}{2}a^{2} \ln {\left| {z^{2} - a^{2}} \right|}}$$
(A.6)

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Ramezani, A., Alasty, A. & Akbari, J. Analytical investigation and numerical verification of Casimir effect on electrostatic nano-cantilevers. Microsyst Technol 14, 145–157 (2008). https://doi.org/10.1007/s00542-007-0409-y

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  • DOI: https://doi.org/10.1007/s00542-007-0409-y

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