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Research Papers

Modeling and Analysis of an Electrically Actuated Microbeam Based on Nonclassical Beam Theory

[+] Author and Article Information
Pierpaolo Belardinelli

Department of Civil and Building Engineering
and Architecture,
Polytechnic University of Marche,
Ancona 60131, Italy
e-mail: p.belardinelli@univpm.it

Stefano Lenci

Professor
Mem. ASME
Department of Civil and Building Engineering
and Architecture,
Polytechnic University of Marche,
Ancona 60131, Italy
e-mail: lenci@univpm.it

Maurizio Brocchini

Professor
Department of Civil and Building Engineering
and Architecture,
Polytechnic University of Marche,
Ancona 60131, Italy
e-mail: m.brocchini@univpm.it

1Address all correspondence to this author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 18, 2013; final manuscript received December 9, 2013; published online February 14, 2014. Assoc. Editor: Carmen M. Lilley.

J. Comput. Nonlinear Dynam 9(3), 031016 (Feb 14, 2014) (10 pages) Paper No: CND-13-1044; doi: 10.1115/1.4026223 History: Received February 18, 2013; Revised December 09, 2013

This work investigates the mechanical behavior of a clamped-clamped microbeam modeled within the framework of the strain-gradient elasticity theory. The governing equation of motion gives proper account of both the effect of the nonlinear midplane stretching and of an applied axial load. An electric-voltage difference, introducing into the model a further source of nonlinearity, is considered, including also a correction term for fringing field effects. The electric force acting on the microbeam is rearranged by means of the Chebyshev method, verifying the accuracy of the proposed approximation. The results show that a uniform error on the whole domain can be achieved. The static solution is obtained by a numerical differential quadrature method. The paper looks into the variation of the maximal deflection of the microbeam with respect to several parameters. Study of the pull-in limit on the high-order material parameters introduced by the nonclassical approach and a comparison with respect to the classical beam theory are also carried out. The numerical simulation indicates that the static response is larger, affected by the use of a nonclassical theory near the pull-in instability regime. The dynamical problem is, finally, analyzed, deriving the multi degree-of-freedom problem through a Galerkin-based approach. The study on the single degree-of-freedom model enables us to note the large influence of the nonlinear terms.

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Figures

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Fig. 1

1D model for the electrically actuated microbeam and the beam cross section

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Fig. 2

The slender microbeam in its undeformed shape, with the main geometrical quantities

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Fig. 3

Convergence rate for a linear distribution of spatial grid nodes (triangle) and for the Chebyshev–Gauss–Lobatto distribution (circle)

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Fig. 4

Convergence of the numerical solution with respect to an increasing number of discretization points. Specific focus is on the maximal deflection convergence for a microbeam with β = α1 = α3 = 0 and α2V2 = 30.

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Fig. 5

Variation of the approximation error of the function 1/(1-χ)2 using the third-order Taylor expansion (dashed and dotted lines) or series Eq. (42) (solid line)

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Fig. 6

Variation of the approximation error of the function 1/(1-χ)2 using the fourth-order Taylor expansion (dashed and dotted lines) or series Eq. (42) (solid line)

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Fig. 7

Variation of the maximum dimensionless static deflection wsmax = ws(x = 1/2), highlighting the influence of α3. For these curves, β = 0 and α1 = 0.

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Fig. 8

Variation of the maximum dimensionless static deflection wsmax = ws(x = 1/2), highlighting the influence of α1. For these curves, β = 0 and α3 = 10-4.

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Fig. 9

Variation of the maximum dimensionless static deflection wsmax = ws(x = 1/2) with respect to β and α2V2 for different values of the axial load N. In these curves, α1=1 and α3=10-5.

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Fig. 10

Variation of the response amplitude for a microbeam with α1 = 1, α3 = 10-5, β = 0.0637. The amplitude of the forcing term is vac = 0.01.

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Fig. 11

Variation of the response amplitude for a microbeam with vac = 0.1 and α2V2 = 10. The solid lines are the curves with β = 0, while for the dashed lines β = 0.0637.

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Fig. 12

Variation of the nonlinear response for a microbeam with respect to different values of α3. The other parameters are α1 = 0, α2V2 = 25, β = 0, N=-10. The amplitude of the dynamic forcing term is vac = 0.01. For the mechanical damping, we used the same graphical arrangement of Fig. 10.

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