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Research Papers

Nonlinear Response of a Microbeam Under Combined Direct and Fringing Field Excitation

[+] Author and Article Information
Prashant N. Kambali

Department of Mechanical
and Aerospace Engineering,
Indian Institute of Technology,
Hyderabad 502205, India

Ashok Kumar Pandey

Department of Mechanical
and Aerospace Engineering,
Indian Institute of Technology,
Hyderabad 502205, India
e-mail: ashok@iith.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 8, 2014; final manuscript received January 29, 2015; published online April 6, 2015. Assoc. Editor: Carmen M. Lilley.

J. Comput. Nonlinear Dynam 10(5), 051010 (Sep 01, 2015) (10 pages) Paper No: CND-14-1069; doi: 10.1115/1.4029700 History: Received March 08, 2014; Revised January 29, 2015; Online April 06, 2015

Microelectromechanical system (MEMS) and Nanoelectromechanical system (NEMS) are mostly actuated by direct forcing due to electrostatic excitation. In general, the electrostatic forcing consists of two main components, the first is the direct forcing which is based on parallel plate capacitance and another is due to the fringing effects. As the size of the beam and its cross section reduces from microscale to nanoscale, the effect of direct forcing diminishes because the overlapping area also reduces. Consequently, the fringing force effect remains the only viable factor to excite the beams electrostatically. In this paper, we present the nonlinear analysis of fixed–fixed and cantilever beams subjected to the direct force excitation, the fringing force excitation, and the combined effect of direct and fringing forces. In the present configuration, while the direct forcing is achieved by applying voltage across the beam and the bottom electrode, the fringing force can be introduced by applying voltage across the beam and the symmetrically placed side electrodes. To do the analysis, we first formulate the equation of motion considering both kinds of forces. Subsequently, we apply the method of multiple scale, MMS, to obtain the approximate solution. After validating MMS with the numerical simulation, we discuss the influence of large excitation amplitude, nonlinear damping, and the nonlinear stiffness under different forcing conditions. We found that fringing force introduces parametric excitation in the system which may be used to significantly increase the response amplitude as well as frequency bandwidth. It is also found that under the influence of the fringing forces from the side electrodes, the pull-in effect can be improved. Furthermore, the present study can be used to increase the sensitivity as well as the operating frequency range of different MEMS and NEMS based sensors under combined forcing conditions.

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Figures

Grahic Jump Location
Fig. 1

(a) A schematic diagram showing the dimensions of the beam, side electrodes, bottom electrode, and the actuation voltages Vs and Vg; (b) schematic of electric field lines under the influence of fringing force and direct force, respectively; (c) sectional side view of the cantilever beam and the bottom electrode; and (d) sectional side view of the fixed–fixed beam and the bottom electrode

Grahic Jump Location
Fig. 2

Variation of approximate static deflection versus the dc-voltage parameter between the beam and the bottom electrode, i.e., η¯dcBb2 for different values of the dc voltage parameter between the beam and the side electrodes η¯dcBs2=0,0.2,0.4,0.6,0.8, and 1 for (a) the fixed–fixed beam and (b) the cantilever beam. Figures shown inside (a) and (b) represent the variation of pull-in point with the dc voltage parameter of the beam and the side electrodes (here, “LP” indicates the start of unstable solution, i.e., the pull-in point).

Grahic Jump Location
Fig. 3

Comparison of solutions based on MMS and the numerical simulation for the fixed–fixed microbeam under (a) the direct force excitation (η¯dcBs = 0,η¯dcBb = 0.02,η¯acBs = 0,η¯acBb = 0.07), (b) the fringing force excitation (η¯dcBs = 0.02,η¯dcBb = 0,η¯acBs = 0.07,η¯acBb = 0), and (c) the combined effect of the direct and fringing force excitation (η¯dcBs = 0.02,η¯dcBb = 0.02,η¯acBs = 0.07,η¯acBb = 0.07). Here, β = 1.84,μNL = 0.9,μL = 0.001, α = 1.

Grahic Jump Location
Fig. 4

Comparison of solutions based on MMS and the numerical simulation for the cantilever microbeam under (a) the direct force excitation (η¯dcBs = 0,η¯dcBb = 0.02,η¯acBs = 0,η¯acBb = 0.2), (b) the fringing force excitation (η¯dcBs = 0.02,η¯dcBb = 0,η¯acBs = 0.2,η¯acBb = 0), and (c) the combined effect of the direct and fringing force excitation (η¯dcBs = 0.02,η¯dcBb = 0.02,η¯acBs = 0.2,η¯acBb = 0.2). Here, β = 0.3,μNL = 0.5,μL = 0.001, α = 1.

Grahic Jump Location
Fig. 5

Frequency response of fixed–fixed microbeam for different ac voltages under the influence of (a) direct force excitation (η¯dcBs = 0,η¯dcBb = 0.2,η¯acBs = 0,η¯acBb = 0.4,0.6,0.8); (b) fringing force excitation (parametric excitation) (η¯dcBs = 0.2,η¯dcBb = 0,η¯acBs = 0.4,0.6,0.8,η¯acBb = 0); and (c) the combined (direct + fringing) force excitation (η¯dcBs = 0.2,η¯dcBb = 0.2,η¯acBs = 0.4,0.6,0.8,η¯acBb = 0.4,0.6,0.8). Here, β = 1.84,μL = 0.001,μNL = 0.5,α = 1. Solid lines represent stable solutions and dash lines represents unstable solutions.

Grahic Jump Location
Fig. 6

Frequency response of cantilever microbeam for different ac voltages under the influence of (a) direct force excitation (η¯dcBs = 0,η¯dcBb = 0.2,η¯acBs = 0,η¯acBb = 0.4,0.6,0.8); (b) fringing force excitation (parametric excitation) (η¯dcBs = 0.2,η¯dcBb = 0,η¯acBs = 0.4,0.6,0.8,η¯acBb = 0); and (c) the combined (direct + fringing) force excitation (η¯dcBs = 0.2,η¯dcBb = 0.2,η¯acBs = 0.4,0.6,0.8,η¯acBb = 0.4,0.6,0.8). Here, β = 0.3,μL = 0.001,μNL = 0.5,α = 1. Solid lines represent stable solutions and dash lines represents unstable solutions.

Grahic Jump Location
Fig. 7

The variation of frequency response of the beam for different values of nonlinear damping parameter μNL under (a) the direct force excitation, (b) the fringing force excitation (parametric excitation), and (c) the combined (direct + fringing) force excitation (here, β = 0,μL = 0.001,α = 1,η¯dcBb = η¯dcBs = 0.2, and η¯acBb = η¯acBs = 0.35)

Grahic Jump Location
Fig. 8

Effect of nonlinear stiffness, β, on the frequency response of the beam under (a) the direct force excitation, (b) the fringing force excitation (parametric excitation), and (c) the combined (direct + fringing) force excitation for the following parameter values μNL = 0.5,μL = 0.001,α = 1,η¯dcBb = η¯dcBs = 0.2,η¯acBb = η¯acBs = 0.8

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