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Modal interactions in a two-nanomechanical-resonator
array
Clément Grenat, Sébastien Baguet, Régis Dufour, Claude-Henri Lamarque
To cite this version:
MODAL INTERACTIONS IN A
TWO-NANOMECHANICAL-RESONATOR ARRAY
C. GRENAT1, S. BAGUET1, R. DUFOUR1and C-H. LAMARQUE2
1Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, F-69621, France
Email: [email protected], [email protected], [email protected] 2Univ Lyon, ENTPE, CNRS UMR5513, LTDS , F-69518, France
Email: [email protected]
ABSTRACT
MEDYNA 2017 25-27 Apr 2017, Sevilla (Spain)
1 INTRODUCTION
Arrays of MEMS or NEMS present complex dynamical behaviors due to electric, magnetic and mechanic nonlinear couplings. Lifshitz and Cross [1] studied the responses of n electrical and mechanical coupled oscillators with parametric resonance in the low nonlinear limit by using a perturbation method. Gutschmidt and Goettlieb worked on arrays with electrical coupling. They focused on the n-beam dynamic behaviour in the region of internal one-to-one, parametric and several internal three-to-one resonances corresponding to low, medium and large DC voltages [2]. Kacem et al. developped a single beam model to investigate the sensitivity of the resonance with respect to the electrostatic forcing. Their researches were carried out using averaging method validated by HBM+ANM [3]. In this paper, an array of n = 2 identical clamped-clamped beams is also considered but coupled only by an electrostatic force in order to study the modal interactions between the two beams due to the electrostatic coupling.
2 ARRAY OF TWO NANOMECHANICAL RESONATORS
A 2-moving-beam array is considered, as sketched in Figure 1. The two beams located at the ends of the array are fixed and serve only as electrostatic actuator. All 4 beams are identical. l, b, h, I, g are the dimensions of the beams, i.e., length, width, height, moment of inertia, gap between two adjacent beams. E, ρ be the Young’s modulus and the material density. Each
0 1 2 3 micro-beam Vdc10 Vac10 Vdc32 Vac32 g O Vdc21 Vac21 h W (x,t)s x O
Figure 1: Array of two clamped-clamped M/NEMS beams.
beam is an electrostatic actuator for its adjacent beams. Vs,s+1 = Vdcs,s+1+ Vacs,s+1cos(Ωt) is the voltage applied between the successive beams s and s + 1 with Vdc, Vac the continuous and alternative voltages. The equation of the beam s in bending is as follows [2].
EI∂ 4w˜ s(˜x, ˜t) ∂ ˜x4 + ρbh ∂2w˜ s(˜x, ˜t) ∂˜t2 −h ˜N + Ebh 2l Z l 0 ∂ ˜ws(˜x, ˜t) ∂ ˜x 2 d˜xi∂ 2w˜ s(˜x, ˜t) ∂ ˜x2 = 0bCn 2 " V2 s,s+1 g + ˜ws+1− ˜ws 2 − V2 s−1,s g + ˜ws− ˜ws−1 2 # (1)
with s = 1, 2. Let 0, Cnbe the dielectric constant and fringing the coefficient respectively. ˜N represents the lineic load along the x-axis. The beams 1 and 2 are clamped-clamped.
3 AVERAGING METHOD
The responses of the beams are more complicated than those of a single beam resonator. A quasi-analytic solution obtained by the averaging method can be used to explain why the elec-trostatic coupling generates additional loops onto the responses. First, the beam lateral deflec-tion is expanded on its fundamental mode only:
w1(x, t) = φ1(x)a11(t), w2(x, t) = φ1(x)a21(t) (2)
First-order Taylor series are then used to simplify the analytic calculation: 1 (1 + ws+1− ws)2 ' 1 − 2(ws+1− ws), 1 (1 + ws− ws−1)2 ' 1 − 2(ws− ws−1) (3) Since the resonators of the 2-beam array have the same boundary conditions, their eigenmodes are identical. Therefore, a Galerkin method is used to eliminate the spatial dependence from the equation of motion (1). Then using the averaging method and considering the solutions a11(t), a21(t) in following forms a11= A11(t) cos(Ωt) + B11(t) sin(Ωt), (4) a21= A21(t) cos(Ωt) + B21(t) sin(Ωt), (5) yield Ω ˙A11 = −B11ω1σ1− 1 2cA11Ω + 3 8β11A 2 11B11+ 3 8β11B 3 11 +1 8β13B11+ 1 2δ11B21+ 1 8δ13B21 (6) Ω ˙B11 = 1 2cB11Ω + 3 8β11A 3 11+ 3 8β13A11+ 1 2γ12+ 3 8β11A11B 2 11 −A11ω1σ1+ 1 2δ11A21+ 3 8δ13A21 (7) Ω ˙A21 = −B21ω2σ2− 1 2cA21Ω + 3 8β21A 2 21B21+ 3 8β21B 3 21 +1 8β23B21+ 1 2δ21B11+ 1 8δ23B11 (8) Ω ˙B21 = 1 2cB21Ω + 3 8β21A 3 21+ 3 8β23A21+ 1 2γ22+ 3 8β21A21B 2 21 −A21ω2σ2+ 1 2δ21A11+ 3 8δ23A11 (9)
where the coefficients βij, γij, δij depend on the beam characteristics and on the applied volt-ages. They are not detailed here for the sake of conciseness. In Equations (6)-(9), the coupling terms (12δ11+ 18δ13)B21 and (12δ11+ 18δ13)A21 represent the influence of the second beam on the first beam and the coupling terms (12δ21+ 18δ23)B11, (12δ21+ 38δ23)A11 the influence of the first beam on the second beam. When ˙A11 = ˙B11 = ˙A21 = ˙B21 = 0 the steady-state mo-tions appear. The corresponding nonlinear algebraic system is solved by an adapted numerical method, the obtained approximated solution is in agreement with a reference solution obtained by HBM+ANM [3] not shown here.
MEDYNA 2017 25-27 Apr 2017, Sevilla (Spain) 22.36 22.38 22.4 22.42 22.44 22.46 22.48 22.5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Ωnon−dimensional Wmax D1 E1 B1 C1 without influence terms
22.39 22.4 22.41 22.42 22.43 0.01 0.015 0.02 0.025 E1 (a) 22.36 22.38 22.4 22.42 22.44 22.46 22.48 22.5 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Ωnon−dimensional Wmax B2 C2 D2 E2 22.385 22.395 22.405 0.08 0.09 0.1 0.11 C2 B2 22.488 22.49 22.492 0.21 0.211
0.212 without influence terms
(b)
Figure 2: Design 1 without added mass, Response by neglecting the coupling terms (blue), complete response (red). (a): first-beam response, (b): second-beam response.
and (8)-(9) form two independent systems of equations. Therefore, the responses of the two beams, (blue response curves in Figure 2) are similar to two single-beam responses.
With coupling terms Equations (6)-(9) are dependent and share the same bifurcation points and stability. When a bifurcation point is present on a response curve, the same bifurca-tion point will also be present at the same frequency on the other beam response. Therefore, the limit points originated from the responses without coupling terms will be present on all the other beam responses with coupling terms. This leads to the appearance of additional limit points on the response curves. In Figure 2, the limit points B1and C1on the first beam response generate at the same frequencies the loop B2 − C2 on the second-beam response. In the same way, D2 and E2on the second-beam response produce the loop D1− E1 on the first-beam response. 4 CONCLUSION
A quasi-analytic analysis with the averaging method of a two-nanomechanical-resonator array has been carried out. The existence of modal interactions between the two beams due to the electrostatic coupling has been enlightened. The appearance of additional loops onto response curves has been explained. The form of the response curves of an electrostatic coupled beam array can be anticipated using the uncoupled single-beam-resonator responses. This research represents an increment towards the comprehension and modeling of resonator arrays for appli-cation in mass sensing.
REFERENCES
[1] R. Lifshitz and M. C. Cross. Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators, pages 1–52. Wiley-VCH Verlag GmbH & Co. KGaA, 2009.
[2] S. Gutschmidt and O. Gottlieb. Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large dc-voltages. Nonlinear Dynamics, 67(1):1– 36, 2012.
[3] N. Kacem, S. Baguet, S. Hentz, and R. Dufour. Computational and quasi-analytical models for non-linear vibrations of resonant mems and nems sensors. International Journal of Non-Linear Mechanics, 46(3):532–542, 2011.
