Supplementary information for: Frequency stabilization in nonlinear micro-mechanical oscillators

Transcription

1 Supplementary information for: Frequency stabilization in nonlinear micro-mechanical oscillators Dario Antonio Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL, USA Damián H. Zanette Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche and Instituto Balseiro 8400 Bariloche, Río Negro, Argentina Daniel López Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL, USA (Dated: April 10, 2012)

2 2 Oscillation frequency! Oscillation amplitude 1/2 A 1 / Driving force amplitude Driving force amplitude F 0 /m 1 Supplementary Figure S1. Oscillation frequency ν of the two coupled modes as a function of the driving force amplitude F 0, for ɛ 1/ω 2 = ɛ 2/ω 2 = 10 4 and j 1/ω2 2 = j 2/ω2 2 = The frequency ratio between the two modes is ω 2/ω 1 = Axes have been rescaled to make the plot independent of the specific values of ω 2 and β. In the horizontal axis, α = ω 3 2 β/(ɛ1/ω 2). Arrows show the hysteresis cycle in an experiment where F 0 is first increased, in such a way that ν overcomes ω 2 and jumps to the upper branch after the stabilization interval, and then decreased. The insert shows the oscillation amplitude A 1 of the first mode as a function of the driving force amplitude.

3 3 a b c A m p litu d e (m V ) F re q u e n c y (k H z ) Supplementary Figure S2. Internal resonance in different resonators. We have detected the coupling between the first mode and higher frequency modes at the internal resonance condition in a variety of micromechanical resonators. Depending on the frequency of the higher modes relative to the first mode, and the detuning reached with the maximum driving strength, one or more internal resonance conditions were detected (indicated by arrows in Supplementary Figure S2). In all cases the lower frequency mode was the first in-plane flexural mode, but the high frequency modes were both in-plane and out of plane, primary and secondary flexural modes, and also torsional modes. Therefore, the possibility of coupling with different types of modes gives great flexibility for the design of oscillators using this stabilization mechanism. All the devices are c-c resonators, actuated and detected using comb-drive electrodes, but differ in their dimensions and in the number of their parallel interconnected beams. The vertical arrows indicate the presence of an internal resonance condition a, Resonator 1, beam length l = 500µm, thickness t = 10µm, wih w = 2µm, number of beams n = 6. b, Resonator 2, same nominal dimensions as resonator 1 but with a coating of 20 nm of chrome and 500 nm of gold. c, Resonator 3, l = 1000µm, thickness t = 25µm, wih w = 2µm, number of beams n = 1.

4 4 SUPPLEMENTARY METHODS Analytical model for frequency stabilization by internal resonance Frequency stabilization by internal resonance in the clamped-clamped micro-oscillator considered in this paper is due to the transfer of energy from the first oscillation mode to the principal torsional mode. In the self-sustaining configuration, increasing the driving force causes a growth of both the amplitude and, due to nonlinear effects, the frequency f 1 of the first mode. When f 1 reaches one third of the frequency f 3 of the principal torsional mode, internal resonance takes place, and energy is transferred from the former to the latter. The coupling between the two modes is mediated by the 3f 1 -component of the first mode oscillations (1:3 internal resonance). As a result of this energy transfer, the growth rate of the amplitude and the frequency of the first mode decreases abruptly and, hence, both become stabilized. To demonstrate this mechanism by means of an exactly solvable analytical model, we study the analogous case of a 1:1 internal resonance, which simplifies the analysis without affecting the nature of the problem, and consider the coupled dynamics of two oscillators, each of them representing one of the participating modes. The first mode is represented by a nonlinear self-sustaining oscillator described by the equation d 2 x 1 m c dx m 1ω1x k 3 x 3 1 = F 0 cos(φ 1 (t) ψ 0 ) + J(x 2 x 1 ). (1) The first term in the right-hand-side is the driving force, with φ 1 (t) the oscillation phase of x 1 (t) and ψ 0 the phase shift of the force. The second term stands for the coupling with the higher frequency mode, with coupling strength J, and derives from a harmonic interaction potential V int = J 2 (x 2 x 1 ) 2. The higher frequency mode, in turn, is represented by a linear oscillator coupled to the first: d 2 x 2 m c dx m 2ω2x 2 2 = J(x 1 x 2 ). (2) We assume that ω 2 > ω 1, in such a way that increasing the driving force amplitude F 0 makes the frequency of the first oscillator approach ω 2, thus producing the resonance. In view of the dependence of the driving force of the first oscillator on its phase, it is convenient to introduce a phase-amplitude representation for the two oscillators, replacing the position x i (t) and the velocity v i (t) = dxi (i = 1, 2) by the phase φ i (t) and the amplitude A i (t). The new variables are defined by x i (t) = A i (t) cos φ i (t), v i (t) = νa i (t) sin φ i (t), (3) where ν is, in principle, an arbitrary constant. Seeking for solutions with time-independent A 1 and A 2, as expected to be the case in the long-time asymptotic oscillations, we get dφ1 = dφ2 = ν, which shows that the two oscillators move synchronously with frequency ν. We take φ 1 (t) = νt and φ 2 (t) = νt ψ 2, with ψ 2 the phase shift between the two oscillators. Separating terms in sin νt and cos νt and neglecting higher-harmonic contributions proportional to cos 3νt, the equations of motion become algebraic equations for the four unknowns ν, A 1, A 2, and ψ 2. For a driving phase shift ψ 0 = π/2, which corresponds to operating the nonlinear oscillator in its resonance peak, the oscillation frequency ν is given by the equation ( ) 2 ν 2 = ω1 2 + j 1 + j 1j 2 (ν 2 ω2 2 j 2 ) ν 2 ɛ (ν2 ω2 2 j 2) 2 + β F 0 /m 1 ν 2, (4) ɛ 1 + ɛ 2j 1j 2 ν 2 ɛ 2 2 +(ν2 ω 2 2 j2)2 with ɛ i = c i /m i, j i = J/m i (i = 1, 2), and β = 4k 3 /3m 1. This is equivalent to a polynomial third-degree equation for ν 2, whose exact solutions can be found analytically. Once it has been solved, the remaining unknowns are given by the relations and A 1 = νɛ 1 + F 0 /m 1 νɛ 2j 1j 2 (5) ν 2 ɛ 2 2 +(ν2 ω2 2 j2)2 j 2 A 1 A 2 exp iψ 2 = ν 2 ω2 2 j. (6) 2 + iνɛ 2

5 Supplementary Figure S1 illustrates the results for ν and A 1 as functions of the driving force amplitude F 0. The stabilization of the oscillation frequency and amplitude, as ν approaches ω 2, is apparent. Arrows show the path of an experiment in which, first, the driving force is increased in such a way that ν becomes stabilized, until it overcomes ω 2 and suddenly jumps to the upper branch. Then, upon decreasing F 0, the oscillation frequency decreases along the upper branch, completing the hysteresis cycle. 5