Noise-induced chaos in the electrostatically actuated MEMS resonators
Highlights
► Nonlinear dynamics of electrostatically micro-resonators are investigated. ► A reduced-order model of the microresonator is developed. ► The necessary conditions for the rising of chaos are obtained.
Introduction
Micro-electromechanical resonators are one of the most commonly used components for various communication and signal processing applications [1]. Electrostatically actuated resonators have the advantages of simple structures that allow easy batch fabrications and they form a major component in many MEMS devices [2], [3]. However, the resonators have to driven close to or even into nonlinear regimes to store enough energy at micro scale [1], [4]. When a resonator is used as a frequency source in MEMS devices, noise plays numerous effects, such as limiting dynamic range and selectivity, causing loss of lock, limiting acquisition and reacquisition capability in phase-locked-loop systems, on their performance [5]. Various nonlinearities will occur and dominate in the resonator dynamic behavior due to the specific resonator design and the operating conditions.
In order to predict the influence of resonator nonlinearities on the performance of MEMS devices, the nonlinear dynamics of the resonators has to be investigated. Carr et al. [6] and Zalalutdinov et al. [7] studied the parametric amplification of the motion of resonators through electrostatic and optical actuation. However, the reported methods should be used to discuss the instability and control strategies. Abdel-Rahman and Nayfeh [8] applied the super-harmonic and combination parametric resonances as suitable excitation methods to minimize electrical “feedthrough”. The study on the array of parametrically excited strings were carried out both experimentally [9] and theoretically [10]. Hu et al. [11] discussed the resonances of electrostatically actuated micro-cantilevers, while Baskaran and Turner [12] demonstrated the coupled modes parametric resonance. Fu et al. [13] analyzed the nonlinear dynamic stability for an electrically actuated viscoelastic microbeam. Krylov et al. [14] investigated the possibility of parametric stabilization of electrostatically actuated microstructures under the effects of AC component and dc component voltages. Younis and Nayfeh [15] and Abdel-Rahman and Nayfeh [8] studied the dynamic responses of an electrostatically resonator to a primary-resonance excitation [15], a superharmonic-resonance excitation of order two [8], and a subharmonic-resonance excitation of order one-half [8]. It can be concluded that these models gave accurate results for small AC amplitudes and hence small motions. Kacem et al. [16] studied the nonlinear dynamics of nanomechanical beam resonators to improve the performance of MEMS-based sensors. Alsaleem et al. [17] investigated the nonlinear phenomena, including primary resonance, superharmonic and subharmonic resonances, in electrostatically actuated resonators both experimentally and theoretically. Zhang and Meng [18] analyzed the nonlinear dynamics of the electrostatically actuated resonant MEMS sensors under parametric excitation. Mestrom et al. [1] modeled the dynamics of a MEMS resonator numerically and experimentally considering the effect of thermal noise. Haghighi and Markazi [19] predicted the chaos in MEMS resonators and presented a robust adaptive fuzzy control method to control the chaotic motion. The chaotic motion was suppressed and came into a periodic motion with a robust fuzzy sliding mode controller when the MEMS resonators had system uncertainties [20]. Rhoads et al. [21] reviewed nonlinear dynamics and its applications in micro- and nanoresonators, and focused on the simple lumped-mass models for individual resonators throughout the review paper. However, when the resonator operates, small fluctuations in the gap (Brownian motion, thermal-mechanical noise, random vibrations and oil-canning of the package) will cause frequency fluctuations [1], [5].
The dynamic response of the electrostatically actuated MEMS resonator subjected to random disturbance due to fluctuations and other uncertainties has not been paid more attention in literature. In this Letter, based on the random Melnikov approach, which is widely used by most researchers [22], [23], [24], [25], [26], the nonlinear dynamic behavior of a clamped–clamped microbeam loaded by a symmetric combined voltage, which is composed of a direct current bias voltage and an alternating current (AC) voltage, is investigated analytically and numerically. Our aim is to predict the analytical shape of the most generic dynamics equation (Duffing-like equation) for an ideally elastic nonlinear micromechanical resonator that has been reduced to a single-degree-of-freedom. This equation could contain a number of nonlinear coefficient, such as nonlinear cubic stiffness, dc bias voltage and AC voltage, and random intensity, each of which having its well definition.
This Letter is organized as follows. Section 2 presents the mathematical model of the clamped–clamped resonator. Section 3 describes the brief introduction of the bounded noise. The threshold of bounded noise amplitude for the onset of chaos in the system is determined by using the random Melnikov method in Section 4. Numerical results and discussion of the nonlinear dynamics and chaos under random disturbance are provided in Section 5, and Section 6 concludes the Letter.
Section snippets
Mathematic model
For the electrostatically actuated clamped–clamped beam resonator [1], as shown in Fig. 1, is subject to a random disturbance, and using the dynamic model for the MEMS resonator derived by Mestrom et al. [1] and Haghighi and Markazi [19], the non-dimensional governing equation of motion can be given by where z is the vertical displacement of the microbeam, and are effective lumped mass, damping coefficient, linear mechanical stiffness and cubic
Bounded noise
Bounded noise is a harmonic function with constant amplitude and random frequency and phases. Stratonovich [24] firstly presented the bounded noise process and defined it as a slowly varying random process. The random process is governed by the following form [22], [23], [24], [26] where σ and are the amplitude and frequency of the random excitation, respectively, and , σ, and δ are the positive constants, is a standard Wiener process, Γ is a random
Random Melnikov analysis
On the assumption that a MEMS resonator has high quality factor and AC voltage is much smaller than the dc bias voltage, which is verified to be valid [19], the governing equation of the dynamic system (4) can be rewritten as where , and , and ε is a small parameter, and .
For , Eq. (9) can be regarded as an unperturbed system and written as
The
Numerical results and discussion
Eq. (9) is a nonlinear one with time varying coefficient, disturbed with a random noise process. Monte Carlo method can be used to numerically analyze the dynamic response of MEMS resonator system subject to random excitation. Eq. (5) can be rewritten as The formal derivative of the unite Wiener process is a Gaussian white noise, which has the power spectrum of a constant and is unrealized in the physical view. The pseudorandom signal can be
Conclusions
In this Letter, noise-induced chaos in the electrostatically actuated MEMS resonators investigated analytically and numerically. The random disturbance is described by a random bounded noise process. The random Melnikov process is derived and used to establish the threshold of bounded noise amplitude for the occurrence of chaos. Effects of the random disturbance, dc bias voltage, AC voltage and cubic nonlinear stiffness on the dynamic responses of the resonator system are investigated. The
Acknowledgements
This work was supported by Shanghai Rising-Star Program under Grant No. 11QA1403400, and the National Science Foundation of China under Grant No. 11072147, and was grateful for the support by Japan Society for the Promotion of Science.
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