Frequency Locking in a Forced Mathieu-van Der Pol-duffing System

Optically actuated Radio Frequency MEMS devices are seen to self oscillate or vibrate under illumination of sufficient strength [1]. These oscillations can be frequency locked to a periodic forcing, applied through an inertial drive at the forcing frequency, or subharmonically via a parametric drive, hence providing tunability. In a previous work [2] this MEMS device was modeled by a three dimensional system of coupled thermomechanical equations requiring experimental observations and careful finite element simulations to obtain the model parameters. The resulting system of equations is relatively computationally expensive to solve which could impede its usage in a complex network of such resonators. In this paper we present a simpler model which shows similar behavior to the MEMS device. We investigate the dynamics of a Mathieu-van der PolDuffing equation, which is forced both parametrically and nonparametrically. It is shown that the steady state response can consist of either 1:1 frequency locking, or 2:1 subharmonic locking, or quasiperiodic motion. The system displays hysteresis when the forcing frequency is slowly varied. We use perturbations to obtain a slow flow, which is then studied using the bifurcation software package AUTO. INTRODUCTION Tunable limit cycle MEMS oscillators and resonators are becoming important components in RF microsystems where they are used as electromechanical filters [3], amplifiers and nonlinear mixers [4]. They also find use in different kinds of scanning probemicroscopes [5,6], as well as biological and chemical sensors [7]. 1 In previous works MEMS devices consisting of a thin, planar, radio frequency resonator [1, 2, 8, 9] were studied. These devices were shown to self-oscillate in the absence of external forcing, when illuminated by a DC laser of sufficient amplitude. This system can also be forced externally either parametrically, by modulating the incident laser, or nonparametrically, by using a piezo drive at the natural frequency of the device. In the presence of external forcing of sufficient strength and close enough in frequency to that of the unforced oscillation, the device will become frequency locked or get entrained by the forcer. A model was presented in [1, 2, 8, 9] which consisted of a third order system of ODE’s. Figure 1 shows the experimentally observed amplitude and frequency response close to entrainment. The limit cycle oscillation is entrained by the forcing, close to the natural frequency of the oscillator. Parametric variation of the entrainment region as a function of forcing amplitude is shown in Figure 1c. The outer boundaries in Figure 1c separate entrained regions from quasi-periodic response. In this paper a simpler model is studied, which still shows all the relevant phenomena seen in the disc resonator, namely limit cycles, parametric excitation and nonparametric excitation. The essential features of the entrainment in a disc resonator can be listed as follows: 1. For a DC laser of sufficient power, the disc resonator starts to self oscillate at constant amplitude. The simplest canonical model which captures this behavior is a van der Pol (vdP) oscillator [10]. It consists of an −εẋ(1− x) term added to a 1D simple harmonic oscillator (SHO), which in the absence of forcing leads to a steady state vibration called a limit cycle. For small values of ε the limit cycle has frequency close to 1, which is the Copyright c © 2007 by ASME frequency of the unforced linear oscillator. 2. The limit cycle in the system can be periodically forced either parametrically, by modulating the laser, or nonparametrically, by using a piezo drive. The Mathieu equation [10], which consists of adding a εαcos(2ωt)x term to a 1D SHO, can model parametric forcing of the system applied at twice the natural frequency of the resonator. This term in the absence of the vdP term, renders the origin unstable when the parametric forcing frequency 2ω is close to twice the frequency of the unforced linear oscillator. Nonparametric forcing can be modeled by a term of form F0 sinωt . 3. When entrained, the system shows a backbone-shaped amplitude vs forcing frequency response. This kind of behavior is typical of the large amplitude response of structures and can be modeled by a Duffing’s equation term [10], εβx, added to the SHO. This paper concerns the following differential equation, which may be thought of as a forced Mathieu-van der PolDuffing equation: ẍ+(1+εα cos(2ωt+φ))x−εẋ(1−x)+εβx = εF cosωt (1) where εα is the magnitude of parametric forcing applied at frequency 2ω, and εF is the magnitude of nonparametric forcing applied at frequency ω while φ is the phase difference between the parametric and the nonparametric forcing. β is the coefficient of the cubic nonlinearity term and ε is a small parameter which will be used in the perturbationmethod. Eq.(1) is combination of a van der Pol (vdP) equation term [10], −εẋ(1− x), a Mathieu equation term [10], εαcos(2ωt)x, and Duffing’s equation term, (εβx), added to a forced SHO. NUMERICAL SIMULATIONS Numerical results and subsequent analysis are presented for two cases. To begin with, only non-parametric excitation is applied to the system. This corresponds to using α = 0 in equation 1. The numerical results in Figure 2 show that as ω is increased (quasistatically) from 1, the response first consists of a quasiperiodic motion, which is a combination of contributions from the limit cycle and from the forcing. As the frequency is swept forward, the relatively-constant amplitude quasiperiodic motion suddenly jumps onto a single frequency response which increases in amplitude with a further increase in the forcing frequency. Beyond a certain forcing frequency (around ω=1.13 in Figure 2) the motion jumps back to the lower amplitude quasiperiodic response. Hysteresis is seen when the frequency is swept back. This response is similar to the experimentally obtained response for the disk resonator. Next the parametric forcing is also switched on and response amplitude as a function of forcing frequency ω for a case with parameters ε = 0.1, α = 1 , F = 0.3, β = 0.0 and φ = 0 is shown 2 in Figure 3. The cubic nonlinearity (β) is not considered here to simplify the subsequent analysis. Quasiperiodic behavior (QP) is observed in the regions located approximately at ω < 0.97 and ω> 1.03. Periodic behavior at the forcing frequency is observed in the rest of the plot, corresponding to entrainment. As we sweep the frequency forward inside the entrained region, the amplitude jumps to a higher value at a frequency ω≈ 1.015. No comparable jump is seen when the frequency is swept backward, indicating hysteresis. We note that the parameters used in Figures 2 and 3 have been chosen to illustrate the hysteresis exhibited by the model system of Equation 1, and are not obtained from the MEMS device referred to in Figure 1. PERTURBATION SCHEME We use the two variable expansion method (also known as the method of multiple scales) to obtain an approximate analytic solution for equation 1. The idea of thismethod is to replace time t by two time scales, ξ = ωt , called stretched time, and η = εt , called slow time. The forcing frequency ω is expanded around the natural frequency of the oscillator (ω= 1), i.e.