Frequency locking in a forced Mathieu–van der Pol–Duffing system

Abstract

Optically actuated radio frequency microelectromechanical system (MEMS) devices are seen to self-oscillate or vibrate under illumination of sufficient strength (Aubin, Pandey, Zehnder, Rand, Craighead, Zalalutdinov, Parpia (Appl. Phys. Lett. 83, 3281-3283, 2003)). 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 (Aubin, Zalalutdinov, Alan, Reichenbach, Rand, Zehnder, Parpia, Craighead (IEEE/ASME J. Micromech. Syst. 13, [1018][1019][1020][1021][1022][1023][1024][1025][1026] 2004)), this MEMS device was modeled by a three-dimensional system of coupled thermo-mechanical 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 that shows similar behavior to the MEMS device. We investigate the dynamics of a Mathieu-van der Pol-Duffing 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, M. Pandey · R. H. Rand ( ) · A. T. Zehnder Department of Theoretical and Applied Mechanics, Cornell University, 207 Kimball Hall, Ithaca, NY 14853-1503, USA e-mail: rhr2@cornell.edu 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.

Entrainment of Micromechanical Limit Cycle Oscillators in the Presence of Frequency Instability

“…In [47], a simpler forced Mathieu-van der Pol-Duffing model was considered, which reproduces the essential features of experimental data in [44]: LCOs (van der Pol term), an amplitude-frequency relationship (Duffing term), and parametric forcing (Mathieu term). Perturbation theory was used to derive the slow flow equations assuming no parametric forcing, and numerical continuation of the slow flow for an amplitudehardening limit cycle indicated partial hysteresis, specifically a distinction between f free-up and f lock-down but not between f free-down and f free-up .…”

“…Units in (1) are as follows: 1) time is nondimensionalized such that the device has linear resonant frequency of 1; and 2) displacement is nondimensionalized by the measured limit cycle amplitude (discussed later). Model equations are a simplification of those presented in [47]. For doubly supported beams that support tension across their length, the linear stiffness is temperature dependent, thus, forcing via laser modulation will parametrically drive the device.…”

“…When forcing is inertial (as in our case), out-ofplane displacement will modulate the absorptive heating, and thus the stiffness, however, this is a second-order effect. Thus, we drop the parametric term from [47].…”

“…In addition, no matter how small the drive amplitude, A D , a drive frequency f D sufficiently close to f LCO will guarantee entrainment. Prior modeling and experimental work on LCOs in MEMS has illustrated hysteresis in the region of entrainment [44]- [47], a tilt toward higher frequency of primary entrainment at higher drive amplitudes in amplitude-hardening limit cycles [45], and explored the regions of subharmonic entrainment in a force relaxation oscillator [48].…”

In-Plane Nonlinear Dynamics of Wind Turbine Blades

“…Other work has been on forced and unforced parametrically excited systems with van der Pol, Rayleigh and Duffing nonlinearities. Rand [14][15][16][17] in collaboration with others have analyzed the dynamics and bifucations of a forced Mathieu equation and properties of superharmonic resonances at 2:1 and 4:1. Belhaq [18] has studied quasi-periodicity in systems with parametric and external excitation.…”