Self-induced parametric amplification in ring resonating gyroscopes

Highlights

• Self-induced amplification in ring and disk resonating gyroscopes is studied.

• Nonlinear modal coupling results in parametric amplification of the sense mode.

• Parametric coupling is caused primarily by electrostatic nonlinearities.

• Amplification can result in significant increase of the gyroscope rate sensitivity.

Abstract

We investigate self-induced parametric amplification that arises from dispersive nonlinear coupling between degenerate modes in systems with circular symmetry that rotate about the axis of symmetry. This phenomenon was first observed in micro-electromechanical ring/disk gyroscopes, where it provided enhanced readout gain using purely passive nonlinear effects [Nitzan et al. [22], 2015]. The goal of this investigation is to provide a fundamental description of this phenomenon, which is an example where nonlinear dynamics can improve the performance of a practical device. To describe this behavior, we consider the in-plane vibrations of a thin ring surrounded by electrodes that rotates about its symmetry axis at a rate Ω much smaller than its vibration frequencies ω_n, as is the case in applications. The focus is on the pair of degenerate elliptical modes, one of which is taken as the drive mode and the other as the sense mode for the sensor. These modes are coupled through both inertial (Coriolis) and geometric nonlinear effects, as described by general forms of the kinetic and potential energies that account for finite deformation kinematics, as well as electrostatic effects. We investigate the specific effects of this coupling on the system performance and its sensitivity when used as a sensor for the spin rate. Specifically, we show that drive mode vibrations with sufficiently high amplitude affect the sense mode dynamic behavior in the form of parametric pumping, which leads to a considerable amplification of the sense mode response. As this response amplitude is proportional to Ω, it results in a substantial increase of the gyroscope sensitivity with respect to the external angular rate. We also illustrate that the effects of the sense mode vibrations on the drive mode dynamics can be neglected in the model when $\Omega /{\omega }_n\ll 1$. Finally, we illustrate the applicability of our results by considering the dynamic response of a representative MEMS gyroscope model and quantifying the predicted benefits of these nonlinear effects.

Introduction

Studies of the linear and nonlinear vibrations of systems with circular symmetry have a long history and include papers on the transverse vibrations of plates, shells, membranes, rods, and tubes as well as the in-plane vibrations of plates and rings; see [1], [2], [3], [4] and the research cited therein for a sampling of these works. This class of systems has applications in a number of areas such as antennae, pipes, and, most relevant to the present work, wine glass vibratory gyroscopes that use Coriolis effects to measure spin rates [5], [6].

In recent decades there has been a desire to develop smaller versions of these gyroscopes, spurred by technological advancements in fabrication techniques and by increasing demands in commercial and military applications, which have led to a number of important advances in this technology space [7]. Prominent among these are developments in micro-electro-mechanical-system (MEMS) vibratory gyroscopes, which have shown great potential due to their small dimensions, favorable power consumption, and high quality factors [8], [9]. Generally, such devices are based on a micro-mechanical resonator with at least two matched resonant modes that interact via Coriolis effects [6]. Specifically, the resonator is forced to oscillate in one of its vibrational modes, called the drive mode, while the external rotation at Ω gives rise to Coriolis coupling between this mode and its symmetric partner, the sense mode, which is not driven by an external input. The vibrations of the sense mode thus have an amplitude proportional to Ω (when it is small as compared to the gyroscope operation frequency), and by measuring the amplitude of the readout signal from the sense mode, one can estimate Ω.

Improving the precision and accuracy of MEMS vibratory gyroscopes is a challenging task involving the precise matching of high-Q modal frequencies [10], [11], [12], compensation of quadrature errors that arise from coupling of the drive and sense modes [13], [14], [15], and optimizing the geometry of the resonator in order to achieve higher Q factors [16], [17], to name a few. Also, all such devices are operated in the linear operating regime, so as to avoid frequency shifts associated with nonlinearity. In this light, flexural-mode ring [13], [18] and disk [19], [20], [21] vibratory gyroscopes offer significant advantages due to the inherent symmetry in their geometries and, consequently, symmetry of their drive and sense modes. Recent work on disk resonating gyroscopes (DRGs) has experimentally demonstrated that the gyroscope sensitivity to the external angular rate can increase significantly when the gyroscope is driven into a nonlinear operating regime [22]. The authors hypothesized that the observed phenomenon is due to parametric amplification [23], [24], [25] arising from nonlinear elastic coupling between the drive and sense modes of the device, which have nearly equal frequencies. In classical nonlinear vibrations, this is an example of autoparametric resonance [26], [27], [28].

The sensitivity S of a rate gyroscope, that is, the ratio of the amplitude of the sense signal to the angular rate Ω, is one the most important characteristics of sensor performance, since it, and the noise levels of the device, quantify the resolution of the sensor in terms of the lower end of the angular velocities that can be detected [6]. Thus, there is strong motivation to understand, from a fundamental point of view, the advantageous effects of the self-induced amplification of the sense signal observed in [22], especially since it appears to be the fortuitous result of passive nonlinear behavior, requiring no additional sensing or actuation. This is precisely the goal of the investigation described in this paper.

Nonlinear modal coupling is a well-known phenomenon in the theory of nonlinear vibrations and it has been thoroughly studied in a wide variety of systems [29], including micromechanical systems [30], [31], [32], [33]. It generally occurs in resonators experiencing vibration amplitudes at which nonlinear strain-displacement relationships, or other nonlinear effects, couple two or more vibrational modes. Furthermore, specific research on the nonlinear vibrations of spinning ring-like geometries has illustrated the rich dynamics associated with the in-plane flexural modes of these structures [34], [35], [36], [37]. In this paper we analyze the dynamic behavior of the elliptical modes in ring/disk resonating gyroscopes to explain and explore self-induced parametric amplification in these systems [22]. In particular, we use a model of the resonator consisting of a thin ring spinning about its axis of radial symmetry with electrostatic forces arising from capacitive actuation/sensing schemes. Using finite deformation kinematics, we show that the elliptical drive and sense modes are nonlinearly coupled through both stiffness (including electrostatic contributions) and inertial terms. Next, we show that the general case of mode-coupled dynamics can be simplified by neglecting the back-action of the sense mode motion on the drive mode (due to their differing amplitudes), and provide conditions for which this approximation holds. In this simplified picture, we discuss the effect of inertial nonlinearities on the drive mode dynamics and show how nonlinear modal interactions lead to parametric amplification of the sense mode, and thus to an increase in the gyroscope sensitivity.

The remainder of the paper is organized as follows. In Section 2, we formulate a model for resonator geometries that support a pair of degenerate (equal frequency) n=2 radial modes. In Section 3 we consider the nonlinear in-plane flexural vibrations of a thin spinning ring in the presence of electrostatic actuation. A detailed analysis of the dynamic behavior of the drive and sense modes is given in Sections 3.2 and 3.3, respectively. Finally, in Section 3.4 we illustrate the applicability of our results to a model of the representative ring resonating gyroscope reported in [18], and concluding remarks are given in Section 4.