تجزیه و تحلیل حساسیت ارتعاشات از ژیروسکوپهای حلقه ویبراتوری MEMS

This paper presents a detailed model for possible vibration effects on MEMS degenerate gyroscopes represented by vibratory ring gyroscopes. Ring gyroscopes are believed to be relatively vibration-insensitive because the vibration modes utilized during gyro operation are decoupled from the modes excited by environmental vibration. Our model incorporates four vibration modes needed to describe vibration-induced errors: two flexural modes (for gyro operation) and two translation modes (excited by external vibration). The four-mode dynamical model for ring gyroscopes is derived using Lagrange's equations. The model considers all elements comprising a ring gyroscope, namely the ring structure, the support-spring structures, and the electrodes that surround the ring structure. Inspection of this model demonstrates that the output of a ring gyroscope is fundamentally insensitive to vibration due to the decoupled dynamics governing ring translation versus ring flexure, however, becomes vibration-sensitive in the presence of non-proportional damping and/or capacitive nonlinearity at the sense electrodes.

Mechanical vibrations potentially degrade the performance of microelectromechanical systems (MEMS) devices because the performance frequently relies on the displacements of or stress in microstructures. The undesirable dynamics induced by external vibration generates errors in the device output. Such vibration-induced output errors, also called vibration sensitivity, have been reported for various MEMS sensors and actuators. MEMS vibratory gyroscopes, because of their high quality factor (Q-factor), are known to be susceptible to vibration. The high Q is beneficial in improving gyro performance but also amplifies vibration amplitudes at certain frequencies and increases output signal distortions. Whereas, degenerate MEMS gyroscopes are conceptually known to be less sensitive to environmental excitation because of inherently symmetric gyro structures [1] and [2]. Degenerate MEMS gyroscopes utilize a degenerate vibration mode pair as the drive and sense modes to maximize the energy transfer between the two modes. A degenerate pair of vibration modes refers to two modes that have distinct mode shapes but identical natural frequencies [3] and [4]. Many degenerate gyroscopes utilize the so-called wine-glass modes of a vibrating shell structure representing two flexural modes (i.e., drive and sense modes). Wine-glass modes arise in several shell-like structures, including common ring gyroscope designs [1], [2] and [5]. The natural frequencies of axisymmetric ring structures arise as degenerate pairs when the structures are fabricated from isotropic materials [4] and [6]. Fig. 1 illustrates a conceptual view of a ring gyroscope. A vibratory ring gyroscope consists of a ring structure, support-spring structures, and electrodes surrounding the ring structure. The electrodes are used for drive, sense, or control of the gyro. The operation of the ring gyroscope relies on two elliptically shaped vibration modes, named the primary and secondary flexural modes, which are also called the drive and sense modes, respectively. The two flexural modes have identical natural frequencies due to the (assumed) symmetry of the ring. Several variations of this design are also reported in [5], [7], [8], [9], [10] and [11], but the basic concept remains the same. Full-size image (41 K) Fig. 1. Conceptual view of a MEMS ring gyroscope. Figure options Ring gyroscopes are known to have a low vibration sensitivity because external vibration excitation couples only weakly (if at all) to the two flexural modes [1], [12] and [13]. This knowledge is based on the vibration modes of axisymmetric ring structures [14] and [15]. Nonetheless, several studies offer qualitative explanations of potential vibration-induced error sources in degenerate gyroscopes including hemispherical resonator gyros [16] and ring gyros [1]. Therefore, it is still crucial to analyze and understand the operation of ideal ring gyros subjected to external vibration. Several studies report rigorous analyses of ring gyro operation [1], [2], [17] and [18] or the ring gyro's response to external vibration [1], [19], [20] and [21]. However, these prior studies only present models that consider only partial components compromising a ring gyroscope or do not include sufficient number of vibration modes needed to analyze both gyro operation and vibration-induced errors. For instance, models exist that consider only the mechanical ring structures and ignore the support-spring structures [17], [18], [19] and [20], consider mechanical structures without electrodes [20] and [21], or do not account for the vibration modes directly excited by external vibration [1], [2] and [10]. The potential energy of the support-spring structure needs to be included because the flexural stiffness of each support spring is not negligible compare to the stiffness of a ring structure [2] and [21]. The electrostatic force from the electrodes is important in evaluating gyro performance and vibration sensitivity [10] and [22]. The gyro read-out circuits often utilize parallel-plate sensing mechanism that contributes a nonlinear behavior between sense capacitance and sense axis displacement. This capacitive nonlinearity generates vibration-induced errors in other types of gyroscopes, such as tuning fork gyroscopes [22]. Thus, it is essential to consider all components of ring gyroscopes to achieve a comprehensive understanding of vibration sensitivity of ring gyroscopes. This paper fills this void by contributing a detailed model that includes all components of ring gyroscopes (including the ring structure, the support-beam structure, the electrodes, and damping). The model describe both gyro operation and its response to external vibration by employing Lagrange's equation and four vibration modes either representing ring gyro operation (named flexural modes) or excited by external linear vibration (named translation modes). This work analyzes ideally fabricated ring gyroscopes and represents a step toward a complete model to understand the vibration sensitivity of MEMS ring gyroscopes. We open with an overview of this model in Section 2 and derive equations governing energies from or work done by each ring-gyro components described from Sections 2, 2.1, 2.2, 2.3, 2.4, 3, 4, 4.1, 4.2, 5, 5.1, 5.2 and 6. The derived equations are analyzed using Lagrange's equation in Section 7. Furthermore, the effect of capacitive nonlinearity at sense electrodes is analyzed in Section 8. In addition, we also briefly discuss the effect of nonideality including non-proportional damping (Section 6.2).

This paper contributes a detailed model of the vibration effects on degenerate gyroscopes as represented by ring gyroscopes. Ring gyroscopes utilize a pair of degenerate flexural modes to detect rotation rate. The two flexural modes (named the drive and the sense modes) induce bending of the ring structure. Meanwhile, vibration modes excited by external vibrations (named translation modes) induce translation of the ring structure. To analyze ring gyroscopes subjected to environmental vibration, we derive a four mode of a ring gyro consisting of, two translation modes and two flexural modes. Lagrange's equations are employed to derive the equations of motion for this four degree-of-freedom (4-DOF) dynamic system. Our analytic model includes all components comprising ring gyroscopes. In particular, this model captures the potential energies due to bending of the ring structure and the support springs, the potential energy due to electrostatic forces at the drive and sense electrodes, and dissipation. The resulting 4-DOF equations of motion contain terms representing (1) the modal mass, damping, and stiffness of the ring gyro mechanical structure, (2) modal coupling induced by the Coriolis forces and by angular acceleration, (3) additional stiffness from centripetal acceleration and electrostatic forces, (4) environmental excitation, and (5) electrostatic actuation. If we assume proportional damping, the 4-DOF model decouples into two two-degree-of-freedom (2-DOF) systems that separately govern the translation modes and flexural modes. This decoupling clearly demonstrates that environmental vibration, which excites the translation modes, has no influence on the flexural modes that govern gyro performance. Meanwhile, if we assume non-proportional damping, the 4-DOF model remains coupled and this creates a pathway for exciting flexural modes via external excitation. Another source of vibration sensitivity of ring gyroscopes is capacitive nonlinearity of the sense electrode. The nonlinearity generates high-order coupling terms that cannot be filtered out. The model developed herein does not capture other potential vibration-induced error sources that are beyond the scope of this paper. The un-modeled error sources include those resulted from high frequency external vibration or from imperfections that couple ring translation and flexure, and may be analyzed in future study. High frequency vibration (with spectral content frequency containing the flexural-mode resonant frequencies) may directly excite the flexural modes leading to undesired responses that cannot be distinguished from the desired responses excited by ring gyro operation. This error mechanism obviously exists even for ideally fabricated ring gyroscopes. On the other hand, vibration-induced errors by fabrication imperfection may occur when the flexural modes are excited by translation modes. The decoupling of flexural and translation modes can arise from the assumed perfect symmetry of the ring gyro. The symmetry may be destroyed by a non-uniform or asymmetric distribution of ring mass and/or stiffness (inertial and/or compliance coupling) as previously noted in analyses of degenerate gyroscopes [1] and [16]. When this symmetry is destroyed (for example, by manufacturing limits), the modes of the imperfect ring may involve coupled ring flexure and translation. This leads to a direct pathway to couple external vibration to ring flexure, thereby introducing gyro output errors. If the frequency of the induced translation motion happens to coincide with the flexural mode frequency, a flexural motion that is indistinguishable from motions caused by rotation rate is developed and leads to vibration sensitivity of ring gyroscopes.