Browsing by Subject "Quadrature error"

Now showing 1 - 4 of 4

Open Access
Analysis of quadrature and frequency split in a MEMS vibrating ring gyroscope with structural imperfections
(IEEE, 2021-08-06) Hosseini-Pishrobat, Mehran; Tatar, Erdinç
Structural imperfections affect the performance of MEMS vibrating ring gyroscopes (VRGs), dominantly in terms of quadrature error and frequency split. We consider a VRG with a ring subjected to width nonuniformity and supported by an imperfect suspension. This reflects the scenario of nonuniform etching, common in microfabrication processes. We show that the ring's width nonuniformity mainly results in a rotation of the mode-shapes with respect to those of a perfect ring while an imperfect suspension induces frequency split between the modes. On this basis, we calculate the quadrature error in the gyroscope's output. As an example, a 4° mode-shape rotation and 15Hz frequency split in a 60kHz VRG generate an approximate quadrature error of 1000°/s.
Open Access
Analytical and experimental study of imperfections, stress, and temperature effects in circular MEMS gyroscopes
(2024-02) Hosseini-Pishrobat, Mehran
Inspired by the outstanding performance of the hemispherical resonator gyroscopes (HRG), MEMS gyroscopes with circularly symmetric structures are a promising candidate for the next generation of high-performance, cost-effective gyroscopes. However, scaling down to MEMS poses certain performance-limiting challenges: increased sensitivity to inevitable fabrication imperfections and environmental variations, chiefly stress and temperature, that perturb the gyroscope’s ideal modal space and result in quadrature/in-phase errors and, more importantly, long-term drift. Understanding these limiting factors is imperative for harnessing the full potential of circular MEMS gyroscopes. This thesis approaches this objective through an analytical modeling viewpoint. Here, “analytical” is meant to connote an approach based on the physics underlying the gyroscopes’ operation as described by the variational principles of solid mechanics. For the experimental evaluations, we use our fabricated double-ring vibrating ring gyroscope (VRG) (3.2 mm-diameter, 57-59 kHz) and 10-ring disk resonator gyroscope (DRG) (3.4 mm-diameter, 41 kHz). We start by calculating the mode shapes of the entire structure of multi-ring gyroscopes in the presence of structural imperfections and elastic anisotropy. By deriving the gyroscope’s nonideal drive-sense transfer function matrix, we provide rigorous definitions for the quadrature and in-phase outputs, highlighting the role of angular gain, frequency split, mode shape rotations, and quadrature leakage into the in-phase due to the sense mode’s phase error. Next, we present a model for the effects of mechanical stresses leading to the concept of stress stiffness, an additional stiffness induced by such stresses through geometric nonlinearity. We carry out an eigenvalue perturbation analysis to obtain the frequency shifts, mode shape rotations, and quadrature/in-phase errors generated by the stress stiffness. Taking advantage of the circular geometry, we have equipped our ring gyroscopes with 16 capacitive stress sensors located 45◦-apart (eight inside and eight outside the main ring), which pick up the local stress at the substrate level. We present an interpolation scheme to reconstruct the substrate’s stress field using the outputs of the stress sensors, providing us with the mechanical stresses responsible for the stress stiffness in the silicon layer. We validate the model based on PCB bending tests. We finally set out a modeling framework for temperature effects in ring gyroscopes. Our temperature experiments gave temperature coefficient of frequencies (TCFs), such as -10 ppm/◦C and -14 ppm/◦C, that are considerably different than the TCF value ~-30 ppm/◦C expected from the ~-60 ppm/◦C temperature dependency of Young’s modulus of silicon. The model revolves around the engendered stiffness and opposing interaction of two fundamental mechanisms of temperature effects: changes in material properties and thermal stresses. The model demonstrates remarkable efficacy in accurately predicting the TCF and sheds light on residual stresses’ role in forming frequency-temperature hysteresis loops. Considering the great potential of integrating stress with temperature for the long-term performance improvement of MEMS gyroscopes, the results of this thesis serve as a building block toward physics-informed drift compensation algorithms.
Embargo
Analytical and experimental study of stress effects in a MEMS ring gyroscope
(Elsevier, 2023-09-09) Hosseini-Pishrobat, Mehran; Erkan, Derin; Tatar, Erdinc
External stress affects the stiffness distribution of a MEMS gyroscope and, along with temperature, is affiliated with long-term drift. Although the detrimental effects of stress on MEMS gyroscopes are well-documented, modeling of such effects is still lacking in the literature. For the first time, we present an analytical model that mathematically describes the stress effects in a ring gyroscope. Our model revolves around the key observation that stress-induced anchor displacements result in variations of electrostatic gaps and nonhomogeneous boundary conditions at the interface between the gyroscope’s suspension system and the anchored internal structure. Our gyroscope is equipped with 16 capacitive stress sensors distributed with 45° symmetry on the inside and outside of the main ring. We use these stress sensors’ measurements to interpolate the strain field across the substrate and deduce the anchor displacements. To capture the stress effects, we show that two fundamental assumptions in the existing literature should be amended: (1) Linearity: the linear engineering strain should be upgraded to the nonlinear Green–Lagrange strain to reveal the stress-induced stiffness through geometric nonlinearity; (2) Inextensibility: for a ring, this stress stiffness is determined by the extensional stress arising from centerline extensibility. We analyze variations of frequencies and mode shapes’ orientation along with the resultant quadrature and in-phase errors. Moreover, we present a fairly general formulation incorporating fabrication-induced imperfections and elastic anisotropy. We validate our model experimentally using extensive bending tests performed on our 59 kHz, 3.2 mm diameter gyroscope.
Open Access
Modeling and analysis of a MEMS vibrating ring gyroscope subject to imperfections
(Institute of Electrical and Electronics Engineers, 2022-05-06) Hosseini-Pishrobat, Mehran; Tatar, Erdinç
We present a new mathematical model for a vibrating ring gyroscope (VRG) in the presence of imperfections, namely, structural defects and material anisotropy. As a novelty, we calculate the mode shapes of the internal suspension structure to enable a more accurate and modular analysis of the VRG’s mass and stiffness distributions. Solving the associated eigenvalue problem shows that imperfections result in the frequency split between the gyroscope’s operating mode shapes, rotating their orientation with respect to the nominal drive and sense axes. We then use perturbation analysis to solve the VRG’s equations of motion and analyze the quadrature error that arises from frequency/damping mismatch between the mode shapes. We use our model to detail the various effects of the etching-related undercuts, structural uncertainties, and Young’s modulus anisotropy–in the form of suitable space-dependent functions–on the mode shapes and the quadrature error for the first time. The results reveal that rings are robust against imperfection, while the straight beams used in the suspension system are most likely responsible for the frequency split and quadrature error. For example, 50 nm (0.5%) width variation in a beam that connects the VRG’s suspension to an anchored internal structure leads to 4700°/s quadrature error. To validate our modeling, using the experimental data from a fabricated 59 kHz VRG, we provide rigorous, comparative simulations against the finite element method (FEM).