Research Papers

Decoupling of a Disk Resonator From Linear Acceleration Via Mass Matrix Perturbation

[+] Author and Article Information
David Schwartz

e-mail: rtm@seas.ucla.edu Department of Mechanical and Aerospace Engineering,  University of California, Los Angeles, CA 90095daves@ucla.edu

Robert T. M’Closkey

e-mail: rtm@seas.ucla.edu Department of Mechanical and Aerospace Engineering,  University of California, Los Angeles, CA 90095

The first and third harmonics of the mass asymmetry are simply the first and third harmonics of the mass/unit length function along the equatorial axis of the hemispherical resonator. The harmonics are considered “balanced” when they have a value of zero. The analysis can be used to show that imbalances in the first and third harmonics of mass asymmetry are responsible for linear acceleration coupling in axisymmetric rings as well.

The denominators of Eq. 7 are factored out of the dynamic parts because the total deviation from symmetry, mi, is much smaller than the total mass of the resonator, meaning that that the inverse of the denominators are effectively constant with respect to the mass asymmetry.

The mode shape used in this derivation is defined by w(φ, t) = 2Acos(2φ – 2Ψ) sin (ωt) and u(φ, t) = A sin (2φ – 2Ψ) sin (ωt) where w and u are the radial and tangential parts of the velocity of a point on the ring at angular position φ [9]. An exaggerated mode shape is visualized as the dotted ellipse in Fig. 1.

J. Dyn. Sys., Meas., Control 134(2), 021005 (Dec 29, 2011) (10 pages) doi:10.1115/1.4005275 History: Received May 26, 2010; Revised August 09, 2011; Published December 29, 2011; Online December 29, 2011

Axisymmetric microelectromechanical (MEM) vibratory rate gyroscopes are designed so the central post which attaches the resonator to the sensor case is a nodal point of the two Coriolis-coupled modes that are exploited for angular rate sensing. This configuration eliminates any coupling of linear acceleration to these modes. When the gyro resonators are fabricated, however, small mass and stiffness asymmetries cause coupling of these modes to linear acceleration of the sensor case. In a resonator postfabrication step, this coupling can be reduced by altering the mass distribution on the resonator so that its center of mass is stationary while the operational modes vibrate. In this paper, a scale model of the disk resonator gyroscope (DRG) is used to develop and test methods that significantly reduce linear acceleration coupling.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Left: The Silicon Disk Resonator Gyroscope (SiDRG) has an 8 mm diameter and motivates the study in this paper. Only one quadrant of the resonant structure is shown here. In an operational gyroscope, electrodes are embedded between the rings to drive and sense the in-plane elliptical Coriolis-coupled modes. Right: The frequency response of the SiDRG exterior ring’s radial velocity to in-plane excitation of the resonator’s central attachment point. The two n = 2 Coriolis-coupled modes exist at two slightly different frequencies, indicating a detuning which is caused by small mass asymmetries. The fact that these modes are observable in this experiment suggests that the Coriolis-coupled modes exhibit coupling to movement at the attachment point, which can impact an operational gyroscope’s performance by allowing linear case acceleration to produce spurious rate signals. By performing mass perturbations on the top surface of the resonator it is possible to reduce this coupling. This paper investigates methods for reducing the coupling and tests these methods on a macro-scale model of the SiDRG (shown in Fig. 2). The two n = 1 modes, which correspond to in-phase motion of the resonator rings, are nominally coupled to linear acceleration and Coriolis forces, but are not generally used for rate detection.

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Figure 2

Left: The experimental setup of the Macro DRG, configured to measure linear acceleration coupling. The resonator hangs from the flexible post so that in-plane movement of the resonator’s attachment point can be measured by the accelerometers. On the right hand side of the resonator, near one of the D2 actuators, two stacks of the NdFeB magnets can be seen. These are used to create a reversible mass perturbation to the resonator. Right: A schematic of the forcer/pickoff arrangement for the Macro DRG. The angular positions for mass perturbations are measured as the angle with respect to the x-axis, which is aligned with the A1 and S1 sensors and intersects the central axis of the post.

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Figure 3

Top: Electromagnetic actuator and capacitative sensor setup. Bottom: Testing block diagram.

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Figure 4

The 2 × 2 frequency response of the accelerometer velocity to a chirp input from each forcer, denoted Hv , is displayed using the dotted traces, while the frequency response of the capacitative sense measurements from the same inputs, denoted Hs , is displayed using the solid traces. If no linear acceleration coupling were present, the accelerometer velocity responses would be nearly linear in this frequency range and correspond to motion of resonator center as a consequence of the cantilever response of the post to which it is attached.

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Figure 5

Block diagram for identification of the coupling matrix, B(ω), from measured responses Hv and Hs

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Figure 6

The real (left) and imaginary (right) parts of the four components of B(ω). The slopes of the real parts and the small imaginary parts are caused by post dynamics in the region of the modes of interest. A constant, real coupling matrix, B¯, is calculated by taking the average of the real parts of B(ω) at the two resonant frequencies.

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Figure 7

An experiment that tests the linearity of ΔB¯ with respect to m. Five separate tests were done in which the value of B¯ was measured as magnets were added to an individual spoke. Magnets were attached in two stacks so that the final test with twelve magnets is performed with two stacks of six magnets. The average of the four channels of ΔB¯ is plotted as the dots whereas the error bars give the standard deviation of the tests. The solid line is a least square fit to the averaged data. The assumption that ΔB¯ is linear with respect to m appears to be appropriate.

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Figure 8

The four components of the perturbation matrix, ΔB¯, plotted against the placement of the perturbing mass. In this case, the coupling matrix, B¯, was measured as a six magnet test mass is placed at each of the spokes as well as the midpoints in between the spokes. The perturbation matrix is taken as the difference between the coupling matrix of the perturbed resonator and coupling matrix associated with the unperturbed resonator. The test was conducted five times and the error bars represent the standard deviation of the data. The dotted trace is given by the first and third harmonic approximation of the averaged data explicitly given in Eq. 6.

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Figure 9

The magnitudes of the discrete Fourier transform of the mean values plotted in Fig. 8. The four bars of each grouping are the ΔB¯11, ΔB¯12 ,ΔB¯21, and ΔB¯22 components, respectively. It is clear that the first and third harmonics are the dominant features of the perturbation function.

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Figure 10

Basic diagram for ring linear acceleration coupling analysis. The dotted shape represents a possible mode shape for the ring, with antinodal axis an angle Ψ from the x-axis. The position of the single attached mass, mi , oscillates with the ring, thereby causing the center of mass, shown here as ‘×’, to oscillate as well.

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Figure 11

A plot of tr(Hv*Hv) before and after the two decoupling methods were implemented. Each of the methods exhibit over a 95% reduction in the H2 norm of the forcers-to-accelerometers transfer functions. For reference, the transfer function Hv measured prior to decoupling is the solid trace plotted in Fig. 4.

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Figure 12

Uncorrelated white-noise inputs with equal variance are applied to the two electromagnetic forcers in a frequency band encompassing the modes of interest for both the original and decoupled cases. The velocities of the post are measured before decoupling (gray) and after decoupling (black) using the spokes method.

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Figure 13

The averaged power spectrum of both accelerometers with uncorrelated band-limited noise inputs with a 50mv/Hz spectral density before (gray) and after (black) decoupling is performed using the spokes method. The modes of interest are located between 1630 and 1650 Hz.

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Figure 14

A ring down test is performed on the resonator before and after decoupling is implemented and the time responses of the peak outputs are plotted. The quality factors are approximated by the slope of linear least square fits to the data. The low frequency mode shows a marked increase in its quality factor.

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Figure 15

A plot of the (1, 1) component of ΔB¯(m0,φ) as measured during the calibrations for the example of the general method. The diameters of the circles scale with the magnitudes of the perturbation used for each calibration step. The first and third harmonic approximation of the perturbation function, [cos φ sin φ cos 3φ sin 3φ]x 11 , is plotted using the solid line. This approximation is used to guide the final two decoupling steps. The dotted line is the first and third harmonic approximation that was shown in Fig. 8. Though it does not precisely match the previous approximation, which utilized more perturbations, the new approximation is still a useful decoupling tool.



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