Research Papers

Frequency Tuning of a Disk Resonator Gyro Via Mass Matrix Perturbation

[+] Author and Article Information
David Schwartz, Dong Joon Kim

Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095

Robert T. M’Closkey1

Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095rtm@obsidian.seas.ucla.edu

For more information on the operation of vibratory angular rate sensors, refer to Ref. 14.

Much of the analysis, however, applies to higher-order Coriolis-coupled modes as well.

The Macro DRG uses electromagnetic actuation instead of electrostatic actuation because electrostatic forces are too weak for effective actuation at the Macro DRG scale. Furthermore, the gaps required for electrostatic actuation are so small that viscous effects dominate the dynamics of the air in the gaps causing significant nonlinear damping, an effect called squeeze-film damping (15).

It can be shown that adding the same mass at 90, or 180, degree angles relative to the current position produces the same mass matrix perturbation for the modes of interest. Thus, the experimental results for the first 90 degree arc can be extrapolated to represent perturbations to any angular location on the structure.

Unlike the example in the previous paragraph, only one magnet is used to calibrate each spoke for each case. This reduces the possibility of placing more magnets than necessary on a spoke. In the case displayed on the fifth row, however, the approximated position of the antinode of the high frequency mode was close enough to spoke 1 that the optimization code called for nine magnets on spoke 1 and zero on spoke 2 total. Since one magnet had already been placed on spoke 2, an additional magnet was placed on spoke 4, which canceled out the effect of the calibration while still treating the perturbations as “irreversible.” Thus the distribution of magnets after the first three steps is nine magnets on spoke 1, one on spoke 2, and one on spoke 4.

In this one case the position of the antinode of the high frequency mode after the first three steps was far enough away from the fine tuning spoke that the additional magnet did not reduce the split. A more complex threshold could easily be derived for cases like this, but have been ignored here for simplicity.


Corresponding author.

J. Dyn. Sys., Meas., Control 131(6), 061004 (Nov 09, 2009) (12 pages) doi:10.1115/1.3155016 History: Received October 13, 2008; Revised May 14, 2009; Published November 09, 2009; Online November 09, 2009

Electrostatic tuning of the resonant modes in microelectromechanical vibratory gyroscopes is often suggested as a means for compensating manufacturing aberrations that produce detuned resonances. In high performance sensors, however, this approach places very stringent requirements on the stability of the bias voltages used for tuning. Furthermore, the bias voltage stability must be maintained over the operating environment, especially with regard to temperature variations. An alternative solution to this problem is to use mass perturbations of the sensor’s resonant structure for resonant mode tuning. This paper presents a new mass perturbation technique that only relies on the sensor’s integrated actuators and pick-offs to guide the mass perturbation process. The algorithm is amenable to automation and eliminates the requirement that the modal nodes of the resonator be identified by direct measurement.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 15

Frequency responses from steps 1 and 2 of the algorithm with their corresponding fit. ○ represents the empirical response from the unperturbed case, ◻ represents the response when magnets are added only to spoke 4, and ◇ represents when magnets are added to spoke 4 and spoke 1. The solid traces represent the frequency responses of the model that was fit to the three data sets. The model associated with this fit is used to determine the number of magnets that are needed on each spoke to achieve a tuned state. The frequency response after the tuning magnets are added can be seen in Fig. 1.

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Figure 6

Left: orientation of Δ1 perturbation. Right: orientation of Δ2 perturbation. Masses are added at four points in each case to achieve the most even possible mass loading. These are the perturbations corresponding to ◻ and ◇ in Figs.  78.

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Figure 7

The two-input/two-output empirical and model frequency response magnitudes used to test the model fitting algorithm. The empirical data for the test with no perturbation is represented by ○ and the data resulting from the Δ1 and Δ2 perturbations (shown in Fig. 6) are represented by ◻ and ◇ respectively. The model fits given by (R0+jωR1)(−(M0+Δk)ω2+K+jωC)−1 of the three data sets are the solid traces. Thus, the change in the frequency response due to the addition of magnets is successfully modeled as a change to only the mass matrix.

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Figure 8

The two-input/two-output empirical and model frequency response phase plots corresponding to the magnitude plots in Fig. 7

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Figure 9

Empirical frequency response of Macro DRG with double the mass perturbation at the Δ1 perturbation locations (○) compared with the frequency response predicted by the model (R0+jωR1)(−(M0+2Δ1)ω2+K+jωC)−1 (solid trace)

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Figure 10

Empirical frequency response of Macro DRG with the mass perturbations in both the Δ1 and Δ2 perturbation locations (○) compared with the frequency response predicted by the model (R0+jωR1)(−(M0+Δ1+Δ2)ω2+K+jωC)−1 (solid trace)

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Figure 11

Left: illustration of the testing setup used to determine the radial deflection of the outermost ring of the Macro DRG resonator. The driving electromagnet and the resonator are fixed on the rotational stage so that the deflection can be measured at 2 deg increments. Right: plot of the radial deflection as a function of angular position. The dots represent the experimental data, while the dotted line is a sine wave fit. The large deviations from the sine wave are most likely caused by the additional stiffness provided by the spokes. One proposed tuning method involves adding mass directly to the resonator at the location corresponding to the antinode of the experimental data. This is the method that has been used in the past to tune other ring shaped devices.

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Figure 12

Left: the five traces represent five experimental attempts to tune the Macro DRG by placing magnets at only one angular location, using locations of 72 deg, 74 deg, 76 deg, 78 deg, and 80 deg. Data were taken when zero, four, six, seven, eight, nine, and ten magnets were added for each case. The frequency splits are determined from the models that are fit to the individual experimentally determined frequency responses. In each case the split is at first reduced but at some point further addition of mass increases the split. It appears that the smallest split would be achieved near a location of 77 deg. It is interesting to note that if the placement is as little as 3 deg away from this location, the minimum split increases to nearly 0.4 Hz. Right: the data taken when tuning is attempted at 76 deg is replicated as ○ in this plot. The solid line fit is done by fitting the empirical data from the first two points to a set {M0,Δ,C,K,R0,R1} and using λ¯M0+(α/4)Δ,K and λ̱M0+(α/4)Δ,K to determine the split for any number of magnets (where α is the number of masses added). This is a good test of the predictive relevance of the model, but also shows how much the minimum achievable split is increased by using a quantized amount of mass.

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Figure 13

Left: mass matrix perturbation, Δ¯, as a function of the counterclockwise angle from the D1 axes, θ, found by fitting the model to nine empirical frequency response data sets. The upper diagonal term is denoted by ◇, the lower diagonal by ○, and the off diagonal term by ◻. Right: illustration of the eight placements of the magnet in the tests. When the experiment is repeated with different gap widths for the actuators and sensors and different initial magnet distributions, the functional relationship varied only slightly. The absolute magnitudes are not important because the model parameter set is scaled by the M0>I constraint in Eq. 6. The relative magnitudes, however, are at the very least interesting. The solid black magnet in the right-hand picture illustrates the perturbation that corresponds to the data points that are solid black in the left-hand figure.

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Figure 14

Illustration of the assumed mode shape of the ring using u=A cos(2(θ−Φ)), where Φ indicates the location of the antinodes of the mode shape, and u is the radial displacement of the mode shape as a function of the angle θ. We will use this notation to approximate the mode shape of the n=2 Coriolis-coupled mode of the Macro DRG.

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Figure 1

Left: photograph of the resonant structure of the Boeing silicon disk resonator gyroscope. The n=2 Coriolis-coupled mode of the resonator is generally utilized for rate detection. Right: the SiDRG frequency response using embedded drive and sense electrodes within a narrow, 100 Hz band encompassing the “fundamental” Coriolis modes. Though the frequency split is small in a relative sense—less than 0.3%—the sensor effectively has no mechanical gain in this state.

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Figure 2

Left: illustration of an n=2 Coriolis-coupled mode in a perfectly axisymmetric ring. This mode appears in a degenerate pair, meaning it can occur with any angular position relative to the axis of symmetry. Middle: when slight mass asymmetries exist, the modes manifest themselves in two fixed angular orientations, 45 deg apart. These modes have two slightly different frequencies and cause the detuning seen in Fig. 1. Right: when the correct amount of mass is added to the antinode of the high frequency mode (the “target for tuning” on the asymmetric detuned ring), a tuned state can be reached. Even though the mass distribution is asymmetric, the important properties of the ideal ring are recovered.

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Figure 3

Left: photograph of the Macro DRG. The two electromagnetic actuators are labeled D1 and D2, and the two capacitive pick-offs, which detect radial deflection of the resonator, are labeled S1 and S2. Small NdFeB magnets are added to create a reversible perturbations of the mass distribution of the resonator. Top right: diagram of the electromagnetic actuator. Bottom right: diagram of the capacitive sense pick-off. This design was used to minimize electromagnetic coupling to the transimpedance amplifier.

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Figure 4

Block diagram of test setup. The filtered drive and sense signals are denoted Di and Si, i=1,2, respectively. Frequency response data is used to construct a two-input/two-output model of the Macro DRG dynamics. The dotted path represents an alternative setup that is used to calibrate gaps between the pick-offs and the resonator.

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Figure 5

Left: the S1/D1 channel of the empirical wideband frequency response of the Macro DRG showing several resonator modes. At this scale there appears to be no split between the n=2 modes (near 1.6 kHz). Right: the narrowband dynamics of all four channels in a neighborhood of the fundamental Coriolis-coupled modes. The data points are represented by ○ while the trace through the points is a model that was fitted using the process described in this paper. Just as in the SiDRG response, the Coriolis-coupled modes of the Macro DRG have a small frequency split despite the fact that the steel resonator is highly symmetric.

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Figure 16

Left: illustration of the identified spokes for mass loading for the example in the text. The gray axes represent the approximated antinodal axes of the unperturbed Macro DRG. The small white circles represent where the two magnets may be placed to calibrate the spoke 1, and the two black circles represent where two more may be added to calibrate spoke 4. Right: The final orientation of the magnets that successfully tunes the Macro DRG so that the antinodal axes are “trapped” between the tuning spokes. The calibrations in the previous steps are used to choose this orientation.

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Figure 17

The empirical frequency response of the unperturbed resonator is again represented by ○, and exhibits a split of 1.52 Hz. Using the model of the data in Fig. 1, weightings of 3.2 magnets on spoke 1 and 7.2 magnets on spoke 4 are predicted to give a tuned state. ◻ represents the empirical frequency response when three magnets are on spoke 1 and seven magnets are on spoke 4. The solid line corresponding to this data is the predicted response using the model. The final split is 0.08 Hz.



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