1. Introduction
The Coriolis vibratory gyroscope (CVG) is a type of inertial device measuring angular velocity through the precession of elastic waves. The CVGs with axisymmetric shell resonators, in particular, are well known for their outstanding capabilities of high accuracy, long durability, considerable reliability, low power consumption, maintenance-free concept, and are widely used in the navigation fields and platform stabilization systems [
1,
2,
3,
4,
5,
6,
7,
8]. For example, the hemispherical resonator gyroscopes (HRGs) have claimed 30 million hours of continuous operation without a single mission failure [
9].
For this type of gyroscopes, there are mainly three types of excitation and detection, including electrostatic, electromagnetic, and piezoelectric methods [
10]. The representative products using electrostatic methods include the Northrop Grumman H130 series [
11] and the Safran HRG Crystal
TM series [
12,
13]. The representative products with piezoelectric transduction include Watson Inc. Pro Gyro
® series [
14] and InnaLabs Inc. GI-CVG series [
15]. Wu et al. proposed a noncontact measurement system using electromagnetic excitation and microphone detection [
16], which is simple as a testing apparatus, but this system can only characterize metal resonators. Due to minimal damping, electrostatic excitation and detection allow high Q factors, as in [
10,
17,
18,
19,
20]. However, electrostatic excitation and detection have lower electromechanical transduction efficiency compared to piezoelectric transduction. There are several means to compensate for this, such as applying high direct current (DC) voltages [
17,
18,
19], using gap closing mechanisms [
20], relying on sub-micron transduction gaps [
21,
22], and adding combs [
23]. On the other hand, piezoelectric transduction allows for lower motional resistance due to the higher electromechanical coupling [
24,
25,
26,
27], meanwhile requires no DC voltage application for operation, which can greatly simplify interfacing electronics. However, piezoelectric materials will inevitably introduce extra loss, which results in lower Q factors. For CVGs with axisymmetric shell resonators made from fused silica, electrostatic excitation, and detection usually outperform the rest for their low impact on the resonator, low power usage, high sensitivity, and high stability.
The axisymmetric shell resonator is the critical component of the CVG, the vibrational characteristics, including the resonant frequency, frequency mismatch, Q factor, and Q factor asymmetry, determine the overall performance of the gyroscope. The mechanical frequency mismatch and Q factor, in particular, directly determine the precision and drift characteristics of the gyroscope [
28]. These vibrational parameters also affect the design of the electrical parameters, and the vibrational characteristics in practice are affected by electrical conditions in turn. Electrostatic tuning has long been recognized as an effective method for on-chip active mode matching [
29,
30,
31,
32,
33,
34]. To give a few examples, Darvishian et al. investigated the electrostatic frequency tuning in a birdbath shell resonator as a function of voltage, capacitive gap between the shell and electrode, electrode span angle, and height, and electrode placement and configuration using a numerical approach [
34]. Ahn et al. investigated the electrostatic tuning for perfect mode-matching of a wineglass mode disk resonator gyroscope [
31]. Zhang et al. investigated the mismatch compensation using electrostatic spring softening and tuning for Microscale Rate Integrating Gyroscopes (MRIGs) to operate in the whole angle mode [
10]. There are also researchers investigating the effect of electrostatic forces on Q factors, but mostly for tuning fork resonators. For example, Zotov et al. electrostatically tuning the reaction force at the anchors caused by fabrication imperfection to increase the Q factor of anti-phase driven tuning fork Micro-electromechanical Systems (MEMS) [
35,
36]. Cheng et al. investigated the effect of polarization voltage on the measured Q factor of a multiple-beam tuning-fork gyroscope [
37].
For CVGs with fused silica cylindrical resonators, the vibrational characteristics of the resonator will also be affected by the electrostatic forces. This paper intends to report the experimental results on the changes of resonant frequency, frequency mismatch, decay time, and decay time split under electrostatic forces, and provide theoretical analysis on these changes.
This paper comprises five sections. The theoretical analysis of the influence of electrostatic forces on the vibrational characteristics of the resonator (called vibrational characteristics in practice, VCPs) has been presented in
Section 2, and comparison is made with the vibrational characteristics without electrostatic influence (called vibrational characteristics in measurements, VCMs). The methods to measure VCMs and VCPs are described in
Section 3. VCMs were measured by the laser Doppler vibrometer (Polytec, Irvine, CA, USA) with acoustic excitation, while VCPs were measured with electrostatic excitation and detection. The results and discussions are presented in
Section 4, and
Section 5 concludes this paper with a summary of the results.
2. Theoretical Analysis
The kinetic energy term in the Lagrangian of a resonator was investigated using the displacement vector components in spherical polar coordinates, as shown in
Figure 1. It is specified by giving the components of the displacement vector of a point
P on the shell middle surface as a function of the spherical polar coordinates
and
. The displacement vector components in spherical polar coordinates are:
where
wc(
t) and
ws(
t) are, respectively, the radial components of the displacement vector at the equator at azimuth angles of 0° and 45°.
Considering the generic CVG equations under ideal conditions, where there is no damping, frequency mismatch or other forces and ignoring the centrifugal terms, the Lagrangian of the oscillating cylindrical shell has the form:
Among them,
is the system angular velocity,
k is the angular gain,
meff is the effective mass and
, where
f(2) = 1.5296,
m is the mass of the shell resonator. More detailed derivations in obtaining the values of
meff are presented in
Appendix A.
When the resonator is driven and read out capacitively, additional forces should be included in the equations of motion. Considering the
kth electrode, which is placed on a spherical surface that is concentric with the shell and centered at
and
, with angular widths of
and
. If we define
E as the electromotive force and
R as the equivalent circuit resistance [
38], we get:
Including the Coriolis and angular acceleration terms, the centrifugal acceleration terms, the damping terms and different natural frequencies of the two modes, the equations of motion satisfied by CVGs have also been listed in
Appendix A [
39,
40,
41,
42]. Therefore, Lagrange equations are
Firstly, we investigated the electrical contributions to the resonant frequency. Considering the additional electrical potential energy term in the Lagrangian, where
, and substituting into (6) and (7), we have:
A represents the items omitted, the explicit expressions of the values of
,
, and
A are presented in
Appendix B. Substituting these expressions into (8) and (9), we have:
Similarly, we have
where
,
,
, and
.
,
are, respectively, the angular frequency of the resonator excited in the low-frequency principal axis and the high-frequency principal axis, while
,
are, respectively, the decay time constant of the resonator excited in the low-damping axis and the high-damping axis.
In addition, the relation between angular frequency and resonant frequency is
, and the relation between the Q factor and decay time is
[
43]. Therefore,
and we let
,
.
and
, in particular, are respectively the resonant frequency and the decay time constant detected in
Section 4.
3. Experiments and Methods
Our research group has reported fused silica cylindrical resonators with the Q factor approaching 10
6 in 2016 [
44] and 3 × 10
6 in 2019 [
45]. In this research, a fused silica cylindrical resonator with a high Q factor was fabricated in the same way. For electrostatic excitation and detection, the outer surface of the resonator was coated with Cr/Au (~20/60 nm) film by magnetron sputtering. The resonator was then fixed on a fused silica base through its supporting rod, and a cylindrical ring with laser-cut electrodes was attached on the base outside the resonator. The gap between the resonator and the ring was nearly 20 μm. The main electrodes were used to excite or detect resonator vibration, while the auxiliary electrodes were grounded to reduce signal interference, as shown in
Figure 2.
Table 1 presents some dimensions of the resonator, as well as some parameters of the electrostatic excitation and detection system, where
L and
l are, respectively, the height of corresponding cylinders, and
h is the width of the resonator. The resonator was characterized in a vacuum chamber with a pressure of 0.01 Pa, and it was placed on an optical table to avoid environmental vibrations.
3.1. Vibrational Characteristics without Electrostatic Influence
For the measurement of the vibrational characteristics without electrostatic influence (VCMs), the resonator system should be isolated from the applied voltage. A laser Doppler vibrometer (Polytec, Irvine, CA, USA) was used to measure the resonant frequency, frequency mismatch, Q factor (decay time), and Q factor asymmetry (decay time split). The resonator was excited by an acoustic source and its vibration detected by the laser Doppler vibrometer. There were material anisotropy and manufacturing errors; therefore, the resonator shows a pair of principal axes of vibration (low-frequency principal axis and high-frequency principal axis), resulting in a natural frequency mismatch. The excitation direction and the low-frequency principal axis had already been aligned in the same orientation before the measurement. The diagram of the experimental setup is shown in
Figure 3, and the testing procedure of the VCMs has been described in detail in [
16,
44,
46].
3.2. Vibrational Characteristics with Electrostatic Influence
For the measurement of the vibrational characteristics in practice (VCPs), the resonator system was tested under electrostatic excitation and detection. Electrostatic excitation is based on parallel plate capacitance where the two charged parallel plates produce an attractive force, and the electrostatic force can be obtained by applying an appropriate voltage on the electrodes. Electrostatic detection is also based on parallel plate capacitance where the two movable plates can charge or discharge, hence producing a measurable current for the following conditioning circuits [
47]. The outside surface of the resonator and the main electrodes formed parallel plate capacitors, which were used to excite or detect the displacement of the resonator from different directions.
Figure 4 shows the schematic of electrostatic detection, including the C/V converter, bandpass filter, analog to digital (AD) converter, and LabVIEW process program. The upper plate represents one of the main electrodes and the bottom plate represents the outside surface of the resonator.
The capacitance for a parallel plate capacitor is [
48]:
where
is the permittivity of the material between two movable plates,
S is the area of the plates,
d is the actual gap when the resonator vibrates,
x0 is the initial gap between two movable plates, and
x is the displacement of the bottom plate. A variation in the gap between two movable plates causes a variation in the capacitance, resulting in the variation of the current. The series expansion of the current
i around
x = 0 is:
where
V is the high DC voltage applied to the resonator. Because the magnitude of the resonant surface
x is far less than the initial gap
x0, the higher-order terms o(
x2) are negligible. Substitute
into (17); the cross term
splits into a DC component and a
frequency component, which can both be eliminated by the bandpass filter. The output current signal is
The output voltage signal is then:
where
Ramp is the resistance.
Using a multifunction I/O device, the actuating capacitors were connected to voltage sources. Excitation signals were generated by the multifunction I/O device with the controlled program designed and operated in the LabVIEW software, and all the relative parameters could be easily adjusted. Detection signals from sensing capacitors were collected by the multifunction I/O device and processed by the LabVIEW program, as shown in
Figure 3.
The testing procedure of the VCPs was as follows. A pair of ring electrodes EA, along with the low-frequency principal axis was used for actuation, while the pair ED in quadrature with EA was used for detection. A sweeping voltage signal was applied to EA and the sweeping frequency data was recorded from ED. The resonant frequency f was then obtained through Fast Fourier transform. As for Q factor measurement, a sinusoidal voltage signal with the resonant frequency was applied to EA, and the signal was then cut off., and the ring-down time was recorded. The measurement for the resonant frequency and decay time was repeated for the high-frequency principal axis; hence, the frequency mismatch and the decay time split were acquired.