Simulations and Experiments on the Vibrational Characteristics of Cylindrical Shell Resonator Actuated by Piezoelectric Electrodes with Different Thicknesses

The resonator is the key element of the Coriolis Vibratory Gyroscope (CVG). The vibrational characteristics of the resonator, including the resonant frequency, vibrational amplitude, andQ factor, have a great influence on CVG’s performance. Among them, the vibrational amplitude mainly affects the scale factor and the signal-to-noise ratio, and the Q factor directly determines the precision and drift characteristics of the gyroscope. In this paper, a finite element model of a cylindrical shell resonator actuated by piezoelectric electrodes with different thicknesses is built to investigate the vibrational characteristics. The simulation results indicate that the resonant frequency barely changes with the electrode thickness, whereas the vibrational amplitude is inversely proportional to the electrode thickness under the same driving voltage. Experiments were performed with four resonators and piezoelectric electrodes of four sizes, and results were consistent with simulations. The resonant frequencies of four resonators changed within 0.36% after attaching the piezoelectric electrodes. Meanwhile, with the same driving voltage, it was shown that the vibrational amplitude decreased with the increase of electrode thickness. Moreover, thinner electrodes resulted in better Q factor and therefore better performance. This study may provide useful reference on electrode design of the CVGs.


Introduction
The cylindrical resonator gyroscope (CRG) is a kind of Coriolis Vibratory Gyroscope (CVG) which measures angular velocity through the precession of elastic waves.The CRG has many advantages, such as high reliability, high stability, long life, low power consumption, and small size, and is widely used in the medium-and low-precision navigation fields and platform stabilization systems [1][2][3][4][5][6].The representative products include the Watson Inc. Pro Gyro5 series [7] and the InnaLabs Inc. GI-CVG series [8].The cylindrical shell resonator is the core element of the CRG, the vibrational characteristics of which determine its overall performance.
Normally, the resonator can maintain the maximum amplitude at the resonant frequency.A higher amplitude is advantageous in scale factor and detecting signals, which brings a better signal-to-noise ratio (SNR) [9][10][11][12]. factor directly determines the precision and drift characteristics of gyroscopes [13].Pan et al. reported a fused silica cylindrical resonator with the  factor approaching 1000000 [14].However, when actuated by piezoelectric electrodes (also called piezoelectrodes) [15], the  factor dropped dramatically.Therefore, it is of great importance to study the vibrational characteristics of resonators with attached piezoelectrodes.
To our knowledge, there was no relevant research on how the thickness of piezoelectrodes influences the vibrational characteristics such as resonant frequency, vibrational amplitude, and  factor.
In this paper, we first presented the mathematical model of the resonator actuated by piezoelectrodes.Then, we built a finite element model through ANSYS to investigate its vibration shape, resonant frequency, and vibrational amplitude of the resonator with attached piezoelectrodes.Finally, we performed experiments on four resonators made of brass and piezoelectrodes made of PZT-5H.Experimental results were consistent with simulations.We demonstrated that thinner electrodes introduce less degradation of  factors and meanwhile provide a larger amplitude with the same  driving voltage, which is preferred for the SNR of the gyroscope.

Mathematical Models
This paper describes a kind of resonators, including a resonant shell, a vibration-conducting shell, a bottom plate, and a stem.Eight piezoelectrodes are attached on the bottom of the resonator symmetrically to actuate the resonator and detect its vibrational signal, as shown in Figure 2. The resonator is actuated by piezoelectrodes to its lowest bending mode, also called the wineglass mode, with 2 degenerate modes that are spatially orthogonal with each other.Usually, The CVG works in the force-to-rebalance mode, in which two driving piezoelectrodes actuate the resonator into the wineglass mode and maintain a stable amplitude, and the Coriolis force stimulates the other degenerate mode.The controlling piezoelectrodes produce a signal to null the secondary mode, the amplitude of which is linearly proportional to the angular rate [16][17][18][19].
The piezoelectrodes were attached on the bottom of the resonator.A simplified model is shown in Figure 1.When the voltage is exerted on the piezoelectrodes, the driving force   and the driving torque   will act on the resonator.The intensity of   and   depend on the driving voltage, the piezoelectric properties, geometric shape, and size of the piezoelectrodes.The driving force   and the driving torque   can be expressed as [20][21][22][23] where   is the driving voltage,  1 is the weight of strain tensor  of the piezoelectrodes at the radial direction,   is the width of the piezoelectrodes, ℎ  is the thickness of the piezoelectrodes, ℎ  is the thickness of the bottom of the resonator, and  11 ,  13 ,  33 ,  31 , and  33 are the weight of the coefficient in the parameter matrix of PZT-5H.
For the cylindrical shell resonator,  factor refers to the ratio of the energy loss in a vibrational cycle to the total energy stored in the resonator, which can be expressed as [24] where  refers to the total energy stored in the resonators and Δ  refers to the energy loss in one cycle caused by No. th factor.
As simulations on the  factor are complicated, this paper investigates the  factor degradation with the thickness of piezoelectrodes in experiment.The  factor was measured and calculated through MATLAB by the ring-down method, which can be expressed as where  is the resonant frequency and  is the decay time, meaning the time during which the vibrational amplitude decreased to its 1/.

Results and Discussion
3.1.Simulation.When the resonator is at resonance, the steady-state vibrational amplitude depends on the driving force and the torque induced by the piezoelectrodes, which are related to the geometric dimension of the piezoelectrodes.However, the numerical equations cannot describe the correlation between them clearly.Therefore, a finite element model was built with ANSYS.The parameters of materials and the piezoelectric constants of PZT-5H were shown in Table 1.In the finite element model, the thicknesses of the piezoelectrodes were changed from 100 m to 800 m with an increment of 100 m.The length and width of the piezoelectrodes were 8 mm and 1 mm, respectively.The distances between the geometric centers of the piezoelectrodes and the center of the resonator were 9 mm.Other geometric dimensions of the resonator are shown in Table 2 and Figure 2. The driving voltages were set at 10 V throughout the simulation.In addition, according to Zhu et al. [25], the epoxy resin adhesives have minor influence on the vibrational characteristics; therefore we did not include epoxy resin adhesive in our model.This simulation demonstrated how the resonant frequency and the amplitude change with the thickness of piezoelectrodes, so as to find an optimal thickness of the electrodes.The resonant frequencies were obtained by modal analysis.In order to simplify the simulation process, the deformations were obtained by static analysis.Since the vibration amplitude is proportional to the static deformation at the resonant frequency, this static deformation can be a direct indicator of the vibration amplitude.
Results are shown in Table 3 and Figure 3.And the results in Table 3 are compared with experimental results in Figure 8.The resonant frequency increased slightly with the increase of the electrode thickness, and the change of the resonant frequency is linear, basically.Thicker piezoelectrodes introduced bigger relative changes (defined as ( 1 −  0 )/ 0 , where  0 and  1 are the wineglass mode frequency before and after the attachment of piezoelectrodes, resp.).
The vibrational amplitude (evaluated at the bottom edge) was inversely proportional to the electrode thickness, that is, the thinner the piezoelectrodes are, the greater the amplitude is.This is because, under the same driving voltage, the thinner the piezoelectrodes are, the stronger the electric field in the piezoelectrodes is, and the more obvious the inverse piezoelectric effect is, which leads to a greater amplitude.That means the vibrational signal can be easier to detect and achieve a better SNR.

Experiments.
Four brass resonators were fabricated to verify the simulation results.Piezoelectrodes in four different thicknesses were attached on the bottom of the resonators by epoxy resin adhesive, as shown in Figure 4.The thicknesses  of the piezoelectrodes were 200 m, 400 m, 600 m, and 800 m, respectively.A laser Doppler vibrometer was used to measure the resonant frequency, amplitude, and  factor.The resonators were actuated by acoustic sources and by piezoelectric electrodes before and after the attachment of the piezoelectrodes under atmospheric pressure, respectively.The measurement system is shown in Figure 5.
Before the measurement, the actuation direction and the principle axes should be aligned in the same orientation.Then, the vibrational shape is obtained through the scanning of the measurement point defined on the bottom of the resonator, as shown in Figure 6.The antinode of the vibrational shape is selected as the subsequent measurement point.The frequency sweeping data was recorded, and the resonant frequency and amplitude were obtained through fast Fourier transform.The decay time is obtained from the exponential fitting to ring-down data through MATLAB, as shown in Figure 7.The fitting function is as follows: Experimental results are shown in Table 4.And they are compared with simulation results in Figure 8.For electrode thicknesses of 200 m, 400 m, 600 m, and 800 m, the relative increases of the resonant frequencies are 0.14%, 0.24%, 0.29%, and 0.36%, respectively, which indicates that thinner piezoelectrodes introduce less change in resonant frequency.Under the same driving voltage (10 V), the vibrational amplitude is 34.18 nm, 22.38 nm, 16.47 nm, and 12.08 nm,

Actuated by
Actuated by acoustic wave piezoelectrodes respectively, which shows that thinner piezoelectrodes actuated the resonator to a larger vibrational amplitude.The reduction rate of the  factor is 9.49%, 15.71%, 20.65% and 21.43%, respectively, which indicate that thinner piezoelectrodes are also advantageous in maintaining a higher  factor.

Conclusions
This paper investigated the influence of the electrode thickness on the vibrational properties of a cylindrical shell resonator used in cylindrical resonator gyroscopes.A finite element model was built to simulate the resonator actuated by piezoelectrodes with varied thicknesses under the same driving voltage.The resonant frequencies and vibration amplitudes were calculated.Then experiments were carried out on four resonators with four different electrode thicknesses.Experimental results were consistent with simulations.It was demonstrated that the resonant frequency increases slightly and linearly with the increase of electrode thicknesses.Under the same voltage, thinner electrodes provide larger amplitudes, while introducing less degradation of  factors.Therefore, it was advantageous to use thinner piezoelectrodes in the actuation of CVG resonators for better noise and drift performance.

Figure 2 :Figure 3 :
Figure 2: The simulation structure of the resonator.

Figure 7 :
Figure 7: The exponential fitting of vibrational amplitude data.

Figure 8 :
Figure 8: Comparison of the relative change of resonant frequencies and vibrational amplitudes under different thickness of piezoelectrodes between experiments and simulations.

Table 1 :
Parameters of materials and piezoelectric constants of PZT-5H.

Table 3 :
The simulation results of resonant frequencies and vibrational amplitudes under different thicknesses of piezoelectrodes.

Table 4 :
The comparison of experimental results between before and after the piezoelectrodes were attached.