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Introduction and Literature review

Types of gyroscopes

  • Optical Gyroscopes
  • Mechanical Gyroscopes
    • Angular momentum based gyroscopes
    • Coriolis Vibratory Gyroscope (CVG)

The gimbal frame of the gyroscope is attached to the vehicle and is free to rotate. The response is proportional to the angular velocity of the frame on which the sensor is mounted.

Figure 1.2: Principle of CVG operation (Reproduced with permission from Apostolyuk  [2016])
Figure 1.2: Principle of CVG operation (Reproduced with permission from Apostolyuk [2016])

Gyroscope for aerospace and space applications

  • Hemispherical resonator gyroscope (HRG)
    • HRG configuration
    • Working principle
  • Quality factor (Q factor)
    • Resolution
    • Scale factor

The angular velocity of the resulting standing wave precession is different from the input velocity. The mechanism for the mechanical noise is the Brownian motion caused by the molecular collisions.

Table 1.1: Typical number of parts of different types of gyroscopes (Xu et al. [2011])
Table 1.1: Typical number of parts of different types of gyroscopes (Xu et al. [2011])

Damping mechanisms

  • Thermo elastic damping (TED)
    • TED in beam resonators
    • TED in ring resonators
    • TED in coated resonators
    • TED in gyroscope resonators
  • Anchor (support) loss
    • Closed form solutions in beam and disk structures
    • Perfectly matched layer (PML)
    • Effect of anchor geometry
    • Resonators with imperfections
    • Anchor loss in hemispherical and cylindrical resonators
  • Surface loss
  • Material internal friction
  • Internal friction in coated resonators
  • Fluid damping
  • Electronics damping

It also depends a lot on the ratio of the beam radius to the sheave radius. Surface damage indicates damage to the surface of the material, such as surface roughness (SR).

Objective of present work

The design of the resonator is made well away from the TED driven Debye peak to achieve ultra high Q factor in the configuration phase. Thus, the demonstration of high Q-factor in the final functional high-precision macroscale resonator configuration is accomplished.

Structure of the Thesis

The second part, Qfreq, is due to the material properties and the geometry of the resonator. Furthermore, the effect of the stem radius on QAnchor is studied and the result is shown in Figure 4.11.

Thermoelastic damping and Anchor loss: Theory and validation

Thermoelastic damping (TED)

  • Theory and problem formulation
  • Analytical solution for beam resonator
  • HRG configuration and TED

ER where  is the tension,  is the relaxation time for stress to relax exponentially at constant tension, ̇ is the degree of stress, ER is the relaxed isothermal Young's modulus,  is the tension,  is the relaxation time for tension to relax exponentially under constant stress and ̇ is the strain rate. The inner stem is an integral part of the hemispherical region and the resonator is a one-piece configuration in the current study. Normalized displacement plot of the functional elliptical mode (N = 2) of HRG resonator is shown in Figure 2.4.

Figure 2.1: Highly coupled four mechanical domains under thermoelastic damping study
Figure 2.1: Highly coupled four mechanical domains under thermoelastic damping study

Anchor loss

  • Theory and problem formulation
  • Analytical solution for beam resonator
  • Anchor loss in HRG

The beam resonator and support structure are made of the same material, and the dimensions of the support in the x-y plane are much larger than those of the beam resonator. Then the 2D elastic wave equation is used to estimate the shear behavior of the support. The sustain quality factor of a beam resonator is independent of the Young's modulus of the resonator's material.

Figure 2.8: Schematic of beam resonator
Figure 2.8: Schematic of beam resonator

Numerical solution validation

  • Validation for TED
  • Validation for anchor loss

The mechanical resonance frequency is 0.63 MHz, while the frequency corresponding to the thermal relaxation time is 0.60 MHz for this beam resonator configuration. The comparison of the analytical and the numerical results is shown in Table 2.5 for three cases of PML size. From table 2.5 it is clear that numerical simulation results for QAnchor are in good agreement with the analytical result with correct choice of modeling parameters.

Table 2.2: Validation of FE simulation with analytical results for TED of beam resonator  Model details  Q TED    Deviation (%)
Table 2.2: Validation of FE simulation with analytical results for TED of beam resonator Model details Q TED Deviation (%)

Summary

Subsequently, the effect of the material properties of the resonator and different dimensions on the QTED is studied. However, increasing the height of the cylindrical region also increases the size of the resonator and sensor. Now the effect of the height of the cylindrical area on the corner gain is studied and the result is shown in Figure 3.26.

Design for low thermoelastic damping and sensitivitystudy

Estimation of Q Eff requirement

Once the mechanical resonator QEff requirement is arrived at, the target specification for the macro size HRG can be framed based on other functional requirements and design constraints. Similarly, frequency separation of the functional resonant mode with respect to neighboring modes is also critical for armature loss. Considering all these aspects, specifications of the mechanical resonator have been arrived at as given in Table 3.1.

Figure 3.1: Relationship between MNER and Q Eff  for a given design and operating  parameters
Figure 3.1: Relationship between MNER and Q Eff for a given design and operating parameters

Ring resonator parametric study

  • Analytical solution for ring resonator
  • Numerical solution for ring resonator
  • Ring configuration for lowest TED

The frequency corresponding to the thermal relaxation time in the circumferential direction is 2.5 milliHz, which is much less than in the radial direction. Initially, the grid density is varied in the circumferential direction until the frequency and QTED converge. The thermal relaxation time in the circumferential direction is much longer than in the radial direction.

Figure 3.2: 3D Finite element model of ring resonator in COMSOL
Figure 3.2: 3D Finite element model of ring resonator in COMSOL

Hemispherical resonator parametric study …

  • Configuration
  • Mesh sesitivity study
  • Modal analysis
  • Effect of resonator material property
  • Effect of operating temperature
  • Effect of resonator geometry

Figure 3.10 (a) and (b) show two plots of the thermoelastic strain induced normalized temperature deviation plot for N = 2 condition, where the circumferential direction, polar direction and radial thickness direction heat transfer paths are marked. But a more predominant effect is the reduction of QTED with increasing E as seen in the simulation. The influence of shell thickness increase on the QTED increase is more compared to the shell mean radius.

Figure 3.8: Cross section view of hemispherical shell and central supporting stem  Table 3.9:  Nomenclature, description and nominal dimensions of the hemispherical shell  configuration with stem
Figure 3.8: Cross section view of hemispherical shell and central supporting stem Table 3.9: Nomenclature, description and nominal dimensions of the hemispherical shell configuration with stem

Sensitivity study of thin film coating on TED

  • Coating studies on ring resonator
  • Effect of coating on basic hemispherical resonator configuration …
  • Effect of coating variationon hemispherical resonator configuration .79

Now the effect of coating thickness on QTED is investigated in detail and the result is tabulated in Table 3.20. Therefore, a sensitivity study is performed to obtain the effect of coating thickness variation on QTED. This is due to the fact that the variation in the coating thickness in the polar direction is symmetrical in the peripheral direction as far as N = 2 mode TH.

Table 3.17: Effect of the circumferential direction mesh density on Q TED     Number of elements
Table 3.17: Effect of the circumferential direction mesh density on Q TED Number of elements

Hybrid functional resonator configuration

  • Modal analysis for the hybridresonator configuration
  • Performance parameters
    • Effective mass
    • Angular gain
  • Coating studies on hybrid configuration resonator
    • Effect of coating on individual regions of resonator
    • Effect of partial coating configuration of resonator
    • Effect of coating thickness on Q TED

The cylindrical part of the resonator can be manufactured with greater precision than the hemispherical part. It can be seen that the addition of the cylindrical region reduces all frequencies due to the additional mass of the cylindrical region. It can be seen that the effect of the angles of the mantle width from the equatorial plane had a significant effect on the QTED.

Figure 3.23: Functional hemispherical-cylindrical hybrid resonator configuration  Hemispherical  region  is  extended  with  a  cylindrical  ring  region  below its  equator to  arrive  at  the  hybrid  configuration
Figure 3.23: Functional hemispherical-cylindrical hybrid resonator configuration Hemispherical region is extended with a cylindrical ring region below its equator to arrive at the hybrid configuration

Summary

The normalized displacement of the substrate in Figure 4.7 is very small compared to the resonator and stem. The effect of average shell radius on anchor loss is studied and the results are shown in Figure 4.9. The inlet side profile of the thin film coated coupon displacement is shown in Figure 8.14.

Anchor loss and sensitivity study

Numerical modeling of anchor loss in hemispherical resonator

A mapped mesh using the quadrilateral elements is generated in the cross section of the resonator, the substrate and the PML in a 2D plane. The normalized displacement graph is shown in Figure 4.5 where the maximum displacement is seen at the edge of the hemispherical resonator. The stem is also connected to the nodal point of the N = 2 mode of the hemispherical resonator.

Figure 4.1: Finite element model of resonator with substrate and PML region
Figure 4.1: Finite element model of resonator with substrate and PML region

Effect of resonator geometric parameters

  • Shell radial thickness
  • Shell mean radius

As the radius of the shell increases, the anchor location moves away from the hemispherical rim region where maximum deformation occurs. This is similar to the effect of length increase on QAnchor for the cantilever beam resonator configuration.

Effect of mode interactions

The effect of stem length is also investigated based on the interaction between the N = 2 shell state and the stem tilt state. The dimensional tolerances of the shell and stem must be strictly controlled to ensure adequate frequency separation. A non-linear increase in QAnchor is seen when the stem radius is less than 3 mm.

Figure 4.10: Effect of the stem length on frequency separation and Q Anchor
Figure 4.10: Effect of the stem length on frequency separation and Q Anchor

Effect of resonator material properties

Effect of support structure

Effect of resonator structure imperfections

  • Hemisphere shell axis is offset with respect to the stem axis
  • Effect of shell tilt with respect to the stem
  • Effect of shell radial thickness variation
  • Effect of hemispherical height variation
  • Single unbalanced mass sensitivity on frequency split and Q Anchor
  • Effect of unbalanced mass profile on frequency split and Q Anchor

It can be seen that the QAnchor decreases as the magnitude of the radial thickness variation increases. Consider the case of a single unbalanced mass in the resonator generating the frequency distribution of the N = 2 modes. Then the effect of the unbalanced mass resonator on frequency distribution is simulated by adding a point mass to the equator plane and the result is tabulated in table 4.10.

Figure 4.13: Effect of the shell axis offset on Q Anchor
Figure 4.13: Effect of the shell axis offset on Q Anchor

Wine glass configuration resonator

Unbalanced mass distribution has a similar effect on the two N = 2 modes for case 3 and frequency splitting is not present. From these cases it is concluded that the pattern of unbalanced mass has a significant effect on the frequency split and the QAnchor mismatch. It is desirable that a frequency split in milliHz be obtained after fine tuning.

Summary

The frequencies of the two degenerate modes and the frequency separation are shown in Table 6.4 during balancing. Measurements were made in the cylindrical and spherical regions of the shell as shown in Figure 7.6. Then, it interfaces with the substrate interface of the sighting mechanism, which is a rotating part.

Resonator fabrication and Metrology

Fabrication of brittle material

  • Fabrication procedure of fused silica high precision resonator
    • Blank selection
    • Fused silica blank preparation
    • Bulk machining
    • Chemical cleaning
    • Bulk etching
    • Precision machining
  • Fabrication of high precision resonator

Each vertical segment of the plot corresponds to 1 mm size of the rod in the radial direction. The mass processing of the shell using ultrasonic technique leaves some residual layers of cracked material throughout the surface as the base material is brittle. The shell has become transparent because most of the damaged surface layer has been removed during the etching process.

Figure 5.1: Birefringence result of 35 mm diameter rod
Figure 5.1: Birefringence result of 35 mm diameter rod

Metrology

The nominal radial thickness is 1 mm, ball mean radius is 14.6 mm and stem diameter is 6.4 mm. Based on the closeness of the dimensional and the geometric measurements with respect to the design values, the lowest precision and the highest precision resonators are identified.

Summary

Microwave sputter deposition technique is used for mass addition at the required locations of the resonator. Nanoindentation is reliable for determining mechanical properties of the thin film coatings (Nazemian and Chamani [2019]). Ion beam sputtering based ultra-thin film coating is done on the bulk surfaces of the resonator as explained earlier.

Resonator characterization

Resonator excitation setup

The arm attached to the edge also undergoes sinusoidal motion which is used to excite the resonator. Proper chemical cleaning of the resonator and the plate should be done and allow enough time to dry. The resonator is bonded to the quartz plate using cyanoacrylate locktite 4014 binder with a small known compressive load.

Figure 6.1: 3D model of resonator excitation set-up
Figure 6.1: 3D model of resonator excitation set-up

Characterization setup

Measurement procedure

  • Measurement procedure of frequency
  • Measurement of frequency
  • Measurement of frequency split

Perform a coarse sine sweep in the range of +/- 25 Hz around fambient to identify the frequency of the N = 2 mode. The natural frequencies of the eight resonators are measured based on the above procedure. The effect of the shell thickness and the stem diameter on N = 2 frequency is plotted in Figures (6.8) and (6.9).

Figure 6.8: Effect of the shell thickness on N = 2 mode frequency
Figure 6.8: Effect of the shell thickness on N = 2 mode frequency

Balancing methods

  • Balancing procedure
  • Balancing by mass removal method

The method to eliminate the frequency splitting is to remove one, two, or four unbalanced masses from each of the antinodes on the low-frequency axis of the unbalanced resonator. It has been found that four-point mass correction of the same amount at 90 degrees from each other in the circumferential direction results in lower frequency separation. Low precision resonator (resonator 1) is balanced by mass removal method from frequency division of 5.95 Hz.

Figure 6.10: Typical LDV responses of two N = 2 modes between two antinodal points  TH-2485_166103023
Figure 6.10: Typical LDV responses of two N = 2 modes between two antinodal points TH-2485_166103023

Summary

Surface damage indicates damage to the surface of the material such as surface roughness (SR). The step-by-step procedure for measuring frequency and quality factor is discussed below. Internal friction from fused silicon is not the limiting contributor to the effective quality factor of the uncoated resonator.

Material internal friction, Surface loss and Fluid damping

Internal friction of resonator material

It was observed that the QMIF of the material caused by internal friction varied between different grades of fused silica. Many of the previously studied fused silica samples were Type III, which is synthetic fused silica. The summary of the studies of various grades of Heraeus Brand Suprasil synthetic fused silica material is presented in Table 7.1.

Internal friction of coating material

Clearly, QMIF of 107 or more is possible for an uncoated shell, considering the internal friction of the material as the only source of dissipation. Therefore, internal friction from fused silica will not be the limiting contributor to the effective quality factor of the uncoated resonator for the few million requirement. However, the internal friction of fused silica increases towards freezing temperatures by over two orders of magnitude from 300 K to 50 K, adversely affecting the effective quality factor.

Surface loss

  • Nanoindentation
  • Theory of nanoindentation
  • Surface characterization results
  • Surface loss estimation

It can be seen that the hardness and reduced Young's modulus is much less compared to the bulk material's nominal hardness of 11 GPa and Young's modulus of 73 GPa due to the formation of the damaged surface layer after USM . The improvement in hardness and reduced Young's modulus over the resonator surface after wet chemical etching is shown in Tables (7.3) and (7.4). It can be seen that the reduced Young's modulus and hardness improve after each chemical etching step due to the removal of most of the damaged surface layer.

Figure 7.1: Berkocich pyramid indenter used for nanoindentation test (Reproduced with  permission from Sattler [2010])
Figure 7.1: Berkocich pyramid indenter used for nanoindentation test (Reproduced with permission from Sattler [2010])

Limiting Q factor estimation

Using equation (7.10), a rough estimate of the QSurface of the present silicon resonator can be made. Since the present resonator uses Suprasil 312 high purity fused silica material, the QMIF must be greater than 107. The QEff of the gold-coated fused silica resonator can be limited by QTED. with coating), QIF coating and QSurface as these losses are in the same order (106).

Q Eff measurement without thin layer coating

  • Measurement of the frequency and Q Eff (for Q Eff less than 10 5 )
  • Measurement of the frequency and Q Eff (for Q Eff more than 10 5 )

The step-by-step procedure for measuring the frequency and the quality factor is mentioned below. Natural frequency of the resonator is obtained using FFT of the velocity signal of the LDV. It can also be seen that the chemical etch removes most of the damaged surface layer which is the main contributor to the surface loss.

Figure 7.8: A typical run-down measurement of velocity at resonator rim using the LDV
Figure 7.8: A typical run-down measurement of velocity at resonator rim using the LDV

Q Eff circumferential variation

Fluid damping

Since the characterization and operation of the sensor is done at 10-3 N/m2, QFluid is not important in QEff. This QEff is smaller than the reported value of 10 million for the space-only hemispherical resonant structure configuration in Rozelle [2009]. Also, further optimizations of fabrication parameters, etching process parameters and chemical cleaning are proposed to minimize the contribution of surface loss as mentioned in the scope for future work in section 9.3.

Summary

Park make XE7 series instrument which is shown in figure 8.6 is used to make sample. Bruker make Contour GTX 3D optical profiler, which is shown in Figure 8.9, is used to measure thin film coating thickness. This experiment is performed in the Hysitron make Ti Premier series instrument, which is shown in Figure 8.12.

Figure 8.1: 3D model of glancing mechanism in coating machine for 3D coating
Figure 8.1: 3D model of glancing mechanism in coating machine for 3D coating

Demonstration of effective Q factor in functional form resonator …153

Balancing and Q Eff of coated resonator

Therefore, balancing should be done by adding equal mass to four anti-nodal points of the high frequency axis. Therefore, magnetron sputtering is preferred as the mass addition thickness is relatively higher for balancing. An order of magnitude improvement in frequency separation is achieved with this precise mass addition technique.

Coating characterization

  • Atomic force microscopy (AFM)
  • Characterization of magnetron coated thin film
    • Coating thickness measurement
    • Nanoscratch testing
  • Characterization of ion beam coated ultra-thin film
    • Coating thickness and roughness measurement
    • Nanoindentation and nano scratch testing

The achieved QEff after deposition is 2.4 × 106, which also meets the specification of over one million. The QEff and frequency division meet the specification requirements for a final functional hybrid resonator as described in Chapters 3 and 4. The normal load obtained during the test is shown in Figure 8.18 and the lateral force obtained during the test is shown in Figure 8.19. Scratch test results show that the magnetron sputtered thin film gold coating is integral with the underlying silicon dioxide.

Summary

Conclusions and Scope for future work

Summary

Conclusions

Scope for future work

Principle of CVG operation

Classification of CVGs

Typical HRG resonator

Typical HRG configuration

Coriolis forces in HRG resonator

Schematic of precession in HRG

Highly coupled four mechanical domains under thermoelastic damping study

Design guideline for resonator operating region for high Q TED where maximum Q -1

Integral hemispherical resonator configuration with inside stem

Functional elliptical mode (N = 2) normalized displacement

N = 2 elliptical degenerate mode 1 showing nodal and anti nodal locations

N = 2 elliptical degenerate mode 2 at circumferential angle 45 deg to mode 1

A typical plot of strain induced normalised temperature deviation from absolute

Schematic of beam resonator

The normalised deformation inside the substrate and the PML when the resonator

Normalized temperature deviation (increase and decrease from absolute equilibrium

FE model of substrate and PML region with fine mesh

Zoomed view of FE model showing thin cantilever beam resonator with mounting

Relationship between MNER and Q Eff for a given design and operating parameters

Different views of overplot of undeformed and normalised deformation plotfor the

Normalized temperature deviation (increase and decrease from absolute equilibrium

Figure

Figure 3: Responses of the two N = 2 modes ( 1  and  2 ) showing the frequencies and the  corresponding axes locations around circumferential direction
Figure 2.2: Design guideline for resonator operating region for high Q TED  where maximum   Q -1  (maximum dissipation) at   = 1
Figure 2.7: A typical plot of strain induced normalised temperature deviation from absolute  equillibium temperature
Table 2.2: Validation of FE simulation with analytical results for TED of beam resonator  Model details  Q TED    Deviation (%)
+7

References

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