Conclusion

In this work, the preliminary conceptual design of the vibratory gyroscope is presented and one feasible operation associated with the adaptive control strategies is proposed. This design is based on the H-type gyroscope, which has the advantage of reducing electrostatic feedthrough and large output signals. Moreover, an effective suppression of leakage electrostatic coupling is proposed by utilizing the design with the intermediate layer GND of H-gyro. The fabrication errors and the mechanical coupling were identified by the adaptive algorithms and null out along the operation of this gyroscope. With consideration for noises arising in the designed gyroscope, a 0.0063deg s resolution can be achieved and the control voltages of dual-axes driving arms are below 10 V in our simulations.

5.2 Future Works

In this thesis, we designed a MEMS PZT gyroscope which is expected to fabricate based on the in-situ fabrication process. However, more complete simulation and experiments must be accomplished in future. There is still room for further investigation in this study. Some future works are summarized below.

1. Mechanical Q-values and resonant frequencies of two axes have a complicated relationship with temperature characteristics. In this paper, the

2. Different piezoelectric materials have different properties, which will influence the constant values of the stiffness matrix, piezoelectric matrix and dielectric matrix. Moreover, different cut angle of the piezoelectric material would change values of the matrix. In other words, there will be an optimal piezoelectric material and cut angle for the proposed gyroscope design.

3. In order to fabricate the proposed PZT gyroscope by so-gel process, the configuration and electric wiring with planar configuration is preferred. One of the planar configurations that can meet our performance requirements is shown in Figure 5.1.

APPENDIX I

Piezoelectric materials are used in many different types of electromechanical transducers. Now we discuss the motion of the bending model. According to the Kirchhoff & Love relations between strain and deflection of piezoelectric materials [21], ignore

(

1+z/R_i

)

term, the relations become

where U、V、W are respectively total deflection of x、y、z coordinates.

(

1 2 z

) (

u 1 2

)

z 1

(

1 2

)

U α ,α , = α ,α + β α ,α (I.7)

(

1 2 z

) (

v 1 2

)

z 2

(

1 2

)

u、v、w are respectively membrane displacement of x、y、z, and β₁、β₂ are defined as shear strain or bending strain. A₁、A₂ are defined as Lame parameters or fundamental form parameters. The fundamental form parameters above are shown in Table 14, which are different due to the different configuration.

Now we consider the planar actuator (sensor), so-called planar actuator (sensor) is that the stresses (T 、₃ T₄ and T ) of the vertical coordinate are smaller extremely ₅ than that (T₁、T₂ and T ) of the horizontal. When these planar actuators are ₆ combined laminated construction, the equations are described as follow：

( )

1 2

(

1

)

1

(

31 32 36

)

differential operator and piezoelectricity function of differential operator [13]；1、2、

3 are respectively x、y、z coordinates.

For a planar double-decked actuator (sensor), the vertical deflection of the planar double-decked actuator (sensor) is larger extremely than horizontal deflection.

Therefore, now we ignore all the effects of the horizontal deflection and membrane effects；let applied force to the double-decked actuator (sensor) be zero (f =0、₃

(

31 32 36

)

3 d d d

L , , =0) and define the electrical and mechanical boundary conditions.

Then we check the Table 8 to get A₁=1、A₂=0、R₁=R₂= ∞ 、 0

And the above equation can be written another as follows.

(

hw

)

We can get simplified equation of motion;

Bending stiffness is

12

Let the general solution of the vertical deflection be

( )

^{x t W}

( )

^{x ej t}

w , = ^ω ；ω is frequency.

Then the equation (I.20) becomes

0

We define electrical boundary conditions

⎪⎩

Then define mechanical boundary conditions

( )

x t 0

so that the equation (I.28) and equation (I.29) become

t

Then we can get the

Therefore we can get the general solution of the vertical deflection^w

( )

^{x,t . Figure}

3.2 shows the bendingmode simulation and confirms the result. We apply V=10 volt on the driving electrode and can get bending resultw

( )

x x₌L_/2 = .29×10⁻⁹, that is confirmed by numerical solution by FEM.

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Table 1 Electromechanical Coupling Coefficients for Driving and Detecting Electrodes (fork type) [2]

Resonance frequency：fo

k² of driving electrode

k² of detecting electrode f_x mode 18.6203(kHz) 0.00584 1.07×10⁻⁵ fy mode 20.6933(kHz) 8.7×10^{−5 0.00389}

Table 2 Electromechanical Coupling Coefficients for Symmetrical Electrode [2]

Driving electrode Detecting electrode Balanced Unbalanced Balanced Unbalanced

fo(kHz) k² fo(kHz) k² fo(kHz) k² fo(kHz) k² fx mode 18.6207 0.0121 18.6204 0.0099 18.6859 0 18.6301 ^21.5ppm fz mode 20.7915 0 20.7901 9.6ppm 20.6782 0.0048 20.6731 0.0066 Configur

-ation of electrode

Table 3 The properties of the four piezoelectric state variables

6 x 1 stress components

6 x 1 strain components

3 x 1 electric field components 3 x 1 electric charge density displacement

components

Table 4 The matrix properties of the different forms

6 x 6 compliance coefficients

6 x 6 stiffness coefficients

3 x 3 electric permittivity

3 x 6 piezoelectric coupling coefficients for d form

3 x 6 piezoelectric coupling coefficients for e form

3 x 6 piezoelectric coupling coefficients for g form

h 3 x 6 piezoelectric coupling coefficients for h form

Table 5 Compliance Constants of LiTaO3

Compliance Constants ^sij

(

^10−12m2 /^newton

)

s11 4.87

s22 －

s 33 4.36

s44 10.8

s 55 －

s 66 －

s12 -0.58

s 13 -1.25

s14 0.64

s 16 －

s 23 －

s 25 －

Table 6 Piezoelectric Strain Constants of LiTaO3

Piezoelectric Strain Constants ^dij

(

^{10−12coulomb}/^newton

)

d11 20.3

d14 －

d 15 24.5

d22 6.0

d 25 －

d 31 －

d 32 5.3

d 33 7.5

d 36 0.9

Table 7 Permittivity Strain Constants of LiTaO₃ Piezoelectric Strain Constants ^dij

(

^{10−12coulomb}/^newton

)

0 11 ε

ε / 41

0 22 ε

ε / －

0 33 ε

ε / 43

†ε₀ =8.854×10⁻¹²farads/m

Table 8 Stiffness Constants of LiTaO3

Stiffness Constants cij

(

10¹⁰newton/m²

)

c11 23.3

c22 －

c 33 27.5

c44 9.4

c 55 －

c 66 －

c12 4.7

c 13 8.0

c14 -1.1

c 16 －

c 23 －

c 25 －

Table 9 Crystal Symmetry Class and Mass Density

Material Lithium

Tantalate

Chemical Formula LiTaO3

Symmetry Class Trig. 3m

Density

(

^{kg m3}

)

⁷⁴⁵⁰

Table 10 Power Spectral Density of Noise Power spectral density of noise PSD of mechanical

noise

2.62 10× -12N Hz PSD of electrical

noise

2.12 10 V× -8 Hz Position noise of

mechanical noise

5 27 10 V. × -12

Position noise of electrical noise

3.59 10 V× -8

Velocity noise of mechanical noise

1 16 10 V. × -6

Position noise of electrical noise

7.9 10 V× -3

Table 11 Parameters of the designed H-gyro Parameters of Approximated 2^nd Order Model

mass 3.036 10 kg× ^-8 Resonant frequency

of Y mode 215 kHz

Resonant frequency

of H mode 220 kHz

Q-value 100

Table 12 Parameters of Approximated 2nd Order Model Parameters of Approximated 2^nd Order Model

Dxx 13509

D_xx 13823

D_xy 11

Kxx 1350900^2

Kyy 1382300^2

K_xy 801796040

Ω (rad/s) 0.33

Table 13 Selected Adaptive Control Gains Selected adaptive control gains

γD ₁₀₀

γR 100

γ_{Ω 150}

γ1 ₁₀₀

γ2 ₁₀₀

Table 14 Parameters of Different Configurations

Configuration A₁ A₂ dα₁ dα₂ R₁ R₂

Ring 1 1 dx Rdθ R ∞

Beam 1 0 dx 0 ∞ ∞

Radial shell 1 R dx dθ ∞ R

Disk 1 R dr dθ ∞ ∞

Tetragon plane 1 1 dx dy ∞ ∞

Figure 1.1 Coupling classifications of gyroscope

Figure 1.2 Conventional fork type piezoelectric gyroscope Driving

electrod

Detecting electrode Detecting

vibration Driving

vibration

Coriolis force

i/p o/p k2

k₃

k1

k₄

Figure 1.3 H-type LT piezoelectric gyroscope [2]

Figure 2.1 Inertia coordinate of the piezoelectric material Y, 2

Z, 3

X, 1 1

6 5

6

2 4 4 3

5

Figure 3.1 bending model concept.

Figure 3.2 Bending mode simulation by ANSYS

Figure 3.3 Displacement vs. frequency of bending mode

Driving electrode

Detecting electrode

GND

V out

V₁ V2

Isolation layer

Figure 3.5 Size of the designed gyroscope model

Figure 3.6 Configuration of gyroscope model (case II)

Figure 3.7 Modal analysis of the designed model by ANSYS (H mode)

Figure 3.9 Lateral view of the designed model by ANSYS (H mode)

Figure 3.10 Lateral view of the designed model by ANSYS (Y mode)

Figure 3.11 Schema of equivalent circuit between driving and detection electrodes

Figure 3.12 Configuration of the designed model by ANSYS

Figure 3.13 Frequency response with intermediate isolation

Figure 3.14 Frequency response without any intermediate limit

Figure 3.15 Frequency response with intermediate planar electrode GND

re la tiv e d isp la c e m e n t(Y m o d e )

-2 -1 0 1 2 3 4

1 6 11 16

n(n=#1~17) Uzn/Uzo

Figure 3.16 Relative displacements of the designed model (Y mode)

relative displacement(H mode)

-2 0 2 4 6 8

1 6 11 16

n(n=#1~17) Uyn/Uyo

Figure 3.17 Relative displacements of the designed model (H mode)

Figure 3.18 Curve fitting of frequency response (H mode)

Figure 3.19 Curve fitting of frequency response (Y mode)

mesh size vs frequency(H mode)

12600 12800 13000 13200 13400 13600 13800 14000

0 2 4 6 8 10 12 14

mesh density(1/mm) frequency(rad/s)

Figure 3.20 Convergence analysis (H mode)

mesh si ze v s freq u en cy (fo rk mo d e)

13400 13600 13800 14000 14200 14400

0 2 4 6 8 10 12 14

mesh density(1/mm) frequency(rad/s)

Figure 3.21 Convergence analysis (fork mode)

Figure 4.1 Trajectories of x and y axes (a) tracking signal of x axis. (b) tracking signals of y axis.

Figure 4.2 (a) tracking error of x axis. (b) tracking error of y axis.

Figure 4.3 Estimations of damping terms. (a)estimation of Dxx. (b)estimation of Dxy. (c)estimation of Dyy.

Figure 4.4 Estimations of R terms. (a)estimation of Rxx. (b)estimation of Rxy.

(c)estimation of Ryy.

Figure 4.5 Estimation of angular velocity.

Figure 4.6 Control input of x axis.

Figure 4.7 Control input of y axis.

Figure 5.1 Planar electrode design of H-gyro.

在文檔中 應用適應控制之H型壓電陀螺儀的設計與控制 (頁 43-77)