Miniature Flat Type Inertial Piezoelectric Motor - IEEE Xplore

10.1109/ULTSYM.2009.0647

Miniature flat type inertial piezoelectric motor Piotr Vasiljev

Dalius Mazeika

Department of Technical Subjects Vilnius Pedagogical University Vilnius, Lithuania [email protected]

Department of Information Systems Vilnius Gediminas Technical University Vilnius, Lithuania [email protected]

Abstract—Study of a novel design miniature linear piezoelectric motor is proposed in the paper. Size of the motor not exceeds 150 mm3. Piezoelectric motor consists of “butterfly” type actuator and slider. Elliptical trajectory of the actuator is achieved using bending oscillations of the plates of actuator when harmonic or short impulse type excitation is used. Contact zone trajectory of motion is analyzed for both type of excitation. Advantages of impulse type excitation are simpler electric circuit scheme, lesser dependence between the rotor torque and excitation frequency. It allows increasing torque of the rotor without increasing dimensions and mass of the actuator. Numerical modeling of the piezoelectric motor was done to analyze natural frequencies, modal shapes of the actuator and to examine actuator response to the harmonic and impulse type excitation. Experimental prototype of the piezoelectric motor was built and measurement contact zone motion was done. Results of numerical and experimental studies are discussed. Keywords-piezoelectric inertial motor; saw tooth excitation, numerical modeling, experimental study

I.

INTRODUCTION

Piezoelectric motors are widely used for high accuracy positioning devices such as scanners, cameras, mobile phones and others [1, 2]. Advantages of these motors are high resolution, short response time, small size and simple design [3, 4, 5]. The main problem of small or miniature size piezoelectric motors is the small-scale driving force or torque [6]. This feature restricts applicability of the piezoelectric motor. We suppose to use impulse type excitation signal for miniature motors instead of the conventional harmonic signal. Electric circuit scheme is significantly simpler for impulse type excitation signal and stability of mechanical output parameters of the motor can be increased. A novel high precision inertial type linear piezoelectric motor is introduced in this paper. Motor has miniature size and relatively simple design. Motor was named as “butterfly” type due to special actuator design. Elliptic trajectory of the contact point’s movement is achieved by using bending oscillations of two plates that are connected with small hooked link. Numerical modeling of the piezoelectric actuator was carried out to analyze modal shapes and contact point moving trajectories when harmonic and impulse type excitation is applied. Prototype actuator has been made and experimental study was performed.

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II.

DESIGN AND OPERATING PRINCIPLE OF PIEZOELECTRIC MOTOR

Configuration of the motor includes “butterfly” type actuator and slider. Design of actuator comprises of brass oscillator and two rectangular piezoceramic plates that are glued to the oscillator (Fig. 1a). Oscillator has two plates connected with small hooked link and looks like butterfly. Polarization of piezoceramic is oriented along thickness of the plates and piezoelectric effect d31 is used for the actuation. Contact zone is located on top surface of the hooked link (Fig. 1b) and obtain elliptical trajectory of motion when actuator achieves operating resonant mode. Inertial movement of the slider is achieved when electrodes of piezoelectric plates are excited by sinusoidal or impulse type electric signal. The phase difference is equal to π/2 on different piezoceramic elements when harmonic excitation is used and the same phase of voltage is used for impulse type excitation. Bending oscillations B20 are transferred from plates to the connecting link where contact zone is located. Elliptical motion of the contact zone appears due to the difference phase of oscillations of the plates. Oscillator

Contact zone

Piezoelements a)

b)

Figure 1. Principle scheme of the actuator: a) 2D view and amplitude diagram; b) 3D view

III.

FEM EQUATIONS FOR MOTOR MODELING

Finite element method (FEM) was used to perform modal frequency and transient dynamic analysis to calculate trajectories of the contact zone. Basic dynamic equation of the piezoelectric actuator are derived from the principle of minimum potential energy by means of variational functionals and can be written as follows [8]:

[M ]{u&&} + [C ]{u&} + [K ]{u} + [T ]{ϕ} = {F }⎫⎪ ⎬, ⎪⎭ [T ]T {u} − [S ]{ϕ} = {Q}

(1)

2009 IEEE International Ultrasonics Symposium Proceedings

where [M], [K], [T], [S], [C] are matrices of mass, stiffness, electro elasticity, capacity, damping respectively, {u}, {φ} , {F}, {Q} are vectors of nodes displacements, potentials, structural mechanical forces and charge. Driving force of the actuator is obtained from piezoceramic elements. Finite element discretization of these elements usually consists of a few layers of finite elements. Therefore nodes coupled with electrode layers have known potential values in advance and nodal potential of the remaining elements are calculated during the analysis. Dynamic equation of piezoelectric actuator in this case can be expressed as follows [10]:

[M ] {u&&} + [C ] {u&} + [K ] {u} + [T1 ] {φ1} + [T2 ]{φ 2 } = {F }⎫ ⎪ [T1 ]T {u} − [S11 ] {φ1} − [S12 ] {φ 2 } = {Q1} ⎬, T T ⎪ [T2 ] {u} − [S12 ] {φ1} − [S 22 ] {φ 2 } = {0} ⎭ here

[T ] = [T1 [S ] = ⎡⎢

T2 ] S 12 ⎤ S 22 ⎥⎦

S 11

⎣S

(2)

T 12

(3)

where {ϕ1}{ϕ 2 } are accordingly vectors of nodal potentials known in advance and calculated during numerical simulation. Natural frequencies and modal shapes of the actuator are derived from the modal solution of the piezoelectric system [10]:

([ ]

)

det K * − ω2 [M ] = {0} ,

{F } = −[T ]{ϕ } .

(5)

1

Refer to (2), (3), (5) the vector of equivalent mechanical forces can be calculated as follows:

{F } = ([T ][S ] [T ] − [T ]) {ϕ } . −1

eq

2

22

T

2

1

(7)

where {Fp} is the vector of external mechanical forces generated by piezoelements; {Fc} is the vector of nodal contact forces; {Ff} is the vector of resistance forces of the slider, {δ}vector of nodal displacements of the slider; [D], [H] are damping and stiffness matrices of contact surface. IV.

RESULTS OF NUMERICAL MODELING

Numerical modeling of piezoelectric actuator was performed to investigate vibration shapes and trajectories of contact point motion through the modal, harmonic response and transient analysis. FEM model was build and following materials were used in the model - bronze was used for oscillator and PZT-8 for piezoceramic plates. No structural boundary conditions were applied. Material damping was assumed in the model. By observing result of the modal analysis it can be noticed that modal shape No. 15 at 33.7 kHz can be used for operation of the motor. Horizontal plates of the actuator have shape closed to the B20 bending shape of the structural plates and contact zone swings during oscillations (Fig. 2). So this modal shape can be used for harmonic response and transient analysis.

(4)

where [K*] is modified stiffness matrix and it depends on nodal potential values of the piezoelements. Transient analysis is carried out applying to find respond of the actuator to the saw tooth type excitation. Structural mechanical loads are not used in our case so {F } = {0} . Equivalent mechanical forces are obtained because of inverse piezoefect and can be calculated as follows [10]: eq

[M ] {u&&} + [C ] {u&} + [K ] {u} = {Fp }+ {Fc }⎫ ⎪ [D] {δ& }+ [H ] {δ} = {Fc } ⎬, ⎪ & & & [m] {δ}+ [c]{δ} = {F f }+ {Fc } ⎭

1

Figure 2. Modal shape of the actuator at 33.7 kHz

A full harmonic response analysis was performed to give an adequate response curve of the contact zone and to calculate parameters of elliptic trajectory at the resonance frequency. Excitation scheme as shown in Fig. 1a was applied and 40V voltage was used. A frequency range from 32 kHz to 36 kHz with a solution at 100 Hz intervals were chosen and adequate response curves of contact zone oscillation amplitudes and phases were calculated. Graphs in Fig. 3 present amplitude – frequency diagrams of contact point where ux, uy and uz are amplitude projections into coordinate axes.

(6)

Results of structural displacements of the piezoelectric actuator obtained from harmonic response analysis are used for determining the trajectory of contact point movement. FEM governing equations of the linear piezoelectric motor can be written as follows:

Figure 3. Amplitude – frequency diagram of contact zone oscillations

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40 Harmonic 33,7 kHz

30

Transient 33,7 kHz

Displacement uz, nm

Harmonic 32,5 kHz

20 10 0 -10 -20 -30 -40 -80

-60

-40

-20 0 20 Displacement ux, nm

40

60

80

Figure 4. Trajectory of contact zone movement under different excitation frequencies

Figure 6. Impedance frequency diagram of the actuator

Fig. 4 present contact zone motion trajectories when harmonic and impulse type excitation is used. Impulse excitation is rectangular type and duration of the impulse is T/5, where T is vibration period at 33.7 kHz. By observing trajectories of motion it can be concluded that semi axis of the ellipses is larger, when at harmonic excitation at the same frequency is used. Rotation angles of the ellipses actually are the same. Smallest ellipses present the trajectory of motion at 32.5 kHz that is intersection point of ux and uz graphs in Fig. 4. Concluding this section it can be noticed that elliptical trajectories with the similar parameters are obtained using harmonic and impulse type excitation. V.

Measurements of the actuator’s top surface’s oscillations were done to verify operating principle of the actuator. Results of measurements are given in Fig. 7. They confirm results of numerical modeling that the elliptical trajectory of the contact point can be achieved using this actuator. The distribution of oscillation amplitudes on the top surface of the actuator is the same as were obtained in numerical simulation. In order to use actuator as stator of the motor, claiming conditions must be define. Fig. 8 shows the principle scheme of the motor where actuator is claimed at contact zone by elastic rod. Linear motion of the slider has been achieved using this kind of motor scheme.

EXPERIMENTAL STUDY

A prototype actuator, made for experimental studies, is shown in Fig. 5. The aims of experiment were to evaluate operating principle of the actuator and motor and to verify results of the numerical modeling. Impedance-frequency characteristics of the actuator were determined with the help of the 4192A LF Impedance Analyzer (Hewlett Packard). Top surface’s oscillations were measured using a vibrometer POLYTEC CLV 3D. The resonant frequency at 35.1 kHz was determined by means of electrical impedance study (Fig. 6). The difference between experimental and numerical resonant frequencies is 3.98 %. Figure 7. Distribution of oscillation amplitudes on top surface of the actuator

Figure 8. Principle scheme of inertial motor: 1 – acturaor, 2 – piezoelements, 3 – clamping rod, 4 – slider Figure 5. Prototype of piezoelectric actuator

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VI.

[2]

CONCLUSIONS

Miniature piezoelectric inertial type motor was developed. Numerical and experimental investigations confirm the possibility to achieve elliptical trajectories of contact zone when harmonic and impulse type excitation is used. Values of the resonant frequency and amplitudes from finite element model are in good agreement with the results from experimental investigation. Claiming conditions of the motor has been defined and operation of the motor has been tested experimentally.

[4] [5] [6]

[7]

ACKNOWLEDGMENT This work has been supported by Lithuanian State Science and Studies Foundation, Project No. B-07017, “PiezoAdapt”, Contract No. K-B16/2009-1. REFERENCES [1]

[3]

[8]

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