444

ACTUATOR 2006, 10th International Conference on New Actuators, 14 – 16 June 2006, Bremen, Germany

B 7.6

FRICTION DRIVE MODELING OF SAW MOTOR USING

CLASSICAL THEORY OF CONTACT MECHANICS

T. Shigematsu, M. K. Kurosawa

Tokyo Institute of Technology, Yokohama, Japan

Abstract:

A friction drive modeling method of a surface acoustic wave (SAW) motor is proposed. The kernel of the model

was the previously proposed elastic point contact model that took account of stick-slip friction. The model kernel

only required normal/tangential stiffness of a stator/slider and a friction coefficient as arguments for quantitative

simulations. Then in this proposing method, the stiffness was conducted by means of the analytical solutions of

three-dimensional contact problems in the classical theory of elasticity. The use of contact mechanics theory, in

addition, made it possible to deduce three-dimensional deformations and stress distributions straightforward.

Namely, we could carry out the friction drive simulation by the simple model kernel i.e., it resulted small

computation time, and when necessary it provided the displacements and stresses interior of the stator and slider

in a manner similar to that of FEM analyses.

Keywords: Piezo Actuator, Ultrasonic Motor, Surface Acoustic Wave, Contact Mechanics

Introduction

The objective surface acoustic wave (SAW) motor is

composed of a lithium-niobate stator and a silicon

slider; the friction drive happens at interface between

them. Since both the materials are brittle ones, the

friction drive arises brittle fractures and as a result

induces wear. The brittle fractures take place when

the surface or internal tensile stresses exceed the

limiting value-failure stress. Hence, the operation

of the motor should be carried out under the

condition that the tensile stresses are less than the

failure stress, or more preferably the frictional

surfaces should be designed to reduce or to be

tolerant to the stresses. We thereby constructed the

contact mechanics model in order to discuss the

stresses and to utilize it as support tool for the

contact surface designing.

Principle of SAW Motor

A schematic view of the SAW motor is illustrated in

Fig. 1. A SAW device (stator) is made of lithium-

niobate substrate. Both end of the stator, interdigital

transducers (IDTs) are placed. RF electric power

(9.6 MHz) is transduced to Rayleigh wave, a kind of

SAW, at the IDT with piezoelectric effect. The

propagating Rayleigh wave energy is transferred to

mechanical slider motion through frictional force.

The slider is made of silicon substrate. At the

frictional surface, many flat-ended cylindrical

projections are fabricated by dry etching so as to

eliminate the squeezed air film disturbance in the

contact with the Rayleigh wave. The projections are

the only contact points of the slider with the wave, so

that all the driving force is produced hereout.

Concept of Modeling

The projections are much smaller than the slider

body, so that they are of high stiffness relative to the

slider body: the internal or overall deformation of the

projections was negligible to the ones of the slider

body. Then, the assumption that the projection is a

rigid body and the stator/slider are elastic half-spaces

enabled us to translate this contact situation into

well-developed rigid-punch indentation problems in

the theory of elasticity [1].

Owing to these problem formulations, the

normal/tangential contact stiffness (spring constant)

of the stator/slider against the projection indentation

was easily conducted, which affords mechanical

lumped parameter modeling of the contact. Since the

deformation of the spring is naturally the

representation of the displacement in the theory of

elasticity, we can deduce three-dimensional

displacements and stress distributions only with the

deformation of the spring. This two-layer structure

modeling enables us to execute complicated friction

drive simulation by the lumped parameter model,

and when necessary it provides the displacements

and stresses interior of the stator and slider in a

manner similar to that of FEM analyses.

F

i

g

. 1:Schema o

f

SAW moto

r

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ACTUATOR 2006, 10th International Conference on New Actuators, 14 – 16 June 2006, Bremen, Germany

Stiffness of Slider

The slider-projection connection may be translated

into rigid-punch indentation problem with adhesive

boundary i.e. no slip. That is, the projection and the

slider body are approximated as rigid punch and

elastic half-space, and the virtual boundary between

them is assumed to be completely adhesive; that is

schematically illustrated in Fig. 2.

Let the coordinate for forces and

displacements is that x-axis corresponds to the

travelling direction of the slider and z-axis

corresponds to the depth direction of the slider as

indicated in Fig. 2. Then, assuming the elastic half-

space as an isotropic material whose Young’s

modulus and the Poisson’s ratio are E

p

and '

p

, we

get the normal stiffness k

pv

i.e., the normal force P

divided by the uniform normal displacement w

p

of a

circular region at an elastic half-space as [2]:

p

p

p

p

pv

aG

w

P

k

'

'

21

)43ln(

4

−

−

==

(1)

where G

p

= E

p

/2(1+'

p

) is the shear modulus and a

denotes the radius of the projection. As well, the

tangential stiffness k

ph

i.e., the tangential force Q

divided by the uniform tangential displacement d

p

is

[3],

)

16

21

8

1

/(

(

'

p

p

p

ph

aG

d

Q

k

−

+==

(2)

where ( = ln(3-4'

p

)/2.

The stress distribution at the slider-projection

boundary is not given by a closed form solution, but

is expressed as derivatives of integrals [2,3], which

can be calculated numerically with small

computation time.

Stiffness of Stator

The stator-projection contact may be translated into

rigid-punch indentation/sliding problem in a similar

fashion. Frictionless boundary condition is assumed

in the normal indentation, and a Coulomb friction

condition, in which the shear traction is taken as a

friction coefficient multiplied by the frictionless

contact pressure, is assumed in tangential sliding.

Let the z’-axis is in the depth direction of the

stator whereas x-axis is unchanged in the slider

coordinate, and the Young’s modulus and Poisson’s

ratio of the stator are E

s

and '

p

. The shear modulus is

thus G

s

= E

s

/2(1+'

s

). The normal stiffness k

sv

is

given by [4],

)1/(4

ss

s

sv

aG

w

P

k'−==

(3)

where w

s

is the uniform normal displacement at the

contact area with the projection. The tangential

stiffness k

sh

is similarly given by [5],

)2/(8

ss

s

sh

aG

d

Q

k'−==

(4)

With these boundary conditions, the

displacements and stress distributions throughout the

elastic half-space are obtained from closed form

solutions [4,5].

Friction Condition

The indentation of elastic half-space by the

projection under Coulomb friction normally yields

partial slip [6,7]. Namely, at each elementary area of

the interface, when the shear traction q exceeds the

pressure p multiplied by the friction coefficient µ,

slip occurs at the area. On the other hand, if the total

shear force |Q| exceeds the total normal force P

multiplied by µ, the slide of the interface happens.

The partial slip is difficult to implement to the

mechanical lumped parameter modeling. Thus, we

assume that the friction condition at the projection-

stator interface has only two state, complete stick

and slide; the slide occurs if |Q| > µP.

The stiffness difference under stick or slide

condition are within 10 % in normal direction and

5 % in tangential direction at any Poisson’s ratio and

friction coefficient; this can be verified by

comparing the equations (1) and (3) or equations (2)

and (4) putting G

p

=G

s

and '

p

='

s

. The stiffness under

partial slip condition is possibly between the ones of

those friction conditions. Hence, we can say that

excluding the partial slip may cause error in stress

distributions in the elasticity layer, however it

produces slight error of the lumped parameters in

mechanical layer of the model.

The stick or slide friction condition is

compatible with that of a friction drive model of the

motor [8]. The model assumes a single point contact

Fig. 2: Schematic view of slider-

p

rojection

connection for contact mechanics modeling

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ACTUATOR 2006, 10th International Conference on New Actuators, 14 – 16 June 2006, Bremen, Germany

and considers the normal and tangential stiffness of

the stator, and corresponding deformations. The

stiffness multiplied by the deformations give the

normal and tangential force, P and Q. Then, if |Q|

exceeds the limiting value µP slip occurs. With the

idea of this model we can simulate the friction drive.

Friction Drive Model

Assuming the stator is elastic half-space that vibrates

in the same way of surface particles of the Rayleigh

wave, i.e., the phase of the wave is same everywhere,

and the slider body is fixed half-space as illustrated

in Fig. 3, we can apply the contact stiffness obtained

from the contact mechanics formulae to the

previously reported friction drive model with

modification. Using the expression in [8], the friction

drive model based on contact mechanics is explained

as follows.

In one period (

2/32/

))− t

) of the

Rayleigh wave of angular frequency , the normal

displacement of the stator may be expressed as:

y = a

v

sin t (5)

where a

v

is the vibration amplitude in the normal

direction. Then, the projection is in contact with the

stator

*

*

−)) t

(

2/2/

*

))−

), the normal

force P is given by

P = k

sv

d

sv

= k

pv

d

pv

where d

sv

and d

pv

are the normal deformation of the

stator and slider, and which satisfy

d

sv

+ d

pv

= a

v

(sin t – sin *)

The average preload N is then given by

)sin)2(cos2(

2

***

−+=

vv

ak

N

(6)

where 1/k

v

=1/k

sv

+1/k

pv

.

The tangential displacement of the stator is

x = - a

h

cos t (7)

where a

h

is the vibration amplitude in the tangential

direction. The tangential force Q is from the

equilibrium,

Q = k

sh

d

sh

= k

ph

d

ph

where d

sh

and d

ph

are the tangential deformation of

the stator and slider. If we define the slider speed is

v = a

h

sin+ (

2/0

+

))

), and if no slide happens,

then the tangential deformations satisfy

d

sh

+ d

ph

= a

h

(cos * – cos t – (t –*) sin +)

When |Q| > µP, the slide occurs and the

tangential force becomes

Q’ = sgn(Q) µP (8)

The tangential deformations are changed to satisfy

this relationship in which the relative slide length l is

given by | d

sh

+d

ph

– µP/k

h

|, where 1/k

h

=1/k

sh

+1/k

ph

.

Simulation

In order to discuss the validity of the model, the

simulation results would be compared with the direct

measurement of the projection’s displacements [10].

The measurement was performed with the slider that

had 25 µm radius projections, which were arranged

at 300 µm pitch and which had total 169 projections.

The Rayleigh wave conditions were 9.61 [MHz] of

vibration frequency, 20 and 18 [nm] of normal and

tangential vibration amplitude, respectively. The

preload to the slider was 15 N, thus the average

preload to one projection was 0.089 [N].

The material of the slider ‘silicon’ is an

anisotropic material. To use the stiffness deduction

equations, we employed an approximated isotropic

(b) tangential direction

Fig. 4: Simulation results of projection’

s

displacement together with the measurements

Fig. 3: Schematic view of friction drive model

(

a

)

normal direction

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ACTUATOR 2006, 10th International Conference on New Actuators, 14 – 16 June 2006, Bremen, Germany

elastic constants E

p

and '

p

by means of Voigt

average, namely E

p

=165.6 [GPa] and '

p

=0.218 [9].

Putting these values to equations (1) and (2), we

obtained the normal stiffness k

pv

and tangential

stiffness k

ph

as 9.10 and 7.78 [MN/m], respectively.

The stator material ‘lithium niobate’ is a

highly anisotropic material [11] so that no

approximation method to the isotropy has reported.

We then approximated to the isotropy from the

propagation velocity of the Rayleigh wave and the

longitudinal wave; substituting those velocities to the

equations of velocities of an isotropic material

extracted the approximated isotropic elastic

constants, which were E

s

=197 [GPa] and '

p

=0.05.

Accordingly from equations (3) and (4), the normal

stiffness k

sv

and tangential stiffness k

sh

became 9.87

and 9.62 [MN/m], respectively.

Putting N=0.089 [N] to the equation (6) and

making use of bisection algorithm, the contact phase

* was obtained as –1.20 (-68.8

o

). The measurement

was conducted under 0-speed condition, hence; the

parameter relating to the speed + was 0. By

changing the only undecided parameter, friction

coefficient µ, the simulations were carried out. The

results of the projection’s displacement were

indicated in Fig. 4 together with the measurements

that were indicated with grey lines.

The time of simulation results were manually

adjusted in such a way that the normal displacement

would be in phase with that of the measurements.

The simulation of normal displacements accurately

represented the measurements. The phase of the

tangential displacement simulations stayed away

from the measurements. However, the increase of the

friction coefficient µ resulted the increase of the

amplitude of the tangential displacement and

conducted the closer phase.

Discussion

As a friction coefficient between silicon and lithium

niobate, the value 0.18 was reported [12]. However,

as shown in Fig. 4, the simulation results indicated

that the measurements were more closely represented

with the higher friction coefficient. We, thereby,

examined the coefficient variation in relation to the

slider projections’ parameter, preload and sliding

speed. The coefficient varied roughly from 0.1 to 1.

The coefficient generally increased with the increase

of the projections’ diameter and pitch of them. The

increase of the sliding speed increased the

coefficient. The definite value of the friction

coefficient is thus useless. Some kind of modeling or

mapping of the coefficient values will be necessary

to precisely simulate the friction drive.

Conclusion

Since the necessary parameter friction coefficient has

been unknown, deterministic simulations of the

friction drive model could not be performed on

present form. However if it is provided somehow, we

can know the displacements and stresses interior of

the stator and slider by means of this proposed

modeling method. This method would be helpful for

designing the contact surface of the SAW motor.

Acknowledgement

This work was partially supported by the Ministry of

Education, Culture, Sports, Science and Technology,

Grant-in-Aid for Scientific Research. Also, this work

was partially supported by the Grant-in-Aid for

Research Fellowships of the Japan Society for the

Promotion of Science for Young Scientist.

References

[1] K. L. Johnson, “Contact Mechanics,” Cambridge

University Press, 1985.

[2] D. A. Spence, “Self similar solutions to adhesive

contact problems with incremental loading,” Proc.

Roy. Soc. A., Vol. 305, pp. 55-80, 1968.

[3] G. M. L. Gladwell, “A Contact Problem For a

Circular Cylindrical Punch in Adhesive Contact

with an Elastic Half-Space: the Case of Rocking,

and Translation Parallel to the Plane,” Int. J.

Engng. Sci., Vol. 7, pp. 295-307, 1969.

[4] I. N. Sneddon, “Boussinesq’s Problem for a Flat-

Ended Cylinder,” Proc. Camb. Phil. Soc., Vo. 42,

pp. 29-39, 1948.

[5] R. A. Westmann, “Asymmetric Mixed Boundary-

Value Problems of the Elastic Half-Space,”

Trans. ASME, J. Appl. Mech., Vol. 32, pp. 411-

417, 1965.

[6] D. A. Spence, “The Hertz contact problem with

finite friction,” J. Elasticity, Vol. 5, pp. 297-319,

1975.

[7] J. R. Turner, “The Frictional Unloading Problem

on a Linear Elastic Half-Space,” J. Inst. Math

Applics, Vol. 24, pp. 429-469, 1979.

[8] K. Asai and M. K. Kurosawa, “Simulation Model

of Surface Acoustic Wave Motor Considering

Tangential Rigidity,” Electronics and

Communications in Japan, Part. 3, Vol. 87, No. 2,

pp. 131-143, 2004.

[9] R. Hull Ed., “Properties of Crystalline Silicon,”

INSPEC, 1999.

[10] T. Shigematsu and M. K. Kurosawa,

“Friction Drive Dynamics of Surface Acoustic

Wave Motor,” in Proc. IEEE Ultraon., Symp., pp.

305-310, 2005.

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