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 stickslip 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
threedimensional contact problems in the classical theory of elasticity. The use of contact mechanics theory, in
addition, made it possible to deduce threedimensional 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 lithiumniobate 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 valuefailure 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 flatended 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 halfspaces
enabled us to translate this contact situation into
welldeveloped rigidpunch 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 threedimensional
displacements and stress distributions only with the
deformation of the spring. This twolayer 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
445
ACTUATOR 2006, 10th International Conference on New Actuators, 14 – 16 June 2006, Bremen, Germany
Stiffness of Slider
The sliderprojection connection may be translated
into rigidpunch indentation problem with adhesive
boundary i.e. no slip. That is, the projection and the
slider body are approximated as rigid punch and
elastic halfspace, 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 xaxis corresponds to the
travelling direction of the slider and zaxis
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 halfspace 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(34'
p
)/2.
The stress distribution at the sliderprojection
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 statorprojection contact may be translated into
rigidpunch 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 xaxis 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 halfspace are obtained from closed form
solutions [4,5].
Friction Condition
The indentation of elastic halfspace 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
446
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 halfspace 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 halfspace 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
447
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 0speed 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,
GrantinAid for Scientific Research. Also, this work
was partially supported by the GrantinAid 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. 5580, 1968.
[3] G. M. L. Gladwell, “A Contact Problem For a
Circular Cylindrical Punch in Adhesive Contact
with an Elastic HalfSpace: the Case of Rocking,
and Translation Parallel to the Plane,” Int. J.
Engng. Sci., Vol. 7, pp. 295307, 1969.
[4] I. N. Sneddon, “Boussinesq’s Problem for a Flat
Ended Cylinder,” Proc. Camb. Phil. Soc., Vo. 42,
pp. 2939, 1948.
[5] R. A. Westmann, “Asymmetric Mixed Boundary
Value Problems of the Elastic HalfSpace,”
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. 297319,
1975.
[7] J. R. Turner, “The Frictional Unloading Problem
on a Linear Elastic HalfSpace,” J. Inst. Math
Applics, Vol. 24, pp. 429469, 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. 131143, 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.
305310, 2005.
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