Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition
IMECE2010
November 12-18, 2010, Vancouver, British Columbia, Canada
IMECE2010-40074
MODELING AND OPTIMIZATION OF AN ELLIPTICAL SHAPE ULTRASONIC MOTOR USING
COMBINATION OF FINITE ELEMENT METHOD AND DESIGN OF EXPERIMENTS
Hamed Sanikhani
School of Mechanical Engineering,
Sharif University of Technology, Tehran, Iran
h.sanikhani@gmail.com
Javad Akbari
Center of Excellence in Design, Robotics, and
Automation, School of Mechanical Engineering,
Sharif University of Technology, Tehran, Iran
akbari@sharif.edu
Ali Reza Shahidi
Research Center for Science and Technology In
Medicine (RCSTIM), Tehran, Iran
a.shahidi@gmail.com
Ali Akbar Darki
School of Mechanical Engineering,
Sharif University of Technology, Tehran, Iran
a.a.darki@gmail.com
ABSTRACT
Standing-wave ultrasonic motors are a modern class of
positioning systems, which are used to deliver a high precision
linear or rotary motion with an unlimited stroke. The design
process should be performed through an effective optimization
algorithm in order to guaranty proper and efficient function of
these motors. An optimization method of ultrasonic motors is
proposed based on the combination of finite element method
and factorial design as a design of experiments in this study.
The results show the ability of this method in optimal design of
ultrasonic motors especially those which have a complex
structure and multi modes operation principle.
1. INTRODUCTION
Standing-wave ultrasonic motors are a modern class of
positioning systems, which are used to deliver a high precision
linear or rotary motion with an unlimited stroke. Significant
advantages such as large thrust force per unit volume, fast
response and good controllability, non magnetic and noiseless
operation, nano-scale positioning accuracy and high braking
force without power consumption makes these devices a
suitable alternative for conventional electromagnetic actuators
[1, 2].
The working principle of ultrasonic motors is production of
an ultrasonic elliptical micro-scale motion at the contact surface
of stator (vibrator) and rotor (slider). This motion is converted
to a macro-scale motion of rotor by friction effect. In the most
of the standing-wave ultrasonic motors, this elliptical motion is
generated by superposition of two orthogonal motions,
resulting from mode shapes of two adjacent natural frequencies.
These mode shapes of the motor are excited by means of
piezoelectric actuators. Simultaneous excitation of two modes
would be feasible and the elliptical motion would be amplified
subjected to resonance effect, if the natural frequencies locate
close together. This effectively increases the ultrasonic motor
performance. Therefore, the coincidence of these mode shapes’
frequencies is the most important objective in the proper design
of these multi modes ultrasonic motors. Furthermore, in order
to achieve noiseless operation, natural frequencies should be
located in the ultrasonic range (over 20 kHz).
Based on the working principle of the ultrasonic motors,
various researches have been performed about the motor
structure design and its simulation, in the recent years. In some
of these works, vibrational behaviors of the motor have been
modeled using exact analytical equations of motion [3, 4].
However, application of the exact methods is limited to models
with simple structure such as bar type or rectangular shaped
motors. In the more practical problems because of complexity
of structure, nonlinear effects of contact and electromechanical
coupling of piezoelectric elements, finite element method
(FEM) is considered as the most general and applicable method
in modeling and simulation of the ultrasonic motors.
Using the trial and error method is not practical and
efficient to optimum design of the ultrasonic motors, especially
for those with complex structures and multi modes working
principle. Thus, design process should be performed through a
proper optimization algorithm. Lu et al. introduced a
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rectangular ultrasonic motor and tried to locate operating mode
shapes at the same frequency by changing the stator length [2].
Ho studied an elliptical ultrasonic motor and minimized
difference of frequencies of two orthogonal mode shapes by
adjusting the ratio of the major axis diameter to the minor axis
diameter of the ellipse [5]. Shiyang and Ming used two
parameters particle swarm optimization (PSO) to design an
ultrasonic motor [6]. Bouchilloux and Uchino implemented a
genetic algorithm (GA) optimization on a two-dimensional
finite element model of an elliptical motor with four design
parameters [7]. Fernandez and Perriard designed a motor with
two symmetric mode shapes by factorial design method [8].
Because of this symmetry, natural frequencies are identical. So,
the optimization process only focused on maximizing
amplitude of motion.
In this work, modeling, simulation and optimization of an
elliptical ultrasonic motor using combination of finite element
and factorial design are proposed. First parameterization of the
model of the motor is performed. A mathematical model is
fitted on the obtained data from finite element modal analysis
based on three-level factorial design, which is a conventional
method of design of experiments (DoE). Next, the values of the
design parameters are optimized using a genetic algorithm in
order to coincidence of frequencies of the two orthogonal
modes in the ultrasonic range (over 20 kHz). Finally a finite
element optimization is applied in the vicinity of the optimum
point. This optimization compensates approximation between
finite element model and the mathematical model. The final
values of design parameters are obtained from these two
optimization processes.
actuators, a vertical sinusoidal motion would be generated on
the tip surface of stator, and a horizontal sinusoidal motion
would be generated, if out-phase voltage applies. These
resultant motions are given by:
sin 2
(1)
0
sin 2
(2)
0
where
and
are natural frequency and phase difference
between excitation voltage and resultant motion, respectively.
Subscripts n and t refer to the normal and the tangential modes.
2. STRUCTURE AND OPERATING MODES OF THE
MOTOR
The schematic of the proposed ultrasonic motor is
illustrated in Fig. 1. Stator is composed of an elliptical shell
which is connected to a fixed base using a flexible mechanism.
This mechanism is designed in a manner that supports the stator
with the minimum restriction of the motor mode shapes.
Moreover, it is used to preloading the stator against the rotor.
The vibrational motion of the stator is excited by two
multilayer piezoceramic rings as motor actuators. These
actuators are fixed and prestressed between the stator shell and
a central mass. This prestressing mechanism increases the life
of actuators. The central mass is installed in order to separation
of the motion of the two actuators. A cylindrical ceramic part is
inserted in the shell as tip of the stator to improvement of
frictional condition of the contact area. This ceramic tip has a
high wear resistance and coefficient of friction and thus
improves the transmission of motion between the stator and the
rotor.
The two orthogonal mode shapes of the motor (normal mode
and tangential mode) are shown in Fig. 2. The elliptical motion
of the motor tip is generated by simultaneous excitation of
these mode shapes. If in-phase sinusoidal voltage applies to the
Fig. 1. Schematic of the ultrasonic motor
Fig. 2. Orthogonal mode shapes of elliptical shell of the stator:
(a) normal mode, (b) tangential mode
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If the dimensions of the motor are determined so that the
two natural frequencies are close together, simultaneous
excitation of two modes would be possible near their
resonances. To achieve this goal, the driving voltages of two
actuators should have a π⁄2 phase difference at equal
frequency. This frequency could be varied between two natural
frequencies of the motor. Using this method of stimulation, the
desired elliptical motion is expected to be achieved by
superposition of two orthogonal motions described as
Equations (1, 2).
3. OPTIMIZATION
3. 1. PARAMETERIZATION OF THE MODEL
The parameterization of the model is the first step in the
optimal design process. The parametric model increases the
design flexibility and also provides possibility of the
implementation of appropriate optimization methods. In this
work, parameterization of the model is carried out using four
influence design parameters. These parameters, that are shown
in Tab. 1, include the ratio of ellipse axis (ER), thickness of the
central mass (Tcm), thickness of the shell (Ts) and height of the
stator (Hs).
model. However, study of the effects of these parameters on the
natural frequencies for finding their optimum values is not
possible in practice, because of the geometrical complexities
and the wide range of parameter variations. Thus, deriving a
mathematical model with the ability of simulation of the motor
modal behavior has significant advantages in such cases. In
addition various optimization methods could also be
implemented on shuch models easily.
A factorial design, which is a conventional method of
design of experiments (DoE), is applied in order to obtain the
mathematical model. In this method data is collected from a
primary model (in this case finite element model) at the certain
points of the variation ranges of the parameters. Then, a
polynomial is fitted on these data. Depending on the problem,
the number of parameters and required accuracy, this
polynomial could be linear, quadratic or cubic (using higher
degree polynomials is not very common). In a three-level
factorial design which is used in this work, each normalized
parameter could have three values: 0, 0.5 and 1 and thus 3P
points are defined in the design space for a P-parameters
problem. The two natural frequencies of the motor are
mathematically modeled with two quadratic polynomials, as
shown in Equations (4) and (5).
Tab. 1. Design parameters
Design
parameter
Parameter
coefficient
Initial range of
parameter
ER
k1
1-1.4
Tcm
k2
11-Hs(mm)
Ts
k3
3-4.5(mm)
Hs
k4
12-14(mm)
The second column of Tab. 1 (parameter coefficient) shows
normalized parameters which are related to design parameters
by Equation (3).
1
2
3
4
(4)
1
2
3
4
(5)
To determine all 81 coefficients of each equation (
and
, 81 sets of data obtained from the finite element analysis
are used. Substituting these data into above equations yields a
set of linear equations for each frequency (Equations (6) and
(7)).
1
1 2 3 4
(3)
1
1 2 3 4
where DP indicates the design parameter, DP
and DP
are
minimum and maximum values of it and k is related parameter
coefficient. Using these normalized coefficients helps to
facilitate modeling and optimization process and also
comparison of the results.
1
1 2 3 4
1
1 2 3 4
(6)
DP
DP
k
DP
1
k
3. 2. DERIVING A MATHEMATICAL MODEL USING
FACTORIAL DESIGN
It is possible to obtain natural frequencies at each point of
the design parameters domain by solving the finite element
(7)
The coefficient matrices
and
could be
determined using standard methods of linear algebra. In this
study, least squares method is used to find the coefficients. This
method provides less average error in the whole of the
approximation domain.
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3. 3. OPTIMIZATION USING GENETIC ALGORITHM
As mentioned before, the main objective of design of the
multi modes ultrasonic motors is the coincidence of the natural
frequencies. In addition, this adjustment should be occurred in
the ultrasonic range to the noiseless operation of the motor. The
objective function which is defined in order to satisfy these
conditions is defined as Equation (8). In this equation, the
frequencies are calculated using the mathematical models.
|
|
20000
,
between the two frequencies that this error has been eliminated
after the final finite element optimization. Results of this
optimization are demonstrated in Tab. 3.
(8)
The minimization of this function could be performed
using standard optimization methods. In the present work, a
genetic algorithm is implemented on this 4-parameters
objective function to find the optimum point. However, this
point may differ slightly from the optimum point driven from
the finite element model because of the approximation error. To
eliminate this error, a final finite element optimization is carried
out in the vicinity of the mathematical model optimum point.
Due to the small variation ranges, it is expected that the
convergence would be achieved after a few number of
iterations.
Fig. 3. Comparison of objective function value at the DoE
points and the optimum point
4. RESULTS AND DISCUSSION
Tab. 3. Results of final finite element optimization
4. 1. OPTIMIZATION RESULTS
The characteristics of the optimum point which have been
obtained by the implementation of GA on the mathematical
model are shown in Tab. 2. The coincidence of the natural
frequencies has been provided at 20743 Hz in the ultrasonic
range. To show the ability of this optimization method, the
objective function value at the factorial design points and the
optimum point is plotted in Fig. 3.
Parameter
coefficient
Optimum
values
k1
0.86413
k2
1
k3
0.089759
k4
1
Tab. 2. Results of GA optimization
Parameter
coefficient
Optimum
values
k1
0.86407
k2
1
k3
0.089745
k4
1
fn
20743
(Hz)
ft
20743
(Hz)
f
0
(Hz)
Objective
function
value
-743
Finite element modal analysis has been carried out with the
optimized parameters. The FE values of the frequencies of the
normal and the tangential modes are equals to 20760 Hz and
20730 Hz, respectively. So, there is about 30 Hz difference
fn
ft
20763
(Hz)
20766
(Hz)
f
3
(Hz)
Objective
function
value
-760
4. 2. VERIFICATION OF THE MATHEMATICAL MODEL
In order to verify the accuracy of the mathematical model,
the values of the objective function and the natural frequencies
have been compared to the finite element results at some
verification points which are selected based on a reasonable
procedure. In this procedure, three of the four parameters are
fixed at their optimum values and the other one is varied from 0
to 1 with step size of 0.2. Considering the existence of four
parameters, 24 verification points are determined. This
verification shows that average relative errors in the
approximation of the frequencies are less than 0.1% and
average error of the objective function value approximation is
less than 40 Hz. The comparison of these results is shown in
Fig. 4.
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4. 3. SENSITIVITY ANALYSIS
Using the mathematical model, variation of the objective
function with each design parameter is determined as a
sensitivity analysis (Fig. 5). The results show that k1 and k3
have greater effect on the value of the objective function. This
analysis would be useful to determine required manufacturing
tolerances.
a) k1 is variable
Fig. 5. Sensitivity analysis using mathematical model
b) k2 is variable
COLCLUSION
c) k3 is variable
In this work, an optimization method of ultrasonic motors
is proposed based on the combination of finite element method
and factorial design as a design of experiments. In this method,
finite element model is used to determine natural frequencies of
the motor and a mathematical model was fitted on data
obtained from FEM using three-level factorial design. The
accuracy of the mathematical model to estimate the natural
frequencies is verified. Genetic algorithm minimization and
sensitivity analysis is implemented on the mathematical model
to find optimum values of the design parameters and effects of
them on the motor natural frequencies. The results show the
ability of this method in the optimal design of the ultrasonic
motors especially those which have a complex structure and
multi modes operation principle.
REFERENCES
[1] Hemsel, T., Mracek, M., Twiefel, J., and Vasiljev, P., 2006,
“Piezoelectric linear motor concepts based on coupling of
longitudinal vibrations,” Ultrasonics, 44, pp. 591-596.
d) k4 is variable
Fig. 4. Comparison of the mathematical model results with the
finite element results at the verification points : a) k1 is
variable, b) k2 is variable, c)k3 is variable, 4) k4 is variable
[2] Lu, C., Xie, T., Zhou, T., and Chen, Y., 2006, “Study of a
new type linear ultrasonic motor with double-driving feet,”
Ultrasonics, 44: pp. 585–589.
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[3] He, S., Chen, W., Tao, X., and Chen, Z., 1998, “Standing
wave Bi-directional linearly moving ultrasonic motor,”
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[6] Shiyang, L., and Ming, Y., 2008, “Particle swarm
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