Proceedings of the euspen International Conference – Zurich - May 2008
A generic modelling approach for studying the contact
mechanism and dynamic behavior of bimodal standing wave
piezomotors
M. Houben, W. Van de Vijver, F. Al-Bender, D. Reynaerts
Katholieke Universiteit Leuven, Department of Mechanical Engineering, Belgium
michael.houben@mech.kuleuven.be
Abstract
Application of bimodal standing wave piezomotors is complicated by problems
related to the non-linear friction drive mechanism between drive point and slider, e.g.
particle contamination, heat generation and structural vibrations. An advanced contact
model based on Kalker’s rolling theory is included in a lumped-parameter system
representing suspension dynamics, piezomotor dynamics and slider/bearing
dynamics. This model allows studying the influence of several design parameters on
setup performance. Simulation results are presented and qualitatively assessed.
1
Introduction
Ultrasonic piezomotors are distinguished from other driving technologies by a unique
combination of characteristics, such as high force density, high accuracy and low
magnetic noise. During the last decades, several operating principles have been
developed [1]. This paper focuses on the bimodal standing wave variant. Generally
speaking, this kind of piezomotor utilizes piezotransducers to excite two orthogonal
eigenmodes of a certain structure, so as to create an elliptic motion of the drive point,
which is located somewhere on this structure. A friction coupling transfers this
mesoscopic, elliptic motion to the macroscopic motion of e.g. a linear slide.
Although already commercialized, the application of such drives is still hampered by
this inherent non-linear friction coupling mechanism. High heat generation and
particle contamination due to wear of the contacting materials hinder the application
in UHV environments, e.g. wafer inspection systems. Other problems are unstable
low-velocity behavior (100 nm/s and below) due to stick-slip phenomena, and
Proceedings of the euspen International Conference – Zurich - May 2008
structural vibrations. In electron microscopy, for example, these last two problems
hinder automated visual data acquisition.
To enlarge the application domain of these motors, a detailed investigation of their
contact mechanics is needed. This paper presents the first results of a novel
modeling strategy based on an advanced (transient) rolling friction model. It allows
studying the dynamic behavior of a bimodal standing wave piezomotor integrated in
a setup with slider and suspension mechanism.
2
Modeling approach
2.1
Friction model
A fundamental question here is: is the contact mechanism developing in a sliding or
a rolling fashion? Looking at FE-analyses of commercially available motors, and
FE-analyses and measurements on in-house developed motors [2, 3], bending modes
appear to cause the drive point to rotate rather than to translate. A rolling friction
model should therefore be selected.
The research that has been done on rolling friction modeling converges in the work
of Kalker [4]. A recent formulation of transient rolling can be found in [5]. A
Winkler spring bed, as illustrated in fig. 3, models the contact between piezomotor
drive point and slider contact material. When comparing with Hertz theory, the
Winkler approach offers greater flexibility for the geometry of the drive point.
Figure 3: Winkler modeling of the contact between drive point and slider material
2.2
Full dynamic model
The contact model illustrated in fig. 3 is implemented in a 6-DOF mass-springdamper model representing piezomotor, suspension and bearing/slider dynamics.
Figure 4 presents this full dynamic model. The sinusoidal input forces Fh and Fv
excite the elliptic microscopic motion of the modal mass mc of the piezomotor. The
Proceedings of the euspen International Conference – Zurich - May 2008
non-linear contact springs khc and kvc are numerically extracted from the transient
rolling contact model. They transfer the elliptic drive point motion to horizontal and
vertical motions of the slider. The other parameters shown in figure 4 may be easily
determined in an analytic or experimental way.
This approach allows studying the influence of a vast number of parameters on the
dynamic behavior of a piezomotor setup. Transient phenomena are also simulated.
Remark that an electrical piezotransducer model, capable of converting a voltage
input to a force output, is not implemented at this time.
Figure 4: Mass-spring-damper model of piezomotor, suspension and slider dynamics
3
Results
Figures 5a and 5b show slider velocity behavior in function of the amplitude of the
applied forces Fh and Fv. Figure 5c shows a qualitative validation obtained with an
in-house developed piezomotor [3]. The model simulates the well-known non-linear
dead-zone behavior which hampers low-velocity control of bimodal standing wave
piezomotors. Simulated velocities have a realistic order of magnitude. Simulations
of force-velocity characteristics also show typical behavior and magnitudes.
Proceedings of the euspen International Conference – Zurich - May 2008
0.25
0.25
0
0
2
0.02
0.04
0.06
Time [s]
(a) Simulation
0.08
0.1
Slider speed [m/s]
10
4
Slider speed [m/s]
Slider speed [m/s]
0.1
0.05
0.2
Fh and Fv amplitude [N]
0.15
0.14
0.12
20
0.2
0.15
0.1
0.05
0
0
Dead zone
0.1
0.08
0.06
0.04
0.02
5
10
15
20
0
0
Dead zone
5
10
15
Fh and Fv amplitude [N]
Voltage amplitude [V]
(b) Simulation
(c) Measurement
20
Figure 5: (a, b) Slider speed – excitation amplitude simulations and (c) validation
Figure 6 shows a prediction of what happens inside the contact patch on a velocity
reversal between the drive point and the slider. In this example, the patch starts
sticking on the left side, the so-called leading edge. As the velocity difference fades,
the stick-zone grows until the entire contact patch is sticking (figs. 6a to 6d).
Figure 6: Prediction of the microslip evolution inside the contact patch between
drive point and slider on a relative velocity reversal. (a, g): gross slip. (d): full stick.
Proceedings of the euspen International Conference – Zurich - May 2008
The relative velocity reverses, and the stick zone disappears in the other direction of
the contact patch (figs. 6d to 6g). Remark that it is impossible to experimentally
validate this prediction due to the ultrasonic working frequency and small geometric
scale.
4
Conclusion
A generic model, build around an advanced rolling contact theory and capable of
simulating the dynamic behavior of a piezomotor setup for a vast number of
parameters, is presented. Several aspects, not all described, can be studied:
−
Influence of control parameters and material properties on performance.
−
Dimensioning of suspension mechanisms and sliders.
−
Hysteretic behavior of the friction coupling between piezomotor and slider.
−
Microslip distribution in the contact patch between piezomotor and slider.
Furthermore, the transient modeling approach allows integration of the model into a
full mechatronics system model, which may, for example, include a controller and a
measurement system. Future research will focus on the link between contact
mechanism and resulting wear and heat generation.
Acknowledgements:
Research funded by a Ph.D grant of the Institute for the Promotion of Innovation
through Science and Technology in Flanders (IWT).
References:
[1] KOC B., UCHINO K., Piezoelectric Ultrasonic Motors, Comprehensive
Composite Materials, 2003, pp.651-661.
[2] VAN DE VIJVER W., REYNAERTS D., VAN BRUSSEL H., Design and
control of a novel piezoelectric drive module for application in a multi-DOF
positioning stage, ISMA2006 Int. Conf. on Noise and Vibration Engineering, 2006.
[3] DE MOT B., HOUBEN M., Ontwerp van een hybride piëzo-elektrische xyaandrijving met resonante en stappende werkingsmode, MsC Thesis, KUL, 2006.
[4] KALKER J., Three-dimensional elastic bodies in rolling contact, Kluwer, 1990.
[5] AL-BENDER F., DE MOERLOOZE K., A model of the transient behavior of
tractive rolling contacts, Advances in Tribology, 2008.