Piezoelectric Actuators [1 ed.] 9781626183209, 9781613241813

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ELECTRICAL ENGINEERING DEVELOPMENTS

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PIEZOELECTRIC ACTUATORS

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ELECTRICAL ENGINEERING DEVELOPMENTS

PIEZOELECTRIC ACTUATORS

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JOSHUA E. SEGEL EDITOR

Nova Science Publishers, Inc. New York

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Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Piezoelectric actuators / editor, Joshua E. Segel. p. cm. Includes bibliographical references.

ISBN:  (eBook)

1. Actuators. 2. Piezoelectric devices. I. Segel, Joshua E. TJ223.A25P54 2011 621--dc22 2011009073

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CONTENTS Preface Chapter 1

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Chapter 2

Chapter 3

Chapter 4

Chapter 5

vii Recent Progress in Thick-Film Piezoelectric Actuators Prepared by Screen-Printing Marina Santo Zarnik, Hana Uršič and Marija Kosec Pressure Control of Vehicle ABS Using Piezoactuator-Based Valve Modulator Juncheol Jeon and Seung-Bok Choi High Power Ultrasonic Actuators Based on the Langevin Transducer: Classical Configurations and Recent Designs Antonio Iula Active Control of the Dynamics of Hinged-Hinged Beam Using Piezoelectric Absorber B. R. Nana Nbendjo¤and P. Woafo Modeling and Control of Piezoelectric Actuator Systems S. N. Huang and K. K. Tan

Index

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1

29

55

97

117 139

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PREFACE Piezoelectric actuators are simple structures with compact sizes that provide quick and precise responses to an electrical stimulus. The response of the piezoelectric actuator can be a mechanical displacement or a blocking force with the magnitude determined by the structure's geometry and the electromechanical properties of the materials used. This book presents topical research in the study of piezoelectric actuators. Topics discussed include thickfilm piezoelectric actuators prepared by screen-printing; pressure control of vehicle ABS using piezoactuator-based valve modulators; ultrasonic actuators based on the Langevin transducer; the dynamics of hinged-hinged beam using piezoelectric absorbers and modeling and control of piezoelectric actuator systems. (Imprint: Nova Press) Chapter 1 - Recent progress in thick-film piezoelectric actuators can be considered to be the result of the growing opportunities offered for miniaturised electromechanical systems by the successful implementation of new functional materials and technologies. Categorised as bending-type actuators, the thick-film actuators should be capable of relatively large displacements. However, the effect of clamping of the active layer to the relatively stiff substrate results in smaller displacements in comparison to substrate-free structures. Due to the mismatch of the thermal expansion of the film and the substrate materials during the processing the properties of the piezoelectric films differ from those of the respective bulk piezoceramics. Furthermore, an additional drawback can be the deterioration of the piezoelectric film’s properties due to its chemical interaction with the substrate. To ensure a sufficient displacement required for a certain application, different solutions have been proposed, such as a reduction of the thickness of the substrate, processing of the thick-film multilayer actuator

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viii

Joshua E. Segel

structures or the preparation of ―substrate-free‖ actuators. In this review the progress in thick-film piezoelectric actuators prepared by the screen-printing method is reported. The whole production line for the actuators is discussed, including the preparation of the piezoelectric pastes for the screen-printing, the processing of the various piezoceramic thick films on the different substrates and the films’ characterisation. Some representative examples of the most commonly investigated thick-film actuator structures and their applications are described. Chapter 2 - This paper presents a new type of the piezoactuator-driven valve system with a pressure modulator which can be applicable for smallsized vehicles ABS (anti-lock brake system) such as green car. As a first step, a flapper-nozzle valve is devised by locating the piezoactuator in one side of a flexible beam which can make a movement required to control a desired pressure. The governing equation of the valve is then derived and the pressure modulator is connected to the valve in order to obtain high pressure level. After confirming excellent pressure tracking controllability of the proposed valve system, a wheel slip control model is established by adopting a quartercar model. Subsequently, a robust sliding mode controller which has inherent robustness to model uncertainty and external disturbance is designed in order to achieve accurate tracking control of the desired wheel slip rate which directly affects performance of the vehicle ABS. Tracking control performances for the desired wheel slip rate are evaluated under two different trajectories: constant and sinusoidal slip rates. Chapter 3 - This chapter is dedicated to the ultrasonic actuators based on the Langevin transducer. The basic theory and the design criterions of the Langevin transducer alone and joined to the most popular ultrasonic concentrators are firstly presented. A recent design, which is based on a flexural displacement amplifier, is then presented and detailed analyzed. Chapter 4 - This chapter deals with the problem of reducing amplitude, inhibiting catastrophic motion and suppressing horseshoes chaos on a hingedhinged beam subjected to transversal periodic excitation. Piezoelectric ceramics have been used as sensors and actuators. After modelling the system under control, we derive the accurate control parameters leading to the efficiency of the control. This is done using analytical approach along with direct numerical simulation of the base equation. The effect of delay between the detection of the structure motion and the restoring action of the control is pointed out. Chapter 5 - In this chapter, common configuration of the piezoelectric actuator (PA) systems, their mathematical models as well as control schemes

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will be discussed. One major source of uncertainties in PA design and application is the hysteresis behavior which yields a rate-independent lag and residual displacement near zero input, reducing the precision of the actuators. For eliminating the effects of the hysteresis phenomenon, we have to build the model of a PA system. Several representative models will be discussed in the chapter: the Maxwell model, the time delay hysteresis model, the JilesAtherton model, the Preisach model and the dynamical LuGre Model. The controller design is based on the model obtained. The simulation study is given to show the effectiveness of the proposed control scheme.

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In: Piezoelectric Actuators Editors: Joshua E. Segel, pp. 1-27

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Chapter 1

RECENT PROGRESS IN THICK-FILM PIEZOELECTRIC ACTUATORS PREPARED BY SCREEN-PRINTING Marina Santo Zarnik1,2, Hana Uršič1 and Marija Kosec1

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Electronic Ceramics Department, Jožef Stefan Institute, Ljubljana, Slovenia 2 HIPOT-RR, Otočec, Slovenia

ABSTRACT Recent progress in thick-film piezoelectric actuators can be considered to be the result of the growing opportunities offered for miniaturised electromechanical systems by the successful implementation of new functional materials and technologies. Categorised as bendingtype actuators, the thick-film actuators should be capable of relatively large displacements. However, the effect of clamping of the active layer to the relatively stiff substrate results in smaller displacements in comparison to substrate-free structures. Due to the mismatch of the thermal expansion of the film and the substrate materials during the processing the properties of the piezoelectric films differ from those of the respective bulk piezoceramics. Furthermore, an additional drawback can be the deterioration of the piezoelectric film’s properties due to its chemical interaction with the substrate. To ensure a sufficient displacement required for a certain application, different solutions have been proposed, such as a reduction of the thickness of the substrate,

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Marina Santo Zarnik, Hana Uršič and Marija Kosec processing of the thick-film multilayer actuator structures or the preparation of ―substrate-free‖ actuators. In this review the progress in thick-film piezoelectric actuators prepared by the screen-printing method is reported. The whole production line for the actuators is discussed, including the preparation of the piezoelectric pastes for the screenprinting, the processing of the various piezoceramic thick films on the different substrates and the films’ characterisation. Some representative examples of the most commonly investigated thick-film actuator structures and their applications are described.

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1. INTRODUCTION Piezoelectric actuators are simple structures with compact sizes that provide quick and precise responses to an electrical stimulus. The response of the piezoelectric actuator can be a mechanical displacement or a blocking force with the magnitude determined by the structure’s geometry and the electromechanical properties of the materials used. These simple active devices can be used in a wide range of applications, such as micro-positioners, to precisely control the positioning in low- to very-heavy-load applications, miniature ultra-sonic motors, and adaptive mechanical dampers [1–3]. In mechanical systems, the piezoelectric actuator can generate forces or pressures under static or high-frequency conditions and so activate a suitable mechanical device [4]. Bimorph bender actuators are employed for applications that require a large displacement output, i.e., fluid control devices [2, 5], robotic systems [2], swing CCD (charge-coupled device) mechanisms [5, 6]. In an optical system an actuator can be used to move a mirror or another optical switch [7, 8], etc. Depending on the application, specific constructions of the piezoelectric actuators are designed, mainly with the appropriate bulk piezoceramic elements. Recently, there has been a growing interest in micrometer-sized piezoelectric actuators, which are particularly attractive for advanced applications in novel research fields, such as micromechanics, robotics and microfluidics. The active piezoelectric elements integrated into micro-electromechanical systems (MEMSs) micromachined in silicon or in the miniaturised ceramic electro-mechanical structures (named ceramic-MEMSs) should be only a few to a few tens of m thick. For that reason, thick-film piezoelectric actuators are considered as a promising solution for future electro-mechanical systems and smart-structure technologies. Moreover, the piezoelectric thick films exhibit distinct characteristics of technological interest that enable the

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development of a really new generation of smart sensor/actuator systems. Driven by the versatility of the conventional thick-film technology, the processing of various functional films on different substrates is possible along with the extended 2-D and 3-D design possibilities. The key functional materials considered are piezoceramics; however, the use of piezoelectric polymers [9, 10] and ceramic-polymer composite structures [11, 12] were also proposed as appropriate solutions for some low-cost applications. This review focuses on the recent achievements in the field of thick-film piezoelectric actuators manufactured by using the screen-printing method for the deposition of the functional films [12–29]. In the preface the thick-film technology was introduced, and the processing of the piezoceramic thick-films and their functional properties are discussed. In the second section the results of the investigations of the most common piezoceramics, Pb(Zr,Ti)O3 (PZT) and Pb(Mg1/3Nb2/3)O3–PbTiO3 (PMN–PT), thick films are presented, including the preparation of the screen-printable pastes, the processing of the thick-film structures on different substrates and the films’ characterisation. The third section looks at the examples of screen-printed thick-film piezoelectric actuators. Some of the most commonly used thick-film actuator structures and their applications are described.

2. PIEZOCERAMIC THICK-FILM STRUCTURES The conventional thick-film production process was introduced almost fifty years ago as a means of producing thick-film hybrid circuits on ceramic substrates. Today, the use of thick-film technology combined with new, modern technologies and functional materials has become attractive for the production of various applications from simple functional structures to more complex systems, such as integrated micro-fluidic systems, voltage converters, transducers [30–33]. Nevertheless, the thick-film production process is almost identical to that used years ago. The technology typically entails the deposition of several successive functional layers onto an electrically insulating substrate by screen-printing. Screen-printing is the most widely used thick-film deposition technique. It is a relatively simple and convenient method for producing films with thicknesses of a few micrometers up to a few tens of micrometers. During this process the paste applied to the upper surface of the screen is forced through the pattern areas on the screen to the surface of the substrate [34, 35]. The film is then dried and sintered at elevated temperatures to yield a dense thick film. In general, the processing of various functional

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thick-film materials is possible. Among the most popular are thick-film materials based on the different piezoceramic compositions. These piezoceramic thick films exhibit many desirable characteristics relevant to the realisation of the various actuator structures. The important advantage of the conventional thick-film technology is in the successive deposition of several layers, which makes feasible thicker piezoelectric films (up to 100 m thick) and functional multilayer structures. These thick films are fired at lower temperatures than the sintering temperatures of the bulk ceramics, i.e., around 900°C. The use of the appropriate temperature profiles allows the co-firing of the functional films together with the electrical contacts and even the processing of threedimensional self-standing structures. The processing of piezoceramic thick films at temperatures around and below 1000°C could lead to interactions between the films and the substrates [18–21], which could strongly influence the structural and electrical characteristics of the film. The use of a thin barrier layer to prevent the film/substrate interaction has been proposed as an appropriate solution for different materials systems [18, 19, 36, 37]. One of the possible drawbacks of the technology is that the resolution of screen-printed films is typically around 200 μm. Generally, the properties of piezoceramic thick films depend not only on the material composition, but also on the compatibility of the functional material with the electrodes and the substrate materials, and a number of technological parameters relating to their processing. Any interaction between the film and the substrate may result in a deterioration of the material’s functional properties. The functional and dielectric properties of piezoceramic thick films are also closely linked to the geometry and boundary effects imposed by the material system. The constraining substrate influences the electro-active response of the film. As a result of the mentioned technological and boundary effects, the properties of the piezoceramic thick films differ from those of bulk piezoceramics or single crystals with the same composition. The dielectric constants and the piezoelectric coefficients of the films are generally lower than those for bulk ceramics. The investigations of piezoelectric thin films showed that the ferroelectric hysteresis loop of the films may be affected by many factors, including the thickness of the film, the mechanical stresses, the preparation conditions, the thermal treatment and the presence of charge defects [38, 39]. The functional films processed on substrates at elevated temperatures and cooled to room temperatures are thermally stressed, due to the mismatch between the thermal expansion coefficients of the film and the substrate materials [40]. The same

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effects are also expected in the processing of thick films. Recent investigations [15] showed that due to the process-induced residual stresses the electrical and structural properties of the piezoceramic thick films can be changed dramatically in comparison to the unstressed films. The designers of thick-film functional structures for certain applications should be aware of all the above-mentioned technological and boundary effects influencing the resulting properties of the piezoceramic thick films. Some helpful information about the processing of piezoceramic thick films and their properties can be found in the following.

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2.1. Processing of Piezoceramic Thick Films The processing of piezoceramic thick films has been discussed in the open literature by many authors. Because of the lack of an assortment of commercially available piezoceramic thick-film materials and no conventional processing procedures the investigations made so far were carried out for thick films with various piezoceramic compositions and different technological procedures were proposed. The piezoceramic pastes for screen-printing are generally made by mixing the fine powder of the functional piezoceramics with different binders and organic carriers to obtain the suitable printability. The selected piezoceramic composition can be designed and processed with a sintering aid to obtain the appropriate film properties. The most frequently reported are lead-based piezoceramics thick-films: Pb(Zr,Ti)O3 (PZT) [13, 14, 16, 17, 21–24, 41–52], (Pb,La)(Ti,Zr)O3 (PLZT) [18–20, 51], Pb(Mg1/3Nb2/3) O3 (PMN) [25, 53] and Pb(Mg1/3Nb2/3)O3–PbTiO3 (PMN–PT) [15, 25–28, 36, 54–57]. There are also a few reports on the thick films of some other leadbased compositions, e.g., Pb(Ni1/3Nb2/3)O3–PbZrO3–PbTiO3 (PNN–PZ–PT) [58] and Pb(Mg1/3Nb2/3)O3–PbZrO3–PbTiO3 (PMN–PZ–PT) [59]. Due to the toxic nature of lead, the investigation of lead-free thick films has recently become an emerging research topic [60, 61]. However, there are not many results on this subject accessible in the open literature. An updated overview of lead-free piezoelectric materials can be found in [62]. The most commonly used method for the deposition of piezoceramic thick films was screen-printing [12–28, 41–45, 47–49, 63]. However, successful experiments with the electrophoretic deposition [50, 55, 56, 64] and the aerosol deposition [44, 59] techniques as well as the hydrothermal process [53] were also reported.

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The proper selection of the material system, including the compatibility of the functional material with the electrode and the substrate materials, is among the most important for the successful processing of piezoelectric thick-film structures. The process-induced residual stresses in the thick-film structure and the possible reactions between the piezoceramic film and the reactive substrate may change considerably the functional properties of the film. In the open literature the results of the investigations of piezoceramic thick films processed on different substrates were discussed. The most common substrate materials used for piezoceramic thick films are silicon [26, 41, 42, 46, 48, 58] and relatively inert polycrystalline Al2O3 (alumina) [13–28, 36, 41, 42–45, 47, 50]. Recently, low-temperature co-fired ceramics (LTCCs) have proven promising for the integration of ceramic MEMS structures and for this reason considered as desirable substrates for piezoceramic thick films [13, 14, 17, 21, 23, 24, 37, 42, 49, 52, 63, 65]. The LTCC materials are particularly interesting for sensor and actuator applications because they have a relatively low Young’s modulus (90–110 GPa) in comparison to the most widely used alumina (215–414 GPa), which enables higher sensitivities. The LTCC materials are based either on crystallisable glass or a mixture of glass and ceramics. They are sintered at relatively low temperatures, i.e., around 850°C, and are compatible with a variety of functional thick-film materials. However, the processing of piezoceramic films on LTCC materials is still a big challenge because the glass phase in the LTCC substrate might interact with the functional film leading to changes in its electrical characteristics. In order to prevent the film/substrate reaction different solutions were proposed, such as the processing of the barrier layer [37] or use of a very dense electrode film between the PZT film and the substrate [42]. Some results of the investigations of the most commonly studied PZT and PMN–PT thick films are summarized below. Thick PZT films have been mainly processed on alumina or silicon substrates; however, some authors also report about the successful processing of PZT films on the relatively reactive LTCC. For an illustration the test patterns used for the characterisation of PbZr0.53Ti0.47O3 (PZT 0.53/0.47) thick films processed on the alumina and the LTCC substrates by Uršič et al. are shown in fig. 1. The samples were prepared by screen-printing the PZT 0.53/0.47 paste on the prefired thick-film gold electrode/substrate structures and fired at the peak temperature of 850°C. The details about the powder and pastes preparation are given in refs. [17, 21, 49]. The printing and firing procedure was performed in several successive steps to reach the appropriate thicknesses of the functional films.

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(a)

(b)

(c) Figure 1. Thick-film piezoceramic structures: a) Schematic of the cross-section, b) PZT 0.53/0.47 thick-film structure with gold electrodes on Al2O3 and LTCC substrates (Du Pont 951), and b) 0.65PMN–0.35PT thick-film test samples on Al2O3 substrates.

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(a)

(b) Figure 2. SEM micrographs of the cross-section of the a) PZT 0.53/0.47 thick film on LTCC substrate where under the electrode layer a PbO-rich region resulting from the interaction between the film and the substrate can be clearly seen, and b) 0.65PMN– 0.35PT thick films on Al2O3 with interposed PZT barrier layer to prevent interactions between the film and substrate.

Finally, on the top of the functional film, the upper gold electrodes were screen-printed and fired. The samples were poled with a DC electric field of 10 kV/cm at around 160°C for 15 minutes and then cooled in the same bias field as reported in [21]. In fig. 2 (a) the cross-section of the PZT 0.53/0.47

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thick-film on an LTCC substrate is shown, from which the result of the reaction between the substrate and the film, i.e., an interfacial layer rich with PbO, can be clearly seen below the bottom electrode. The (1−x)Pb(Mg1/3Nb2/3)O3–xPbTiO3 (PMN–PT) compositions exhibit very good functional properties and were recently considered as appropriate materials for piezoelectric thin- and thick-films. The processing of thick PMN–PT films was investigated by several research groups [15, 25–28, 36, 54–56]. As in the case of the PZT thick films, the substrates used for thick PMN–PT films are mainly alumina or silicon [15, 25–28, 36]. In some articles PMN–PT films on Pt [15, 54, 55], AlN [15] and PMN–PT [15] substrates were also discussed. Uršič et al. [15] presented the processing and characterisation of 0.65Pb(Mg1/3Nb2/3)O3–0.35PbTiO3 (0.65PMN–0.35PT) thick films on alumina substrates (fig. 1(c)), which were processed as follows. The piezoceramic pastes for screen-printing were prepared from 0.65PMN–0.35PT powders and an organic binder, as described in [27]. In order to prevent the chemical interactions between the film and the substrate a PZT0.53/0.47 barrier layer was processed between the substrate and the bottom platinum electrode [36]. The thick 0.65PMN–0.35PT film was fired with an appropriate temperature profile in a PbO-rich atmosphere at 950oC. Finally, the top electrodes were prepared by sputtering the thin gold film. The samples were poled with a DC electric field of 2.5 kV/cm at around 160°C for 5 minutes and then cooled in the same bias field as reported in [28, 54]. The cross-section of the 0.65PMN–0.35PT thick film with the barrier layer is presented in fig. 2 (b). It has also been shown that under similar processing conditions the microstructures, phase compositions and dielectric properties strongly depended on the stresses in the thick films due to the different temperature expansion coefficients of the thick films and the substrates [15]. The 0.65PMN–0.35PT thick films under compressive stresses exhibited a tetragonal phase in addition to the monoclinic one. The films were sintered to a high density, having a coarse microstructure. These microstructures favour a high dielectric constant. Therefore, with the proper selection of the substrate material, the structural and electrical properties of the films can be controlled.

2.2. Characterisation of Piezoelectrc Thick Films In piezoelectric materials the mechanical stress {T} and the strain {S} are related to the dielectric displacement {D} and the electric field {E}, as stated in the constitutive equations:

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S   s E  T  d E

(1)

D  dT T  εT E

(2)

where [sE] is the compliance matrix evaluated at a constant electric field, [T] is the permittivity matrix evaluated at a constant stress and [d] is the matrix of the piezoelectric coefficients. The successful design of piezoelectric thick-film structures for various applications can take place only with a thorough knowledge of the electrical and electromechanical properties of the piezoelectric film. Since the effective material properties of the piezoceramic thick film depend not only on the material composition but also on its compatibility with the electrodes and the substrate, and the technological parameters relating to the film processing, the characterisation of the piezoelectric thick films is required before designing. Because of a lack of standard procedures for characterising piezoceramic thick films, special attention has to be paid to providing the actual material parameters. In order to obtain proper material parameters some unconventional characterisation approaches were used as a nano-indentation test for the evaluation of the compliance parameters [22, 23, 54]. As already mentioned, the piezoelectric coefficients of the piezoceramic thick films differ from the coefficients of the bulk or single crystals with the same composition. One of the main reasons is that the films are constrained by the substrates. For a constrained film the ratio D3/T3 does not represent the piezoelectric coefficient d33 of the free sample, but an effective piezoelectric coefficient d33,eff [66]

d 33, eff

νs E  s13 Ys  d 33  2d 31 E , E (s11  s12 )

(3)

where d33 and d31 are the direct and the transverse piezoelectric coefficients respectively (C/N), sE13, sE11, sE12 are the elastic compliance coefficients at a constant electric field (m2/N), νs is the Poisson`s ratio of the substrate, Ys is the Young`s modulus of the substrate (N/m2).

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Since in eq. (3) for most piezoceramic materials d31 < 0, s13 < 0 and d31 is relatively large, the effective coefficient measured for the films is generally lower than those of the unclamped materials (d33,eff< d33). The commonly used equipment for measurements of d33,eff of the films are an interferometer [17, 25, 67, 68], a piezo-force microscope (PFM) [17, 68– 71] and a fiber-optic probe (Fotonic sensorTM, MTI Inc. USA) [72]. The Berlincourt piezometer, which was designed for measurements of the piezoelectric coefficient d33 of bulk piezoceramics, was also found to be applicable for measurements of the d33,eff of thick films [16, 17, 27, 44]. It was shown in [17] that the results obtained by different methods, i.e., by using the interferometer, the Berlincourt piezometer and the PFM are comparable. The experimentally obtained d33,eff coefficients of the piezoceramic thick films are given by numerous authors [16, 17, 21–23, 25, 27, 28, 44]. Generally, the characteristics of thick-film bending actuators mainly depend upon the transverse piezoelectric coefficient. For this reason an evaluation of the d31,eff is even more important for a proper design. Unfortunately, the experimentally obtained d31 is not so often reported. The effective transverse piezoelectric coefficient d31,eff of the thick films can be evaluated from the results of substrate-flexure measurements. An evaluation procedure which takes into account the thickness of the film was presented in [22]. Table 1. The elastic and piezoelectric properties of the bulk PZT 0.52/0.48 and the PZT 0.53/0.47 thick films (TF-PZT 0.53/0.47) on the Al2O3 and the LTCC (Du Pont 951) substrates. coeff.

unit

s11E s33E s12E s13E s44E s66E d31 d33

10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (C N-1) 10-12 (C N-1)

PZT 0.52/0.48 bulk [73] 13.8 17.1 -4.07 -5.08 48.2 38.4 -93.5 223

TF-PZT 0.53/0.47 on LTCC [23] 70.4 87.2 -20.7 -29.6 245.9 195.9 - 8.6 70 [17, 21, 23]

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TF-PZT 0.53/0.47 on Al2O3 [22] 33.9 42.1 -10.0 -14.3 11.9 11.9 -29 170 [22], 130 [17, 21]

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For an illustration, the material parameters reported in the open literature for the PZT and PMN–PT thick films processed on two different substrates (Al2O3 and LTCC) are collected in Table 1 and Table 2. As is evident from those data, the elastic compliance of the piezoceramic thick films was higher than those of the bulk for all the film/substrate material combinations considered, while the piezoelectric coefficients d31 and d33 were smaller in comparison with the bulk coefficients. A similar discrepancy between the material parameter obtained for the thick films and the bulk piezoceramic was reported for a commercially available thick-film PZT paste (TF 2100, Insensor A/S) based on Pz26 (Ferroperm) [24, 51].

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Table 2. The elastic and piezoelectric properties of the bulk 0.655PMN– 0.345PT [74] and the thick films of the similar composition (TF0.65PMN–0.35PT) processed on Al2O3 substrates. coeff.

unit

s11E s33E s12E s13E s44E s66E d31 d33

10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (m2 N-1) 10-12 (C N-1) 10-12 (C N-1)

0.655PMN–0.345 PT TF-0.65PMN–0.35PT bulk [74] on Al2O3 [54] 13.5 23.1 14.5 24.8 -4.8 -8.20 -5.9 -10.1 31.0 53 36.6 62.5 -223 -100 480 170–190 [25, 27, 54], 140 [28]

3. PIEZOELECTRIC THICK-FILM ACTUATORS Driven by the versatility of conventional thick-film technology, various designs of thick-film piezoelectric actuators are possible. The simplest thickfilm piezoelectric actuator design is a free-standing cantilever beam that can be realized as a bimorph or multimorph multilayer structure. In addition to cantilever-type actuators, there are also the bridge- and the membrane-type actuators. In combination with the materials and technologies enabling 3D structuring, even arbitrarily shaped thick-film actuator structures can be feasible. According to the type of displacement the thick-film piezoelectric actuators are categorised as bending actuators, which are generally capable of

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larger displacements but exert a weak generative force. Due to the clamping to the substrate they have smaller displacements in comparison to the substratefree structures. Furthermore, the properties of the piezoceramic films differ from those of the respective bulk ceramics; in particular the piezoelectric properties are weaker and can even be reduced by an interaction with the reactive substrates. All these effects should be considered in the design of the structure and the technological procedure. To ensure a sufficient displacement required for a certain application, different design solutions for thick-film actuators are possible. According to the general characterization of the piezoelectric actuators by the piezoelectric coefficients that they exploit, i.e., the mode of the operation, the thick-film actuators are classified as d31-mode actuators producing a displacement perpendicular to the polarization direction. The d33-mode piezoelectric actuator structures, which produce displacements in the same direction as the electric field applied in parallel to the piezoelectric film polarization direction in the so-called interdigitated electrodes design, are more convenient for thinfilm actuators [75, 76]. However, by having a good resolution of the printed thick films the d33-mode thick-film actuators can also be feasible. Recently, different thick-film actuator structures aimed at different applications have been reported, i.e., micropositioners, micropumps, active optical devices, and high-frequency ultrasound transducers, piezoelectric sensors for pressure, force and strain measurements and piezoelectric generator elements, usable for energy harvesting and battery substitutes for low-power circuits. In the following, the most commonly used cantilever- and membrane-type thick-film actuator structures are reviewed.

3.1. Cantilever-Type Actuators The cantilever-type thick-film actuators were prepared by the screenprinting of different piezoceramic thick-film materials on different substrates. The most common structures reported are one-layer bending-type actuators. A schematic representation of the cross-sections of a typical thick-film actuator is shown in fig. 3.

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Figure 3. Scheme of the cross-section of a typical thick-film actuator structure.

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(a) (b)

(c) Figure 4. a) The test samples of the thick-film PZT actuator on LTCC substrates. The length of the thick-film PZT patch with the 3-µm-thick Au electrodes on the top and bottom is 12 mm, the thickness of the LTCC substrates is 200 µm, and the thickness of the PZT film is about 45 µm, b) Displacement of the actuator on LTCC substrate in the first mode resonance (at 10 V), c) Tip displacement versus electric field characteristics.

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In the open literature the successful implementations of the PZT thick-film actuator structures on Al2O3 substrates [22, 23, 43–45, 77] and on silicon [44, 46] were presented. Recently, the excellent possibilities of the LTCC materials and technology for manufacturing even relatively complex 3D structures increased the interest in integrating the piezoelectric thick-film actuators in ceramic MEMS [23, 24, 42, 77]. The PMN–PT thick-film actuators prepared by screen-printing [54] were found to be good competitors with the PMN–PT bulk actuators prepared by a tape-casting procedure [2, 78]. Recent investigations in the field of ceramic MEMS based on laminated LTCC materials and technologies showed that in spite of the technological obstacles in the processing of PZT films on the reactive LTCC substrate, useable performances of the thick-film PZT actuators can be achieved. Experimental and numerical analyses performed by Zarnik et al. [23] showed that even with weaker piezoelectric properties of the PZT thick films on the LTCC, comparable displacements can be achieved for the actuators processed on the alumina and the pre-fired LTCC substrates. For an illustration the thickfilm PZT actuators on the LTCC substrates and their characteristics are presented in fig. 4.

3.1.1. Thick-Film Multilayer Actuator Structures One of the great advantages of the conventional thick-film technology is that it provides excellent capabilities for fabricating passive as well as functional multilayer structures. In this way screen-printing was found to be an appropriate deposition method for manufacturing active multilayer structures with piezoelectric thick films. Harris et al. [44] showed that the two-layer thick-film piezoelectric actuators perform well as an acoustic source for silicon MEMS applications. It was shown that multilayer structures can be considered as a viable alternative to bonded bulk PZT actuators on silicon for MEMS structures with an additional advantage of being suitable for batch processing on a wafer scale. Zhu et al. [45] reported on the successful processing of a substrate-free piezoelectric multilayer actuator structure by using a sacrificial layer. The 11mm-long and 3-mm-wide multimorph actuators with 7–15 layers of 30-μmthick PZT films were manufactured. Under a driving voltage of 10 V, the actuators produced a displacement of 1.57 μm at the corresponding resonance frequency of 56.9 kHz. Moilanen et al. [79] developed a new, double-paste printing method that was used for the deposition of a six-layer thick-film actuator structure on an

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alumina substrate. The multilayer micro-movement actuator was successfully integrated into a hybrid thick-film actuator/force sensor.

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3.1.2. Substrate-Free Actuator Structures Recently, a novel approach to manufacturing large-displacement 0.65PMN–0.35PT/Pt (PMN–PT/Pt) free-standing actuators was presented by Uršič et al. [54]. In contrast to [45] the actuators were prepared by screenprinting the PMN–PT film over the thick-film Pt electrode layer directly onto an Al2O3 substrate (without using any sacrificial layer). This results in a very poor adhesion between the bottom Pt electrode and the Al2O3 substrate, enabling the PMN–PT/Pt thick-film composite structure to be simply separated from the substrate. In this way a ―substrate-free‖ actuator structure was manufactured. In fig. 5 the scheme of the cross-section of the PMN–PT/Pt actuator and a photograph of the actuator are shown.

(a)

(b) Figure 5. a) The scheme of the cross-section of the PMN–PT/Pt actuator and b) the actuator with dimensions 1.8 cm × 2.5 mm × 60 µm (with 50-µm PMN–PT film and 10-µm Pt electrode) during the measurement of the displacement.

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In order to compare the performances of the substrate-free cantilever-type actuators made by using different piezoceramic materials and technological procedures the normalized tip-displacements (the tip-displacement per unit length, ND) of the actuators were calculated. It was shown that the ND of the PMN–PT/Pt free-standing actuator [54] is much higher than the ND of the PZT multilayer actuator [45], i.e., 55 µm/cm at 3.6 kV/cm and 1.4 µm/cm at 3.3 kV/cm, respectively. This difference in the NDs of the PZT and PMN– PT/Pt actuators mainly comes from the different materials used for the active layer and the different processing conditions. Note that the measurements of the displacement of the PZT multilayer actuators discussed in [45] were performed at a resonance frequency of 56.9 kHz, while the measurements of the PMN–PT/Pt were made under static (DC) conditions, although at the same electric field intensity (at around 3.5 kV/cm). To summarize this paragraph, by demonstrating the influence of the relatively stiff substrates on the bending capabilities of the thick-film actuators the tip displacements of the different cantilever-type actuators are compared. The comparison between the NDs vs. applied electric field that were measured under static conditions for the PZT actuators on the Al2O3 and LTCC substrates [22, 77] and PMN–PT/Pt substrate-free actuators [54] is shown in fig. 6.

Figure 6. The tip displacement normalized per unit length of the actuator for the 0.65PMN–0.35PT/Pt actuators and PZT actuators on Al2O3 and LTCC substrates vs. applied electric field.

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As is clear from fig. 6 the ND of the ―substrate-free‖ PMN–PT/Pt actuator is several times higher than the displacement of the PZT actuators on the Al2O3 and LTCC substrates.

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3.2. Membrane-Type Actuators Membrane-type thick-film piezoelectric actuators were generally considered as the active parts of different fluidic systems and sensors for pressure and load measurements. Because of the clamping of the membrane to the rigid base the maximum deflection vs. blocking force characteristics of the membrane-type actuators is generally less steep in comparison to the characteristics of the cantilever-type actuators. Accordingly, the thick-film membrane-type actuators are capable of producing relatively small displacements. However, there were many design ideas proposed for the particular applications and this paper reviews some of them. The early reports Koch et al. presented a novel design of a micromachined MEMS micropump with a screen-printed thick-film PZT membrane-type actuator [80–82]. The tests of the first generation of their thick-film actuatordriven micropumps with the passive dynamic cantilever valves showed characteristics with up to approximately 30% of the performance of the surface-mounted bulk devices. Successful fabrication of the thick-film actuators by screen-printing paste on the basis of Pb(Al,Nb)O3–Pb(Zr,Ti)O3 (PAN-PZT) on the micromachined silicon membrane was reported in the review [83]. The results presented showed that the combination of the MEMS process and the screen-printing of the piezoceramic films can be promising even for the mass production of the membrane-type microactuators. Silicon micromachining techniques have also been applied to the fabrication of membrane actuators described by Futakuchi et al. [84]. On the Si membrane the screen-printed actuator structure with the bottom platinum electrode, 25μm ferroelectric film composed of Pb[Zr0.2Ti0.3(Mg1/3Nb2/3)0.3(Zn1/3Nb2/3)0.1 (Mg1/2W1/2)0.1]O3 and silver top electrode were processed. For the actuator membrane with an area of 3×10 mm2 and a thickness of 100 μm the maximum displacement equalled approximately 0.3 μm for an applied voltage of 25 V. As examples of the use of membrane-type thick-film actuators processed on ceramic substrates, different configurations of piezoelectric thick-film resonant pressure sensor have been reported in the literature. In the early investigation of Moilanen et al. [79] a special double-paste printing technique was developed and employed in the fabrication of the piezoelectric multilayer

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micro-movement actuator. The six-layer PZT thick-film actuator structure processed on the alumina substrate was successfully integrated into the actuator/sensor module with a sensitivity of about 1 mV/N at the supply voltage of 40 V. Later, Morten et al. described an innovative design of resonant gaspressure sensor based on the modulation of the resonant frequency of the sensor by pressure [85]. The vibrations of the sensor’s diaphragm were actuated by a thick-film PZT actuator structure with annular (ring-shaped) interdigitated (IDT) electrodes. The IDT electrodes were buried into the 50-μm PZT film on the clamped alumina diaphragm in such a position that the third mode of the vibration of the diaphragm was excited. In the central part of the diaphragm a thick-film PZT structure was placed, aimed at detecting the induced strain vibrations. A pressure applied to the diaphragm affects the resonant frequency of the system and changes the phase relations that are controlled by phase-locked loop PLL circuit. Recently, realisations of piezoelectric resonant pressure sensors with a membrane-type thick-film actuator processed on pre-fired, three-dimensional LTCC structures were reported [24, 63, 65]. As distinguished from [85], the operation of those sensors is based on the actuation of the LTCC membrane with the thick-film PZT structure sandwiched between the bottom and the top electrodes. The schematic of the cross-section of the piezoelectric resonant pressure sensor structure [65] is presented in fig. 7. The diaphragm is driven to oscillate with the natural resonant frequency (in the fundamental oscillating mode), which is modulated by the applied pressure as a function of the changed tensile state.

Figure 7. The scheme of the cross-section of the thick-film actuator/sensor structure on the LTCC membrane.

At the same time the thick-film PZT structure on the membrane acts as a sensor (utilising the reverse piezoelectric effect). In [24] the feasibility of the

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resonant pressure sensor by using two different thick-film piezoceramic materials was discussed (the undoped PZT 0.53/0.47 and the TF 2100, Insensor A/S). The modified design with the separated actuator and the sensor part of the thick-film structure on the membrane was presented (fig. 8a). A schematic diagram of the actuator/sensor structure in an oscillator circuit is presented in fig. 8b. It was shown that although the PZT thick films processed on LTCC have poor piezoelectric properties in comparison to the bulk piezoceramic, the thick-film actuator structure can be successfully used for the realization of resonant sensors.

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(a)

(b) Figure 8. Piezoelectric resonant pressure sensor [24]; a) the prototype, b) the sensor/actuator structure in the oscillator circuit.

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CONCLUSION There is a continuous, global increase of interest in the miniaturisation of devices, materials and system integration. Piezoelectric thick-film actuators are a good example of the opportunities offered for the miniaturization of electromechanical systems by the successful implementation of new functional materials and technologies. Driven by the versatility of the conventional thickfilm technology the processing of functional structures with piezoelectric thick films on different substrates is possible, along with many design possibilities. The drawback of the technology is the limited minimum resolution of the screen-printed films, which reduces the design possibilities to some extent. An important obstacle for rapid progress in the research field is the high risk for an industry to start with the development of new products because of the lack of commercial piezoelectric thick-film materials and conventional processing procedures. Based on the so-far reported results of investigations, the most research studies were carried out for thick films of the lead-based piezoceramics Pb(Zr,Ti)O3 (PZT), (Pb,La)(Ti,Zr)O3 (PLZT), Pb(Mg1/3Nb2/3) O3 (PMN) and Pb(Mg1/3Nb2/3)O3–PbTiO3 (PMN–PT). However, due to the toxic nature of the lead, investigations of lead-free piezoceramic thick-films have become an emerging research topic. The most commonly used substrate materials were silicon and alumina. Recently, relatively reactive LTCC materials were proven to be promising for the integration of different ceramicMEMS and in such a manner evaluated as the desired substrates for piezoceramic thick-films as well. An important role in the processing of piezoceramic thick films is also played by the electrode material and its processing. An appropriate dense electrode can prevent substrate/film interaction. From all the latest discoveries in the field we can conclude that a proper selection of the compatible material system is of key importance for the successful processing and integration of thick-film piezoelectric structures. An appropriate design of the piezoelectric thick-film actuators for a certain application can take place only with a thorough knowledge of the electrical and electromechanical properties of the piezoelectric films. The properties of the piezoceramic thick films are closely linked to the geometry and boundary effects imposed by the material system and for this reason differ from the properties of bulk ceramics or single crystals with the same composition. The process induced residual stresses in the thick-film structure, and the possible reactions between the piezoceramic films and the substrate materials, may also change considerably the functional film properties. Finally, the resulting dielectric constants and the piezoelectric coefficients of

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the piezoceramic thick films are generally lower than for bulk ceramics. However, since the effective material properties of the piezoceramic thick film depend on the material system and the processing parameters, a characterisation of the piezoelectric thick films is necessary before the designing. The thick-film technology makes possible various designs of bendingactuator structures. The most commonly used are cantilever and membranetype actuators. Generally, thick-film piezoelectric actuators have smaller displacements and exert weaker forces in comparison to their bulk relatives. This is because a stiff and relatively thick substrate in comparison to the active piezoceramic film always reduces the bending ability of the thick-film actuator structure. However, many examples of successfully implemented thick-film actuators have been described by several authors. For the applications where large displacements are required, thick-film piezoceramic/electrode structures separated from the substrates have been suggested. Some inventive approaches for the processing of such substrate-free actuators that have been proposed are a procedure based on processing the functional thick-film structure with a very weak adhesion to the substrate or the use of a sacrificial layer. The results presented by several research groups showed that a combination of the MEMS micromachining process and the screen-printing of piezoceramic films can be promising even for the mass production of the membrane-type microactuators for microfluidic applications. The processing of piezoelectric thick-film actuators on ceramic membranes was shown to be appropriate for the realisation of different configurations of resonant pressure sensors. Piezoceramic thick-film actuators have many potential applications, although their production faces many challenges arising from the successful integration of different material systems. The state of the art in the processing of thick-film actuators is the development of new, effective functional materials and the investigation of innovative designed solutions. There are still a number of challenges to be faced in the production of thick-film piezoelectric actuators and the wide possibilities for further improvements of their performances to meet the industrial demands for mass production. However, the basis for future investigations in this field would seem to be the lead-free materials and green technologies.

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ACKNOWLEDGMENTS The financial support of the Slovenian Research Agency in the frame of the program Electronic Ceramics, Nano-, 2D and 3D Structures (P2-0105) is gratefully acknowledged.

REFERENCES [1] [2] [3] [4]

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[5] [6] [7]

[8] [9] [10] [11] [12] [13]

[14]

Allahverdi, M.; Hall, A.; Brennan, R.; Ebrahimi, M.E.; Marandian, H. N.; Safari A. J. Electroceram. 2002, 8, 129–137. Hall, A.; Akdogan, E.K.; Safari, A. J. Appl. Phys. 2006, 100, 094105 1– 7. Takagi, H.; Sakata, K.; Takenaka, T. Jpn. J. Appl. Phys. 1993, 32, 4280– 4283. Arbogast, D. J.; Calkins, F. T. Electrical system for electrostrictive bimorph actuator, Patent US 6,888,291 B2. Ahn, B.G.; Lee, D.K.; Han, D.Y.; Kang, C.Y.; Choi, J.W.; Kim, H.J.; Yoon, S. J. J. Korean. Ceram. Soc. 2003, 40, 225–228. Tanuma, C. Jpn. J. Appl. Phys. 1999, 38, 5603–5607. Rodrigues, G.; Bastaits, R.; Roose, S.; Stockman, Y.; Gebhardt, S.; Schoenecker, A.; Villon, P.; Preumont, A. Opt. Eng. 2009, 48, 034001 1–7. Yang, E. H.; Shcheglov, K.; Trolier-McKinstry, S. Proc. SPIE 2003, 4983, 326–333. Fukada, E. IEEE Trans. Ultrason. Ferroelect. Freq. Control 2000, 47, 1277–1297. Su, J.; Zhang, Q. M.; Ting, R. Y. Appl. Phys. Lett. 1997, 71, 386–388. Choi, J. J.; Hahn, B. D.; Ryu, J.; Yoon, W. H.; Lee, B. K.; Park, D. S. Sensors Actuat. A 2009, 153, 89–95. Torah, R. N.; Beeby, S. P.; Tudor, M. J.; White, N. M. J. Electroceram. 2007, 19, 95–110. Belavič, D.; Hrovat, M.; Zarnik, M. S.; Holc, J.; Cilenšek, J.; Jerlah, M.; Maček, S.; Uršič, H.; Kosec, M. 2nd Electronics Systemintegration Technology Conference, Greenwich, UK. 2008, 2, 989–994. Hrovat, M.; Belavič, D.; Uršič, H.; Kita, J.; Holc, J.; Drnovšek, S.; Cilenšek, J.; Kosec, M.; Moos, R. 2nd Electronics Systemintegration Technology Conference, Greenwich, UK. 2008, 1, 339–345.

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[15] Uršič, H.; Hrovat, M.; Holc, J.; Tellier., J.; Drnovšek, S.; Guiblin, N.; Dkhil, B.; Kosec, M.; J. Eur. Ceram. Soc. 2010, 30, 2081–2092. [16] Torah, R. N.; Beeby, S. P.; White, N. M. J. Phys. D: Appl. Phys. 2004, 37, 1074–1078. [17] Uršič, H.; Lowe, M.; Stewart, M.; Hrovat, M.; Belavič, D.; Holc, J.; Zarnik, M. S.; Kosec, M.; Cain, M. J. Electroceram. 2008, 20, 11–16. [18] Holc, J.; Hrovat, M; Kosec, M. Mat. Res. Bull. 1999, 34, 2271–2278. [19] Kosec, M.; Holc, J.; Malič, B.; Bobnar, V. J. Eur. Ceram. Soc. 1999, 19, 949–954. [20] Kosec, M.; Murko, D.; Holc, J.; Malič, B.; Čeh, M., Hauke, T.; Beige, H. Z. Met.kd. 2001, 92, 97–104. [21] Uršič, H.; Hrovat, M.; Belavič, D.; Cilenšek, J.; Drnovšek, S.; Holc, J.; Zarnik, M. S.; Kosec, M. J. Eur. Ceram. Soc. 2008, 28, 1839–1844. [22] Zarnik, M. S.; Belavič, D.; Maček, S. Sensors Actuat. A 2007, 136, 618– 628. [23] Zarnik, M. S.; Belavič, D.; Uršič, H.; Maček, S. J. Electroceram. 2008, 20, 3–9. [24] Zarnik, M. S; Belavič, D.; Maček, S.; Holc, J. Int. J. Appl. Ceram. Technol. 2009, 6, 9–17. [25] Gentil, S.; Damjanovic, D.; Setter, N. J. Electroceram. 2004, 12, 151– 161. [26] Gentil, S.; Damjanovic, D.; Setter N. J. Eur. Ceram. Soc. 2005, 25, 2125–2128. [27] Kosec, M.; Holc, J.; Kuščer, D.; Drnovšek, S. J. Eur. Ceram. Soc. 2007, 27, 3775–2778. [28] Kuščer, D.; Skalar, M.; Holc, J.; Kosec, M. J. Eur. Ceram. Soc. 2009, 29, 105–113. [29] Lozinski, A.; Wang, F.; Uusimaki, A.; Leppavuori, S. Meas. Sci. Technol. 1997, 8, 33–37. [30] Lilliehorn, T.; Johansson, S.; J. Micromech. Microeng. 2004, 14, 702– 709. [31] White, N. M.; Turner, J. D. Meas. Sci. Technol. 1997, 8, 1–20. [32] Prudenziati, M. (editor), Handbook of Sensors and Actuators Vol. 1; Thick film sensors, Elsevier, 1994, 209–242 [33] Setter, N. Electroceramic-Based MEMS: Fabrication-Technology and Applications; Kluwer int. series Electronic Materials: Science and Technology, Springer Science+Busines Media, Inc., 2005. NY10013, USA, 49–79.

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[34] Pitt, K. E. G. Handbook of thick film technology, Electrochemical Publication Ltd. 2005. [35] Harper, C. A. Handbook of thick film hybrid microelectronics; McGrawHill, Inc. USA 1974. [36] Uršič, H.; Škarabot, M.; Hrovat, M.; Holc, J.; Skalar, M.; Bobnar, V.; Kosec, M.; Muševič, I. J. Appl. Phys. 2008, 103, 124101–1. [37] Hrovat, M.; Holc, J.; Drovšek, S.; Belavič, D.; Cilenšek, J.; Kosec, M. J. Eur. Ceram. Soc. 2006, 26, 897–900. [38] Damjanovic, D. Rep. Prog. Phys. 1998, 61, 1267–1324. [39] Choi, K. J.; Biegalski, M.; Li, Y. L.; Sharan, A.; Schubert, J.; Uecker, R.; Reiche, P.; Chen, Y. B.; Pan, X. Q.; Gopalan, V.; Chen, L. Q.; Schlom, D. G.; Eom, C. B. Science 2004, 306, 1005–1009. [40] Ohring, M. The material science of thin films; Academic Press, San Diego, 1992. [41] Lou-Moeller, R.; Hindrichsen, C. C.; Thamdrup, L. H.; Bove, T.; Ringgaard, E.; Pedersen, A. F.; Thomsen, E. V. J. Electroceram. 2007, 19, 333–338. [42] Gebhardt, S.; Seffner, L.; Schlenkrich, F.; Schönecker, A. J. Eur. Ceram. Soc. 2007, 27, 4177–4180. [43] Walter, V.; Deobelle, P.; Moal, P. L.; Joseph, E.; Collet, M. Sensors Actuat. A 2002, 96, 157–166. [44] Harris, N. R.; Hill, M.; Torah, R.; Townsend, R.; Beeby, S.; White, N. M.; Ding, J. Sensors Actuat. A 2006, 132, 311–316. [45] Zhu, W.; Yao, K.; Zhang, Z. Sensors Actuat. A 2000, 86, 149–153. [46] Lebedov, M.; Akedo, J; Akiyama, Y. Jpn. J. Appl. Phys. 2000, 39, 5600–5603. [47] Tran-Huu-Hue, P.; Levassort, F.; Meulen, F. V.; Holc, J.; Kosec, M.; Lethiecq, M. J. Eur. Ceram. Soc. 2001, 21, 1445–1449. [48] Chen, H. D.; Udayakumar, K. R; Cross, L. E.; Bernstein, J. J.; Niles, L. C. Appl. Phys. 1995, 77, 3349 –3353. [49] Hrovat, M.; Holc, J.; Drnovšek, S.; Belavič, D.; Bernard, J.; Kosec, M.; Golonka, L.; Dziedzic, A.; Kita, J. J. Mater. Sci. Lett. 2003, 22, 1193– 1195. [50] Kuščer, D.; Kosec M. Key Eng. Mat. 2009, 412, 101–106. [51] Wolny, W. W. Proceedings 2000 12th IEEE International Symposium on Applications of Ferroelectrics 2000, 1, 257–262. [52] Belavič, D.; Hrovat, M.; Zarnik, M. S.; Holc, J.; Kosec, M. J. Electroceram. 2009, 23, 1–5. [53] Chen, X.; Fan, H.; Liu, L.; Ke, S. Ceram. Int. 2008, 34, 1063–1066.

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[54] Uršič, H.; Hrovat, M.; Holc, D.; Zarnik, M. S.; Drnovšek, S.; Maček, S. Sensors Actuat. B 2008, 133, 699–704. [55] Chen, J.; Fan, H. Q.; Chen, X. L.; Fang, P.; Yang, C.; Qiu, S. J. Alloy. Compd. 2009, 471, L51–L53. [56] Kuščer, D.; Kosec, M. J. Eur. Ceram. Soc. 2010, 30, 1437–1444. [57] Kosec, M.; Uršič, H.; Holc, J.; Hrovat M.; Kuščer, D.; Malič, B. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2010, 57, 2205–2212. [58] Akiyama, Y.; Yamanaka, K.; Fujisawa, E.; Kowata Y. Jpn. J. Appl. Phys. 1999, 38, 5524–5527. [59] Futakuchi, T.; Matsui, Y.; Adachi, M. Jpn. J. Appl. Phys. 1999, 38, 5528–5530. [60] Wang, L.; Yao, K.; Ren, W. Appl. Phys. Lett. 2008, 93, 1–3. [61] Ryu, J.; Choi, J.; Hahn, B.; Park, D.; Yoon, W.; Kim, K. Appl. Phys. Lett. 2007, 90, 1–3. [62] Rödel, J.; Jo, W.; Seifert, K.; Anzon, E. M.; Granzow, T.; Damjanovič, D. J. Am. Ceram. Soc. 2009, 92, 1153–1177. [63] Partsch, U.; Arndt, D.; Keitel, U.; Otschik, P. Proceedings of European Microelectronic and Packaging Conference Germany 2003, 331–335. [64] Dorey, R. A.; Whatmore, R. W. J. Electroceram. 2004, 12, 19–32. [65] Zarnik, M. S.; Belavič, D.; Maček, S. Microelectron. Int. 2008, 25, 31– 36. [66] Lefki, K.; Dormans, G. J. M. J. Appl. Phys. 1994, 76, 1764–1767. [67] Li, J. F.; Moses, P.; Viehland, D. Rev. Sci. Instrum. 1995, 66, 215–221. [68] Christman, J. A.; Woolcott, R. R.; Kingon, A. I. Jr.; Nemanich, R. J. Appl. Phys. Lett. 1998, 73, 3851–3853. [69] Burianova, L.; Bowen, C. R.; Prakopova, M.; Sulc, M. Ferroelectrics 2005, 320, 161–169. [70] Zavala, G.; Fendler, J. H.; Trolier-McKinstry, S. J. Appl. Phys. 1997, 81, 7480–7491. [71] Gruverman, A.; Kholkin, A. Rep. Prog. Phys. 2006, 69, 2443–2474. [72] Vyshatko, N. P.; Brioso, P. M.; Cruz, J. P.; Vilarinho, P. M.; Kholkin, A. L. Rev. Sci. Instrum. 2005, 76, 085101 1–6. [73] Jaffe, B.; Cook, W. R.; Jaffe, H. Piezoelectric Ceramics; Academic Press inc., London/New York, 1971, Chapter 7, 146. [74] Alguero, M.; Alemany, C.; Pardo, L.; Thi, M. P. J. Am. Ceram. Soc. 2005, 88, 2780–2787. [75] Zhang, Q. Q.; Gross, S. J.; Tadigadapa, S.; Jackson, T. N.; Djuth, F. T.; Troiler-McKinstry, S. Sensors Actuat. A 2003, 105, 91–97.

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[76] Jeon, Y. B.; Sood, R.; Jeong, J. H.; Kim, S. G., Sensors Actuat. A 2005, 122, 16–22. [77] Belavič, D.; Zarnik, M. S.; Holc, J.; Hrovat, M.; Kosec, M.; Drnovšek, S.; Cilenšek, J.; Maček, S. Int. J. Appl. Ceram. Technol. 2006, 3, 448– 454. [78] Hall, A.; Allahverdi, M.; Akdogan, E. K.; Safari, A. J. Eur. Ceram. Soc. 2005, 25, 2991–2997. [79] Moilanen, H.; Lappalainen, J.; Leppavouri, S. Sensors Actuat. A 1994, 43, 357–365. [80] Koch, M.; Harris, N.; Maas, R.; Evans, A. G. R.; White, N. M.; Brunnschweiler, A. Meas. Sci. Technol. 1997, 8, 49–57. [81] Koch, M.; Evans, A. G. R.; Brunnschweiler, A. J. Micromech. Microeng. 1998, 8, 119–122. [82] Koch, M.; Harris, N.; Evans, A. G. R.; White, N. M; Brunnschweiler, A. Sensors Actuat. A 1998, 70, 98–103. [83] Kim, S. J.; Kang, C. Y.; Choi, J. W.; Kim, D. Y.; Sung, M. Y.; Kim, H. J.; Yoon, S. J. Mater. Chem. Phys. 2005, 90, 401–404. [84] Futukachi, T.; Yamano, H.; Adachi, M. Jpn. J. Appl. Phys. 2001, 40, 5687–5689. [85] Morten, B.; De Cicco, G.; Prudenziati, M. Sensors Actuat. A 1991, 31, 153–158.

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In: Piezoelectric Actuators Editors: Joshua E. Segel, pp. 29-54

ISBN: 978-1-61324-181-3 ©2011 Nova Science Publishers, Inc.

Chapter 2

PRESSURE CONTROL OF VEHICLE ABS USING PIEZOACTUATOR-BASED VALVE MODULATOR Juncheol Jeon and Seung-Bok Choi*

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Department of Mechanical Engineering, Inha University, Incheon, Korea

ABSTRACT This paper presents a new type of the piezoactuator-driven valve system with a pressure modulator which can be applicable for small-sized vehicles ABS (anti-lock brake system) such as green car. As a first step, a flapper-nozzle valve is devised by locating the piezoactuator in one side of a flexible beam which can make a movement required to control a desired pressure. The governing equation of the valve is then derived and the pressure modulator is connected to the valve in order to obtain high pressure level. After confirming excellent pressure tracking controllability of the proposed valve system, a wheel slip control model is established by adopting a quarter-car model. Subsequently, a robust sliding mode controller which has inherent robustness to model uncertainty and external disturbance is designed in order to achieve accurate tracking control of the desired wheel slip rate which directly affects performance of the vehicle ABS. Tracking control performances

* e-mail: [email protected]

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Juncheol Jeon and Seung-Bok Choi for the desired wheel slip rate are evaluated under two different trajectories: constant and sinusoidal slip rates.

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INTRODUCTION Several electronic control systems have been developed for improving safety of vehicle during driving or braking. For example, anti-lock brake system (ABS) has been developed to improve the vehicle steer-ability and the stability by preventing the wheel lock under critical circumstances such as a slippery road condition. In order to prevent the wheel locking, currently existing vehicle ABS calculates the wheel slip and controls the braking pressure by using a solenoid valve. However, most existing solenoid valves used in the ABS have some undesirable features: complicated structure, largesize, pulsatory motion, accuracy limitation of the pressure control and maintenance complex, etc. Therefore, it is desirable to introduce an alternative actuating valve mechanism for the vehicle ABS. One of attractive approaches is to use smart material actuators such as electrorheological (ER) fluids, piezoelectric materials and shape memory alloys (SMA), etc. When ER fluid is used in the valve or servo valve system, the pressure drop of control volume can be continuously controlled by controlling the intensity of the electric field to be applied to the fluid domain. In this case, design simplicity can be achieved since there is no moving part. This inherent feature of the ER fluid has triggered considerable research activity in the development of valve devices [1-4]. However, practical realization of the ER valve system is limited due to very high voltage input (2-5 kV). SMA actuator exhibits recoverable and repeatable movements with the temperature cycling. Since the SMA actuator can produce large deformation (or equivalent force) relative to other actuators, miniaturization of the servo valve can be easily accomplished. The micro valves activated by the SMA actuator fill a niche for compact, low power and moderate speed. These advantages of the SMA actuator have triggered a large amount of research activity in the development of the micro valves for hydraulic or pneumatic systems [5-8]. However, the SMA-based valve is not a good candidate for vehicle ABS since the pressure level is relatively low and the response time is very slow. On the other hand, the piezoelectric actuator which has inherent fast response has been successfully applied to various servo valve mechanisms or systems [9-13]. However, the research on the piezoactuator-driven valve for vehicle ABS is rare. This is because that the pressure level and the controllable pressure range generated from the piezoactuator valve are not enough to meet the requirement

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Pressure Control of Vehicle ABS …

31

for vehicle ABS. Moreover, the inherent hysteretic characteristics of the piezoactuator absolutely require the development of robust control systems. In this work, these problems will be resolved. Consequently, the main contribution of this work is to propose a new type of valve system which can produce the pressure level required for ABS equipped small vehicles such as hybrid car. In order to achieve this goal, a flapper-nozzle valve is devised by locating the piezoactuator in one side of a flexible beam which can make a movement required to control the desired pressure. This valve is then directly connected to the pressure modulator to amplify the controllable pressure range. After formulating the governing equation of the piezostack-based valve system (piezovalve system in short), pressure controllability is evaluated by showing pressure tracking control performance. Subsequently, the proposed valve system is applied to vehicle ABS by adopting a quarter-car model. The governing equation of the slip rate control model is derived and integrated with the piezovalve system. A sliding mode controller is then designed to guarantee the robustness of the vehicle ABS against the hysteretic behaviour of the piezoactuator and the uncertainty of supplying pressure. The sliding mode controller is implemented via MatlabTM Simulink and control performances are evaluated under two different slip rates. The slip rate tracking controllability, braking force, vehicle and wheel velocity and input control voltage for the piezoactuator are presented in time domain.

PIEZOVALVE SYSTEM The configuration of the proposed piezovalve system for pressure tracking control is shown in Fig. 1. The system consists of a piezostack actuator, a flapper, a pneumatic circuit (nozzle, air input/output) and a couple of pistons. When the flapper is pushed to the left side of the nozzle by the piezoactuator, the left side of the piston has higher pressure than the right side. Then, the piston will move to the right side and the output pressure PB will be increased. On the contrary, if the flapper is pulled to the right side of the nozzle, the output pressure will be decreased. By using this mechanism, the output pressure can be controlled to drive the vehicle ABS by controlling the applied voltage to the piezostack actuator. In this section, dynamic models of the piezovalve, the flapper and the pressure modulator are derived,

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Juncheol Jeon and Seung-Bok Choi

32

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respectively and integrated to achieve the governing equation of the proposed piezovalve system.

Figure 1. Schematic configuration of the piezovalve system.

Ps

Ps

Qao

Qbo

Orifice a

Orifice b

P1 Q1

Qan

Qbn

x

xm Figure 2. Mathematical model of the flapper-nozzle valve.

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P2 Q2

Pressure Control of Vehicle ABS …

33

A pneumatic valve activated by the piezoactuator associated with the flapper is considered as shown in Fig. 2. The pneumatic circuit consists of a couple of nozzle and orifice. These are symmetrically positioned with respect to the fixed flapper. The pressure control at the nozzle can be achieved by controlling the displacement of the flapper. This is possible by applying the control voltage to the piezoactuator. Figure 2 also indicates the flow direction of the valve system. The pressure at each nozzle due to the supplying pressure

Ps can be obtained as follows: [14]    2( Ps  P1 ) 2 P1 P1   C qn Aan  Q1  C qo Ao Va  f f     2( Ps  P2 ) 2 P2 P2   C qn Abn  Q2  C qo Ao Vb  f f 

(1)

where  is the bulk modulus of air, Va and Vb are control volumes at a and

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b. In this work, the load flow-rates Q1 and Q 2 are neglected, because they are much smaller than Qan , Qbn . C qn and C qo are experimental flow coefficients at the nozzle and orifice, respectively.  f is the density of the air and Ao is the area of the orifice. Aan and Abn are curtain areas at the nozzles according to the location of the flapper and are determined as follows:

Aan  ( xm  x)Dn ,

Abn  ( xm  x)Dn

(2)

In the above, D n is the diameter of the nozzle and x m is the maximum displacement of the flapper. The nonlinear equations (1) and (2) can be linearized about the equilibrium point x  0 as follows: [15]

P  C1 P  2C 2 x where,

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(3)

Juncheol Jeon and Seung-Bok Choi

34

P  P1  P2 ,

C1 

 

C qo Ao

V  2  f ( Ps  Po ) 



C qnDn xm  , 2  f Po 

C 2  C qnDn

2 Po

f

(4)

However, in practice, it is very difficult to keep the supplying pressure Ps constant even though the filter, regulator and lubricator (FRL) are normally employed. Consequently, in this work this is treated as the uncertainty in pressure control as follows:

Ps  Ps ,n  Ps  Ps ,n (1   1 ),

 1  1  1

(5)

l

r

l2

l1

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y( r ,t ) Fb

F2

Mb F1 Figure 3. Mathematical model of the flapper.

In this piezovalve system, the flapper shown in Fig. 3 (upper) is actuated by the piezostack actuator located in one side of the flapper. In order to derive the dynamic model of the flapper, the free-body diagram of the flapper shown in Fig. 3 (below) is used. The kinetic energy T , potential energy V and virtual work W are obtained as follows: 2

2

l

2T   0

 y     dr , 2V   t 

l

0

 2 y  EI  2  dr ,  r 

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Pressure Control of Vehicle ABS …

35

W  F1 (r  l1 ) y  F2 (r  l 2 ) y

(6)

In the equations above,  and EI are the mass per unit length and the flexural rigidity of the flapper. The force F1 produced from the piezoactuator by applying voltage V (t ) and the force F2 due to the flow-rate at the nozzle are calculated by F1   f p   AT  Acd 33 E  AcS  Acd 33 F2 

Dn2 4

n Ac V (t )   (l1 )   V (t )  k p (l1 )  f a  k p (l1 ) l pzt l pzt

2 ( P2  P1 )  4C qn [ P2 ( xm  x ) 2  P1 ( xm  x ) 2 ]

(7) where f p is the actuating force by the piezoactuator, A is the area of the piezodisk, l pzt is the length of the piezoactuator, V (t ) is the applying voltage,

k p  Ac / l pzt is the spring coefficient of the piezoactuator, f a ( V ) is the

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force generated from the piezoactuator and

 ( Acd33n / l pzt ) is a

proportional gain of the piezoactuator. However, the relationship between actuating force and input voltage is not linear in practice due to the inherent hysteresis behavior of the piezoactuator. In other words, the  may vary in limited range. Therefore, in this work the gain  is treated as the uncertainty in pressure control as follows :

   n     n (1   2 ),

 2  2  1

(8)

In equations (5) and (8),  n and Ps , n are the nominal values of the moment constant and the supplying pressure, respectively, under the condition of all system parameters known.  and Ps are corresponding possible deviations. The variation limits of the uncertainties  and Ps need to be known in the synthesis of the controller by tuning factor;

 i . In equation (7),

 (l1 ) is the displacement of the flapper at l1 which can be obtained by moment equation. The moment equation of the flapper is given by

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Juncheol Jeon and Seung-Bok Choi

36

 F1 (l1  x )  F2 (l2  x ) M ( x)    F2 (l2  x )

(0  x  l1 )

(9)

(l1  x  l2 )

Thus, the displacement of the flapper at x  l1 is obtained as follows:

 (l1 ) 

1 1 1 1 3 3 2  F2l1  F2l1 l2  F1l1  EI  6 2 3 

(10)

Now, the governing equation and the boundary conditions for the flapper are obtained from Hamilton’s principle as follows:

EI EI

4 y r 4 2 y r 2



2 y t 2

 F1 ( r  l1 )  F2  ( r  l 2 )

 M b  F2l2  F1l1 r 0

y r 0  0 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

EI

EI

y r

2 y r 2

0

(11) EI

r 0

 F2l2 r  l1

EI

3 y r 3 3 y r 3

 Fb  F1  F2 r 0

  F2 r  l1

(12) In this work, approximate model of the flapper is employed for controller design by using the assumed mode method. Accordingly, the displacement y(r , t ) of the flapper can be expressed as follows: n

y ( r , t )    i ( r ) q i (t )

(13)

i 1

where

 i (r ) is the mode shape and q i (t ) is the modal coordinate. The

decoupled modal equation can be written as

qi  2 i  i q i   i2 q i  where

1 f i , f i  i (l1 ) F1  i (l 2 ) F2 M ei

(14)

 i is the damping coefficient, i is the natural frequency and M ei is

the equivalence mass. These are expressed as follows:

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Pressure Control of Vehicle ABS … l

M ei   0  ( i ) 2 dr  i2 

37

1  l   2   EI  i  dr M ei  0   

(15)

In this work, it is designed so that the first mode natural frequency of the flapper (114 Hz) is much higher than the dynamic bandwidth of the valve system (10 Hz). By designing the flapper like this, the first mode can be considered to control the pressure at the nozzle. Thus, the displacement of the flapper at r  l 2 is given by

x(t )  y (l 2 , t )  1 (l 2 )qi (t )

(16)

The governing equation of the piezoactuator flapper given by equation (14) can be rewritten using the variable x(t ) as follows:

x  211 x  12 x  Cv1V (t )  C f F2  Cv 2

(17)

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where Cv1 

  1 (l1 )  1 (l2 ) M ei

,

Cv 2 

k p  1 (l1 )  1 (l2 )  (l1 ) M ei

,

Cf 

1 (l2 ) 2 (18) M ei

Now, equations (3) and (15) can be combined to yield the control model of the proposed valve system as follows:   (2  C   2 )P   2C P  2C C V (t )  2C C F  2C C P  (211  C1 )P 1 1 1 1 1 1 2 v1 2 f 2 2 v2

(19)

In this work, it is, of course, designed so that controlled valve system is stable which has one real pole and two complex poles. Since the complex poles are much dominant to operate the valve system in a sinusoidal motion, the real pole is assigned to be located far away from the imaginary axis. Then the system (19) can be reduced as follows:   2   P   2  P  2C 2 C V (t )  C F  C  P 1 1 1 v1 f 2 v2 C1

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(20)

Juncheol Jeon and Seung-Bok Choi

38

Table 1. System parameters of the piezovalve system Component

Air Circuit

Specification Bulk Modulus of Air

Symbol Value  105

Flow Coefficient at the Orifice

C qo

0.6

Flow Coefficient at the Nozzle

C qn

0.6

Area of the Orifice

Ao

0.03mm2

Density of the Air

f

1.23kg/m3

Maximum Displacement of the Flapper

xm

2mm

Control Volume of the Air Circuit

V Ps ,n

100mm3

Nominal Value of the Supplying Pressure

Tuning Factor for Uncertainty of Supplying 1 Pressure

Dn

0.5 mm

Spring Coefficient

Kp

106 N/m

n

3.2

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Tuning Factor for Piezoactuator Uncertainty 1st Natural Frequency st

1 Damping Coefficient

Modulator

0.2

Diameter of Nozzle

Piezoactuator Nominal Value of Gain of Piezoactuator

Flapper

8 bar

Flexural Rigidity of the Flapper

2 1 1

0.2 114 Hz 0.058

410 5 Nm2

Position of Nozzle

EI l2

80 mm

Position of Piezoactuator

l1

3 mm

Large Area of Piston Rod

A1

6.2 Cm2

Small Area of Piston Rod

A2

0.8 Cm2

It is generally known that the pressure of the vehicle ABS needs to be controllable from 100 bar to 180 bar [16]. However, by using the proposed piezovalve with the parameters represented in Table 1, it generates about ±5 bar (refer to Fig. 6) which is not enough pressure for vehicle ABS. Therefore, in order to obtain the required pressure for controlling the ABS, a pressure modulator is introduced and integrated with the piezovalve. The pressure modulator plays two kinds of important roles: increasing the mean value of

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Pressure Control of Vehicle ABS …

39

pressure level by supplying the initial braking pressure generated by master cylinder, and amplifying the controllable pressure range. When the initial supplied pressure by the master cylinder is 140 bar which satisfies the output pressure condition of vehicle ABS, the controllable range of the braking pressure should be ±40 bar. In order to achieve this, the pressure modulator is designed as shown in Fig. 4. In the configuration of the pressure modulator, the modulator consists of a dual-type cylinder filled with different substance (fluid and gas) and a piston rod moving vertical axis to transmit the pressure. By ignoring the frictional force and mass of the piston rod, the force equilibrium equation is written as follows:

P1 A1  P2 ( A1  Arod )  PB ( A2  Arod )  PB A2  0 *

(21)

*

where PB presents the output pressure and PB is the pressure supplied by *

master cylinder. The PB makes the output pressure level high. A1 , A2 and

Arod are the area of the large piston, the small piston and the piston rod, respectively. If A1 and A2 are much larger than Arod , equation (21) can be Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

rewritten by

PB 

A1 A2

P  PB

*

(22)

It is obvious from the configuration that the controllable output pressure from the piezostack-based valve can be amplified from the feature of different *

area between each piston. Assuming the supplied pressure PB is constant, the *

differential value of PB can be neglected. Thus, substituting equation (22) into equation (20), the governing equation of the proposed piezovalve system is derived by

  2  P   2 P  2C 2 A1 C V (t )  C F  C    2 P * (23) P B 1 1 B 1 B v1 f 2 v2 1 B C1 A2

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40

Juncheol Jeon and Seung-Bok Choi

Figure 4. Configuration of the pressure modulator.

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PRESSURE TRACKING CONTROL In this section, pressure controllability of the piezovalve system is evaluated in time domain. In order to this, a sliding mode controller (SMC) is designed to guarantee robust pressure tracking control performance against the imposed uncertainties: the hysteresis behaviour of the piezoactuator and the variation of the supplying pressure. The control issue is to enforce the pressure output PB to the desired pressure trajectories by activating the piezoactuator. Thus, the pressure tracking error is firstly defined as follows: * ev1  PB  PB  Pd ev 2  PB  Pd

(24)

where, Pd represents the desired pressure to be tracked. The first step to formulate the SMC is to establish a sliding surface that guarantees stable sliding mode motion on the surface. Since there is only one control input, a single sliding surface is defined for the system (23) as follows:

Sv  g v  ev1  ev 2 ,

Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(25)

Pressure Control of Vehicle ABS …

41

where g v is the slope of the sliding surface. It is known that a sliding mode exists on the sliding surface whenever the distance to the surface and the velocity of its change are of opposite sign. Thus, the condition for existence of the sliding mode motion is given by [17]

Sv  Sv  0

(26)

Now, from the sliding mode condition, the following SMC is obtained by V (t )  

C1 A2  2A C 2A C 2 2 *    K sgn(S )  g v ev 2  211 PB  1 PB  1 2 C f F2  1 2 Cv 2  1 PB  P d  v 2C2Cv1 A1  A2C1 A2C1 

(27)

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where K v is the discontinuous gain of the SMC. The controller (27) has been designed by assuming that the uncertainties do not exist in the control model (23). However, as mentioned earlier, the uncertainties originated from the supplying pressure and the hysteresis of the piezoactuator should be considered. Now, substituting equations (5) and (8) into equation (23) yields the following control model which contains the imposed uncertainties:   2  P   2 P  2C 2 A1 (1   ) C (1   )V (t )  C F  C    2 P * (28) P B 1 1 B 1 B 2 v1 1 f 2 v2 1 B C1 A2

Using the same procedure to satisfy the sliding mode condition, (26), the following SMC is designed as follows: V (t )   

 C1 A2 1 2A C 2  g v ev 2  211 PB  1 PB  1 2 (1   2 ) C f F2 2C 2 C v1 A1 (1  1 )(1   2 )  A2 C1

(29)

2 A1C 2 2 *   K  sgn( S ) (1   2 ) C v 2  1 PB  P d v A2 C1 

The system parameters employed in this work are listed in Table 1 and control parameters used in the realization of the sliding mode controller are as follows : g v  1800 , K v  6  10 3 . The proposed control system is realized by using the the MatlabTM Simulink. The control structure for the pressure

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Juncheol Jeon and Seung-Bok Choi

42

tracking control is shown in Fig. 5. Prior to demonstrating controllable pressure range of the piezovalve system, the piezovalve system without the modulator is compared to the piezovalve system with the modulator in Fig. 6. It is seen that the pressure output of the piezovalve system without the pressure modulator can be controlled in the range of ±5 bar, while that of the piezovalve system with the pressure modulator can be controlled in the range of ±40 bar. It is clear that the pressure modulator amplifies the pressure range about 8 times.

Figure 5. Pressure control structure of the proposed piezovalve system.

Pressure (bar)

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60

with modulator without modulator

40 20 0 -20 -40 0.0

0.2

0.4

0.6

0.8

1.0

Time (s) Figure 6. Pressure output from the piezovalve system.

Figure 7 presents the sinusoidal pressure tracking control results of the proposed piezovalve system with 140 bar of initial supplied pressure. As

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Pressure Control of Vehicle ABS …

43

mentioned earlier, the pressure level required for vehicle ABS is in the range of 100-180 bar. desired,

Pressure (bar)

180

actual

160 140 120 100 0.0

0.2

0.4

0.6

0.8

1.0

Time (s) (a) Tracking result 2.0

Error (bar)

1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0.2

0.4

0.6

0.8

1.0

Time (s) (b) Tracking error 300

Control input (v)

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

-2.0 0.0

200 100 0 -100 -200 -300 0.0

0.2

0.4

0.6

0.8

1.0

Time (s) (c) Control input voltage

Figure 7. Sine pressure tracking control with the initial pressure of 140 bar.

Since the controllable range of the proposed piezovalve system is 40 bar, the initial supplied pressure is chosen by 140 bar to satisfy the required pressure level for vehicle ABS. In Fig. 7(a), the desired pressure is set at

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44

Juncheol Jeon and Seung-Bok Choi

Pd  140  40 sin(20 t ) bar, because conventional ABS actuated by the solenoid valve is operated around 100~180 bar of pressure and 10Hz frequency. It is clearly observed from Fig.7 that the desired pressure is well tracked by the proposed system. The tracking error shown in Fig. 7(b) is less than ±2 bar. It is also noticed from Fig.7(c) that control input to the piezostack actuator is less than 300V which is suitable for most of piezostack actuators. It is expected from the results that the proposed valve system can be effectively applicable to vehicle ABS.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

WHEEL SLIP CONTROL FOR VEHICLE ABS

Figure 8. Vehicle ABS integrated with the piezovalve system.

Fig. 8 presents the schematic configuration of vehicle ABS integrated with the proposed piezovalve system. When a driver presses the brake pedal, master cylinder generates the initial braking pressure. Because the initial pressure does not satisfy the anti-lock condition, the piezovalve system generates the additive/subtractive pressure for ABS. In other words, the proposed piezovalve system does not supply the whole of required braking pressure but supplies

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Pressure Control of Vehicle ABS …

45

only a changing amount of the additive/subtractive pressure on the basis of the braking pressure from the master cylinder. It makes the change of braking pressure fast and continuous. In order to achieve this goal, a quarter vehicle is adopted and wheel slip model is established. In order to obtain the dynamic equations of the vehicle, several assumptions should be made. Firstly, the lateral and vertical motions are neglected. Only longitudinal vehicle dynamics are considered. Secondary, rolling resistance force is ignored, because it is very small due to braking. Lastly, there is no interaction between the four wheels of the vehicle [18]. Then, using Newton’s second law, the equation of the simplified vehicle motion can be expressed as follows:

mV   Ft  Fa

(31)

where m and V are the mass and velocity of the vehicle, respectively. Ft is the road frictional force and Fa is the aerodynamic force which depends on the shape, size, and instantaneous linear velocity of the vehicle and this Fa can be

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

expressed as follows

Fa 

1   air  Cd A f V 2   4 2 

In the above,

(32)

 air is the density of the air, C d is the aerodynamic

resistance coefficient, A f is the frontal area of the vehicle. Ft , the road frictional force which can be given by the Coulomb law as follows:

Ft   ( ,V ) FN

(33)

where FN is the total normal load and the  ( , V ) is the road frictional coefficient which is a nonlinear function of some physical variable, including the vehicle velocity V and wheel slip  . Fig. 9 shows this relationship between friction coefficient and wheel slip at various velocities varying 6 to 30 m/s on dry road surface. The peak values of the curves are used as the reference wheel slip values in the proposed control system.

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Juncheol Jeon and Seung-Bok Choi

Frictional Coefficient ()

46

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

30m/s

6m/s

0.2

0.4

0.6

0.8

1.0

Wheel Slip ()

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Figure 9.

 

relation for several velocity values.

The braking effect is owing to the frictional coefficient between tire and road surface. And, the frictional coefficient varies with the condition of the road surface as a function of wheel slip and the condition of the vehicle as a velocity. During deceleration, the wheel slip  and the differential equation of wheel slip  are defined by

  

V x  Vw



Vx  Rw

Vx Vx (Vx  Rw )Vx  (Vx  Rw )Vx

Vx 2



1 Vx

(1   )V

x

  Rw 

(34)

where Vx is the vehicle speed in the axial direction, Vw is the wheel speed,

Rw is the wheel radius and  is angular wheel velocity. When the wheel slip condition is zero, the wheel and vehicle velocity are same. On the contrary, when the wheel slip condition is one, the tire is not rotating and the wheels are skidding on the road surface, i.e., the vehicle is no longer steerable. The road adhesion coefficient is a nonlinear function of some physical variables,

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Pressure Control of Vehicle ABS …

47

including the velocity of the vehicle and wheel slip. Thus, wheel angular velocity and acceleration are obtained as follows:



V Rw

 (1   )

Vx Rw

Vx

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V   (1   )   x Rw Rw

(35)

Figure 10. Wheel rotational model.

During deceleration, a braking torque is applied to the wheels. It causes wheel and vehicle speeds to decrease. The rolling resistance force of the wheel is much smaller than the frictional force between the wheel and road, and hence, it can be neglected. In addition, the variables of TD , TB and Ft represent the driving torque, the braking torque and the frictional force of the wheel, respectively. From the wheel rotational model shown in Fig. 10, the following equations are obtained by using Newton’s second law.

I w r  Ft Rw  TD  TB where

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(36)

Juncheol Jeon and Seung-Bok Choi

48

TB  Awc m BF rr P B

(37)

I w is the moment of inertia of the wheel, rr is the mean effective radius of the rotor, Awc is the piston area of the wheel cylinder,  m is the mechanical efficiency and BF is the brake factor. By neglecting the driving torque TD during the braking operation, the following dynamic models for one wheel featuring the piezovalve system with uncertainty are obtained:

  a11  a12 PB  f1

(38)

  a P  a P  b V (t )  f P B 21 B 22 B 2 2 where

a11  

Vx , Vx

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a 21  211 , f2  

a12  a 22  1 , 2

R Awc m B F rr , Vx I w b2 

f1 

2 Vx Rw Ft  Vx Vx I w

2C 2 A1C v1 (1   1 )(1   2 ), C1 A2

2C 2 A1 2 * (1   2 ) C f F2  C v 2   1 PB C1 A2 (39)

Assuming that the acceleration of the vehicle Vx and the frictional force of rear wheel

Fxw

are constant during the wheel slip control,

a11 , a12 , f1 , a11 , a12 and f1 will be obtained by a11  a11 ,

a12  a12 a11 ,

f1  f1a11

a11  2a11 ,

a12  2a12 a11 ,

f  2 f a 2 1 1 11

2

3

2

(40)

The control issue is to obtain a desired wheel slip by operating the proposed piezovlave system, Thus, the tracking error is defined as follows:

e1    d ,

e 2    d ,

e 3    d

Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

(41)

Pressure Control of Vehicle ABS …

49

where  d is the desired wheel slip which has generally a maximum frictional coefficient. From equations (38), (39) and (40), the time derivatives of the error can be rewritten as follows:

e 1  e 2 e 2  e 3 (42)

3 2 e 3  2 a11   2 a11   a11  ( 2 a12 a11  a12 a 22 ) P 2   ( 2 a12 a11  a12 a 21 ) P  a12b2V (t )  2 a11 f1  a12 f 2   d

The sliding surface which guarantees stable sliding mode motion is chosen as follows:

S a  g1e1  g 2e 2  e 3

(43)

where g1 and g 2 are the gradient of the sliding surface. In order to guarantee that the tracking variables e1 , e 2 and e 3 of the system are

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constrained to the sliding surface during the sliding motion, the following sliding mode condition should be satisfied.

S a  Sa  0

(44)

From equations of (42) and (44), the SMC is designed as follows: V (t )  

1 a12b2

*

g e

1 2

3 2  g 2 e 3  2 a11   2 a11   a11  a12 ( 2 a11  a 22 ) P



(45)

2 *   K sgn(s )  a12 ( 2 a11  a21 ) P  2 a11 f1  a12 f 2   d a

where

b2  *

f2

*

2C 2 A1C v1 (1  1 )(1   2 ), C1 A2





2C 2 A1 2 *  (1   2 ) C f F2  C v 2  1 PB C1 A2

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(46)

Juncheol Jeon and Seung-Bok Choi

50

K a is the control gain. It is not desirable to use the discontinuous control law due to the chattering. Therefore, the discontinuous control law is approximated by a continuous one inside the boundary layer width. This can be accomplished by replacing the signum function, sgn(S ) , in equation (45) with the saturation function. It can be verified that the control system (42) with the SMC (45) satisfies the sliding mode condition (44) as follows : SS   g1e 2  g 2 e 3 S 



(1   1 )(1   2 ) 3 2 g1e 2  g 2 e 3 S  2 a11   2 a11   a11  ( 2 a12 a11  a12 a 22 ) P (47) (1  1 )(1   2 )





2   (1   1 )(1   2 ) 2 a 3  2 a 2   a   ( 2 a12 a11  a12 a 21 ) P  2 a11 f1  a12 f 2   d 11 11 11 (1  1 )(1   2 )



 S  K S  a12 ( 2 a11  a 22 ) P  a12 ( 2 a11  a 21 ) P  2 a11 f1  a12 f 2   d a 2

*

0

Computer simulations are performed in order to demonstrate the effectiveness of the proposed valve system for vehicle ABS. The structure of the proposed wheel slip control system associated with the proposed piezvalve system is shown in Fig. 11. Table 2 shows system specifications of the vehicle used in the simulation work and control parameters used in the realization of the sliding mode controller are g1  10 , g 2  10 , K a  41 10 4 . The Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

4

*

initial supplied pressure PB is pressurized to 140 bar constantly. According to Fig. 9, when value of the wheel slip is 0.2, most of the frictional coefficient has the highest value. Therefore, in this work, the desired wheel slip  d is set by 0.2 and 0.2  0.05 sin(10 t ) . If the braking pressure is constant (uncontrolled system) at 140 bar, the vehicle wheel will be locked (   1 ). When the wheel is locked, the vehicle is no longer steerable and has smaller frictional coefficient (frictional force) than the system which has 0.2 of wheel slip condition. It means that the velocity of the controlled system is more quickly decreased than the uncontrolled system. In other words, the braking distance can be reduced by using the slip rate controller. Fig. 12(a) and Fig. 13(a) present the wheel slip of the wheel when

 d is 0.2 and

  0.2  sin(10 t ) , respectively. It is clearly evident that the desired wheel slip is well tracked by the proposed piezovalve system. Fig 12(b) and Fig. 13(b) present the velocity of the vehicle and the wheel. Fig. 12(c) and Fig. 13(c) present the braking pressure at each condition. It is seen from these results that the braking pressure is well controlled with appropriate vehicle and

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Pressure Control of Vehicle ABS …

51

wheel velocities. It is also observed from Fig. 12(d) and Fig. 13(d) that control input for the piezoactuator is less than 300V which is suitable for practical realization of piezostack actuators.

Figure 11. Control structure for the wheel slip rate of the vehicle ABS.

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Table 2. System parameters of the vehicle Specification

Symbol

Value

Density of the Air

 air

1.23 kg/m3

Aerodynamic Resistance Coefficient

Cd

0.54

Frontal Area of the Vehicle

Af

2.04 m2

Total Normal Load

FN

4900 N

Wheel Radius

Rw

0.36 m

Moment of Inertia of the Wheel

Iw

1.13 kg·m2

Piston Area of the Wheel Cylinder

Awc

0.003 m2

Mean Effective Radius of the Rotor

rr

0.13 m

Mechanical Efficiency

m

0.8

Brake Factor

BF

0.6

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Juncheol Jeon and Seung-Bok Choi

52

0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

Vehicle Wheel

30

Velocity (m/s)

Slip Rate

35

Desired Actual

0.25

25 20 15 10 5 0 0

5

1

2

Time (s)

(a) Wheel slip Pressure (Bar)

4

5

(b) Velocity of vehicle and wheel

200

300 200

Voltage (V)

150 100 50 0 0

3

Time (s)

1

2

3

4

100 0 -100 -200 -300 0

5

1

2

Time (s)

3

4

5

Time (s)

(c) Braking pressure

(d) Control input voltage

Velocity (m/s)

0.25

Slip Rate

35

Desired Actual

0.30

0.20 0.15 0.10 0.05 0.00 0.0

0.5

1.0

1.5

Vehicle Wheel

30 25 20 15 10 5 0 0

2.0

1

Time (s)

(a) Wheel slip

3

4

5

(b) Velocity of vehicle and wheel 300 200

Voltage (V)

150 100 50 0 0

2

Time (s)

200

Pressure (Bar)

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

Figure 12. Braking performances: constant slip (  d =0.2).

1

2

3

4

5

100 0 -100 -200 -300 0

1

Time (s)

(c) Braking pressure

Figure 13. Braking performances: variable slip (  d

2

3

4

5

Time (s)

(d) Control input voltage

 0.2  0.05 sin(10 t ) ).

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Pressure Control of Vehicle ABS …

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CONCLUSION In this work, a new piezovalve system which can be applicable to vehicle ABS was devised and its feasibility for practical application was investigated showing wheel slip control performance. After formulating the piezovalve system subjected to uncertainties such as the variation of the supply pressure, a sliding mode controller was designed to demonstrate the pressure tracking controllability. It has been shown that desired pressure could be achieved by operating the piezovalve system with high accuracy and enough pressure level required for vehicle ABS. Subsequently, the dynamic model for wheel slip control was derived and a sliding mode controller was designed to achieve desired wheel slip. It has been demonstrated that two different types of desired wheel slip : constant wheel slip and sinusoidal wheel slip are accurately achieved by operating the proposed piezovalve system. The control results shown in this work are self-explanatory justifying that high performance of vehicle ABS could be devised and realized by adopting various types of piezoactuator-based valve system.

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REFERENCES [1]

Whittle, M.; Firoozian, R.; Bullough, W.A. J Intell M ater Systems Structs. 1994, 5, 105-111. [2] Choi, S.B.; Park, D.W.; Cho, M.S. Mechatronics. 2001, 11(2), 157-181. [3] Choi, S.B.; Cho, M.S. Int J Vehicle Design. 2005, 38, 196–209. [4] Choi, S.B.; Lee, T.H.; Lee, Y.S.; Han, M.S. Smart Mater Struct. 2005, 14, 1483–1492. [5] Chonan, S.; Tani, Z.W.; Tanim, J.; Orikasa, S.; Tanahashi, Y.; Takagi, T.; Tanaka, M.; Tanikawa, J. Smart Mater Structs. 1997, 6, 410-414. [6] Kohl, M.; Skrobanek, K.D.; Miyazaki, S. Sensors and Actuators. 1999, 72, 243-250. [7] Pemble, C.M.; Towe, B.C. Sensors and Actuators A. 1999, 77, 145-148. [8] Matthew, E.P.; Bruce, C.T. Sensors and Actuators A. 2006, 128, 344349. [9] Dong, S.; Du, X.H.; Bouchilloux, P.; Uchino, K. Journal of Electroceramics. 2002, 8, 155-161. [10] Lee, D.G.; Or, S.W.; Carman, G.P. J Intell M ater Systems Structs. 2004, 15, 107-115.

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54

Juncheol Jeon and Seung-Bok Choi

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[11] Meada, R.; Tsaur, J.J.; Lee, S.H.; Ichiki, M. Journal of Electroceramics. 2004, 12, 89-100. [12] Kaoru, H.; Masayoshi, E. Sensors and Actuators A: Physical. 2002, 97, 33-38. [13] Mauck, L.D.; Lynch, C.S. J Intell Mater Systems Structs. 2000, 11, 758764. [14] Watton, J. Modeling, Monitoring and Diagnostic Techniques for Fluid Power System; Springer: London, UK, 2007; Vol. 1, pp 48-65. [15] Choi, S.B.; Yoo, J.K. Proc. Instn Mech. Engrs Part C : J. Mechanical Engineering Scince. 2004, 218, 83-91. [16] Erjavec, J. Automotive Brakes; Thomson Learning, Inc.: New York, US, 2004; Vol. 1, pp 359-429. [17] Edwards, C.; Spurgeon, S. K. Sliding Mode Control Theory and Applications; Taylor & Francis: Lodon, UK, 1998; Vol. 1, pp 1-18. [18] Kayacan, E.; Oniz, Y.; Kaynak, O. IEEE Transaction on Industrial Electronics. 2009, 56(8), 3244-3252.

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In: Piezoelectric Actuators Editor: Joshua E. Segel, pp. 55-95

ISBN 978-1-61324-181-3 c 2011 Nova Science Publishers, Inc.

Chapter 3

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H IGH P OWER U LTRASONIC ACTUATORS B ASED ON THE L ANGEVIN T RANSDUCER : C LASSICAL C ONFIGURATIONS AND R ECENT D ESIGNS Antonio Iula∗ University of Basilicata

Abstract This chapter is dedicated to the ultrasonic actuators based on the Langevin transducer. The basic theory and the design criterions of the Langevin transducer alone and joined to the most popular ultrasonic concentrators are firstly presented. A recent design, which is based on a flexural displacement amplifier, is then presented and detailed analyzed.

PACS 05.45-a, 52.35.Mw, 96.50.Fm. Keywords: Ultrasonic actuators, piezoelectricity, Langevin transducer, flexural amplifier. AMS Subject Classification: 53D, 37C, 65P.

1.

The Langevin Transducer

The Langevin transducer is the basic component of the great part of the high power ultrasonic actuators based on the piezoelectric effect. Starting from Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, ∗

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56

Antonio Iula

some historical recalls, this paragraph briefly describes the working principle, the fundamental configurations and some simple design criterions of the Langevin transducer.

1.1.

Historical background

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A transducer is a device that converts one form of energy into another. In ultrasonics the most typical transducers are piezoelectric. Piezoelectricity describes the phenomenon of the generation of an electric charge in a substance, which is proportional to an applied mechanical stress and, conversely, a dimensional change proportional to an applied electric field. An essential requirement for piezoelectric interaction is that certain axes of the material possess polarity. The piezoelectric effect was discovered in 1880 by Pierre and Jacques Curie and it was not used in practical transducers until the First World War, when Paul Langevin tried to apply crystal quartz as a means of locating German submarines. The transducer developed by Langevin was not ready before the end of the war and it was used as a depth-sounding device after the war. The Langevin transducer is basically composed of one or more pairs of piezoelectric rings sandwiched between two metal masses. It can be excited to resonate in length-extensional mode at low frequency, avoiding the need for high-driving voltages. The structure is suitable to absorb high electrical power; to increase the mechanical strength of piezoceramic elements, it is usually prestressed by inserting a bolt along its principal axis (see Figure1). In underwater reporting applications a low frequency resonator is required (10÷100 kHz), due to the specific characteristics of water as a medium of acoustic wave propagation. In fact, the water introduces attenuation losses proportional to the square of the frequency of the wave that propagates. As the working frequency is inversely proportional to the length of the transducer, the final trend was to use a transducer sufficiently long to ensure resonances below 100 kHz. At the beginning, the only piezoelectric material used was the quartz; to realize transducers only with the active material was an impractical idea because the crystal was very expensive and, furthermore, the transducer had a very high electrical input impedance. Paul Langevin realized the transducer using thin layers of quartz as the active material placed between two masses of steel, achieving the double benefit of lowering the resonant frequency of the whole structure and saving the active material. This configuration is known as Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

High Power Ultrasonic Actuators Based on the Langevin Transducer 57

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Figure 1. Elements of a two ceramics transducer (a) and assembled transducer (b). a sandwich structure. With the first use of piezoelectric ceramics in the field of low-frequency applications, the transducers were simple composed of a rectangular or cylinder sample of piezoelectric material, given the low cost of barium titanate ceramics compared to quartz (see Figure 2). However, this simple structure was inherently very fragile, as the tensile strength of ceramics is very low. Due to these constraints the project of the Langevin sandwich was reconsidered and difficulties were overcome through simple pre-stress processes. The two basic structures of a Langevin ultrasonic transducer are summarized in Figure 3 and 1. The transducer of Figure 3 is realized with a single ceramic; the bolt-tensioning must be insulated in order to avoid contact between the two masses, which would create a short-circuit. For the same reason an insulating layer is placed between the piezoceramic ring and one of the masses, requiring for a significant constraint on the maximum prestress force applied to the structure itself. This disadvantage is overcome by using a pair of ceramics polarized in the opposite direction (see Figure 1a). The hot electrode is positioned between the piezoelectric rings and the masses are both at zero potential. The primary of this sandwich structure can Ebook be summarized as follows: Piezoelectricadvantages Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Central,

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Figure 2. First transducers composed of only active material; cylinder shape (a) parallepiped shape (b). 1. The possibility of using ceramics with reduced thickness involves: (a) a reduction in production costs, (b) a high capacity and low impedance of the piezoceramic element, both favorable characteristics at low frequencies where these transducers typically operate; 2. The prestress of the structure increases the mechanical efficiency of the transducer. In fact, the performances of the sandwich structure are limited by the tensile strength 1 T of the ceramic; the application of a static pre-stress S, opposed to pulling force, increases the resistance to T + S. 3. The use of metallic masses is extremely advantageous because: (a) by properly matching their surfaces to the ceramics, an effective heat dissipation is achieved, allowing the device to be driven with very high power; (b) other metallic elements (ultrasonic concentrators) can be easily added. 1 It is defined as the maximum stress that the material can withstand without damage to its lattice structure

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High Power Ultrasonic Actuators Based on the Langevin Transducer 59

Figure 3. Samples of first low frequency ultrasonic transducers (a) and assembled transducer (b).

1.2.

Design criterions

Let us consider the symmetrical Langevin transducer of Figure 1b, schematically represented in Figure 4. The vibrational behaviour of this structure is rather complicate because at least two vibration modes, the thickness-extensional mode and the radial mode, should be taken into account for each cylinder shaped piezoceramic or metal element [1], [2]. Analytical multidimensional modeling of piezoceramic structures is rather complex due to the unsolvable differential coupled equa-

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tions system, which describes the element vibration [3]-[5]. Some attempts in this direction have been made [6], [7], but today such structures are mainly simulated by means of the finite element approach [8]-[11]. The Langevin structure can be analyzed with the classical one-dimensional theory [12]; this approach is able to describe only the thickness-extensional modes and therefore does not take into account the unavoidable lateral vibrations of both the piezoelectric ceramic and the loading masses. This theory can be applied whenever the length of the transducer is much higher than its diameter. Under this hypothesis, assuming that the transducer resonates in its fundamental length extensional mode, the total length of the transducer is half wavelength (λ/2); consequently, the nodal plane of the displacement divides the structure in two equal parts, each of length λ/4. Due to the symmetry we can study only one half.

Figure 4. Schematic representation of the sandwich structure with two ceramics; the distribution of the longitudinal displacement along the axis direction is also shown. The analysis is carried out under the hypothesis that no load is applied to the transversal sections of the metal masses. Assuming that a plane wave propagates along the z direction, applying the resonance conditions at the ends A and C, and imposing the continuity of force and velocity at the interface B between piezoceramic and metal, we obtain: tan θc tan θl =

Zc = R Zl

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(1)

High Power Ultrasonic Actuators Based on the Langevin Transducer 61 being

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θc =

ωc ωa , θl = , Zc = ρc vc Ac , Zl = ρl vl Al vc vl

(2)

where θ is the phase length, v is the propagation velocity of the acoustic wave, ρ is the mass density, Z is the acoustic impedance and Ac and Al are the areas of the sections of the ceramic and the masses, respectively. Equation 1 relates the resonance frequency of the transducer ω with the physical dimensions a and c and with the impedance ratio R. Equation 1 also shows that different dimensions of the elements and combinations of materials may lead to the same resonant frequency. This behaviour enables designers to optimize the structure for different design specifications; for example, the electrical impedance can be modified by opportunely choosing the thickness of the piezoceramic rings. After choosing the material, the resonance frequency and two of the three characteristic dimensions of the transducer, Equation 1 allows to determine the unknown dimension. In analysis problems, i.e., when the transducer has a given geometry, it is not possible to achieve the resonance frequency of the structure from Equation 1 in an analytic form. To this end the universal design curves shown in Figure 5 can be exploited. These curves are obtained by Equation 1 and provide the relations between the phase lengths of the two materials at several values of the impedance ratio R. The resonance frequency of the transducer can be obtained by plotting a straight line with slope θl avc = . θc cvl

(3)

The coordinates of the point of intersection of this line with the curve with the imposed impedance ratio R allow to compute the resonance frequency by means of Equations 1 and 2.

2.

Ultrasonic Concentrators

This paragraph presents the fundamental elements of the theory of the ultrasonic concentrators, calculating the most important parameters of some popular profiles.

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Figure 5. Universal design curves: graphic solution of the Langevin equation.

2.1.

Theory

The Langevin type transducers can generate movement at their ends, characterized by the parameters of force and velocity, which is often not sufficient in the current uses as actuators. In this case it is necessary to add to the transducer a supplementary mechanical structure, called Ultrasonic Concentrator or Ultrasonic Horn. Basically, an ultrasonic concentrator is a resonant structure that is able to provide an amplification between output and input displacements. This displacement ampification can be achieved by decreasing its cross section with several different profile. Most commonly used ultrasonic concentrators have a longitudinal profile of type [13]-[15]: • conical, • exponential, • catenoidal.2 2 In

physics and geometry, the catenary is the curve that an idealized hanging chain or cable assumes when supported at its ends and acted on only by its own weight; assuming the y axis coincident with the axis of symmetry and the x axis at a distance c from the vertex, the Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

High Power Ultrasonic Actuators Based on the Langevin Transducer 63

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Other profiles that promise improved performances are currently under investigation [16]. In the following we analyze the classical profiles through a study of the longitudinal vibrations of an element with variable cross-section, assuming a uniform distribution of stress on the generic section of the transformer and that a plane wave propagates in the material. This hypothesis is verified when the longitudinal dimension of the concentrator is much greater than the diameter of its cross section. With reference to Figure 6, only the Tzz component of the

Figure 6. Generic structure of an ultrasonic concentrator. stress tensor is not zero, and the equation that describes the displacement of the structure under investigation is: ρ · S (z) · dz

∂2 uz ∂Tzz ∂S (z) = · S (z) · dz + · Tzz · dz, 2 ∂t ∂z ∂z

characteristic equation of the catenary is: x  x  c x y = exp + exp − = c cosh 2 c c c

(5)

(4)

Consequently, the catenoid is a three-dimensional shape made by rotating a catenary curve around the x axis. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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where ρ is the mass density e S(z) is the function that describes the profile of the concentrator. Considering harmonic oscillations and exploiting the Hooke law, equation (5) become: u00 +

1 · S0 (z) · u0 + k2 u = 0, S (z)

where k2 =

ω2 ρ ω2 = 2 . c vl

(6)

(7)

In (7) ω is the angular frequency of the vibration and vl is the propagation velocity of the longitudinal wave. Note that c is the elastic constant of the material. For the displacement velocity we have an expression similar to Eq. 39, being v(z) = jω · u(z):

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v00 +

1 · S0 (z) · v0 + k2 u = 0. S (z)

(8)

Equation (8) has to be solved by imposing the following boundary conditions: • for the section of plane z2 , assuming stress free boundary condition, we have: v0 (z2 ) = 0 , (9) • for the section of plane z1 , assuming a perfect match between concentrator and driving transducer, we have: v0 (z1 ) = 0 .

(10)

When conditions (9) and (10) are satisfied, the concentator is a resonant structure and it is transparent for the driving transducer. Therefore, also relation v(z1 ) = v0 (11)

holds, where v0 is the amplitude of the velocity of the external surface of the driving transducer at the resonance frequency, in absence of the concentrator. Finally, let’s define the ratio between the radii of the external surfaces of the structure: R1 N = , (12) R2 Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, which is a characteristic parameter of all ultrasonic concentrators.

High Power Ultrasonic Actuators Based on the Langevin Transducer 65

2.2.

Conical concentrator

For concentrator with conic profile, the area of a generic section is defined by the following function S(z): S (z) = S1 (1 − αz)2 , where α=

R1 − R2 , R1 · l

(13)

(14)

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and l is the length of the concentrator.

Figure 7. Concentator with conical profile. By substituting equation (13) in (8) we obtain the equation of the velocity in the concentrator: 2 0 v00 + v + k2 v = 0. (15) 1 z− α The solution of this equation is: 1  [A1 ·Cos (k · z) + A2 · Sin (k · z)] . v (z) =  1 z− α

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(16)

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By applying the boundary conditions described in equation (10) and (11) we have the relations between the coefficients A1 e A2 :   k(1 − αl) sinkl − α cos kl A2 = A1 · k(1 − αl) coskl + α sin kl α A2 = A1 · , (17) k and hence the final equation of velocity:   v0 α v (z) = Cos [k · z] + · Sin [k · z] . (1 − z · α) k

(18)

By exploiting equation (13), from (18) we obtain the following equation:

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tan kl =

kl (kl)2N (1−N)2

+1

,

(19)

which gives the values of kl, as a function of N, for which we have longitudinal resonant modes in the concentrator. Note that the phase velocity of the conical structure is equal to the one that would be achieved with a cylinder structure. But, differently from the cylinder structure, its length is not equal to λ/2 and the high order resonance frequencies are not multiple of the fundamental one. In the limit case N → ∞ (R2 → 0), from equation (19) we obtain the following relation: tan(kl)∞ = (kl)∞ (20) and its roots are: (kl)∞ = 4.493, 7.725, 10.9 . . . The velocity amplification of the conical structure is:   vl = N cos kl − N − 1 sinkl , v Nkl 0

(21)

(22)

which clearly shows that the conical profile in not effective as a ultrasonic concentrator as vv0l < N. Furthermore, it can be easily demonstrated that, by increasing N, the velocity amplification tends to the limit value: q vl lim = 1 + (kl)2∞ . (23) N→∞ v0 From relation (21) we have that a conical concentrator at its fundamental resonance frequency provide a maximum velocity amplification of ' 4.6.

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High Power Ultrasonic Actuators Based on the Langevin Transducer 67

2.3.

Exponential concentrator

The shape of an exponential concentrator is defined by the function: S (z) = S1 · e−2·β·z ; where β=

ln(N) . L

(24)

(25)

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The equation for the velocity is:

Figure 8. Concentrator with exponential profile. v00 − 2 · β · v0 + k2 v = 0

(26)

that has the following solution:

where


v (z) = eβz A1 cos k0 z + A2 sink0 z , k0 =

p k 2 − β2 .

(27)

(28)

Relation (28) highlight that the phase velocity of longitudinal waves is a function of the attenuation factor β of the exponential profile and, furthermore, it Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, is higher than that of a structure with constant section.

68

Antonio Iula

By applying the boundary conditions (9) and (10) we obtain the expression of the displacement velocity along z:   β v = v0 eβz cos k0 z − 0 sink0 z . (29) k Similarly to the previous case we achieve the following relations: sink0 l = 0 ⇒ k0 l = nπ ⇒ l = n

λ0 , 2

(30)

which show that the length of an exponential concentrator is equal to an integer multiple of half-wavelength. The velocity amplification of an exponential concentrator is: vl = N, (31) v 0

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i.e., it depends on the ratio between the radii of the two external sections and is not dependent on the length.

2.4.

Catenoidal concentrator

For a concentrator with catenoidal profile the funtion S(z) is:

where γ =

S(z) = S2 cosh2 γ(l − z),

(32)

1 R1 1 arccos h = arccos h N. l R2 l

(33)

Substituting equations (32) and (33) in the general equation (8) we obtain the expression of the velocity: v (z) =


1 A1 cosk0 z + A2 sink0 z ch(l − z)

(34)

with k0 =

p k 2 − γ2 .

(35)

By applying boundary conditions (9) e (10) we have: v =

v0 cosh γl

cosk0 z −

γ

tanh γl sink0 z .

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0

(36)

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High Power Ultrasonic Actuators Based on the Langevin Transducer 69

Figure 9. Concentrator with catenoidal profile. From equations (36) it is possible to get the values of k0 l in resonance conditions: r 1 0 0 k l tank l = 1 − 2 arccos h N. (37) N The velocity amplification of an catenoidal concentrator is: vl = N . (38) v cosk0 l 0

Expression 38 shows that designing the concentrator in such a way that cos(k0l) is very lower than 1, very high velocity amplification can be achieved. In practical applications, the catenoidal concentrator is the most employed whenever very high velocity amplifications are required, while conical concentrators are preferred for low amplification applications, because of their simplicity of realization.

2.5.

Sectional concentrator

Ultrasonic actuators are often realized by exploiting the so called sectional concentrators; they are composed of a rod of variable cross section interposed

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between two rods with constant cross section as shown in Figure 10.

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Figure 10. General structure of a sectional concentrator. The profile of the intermediate element can be one of those described in previous sections. Sectional concentrators make it possible to obtain considerably larger amplifications factors in comparison with concentrators having simple forms (given identical base dimensions), so they are widely employed whenever large oscillation amplitudes and deformations are required. Considering still valid the plane wave approximation, the analysis can be carried out starting from equation (39), which describes displacement distribution z direction that we report here for convenience: u00 +

1 · S0 (z) · u0 + k2 u = 0. S (z)

(39)

Solutions can be written in the form: u1 (z) = A1 cos kz + B1 sinkz 0

when −l1 < z < 0 0

u2 (z) = F(z)(A2 cos k z + B2 sin k z) when 0 < z < l2 u3 (z) = A3 cos kz + B3 sinkz

(40)

when l2 < z < l2 + l3

where F(z) = 1/r(z) e r(z) indicates the radius of the generic radius of the variable cross section element. The boundary condition for the displacement u and the deformation u0 are: for z = l2 + l3

u3 = uout

u03 = 0

for z = l2

u2 = u3

u02 = u03

for z = 0

u1 = u2

u01 = u02

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for z = −l1

u1 = uin

0

cS1 u = −Fin

(41)

High Power Ultrasonic Actuators Based on the Langevin Transducer 71 where in the last of (41) c the elastic constant of the material. By solving the equation system (40) and applying the boundary conditions (41), we obtain the coefficients A1 , B1 ,A3 , B3 : A1 = uin

∆1 , ∆1 coskl1 − ∆2 sinkl1

B1 = uin

∆2 ∆1 coskl1 − ∆2 sinkl1

(42)

A3 = uin

N cosk(l2 + l3 ) , ∆1 coskl1 − ∆2 sinkl1

B3 = uin

N sink(l2 + l3 ) ∆1 coskl1 − ∆2 sinkl1

(43)

where we set: N

=

F(l2 ) R1 = F(0) R2

k sinkl3 sink0 l2 (44) k0 = sinkl3 (cos k0 l2 + δ1 sink0 l2 ) − cos kl3 (δ2 cos k0 l2 − δ3 sink0 l2 )

∆1 = cos kl3 (cosk0 l2 + δ4 sink0 l2 ) −

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∆2

and δ1 , δ2 , δ3 , δ4 only depends on the profile of the variable cross section region. The coefficients A2 , B2 are obtained from relations: A1 = F(0)a2,

kB1 = F 0 (0)A2 + F(0)k0B2 .

(45)

The equation for computing the resonance frequency of the concentrator is:

∆2 , (46) ∆1 while the amplification factor of the displacement M is given by the following relation: uout N , M= = (47) uin ∆1 cos kl1 − ∆2 sinkl1 tankl1 = −

which, at the resonance frequency, become: N M p = coskl1 . ∆1

(48)

An important parameter in the design of the concentrator is the input impedance which is defined as: Zin

= j

∆1 tankl1 + ∆2

,

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(49)

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Antonio Iula

where

ckS1 . (50) ω These expressions are quite general and can be applied for the analysis of different types of concentrators. The profile of the variable section element is chosen among those described in the previous sections. Z0 =

2.5.1.

Stepped horn

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Let us consider a particular configuration of the section concentrator: it is simply obtained by the junction of two rods with different cross section (see Figure 11).

Figure 11. Stepped horn. This structure is called stepped horn, and can be considered as a sectional concentrator with an exponential shaped intermediate element with length that tends to zero. By applying equation (46) to the exponential profile we have: δ1 = δ4 =

β kc

δ2 = 0 δ3 =

k ; kc

(51)

and when l2 tends to zero we get the expression to determine the resonance frequency of the stepped horn:

1 tankl3 . (52) N2 The expression of the amplification factor of the displacement M p at the resonance frequency in this particular case becomes: coskl1 1 +Ebook (N 4Central, − 1) sin2 kl1 ; (53) M p = Incorporated, 2011. = ProQuest Piezoelectric Actuators, Nova Science Publishers, tan kl1 = −

High Power Ultrasonic Actuators Based on the Langevin Transducer 73 it is possible to plot the design curves corresponding to relations (52) and (53), obtaining the results shown in Figure 12. As can be seen, the maximum amplification, equal to N 2 , is obtained when kl1 = kl3 = π/2.

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Figure 12. Design curves for a stepped horn.

3.

Langevin Transducer with Ultrasonic Concentrator

This paragraph illustrates the basic design criterion of an ultrasonic actuator composed of a Langevin transducer and a displacement amplifier. An overview of the main applications of such transducers is then presented.

3.1.

Design criterion in vacuum

In all applications where the longitudinal displacements generated by the Langevin transducer are not sufficient, an ultrasonic concentrator acting as a displacement amplifier is usually joined to it. The Langevin transducer therefore works as a driver for the whole actuator. In the following we illustrate the basic design criterion under the assumption that the actuator works in vacuum (which is a very good approximation also for air). In general, the actuator has to be designed to provide the maximum displacement at one of its ends. This is achieved by dimensioning the concentrator to provide its maximum displacement amplification.

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The following analysis is restricted to a Langevin transducer joined to a stepped horn, which is widely used in the practice for its good amplification and the easy of fabrication. As shown in a previous paragraph, the stepped horn has maximum amplification when the longitudinal dimension of the two cylinder shaped elements are both length equal to λ/4 (see Figure 13). In this case the displacement amplification is N 2 , where N is the ratio between the radii of the two sections. Let us assume that the working frequency is f w .

Figure 13. Langevin transducer with stepped horn.

Dimensions of L3 e L4 (see Figure 13) of the stepped horn are immediately fixed because they have to be both equal to λ/4 to achieve the maximum amplification. Then the Langevin transducer is separately designed to resonate in vacuum at the same frequency f w. Note that the end section of mass M2 of the transducer is loaded by the input acoustical impedance of the concentrator, indicated in the Figure as Zacu . As we are assuming that the actuator operates in vacuum, such impedance has only the imaginary part, which is zero at the working and resonance frequency f w . Figure 14 shows the simulated input electrical impedance (a) and the longitudinal displacement at the end of the actuator as a function of frequency. In this example the actuator was designed to work at f w =130 kHz. As can be seen the working frequency, which is the fundamental one for the Langevin transducer, corresponds to the second harmonic for the whole actuator. Figure 15 shows the profile of longitudinal displacement along the actuator computed at the working frequency, clearly highlighting the amplification function of the concentrator. It should be noted that with a correct design the nodes thePublishers, displacement are inProQuest correspondence Piezoelectrictwo Actuators, Novaof Science Incorporated, 2011. Ebook Central, of the step of the concen-

High Power Ultrasonic Actuators Based on the Langevin Transducer 75

Figure 14. Modulus of the electrical input impedance (a) and longitudinal displacement at the end of the actuator(b) vs frequency.

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trator and of the middle plane of the piezoceramic stack. This last condition guarantees the best performances of the Langevin transducer.

Figure 15. Longitudinal displacement along the transducer. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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3.2.

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Applications

The Langevin transducer has been used for several decades as a power vibration generator in a large variety of industrial and biomedical applications. As described in previous paragraphs, in its basic configuration the Langevin transducer is a narrow band high intensity pressure generator. During the years, several modifications have been made to the basic configuration depending on the specific application. So, whenever high accuracy and high resolution sonar systems are imperative needs, broadband transducers are required. In this case the transducer is usually known as tonpilz transducer or acoustic mushroom [17]-[20]. The bandwidth of the tonpilz transducer may be improved to a limited extent by reducing the thickness or increasing the front diameter of the cone head. Both of these approaches may lead to the appearance of flexural resonances of the head. The flexural vibration of the head, often referred to as head flapping, has usually been considered to have a negative influence on the performance, but some investigation has shown that it is possible to use the flexural motion to a positive effect. The transducer has also been used as an ultrasonic cleaner by exploiting the cavitation effect. Ultrasonic cleaners are nowadays widely used to clean jewellery, lenses and other optical parts, watches, dental and surgical instruments, fountain pens, industrial parts and electronic equipment [21]-[23]. In other applications there is the need to generate ultrasonic energy in air or in gases. In this case the main problems are related to the low specific acoustic impedance and high absorption of the medium. Therefore, in order to obtain an efficient ultrasonic transmission and to produce high pressure levels, it is necessary to achieve good impedance matching between the transducer and the gas, large amplitudes of vibration and highly directional beams. To this end a stepped-plate transducer has been developed in which these prerequisites have been attained. It essentially consists of an extensive circular plate of stepped shape driven at its centre by Langevin transducer with an ultrasonic concentrator [24], [25]. The Langevin transducer has been widely exploited also as a mechanical actuator, i.e., as a transducer able to convert the electrical energy into some kind of motion. When used with an ultrasonic concentrator the transducer has been employed in a wide variety of fields including surgical ophthalmic applications as an ultrasonic bistoury in cystectomy operations [26], oral implantation applications to drill holes in bone and to facilitates the insertion of

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High Power Ultrasonic Actuators Based on the Langevin Transducer 77 the implant in the created cavity [27], [28], space applications as an ultrasonic driller to allow probing and sampling of rocks for in-situ planetary analysis [29], [30]. A major application of the Langevin transducer joined to a mechanical ampifier is the ultrasonic welding, which is an industrial technique whereby high-frequency ultrasonic acoustic vibrations are locally applied to workpieces being held together under pressure to create a solid-state weld [31][33]. It is commonly used for plastics, and especially for joining dissimilar materials. In ultrasonic welding, there are no connective bolts, nails, soldering materials, or adhesives necessary to bind the materials together. The applications of ultrasonic welding are extensive and are found in many industries including electrical and computer, automotive and aerospace, medical, and packaging. Wires, microcircuit connections, sheet metal, foils, ribbons and meshes are often joined using ultrasonic welding. Ultrasonic welding is a very popular technique for bonding thermoplastics as well. In general the Langevin transducer with or without concentrators has been employed to generate high intensity mechanical vibration in a wide variety of industrial application, including ultrasonic lubrication of metal casting [34] and heatsensitive foods drying in food industry processing [35], [36]. Among them ultrasonic motors are probably the most important and popular. Ultrasonic motors have attracted the interest of a large number of investigator in the last two decades, and there are several industrial applications where they can compete with conventional electromechanical motors including high accuracy positioning, manufacturing process control, pick-and-place assembly, camera autofocus [37]. Whenever there is the need for motors able to provide high torque, the Langevin transducer is generally used as vibration generator [38][42]. Main applications of high power ultrasonic motors are in the automotive field (window lifts, windshield wipers, or seat movers) and the aerospace field (to move robot arms or rover board instrumentations in planetary environments).

4.

A New Flexural Ultrasonic Actuator

This paragraph describes a new type of ultrasonic actuator, which is able to provide higher displacements than classical ultrasonic actuators based on sectional concentrators. It is composed of a symmetrical Langevin transducer working in a length–extensional mode and of a displacement amplifier that Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, is designed to vibrate in flexural mode at the same working frequency of the

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Langevin transducer [43], [44]. The working principle of the actuator is firstly described. Then two different actuators are analyzed and experimentally evaluated. The first one is based on a very basic design and is used to compare its performance with those of a classical actuator, based on a stepped horn concentrator. The second one is designed to work at a lower frequency in order to produce a very high displacement amplification and in such a way that the whole actuator has 2 close working frequencies, with similar flexural deformations of the front surface. Such an actuator could be exploited in several applications as a multifrequency ultrasound power generator, but also as a mirror with variable bend radius to deflect laser beams in optical applications.

4.1.

Working principles

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Figure 16a schematically shows the proposed actuator: it is composed of a symmetrical Langevin transducer joined to a displacement amplifier designed to work in a flexural mode. The Langevin transducer is composed of a couple

Figure 16. a) Schematic view of the proposed actuator, b) schematic view of a classical high displacement actuator.

of piezoceramic disks with radius a and thickness tc , poled along z direction but with opposite polarities, and of two cylinder shaped steel masses, which Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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High Power Ultrasonic Actuators Based on the Langevin Transducer 79 have radius identical to that of the disks and thickness t. The displacement amplifier is composed of a cylinder shaped base with the same radius of the Langevin transducer and thickness b1 , of a number of arms (each with length b2 and section S), and of a thin disk (thickness c) acting as a displacement collector. This kind of structure is able to transform the almost flat displacement provided by the Langevin transducer into a flexural deformation of the arms. This deformation is transmitted to the collector. In this case, the maximum value of the axial displacement occurs at the center of the collector. Figure 16b shows the classical structure commonly used to generate high displacements, which is composed of a Langevin transducer and a stepped horn. As shown in previous paragraphs, the Langevin transducer and the stepped horn are designed to work at the same frequency, and the displacement amplification provided by the horn is equal to the ratio between the areas of the two sections. Also for the proposed actuator the design criterion that has been followed was to separately design the Langevin transducer and the flexural amplifier at the same working frequency. However, due to its complexity the flexural amplifier cannot be designed by using an analytical approach. Furthermore, due to the high number of variables, it is to expect that there are several configurations able to generate a flexural motion at the same frequency. Due to the complexity of the physical structure, FE method was preferred to the analytical approach to model and analyse the flexural amplifier. Nevertheless, some analytical models of the flexural vibration of rods [45], [46] and of plates [47] were exploited as a helpful aid to the FE design.

4.2.

Finite element model

FE analysis was performed by using a commercial package (ANSYS). The symmetrical Langevin transducer was modelled by imposing the continuity of the displacements both in radial and in axial directions at the interfaces between the pieoceramic disks (PZ26 by Ferroperm) and the steel masses (mass density ρ = 7800 kg/m3 , Young modulus E = 2.06 · 1011 N/m2 , Poisson ratio σ = 0.3), as well as at the interface between each couple of piezoceramic disks. Furthermore, for the piezoceramic elements, electrodes was simulated by setting all nodes belonging to each flat surface at the same electrical potential. Next step was to model the new displacement amplifier. Due to the complexity of the structure, a trade off between two main requirements was to be Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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found: a sufficiently dense and regular mesh, in order to guarantee accuracy in the results, and a relatively low number of nodes, in order to contain computational time. The structure is not axial symmetric but it is symmetric with respect to the x = 0 and y = 0 planes (see section ll’ in Figure 16a); so we exploited these symmetries and modelled only a quarter of the whole structure. In order to compare the performances of the proposed amplifier with classical amplifiers, we also modelled a stepped horn. The comparison of the performances of the amplifiers was carried out by setting the end section of the stepped horn equal to the total area 4S of the arms. A modal analysis of these structures was then performed, in order to find their natural resonance frequencies and the corresponding modal shapes; in this way the mechanical amplification of the amplifiers has been computed. For the two actuators, obtained by joining the Langevin transducer to the two amplifiers, harmonic analyses were then performed: an ideal harmonic current generator with variable frequency was applied to the piezoceramic disks (electrically connected in parallel), in order to compute the electrical input impedance and the output displacement. To compare simulation results to experiments, the electrical input impedances were also computed by driving the two actuators with an ideal voltage generator. In all harmonic analyses, internal structural losses of the transducer have been taken into account by applying an appropriate damping ratio, defined as the ratio between loss energy and kinetic energy.

4.3. 4.3.1.

Basic design Numerical analysis

A modal analysis of the Langevin transducer, used as a vibration driver for both the mechanical amplifiers, was first carried out. The radius of both masses and piezoceramic disks was a = 5 mm, the thickness of the piezoceramic tc = 2 mm, and masses’ length t = 21.5 mm. The fundamental lengthextensional resonance frequency obtained by simulation was f = 50.41 kHz. In order to model a flexural amplifier resonating at the same frequency of the Langevin transducer, four variables are to be determined (b1 , b2 , c and S) (see Figure 16a), the diameter 2a being equal to that of the driver. Another variable is the number of arms; in our analysis we fixed it to four. As mentioned previously, an analytical model able to design the whole structure is not present in literature. The design of the amplifier was carried out by exploiting an anaPiezoelectriclytical Actuators,model Nova Science ProQuest [45], Ebook Central, of Publishers, an arm Incorporated, flexural 2011. vibration [46], in order to determine the

High Power Ultrasonic Actuators Based on the Langevin Transducer 81

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length b2 and the section S of the initial solution. Then an iterative FE analysis was carried out in order to find a solution that minimize the difference between the flexural resonance frequency of the amplifier and that of the driver. Several solutions were found. Among them the configuration that exhibits best performances was: b1 = 5 mm, b2 = 8 mm, c = 1 mm and S = 3.14 mm2 . Figure 17a shows the deformed shape of the flexural mode of the amplifier, which clearly shows that the almost flat displacement at the base of the amplifier is converted in a flexural motion of the arms and, hence, of the collector. A mechanical amplification, defined as the ratio between the maximum dis-

Figure 17. a) Deformed shape of the proposed amplifier, b) deformed shape of a stepped horn amplifier.

placement of the front and the back sections, of about 9.1 was obtained. The deformed shape of a stepped horn with end section equal to 4S (a’=2 mm), designed to work at the same frequency, was also computed (see 17b). In this case a mechanical amplification of about 6.25 was obtained, in agreement with the classical one dimensional theory [14]. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, Hence, the flexural amplifier exhibits an amplification that is about 50%

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greater than the stepped horn. Performances of the whole actuator have been evaluated by joining the proposed amplifier to the symmetrical Langevin transducer; a harmonic analysis was performed by driving the actuator with an ideal current generator. In this case, the working frequency of the Langevin transducer, i.e., the frequency of maximum displacement, practically coincides with the parallel resonance frequency f p . Figure 18 shows the displacement computed at the center of the end sections of the two actuators as function of frequency. As it can be seen, at the working frequency f p , the displacement on the front section of the proposed actuator is about twice the displacement of the front section of the classical one. Furthermore, base displacement of the proposed actuator is higher than that of the classical one. The mechanical amplifications, defined as the ratio between front and back sections displacement, are in agreement with results of the modal analysis. This behaviour can be explained by observing the plot of the electrical impedances shown in Figure 19a. The two actuators are driven with the same current; being the electrical impedance at the working frequency f pF higher than that at the working frequency f pL , the proposed actuator is able to absorb a higher electrical power than the classical one. Figure 19b shows the electrical input impedance computed by driving the two actuators with an ideal voltage generator. When a Langevin transducer is driven with a voltage generator its working frequency (i.e., the frequency of maximum displacement) practically coincides with the series resonance f s . Consequently, in order to trim f s to the resonance frequency of the amplifiers, the length t of the masses (see Figure 16) was reduced from 21.5 mm to 20 mm. 4.3.2.

Experimental results

In order to experimentally validate results obtained by simulations, two actuators were manufactured by following FE design. However, as it is well known, Langevin transducers need to be prestressed; therefore, two couples of piezoceramic rings (inner diameter 5 mm) were used instead of disks in order to allow a bolt to pass through them. Prestress operation was accomplished by means of a torque wrench, in order to ensure its repeatability. A torque of 4 N·m was applied to both actuators. As in the experiments an electronic circuit that works like an almost ideal voltage generator was used to drive the transducers, Langevin transducers with masses’ length of 20 mm were realized. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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High Power Ultrasonic Actuators Based on the Langevin Transducer 83

Figure 18. Displacements computed at the center of the end sections of the two actuators as a function of frequency.

Figure 19. Modulus of the electrical input impedances computed for the two actuators as a function of frequency: a) actuators driven with a current generator, b)actuators driven with a voltage generator.

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Figure 20 shows a photo of the manufactured actuators. Figure 21 shows the electrical input impedance of the two actuators measured with a HP4194A impedance analyzer. As it can be seen, while the working frequency f sL of the classical actuator is in a quite good agreement with that predicted by simulations (see Figure 19b), for the proposed actuator the working frequency f sF is slightly higher than the simulated one (about 4%). This behaviour can be explained by observing that the right corners between arms and collector and between arms and base (see Figure 16a) are rounded off by mechanical machining (see Figure 20). Consequently, the realized actuator results more stiff than the simulated one. It can also be observed that the modulus of the impedance of the proposed actuator is lower than that of the classical actuator at the working frequencies f sF and f sL , respectively. Hence, the proposed actuator is able to absorb a higher electrical power than the classical one. In order to evaluate performances of the two actuators, a driving electronic circuit has been realized. Basically, it is composed of a power amplifier (halfbridge technology) and an LC card to match actuator electrical impedance. The input is a square wave at the working frequency of the transducer. Power stage output is a square wave with an amplitude equal to Vcc. Filtering stage provides each transducer to be driven by a sinusoidal signal. Displacement at the center of the front section of each transducer has been measured by means of an interferometer technique. The interferometer system is composed of a laser interferometer (Polytec OFV302) that is connected to a control and elaboration unit (Polytec OFV3000). Figure 22 shows the results obtained with several values of applied voltage. As can be seen, the proposed actuator exhibits a displacement about 50% greater than the classical one.

4.4. 4.4.1.

A flexural multifrequency actuator Numerical analysis

A second flexural actuator was designed following the same approach previously shown, with increased output displacement and with two close operating frequencies. It is know that, in vibrating systems, the amplitude of the displacement is inversely proportional to the working frequency. The working frequency was therefore chosen in the range 17-20 kHz, in order to have a high displacement but at the same time in order to work out of the audible range. The diameter of the amplifier was fixed to 2a= 30 mm, according to the Piezoelectricdiameter Actuators, Nova Publishers, Incorporated, 2011.will ProQuest Central, ofScience the piezoceramics that be Ebook used. A flexural amplifier vibrating

High Power Ultrasonic Actuators Based on the Langevin Transducer 85

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Figure 20. Photo of the manufactured prototypes.

Figure 21. Comparison between measured electrical impedances of the two actuators. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Figure 22. Measured displacement of the two actuators as a function of applied voltage.

at f r = 19.2 kHz was obtained with the following parameters: b1 = 8 mm, b2 = 14 mm, c = 0.5 mm and S ≈ 300 mm2 . A symmetrical Langevin transducer working at 19.2 kHz was obtained by setting the radius of piezoceramic disks and loading masses to 15 mm, piezoceramic thickness to 2 mm, and loading masses length to 50 mm. Performances of the whole actuator have been evaluated by joining the proposed amplifier to the symmetrical Langevin transducer. Figure 23 shows the computed electrical input admittance. As can be seen, besides the working frequency at 19.2 kHz, the transducer exhibits another resonance frequency at 17.4 kHz, which correspond to the fundamental longitudinal mode of the whole transducer. It is to highlight a first important difference with respect to the classical actuator based on a stepped horn. In that case the working frequency is twice the fundamental resonance frequency (see Figure f:trasdconcimpspost), while for the proposed actuator the difference between the first two resonance frequencies depends on the design parameters of the flexural amplifier. The plot also shows that the value of the admittance at Piezoelectricthe Actuators, Science Publishers, Incorporated, ProQuest Ebookthan Central, firstNova resonance frequency is 2011. much higher that observed at the second

High Power Ultrasonic Actuators Based on the Langevin Transducer 87

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Figure 23. Computed electrical admittance of the flexural actuator. resonance frequency. Also this behaviour is different from that observed for the classical actuator, for which admittance values at the two first resonance frequencies are comparable. In order to further investigate the vibrational behaviour of the actuator at both these frequencies we computed the axial displacement Uz along the axis of the actuator and along a diameter (at x=0) on the front end section. The actuator was driven with a harmonic voltage with amplitude of 60 V. The plot of Uz versus z and r computed at the two resonance frequencies are shown in Figure 24a and 24b, respectively. As can be seen, at the two working frequencies the maximum output displacement is almost the same; nevertheless, the vibrational behaviour is somehow different. Figure 24a shows that the axial displacement along the Langevin transducer at 17.4 kHz is much higher than at 19.2 kHz, again highlighting the load effect of the amplifier at the second resonance frequency. At this frequency the lower axial displacement is almost completely compensated by the higher amplification of the flexural amplifier. It can also be noted that, as expected, at the first resonance frequency the rest position is not exactly at the centre of the piezoceramic stack. Figure 24b shows that at the first frequency the vibration mode is a combination of a longitudinal mode and of a flexural mode, while at the second frequency an almost pure flexural mode can be recognized.

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Figure 24. Computed axial displacement a) along z and b) along a diameter (at x=0) on the front end surface. The amplitude of the applied voltage is 60 V.

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High Power Ultrasonic Actuators Based on the Langevin Transducer 89 4.4.2.

Experimental results

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Figure 25 shows a photo of a prototype of the actuator manufactured by following FE design. Also in this case piezoceramic rings (inner diameter 8 mm)

Figure 25. Photos of the manufactured prototype: a) global view; b) detail of the mechanical machined region.

were used instead of disks in order to allow pre-compression operations. A static torque of 35 Nm was applied. Figure 26 shows the plot of the electrical input impedance of the actuator. As can be seen, the value of the first resonance frequency is in a quite good agreement with that obtained by simulations. As far as the second working frequency is concerned, it is sensibly higher than that predicted by simulation and, furthermore, it is not much evident in the admittance plot. As already mentioned, both these behaviours could be explained by observing that the mechanical machining of the prototype (see Figure 25b) does not reproduce in a perfect way the simulated model. Figure 27 shows the plots of the axial displacement measured on a diameter of the front section of the actuator at the two working frequencies when the driving voltage is Vcc=150 V. As can be seen the deformed shape measured at the first frequency is in a good qualitative agreement with those obtained by numerical simulations (see Figure 24b), while at the second frequency some differences with respect to simulations, probably due to the imperfections in mechanical machining, can be appreciated. In particular, the displacement Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, in the proximity of the borders is not null, and the maximum displacement

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Figure 26. Measured electrical admittance of the whole actuator. is lower, in percentage on that of the first frequency, than that predicted by simulations. From the plot the maximum deflection angle of a laser beam that normally hits the surface of the actuator can be calculated. By driving the actuator with a maximum voltage of 150 V at the first resonance frequency and of 240 V at the second resonance frequency, deflection angles of about 2.4◦ and 2◦ , respectively, are obtained.

References [1] Don A. Berlincourt, Daniel R. Curran, Hans Jaffe, “Piezoelectric and Piezomagnetic Materials and Their Function in Transducers” in Physical Acoustics, W.P. Mason, Ed., Academic Press, New York, 1964. [2] H. F. Tiersten, Linear Piezoelectric Plate Vibration, New York: Plenum Press, 1969. [3] M. Brissaud,“Characterization of Piezoceramic”, IEEE Trans. on Ultrason., Ferroelect., Freq. Contr., Vol. 38, number 6, November 1991.

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Figure 27. Measured axial displacement along the front diameter. Vcc=150 V. [4] M. Brissaud,“Three-Dimensional Modeling of Piezoelectric Materials”, IEEE Trans. on Ultrason., Ferroelect., Freq. Contr., Vol. 57, number 9, September 2010. [5] A. Iula, N. Lamberti, M. Pappalardo, “An approximated 3-D model of Cylinder-Shaped Piezoceramics Elements for Transducers Design”, IEEE Trans. on Ultrason. Ferrelect., Freq. Contr., vol. 45 July 1998. [6] G. Hayward and D. Gilles, “Block diagram modeling of tall, thin parallelepiped piezoelectric structures,” J. Acoust. Soc. Am., vol. 86, no. 5, pp. 16431653, 1989. [7] A. Iula, R. Carotenuto, M. Pappalardo, N. Lamberti, “An approximated 3-D model of the Langevin transducer and its experimental validation”, J. Acoust. Soc. Am., vol. 111, no. 6, pp. 26752680, June 2002. [8] R. Lerch, “Simulation of the Piezoelectric Devices by Two- and ThreeDimensional Finite Element”, IEEE Trans. on Ultrason. Ferrelect., Freq. Contr., vol. 37,no 2, May 1990. [9] X. Xian, S. Lin, “Study on the compound multifrequency ultrasonic transducer in flexural vibration,” Ultrasonics, vol 48, pp. 202-208, 2008. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[10] D. W. Hawkins and P.T. Gough, “Multiresonance design of a Tonpilz transducer using the finite element method,” IEEE Trans. on Ultrason., Ferroelect., Freq. Contr., vol. 43, no.5, pp. 782-790, 1998 . [11] A. Iula, F. Vazquez, M. Pappalardo, J. A. Gallego, “Finite element threedimensional analysis of the vibrationalbehaviour of the Langevin-type transducer,” Ultrasonics, vol. 40, pp. 513-517, 2002. [12] E. A. Neppiras, “The pre-stressed piezoelectric sandwich trasducer”, in Ultrasonic International 1973 Conference Proceeding, pp. 295-302, 1973. [13] L. G. Merkulov, “Design of ultrasonic concentrations”, Soviet Phys. Acoust., vol. 3, no. 3, pp. 246-255, 1957.

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[14] L. G. Merkulov and A. V. Kharitonov, “Theory and analysis of sectional concentrators”, Soviet Phys. Acoust., vol. 5, pp. 183-190, 1959. [15] E. Eisner, “Design of Sonic Amplitude Transformers for High Magnification”, in Journal of the acoustical society of America vol. 35, no. 9, pp. 1367-1377, 1963. [16] D.A. Wang, W.Y. Chuang, K. Hsu, H.T. Pham, “Design of a Bzier-Profile Horn for High Displacement Amplification”, Ultrasonics, Volume 51, Issue 2, February 2011, Pages 148-156. [17] Q. Yao and L. Bjorno, “Broadband Tonpilz Underwater acoustic transducers based on multimode optimization,” IEEE Trans. on Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 5, pp. 1060-1066, 1997. [18] D. Rajapan, “Performance of a low-frequency, multi-resonant broadband Tonpilz transducer”, J. Acoust. Soc. Am., vol. 111, no. 4, pp. 1692-1694, June 2002. [19] H. Xiping, H. Jing, “Study on the broadband tonpilz transducer with a single hole ”, Ultrasonic, vol. 49, pp. 419-423, 2009. [20] K. Saijyou, T. Okuyama, “Design optimization of wide-band Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface”, J. Acoust. Soc. Am., vol. 127, no. 5, pp. 2836-2846, May 2010.

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High Power Ultrasonic Actuators Based on the Langevin Transducer 93 [21] C. Zentgraf, H. Hielscher, “High ultrasonic power for clean wires”, Wire 48 (4), pp. 16-17, 1998. [22] J.O. Kim, S. Choi, J.H. Kim, “Vibroacoustic characteristics of ultrasonic cleaners”, Applied Acoustics 58 (2), pp. 211-228, 1999. [23] B. Niemczewski, “Proposal of a test procedure for ultrasonic cleaners”, Trans. of the Inst. of Metal Finishing 81 (1), pp. 28-31, 2003. [24] J.A. Gallego-Juarez, G. Rodriguez-Corral, L. Gaete-Garreton, “An ultrasonic transducer for high power applications in gases”, Ultrasonics 16 (6), pp. 267-271, 1978.

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[25] J.A. Gallego-Jurez, “Piezoelectric ceramics and ultrasonic transducers”, Journal of Physics E: Scientific Instruments 22 (10), art. no. 001, pp. 804-816, 1989. [26] A. Iula, S. Pallini, F. Fabrizi, R. Carotenuto, N. Lamberti, M. Pappalardo, “A high frequency ultrasonic bistoury designed to reduce friction trauma in cystectomy operations,” in: Proc. of IEEE Ultrason. Symp., 2001, pp. 1331-1334. [27] Andrea Cardoni, “Enhancing Oral Implantology With Power Ultrasonics”, IEEE Trans. on Ultrason. Ferrelect., Freq. Contr., vol. 57,no 9, September 2010. [28] A. Cardoni, M. Lucas, M. Cartmell, F. Lim, “A novel multiple blade ultrasonic cutting device” Ultrasonics 42, pp. 69-74, 2004. [29] X. Bao, Y. Bar-Cohen, Z. Chang, B.P. Dolgin, S. Sherrit, D.S. Pal, S. Du, T. Peterson, “Modeling and computer simulation of ultrasonic/sonic driller/corer (USDC)” IEEE Trans. on Ultrason. Ferrelect., Freq. Contr., vol. 50 ,no 9, pp. 1147-1160, September 2010. [30] A. Cardoni, P. Harkness, M. Lucas, “Ultrasonic rock sampling using longitudinaltorsional vibrations” Ultrasonics 50, pp. 447452, 2010. [31] L. Parrini, “Design of Advanced Ultrasonic Trasdcers for Welding Devices”, in IEEE Transactions on ultrasonics, ferroelectrics, and frequency control, vol. 48, no. 6, pp.1632-1639, 2001. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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[32] S. W. Or, H. L. W. Chan, V.C. Lo, C. W. Yuen, “Dynamics of an Ultrasonic Trasducer Used for Wire Bonding,” IEEE Trans. on Ultrason., Ferroelect, Freq. Contr., vol. 45, no. 6, pp. 1453-1460, Nov. 1998. [33] J. Tsujino, H. Yoshihara, K. Kamimoto, and Y. Osada, “Highfrequency longitudinal-transverse complex vibration systems for ultrasonic wire bonding, in 1997 IEEE Ultrason. Symp. Proc., p. 849. [34] A. Iula, G. Caliano, A. Caronti, M. Pappalardo, “A Power Transducer System for the Ultrasonic Lubrification of the Continuous Steel Casting,” IEEE Trans. on Ultrason., Ferroelect, Freq. Contr., vol. 50, no. 11, pp. 1501-1508, Nov. 2003.

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[35] S. de la Fuente-Blanco, E. Riera-Franco de Sarabia, V.M. AcostaAparicio, A. Blanco-Blanco, J.A. Gallego-Jurez, “Food drying process by power ultrasound”, Ultrasonics 44 (SUPPL.), pp. e523-e527, 2006. [36] J.A. Gallego-Jurez, E. Riera, S. de la Fuente Blanco, G. RodrguezCorral, V.M. Acosta-Aparicio, A. Blanco, “Application of high-power ultrasound for dehydration of vegetables: Processes and devices”, Drying Technology 25 (11), pp. 1893-1901, 2007. [37] S. Ueha, Y. Tomikawa, M. Kurosawa, K. Nakamura, Ultrasonic MotorsTheory and Applications, Clarendon Press, Oxford, 1993. [38] A. Iula, A. Corbo, M. Pappalardo, “FE Analysis and Experimental Evaluation of the Performance of a Travelling Wave Rotary Motor driven by High Power Ultrasonic Transducers”, Sensors and Actuators A, Volume 160, Issues 1-2, May 2010, Pages 94-100. [39] A. Iula, and M. Pappalardo, “A High Power Travelling Wave Ultrasonic Motor,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol. 53, no. 7, pp.1344-1351, 2006. [40] M. K. Kurosawa, O. Kodaira, Y. Tsuchitoi, T. Higuchi, “Transducer for High Speed and Large Thrust Ultrasonic Linear Motor Using Two Sandwich–Type Vibrators”’, IEEE Trans.Ultrason., Ferroelect., Freq. Contr., 45 (5) (1998) pp. 1188–1195.

[41] X. Li , W. Chen, T. Xie, J. Liu, “Novel high torque bearingless two-sided rotary ultrasonic motor”’, Journal of Zhejiang University SCIENCE A, 8 Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, (5)(2007) pp. 786-792

High Power Ultrasonic Actuators Based on the Langevin Transducer 95 [42] J. Oh, H. Jung, J. Lee, K. Lim, H. Kim, B. Ryu, D. Park, “Design and performances of high torque ultrasonic motor for application of automobile”, J Electroceram 22, (2009) pp. 150155. [43] A. Iula, L. Parenti, F. Fabrizi, M. Pappalardo, “A high displacement ultrasonic actuator based on a flexural mechanical amplifier,” Sensors and Actuators A, vol 125, pp. 118-123, 2006. [44] A. Iula, “Design and Experimental Characterization of a Multifrequency Flexural Ultrasonic Actuator”, IEEE Trans.Ultrason., Ferroelect., Freq. Contr., vol. 56, p. 1725-1730, (2009) [45] A. Iula, R. Carotenuto, N. Lamberti, M. Pappalardo, “A Matrix Model of the Axle Vibration of a Piezoelectric Motor,” Ultrasonics, vol. 38, pp. 41-45, 2000.

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[46] A. Iula, M. Pappalardo, “A General Model of the Axle Vibration in Piezoelectric Motors,” Ultrasonics, vol. 42, pp. 291-296, 2004. [47] A. W. Leissa, Vibration of Plates, Washington: US Government Printing Office, 1970, NASA SP-160.

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In: Piezoelectric Actuators Editor: Joshua E. Segel, pp. 97-115

ISBN 978-1-61324-181-3 c 2011 Nova Science Publishers, Inc.

Chapter 4

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ACTIVE C ONTROL OF THE DYNAMICS OF H INGED -H INGED B EAM U SING P IEZOELECTRIC A BSORBER B. R. Nana Nbendjo∗ and P. Woafo Laboratory of Modelling and Simulation in Engineering and Biological Physics, University of Yaound´e I, Box 812, Yaound´e, Cameroon

Abstract This chapter deals with the problem of reducing amplitude, inhibiting catastrophic motion and suppressing horseshoes chaos on a hingedhinged beam subjected to transversal periodic excitation. Piezoelectric ceramics have been used as sensors and actuators. After modelling the system under control, we derive the accurate control parameters leading to the efficiency of the control. This is done using analytical approach along with direct numerical simulation of the base equation. The effect of delay between the detection of the structure motion and the restoring action of the control is pointed out.

PACS 05.45-a, 52.35.Mw, 96.50.Fm. Key Words: Active control, piezoelectric ceramics, hinged-hinged beam, delay, harmonic balance method, Melnikov theory AMS Subject Classification: 53D, 37C, 65P.

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E-mail address: [email protected]

98

1.

B. R. Nana Nbendjo and P. Woafo

Introduction

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The study of vibrating structures has been a subject of particular interest in recent years. This is due to the fact that structures under harmonic excitation appear in various fields of fundamental and applied sciences [1-3]. Among theses studies, particular attention had been paid to the dynamical behavior of beams. It was shown that when the beam is not highly loaded, its dynamics could be explained by the classical Duffing oscillator [2]. In [3] the authors used the nonlinearity of a foundation and showed that the behavior of the beam could be expressed by a φ6 potential. On the first hand of this chapter, we consider the geometrical nonlinearity of the beam to show that the dynamics of a hinged-hinged beam under transversal excitation is described by an unbounded single well φ6 potential. In the second hand one of the main challenges in the design of structures is to reduce noises and vibrations. In fact, noises and vibrations lead to fatigue and damage which can result in the reduction or even loss of the system’s performance. So it has become imperative to integrate dynamic analysis into the design process in the form of a controlled strategy. For these purposes, active control of vibration in structures has been investigated by an increasing number of researchers [4-8]. There has been a great deal of theoretical and experimental work examining the use of point forces for vibration control and more recently the use of piezoelectric crystals laminated to the surfaces of structures. In 1985, Bailey et al.[9] introduced piezoelectric actuators for active vibration control. They used the actuators bonded to the surface of a cantilever beam in their feedback vibration damping design. Demitriadis et al.[10] in 1991 performed a two dimensional extension of Crawley and Deluis work [11], applying a pair of laminated piezoelectric actuators to a plate. They demonstrated that the location and shape of the actuator dramatically affect the vibration response of the plate. Recently in 2002, Morgan et al.[12,13], proposed a semi active piezoelectric absorber with varying frequency in a structure with linear dynamics; they also presented an experimental setup to validate numerical prediction. The aim of the present work is threefold. First, it shows that by taking into account higher nonlinear contributions of the induced stretched forces, the dynamics of a hinged-hinged beam under external excitation is described by a single well φ6 potential. Secondly we present and describe the beam dynamics under control. The third part deals with the optimization of control process. Here, we determine the range of the control parameters leading to increase the stability domain of the structure Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, and that can conveniently reduce the amplitude of vibration. Moreover, we

.

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99

emphasize on the external excitations that can produce the catastrophic failure of the beam or escape from a potential well [14] . In [15] the authors proposed an approximate condition to escape from the potential well by comparing the maximum energy of the motion with the barrier of the potential. This method is used here to determine the condition for the appearance of a catastrophic or unbounded motion. To complement our results, the numerical integration of the resulting modal equation of motion is also performed. We conclude our work in the last section.

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2.

Modelling of the Dynamics of a Hinged-Hinged Beam under Transversal Excitation by an Unbounded Single Well φ6 Potential

The Physical model represented in figure 1 is a hinged-hinged beam of length l subjected to the action of transversal excitation f , axial load P0 and to substrate reaction f s. Its transversal and longitudinal displacements are respectively w and u. The beam has Young’s modulus E, a mass density ρ and a crosssectional area S. These Physical and geometrical characteristics are assumed constant. Let us consider an element of length dx of the beam at rest. Its corresponding value when the beam is deflected is given by 1

ds = [(1 + ux )2 + w2x ] 2 dx,

(1)

where ux and wx are the partial derivatives with respect to x. The unit vector parallel to the deflected element can be expressed as

Figure 1. Hinged-hinged beam under transversal excitation. Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

100

B. R. Nana Nbendjo and P. Woafo δ = [(1 + ux)i + wx j]

dx , ds

(2)

Consequently, the effective instantaneous tension of the beam becomes Pi = P0 + ES

(ds − dx) , dx

(3)

Taking into account the presence of a transversal viscous damping of coefficient ζ, the beam dynamics is described by the following momentum equations ∂ (Pi δ)i ∂x ∂4 w ∂ ρSwtt = − (Pi δ)j + EI 4 − ρSζwt + f + f s ∂x ∂x ρSutt = −

(4)

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where I is the moment of inertia of the beam cross section. the subscripts t, tt, x and xx stand for the nth order derivatives with respect to t and x respectively. We then carry out the development of dx ds up to the second order and obtain the following expression. 1 3 dx = 1 − (2ux + u2x + w2x ) + (2ux + u2x + w2x )2 , ds 2 8

(5)

Inserting this expression in equation (6), we obtain 1 ∂ 3 ρAutt − ESuxx = (ES − P0 ) [(u2x + w2x − 2ux w2x ) − (u2x + w2x ) 2 ∂x 4 3 3 2 2 2 2 2 +5ux + 3ux (ux + wx ) + ux (ux + wx )] 4 ∂ ρSwtt + P0 wxx + EIwxxxx = (P0 − ES) (ewx ) − ρSζwt ∂x + f + fs (6) with 1 3 3 e = ux − u2x + w2x − ux(u2x + w2x ) − (u2x + w2x )2 , 2 2 8

(7)

Note that these equations are valid when the wavelength of the transverse vibrations is greater that the radius of gyration of the cross section of the beam. Otherwise the effects of the shear forces and that of the inertia rotation cannot be neglected. The system of equations (6) contains the first, second third and Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, fourth order power of u arising from the deflection induced by the motion.

Active Control of the Dynamics of Hinged-Hinged Beam. . .

101

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Figure 2. Unbounded monostable φ6 potential

Only the first order approximation of u will be considered here. This means that the transversal displacement is more important than the longitudinal displacement. Under that circumstance it is appropriate to consider u to be 0(w4 ). We therefore neglected the following terms u2x , ux w2x , u3x , u2x (u2x + w2x ) can be neglect and equation (7) becomes 1 3 e = ux + w2x − w4x , 2 8

(8)

We stress here the presence of the fourth order term w4x in equation (10). Moreover, we consider the following boundaries conditions: u(0,t) = u(l,t) = 0

(9)

Thus after neglecting axial load P0 and substrate reaction f s, the general equation governing the behavior of hinged-hinged beam is given by (after extracting u from eq. (6), see ref.[2])  Z l  Z 1 3 l 4 2 ρSwtt + δwt + EIwxxxx − ES w dx − w dx wxx = P(x,t) (10) 2l 0 x 8l 0 x

which differs from the classical one [2] by the − 8l3 0l w4x dx component. Since in the pioneering work of Holmes [1], investigations had always nePiezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, glected this term and finally arrived to the classical bounded Duffing equation R

102

B. R. Nana Nbendjo and P. Woafo

with one or two wells for the single mode dynamics. As it will appear later in this work, by taking this term into consideration, one arrives at the extended Duffing oscillator with unbounded monostable (see figure 2). These configuration seem more realistic since they may explain the catastrophic failure of beams under the action of high loads. Moreover when the amplitude of motion is not high, these configurations recover the dynamics of the classical Duffing oscillator with the φ4 potential reported in [2].

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Figure 3. Hinged-hinged beam under piezoelectric absorber.

3. 3.1.

Description of the Model and General Mathematical Formalism Description of the physical model

The physical model presented in figure 3 is an isotropic hinged-hinged beam with a piezoelectric actuator. The local vibration in the structure is monitored using a piezoelectric sensor. The configuration integrates piezoelectric materials with an active voltage source, a passive resistance and inductance shunting circuit. On one hand, the structural vibration energy can be transferred and dissipated in the tuned shunting circuit passively. On the other hand, the control voltage will drive the piezo-layer, through the circuit, and actively suppress vibration in the host structure [16]. The passive inductance of the shunt circuit L p , is selected so that the absorber is tuned to the nominal or expected excitation frequency. No resistance is intentionally added to the circuit, however the passive inductor may have significant internal resistance which is represented by R p . An important element of any practical control system are the transducers used for implementation of the control. Thus, the Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, Sensors are needed for measurements which can be used to estimate important

Active Control of the Dynamics of Hinged-Hinged Beam. . .

103

disturbance and system variables. Actuators are used to apply control signals to the system in order to change the system response in the required manner. In general, sensors provide information to the controller to determine the performance of the controlled system or to provide signals related to the system response. Thus, sensors and actuators provide the link between the controller and the physical system to be controlled and their design and implementation is of prime importance.

3.2.

General mathematical formalism

To derive the system equations, let us assume that the rotational inertia is negligible, the piezoelectric Lead Zirconium Titanate (PZT) layers are said to be short compared to the beam. Thus, it is assumed that the model of the structure and the piezoelectric absorber can be obtained and it is given by [19]

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ρSwtt + δwt + EIwxxxx  Z Z 1 l 2 3 l 4 −E wx dx − wx dx wxx + kc Q 

2l

8l

0

= P(x,t)

0

L p Qtt + R p Qt +

1 Q − kc w = Vc Cp

(11)

L p and R p are the passive inductance and resistance of the shunt circuit, Q is the charge on the piezoelectric material, Cp is the capacitance of the piezoelectric under constant strain and Vc is the control voltage. The control voltage cannot modify the coupling term in the structure equation, but it can be used to increase the effective coupling in the circuit equation. l is the length of the beam and kc the coupling coefficient, representing the conversion from mechanical energy to electrical energy and vice-versa. Using the dimensionless variables defined as follows: W=

w , l

x z= , l

l∗ =

l L

and q =

Q q0

where l ∗ and q0 are respectively the reference length and charge. Setting, τ = ω0 t, ω20 = a=

ρSl 2

, V=

Rp EI 1 δ , λ1 = , λ2 = , ω202 = , 4 ρSl ρSω0 L p ω0 L p Cp ω20 Vc

, α=

ES

, γ=

kc Q0

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2

2 2

2

, F(z, τ) =

P(x,t) 2

104

B. R. Nana Nbendjo and P. Woafo

Therefore the above two differential equations are reduced to the following set of non dimensional differential equations

−α



Z l? 1

2l ?

0

Wz2 dz −

Wττ +Wzzzz + λ1 Wτ  4 Wz dz Wzz + γq = F(z, τ)

Z l? 3

8l ?

0

qττ + λ2 qτ + ω202 q − aγW

= V

(12)

Taking into account the boundary conditions given by W (z, τ) = W 00 (z, τ) = 0

(13)

at z = 0 and z = 1. We set n

W (z, τ) =

∑ y j (τ)sin(π jz)

(14)

j=1

Consider the case of localized force given by Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

F(z, τ) = f 0 cos(Ωτ)δ(z − z0 )

(15)

where z0 is the axial point of application. For the first mode of vibrations on which the major part of the energy is concentrated. Inserting equations (14) and (15) into equation (12), multiplying by sin(πz) and performing the integration form 0 to 1, it becomes 4γ q = 2 f 0 cos(Ωτ)sinπz0 π πaγ q¨ + λ2 q˙ + ω202 q − y1 = V 4

y¨1 + λ1 y˙1 + ω201 y1 + cy31 + dy51 +

(16)

9 where ω201 = π4 , c = 21 απ4 and d = − 64 απ6 . With this equation, the optimization process can be achieved by detecting the goog physical characteristic of coupling parameter γ which stands for the conversion from mechanical energy to electrical energy and vice-versa and also on the characteristic of voltage source V .

3.3.

Optimization of control process

3.4.

Stability of active structural control

Rather than reinforcing the stability of the system, the control design can destabilize the system. For that aim it is necessary to look for the condition

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Active Control of the Dynamics of Hinged-Hinged Beam. . .

105

fulfilled by the control gain parameter so that the whole structure should be stable. Following the classical local stability analysis of Lyapunov [17], we examine the fixed points of the system. Considering the system of equation (16), one finds that u0 = (0, 0, 0, 0) is a fixed point for all parameter values. Local stability can be determined by investigating the linearized system U˙ = D f (u0 ).U

(17)

where D f (u0 ) is the jacobian of f at u0 . The characteristic equation of the jacobian matrix is then written as

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s4 + (λ1 + λ2 )s3 + (ω201 + λ1 λ2 + ω202 )s2 + (λ1 ω202 + λ2 ω201 )s +ω201 ω202 + aγ2 = 0

(18)

The Lyapunov analysis stipulates that the fixed points are stable if the real parts of the roots of the characteristics equation are all negative. Otherwise (if at least one root has a positive part), the fixed point is unstable. Using the Routh-Hurwitz criterion [18], we derived the following condition for stability of the system under control γ2 ≺

(λ1 + λ2 )[(ω201 + ω202 + λ1 λ2 )(λ1 ω202 + λ2 ω201 ) − ω201 ω202 ] a(λ1 + λ2 )2 (λ1 ω202 + λ2 ω201 )2 − a(λ1 + λ2 )2

(19)

The above simple analysis of fixed points and local stability leads to the conclusion that, if the physical characteristics γ of the piezoelectric elements do not respect the condition given by equation (9), it will destabilize the mechanical system leading to catastrophic consequences. To illustrate this effect, let us consider an example of steel beam having the following parameters E = 200 × 109 N/m, ρ = 7850kg/m3 , l = 2m, S = 0.05 × 0.1m2 and δ = 80Ns/m. The physical characteristics of the controlled elements are given by R p = 1000Ω, L p = 15H and Cp = 12µF (see ref[13]). Using the relation between the dimensional and the non dimensional parameters which we obtained before. The following dimensionless parameter ω0 = 71.38hz, λ1 = 0.028, λ2 = 0.93, α = 6.36, a = 5.23, ω01 = 9.85 and ω02 = 1.04 are derived. This is obtained for the reference charge Q0 = 2c. We then compute the critical value of the control gain parameter for which control system is stable. Using Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, the condition given by equation (9), the control system is stable if γ < 0.71.

106

B. R. Nana Nbendjo and P. Woafo

3 Without delay with delay

2.5

β

2

1.5

1

0.5

0

0

1

2

3

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Ω

Figure 4. Domain in space parameters (Ω, β) where the control is efficient.

Taking into consideration the non dimensional expression it appears that the whole structure is stable if kc < 141.98 Vcm−1 . In the contrary, the control design will be the source of instability on the structure leading to catastrophic consequences.

3.5.

Effects of the control on the amplitude of Harmonic oscillations

The governing modal equations of the mechanical system under control can be rewritten in the following form y¨1 (τ) + λ1 y˙1 (τ) + ω201 y1 (τ) + cy31 (τ) + dy51 (τ) + βq(τ) = q(τ) ¨ + λ2 q(τ) ˙ + ω202 q(τ) π2 a

f 1 cos(Ωτ)

= µβy1 (τ) (20)

Where V = 0, f 1 = 2 f 0 sinπz0, β = 4γ π and µ = 16 . The analysis of the control design can not be well achieved if the inevitable time delay is not taken into

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107

account. The delay is materialized by the fact that the control system doesn’t act at the same time with the excited structure. Mathematically, this effect is taken into account by using the retarded functional differential equation [14,20]. Thus, for a control system with delay, the differential equation given by (20) becomes: y¨1 (τ)+λ1 y˙1 (τ)+ω201 y1 (τ)+cy31 (τ)+dy51 (τ)+βq(τ) = f1 cos(Ωτ) q(τ) ¨ + λ2 q(τ) ˙ + ω202 q(τ) = µβy1 (τ − τq ) (21) where τq is the time delays for displacement in the system. In the linear limit (c = d = 0), the amplitude of the harmonic oscillation of the controlled system is obtain using harmonic balance method and given by Ac =

f1 1 2 2 [(ω01 − Ω − βη1 )2 + (Ωλ1 − βη2 )2 ] 2

(22)

where  (ω202 − Ω2 )sinΩτq + λ2 ΩcosΩτq η1 = µβ (ω202 − Ω2 )2 + (λ2 Ω)2  2  (ω02 − Ω2 )cosΩτq − λ2 ΩsinΩτq η2 = µβ (ω202 − Ω2 )2 + (λ2 Ω)2

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Comparing Ac (amplitude of controlled system) with the amplitude Anc of the vibrations of the uncontrolled system, we see that the control is efficient if Ac < Anc. This means that the control parameters satisfy the following condition: η1 β[η1 β − 2(ω201 − Ω2 )] + η2 β[η2 β − 2λ1 Ω]  0

(23)

Figure 4 displays in the space parameters (Ω, β) this region. The curve with a thin line represents the case where the delay is not considered and the region below this curve represents the domain where the amplitude of vibration of the mechanical structure is reduced. Taking into account the effects of delay, we present in the same graph the case where the delay is given by τq = 0.1 and we arrive at the following conclusion the region of efficiency decrease because of time delay. In the non linear case, the amplitude A of harmonic vibrations is derived using the Harmonic balance method and obeys to the following non linear algebraic equation: 25

d 2 A10 +

15

cdA8 +

9

5 c2 + d(ω2 − Ω2 + βη1 ) A6

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108

B. R. Nana Nbendjo and P. Woafo

200 (a)

Ab

150 100 50 0

0

1

2

3

β

33

(b) 32

f1

31 30 29 28 Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

27

0

1

2 τ θ

3

4

Figure 5. (a): Boundary of the domains in the space parameters (β, Ab ), where the control of amplitude is efficient (b): Evolution of amplitude as function of time delay τq for β = 0.1

3 + c(ω201 − Ω2 − βη1 )A4 + [(ω201 − Ω2 + βη1 )2 2 +(Ωλ1 − βη2 )2 ]A2 − f 12 = 0

(24)

We remind the reader that η1 and η2 are functions of τq as given above. Assuming that at the frontier separating regions of efficiency and inefficiency of the control, the amplitudes of both the controlled and uncontrolled (β = 0) systems are equal, it results that the amplitude of the oscillations at this limit is given by A2b =

−3cβη1 − 2ψ2 5dβη1

(25)

where ψ = 49 c2 η21 β2 − 5dβη1 (βη1 (βη1 + 2(ω201 − Ω2 )) + βη2 (βη2 − 2λ1 Ω)) Inserting equation (25) in equation (24) (with A = Ab ), we obtain the boundary

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separating the domain where the control is efficient (reduction of amplitude of vibration) to the domain where it is inefficient. Figure 5a presents the evolution of Ab as a function of β with τq = 0 and Ω = 9.85 (we remind the reader that 9.85 is the frequency at the primary resonance). The domain of the efficiency of the control is below the curve. It appears that as the excitation amplitude increases, we need greater values of β to reduce the vibration of the structure. In figure 5b, we have plotted this boundary in the (τq , f 1 ) plane along with the case where delay is not taken into account (solid horizontal line). This is done for β = 0.1. It is found that f 1 is a periodic function of τq . Thus, with a good choice of time delay, a better protection of the structure can be obtained. However for some values of time-delay the control is affected in the bad direction. All this result is obtained taking into account the fact that there is no external source voltage acting on the system. If the voltage source if different from zero (that means constant or periodic or else) the amplitude response of the mechanical structure should be affected (see ref[19] for explanation).

3.6.

Effects of the control on the appearance of unbounded motion

Depending on the value of the external force and the values of the other parameters, the system initially moving inside the potential well can cross the barrier of the potential to exhibit unbounded motions leading to failure. It is important to analyze the effects of the control parameters on the condition for the escape from the potential well. We use the method of energy [20] as described in the introduction. We thus find that the amplitude of the excitation at the frontier between catastrophic and bounded motions is given by f c2

=



5 5 3 3 dA + cA + (ω201 − Ω2 βη1 )Ab 8 b 4 b

2

+ (Ωλ1 + βη2 )2 A2b

with A2b =

6bq2c + 3cq4c + 2dq6c − 6(ω201 − Ω2 )Xm − 3cXm2 − 2dXm3 6Ω2

where −c −

q

c2 − 4d(ω201 − Ω2 )

2 Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Xm =

and qc =

−c −

q

c2 − 4dω201

(26)

110

B. R. Nana Nbendjo and P. Woafo

8.112 Without delay With delay

8.111

fc

8.11

8.109

8.108

8.107

8.106

0

1

2

3

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β

Figure 6. Evolution of critical force leading to catastrophic motion as function of control gain parameters β.

Equations (26) give an approximate expression for the critical forcing above which a catastrophe can occur. Its variation as a function of β is plotted in figure 6 taking into account the case without delay (thin line). We find that f c increases with β. This means that the better protection of the structure should take into account this consideration. In other analyze the effects of time delay we display in the same figure 6 the variation of the critical force as function of β for τ = 0.005. We find that for each value of control gain parameter β the critical force f c leading to catastrophic motion decrease because of delay. We can conclude that the time delay reduce the efficiency of the control.

3.7.

Effects of the control on the basin of stability

In this section, we are interested in the study of global bifurcation before and after loss of stability [21,22]. Since this condition can be detected by means

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Active Control of the Dynamics of Hinged-Hinged Beam. . .

111

6

Initial velocity

4

2

0

-2

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-4

-0.5

0 Initial position

0.5

Figure 7. Fractal basin boundary in the uncontrolled case for f 1 = 6.5 and Ω = 9.85 .

of basin of attraction, it is important to obtain the condition for theoretically preventing chaotic dynamics. This may imply the existence of fractal basin boundaries and the so-called horseshoes structure of chaos. To deals with that condition, we have simulated numerically the system of equations (20) using fourth order Runge kutta algorithm to look for the effects of the control parameters on the onset of the fractality in the basin of attraction. First Considering the case of the system without control, figure 7 shows for example a fractal basin boundary for f 0 = 6.5. Now taking into account the presence of the control without delay, we consider only the effects of β on the critical value. Figure 8 shows that with the same parameters using to obtain the figure 7, the system become regular for β = 0.5 and this is accompanied by an enlargement of the basin of attraction. This means that by making a good choice of the coupling parameters we can conveniently increase the domain of safety of the Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, mechanical structures. Turning our interest on the effects of the time delays,

112

B. R. Nana Nbendjo and P. Woafo

6

4

Initial velocity

2

0

-2

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-4

-6 -0.5

0 Initial position

0.5

Figure 8. Regular shape of basin because of control for f 1 = 6.5, Ω = 9.85 and β = 0.25.

we find that the fractality appears more early (see refs [15]). This means that the best estimation of the optimal parameters for the efficiency of the control should not neglect the effects of time-delays.

4.

Conclusion

In this work, by considering a higher nonlinear term of the geometric nonlinearity, it is found that the dynamics of an hinged-hinged beam under transversal excitation is described by a more realistic catastrophic single well φ6 potential. The possibility of using piezoelectric ceramics to reduce the amplitude response of hinged-hinged beam has been shown. Basic non Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, linear equations for the system under control have been derived. Analytical

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treatment of this equation permitted in the space parameters of the systems the domain of control gain parameters leading to reinforce the stability of structural control, leading to the reduction or the amplification of the amplitude of the system under control. The dynamics of the resultant system under control (harmonic response, escape from a potential well and basin of attraction) have been studies. The effects of time-delay in the approximate critical force leading to reduction of amplitude and escape from a potential well appear to be important and should be taken into account for the designing of control devices. The analytical results have been complemented by the numerical simulation of the original non-linear equation and metamorphoses of the basin of attraction have been observed.

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Acknowledgments Part of this work was completed at the Max Planck Institute for the Physics of Complex Systems (MPIPKS), Germany during a short research visit of B.R. Nana Nbendjo. He is grateful to MPIPKS for financial support. The authors wish to thank Dr R. Tchoukuegno, for useful discussions and contribution to this work.

References [1] P. Holmes, ”A nonlinear oscillator with a strange attractor”, Philos. Trans. R. soc. London ser A282 (1979) 419-448 [2] A. H. Nayfey, D. T. Mook, ”Nonlinear Oscillations”, Wiley, New York (1979). [3] S. Lenci, G. Menditto, A. M.Tarantino, ”Homoclinic and heteroclinic bifurcations in the nonlinear dynamics of beam resting on an elastic substrate”, Int. J. Nonlinear Mechanics 34 (1999) 615-632. S. Lenci, A.M. Tarantino, ”Chaotic dynamics of an elastic beam resting on a Winkler-type soil”, Chaos, Solitons and Fractals 7 (1996) 16011614. [4] T.T. Soong, Active structural control theory and practice, John wiley and Sonc, Newy york, 1950. [5] C.R. Fuller, S.J. Eliot, P.A. Nelson, Active control of vibration, London Academic, 1997.

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[6] D. Sun, L. Tong, Modeling and analysis of curved beams with debonded piezoelectric sensor/actuator patches, International Journal of Mechanical Science. 44 (2002) 1755-77. [7] J.X. Gao, Y.P. Shen, Active control of geometrically nonlinear transcient vibration of composite plates with piezoelctric actuators, Journal of Sound and Vibration. 267 (2003) 911-28. [8] R. Tchoukuegno, B.R. Nana Nbendjo, P. Woafo, Linear feedback and parametric controls of vibration and chaotic escape in a φ6 potential, International Journal of Nonlinear Mechanics. 18 (2003) 531-41. [9] T. Baily, J.E. Hubbard, Distributed piezoelectric-polymer active vibration, Journal of Guidance, Control and Dynamics. 8 (1985) 605-11.

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[10] E.K. Dimitiadis, C.R. Fuller, Investigation on active control of sound transmission through elastic plates using piezoelectric actuators, AIAA Paper. 89 (1989) 1062. [11] E.F. Crawley, J. Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA Journal. 25 (1987) 1373-85. [12] R.A. Morgan, R.W. Wang, it An active-passive piezoelectric absorber for structural vibration control under harmonic excitations with time-varying frequency, part1: Algorithm developpment and analysis, Journal of Vibration and Acoustics. 124 (2002) 77-83. [13] R.A. Morgan, R.W. Wang, An active-passive piezoelectric absorber for structural vibration control under harmonic excitations with time-varying frequency, part2: Experimental validation and parametric study, Journal of Vibration and Acoustics. 124 (2002) 84-89. [14] B. R. Nana Nbendjo, Y. Salissou and P. Woafo, ”Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator” Chaos, Solitons and fractals 23 (3)(2005) 809-816. [15] B. R. Nana Nbendjo. ”Dynamics and active control with delay of the dynamics of unbounded monostable mechanical structures with φ6 potentials” PhD Dissertation, Univesity of Yaound I. Cameroon 2004.

[16] M.S. Tsai, K.W. Wang, On the damping caracteristics of active piezoelectric actuator with passive shunt, Journal of Sound and Vibration. 121 Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, (1999) 1-22.

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[17] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D. 16 (1985) 285-317. [18] C. Hayashi, Non-linear oscillations in physical systems, McGraw-Grill, New-York, 1964. [19] B. R. Nana Nbendjo.” Amplitude control on hinged-hinged beam using piezoelectric absorber: Analytical and numerical explanation” International Journal of Non-Linear Mechanics 44 (2009) 704 - 708. [20] L. N. Virgin, R. H. Plaut, C. C. Cheng, ”Prediction of escape from a potential well under harmonic excitation”, Int. J. of Nonlinear Mechanics 27 (1992) 357-367. [21] M. Soutif, ”Vibration, propagation, diffusion”, Dunod, Paris, 1970

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[22] V. K. Melnikov,”On the stability of the center for time periodic perturbations” Trans, Moskow Math. Soc.12 (1963) 1.

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In: Piezoelectric Actuators ISBN 978-1-61324-181-3 c 2011 Nova Science Publishers, Inc. Editor: Joshua E. Segel, pp. 117-138

Chapter 5

M ODELING AND C ONTROL OF P IEZOELECTRIC ACTUATOR S YSTEMS

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S. N. Huang and K. K. Tan National University of Singapore Singapore 117576

Abstract In this chapter, common configuration of the piezoelectric actuator (PA) systems, their mathematical models as well as control schemes will be discussed. One major source of uncertainties in PA design and application is the hysteresis behavior which yields a rate-independent lag and residual displacement near zero input, reducing the precision of the actuators. For eliminating the effects of the hysteresis phenomenon, we have to build the model of a PA system. Several representative models will be discussed in the chapter: the Maxwell model, the time delay hysteresis model, the Jiles-Atherton model, the Preisach model and the dynamical LuGre Model. The controller design is based on the model obtained. The simulation study is given to show the effectiveness of the proposed control scheme.

1.

Introduction

In precision actuation systems, the piezoelectric actuator (PA) plays a key role in high-precision positioning. The main benefits of a PA include low thermal losses and, most importantly, the high precision and accuracy achievable Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, consequent of the direct drive principle. The industrial applications of the

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PA in various optical fibre alignment, mask alignment, and medical micromanipulation systems are self-evident testimonies of the effectiveness of the PA. In spite of the benefits of a PA in these application domains, there are challenges in modeling and control of these devices. Various approaches to modeling and control of the PA system have been reported over the last two decades (see [1, 2, 3, 4, 5, 8, 7, 9, 10, 11]. These results show that PAs have performance attributes and properties which can be valuable in precision positioning and vibration control etc while using their proposed approaches. Based on the linear mapping between the driving charge and displacement of the PA, the charge-driven technique is proposed by Newcomb and Flinn [1]. In Hagood et al. [2], an active structure control is proposed which is based on a linear transfer function model. In Cole and Clark [3] and Vipperman and Clark [4], the least mean square (LMS) algorithm is used to adaptively compensate the nonlinear dynamics affecting the PA. The Preisach model is used to compensate for the tracking error of piezoelectric actuators as proposed by Ge and Jouaneh [5]. In Stepanenko and Su [6], the Duhem model-based intelligent controller is designed. In Choi et al. [8], the Maxwell slip model is used to develop a feedforward compensator for a piezoelectric actuator. In Cru-Hernandez and Hayward [7], the hysteresis nonlinearity is treated as a constant phase lag and accordingly, a phase-lead model is used to design a compensator. In Santa et al. [9], a neural network model is established which is then used in the controller design. In our work [10] and Leang and Devasia [11], iterative learning method is used to compensate the effects of system uncertainties. In Low and Guo [13], modeling of a three-layer piezoelectric bimorph beam with hysteresis is built. In Goldfarb and Celanovic [14], a mechanical modeling with circuit for piezoelectric stack actuators is derived and a controller is designed based on the model. In Abidi and Sabanovic [12], discrete sliding mode control is applied to a piezostage for rejecting disturbance in the system. A latest literature [17] shows an interesting application of the PA system in biomedical engineering.

The goal of this chapter is to design and analyze the control scheme in the PA system. As discussed above, the hysteresis is a main obstacle to progress. For eliminating or reducing the effects of the hysteresis nonlinearity, the model of a PA system is built. The controller design is based on the model obtained. First, the normal sliding mode control is discussed. Secondly, the modelbased adaptive control is presented incorporating with nonlinear approximator. Thirdly, the modified adaptive control is designed so as to prevent the paramPiezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, eters going to infinity. Finally, the detailed simulation studies are presented in

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the chapter.

2.

Piezoelectric Actuator

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Piezoelectric materials change dimension when an electric field is applied. Piezo actuator is made of piezoelectric materials and electrodes. The most common design for PAs is a stack of ceramic layers separated by thin metallic electrodes (see Figure 1). Such devices are capable of achieving highprecision displacements and holding forces. Commercial products for standard designs are available from Physik Instrumente, which can generate forces of several thousand Newton and realize precise positioning in micro seconds. Since expansion of PA ceramic is nearly linear with voltage, hysteresis must be eliminated by precise closed loop control. Hysteresis is an input/output nonlinearity with effects of non-local memory, i.e., the output of the system depends not only on the instantaneous input, but also on the history of its operation. In general, hysteresis can be observed through an open-loop operation. A PA exhibits nonlinear characteristics, most notably in the form of a hysteretic behavior, which yields a rate-independent lag and residual displacement near zero input. Figure 2 shows the real-time open-loop response of a PA manufactured by Physik Instrumente (PI). This behavior represents a strong uncertainty to the control system. It is a challenging work to design a controller handling this uncertainty.

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Figure 2. Hysteresis phenomenon

3.

Mathematical Modelling of Hysteresis

Various methods have been developed to model hysteresis in piezoceramic actuators, such as the Maxwell model, the time delay hysteresis model, the Jiles-Atherton model, the Preisach model and dynamical LuGre Model. In this section, we briefly review these models and their main features.

3.1.

The Maxwell model

Hysteresis behavior could be modeled by combining a spring with a Coulomb friction element [14]. This analogy is the basis for describing the hysteresis shown in a PA system. Figure 3 shows an element which is comprised of a spring and a block that has a Coulomb friction. The mathematical equation of this behavior can be given by  k(x − xb ) if |k(x − xb )| < f F= (1) f sgn(x) ˙ and xb = x − kf sgn(x)else ˙ where x is the displacement, F is the output force, k is the stiffness of the spring, f is the breakaway friction force of the block, and xb is the position

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of the block. This is a classical model for descrbing the hysteresis. However, for the control design, it is quite difficult to be analyzed, for it contains strong nonlinearities.

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Figure 3. Maxwell slip element

3.2.

The time delay model

It has been experimentally observed that a time-delay should be introduced into the hysteresis model to capture memory. Physically, this behavior can also be found in friction and this memory is the result of state in the interface which must adjust to the new sliding condition before the force will attain its new value. Hess and Soom [18] presented a simple time-delay model for this behavior, i.e., F(t) = c0 x(t ˙ − h) +

c1 sgn(x(t)), ˙ 1 + c2 x˙2 (t − h)

(2)

where h is the time delay, and c0 , c1 , c2 are positive constants. The time delay h can be estimated from the experiment. However, c0 , c1 , c2 are not easy to estimate, since they can be nonlinearly-occurring parameters with respect to friction. This is a typical static model developed in early friction research and is also suitable for hysteresis researches.

3.3.

The Jiles-Atherton model

In modelling magnetic components, it is required to accurately represent the hysteresis behavior of the magnetic core material used in these components in Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, the simulation model. One model that has been used quite widely is the Jiles

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and Atherton theory [19] which shows how the parameters for the model may be extracted from a set of measured data for a major hysteresis loop but does not consider arbitrary loop sizes. In its original form the Jiles-Atherton model leads to the following expression for the differential susceptibility [20]: dM 1 − c¯ Man − M c¯ ∂Man = + ∂M ∂M ˜ an an dH ∂He ¯ an − M) 1 − c¯α¯ 1 − c¯α¯ ∂He δk − α(M ∂He

(3)

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˙ the effective magnetic field strength He = H + αM, and anwith δ = signH, hysteretic magnetization Man . The latter is usually modeled by a Langevin or Ising model.     He A Man = MS coth (4) − A He where MS is model parameter that represents the saturation magnetization and A is model parameter that characterizes the shape of the anhysteretic magnetization. c, ¯ α¯ and k˜ are model parameters that can be determined experimentally. The Jiles-Atherton model is often used for describing hysteresis, for it can be easily implemented. Moreover, the Jiles-Atherton model requires few memory storage, as its status is completely described by only five parameters.

3.4.

The Preisach model

For the representation of hysteresis, behavioral models are also often used to approximate hysteresis independently of the physical background, which makes them general. As a result their model parameters have no clear physical meaning. The best-known representative of such class are the Preisach type models. Consider hysteresis operators [21] as shown in Figure 4, where α and β are the values of the input u, at which the output switches between unity and zero. A rectangular loop on the input/output graph represents one operator. As the input u is monotonically increased from zero, the ascending branch abcde is followed. However, when the input is monotonically decreased from +∞ to 0, the output follows along the descending branch ed f ba. Associated with each of these operators, γˆ αβ is an arbitrary weight function µ(α, β), termed the Preisach function. The Preisach operator is constructed as a weighted superposition of relays, that is f (t) =

Z

α≥β

Z

µ(α, β)ˆγαβu(t)dαdβ

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(5)

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Figure 4. Hysteresis operator

3.5.

The LuGre model

In this subsection, in order to consider the hysteresis, the LuGre model presented in [16] is discussed. It is a dynamical description of the friction-induced hysteresis. This model considers two rigid bodies that make contact through elastic bristles. When a tangential force is applied, the bristles will deflect like springs which gives rise to the friction force. The contact surface effects are lumped into an average asperity deflection z that is modeled by: z˙ = x˙ −

|x| ˙ z. h(x) ˙

(6)

The first term gives a deflection that is proportional to the integral of the relative velocity. The second term asserts that the deflection z approaches a steady state value zs given by zs = h(x)sgn( ˙ x), ˙

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while x˙ is constant.

(7)

124

S. N. Huang and K. K. Tan Lemma 1 [16] The internal state z is bounded. The function h is given by 2

h(x) ˙ =

˙ x˙s ) Fc + (Fs − Fc )e−(x/ , σ0

(8)

where Fc is the Coulomb friction level, Fs is the level of the striction force, and x˙s is the Stribeck velocity. The force generated from z is described as F = σ0 z + σ1 z˙ + σ2 x, ˙

(9)

where σ0 the stiffness,σ1 the damping coefficient and σ2 the viscous friction coefficient. Though the LuGre model is developed for the frictional force, it can represent hysteretic phenomena. In this chapter, we would like to use the LuGre model for the controller design.

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4.

Modeling of the Piezoelectric Actuator

The use of the PA systems for an accurate control depends greatly on the built model of the control system. The models of the hysteresis discussed above only represent some nonlinear parts in the PA systems and the electrical signals in the PA systems are not involved in those models. The purpose of the model presented in this section is to map the relationship between input and output at the electrical port of the PA system in a special form that can be represented by a set of equations. There are three classes of PA models: The linearized constitutive model [2], mechanical model combined with circuit [14, 22] and motion model [13, 14]. Since the mechanical model presented in [13] is a traditional differential equation, it is more easy to be used in the controller design. In the following contents, this model will be discussed. For the PA motion system, Newton’s second law can be applied it to get the following model: mx¨ = −Kg x − K f x˙ + Ke (u(t) − F),

(10)

where u(t) is the time-varying actuator terminal voltage, x(t) is the position of the piezoelectric element, Kg is the mechanical stiffness, K f is the damping coefficient produced by the actuator, Ke is the input control coefficient, m is the effective mass, and F is the system nonlinear hysteresis. Utilizing the Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, input(u)/output(x) signals, the rough estimates for the parameters m, Kg , K f can

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be obtained by using traditional identification techniques without considering the effects of the nonlinear disturbance. For describing the hysteresis phenomenon, the LuGre model (9) is used in comparison with other existing models, for it is a dynamical model. From the dynamical model (9), the force F can also be written as σ1 |x|z ˙ h(x) ˙ = (σ1 + σ2 )x˙ + Fd (z, x) ˙

F = (σ1 + σ2 )x˙ + σ0 z −

(11)

The first part (σ1 + σ2 )x˙ is a simple function of the velocity, while the secσ |x| ˙ ond part (σ0 − h(1 x) ˙ )z is scaled by the z due to the dynamical perturbations in hysteresis. Substituting (11) into (10) yields mx¨ = −Kg x − [K f + Ke (σ1 + σ2 )]x˙ + Ke u − Ke Fd (z, x) ˙

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5.

Controller Design

Controlling a PA can be defined as influencing it in such a way as to force it to operate in according with certain assumed requirements. Moreover, the control objective of the system (10) is to find a control mechanism for every bounded smooth desired output with bounded time derivatives so that the controlled output converges to the desired output as closely as possible. In this section, the control design will be discussed in details. We first define the tracking error as e = xd − x where xd is the desired output. In a practical process, the filtered tracking error is more commonly used, given by r = KI

Z t 0

e(τ)dτ + Kp e + e, ˙

(12)

where KI , Kp > 0 are chosen such that the polynomial s2 +Kp s+KI is Hurwitz. Differentiating r(t), (1) may be written in terms of the filtered tracking errors as m r˙ = Ke

m Kg (KI e + Kp e˙ + x¨d ) + x Ke Ke K f + Ke (σ1 + σ2 ) Piezoelectric Actuators, Nova Science Publishers,+ Incorporated, 2011. ProQuest Ebook x˙Central, u + Fd (z, x). ˙

(13)

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Since z and h(x) ˙ are bounded, |Fd (z, x)| ˙ = |(σ0 − σ1

|x| ˙ )z| ≤ s1 + s2 |x|, ˙ h(x) ˙

(14)

where s1 and s2 are constants. So we define the control input as u = s¯1 sgn(r) + s¯2 xsgn(xr) + s¯3 xsgn( ˙ xr) ˙

(15)

where s¯1 , s¯2 , and s¯3 are the upper bounds of some functions. This is a sliding mode type controller [15]. Consider the Lyapunov function Vr = 12 Kme r2 . Taking the time derivative of Vr , it follows that m r˙r Ke m Kg ≤ |(KI e + Kp e˙ + x¨d )r| + |xr| Ke Ke K f + Ke (σ1 + σ2 ) + |xr| ˙ + s1 |r| + s2 |xr| ˙ Ke −s¯1 ||r| − s¯2 |xr| − s¯3 |xr| ˙     m Kg ≤ − s¯1 − |(KI e + Kp e˙ + x¨d ) − s1 |r| − s¯2 − |xr| Ke Ke   K f + Ke (σ1 + σ2 ) − s¯3 − ˙ − s2 |xr| Ke

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V˙r =

(16)

K K +K (σ +σ ) If s¯1 > Kme |(KI e + Kp e˙ + x¨d ) + s1 , s¯2 > Kge , s¯3 > f eKe 1 2 + s2 , V˙ < 0. This shows that the controller (15) can drive r(t) asymptotically to converge to zero. Based on the analysis, the following theorem is established.

Theorem 1. Consider the plant (10) and the control objective of tracking the desired trajectories, xd , x˙d , x¨d . The control law given by (15) ensures that r(t) asymptotically converges to zero if choosing appropriate parameters s¯1 , s¯2 , s¯3 .

In Theorem 1, the parameters s¯1 , s¯2 , s¯3 are required to be chosen appropriately. Obviously, this condition is restrictive. Sometime, it is quite difficult to obtain these bound values, especially for s¯1 and s¯3 since they are related to nonlinear hysteresis. In what following, the polynomial expansion is adopted to approximate the function h(x). ˙ Here, the polynomial function is a generalized concept, including Taylor series [24], orthogonal series [25], fuzzy logic Piezoelectric Actuators, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central, system [26] and neural networks [23]. The main property of a polynomial

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function used for estimation purpose is the function approximation property. It has been proved that any continuous functions can be approximated by a polynomial function. The approximation property can be stated as follows: for a given continuous function f (x), there exist constant weight vector W such that f (x) can be represented as f (x) = W Φ(x) + ε(x), |ε(x)| < M

(17)

where Φ(x) are nonlinear basis functions and ε(x) is the error term and it is bounded. Recalling the force (11) F = (σ1 + σ2 )x˙ + Fd (z, x) ˙

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the function Fd (z, x) ˙ in F is given by   |x| ˙ σ1 zM |Fd (z, x)| ˙ = σ0 − σ1 z ≤ s1 + |x| ˙ h(x) ˙ h(x) ˙

(18)

(19)

M with the bound value zM of z. Since σh(1 zx) ˙ is a continuous function, it can be approximated by σ1 zM = W Φ(x) ˙ + ε(x), ˙ |ε(x)| ˙