Design & Build a Piezoelectric Motor

Transcription

1 School Of Mechanical Engineering Design & Build a Piezoelectric Motor Final Year Project Final Report AUTHORS: Sam Weckert & Andrew Meyer SUPERVISOR: Dr Ben S. Cazzolato

2 EXECUTIVE SUMMARY AIM The aim of this project is to determine the method of operation of the Elliptec X15G Ultrasonic Motor. In particular, an understanding of the geometry of the resonator that creates elliptical shaped tip motion is sought. Using the information gained from the Elliptec ultrasonic motor, a new motor will be designed and built. This motor will be capable of twin-axis bi-directional motion and still use a single piezoceramic element. SCOPE This research intends to investigate the Elliptec X15G ultrasonic motor using two different techniques. The first of these is to develop an accurate simulation of the motor using ANSYS finite element analysis software. Using this technique means that the design of a new motor can be achieved much more quickly, without having to build and test the model. The second method of analysis is experimental. Using this method we expect to analyse an Elliptec X15G motor to gain verification of our ANSYS simulation. One area of this research that is not being undertaken is an analytical study of the motor. This study would help to define the geometrical relationship that causes the elliptical motion at the tip. However, due to the complex geometry involved, this analysis would be extremely difficult. SIGNIFICANCE This report forms the final part of assessment for a Final Year Mechatronic Engineering research project. It has been written to show the outcomes that have been achieved from the undertaking of this project. It has also been written to form a concise reference for all of those who are interested in continuing research in the area of biaxial standing wave piezoelectric motors. RESULTS TO DATE This project was significantly more difficult than initially anticipated. At the outset, it was anticipated that developing an understanding of the operation principles behind Written by: Sam Weckert & Andrew Meyer i

3 the Elliptec motor was going to be relatively easy and this would only be a small part of the project. The majority of the project was expected to be extending the Elliptec design to a dual-axis bi-directional ultrasonic motor. In reality, much of the time has been spent determining the principle of operation and attempting some elementary optimisation in ANSYS. One early limitation in this research was the lack of highspeed data acquisition and measurement equipment. The National Instruments Data Acquisition Card and the Laser Vibrometer have proved invaluable tools in the collection of Elliptec motor data. This has allowed the production of accurate transfer functions of the Elliptec motor, resulting in a clear understanding of its operational principles. The project has had many successes. The conditions required to generate elliptical motion are understood and as such it is now possible to predict the operating frequencies from an inspection of the Bode plot. A successful ANSYS model for the Elliptec motor has been generated. The results from this model have been compared to experimental data and they have compared favourably. Using the ANSYS optimiser several elementary optimisations of the Elliptec motor were completed. These have resulted in alternative geometry designs, which would require further testing to confirm their validity. Due to time constraints a biaxial motor was manufactured before an ANSYS model could be completed. This design was then put through significant experimental testing to determine the validity of the biaxial concept. Although motion in both directions was achieved at both driving points, it was not possible to decouple the operating modes for individual operation at each driving point. This is one area of this research that could be continued in the future for this concept to be turned into a reality. Written by: Sam Weckert & Andrew Meyer ii

4 ACKNOWLEDGEMENTS This report is the result of many hours of hard work. There have been many people who have made the work much more enjoyable and have helped when we have had a problem. First and foremost, we must thank our supervisor Dr. Ben Cazzolato. Ben has helped us with almost all aspects of the project. Many of the outcomes that have been achieved would not have been completed without his support. Mr Pierre Dumuid has helped us with much of the Matlab coding used to process the results. Without his help we would still have a huge file filled with numbers and no way to graphically view our results. We also appreciate Pierre for his constant conversational ability; it has made working in the Vibrations laboratory much more fun (We apologise if you are three months behind schedule for your PhD Pierre!) Mr Rick Morgans has been a tremendous help with many aspects of the ANSYS modelling. Although ANSYS might be one of the most powerful Finite Element Analysis software packages available it sure isn t the most user friendly. Rick has helped us to generate the Elliptec motor model, which is comparable with the experimental results. The Workshop Staff and the Electronics and Instrumentation staff have made our lives much easier by helping with the manufacture of many of the components used in our testing. We still cannot believe that it is possible to drill and tap a 1mm hole so that it remains square! Thank you also to our friends and families, we appreciate your support. Sam Weckert & Andrew Meyer 2003 Written by: Sam Weckert & Andrew Meyer iii

5 TABLE OF CONTENTS EXECUTIVE SUMMARY...i ACKNOWLEDGEMENTS...iii TABLE OF CONTENTS...iv LIST OF FIGURES...vii LIST OF TABLES...x 1. INTRODUCTION LITERATURE REVIEW Types of Ultrasonic Motors Standing Wave Ultrasonic Motor Propagating Wave Ultrasonic Motor Elliptec X15G Ultrasonic Motor Benefits of the Elliptec X15G Ultrasonic Motor Limitations of the Elliptec X15G Ultrasonic Motor Other Ultrasonic Motors Mode Coupled Motors New Pathways for Research FINITE ELEMENT ANALYSIS (FEA) MODELLING Elliptec Motor Analysis Techniques Resonator Constraining Spring Piezoceramic element Final Model Comparison of FEA vs. Experimental Results Optimiser Terminology Two-Dimensional Test Case Elliptec Motor Optimisation Cost Function Problems with the Optimiser...36 Written by: Sam Weckert & Andrew Meyer iv

6 3.3.5 Biaxial Motor Possible Future Research for the FEA modelling EXPERIMENTAL WORK Workshop Manufactured Replica Resonators Damping Measurement Piezoelectric Element Determination Elliptec X15G Motor Analysis Data Collection Using a Tektronix Oscilloscope Data Collection Using a Spectrum Analyser Data Collection Using a National Instruments Data Acquisition Card (PCI ) Time Series Sampling Data Collection Using a National Instruments Data Acquisition Card (PCI ) Motor Frequency Operational Range Motor Speed and Temperature Characteristics Damaged Motor Revival Suggested Theory of Operation Biaxial Motor Testing Experimental Validation Laser Vibrometer Principle of Operation Safety PRELIMINARY COSTING Prototype Motor Mass Produced Motor CONCLUSION Overview of Completed Research Future Research...78 APPENDIX A CAD DRAWING OF ELLIPTEC REPLICA RESONATOR..81 APPENDIX B CAD DRAWING OF NEW BIAXIAL RESONATOR...82 Written by: Sam Weckert & Andrew Meyer v

7 APPENDIX C MATLAB CODE USED TO GENERATE RESULTS...83 APPENDIX D BIAXIAL MOTOR TESTING RESULTS...96 APPENDIX E EXPERIMENTAL VALIDATION RESULTS APPENDIX F ANSYS CODE Written by: Sam Weckert & Andrew Meyer vi

8 LIST OF FIGURES Figure 2.1: General Diagram of an Ultrasonic Motor [1]...3 Figure 2.2: Principle of Standing Wave Type Motor [1]...4 Figure 2.3: Principle of Propagating Wave Type Motor [1]...5 Figure 2.4: The Elliptec X15G Ultrasonic Motor [2]...5 Figure 2.5: Influence of Motor Cooling on Speed [3]...8 Figure 2.6: Tip Wear on a Motor after extended running periods mounted off-centre to the driven object [3]...8 Figure 2.7: Mode Coupling for a Rectangular Vibrator [10]...10 Figure 3.1: The Resonator (Top Left), Piezoelectric Crystal (Bottom) and Constraining Spring (Top Right) of the Elliptec X15G...13 Figure 3.2: The Defeatured Resonator...14 Figure 3.3: The CAD Generated Resonator...14 Figure 3.4: The CAD Generated Solid Model...15 Figure 3.5: The modelled Elliptec motor showing the final spring configuration...18 Figure 3.6: The Elliptec Motor showing the applied point forces used in modelling.19 Figure 3.7: Elliptec Motor showing the surfaces of the piezoceramic element to which the voltage potential was applied for a coupled field analysis...20 Figure 3.8: A comparison of the Bode Plots generated using the Full Harmonic Solution Method (Left) vs. the Modal Superposition Method (Right)...21 Figure 3.9: The final Elliptec X15G motor model after a harmonic analysis...21 Figure 3.10: Bode plot of the results from the final ANSYS model...22 Figure 3.11: Bode Plots of FEA and Experimental Results superimposed...23 Figure 3.12: Final Model Animation showing the path traced by the tip...24 Figure 3.13: A generic Elliptec motor indicating the variables that were used for optimisation...27 Figure 3.14: Optimised model shape...29 Figure 3.15: Optimised model amplitude frequency response...30 Figure 3.16: Optimised model phase frequency response...30 Figure 3.17: Optimised model shape...31 Figure 3.18: Optimised model amplitude frequency response...31 Figure 3.19: Optimised model phase frequency response...32 Figure 3.20: Optimised model shape...33 Written by: Sam Weckert & Andrew Meyer vii

9 Figure 3.21: Optimised model amplitude frequency response...33 Figure 3.22: Optimised model phase frequency response...34 Figure 3.23: Optimised model shape...35 Figure 3.24: Optimised model amplitude frequency response...35 Figure 3.25: Optimised model phase frequency response...36 Figure 3.26: The Prototype Biaxial Resonator...37 Figure 3.27: Results of theoretical testing into the length of the long resonator finger vs. vibration amplitude...39 Figure 4.1: The Workshop Manufactured Resonators (The centre resonator was anodised blue for aesthetic purposes)...41 Figure 4.2: Manufactured resonator showing the piezoceramic crystal restraining screw...43 Figure 4.3: The completed replica motor...44 Figure 4.4: Diagram of damping calculation...45 Figure 4.5: Material Properties for PZT-4 [16]...46 Figure 4.6: Motor Mounting Configuration...48 Figure 4.7: The Experimental Setup...49 Figure 4.8: Results of the time series sampling...52 Figure 4.9: Z Direction Transfer Function...53 Figure 4.10: Y Direction Transfer Function...54 Figure 4.11: X Direction Transfer Function...54 Figure 4.12: Elliptical shaped tip motion over a range of frequencies (Forward)...56 Figure 4.13: Elliptical shaped tip motion over a range of frequencies (Reverse)...56 Figure 4.14: Temperature Response of an Elliptec Piezoelectric Motor...59 Figure 4.15: Picture of deformed Elliptec Resonator (Left) and Untampered Resonator (Right)...60 Figure 4.16: Bode Plot for Damaged Motor...61 Figure 4.17: Bode Plot Comparison between Untampered Motor and Damaged Motor...62 Figure 4.18: The Effect of Phase Shift on Elliptical Shape...64 Figure 4.19: The New Biaxial Resonator...65 Figure 4.20: The New Biaxial Resonator showing tapped hole...65 Figure 4.21: Biaxial Motor Test Results Z Direction Long Finger...67 Figure 4.22: Biaxial Motor Test Results Z Direction Short Finger...67 Written by: Sam Weckert & Andrew Meyer viii

10 Figure 4.23: Results for both fingers with 14mm removed from the long finger...68 Figure 4.24: Results for both fingers with 6mm removed from the long finger...68 Figure 4.25: Mass Weighted Resonator Z Direction...69 Figure 4.26: Mass Weighted Resonator Y Direction...70 Figure 4.27: Schematic top view of the sensor head, the laser beams and the coordinate systems [14]...73 Figure 4.28: Schematic side view of the sensor head, laser beams and the coordinate system [14]...73 Figure 4.29: Laser Vibrometer with Laser Shielding...74 Figure D.1: Biaxial Motor Testing Z Direction Long Finger...96 Figure D.2: Biaxial Motor Testing Y Direction Long Finger...96 Figure D.3: Biaxial Motor Testing X Direction Long Finger...97 Figure D.4: Biaxial Motor Testing Z Direction Short Finger...97 Figure D.5: Biaxial Motor Testing Y Direction Short Finger...98 Figure D.6: Biaxial Motor Testing X Direction Short Finger...98 Figure D.7: Biaxial Motor Z Direction 14mm Removed from Long Finger...99 Figure D.8: Biaxial Motor Z Direction 12mm Removed from Long Finger...99 Figure D.9: Biaxial Motor Z Direction 10mm Removed from Long Finger Figure D.10: Biaxial Motor Z Direction 8mm Removed from Long Finger Figure D.11: Biaxial Motor Z Direction 6mm Removed from Long Finger Figure D.12: Biaxial Motor Z Direction 4mm Removed from Long Finger Figure D.13: Biaxial Motor Z Direction 0mm Removed from Long Finger Figure E.1: Background Board Vibration Figure E.2: Simultaneous Sampling Testing Figure E.3: Direction Testing X Direction Figure E.4: Direction Testing Y Direction Figure E.5: Direction Testing Z Direction Written by: Sam Weckert & Andrew Meyer ix

11 LIST OF TABLES Table 2.1: Precision comparison DC-motor and Elliptec motor [2]...6 Table 3.1: Resonant frequency comparison between simplified and CAD model...16 Table 4.1: PZT-4 material properties [16]...46 Table 4.2: Operating frequency ranges for the five Elliptec motors...57 Table 4.3: Drive Frequencies of the Biaxial Motor...66 Table 5.1: Prototype Motor Costing...75 Table 5.2: Mass Produced Motor Costing...76 Written by: Sam Weckert & Andrew Meyer x

12 1. INTRODUCTION Ultrasonic motors have long been thought of as expensive and as having too many problems to be suitable for many commercial applications. In 1987, Canon became the first company to successfully use an ultrasonic motor in their Lens drive systems. Since that time many other companies have begun to release ultrasonic motors that have been successfully used in many different products. The motors released up until now however, have suffered from two main limitations. These are: 1) They have been expensive, making them unsuitable for many low-cost applications 2) They are complex to produce and to drive preventing many smaller companies from making the change to ultrasonic motor drive With the rapid decrease in the price of suitable microcontrollers, the problem of driving an ultrasonic motor has diminished, but they still have only been suitable for high-cost applications. In 2002, Elliptec AG, a Siemens Technology-To-Business Centre start-up released an ultrasonic motor set to change the ultrasonic motor market. Unlike ultrasonic motors produced previously, the Elliptec motor consists of only three main components. All of these components can be easily produced, finally making the ultrasonic motor suitable for low cost applications. The motor can easily be driven using a low-cost microcontroller and by varying the drive frequency, the direction of the motor can be easily reversed. Although the Elliptec motor has many benefits, there are also some limitations that need to be addressed. This project aims to address some of the limitations of the Elliptec Ultrasonic motor and to extend its principle to make it suitable for a larger number of commercial applications. Written by: Sam Weckert & Andrew Meyer Page 1 of 127

13 2. LITERATURE REVIEW While electromagnetic motors still dominate in industry, it is unlikely that any drastic improvements will be made. Advances in technology demand smaller, more powerful devices. Using traditional electromagnetic motor technology it is very difficult to produce devices with the required energy efficiency [1]. Ultrasonic motors made with piezoceramic elements are unique in that their efficiency is not affected by their size [1]. Thus, ultrasonic motors are the obvious choice for making micro-motors. Ultrasonic motors have many advantages over traditional electromagnetic motors. These advantages include, low speed & high torque, silent operation, precise positioning and high mechanical efficiency [1]. Added to these advantages, the ultrasonic motor usually consists of a simple structure making them ideal for mass production. The following sections give an introduction to the different types of Ultrasonic motors and their methods of operation. A detailed introduction to the Elliptec X-15G is also given. This introduction aims to give an understanding of the reason for this research. 2.1 Types of Ultrasonic Motors Ultrasonic motors can be roughly divided into two general categories, based on the wave characteristic excited within the vibrational element. These are the standing wave and propagating wave type ultrasonic motor. The general principle of operation of an ultrasonic motor is summarized in Figure 2.1. Written by: Sam Weckert & Andrew Meyer Page 2 of 127

14 Figure 2.1: General Diagram of an Ultrasonic Motor [1] Standing Wave Ultrasonic Motor The standing-wave type is sometimes referred to as a vibratory-coupler type or a "woodpecker" type, where a vibratory piece is connected to a piezoelectric driver and the tip portion generates flat-elliptical movement. Attached to a rotor or a slider, the vibratory piece provides intermittent rotational torque or thrust. The standing-wave type has, in general, high efficiency, but lack of control in both clockwise and counter clockwise directions is a problem. The operational principle of a standing wave type ultrasonic motor is shown in Figure 2.2. Written by: Sam Weckert & Andrew Meyer Page 3 of 127

15 Figure 2.2: Principle of Standing Wave Type Motor [1] Propagating Wave Ultrasonic Motor The propagating-wave type (a surface-wave or "surfing" type) combines two standing waves with a 90-degree phase difference both in time and in space, and is controllable in both rotational directions. By means of the travelling elastic wave induced by the thin piezoelectric ring, a ring-type slider in contact with the "rippled" surface of the elastic body bonded onto the piezoelectric is driven in both directions by exchanging the sine and cosine voltage inputs. Another advantage is its thin design, which makes it suitable for installation in cameras as an automatic focusing device. A typical implementation of a propagating wave ultrasonic motor is shown in Figure 2.3. Written by: Sam Weckert & Andrew Meyer Page 4 of 127

16 Figure 2.3: Principle of Propagating Wave Type Motor [1] 2.2 Elliptec X15G Ultrasonic Motor The Elliptec X15G Ultrasonic Motor is an example of a propagating wave motor. The motor combines two standing wave vibrations in order to achieve the elliptical motion at the tip of the vibrating element. By generating this elliptical shape the motor is able to operate using the micro-push method. This method advances the driven object by using thousands of small steps. A diagram of the Elliptec X15G motor is shown below in Figure 2.4. Figure 2.4: The Elliptec X15G Ultrasonic Motor [2] Written by: Sam Weckert & Andrew Meyer Page 5 of 127

17 The vibration of the motor is excited using a stacked piezoceramic element. Piezoceramics extend and contract when an electric field is applied. Stacked layers of piezoceramics are used to generate adequate deflections at low voltages. The ceramic reacts very quickly to the applied voltage and generates high forces. Stacked piezoceramic elements expand and contract in the order of 0.1% [2]. This is not enough to use them alone as a motor. Thus, the vibrating element or resonator is used to amplify the resulting excitation [2]. 2.3 Benefits of the Elliptec X15G Ultrasonic Motor The Elliptec X15G Ultrasonic Motor has many advantages over traditional electromagnetic motors. It offers advantages such as precision and a controllable speed range from 0 to 30 mm/sec with silent operation. A comparison of the Elliptec motor and a standard electromagnetic motor is shown in Table 2.1 Table 2.1: Precision comparison DC-motor and Elliptec motor [2] The motor weighs one-twelfth and has one-fifth the space requirement of a standard electromagnetic motor. Additionally, due to the use of the stacked piezoceramic element, a wide supply voltage range can be used. The motor construction is simple with only three parts (piezoelectric element, mounting spring and vibration frame). It does not require a gearbox, as the motor can produce slow motion directly. This is especially advantageous in applications where there is little room. This feature also has the added benefit of lowering the overall system cost. The speed of the motor can be easily varied using a Pulse Width Modulated (PWM) signal. Written by: Sam Weckert & Andrew Meyer Page 6 of 127

18 While the basic technology behind the motor dates back more than 35 years [2], previous piezoelectric motors required very high voltages and multiple piezoceramic elements to operate making them too expensive. Compared to present piezoelectric motors, the Elliptec X15G construction is simpler and does not require tight tolerances, making the motor substantially cheaper. 2.4 Limitations of the Elliptec X15G Ultrasonic Motor There are several limitations of the Elliptec X15G Ultrasonic motor. The most significant of these is the short lifespan of the motor. In tests completed by Visser [3] it was found that the average lifespan of the motor was 5 hours. This was equivalent to 5 kilometres of running distance (assuming that the motor was run at full speed). There were several factors that influenced this short lifespan. The high running temperature of the motor reduces the life of the piezoceramic element. This temperature can easily reach the Curie temperature of the piezoceramic material. Sustained periods of operation near this temperature degrade the performance of the piezoceramic material. The reason for this high temperature is the resonance condition at which the motor operates, and also the high frequency square wave signal used to drive the motor. The drive signal increases the temperature because the high frequency harmonics contribute to the energy input but they do not cause the motor to move, this is caused by the lower frequency components around the operating resonance frequency of the motor. The operating temperature of the motor has a significant effect on the maximum speed reached. The influence of motor cooling on speed for both natural and forced convective operation is shown in Figure 2.5. Written by: Sam Weckert & Andrew Meyer Page 7 of 127

19 Figure 2.5: Influence of Motor Cooling on Speed [3] Wear is also another factor that significantly reduced the lifespan of the motor. Work completed by Visser [3] indicated that if the motor was run at full speed for extended periods of time, tip wear became a serious problem. This effect could be exacerbated if the motor was run off-centre to the driven object. Figure 2.6 shows the effect of continuous running of an off-centre motor. Figure 2.6: Tip Wear on a Motor after extended running periods mounted offcentre to the driven object [3] Written by: Sam Weckert & Andrew Meyer Page 8 of 127

20 These two limitations lead to serious questions concerning the reliability of the Elliptec Ultrasonic motor. Visser s work found that although the Elliptec Ultrasonic motor is not robust, it might be suitable for certain applications provided that the design is improved. Visser further recommends that piezoelectric actuators are a good option for active control applications, giving good positional response and disturbance rejection of up to 50% over a narrow bandwidth. 2.5 Other Ultrasonic Motors The Elliptec Ultrasonic motor is unique in that it requires only one piezoceramic element, and operates on a relatively low voltage. Other similar designs use multiple piezoceramic elements to produce an elliptical motion at the tip of the resonator [4-6]. Although the designs appeared to have much simpler geometry and design procedure, the cost of producing these would be greatly increased due to the multiple piezoceramic elements that are required. Since these designs do not require modal coupling of two resonant frequencies to be excited, the geometry is less critical than the Elliptec Ultrasonic motor. Consequently, it is possible to analyse these motors analytically. Two of these designs use a similar configuration to the Elliptec piezoelectric motor but use three piezoceramic elements to generate the elliptical motion [5, 6]. Effectively, these designs have two resonance frequencies, one that excites a longitudinal mode and another that excites a bending mode. By applying these frequencies at two separate piezoceramic elements, elliptical motion can be generated at the tip. By adjusting the phase between the two signals it is possible to alter the shape and orientation of the generated ellipse. Another motor frequently encountered in research papers was the Linear Propagating Wave type. These motors are the only type encountered that use a piezoceramic stack [7, 8]. Using a piezoceramic stack meant that the motor could be run on voltages as low as 3 V P-P [7]. These motors operate on the superposition of two standing waves. 2.6 Mode Coupled Motors Several of the research papers reviewed dealt with motors that operated similarly to the Elliptec Ultrasonic Motor. These motors required that the geometry be such that two vibration modes coupled together. These analyses were completed using a rectangular vibrator [9, 10]. This research found that for all practical cases the width Written by: Sam Weckert & Andrew Meyer Page 9 of 127

21 of the vibrator does not need to be considered [10]. In this analysis the ratio of the coupled frequencies is plotted against a length ratio of the vibrator. This information is shown in Figure 2.7. Figure 2.7: Mode Coupling for a Rectangular Vibrator [10] This research found that the 1 st Longitudinal and the 2 nd bending mode coincide when the length ratio (d/l) is 0.28 [10]. The two vibration modes were chosen because they allowed support of the vibrator at the central nodal point and they had sufficient amplitude to cause significant motion. Using this technique it should be possible to couple together any two bending and longitudinal resonance modes [10]. This research also concluded that to couple the modes successfully, external asymmetric geometry or internal non-linearity must be added to the vibrating element although this could not be proven analytically. 2.7 New Pathways for Research As can be seen from the information presented above, there has been a large amount of research done on the general operation theories of ultrasonic motors. Due to the geometric simplicity of motors that use multiple piezoceramic elements there has been considerable effort invested into determining accurate analytical models for the Written by: Sam Weckert & Andrew Meyer Page 10 of 127

22 purposes of design. There has been very little research into the phenomenon of modal coupling of two resonances at a particular frequency. There was no literature found which discussed the possibility of a motor being reversible at a second independent frequency. Obviously, the understanding of this modal coupling and how to design geometry to cause this effect is a primary objective. There was also no mention in the literature reviewed of an ultrasonic motor capable of twin-axes motion. This type of motor would be especially useful in applications that are very cost sensitive. A twin-axis ultrasonic motor based on the mode coupling principle would be economical to produce and would take up less space than two conventional electromagnetic motors. This device would be even more cost effective if a single piezoceramic element could be used as an actuator. The feasibility and design of a twin-axis motor is another primary objective of this project. The information presented above has shown that the Elliptec X15G ultrasonic motor has many limitations that need to be addressed before the motor will be suitable for inclusion into applications. The primary problem is the heat generated by the entire vibrational assembly. By altering the geometry of the vibrator it should be possible to obtain geometry capable of much more efficient heat dissipation. This would only be possibly with an increased understanding of the mode coupling principle. A secondary problem of the Elliptec X15G is the wearing of the tip under normal operating conditions. Again, if the operating theory were known it would be possible to change the tip material so that the wear would not be a significant issue. In conclusion, this project is required to determine the method of operation and in particular the theory behind the modal coupling phenomenon. Using this information it is hoped that many of the limitations of the current Elliptec X15G ultrasonic motor can be removed. This research also intends to determine the feasibility of a twin-axes bi-directional ultrasonic motor. Written by: Sam Weckert & Andrew Meyer Page 11 of 127

23 3. FINITE ELEMENT ANALYSIS (FEA) MODELLING The Finite Element Analysis software package, ANSYS, was used for the computermodelling component of the project. This software was chosen because it is a product that is widely used in industry and consequently there is a significant amount of support available. It is capable of performing many different types of analyses, however, for this research we were particularly interested in ANSYS because of its ability to analyse coupled electro-mechanical systems. 3.1 Elliptec Motor Analysis Techniques Analytical solutions for the resonance frequencies of a structure can only be easily calculated for simple shapes such as a cantilever beam with constant cross section. Although the geometry of the Elliptec motor is still relatively simple, the mathematics to find an analytical solution for the resonance frequencies would be extremely complex. Consequently a Finite Element model was employed. The aim was to generate results that compared favourably with experimental results. This would help gain insight on the operating principles behind the motor and also give confidence for the validity of future ANSYS models using our own designs. The Elliptec motor can be dissected into three main components. They are: 1. The Resonator 2. The Constraining Spring 3. The Piezoceramic Element These components are shown in Figure 3.1. Written by: Sam Weckert & Andrew Meyer Page 12 of 127

24 Figure 3.1: The Resonator (Top Left), Piezoelectric Crystal (Bottom) and Constraining Spring (Top Right) of the Elliptec X15G Resonator Two versions of the resonator were employed during the progression towards the final FEA model for the Elliptec motor. In the beginning stages of the project a simplified geometry model of the Elliptec motor was generated with the aid of some code written by the group s supervisor, Dr Ben Cazzolato [1]. The resonator was constructed simply using two rectangular blocks glued onto an annulus. This represented a de-featured model of the actual resonator. Figure 3.2 shows a picture of the defeatured resonator. Written by: Sam Weckert & Andrew Meyer Page 13 of 127

25 Figure 3.2: The Defeatured Resonator The second version was derived from a CAD model and consequently had more detail included. The CAD generated model is shown in Figure 3.3. Figure 3.3: The CAD Generated Resonator The CAD model of the Elliptec motor was generated from drawings in the Elliptec manual in order to have replica resonators manufactured by the workshop. Due to the simplified geometry of the initial ANSYS model it was considered advantageous to use this CAD drawing to generate a geometrically more accurate model. The solid model generated in AutoCAD is shown in Figure 3.4. Written by: Sam Weckert & Andrew Meyer Page 14 of 127

26 Figure 3.4: The CAD Generated Solid Model Licensing restrictions prevented the CAD drawing from being converted into a Para- Solid file. Consequently the CAD drawing was converted to an IGES file and this was imported into the ANSYS workspace. This process presented many problems. ANSYS would not generate a solid model from the imported IGES file. The only useful part of the imported model was the lines representing the outline of the resonator. Areas were created from the corresponding keypoints and then volumes were generated from these areas. This was a time consuming process, however the new model contained many geometric features missing from the original model. The cutout on the tip, the holes for the spring attachment and the non-circular nature of the centre hole represent the main added features. An additional problem was that the imported IGES file always came into ANSYS in mm regardless of the IGES conversion settings. Non-SI units were persevered with for a while, however difficulties soon arose when specifying material properties and spring constants. Consequently, using the copy and scale command an SI model 1/1000 th the size was generated. The larger model was then deleted. Written by: Sam Weckert & Andrew Meyer Page 15 of 127

27 Simplified Geometry Model CAD Drawing Based Model Number Frequency (Hz) Frequency (Hz) % Change N/A 2 ~0 ~0 N/A 3 ~0 ~0 N/A E E E E E E E E E E E E E E E E Table 3.1: Resonant frequency comparison between simplified and CAD model Once the models had been constructed in ANSYS a modal analysis was conducted on both. The results illustrating the differences between the two models are shown in Table 3.1. The changes in the resonant frequencies were significant, especially in the higher frequencies, and consequently the progression to the geometrically more accurate model was a valid development. In meshing the resonator it was important to have sufficient mesh density to ensure accurate results however an excessive number of elements would lead to large solving times and an increase in disk space required for the solution. Additionally the number of elements available was restricted to by the version of ANSYS available. The final mesh used satisfied the necessary six elements per wavelength resolution mandatory for a Finite Element Modal Analysis [11], however was slightly short of the recommended 20 elements per wavelength [12]. Rerunning the analysis with a finer mesh checked the mesh density of the model. The results did not change, hence indicating that the original mesh was of sufficient density. It was important to keep solution times as low as possible as any gains here would be even more beneficial when using the optimiser where multiple iterations would be performed. By modifying the solution settings solution times could have been reduced even further. Written by: Sam Weckert & Andrew Meyer Page 16 of 127

28 Many of the nodal outputs such as stress and strain were not required and hence their exclusion would have saved both solution time and disk space. Another important factor that needed to be included in the Finite Element Analysis was the damping ratio of the resonator and its surrounding structure. By testing five Elliptec X15G motors an average value of damping of 1.75% was determined. The procedure used to determine this value is covered in Section Constraining Spring Many approaches were attempted and a considerable amount of time was spent before a satisfactory representation for the constraining spring was achieved. Initially threedimensional longitudinal springs were used to approximate the behaviour of the constraining springs. These were attached to each corner node on the resonator and the anchor points for each spring were fully constrained. Each anchor node was located by translating the corner node a distance of 1mm in all three orthogonal directions (X, Y, Z) such that the spring s orientation was comparable to the inside diagonal of a cube. The four three-dimensional longitudinal springs should have fully constrained the model however a modal analysis revealed three free body modes. Four three-dimensional torsional springs were overlaid on top of the existing longitudinal springs but this caused an error message during the modal analysis and no solution was obtained. In order to understand why using the four three-dimensional longitudinal springs resulted in free body modes, various simplified two-dimensional models were generated to help solve the problem. Through this, it was discovered that the threedimensional longitudinal springs only acted along the line of application and consequently were essentially equivalent to one-dimensional springs with their nodal coordinate systems rotated. This explained the three free body modes in the previous model. One-dimensional X, Y and Z longitudinal springs were attached using coincident nodes at the four locations shown in Figure 3.5. Using these twelve springs the modal was fully constrained with no free body modes. Written by: Sam Weckert & Andrew Meyer Page 17 of 127

29 Figure 3.5: The modelled Elliptec motor showing the final spring configuration Modelling the constraining spring with a series of beam elements was also attempted. The properties of spring steel and geometry similar to the springs on the Elliptec motor were used. However a modal analysis revealed a high number of resonant frequencies associated with the springs and consequently this concept was abandoned Piezoceramic element Two approaches were attempted in modelling the piezoceramic element. The first approach consisted of applying oscillating point forces to represent the piezoceramic element expanding and contracting. These were located at the centroid of the contacting area between the piezoceramic element and the resonator. Figure 3.6 shows the motor with the applied point forces. Written by: Sam Weckert & Andrew Meyer Page 18 of 127

30 Figure 3.6: The Elliptec Motor showing the applied point forces used in modelling The piezoceramic element was meshed with the same type of elements used to mesh the resonator but with different material properties. The piezoceramic material is anisotropic and consequently the material properties for each orthogonal direction were derived from a material properties matrix as discussed in Section 4.3. The second approach utilised Solid 98 Tetrahedral Coupled-Field elements to mesh the piezoceramic block. These allowed the application of a voltage potential across the piezoceramic element to simulate the expansion and contraction of the block. The surfaces on which the potential was applied are shown in Figure 3.7. Written by: Sam Weckert & Andrew Meyer Page 19 of 127

31 Figure 3.7: Elliptec Motor showing the surfaces of the piezoceramic element to which the voltage potential was applied for a coupled field analysis Two different solution methods were used to generate the spectral information from the motor model. The initial technique used was the modal superposition method. This method uses the natural frequencies and mode shapes from the modal analysis to characterize the dynamic response of a structure to harmonic excitations. To complete an analysis using this method it is necessary to firstly complete a modal analysis. The resulting modes are then combined and used to give the results of the harmonic analysis. The major benefit of this method was that it was extremely fast, however, it was unable to be used in cases where a coupled analysis was required. The second solution method was a full harmonic analysis. This method generated all of the equations of motion of the system and then solves them for each particular frequency step. Because of the significant amount of computation required this method was much slower than the modal superposition method and required large amounts of disk space. However, this method was the only one capable of completing coupled field analyses. In the case of the coupled field analysis it was necessary to perform a full harmonic analysis on the model. Consequently the point forces approach using the modal Written by: Sam Weckert & Andrew Meyer Page 20 of 127

32 superposition method was employed as the preferred choice for modelling the piezoceramic element. A check was conducted to ensure that the results using the modal superposition method were the same as those obtained using the full harmonic analysis. The results of this test are shown in Figure 3.8. Figure 3.8: A comparison of the Bode Plots generated using the Full Harmonic Solution Method (Left) vs. the Modal Superposition Method (Right) Final Model The final Finite Element Analysis model for the Elliptec motor showing the three components discussed in Sections 3.1.1, and is shown in Figure 3.9. Figure 3.9: The final Elliptec X15G motor model after a harmonic analysis Written by: Sam Weckert & Andrew Meyer Page 21 of 127

33 The ANSYS results obtained from the final model are shown in Figure Figure 3.10: Bode plot of the results from the final ANSYS model 3.2 Comparison of FEA vs. Experimental Results It was important to verify that the FEA model of the Elliptec motor was an accurate representation of the physical system. The results from the FEA model compared very favourably against experimental results obtained using the laser vibrometer. Written by: Sam Weckert & Andrew Meyer Page 22 of 127

34 Figure 3.11: Bode Plots of FEA and Experimental Results superimposed As can be seen in Figure 3.11 the phase of the FEA and experimental results are very similar, especially within the motors operating range. The resonance peaks for the two operating frequencies are evident in both sets of data however in the FEA results these peaks are slightly more spread apart. It was proposed that this difference was due to neglecting the mass of the spring. This situation could be rectified by adding inertial point masses to the spring locations in the FEA model. An approximate magnitude for the mass of the spring was calculated using the density of spring steel and the geometry of the spring. This was then equally distributed to the four locations on the resonator. This modification did not significantly change the position of the two resonance peaks so the inertial point masses were increased in magnitude, each time by an order of magnitude. A trend soon emerged that increasing the magnitude of the inertial point masses actually caused the two resonance peaks to spread even further. This was not what was hoped for and hence this idea was abandoned. Other possible reasons to account for the difference between the FEA and experimental results may include: 1. Neglecting the effects of temperature in the FEA model. 2. Small inaccuracies in the geometry of the model. 3. Incorrect material properties used for the aluminium in the resonator. Written by: Sam Weckert & Andrew Meyer Page 23 of 127

35 4. Neglecting the added weight of solder and wires on piezoceramic element. 5. Incorrect material properties for the piezoceramic block. Realistically there will always be some discrepancy between theoretical and experimental results so the comparison between the two sets of results was still considered favourable. The FEA model was driven at one of the expected operating frequencies and an animation of elliptical motion at the tip was generated. A still frame from this animation sequence is shown in Figure 3.12 Figure 3.12: Final Model Animation showing the path traced by the tip This gave confidence in the Finite Element method and provided some assurance that future models designed in ANSYS that exhibited elliptical motion would actually do so when physically manufactured and tested using the laser vibrometer. It also confirmed that a contact analysis was not necessary. Written by: Sam Weckert & Andrew Meyer Page 24 of 127

36 3.3 Optimiser In order to achieve the ultimate goal of a dual axis motor, elliptical motion would be required at one tip at two frequencies (forwards and reverse), and additionally at another tip at two different frequencies (forwards and reverse). Changing the geometry to achieve these characteristics using only trial and error would be an extremely time consuming task. Fortunately, the ANSYS software offers an optimisation routine to accommodate such a problem Terminology Some important terminology used in the ANSYS design optimisation will now be defined for future reference [12]. Design Variables (DVs) are independent quantities that are varied in order to achieve the optimum design. Upper and lower limits are specified to serve as constraints on the design variables. These limits define the range of variation for the DV. Typically, design variables include dimensions of the geometry of a structure. State Variables (SVs) are quantities that constrain the design. They are also known as dependent variables, and are typically response quantities that are functions of the design variables. A state variable may have a maximum and minimum limit, or it may be single sided, having only one limit. Maximum stress or strain constraints on a design are usually incorporated into the design in this way. The Objective Function (or Cost Function) is the dependent variable that is to be minimised. It should be a function of the DVs, that is, changing the values of the DVs should change the value of the objective function. A Design set is simply a unique set of parameter values that represents a particular model configuration. Typically, a design set is characterized by the optimisation variable values; however, all model parameters (including those not identified as optimisation variables) are included in the set. A feasible design is one that satisfies all specified constraints - constraints on the SVs as well as constraints on the DVs. If any one of the constraints is not satisfied, the design is considered infeasible. The best design is the one that satisfies all constraints and produces the minimum objective Written by: Sam Weckert & Andrew Meyer Page 25 of 127

37 function value. If all design sets are infeasible, the best design set is the one closest to being feasible, irrespective of its objective function value Two-Dimensional Test Case To gain familiarity with this section of the software, a two-dimensional test case was attempted. A simple two-dimensional cantilever beam was generated and it was desired to maximise the displacement amplitudes at the upper end node of the beam in both translational directions for two particular excitation frequencies. Additionally it was required to have the ability to designate specific phase differences between the two displacements at each frequency. This problem was very similar to what was required to optimise a single axis motor and consequently the code could be modified and reused in future work. The geometry of the two-dimensional beam (length and width) was parametised. These parameters (length and width) represented the design variables. This would allow the ANSYS optimiser to perform multiple iterations changing the geometry each time until the problem was optimised. Harmonic analyses were performed at the two particular excitation frequencies and the complex displacements at the node of interest were retrieved from the results file and assigned to parameters for use in the cost function. The major hurdle in retrieving the data was realising that ANSYS stores the real part of a complex variable in one data set and the imaginary part in a subsequent data set. The displacement amplitudes and phase information were incorporated into the cost function using squared error terms such that increasing amplitudes would decrease the cost function Elliptec Motor Optimisation After successfully completing the two-dimensional optimisation test case, the code was to be extended to enable the optimisation of a single axis motor. From the operating frequency ranges obtained experimentally for the five Elliptec motors, the mean averages were: 79.3kHz for one direction and 97.65kHz for the other direction. The determination of these results is discussed in Section 4.5. It was envisaged to try optimising the Elliptec FEA model for these two frequencies (80kHz and 100kHz for simplicity) and hopefully obtain geometry similar to the actual motors, or Written by: Sam Weckert & Andrew Meyer Page 26 of 127

38 alternatively come up with bigger displacements than the existing Elliptec motor achieves. It was necessary to build the model from scratch to enable parametisation of the geometry. It was not possible to simply import the model from a CAD drawing as had been done in the past. Furthermore it was not possible to refer to node, line, area, volume or element numbers in the code when applying constraints, forces, etc as these numbers would change as the design variables and consequently the geometry changed. To overcome this, a select by location method was employed and also various important nodes such as at the tip and spring locations were renumbered with specific values. Figure 3.13 shows how the geometry was parametised, resulting in five design variables, with the fifth being the thickness of the resonator into the page. W 1 W 1 L 1 L 1 Figure 3.13: A generic Elliptec motor indicating the variables that were used for optimisation In practice any optimised single axis or dual axis motor that was designed would need to use the same piezoceramic element as the Elliptec motor because custom shaped piezoceramic elements would be expensive to purchase in the small volumes required for prototyping. Consequently, in the optimisation routine the inner radius of the circular crystal cavity needed to remain constant at the value used by the Elliptec motor. Furthermore the outer radius was also made constant. It was thought that the method of insertion of the piezoceramic block into the resonator cavity involved elastically deforming the resonator. If the cavity wall was too thin or too thick then this operation may not be possible and hence the outer radius used in the Elliptec motor was also retained. In hindsight the outer radius could have been included as a Written by: Sam Weckert & Andrew Meyer Page 27 of 127

39 design variable because an alternate method of inserting the crystal was later employed. Due to problems with poor coupling between the resonator and the piezoceramic element this alternate method involved using a small screw to hold the piezoceramic element under compression. Point forces were applied at the centroid of the contact area between the crystal and resonator to represent the crystal expanding and contracting. This was opposed to using piezoelectric (coupled field) elements. Coupled field elements would significantly increase solving time and may not even work with the optimiser. Because the optimiser performs multiple iterations of the code it was important to keep the solving time of each iteration to a minimum. ANSYS offers two main methods for the design optimisation. The subproblem approximation method is an advanced zero-order method that uses approximations (curve fitting) to all dependent variables (SVs and the objective function). The first order method uses derivative information, that is, gradients of the dependent variables with respect to the design variables. A user defined optimising algorithm can also be used. The subproblem approximation method was chosen as it generally less computationally intense and more likely to find a global minimum rather than just a local minimum, which the first order method can be prone to Cost Function The phase requirements for operation of the Elliptec motor were not well understood and consequently a variety of optimisations were performed using different phase constraints. The basic cost function excluding any phase constraint is shown in Equation 3.1. J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 (3.1) Where: J = the cost function X 1 = X displacement at the tip when excited at 80kHz Z 1 = Z displacement at the tip when excited at 80kHz X 2 = X displacement at the tip when excited at 100kHz Z 2 = Z displacement at the tip when excited at 100kHz Written by: Sam Weckert & Andrew Meyer Page 28 of 127

40 Modifications were then made to this to incorporate various phase constraints: 1) Positive 90 degrees phase difference at 80 khz and -90 degrees phase difference at 100 khz. The cost function is shown in Equation 3.2. J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 + ( (90-1 )/90) 2 + ((-90-2 )/90) 2 (3.2) Where: 1 = the phase difference between the bending and longitudinal vibrations at 80kHz, 2 = the phase difference between the bending and longitudinal vibrations at 100kHz The model shape and amplitude and phase frequency response plots for the optimal solution using this cost function are shown in figures 3.14, 3.15 and 3.16 respectively. Figure 3.14: Optimised model shape Written by: Sam Weckert & Andrew Meyer Page 29 of 127

41 Figure 3.15: Optimised model amplitude frequency response Figure 3.16: Optimised model phase frequency response 2) Negative 90 degrees phase difference at 80 khz and +90 degrees phase difference at 100kHz. The cost function is shown in Equation 3.3. Written by: Sam Weckert & Andrew Meyer Page 30 of 127

42 J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 + ( (90-1 )/90) 2 + ((-90-2 )/90) 2 (3.3) The model shape and amplitude and phase frequency response plots for the optimal solution using this cost function are shown in figures 3.17, 3.18 and 3.19 respectively. Figure 3.17: Optimised model shape Figure 3.18: Optimised model amplitude frequency response Written by: Sam Weckert & Andrew Meyer Page 31 of 127

43 Figure 3.19: Optimised model phase frequency response 3) Phase difference at 80kHz with an opposite sign to the phase difference at 100kHz. This was achieved by multiplying the two phase differences together to form a state variable and then setting the maximum for this state variable to zero so that it would always be negative for a feasible design. This cost function is shown in Equation 3.4. J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 (3.4) The model shape and amplitude and phase frequency response plots for the optimal solution using this cost function are shown in figures 3.20, 3.21 and 3.22 respectively. Written by: Sam Weckert & Andrew Meyer Page 32 of 127

44 Figure 3.20: Optimised model shape Figure 3.21: Optimised model amplitude frequency response Written by: Sam Weckert & Andrew Meyer Page 33 of 127

45 Figure 3.22: Optimised model phase frequency response 4) No phase constraint. The cost function is shown in Equation 3.5 J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 (3.5) The model shape and amplitude and phase frequency response plots for the optimal solution using this cost function are shown in figures 3.23, 3.24 and 3.25 respectively. Written by: Sam Weckert & Andrew Meyer Page 34 of 127

46 Figure 3.23: Optimised model shape Figure 3.24: Optimised model amplitude frequency response Written by: Sam Weckert & Andrew Meyer Page 35 of 127

47 Figure 3.25: Optimised model phase frequency response Problems with the Optimiser With only the four displacement amplitudes in the cost function and no phase constraints in either the cost function or as a state variable, the optimiser was run and returned geometry different to the Elliptec motor as its optimal solution. A separate harmonic analysis was then performed using the Elliptec motor dimensions to compare the displacement amplitudes between the so-called optimised model and the Elliptec motor model. Unfortunately, the Elliptec motor model achieved higher amplitudes and the cost function value calculated using these amplitudes was actually lower and consequently represented a better design compared with that for the optimised model. A model using the Elliptec motor dimensions represented a possible design set in the optimisation and consequently should have come up as the optimised solution. It was deduced that this problem was due to the subproblem approximation method not being the most suitable for this problem. The group s supervisor, Dr Ben Cazzolato [11], recommended using the first order method instead. This method was prone to finding only local minima. Consequently, in order to get the optimiser to converge on a solution with geometry the same or similar to the Elliptec motor it would be necessary to have the initial design set close to the Elliptec motor model. For future optimising it would be very difficult to obtain an optimal solution using Written by: Sam Weckert & Andrew Meyer Page 36 of 127

48 this method, as it requires an approximate optimal solution to be known beforehand. A better method would be to generate various random design sets over the design space and then use each of these design sets as a starting point for the first order method. Due to time constraints the group was unable to pursue this area any further Biaxial Motor Having been unsuccessful in mastering the ANSYS optimiser, it was decided to use the optimiser with only one design variable to help aid the design of a dual axis motor. A second set of resonators was manufactured by the workshop with the geometry as shown in Figure These were based on the Elliptec motor with an extra finger added onto the short end. tip2 tip1 Figure 3.26: The Prototype Biaxial Resonator The single design variable for the simplified optimisation was chosen as the length of this extra finger in an attempt to determine the optimum length. The sweep function in the ANSYS optimiser was used. This subdivided the length into ten increments between no extra finger present (geometry the same as the Elliptec motor) and full finger present (geometry as shown in Figure 3.26). The tip displacements and corresponding cost function were evaluated for each case. The aim was to maximize the tip1 displacements at 40 khz and 60kHz, representing the possibility for forwards Written by: Sam Weckert & Andrew Meyer Page 37 of 127

49 and reverse motion, and then the tip2 displacements at 80kHz and 100kHz. No phase constraints were included. Furthermore tip2 displacements at 40kHz and 60kHz and tip1 displacements at 80kHz and 100kHz were not penalised. The cost function used is shown in Equation 3.6. J = (1- X 1 ) 2 + (1- Z 1 ) 2 + (1- X 2 ) 2 + (1- Z 2 ) 2 + (1- X 3 ) 2 + (1- Z 3 ) 2 + (1- X 4 ) 2 + (1- Z 4 ) 2 (3.6) Where X 1 = X displacement at tip1 when excited at 40kHz Z 1 = Z displacement at tip1 when excited at 40kHz X 2 = X displacement at tip1 when excited at 60kHz Z 2 = Z displacement at tip1 when excited at 60kHz X 3 = X displacement at tip2 when excited at 80kHz Z 3 = Z displacement at tip2 when excited at 80kHz X 4 = X displacement at tip2 when excited at 100kHz Z 4 = Z displacement at tip2 when excited at 100kHz The optimum solution for the given cost function was obtained with the length of the extra finger set to 6.09mm. The graph shown in Figure 3.27 illustrates how the tip displacements changed as the length of the extra finger was increased. Written by: Sam Weckert & Andrew Meyer Page 38 of 127

50 Figure 3.27: Results of theoretical testing into the length of the long resonator finger vs. vibration amplitude The results appeared promising for 6mm and 13mm lengths of the extra finger. The plan was to trim the extra finger to these lengths and test the resulting motor with the laser vibrometer. 3.4 Possible Future Research for the FEA modelling Attempts to design a dual axis motor have so far been plagued with decoupling problems. When elliptical motion has been present at one tip it has also unfortunately been present at the other tip at the same frequency. It may be necessary to penalise amplitudes in the cost function to try and decouple the motion of the two tips. Ideally when there is elliptical motion at one tip there should be no elliptical motion at the other tip, however it would be sufficient to still allow the right conditions for elliptical motion at the second tip but just ensure the amplitudes were sufficiently low to prevent any motion. Greater experience using the different optimiser methods is recommended. This would lead to improved results using the ANSYS optimiser. Generating various random design sets over the design space and then using each of these design sets as a starting point for the first order method would hopefully prove a successful method Written by: Sam Weckert & Andrew Meyer Page 39 of 127

51 for optimising the single axis and also dual axis motors. Multiple computers (one computer per random design set) could be used for this method in order to obtain an optimised model in a realistic timeframe. The various possible reasons for the slight difference between the FEA and experimental results could be further investigated and this would hopefully lead to an improved model for the Elliptec motor. Written by: Sam Weckert & Andrew Meyer Page 40 of 127

52 Final Report 4. EXPERIMENTAL WORK 4.1 Workshop Manufactured Replica Resonators Three resonators were manufactured by the workshop using the CAD drawings of the Elliptec resonator. These resonators are shown in Figure 4.1. By using the crystal and spring from an Elliptec motor it was hoped to construct a working motor using one of these replica resonators. The type of aluminium alloy used by Elliptec was unknown, however the density and Young s modulus of elasticity were found not to vary significantly between the different aluminium alloys [13]. These were the only material properties assumed to be critical to the performance of the motor and consequently the type of aluminium alloy used by the workshop was unimportant. Figure 4.1: The Workshop Manufactured Resonators (The centre resonator was anodised blue for aesthetic purposes) Transplanting the spring from the Elliptec motor to the replica resonator was a relatively simple procedure. The crystal however posed many problems. Careful inspection of the replica resonator geometry revealed slight differences when Written by: Sam Weckert & Andrew Meyer Page 41 of 127

53 compared with the Elliptec motor resonator. In particular, the cavity for the piezoceramic material was slightly larger. The CAD drawings upon which the replica resonators were manufactured were based on dimensioned scale drawings from the Elliptec motor technical manual. In hindsight it would have been advantageous to actually measure the critical dimensions, rather than blindly relying on the accuracy of the drawings in the technical manual. The piezoceramic material was carefully removed from the Elliptec motor by gently squeezing the resonator lengthwise. Pieces of thin brass sheet were used to pack the piezoceramic material into the slightly oversized hole of the replica resonator. Due to the higher density and Young s modulus of elasticity of brass, it would have been better to use aluminium sheet to the pack the crystal, but this was unavailable at the time. The motor was swept slowly over the expected operating frequency range, however the tip would not turn the rotor in either direction. It was then noticed that although the piezoceramic material seemed tightly packed when the motor was not operating, when supplied with an excitation signal the piezoceramic material became quite loose. This was due to the expansion and contraction of the piezoceramic material and consequently emphasized the importance of having the piezoceramic ceramic material held initially in compression in the cavity. As a next attempt, the replica resonator was lightly deformed so the piezoceramic material was flush with the cavity edge. Superglue was then used to secure the piezoceramic material to the cavity s inner surface. The motor was slowly swept over the expected operating frequency range, however no movement transpired. The replica resonator was then further squeezed in the vice such that the piezoceramic material was held in compression and did not require any superglue. Again the motor failed to operate. At this stage the geometry was now significantly different to that of the Elliptec motor resonator and hence the failure of the tip to turn the rotor was of no surprise. A second manufactured resonator was drilled and tapped so that a 1mm screw was able to be placed in the spring end of the resonator, as shown in Figure 4.2. Written by: Sam Weckert & Andrew Meyer Page 42 of 127

54 Figure 4.2: Manufactured resonator showing the piezoceramic crystal restraining screw By carefully aligning the piezoceramic crystal and tightening the screw it was possible to obtain good mechanical coupling between the piezoceramic crystal and the resonator. The completed resonator is shown in Figure 4.3. Initial testing of the manufactured resonator was disappointing as no rotation of the driven surface occurred. The manufactured resonator was analysed using the three-dimensional laser vibrometer. Examination of the Bode plot generated through this procedure showed that there was similar phase and frequency components. However, the amplitude components were significantly less than the commercial Elliptec motors. This problem implied that there was insufficient coupling between the resonator and the piezoceramic element. Consequently, it was not possible to cause high enough amplitude elliptical shaped vibration to produce motion on the driven object. By tightening the screw holding the piezoceramic element in place it was possible to increase the coupling so that the vibrations in the resonator would be of sufficient amplitude to allow operation. After the screw was tightened, hence increasing the coupling, the motor was reanalysed. In this instance the motor operated successfully in both directions at a Written by: Sam Weckert & Andrew Meyer Page 43 of 127

55 frequencies similar to the commercial Elliptec motor. A further discussion of the results of this analysis are presented in Section Figure 4.3: The completed replica motor 4.2 Damping Measurement The mechanical damping of the motors structure was required for use in the ANSYS modelling and optimisation. The damping was calculated from the resonance peaks in the frequency domain from the experimentally generated Bode plots. Firstly one of the resonance frequencies was determined. By measuring 3dB from the peak amplitude of the resonance (a factor of 2), it is possible to find a Quality Factor for the motor. This relationship is shown in Figure 4.4. The Quality Factor is directly related to the mechanical damping of the structure, as shown in Equations 4.1 and 4.2. Written by: Sam Weckert & Andrew Meyer Page 44 of 127

56 Figure 4.4: Diagram of damping calculation Q = 1 Q 2 (4.1) (4.2) Using this principle over the range of motors tested an average value of damping of 1.75% was found. This value was directly used in subsequent ANSYS modelling and optimisation as a constant damping ratio for the motor. Further discussion of this aspect of the modelling is given in Chapter Piezoelectric Element Determination In the initial ANSYS model the piezoelectric element was modelled with the same properties as aluminium. Further analysis showed that although the Young's modulus of the piezoelectric material was approximately the same as aluminium, the majority of the other material properties were significantly different. In the subsequent simulations the material properties were replaced with PZT-4, a commonly available piezoceramic material. The material properties of this material are shown in Figure 4.5. Written by: Sam Weckert & Andrew Meyer Page 45 of 127

57 Figure 4.5: Material Properties for PZT-4 [16] From Figure 4.5 it was possible to deduce the material constants for PZT-4. These values are shown in Table 4.1. In the FEA analysis the Z direction was the direction of polarization in the piezoceramic element. For uniformity, the material properties shown in Table 4.1 use the Z - direction as the polarization direction. PZT-4 Material Properties Shear Modulii (GPa) G(yz) G(xz) G(xy) Young's Modulii (GPa) E(x) 81.3 E(y) 81.3 E(z) Poisson's Ratio v(xy) v(xz) v(yz) Table 4.1: PZT-4 material properties [16] To confirm that the Elliptec piezoceramic element was indeed PZT-4, a piezoceramic element was removed from the resonator for examination. The mass of the individual element was 0.3 grams. This value was compared to an expected value using the Written by: Sam Weckert & Andrew Meyer Page 46 of 127

58 density of the piezoceramic material PZT-4. The expected mass was 0.27 grams. The actual mass was expected to be slightly higher than the theoretical value since all of the solder on the electrode sides of the crystal were unable to be removed. 4.4 Elliptec X15G Motor Analysis Data Collection Using a Tektronix Oscilloscope Initial measurements were taken using a Tektronix 4-channel digital oscilloscope. This provided points of data that were sampled at 250 khz. This gave an overall sample time of 0.4 seconds. The spectral results were calculated using a point Fast Fourier Transform (FFT), applying a Hanning window and a 75% overlap for improved resolution. The Matlab code used to generate the results can be seen in Appendix C. The motor was mounted inverted on the driving board and the laser vibrometer was positioned so that the beams converged on the end of the drive tip. In this configuration the Z-Axis runs along the centreline of the motor, the Y-Axis is vertically perpendicular to the drive tip of the motor and the X-Axis is horizontally perpendicular to the drive tip of the motor. Figure 4.6 shows the motor mounted in this configuration and the directions of measurement. Written by: Sam Weckert & Andrew Meyer Page 47 of 127

59 Final Report Z - Axis Y - Axis X - Axis Figure 4.6: Motor Mounting Configuration The motor was excited using white noise. The results obtained from this experiment were poor. Examining the coherence showed that the coherence in all three directions was very low. Examining the cross-spectral densities of the response in each direction (X, Y & Z) showed that the calculated transfer functions contained a very significant transient component. This component was due to the high quality factor (Q) and hence low damping of the vibrator assembly. This implied that the data would need to be collected over a longer time period. It was also decided that a better method of exciting the motor was using a swept sine wave. This would give the transient response enough time to decay to a low enough level so that the true transfer function could be calculated Data Collection Using a Spectrum Analyser Following the collection of the initial results, a spectrum analyser was used to see whether the coherence of the results would improve if the data were sampled over a longer period of time. The spectrum analyser was only able to collect 2048 points of data, to sample more data the spectrum analyser was set to take 50 samples and continuously average them to get a final result. Unfortunately, the Spectrum analyser owned by the Department of Mechanical Engineering would only sample to a Written by: Sam Weckert & Andrew Meyer Page 48 of 127

60 Final Report maximum of 102 khz meaning that our results would only be accurate up to approximately 50 khz. However, this procedure was primarily meant to confirm a hypothesis and not to collect data. This time the data collected was of a much higher standard. The coherence was close to unity along the entire frequency range that was used to drive the motor. This indicated that our previous hypothesis was correct and that the data must be sampled over a longer time period Data Collection Using a National Instruments Data Acquisition Card (PCI ) The first relevant experimental results were taken were with a National Instruments PCI-4451 Data Acquisition Card. This card was used because it was able to sample data continuously on two channels, each at 200 khz. The card also was also able to sample frequencies up to 90 khz without any signal roll-off. Since it was only possible to sample two channels at once, a signal switchbox was required to avoid manual connection of the individual channels to the data acquisition board. This switchbox could be controlled using the parallel port on the computer (LPT1). The motor was excited using a swept sine signal generated by a Hewlett Packard function generator. The experimental setup is shown in Figure 4.7. Computer & NI Card Function Generator Laser Vibrometer Signal Switchbox PSU Motor & Control Board Oscilloscope Sam Figure 4.7: The Experimental Setup Written by: Sam Weckert & Andrew Meyer Page 49 of 127

61 The range examined was from 10 khz to 100 khz. The sweep time was 30 seconds. This time was chosen for a number of reasons. Initially, it was uncertain how much data the board would be able to collect before the computer suffered from a buffer overrun. Using trial and error the maximum sampling time was found to be 50 seconds, which gave 10 million samples. An attempt was made to calculate the spectral information of the data, however, Matlab had insufficient memory to complete this task. The same situation occurred when a sample time of 40 seconds was used. Finally, it was decided to use a 30 second sample time and complete more averaging. Once a sample had been collected the spectral information was calculated. This was completed using a 1024-point FFT with 75% overlap. A Hanning window was applied over the data. This window was used because it gave the most attenuation of sidelobe leakage whilst maintaining the overall amplitude of the signal. The result of the calculation was four functions for each channel, these were: 1) P xx The Power Spectral Density of the Swept Sine Wave 2) P xy The Cross Spectral Density of the Swept Sine to the Output of the Motor 3) P yy The Power Spectral Density of the Output of the Motor 4) T xy The Transfer Function of the Motor output to the Swept Sine Input (T xy = P xy /P xx ) The transfer function T xy can be calculated using one of two methods. The method chosen was best for situations where the output signal is more contaminated by noise than the input signal. This was the situation in this case since there would be minimal noise from the swept sine signal. The coherence of the signal could be directly calculated from the four functions. This was given by: 2 ( f ) = P P xx xy ( f ) ( f ) P yy 2 ( f ) (4.3) Four linear averages were calculated for each of the functions collected. This gave a total data acquisition period of 120 seconds. At this point the collected data was still in terms of a left, right and top channel as collected from the Laser Vibrometer. The next stage was to covert the data from this form into the individual X, Y and Z components. This process could only be performed on the transfer function Txy as the Written by: Sam Weckert & Andrew Meyer Page 50 of 127

62 power spectral densities Pxx and Pyy have units Volts 2 /Hz and do not multiply or add linearly. More information on the conversion can be found in Section 4.5, which discusses the Laser Vibrometer and the conversion in detail. Following this procedure the transfer function of the motor could be plotted for the X, Y, and Z directions. This procedure was completed for 5 motors so that the commonalities could be observed between them. This also gave some indication of the variation that was occurring between the motors. The biggest limitation of these results was that they only showed the first resonance peak of operation. A much higher sampling rate was required to enable both resonance peaks to be seen in the Bode diagram Time Series Sampling As a further test of the validity of the results generated from the data collected using the PCI 4451 National Instruments Data Acquisition Card, a time series of data was collected in the Y and Z directions. The motor was driven at the optimal frequencies for both operating directions and a sample of 0.5 milliseconds was taken. The two sets of data were plotted against each other to show the relative motion of the tip of the resonator. An ellipse was generated and is shown in Figure 4.8. These results also confirmed that the elliptical motion was possible without an external influence from the driven object. This would significantly reduce the complexity of the ANSYS modelling since this testing proved that a contact analysis was not necessary. Written by: Sam Weckert & Andrew Meyer Page 51 of 127

63 Figure 4.8: Results of the time series sampling Data Collection Using a National Instruments Data Acquisition Card (PCI ) This card had several major benefits over the PCI Firstly, it provided data acquisition at a rate of up to 5 MS/s. Consequently, it was possible to sample data at a much higher rate than the PCI This enabled the verification of results that were collected at the lower sampling rate. The second benefit of this card was that it provided four analog input channels. This meant that all three orthogonal directions (X, Y and Z) plus the driving signal could be sampled simultaneously. This greatly simplified the experimental setup since the signal switchbox was not required. A further simplification was found when it was realised that the conversion unit in the laser vibrometer was capable of processing data up to 250kHz. This resulted in a significant decrease in the amount of Matlab code required to process and display the data. The motor was driven by a swept sine wave between 10kHz and 150kHz. With this arrangement the data was sampled at 500kHz. This gave reliable data up to 250kHz, Written by: Sam Weckert & Andrew Meyer Page 52 of 127

64 well within the range of the swept sine signal. Two samples of 10 seconds each were collected for each channel. This sampling time value was calculated through trial and error. A larger value would cause Matlab to return a buffer overrun error. Thus a total data collection period of 20 seconds was used. Once the data had been collected the spectral data was calculated and then averaged using Matlab. This was completed using a 1024-point FFT with 75% overlap. A Hanning window was applied over the data to reduce sidelobe leakage. The results that were produced verified the data collected at the lower sampling rate of the PCI 4451 Data Acquisition Card. For the first time it was also possible to see both of the resonance peaks that occur during operation. Figures 4.9, 4.10 and 4.11 show the results of the 5 test motors plus the replica motor for each individual component (X, Y & Z). Figure 4.9: Z Direction Transfer Function Written by: Sam Weckert & Andrew Meyer Page 53 of 127

65 Figure 4.10: Y Direction Transfer Function Figure 4.11: X Direction Transfer Function As can be seen from the Z and Y directions there were many similarities between the five motors. Of particular interest were the characteristics of each motor around the operational points. Each motor showed very distinct resonance peaks at these points Written by: Sam Weckert & Andrew Meyer Page 54 of 127

66 and there was a significant phase shift of approximately 180 between the two resonance points. The results in the Y and Z directions also showed some variation in the operational point of each motor. These differences are due to differences in the operating temperature, geometry and construction material [2]. Although the operating frequency varied slightly between motors the general shape of the transfer function remained constant. Results in the X direction showed very few similarities. However, this was expected since the elliptical components of motion are excited in the Y and Z directions. The resonance amplitudes in the X direction were significantly smaller than the equivalent amplitudes in the Y and Z directions. There were many more resonance peaks in the experimental results compared to those results generated using the ANSYS model. However, these additional peaks had only very small amplitudes. These peaks existed because of cross coupling that was occurring between the bending vibrations occurring in the X-axis and the Y and Z- axes. The next part of the research was to obtain the value of the phase shift between the Y and Z components at the operational points. Using this result it was possible to plot the motion of the tip. This was completed using code written in Matlab. The resulting phase difference was significantly different to what was expected. The initial theory of operation suggested that the phase difference during motor operation was approximately 90, resulting in a nearly circular ellipse. The actual results indicated that the operational phase shift was very small, in the order of This range of values was considered reasonable, as the resulting ellipse was very flat resulting in a maximum distance travelled for each cycle and consequently a very efficient motor. The generated ellipses for a range of frequencies around the operating points are shown in Figures 4.12 and Written by: Sam Weckert & Andrew Meyer Page 55 of 127

67 Figure 4.12: Elliptical shaped tip motion over a range of frequencies (Forward) Figure 4.13: Elliptical shaped tip motion over a range of frequencies (Reverse) Written by: Sam Weckert & Andrew Meyer Page 56 of 127

68 4.5 Motor Frequency Operational Range Using the data from the laser vibrometer, Bode plots were generated for each of the five untampered Elliptec motors. It was desired to plot ellipses from the Bode plot data at the expected operating frequencies, so it was necessary to measure the operating frequency range for the forward and reverse directions for each of the motors. A sine wave, amplitude 5V, was used to drive the motors. Once the rotor was turning, the frequency was slowly increased or decreased until the motor stopped. These values represented the upper and lower limits for the operating frequency range in a particular direction. Table 4.2 shows the operating frequency ranges for the five untampered Elliptec motors. Operating Frequency Range (khz) Motor No. Forward Reverse Table 4.2: Operating frequency ranges for the five Elliptec motors The first motor had a range of 7683kHz for the forward direction. However supplying the motor with a 76kHz signal would not cause it to move. This is the frequency when the movement stopped after the motor had previously been excited around the optimal frequency. Additionally the mean value of the upper and lower limits for each range did not appear to necessarily correspond to the optimal frequency. Written by: Sam Weckert & Andrew Meyer Page 57 of 127

69 The frequency ranges were also found to change slightly depending on how long the motor had been running. This was assumed to be due to thermal expansion of the resonator from heat generated by the crystal. 4.6 Motor Speed and Temperature Characteristics The next logical step after the determination of the operational speed ranges of the motor was to determine an optimal speed for each motor. The motor speed was recorded using an optical tachometer that was connected to a computer via a DSpace Control board. A Hewlett-Packard function generator was used to sweep through each set of running frequencies. Using the obtained data, the optimal frequency was defined as the frequency that resulted in the highest rotational speed from the motor. This allowed the operation characteristics at the optimal frequency to be determined. Using these characteristics the operating ellipse shapes at the drive tip for each motor were plotted over a range of frequencies. These results confirmed that the ellipses were being generated even though the resonator was not in contact with the driven object. The generated ellipses for one of the tested motors can be seen in Figures 4.12 and 4.13, the ellipses are shown for both forward and reverse motion. The main drawback of the Elliptec X15G was its very short lifespan due to the depolarisation of the piezoelectric element. This was occurring due to the extremely high operating temperature of the motor. An Optrex Thermohunter PT-3S non-contact thermometer was used to measure the temperature on the surface of the piezoelectric element during operation. The collected results are shown in Figure The steady state operating temperature of the motor was approximately 120ºC. This was significantly different to the Curie temperature of 360ºC [16]. It was believed that the internal temperature of the piezoelectric element was probably higher than the surface temperature, and that the constant operation at this temperature was damaging to the microstructure of the piezoelectric element. Written by: Sam Weckert & Andrew Meyer Page 58 of 127

70 Figure 4.14: Temperature Response of an Elliptec Piezoelectric Motor A large rise in temperature occurred when the motor stalled. It was believed that the micro-motion of the tip generated sufficient airflow around the motor to effectively provide some forced convection. When the motor stalled, the elliptical motion stopped allowing only natural convection to occur. 4.7 Damaged Motor Revival As a test of the robustness of the Elliptec X15G a new piezoceramic element was reinserted into a previously used resonator. As a result of the extensive manipulation of removing and inserting the piezoceramic material several times, the chosen resonator was now noticeably distorted in shape. This was particularly evident around the circular section of the piezoceramic cavity. The difference between the two motors can be seen in Figure Written by: Sam Weckert & Andrew Meyer Page 59 of 127

71 Figure 4.15: Picture of deformed Elliptec Resonator (Left) and Untampered Resonator (Right) The motor was swept slowly over a 60kHz to 120kHz frequency range. It showed no signs of movement at all, not even at the 75kHz frequency it had previously worked at. Using the laser vibrometer, data was collected by driving the motor with a swept sine wave from 10kHz to 100kHz. It would have been beneficial to go higher than 100kHz however the PCI 6110 Data Acquisition Card was not available at this time. Inspection of the resulting Bode plots generated in MATLAB revealed amplitude peaks at approximately 52kHz and also a considerable phase shift, indicating a resonance around this frequency. The Bode plot generated is shown in Figure Written by: Sam Weckert & Andrew Meyer Page 60 of 127

72 Amplitude (db) X Y Z Bode Plot - Elliptec Piezo Motor - Results Damaged Motor Frequency (Hz) x X Y Z Phase (deg.) Frequency (Hz) x 10 4 Figure 4.16: Bode Plot for Damaged Motor A signal at this frequency was supplied to the motor and it reluctantly turned. This was very exciting and illustrated how the data obtained from the laser vibrometer could be used to predict the operating frequency of the motor. Written by: Sam Weckert & Andrew Meyer Page 61 of 127

73 Amplitude (db) Bode Plot - Elliptec Piezo Motor - Results Damaged Motor & Untampered Motor Y - Damaged Z - Damaged Y - Untampered Z - Untampered Frequency (Hz) x 10 4 Phase (deg.) Y - Damaged Z - Damaged Y - Untampered Z - Untampered Frequency (Hz) x 10 4 On the damaged motor a similar amplitude and phase shift was observed. It was predicted that motion would occur at this point On an undamaged Elliptec motor movement would be expected here Figure 4.17: Bode Plot Comparison between Untampered Motor and Damaged Motor Referring to Figure 4.17, the amplitude of the damaged motor resonance peak was approximately -5.98dB. This was considerably lower than the amplitude of the peak at the first operating frequency of the untampered motor (+0.73dB). This explained why the motor only turned reluctantly at this frequency. Furthermore, the laser vibrometer data had been obtained with no contact on a driven object. This confirms that a contact analysis between the tip and the rotor was not required to generate a successful ANSYS model. This experiment showed the robustness of the Elliptec X15G and also showed that operation was still possible even with significantly different geometry to the original motor. Furthermore, it was shown that the most critical aspect to the operation of the Written by: Sam Weckert & Andrew Meyer Page 62 of 127

74 motor was the amplitude of the resonance point. If the amplitude were not high enough the motor would have had insufficient power to move the driven object. 4.8 Suggested Theory of Operation The experimental data collected gave an insight into the operation principle behind the Elliptec piezoelectric motor. Data collected using the Laser Vibrometer showed two distinct peaks at the operational frequencies. This data confirmed that the motor was operating at a resonance condition. The phase information showed that the motor was operating because of a phase difference between the two vibration modes. The most surprising result to come from the experimental work was the magnitude of the phase shift. Initially, it was thought that the phase shift would be around 90, resulting in a nearly circular ellipse. In reality, the phase shift was significantly smaller than this value, in the order of This resulted in a flat ellipse, which maximized the distance travelled for each cycle of the motor. This made the motor very mechanically efficient since minimal slippage occurred during each cycle. The difference between the ellipses of a range of phase differences is shown in Figure It was also found that the most critical feature to ensure the motors operation was the amplitude of the resonance response of the motor. This emphasized the importance of the coupling between the piezoceramic element and the resonator. The piezoceramic element was mounted off-centre to the resonator simply to excite the bending vibrations to create the elliptical shaped motion. This suggested theory of operation was then used as the basis for the optimisation in ANSYS. This is further discussed in Chapter 3. Written by: Sam Weckert & Andrew Meyer Page 63 of 127

75 Figure 4.18: The Effect of Phase Shift on Elliptical Shape 4.9 Biaxial Motor Testing Due to time constraints a second set of resonators had to be manufactured before the biaxial motor ANSYS model could be fully optimised. Thus, to simplify the ANSYS analysis and allow new resonators to be manufactured, it was decided to create a geometry that consisted of a single variable. This variable was chosen as the length of the second resonator finger. The CAD image of the new design is shown in Appendix B. A picture of the new resonator is shown in Figure Written by: Sam Weckert & Andrew Meyer Page 64 of 127

76 Short Finger Long Finger Figure 4.19: The New Biaxial Resonator The new resonators were designed so that there was an interference fit between the piezoceramic element and the resonator. This would simplify the procedure required to hold the piezoceramic element in place inside the resonator. In reality, this interference was not sufficient to ensure proper placement inside the resonator. To overcome this limitation the shorter end of the biaxial resonator was tapped to accept a 1mm custom-made screw. This ensured proper placement of the piezoceramic element inside the resonator. This arrangement can be seen in Figure Figure 4.20: The New Biaxial Resonator showing tapped hole Written by: Sam Weckert & Andrew Meyer Page 65 of 127

77 Once the piezoceramic element was secured, the motor was affixed to a modified test stand that enabled each resonator finger contact with an independent drive wheel. The motor was tested by sweeping through a range of drive frequencies between 10kHz and 150kHz. It was found that the motor operated at multiple frequencies in both directions. It was also found that when one tip was operating the other tip would also drive. Table 4.3 shows the frequencies at which the motor operated. FORWARD FREQUENCIES (Hz) REVERSE FREQUENCIES (Hz) Short Finger Long Finger Table 4.3: Drive Frequencies of the Biaxial Motor Since the motor had not been optimised for biaxial motion a simple scheme was devised to determine the optimal geometry. The new finger added to the resonator was divided into eight equal sections. The motor was run and analysed using the laser vibrometer on both fingers. After each test, a section of the divided finger was removed and new geometry was tested. This was completed until all that remained was a geometry that resembled the original Elliptec motor. The spectral results were calculated and averaged using code generated in Matlab. This was completed using a 1024-point FFT with 75% overlap. A Hanning window was applied over the data to reduce sidelobe leakage. Figures 4.21 and 4.22 show the results for each finger in the Z direction. The results for the other directions can be seen in Appendix D. Written by: Sam Weckert & Andrew Meyer Page 66 of 127

78 Figure 4.21: Biaxial Motor Test Results Z Direction Long Finger Figure 4.22: Biaxial Motor Test Results Z Direction Short Finger These results showed that as the length of the long finger decreased, the operational frequencies increased. This was as expected since the natural resonant frequency is proportional to the inverse square root of the mass. The most discouraging aspect of Written by: Sam Weckert & Andrew Meyer Page 67 of 127

79 these results was that when the short finger and long finger results were overlaid, it became apparent that the resonance peaks were occurring at the same frequencies. This implied that the design of the resonator was such that it did not allow the decoupling of the two operation modes. Figures 4.23 and 4.24 show the overlaid results for both fingers. The rest of the results for these cases can be seen in Appendix D. Figure 4.23: Results for both fingers with 14mm removed from the long finger Figure 4.24: Results for both fingers with 6mm removed from the long finger Written by: Sam Weckert & Andrew Meyer Page 68 of 127

80 This experiment showed that it was not possible to decouple the operating modes simply by changing the length of the second resonator finger. It was proposed that this was caused by both the resonator fingers having the same cross sectional area. As an experiment to test this theory it was decided to load the longer resonator finger with a small mass to see if this had any effect on the problem of decoupling the operating modes. A small nut of mass 0.8 grams was affixed to the centre of the long finger. The results for this experiment in the Y and Z directions are shown in Figures 4.25 and Figure 4.25: Mass Weighted Resonator Z Direction Written by: Sam Weckert & Andrew Meyer Page 69 of 127

81 Figure 4.26: Mass Weighted Resonator Y Direction Upon closer examination of the results it was found that the resonance operating frequencies of the mass weighted resonator had shifted slightly relative to the unweighted resonator. This was as expected since the resonance frequency of a structure is inversely proportional to the square root of the mass. However, when Figures 4.25 and 4.26 were examined it was found that the operating frequencies for both of the fingers had decreased. It had been hoped that adding the mass to one finger would effectively decouple the resonance frequencies for each finger, hence allowing an independent operating mode for each resonator to be determined. Adding a larger mass may have caused this phenomenon to occur. Due to time constraints, no further investigation could be done into this area Experimental Validation Due to the research nature of the project it was often not known exactly what to expect from the experimental results. Consequently, various other experiments were completed to help verify the validity of the data obtained. The transfer function obtained experimentally for the Elliptec motor seemed excessively complicated, with too many poles for the simple resonator spring combination. It was suggested that this was possibly due to additional modes generated by the supporting structures [11]. These supporting structures included the Written by: Sam Weckert & Andrew Meyer Page 70 of 127

82 mounting board as well as the working platform of the trolley on which the motor was positioned for testing. With the motor running, the laser vibrometer was directed at the mounting board and a transfer function was generated for the vibration of the board. From this experiment it was found that the vibration of the board had no major effect on the motor transfer function. Most of the vibration was well below -60dB and consequently would have very little impact on the generated motor transfer function. This was an important step because it allowed a legitimate comparison to be made between the ANSYS computer simulated results and the experimental results. The results from this test can be seen in Appendix E. The elliptical motion generated at the tip of the resonator was initially expected to be the combination of longitudinal and vertical bending motion. It was not anticipated to have any significant horizontal bending vibrations, perpendicular to the two previous directions. However the transfer functions generated did not support this hypothesis with significant horizontal bending vibrations evident in the results. Either a mistake had been made with the coordinate system orientation or possibly there was extensive cross coupling between the orthogonal directions. A shaker and stinger setup was used to generate vibration purely in one direction. This uni-direction vibration was aligned with each of the orthogonal coordinate system axes of the laser vibrometer and the vibrations measured. The results were used to establish the correct coordinate system orientation and additionally indicate the level of cross talk between the orthogonal components. This resulted in the correction of results obtained and presented in the preliminary report. The results from this test can be seen in Appendix E. The phase diagram for the transfer function showed a constant linear slope outside any major phase shifts. This is characteristic of a time delay between the input and output signals and could be caused by non-simultaneous sampling of the two data channels. This was tested by sampling the same signal with both channels of the data acquisition card. If simultaneous sampling was used, resulting in no time delay between measurements, then the phase diagram should have been a flat horizontal line of zero slope. This is what was observed during this testing. These results can be seen in Appendix E. Written by: Sam Weckert & Andrew Meyer Page 71 of 127

83 4.11 Laser Vibrometer Principle of Operation The three-dimensional laser vibrometer measures vibration velocities in all three orthogonal spatial directions (X, Y, and Z) at a single point. Three helium-neon laser beams are pointed at the vibrating object and scattered back from it. Each laser beam is subjected to a small frequency shift due to the Doppler effect, where the Doppler frequency, f D, is a function of the velocity component, V, in the direction of the laser beam [14]. f 2. V D = (4.3) Where is the wavelength of the laser emission. This principal is also used in a police radar gun, where microwaves are used instead of laser emissions. Superimposing an object beam with an internal reference beam generates beating, where the beat frequency, f B, is equal to the Doppler frequency. This frequency is however independent of the direction of the velocity. The direction is realized by introducing an additional fixed frequency shift in the interferometer. This is added to the Doppler frequency to generate an unambiguous signal form, where the resulting frequency shift, f, is [14] V f = f B + 2. (4.4) The sensor head generates three laser beams: top, left and right, which measure the velocity components V T, V L and V R respectively. The measurement geometry is displayed in Figures 4.27 and Written by: Sam Weckert & Andrew Meyer Page 72 of 127

84 Top View Figure 4.27: Schematic top view of the sensor head, the laser beams and the coordinate systems [14] Side View Figure 4.28: Schematic side view of the sensor head, laser beams and the coordinate system [14] The velocity components V X, V Y, and V Z in the orthogonal coordinate system can be calculated using the relations [14]: v v v X Z Y VR VL = 2sin VR + VL = 2cos VT ( VZ cos ) = sin (4.5) (4.6) (4.7) Where V T, V L and V R are the velocity components in the Top, Left and Right directions respectively. is the angle between each of the Laser beams (This was 12 in the case of the Polytec Laser Vibrometer) The decoder is capable of performing these coordinate system transformations to a maximum frequency of 250kHz. Consequently, it is possible to use the decoder module to process the data resulting in significantly shorter data processing times Written by: Sam Weckert & Andrew Meyer Page 73 of 127

85 Safety The Polytec Three-Dimensional laser vibrometer contains a Class 2 laser and consequently the laser output power is 1mW maximum. The prototype board for running the motors had tracks that were highly reflective. As a safety measure, these tracks were blacked out to reduce any reflection from the laser. Additionally a cardboard box was made to encase the board with a small cutout to allow the laser beams through. This safety equipment can be seen in Figure 4.29 Figure 4.29: Laser Vibrometer with Laser Shielding Written by: Sam Weckert & Andrew Meyer Page 74 of 127

86 5. PRELIMINARY COSTING The most significant cost in the construction of any ultrasonic motor was found to be the excitation element. In this particular case this was the piezoceramic element. As expected the price of this element was highly variable, dependent on the quantity of elements required. The price of the resonator and spring assembly was insignificant relative to the price of the piezoceramic element. However, as the production quantity increased the price of the resonator and spring assembly would approach the cost of the raw material. The following analysis shows the predicted price for producing a single prototype motor and the cost of mass-producing motors commercially. 5.1 Prototype Motor For this analysis it was assumed that all of the required production would be outsourced, no machinery would be purchased and only manual assembly of the parts would be completed. The piezoceramic element would be obtained from Noliac A/S. The resonator was to be cut from Aluminium plate using Electric Discharge Machining (EDM). The spring was to be wound from commercially available spring steel. Item Material Quantity Price (AUD) Piezoceramic Element S1 Piezoceramic 1 $42.00 Resonator Aluminium 1 $ Spring Oil Treated Spring Steel 1 $ Leads Plastic Coated Hookup Wire 2 $1.00 TOTAL $ Elliptec X15G Motor COMPLETE ASSEMBLY 1 $60.00 Table 5.1: Prototype Motor Costing Table 5.01 shows that it was cheaper to purchase a new Elliptec X15G motor and remove the piezoceramic element and spring for use in the new motor. The biaxial motor was produced at a similar cost, since the price of producing the resonator elements was almost identical to the cost incurred for the production of the replica resonator. All other costs remained the same. Written by: Sam Weckert & Andrew Meyer Page 75 of 127

87 5.2 Mass Produced Motor This section deals with producing the prototype motors in the order of 100 million units per year. In this analysis it was also assumed that no machinery was purchased. The cost of manually assembling the completed components was also not included since at these volumes it would be more efficient to automatically assemble the motors. When the components are produced in such large quantities, the price of the manufacturing becomes negligible and the overall cost of the component is the price of the material used in the component. Thus, to calculate the price of each of the components used in the motor, the volume is calculated and then used to calculate an overall component price. This analysis also ignores assembly costs. It was assumed here, that the Aluminium resonators would be stamped, resulting in a very low manufacturing time. The springs would also be automatically manufactured, resulting in a very low overall cost. Item Material Quantity Price/kg (AUD) Price Piezoceramic Element S1 Piezoceramic 1 N/A $ Resonator Aluminium 1 $2.10 $ Spring Oil Treated Spring Steel 1 $1.08 $ Leads Plastic Coated Hookup Wire 2 N/A $ TOTAL $ Table 5.2: Mass Produced Motor Costing Table 5.2 shows that it is theoretically possible to produce the ultrasonic motors for a cost that is comparable to a standard electromagnetic motor. However, there are many more parts present in an electromagnetic motor and theoretically a motor of this type should be able to be produced for much less. The new biaxial motor should be able to be produced for a price that is almost identical to the price of the mass produced replica shown in Table 5.2. The extra amount of Aluminium used in the resonator is almost negligible and consequently would result in no significant price rise. Written by: Sam Weckert & Andrew Meyer Page 76 of 127

88 6. CONCLUSION 6.1 Overview of Completed Research An ANSYS finite element model for the Elliptec motor was generated. A modal analysis of the model was completed and the resulting mode shapes obtained agreed with those expected for the physical system. A harmonic analysis on the ANSYS model was performed. Using the results from the harmonic analysis a transfer function for motion at the tip was generated from the model and the resulting data was compared with experimental results. Although the resonance peaks were slightly more spread apart, the phase information compared favourably with the experimentally generated spectrums. Using ANSYS, several optimisation cases of the Elliptec motor were investigated. These optimisation cases attempted to maximize the amplitude of the drive tip vibration whilst maintaining a desired phase shift between the longitudinal and bending vibrations. Several of these cases resulted in geometries that were significantly different to the original Elliptec piezoelectric motor. To further investigate the results of these optimisations it would be necessary to construct and test the resonators that were designed through the optimisation process. Unfortunately, due to time constraints this was unable to be completed. Five Elliptec motors were tested and a transfer function between the harmonic driving function and the output tip vibrations was generated. This information was analysed to determine the underlying principle behind the method of operation of the Elliptec motor. It was found that the motor operates at a resonant condition and that the magnitude of the phase shift between the bending and the longitudinal vibration modes was between This condition resulted in a very flat ellipse that was very mechanically efficient. A time series sampling experiment was used to determine that no contact analysis experiment was required. A replica Elliptec motor was manufactured and the motor was made to operate by using a small screw to increase the coupling between the piezoceramic element and the resonator. This experiment showed that the Elliptec design requires a minimum Written by: Sam Weckert & Andrew Meyer Page 77 of 127

89 level of vibration amplitude to operate, hence reinforcing the importance of the coupling between the motor and the piezoceramic element. Finally, a prototype biaxial motor was extensively tested. A similar arrangement to the one used in the replica motor was used to hold the piezoceramic element in place. Bidirection motion was obtained at both drive points, however, the motion was unable to be decoupled to allow independent control at each drive tip. This would be one of the major research areas if someone was to continue researching this type of biaxial drive. 6.2 Future Research Many of the original aims that were set have been achieved and consequently there are only a few areas that require further research. The area with the most scope for further research is an investigation into methods for decoupling the biaxial operating modes to allow independent biaxial control. One possible method for doing this would be to investigate the effect of varying the thickness of one resonator finger whilst keeping the other constant. This experiment could potentially be completed in ANSYS if the biaxial motor could be successfully modelled in ANSYS. Another potential research area involves further investigation of the ANSYS optimising procedures to aid in the decoupling of the operation modes for the biaxial design and to improve the design of the current Elliptec piezoelectric motor. The research completed in this project gives a fundamental understanding of the principles behind the operation of the Elliptec piezoelectric motor. It would now be possible to continue ANSYS modelling with a known set of operational requirements. Written by: Sam Weckert & Andrew Meyer Page 78 of 127

90 REFERENCES [1] K. Uchino and B. Koc, "Compact piezoelectric ultrasonic motors," Ferroelectrics: Proceedings of the nd Asian Meeting on Ferroelectricity (AMF-2),Dec 7-Dec , vol. 230, pp. 375/73-388/86, [2] Anonymous, Technical Manual, Version 1.2 ed. Dortmund: Elliptec Resonant Actuator AG, [3] E. Visser, "Active mirror control using a piezoelectric motor," University of Adelaide, Adelaide April [4] B. Andersen, M. Blanke, and J. Helbo, "Two mode resonator and contact model for standing wave piezomotor," presented at Proceedings of the 2001 ASME Design Engineering Technical Conferences & Computers and Information in Engineering Conference, [5] M. Fleischer, D. Stein, and H. Meixner, "Novel ultrasonic motors with monoand bimodal drives," Sensors-and-Actuators-A-Physical., vol. A21(1-3):, pp , [6] F. Rong Fong, Y. Chih Min, and C. Dong Guey, "Dynamic and contact analysis of a bimodal ultrasonic motor," [7] H. Saigoh, M. Kawasaki, N. Maruko, and K. Kanayama, "Multilayer piezoelectric motor using the first longitudinal and the second bending vibrations," in Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes & Review Papers. Proceedings of the 15th Symposium on Ultrasonic Electronics,Nov 28-Dec , vol. 34. Kyoto,Jpn: JJAP,Minatoku,Japan, 1995, pp [8] T. Funakubo, T. Tsubata, Y. Taniguchi, K. Kumei, T. Fujimura, and C. Abe, "Ultrasonic linear motor using multilayer piezoelectric actuators," Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes & Review Papers. Proceedings of the 15th Symposium on Ultrasonic Electronics,Nov 28-Dec , vol. 34, pp , [9] Y. Tomikawa, T. Takano, and H. Umeda, "Thin rotary and linear ultrasonic motors using a double-mode piezoelectric vibrator of the first longitudinal and second bending modes," Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes. 9th Meeting of the Ferroelectic Materials and their Applications - FMA-9,May , vol. 31, pp , [10] M. Aoyagi, Y. Tomikawa, and T. Takano, "Ultrasonic motors using longitudinal and bending multimode vibrators with mode coupling by externally additional asymmetry or internal nonlinearity," Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes. 9th Meeting of the Written by: Sam Weckert & Andrew Meyer Page 79 of 127

91 Ferroelectic Materials and their Applications - FMA-9,May , vol. 31, pp , [11] B. Cazzolato, "Personal Communication." Adelaide, [12] Anonymous, Dynamics - User Guide for Revision 5.0. Houston, USA: Swanson Analysis Systems, [13] I. Brown, "Personal Communication." Adelaide, [14] Anonymous, Three-Dimensional Laser Vibrometer - Preliminary User Manual. Waldbronn, Germany: Polytec GMBH, [15] O. Wibom, Personal Communication. Adelaide, 2003 [16] Engineering Fundamentals, 2003, Lead Zirconate Titanate (PZT-4) Material Properties, viewed 28 th October 2003, < Material_ID=PZT-4> Written by: Sam Weckert & Andrew Meyer Page 80 of 127

92 APPENDIX A CAD DRAWING OF ELLIPTEC REPLICA RESONATOR Written by: Sam Weckert & Andrew Meyer Page 81 of 127

93 APPENDIX B CAD DRAWING OF NEW BIAXIAL RESONATOR Written by: Sam Weckert & Andrew Meyer Page 82 of 127

94 APPENDIX C MATLAB CODE USED TO GENERATE RESULTS The following code was used to collect the results from the National Instruments PCI- 6110: % Define some parameters clear all close all fs_in = 500e3 fs_out = 51E3 duration = 10; %10 second acquisition nfft = 2^10; PxxX = zeros(nfft/2+1,1); PxyX = zeros(nfft/2+1,1); PyyX = zeros(nfft/2+1,1); TxyX = zeros(nfft/2+1,1); PxxY = zeros(nfft/2+1,1); PxyY = zeros(nfft/2+1,1); PyyY = zeros(nfft/2+1,1); TxyY = zeros(nfft/2+1,1); PxxZ = zeros(nfft/2+1,1); PxyZ = zeros(nfft/2+1,1); PyyZ = zeros(nfft/2+1,1); TxyZ = zeros(nfft/2+1,1); Num_Samples = 2 for i = 1:Num_Samples AI = analoginput('nidaq',1) chan = addchannel(ai,[0,1,2,3]) set(ai) %AO = analogoutput('nidaq',1) %chanout = addchannel(ao,[0]) %set(ao,'triggertype') %set input parameters set(ai,'samplerate',fs_in) ActualRate_in = get(ai,'samplerate') set(ai,'samplespertrigger',duration*actualrate_in) set(ai,'triggertype','manual') blocksize = get(ai,'samplespertrigger'); Fs = ActualRate_in; % T1 = linspace(0,duration,duration*actualrate_in)'; % Add a delay so that IO are synched % set(ai,'triggerdelayunits','samples') % set(ai,'triggerdelay',-925) % % %set output parameters % set(ao,'samplerate',fs_out) % ActualRate_out = get(ao,'samplerate') % set(ao,'triggertype','manual') % T = linspace(0,duration,duration*actualrate_out)'; % data = sin(1000*pi*t); % %data = sin(linspace(0,2*pi*10,duration*actualrate_out))'; % putdata(ao,[data]) Written by: Sam Weckert & Andrew Meyer Page 83 of 127

95 %Start start([ai]) trigger([ai]) newdata = getdata(ai); % Plot data %subplot(2,1,1) %plot(data) %subplot(2,1,2) [Px,Fx] = spectrum(newdata(:,1),newdata(:,2),nfft,0.75*nfft,hanning(nfft),fs); PxxX = (PxxX + 1/Num_Samples*Px(:,1)); PxyX = (PxyX + 1/Num_Samples*Px(:,3)); PyyX = (PyyX + 1/Num_Samples*Px(:,2)); TxyX = (TxyX + 1/Num_Samples*Px(:,4)); [Py,Fy] = spectrum(newdata(:,1),newdata(:,3),nfft,0.75*nfft,hanning(nfft),fs); PxxY = (PxxY + 1/Num_Samples*Py(:,1)); PxyY = (PxyY + 1/Num_Samples*Py(:,3)); PyyY = (PyyY + 1/Num_Samples*Py(:,2)); TxyY = (TxyY + 1/Num_Samples*Py(:,4)); [Pz,Fz] = spectrum(newdata(:,1),newdata(:,4),nfft,0.75*nfft,hanning(nfft),fs); PxxZ = (PxxZ + 1/Num_Samples*Pz(:,1)); PxyZ = (PxyZ + 1/Num_Samples*Pz(:,3)); PyyZ = (PyyZ + 1/Num_Samples*Pz(:,2)); TxyZ = (TxyZ + 1/Num_Samples*Pz(:,4)); % % Display the difference in triggering times % AItime = AI.InitialTriggerTime % AOtime = AO.InitialTriggerTime % delta = abs(aotime - AItime); % sprintf('%d',delta(6)) stop([ai]) delete(ai) clear AI end % delete(ao) % clear AO save('c:\documents and Settings\piezo\Desktop\AllData.mat','Fx','Fy','Fz','PxxX','PxyX','PyyX','TxyX','PxxY','P xyy','pyyy','txyy','pxxz','pxyz','pyyz','txyz') beep pause(1) beep pause(1) beep Once the data for all the five motors tested was analysed, the following code was used to plot all of the X, Y and Z components on to separate graphs % Code to generate results from PCI-6110 % Andrew Meyer a % Data analysis for results taken on clear all close all fnames = {'AllData1.mat' 'AllData2.mat' 'Alldata3.mat' 'Alldata4.mat' 'Alldata5.mat' 'AlldataR.mat'}; ftitles = {'Motor 1' 'Motor 2' 'Motor 3' 'Motor 4' 'Motor 5' 'Motor Replica'}; styles = {'r' 'g' 'b' 'k' 'y' 'c'} %Now plot the results for each of the motors for ii = 1:6 load(fnames{ii}) figure Written by: Sam Weckert & Andrew Meyer Page 84 of 127

96 subplot(2,1,1) plot(fx,20*log10(abs([txyy TxyZ]))) axis([ ]) legend('y','z',-1) xlabel('frequency (Hz)') ylabel('amplitude (db)') title(['bode Plot Elliptec Piezoelectric ' ftitles{ii}]) subplot(2,1,2) plot(fx,180/pi*(unwrap(angle([txyy TxyZ]))),Fx,(180/pi)*(unwrap(angle(TxyY))- unwrap(angle(txyz)))) axis([ ]) legend('y','z','y-z',-1) xlabel('frequency (Hz)') ylabel('phase (deg.)') freq_max = % Maximum frequency point measured freq_min = % Minumum frequency point measured plot_freq = 2000 % Plot an ellipse every 2000 Hz motor_no = ftitles{ii} % The number of the motor being tested direction = 'Forward' Tip_Viewer freq_max = % Maximum frequency point measured freq_min = % Minumum frequency point measured plot_freq = 2000 % Plot an ellipse every 2000 Hz motor_no = ftitles{ii} % The number of the motor being tested direction = 'Reverse' Tip_Viewer end %Now plot the results for each direction - Y Direction figure ax1 = subplot(2,1,1) hold on ax2 = subplot(2,1,2) hold on for ii = 1:6 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyy])),styles{ii}) axis([ ]) legend(ftitles,-1) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Elliptec Piezoelectric Motors - Y-Direction') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyy]))),styles{ii}) axis([ ]) legend(ftitles,-1) xlabel('frequency (Hz)') ylabel('phase (deg.)') end hold off %Now plot the results for each direction - Z Direction figure ax1 = subplot(2,1,1) hold on ax2 = subplot(2,1,2) hold on for ii = 1:6 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{ii}) axis([ ]) legend(ftitles,-1) Written by: Sam Weckert & Andrew Meyer Page 85 of 127

97 xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Elliptec Piezoelectric Motors - Z-Direction') if ii == 4 dataz = unwrap(angle(txyz)) + 2*pi; else dataz = unwrap(angle(txyz)); end if ii == 5 dataz = unwrap(angle(txyz)) + 2*pi; end axes(ax2) plot(fx,180/pi*(dataz),styles{ii}) axis([ ]) legend(ftitles,-1) xlabel('frequency (Hz)') ylabel('phase (deg.)') end hold off %Now plot the results for each direction - X Direction figure ax1 = subplot(2,1,1) hold on ax2 = subplot(2,1,2) hold on for ii = 1:6 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyx])),styles{ii}) axis([ ]) legend(ftitles,-1) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Elliptec Piezoelectric Motors - X-Direction') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyx]))),styles{ii}) axis([ ]) legend(ftitles,-1) xlabel('frequency (Hz)') ylabel('phase (deg.)') end hold off The following code was used to generate the ellipses that were occurring at the tip of the Elliptec motor during operation. This m-file was required to be given a set of input variables to operate successfully. % This program calculate the ellipses generated at the tip of the elliptec % resonator over a specified frequency range % Andrew Meyer a % close all % clear all % GENERAL INFORMATION % freq_max = % Maximum frequency point measured % freq_min = % Minumum frequency point measured % plot_freq = 2000 % Plot an ellipse every 500 Hz % motor_no = 1 % The number of the motor being tested % direction = 'Forward' figure Written by: Sam Weckert & Andrew Meyer Page 86 of 127

98 loop_num = (freq_max-freq_min)/plot_freq; F = Fx Txy_Y = TxyY Txy_Z = TxyZ Txy_X = TxyX clear AmpPhaseMatrix % So we do a loop finding the ellipse generated at each time frequency for I = 0:loop_num end search_freq = freq_min+(plot_freq*i); Mat_Entry = find(f>search_freq); Mat_Entry = Mat_Entry(1,1); AmpY = abs(txy_y(mat_entry,1)); AmpZ = abs(txy_z(mat_entry,1)); PhaseY = angle(txy_y(mat_entry,1)); PhaseZ = angle(txy_z(mat_entry,1)); PhaseDiff = abs(phasey - PhaseZ); AmpPhaseMatrix(I+1,1) = F(Mat_Entry,1); AmpPhaseMatrix(I+1,2) = AmpY; AmpPhaseMatrix(I+1,3) = AmpZ; AmpPhaseMatrix(I+1,4) = PhaseDiff; syms t for I = 1:loop_num+1 hold on ezplot(ampphasematrix(i,3)*sin((2*pi*ampphasematrix(i,1)*t)+ampphasematrix(i,4)),a mpphasematrix(i,2)*sin(2*pi*ampphasematrix(i,1)*t),[0,1/ampphasematrix(i,1)]) end % Change the friggen colors of the lines on the graph ch = get(gca,'children'); for i = 1:length(ch) set(ch(length(ch)+1-i),'color',[rand rand rand]); end % Make a legend legend(num2str([ampphasematrix(:,1) ((180/pi)*AmpPhaseMatrix(:,4))])) % Make a title title(['ellipical Shapes Generated at the Tip Over A Range of Frequencies - Motor ' motor_no ' - ' direction]) xlabel('uncalibrated Displacement') ylabel('uncalibrated Displacement') This code was used to analyse all of the biaxial results obtained during the resonator shortening experiement. % Biaxial Motor Test Results (Shortened Resonator) % Andrew Meyer % 3/09/03 clear all close all fnames = {'AllData-0L.mat' 'AllData-0S.mat' 'AllData-4L.mat' 'AllData-4S.mat' 'AllData-6L.mat' 'AllData-6S.mat' 'AllData-8L.mat' 'AllData-8S.mat' 'AllData-10L.mat' 'AllData-10S.mat' 'AllData-12L.mat' 'AllData-12S.mat' 'AllData-14L.mat' 'AllData- Written by: Sam Weckert & Andrew Meyer Page 87 of 127

99 14S.mat'}; ftitles = {'-0mm Long' '-0mm Short' '-4mm Long' '-4mm Short' '-6mm Long' '-6mm Short' '-8mm Long' '-8mm Short' '-10mm Long' '-10mm Short' '-12mm Long' '-12mm Short' '-14mm Long' '-14mm Short'}; styles = {'r' 'g' 'b' 'k' 'y' 'c' 'm'}; legendtemp = {'' '' '' '' '' ''}; % Plot All of the individual results % for ii = 1:14 % eval(['load ' fnames{ii}]) % figure % subplot(2,1,1) % plot(fx,20*log10(abs([txyx TxyY TxyZ]))) % axis([ ]) % xlabel('amplitude (db)') % ylabel('frequency (Hz)') % title(['shortened Biaxial Resonator - ' ftitles{ii} ' ']) % subplot(2,1,2) % plot(fx,180/pi*(unwrap(angle([txyx TxyY TxyZ])))) % xlabel('phase (degrees)') % ylabel('frequency (Hz)') % axis([ ]) % end % Now plot the results for each direction and resonator tip firstly X % (SHORT) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 2:2:14 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyx])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - X - Direction - SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyx]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot the results for each direction and resonator tip firstly Y % (SHORT) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); Written by: Sam Weckert & Andrew Meyer Page 88 of 127

100 hold on stylecount = 1; for ii = 2:2:14 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyy])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Y - Direction - SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyy]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot the results for each direction and resonator tip firstly Z % (SHORT) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 2:2:14 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Z - Direction - SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyz]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot the results for each direction and resonator tip firstly X Written by: Sam Weckert & Andrew Meyer Page 89 of 127

101 % (LONG) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:13 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyx])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - X - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyx]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot the results for each direction and resonator tip firstly Y % (LONG) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:13 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyy])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Y - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyy]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); Written by: Sam Weckert & Andrew Meyer Page 90 of 127

102 hold off % Now plot the results for each direction and resonator tip firstly Z % (LONG) figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:13 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Z - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyz]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off legendtemp = {'' ''}; % Now plot the results for both fingers at a set length % (LONG & SHORT) for kk = 1:2:13 figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = kk:kk+1 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Z - Direction - LONG & SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyz]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; Written by: Sam Weckert & Andrew Meyer Page 91 of 127

103 end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off end The following results were used to obtain all of the results to the mass-weighted biaxial resonator experiments. % Biaxial Motor Test Results (Mass Weighted Resonator) % Andrew Meyer % 3/09/03 clear all close all fnames = {'AllData-0L.mat' 'AllData-0S.mat' 'AllDataML.mat' 'AllDataMS.mat'}; ftitles = {'Biaxial Long' 'Biaxial Short' 'Mass Weighted Long' 'Mass Weighted Short'}; styles = {'r' 'g' 'b' 'k'}; legendtemp = {'' ''}; % Plot All of the individual results for ii = 1:4 eval(['load ' fnames{ii}]) figure subplot(2,1,1) plot(fx,20*log10(abs([txyx TxyY TxyZ]))) axis([ ]) xlabel('amplitude (db)') ylabel('frequency (Hz)') legend('x', 'Y','Z',-1) title(['shortened Biaxial Resonator - ' ftitles{ii} ' ']) subplot(2,1,2) plot(fx,180/pi*(unwrap(angle([txyx TxyY TxyZ])))) xlabel('phase (degrees)') ylabel('frequency (Hz)') legend('x', 'Y','Z',-1) axis([ ]) end % Now plot both of the results for one end % (SHORT) X figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 2:2:4 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyx])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - X - Direction - SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyx]))),styles{stylecount}) axis([ ]) Written by: Sam Weckert & Andrew Meyer Page 92 of 127

104 xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot both of the results for one end % (SHORT) Y figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 2:2:4 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyy])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Y - Direction - SHORT') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyy]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot both of the results for one end % (SHORT) Z figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 2:2:4 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Z - Direction - SHORT') Written by: Sam Weckert & Andrew Meyer Page 93 of 127

105 axes(ax2) plot(fx,180/pi*(unwrap(angle([txyz]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % NOW THE LONG END % Now plot both of the results for one end % (LONG) X figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:3 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyx])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - X - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyx]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot both of the results for one end % (LONG) Y figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:3 load(fnames{ii}) axes(ax1) Written by: Sam Weckert & Andrew Meyer Page 94 of 127

106 plot(fx,20*log10(abs([txyy])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Y - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyy]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off % Now plot both of the results for one end % (LONG) Z figure ax1 = subplot(2,1,1); hold on ax2 = subplot(2,1,2); hold on stylecount = 1; for ii = 1:2:3 load(fnames{ii}) axes(ax1) plot(fx,20*log10(abs([txyz])),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('amplitude (db)') title('bode Plot - Biaxial Motor - Z - Direction - LONG') axes(ax2) plot(fx,180/pi*(unwrap(angle([txyz]))),styles{stylecount}) axis([ ]) xlabel('frequency (Hz)') ylabel('phase (deg.)') legendtemp{stylecount} = ftitles{ii}; stylecount = stylecount + 1; end axes(ax1) legend(legendtemp,-1); axes(ax2) legend(legendtemp,-1); hold off Written by: Sam Weckert & Andrew Meyer Page 95 of 127

107 APPENDIX D BIAXIAL MOTOR TESTING RESULTS Figures A.1 through to A.6 are results showing the relative effect of changing the length of the long finger of the biaxial resonator Figure D.1: Biaxial Motor Testing Z Direction Long Finger Figure D.2: Biaxial Motor Testing Y Direction Long Finger Written by: Sam Weckert & Andrew Meyer Page 96 of 127

108 Figure D.3: Biaxial Motor Testing X Direction Long Finger Figure D.4: Biaxial Motor Testing Z Direction Short Finger Written by: Sam Weckert & Andrew Meyer Page 97 of 127

109 Figure D.5: Biaxial Motor Testing Y Direction Short Finger Figure D.6: Biaxial Motor Testing X Direction Short Finger Written by: Sam Weckert & Andrew Meyer Page 98 of 127

110 Figures A.7 through to A.14 show both resonator finger spectral responses for a particular geometry case. Figure D.7: Biaxial Motor Z Direction 14mm Removed from Long Finger Figure D.8: Biaxial Motor Z Direction 12mm Removed from Long Finger Written by: Sam Weckert & Andrew Meyer Page 99 of 127

111 Figure D.9: Biaxial Motor Z Direction 10mm Removed from Long Finger Figure D.10: Biaxial Motor Z Direction 8mm Removed from Long Finger Written by: Sam Weckert & Andrew Meyer Page 100 of 127

112 Figure D.11: Biaxial Motor Z Direction 6mm Removed from Long Finger Figure D.12: Biaxial Motor Z Direction 4mm Removed from Long Finger Written by: Sam Weckert & Andrew Meyer Page 101 of 127

113 Figure D.13: Biaxial Motor Z Direction 0mm Removed from Long Finger Written by: Sam Weckert & Andrew Meyer Page 102 of 127

114 APPENDIX E EXPERIMENTAL VALIDATION RESULTS Figure F.1 shows the results obtained from the experiment investigation the effect of the background driver board vibration on the generated Elliptec motor transfer function. Figure E.1: Background Board Vibration Figure F.2 shows the results from the testing to investigate whether the data acquisition board used simultaneous sampling. Figure E.2: Simultaneous Sampling Testing Written by: Sam Weckert & Andrew Meyer Page 103 of 127

115 Figures F.3, F.4 and F.5 show the results of the orthogonal direction testing of the Laser Vibrometer. Figure E.3: Direction Testing X Direction Figure E.4: Direction Testing Y Direction Written by: Sam Weckert & Andrew Meyer Page 104 of 127