The Pennsylvania State University. The Graduate School. Department of Aerospace Engineering ACTUATION OF AN ACTIVE GURNEY FLAP

Transcription

1 The Pennsylvania State University The Graduate School Department of Aerospace Engineering ACTUATION OF AN ACTIVE GURNEY FLAP FOR ROTORCRAFT APPLICATIONS A Thesis in Aerospace Engineering by Michael Thiel 2006 Michael Thiel Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2006

2 I grant The Pennsylvania State University the nonexclusive right to use this work for the University's own purposes and to make single copies of the work available to the public on a not-for-profit basis if copies are not otherwise available. Michael Thiel The thesis of Michael Thiel was reviewed and approved* by the following:

3 iii George Lesieutre Professor of the Department of Aerospace Engineering Head of the Department of Aerospace Engineering Thesis Advisor Mark Maughmer Professor of the Department of Aerospace Engineering Gary Koopmann Distinguished Professor of the Department of Mechanical Engineering *Signatures are on file in the Graduate School

4 ABSTRACT iv The goal of this research is to develop actuation methods for an actively-deployed Gurney flap with applications to rotorcraft. Previous research has shown that Gurney flaps can offer significant performance gains when correctly deployed, but actuation methods remain to be developed. The actuator system for an active Gurney flap should be lower in weight and require less power to actuate than actuators developed to drive traditional trailing-edge flaps, due to smaller hinge moments. Actuation methods with operating frequencies of greater than 30 Hertz were investigated. Two distinct actuation methods were explored to actuate a Gurney flap located at 90% of the airfoil chord with a height of 1% of the chord. First, a concept employing a voice coil was developed. A mathematical model of this concept was developed to simulate dynamic performance and to ensure that the design could operate under the centrifugal (CF) loads experienced in a rotor blade. A prototype was built and bench testing was completed. This design met the deflection and frequency requirements (3 mm and 30 Hz, respectively); however, there were nonlinearities in the response that were not modeled. A second-generation voice coil driven actuation system was designed to have more linear performance, which also met the requirements. A third design, involving the use of a tapered piezoelectric bender augmented by CF loading as the actuation method, was developed and optimized. A genetic algorithm that employed a finite element model developed in MATLAB was used to design the piezoelectric bender. The dynamics of the piezoelectric bender, under no loading, were tested and the model was validated for the no-load case. Initial results show that a CF-augmented bimorph may be a feasible actuation method for an active

5 Gurney flap. Actuators for an actively-deployed Gurney flap will continue to be v developed with the long term goal of a flight quality actuation system that will enhance the performance of rotorcraft.

6 TABLE OF CONTENTS vi LIST OF FIGURES...ix LIST OF TABLES...xii ACKNOWLEDGEMENTS...xiii Chapter 1 Introduction Motivation Gurney Flaps Goal of Current Research...3 Chapter 2 Background The Gurney Flap Gurney Flap Aerodynamics Performance Benefits Previous Actuator Work Actuation Methods Piezoelectric Ceramics Introduction Piezoelectric Modeling Voice Coils Development of Traditional Trailing-Edge Flaps for Rotorcraft...18 Chapter 3 Development of Voice Coil Driven Actuation Concepts Introduction to Concepts Voice Coil Testing First-Generation Voice Coil Driven Actuation System Concept Description Modeling of Initial Concept MATLAB Simulation of Initial Concept Centrifugal Loading Calculations Fabrication and Testing of Initial Voice Coil Concept Fabrication and Testing of Second-Generation Voice Coil Driven Actuation System...41 Chapter 4 Piezoelectric Bimorph Driven Concept Development Introduction Finite Element Modeling of Bimorph Concept Parametric Study of Tapered Bimorph...50

7 4.4 Optimization of Piezoelectric Tapered Bimorph Initial Results of Optimization Resizing of Flap System Preliminary Testing of Piezoelectric Bimorph Testing of Bimorph Simulation of Frequency Response Function Comparison of the Experimental and Simulated Frequency Response...62 Chapter 5 Conclusions and Recommendations Conclusions Recommendations for Future Work...67 Bibliography...69 Appendix A MATLAB Simulation of Initial Voice Coil Concept...72 A.1 Physical Dimensions...72 A.2 State-Space Modeling and Plotting...73 A.3 CF Load Calculations...74 Appendix B Data and MATLAB Code Used to Generate Figure B.1 MATLAB Code to Generate Frequency Response Plot of Data...77 B.2 Cable Data Set...79 B.3 2 nd Generation Data Set...89 Appendix C Base FEM Code Used for Analysis of CF-Augmented Bimorph...93 C.1 Dimensions and Parameters...93 C.2 Area and Inertia Calculations...94 C.3 Assembly of the Property, Axial Load, and Force Vector...94 C.4 Assembly of Global Matrices...95 C.5 Enforcement of Boundary Conditions...96 C.6 Static Solution...96 C.7 Eigenvalue Problem...97 Appendix D Flap System Inertia Calculations Used in Piezoelectric Bimorph Design...98 D.1 Base Calculations...98 D.2 Modifications Made to Include in Optimizer...99 Appendix E Genetic Algorithm Used to Optimize Piezoelectric Bimorph E.1 real_coded_sim_taper.m vii

8 E.2 limits_2.m E.3 PNX.m E.4 spike.m E.5 tourney_min.m E.6 PZT_FEM_Taper.m Appendix F Analysis of Spectral Data and FEM Code for Frequency Response Synthesis F.1 Modified FEM for Testing and Frequency Response Synthesis F.2 Code Used for Comparison of Data to Simulation viii

9 LIST OF FIGURES ix Figure 1-1: The Gurney Flap...2 Figure 2-1: Formation of vortices around a Gurney flap...5 Figure 2-2: Concept of distributed Gurney flaps....6 Figure 2-3: CFD results of a VR-12 airfoil with a fixed 1% Gurney flap at 1.0c....7 Figure 2-4: Lift coefficient variation as location and height are varied....7 Figure 2-5: Deployment schedule for a Gurney flap on a VR-12 airfoil...8 Figure 2-6: Power curves for a rotor with and without Gurney flaps...10 Figure 2-7: Deployment schedule for an active Gurney flap...11 Figure 2-8: Schematic of Gurney flap concept by Thepvongs Figure 2-9: Fabricated active Gurney flap developed by Thepvongs...12 Figure 2-10: MiTEs used for flutter suppression...13 Figure 2-11: Test setup used to show flutter suppression capabilities of MiTEs Figure 2-12: Piezoelectric sheet with voltage across the thickness Figure 2-13: Schematic of piezoelectric bimorph, note elements 1 and 2 are poled in the same direction causing forces to be generated in opposite directions...16 Figure 2-14: Moments generated by a piezoelectric bimorph Figure 2-15: The H2W Technologies NCC X voice coil Figure 2-16: Smart rotor blade using trailing edge flaps, the flap is in yellow Figure 2-17: Flap deflections of the SMART rotor systems for different voltages and driving frequencies normalized by rotor speed...19 Figure 2-18: Active flap designed by Bao et al. for rotor control...20 Figure 2-19: Flap deflections using actuation method shown in Figure Figure 2-20: Active flap rotor using multiple trailing edge flaps....21

10 Figure 2-21: Bench top model of a trailing edge flap using a RAS...22 x Figure 3-1: Schematic of setup used to test the voice coil...24 Figure 3-2: Data from initial testing of the voice coil Figure 3-3: Initial design concept profile...27 Figure 3-4: Initial design concept perspective...27 Figure 3-5: Schematic used to model the initial voice coil concept Figure 3-6: Simulated response using MATLAB code in Appendix A Figure 3-7: Simulated frequency response using MATLAB code in A Figure 3-8: Schematic of arms and housing used for modeling the effects of CF loading Figure 3-9: Schematic of bracket used for modeling the effects of CF loading...34 Figure 3-10: Results for CF loading of arm/housing combination shown in Figure 3-8. The squares represent the deflections and the triangles represent maximum stress Figure 3-11: Results for CF loading of the bracket shown in Figure 3-9. The squares represent the deflections and the triangles represent maximum stress...36 Figure 3-12: Side view of the fabricated, initial concept...37 Figure 3-13: Top view of the initial concept Figure 3-14: Top view of the initial concept Figure 3-15: Sample data set from initial prototype Figure 3-16: Frequency response of initial prototype...40 Figure 3-17: Fabricated second generation concept Figure 3-18: Experimental setup used for the testing of both concepts...42 Figure 3-19: Sample data from the 2 nd generation prototype Figure 3-20: Compiled deflection data from 2 nd generation concept Figure 4-1: Gurney flap concept with CF-augmented bimorph...46

11 Figure 4-2: Schematic used to determine effective mass of the flap system...49 xi Figure 4-3: Tapered bimorph deflection with and without the effect of CF loading...51 Figure 4-4: Dependence of maximum tip deflection on thickness variations Figure 4-5: Dependence of the 1 st natural frequency on thickness variations Figure 4-6: Implementation of genetic algorithm...55 Figure 4-8: Power spectrum for tapered bimorph...60 Figure 4-9: Magnitude of simulated- and experimentally-determined frequency response functions. Data cursors mark the frequency (X) and amplitude (Y) of the peaks...63 Figure 4-10: Phase of simulated- and experimentally-determined frequency response functions Figure 4-11: Coherence from data set used to generate frequency response data...65

12 LIST OF TABLES xii Table 4-1: Optimized CF-augmented bimorph...56 Table 4-2: Results from optimized flap system...58 Table 4-3: Properties of bimorph used for testing Table 4-4: Damping Ratios for the first and second bending modes...60

13 ACKNOWLEDGEMENTS xiii I would like to thank the Penn State Rotorcraft Center of Excellence for funding of this project. Also, I would like to acknowledge Dr. Gary Koopmann and the Penn State Center for Acoustics and Vibration for the use of equipment and laboratory space. Dr. Mark Maughmer has provided me with insight into the background and benefits of Gurney flaps. The project is supported through the NASA Graduate Student Researcher Program (GSRP) under the direction of William Warmbrodt and Preston Martin at NASA Ames Research Center. Finally, I would like to thank Dr. George Lesieutre for giving me the opportunity to work on this project.

14 Chapter 1 Introduction 1.1 Motivation The use of trailing edge flaps, distributed spanwise, on helicopter rotor blades allows for the control of the sectional aerodynamic properties of the blade. Having the ability to control the aerodynamics of a blade section is beneficial for increasing the performance of the vehicle. Trailing edge flaps can also be deployed to suppress vibrations of the rotor blades (which are in turn transmitted to the vehicle), thus increasing passenger comfort and reducing fatigue life of components. There has been much research into the control of these vibrations using traditional trailing edge flaps [1]- [3]. An additional possibility for control of the spanwise sectional aerodynamics of the rotor blade is the use of Gurney flaps distributed along the rotor blade. Gurney flaps have a much smaller wetted area and considerably less inertia than a traditional flap, thus smaller forces should be required to actuate it as compared to a traditional flap. Extensive work has been conducted to demonstrate the potential benefits Gurney flaps can provide for rotorcraft [4],[5]. The current research addresses the development of actuation systems for a full-scale active Gurney flap to be used for the performance enhancement of rotorcraft as well as the possible suppression of vibrations of the rotor blade.

15 1.2 Gurney Flaps 2 Gurney flaps are small plates placed perpendicular to the flow near the trailing edge of an airfoil. Figure 1-1 illustrates the concept. These types of flaps generate lift due to an effective increase in the camber of the airfoil. Two factors affect the overall performance of the airfoil with a Gurney flap: the height of the flap and its location. As the flap size is increased, increases in both lift and drag occur. Also, as the location of the flap is moved closer to the leading edge of the airfoil, the Gurney flap becomes less effective. Typical values for height and location of the Gurney flap (with respect to chord length) are 0.5% to 2% high and located between 0.9c and 1.0c. Figure 1-1: The Gurney Flap When used on rotorcraft to enhance the performance of a rotor blade, the flap would typically be deployed when the blade is retreating. In forward flight, the retreating side of the rotor experiences a lower velocity and blade stall becomes a concern [6]. Currently, to prevent the blade from stalling, the pitch of the entire rotor blade is adjusted or traditional trailing edge flaps can be used. These two solutions require relatively high actuator power to be effective. A large force is required to rotate the entire blade and a large hinge moment is required to actuate a traditional flap. Other challenges to rotorcraft development arise from the unsteady nature of the aerodynamics, including noise produced by blade-vortex interactions (BVI), and

16 vibrations of rotor blades themselves which are then transmitted to the vehicle. The 3 majority of the vibrations transmitted to the vehicle which cause passenger discomfort originate from the vibrations of the main rotor blades. Much of the blade vibrations are caused by the dynamic stall of the rotor blade on the retreating side [7]. Since dynamic stall happens once per revolution, the hub experiences vibrations at harmonics of the frequency N b /rev, where N b is the number of blades. 1.3 Goal of Current Research The goal of this research is to develop full-scale prototypes of an active Gurney flap actuation system. Different designs involving different actuation methods will be built and tested. The main challenges include the size constraints presented by an airfoil, the weight and balance of the actuator system, and the frequency requirements of the application. The airfoil chosen for this research is a VR-12 airfoil with a 14-inch chord, and a Gurney flap located at 90% of the chord having a height of 1% of the chord. These dimensions were chosen to allow eventual wind-tunnel tests to be conducted and comparison of the results to the previous CFD work [4],[8]. The small size of the airfoil restricts the size of the actuator system, but also allows the design to be applied to larger rotorcraft. The final design should minimize actuator weight and consider its placement so that the performance and dynamics of the rotor blade are not greatly affected. Another factor in the design is the operating frequency range, since the actuator must be able to operate at a high frequency to provide the control necessary for performance gains (1/rev) and vibration reduction (N b /rev). The actuator system must also be able to withstand the

17 quasi-static centrifugal forces in the blade, which are typically on the order of hundreds 4 of g s. Different actuation methods were investigated, including voice coils and piezoelectric benders. Both types of actuators are small enough to fit within the airfoil and can be operated at the necessary frequencies. In order to determine the most effective actuation method, different prototypes were built, each using a different actuation method, with the final goal of a concept demonstration.

18 5 Chapter 2 Background 2.1 The Gurney Flap Gurney Flap Aerodynamics Gurney flaps increase the maximum lift of an airfoil due to an effective increase in the camber. The effective increase in camber arises from the formation of two vortices downstream of the flap, as shown in Figure 2-1 [9]. Like traditional trailing edge flaps, Gurney flaps can be distributed spanwise along the rotor blade, as shown in Figure 2-2, to control the local aerodynamic properties (C l, C d, C m ) of the rotor blade. Figure 2-1: Formation of vortices around a Gurney flap.

19 6 Figure 2-2: Concept of distributed Gurney flaps. Previous computational fluid dynamics (CFD) studies and wind-tunnel testing have addressed different Gurney flap configurations [4],[5], [8],[10],[11]. Much of the work addressed the capabilities of Gurney flaps under typical rotorcraft flight conditions. Kinzel et al. performed CFD-based research using a VR-12 helicopter airfoil. Both steady and unsteady aerodynamics were considered as well as the effects of dynamic stall. Figure 2-3 shows the effect of a 1% Gurney flap located at 1.0c on a VR-12 airfoil at a Mach number of 0.2 [4]. The results show that the Gurney flap increases the lift coefficient, C L, at all angles of attack, and has a larger negative moment coefficient, C M, at all angles of attack. The Gurney flap also increases the drag of the airfoil. The airfoil stalls at a higher lift coefficient when the Gurney flap is deployed, which allows for a greater range of operation. These results are for a Gurney flap located at 1.0c; however, for practical purposes, the flap should be placed upstream of the trailing edge in order to provide space for flap retraction. To generate comparable performance gains at an upstream location, Figure 2-4 is used to select the flap location. The data in Figure 2-4 was obtained from windtunnel experiments on a S903 airfoil, which is comparable to the VR-12 airfoil. For

20 example, Figure 2-4 shows that a Gurney flap located at 0.9c with a height of 0.02c 7 generates the same change in C l, max as one located at 1.0c with a height of 0.01c. Figure 2-3: CFD results of a VR-12 airfoil with a fixed 1% Gurney flap at 1.0c. Figure 2-4: Lift coefficient variation as location and height are varied. Figure 2-5 shows the optimal (quasi-static) deployment schedule of a Gurney flap. In the figure, the solid line represents a baseline VR-12 airfoil, the dashed line a VR-12 with a fixed Gurney flap, and the X s represent the deployment schedule. Close

21 inspection of the figure shows that the flap actuates to stay on the outside of the drag 8 polar, which yields the minimal drag for the same lift. Thus, the flap is only actuated at higher angles of attack. For rotor blades in forward flight, high angles of attack correspond to the retreating side of the rotor where stall limits the rotor performance. From this fact, research was conducted to examine the possible performance benefits to rotorcraft. Figure 2-5: Deployment schedule for a Gurney flap on a VR-12 airfoil Performance Benefits Since Gurney flaps offer an advantage at higher angles of attack, deploying a Gurney flap as the blade retreats offers an increase in lift. In order to maintain rotorcraft roll stability, a subsequent increase in lift on the advancing side must be generated to

22 9 counteract the increase of in lift on the retreating side. This means that the onset of stall will not be as great a limit on the design of rotor blades. An overall increase in lift can also be obtained, which allows for many possibilities when designing a rotor. For an increase in aerodynamic performance, the flap needs to be actuated once per revolution (on the retreating side) and, for a typical rotor, this is between four and six Hz. This possible increase in lift allows for several benefits, including an increase in payload, smaller blades, and/or faster speeds. Several performance characteristics were calculated, both for a rotor using optimally-placed and -deployed Gurney flaps and a baseline rotor, and then compared [5]. From these results, an increase in figure of merit (a measure of efficiency in hover [6]) at higher thrust coefficients can be achieved. Figure 2-6 shows the effect of Gurney flaps on the power required for a nominal helicopter and then the same helicopter with a higher gross take off weight (GTOW). The curves labeled as MiTE correspond to the deployed Gurney flap. (MiTE is short for Miniature Trailing Edge Effector, a term which is used interchangeably with active Gurney flap. ) The lower curves represent the helicopter with a base gross weight, and a small decrease in power required is seen at higher flight speeds. The effect of the Gurney flap is more pronounced when the GTOW is increased. This is due to the increase in lift that the Gurney flap provides. Similar curves are presented for different cases in Ref. 11 (i.e. different altitudes and Gurney flaps scheduled for maximum range or speed). Figure 2-7 shows the required Gurney flap deployment required to generate the results shown in Figure 2-6. The Gurney flap is only needed on the retreating side of the rotor. Note that the code that was used allowed for a spanwise variation in airfoil cross section. For this case, a VR-15 airfoil, which is thinner than a VR-12, is used on the

23 10 outboard section of the rotor. A thinner airfoil is beneficial since it will produce less drag at higher Mach numbers, towards the tip of the advancing side of the rotor, which will reduce the required profile power. Normally, this is done with a compromise in maximum lift on the retreating side, but with the deployment of a Gurney flap, higher lift values can be obtained. Figure 2-6: Power curves for a rotor with and without Gurney flaps. Even though active Gurney flaps have been shown to increase the performance of rotorcraft, the ability to control rotor vibrations with Gurney flaps has not been explored. The operating frequencies for vibration control on a four-bladed rotor would be on the order of 16 to 24 Hz.

24 11 Figure 2-7: Deployment schedule for an active Gurney flap Previous Actuator Work An initial study by Thepvongs into the actuation of a Gurney flap using a piezoelectric bender was conducted to determine if the necessary displacements and actuation frequencies could be achieved [12]. The design was chosen to use a tapered bender which is similar to design used by Hall et al. [13]. A finite element model was developed to predict the deflection and natural frequencies of the system. A full-scale prototype was built and the schematic can be seen in Figure 2-8, along with the fabricated design in Figure 2-9.

25 12 Figure 2-8: Schematic of Gurney flap concept by Thepvongs. Figure 2-9: Fabricated active Gurney flap developed by Thepvongs. This design was tested and proven to meet the required displacements and actuation frequencies. For the case studied, a deflection of 0.36 in was required based on a 0.02c-high Gurney flap located at 0.9c. The airfoil considered was the S903. The design achieved a maximum displacement of approximately 0.80 inches at the desired operating frequency of 18.5 Hz. This work showed that a Gurney flap could be actuated on the scale required in rotorcraft applications.

26 13 There has also been research into the use of active Gurney flaps to control flutter on fixed wing aircraft [14]. Figure 2-10 shows the Micro-Trailing Edge Effectors (MiTEs) used to control aerodynamic loading. Bieniawski et al. placed the flap at the trailing edge of the wing since the goal of the work was flutter suppression, and the increased drag associated with a blunt trailing edge was not a concern [15]. Wind-tunnel tests were performed and the MiTEs were shown to successfully suppress flutter of the test wing, shown in Figure 2-11, when a closed loop control system was implemented. Figure 2-10: MiTEs used for flutter suppression. Figure 2-11: Test setup used to show flutter suppression capabilities of MiTEs.

27 2.2 Actuation Methods Piezoelectric Ceramics Introduction Piezoelectric ceramic, or PZT (lead zirconic titanate), has the ability to convert mechanical energy to electrical energy through the piezoelectric effect. This is exploited in sensing applications. For actuators, the reverse piezoelectric effect converts electrical energy (voltage) into mechanical energy (strain) [16]. Polycrystalline piezoelectric ceramics are first poled to develop the desired electromechanical coupling. For this research, the piezoelectric ceramic sheet was poled in the 3 (z) direction, which results in a strain in the 1 (x) (and 2 (y)) direction. Figure 2-12 shows a piezoelectric sheet, where a voltage across the thickness, t c, will produce an electric field in the 3 direction, causing a strain in the 1 direction. Figure 2-12: Piezoelectric sheet with voltage across the thickness. When PZT material is bonded to either side of a shim, opposite fields can be applied to each piezoelectric patch to induce bending in the beam. This type of actuator

28 is characterized as a piezoelectric bimorph or bender. A piezoelectric stack actuator 15 consists of several PZT elements bonded on top of each other, with the stroke being the sum of the displacements induced in each individual layer. Typically, piezoelectric stacks generate higher forces and a smaller stroke than a bimorph (for a similar amount of piezoelectric material) Piezoelectric Modeling When modeling a piezoelectric actuator, an expression for the induced force (or moment) in terms of the geometric and electrical properties of the actuator is needed. This section addresses the modeling of a piezoelectric bimorph and will be referenced in a later chapter. Several assumptions are made to simplify the model and reduce the computational cost of a finite element analysis of the system. An Euler-Bernoulli beam is assumed, and the strain throughout the shim and piezoelectric material is taken to be linear. The shear lag in the bond layer is ignored and a perfect bond assumed, which results in a concentrated force/moment at the boundary of the piezoelectric material [17]. A complete derivation of the equations of motion for a piezoelectric material bonded to a substructure is available in reference [17]. For a piezoelectric bimorph, the shim can act as the electrical ground, and when a positive voltage is applied to the PZT elements, forces are generated as shown in Figure Eq. 2-1 is the expression for the force generated at each end of the piezoelectric element, with E c the Young s modulus of the PZT, b the width of the F Λ = E c bt c Λ free 2-1

29 16 structure, t c the thickness of the piezoelectric element, and Λ free the induced strain in the piezoelectric when the PZT patch is unconstrained. The induced strain is defined in Eq. 2-2 (d 31 is the piezoelectric constant defined in meters/volt). Λ free = d 31 V t c 2-2 Figure 2-13: Schematic of piezoelectric bimorph, note elements 1 and 2 are poled in the same direction causing forces to be generated in opposite directions. When voltage is applied as shown in Figure 2-13, there are moments (+Μ Λ and - Μ Λ ) applied at the center of the shim (of thickness, t b ) at locations corresponding to the edges of the PZT material. The moments are equal and opposite and defined in Eq M Λ = E c bt c ( tb + tc ) Λ free 2-3

30 The expression in Eq. 2-3 is easily implemented in a beam finite element code by 17 including it in the force vector. Figure 2-14 is the result when the PZT patches are perfectly bonded to the substructure. Figure 2-14: Moments generated by a piezoelectric bimorph Voice Coils Voice coils are small, linear, direct drive electric motors and have the ability to operate at high frequencies with a stroke in the required range. Figure 2-15 illustrates a representative voice coil chosen for this research. Voice coils provide large displacements but relatively small force when compared to the piezoelectric stack actuators. Individually, voice coils will not be as heavy as piezoelectric stacks that have the same stroke length [18]. Voice coils that are compact enough to fit inside an airfoil cross-section should be able to provide an adequate force and stroke without the need for an extensive motion amplification system that a concept employing a piezoelectric stack would require.

31 18 Figure 2-15: The H2W Technologies NCC X voice coil. 2.3 Development of Traditional Trailing-Edge Flaps for Rotorcraft Over the last decade, there has been much research into the development and testing of traditional trailing-edge flaps for the control of rotor blade vibrations [1]-[3], [7], [19]. Straub et al. developed and tested a full scale active flap rotor system incorporating two piezoelectric stack actuators as part of the SMART rotor program [1]. Figure 2-16 shows the full scale rotor blade with the active flap system. Full-scale testing was completed on a whirl tower rig, and the required deflections to control vibrations (only 2 degrees were necessary in forward flight) were attained with results shown in Figure Another important development in this research was the fabrication methods developed to integrate the actuator into the blade. This included both the structural optimization of the blade/flap interface and the subsequent transferring of data and power to the flap system. The program was deemed worthy of forward flight tests.

32 19 Figure 2-16: Smart rotor blade using trailing edge flaps, the flap is in yellow. Figure 2-17: Flap deflections of the SMART rotor systems for different voltages and driving frequencies normalized by rotor speed.

33 20 Other research on the development of a single, Mach-scale trailing edge flap was conducted by Bao et al [3]. In this work, a tapered piezoelectric bender [13] was selected as the actuation method as shown in Figure This method uses a rod and cusp to amplify the motion of the piezoelectric bender. This design was built and tested in a vacuum chamber to test the structural integrity of the flap/blade system. The results from hover tests in a vacuum are presented in Figure 2-19, and show that the flap system can be successfully operated under centrifugal loading. The system has yet to be tested under aerodynamic loads. Figure 2-18: Active flap designed by Bao et al. for rotor control.

34 21 Figure 2-19: Flap deflections using actuation method shown in Figure There has also been research conducted by Kim et al. using multiple trailing edge flaps (Figure 2-20) [2],[19]. This uses mechanical and electrical tuning to increase the operating bandwidth. For this actuator design, a piezoelectric shear tube [20] was used. The bench top tests of the Resonant Actuation System (RAS) (seen in Figure 2-21) were conducted and deflection requirements were met. There has also been recent numerical work to show the potential of using multiple active trailing edge flaps to control the loading on the rotor blade. Figure 2-20: Active flap rotor using multiple trailing edge flaps.

35 22 Figure 2-21: Bench top model of a trailing edge flap using a RAS. The previous research into development of traditional, trailing edge flaps for rotorcraft applications have shown that trailing edge devices can provide a benefit to rotorcraft. For an active Gurney flap concept to be successful, it must provide the same benefits that traditional trailing edge flap, while requiring less power and a lower weight. Many of the same design issues will arise in the development of an active Gurney flap concept and similar testing will need to be conducted. For example, the centrifugal loading must be accounted for and eventually blade integration methods must be developed. The research into traditional trailing edge flaps will provide a loose roadmap for the development of an active Gurney flap concept.

36 Chapter 3 Development of Voice Coil Driven Actuation Concepts 3.1 Introduction to Concepts For the initial concept development, a voice coil was chosen as the actuation method. The specific voice coil can be seen in Figure The NCC X is rated for maximum efficiency at 1 Amp and 3.3 Volts. However, the actuator has the ability to operate effectively up to 2 Amps and 6.6 Volts. When operating at the maximum values, a maximum stroke of 6.35 millimeters and a maximum force of ~13.3 Newtons can be obtained. The total mass is kilograms with a moving mass of only kilograms, since most of the mass is located in a stationary magnet. To integrate the voice coil into an actuation system, the linear motion of the voice coil must be translated into the vertical motion of the flap. For each of the two concepts, the voice coil was mounted as far forward in the airfoil as space would allow. This was done to ensure the aeroelastic stability of the airfoil section. Since the coil assembly is not attached to the magnet, it must be centered for maximum efficiency and to avoid any damage. To accomplish this, a linear bearing is employed which serves the purpose of both supporting and centering the coil.

37 3.2 Voice Coil Testing 24 The goal of initial testing was to obtain the dependence of the force on the current of the coil, to be used later in the modeling of the coil. Figure 3-1 provides a schematic of the setup used to test the NCC actuator. The voice coil drives a shaft which is connected to a spring that provides a mechanical load. The shaft and bearing used were also chosen for use in the flap system. The bearing is frelon-lined and requires no lubrication. It should not be affected by the high g-force loads that are present in helicopter rotor blades. The bearing functions by transferring a small amount of frelon to the shaft and forming a frelon-on-frelon interface that has very low friction. The shaft is made of hardened anodized aluminum and is recommended for use with the chosen bearing. The load provided by the spring can be varied by adjusting a set screw. A faceplate had to be added to provide a mounting point for the shaft at the center of the coil, due to the fact that the coil did not have a preexisting hole on center. Figure 3-1: Schematic of setup used to test the voice coil.

38 25 The displacement of the faceplate was obtained by integrating the signal from a laser velocimeter. The force was then determined by applying Hooke s Law (F = kx) and accounting for an initial displacement of the spring. Figure 3-2 presents the data as well the linear fit used to model the voice coil after it is integrated into the flap system. For this test, the voltage was held constant and the drive frequency was swept from 4 to 5.5 Hertz. The current, I, is calculated by taking the magnitude of Eq. 3-1 where V and f are the drive voltage and frequency in Hertz, and R and L are the resistance and inductance of the coil. It should be noted that there is a non-zero y-intercept, which accounts for any friction that must be overcome to actuate the coil. I = V R + i( 2πf )L 3-1 Figure 3-2: Data from initial testing of the voice coil.

39 First-Generation Voice Coil Driven Actuation System Concept Description The initial concept was chosen to have a Gurney flap located at 0.90c and a height of 0.01c. Again, these dimensions were chosen so that the assembly can eventually be tested in a wind tunnel. The initial design for an active Gurney flap can be found in Figures 3-3 and 3-4. The key features of this design are the cable and housing assembly and the use of frelon-lined bearings. The cable and housing are used to transfer the motion from the voice coil into that which rotates the flap arms. The cables also allow for flexibility in the assembly, which is advantageous in the high vibration environment of a rotor blade. All shafts are made of anodized aluminum; both the cable and housing are steel, and the remainder of the material is 6061 aluminum. The bearings allow for the tight tolerances that are required in order to hold the moving coil assembly of the voice coil in place. The bearings also must not bind under the high centrifugal loads experienced in the helicopter blade, which is why a frelon bearing was chosen over a ball bearing. Lastly, the voice coil is placed as far forward in the cross-section as possible to locate the center of gravity of the assembly close to the aerodynamic center (for aeroelastic stability).

40 27 Figure 3-3: Initial design concept profile. Figure 3-4: Initial design concept perspective Modeling of Initial Concept A linear state-space model of the design was developed and used to simulate the response of the flap/actuator system. The simulation was developed to predict the rotation and corresponding flap deflection given an arbitrary input signal. The voltage input to the coil results in a displacement of the bracket and motion of the cable, which

41 28 causes the arms to rotate the flap. The size of the flap was selected using the simulation by ensuring the deflection requirements were met for a given flap weight. In the model, all parts are considered rigid except for the cable, which has a finite axial stiffness. Viscous damping is introduced in the bearings, and the electrical dynamics of the voice coil are accounted for as well. The aerodynamic drag on the flap was modeled as a constant force equal to the maximum force calculated from the CFD results. (The normal force on the flap is ignored since the values are within the error of the CFD calculations.) Figure 3-5 shows the important elements of the model along with their associated parameters (e.g. M s1 represents the mass of shaft 1, L A the length of the arms, c TB the damping coefficient of the thrust bearing, etc.). The current and voltage (i(t) and V(t)) result in an actuation force generated by the coil, F ACT. This force results in a linear displacement, u(t), of the bracket connecting the cables. The cable then causes a rotation of the arms, θ(t), about shaft 2, which causes the flap to deflect upwards and achieve the desired deflection. There are four key lengths, l 1 l 4, which govern the location of the pivot point and the moment generated by the cable. Figure 3-5: Schematic used to model the initial voice coil concept.

42 The actuator force is modeled using Eq. 3-2 where (FI) is the slope of the curve fit from Figure 3-2. The stiffness of both cables is given in Eq. 3-3 where A is the cross- 29 F ACT = ( FI ) i( t) 3-2 k = c 2 AE L c 3-3 sectional area of a single cable, L c is the length of the cable, and E is the modulus of elasticity. For simplicity in modeling, Eqs. 3-4 and 3-5 are used as expressions for the M + M + M m = M c + M s1 I = I + I eff A s2 FP B moving mass, M m, and the effective inertia, I eff. In the model, the voltage is used as the input and is expressed in terms of the current in Eq ( t) = Li& ( t) Ri( t) V Three equations are used to represent the dynamics of the system. The first is the equation of motion of the moving mass, the second is the equation of motion for the effective inertia and the third is an equation for the current. The three equations are given in Eqs. 3-7, 3-8, and 3-9. Five states are used, the position and velocity of the bracket, M m u& ( t) = ( FI ) i( t) c u& ( t) k l θ ( t) TB c 2 3-7

43 30 the angle and velocity of the arm, and the current. The state vector and its derivative are given in Eq Eq shows the full state-space model which is used to simulate the response for an arbitrary input voltage MATLAB Simulation of Initial Concept A state space model was created using MATLAB s control system toolbox. This was done by first defining A, B, C, and D matrices using the form shown in Eq () ( ) ( ) ( ) ( ) θ θ θ θ l Sp Drag t c t u l k t l k t I TB c c eff + + = & & & 3-8 () ( ) () t i L R L t V t i = & 3-9 {} = i u u x θ θ & &,{ } = i u u x & & & && && & θ θ 3-10 {} ( ) ( ) ( ) {} () t V L x L R I l Sp Drag l k I l k I c M FI M l k M k M c x eff c eff c eff TB m m c m c m TB + + = & 3-11

44 31 & a) { x } = [ A]{ x} + [ B]u b) { y } = [ C]{ x} + [ D]u 3-12 Eq is already in the proper form and the gain matrix, [D], is set to zero. The observation matrix, [C], can be chosen to select the output of the model. The flap is fully deployed when it reaches a deflection of approximately 3.5 millimeters which corresponds to an angle of 3 degrees. Eq b) becomes the following (Eq ) when the desired output is the flap angular deflection. { y} = [ ]{ x} 3-13 Appendix A.1 contains all the dimensions and properties. The MATLAB code found in Appendix A.2 uses the LTI function, ss to create a state-space model. The command lsim generates the response data for a 4 Hz sine wave with a peak-to-peak voltage of 5 Volts. Different values of [C] are used to generate the time history of the other states. The current values are used to then calculate the power for reference. Figure 3-6 shows the actuator s simulated response to a sine wave input with a peak voltage of 5 and a frequency of 4 Hz. The simulation also showed that there were no resonances in or below the operating range of the flap (1/rev to 4/rev) so that nominal operation is quasi-static. Figure 3-7 was created using the command bodemag which creates a frequency response plot from the LTI model.

45 32 Figure 3-6: Simulated response using MATLAB code in Appendix A.2 Figure 3-7: Simulated frequency response using MATLAB code in A.2

46 3.3.4 Centrifugal Loading Calculations 33 An arm/cable housing combination and the bracket connecting the cables to the arms were analyzed to ensure that centrifugal (CF) loading due to the blade rotation would not adversely affect the performance of the concept. The arm and housing combination was considered since those parts are the thinnest and most apt to deform under the spanwise CF loading. The bracket is most likely to deform under the chordwise CF loading due to its orientation in the airfoil section. For the calculations, the CF loading was assumed to be 700 g s in the span-wise direction and 50 g s in the chordwise direction. The arm/cable housing combination was divided into three sections; Arm Section 1 represents the arm from the thrust bearing to the flap, Arm Section 2 connects the thrust bearing to the housing, and Housing represents the combined stiffness of the cable and housing. Figure 3-8 shows the model used, with the thrust Figure 3-8: Schematic of arms and housing used for modeling the effects of CF loading. bearings acting as fixed boundary conditions for both arm sections and the housing/cable connection with the arm modeled as a pin joint (this boundary condition could result in an underestimate of the deflection and an overestimate in the stresses). Figure 3-9 is the model use for the bracket with the pinned boundaries representing the cable attachment

47 34 Figure 3-9: Schematic of bracket used for modeling the effects of CF loading. points. For each section, Eq is solved for to determine the internal moment, M(x), (Eq. 3-15) and displacement, w(x), (Eq. 3-16). The load, p(x), is constant and is ( EI w ) = p( x) 2 x M + 2 ( x) = EIw = p0 + C3x C x x x w EI ( x) = p0 + C3 + C 2 + C1x C represented as p 0. The four constants, C 0 C 3, are determined from the boundary conditions. The displacement and stress (calculated from the moment) were calculated for each section and assembled and plotted using the code found in A.3. The internal stress and displacement for the arm/housing sections are plotted together in Figure The zero is the flap location and the discontinuity represents the thrust bearing. The results are as expected with the maximum stresses occurring at the fixed boundary conditions and the maximum values of the displacement occurring at the locations of the minimum stress. Figure 3-11 presents similar results for the bracket.

48 The stresses are well below the yield strength of aluminum (~25x10 7 N/m 2 ). Deflections reach a maximum of less than a millimeter for both cases as well. 35 Figure 3-10: Results for CF loading of arm/housing combination shown in Figure 3-8. The squares represent the deflections and the triangles represent maximum stress.

49 36 Figure 3-11: Results for CF loading of the bracket shown in Figure 3-9. The squares represent the deflections and the triangles represent maximum stress Fabrication and Testing of Initial Voice Coil Concept To facilitate fabrication, the concept was redesigned for the bench top (i.e. the size of brackets and supports were not considered) but will still fit within a profile of the VR-12 airfoil. Figures 3-12 through 3-14 show the initial, fabricated prototype. Other changes had to be made as well. First, the flap arms were increased in diameter to ensure a secure attachment to the flap. Consequently, the shaft and thrust bearings had to be enlarged to accommodate the larger arm diameter. Also, the inner set of thrust bearings were removed since alignment of the flap arms was easily attained. The voice coil is driven by an Audio Centron RMA-250 Stereo Power Amplifier with an Agilent 35670A

50 Dynamic Signal Analyzer (DSA) used as a signal generator. The DSA is also used to 37 monitor the input voltage to the coil. Figure 3-12: Side view of the fabricated, initial concept. Figure 3-13: Top view of the initial concept 1.

51 38 Figure 3-14: Top view of the initial concept 2. The vertical flap velocity was measured using a Polytec OFV 502 Fiber Interferometer and Polytec OFV 2600 Vibrometer Controller. The DSA was used to acquire both the velocity data and the input voltage data. Since the laser directly measures the velocity of the flap, the velocity signal is integrated in MATLAB to obtain the time history of the flap position (only a third of the signal is integrated to save computational time). This position was then converted to an angular deflection. A complete set of data was taken for five different drive frequencies (5 Hz 25 Hz) and four voltages (3.5 V 6.5 V). The MATLAB code used to process the data is presented in Appendix B.1 and the complete data set can be found in Appendix B.2. It should be noted that some of the data showed a steady increase in the position and angle plots. This is due to a slight offset in the velocity data. When the data was compiled, each set was inspected and a peak-to-peak value was used.

52 39 Figure 3-15 shows a sample data set which includes the input signal used to drive the coil and the velocity, position, and angular deflection of the flap itself. The data revealed some obvious non-linearities. This was due to several factors, including the coil bottoming out in some instances. The arms would occasionally make contact with the cable housing as well. These factors are believed to be the cause of the sharp spikes in the velocity signal. The cable also was not kept perfectly in tension or compression and tended to buckle slightly. These non-linear effects were not modeled, which meant the theory was unable to be verified with the experimental data. The friction in the coil itself as well as the minimum voltage required for movement, was not modeled either. Figure 3-15: Sample data set from initial prototype. The results were compiled and can be found in Figure The data shows that the actuator system is able to attain higher deflections for increasing voltages as well as

53 increasing the drive frequency. The curves corresponding to the all but the highest 40 operating voltage have similar slopes and also show roughly the same increase in deflection as the voltage is increased. When the coil was driven at maximum voltage, the nonlinearities had the greatest affect on the performance (i.e. at lower voltages, the coil tended to bottom out less). However, the data shows that at higher frequencies and higher voltages, the required deflection (3 degrees) can be obtained. Figure 3-16: Frequency response of initial prototype. The initial design proved to be inconsistent. The main inconsistency was due to the fraying and eventual failure of the cables, which would result in have to realign the arms and bracket. When reassembling the system, it was impossible to attach the cables

54 41 so that they were identical lengths. Thus, the actuation of each of the arms was slightly out of phase. Even before complete failure of the cable, as individual strands were broken, the operation of the device suffered. These factors, along with the aforementioned non-linearity, led to a second-generation prototype being built and tested. 3.4 Fabrication and Testing of Second-Generation Voice Coil Driven Actuation System A more linear design than that of the first-generation was then developed, wherein the cable and housing were replaced by a rigid link. Figure 3-17 shows the second concept. The base plate, coil and bearing assembly, and flap assembly were not changed in the second design. The linkage is pinned at both ends to allow rotation. The linkage was designed to statically offset the coil slightly to help prevent the coil from bottoming out. Washers were fabricated out of Teflon to keep the link aligned properly. A small piece of foam was placed between the coil and the magnet to damp the effect of the coil bottoming out. Also, the base plate had to be slightly modified to ensure the flap arms would not contact the base. The disadvantage of using this rigid design is the tendency for the system to bind when loaded perpendicularly to the motion (i.e. from the CF loads in a rotor blade). The cable from the first generation allowed some flexibility in the system.

55 42 Figure 3-17: Fabricated second generation concept. The data for the second concept was taken in a manner identical to that used for the first concept (Figure 3-18 shows the setup used for both concepts). Figure 3-19 Figure 3-18: Experimental setup used for the testing of both concepts.

56 shows sample data taken at the same location on the flap as that from Figure Again, the flap angle, deflection, and velocity, as well as the input signal, are given. The data more closely resembles that of a linear system with the output (flap angle) more closely resembling the input voltage. Figure 3-19: Sample data from the 2 nd generation prototype. The operation, however, was still fairly inconsistent. At lower frequencies, the concept was unable to be operated with any reliability above 2 Volts. At the lower velocities associated with the lower frequencies, the Teflon bearings did not operate well, and the initial friction was harder to overcome than at higher frequencies. This is why, in Figure 3-20, which shows a compilation of the data, there are no data points at higher operating voltages for the 5 Hz input. The second generation concept employing a voice

57 coil also met the deflection and frequency requirements. This second generation concept proved to be more reliable than the initial concept at higher the operating frequency. 44 Figure 3-20: Compiled deflection data from 2 nd generation concept.

58 Chapter 4 Piezoelectric Bimorph Driven Concept Development 4.1 Introduction A bimorph (piezoelectric bender) augmented with centrifugal (CF) loads was developed as an actuation method. In this concept, the bimorph is oriented in the blade section in such a way that the CF loads place the actuator in compression. (Figure 4-1 illustrates the concept.) When a beam is placed in compression, the effective lateral stiffness of the beam is lowered and thus, given the same actuation force, larger displacements are expected [21]. However, a simultaneous reduction in the natural frequency of the beam occurs, due to the decrease in stiffness. This concept is appealing since piezoelectric actuators exhibit relatively low deflections, and an actuator without the need for motion amplification should have higher efficiency. This concept of placing a bimorph in compression was explored by Lesieutre et al. and increases in deflection and effective coupling coefficients were demonstrated [22]. Barret et al. implemented the concept in actuators developed for flight control of a UAV [23]. The bimorph actuator was designed to meet a deflection requirement of 0.7 mm with a first natural frequency greater than 30 Hz. The deflection requirement is equal to the linear motion of the rigid link from the second-generation voice coil concept since the flap assembly is the same for both actuation methods to allow for comparisons.

59 46 Figure 4-1: Gurney flap concept with CF-augmented bimorph. 4.2 Finite Element Modeling of Bimorph Concept A finite element code was developed in MATLAB employing four degree-offreedom beam elements (two-nodes: displacement and rotation at each node) to model the CF-augmented bimorph/flap system. The code, found in Appendix C, returns the free displacement, blocked force, and first natural frequency of the bimorph given a set of input parameters (vector X in the code). Both the static (Eq. 4-1 ) and eigenvalue (Eq. 4-2) problems are solved using standard methods. This section addresses the

60 { F} = G [ K] G { w}, { } [ ] 1 w = K { } G F G 2 [ ω [ M ] + [ K] ] = 0 n G G inclusion of the piezoelectric material and the centrifugal loading into the calculations. This code was used as the basis for parametric studies and was also used in a genetic algorithm to optimize the design. The force vector includes the moments induced by the piezoelectric material (see Section for development of the induced moment); the elemental force vectors for both the first and second layer of the bimorph are shown in Eq The 0 M 0 M (1) Λ1 { F} =, { F} el Λ1 (2) el 0 M = 0 M Λ2 Λ2 4-3 value of the induced moment for each layer is different and the expressions for both M Λ1 and M Λ2 are shown in Eq (Note: t c1 and t c2 correspond to the thicknesses of the M Λ M Λ = be 1 b + ( t t ) d V c c1 31 ( t + t t ) d V = be 2 b 2 c + c 1 c first and second layers, respectively.) The vectors NP and NP2 contain the elemental locations of the first and second layers of PZT material, and control which elemental force vector is used when the global force vector is assembled (see Appendix C.4). The two elemental force vectors are summed if, within the loop, the current element location

61 is equal to a value of NP2. After assembling the global force vector, non-zero values 48 should only remain at locations corresponding to the ends of the PZT material. The effect of the piezoelectric ceramic is also accounted for by varying the elemental properties of the beam (EI, ρa). Three values of EI and ρa are calculated: one corresponding to the center shim alone, one for the shim plus one layer of PZT, and one for the shim plus two layers of PZT. The parallel axis theorem is applied to calculate the moments of inertia for elements involving PZT material. The result is two vectors (labeled EI1 and rhoa1 in the code found in Appendix C.3) containing the property variations along the span of the bimorph. These values are used when the global mass and stiffness matrices are assembled (Appendix C.4). The centrifugal loading is accounted for by modifying the stiffness matrix of each element, [k], through the introduction of a geometric stiffness matrix, [k] geom., which is proportional to the axial load induced by the CF loads. The expression in Eq. 4-5 is used [ K] el 12 6L 12 6L 36 3L 36 3L 2 2 ( ) L L L L ( Paxial ) 2 2 L L L L EI el el = + 3 L 12 6L 12 6L 30L 36 3L 36 3L L L L L L L L L [ k ] [ k ] geom 4-5 when assembling the global stiffness matrices. The axial load, P axial, varies along the span of the beam since the load at each element depends on the mass of all outboard elements. The incremental axial load due to an individual element, i, is defined in Eq. 4-6, with Ω and R representing the rotational frequency of the rotor and radial location of P ( i) axial = ρ ( i) ( i) 2 ( A) ( L) Ω R 4-6

62 the flap. (The negative sign indicates that the beam is in compression.) To obtain a 49 vector of axial load values that can be used in Eq. 4-5, Eq. 4-7 is implemented in a for n ( P ) = max axial el i= n P ( i) axial, n = element number 4-7 loop in Appendix C.3 for the number of elements. The bimorph must also support the CF loading due to the mass of the bar attached to the tip and the corresponding finite element is modified to include this effect. The inertia of the flap assembly is modeled as an additional tip mass attached to the beam. The effective mass of the flap system is calculated based on the dimensions of the second-generation voice-coil-driven actuation concept. Figure 4-2 labels the key features of the flap system that the code found in Appendix D.1 uses to calculate the tip Figure 4-2: Schematic used to determine effective mass of the flap system. mass used in the FEM code. (The code was also used to optimize the dimensions in a later study.) The effective mass is calculated by determining the kinetic energy of both the bar, with mass m b, and the rotating elements, with inertia I eff, and equating that to the

63 total system kinetic energy in the x-direction (at the tip of the bimorph). The total kinetic energy is given Eq. 4-8 and after substituting for θ, (x = θ d 1 ), and the effective mass is & T = meff x = mb x + & 1 2 I eff θ & as shown in Eq The effective mass is then added to the displacement degree of freedom in the global mass matrix. m = m eff b + I eff 2 d Parametric Study of Tapered Bimorph A multi-layered tapered bimorph (similar to that in Ref. [13]) was chosen to drive the flap system. Several geometric parameters of the bimorph can be varied in the design process, including the thickness of each layer (t c1 and t c2 ), thickness of the shim (t b ), width of the bimorph (b), total length (L tot ), and the starting and ending locations of each piezoelectric layer (P start, P finish1, and P finish2 ). Only the total length and the width have physical constraints corresponding roughly to the span of the flap and the thickness of the airfoil, respectively. An initial study was done to verify the gain in deflection by augmenting the bimorph with CF loads. Figure 4-3 shows the effect of the CF loading on a two-layered bimorph with dimensions that were selected to meet the deflection requirements. The FEM code found in Appendix C was modified to output the static deflections to generate the plot. The results show an increase in deflection of

64 approximately 10% due to the CF loading with the actuator located at a radial location of 4.5 meters and with a rotational frequency of five hertz. 51 Figure 4-3: Tapered bimorph deflection with and without the effect of CF loading. Another study was conducted to determine the effect of each PZT layer s thickness on the maximum deflection and natural frequency of the bimorph. Figure 4-4 shows the effect on the maximum tip deflection of varying the thickness of the 1 st layer of piezoelectric material from 0.5 mm to 3 mm, for discrete thickness values of the 2 nd layer. Figure 4-5 shows the effect on the natural frequency for the same thickness variation used in Figure 4-4. All other variables are fixed to the same value used to generate Figure 4-3. The values of natural frequency and maximum deflection exhibit opposite trends as the

65 52 Figure 4-4: Dependence of maximum tip deflection on thickness variations. Figure 4-5: Dependence of the 1 st natural frequency on thickness variations.

66 53 thicknesses are varied: as the thicknesses were increased, the tip deflection decreased and need for optimization. 4.4 Optimization of Piezoelectric Tapered Bimorph To optimize the CF-augmented bimorph, a genetic algorithm was used. Tournament selection was employed to select the best population members (parents) to be used to create the next population generation (offspring). The design variables used are the thicknesses of the shim and PZT layers, the total width, total length, and locations of the two layers. The thicknesses are allowed to vary from half to three millimeters and there is no restriction on the location of the PZT. When the elemental locations of the PZT are passed to the finite element code, their values are limited to only integers and the location of the second layer is always a subset of the location of the first layer. The objective function was defined to maximize the natural frequency given a constraint on the maximum deflection. If the deflection constraint is violated, a large penalty is incurred. The physical constraints are imposed simply by not allowing the optimizer to choose values greater than the limits set by the user. The genetic algorithm (GA) is implements the process shown in Figure 4-6. First, the design parameters, limits, and number of generations are selected using the code found in Appendix E.1 labeled real_coded_sim_taper.m. To begin, a random population is created which has a size of three times the number of parameters. (This was determined to be the most efficient value.) These parameters are passed to

67 54 limits_2.m (Appendix E.2) which ensures the values are within the limits and, if not, sets the parameters to the minimum or maximum as appropriate. The parameters corresponding to the starting and finishing locations of the piezoelectric material are also set to integers within limits_2.m. PNX.m (Appendix E.3) is used to create the children from selected parents. The number of children created is set to ten times the population size. The children are created around the parents using a Gaussian distribution. The width of the distribution can be controlled via the parameter eta and a value of 1.5 was determined to yield the best results. Random noise is then added to the population using spike.m (Appendix E.4 ) to help prevent the algorithm from converging to a local minimum. Finally, tourney_min.m (Appendix E.5) uses a tournament selection method to select the best children to be used as parents for the next generation. The fitness is evaluated using PZT_FEM_Taper.m (Appendix E.6 ), which uses the FEM code developed in Section 4.2, but only returns the value of the fitness function. This process is repeated for either a set number of generations, or until a goal is obtained, which is controlled by the parameter max_error.

68 55 Figure 4-6: Implementation of genetic algorithm. 4.5 Initial Results of Optimization Initial optimization yielded natural frequencies of ~18 Hz and the required deflection of 0.7 mm. This performance is acceptable if the goal of the active Gurney flap is solely to increase the performance of the rotor, without regard for vibration reduction. In order to control vibrations of the rotor blade, however, natural frequencies on the order of 30 Hz are desired. A natural frequency of 30 Hz could not be obtained given the use of the existing flap assembly. Therefore, to increase the natural frequency without sacrificing tip deflection, the inertia associated with the flap assembly needed to be reduced.

69 56 The optimizer was then run with the objective of maximizing the available inertia, with a penalty now associated with a natural frequency of less than 30 Hz. Table Table 4-1 shows the results for a two-layer tapered bimorph, and FigureFigure 4-7 shows a schematic. The available flap inertia is approximately 25% of that of the current flap assembly. (Note that the flap inertia is referenced to the tip mass used in the model.) Therefore, changes to the flap assembly need to be made in order to control vibrations with the CF-augmented bimorph. Table 4-1: Optimized CF-augmented bimorph. DESIGN VARIABLES Shim Thickness, m st Layer PZT Thickness, m nd Layer PZT Thickness, m Width, m Length, m PZT Start (Element # out of 30) 1 1 st Layer Finish (Element # out of 30) 29 2 nd Layer Finish (Element # out of 30) 16 RESULTS Natural Frequency, Hz Displacement, m e-04 Blocking Force, N Available Flap Inertia, kg Figure 4-7: CF-Augmented Bimorph with dimensions from TableTable 4-1.

70 4.6 Resizing of Flap System 57 The MATLAB code found in Appendix D.1 was modified to allow variations in the sizes of the shaft, bar, and flap as well as allowing for changes in the distance between the pin and shaft (d 1 ) and the length of the arm from the shaft to flap (d 2 ). These modifications are found in Appendix D.2, as well as a fitness value, fitness, to be used in the genetic algorithm described in Section thicknesses are varied: as the thicknesses were increased, the tip deflection decreased and need for optimization The optimizer is allowed to vary both d1 and d2 and the values of A, B, and C, which correspond to the fraction of the original mass of an element (shaft, bar, or flap). The dimensions of the bracket were fixed, as were the diameters of the arms (the mass of the arm is affected by d2 ). The fitness function is computed in two ways, either set to a penalty if the value of d1 is too large, or set to the negative of the sum of the ratios (A, B, and C) and d2 (this serves the purpose to maximize the ratios and d2 ). The fitness value is also subject to the constraint that the inertia must be less than or equal to the available flap inertia found in TableTable 4-1. Table 4-2 presents the optimized solution as well as the original values of d 1 and d 2. The value of d 1 is as large as possible since the effective mass is proportional to the inverse of d 1 (see Eq. 4-9). The increase in d 2 lowers the inertia of the flap while simultaneously increasing the mass of the arms. The results show that the shaft needs no modifications since A is equal to one. This was expected since the shaft is the axis of rotation and does not contribute as heavily to the total inertia. The mass of the bar has to be reduced slightly and the flap should be

71 reduced to 40% of its current mass (0.4 was set as the minimum allowable ratio for all 58 three ratios). Table 4-2: Results from optimized flap system Parameter Optimized Original d 1, m d 2, m A 1 - B C (Floor) Preliminary Testing of Piezoelectric Bimorph Preliminary testing was conducted to verify the finite element analysis used in the design of the piezoelectric bimorph. The bimorph was built using the piezoelectric ceramic material, PZT-850, supplied by APC International. Table 4-3 provides the piezoelectric constant, short circuit Young s modulus and density of PZT-850 (labeled E c and ρ c ), and physical characteristics of the bimorph. The frequency response characteristics of the bimorph were analyzed.

72 59 Table 4-3: Properties of bimorph used for testing. Property Value d 31, m/volt 175e-12 E c, N/m 2 6.3e10 ρ c, kg/m b, m 16.4e-3 t c1, m 1.59e-3 t c2, m 1.01e-3 t b, m L tot, m 91.86e-3 P start, element # (out of 30) 1 P finish1, element # (out of 30) 21 P finish2, element # (out of 30) Testing of Bimorph The initial test was designed to measure the velocity near the tip of the tapered bimorph with clamped-free boundary conditions. The same laser vibrometer and digital signal analyzer described in Section was used acquire the data. A Hewlett Packard 33120A signal generator and a Piezo Systems Inc. Model ESA-208 piezoelectric amplifier were used to drive the actuator. Electrical leads were connected to the outer piezoelectric layers and the shim acted as the other electrode, thus causing the voltage across both the inner and outer layers to be equal. The actuator was given a chirp signal as an input that varied from 0 Hz to 1600 Hz, to excite the first two bending modes. The frequency response data was measured over 25 averages (after applying a Hanning window) and saved along with the coherence and power spectrum (to make damping ratio calculations).

73 The power spectrum can be seen in Figure 4-8 and shows resonant peaks (f n ) at 163 Hz and 1306 Hz. From this plot the damping ratios, ζ, for each peak were calculated using the half power method, shown in Eq. 4-10, where f 2 and f 1 are the frequencies corresponding to the half power points. The damping ratios for each mode are shown in Table 4-4. These values were used to match the calculated frequency response functions with the experimental data. 1 f 2 f1 ζ = f n 60

74 61 Table 4-4: Damping Ratios for the first and second bending modes. Frequency, Hz Damping Ratio Figure 4-8: Power spectrum for tapered bimorph Simulation of Frequency Response Function The frequency response function was synthesized using the code found in Appendix F.1. In order to match the natural frequencies, the boundary condition had to be altered from the original clamped-free condition. The fabricated bimorph was clamped on the shim only with a very small gap between the piezoelectric material and

75 the clamp. Therefore, to remove the rotational degree of freedom in the model at the 62 boundary would be inappropriate. The best agreement between data and theory occurred when the boundary condition was modeled as pinned with a rotational spring. Shifting the piezoelectric material outboard by a single element was also modeled, but the agreement between the data and model was not as good around the second natural frequency. A frequency response function of unit velocity per unit induced moment was generated for three different inputs then summed. A frequency response function needed to be generated corresponding to each the locations where the PZT induced a moment: the root of the beam and at the end of the inner and outer layers. When summing the three transfer functions, each are multiplied by the magnitude of the corresponding induced moment to obtain a physical moment instead of the unit moment used in the theory. Also, the transfer function associated with the induced moment at the root is taken as negative since the moment acts in the opposite direction from the other two induced moments. Equation 4-11 shows the form of the transfer function used to model the system, where m is the number of modes to include in the synthesis, φ R is a matrix of the eigenvectors, q in and q out are the degrees of freedom corresponding to input and output of the transfer function, and ζ r and ω r are the damping and frequency of mode r. The frequency response is then calculated over a desired frequency range (Ω) by substituting iω for s. The magnitude and phase are then computed from these values. q& M out Λ ( s) () s = m r = 1 ( q ) φ ( q ) φ R in R out s s 2ζ rωr s ωr 4-11

76 4.7.3 Comparison of the Experimental and Simulated Frequency Response 63 Figure 4-9 shows the magnitude of the frequency response function obtained experimentally along with the synthesized frequency response function and Figure 4-10 plots the phase of the data and simulation. The simulation and data show very good agreement at the first natural frequency with the predicted value differing by only a half of a Hertz. The frequencies of the second resonant peak differ by 56 Hertz. The magnitude of the response at both of the peaks is on the same order of magnitude. The relative difference in heights between the first and second peaks is captured by the simulation. The simulation also shows the same trends as the data on both the magnitude and phase plots. The discrepancies at lower frequencies occur because the coherence (shown in Figure 4-11) at those frequencies was almost zero.

77 Figure 4-9: Magnitude of simulated- and experimentally-determined frequency response functions. Data cursors mark the frequency (X) and amplitude (Y) of the peaks. 64

78 Figure 4-10: Phase of simulated- and experimentally-determined frequency response functions. 65