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MS thesis

  1. 1. The Pennsylvania State University The Graduate School College of Engineering AN EXPERIMENTAL AND THEORETICAL CORRELATION OF AN ANALYSIS FOR HELICOPTER ROTOR BLADE AND DROOP STOP IMPACTS A Thesis in Aerospace Engineering by Jonathan Allen Keller © 1997 Jonathan Allen Keller Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science December, 1997
  2. 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. Jonathan Allen Keller
  3. 3. We approve the thesis of Jonathan Allen Keller. Date of Signature Edward C. Smith Assistant Professor of Aerospace Engineering Thesis Advisor George A. Lesieutre Associate Professor of Aerospace Engineering Farhan S. Gandhi Assistant Professor of Aerospace Engineering Dennis K. McLaughlin Professor of Aerospace Engineering Head of the Department of Aerospace Engineering
  4. 4. iii ABSTRACT A theoretical and experimental study of the transient response of an articulated rotor blade experiencing a droop stop impact was conducted. The rotor blade is modeled using the finite element method and the droop stop is simulated using a conditional rotational spring. The application of linear versus nonlinear beam bending theories was examined. During a rotor blade and droop stop impact, the boundary conditions of the blade change from a hinged condition to a cantilevered condition. Three different methods of integrating the equations of motion in time for a rotor blade droop stop impact event were studied. The first method was a direct integration of the physical space equations of motion. The second method was an integration in modal space using only the hinged modes of the blade. The third method was an integration in modal space using the appropriate modes of the blade depending on blade/droop stop contact. A 1/8th scale model articulated rotor blade was constructed. A series of modal parameter identification tests were performed on the model blade to determine its natural frequencies and modal damping ratios. Given a range of initial flap hinge angles, drop tests of the model rotor blade were conducted at zero rotational speed. The transient response of the tip deflection, flap hinge angle, and strain at three locations were measured. Good correlation existed between the experimental data and all three analytic methods. The computational efficiency of both modal integration techniques was compared to the physical space integration, which required the most time to complete. The modal space integration of the
  5. 5. iv equations of motion using only the hinged modes of the blade was 3 times faster than the physical space integration. The modal space integration of the equations of motion using either the hinged or cantilevered modes of the blade depending on rotor/blade droop stop contact was 5.4 times faster than the physical space integration.
  6. 6. v TABLE OF CONTENTS LIST OF TABLES ....................................................................................................... viii LIST OF FIGURES........................................................................................................ix LIST OF SYMBOLS..................................................................................................... xii ACKNOWLEDGMENTS ............................................................................................. xv Chapter 1 INTRODUCTION....................................................................................1 1.1. Background and Motivation ..........................................................................1 1.2. Literature Review - Beam Impacts.................................................................5 1.2.1. Unconstrained Modes Only.............................................................7 1.2.2. Both Unconstrained and Constrained Modes...................................8 1.2.3. Approach Comparisons...................................................................8 1.2.4. Summary ........................................................................................9 1.3. Scope of the Present Research.....................................................................10 Chapter 2 MODELING AND ANALYSIS OF DROOP STOP IMPACTS .............17 2.1. Derivation of the Blade Equations of Motion...............................................17 2.1.1. Variation of the Kinetic Energy.....................................................18 2.1.2. Variation of the Total Strain Energy..............................................19 2.1.2.1. Variation of the Blade Elastic Strain Energy ...................19 2.1.2.2. Variation of the Droop Stop Strain Energy.....................21 2.1.3. Variation of the External Work.....................................................23 2.2. Finite Element Discretization.......................................................................23 2.3. Modal Analysis............................................................................................26 2.4. Transient Response Analysis........................................................................29 2.4.1. Modal Swapping Off Technique....................................................29 2.4.2. Modal Swapping On Technique ....................................................30
  7. 7. vi Chapter 3 EXPERIMENTAL TESTING................................................................36 3.1. Model Blade Construction Methodology .....................................................36 3.2. Modal Parameter Identification....................................................................40 3.2.1. Experimental Procedure................................................................40 3.2.2. Frequency Response and Coherence Functions..............................42 3.2.3. Dimensional Natural Frequencies ..................................................43 3.2.4. Nondimensional Natural Frequencies ............................................44 3.2.5. Modal Damping Ratios .................................................................45 3.2.6. Mode Shapes................................................................................46 3.2.7. Modal Assurance Criteria..............................................................47 3.3. Droop Stop Impact Tests ............................................................................48 3.3.1. Test Procedure .............................................................................49 3.3.2. Instrumentation.............................................................................50 3.3.3. Calculation of Measured Tip Deflection ........................................51 Chapter 4 RESULTS AND DISCUSSION.............................................................68 4.1. Comparison of Linear and Nonlinear Bending Theory..................................68 4.1.1. Newton-Rapson Iteration Method.................................................69 4.1.2. Validation with Experimental Data................................................71 4.1.3. Nonlinear Analysis of Model Beam ...............................................72 4.2. Comparison of Correct and Incorrect Modal Swapping Algorithm...............73 4.3. Modal Damping Ratios................................................................................73 4.4. Drop Test Results .......................................................................................74 4.4.1. Results from 2.6º Drop Angle .......................................................75 4.4.2. Results from 4.0º Drop Angle .......................................................77 4.4.3. Results from 5.2º Drop Angle .......................................................78 4.4.4. Results from 7.6º Drop Angle .......................................................79 4.4.5. Results from 9.7º Drop Angle .......................................................80 4.4.6. Summary of Drop Tests................................................................81 4.5. Comparison of Computational Efficiency.....................................................82 Chapter 5 CONCLUSIONS AND RECOMMENDATIONS.................................108 5.1. Conclusions...............................................................................................109 5.2. Recommendations .....................................................................................110
  8. 8. vii APPENDIX .............................................................................................................112 BIBLIOGRAPHY .......................................................................................................134
  9. 9. viii LIST OF TABLES 3-1 Physical Characteristics of H-46 Blade and Model Blade ....................................54 3-2 Dimensional Natural Frequency Comparison of Model Blade .............................55 3-3 Nondimensional Natural Frequency Comparison.................................................56 3-4 Modal Damping Ratios.......................................................................................57 3-5 MAC Matrix for Hinged Condition and Cantilevered Condition..........................58 4-1 Parameters Used in Finite Element Model ..........................................................85
  10. 10. ix LIST OF FIGURES 1-1 “Tunnel Strike” and “Tailboom Strike” Depictions .............................................11 1-2 Droop Stop Schematic .......................................................................................12 1-3 H-46 Droop Stop Mechanism.............................................................................13 1-4 Measured H-46 Engagement RPM and Simulated H-46 Rotor Engagement Tip Deflection and Flap Hinge Angle Time Histories (from Ref. 15)...................14 1-5 Articulated Rotor Blade Bending Modes ............................................................15 1-6 Schematic of Heat Exchanger Shaft (from Ref. 21) and Industrial Relay (from Ref. 17)....................................................................................................16 2-1 Beam Coordinate System ...................................................................................32 2-2 Finite Element Representation............................................................................33 2-3 Flap Hinge and Droop Stop Assembly................................................................34 2-4 Incorrect and Correct Modal Swapping On Algorithms ......................................35 3-1 Nondimensional Mass and Flapwise Stiffness Distributions.................................59 3-2 Driving Point Frequency Response Function Magnitude, Phase, and Coherence for the Hinged Test ....................................................................60 3-3 Blade Tip Frequency Response Function Magnitude, Phase, and Coherence for the Hinged Test ....................................................................61 3-4 Driving Point Frequency Response Function Magnitude, Phase, and Coherence for the Cantilevered Test ............................................................62 3-5 Blade Tip Frequency Response Function Magnitude, Phase, and Coherence for the Cantilevered Test ............................................................63
  11. 11. x 3-6 First through Sixth Bending Mode Shapes in the Hinged Condition ....................64 3-7 First through Sixth Bending Mode Shapes in the Cantilevered Condition............65 3-8 Model Blade Schematic and Experimental Apparatus .........................................66 3-9 Estimation of Error in Calculated Tip Defections due to Accelerometer Drift ...........................................................................................67 4-1 Comparison of Nonlinear Bending Theory with Experimental Results (from Ref. 27)........................................................................................86 4-2 Comparison of Nonlinear Bending Theory with Experimental Results (from Ref. 27)........................................................................................87 4-3 Nonlinear Analysis for Model Beam ...................................................................88 4-4 Flap Hinge Angle, Tip Deflection, and Strain at x/L = 0.20 Results for Incorrect and Correct Modal Swapping Algorithms When Dropped from 2.6º ...................................................................................89 4-5 Flap Hinge Angle, Tip Deflection, and Strain at x/L = 0.20 Results for Incorrect and Correct Modal Swapping Algorithms When Dropped from 5.5º ...................................................................................90 4-6 Flap Hinge Angle, Tip Deflection, and Strain at x/L = 0.20 for a Simulated Drop Test When Dropped from 5.5º with and without Measured Modal Damping Ratios..........................................................91 4-7 Convergence Histories for Modal Swapping Off Technique and Modal Swapping On Technique..........................................................................92 4-8 Model Blade Hinge Angle and Tip Deflection When Dropped from 2.6º.............93 4-9 Model Blade Strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 When Dropped from 2.6º ...................................................................................94 4-10 Model Blade Hinge Angle and Tip Deflection When Dropped from 4.0º.............95 4-11 Model Blade Strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 When Dropped from 4.0º ...................................................................................96
  12. 12. xi 4-12 Model Blade Hinge Angle and Tip Deflection When Dropped from 5.2º.............97 4-13 Model Blade Strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 When Dropped from 5.2º ...................................................................................98 4-14 Model Blade Hinge Angle and Tip Deflection When Dropped from 7.6º.............99 4-15 Model Blade Strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 When Dropped from 7.6º .................................................................................100 4-16 Model Blade Hinge Angle and Tip Deflection When Dropped from 9.7º...........101 4-17 Model Blade Strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 When Dropped from 9.7º .................................................................................102 4-18 Maximum Tip Deflections and Maximum Strains at x/L = 0.20.........................103 4-19 Maximum Strains at x/L = 0.30 and x/L = 0.40 .................................................104 4-20 Typical Drop Test and Required CPU Clock Time ...........................................105 4-21 Typical H-46 Rotor Engagement (from Ref. 15) and Required CPU Clock Time..............................................................................................106 4-22 Example Safe Engage Envelope for the H-46 Sea Knight (from Ref. 15) ..........107
  13. 13. xii LIST OF SYMBOLS A Blade cross sectional area C Global damping matrix e Flap hinge offset E Modulus of elasticity f Elemental load vector F Global load vector h Blade thickness H Vector of Hermitian shape functions I Area moment of inertia I Identity matrix k Elemental stiffness matrix K Global stiffness matrix Kβ Droop stop spring stiffness L Blade length m Elemental mass matrix M Global mass matrix Nel Number of finite elements Nm Number of modes q Global vector of modal coordinates
  14. 14. xiii s Blade deformed length coordinate t Time T Kinetic energy Tp Data acquisition window length U Strain energy w Flapping deflection w Global vector of flapping displacement W External work x Undeformed length coordinate α Proportional damping constant β Flap hinge angle δ Variational operator Φ Mode shape η Proportional damping constant θ Blade rotation angle ρc Instantaneous blade curvature ρs Blade mass density τ2 Force-exponential window damping constant ω Blade natural frequency ζ Modal damping ratio Π Total energy
  15. 15. xiv ( )& ( )∂ ∂t ( )&& ( )∂ ∂2 2 t ( )′ ( )∂ ∂s ( )* Nondimensional quantity ( ) Modal space quantity ( )a Inboard node of finite element ( )b Outboard node of finite element ( )B Contribution from blade ( )c Cantilevered condition ( )DS Contribution from droop stop ( )h Hinged condition ( )i ith finite element ( )NL Contribution from nonlinear effects ( )T Transpose of a vector or matrix
  16. 16. xv ACKNOWLEDGMENTS While in my Junior year of studies, a first-year professor full of excitement actually attempted to interest me, a student who often fell asleep in the said professor’s class, in the field of rotorcraft. Over the next two years, he kept signing me up for field trips and meetings - try as I might to avoid them. When I was nearing graduation, he convinced me to enter graduate school and study rotorcraft under his tutelage even though I was reluctant. Looking back, it was one of the best decisions that I have ever made. Graduate school has truly been an eye-opening experience. I would like to express my sincere gratitude to my advisor, Dr. Edward C. Smith. I am indebted to Bill Geyer of the Dynamic Interface Team of the Naval Airwarfare Center, for beginning the research on the H-46 Sea Knight Tunnel Strike Phenomenon, for his hospitality, and for his help over the past two years. Working with him has been a pleasure. The hours spent in conversation and exploring the H-46 up close were invaluable. I would also like to thank the Dynamic Interface Team of the Naval Airwarfare Center at Patuxent River, Maryland; the NASA/Army National Rotorcraft Technology Center; the USMC Depot at Cherry Point, North Carolina; and the Vertical Flight Foundation for financing my graduate education. I would also like to thank Dr. George A. Lesieutre for his very astute observation of an analytical mistake I was making. Also, three other students, Eric Ruhl, Christopher
  17. 17. xvi Knarr, and Christopher McLean were especially helpful in building the experimental apparatus and instrumentation that I used. Finally, I would like to thank all my friends and especially my parents for always supporting my decisions and patiently listening to my rantings and ravings. Their support, both emotional and financial, was invaluable.
  18. 18. 1 Chapter 1 INTRODUCTION 1.1. Background and Motivation Since 1943, the U.S. Navy has operated rotorcraft aboard ships, where unique and often hazardous conditions can be encountered. One of the more troublesome problems that can happen when rotorcraft are operated from ships occurs during the engagement and disengagement of the rotor system while the aircraft is on the flight deck. Excessive aeroelastic flapping of the rotor due to high wind over deck conditions and high sea states can occur at low rotor speeds when the centrifugal stiffening is low, usually less than 20% of the operational rotor speed [1]. This can cause the rotor blade tips to deflect several feet and contact either the fuselage of the helicopter, which is called a “tunnel strike” for tandem rotor configurations; or the tailboom of the helicopter, which is called a “tailboom strike” for single rotor configurations. Both situations are shown schematically in Figure 1-1. The H-46 Sea Knight, a tandem rotor helicopter with articulated rotor blades, is particularly susceptible to the tunnel strike problem [2-4]. Over 100 tunnel strike incidents have occurred during the H-46’s operational history since 1964, ranging from little or no damage to the complete loss of the helicopter [5-6]. The cost in dollars can
  19. 19. 2 range from only man hour costs when inspections are required for minor tunnel strikes, to approximately $500,000 for a tunnel strike which is severe enough to suddenly stop the drive train system [1]. If the damage is severe enough to render the helicopter unrepairable, the cost in dollars cannot be calculated because the H-46 is no longer in production and cannot be replaced. When the rotor system is at rest, an H-46 rotor blade droops under its own weight a distance, δs, of approximately 15 inches or 5% of its length as shown in Figure 1-2 [7]. A tunnel strike occurs during an engagement or disengagement when the rotor blade deflects elastically an additional distance, δe, of 40 inches or 13% of its length measured from the plane of the rotor disk. Two rotor system features of articulated rotor blades that are very important to the engagement and disengagement processes are the droop stop and flap stop mechanisms. The droop stop supports the blade weight by restraining the downward rotation of the flap hinge while the rotor system is at rest and at low rotor speeds. The flap stop limits excessive upward flapping motion of the blade using the same type of mechanism. A schematic of the droop stop is shown in Figure 1-2 and a picture of the actual H-46 droop stop is shown in Figure 1-3. The position of the droop stop, either extended or retracted, is governed by two opposing forces acting on a counterweight. When the outward pulling of the centrifugal force is less than the inward pulling of the spring force, the droop stop extends and prevents the rotor blade from flapping below the droop stop angle. When the outward pulling of the centrifugal force is greater than the inward pulling of the spring force, the droop stop retracts allowing the blade to flap freely. During a rotor engagement or
  20. 20. 3 disengagement, high wind over deck conditions can cause the rotor blade to repeatedly impact the flap and droop stops resulting in large elastic deformations of the blade. Since 1985, tailboom strikes have been the subject of both analytical and experimental investigation by Newman at the University of Southampton [8-12]. The transient blade flapping response of a hingeless rotor system during engage and disengage sequences was predicted using an elastic rotor code. The rotor system RPM time histories were specified with analytic functions. Quasi-steady aerodynamic theory including a Kirchoff trailing edge separation model was used. The ship airwake environment included ship roll motion effects and simple deterministic gusts, which were developed from model scale wind tunnel surveys and were correlated with full scale ship airwake data. The model was later improved by including aerodynamic flap-torsion coupling and the deterministic gust model was correlated with additional wind tunnel experiments [9]. In 1992, the elastic flap code was modified for articulated rotor systems including mechanical droop and flap stops. These stops were modeled using high rate linear springs that apply a restraining force only when the blade flap deflection at the spring location is large enough to cause contact. These high rate springs then appeared as additional point loads in the forcing term. When an articulated blade is in contact with the stop it acts like a cantilevered beam; however when the blade is not in contact with the stop it acts like a hinged beam. Only the articulated modes of the rotor were used in the modal analysis, whether or not the blade was in contact with the stop [10]. Recent studies were conducted to validate the rotor analysis in which a model rotor system with rigid, teetering blades was placed aboard a scaled ship deck and tested
  21. 21. 4 in a wind tunnel [11-12]. Simulated engagements and disengagements of the model rotor system were performed at different locations on the scaled ship deck. The influence of the ship’s structure on the surrounding airwake was proven to be very important to the blade’s overall behavior. Researchers at Penn State University have also been investigating the transient aeroelastic response of a rotor during shipboard engage and disengage operations [13-15]. This model features seven degree of freedom finite elements for elastic bending and torsion of an articulated rotor blade. A high rate rotational spring located at the flap hinge is used to simulate contact between the rotor blade and droop stop. For a given rotor speed variation in time and ship airwake environment, the equations of motion can be integrated in either physical or modal space using a fourth order Runge-Kutta integrator. Validation with the wind tunnel tests in Ref. 11-12 was performed [15]. An example of the results from the analysis in Ref. 15 is shown in Figure 1-4. The top graph shows the experimentally measured H-46 Sea Knight rotor engagement RPM profile in which the rotor is under 10%NR for approximately 10 seconds. The bottom graph shows the time history of the tip deflection and flap hinge angle for the simulated rotor engagement. The specified angle for contact between the rotor blade and droop stop was -1.25° and the specified angle for flap stop contact was 1.5°. Note that the largest tip deflections occur when the blade elastically bends after repeatedly impacting the flap and droop stops. Correctly modeling the effects that rotor blade and droop stop interactions have on the blade response is essential to an accurate transient response analysis.
  22. 22. 5 The boundary conditions of the blade change dramatically during an impact between the rotor blade and droop stop. When the blade is not in contact with the droop stop, the flap hinge is free to rotate and the blade acts as a hinged beam. When the blade is in contact with the droop stop, the flap hinge is restrained from further downward rotation and the blade acts as a cantilevered beam. Therefore, the rotor blade can exhibit two sets of distinct bending modes depending on whether the blade is or is not in contact with the droop stop. Figure 1-5 depicts the different vibration behavior of the rotor blade for each condition. A high rate spring is typically used to analytically model the droop stop, freezing the hinge angle during an impact. 1.2. Literature Review - Beam Impacts Early research specifically dealing with rotor blade and droop stop impacts was conducted by Leone at Boeing Vertol [16]. The bending moment distribution along an articulated rotor blade was derived from the solution of the homogenous integro-partial differential equation governing the transient flap-bending behavior during a droop stop impact. Only the fundamental cantilevered mode was considered in the analytic formulation. Using a full scale aircraft operating at full rotor speed, an excessive cyclic pitch input to the rotor system was used to cause an impact between the rotor blade and droop stop. Experimental measurements of the rotor blade flapping motion and root bending moment for varying blade flapping impact velocities were recorded with an
  23. 23. 6 oscilloscope. Good agreement existed between the predicted and the measured root bending moment over a wide range of measured impact velocities. Impacts of elastic beams with rigid constraints has been the subject of much investigation, a few specific examples are included in Refs. 17-25. Much of this research was motivated by the vibration of beams or rotating shafts between supports, a situation often found in heat exchangers; or by the high frequency chattering of beams against rigid stops, a situation often found in industrial relays. Both examples are shown schematically in Figure 1-6. Normally, the vibration of such structures is calculated using the structure’s eigenfunctions, or mode shapes. If this procedure is followed, the constraint can be treated with two different approaches. In the first approach the structure’s vibration can be described in terms of the modes of the unconstrained structure and the constraint treated as an applied force. In the second approach the constraint can be treated as an integral part of the structure. The unconstrained modes can be used to describe the motion of the structure when it is not in contact with the restraint; and the constrained modes can be used to describe the motion of the structure when it is in contact with the restraint. The next two sections will describe the related research into each of the two methods. The third section reviews comparisons of the two approach methods.
  24. 24. 7 1.2.1. Unconstrained Modes Only Fathi and Popplewell in Ref. 17 studied a cantilever beam contacting an undamped, stiff linear stop using only unconstrained modes. The contact force between the beam and the stop, the tip acceleration, the tip velocity, and the tip displacement were calculated for varying numbers of unconstrained modes used in the analysis. The tip displacement required the least number of modes while the contact forces required the most number of modes to obtain a converged solution. In Ref. 18 Lo considered the problem of a cantilevered beam chattering against a rigid stop. The deflected shape of the beam, the contact force between the beam and the rigid stop and the points in time at which the beam was in contact with the stop, or chatter pattern, were calculated. The deformed shape of the beam was photographically measured using a multi-flash strobe light and the chatter pattern was experimentally measured with an oscilloscope and agreed well with the analytical values. Similar to Ref. 17, the deflected shape of the beam was predicted accurately with a lower number of modes than the contact force or chatter pattern. Both Molnar et al. in Ref. 19 and Shah et al. in Ref. 20 considered the problem of the response of gap restrained piping systems using only the unconstrained modes of the system. Solution efficiencies were compared either for the direct integration of the equations of motion in physical space coordinates or the integration of the equations of
  25. 25. 8 motion in modal space coordinates. Significant time savings were realized using the modal superposition method to integrate the equations of motion. 1.2.2. Both Unconstrained and Constrained Modes In Ref. 21 Chen et al. use both the unconstrained and constrained modes to describe the vibration of a simply supported beam, constrained between two undamped stops and forced by a sinusoidal distributed load. The steady state deflection shape of the beam and dynamic stresses in the beam were calculated for varying forcing frequencies. Rogers and Pick present finite element approaches to the problem of a heat exchanger tube vibrating inside an annulus [22-23]. Combinations of the constrained and unconstrained modes were used to compare the simulated and experimentally measured vibration response of the beam. In Ref. 22 the rms contact forces generated in an impact between the beam and the clearance supports were measured and showed good correlation with the predicted values. In Ref. 23 the time histories of the deflection, velocity, acceleration, and contact forces of a single heat exchanger tube forced at its fundamental natural frequency were predicted and show good correlation with experimental data. 1.2.3. Approach Comparisons Davies and Rogers compared using either only the unconstrained or only the constrained mode shapes to describe the motion of a cantilevered beam contacting a rigid
  26. 26. 9 linear stop [24]. The constrained mode shapes were obtained in terms of the unconstrained modes and shown to be orthogonal. In addition, the unconstrained modes were proven to form a complete set of modes for the constrained structure. The undamped motion of the structure could be described in terms of either the constrained or the unconstrained modes; but the two descriptions were equivalent only if the modal damping ratio was independent of the mode number. However, the two descriptions only differed by a significant amount at a resonance condition. 1.2.4. Summary Refs. 17-25 demonstrate that a structure’s vibration can be successfully described using either only the unconstrained modes of the structure, or using the unconstrained modes of the structure when it is not in contact with the restraint and the constrained modes of the structure when it is in contact with the restraint. Preliminary results examining both solution approaches when applied to an impact between an articulated model rotor blade and a droop stop were published in an earlier conference paper [25]. Problems in the solution approach using both the unconstrained and the constrained mode shapes were found, which was inconsistent with the research published in Refs. 21-24. The source of the inconsistencies in Ref. 25 has been identified and corrected using a method similar to Refs. 21-24. This correction will be discussed in Chapter 2. In addition to the correction of the results published in Ref. 25, this thesis contains the results of additional experiments performed on the model beam to identify its
  27. 27. 10 modal parameters. All of the objectives of the current research are contained in the following section. 1.3. Scope of the Present Research The objectives of the current research are as follows: (1) to predict the transient response of an elastic model rotor blade undergoing a droop stop impact using the finite element method, (2) to validate the algorithms using either only the unconstrained modes or using both the unconstrained and the constrained mode shapes, (3) to investigate and to ensure the algorithms’ computational efficiency for use in an aeroelastic rotor analysis (4) to design and fabricate a Froude and vibration scaled articulated model rotor blade, and (5) to measure the time histories of the tip deflection, flap hinge angle, and strain of the model rotor blade during a droop stop impact event.
  28. 28. 11 Figure 1-1: "Tunnel Strike" (top) and "Tailboom Strike" (bottom) Depictions
  29. 29. 12 αs δe δs Tunnel Aft RotorR Droop Stop Centrifugal Force Counter Weight Rotor Shaft Figure 1-2: Droop Stop Schematic
  30. 30. 13 Figure 1-3: H-46 Droop Stop Mechanism Droop Stop Counterweight Flap Hinge
  31. 31. 14 0 5 10 15 20 25 30 0 5 10 15 Measured Curvefit Ω(t)(%NR) Time (s) -80 -60 -40 -20 0 20 40 60 -10 0 10 20 30 40 50 60 0 5 10 15 Tip Deflection Hinge Angle TipDeflection(in) Time (s) Figure 1-4: Measured H-46 Engagement RPM (top) and Simulated H-46 Rotor Engagement Tip Deflection and Flap Hinge Angle Time Histories (bottom)
  32. 32. 15 Rotor Shaft No Droop Stop Contact Hinged Mode Droop Stop Contact Rotor Shaft Cantilevered Mode Figure 1-5: Articulated Rotor Blade Bending Modes
  33. 33. 16 Figure 1-6: Schematic of Heat Exchanger Shaft from Ref. 21 (top) and Industrial Relay from Ref. 17 (bottom)
  34. 34. 17 Chapter 2 MODELING AND ANALYSIS OF DROOP STOP IMPACTS This chapter develops the model and discusses the analysis that is used to simulate an impact between an elastic rotor blade and a droop stop mechanism. The first section develops the equations of motion for the blade and droop stop mechanism. The second section discusses their spatial discretization using the finite element method. The third section describes a modal analysis of the resulting equations of motion. The fourth section discusses different time integration techniques of the physical space and modal space equations of motion related to the droop stop impact problem. 2.1. Derivation of the Blade Equations of Motion A schematic diagram of the elastic blade and coordinate system is shown in Figure 2-1. The equations governing the transverse motion of the elastic blade are derived using the generalized Hamilton’s Principle ( )δ δ δ δΠ = − + =∫ T dt t t 1 2 U W 0 (2.1)
  35. 35. 18 where δT is the variation of the kinetic energy, δU is the variation of the strain energy, and δW is the virtual work due to external forces. Expressions for these quantities will be derived in the following sections. 2.1.1. Variation of the Kinetic Energy, δδT The kinetic energy is the energy of the blade due to its velocity. The equation that describes the kinetic energy of the blade is T dAdss b b A L = ⋅∫∫∫ 1 2 0 ρ V V (2.2) where ρs is the mass density of the blade, and Vb is the blade velocity relative to the hub. In this analysis, the blade velocity is only due to flapping motion and is expressed as Vb w k= & $ (2.3) where &w is the flapping velocity. Substituting Eqn. 2.3 into Eqn. 2.2 and taking the variation yields the variation of the blade kinetic energy δ ρ δT w w dA dss A L = ∫∫∫ & & 0 (2.4)
  36. 36. 19 2.1.2. Variation of the Total Strain Energy, δδU The variation of the strain energy can be expressed as a summation of the contributions from the elastic bending of the rotor blade and the deformation of the droop stop spring. The variation of the strain energy is then expressed as δ δ δU U UB DS= + (2.5) where δUB is the variation of the blade elastic strain energy and δUDS is the variation of the droop stop strain energy. Both components of Eqn 2.5 will be formulated in the following sections. 2.1.2.1. Variation of the Blade Elastic Strain Energy, δδUB Since the largest tip deflections to be explored in this study were to be 18% of the length of the beam, a nonlinear beam bending theory was utilized in the formulation of the blade strain energy due to elastic bending. The effect of the inclusion of the nonlinear terms will be discussed in Chapter 4. From Ref. 26, the strain energy due to the elastic bending of the blade can be expressed as U B c L EI ds=      ∫ 1 2 1 2 0 ρ (2.6)
  37. 37. 20 where s is the length coordinate of the deformed blade. Following the procedure in Ref. 27, the term ρc, the instantaneous radius of curvature of the beam, can be determined by examining the geometry in Figure 2-1 1 ρ ∂θ ∂c s = (2.7) The angle θ is also defined in Figure 2-1 as θ ∂ ∂ =       − sin 1 w s (2.8) Taking the partial derivative of θ with respect to s, the deformed length coordinate, yields ( ) ( ) ∂θ ∂ ∂ ∂ ∂ ∂s w s w s = − 2 2 2 1 (2.9) Substituting Eqn. 2.9 into Eqn. 2.6 yields ( ) U B L EI w s w s ds=       −        ∫ 1 2 2 2 2 20 1 1 ∂ ∂ ∂ ∂ (2.10) In order to further simplify Eqn. 2.10 a power series expansion is used to simplify the term containing the slope of the beam, ∂ ∂w s
  38. 38. 21 ( )f a a a a a for a= − = + + + + − < < 1 1 1 1 12 3 L (2.11) Substituting the proper terms into Eqn. 2.10 and splitting the equation into two parts yields U EI w s ds EI w s w s w s w s dsB L Linear L Nonlinear =       +             +             +         ∫ ∫ 1 2 1 2 2 2 2 0 0 2 2 2 2 2 2 2 4 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 2444 3444 L 1 2444444444 3444444444 (2.12) The first term in Eqn. 2.12 is the traditional expression for the strain energy utilized in linear bending theory. The second term in Eqn. 2.12 is the additional strain energy due to nonlinear effects and is proportional to even powers of the slope of the beam. Taking the variation of Eqn. 2.12 yields δ ∂ ∂ δ ∂ ∂ δU EI w s w s ds Linear U Nonlinear B L BNL =                 +∫ 2 2 2 20 1 24444 34444 123 (2.13) The derivation of the nonlinear terms is presented in the Appendix. 2.1.2.2. Variation of the Droop Stop Strain Energy, δδUDS Now that the strain energy due to the elastic bending of the blade has been determined, the additional strain energy due to the droop stop spring must be accounted for. All articulated rotor blades require some form of a droop stop to support the blade weight at low rotor speeds and at rest. The analysis uses a conditionally applied
  39. 39. 22 rotational spring based on the instantaneous value of the hinge angle to model the droop stop. The strain energy due to the elastic bending of a rotational spring is given by ( ) ( )( )U DS hinge DSK t t= − 1 2 2 β β β (2.14) where Kβ(t) is the conditional spring stiffness, βhinge(t) is the flap hinge angle, and βDS is the specified angle for droop stop contact. The hinge angle, βhinge(t), is calculated directly from the difference in the slope of the blade on either side of the flap hinge ( ) ( )βhinge t w e t= ′∆ , (2.15) where e is the flap hinge offset. For hinge angles above the specified setting at which a droop stop impact occurs, β DS , the spring stiffness is zero. For hinge angles equal to or below the specified setting at which a droop stop impact occurs, the spring stiffness is set large enough to restrain the hinge angle to less than .001º of rotation. To meet this requirement, it was necessary to set the spring stiffness to one million times the static weight moment of the blade about the flap hinge, per radian of rotation. The spring stiffness, Kβ(t), can then be expressed as ( ) ( ) K t if t K t x m g s ds if t hinge DS e L hinge DS β β β β β β = > = ≤∫ 0 1 106 ( ) ( ) (2.16)
  40. 40. 23 Taking the variation of the droop stop strain energy yields ( ) ( )( )δ β β δββU K t tDS hinge DS hinge= − (2.17) 2.1.3. Variation of the External Work, δδW The variation of the external work can be expressed as being simply from gravitational forces. Aerodynamic forces have been neglected in this study because the rotational speed is zero. The virtual work due to gravity is expressed as δ ρ δW g dA w dss A L = − ∫∫∫0 (2.18) 2.2. Finite Element Discretization The energy expressions are spatially discretized using the finite element method. Cubic Hermitian shape functions, H, are used to expand the flapping degrees of freedom within each element ( ) ( ) ( )w s t s ti i i, = H w (2.19) where si is the deformed length coordinate within the beam element and wi are the elemental degrees of freedom defined in the Appendix. The beam is represented by four degree of freedom finite elements as shown in Figure 2-2. The virtual energy expression
  41. 41. 24 in terms of the elemental mass matrix mi, the elemental stiffness matrix ki, the elemental load vector fi, and the nonlinear force vector iNLf becomes ( )( )δ δΠΠ = ∑ = + + − =∫ w m w k w f w fi T i N i i i i NL i i it t el dt 11 2 0&& (2.20) where Nel is the number of elements. Note that all of the nonlinear terms from the blade strain energy equation have been expressed in terms of a nonlinear forcing vector. Expressions for each of the elemental matrices and vectors are also included in the Appendix. Summation of the elemental matrices and vectors for all the elements in the finite element model yields the global form of the virtual energy expression ( )( )δ δΠ = + + − =∫ w M w Kw F w FT NL t t dt&& 1 2 0 (2.21) where the global matrices and vectors are given by ( ) ( ) M m K k F w f w F f w w = = = = = = = = = = ∑ ∑ ∑ ∑ ∑ i i N i i N NL NL i i i N i i N i i N el el el el el 1 1 1 1 1 (2.22)
  42. 42. 25 When assembling the global equations of motion, special attention must be taken at the flap hinge location because two distinct rotational degrees of freedom exist. If the flap hinge is located between elements i-1 and i, the elemental degrees of freedom ( )′ − wb i 1 and ( )′wa i become separate global degrees of freedom. In addition to the uncoupling of the rotational degrees of freedom at the flap hinge, the effect of the droop stop spring must be accounted for in the stiffness matrix and load vector. The motion dependent terms in Eqn. 2.17 are added to the stiffness matrix and the motion independent terms are added to the load vector. The addition of the droop stop spring to both the stiffness matrix and load vector is shown in Figure 2-3. Once the global mass and stiffness matrices and load vector have been assembled, the discretized equations of motion become ( ) ( ) ( ) M w K w F F w K K F F K = K F F && + = − = =    > =    ≤ NL h h hinge DS c hinge DS if t if t β β β β c (2.23) Also, if a proportional damping model is chosen, the global damping matrix can be assumed to be a linear combination of the global mass and stiffness matrices C M K= +α η (2.24)
  43. 43. 26 The global equations of motion are then ( )M w Cw K w F F w&& &+ + = − NL (2.25) Several quantities of interest to the present problem can be calculated from the global degrees of freedom. The deflection of the tip of the blade, wtip, is simply ( ) ( )( )w t w ttip b Nel = (2.26) If the flap hinge is located between elements i-1 and i, the flap hinge angle, βhinge, is ( ) ( )( ) ( )( )β hinge a i b i t w t w t= ′ − ′ −1 (2.27) The strain at any location along an element is given by ( ) ( ) ( ) ( )ε ss is ,t h s s ti i i= − ′′ 2 H w (2.28) 2.3. Modal Analysis Because the number of degrees of freedom may become large, the matrices and vectors in Eqn. 2.25 can become very large. For computational efficiency, it is common to transform Eqn. 2.25 into a generalized, or modal, coordinate system. The first step in this process is to examine the undamped, free vibration response of the beam. Eqn. 2.25 then reduces to
  44. 44. 27 Mw Kw 0&& + = (2.29) A general solution for Eqn. (2.29) is given by w = ΦΦei tω (2.30) where ΦΦ is a vector representing the amplitude of the mode shape and ω is the natural frequency of the mode. Substituting Eqn. (2.30) into Eqn. (2.29) yields K M 0−      =ω ω2 ΦΦei t (2.31) This is an eigenvalue problem. Eqn. (2.29) can then be expressed in terms of the eigenvalues and eigenvectors as M Kω r r r dofr N2 12ΦΦ ΦΦ= = , ,..., (2.32) where Ndof is the number of degrees of freedom. The modal matrix can then be assembled from the individual modal vectors [ ] [ ]ΦΦ ΦΦ ΦΦ ΦΦ= ≤1 2 L N m dofm N N (2.33) where Nm is the number of modes used in the analysis. Recall that two different stiffness matrices are used in this transient analysis - one corresponding to the cantilevered condition of the beam and one corresponding to the hinged condition of the beam; therefore, two independent sets of eigenvalues and eigenvectors are generated
  45. 45. 28 ω ω ω ω = =    = = =    = h h h c c c if if ΦΦ ΦΦ ΦΦ ΦΦ K K K K (2.34) It is also convenient to mass normalize the mode shapes so that ΦΦ ΦΦr T r mr NM = =1 12, ,..., (2.35) The response of each mode of the beam combines linearly according to the equation w q= ΦΦ (2.36) where q is the vector of modal amplitudes. Substitution of Eqn. (2.36) into Eqn. (2.25) and premultiplying by ΦΦT yields an uncoupled set of Nm equations ( ) [ ] [ ] ( ) Mq Cq Kq F F w M M I C C K K F F F F w && &&+ + = − = = = = = = = = NL T T n T n T NL T NL ΦΦ ΦΦ ΦΦ ΦΦ ΦΦ ΦΦ ΦΦ ΦΦ 2 2 ζω ω (2.37) where ζ is the modal damping ratio and ωn is the natural frequency.
  46. 46. 29 2.4. Transient Response Analysis In this particular study, the equations of motion can be solved using three different techniques. The first and simplest technique is a direct solution of Eqn. (2.25), called a physical space integration. The second technique is a modal space solution of Eqn. (2.37) using only the hinged modes of the beam regardless of droop stop contact, called a modal swapping off integration. The third and most complicated technique is a modal space solution of Eqn. (2.37) using either the hinged or cantilevered modes depending on droop stop contact, called a modal swapping on integration. In each method the high rate rotational spring, Kβ(t), must be added to or removed from the equations of motion depending on whether the blade is in contact with the droop stop. Once the solution technique was chosen, the Newmark method, described in Ref. 28, was used to time integrate the appropriate equations of motion. Further descriptions of the modal space solution methods follow for clarification. 2.4.1. Modal Swapping Off Technique The second method of integration is a modal space integration of Eqn. (2.37). In the second method, for sake of simplicity, only the hinged mode shapes are used throughout the integration whether the blade is in contact with the droop stop or not. The appropriate modal matrix in this method is then
  47. 47. 30 ( ) ( ) ΦΦ ΦΦ ΦΦ ΦΦ = > = ≤ h hinge DS h hinge DS if t if t β β β β (2.38) Recall that this is the solution method used in Refs. 10-12 and Refs. 17-20. 2.4.2. Modal Swapping On Technique In the third and most complex method, the hinged modes of the rotor are used when the rotor is not in contact with the droop stop and the cantilevered modes of the rotor are used when the rotor is in contact with the droop stop. The appropriate modal matrix in this method is then ( ) ( ) ΦΦ ΦΦ ΦΦ ΦΦ = > = ≤ h hinge DS c hinge DS if t if t β β β β (2.39) This technique of switching the modes of the rotor blade from hinged to cantilevered during a transient analysis has been referred to as “modal swapping” [13-15, 25]. However, if only the mode shapes are switched during a droop stop impact, which is the procedure followed in Refs. 13-15 and Ref. 25, then problems with discontinuities in the solution may arise. Upon further investigation, it was determined that if the modal matrix ΦΦ is switched during a droop stop contact, then the modal space amplitudes q must also be adjusted to maintain the physical deflection shape of the beam. The procedure accomplishing this is described below and was first outlined in Ref. 29. The deflection of the beam can be approximated by either set of modal matrices and modal amplitudes
  48. 48. 31 w q q≅ ≅ΦΦ ΦΦh h c c (2.40) During a transient analysis, if a switch from the hinged mode shapes to the cantilevered mode shapes is performed, then not only must the mode shapes be switched, but also the modal amplitudes must be adjusted. Premultiplying Eqn. 2.40 by ΦΦc T M yields ΦΦ ΦΦ ΦΦ ΦΦc T c c c T h hM q M q= (2.41) Because the mode shapes have previously been mass normalized ΦΦ ΦΦ ΦΦ ΦΦ c T c c c T h h M I q M q = = (2.42) Similarly, when a switch from the cantilevered mode shapes to the hinged mode shapes is performed ΦΦ ΦΦ ΦΦ ΦΦh T h h h T c cM q M q= (2.43) Simplifying Eqn. 2.43 yields ΦΦ ΦΦ ΦΦ ΦΦ h T h h h T c c M I q M q = = (2.44) Figure 2-4 shows the incorrect modal swapping on procedure, used in Refs. 13-15 and Ref. 25, and also the corrected modal swapping procedure, used herein.
  49. 49. 32 x s e ρc dθ ds θ ds dw w(x,t) Lz dx Figure 2-1: Beam Coordinate System
  50. 50. 33 Conditionally Applied Spring Kβ Rigid Shaft wa wb 4 DOF Beam Elements w'b w'a Beam Element i si Li Figure 2-2: Finite Element Representation
  51. 51. 34 Kβ(t) -Kβ(t) -Kβ(t) Kβ(t) wa(t) w'a(t) wb(t) w'b(t) wa(t) w'a(t) wb(t) w'b(t) wa(t) w'a(t) wb(t) w'b(t) wa(t) w'a(t) wb(t) w'b(t) Element i-1 degrees of freedom Element i degrees of freedom -Kβ(t) Kβ(t) wa(t) w'a(t) wb(t) w'b(t) w'a(t) wb(t) w'b(t) Element i-1 degrees of freedom Element i degrees of freedom Load Vector Assembly Stiffness Matrix Assembly wa(t) Figure 2-3: Flap Hinge and Droop Stop Assembly
  52. 52. 35 q = ΦΦc T M ΦΦh qh Let w = ΦΦ q ΦΦ = ΦΦh M q + C q + K q = F•• • M w + C w + K w = F •• • Droop Stop Contact ? YesNo ΦΦ = ΦΦc Were modes switched? Were modes switched? q = ΦΦh T M ΦΦc qc Yes Yes NoNo Shape Is Continuous! wh = ΦΦhqh = ΦΦcqc = wc Figure 2-4: Incorrect (top) and Correct (bottom) Modal Swapping On Algorithms FKwwCwM =++ &&& Let w = ΦΦq Droop Stop Contact? Φ = ΦΦ = ΦcΦ = ΦΦ = Φh FqKqCqM =++ &&& YesNo
  53. 53. 36 Chapter 3 EXPERIMENTAL TESTING This chapter describes all the experimental procedures utilized and testing performed on the model blade. The first section details the construction methodology used to design the model blade. The second section describes the modal parameter identification tests performed to identify the model blade natural frequencies, the modal damping ratios, and the mode shapes. The third section describes the experimental procedures and equipment used in the drop tests to investigate an impact between the model blade and droop stop impact. 3.1. Model Blade Construction Methodology In order to validate the analytical techniques used to simulate an impact between a rotor blade and a droop stop, a 1/8th length-scaled structural model of an H-46 rotor blade was constructed. The material chosen for the blade was 6061-T6 Aluminum because it was readily obtainable and easily machined. Two small steel blocks were used to make the droop stop because they were much stiffer than the Aluminum and would deform only a small amount when the Aluminum blade impacted the droop stop. Once the materials for the blade and droop stop were chosen, the distributions of the mass and flapwise stiffness along the length of the blade had to be determined to ensure that the vibration
  54. 54. 37 characteristics of the model blade were similar to the H-46 blade. In order to do so, the governing differential equation for the transverse bending of an unforced beam must be examined ( ) ( ) ( ) ( )ρ ∂ ∂ ∂ ∂ ∂ ∂ A x w x t t x EI x w x t x 2 2 2 2 2 2 0 , , +       = (3.1) where w is the flapwise deflection, x is the undeformed length coordinate, ρA is the mass distribution, and EI is the flapwise stiffness distribution. If the beam is assumed to oscillate harmonically, the flapwise deflection can be written as ( ) ( )w x t w x ei t , = ω (3.2) Substituting Eqn. (3.2) into Eqn. (3.1) yields ( ) ( ) ( ) ( )− +       =ω ρ ∂ ∂ ∂ ∂ 2 2 2 2 2 0A x w x x EI x w x x (3.3) The quantities in Eqn. (3.3) are nondimensionalized as shown below ( ) ( ) ( ) x x L w x w x L A x A x A EI x EI x EI * , * * , * * , * *=       =       =       =ρ ρ ρ 0 0 (3.4) where the values EI0 and ρA0 are evaluated at the root (x/L = 0) location. Substituting these definitions into Eqn. 3.3 yields the nondimensional governing differential equation for beam bending
  55. 55. 38 −             +                      = ρ ω ρ ∂ ∂ ∂ ∂ A L EI A x w x x EI x w x t x 0 2 4 0 2 2 2 2 0* * * * * * * * * , * (3.5) A constant called the nondimensional natural frequency, ω* is defined as ω ρ ω*2 0 2 4 0 = A L EI (3.6) If the nondimensional natural frequency, the nondimensional mass distribution, and the nondimensional mass distribution are identical for the model blade and the full scale blade, then the nondimensional flapwise bending, w* , of the model blade will be identical to the full scale blade. The static deflection behavior of the model blade should also be similar to the H- 46 blade. So the governing differential equation for beam bending, Eqn. 3.1, must be reformulated for the static case ( ) ( ) ( ) ∂ ∂ ∂ ∂ ρ 2 2 2 2 x EI x w x t x A x g ,      = − (3.7) Substituting the nondimensional quantities defined in Eqn. 3.4 into Eqn. 3.7 yields ∂ ∂ ∂ ∂ ρ ρ 2 2 2 2 0 3 0x EI x w x t x A gL EI A x * * * * * , * * *                     = −       (3.8)
  56. 56. 39 The constant term in Eqn. 3.8 is defined as the Froude number of the beam Fr A gL EI = ρ 0 3 0 (3.9) By examining Eqns. 3.5-6 and 3.8-9 the following conclusion can be made. If the nondimensional mass and flapwise stiffness distributions, the nondimensional natural frequency, and the Froude number of the model blade are the same as the full scale blade, then the both the static and dynamic vibration behavior of the model blade will be similar to the full scale blade. In the construction of the model blade, these quantities were matched as closely as possible to the H-46 blade properties, which were taken from Ref. 30. The nondimensional mass distribution and flapwise stiffness distribution of the model blade and the H-46 blade are shown in Figure 3-1. The relevant physical properties of the H-46 blade and the model blade are presented in Table 3-1. Because of the mechanical complexity of the hub region of the H-46 blade, the exact mass and stiffness distributions in this region could not be easily incorporated into the model blade. Therefore, the nondimensional natural frequencies and the Froude number of the model blade are similar to but not identical to the full scale H-46 blade.
  57. 57. 40 3.2. Modal Parameter Identification Because the mass and stiffness distributions were not exactly matched between the model and full scale blade, a series of experimental modal analysis tests were performed on the model blade to determine the nondimensional natural frequencies for the first six modes in both its hinged and cantilevered states. The following sections describe the series of modal analysis tests and results. In addition to the measurement of the nondimensional natural frequencies of the model blade, the modal analysis tests determined the structural damping ratios of the model blade. These measured structural damping ratios can then be used instead of an approximate proportional damping model in the transient analytic solutions for the model blade. Even though the static and dynamic behavior of the model and full scale blades are not exactly identical, there is sufficient correlation between the two to allow conclusions to be made about the H-46 blade behavior from the model blade behavior. 3.2.1. Experimental Procedure The modal parameter identification process was performed by a common roving input/fixed response test [31]. The excitation was supplied by a modally tuned impact hammer mounted with a load cell and interchangeable hammer tips of different hardnesses. The response was measured with a piezoelectric accelerometer mounted to the model beam with wax. To determine the most reliable manner in which to collect data, a comprehensive pretest was performed on the beam in both its hinged and cantilevered
  58. 58. 41 conditions. After some trial and error, it was determined that the best method to test the beam in the hinged condition was to remove the droop stop and to clamp the root end in a vice allowing the beam to hang vertically. In this configuration, the hinge was completely free to rotate without any interference from the droop stop, and the effect of any “play”, or extra tolerance, in the hinge was be minimized. The best method to test the beam in the cantilevered condition was to include the droop stop and to attach small aluminum bars to the sides of the hinge with screws, thereby locking the hinge into place. The beam was then clamped to a table at the root end to fully simulate the cantilevered condition. A further complication to the modal parameter test was encountered because of the wide range of natural frequencies that were to be measured (approximately 3 Hz for the first cantilevered mode to 280 Hz for the sixth hinged mode). A single hammer tip was not found that could satisfactorily excite such a wide range of frequencies. To circumvent this problem, two separate sets of tests were performed on the beam. The first set of tests were used to determine the only first bending mode in both the hinged and cantilevered conditions. In these tests a soft tip, which imparted the most energy to the lower frequencies thereby exciting the lower modes, was mounted on the hammer. Three grid points along the width of the beam at four spanwise locations were used which provided more than enough resolution to obtain the first mode shape. The second set of tests was used to determine the second through sixth bending modes. In these tests a hard tip, which excited a wider range of frequencies and modes, was mounted on the hammer. Three grid points along the width of the beam at eleven spanwise locations provided enough resolution to capture up to the sixth mode.
  59. 59. 42 The frequency response functions were collected using WAVEPAK® data acquisition software [32]. Five ensemble averages were used at each grid point for each frequency response function. Because there was only a small amount of material damping in the beam, a 25% Force-Exponential window was applied to the input hammer and output accelerometer signals. The frequency resolution was 0.5 Hz for the first set of tests and 1.0 Hz for the second set of tests. The frequency response functions measured via WAVEPAK® were analyzed using STARModal® software [33]. The results from STARModal® are split into example frequency response and coherence functions, dimensional natural frequencies, nondimensional natural frequencies, damping ratios, mode shapes, and modal assurance criteria. 3.2.2. Frequency Response Functions The measured frequency response functions for magnitude, phase, and coherence in the hinged condition are shown at the driving point in Figure 3-2 and at the blade tip in Figure 3-3. The measured frequency response functions for magnitude, phase, and coherence in the cantilevered condition are shown at the driving point in Figure 3-4 and at the blade tip in Figure 3-5. The driving point frequency response functions were chosen because they display sharper resonances and anti-resonances than any other test point. Also, an accurate measure of the driving point frequency response function is important because it is used in the synthesis of the frequency response functions for all other test points in a roving input/fixed response test [31]. Because the driving point was not
  60. 60. 43 located on the centerline of the beam, both bending and torsion natural frequencies appear. The natural frequencies of the beam can easily be picked out in the magnitude graph of the frequency response functions. The coherence functions, which give an indicator of the degree of confidence in the frequency response function measurement, are very close to one except at the very low frequencies (< 5 Hz) and the anti-resonance frequencies; consequently, modes with very low natural frequencies will be difficult to measure. Since the coherence is nearly equal to one at even the highest frequencies, the hammer tips used in the modal analysis tests imparted sufficient energy in this frequency range. 3.2.3. Dimensional Natural Frequencies The natural frequencies calculated in the finite element analysis of the beam and the measured in the modal analysis tests are presented in Table 3-2 for both the hinged and cantilevered conditions. The zero frequency rigid body mode of the beam in its hinged condition could not be measured with the WAVEPAK® software. In general, there is excellent correlation between the finite element results and the modal analysis tests. All of the measured natural frequencies, except for the fifth and sixth cantilevered bending modes, were within an average of 5% of the predicted natural frequencies. Two extra mechanical complications in the cantilevered set of tests, namely the attachment of the droop stops under the flap hinge and the attachment of the aluminum bars that locked the flap hinge into place, are probably to blame for the error in the fifth and sixth cantilevered bending modes.
  61. 61. 44 3.2.4. Nondimensional Natural Frequencies As stated earlier, in order to ensure that the vibration behavior of the model blade emulates the H-46 blade, both the mass and stiffness distributions of the blades should be matched along with the nondimensional natural frequencies. Since the exact mass and stiffness distributions were not matched because of the mechanical complexity in the hub regions of the full scale blade, the nondimensional natural frequencies are not identical. Table 3-3 compares the nondimensional natural frequencies of the H-46 blade, and the predicted and measured nondimensional natural frequencies of the model blade. In general the model blade nondimensional natural frequencies are lower than the H-46 nondimensional natural frequencies; however, there is still good correlation between them. . This difference can be explained by reexamining Figure 3-1. The inboard region of the model blade is stiffer than the H-46 blade; therefore, the dimensional natural frequencies of the model blade are lower than they would be if the stiffness distribution exactly matched the full scale blade. Consequently, this causes the nondimensional natural frequencies for the model blade to be lower than the full scale blade. The measured nondimensional natural frequencies of the model blade are within a maximum of 20% of the predicted H-46 nondimensional natural frequencies in both conditions.
  62. 62. 45 3.2.5. Modal Damping Ratios Another reason for performing the modal parameter tests was to measure the modal damping ratios for use as inputs into the drop test simulations and for comparison to the proportional damping model. The damping ratios of the beam in both the hinged and cantilevered conditions are presented in Table 3-4. The first column in Table 3-4 is the measured damping ratios calculated directly in STARModal® ; however, these measured damping ratios include the extra numerical damping added by using a 25% Force-Exponential data acquisition window in WAVEPAK® . The second column in Table 3-4 is the corrected damping ratios, which have the extra numerical damping from the 25% Force-Exponential data acquisition window removed from them. The conversion from measured to corrected damping ratios is given by [31-32] ζ ζ τω corrected measured p nT = − 1 (3.10) where Tp is the length of time of the data acquisition window, τ is the numerical damping constant used in the Force-Exponential data acquisition window, and ωn is the natural frequency of the mode of interest. Note that the corrected modal damping ratio is a function of the natural frequency; consequently, any natural frequencies that were inaccurately measured will also have less accurate damping ratios. The damping in the first bending mode of the hinged beam is the largest due to friction in the hinge. The first
  63. 63. 46 bending mode of the cantilevered beam was actually calculated to have a negative damping ratio, which is not physically possible. This mode had a very low natural frequency of 3.1 Hz, which was difficult to measure as stated earlier and as seen in Figure 3-4. The first and sixth corrected damping ratios for the cantilevered case, actually calculated to be less than zero because neither mode was well measured, were set equal to zero in all the subsequent analytic simulations. The third column in Table 3-4 is the calculated modal damping ratios when using a proportional damping model. Note that the proportional damping constants have different values for the hinged and cantilevered cases. The calculation of the modal damping ratios using proportional damping constants is given by ς α ω ηω = + 2 2n n (3.11) Again, these calculated modal damping ratios are only used when the equations of motion are integrated in physical space rather than modal space. The constants α and η were chose to approximate the measured modal damping ratios. 3.2.6. Mode Shapes The frequency response functions measured in WAVEPAK® and analyzed in STARModal® were used to identify the first six mode shapes of the beam. The measured mode shapes are shown in Figure 3-6 for the hinged condition and Figure 3-7 for the
  64. 64. 47 cantilevered condition. The first five mode shapes were easily measured for the hinged condition, but the resolution starts to degrade at the sixth mode shape. More grid points along the length of the beam are needed for a more accurate prediction. Mode shapes two through 5 were well measured in the cantilevered condition. The amplitude of the mode shape at the tip of the beam for the first cantilevered mode shape was not well measured. Again, this was probably because this mode had such a low natural frequency (3.1 Hz). In this region the coherence function is quite noisy, so it is no surprise that the mode shape was poorly predicted. Like the sixth hinged mode, the resolution of the sixth cantilevered mode shape is again degraded because not enough grid points were used. 3.2.7. Modal Assurance Criteria The last set of results from the modal parameter identification tests are the modal assurance criterion (MAC) matrices. MAC matrices are useful because the allow a quantitative method to evaluate the orthogonality of experimentally measured modes and allow comparison to finite element results [31]. The definition of the MAC matrix is given by MAC i j N i j i j m= = Φ Φ Φ Φ 2 2 2 1, , ,K (3.12) All elements of the MAC matrix are bounded between zero and one. A MAC value that approaches zero implies that the modal vectors are orthogonal. A MAC value that
  65. 65. 48 approaches one implies that the modal vectors represent the same mode shape. Generally, a MAC value less than or equal to 0.05 implies orthogonality, while a MAC value greater than or equal to 0.90 implies the modes are the same. The MAC matrices were generated in STARModal® and are shown in Table 3-5. As previously mentioned, the first bending mode in both the hinged and cantilevered conditions was determined in an independent test; therefore, only the higher mode test MAC matrices are presented. Upon inspection, the hinged MAC matrix listed in Table 3-5 indicates very minimal interaction between the 2nd & 3rd , 2nd & 5th ,and 3rd & 6th modes. No interaction was found for the cantilevered modes. 3.3. Droop Stop Impact Tests In order to focus on the impact between the rotor blade and the droop stop, a series of drop tests were conducted. Each drop test was performed at zero rotational speed to minimize aerodynamic effects and to simplify the experimental apparatus. The investigation of droop stop impacts at zero rotor speed is justified because most tunnel strikes for the H-46 occur at less than 20% of the full rotor speed. In this low speed region, the centrifugal stiffening is less than 4% than at full rotor speed. The procedure for the droop stop impact tests or “drop tests” are described below.
  66. 66. 49 3.3.1. Test Procedure Each drop test was performed in the following manner. The root end of the blade inboard of the flap hinge was clamped between vice grips. Then the section of the blade outboard of the flap hinge was rotated upward and given an initial flap hinge angle ranging from 2º to 10º. Once the desired flap hinge angle was reached, the blade was held in place by an electromagnet located just beyond the flap hinge. Each drop test commenced when the electromagnet was shut off. To ensure that the magnetic force created by the electromagnet dissipated quickly, a 12V Zener diode was placed in series with the electromagnet. Without anything to support it, the blade section outboard of the flap hinge rotated downward about the flap hinge freely until it contacted the droop stop located directly underneath the flap hinge. Once the blade contacted the droop stop, it cannot rotate downward any further; however, the inertia of the blade causes the tip to continue bending downwards elastically. During this process the kinetic energy of the blade is converted to strain energy. Once the blade tip reached its point of maximum downward deflection, it has no kinetic energy but has stored strain energy. After this point the excess strain energy is converted back into kinetic energy in the upward rebounding of the blade tip. The motion of the blade was measured for one full cycle of downward rotation and bending and upward rebounding. This process was repeated four times for each initial flap hinge angle, ranging from 2º to 10º, to determine the repeatability of the data. Correlation between the experimentally measured data and the analytic predictions will be presented and discussed in Chapter 5.
  67. 67. 50 3.3.2. Instrumentation A schematic of the model blade is shown in Figure 3-8. The model blade was instrumented with several measurement devices. A PCB® piezoelectric accelerometer, rated at +/-50g’s, was attached to the blade tip. Three 120Ω Micro Measurements® strain gages were attached to the upper surface of the model blade at locations of 20%, 30%, and 40% of the total blade length. A 10kΩ Maurey Instrument Company® linear motion potentiometer was located just outboard of the flap hinge. The linear motion potentiometer was used to measure a vertical displacement of the hub section located just outboard of the flap hinge. The measured vertical displacement of the hub was then converted to a flap hinge angle by assuming the hub to be rigid. The linear motion potentiometer was located approximately 1 inch outboard of the flap hinge, or at a location of x/L = 4.2%. As seen in Figure 3-1, this radial location is located well within the very stiff part of the hub so the rigid hub assumption was felt justified. Each of the measurement devices simply output a raw voltage signal during each drop test. All of the raw voltages were acquired at a rate of 1 kHz using a Techkor™ MEPTS-9000 Data Acquisition System. To give the raw voltages meaningful physical results, each device must be calibrated. The accelerometer was pre-calibrated by PCB Piezoelectronics® , and had a calibration constant of 109 mV/g. The strain gages were calibrated through an electric shunt device included in the MEPTS-9000 data acquisition system. The linear motion potentiometer was calibrated by manually lifting the blade to a several known flap hinge angles, which were measured with a protractor, and recording
  68. 68. 51 the resulting voltages from the MEPTS-9000 system. In this manner a voltage versus flap hinge angle conversion was made. It is estimated that this process was accurate to within ±1º. A PC equipped with Labtech Notebook™ was used for the post processing of the data. A schematic of the experimental apparatus is also shown in Figure 3-8. 3.3.3. Calculation of Measured Tip Deflection At the beginning of each drop test, the initial shape of the beam is a combination of the static droop under its own weight and a known rotation of about the flap hinge. This shape is shown graphically in Figure 3-8. The initial condition for the flap deflection at the tip of the model blade is given by ( ) ( ) ( )( )w L e wtip hinge sd0 0= − +sin β (3.13) where L is the length of the blade, e is the flap hinge offset, βhinge(0) is the initial flap hinge angle as measured by the linear motion potentiometer, and wsd is the static deflection of the tip. Since no direct measurement of static deflection of the blade tip was possible, it was assumed to be exactly the same as predicted by the finite element model. The initial velocity was set equal to zero since the blade was released from rest. Once the initial position and velocity of the blade tip have been specified, the measured tip accelerations can be used to calculate the measured displacement and velocity of the tip at any time, t, as given by
  69. 69. 52 ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] & && && & & w t t w t w t t w t t w t w t t tip tip tip tip tip tip = + − = + − ∆∆ ∆∆ ∆∆ ∆∆ 2 2 (3.14) where &&w tip is the measured tip acceleration and ∆t is the time interval between measurements, or 1 ms for a 1kHz data acquisition rate. Some error in each tip acceleration measurement exists. When integrated twice in time, this error can have a measurable impact on the tip deflections. Upon further investigation, even when the accelerometer was placed upon a motionless table a ±5 mV oscillation in the accelerometer signal was identified, hereafter referred to as “accelerometer drift”. As stated earlier, the calibration constant for the accelerometer was 109 mV/g; therefore, the ±5 mV drift translates into an uncertainty of ±0.046g in the acceleration measurement. The effect of the this drift on the measured tip deflections will now be examined. The equation for the additional displacement due to this drift is given by ( )∆ ∆w t at= 1 2 2 (3.15) where ∆a is the measured acceleration drift of 0.046g. Substituting this yields ( )∆w t gt= 0 046 2 2. (3.16) The variation of ∆w(t) is presented in Figure 3-9. Note that at t = 0.25 seconds, ∆w/L is approximately 0.015, but at t = 0.5 seconds ∆w/L grows to 0.055. The maximum tip
  70. 70. 53 deflection is the main parameter of interest in the drop tests and for each test it approximately occurs at t = 0.25 seconds. The range of maximum tip deflections to be measured in the drop tests ranges from 0.08 for the lowest initial hinge angle to 0.18 for the highest hinge angle. So the percent error in the maximum tip deflection due to the accelerometer drift is 19% for the lowest initial flap hinge angle to 8% for the highest flap hinge angle. Good correlation between the analytic solutions and experimental data will be more difficult to obtain the longer in time the simulation is predicted. An attempt at minimizing the effect of the drift was made in the calculation of the measured tip deflection. Before each drop test while the blade was sitting motionless, the measured acceleration was recorded. As stated earlier, these readings were in the range of ±0.046g. The average of this initial reading was then subtracted from all the measured accelerations before the tip deflection was calculated.
  71. 71. 54 Table 3-1: Physical Characteristics of H-46 Blade and Model Blade H-46 Blade Model Blade E (lb/in2) 10x106 ρ (lb/in3) 0.0955 h (in) .125 L (ft) 25.5 3.28 EI0 (lb-in2 ) 569.0x106 0.089x106 ρA0 (lb/in) 6.79 0.275 Fr 0.342 0.181 wsd (wtip/L) 0.05 0.04 βDS (deg) -1.25º 0.0º
  72. 72. 55 Table 3-2: Dimensional Natural Frequency Comparison of Model Blade Hinged Cantilevered Mode Predicted ωn (Hz) Measured ωn (Hz) Error (%) Predicted ωn (Hz) Measured ωn (Hz) Error (%) Rigid Body 0.0 * * N/A N/A N/A 1st Bending 9.3 9.3 0.0 3.0 3.1 3.2 2nd Bending 28.4 27.3 4.0 19.9 19.2 3.6 3rd Bending 64.7 61.3 5.5 57.3 55.1 4.0 4th Bending 121.8 116.6 4.5 111.9 107.9 3.7 5th Bending 199.5 189.8 5.1 165.3 198.6 16.8 6th Bending 297.0 283.7 4.7 211.5 246.8 14.3 *Not measurable in WAVEPAK®
  73. 73. 56 Table 3-3: Nondimensional Natural Frequency Comparison Hinged Cantilevered Mode Number H-46 (Predicted) Model (Predicted) Model (Measured) H-46 (Predicted) Model (Predicted) Model (Measured) 1st Bending 9.3 8.1 8.1 3.1 2.6 2.7 2nd Bending 29.9 24.7 23.7 18.8 17.3 16.7 3rd Bending 66.9 56.2 53.3 48.7 49.8 47.9 4th Bending 116.1 105.9 101.3 90.1 97.2 93.8 5th Bending 179.5 173.4 165.0 142.3 143.7 172.6 6th Bending 256.0 258.2 246.6 208.8 183.9 214.5
  74. 74. 57 Table 3-4: Modal Damping Ratios Hinged Cantilevered Mode Measured ζ (% ζcrit) Corrected ζ (% ζcrit) Physical Space ζ (α=1, β=1x10-4 ) (% ζcrit) Measured ζ (% ζcrit) Corrected ζ (% ζcrit) Physical Space ζ (α=1x10-4 , β=1x10-4 ) (% ζcrit) Rigid Body * * N/A N/A N/A N/A 1st Bending 7.9 4.5 1.1 6.0 -4.3≅0.0 0.1 2nd Bending 3.2 0.9 1.2 3.7 0.4 0.6 3rd Bending 1.2 0.2 3.8 2.1 0.9 3.4 4th Bending 0.8 0.3 6.0 4.5 3.9 5.6 5th Bending 0.9 0.6 8.9 1.0 0.7 7.6 6th Bending 0.8 0.6 12.3 0.1 -0.2≅0.0 9.4 *Not measurable in WAVEPAK ®
  75. 75. 58 Table 3-5: MAC Matrix for Hinged Condition (top) and Cantilevered Condition (bottom) Hinged 2nd Bending 3rd Bending 4th Bending 5th Bending 6th Bending 2nd Bending 1.00 .06 .01 .06 .01 3rd Bending .06 1.00 .01 .00 .05 4th Bending .01 .01 1.00 .01 .03 5th Bending .06 .00 .01 1.00 .04 6th Bending .01 .05 .03 .04 1.00 Cantilevered 2nd Bending 3rd Bending 4th Bending 5th Bending 6th Bending 2nd Bending 1.00 .01 .01 .00 .02 3rd Bending .01 1.00 .03 .01 .03 4th Bending .01 .03 1.00 .02 .01 5th Bending .02 .03 .01 1.00 .01 6th Bending .00 .01 .02 .01 1.00
  76. 76. 59 Figure 3-1: Nondimensional Mass (top) and Flapwise Stiffness (bottom) Distributions
  77. 77. 60 1.0E-02 1.0E-01 1 . 0 E + 0 0 1 . 0 E + 0 1 1 . 0 E + 0 2 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Magnitude -270 -180 -90 0 9 0 1 8 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Phase(deg) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Coherence Figure 3-2: Driving Point Frequency Response Function Magnitude (top), Phase (middle), and Coherence (bottom) for the Hinged Test
  78. 78. 61 1.0E-01 1 . 0 E + 0 0 1 . 0 E + 0 1 1 . 0 E + 0 2 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Magnitude -270 -220 -170 -120 -70 -20 3 0 8 0 1 3 0 1 8 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Phase(deg) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Coherence Figure 3-3: Blade Tip Frequency Response Function Magnitude (top), Phase (middle), and Coherence (bottom) for the Hinged Test
  79. 79. 62 1.0E-02 1.0E-01 1 . 0 E + 0 0 1 . 0 E + 0 1 1 . 0 E + 0 2 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Magnitude -270 -180 -90 0 9 0 1 8 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Phase(deg) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Coherence Figure 3-4: Driving Point Frequency Response Function Magnitude (top), Phase (middle), and Coherence (bottom) for the Cantilevered Test
  80. 80. 63 1.0E-02 1.0E-01 1 . 0 E + 0 0 1 . 0 E + 0 1 1 . 0 E + 0 2 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Magnitude -270 -220 -170 -120 -70 -20 3 0 8 0 1 3 0 1 8 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Phase(deg) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 Frequency (Hz) Coherence Figure 3-5: Blade Tip Frequency Response Function Magnitude (top), Phase (middle), and Coherence (bottom) for the Cantilevered Test
  81. 81. 64 Figure 3-6: First through Sixth Bending Mode Shapes in the Hinged Condition
  82. 82. 65 Figure 3-7: First through Sixth Bending Mode Shapes in the Cantilevered Condition
  83. 83. 66 Accelerometer Strain Gages Droop Stop Linear Motion Potentiometer βhinge(0) wtip(0)(L-e)sin[βhinge(0)] wsd=K-1F+ - Electromagnet Linear Motion Potentiometer Strain Gages Tip Accelerometer Model Beam MEPTS-9000 Post Processing Tip Deflection Strains Flap Hinge Angle Figure 3-8: Model Blade Schematic (top) and Experiment Apparatus (bottom)
  84. 84. 67 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.0 0.1 0.2 0.3 0.4 0.5 Time (s) ∆w/L Figure 3-9: Estimation of Error in Calculated Tip Deflections due to Accelerometer Drift
  85. 85. 68 Chapter 4 RESULTS AND DISCUSSION The previous chapters developed the theory used to model the blade and droop stop and described the test procedure that was followed to validate the analysis method for an impact between the rotor blade and the droop stop. There are several sections in this chapter. The first section of this chapter describes the differences in linear and nonlinear beam bending theories, which were derived in Chapter 2, with respect to the drop tests and tunnel strike phenomenon. The second section of this chapter describes the differences obtained in the analytic solution for a drop test if the modal swapping algorithm using both sets of mode shapes, also described in Chapter 2, is not performed correctly. The third section of this chapter discusses the differences in the analytic solution for a drop test with and without the inclusion of the measured modal damping ratios, which were presented in Chapter 3. The fourth and final section presents the results of the drop tests described in Chapter 3 and discusses their significance. 4.1. Comparison of Linear and Nonlinear Beam Bending Theory Remember that a tunnel strike on an H-46 Sea Knight occurs when the rotor blade tip deflects 18% of its length. In addition, the experimental drop tests were structured so that the model beam would deflect a maximum of 18% of its length. Since this is such
  86. 86. 69 large deflection, the validity of applying linear beam bending theory to both problems was called into question. This study will quantify the differences in applying linear and nonlinear bending theories to the problem at hand. In Section 2.1.2.1., the linear and nonlinear expressions for the strain energy of the blade due to elastic bending were derived. By using a nonlinear beam bending theory, the global equations of motion were given by Eqn. 2.22. For convenience Eqn. 2.22 is restated here ( )M w Cw K w F F w&& &+ + = − NL (4.1) The elemental forms of the mass damping matrices along with the linear and nonlinear force vectors are given in the appendix. Since the only nonlinear terms occur in the forcing vector and not in the mass and damping matrices, the effect of the nonlinearities can be assessed by examining the static solution of Eqn. 4.1. In the absence of the inertia and damping forces, Eqn. 4.1 reduces to ( )K w F F w= − NL (4.2) 4.1.1. Newton-Raphson Iteration Method Since Eqn. 4.2 is nonlinear, the Newton-Raphson iteration method was employed for the solution [34]. Using this method, a vector-valued function, G, is defined as ( ) ( )G w K w F F w= − + =NL 0 (4.3) The function, G, is then expressed as a first order Taylor series
  87. 87. 70 ( ) ( ) ( ) ( ) ( )G w G w w G w G w w w w w w = − = − + = 0 0 2 0 ∆ ∆ ∆ ∂ ∂ O (4.4) Substitution of the proper terms from Eqn. 4.3 into Eqn. 4.4 yields ( ) ( )K F w w w Kw F F w w w +         = − + = ∂ ∂ NL NL 0 0 0∆ (4.5) All terms in Eqn. 4.5, except for the increment ∆w, are known quantities. Solving for the unknown increment, ∆w, yields ( ) ( ){ }∆w K F w w Kw F F w w w = +         − + = − ∂ ∂ NL NL 0 1 0 0 (4.6) The next approximation for the desired vector is then equal to w w w= −0 ∆ (4.7) This process is repeated until a sufficiently converged solution for w is obtained.
  88. 88. 71 4.1.2. Validation with Experimental Data In Ref. 27 a series of tests were performed to examine the nonlinear transverse bending characteristics of thin beams. Two beams were constructed from high tensile steel, were 36 inches long and had thicknesses of 0.032 inches and 0.065 inches. Both beams were tested with simply supported boundary conditions and were loaded with a range of point loads applied at locations of 1/6, 1/4, 1/3, and 1/2 of the beam length. The bending deflections were measured at the point of load application with a probe accurate to within ±0.005 inches (.00014 %L) over a range of 15 inches (41.7 %L). Figure 4-1 shows the results of the tests in which the point load was applied at both 1/6 and 1/3 of the length of the span. Figure 4-2 shows the results of the tests in which the point load was applied at both 1/4 and 1/2 of the length of the span. In each figure, the results from Ref. 27 are shown on the top graph, the analytical result being indicated by a solid line and the experimental results represented by data points (x and +). The results from the present analysis are shown in the bottom graph. Both the linear bending theory and increasingly accurate nonlinear theories, up to terms proportional to ( )∂ ∂w s 6 , are shown. Note that for the simply supported case, terms proportional to the slope of the beam to the sixth power must be included in the expression for the strain energy of the beam for good correlation with the experimental results. In each graph, also note that the point at which linear bending theory begins to deviate from the more exact
  89. 89. 72 nonlinear bending theory is dependent on the point along the beam length at which the load is applied. 4.1.3. Nonlinear Analysis of the Model Beam The analysis described in Section 4.1.1 was repeated for the model beam described in Chapters 2 and 3. There are two important differences between the present study for the model beam and the analysis described in the previous section. The first is that in the drop tests the model beam is in a cantilevered condition rather than in a simply supported condition. The second is that in the drop tests gravity is the only external force acting on the model beam. So a range of distributed vertical loads, scaled in multiples of the acceleration due to gravity, instead of a range of point loads were applied to the model beam. Figure 4-3 illustrates the results of the analysis for the model beam. The top graph displays the calculated tip deflection for the beam with varying multiples of gravity. For deflections of less than 18% of the length of the beam, the linear and nonlinear deflection theories are almost identical. The bottom graph displays the error between linear and nonlinear deflection theories as a function of the tip deflection. For the region of interest in this study, the error between the two solutions is less than 4% for tip deflections less than 18% of the blade length. Linear bending theory was deemed sufficient for both the tunnel strike problem and the model beam drop tests.
  90. 90. 73 4.2. Comparison of Correct and Incorrect Modal Swapping Algorithms Section 2.4.2. described a technique to integrate the equations of motion in modal space coordinates in which the appropriate set of modes, either hinged or cantilevered, is used depending on whether the blade is in contact with the droop stop or not. If this analysis is not performed correctly, as in Refs. 13-15 and 25, discontinuities in the transient response solution can arise. As stated in Chapter 2, the discontinuities are the result of switching the mode shapes without readjustment of the modal amplitudes. To quantify the differences between the correct and incorrect algorithms, the results of two example drop tests using both algorithms are shown in Figures 4-4 and 4-5. Note the physical impossibility of the results using the incorrect algorithm in both figures. Not only does the flap hinge angle rebound to an angle higher than it was dropped, but it also goes below the droop stop angle for short periods of time. There are also discontinuities in the tip deflection solution, in which the tip seems to discontinuously “jump” upwards and downwards at a high frequency. Also note the large discrepancy in the predicted strain values. 4.3. Modal Damping Ratios Section 3.2.5. discussed the experimental determination of the modal damping ratios. This section details the differences in the results of drop test transient analysis with and without the inclusion of modal damping ratios. Figure 4-6 displays the results of an
  91. 91. 74 analysis for a simulated drop test of the model blade from 5.5º. For the fully undamped blade, the flap hinge angle can be seen to impact and bounce off of the droop stop many times as might be expected for a totally undamped collision. For the damped blade, note that all of the smaller rebounds off of the droop stop have been eliminated, while the two largest rebounds remain. The inclusion of the modal damping ratios has a minimal effect on the tip deflection time history. The two simulations only deviate a small amount from each other after a full cycle of downward tip bending and upward rebounding. Inclusion of the modal damping ratios has a large effect on the strain time histories for the undamped and damped blades. Not only was the magnitude of the maximum strain reduced by 20%, but also the high frequency content of the strain prediction was reduced. In summary, the inclusion of the modal damping ratios has a minimal effect on the tip deflections, but has a major effect on the flap hinge angle and strain calculations. 4.4. Drop Test Results This section will compare the calculated results from the transient response analyses with the experimental data generated from the drop tests performed on the model blade. Example time histories from each different drop angle will be discussed along with a summary for all the drop angles. Both the accuracy and the computational efficiency of each analytical method will be addressed. The parameters utilized in the generation of the converged analytic solutions are presented in Table 4-1. Convergence histories for the maximum strain at x/L = 0.20 in a drop test from 9.7º are presented in Figure 4-7. The
  92. 92. 75 Modal Swapping Off technique converged for 6 modes, while the Modal Swapping On technique, because it more accurately represents the boundary conditions at all times, converged for 4 modes. Also note that the Modal Swapping On technique utilized a time step twice as large as the Modal Swapping Off technique. From Ref. 17, the maximum allowable time step is constrained by ∆∆t n max max ≤ 1 2ω (4.8) where ω nmax is the largest natural frequency used in the simulation. The time steps used in all the simulations were set equal to one-fourth of the value in Eqn. 4.8 for complete accuracy. Since the Modal Swapping Off technique needed more modes with higher natural frequencies than the Modal Swapping On technique to obtain a converged solution, the Modal Swapping Off technique also required a smaller time step. 4.4.1. Results from 2.6º Drop Angle Figure 4-8 displays the predicted and measured time histories of the flap hinge angle and tip deflection, while Figure 4-9 shows the results of the strain at locations of x/L = 0.20, x/L = 0.30, and x/L = 0.40 during a drop test from an initial flap hinge angle of 2.6°.
  93. 93. 76 In Figure 4-8, the first rebound of the flap hinge after droop stop contact of 1.0º at t = 0.06 seconds is predicted very well. While the blade is in contact with the droop stop, the experimental points seem to oscillate by a very small amount about 0º. As mentioned in Chapter 3, the linear motion potentiometer was located just outboard of the flap hinge and measures a vertical displacement of the hub. The vertical displacement of the hub was then converted to a flap hinge angle by assuming that the hub was rigid. Some of the elastic bending of the hub, which is not perfectly rigid, appears as a change in the flap hinge angle during the drop test. There is also good correlation between the measured and predicted tip deflection time histories as seen in Figure 4-8. The “snap-up” of the blade tip of 0.01 wtip/L at t = 0.02 seconds, caused by the flexing motion of the blade after the electromagnet is turned off, is predicted very well. This flexing motion is shown schematically alongside the tip deflection. Also, the maximum tip deflection of -0.085 wtip/L is captured by the analytic simulations. However, the experimental data lags behind the analytical predictions by approximately 0.02 seconds at the point of maximum deflection and by 0.04 seconds at the point of maximum rebound. The reason for this is still under investigation. Figure 4-9 presents the results of the three strain gages during the same drop test. Except for the same amount of time lag between the experimental data and the analytical methods as in the tip deflection measurement, there is excellent correlation. The maximum strains of 725, 520, and 440 µε are matched very well. The majority of the peaks and valleys of the strain measurements are predicted along with the maximum strain. Also, note that the strain actually becomes negative while the blade is in freefall, meaning
  94. 94. 77 that it is bent upwards at those times. This is further evidence of the flexing of the blade during freefall, which also causes the “snap-up” of the blade tip at the beginning of the drop test. 4.4.2. Results from 4.0º Drop Angle Figure 4-10 shows the predicted and measured time histories of the flap hinge angle and tip deflection, while Figure 4-11 shows the results of the strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 during a drop test from an initial flap hinge angle of 4.0°. In Figure 4-10 the note that the first rebound of 1.4º and second rebound of 1.2º of the flap hinge off of the droop stop are predicted best by the Physical Space and Modal Swapping On integrations. The Modal Swapping Off integration did not rebound as far off the droop stop, probably because the use of only hinged modes could not accurately describe the shape of the beam at these points. Additional simulations showed that even inclusion of more than 6 modes did not predict this rebound more accurately. The “snap-up” of 0.01 wtip/L of the blade tip can again be seen in Figure 4-10, however, the point of maximum tip deflection is not predicted as accurately as in the previous drop test. All three analytic methods predict a maximum deflection of -0.11 wtip/L while the experimental data shows a maximum wtip/L of -0.115, a 5% difference. Figure 4-11 shows the results for the strain gage measurements during the drop test. Again, there is excellent correlation between the peaks and valleys and the maximum strain for all three gages. The maximum strains of 900, 650, and 550 µε are well matched.
  95. 95. 78 The same amount of time lag of 0.04 seconds in the analytical measurements and the experimental data is again evident. 4.4.3. Results from 5.2º Drop Angle Figure 4-12 shows the predicted and measured time histories of the flap hinge angle and tip deflection, while Figure 4-13 shows the results of the strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 during a drop test from an initial flap hinge angle of 5.2°. In Figure 4-12 the note that the first rebound of the flap hinge off of the droop stop has decreased to 1.2º and second rebound has increased to 2.5º for the analytic predictions. The experimental data matches the first rebound very well, but only reaches to 2.0º for the second rebound. The “snap-up” of the blade tip in Figure 4-12 is again predicted well. The maximum tip deflection for all three analytic methods is -0.13 wtip/L, while the experimental data shows -0.14 wtip/L, an 8% difference. The difficulty in integrating an acceleration signal to derive tip deflections becomes evident here. The experimental data points tend to diverge from the analytical predictions at the later points in time. This is caused by buildup of the drift error, described in Chapter 3, in the tip acceleration measurement. Even a small amount of error in each individual tip acceleration measurement can build up into a substantial error in the resulting tip deflection measurement after it is integrated twice in time.
  96. 96. 79 The three strain gage measurements are shown in Figure 4-13. Again note the overall excellent correlation between the experimental data and the analytic methods. The three maximum strains of 1000, 780, and 650 µε are all captured, except that the maximum strain in the third strain gage for the Modal Swapping On technique is a slightly higher 700 µε, a 7% difference. The same time lag of 0.04 seconds is again evident in all of the data. 4.4.4. Results of 7.6º Drop Angle Figure 4-14 presents the predicted and measured time histories of the flap hinge angle and tip deflection, while Figure 4-15 shows the results of the strain at x/L = 0.20, x/L = 0.30, and x/L = 0.40 during a drop test from an initial flap hinge angle of 7.6°. The character of Figure 4-14 has changed from the earlier plots. What appeared before as a rebound of the flap hinge angle off of the droop stop at t = 0.8 seconds has changed. At the beginning of the drop test the blade rotates freely downward to 1.0º; however, at t = 0.8 seconds the flap hinge angle suddenly stops, reverses directions and moves upwards to 2.0º before reversing direction again and finally contacting the droop stop. This type of event will hereafter be called a “knee” in the flap hinge angle. It is obvious that the first rebound seen in the earlier graphs, the “knee” in Figure 4-14, the “snap-up” of the blade tip, and the negative strain measurements are caused by the flexing of the blade as it falls. The major rebound of the flap hinge angle is seen at t = 0.4 seconds. The analytic methods predict the flap hinge to rebound to 5.0º but the

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