Development and Validation of a Numerical Blade Element Helicopter Model in Support of Maritime Operations

Transcription

1 Master of Science Thesis Development and Validation of a Numerical Blade Element Helicopter Model in Support of Maritime Operations W.R.M. Van Hoydonck B.Sc. August 18, 26

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3 Development and Validation of a Numerical Blade Element Helicopter Model in Support of Maritime Operations Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology W.R.M. Van Hoydonck B.Sc. August 18, 26 Faculty of Aerospace Engineering Delft University of Technology

4 Delft University of Technology Copyright c W.R.M. Van Hoydonck B.Sc. All rights reserved.

5 Delft University Of Technology Department Of Design, Integration and Operations of Aircraft and Rotorcraft The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled Development and Validation of a Numerical Blade Element Helicopter Model in Support of Maritime Operations by W.R.M. Van Hoydonck B.Sc. in partial fulfillment of the requirements for the degree of Master of Science. Dated: August 18, 26 Head of department: prof.dr.ir. Th. van Holten Supervisor: dr. M. D. Pavel Reader: ir. G. M. Voorsluijs

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7 Acknowledgements First of all, I would like to express my deepest gratitude and appreciation to dr. M. D. Pavel who has been my adviser during this thesis research. She provided encouragement, inspiration and sound advice whenever I encountered problems. For his invaluable advise on the implementation and validation of the SYCOS pilot model, I would like to thank prof. R. Bradley from Glasgow Caledonean University. ir. G. M. Voorsluijs provided me with the trim data of the Puma helicopter, for which I am very grateful. Special thanks goes to ing. M. Haanschoten who made it possible for me to run Linux on my workstation and who magically fixed the printers whenever they stopped working. All fellow graduate students in lecture room F that have been my friends, I thank you all dearly for the pleasant and inspiring atmosphere. Finally, I would like to thank my family and friends for their moral support and interest in my work. From now on, I will take more time to make a move, so be prepared for the next game of chess. Delft, University of Technology August 18, 26 W.R.M. Van Hoydonck B.Sc. v

8 vi Acknowledgements

9 Summary As part of a joint research project between Romania and the Netherlands, this thesis project started with a literature survey on the simulation and handling qualities of helicopters operating near ship decks. Topics covered in this survey include the accuracy of the main rotor inflow model, dynamic ground effect, turbulence modelling, ship motion modelling and models of mathematical helicopter pilots. It was concluded that in order to increase the fidelity of the model and allow for future extensions, a numerical approach should be taken to model the main rotor blade aerodynamics. In addition, the nonuniform Pitt-Peters dynamic inflow model is used for the inflow through the rotor disk. It is derived in the wind axis system, so it does not allow for sideward flight. Since one of the most often used landing procedures for helicopters stationed aboard ships includes a lateral repositioning ending in stationkeeping above the flight deck, the basic Pitt-Peters inflow model was modified to make this possible. The blade flapping dynamics are second order in lateral tilt, longitudinal tilt and coning, this means that the helicopter model has a total of nine structural degrees of freedom. In the past, another blade element model has been used for research at the Faculty of Aerospace Engineering. However, problems were encountered with it due to its implementation of the main rotor. Its trim routine only trimmed the fuselage, leaving the derivatives of the main rotor states nonzero at the start of a time simulation. This resulted in transients in the initial response of the helicopter in trimmed flight. By implementing a separate trim routine for the main rotor, the tail rotor and the fuselage, the complete helicopter can be trimmed correctly. The trim algorithm is capable of trimming the helicopter in any straight flight condition, including sideward flight. This in turn revealed a problem with the aerodynamics model of the vertical fin. In sideward flight, the angle of sideslip of the vertical fin is near 9 degrees. Using a linear relation between the aerodynamic forces and the angle of sideslip results in extremely large values for the drag of the fin. The tail rotor pitch angle would quickly diverge to minus 9 degrees to counter the drag. The same large angle approximation as used on the main rotor blades corrects this problem. The model has been validated in trim for the Puma and Bo15 helicopter against data found in literature. Nonuniform inflow improves the trim prediction of the rotor lateral tilt angle when compared to uniform inflow. Concerning the influence of nonuniform inflow on the response of the Puma helicopter to disturbances and control inputs, the use of nonuniform inflow strengthens the coupling between the longitudinal and lateral modes considerably when compared with uniform inflow in hover and low-speed forward flight. The phugoid becomes almost twice as unstable in hover while the short period mode increases its damping somewhat. Furthermore, time vii

10 viii Summary simulations with doublet inputs on the controls show that there are notable differences between the initial response of the helicopter and the long-term steady-state response. To simulate a complete landing manoeuvre, the SYCOS (Synthesis Through Constrained Simulation) mathematical pilot model developed at the Glasgow Caledonean University is used instead of classical PID-controllers. The control structure of this pilot model consists of two in series placed parts, the first being a crossover element, the output of which is processed by the second part, a linear inverse of the helicopter that generates the necessary corrective actions. The pilot model uses earth-oriented velocities and rate of change of heading angle as input. The constrained linear inverse model must follow these four constraints, so there are four eigenmodes with zero eigenvalues in addition to the zero associated with the heading angle. The four remaining eigenvalues are associated with two types of oscillatory behaviour of the helicopter around its centre of gravity, namely pitch and roll motion. These are the zero dynamics of the constrained system. The former is only marginally stable, resulting in stable but only lightly damped pilot control actions. The addition of a PD-controller in the pitch channel improves the damping of the pilot control activity sufficiently. Control of the heading angle is inadequate with this pilot model, so an additional PID-controller was added to keep it within acceptable ranges. The pilot model used here is capable of guiding an helicopter along an approach trajectory consisting of three parts. The first is a decelerated approach along a three degree glide path to a point alongside the ship. The second is a lateral repositioning ending in stationkeeping above the flight deck. In the final part, the helicopter lands on the ship deck. The blade element main rotor was specifically implemented with future extensions in mind. These include but are not limited to Peters-He finite state inflow modelling and the extensions based on this model for partial and dynamic ground effect. In the blade flappings dynamics, the influence of the linear velocity on the total kinetic energy was neglected. Therefore, in high speed flight, discrepancies between flight test results and simulation results will increase. To make the model valid for the complete flight envelope, the main rotor dynamics may have to be upgraded. Also, the impact of its wake may alter the pitch equilibrium of the helicopter, but this and other aerodynamic interferences between the different parts of the helicopter was not taken into account in this thesis.

11 Table of Contents Acknowledgements Summary List of Figures List of Tables List of Symbols Acronyms v vii xv xix xxi xxix 1 Introduction Background Thesis Objectives Report Outline Literature Survey Introduction Helicopter Simulation Model Inflow and Wake Models Pilot Model Related work done at the Faculty of Aerospace Engineering in Delft Helicopter Model General Requirements for the Helicopter Analysis and Simulation Program Main Rotor Azimuth Angle Definition Blade geometry Blade flapping dynamics Blade Feathering Pitch-Flap Coupling Reference Frames ix

12 x Table of Contents Body Fixed Frame of Reference Inertial Frame of Reference From Centre of Gravity to Blade Axis Rotor Reference Frames Tail Rotor Vertical Fin and Horizontal Stabilizer Fuselage Helicopter Equations of Motion Simulated Helicopters Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics Introduction Momentum Analysis Momentum Analysis in Hover Unsteady Momentum Theory - Dynamic Inflow Dynamic Inflow - Coordinate System and Definitions Tip path plane velocity components Nonlinear Dynamic Inflow Dynamic Inflow - Adaptation for Sideward Flight Pitt-Peters Non-Uniform Dynamic Inflow for an Analytical Main Rotor Wake Distortion Dynamics Introduction Dynamic Wake Distortion Model Augmented Pitt-Peters Dynamic Inflow Model Helicopter Trim Introduction The Most General Steady Motion Helicopter Trim in Straight Flight Kinematic Constraints Trim Routine - Theory Trim Results Forward Flight Sideward Flight Influence of First Blade Azimuth Position Inverted Trim

13 Table of Contents xi 6 Linearisation Introduction Linearisation Routine Theory Natural Modes Results Steady-state Main Rotor and Tail Rotor Influence of Nonuniform Inflow on the Stability and Control Derivatives Uncoupled and Coupled Eigenmodes Influence of Unsteady Main Rotor and/or Tail Rotor on Trim Results Case 2: Unsteady Main Rotor and Unsteady Tail Rotor Case 3: Unsteady Main Rotor and Steady Tail Rotor Case 4: Unsteady Tail Rotor and Steady Main Rotor Time Response of the Linear Models Influence of Wake Distortion Dynamics on Rotorcraft Responses Time Simulation - Ship Approach and Landing Introduction Pilot Model Overall Transfer Function Inverse Component Inverse Model Validation Simulation Flight Path References Phase I: Closure to Ship Phase II: Lateral Repositioning Phase III: Station Keeping and Landing Helicoper Stabilisation & Flight Path Reference Corrections Causes of Instabilities Stability Augmentation Position Corrections Results Spatial Position of the Helicopter and Ship in Time Velocity references and Controls Robustness of the Pilot Model Conclusions and Recommendations Conclusions Recommendations References 99

14 xii Table of Contents A Main Rotor Mathematical Model 15 A.1 Blade Element Velocity Components A.1.1 Assumptions and Restrictions A.1.2 Rotor System A.2 Blade Element Forces A.2.1 Assumptions A.2.2 Blade Segment Aerodynamics A.3 Derivation of the Blade Flapping Equation A.3.1 Derivation in the Rotating Frame of Reference A.3.2 Conversion from Rotating to the Non-Rotating Frame of Reference. 114 A.4 Rotor Aerodynamic Forces and Moments A.4.1 Rotor Forces and Moments at the Hub A.4.2 Main Rotor Forces and Moments at the Centre of Gravity A.5 Tip Path Plane Frame of Reference A.5.1 Exact Transformation Matrix A.5.2 Euler Transformation Approximation B Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage 121 B.1 Force and Moment Contributions of Empennage and Tail Rotor B.1.1 Horizontal Stabiliser B.1.2 Vertical Fin B.1.3 Tail Rotor B.2 Force and Moment contributions of the Helicopter Fuselage B.2.1 Drag Model B.2.2 Drag Model B.2.3 Moment contribution B.2.4 Problems C Additional Trim Results 131 C.1 Trim Results for the Bo C.1.1 Data Conversions C.1.2 Trim Results in Forward Flight C.1.3 Influence of Pitch-Flap Coupling C.2 Trim Routine Accuracy D Stability and Control Derivatives 137 D.1 Puma Helicopter D.2 Bo15 Helicopter E Helicopter Parameters 147 F System Output Filtering 149

15 Table of Contents xiii G Level 2 C-Mex Gateway S-Function to Fortran 153 G.1 Introduction G.2 C-MEX Gateway for a Continuous-Time State-Space System G.2.1 Mixed-Language Programming and Compiler Compatibility G.2.2 Fortran Continuous-Time State-Space System G.2.3 C-MEX S-function Contents G.3 Helix C-MEX Gateway Interface G.3.1 Introduction G.3.2 Interface: Fortan-side G.3.3 Interface: C-side G.4 Fortran Continuous-Time State-Space System G.5 C-MEX Gateway Interface H Helix User Guide 167 H.1 Introduction H.2 Prerequisites H.3 Installation and Compilation on Linux H.3.1 Compiler Installation H.3.2 Program Compilation, Execution and Command Line Options H.3.3 Simulink Interface Compilation H.4 Installation and Compilation on Windows H.5 Program Control File H.5.1 Flight Dynamics Model H.5.2 Trim Options H.5.3 Linearisation Options H.5.4 Flight Condition H.5.5 Plot Options H.6 Source Code Structure

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17 List of Figures 3.1 Azimuth angle definition of main rotor blades Geometry for the Puma main rotor blade Blade flapping angles in the non-rotating frame Pitch-flap coupling through control system geometry Schematic sideview of the Puma helicopter Rotor reference frames orientation of a blade in the hub plane Tip path plane orientation wrt the hub plane frame of reference Different operating conditions of the tail rotor viewed from above French military variant of the Aérospatiale AS 33 Puma South African civil version of the MBB Bo Velocity definitions in the tip path plane and wind axes system Trimmed flight with non-zero flight track angle χ Trimmed flight without flight track angle χ Fuselage pitch angle in trimmed forward flight Fuselage roll angle in trimmed forward flight Main rotor collective pitch angle in trimmed forward flight Main rotor longitudinal cyclic in trimmed forward flight Main rotor lateral cyclic in trimmed forward flight Tail rotor collective pitch in trimmed forward flight Main rotor inflow and flapping angles in trimmed forward flight Main rotor tilt angles with respect to the no feathering plane Trim results in sideward flight Influence of blade azimuth ψ 1 on trimmed controls in forward flight Flow diagram of the linearisation routine Loci of uncoupled eigenvalues (nonuniform inflow, steady-state rotors) Loci of coupled eigenvalues for Puma with steady rotors Loci of eigenvalues for Puma with unsteady rotors Loci of eigenvalues for Puma with unsteady main rotor xv

18 xvi List of Figures 6.6 Loci of eigenvalues for Puma with unsteady tail rotor Two degree amplitude doublet inputs for the nonlinear model Time response to main rotor collective doublet input Time response to tail rotor collective doublet input Time response to main rotor longitudinal cyclic doublet input Time response to main rotor lateral cyclic doublet input Influence of wake distortion dynamics on time responses of the V = 5 m/s Schematic overview of the sycos pilot structure Overall transfer function of the pilot-system-output chain Pilot modeled as a crossover element and the inverse of the system Normal and constrained system V = 2 m/s Kinematic profiles for the linear model time simulation Pilot controls for the linear inverse time simulation Standard forward-facing landing procedure [Lumsden et al., 1998] Spatial position profile during closure to ship Vertical acceleration profile parameters during phase I, closure to ship Horizontal kinematic references during phase I, closure to ship Vertical kinematic references during phase I, closure to ship Lateral acceleration profile for the sidestep manoeuvre Kinematic references during phase II, lateral repositioning Kinematic references during phase III, landing Pilot control activity during phase I of the simulation: closure to ship Velocities and velocity references during phase I Uncorrected positional and heading errors during phase I: closure to ship Positional and heading errors during phase I: closure to ship Spatial position of the Puma and frigate Velocities and velocity references during the complete manoeuvre Pilot control activity during the complete manoeuvre Pilot control activity during the last two phases Overview of the Simulink implementation of the SYCOS pilot model A.1 Rotor axes transformations A.2 Rotor blade sideview: definition of blade forces A.3 Lift and drag coefficient as a function of angle of attack A.4 Rotor hub top view: definition of rotor forces and moments A.5 Tip path plane orientation wrt the hub plane reference frame A.6 Tip path plane orientation wrt the hub plane reference frame B.1 Angle of attack α of the horizontal stabiliser

19 List of Figures xvii B.2 Angle of sideslip β of the vertical fin B.3 Sign conventions for the tail rotors of the Puma and the Bo B.4 Angle of attack α and sideslip β of the fuselage B.5 Simple fuselage drag force model B.6 Simple fuselage drag force model C.1 Top level trim variables for the Bo15 compared with DLR flight test data C.2 Fuselage attitude angles in trimmed forward flight C.3 Rotor trim variables for the Bo15 compared with flight test data C.4 Influence of pitch-flap coupling on the trim variables C.5 Convergence of trim routine D.1 Puma longitudinal stability derivatives as a function of airspeed D.2 Puma lateral stability derivatives as a function of airspeed D.3 Puma lateral into longitudinal stability derivatives as a function of airspeed. 139 D.4 Puma longitudinal into lateral stability derivatives as a function of airspeed. 14 D.5 Puma main rotor longitudinal control derivatives as a function of airspeed.. 14 D.6 Puma main rotor lateral control derivatives as a function of airspeed D.7 Puma tail rotor control derivatives as a function of airspeed D.8 Bo15 longitudinal stability derivatives as a function of airspeed D.9 Bo15 lateral stability derivatives as a function of airspeed D.1 Bo15 lateral into longitudinal stability derivatives as a function of airspeed. 143 D.11 Bo15 longitudinal into lateral stability derivatives as a function of airspeed. 143 D.12 Bo15 main rotor longitudinal control derivatives as a function of airspeed D.13 Bo15 main rotor lateral control derivatives as a function of airspeed D.14 Bo15 tail rotor control derivatives as a function of airspeed F.1 Frequency response of a seventh-order Butterworth filter F.2 Butterworth filter applied to pitch rate G.1 Fortran continuous-time state space dynamic system G.2 Simulation output of the state space system G.3 Schematic overview of the Fortran-Simulink gateway interface H.1 Sample graphical output from Plplot with Gnome Canvas Widget driver H.2 Graphical view of the Fortran source code

20 xviii List of Figures

21 List of Tables 4.1 Inflow time constants for the main and tail rotors of the Puma and Bo Uncoupled longitudinal eigenvalues and eigenvectors of the Puma in hover Uncoupled longitudinal eigenvalues and eigenvectors of the Puma at 6 m/s Uncoupled lateral eigenvalues and eigenvectors of the Puma in hover Uncoupled lateral eigenvalues and eigenvectors of the Puma at 6 m/s Phugoid eigenmode for the Puma in hover and forward flight Dutch roll eigenmode in hover and forward flight Roll subsidence eigenmode in hover and forward flight Spiral subsidence eigenmode in hover and forward flight Short period eigenmode in forward flight Pitch and heave subsidences and short period eigenmode in hover Eigenmode characteristics of matrix A V = 2 m/s Phase I (closure to ship) initial conditions Phase I (closure to ship) derived parameters Phase II (lateral repositioning) kinematic parameters Phase III (landing) kinematic parameters Values of the gains used in the heading stabiliser Parameter variations used to test the robustness of the pilot model B.1 Fuselage parasitic drag area calculation C.1 Trim accuracy and perturbation magnitude of the trim routines D.1 Dimensions of the stability and control derivatives E.1 Parameters of the Puma and Bo15 helicopters H.1 Versions of the programs and libraries used to develop Helix H.2 Helix command line options xix

22 xx List of Tables

23 List of Symbols Latin Symbols A rotor disk area [m 2 ] A a A c state matrix of a linear system real part of an complex number state matrix of constrained linear system a x earth-referenced X-axis [m/s 2 ] a y1 maximum lateral acceleration [m/s 2 ] a z1 maximum vertical acceleration during phase I [m/s 2 ] B b C control matrix of a linear system complex part of an eigenvalue output matrix of a linear system c main rotor blade chord [m] C output matrix of a linear system c d zero-thrust drag coefficient [-] C Dhs C Dhs drag coefficient of the horizontal stabilizer drag coefficient of the vertical fin c lα lift curve gradient [1/rad] C Lhs C Lvf lift coefficient of the horizontal stabilizer lift coefficient of the vertical fin c t time factor [s] C T thrust coefficient [-] C Θ D D Abbreviation for cosine function, where the subscript is the argument direct transmission matrix of a linear system direct transmission matrix of a linear system D fus fuselage drag [N] d type segment discretization type e flapping hinge offset [m] xxi

24 xxii List of Symbols ee root cut-out [m] F fuselage drag-surface coefficient F x fuselage equivalent plate area as seen from afront [m 2 ] F y fuselage equivalent plate area as seen from aside [m 2 ] F z fuselage equivalent plate area as seen from above [m 2 ] F t tail rotor fin blockage factor [-] f x x-offset of centre of gravity wrt body-fixed frame of reference [m] f y y-offset of centre of gravity wrt body-fixed frame of reference [m] f z z-offset of centre of gravity wrt body-fixed frame of reference [m] f dimensionless x-offset of cg with respect to rotor head [-] f 1 dimensionless y-offset of cg with respect to rotor head [-] h altitude above sea level [m] H(s) open-loop transfer function between error and output h dimensionless z-offset of cg with respect to rotor head [-] I bl rotor blade moment of inertia [kg m 2 ] I x moment of inertia about x-axis [kg m 2 ] I y moment of inertia about y-axis [kg m 2 ] I z moment of inertia about z-axis [kg m 2 ] J xz product of inertia about z-axis [kg m 2 ] k coefficient of fuselage pitching moment [m] K β equivalent spring constant [Nm/rad] K fus correlation factor fuselage pitching moment K pf proportional feedback of pitch-flap coupling [-] K Ψ1 K Ψ2 K q K r K Θ proportional feedback gain for heading angle Ψ integral feedback gain for heading angle Ψ derivative feedback gain for the pitch angle Θ derivative feedback gain for heading angle rψ proporitional feedback gain for the pitch angle Θ L p,... moment derivative with respect to angular velocity [1/s] L u,... moment derivative with respect to linear velocity [rad/(m s)] [L] [ L] inflow gain matrix in Pitt-Peters dynamic inflow complete inflow gain matrix associated with augmented Pitt- Peters dynamic inflow m helicopter mass [kg] m bl rotor blade mass [kg] M θ,... force derivative with respect to control [1/s 2 ] [M] apparent mass matrix in Pitt-Peters dynamic inflow ṁ mass flow through the rotor in momentum theory [kg/s]

25 List of Symbols xxiii n n bl n s total number of states main rotor number of blades number of blade segments P period of an eigenvalue [s] p helicopter roll rate [rad/s] p nondimensional roll rate [-] q helicopter pitch rate [rad/s] q nondimensional pitch rate [-] R main rotor radius [m] r helicopter yaw rate [rad/s] r x x-offset of gearbox centre wrt body-fixed frame of reference [m] r y y-offset of gearbox centre wrt body-fixed frame of reference [m] r z z-offset of gearbox centre wrt body-fixed frame of reference [m] S wake spacing in wake distortion dynamics [-] S Θ Abbreviation for sine function, where the subscript is the argument S vf vertical fin area [m 2 ] T dimensional rotor thrust [N] t initial time [s] T 1/2 time to half amplitude [s] T 2 time to double amplitude of an eigenvalue [s] T Θ Abbreviation for tangent function, where the subscript is the argument T tr tail rotor thrust [N] u control vector of a (linear) system u helicopter linear velocity along X-axis [m/s] U P perpendicular velocity component relative to blade segment [m/s] U T tangential velocity component relative to blade segment [m/s] u vector containing pilot controls V magnitude of the linear velocity of the helicopter [m/s] V modal matrix of a linear system, contains all eigenvectors as columns v helicopter linear velocity along Y-axis [m/s] V initial horizontal velocity [m/s] v heli v mr v tr trim variable vector for the top level (fuselage) trim routine main rotor trim variable vector tail rotor trim variable vector v i rotor induced flow [m/s]

26 xxiv List of Symbols v i V m i-th eigenvector belonging to the i-th eigenvalue λ i mass flow parameter associated with the mean inflow through the TPP V s ship speed [m/s] v ymax maximum lateral velocity [m/s] V mass flow parameters associated with the first order harmonics [-] of the inflow distribution [V ] mass flow parameter matrix in Pitt-Peters dynamic inflow V ol fus volume of an body equivalent to the fuselage with circular cross-sections w helicopter linear velocity along Z-axis [m/s] X azimuth angle of the speed vector V [rad] X wake skew used in Pitt-Peters dynamic inflow [-] x helicopter state x x-coordinate [m] X end total distance flown at the end of a phase [m] x i i-th state of a system of differential equations X p,... force derivative with respect to angular velocity [m/(rad s)] x t trim value of a helicopter state X θ1s,... force derivative with respect to control [m/(s 2 rad)] X u,... force derivative with respect to linear velocity [1/s] ω damped natural frequency of an eigenvalue [rad/s] y y-coordinate [m] z z-coordinate [m] Z initial altitude [m] Z d desired final altitude [m] z solution vector of a system of nonlinear differential equations [-] [m 3 ] Greek Symbols α fuselage angle of attack [rad] α hs geometric angle of attack of the horizontal stabilizer [rad] α hs angle of attack of the horizontal stabilizer [rad] β blade flapping angle [rad] β fuselage angle of sideslip [rad] β coning angle [rad] β vf geometric angle of sideslip of the vertical fin [rad] β 1c longitudinal tilt angle [rad]

27 List of Symbols xxv β 1c nondimensional rotor disk longitudinal flapping rate [-] β 1s lateral tilt angle [rad] β 1s nondimensional rotor disk lateral flapping rate [-] β p blade precone angle [rad] β vf angle of sideslip at the vertical fin [rad] γ flight path angle [rad] γ d desired flight path angle [rad] sideslip angle at the TPP [rad] H altitude difference [m] s rotor shaft tilt with respect to body fixed frame of reference, positive aft [rad] t total time flown at the end of a phase [s] t I total time flown during phase I [s] t II total time flown during phase II [s] t III total time flown during phase III [s] X initial horizontal separation between helicopter and ship [m] y lateral displacement during phase II [m] δ 3 inclination of flapping hinge [rad] δ p pass band ripple [-] δ s stop band attenuation [-] δx [ ] [ L] [ L 1 ] [ L 2 ] [ L 3 ] perturbation of a helicopter state transformation matrix between TPP and WA reference frames additional inflow gain matrix associated with wake distortion dynamics wake curvature and rotor mean loading coupling matrix associated with wake distortion dynamics wake curvature, wake skew and mean loading coupling matrix associated with wake distortion dynamics wake skew, wake curvature and cyclic loading coupling matrix associated with wake distortion dynamics ζ damping ratio of an eigenvalue [-] Θ fuselage pitch angle [rad] θ main rotor collective pitch angle [rad] θ tr tail rotor collective pitch angle [rad] θ 1c main rotor lateral cyclic pitch angle [rad] θ 1s main rotor longitudinal cyclic pitch angle [rad] θ pf main rotor blade pitch due to flapping [rad]

28 xxvi List of Symbols θ tw main rotor blade twist [rad] κ c longitudinal wake curvature [-] κ s lateral wake curvature [-] Λ λ λ 1c λ 1s λ i eigenvalue matrix of a linear state-space system nondimensional uniform inflow component through TPP in wind axes system nondimensional longitudinal inflow through TPP in wind axes system nondimensional lateral inflow component through TPP in wind axes system i-th right eigenvalue of a linear system λ i nondimensional induced flow or induced inflow ratio [-] µ aerodynamic roll angle [rad] µ ar advance ratio or in-plane velocity in TPP [-] µ ir inflow ratio at the TPP [-] µ op nondimensional out-of-plane velocity at TPP [-] µ par dimensionless airspeeds parallel to the (tail) rotor disk µ per dimensionless airspeeds perpendicular to the (tail) rotor disk µ x nondimensional x-component of relative velocity in TPP [-] µ y nondimensional y-component of relative velocity in TPP [-] µ z nondimensional z-component of relative velocity in TPP [-] ν damping of an eigenvalue [1/s] ρ air density [kg/m 3 ] τ time constant of an eigenvalue [s] τ time delay [s] [τ D ] nondimensional time constant matrix associated with wake distortion dynamics τ λ time constant in quasi-dynamic inflow [s] Φ fuselage roll angle [rad] χ flight track angle, difference between azimuth angle of speed vector and helicopter X-axis [-] [-] [-] [rad] Ψ fuselage azimuth angle [rad] ψ blade azimuth angle [rad] Ω magnitude of the angular velocity of the helicopter [rad/s] Ω main rotor angular velocity [rad/s] ω cutoff frequency, the frequency at which the magnitude response of the filter is 1/ 2 of its nominal value at the pass band [rad/s] ω n undamped natural frequency of an eigenvalue [rad/s]

29 List of Symbols xxvii ω p pass band edge frequency [rad/s] ω s stop band edge frequency [rad/s] Subscripts qs t quasi-steady value trim value

30 xxviii List of Symbols

31 Acronyms CL CP DUT FUS HP HS NFP MR PD PID RH stdout TPP TR VF WA Centre Line Control Plane Delft University of Technology Fuselage Hub Plane Horizontal Stabilizer No Feathering Plane Main Rotor Proportional Derivative Proportional Integral Derivative Rotor Hub Standard Out the default output stream (normally, to the screen) Tip Path Plane Tail Rotor Vertical Fin Wind Axis xxix

32 xxx Acronyms

33 Chapter 1 Introduction 1.1 Background In 1989, the Romanian military joined NATO. To comply with its standards, the Romanian Marine acquired two former Royal Navy Type 22 frigates in December 22. Both were regenerated and modernised by BAE Systems. The second frigate Regina Maria, was commissioned in April of 25 after successfull sea trials in March of that year. It has not yet been decided which helicopters will be stationed aboard these frigates, but one possible candidate is the AS-33 Puma built by the Romanian helicopter manufacturer IAR S.A. Brasov. Since the Puma was originally not designed for naval operations, research has been started to investigate the possibilities and limitations of using the Puma helicopters for shipboard operations. The behaviour of large helicopters operating in a maritime environment is different from helicopters performing land-based operations because of the interaction with the (moving) landing platform. The dynamic interface environment, particularly in the presence of wind, turbulence, ship motion and low visibility can result in a high workload for the pilot. In addition, helicopters are notorious for strongly coupled off-axis responses during manoeuvring flight, which increases the workload even more. To ensure that a certain helicopter/ship combination is compatible, a set of limits is established within which save launch and recovery can take place. Normally, these limits are determined by conduction tests at sea [Carico et al., 23] for the helicopter/ship combination of interest. An important goal of these tests is the establishment of wind-over-deck flight (WOD) envelopes for various meteorological conditions. These envelopes determine the combinations of relative wind direction and wind strength in the neighbourhood of the landing platform for which a save helicopter launch or recovery can take place. The conventional method by actual flight tests requires a significant amount of time and resources and it is limitid by the weather and sea conditions. So, in recent years, numerous efforts have been devoted to develop modelling and simulation tools for rotorcraft/ship dynamic interface testing [He et al., 22]. In a complete simulation tool, the following topics may be covered: 1. modelling the helicopter flight dynamics, 1

34 2 Introduction 2. modelling the pilot feedback loop in off-line simulations, 3. defining the trajectory to be flown by the pilot, 4. for a given sea state, model the ship motion and its turbulent airwake, 5. modelling the interaction between the wake of the helicopter and the ship. 1.2 Thesis Objectives The objective of this thesis is to develop and validate a helicopter model that can be used to define critical situations for helicopters operating in a maritime environment. The main focus will be on the first three items in the above list. Based on a literature survey, the major shortcomings of the six degrees of freedom model currently in use at the chair of Design, Operation and Integration of Aircraft and Rotorcraft will be determined. The upgraded model will then be validated statically against available trim data. Then, the influence of the extensions on the dynamics of the model will be examined by means of an linearised model of the helicopter. Finally, a pilot model will be build that is capable of performing a prescribed landing manoeuvre. 1.3 Report Outline The structure of this report is as follows. In Chapter 2, an summary is given of the literature survey that was carried out at the start of this thesis. An overview of the mathematical helicopter model that was derived and some general implications and restrictions is given in Chapter 3, followed by a discussion of Pitt-Peters dynamic inflow model and a wake distortion dynamics extension to that model in Chapter 4. The implementation of a trim routine and the trim results for the Aerospatiale AS33 Puma helicopter can be found in Chapter 5. The sixth chapter discusses the linearisation routine, followed by an eigenmode analysis for the Puma helicopter with both uniform and nonuniform inflow. In Chapter 7, the pilot model, the flight path references and the results for the inverse time simulation are discussed. Finally, the conclusions of this thesis and recommendations for future work can be found in chapter 1. An in-depth derivation of the mathematical model of the helicopter can be found in Appendices A and B. The main rotor is discussed in the former and the tail plane surface models, tail rotor model and fuselage model in the latter. Additional trim results are discussed in Appendix C and the stability and control derivatives of the Puma (and Bo15) are given in Appendix D. Parameters for both helicopters are listed in Appendix E. Background on the C-MEX gateway interface between Fortran and Simulink is given in Appendix G and in the last appendix, a user guide is presented for the simulation program.

35 Chapter 2 Literature Survey 2.1 Introduction This chapter contains a summary of the literature survey [Van Hoydonck, 26] that was carried out at the start of this thesis research. The goal of this survey was to gain insight in issues that surround simulation and handling qualities investigations for helicopters operating near ship decks. It focused mainly on the following three topics, 1. the helicopter simulation model, 2. the ship simulation model and 3. the interaction between the helicopter and the environment in the neighbourhood of the ship; the rotorcraft/ship dynamic interface. In the following section, the first item is discussed since that became the major focus of this thesis. Divided in two parts, (dynamic) inflow and pilot models are discussed. Thereafter, some examples of related projects and research at the Faculty of Aerospace Engineering are listed. 2.2 Helicopter Simulation Model The focus of this part was on inflow and wake models that have been developed in the last 5 years. Traditionally, two different approaches exist to model the rotor wake: inflowtype models where only a velocity distribution at the rotor disk is modeled and flow-type models where the dynamics of the entire wake is captured in the simulation. A short overview of the former is discussed here Inflow and Wake Models In introductory textbooks about helicopter performance and dynamics [van Holten & Melkert, 1994], the rotor inflow is calculated with Glauert s hypothesis. It assumes a uniform inflow through the rotor disk and while this is acceptable for 3

36 4 Literature Survey performance calculations, it doesn t give realistic results for blade dynamics or vibratory loads. The model fails because in forward flight most of the wake (and thus also the tip vortices) is located below the first and fourth quadrant of the rotor. The induced velocity associated with these tip vortices is therefore much higher in the first and fourth quadrant with a reduced angle of attack as a result. This gives a smaller lift in the aft region of the rotor disk when compared to the uniform inflow model. Until the mid 196 s improvements for Glauert s classical inflow model were mainly analytical and empirical [Wheatley, 1935], but starting from the mid 196 s (numerical) approaches based on the Biot-Savart law became popular [Coleman et al., 1945; Castles Jr. & de Leeuw, 1954; Heyson & Katzoff, 1957]. More recently, research focus has shifted to include the inertia of the air surrounding the rotor in the inflow models. The first one of these dynamic extensions is due to Carpenter & Fridovitch [1953], who extended momentum theory by including an apparent mass term which models the inertia of the air surrounding the rotor. First order inflow models have seen wide use in literature, both steady [Chen, 1989] and unsteady variants [Goankar & Peters, 1988]. The first model of dynamic inflow useful for both hover and forward flight is that of Pitt & Peters, [1981; 1983]. As normally formulated, the model is a perturbation theory written in the wind axis reference system for zero hub motions while for practical applications, one would rather write it in the rotor reference frame that allows for sideward flight. These more usable forms are presented in [Peters & HaQuang, 1988]. Later, Peters & He [1989] and Peters et al. [1989] showed how this model could be generalised to a theory with an arbitrary number of inflow harmonics and an arbitrary number of radial shape functions per harmonic; the model which is now known as the Peters-He finite state inflow model. During the late nineties the Peters-He model was further sophisticated to include partial and inclined ground effect [Xin, Prasad, & Peters, 1999; Xin, Prasad, Peters, Itoga, et al., 1999; Xin, 1999; Xin et al., 2]. In these extensions, the influence of the ground plane is represented as a source-like pressure perturbation in the flow field. The total pressure perturbation which determines the induced inflow at the rotor disk is obtained as the superposition of the contributions due to the rotor and the ground plane. A shortcoming of the Pitt-Peters and Peters-He models is that they treat the rotor wake as quasi-steady in which wake bending due to rotor pitch and/or roll takes place instantaneously, while in reality, it does take some finite time to develop the wake curvature. Similarly, the wake skew and spacing also develop gradually instead of instantaneously during manoeuvres. For instance, a roll motion affects the lateral distribution of the vortices emitted by the rotor blades. Therefore, the lateral gradient in the induced velocity field is also affected and so are the lateral distributions of the angles of attack and airloads. Since the rotor acts as a gyroscope, these lateral variations contribute to the off-axis pitch response. As a result, these models are only valid for steady manoeuvring flight. To overcome this limitation, several aerodynamic tools were developed in recent years to study rotor dynamic wake distortion effects during manoeuvring flight.

37 2.2 Helicopter Simulation Model 5 The first correction for dynamic inflow theory that includes wake distortion dynamics was proposed by [Arnold et al., 1995], [Keller, 1995] and [Keller & Curtiss, 1996]. In this model, the effect of pitch and roll rates are included as extra forcing functions for the lateral and longitudinal components of the inflow dynamics. Prasad et al. [21] use a comprehensive free wake analysis to investigate self-induced wake distortion effects of which the results are then used to augment a dynamic inflow model with additional states for wake curvature and skew. The same authors develop a four-state reduced order model that captures dynamic wake distortion effects during transitional, manoevring and forward flight using vortex tube analysis in [Zhao et al., 22] and [Prasad et al., 22] and are combined with the Pitt-Peters and Peters-He inflow models. The model can be expressed as a compact set of ordinary differential equations in four variables, namely the wake skew, wake spacing and longitudinal and lateral wake curvatures This dynamic wake distortion model is extended in [Zhao et al., 23] to account for the cyclic loading effect and the nonlinear couplings between wake curvature and wake skew on inflow across the rotor disk during manoeuvring flight Pilot Model To be able to control the helicopter in a realistic manner during off-line simulations, some kind of control is necassary. The classical way to do this is by designing a controller for every control input, e.g. a collective (PID) controller that governs the vertical speed of the rotorcraft. In Pavel [21, App. E3], an example of such a classical controller is given. Most helicopters are also equipped with a Stability Augmentation Systems which applies artificial damping to for example the pitch rate to reduce pilot workload. The development of a helicopter pilot model that is used to pilot the Coupled Rotor Fuselage Model (CRFM) at Westland Helicopters is described in Hamm [1994]. This model consists of integration subroutines that connect the pilot model logic to the vehicle and diplay software, a perception model, top level logic which models the conscious thought process of the pilot, intermediate level handling subroutines that model the subconscious pilot processes and feedback controllers that simulate the pilot s motor skills and generate the control displacements. Another way of controlling a helicopter with a mathematical pilot is by inverse simulation. Here, the actions of the pilot correct errors between the actual output and the desired reference. In this kind of simulation, one of the four required references is the flight path defined by the position of the centre of mass of the helicopter as a function of time. In addition, the heading angle or angle of sideslip is also given to make up the four reference values defining uniquely the four control positions. Turner et al. [2] and Bradley & Brindley [22] describe a generic helicopter pilot model (Synthesis through Constrained Simulation, sycos) that is capable of flying a helicopter along a prescribed flight path. The aim of this model is to get control activity similar to that of a real pilot and ultimately extract workload ratings from the simulations, not just reducing the gap between the output and the desired reference. The pilot model consists of an inverse of the system to be controlled and a crossover element incorporating

38 6 Literature Survey a gain, time delay and an integration. Within the pilot model, the inverse component is the pilot s adaption to the dynamics of the controlled system. It is the inverse of a simple vehicle dynamics model which represents the pilot s perception of the vehicle dynamics. A similar model has been used by van der Vorst [1998] with the difference that here the inverse dynamics model is replaced by a neuromuscular system which simulates the limbmanipulator dynamics and the proprioceptive feedback of the muscle spindles. The central nervous system is modeled by the crossover model. This model is also more interesting than a plain crossover model since it reflects the information processing in the human body. In [Hess et al., 22] a closed-loop pilot/vehicle modelling procedure is applied to the inverse dynamic analysis of a manoeuvre (lateral repositioning with and without slung load) from ADS-33. The analysis procedure permits the examination of task requirements in terms of aggressiveness, pilot/vehicle performance and resulting handling qualities. The pilot model is the Structural Pilot Model from [van der Vorst, 1998] which was discussed in the previous paragraph. 2.3 Related work done at the Faculty of Aerospace Engineering in Delft The model van Gool [1997] developed for his research in the field of helicopter responses to atmospheric turbulence has a numerical blade element main rotor which utilizes 2D section airfoil data to compute the section forces. The tail rotor is modeled with a modified Bailey tail rotor model. The inflow model uses harmonic inflow components determined by Glauert s distribution. Ypma [1996] derived a coupled rotor model for the prediction of blade flapping angles for her Masters Thesis at the Flight Mechanics and Propulsion chair. This model utilises a first harmonic static inflow distribution to model the inflow at the rotor disc. The model is compared with experimental data available in literature. One of the main conclusions drawn from this thesis work is that... a first harmonic inflow model with longitudinal varying inflow is always preferable over a uniform inflow model. The use of a longitudinal varying inflow component improves the estimation of the lateral flapping angle b 1 compared to purely uniform inflow. In the literature survey of her Doctoral thesis, Pavel [21] mentions the significance of inflow dynamics on helicopter stability and control characteristics, but they are not included in the development of the Critical Pole Distance Method. The uniform inflow of both the main and tail rotor are modeled as a quasi-steady dynamic inflow by means of a time constant with a value between.1 and.5 seconds. At the Control and Simulation Chair, de Back [1994] graduated in August 1994 on the effects of non-uniform dynamic inflow on helicopter flight responses. It was an investigation of the by him erroneously called Pitt-Peters model 1. de Back concluded 1 The actual implementation was the model developed by He during his Doctoral thesis research, the Peters-He model

39 2.3 Related work done at the Faculty of Aerospace Engineering in Delft 7 that the inclusion of non-uniform dynamic inflow affects all rotorcraft model responses and that the non-uniformity strengthens the cross-coupling between the symmetrical and asymmetrical modes. However, this model has not been used in further research. For his internship at the Boeing Company in Mesa, Arizona, de Bruijn [22] implemented the Peters-He finite-state inflow model with dynamic wake distortion in the Apache FLY Real Time (FLYRT) simulation code. Hoencamp [23] graduated on an implementation of the Peters-He finite-state inflow model for the Apache helicopter operating near ship decks. The apache simulation code FLYRT, was extended with the ground-effect extension for the Peters-He inflow model as developed by Xin [1999]. The influence of the fuselage is taken into account by adding two partially overlapping pressure distributions in the flow at a normalised height of.25h below the rotor. Voorsluijs [23] developed a modular generic helicopter model, suitable for real-time simulation in the human-machine laboratory and the SIMONA research simulator. The model is based on previous work by van Gool [1997] and IJsselmuiden [21]. This model is further being developed by Voorsluijs for research in user-friendly control of a helicopter UAV.

40 8 Literature Survey

41 Chapter 3 Helicopter Model The first section lists some general design requirements for the helicopter analysis and simulation program Helix that was developed as part of this thesis. Then, some details are given about the actual implementation; sign conventions are discussed, reference frames are defined and the level of detail of all parts of the helicopter are discussed together with the implications this has. The actual derivations of the mathematical models can be found in two appendices, for the main rotor mathematical model, it is given in Appendix A and the models of the other parts (fuselage, tail rotor and tail surfaces) are given in Appendix B. 3.1 General Requirements for the Helicopter Analysis and Simulation Program While doing literature research, the following list of general requirements for the helicopter simulation program was compiled. It is mainly based on the recommendations in Voorsluijs [23, App. A]. 1. It must be possible (and easy) to change the rotor rotational direction so that both the Puma and Bo15 helicopters can be simulated, 2. related to this, the tail rotor must work correctly for helicopters with both types of rotor. 3. The centre of gravity of the helicopter should not be used as a reference point for the geometric positions of the different parts of the helicopter, since this prohibits one to change the position of the centre of gravity during a simulation. 4. It should be possible to trim the helicopter completely, without suffering from transient effects of the numerical blade element model as encountered by [Voorsluijs]. 5. An interface with Simulink should be available for controller design. 6. The use of a harmonic integration scheme for the blade flapping dynamics as suggested in Johnson [198] and used in Voorsluijs [23] should be avoided, all state 9

42 1 Helicopter Model derivatives should be integrated with the standard ODE solvers that are shipped with Simulink. 7. It must run fast enough on modern computer hardware. Initially, the flight dynamics model was implemented as Matlab M-code. Trim runs and time simulations indicated that the use of a numerical algorithm to calculate the aerodynamic forces on the rotor blades makes the code execution extremely time consuming. Therefore, the flight dynamics model including the trim and linearisation routines was manually converted to Fortran 95. This made the program approximately 1 times as fast. The interface between Fortran and Simulink is discussed in Appendix G. The sixth requirement implies that during time simulations, only the states are available to calculate the state derivatives. The equations of motion, which in general have the following form, ẏ = f(ẏ, y, u) (3.1) should be written explicitly. For a mathematical rotor model with rigid blades, this can be done without too much effort. 3.2 Main Rotor Azimuth Angle Definition The angular position of the blades in the hub plane are defined by one coordinate, the blade azimuth angle ψ 1. The standard counterclockwise definition is used for both types of rotor (see Fig. 3.1). The major advantage is that this means that no sign changes are needed in the equations for the blade flapping dynamics for either type of rotor. As a downside, counterclockwise rotating rotors such as found on the Bo15 have negative rotor rotational speed and clockwise rotors have a positive rotor rotational speed. As a consequence, care must be taken when dividing certain parameters by e.g. the tip speed ΩR Blade geometry The helicopter blades have a rectangular planform with an optional root cut-out. They are divided into a number of blade segments, for which two different distributions are available. These are the equal annuli segment distribution [Houck et al., 1977] and a constant width segment distribution. The equal annuli distribution has the advantage that the segments are more densely packed towards the region with greater change in dynamic pressure. Both distributions are depicted in Fig. 3.2 for a blade of the Puma helicopter with 1 segments 2. 1 The azimuth angle of the first blade, ψ 1 is used as a state in the simulation program to keep track of the position of all the blades. 2 By default, number of segments for both helicopters is 15 in the simulations.

43 3.2 Main Rotor 11 X hp ψ = 18 Ω P uma Ω Bo15 ψ = 9 ψ = 27 Y hp ψ = Figure 3.1: Azimuth angle definition of main rotor blades (a) Equal annuli segment distribution (b) Constant width segment distribution Figure 3.2: Geometry and segment distributions for the Puma helicopter main rotor blade Blade flapping dynamics To releave the blade root and the hub from the moments produced by the fluctuating forces acting on the rotor, flapping and feathering hinges located near the rotor hub are used 3. The blades flap due to the azimuthally varying lift and drag forces. Lead-lag hinges allow the blades to respond to the Coriolis forces that are produced when the radius of gyration changes due to the flapping motion. The flapping motion is highly damped and mainly affects the control response characteristics and much less the aeroelastic stability characteristics of the rotor. The lagging dynamics on the other hand are only lightly damped and mainly affect the aeroelastic stability characteristics [Pavel, 21]. Therefore, only the flapping dynamics are taken into account as they are much more important for flight dynamics analysis and simulation. 3 Actual hinges in an articulated rotor or hingeless flexures in a hingeless or bearingless rotor design.

44 12 Helicopter Model In the non-rotating frame of reference, the blade flapping angle β is represented by the following truncated Fourier series, β(ψ) = β + β 1s sin ψ + β 1c cos ψ (3.2) A graphical representation of the three constant terms in this equation is given in Fig. 3.3 and a physical explanation of these three characteristic flapping terms is given in the three subsequent paragraphs. β Ω T P P Ω T P P β β 1c Xhp β 1s Y hp Z hp (a) View along advancing blade, showing positive longitudinal tilt β 1c Z hp (b) View from behind, with positive lateral tilt β 1s Figure 3.3: Blade flapping angles in the non-rotating frame Coning Angle β The average or mean part of the flapping motion is represented by the coning angle β (see Fig. 3.3). This angle is a result of the aerodynamic balance around the flapping hinge of the rotor blade. The centrifugal force is a function of the rotor rotational speed which means that it can be assumed constant as long as the rotational speed is kept constant. As a result, the coning angle β is a function of the magnitude and distribution of the lift on the rotor blades. Longitudinal Flapping β 1c The coefficient β 1c represents the amplitude of the pure cosine flapping of the rotor blades or the fore-aft tilt of the tip path plane (TPP) as is depicted in Fig. 3.3a. The rotor has the tendency to tilt back in forward flight due to the dissymmetry in lift distribution between the advancing and retreating side of the rotor. On the advancing side, the blade segments experience a higher relative velocity and dynamic pressure than at the back and front of the rotor (ψ = and ψ = 18 ). The extra lift will cause the blade to flap upward as it advances towards the front of the rotor. The maximum flap angle will be reached at ψ = 18 and here, the flap velocity equals zero. On the retreating side, the opposite happens. The reduced dynamic pressure causes the blade to flap downward again. The downward motion increases the angle of attack α of the blade again, counteracting the negative flapping motion and increasing the blade lift again until at ψ =, the blade flapping is again zero.

45 3.2 Main Rotor 13 In the final equilibrium condition, the tip path plane will be tilted backward over an angle β 1c. The aerodynamic forcing function in this case has its maximum at ψ = 9 whereas the resultant flap motion has its maximum at ψ = 18, which is a phase delay of 9 degrees. Lateral Flapping β 1s The lateral flapping or pure sine motion is represented by β 1s (Fig. 3.3b). A counterclockwise rotating rotor has the tendency to flap to the right 4. The cause of this flapping lies in the coning of the rotor. For a rotor in forward flight with a positive coning angle, the angle of attack of the blades at the back is decreased and at the front, the angle of attack reaches a maximum. As a result, a second forcing function is produced, but this time it is 9 degrees out of phase with the forcing function causing the longitudinal flapping. The blade experiences a maximum lift increase when the angle of attack is maximal and the subsequent maximal displacement happens 9 degrees later on, tilting the tip path plane to the right as seen from behind. The phase delay of 9 degrees between the forcing function and the resulting flap displacement means that the rotor is excited at or near the natural frequency of blade flapping. Higher-order Harmonics The Fourier representation of the blade flapping dynamics (Eq. 3.2) does not include the higher order harmonics (β 2s, β 2c,...). These higher order harmonics cause a small warping of the tip path plane (TPP) when added, but this effect is negligible for trim and performance analysis. However, their effect on vibration and aeroelastic analysis is significant Blade Feathering Blade feathering is the motion of the blade around the feather axis. Using a truncated Fourier series, this motion can be described in the nonrotating frame of reference as θ(r, ψ) = θ + θ 1s sin ψ + θ 1c cos ψ + θ tw r R + θ pf (3.3) In here, θ tw is the pitch due to the blade twist and θ pf is the pitch due to blade flapping. There are two sources of input to this equation [Leishman, 22], 1. swashplate, the orientation of which is controlled by the pilot, 2. elastic deformation of the blade itself and the control system In this thesis, only the first item is considered. The collective pitch θ controls the average blade pitch angle, the average lift and therefore the average rotor thrust. It is the control left. 4 For a helicopter with a clockwise rotating rotor, such as the Puma, the lateral flapping will be to the

46 14 Helicopter Model used by the pilot to alter the vertical motion of the helicopter. The cyclic pitch angles θ 1s and θ 1c control the orientation or tilt of the rotor disk, thereby changing the direction in which the rotor thrust points. The swashplate consist of two disks concentric with the shaft. They are connected to each other by means of a set of bearings, which allows the upper disk to rotate with the rotor blades. The lower disk is connected to the control system and does not rotate. The connecting between the swashplate and the rotor blades is accomplished by push-pull rods and pitch horns. The pilot controls the vertical position of the swashplate with the collective input. The two cyclic inputs control the lateral and longitudinal tilt of the swashplate. For a rotor with central flapping hinges, θ 1c controls the lateral tilt of the disk. Due to the 9 degree phase angle between the applied forcing function and its response, a pure cosine input of θ 1c will give the blade at ψ = a positive pitch input, resulting in a maximum positive flap displacement 9 degrees later, tilting the rotor disk to the left (i.e. a positive β 1s tilt). Similarly, a sine input of θ 1s will cause the tip path plane to tilt backward, which is equivalent to a negative β 1c tilt. Hence, the term θ 1s is called the longitudinal cyclic and the term θ 1c is called the lateral cyclic. From the above, it should be clear that there is an analogy between feathering and flapping for centrally hinged rotors. This will be discussed in some more detail in section 3.3, where the different frames of reference are discussed that are used in the simulation program Pitch-Flap Coupling The hingeless rotor design of the Bo15 is approximated with the same centrally hinged articulated rotor as the Puma. The main difference is the much higher hub stiffness and the addition of pitch-flap coupling. This is introduced through the control system geometry. Fig. 3.4 shows a top view of the location of the push-pull rod with respect to the flapping hinge. The pitch bearing or feather hinge is located outboard of the flap hinge which means that if the pitch link is not in line with the axis of the flapping hinge, a flap displacement will cause the blade to pitch. δ 3 Ω feather hinge flapping hinge Figure 3.4: Pitch-flap coupling through control system geometry This is a form of proportional feedback of the flap motion on the feather motion and it is governed by the following equation [Johnson, 198], θ pf = K pf β bl (3.4)

47 3.3 Reference Frames 15 where K pf is a proportional feedback, depending on the angle δ 3 (Fig. 3.4) as follows K p = tan δ 3 (3.5) For the Bo15, the value of δ 3 is set to 45, as suggested in Padfield [2]. No pitch-flap coupling is added for the rotor of the Puma, so its value of δ 3 equals zero. When δ 3 has a positive value, pitch-flap coupling will act as aerodynamic damping term, reducing the blade pitch for a positive flap angle, thereby reducing the lift on the blade. 3.3 Reference Frames In this section, an overview of the reference frames is given that were used for the helicopter simulation program Body Fixed Frame of Reference The fuselage axes are aligned with the frame-station/butt-line/waterline reference. This reference system, adopted from ship design, uses these terms to depict positions in the X, Y and Z axes respectively (as suggested in Jensen & Curtiss Jr. [1991]). Normally, the waterline position defines a horizontal plane where the ground or a plane below the ground is given a value of zero. Every position on the aircraft then has a positive waterline value that indicates the distance above the zero level. Since there is no data available about the height of the Puma landing gear in unloaded position, another plane had to be chosen. The cabin floor level was selected as reference level for all measurements in the Z-direction, where all positions above this reference plane have a negative value as is normal in flight dynamics. X ro X hp Z hp h r R tr z hs X so Z so r z z vf z tr X bf r x Z ro cabinfloorlevel Z bf x vf x tr = x hs z hs Figure 3.5: Schematic sideview of the Puma helicopter The frame-station is similarly defined for the position along the longitudinal axis. In this case, the zero position is taken at the aft side of the (left) cabin door. Finally, the buttline system defines left to right positions where the centreline of the helicopter is zero. All linear dimensions of the helicopter are given with respect to this reference system, called

48 16 Helicopter Model the body fixed frame of reference, { E bf }. This way, one could easily vary the position of the centre of gravity during a simulation, e.g. to simulate fuel usage by the engine(s). A schematic side view of the Puma helicopter depicting the most important dimensions is given in Fig. 3.5 and a complete overview of all geometric and inertial parameters of both helicopters is given in Appendix E. Since the rotor forces and moments are initially resolved into the shaft axis system, a body fixed frame of reference that coincides with the shaft axis system may look more suitable. This model, however, is designed to take a shaft tilt ( s, positive aft) into account, just like on the real Puma helicopter Inertial Frame of Reference The orientation of the helicopter with respect to the flat earth is described by the three Euler angles Ψ, Θ and Φ. The transformation between the body-fixed frame of reference and the inertial frame of reference {E e } can be written as { Ebf } = [Φ][Θ][Ψ] {Ee } (3.6) with C Ψ S Ψ C [Ψ] = S Ψ C Ψ, [Θ] = Θ S Θ 1 1 S Θ C Θ 1 and [Φ] = C Φ S Φ (3.7) S Φ C Φ Note that throughout this thesis, sine, cosine and tangent functions may be abbreviated to e.g. S Θ, C Θ and T Θ, where the subscript is the argument. The inverse of Eq. 3.6 is used to determine the position of the helicopter in the inertial frame of reference during time simulations, defined by the three Cartesian coordinates x, y and z. Using the Euler angles to keep track of the orientation of the helicopter has the advantage that only three physically meaningful numbers are required. The biggest disadvantage of Euler angles is that they suffer from gimbal lock, a condition where 2 axes effectively line up, resulting in a temporary loss of a degree of freedom. This happens when the pitch angle approaches (or equals) plus or minus 9 degrees. At this point, a change in roll angle Φ will be indistinguishable from a change in azimuth angle Ψ. Also, the derivatives of the heading angle and the roll angle ( Ψ and Φ) approach infinity. If it is required to simulate high pitch angle manoeuvres, Euler angles cannot be used to keep track of the orientation of the helicopter. Instead, quaternions should be used [see IJsselmuiden, 21, App. D]. These were not implemented in the simulation program, so it is not advisable to simulate high pitch manoeuvres From Centre of Gravity to Blade Axis In helicopter dynamics [Padfield, 2] and flight dynamics in general [Cook, 1997] mass is assumed to be distributed symmetrically with respect to the butt-line of the aircraft. As a consequence, the products of inertia I yz and I xy are zero. Nevertheless, the

49 3.3 Reference Frames 17 centre of gravity of the helicopter is allowed to have three offsets (f x,f y, f z ) with respect to the body fixed frame of reference, depending on the weight distribution of the payload of the helicopter 5. The centre of the rotor origin frame of reference {E ro } (see Fig. 3.5) is located near the gearbox of the helicopter, at the point (r x,r y and r z ) in the body fixed frame of reference. For a normal helicopter, the offset in the y-direction is normally zero due to symmetry. Due to the forward tilt of the main rotor shaft with respect to the body axis, the value of the shaft tilt angle s is negative in the simulation. The relation between the unit vectors of the rotor origin frame of reference and the shaft origin frame of reference is {E so } = [ s ] {E ro } (3.8) with s defined as C s S s [ s ] = 1 (3.9) S s C s A translation h r along the positive Z-axis of the shaft origin frame of reference (again see Fig. 3.5) defines the position of the hub plane frame of reference { E hp }. With normal helicopters, the hub is located above the gearbox, so h r has a negative value. With a rotation ψ around the negative Z sa -axis, the unit vector of the hub plane frame of reference is transformed into the projection axis frame of reference. From there, a rotation β around the positive Y pa -axis leads to the blade axis system (see Fig. 3.6), with {E ba } = [β][ψ] { E hp } C β S β [β] = 1 S β C β (3.1) C ψ S ψ and [ψ] = S ψ C ψ (3.11) 1 These last two reference frames define the position of the blades with respect to the hub plane Rotor Reference Frames Basically, there are three reference frames in use to describe the rotor dynamics [Padfield, 2]. These are the Hub Plane (HP) frame of reference, the Tip Path Plane (TPP) and the No Feathering Plane (NFP). The latter two frames of reference are used in the textbook of Bramwell since they greatly simplify the expressions for the rotor forces. 5 In the actual implementation, the values of the three offsets of the centre of gravity are given with respect to the centre of the hub plane frame of reference, as done by Pavel [21]. Internally, these offsets are then converted to the body fixed frame of reference.

50 18 Helicopter Model X hp X ba β Y hp X pa Y pa ψ X pa (a) Top view of the rotor showing the Hub Plane and the Projection Axis frames of reference β Y pa Z ba (b) The transformation from the Projection Axis to the Blade axis Figure 3.6: Rotor reference frames orientation of a blade in the hub plane Hub Plane The reference frame that is used to describe the flapping dynamics in this thesis is the hub plane frame of reference, oriented perpendicular to the rotor shaft. In recent textbooks [Padfield, 2; Leishman, 22], it is the plane of preference to derive and describe the motions of the blades in. With respect to this reference frame, the blades both have nonzero flapping and feathering. It is the most natural plane to describe the dynamics of the blade in since it is linked to a physical part of the helicopter. PSfrag Tip Path replacements Plane β 1s β 1c β 1c j tpp e 1 k tpp Y tpp β 1s Y hp Z tpp Z hp X hp Figure 3.7: Tip path plane orientation wrt the hub plane frame of reference The expression for the blade flapping in the nonrotating frame (Eq. 3.2) describes the boundaries of a plane in space through which the tips of the blades pass, named the Tip

51 3.3 Reference Frames 19 Path Plane. An observer in this plane will not see a variation in blade flapping. The nonuniform Pitt-Peters inflow and quasi-dynamic inflow models are all derived with respect to this plane. The exact transformation matrix between the hub plane and the tip path plane can be written as 6 where { Etpp } = [β1s β 1c ] { E hp } [β 1s β 1c ] = C β1c norm S β1s S β1c C β1s norm C 2 β 1s S β1c norm C β1s S β1s S β1c C β1s norm C β1c S β1s norm C β1c C β1s norm (3.12) (3.13) and norm = C 2 β 1s S 2 β 1c + C 2 β 1c (3.14) Eq is used in the simulation program to convert the aerodynamic rotor forces and moments in matrix notation from the hub plane to the tip path plane. No Feathering Plane With respect to the NFP, there is no 1/rev pitch variation, both θ 1s and θ 1c are zero. There is however a cyclic variation in β 1s and β 1c. This plane is sometimes used for performance analysis. Equivalence between Flapping and Feathering As already stated, there is an equivalence between flapping and feathering for a rotor with centrally placed flapping hinges. It is important to appreciate this equivalence since it allows one to convert an analysis in one frame of reference to another frame of reference. Imagine a rotor which only exhibits lateral flapping (β 1s ) and no feathering (θ 1s = θ 1c = ). When a blade passes the back of the rotor (ψ = ), the flapping angle with respect to the hub plane will be zero. The pitch (or feathering) angle with respect to the tip path plane equals the flapping with respect to the no feathering plane. This can be extended to the more general case where a rotor has both nonzero flapping and feathering with respect to the hub plane [Leishman, 22]:... the amount of blade pitch (feathering) with respect to the TPP is equal to the amount of blade flapping with respect to the NFP. Fore and aft (longitudinal) flapping or β 1c with respect to the NFP is equivalent to lateral feathering (pitch) with respect to the TPP See App. A.5 for the derivation and a comparison with an Euler approximation.

52 2 Helicopter Model In older literature, flapping angles are generally given with respect to the NFP instead of the HP as is done here. For the Puma, the following conversion should be used, (β 1c ) NF P = β 1c + θ 1s (3.15) (β 1s ) NF P = β 1s θ 1c (3.16) whereas for the Bo15, (β 1c ) NF P = β 1c θ 1s (3.17) (β 1s ) NF P = β 1s + θ 1c (3.18) holds true. In trimmed hover, (β 1c ) NF P and (β 1s ) NF P are both zero. 3.4 Tail Rotor The purpose of the tail rotor in normal helicopter configurations is twofold. On the one hand, the tail rotor counteracts the torque reaction of the main rotor on the fuselage. On the other hand, it provides the pilot with directional control around the top axis of the helicopter. The tail rotor is mounted near the vertical fin and must operate in an aerodynamically complex environment, influenced by the wakes of the main rotor, hub and fuselage. It must be able to produce thrust with relative wind coming from practically any direction. One such condition is sideward flight or hovering with a sidewind where the relative wind blows in the opposite direction as the induced flow of the tail rotor itself, generating a complex unsteady flow field, known as the vortex ring state. It can also occurs on the main rotor when the helicopter performs a steep or vertical descent. Fig. 3.8 show schematically what happens in the normal operating condition (3.8a), the vortex ring state (3.8b) and the windmill state (3.8c), where the rotor extracts energy from the surrounding air. In this thesis, these interactional aerodynamics are not modelled, except for the interaction with the vertical fin, which is modelled with an empirical fin blockage factor F t, relating the ratio of the vertical fin area S vf to the tail rotor area [Padfield, 2], F t = S vf πr 2 tr (3.19) The tail rotor models of both helicopters are of the pusher type, accelerating air away from the vertical fin. Through experimentation, it has been found that this configuration tends to have a higher overall efficiency [Leishman, 22]. The tail rotor is modelled as an actuator disk, in which all blade flapping dynamics is neglected. The inflow through the tail rotor disk is used as a state in the simulation program, its derivative is calculated from the difference between the blade element and Glauert thrust coefficient, multiplied by a constant. This model is called the quasi-dynamic inflow. The complete mathematical model of the tail rotor is given in Appendix B.

53 3.5 Vertical Fin and Horizontal Stabilizer 21 v y v y v i v y 2v i v y v y (a) Tail rotor in normal working state, v i > v y v y v y v i v y v y v y (b) Tail rotor in vortex ring state, v i v y v y v y 2v i v y v i v y v y (c) Tail rotor in windmill state, v i < v y Figure 3.8: Different operating conditions of the tail rotor viewed from above 3.5 Vertical Fin and Horizontal Stabilizer The vertical fin and horizontal stabilizer aid the pilot in stabilizing and controlling the helicopter in flight. The vertical fin produces a side force in forward flight which may alleviate the tail rotor thrust requirements. The horizontal stabilizer has a similar purpose, counteracting the negative stability characteristics of the fuselage and main rotor. During low speed forward flight, the aerodynamic conditions experienced by the horizontal stabilizer may vary rapidly as the helicopter increases its velocity. For the Puma in hover, the wake of the main rotor does not impinge the horizontal stabilizer. As speed is increased, the rotor wake will hit the tail rotor with a speed of approximately 1 m/s, which will suddenly generate a large downforce and therefore a nose up pitching moment. The horizontal tail plane of the Bo15 is located much more to the front, which prevents the sudden changes in aerodynamic loading since it is immersed in the main rotor wake in the lower part of the flight envelope. Although important, the influence of the wake of the main rotor on the tail surfaces is not taken into account in the simulation program. The mathematical models for both aerodynamic surfaces is given in Appendix B.

54 22 Helicopter Model 3.6 Fuselage The fuselage is the largest component of the helicopter and therefore generates a lot of drag. Blunt aft fuselages may cause the flow to separate, thereby increasing the drag and causing turbulent flow at the tail surfaces and tail rotor. Combined, the rotor hub assembly and shaft are another source of drag. In the trim analysis, the effect of two different drag models is investigated, the first one is characterised by a drag force that only depends upon the magnitude of the relative free stream velocity. The second model also depends upon the angle at which the relative free stream velocity hits the fuselage. Since most helicopter have a greater fuselage area in front of the centre of gravity, the fuselage generates a destabilizing pitch moment, for which the same model as used by Pavel [21] is used in this thesis. The mathematical model of the fuselage is given in Appendix B. 3.7 Helicopter Equations of Motion The equations of motion and the kinematic equations that are used to track the orientation and position of the helicopter in time are listed here. The fuselage is modelled as a rigid body with six degrees of freedom, which are governed by the following six differential equations 7, u = F x m g S Θ + vr wq v = F y m + g C Θ S Φ + wp ur ẇ = F z m + g C Θ S Φ + uq vp (3.2a) (3.2b) (3.2c) In Eqs. (3.2), the forces F x, F y and F z consist of the aerodynamic contributions of the various subsystems of the helicopter that were described earlier in this chapter. The angular accelerations follow from ṗ = I z(l + q r (I y I z ) + p q J xz ) + J xz (N + p q (I x I y ) q r J xz ) I x I z Jxz 2 q = M + p r (I z I x ) + J xz (r 2 p 2 ) I y ṙ = I x(n + p q (I x I y ) q r J xz ) + J xz (L + q r (I y I z ) + p q J xz ) I x I z Jxz 2 (3.21a) (3.21b) (3.21c) These { } last three equations define the rotation rates in the body fixed frame of reference Ebf. The relation between the angular velocities and the time derivatives of the Euler angles can be written in matrix notation as [Cook, 1997], S Ψ Φ C Φ Θ = CΘ CΘ p C Φ S Φ q (3.22) Φ 1 S Φ T Θ C Φ T Θ r 7 For a complete derivation, see e.g. Ruijgrok [199] or Padfield [2].

55 3.8 Simulated Helicopters 23 The velocity of the helicopter in an inertial frame of reference can be determined by using the inverse transformation matrix as defined by Eq. 3.7, ẋ u ẏ = [Ψ] 1 [Θ] 1 [Φ] 1 v (3.23) ż w 3.8 Simulated Helicopters Manufactured by Aérospatiale, the first production model of the AS 33 Puma helicopter (Fig. 3.9) flew in It is a twin-engine, all-weather, support helicopter in the 6 tonnes class. Variants of this helicopter were licence-built in South Africa, Romania and Indonesia. It is or was in service with a number of civil operators and armed forces around the world. The Eurocopter Deutschland (formerly MBB) Bo 15 (Fig. 3.1) is a twin-engine multipurpose utility helicopter in the 2.5 tonnes class. As the first light twin-engined helicopter in commercial service, it gained widespread use with as a police and trauma helicopter. Apart from their size, the biggest difference between these two helicopters lies in the design of the main rotor. The Puma features a classical four-bladed articulated main rotor whereas the Bo15 has a four-bladed hingeless main rotor with fibre-reinforced composite rotor blades, ensuring high manoeuvrability. Relevant parameters of these helicopters are listed in Appendix E. Figure 3.9: French military variant of the Aérospatiale AS 33 Puma

56 24 Helicopter Model Figure 3.1: South African civil version of the MBB Bo15

57 Chapter 4 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics 4.1 Introduction In this chapter, the basics of the dynamic inflow model of Pitt and Peters is given. The first section shortly discusses momentum theory and its first unsteady extension, which ultimately resulted in the Pitt-Peters dynamic inflow. The original model [Pitt & Peters, 1981] is derived with respect to the wind axes system, which does not allow for sideward flight. The adaptations necessary for sideward flight are also discussed here. The last section lists the formulae for the wake distortion dynamics model of Zhao [25]. The model of Zhao is included since Pitt-Peters dynamic inflow alone does not improve off-axis helicopter response to cyclic inputs. 4.2 Momentum Analysis The fundamental assumption in momentum theory is that the rotor can be idealised as an infinitesimally thin actuator disk over which a pressure difference exists. This is equivalent to saying that the rotor is made up of an infinite number of blades with zero thickness Momentum Analysis in Hover Starting from the conservation laws of fluid mechanics (conservation of mass, conservation of momentum and conservation of energy), it is possible to derive the following equation relating the thrust T produced by a hovering rotor to the induced flow v i through that rotor, T = 2ṁv i = 2(ρAv i )v i = 2ρπR 2 v 2 i (4.1) 25

58 26 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics where ṁ is the mass flow through the rotor, ρ is the air density and A is the area of the rotor disk. Making this equation nondimensional results in C T = 2ρπR2 v 2 i ρπr 2 Ω 2 R 2 = 2λ2 i (4.2) Inhere, λ i is the induced inflow ratio defined as λ i = v i ΩR (4.3) Substituting the thrust by the helicopter weight mg in Eq. (4.1) and solving for the inflow v i results in a simple relation between the magnitude of the induced velocity in hover on one side and the helicopter weight and rotor geometry on the other side, v i = mg 2ρπR 2 (4.4) For the Puma and Bo15 1, the magnitude of the induced flow in hover equates to Puma: v i = = m/s (4.5) π Bo15: v i = = 1.78 m/s (4.6) π Unsteady Momentum Theory - Dynamic Inflow The study of unsteady rotor aerodynamics can be done on two levels, the first one is locally, by modelling the unsteady aerodynamic effects found on the airfoil itself, or globally, by accounting for unsteady effects on the resulting rotor forces and moments. This last method is called dynamic inflow and is a means to take the inertia of the mass surrounding the rotor as a whole into account. Locally, unsteady aerodynamic effects manifest as a lag in the build-up of lift in response to a change in angle of attack. Its global equivalent is a lag in the build-up of rotor thrust in response to a change in (collective) pitch. Carpenter & Fridovitch [1953] extended momentum theory by including an apparent mass term which models the inertia of the air surrounding the rotor. They added an accelerated mass of air occupying 63.7% of the air mass of the circumscribed sphere of the rotor to the thrust (Eq. (4.1)), T =.637ρ 4 3 πr3 v i + 2πR 2 ρv i (4.7) In coefficient form, this equation can be written as C T =.637(4/3) Ω λ i + 2λ 2 i =.849 Ω λ i + 2λ 2 i (4.8) Linearising the latter equation around a trim condition using λ i = λ i + δλ i and C T = C T + δc T gives, C T + δc T =.849 Ω δ λ i + 2( λ 2 i + 2 λ i δλ i + (δλ i ) 2 ) (4.9) 1 Using the parameters as listed in Appendix E

59 4.2 Momentum Analysis 27 By substituting Eq. (4.2) in Eq. (4.9) and neglecting higher order terms (O(δλ i ) 2 ) leads to ( ) ( ).849 δ 4Ω λ λ 1 i + δλ i = δc T (4.1) i 4 λ i This is a first-order differential equation in δλ i with a time constant τ λ of τ λ =.849 (4.11) 4 λ i Ω For a typical main rotor, τ λ is of the order of.1 seconds. For the main rotors of the Puma and Bo15, the value of this time constant in hover equals.14 seconds and.1 seconds, respectively. For both helicopters, Eq. (4.11) can also be used to calculate the inflow time constants for the tail rotors in hover. This time, the induced inflow values are not calculated analytically. Instead, the values below are a result of the trim routine discussed in Chapter 5 2. The (dimensional) induced inflow values for the Puma and the Bo15 helicopter are m/s and m/s, respectively. These are made nondimensional by dividing them by the tail rotor tip speed Ω tr R tr. Using Eq. (4.11) and substituting the parameters with the values listed in Appendix E, the tail rotor inflow time constant for the Puma and Bo15 are respectively.24 second and.14 second. Because these values are significantly smaller than the time constants for the main rotors, a direct comparison is not possible. Instead, one may compare the distance a blade covers in these time periods. They are all listed in Table 4.1 together with the time constants itself and the inflow velocities for the main and tail rotors of both helicopters. Puma Bo15 Main Rotor Tail Rotor Main Rotor Tail Rotor Dimension v i [m/s] ΩR [rad m/s] λ i [-] τ λ [s] revs in τ λ [-] Table 4.1: Inflow time constants for the main and tail rotors of the Puma and Bo15 It follows for all rotors that the distances the blades cover in the time period τ λ are approximately the same, they vary between half a revolution and three-quarter of a revolution. The values for the time constants presented here are used in the quasi-dynamic inflow models in the time simulations. The idea presented in this section has been extended for a first order nonuniform inflow model by Pitt & Peters [1981]. The basics of the resulting model is discussed in the following sections. 2 An analytical calculation for the tail rotor thrust and induced inflow as can be found in [van Holten & Melkert, 1994] is not extremely accurate, the helicopter lateral trim equilibrium in hover is more complex than what is depicted there.

60 28 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics 4.3 Dynamic Inflow - Coordinate System and Definitions This section deals with the coordinate systems and some definitions which are specifically needed for the dynamic inflow model Tip path plane velocity components The coordinate system of the tip path plane 3 is depicted in Fig The three nondimensional relative velocity components in the three axes of this coordinate system are µ x, µ y and µ z. The relative velocity in the tip path plane is called the advance ratio µ ar or in-plane velocity defined as µ ar = µ 2 x + µ2 y (4.12) The velocity component perpendicular to the tip path plane (µ z ) is called the out-of- C Ltpp X tpp λ(ψ, r) wa C T µ x C lwa µ z = µ op µ y µ ar X wa C mwa C Mtpp Y tpp Z tpp = Z wa Y wa Figure 4.1: Velocity definitions in the tip path plane and wind axes system plane velocity µ op. Together, they define the orientation of the wind axes system with respect to the tip path plane system. The nondimensional inflow through the rotor is defined perpendicular to the tip path plane and given by r λ(r, ψ) wa = λ + λ 1s R sin ψ + λ r 1c cos ψ (4.13) R where ψ is the azimuth angle measured counterclockwise starting from the negative X-axis of the wind axes system. At the centre of the tip path plane, the total inflow is equal to λ, positive down. It follows that the total relative velocity at that point, the inflow ratio µ ir, equates to µ ir = λ µ op (4.14) 3 The transformation matrices needed to convert to and from the tip path plane frame of reference are derived in Appendix A

61 4.3 Dynamic Inflow - Coordinate System and Definitions 29 The steady wake skew angle χ is calculated as follows ( ) ( ) χ = tan 1 µip = tan 1 µip λ µ op µ ir (4.15) Nonlinear Dynamic Inflow The induced velocity is assumed to have the form of Eq. (4.13) in the wind axes system, where λ, λ 1s and λ 1c are the uniform, lateral and longitudinal components of the inflow, respectively. The time histories of these inflow states are governed by the following firstorder differential equation [Zhao, 25], λ [M] λ 1 1s + [V ][ L] λ 1c λ λ 1s λ 1c C T = C L C M wa (4.16) The apparent mass matrix [M], the mass flow parameter matrix [V ] and the inflow gain matrix [L], respectively, can be written as π [M] = 16 45π (4.17) 16 45π V m [V ] = V (4.18) V and [ L] = π 64 X 2(1 + X 2 ) 15π 64 X 2(1 X2 ) where X is the wake skew [Zhao, 25] defined as (4.19) X = tan χ 2 (4.2) The mass flow parameter V m is associated with the mean inflow through the rotor tip path plane and can be written as V m = µ 2 ar + µ2 ir (4.21) V is associated with the first order harmonics of the inflow distribution, V = µ2 ar + µ ir (µ ir + λ ) V m (4.22)

62 3 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics For the simulation program, Eq. (4.16) is written in the following form and added to the equations of state, λ λ 1s = [M] 1 C T λ 1 C L [V ][ L] λ 1s (4.23) C M λ 1c λ 1c 4.4 Dynamic Inflow - Adaptation for Sideward Flight The dynamic inflow model as outlined in the previous section is defined in the wind-axis frame of reference. As a consequence, it is not possible to define sideward flight, since the X-axis of the wind axes system always points in the direction of the relative wind. The adaptations given here resemble the ones derived by Peters & HaQuang [1988], the main difference being the choice of the positive directions of the wind-axis system and hub axis system. Since the wake skew angle χ is calculated in the XZ-plane of the wind axes system, all parameters in the gain matrix [L] (Eq. (4.19)) and flow matrix [V ] (Eq. (4.18)) are automatically calculated in the wind-axis system. If we now assume that the inflow distribution given by Eq. (4.13) is also given in the wind-axis system, the only remaining thing to do is convert the forcing moments from the tip path plane frame of reference to the wind-axis frame of reference. Again referring to Fig. 4.1, the angle 4 defines the orientation of the wind-axis system with respect to the tip path plane system and is calculated as ( ) = tan 1 µy (4.24) µ x Then, the transformation matrix [ ] is 1 [ ] = C S (4.25) S C and C T C L C M wa = [ ] C T C L C M tpp (4.26) The azimuthal position ψ of a blade is given with respect to the hub plane frame of reference. If this angle is directly plugged into Eq. (4.13) to get the value of the inflow at a particular blade element of the blade in question, an error is made. One has to correct for the angle to take sideward flight into account, r λ(r, ψ) = λ + λ 1s R sin(ψ + ) + λ r 1c cos(ψ + ) (4.27) R This way, the model is also valid for sideward flight. 4 One could define this angle as the angle of sideslip at the tip path plane.

63 4.5 Pitt-Peters Non-Uniform Dynamic Inflow for an Analytical Main Rotor Pitt-Peters Non-Uniform Dynamic Inflow for an Analytical Main Rotor Extending an analytically derived main rotor model such as described in Ypma [1996] and Pavel [21] with Pitt-Peters non-uniform dynamic inflow can be done as follows. The right-hand side of Eq contains three aerodynamic forcing functions generated by the main rotor. These can be calculated analytically by nondimensionalising the following equations [Leishman, 22] T mr = N bl 2π M xmr = N bl 2π M ymr = N bl 2π 2π R 2π R 2π R df z dψdr (4.28) ydf z sin ψdψdr (4.29) ydf z cos ψdψdr (4.3) where df z is the force perpendicular to the rotor disk (Leishman), df z = 1 2 ρcc l α (θu 2 T U P U T )dy (4.31) the terms U T and U P are both a function of the relative velocity with respect to the blade segment. The three inflow terms appear in the equation for the perpendicular velocity U P. 4.6 Wake Distortion Dynamics Introduction The development of flight dynamic models for helicopters has matured to the point where the prediction of the primary response of single rotor helicopters to small control inputs, or the on-axis response, is fairly well established. However, accurate prediction of the off-axis response is still problematic. This discrepancy has been shown to exist for a wide range of helicopter, which has led to the belief that their is some fundamental phenomena missing or not treated correctly. Several attempts have been undertaken to explain this off-axis response discrepancy, most of them can be classified as - gyroscopic forces due to the angular momentum of the rotor wake, - aerodynamic interaction between the rotor and the fuselage, - dynamic twisting due to gyroscopic feathering moment, - wake distortion/curvature effect. A key source of the off-axis response sign reversal arises from the interaction between rotor flapping dynamics and rotor inflow dynamics during manoeuvring flight. A pitch or roll manoeuvre or a combination of both will cause additional distortion to the wake

64 32 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics geometry. This distortion will in turn change the induced inflow across the rotor disk which affects the blade loads, rotor blade dynamic flapping response and ultimately the vehicle response characteristics. To account for this discrepancy, the Pitt-Peters dynamic inflow model is augmented with the dynamic wake distortion model developed by Zhao [25]. Here, only the basic background and the resulting formulae of the model will be given. The interested reader should consult Zhao for a more in-depth discussion Dynamic Wake Distortion Model Zhao uses a vortex tube method to determine the influence of distortions in the geometry of the wake at the rotor disk. In the vortex tube method, the shed tip vortices are assumed to all lie on a tube of continuous vorticity, which represents the outer surface of the wake. The Biot-Savart law is then used to obtain the inflow at the rotor disk. The effects of three different disturbances (step changes in wake bending, wake skew and wake spacing) are examined and it is concluded that they all exhibit a first order behaviour with time. The effect of dynamic wake distortions on the inflow across the rotor disk can then be represented by a set of first-order differential equations as Ẋ X X Ṡ S S [τ D ] + = (4.32) κ c κ c κ c κ s κ s κ s qs where X, S, κ c and κ s are wake skew, wake spacing, longitudinal and lateral wake curvatures, respectively, subscript qs denotes quasi-steady values. Matrix [τ D ] contains the nondimensional time constants associated with the dynamic wake distortion effects. In general, this matrix is fully populated, which means that wake skew, wake spacing and wake curvatures are fully coupled. Here, the coupling effects are neglected which results in the following diagonal form for the time constant matrix, τ X [τ D ] = τ S τ R (4.33) τ R The right-hand side of Eq corresponds to the quasi-steady wake skew, wake spacing, longitudinal and lateral curvatures, given by (X) qs = tan χ 2 (S) qs = 2πV m (κ c ) qs = q β 1c λ + V c (κ s ) qs = p β 1s λ + V c (4.34a) (4.34b) (4.34c) (4.34d)

65 4.6 Wake Distortion Dynamics 33 In the above equations, p and q denote the nondimensional roll and pitch rates. β 1s are the nondimensional rotor disk longitudinal and lateral flapping rates 5. β 1c and The states of this system are coupled with the Pitt-Peters dynamic inflow model through some additional gain matrices. This will be discussed in the next section Augmented Pitt-Peters Dynamic Inflow Model The inflow gain matrix [ L] of Eq is augmented with an additional gain matrix [ L] which is further decomposed in [ L] = [ L 1 ] + [ L 2 ] + [ L 3 ] (4.35) Inhere, [ L 1 ] denotes the coupling between wake curvature and rotor mean loading. [ L 2 ] accounts for the coupling effect between wake curvature, wake skew and rotor mean loading, the last gain matrix, [ L 3 ], represents the coupling effect between the wake skew, wake curvature and the cyclic loading. They are written as follows, [ L 1 ] = K Re κ s 2 (4.36) κ c 2 [ L 2 ] = K Re 3 4 κ sx 2 (4.37) 3 4 κ cx 2 and where [ L 3 ] = K Re 5 4 µκ cx l κ sx (4.38) 5 4 µκ sx l κ cx l 22 = 5 2 κ cx 3µ 2 κ s( X2 ) (4.39) l 32 = 5 2 κ sx 3µ 2 κ c(1 3 2 X2 ) (4.4) In Eqs. (4.36) to (4.38), the parameter K Re is an (empirical) wake curvature parameter. In literature, its value varies between 1. and 2., depending on the method used to derive the wake distortion dynamics model. In this thesis, a value of 1.5 is used. The complete inflow gain matrix [ L] can then be written as [ L] = [ L + L] = [ L + L1 + L2 + L3] (4.41) The resulting complete inflow gain matrix [ L] cannot be inverted easily, so in the simulation program, this is done numerically with an LU-decomposition. 5 Nondimensionalised by Ω, i.e. β 1s = β 1s Ω, p = p Ω,...

66 34 Pitt-Peters Dynamic Inflow and Wake Distortion Dynamics

67 Chapter 5 Helicopter Trim 5.1 Introduction The trim condition of an aircraft or helicopter is a combination of states and control inputs for which the forces and moments are in equilibrium and at the same time satisfy a given flight condition. This chapter deals with the algorithm that calculates the trim condition for the helicopter simulation. Starting from the equations for the most general steady motion, the kinematic constraints for straight, trimmed flight are derived. The kinematic constraints will then be simplified to the flight condition where the projections on the horizontal plane of both the airspeed vector and the X-axis of the body fixed frame of reference coincide. After that, the theoretical background and practical implementation of the actual trim routine is discussed. Lastly, trim results are presented for a variety of flight conditions and compared with results found in literature. 5.2 The Most General Steady Motion Steady motion requires that the time derivatives of all state variables involved in the equations of motion, Eqs. (3.2) and (3.21), are zero (i.e., u = v = ẇ = ; ṗ = q = ṙ = ; and Θ = Φ = ). This condition leads to the following set of equilibrium equations, X = mg S Θ + m(wq vr) (5.1) Y = mg C Θ S Φ + m(ur wp) (5.2) Z = mg C Θ C Φ + m(vp uq) (5.3) L = I xz pq (I y I z )qr (5.4) M = I xz (r 2 p 2 ) (I z I x )rp (5.5) N = I xz qr (I x I y )pq (5.6) It follows that only the angular velocity Ψ may have a nonzero value. Then, the resultant angular velocity of the airplane is around an earth-fixed vertical axis. The kinematic 35

68 36 Helicopter Trim relations for the body-fixed angular velocities reduce to p = Ψ S Θ q = Ψ C Θ S Φ r = Ψ C Θ C Φ (5.7) (5.8) (5.9) The complete angular velocity Ω then equates to Ψ 1. In other words, both the direction and the magnitude of the resultant angular velocity vector stay the same. In terms of the Euler angles defining the air path axis [see Ruijgrok, 199, Chap. 3], we get Ω = Ẋ and γ = µ = (5.1) Therefore, the flight path angle γ and the aerodynamic angle of roll µ are also independent of time. As a further consequence, the same holds true for the fuselage angle of attack and angle of sideslip, α and β, since at a given velocity the aerodynamic force solely depends on the attitude of the airplane relative to the velocity vector and this orientation is fully determined by these angles. Another observation which follows from the above is that Ψ, namely Ẋ has the same magnitude as Ψ = Ẋ = Ω (5.11) So, during trim, the difference between Ψ and Ẋ is zero. As a consequence, the difference between Ψ and X is a (nonzero) constant. This difference, named χ, is used as an extra parameter defining the track angle. In Fig. 5.1, a graphical representation of the relationship between the angles Θ, Φ, α, β, γ, χ, X and Ψ in trimmed flight is given, with all angles drawn on a sphere with radius V. X e and Z e are two of the three axes of the moving Earth axis system or local horizon system. X cg is the x-axis of the centre of gravity frame of reference of the helicopter. Now that the proper relation between the flight path frame of reference and the reference frame that defines the orientation of the helicopter is established, the airspeed vector is transformed to the latter by V Ψ = (V C γ C χ, V C γ S χ, V S γ ) {E Ψ } (5.12) With two more Euler transformations over the pitch angle Θ and the roll angle Φ, the velocity components in the centre of gravity frame of reference become u = V ( C Θ C γ C χ + S Θ S γ ) v = V ( S Φ ( S Θ C γ C χ C Θ S γ ) + C Φ C γ S χ ) w = V ( C Φ ( S Θ C γ C χ C Θ S γ ) S Φ C γ S χ ) (5.13a) (5.13b) (5.13c) 1 Ω p p 2 + q 2 + r 2 = Ψ

69 PSfrag5.3 replacements Helicopter Trim in Straight Flight 37 α X cg β Θ Φ Z e V γ X a X χ X Ψ Ψ X e Figure 5.1: Trimmed flight with non-zero flight track angle χ 5.3 Helicopter Trim in Straight Flight Kinematic Constraints In steady, straight, climbing flight, the helicopter state can be prescribed by a certain airspeed V, a flight path angle γ (positive up), the altitude h and the flight track angle χ. The angular velocities p, q and r are zero p = q = r = (5.14a) (5.14b) (5.14c) which together with Eq. (5.13) form the kinematic constraints for this flight condition. In [Voorsluijs, 23], a similar approach is taken as the one described here, the only difference is that Voorsluijs implicitly sets the flight track angle χ equal to zero. The resulting kinematic relationships between the angles Θ, Φ, α, β, γ, and Ψ are depicted in Fig. 5.2, which resembles Fig. 5.1, only here the angle χ is set to zero. This is the trim condition for conventional forward flight. By examining this figure, it is clear that when the sideslip angle β is zero, the roll angle Φ of the helicopter must also be zero and Θ = α + γ Trim Routine - Theory The first order Newton iteration method is used to find the trim values of the states and controls. The method is illustrated here for the case of two functions with two variables. Suppose we have a system of two nonlinear differential equations ẋ 1 = f(x 1, x 2 ) (5.15) ẋ 2 = g(x 1, x 2 ) (5.16)

70 38 Helicopter Trim X cg Θ α Φ V β X e Z e γ X a Ψ = X X Ψ Figure 5.2: Trimmed flight without flight track angle χ We are interested in finding the roots or solution z of this system, i.e. the values for x i for which the following condition holds, f(x 1, x 2 ) = (5.17) g(x 1, x 2 ) = (5.18) Using Taylor s series expansion of the two functions near (x 1, x 2 ) we find f(x 1 + h, x 2 + k) = f(x 1, x 2 ) + h f x 1 + k f x 2 + O(h 2 + k 2 ) g(x 1 + h, x 2 + k) = g(x 1, x 2 ) + h g x 1 + k g x 2 + O(h 2 + k 2 ) and if we only keep the first order terms, we are looking for a couple (h, k) such that (5.19) f(x 1 + h, x 2 + k) = f(x 1, x 2 ) + h f + k f x 1 x 1 g(x 1 + h, x 2 + k) = g(x 1, x 2 ) + h g + k g (5.2) x 1 x 1 The right-hand of Eq. (5.2) can be written in matrix form as ( ) ( ) ( ) f f x 1 x 2 h f(x1, x 2 ) = k g(x 1, x 2 ) g x 1 g x 2 (5.21) The two-by-two matrix is called the Jacobian matrix and contains the partial derivatives of the functions f and g and is normally written in shorthand notation as J(x 1, x 2 ). If we now define the iterative process ( ) ( ) ( ) x 1n+1 x 1n = J 1 f(x 1n, x 2n ) (x 1n, x 2n ) (5.22) g(x 1n, x 2n ) x 2n+1 x 2n Starting from an initial guess (x, y ), this process will converge to the roots of the system if the initial conditions are chosen sufficiently close to the final values.

71 5.3 Helicopter Trim in Straight Flight 39 For a trim problem, the solution criterion z is equal to zero. The trim variable vector v contains the smallest number of state variables (x 1 and x 2 in the above example) that can be chosen freely such that all state variables of the system can be determined. The iterative trim process as illustrated above is implemented in a nested way, one top level trim routine drives the rigid body accelerations to zero and two subroutines, one for the main rotor and the other for the tail rotor. The division between rotors and fuselage stems from the fact that under normal operating conditions, both rotors can be trimmed for a given combination of fuselage states and controls. Top Level Helicopter Trim As already stated in section 5.3.1, in trimmed flight, the angular velocities must be zero and the linear velocities depend on the desired flight (i.e. V, γ, χ and h) condition and the pitch and roll angles Θ and Φ, respectively. This leaves the four pilot controls θ, θ 1s, θ 1c and θ tr free to vary. The trim variable vector v heli for the top level (fuselage) trim routine then becomes v heli = (Θ Φ θ θ 1s θ 1c θ tr ) T (5.23) The solution criterion vector z is based on the rigid body accelerations, z heli = ( u v ẇ ṗ q ṙ) T (5.24) For straight trimmed flight, the solution criterion simply is z heli = ( ) T (5.25) For stability reasons and to improve convergence accuracy of the fuselage velocity derivatives, an extra variable was added to the top level trim routine. The tail rotor uniform inflow was added to the above trim variable vector, and the solution criterion vector was expanded with its derivative, the tail rotor uniform inflow derivative. Main Rotor Trim The main rotor trim routine drives all nine main rotor state derivatives to zero such that the top level trim routine only sees the steady-state behaviour of the main rotor. With the non-uniform variant of the Pitt-Peters dynamic inflow model, the trim variable vector v mr is defined as v mr = (λ λ 1s λ 1c β β 1s β 1c β β1s β1c ) T (5.26) and the solution criterion vector z mr consists of their respective derivatives z mr = ( λ λ1s λ1c β β1s β1c β β1s β1c ) T (5.27) In a steady condition, all main rotor derivatives must be zero, z mr = ( ) T (5.28)

72 4 Helicopter Trim In the case of uniform main rotor inflow, λ 1s and λ 1c and their time derivatives are left out of the trim variable vector and the solution vector. The inclusion of the wake distortion dynamics model of Zhao [25] in the mathematical model of the helicopter rotor has no influence on the trim results. However, two of its states (wake spacing X and wake skew S) have nonzero values in trim, so if wake distortion dynamics is included, they are added to the vector v mr. The solution vector z mr is then augmented with the derivatives Ẋ and Ṡ. It should be noted that the influence of ṗ, q and ṙ on the flapping derivatives was not included in the trim routine from the start. This has a small influence on the accuracy of the trim results, which will be discussed in Appendix C.2. Tail Rotor Trim The dynamics of the tail rotor inflow is modeled using the uniform component of the Pitt-Peters dynamic inflow λ T R or quasi-dynamic inflow, so the trim variable v tr is v tr = λ T R (5.29) The solution variable z tr is the time derivative of the uniform inflow, z tr = λ T R (5.3) which must be zero. 5.4 Trim Results Here, the trim results for the Puma helicopter are discussed. First, conventional forward flight is discussed for airspeeds from hover up to the never-exceed speed of the Puma, which is 15 kts or approximately 8 m/s. In the section thereafter, trim results in sideward flight are discussed. A short discussion on the influence of the azimuthal blade position on the trim results is also given Forward Flight Fuselage Attitude and Pilot Controls In Figs. 5.3 to 5.8, the trim values of the top level trim routine variables (fuselage pitch and roll angles Θ and Φ and pilot controls θ, θ 1s, θ 1c and θ T R ) are shown for sea level forward flight. They are compared with trim data generated at the Faculty of Aerospace Engineering with both an analytical [Pavel, 1996] and a numerical [Voorsluijs, 23] helicopter model. Furthermore, data from a simulation program developed at the Bureau of Mechanical Engineering in Stellenbosch and trim measurements performed by the South African Air Force were obtained through Voorsluijs. The two numerical blade element simulation models MoGeHM and Helix shows good correlation below 5 m/s. The difference between these two models becomes bigger above

73 5.4 Trim Results 41 this airspeed. This is in part due to the use of different main rotor aerodynamics models 2. But there are also differences in the formulation of the rotor geometry. Helix uses a rotor with centrally located flapping hinges and as a consequence, there is no resulting moment at the rotor hub due to forces on the rotor blades. The only moment that is generated at the flapping hinge is due to the equivalent spring stiffness. MoGeHM on the other hand, uses a rotor with hinge offsets, which does allow for resulting moments at the centre of the hub. The inclusion of the linear velocities in the calculation of the total kinetic energy of the main rotor blades may also reduce the gap between the two models at high speed forward flight. The most striking difference for the fuselage pitch angle Θ (see Fig. 5.3) is the difference between the analytical model and the other models. This is due to the fact that the analytical model does not include a shaft tilt. The fuselage roll angle Φ (see Fig. 5.4) is overestimated by all simulations with respect to the SAAF measurements. But since these measurements are fairly scattered, it is difficult to say how accurate they are. Throughout the entire velocity range, all simulations display the same trends for the pitch angles of the main rotor and tail rotor. For the main rotor collective pitch angle θ (Fig. 5.5), the differences between the analytical model and the two numerical blade element models is mainly due to the inclusion of blade twist in the latter two. For the longitudinal cyclic pitch θ 1s displayed in Fig. 5.6, all simulation data coincides fairly well, but there is a substantial difference with the SAAF measurements. The exact reason for this error is not known, but it could be due to the place where the pilot controls were measured. In the ideal case, all control angles are measured directly at the blade pitch bearings. If they are measured close to the pilot itself (as is the case with Bo15 flight test data generated by DLR), errors could be introduced when converting pilot stick displacements to pitch bearing angles. The simulation result for the lateral cyclic pitch θ 1c (Fig. 5.7) all show the typical bump at around 15-2 m/s. However, this feature is not clearly visible in the SAAF data. Lastly, the values for the tail rotor collective pitch angle θ tr (see Fig. 5.8) agree well with each other at low speeds. The differences between the results produced with the simulation models and the SAAF measurements at high speed could be due to complex interaction between the main rotor wake and the tail rotor. Main Rotor States The trim values of the main rotor states are shown in Fig The uniform inflow, depicted in Fig. 5.9b, shows the dimensional form of the typical curve that is obtained with momentum analysis [Leishman, 22, Chap. 2]. In hover, the induced flow through the rotor is uniform. As the airspeed increases, the longitudinal inflow component increases sharply until it reaches a maximum value of approximately 7 m/s at an airspeed of 15 m/s (advance ratio µ =.7). After this point, it shows the same trend as the uniform inflow component. At speeds above 4 m/s, the magnitude of the longitudinal inflow component 2 MoGeHM uses a steady-state model with Mach correction whereas Helix uses a steady-state model without Mach correction

74 42 Helicopter Trim Helix trim MoGeHM trim 6 dof analyt. trim Helper SAAF measurements Θ [deg] V [m/s] Figure 5.3: Fuselage pitch angle in trimmed forward flight 6 Helix trim 5 MoGeHM trim 6 dof analyt. trim 4 Helper SAAF measurements Φ [deg] V [m/s] Figure 5.4: Fuselage roll angle in trimmed forward flight

75 5.4 Trim Results Helix trim MoGeHM trim 6 dof analyt. trim Helper SAAF measurements θ [deg] V [m/s] Figure 5.5: Main rotor collective pitch angle in trimmed forward flight Helix trim MoGeHM trim 6 dof analyt. trim Helper SAAF measurements θ1s [deg] V [m/s] Figure 5.6: Main rotor longitudinal cyclic in trimmed forward flight

76 44 Helicopter Trim 5 Helix trim 4 MoGeHM trim 6 dof analyt. trim 3 Helper SAAF measurements θ1c [deg] V [m/s] Figure 5.7: Main rotor lateral cyclic in trimmed forward flight Helix trim MoGeHM trim 6 dof analyt. trim Helper SAAF measurements θtr [deg] V [m/s] Figure 5.8: Tail rotor collective pitch in trimmed forward flight

77 5.4 Trim Results 45 (Fig. 5.9d) is larger than the magnitude of the uniform component, which means that at the front most part of the rotor, the induced flow will be upward β [deg] λ [m/s] V [m/s] (a) Main rotor coning V [m/s] (b) Main rotor uniform inflow β1c [deg] -1 λ1c [m/s] V [m/s] (c) Main rotor longitudinal tilt V [m/s] (d) Main rotor longitudinal inflow.1 β1s [deg].5 λ1s [m/s] V [m/s] (e) Main rotor lateral tilt V [m/s] (f) Main rotor lateral inflow Figure 5.9: Main rotor inflow and flapping angles in trimmed forward flight At first sight, the predictions of the lateral and longitudinal tilt (Figs. 5.9e and 5.9c) angles of the main rotor are completely different from those found in Harris [1972]. The lateral tilt angle β 1s should show the same bump as the lateral cyclic pitch angle θ 1c. The difference is due to a different choice of reference frames. In this thesis, all flapping angles in the rotating and nonrotating frames of reference are measured with respect to the hub plane. In Harris [1972], the blade angles are measured with respect to the NFP, normally used for performance analysis. The transformation between these two reference

78 46 Helicopter Trim frames is defined for the Puma as (β 1c ) NF P = β 1c θ 1s (5.31) (β 1s ) NF P = β 1s + θ 1c (5.32) For hover, the values of both (β 1c ) NF P and (β 1s ) NF P should be zero. In Fig. 5.1, the resulting tilt angles are shown for uniform and nonuniform inflow. In contrast to the lateral tilt angle (Fig. 5.1b), the longitudinal tilt angles are not influenced by the inflow model (Fig. 5.1c). Since nonuniform Pitt-Peters inflow mainly adds a longitudinal inflow component in trim, changes in rotor trim response can be expected to occur laterally due to the 9 degree phase lag of the rotor. Uniform inflow does not add any lateral excursion to the tilt, so it may be concluded that nonuniform inflow improves the prediction of the lateral tilt angle. Nonuniform inflow Uniform inflow 2 3 (β1c)nf P [deg] (β1s)nf P [deg] V [m/s] (b) Main rotor longitudinal tilt wrt NFP V [m/s] (c) Main rotor lateral tilt wrt NFP Figure 5.1: Main rotor tilt angles with respect to the no feathering plane for uniform and nonuniform inflow Sideward Flight The trim algorithm that was discussed earlier is capable of trimming the helicopter in any straight flight condition, including climbing and sideward flight. In this section, the results generated for sideward trimmed flight will be discussed. The quality of the trim results in sideward (and backward) flight depend for a large part on the aerodynamic models of the tail surfaces. At first, the lift coefficient was related in a linear way to the angle of attack. This is valid for small angles of attack or sideslip (say 2 (α hs, β vf ) 2 ), but at angles beyond this range, stall occurs so that the lift coefficient can no longer be calculated using a simple, linear relation. In sideward flight, the result then is that the lift and drag coefficient of the vertical stabilizer increase quadratic with airspeed and are in fact unbounded. The collective pitch angle of the tail rotor reacted in a similar quadratic way,

79 5.4 Trim Results 47 increasing rapidly to -9 degrees and beyond 3. Using the same large angle approximation as for the main rotor blade segments, this unrealistic behaviour was solved. The resulting fuselage attitudes and pilot control settings are shown in Fig for sideward flight (to the right, χ = 9 ) for speeds from to 4 m/s. In App. B, two different fuselage drag models are derived, these are compared here. The second drag model (white dots) gives the highest drag in sideward flight. As a result, the roll angle will increase more rapid with airspeed (Fig. 5.11c). The increased drag is mainly equalised by the main rotor thrust, which means that the collective pitch will increase considerably for this model, as can be seen from Fig. 5.11d. For both models, it shows the same behaviour as in forward flight. The lateral cyclic pitch θ 1c (Fig. 5.11f) in sideward flight shows the exact same behaviour as the longitudinal cyclic θ 1s in forward flight. The tail rotor collective pitch (Fig. 5.11g) shows a discontinuity when the airspeed is approximately 26.5 m/s. In Chapter 3, the vortex ring state phenomenon was briefly mentioned. At this lateral speed, the tail rotor enters the vortex ring state since the tail rotor induced flow is approximately equal to 26.5 m/s. The jump in the control settings are a result of the mathematical model of the tail rotor, which does not take this phenomenon into account. It is therefore advisable to avoid flight conditions with the simulation where this occurs, since it may result in unrealistic helicopter responses with discontinuities Influence of First Blade Azimuth Position Due to the numerical implementation of the main rotor, the trim solution depends on the position of the main rotor blades. Fig shows what happens with the trim values of the pilot controls when the main rotor is trimmed for multiple azimuth positions ( tot 9 degrees). As can be seen from this figure, after 9 degrees, the exact same trim values are found for the controls. This is due to the number of blades (4) of the Puma helicopter and gives rise to a small but persistent undamped vibration in free flight. The amplitude of this vibration is near zero in hover and enlarges with flight speed. In Appendix F, a filter is designed to suppress this vibration before sending the state signals to the mathematical pilot Inverted Trim As already stated in section 5.3.2, the numerical Newton iteration method needs an initial guess. If this initial guess is chosen sufficiently close to the trim value, the Newton iteration method will converge. Normally, the initial values for the fuselage attitudes and the pilot controls have values within the limits of normal trim. However, if the initial fuselage roll angle Φ is set to 18 degrees, the resulting fuselage attitude will be upside down (for speeds below 7.5 m/s, no trim solution is found at all). 3 Note that in sideward flight (χ = 9 ), the angle of sideslip β vf is approximately 9 degrees, the vertical stabilizer drag points to the left. This large lateral force is mainly balanced by the tail rotor thrust which requires large negative pitch angles.

80 48 Helicopter Trim Fuselage model 2 Fuselage model Θ [deg] 2-2 Φ [deg] V [m/s] (b) Fuselage pitch PSfrag in sideward replacements flight V [m/s] (c) Fuselage roll in sideward flight -2 θ [deg] θ1s [deg] V [m/s] (d) Main rotor collective pitch in sideward flight V [m/s] (e) Main rotor longitudinal cyclic pitch in sideward flight 1 θ1c [deg] θtr [deg] V [m/s] (f) Main rotor lateral cyclic pitch in sideward flight V [m/s] (g) Tail rotor collective pitch in sideward flight Figure 5.11: Top level trim results in sideward flight (χ = 9, h = 12m), showing a discontinuity in the tail rotor collective pitch θ tr, which is a consequence of the tail rotor entering the vortex ring state. This is not completely surprising, since in both the normal trim state and the inverted trim state, the rotor disc is more or less parallel to the earth surface and as a consequence, the rotor thrust opposes the weight of the helicopter.

81 5.4 Trim Results 49 θ [deg] θ1s [deg] θ1c [deg] θtr [deg] ψ 1 [deg] Figure 5.12: Influence of first blade azimuth position ψ 1 on values of trimmed controls at V = 6[m/s]

82 5 Helicopter Trim

83 Chapter 6 Linearisation 6.1 Introduction By using small perturbation theory, the helicopter simulation model can be linearised around a previously trimmed flight condition. The helicopter state x can then be described as a perturbation δx from the trim value x t, x = x t + δx (6.1) which makes it possible to study the characteristic motions of the helicopter using standard control theory. In this chapter, the implementation details and results of the numerical linearisation routine are discussed. By using two slightly different algorithms for the linearisation routine, it is demonstrated that there is a significant difference between the initial response of the helicopter to a control input and the steady-state response of the helicopter. The first one emulates a steady-state main and tail rotor by trimming the rotors at the disturbed flight condition before summing the resulting forces and moments of the different helicopter components ((1) in Fig. 6.1). The second method skips this step and directly calculates the rotor forces and moments without trimming them first ((2) in Fig. 6.1). Furthermore, the influence of Pitt-Peters dynamic inflow on the steady-state dynamics of the helicopter is investigated by comparing the linearisation results with those obtained with uniform inflow. 6.2 Linearisation Routine This section discusses the linearisation routine as used in the simulation program to extract the reduced order linear model. It is based on the linearisation algorithm described in [Voorsluijs, 23] Theory The general structure of a linear helicopter model can be written as ẋ Ax = Bu (6.2) 51

84 52 Linearisation Main Rotor 2 Trimmed States & Disturbances Tail Rotor 1 2 Trim Σ Disturbed Forces & Moments FUS VF HS Stability Derivatives Nondimensionalisation & Trim Corrections Figure 6.1: Flow diagram of the linearisation routine, with (1) the steady-state rotor emulation and (2) the unsteady rotors To be able to study the decoupled longitudinal and lateral dynamics, the following order is used for the state vector x [Padfield, 2], x = (u w q Θ v p Φ r Ψ) T (6.3) Inhere u, v and w are the translational velocities in the body fixed frame of reference, p, q and r are the angular velocities and Ψ, Θ and Φ are the three Euler angles defining the orientation of the helicoper in an inertial frame of reference. Since a flat-earth approximation is used for the mathematical model, the heading angle Ψ will not influence the stability derivatives. However, it is included here since it could be of use for controller design [Hess, 25] or inverse simulations [Bradley & Thomson, 25]. The first four variables in Eq. (6.3) represent the longitudinal fuselage dynamics and the four subsequent variables represent the lateral fuselage dynamics of the helicopter. The control vector u consists of the four pilot controls, u = (θ θ 1s θ 1c θ tr ) T (6.4) The state matrix A in Eq. (6.2) can be written as X u X w X q w t g C Θt X v X p X r + v t Z u Z w Z q + u t g C Φt S Θt Z v Z p v t g S Φt C Θt Z r M u M w M q M v M p M r C Φt S Φt A = Y u Y w Y q g S Φt S Θt Y v Y p + w t g C Φt C Θt Y r u t L u L w L q L v L p L r S Φt T Θt 1 C Φt T Θt N u N w N q N v N p N r S Φt / C Θt C Φt / C Θt

85 6.2 Linearisation Routine 53 inhere, the subscript stands for trim. The control matrix B can be expressed as X θ X θ1s X θ1c X θtr Z θ Z θ1s Z θ1c Z θtr M θ M θ1s M θ1c M θtr B = Y θ Y θ1s Y θ1c Y θtr L θ L θ1s L θ1c L θtr N θ N θ1s N θ1c N θtr (6.5) (6.6) As already stated, the stability and control derivatives are calculated with a routine comparable to the routine used to trim the helicopter. The forces and moments y generated by the five helicopter components are symmetrically perturbed by each of the states x i using [see van Kan, 2, Chap. 2], y i+1 y i 1 2h = dy dx + O(h 2 ) (6.7) xi where h is the magnitude of the perturbation. To ensure the best overall linearisation, the linear velocities are given a perturbation of 1 m/s while controls, attitudes and angular velocities are given a perturbation of the order of.1 rad, as suggested in Padfield [2]. The resulting derivatives are written in semi-normalised form before they are summed and inserted in Eqs. (6.5) and (6.6). The forces are divided by the aircraft mass and the moments are multiplied by the inverse inertia matrix [de Leeuw, 1991], X i = df x i m Y i = df y i m Z i = df z i m L i = I z dm xi + J xz dm zi I x I z J 2 xz M i = dm y i I y N i = J xz dm xi + I x dm zi I x I z J 2 xz (6.8) (6.9) (6.1) (6.11) (6.12) (6.13) Depending on the required output of the state space system, two more matrices are added to complete the state-space system, these are the output matrix C and the direct transmission matrix D, ẋ = Ax + Bu y = Cx + Du (6.14) (6.15)

86 54 Linearisation In Chapter 7, contents for the output matrices is derived for use with the linear model inverse of the sycos pilot. The linear velocities are perturbed with values of 1 m/s. For trim velocities below this value, this gives problems with the Pitt-Peters inflow model corrected for sideward flight. Due to the two-sided symmetrical perturbation (Eq. (6.7)), the sideslip angle at the tip path plane will change approximately 18 for a negative perturbation (i.e. backward flight). As a consequence, the aerodynamic roll and pitch moments at the rotor are negated due to the transformation. Then, two coupling stability derivatives (Y u and L u ) change abruptly between and 1 m/s. To solve this, the linearisation routine in hover uses a one-sided positive perturbation for the body states, y i y i 1 = dy h dx + O(h 2 ) (6.16) xi A third stability derivative (L q ) also shows a suspicious jump between hover and low speed forward flight, however the above one-sided linearisation does not correct this. Therefore, something else must be causing this abnormality Natural Modes The free helicopter motion with the states fixed at their trim values x is governed by the left hand side of Eq. (6.2), ẋ Ax = (6.17) The characteristic equation is given by equating the characteristic polynomial to zero (s) = si A = (6.18) The roots or zeros of this equation are the eigenvalues λ i of the state matrix A. A right eigenvalue λ i and its corresponding eigenvector v i satisfy Av i = λ i v i (6.19) The modal matrix V contains all eigenvectors as columns, V = [v 1 v 2... v n ] (6.2) where n equals the number of states. From Eqs. (6.19) and (6.2), it then follows that λ 1 λ 2 AV = V. = V Λ (6.21).. λ n

87 6.2 Linearisation Routine 55 This defines a similarity transformation between the state matrix A and the eigenvalue matrix Λ. In this form, the state equations are said to be in modal form. The solution of the initial value problem can then be written as e λ 1t e λ 2t x(t) = V. V 1 x.. = y(t)x (6.22) e λnt This means that the free helicopter motion is a linear combination of the natural modes, each with an exponential character in time defined by the eigenvalue, and a distribution among the states defined by the eigenvector [Padfield, 2]. Uncoupled Longitudinal Eigenmodes The helicopter dynamics in the longitudinal plane are governed by a pitch motion (Θ and q), a vertical motion (w) and a longitudinal motion (u). If we assume that the longitudinal and lateral dynamics are only weakly coupled (or not at all), then the above analysis can be applied to the upper left 4 by 4 submatrix of the state matrix A, X u X w X q w t g C Θt A long = Z u Z w Z q + u t g C Φt S Θt M u M w M q (6.23) C Φt Just as with airplanes, the longitudinal eigenmotions are the phugoid and the short period 1. The phugoid is unstable in hover and will become (more) stable as airspeed is increased. This is mainly due to the increased effectiveness of the horizontal stabilizer with forward flight. The short period mode is a rapid change in attitude and angle of attack without much changes in the flight path and airspeed. In hover, its eigenvalues are real and form two different modes, the pitch subsidence and the heave subsidence. As speed increases, the root loci of these modes merge and move into the complex plane, which is again due to the presence of the horizontal stabilizer. Without it, two stable subsidences are present for the complete linearisation range. Uncoupled Lateral Eigenmodes The same can be done for the lateral side, the dynamics of which are governed by a lateral motion (v), a roll motion (p and Φ) and a yaw motion (r). The following 4 by 4 matrix defines the uncoupled lateral dynamics of the helicopter, Y v Y p + w t g C Φt C Θt Y r u t A lat = L v L p L r 1 C Φt T (6.24) Θt N v N p N r 1 The similarities between the eigenmotions of an aircraft and a rotorcraft reduce significantly if the two tail plane surfaces are omitted.

88 56 Linearisation The lateral modes are the Dutch roll oscillation and two aperiodic modes, the roll subsidence and spiral subsidence. Coupled Eigenmodes For the coupled eigenmodes, the upper left 8x8 submatrix in Eq. 6.5 is used. The last row, containing the influence of the heading angle Ψ on the helicopter stability, is not included here, since it only adds a zero to an eigenvalue plot 2. Physical Interpretation of Poles Once all eigenvalues are calculated using Eq. (6.18), some parameters can be calculated that give some insight in the natural motion of the helicopter. In general, a complex conjugate pair of eigenvalues can be written as λ 1,2 = a ± ib (6.25) where i (or j) is the complex unit, a is the value of the real part and b is the value of the complex part of the eigenvalue. The damped natural frequency ω is defined as ω = Im(λ) (6.26) The amount of damping ν of the eigenvalue is the negative of the real part of the eigenvalue so that a positive value denotes positive damping or decay in time, ν = Re(λ) (6.27) The undamped natural frequency ω n is the magnitude of the eigenvalue and follows from Eqs. (6.26) and (6.27) as ω n = ν 2 + ω 2 (6.28) and the damping ratio ζ is defined as ζ = Re(λ) ω n (6.29) For complex eigenvalues, the period P is the inverse of the damped natural frequency, P = 2π ω (6.3) For (complex) stable eigenvalues, located in the left half of the complex plane, the time to half amplitude of an eigenvalue T 1/2 is calculated with T 1/2 = ln 2 Re(λ) (6.31) 2 The associated eigenvector is ( 1) T

89 6.3 Results Steady-state Main Rotor and Tail Rotor 57 and for unstable eigenvalues, the time to double amplitude T 2 equals T 2 = ln 2 Re(λ) (6.32) Lastly, the time constant τ for real eigenvalues is τ = 1 λ (6.33) 6.3 Results Steady-state Main Rotor and Tail Rotor This section discusses the results in case of a steady-state main rotor and tail rotor for the Puma helicopter Influence of Nonuniform Inflow on the Stability and Control Derivatives The use of a first order inflow distribution instead of a uniform inflow distribution mainly results in changes in the off-axis stability and control derivatives. This can be illustrated with an example as follows. Disturbing the longitudinal velocity component (u) (i.e. changing the advance ratio) will result in a different equilibrium for the steady-state main rotor; the wake skew angle and the longitudinal inflow component (λ 1c ) will change. The longitudinal blade incidence variation will cause the lateral flapping to change due to the 9 phase lag of the rotor response. As a result, the rotor rolling moment will change. The stability and control derivatives for the Puma helicopter with uniform and nonuniform inflow are displayed in Appendix D, in Figs. D.1 to D.6. The data points generated with the mathematical model derived for this thesis are shown with white and black circles for the nonuniform and uniform inflow, respectively. All stability and control derivatives are shown in the same order as the graphs in Padfield [2], on pages 269 to 275. From these figures, it is clear that the addition of nonuniform inflow results in larger changes of the values of the coupling derivatives as a function of airspeed (see e.g. X v, X p, M v and M p in Fig D.3 and Y u, L u, Y w and Y w in Fig D.4). As a consequence, the difference between coupled and uncoupled eigenvalues will become larger when using nonuniform inflow. These changes are most notable in the lower flight regime. The impact on the frequency and damping of the actual eigenvalues will be discussed in the next section Uncoupled and Coupled Eigenmodes As already stated in the previous section, nonuniform inflow mainly changes coupling derivatives. Here, the actual influence on the eigenvalues in terms of frequency and damping will be discussed.

90 58 Linearisation Uncoupled Eigenmodes The root-loci of the uncoupled longitudinal and lateral eigenmodes for the Puma helicopter in its base configuration (nonuniform inflow) are displayed in Figs. 6.2a and 6.2b for forward speeds between m/s and 6 m/s V = m/s V = 6 m/s eigenvalue V = m/s V = 6 m/s eigenvalue Im(λ)[rad/s] Im(λ)[rad/s] Re(λ)[1/s] (a) Uncoupled longitudinal eigenvalues Re(λ)[1/s] (b) Uncoupled lateral eigenvalues Figure 6.2: Loci of uncoupled eigenvalues as a function of forward speed (V = to 6 m/s), steady-state main rotor with nonuniform inflow Longitudinal Eigenmodes In the longitudinal plane, there are four poles, two are associated with the long period oscillation (or phugoid) and these are slightly unstable. They become more stable in forward flight. Two non-harmonic stable poles form the heave and pitch subsidence and in forward flight, they become harmonic and are then called the short period oscillation. The eigenvalues and eigenvectors for the hover and forward flight cases can be found in Tables 6.1 and 6.2, respectively. In hover, the phugoid and the pitch subsidence have a similar character if one examines their eigenvectors, both motions are strong in pitch angle Θ and pitch rate q. In high speed forward flight, the phugoid has become a stable motion. The influence of the vertical speed w has increased whereas the pitch motion is less pronounced. λ(hover) ±.358i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] Table 6.1: Uncoupled longitudinal eigenvalues and eigenvectors of the Puma in hover 3 The root-locus plot of the helicopter with uniform inflow is indistinguishable from the one in Fig. 6.2, so it is not displayed here.

91 6.3 Results Steady-state Main Rotor and Tail Rotor 59 λ(6m/s).87 ± 1.663i.23 ±.128i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] Table 6.2: Uncoupled longitudinal eigenvalues and eigenvectors of the Puma at 6 m/s Lateral Eigenmodes In the lateral plane in hover (Table 6.3), one can again see one unstable harmonic eigenmode and two stable subsidences. The first one is called the roll subsidence due to the strong effect in p and the second one is the spiral subsidence. The oscillatory mode is a harmonic motion in roll and yaw which is called the Dutch roll. λ(hover) ±.517i.435 T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.3: Uncoupled lateral eigenvalues and eigenvectors of the Puma in hover In high speed forward flight, the above lateral modes have changed a fair amount, see Table 6.4. The roll subsidence now has a much stronger effect in both roll rate and roll angle. The spiral subsidence has changed its character, now it is mainly a motion in roll angle. The Dutch roll has become a high-frequency (wrt hover) stable oscillation. λ(6m/s) ± 3.144i.58 T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.4: Uncoupled lateral eigenvalues and eigenvectors of the Puma at 6 m/s

92 6 Linearisation Coupled Eigenmodes The coupled eigenmodes with nonuniform inflow are shown in Fig. 6.3a and those with uniform inflow are shown in Fig. 6.3b. 4 3 V = m/s V = 6 m/s eigenvalue 2 Im(λ)[rad/s] Re(λ)[1/s] (a) Nonuniform inflow 4 3 V = m/s V = 6 m/s eigenvalue 2 Im(λ)[rad/s] Re(λ)[1/s] (b) Uniform inflow Figure 6.3: Loci of coupled eigenvalues for the Puma with steady rotors as a function of forward speed (V = to 6 m/s) The following general observations regarding the influence of coupling on the model with nonuniform inflow can be made. The phugoid is a lot more unstable near hover than was the case for the uncoupled phugoid and as speed is increased, it becomes rapidly more stable. The Dutch roll is slightly more stable when compared with the uncoupled case and

93 6.3 Results Steady-state Main Rotor and Tail Rotor 61 it has the same large change in frequency and damping as speed is increased. The pitch and heave subsidence in hover do not exist any more, their hover eigenmodes are located in the complex plane. The damping of this last eigenmode is a lot bigger than for the uncoupled case. The left-most aperiodic mode (roll subsidence) shows a large variation in damping as speed is changed and its hover and high speed eigenvalues are reversed with respect to the uncoupled case. The spiral subsidence on the other hand is more compact (less change in damping). For the uniform case, less coupling is visible. The frequencies and damping of the phugoid are more or less the same as in the uncoupled case. Here, the pitch and heave subsidence do exist, but their eigenvalues are located closer together and as speed increases, their loci merge and the mode transforms into the short period oscillation. The influence of couping is strongest near hover for this mode. Just as with the nonuniform case, the roll subsidence reverses in terms of damping. Phugoid In forward flight, the content of the uniform and nonuniform eigenvectors are quite similar (see Table 6.5). In hover, the coupling due to nonuniform inflow makes the phugoid more rapid (higher frequency, smaller time to double amplitude, etc). Furthermore, the motion becomes less pronounced in forward velocity and more pronounced in pitch and pitch rate. The magnitude of the lateral states in the eigenvector suggest that this longitudinal mode is highly coupled with the lateral states, especially with nonuniform inflow. Uniform Nonuniform V = m/s V = 6m/s V = m/s V = 6m/s λ.125 ±.334i.2 ±.12i.37 ±.51i.2 ±.119i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.5: Phugoid eigenmode for the Puma in hover and forward flight Dutch Roll As stated earlier, the use of nonuniform inflow makes the Dutch roll (Table 6.6) less unstable in hover, the magnitude of the damping becomes four times smaller while the frequencies approximately stay the same. In terms of eigenvector content, the off-axis states become more important with nonuniform inflow, just as was the case with the phugoid.

94 62 Linearisation Uniform Nonuniform V = m/s V = 6m/s V = m/s V = 6m/s λ.67 ±.539i.937 ± 3.153i.18 ±.548i.931 ± 3.152i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.6: Dutch roll eigenmode in hover and forward flight Roll Subsidence With uniform inflow, the magnitudes of the dominant lateral states (p, Φ and r) are more or less equal for the roll subsidence (Table 6.7). This changes completely with nonuniform inflow, here the roll subsidence is primarily a motion in roll rate. The influence of lateral velocity is bigger than for uniform inflow. Uniform Nonuniform V = m/s V = 6m/s V = m/s V = 6m/s λ T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.7: Roll subsidence eigenmode in hover and forward flight Spiral Subsidence By inspecting the contents of the eigenvectors in hover, this lateraldirectional mode (Table 6.8) shows less coupling with the longitudinal states for nonuniform inflow when compared with the uniform inflow case. This is compensated with increased magnitude of all four on-axis states. In forward flight, the spiral subsidence doesn t show significant differences between uniform and nonuniform inflow, the roll angle Φ is the dominant state for both configurations, followed by the yaw rate r.

95 6.3 Results Steady-state Main Rotor and Tail Rotor 63 Uniform Nonuniform V = m/s V = 6m/s V = m/s V = 6m/s λ T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.8: Spiral subsidence eigenmode in hover and forward flight Short Period Mode and Pitch and Heave Subsidences In forward flight, the short period mode shows less coupling with the lateral states (Table 6.9). For nonuniform inflow, the short period in hover is mainly a motion in pitch and pitch rate (Table 6.1) whereas in forward flight, the influence of the vertical velocity component becomes more pronounced. The pitch and heave subsidences both have a strong coupling with yaw rate r, whereas for the short period in hover, the off-axis coupling is more evenly distributed over the lateral states. Uniform Nonuniform V = 6m/s λ.755 ± 1.73i.777 ± 1.696i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase u [m/s].6.61 w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.9: Short period eigenmode in forward flight

96 64 Linearisation Pitch Subsidence Heave Subsidence Short Period Uniform Nonuniform λ ±.349i eigenvector mag phase mag phase mag phase T 1/2 T ω ω n ζ τ u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 6.1: Pitch and heave subsidences and short period eigenmode in hover 6.4 Influence of Unsteady Main Rotor and/or Tail Rotor on Trim Results In Helix, two major options are available that influence the linearisation results. These are switches that retrim the main and/or tail rotor after a disturbance is applied to one of the states, as depicted in Fig. 6.1 at the beginning of this chapter. If these switches are turned on, the main rotor transforms from one with unsteady inflow and second order blade flapping dynamics into a steady-state main rotor and the tail rotor quasi-steady dynamic inflow becomes steady-state inflow. Therefore, this method could (in theory) be used to perform time simulations with a steady-state main and tail rotor. The following four cases can be distinguished: 1. both steady-state main rotor and tail rotor, 2. steady-state main rotor and unsteady tail rotor, 3. both unsteady main rotor and tail rotor and 4. unsteady main rotor and steady-state tail rotor. The first case was the subject of section 6.3. Hereafter, the last three cases are shortly discussed. Insufficient time was available to do a complete analysis as given in the previous section, so only the root-locus plots are given for the coupled and uncoupled cases of the helicopter with nonuniform inflow. The contents of the eigenvectors are not displayed Case 2: Unsteady Main Rotor and Unsteady Tail Rotor The root-loci of the longitudinal uncoupled eigenmodes are displayed in Fig. 6.4a. Some minor changes in damping and frequency are visible with respect to the baseline case

97 6.4 Influence of Unsteady Main Rotor and/or Tail Rotor on Trim Results 65 (Fig 6.2a), but the motions have the same character. In hover, there are two stable subsidences (pitch and heave) and one slightly unstable oscillatory mode (phugoid). In high speed flight, the former two transform into the short period oscillation and the latter becomes more stable. Im(λ)[rad/s] V = m/s V = 6 m/s eigenvalue Re(λ)[1/s] (a) Longitudinal uncoupled Im(λ)[rad/s] 4 3 V = m/s V = 6 m/s eigenvalue Re(λ)[1/s] (b) Lateral uncoupled 4 V = m/s V = 6 m/s eigenvalue 3 2 Im(λ)[rad/s] Re(λ)[1/s] (c) Coupled Figure 6.4: Loci of eigenvalues for the Puma (with nonuniform inflow) as a function of forward speed (V = to 6 m/s) with unsteady rotors Except for some similarities in hover, the lateral uncoupled eigenmodes shown in Fig. 6.4b are completely different from their steady-state counterparts in Fig. 6.2b. They show similar changes in frequency and damping as the longitudinal uncoupled eigenmodes and may therefore be called lateral phugoid and lateral short period. One could conclude that the initial response of the helicopter to a disturbance (before the rotors reach a steady state) is similar in the longitudinal and lateral plane. As was shown in the previous section, nonuniform inflow mainly influences the off-axis response of the helicopter, with especially large changes in damping for the phugoid. With this unsteady configuration, the wake skew does not change so that the lateral flapping

98 66 Linearisation and rolling moments stay (more or less) the same. As a results, almost no changes happen in terms of damping for the (longitudinal) phugoid. For the lateral short period, the most visible changes happen in low forward speed. The two hover subsidences merge at a point with a lower damping (ν.8 instead of ν 1.1) Case 3: Unsteady Main Rotor and Steady Tail Rotor Im(λ)[rad/s] 4 V = m/s V = 6 m/s 3 eigenvalue Re(λ)[1/s] (a) Longitudinal uncoupled Im(λ)[rad/s] 4 V = m/s V = 6 m/s 3 eigenvalue Re(λ)[1/s] (b) Lateral uncoupled 4 V = m/s V = 6 m/s eigenvalue 3 2 Im(λ)[rad/s] Re(λ)[1/s] (c) Coupled Figure 6.5: Loci of eigenvalues for the Puma helicopter (with nonuniform inflow) as a function of forward speed (V = to 6 m/s) with unsteady main rotor With an unsteady main rotor and a steady tail rotor, the biggest changes with respect to case 2 are to be expected in the lateral plane. And indeed, the longitudinal uncoupled eigenvalues shown in Fig. 6.6a are the same as the ones in the previous section. The lateral uncoupled eigenvalues (Fig. 6.5b) has a similar Dutch roll as in case 1 (Fig. 6.2b), but the other eigenmodes are different. In hover, two stable subsidences are present which merge and move into the complex plane. The undamped natural frequency ω n

99 6.4 Influence of Unsteady Main Rotor and/or Tail Rotor on Trim Results 67 rapidly becomes smaller as speed PSfrag is replacements increased until in high speed forward flight, they almost become aperiodic. This last eigenmode has the biggest changes in the coupled case (Fig. 6.5c), the two subsidences become a complex conjugate pair of eigenvalues Case 4: Unsteady Tail Rotor and Steady Main Rotor Im(λ)[rad/s] V = m/s V = 6 m/s eigenvalue Re(λ)[1/s] (a) Longitudinal uncoupled Im(λ)[rad/s] 4 3 V = m/s V = 6 m/s eigenvalue Re(λ)[1/s] (b) Lateral uncoupled 4 V = m/s V = 6 m/s eigenvalue 3 2 Im(λ)[rad/s] Re(λ)[1/s] (c) Coupled Figure 6.6: Loci of eigenvalues for the Puma helicopter (with nonuniform inflow) as a function of forward speed (V = to 6 m/s) with steady-state main rotor and unsteady tail rotor The uncoupled longitudinal eigenvales in the last case (Fig. 6.6a), are the same as in the first case. In the lateral direction (Fig. 6.6b) in hover, one subsidence has a time constant τ of approximately.5 (the one located to the far left). The also happened in case 2 and therefore, one may conclude that this is caused by the unsteadiness of the tail rotor. The large changes in tail rotor thrust between the steady-state and unsteady cases make that the quickness of the response around the top axis changes dramatically between initial

100 68 Linearisation response and steady-state response. The locus of the Dutch roll oscillation does not change its frequency as speed is increased until 9 m/s. At that point, they have approximately the same frequency and damping as the other eigenmode (eigenvalues located closely together). As speed is increased beyond 1 m/s, the eigenvalues separate again and the eigenvalues of the Dutch roll move further into the complex plane, with big changes in frequency. Due to the increased off-axis coupling as a results of nonuniform inflow, the eigenvalues of the phugoid in hover move to the right for the coupled case (Fig. 6.6c). Two stable subsidences can be distinguished due to the coupling. The Dutch roll mode splits up in two, its low-speed part (below 9 m/s) connects with the high speed part of the longitudinal short period and vise versa Time Response of the Linear Models The eigenvalue plots in the previous sections do not shed any light on the time response of the different linear models to a control input with respect to the response of the nonlinear model. The models of case 1 (both steady-state main and tail rotor) and case 2 (both unsteady main and tail rotor) are used and compared with the nonlinear Puma model at a forward flight speed of 4 m/s. Starting from trimmed flight, the controls are held fixed at their trim values for two seconds. Then, a doublet signal (see Fig with an amplitude of two degrees lasting for two seconds is added to one of the controls at a time. Thereafter, the controls are held for 12 seconds in their trim positions. When utilizing the numerical blade element main rotor, the helicopter model MoGeHM developed by [Voorsluijs] suffers from a transient in time simulations at the start. The cause of this problem lies in its trim routine which only trims the fuselage states and simulates But still... the modules separately with constant inputs long enough for the transients to subside.... the nonlinear system starts with small perturbations of roll and yaw which result in a bias in the roll and yaw rates. By explicitly trimming the main rotor in a trim run, Helix does not suffer from these transients. This is clearly visible in Figs. 6.8 to 6.11, which show the response of the nonlinear model and the two linear models. For the time responses to the main rotor cyclic inputs, there is a clear difference between the initial response of both linear models. The unsteady linear model follows the initial 4 Note that all inputs are first positive and afterwards negative. This means for the longitudinal cyclic θ 1s that the blade at ψ = 9 will get a pitch increase. And because the rotor of the Puma rotates counterclockwise, this is the retreating blade. Therefore, this input will tilt the rotor disk forward, with a negative pitch rate as a consequence. For the Bo15, the opposite would happen.

101 6.5 Influence of Wake Distortion Dynamics on Rotorcraft Responses θ θ1s -5 5 θ1c -5 5 θtr Time [s] Figure 6.7: Two degree amplitude doublet inputs for the nonlinear model trends of the states whereas for the steady linear model, the long term responses of the helicopter are followed. This difference is less for the collective inputs of the main and tail rotor (see for example Fig. 6.8a) but still too much to call it useful. From Fig. 6.8, it is clear that a positive increase in main rotor collective θ (first part of the doublet input) does not solely result in an vertical velocity increase. All axes are affected, which suggests that strong coupling exists. Due to the increase in main rotor collective input, the overall torque, drag and sideforce of the rotor increase and since these increases are not corrected for by the other controls, the helicopter starts to develop a positive pitch rate q, a negative yaw rate r and a positive sway motion v next to the intended negative heave motion. Similarly, a increased tail rotor collective pitch θ tr (see Fig. 6.9) increases the tail rotor thrust to the left. Since the tail rotor is located above and behind the centre of gravity, a negative sway velocity v is developed. The helicopter also starts to roll to the left and yaw to the right. Overall, the steady linear model performs best and shows the same long-term trends as the nonlinear model. 6.5 Influence of Wake Distortion Dynamics on Rotorcraft Responses As already stated in Chapter 4, inflow models alone do not improve the off-axis response to control inputs. Therefore, the wake distortion dynamics augmentation for the Pitt- Peters inflow model developed by Zhao [25] was included. Due to time constraints, the

102 Sfrag replacements 7 Linearisation Nonlinear Steady linear Unsteady linear Nonlinear Steady linear Unsteady linear u [m/s] Sfrag replacements w [m/s] v [m/s] Time [s] (a) Linear velocities u [m/s] v [m/s] w [m/s] Time [s] (a) Linear velocities p [rad/s] Sfrag replacements q [rad/s] r [rad/s] Time [s] (b) Angular velocities p [rad/s] q [rad/s] r [rad/s] Time [s] (b) Angular velocities Ψ [deg] Θ [deg] Φ [deg] Ψ [deg] Θ [deg] Φ [deg] Time [s] (c) Attitudes Time [s] (c) Attitudes Figure 6.8: Time response to main rotor collective doublet input Figure 6.9: Time response to tail rotor collective doublet input

103 6.5 Influence of Wake Distortion Dynamics on Rotorcraft Responses 71 Nonlinear Steady linear Unsteady linear Nonlinear Steady linear Unsteady linear u [m/s] w [m/s] v [m/s] Time [s] (a) Linear velocities u [m/s] v [m/s] w [m/s] Time [s] (a) Linear velocities p [rad/s] q [rad/s] r [rad/s] Time [s] (b) Angular velocities p [rad/s] q [rad/s] r [rad/s] Time [s] (b) Angular velocities Ψ [deg] Θ [deg] Φ [deg] Ψ [deg] Θ [deg] Φ [deg] Time [s] (c) Attitudes Time [s] (c) Attitudes Figure 6.1: Time response to main rotor longitudinal cyclic doublet input Figure 6.11: Time response to main rotor lateral cyclic doublet input

104 72 Linearisation effects on the helicopter responses were not fully investigated. But to show that there are indeed clear differences, Fig shows the response of the Puma to a longitudinal cyclic doublet (θ 1s ) with an amplitude of 1 degree at a forward speed of 5 m/s 5. It should be noted that at this velocity, the PSfrag Puma replacements helicopter is fairly unstable, which results in clear differences between the linear and and nonlinear model. No clear differences are visible before the first four seconds, it is only thereafter that the baseline and augmented model start to show differences. Overall, the augmented model is more stable, which suggests that part of the extra destabilisation introduced by Pitt-Peters nonuniform inflow is reduced again by the inclusion of wake distortion effects. During the first six seconds after the end of the doublet input (4 1 s), the off-axis response (Φ, Ψ, p and r) shows clear sign changes. Initial response of the on-axis states Θ and q is more or less the same, while in the long term, difference between the two models increase. u [m/s] v [m/s] w [m/s] Time [s] (a) Linear velocities Ψ [deg] Θ [deg] Φ [deg] p [rad/s] q [rad/s] r [rad/s] Time [s] (c) Attitudes Time [s] (b) Angular velocities Figure 6.12: Influence of wake distortion dynamics on time responses of the V = 5 m/s 5 At a speed of 4 m/s, differences are between the model with and without wake distortion dynamics are almost indistinguishable

105 Chapter 7 Time Simulation - Ship Approach and Landing 7.1 Introduction The topic of this chapter is the time simulation in which a landing manoeuvre is performed. The Puma helicopter first flies with high speed, far away from the ship to a point alongside the ship deck. Then, a sidestep is executed, ending in stationkeeping above the deck. In the last part, a vertical descend is initiated ending in touchdown on the ship deck. Instead of using classical PID-controllers to model the stabilisation and navigation loops of a mathematical pilot, the sycos (SYnthesis through COnstrained Simulation) pilot developed in Turner et al. [2] and Bradley & Brindley [22] is used as an alternative way to achieve the above goal. The pilot model itself is explained in section 7.2. As input, this model uses the intended flight path which is discussed in section 7.3. In the fourth section, the applied stability augmentation and reference corrections are discussed. In the final section, the results are discussed. 7.2 Pilot Model This section will shortly explain the background of the sycos pilot model. This pilot model can be used in off-line helicopter simulations for evaluation of rotorcraft performance and handling qualities. It overcomes some of the precise, open-loop control of pure inverse simulations, by using a corrective control structure to correct control settings when departures from the intended flight path are detected. The control structure consists of two in series placed parts, the first being a crossover element, the output of which is processed by the second part, a learned response that generates the necessary corrective actions. A schematic overview of the resulting control structure is given in Fig In the following two sections, the separate building blocks of the pilot model will be discussed, followed by some simple checks that can be used to asses whether the model was implemented correctly. 73

106 74 Time Simulation - Ship Approach and Landing learned response linear inverse y ref + error ke τs + D 1 u Nonlinear Model Output y C x (si A) 1 B Figure 7.1: Schematic overview of the sycos pilot structure, showing (fltr) the crossover component, the learned response or inverse system and the nonlinear model with its output Overall Transfer Function In helicopter simulations which employ a mathematical model for the pilot, the pilot applies corrective actions to stabilise the helicopter around its current flight state. McRuer & Krendel [1974] state that the pilot adapts to the system in such a manner that the open-loop transfer function H(s) between the error and the output is of the form H(s) = ke τs s (7.1) where k and τ are the crossover frequency and time delay with typical values of 2 and.2, respectively. This transfer functions models the general behaviour of the pilot in two parameters rather than trying to represent all internal dynamics of the pilot such as is the case when the limb-manipulator (arm-stick) dynamics is modeled. Fig. 7.2 shows how the above transfer function can replace the pilot-system-output chain. y ref error + Pilot System Output y equivalent to y ref + error ke τs s y Figure 7.2: Overall transfer function of the pilot-system-output chain If we now want to use the crossover model as a part of the pilot, we must ensure that the open-loop transfer function between the error and the output stays the same. The easiest way to accomplish this is by adding the inverse of the system plus output between the crossover element and the system block. This means that the input of the inverse block must be the same as the output of the system, since they cancel each other out. The resulting flow diagram is given in Fig The two blocks on the left are part of the pilot, the first block is the crossover element, the output of which is processed by the

107 7.2 Pilot Model 75 inverse system component that generates the required corrective control inputs. In the following section, the building blocks of the inverse component are derived. y ref + error ke τs s y Inverse of System + Output u System + Output y Figure 7.3: Pilot modeled as a crossover element and the inverse of the system Inverse Component As references, the three earth referenced velocities (ẋ,ẏ,ż) and the rate of change of azimuth angle ( Ψ) are used. As already stated in the previous section, the inverse component represents the adaptation of the pilot to the system he or she controls. Since the pilot has only a limited knowledge of the behaviour of the helicopter, the inverse component must not be exact. For this reason, the linear model derived in Chapter 6 can be used, ẋ = Ax + Bu (7.2) Here again, x is the vector containing the states, x = (u w q Θ v p Φ r Ψ) T (7.3) and u is the control vector u = (θ θ 1s θ 1c θ tr ) T (7.4) The following output 1 equation is used y = Cx (7.5) The earth referenced velocities and the azimuth angle are used as a basis for the linear inverse, y = (ẋ ẏ ż Ψ) T (7.6) Using the Euler transformation between the body referenced linear velocities u, v and w and the earth referenced linear velocities ẋ, ẏ and ż, matrix C can be written as C Ψ C Θ C Ψ S Θ C Φ + S Ψ S Φ C Ψ S Θ S Φ S Ψ C Φ C = S Ψ C Θ S Ψ S Θ C Φ C Ψ S Φ S Ψ S Θ S Φ + C Ψ C Φ S Θ C Θ C Φ C Θ S Φ (7.7) 1 1 Note that this is not completely the same output as the output y in the figures shown in the previous section.

108 76 Time Simulation - Ship Approach and Landing Feedback linearisation is used to obtain the inverse. The output equation (7.6) is differentiated and the state equation (7.2) is substituted, ẏ = Cẋ = C(Ax + Bu) = CAx + CBu (7.8) The last row of matrix CB is equal to zero which makes this matrix singular. This means that the rate of change of azimuth ( Ψ) is not directly influenced by the controls. One more differentiation gives the following result, ÿ = CAẋ + CB u = CA 2 x + CABu + CB u (7.9) Since the last row of matrix CB equals zero, the time derivatives of the controls ( u) do not appear in the equation of Ψ. As a consequence, it is possible to assemble an output equation of the form ȳ = Cx + Du (7.1) where the output vector consists of accelerations, ȳ = (ẍ ÿ z Ψ) T The matrices C and D equate to [ ] [ ] CA(1 : 3, :) CB(1 : 3, :) C = and D = CA 2 (4, :) CAB(4, :) (7.11) (7.12) Since the inverse of Eq. (7.1) with the controls u as output and ȳ (Eq. (7.11)) as input needs the derivatives of the references that are tracked by the pilot, an additional derivative block is needed between the crossover element and the model inverse block. After combining the integrator action of the crossover element with the derivative block, the system as shown in Fig. 7.1 is obtained Inverse Model Validation This section describes some simple checks that can be done to verify wether the inverse model was derived and implemented correctly. First, it is verified that the eigenvalues of the constrained system are all zero except for the zero dynamics. Thereafter, the nonlinear model is replaced with the linear model. In this configuration, the output should perfectly track the references with a small time delay. Eigenvalues of the Constraint System Using Eq. 7.1, the values for the controls u that will be fed to the nonlinear model follow as u = D 1 (ȳ Cx) (7.13) Substitution Eq. (7.13) in Eq. (7.2), the state matrix A c of the constrained or inverse model equals A c = A B D 1 C (7.14)

109 7.2 Pilot Model 77 This model must follow four constraints, so there are four modes with zero eigenvalues. The fifth zero mode is caused by the heading angle Ψ, which does not add any dynamics to the system. The last four eigenvalues two pairs of complex conjugate eigenvalues are associated with two types of oscillatory behaviour of the helicopter around its centre of gravity, namely pitch and roll motion. These are the zero dynamics of the constrained system. Fig. 7.4 shows the positions of the eigenvalues in the complex plane for a velocity of 2 m/s. At this speed, both pairs of eigenvalues are located in the left half of the complex plane 2. From Table 7.1, it is clear that the slow eigenmode (high value for T 1/2 ) is associated with the pitch oscillation and the fast eigenmode is associated with roll oscillation. The former is only marginally stable with a damping of.58. The consequences of the stability of the constrained linear model is discussed some more in section original system eigenvalues 4 + constrained system eigenvalues Im (λ) [rad/s] Re (λ) [1/s] Figure 7.4: Normal and constrained system V = 2 m/s Linear Model Inverse Simulation If the nonlinear helicopter model block is replaced with the same linear model as used in the inverse component, the simulation should track the reference velocities perfectly with a time delay equal to the reciprocal value of the gain k of the crossover element, i.e. a time delay of.5 seconds. A simple horizontal deceleration profile lasting 125 seconds is used to test this. During the first 25 seconds, the x-acceleration is smoothly decreased until a value of -.4 m/s 2 is reached. Then, for 75 seconds, this deceleration is held fixed and thereafter, it is 2 After inspection of the constrained eigenmodes between hover and 4 m/s, it was chosen to use the linear model around 2 m/s since this resulted in the most stable constrained system.

110 78 Time Simulation - Ship Approach and Landing λ (V = 2 m/s).742 ± 4.53i.132 ± 2.271i T 1/2 T ω ω n ζ τ eigenvector mag phase mag phase u [m/s] w [m/s] q [deg/s] Θ [deg] v [m/s] p [deg/s] Φ [deg] r [deg/s] Table 7.1: Eigenmode characteristics of matrix A V = 2 m/s smoothly increased again to zero in a timespan of 25 seconds. The resulting earthreferenced acceleration, velocity and position profiles are depicted in Fig 7.5. The inset in the second subplot shows that the output velocity (dashed line) lags.5 seconds in time with respect to the reference velocity. Acceleration [m/s2] Velocity [m/s] Position [m] Time [s] Figure 7.5: Deceleration, velocity and position profiles for the linear model time simulation, inset shows the output velocity with a dashed line with respect to the reference speed For this case, the nonlinear model was linearised around 2 m/s and the initial x-velocity is set to 4 m/s. The initial conditions for the states of the linear model are then nonzero. Substracting the trim states at the linearisation point from the trim states at the initial condition point gives a fairly good guess of the needed initial conditions. However, this

111 7.3 Simulation Flight Path References 79 results in big control fluctuations at the beginning of the time simulation, so a trim routine is needed. It employs the same variables as the top level trim routine discussed in Chapter 5. Here, the kinematic constraints are the differences between the earth-oriented velocities (and rate of change of heading angle) at the linearisation point and those at the initial trim PSfrag point. replacements The resulting time histories of the four controls are shown in Fig Control Position [deg] 1 5 θ θ 1s θ 1c θ tr Time [s] Figure 7.6: Pilot controls for the linear inverse time simulation 7.3 Simulation Flight Path References Here, the flight path references for the shipboard approach will be derived. For the whole time simulation, it is assumed that the ship and helicopter sail and fly north (i.e. the azimuth angle Ψ and its derivative equal zero). The shipboard approach and landing manoeuvre is divided in the following three phases: 1. ship closure, bring the helicopter from far away to a point alongside the landing platform of the ship, with the same velocity as the ship, 2. lateral repositioning to a point above the ship flight deck, 3. vertical descent ending in a landing on the ship flight deck. The kinematic profiles needed to generate the references during these three phases will be derived in the following sections. The profiles for the first and second phase are similar to the ones described in Xin & He [22] Phase I: Closure to Ship During this phase, the pilot performs the necessary control actions to bring the helicopter from far away to a point alongside the ship. The initial forward flight speed must be

112 8 Time Simulation - Ship Approach and Landing Figure 7.7: Standard forward-facing landing procedure [Lumsden et al., 1998] reduced to the speed of the ship, and the altitude is also reduced during this phase. This manoeuvre is conducted along a relatively constant glide slope and can be subdivided in three parts: 1. Initial transition: from a steady level flight condition, accelerate vertically to develop an initial descent rate and develop a desired horizontal deceleration; 2. Steady descent and deceleration: along a constant glide slope, reduce the horizontal and vertical velocities; 3. Final transition: reduce descent rate to zero and end with the same speed as the ship. H V γ D Z X X end X Figure 7.8: Spatial position profile during closure to ship The time instants at the end of these three parts are denoted as t 1, t 2 and t 3. The initial time t equals zero and the time increments are defined as t 1 = t 1 t t 2 = t 2 t 1 t 3 = t 3 t 2 (7.15) The total time is the sum of the parts, t I = t 1 + t 2 + t 3 = t 3 t (7.16)

113 7.3 Simulation Flight Path References 81 Kinematic Profile The kinematic parameters for the approach are the initial horizontal velocity V, the initial altitude Z, the constant ship speed V s, the initial horizontal separation between the helicopter and the ship X, the desired flight path angle γ d and the desired final altitude Z d. The numerical values for these parameters are listed in Table Parameter description Symbol Value Unit Initial horizontal velocity V 4 [m/s] Initial altitude H 12 [m] Ship speed V s 5 [m/s] Initial horizontal separation X 23 [m] Desired flight path angle γ d -3 [deg] Desired final altitude H d 2 [m] Table 7.2: Phase I (closure to ship) initial conditions The kinematic acceleration along the earth referenced X-axis a x is set to be ( )).5 a x2 (1 cos π(t t ) t 1 t t < t 1 a x (t) = a x2 t 1 t t 2 (7.17) ( )).5 a x2 (1 + cos π(t t ) t 1 t 2 < t t 3 which is the same as the deceleration profile depicted in Fig The vertical acceleration profile is somewhat more complex and because of that it is also shown graphically in Fig. 7.9, ( )).5 a z1 (1 cos 2π(t t ) t 1 t t < t 1 ( ( )) 2 a z2 +.5 (a z1 + a z2 ) 1 cos 2π(t t ) t12 t a z (t) = 1 t < t 1 (7.18) a z2 t 1 t t ( )) 2.5 a z2 (1 + cos π(t t2 ) t 3 t 2 < t t 3 Kinematic Constraints At the end of this phase, it is required that the X-coordinate of the helicopter and the ship coincide, X end = X + V s t (7.19) It is assumed that the first and third time segments are of equal length, t 1 = t 2 = c t t t 2 = (1 2c t ) t (7.2)

114 82 Time Simulation - Ship Approach and Landing a z t 1 t 2 t 3 a z1 a z2 t Figure 7.9: Vertical acceleration profile parameters during phase I, closure to ship As a consequence of Eq. (7.2), the X-deceleration profile is symmetrical in time and its integral the X-velocity profile is anti-symmetrical in time (see Fig. 7.5). Then, the total distance covered is simply the area under the triangle with base t and height V V s plus the area of the rectangle with height V s and base t, X end = V V s t + V s t (7.21) 2 Solving Eqs. (7.19) and (7.21) for the unknown variables X end and t gives t = 2 X V + V s (7.22) X end = V + V s V V s X (7.23) The required longitudinal deceleration a x can be calculated as a x = V s V t 1 + t 2 = f(c t ) (7.24) which is a function of the unknown variable c t. The vertical deceleration during the second time segment (a z2 ) depends on the horizontal deceleration a x and the desired glide slope γ d, a z2 = a x tan γ d = f(c t ) (7.25) To ensure that the vertical velocity at t = t end equals zero, the area under the vertical acceleration profile should be equal to zero, t3 t a z (t)dt = (7.26) The desired final altitude is reached if the area under the vertical velocity profile equals the altitude difference H, t3 t v z (t)dt = H = H H d (7.27)

115 7.3 Simulation Flight Path References 83 These last two constraints are a function of the variable c t and the maximum vertical acceleration during the first phase (a z1 ). They are used as free variables in a numerical Newton iteration to ensure that the constraints imposed by the above two integrals are met. The resulting kinematic profiles for the initial conditions given in Table 7.2 are depicted in Figs 7.1 and The values of the derived parameters are displayed in Table 7.3. Parameter description Symbol Value Unit Total horizontal distance flown X end [m] Total time t I [s] Initial transition time length t [s] Steady descent time t [s] Vertical deceleration on the glide slope a z2.168 [m/s 2 ] Maximum vertical acceleration a z [m/s 2 ] Maximum horizontal acceleration a x [m/s 2 ] Table 7.3: Phase I (closure to ship) derived parameters Acceleration [m/s 2 ] Velocity [m/s] Position [m] Time [s] Figure 7.1: Horizontal kinematic references during phase I, closure to ship Phase II: Lateral Repositioning At the end of phase one, the helicopter is advancing with the same speed as ship. An acceleration in the positive Y-direction is initiated until a maximum value a y1 is reached.

116 84 Time Simulation - Ship Approach and Landing Acceleration [m/s 2 ] Velocity [m/s] Position [m] Time [s] Figure 7.11: Vertical kinematic references during phase I, closure to ship The helicopter reduces the lateral acceleration back to zero to obtain a constant sideward velocity. This velocity is held for some time until the opposite kinematic profile is executed, which ends with stationkeeping above the landing platform. This lateral repositioning manoeuvre is similar to the sidestep manoeuvre as described in [ADS-33E-PRF], but here, an additional constant forward velocity is present. Kinematic Profile The equations that define the acceleration profile of this manoeuvre are displayed in Eq and a graphical representation is given thereafter, in Fig ( )).5 a y1 (1 cos 2πt t4 t 5 t 4 t < t 5 a y (t) = t 5 t < t 6 (7.28) ( )).5 a y1 (1 cos 2πt t4 t 5 t 6 t < t 7 The begin and end time instants of this phase are t 4 and t 7, with t 5 and t 6 marking the begin and end of the middle part, respectively. All time increments are assumed to be equal in length, t 5 = t 5 t 4 = t 6 = t 6 t 5 = t 7 = t 7 t 6 (7.29) Kinematic Constraints The constraints for this manoeuvre are the maximum allowable lateral acceleration a ymax and the total displacement in the y-direction, y. By means of analytical integration of

117 7.3 Simulation Flight Path References 85 a y t 5 t 6 t 7 a y1 t 4 t 5 t 6 t 7 t Figure 7.12: Lateral acceleration profile for the sidestep manoeuvre Eq. (7.28), the velocity and positional kinematic profiles are obtained. Solving them for the unknown maximum lateral velocity and total time (v ymax and t II, respectively), the following simple relations are obtained, v ymax =.5 a ymax y (7.3) y t = (7.31) a ymax PSfrag A replacements graphical representation of the resulting acceleration, velocity and positional profiles is given in Fig By comparing Figs. 7.1 and 7.13, it is clear that the acceleration profile define here is the derivative of the longitudinal deceleration profile of phase I. Acceleration [m/s 2 ] Velocity [m/s] Position [m] Time [s] Figure 7.13: Kinematic references during phase II, lateral repositioning The lateral distance traversed by the helicopter (from helicopter centerline to ship centerline) equals two rotor diameters or 3 metre. The maximum lateral acceleration is set to.5 m/s 2. The values of all kinematic parameters used for this flight phase are listed in Table 7.4.

118 86 Time Simulation - Ship Approach and Landing Parameter description Symbol Value Unit Maximum lateral acceleration a ymax.5 [m/s 2 ] Lateral distance flown y 3 [m] Maximum lateral velocity V ymax 1.94 [m/s] Total time t II 23.1 [s] Table 7.4: Phase II (lateral repositioning) kinematic parameters Phase III: Station Keeping and Landing Phase II ends at the moment that the helicopter hovers above the flight deck. The current altitude of the helicopter is the same as the altitude reached at the end of phase I, H d. The height above the water line of the flight deck of a Type 22 frigate is 4.71 m [Booij, 25], which means that in terms of rotor diameters, the helicopter hovers approximately one rotor diameter above the flight deck. The landing is performed in one stage, without intermediate hover. The two-stage landing is common when operating in rough weather/high sea state, but this was not modeled due to the small time this stage takes relative to the total length of the simulation 3. Kinematic Profile and Kinematic Constraints The acceleration profile during this stage is a simple sine curve. During the first half of the manoeuvre, the helicopter has a positive vertical acceleration to build up some vertical speed. This is reversed during the second half of the manoeuvre which ends in touchdown with a near-zero vertical velocity. The acceleration profile is ( ) 2πt t8 a z (t) =.5 a zmax sin t 8 t < t 9 (7.32) t 9 The total duration of this segment is t III = t 9 t 8. There are two constraints for this phase, the total height h 3 and the mean vertical velocity, v zmean. The velocity and position profiles are obtained by means of analytical integrating of Eq. (7.32). These are then used to calculate the maximum vertical acceleration a zmax and the time increment t III, a zmax = h 3 v zmean (7.33) t III = 2πv z mean h 3 (7.34) The resulting kinematic profile is depicted in Fig and the values of the for this stage relevant parameters are listed in Table It would be appropriate to include the motions of the ship and wind/turbulence if the sole objective of the time simulation is to model the stationkeeping and landing phase as is done in Hess [25]. 4 This value is corrected for the vertical distance between the landing gear and the centre of gravity.

119 7.4 Helicoper Stabilisation & Flight Path Reference Corrections 87 Parameter description Symbol Value Unit Mean vertical velocity v zmean 1. [m/s] Vertical distance covered h [m] Maximum vertical acceleration a zmax.49 [m/s 2 ] Total time t III 12.8 [s] Table 7.5: Phase III (landing) kinematic parameters Acceleration [m/s 2 ] Velocity [m/s 2 ] Position [m] Time [s] Figure 7.14: Kinematic references during phase III, landing 7.4 Helicoper Stabilisation & Flight Path Reference Corrections During a manoeuvre, the pilot must accomplish two tasks, the first is following the intended flight path and the second one keeping the helicopter stable around the flight path. The pilot model described in section 7.2 is able to guide the helicopter along the intended flight path, but it is not guaranteed that it will also keep the helicopter completely stable around the flight path. The causes of instability and the measures that were needed to reduce them are discussed in the following three sections Causes of Instabilities The prime cause of instabilities is the use of the time delay τ in the crossover element. A simulating of phase I with the standard sycos pilot becomes completely unstable after 15 seconds and it crashes at 2 seconds. Removing the time delay results in a slightly unstable simulation. The deceleration phase becomes unstable as the simulation

120 88 Time Simulation - Ship Approach and Landing progresses, but it is not until 125 seconds that it crashes completely. Looking at the resulting Euler angles reveals that the heading Ψ is not well controlled. It shows great fluctuations, with amplitudes peaking at ± 2 degrees. The behaviour of the yaw rate shows a clear unstable oscillation. Furthermore, looking at the velocity references and the actual velocities reveals that the horizontal component of the earth-referenced velocity (ẋ) also shows a divergent oscillation. On average, the references are followed but not in a stable manner Stability Augmentation The tail rotor control value θ tr calculated by the linear inverse is not capable of keeping the heading angle within reasonable limits 5. Therefore, additional feedback is needed to prevent the heading from diverging from its intended value. An inner-loop corrective control action θ tr is added to the controls of the pilot in the form of a PID-controller. It has the following general form [Pavel], t θ tr = K r r + K Ψ1 (Ψ des Ψ) + K Ψ2 (Ψ des Ψ)dτ (7.35) for the current manoeuvre, the desired heading angle Ψ des is always. The values of the three gains are listed in Table 7.6. Gain Value K r -.75 K Ψ1 -.7 K Ψ2 -.1 Table 7.6: Values of the gains used in the heading stabiliser With this additional controller, the simulation is completed successfully. Therefore, one may conclude that as already stated the heading control (or more precisely lack thereof) is one of the prime reasons for the instabilities, together with the time delay. The resulting control action for the first phase (deceleration from 4 m/s to 5 m/s) is shown in Fig The velocity references and velocities are shown in the figure thereafter, Fig On top of the mean value, all controls show an additional slowly varying oscillation with a fairly large amplitude. If given enough time, they will converge to a steady-state value, but much too slow. These control fluctuations are caused by the marginal stability of the pitch oscillation of the constrained linear system (Fig. 7.4 and Table 7.1). So, additional feedback is needed to increase the stability of the linear model. The following PD-feedback law is used inside the linear model to stabilize the pitch oscillation, θ 1s = K q q + K Θ Θ (7.36) 5 This is not completely surprising since the rate of change of heading Ψ is controlled by the pilot and not the heading itself.

121 7.4 Helicoper Stabilisation & Flight Path Reference Corrections θ [deg] 12 θ1s [deg] θ1c [deg] θtr [deg] Time [s] Time [s] Figure 7.15: Pilot control activity during phase I of the simulation: closure to ship 45 ẋ [m/s] 3 15 ẏ [m/s] ż [m/s] Ψ [deg/s] Time [s] Figure 7.16: Velocities and velocity references during phase I of the simulation: closure to ship

122 9 Time Simulation - Ship Approach and Landing where K q and K Θ have values of.1 and.5, respectively. The inclusion of this PDfeedback results in smooth controls without any additional low-frequency damping or instabilities due to the time delay Position Corrections Due to the time delay and gain of the crossover element, there is a small delay between the intended velocities and the actual velocities at every time instant. As a result, there will be a difference between the actual position of the helicopter and the intended position at the end of the deceleration. The second plot in Fig shows the lateral velocity as a function of time. It would be a coincidence if the area under that curve would add up to exactly zero. An indeed, the resultant is not zero, but positive, i.e. the helicopter has moved somewhat to the right. The same holds true for the longitudinal position, the helicopter is flying some metre too far ahead. The positional errors are shown in Fig At the end of the deceleration, the helicopter is flying approximately 25 metre too far ahead, 1 metre to the right and 1 metre too low. The heading is correct, thanks to the feedback of Eq These errors are corrected at their source. The actual positions and velocities of the helicopter are fed back to the subsystem that generates the references and used to calculate a correction for the velocity reference. The corrections applied here try to replicate the following pilot behaviour: low priority is given to positional errors when flying at high speeds and corrective actions are given higher priority as speed is reduced. This behaviour can be achieved with the following controller, V corr = K p (x x ref ) tanh V V (7.37) inhere, x and x ref are the three actual positions and position references and V is the total airspeed of the helicopter. The value of tanh V V is 1 for hover and decays to zero as speed is increased. If the above function is used in the time simulations with a value of.35 for the gain K p, the results as depicted in Fig are obtained. The upper plot shows the velocity corrections applied to the reference velocities and in the lower part, one can see the resulting positional errors. Starting from 8 seconds (V 23 m/s), the corrections start to increase, resulting in reduced errors in position when compared with the uncorrected case (Fig. 7.17). At 125 seconds (V 5 m/s), the corrections and errors both reach a maximum, after which they both reduce to zero. In the end, a position is reached within 1 m from the intended spatial position. At this point, all necessary additions to the bare sycos piloting structure have been explained. The actual results are discussed in the next section.

123 7.4 Helicoper Stabilisation & Flight Path Reference Corrections 91 Ψ-error Positional errors [m, deg] x-error y-error z-error Ψ-error Time [s] żfigure corr. 7.17: Uncorrected positional and heading errors during phase I: closure to ship z-error Reference velocity corrections [m/s] Positional errors [m] ẋ corr. ẏ corr. ż corr. x-error y-error z-error Time [s] Figure 7.18: Positional and heading errors during phase I: closure to ship

124 92 Time Simulation - Ship Approach and Landing 7.5 Results The model structure described in the above sections was implemented as a Simulink model. The general structure is the same as the graph in Fig. 7.1 and it is shown at the end of this chapter, in Fig Spatial Position of the Helicopter and Ship in Time In Fig. 7.19, a spatial plot (with some time instants) is given of the position of the helicopter and the ship in time. The ship starts 23 metre ahead and 3 metre to the right of the helicopter. The initial altitude of the helicopter is 12 metre, and during phase I, it will descent to 2 metre and reduce its speed from an initial 4 m/s to 5 m/s. After that, a lateral repositioning is executed, ending in stationkeeping above the flight deck of the ship. A descent is initiated, ending in touchdown on the deck with a small vertical velocity. The total simulation takes about 19 seconds vertical position (h) [m] lateral position (y) [m] longitudinal position (x) [m] Figure 7.19: Spatial position of the Puma and frigate During the first phase, there are two peaks in the lateral-positional error, one just after the initial transition (29 seconds mark) and the last one at the end of the final transition of phase I (121 seconds mark).

125 7.5 Results Velocity references and Controls Similar to Fig. 7.16, Fig. 7.2 shows the final velocities and velocity references as a function of time. Note that in the latter figure, the reference velocities may differ from the ones shown in the former due to the controller discussed in section The helicopter follows the horizontal references better than was the case without the position corrections. Two control regions can be distinguished from Fig. 7.21, the first one shows slowly varying controls during phase I, and more abrupt controls during the last two phases, the lateral repositioning and the vertical descent. The first part is a fairly gentle manoeuvre because the transition periods take a relatively long time ( 22 seconds). The pilot control activity during the last two phases is shown in more detail in Fig It is clear from these figures that the control activity of the pilot firstly depends upon the aggressiveness of the task at hand and to a lesser extend on the stability and damping characteristics of the constrained system. If they do not have adequate damping, the controls will be sluggish and the errors will become big (as could be seen in Fig. 7.15). To increase the similarity between the controls produced with the sycos pilot model and a real pilot, nonlinear elements such as dead-band and hysteresis loops should be added to the pilot control structure. Then, control output would be more discrete with a stepped appearance Robustness of the Pilot Model In order to assess the robustness of the pilot model in case of uncertainties in the nonlinear model parameters, the following three parameters were varied in a number of combinations which are given in Table. 7.7, position of the centre of gravity 6, mass of the helicopter and moments of inertia and product of inertia of the fuselage. An mass increase of approximately 22 kg without increasing the inertia of the helicopter results in a reduced performance, with similar fluctuations in the controls as in Fig If the inertia moments are increased (3 and 4) accordingly, these fluctuations disappear again. Reducing the mass (2) increases the performance of the helicopter. Displacing the centre of gravity too much in the longitudinal direction (5, 7, 8 and 9), reduces the stability of the helicopter and the ability of the pilot to stabilize the helicopter significantly. A lateral displacement of the centre of gravity (1 and 11) influences the lateral-directional stability of the helicopter during the simulation somewhat, but less significant than in the longitudinal direction. To conclude, the sycos pilot model is able to cope with mass, inertia and centre of gravity changes as long as they stay within reasonable values. The most critical change in centre of gravity location is the fore-aft direction, the range of which may not be bigger than.5 metre. 6 The positions are given with respect to the hub frame of reference, as in Pavel [21].

126 94 Time Simulation - Ship Approach and Landing 45 ẋ [m/s] 3 15 ẏ [m/s] ż [m/s] Ψ [deg/s] Time [s] Figure 7.2: Velocities and velocity references during the complete manoeuvre θ [deg] 12 θ1s [deg] θ1c [deg] 1.5 θtr [deg] Time [s] Time [s] Figure 7.21: Pilot control activity during the complete manoeuvre

127 7.5 Results θ [deg] θ1s [deg] θ1c [deg] θtr [deg] Time [s] Time [s] Figure 7.22: Pilot control activity during the last two phases, lateral repositioning and landing config. m I x I y I z J xz x cg y cg z cg notes default config. 1 8 increased mass 2 45 decreased mass % +5% +5% 5 % increased inertia % +1% +1% 1 % increased inertia 5.75 cg.75 m in front of RH 6.5 cg.5 m in front of RH cg.4 m behind RH cg.2 m behind of RH cg.11 m in front of RH 1.6 cg.6 m to the right of CL cg.6 m to the left of CL Table 7.7: Parameter variations used to test the robustness of the pilot model

128 96 Time Simulation - Ship Approach and Landing Load trim and Load Initial linearisation data Control Values Generate Landing Pattern velo_earth velo_ref pos_in Accelerations, Velocities, Positions & Azimuth angle Nonlinear Model Inverse Time Simulation Wim Van Hoydonck Thu Jul 2 17:17: Transport Delay 2 2 Crossover frequency Crossover frequency K*u Gain init_controls controls at trim condition K*u Gain1 x. x = Ax + Bu linear system sfunction In1 linear system states output trim controls linear model controls liner model controls R2D R2D sas controls2 controls../postpro/nlin_ctrls2_final.mat complete controls lin_vel_body SAS Helix Fortran Helicopter Model ang vel euler lin_vel_body ang_vel euler euler_dot output position pos+psi vel+dpsi vel_earth accel_earth + ddpsi System output grouping & filtering Figure 7.23: Overview of the Simulink implementation of the SYCOS pilot model sas controls ostpro/lin_vel_ang_vel2_final.m lin_vel R2D ang_vel R2D euler position pos+psi../postpro/nlin_positions_final.mat To File3 pos Simulation End trigger references1 references /postpro/nlin_refs2_final.m

129 Chapter 8 Conclusions and Recommendations 8.1 Conclusions The goal of this study was to develop and validate a helicopter model in support of maritime operations. A literature survey indicated that the use of nonuniform dynamic inflow through the main rotor can improve the fidelity of the helicopter model as a whole. To allow for future extensions, the rotor aerodynamics are modeled with numerical blade element theory. Originally, the model was implemented as Matlab M-code. However, this makes time simulations extremely time-consuming. Therefore, the source code including the trim and linearisation routine was converted to Fortran 95, making the model approximately 1 times as fast. The mathematical model developed can be to simulate helicopters with a clockwise or counterclockwise rotating main rotor. This made it possible to validate the model for both the Puma and the Bo15. Overall, the trim results show good comparison with results found in literature. The inclusion of Pitt-Peters nonuniform inflow improves the lateral tilt prediction of the main rotor in trim when compared with a uniform inflow model. Small additions such as pitch-flap coupling and a precone angle for the Bo15 improve the prediction of the coning angle significantly. By implementing a separate trim routine for the main rotor, the tail rotor and the fuselage, the helicopter including the the rotor states can be trimmed completely. Regarding the linearisation of the helicopter, nonuniform inflow only influences the coupling derivatives in the case of the Puma, for the Bo15, the uncoupled derivatives are also affected. Its influence is mainly visible in hover and low-speed flight. The phugoid becomes more unstable and the damping of the short period mode in and near hover is somewhat larger. It was demonstrated that large differences exist between initial and long-term response of the 6-DOF linear helicopter. By using doublet inputs for all controls, it was furthermore shown that these differences mainly exist for the main rotor cyclic inputs and to a lesser extend for the collective inputs for the main and tail rotors. For long time simulations, it is best to use a linear model in which the rotors are trimmed after a disturbance is applied. The inclusion of wake distortion dynamics results in sign changes in the off-axis response after a doublet input. 97

130 98 Conclusions and Recommendations A pilot model consisting of a crossover element and an inverse of the linearised model was used to simulate a helicopter performing a ship landing. The constrained linear model proved to be too lightly damped in pitch which made it necessary to include some feedback in the linear model. Heading control without an additional PID-controller proved inadequate, the exact cause is however uncertain. Overall, the resulting control structure utilised here proved extremely versatile. The exact same pilot model was used to execute three completely different manoeuvres: a decelerated approach along a three degree glide slope to a point alongside the ship, a sidestep ending in stationkeeping above the ships flight deck and a landing. Using a simple parameter variation, it was shown that the pilot model performs well if the mass properties of the helicopter are not known accurately. 8.2 Recommendations The blade element main rotor was specifically implemented with future extensions in mind. These include but are not limited to Peters-He finite state inflow modelling and the extensions based on this model for partial and dynamic ground effect. Skewed flow at the blade elements may further improve the prediction of lateral rotor tilt in trim. In the blade flappings dynamics, the influence of the linear velocity on the total kinetic energy was neglected. To make the model valid for the complete flight envelope, the main rotor dynamics have to be upgraded. A wake distortion dynamics model was included in the main rotor model. However, its influence on the helicopter responses was not investigated extensively. This should be done with more care. Interactional aerodynamics between the different parts of the helicopter were completely neglected here, influence of main rotor inflow on forces and moments generated by the tail plane surfaces and tail rotor can be significant. Related to this, the aerodynamics model of the fuselage needs upgrading. It was assumed that all forces and moments produced by the fuselage originate at the centre of gravity. This is wrong since they originate at the centre of pressure of the fuselage. It was further assumed that the fuselage only produces a pitch moment and a drag force, neglecting the roll and yaw moments. Table look-up models generated from wind tunnel tests should be used for that. It was shown that the sycos pilot model can be used to simulate a landing manoeuvre on a ship deck. The performace of the pilot model itself should be investigated by simulating (high agility) standard manoeuvres from ADS-33. In place of continuous control inputs, pilots make discrete adjustments to control inputs (stepped appearance). Nonlinear elements such as dead-band and a hysteresis can be added to the pilot control structure to mimic this behaviour. Regarding the simulation program itself, the analysis parts should be decoupled completely from the dynamics model. This is of utmost importance for future extensions that add states and/or degrees of freedom to the model. E.g. the core of the trim routine consists of three subroutines, which all implement a Newton iteration to trim a certain part of the helicopter model. This should be generalized to one trim routine, flexible enough to trim any part, independent of the number of states or degrees of freedom.

131 References Anon.(22). Using MATLAB. Natick: The MathWorks, Inc. Arnold, U. T. P., Keller, J. D., Curtiss, H. C., & Reichert, G. (1995, Aug.). The Effect of Inflow Models on the Dynamic Response of Helicopters. In Proceedings of the 21st European Rotorcraft Forum (Vol. 2, pp. VII.8.1 VII.8.14). St. Petersburg, Russia. Booij, P.(25, Nov.). Centre of Knowledge Romania The Netherlands, Helicopter-Ship Qualification. (Presentation NLR) Bradley, R., & Brindley, G. (22, Sept.). Progress in the Development of a Robust Pilot Model for the Evaluation of Rotorcraft Performance, Control Strategy and Pilot Workload. In Proceedings of the 28th European Rotorcraft Forum. Bristol, UK. Bradley, R., & Thomson, D. G. (25, Sept.). Helicopter and Tilt-Rotor Inverse Simulation: Methods, Features, Problems and Cures. In Proceedings of the 31st European Rotorcraft Forum. Florence, Italy. Carico, G. D., Fang, R., Finch, R. S., Geyer Jr., W. P., Krijns, C., & Long, K. (23, Feb.). Helicopter/ship qualification testing (RTO-AG-3 Vol. 22). Neuilly-sur-Seine: NATO. Carpenter, P. J., & Fridovitch, B. (1953, Nov.). Effect of Rapid Blade-Pitch Increase on the Thrust and Induced Velocity Response of a Full-Scale Helicopter Rotor (NACA TN-344). Castles Jr., W., & de Leeuw, J. H.(1954). The Normal Component of the Induced Velocity in the Vicinity of a Lifting Rotor and Some Examples of its Application (NACA Report No. 1184). Chen, R. T. N.(1989, Sept.). A Survey of Nonuniform Inflow Models for Rotorcraft Flight Dynamics and Control Applications. In Proceedings of the 15th European Rotorcraft Forum. Amsterdam. Coleman, R. P., Feingold, A. M., & Stempin, C. W. (1945). Evaluation of the Induced- Velocity Field of an Idealized Helicopter Rotor (NACA-WR-L-126). Langley Memorial Aeronautical Laboratory, Langley Field, Virginia. Cook, M. V.(1997). Flight Dynamics Principles. Arnold. de Back, E. (1994). Effects of Non-Uniform Dynamic Inflow on Helicopter Flight Responses An Investigation of the Pitt-Peters Model. M.Sc. Thesis, Delft University of Technology, Delft. 99

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133 References 11 Jensen, P. T., & Curtiss Jr., H. C. (1991, Aug.). An Analytically Linearized Helicopter Model with Improved Modeling Accuracy - Interim Report, August 1, 199 August 1, 1991 (No. NASA-CR ). Princeton: Princeton University, Department of Mechanical and Aerospace Engineering. Johnson, W. (198). Press. Helicopter Theory. Princeton, New Jersey: Princeton University Keller, J. D. (1995, Aug.). An Investigation of Helicopter Dynamic Coupling Using an Analytical Model. In Proceedings of the 21st European Rotorcraft Forum (Vol. 2, pp. VII.14.1 VII.14.14). St. Petersburg, Russia. Keller, J. D., & Curtiss, H. C. (1996, June 4 6). Modeling the Induced Velocity of a Maneuvring Helicopter. In Proceedings of the 52nd Annual Forum of the American Helicopter Society. Washington, D.C. Kim, F. D., Celi, R., & Tischler, M. B. (1993, Nov. Dec.). Forward Flight Trim and Frequency Response Validation of a Helicopter Simulation Model. Journal of Aircraft, 3 (6). LAPACK Linear Algebra PACKage. (26, July 12). lapack/. Leishman, J. G. (22). Principles of Helicopter Aerodynamics. Cambridge: Cambridge University Press. Lumsden, R. B., Wilkinson, C. H., & Padfield, G. D. (1998, Sept.). Op2 Challenges at the Helicopter-Ship Dynamic Interface. In Proceedings of the 24th European Rotorcraft Forum. Marseilles, France. McRuer, D. T., & Krendel, E. S. (1974, Jan.). Mathematical Models of Human Pilot Behaviour (AGARDograph No. AG 188). NATO Advisory Group for Aerospace Research and Development. Padfield, G. D.(2). Helicopter Flight Dynamics. Oxford: Blackwell Science Ltd. Padfield, G. D., McCallum, A. T., Haverdings, H., Dequin, A., Haddon, D., Kampa, K., et al. (1996, Sept.). Predicting Rotorcraft Flying Qualities Through Simulation Modelling A Review of Key Results from GARTEUR AG6. In Proceedings of the 22nd European Rotorcraft Forum (Vol. 2, pp ). Brighton. Pavel, M. D. (1996). Six Degrees of Freedom Linear Model for Helicopter Trim and Stability Calculation (No. Memorandum M-756). Delft: Delft University of Technology. Pavel, M. D. (21). On the Necessary Degrees of Freedom for Helicopter and Wind Turbine Low-Frequency Mode Modeling. Ph.D. Thesis, Delft University of Technology, Delft. Peters, D. A., & HaQuang, N.(1988, Oct.). Technical Note Dynamic Inflow for Practical Applications. Journal of the American Helicopter Society, pp

134 12 References Peters, D. A., & He, C. J.(1989, May). Correlation of Measured Induced Velocities with a Finite-State Wake Model. In Proceedings of the 45th Annual Forum of the American Helicopter Society. Boston, Mass. Peters, D. A., He, C. J., & Boyd, D. D.(1989, Oct.). A Finite-State Induced-Flow Model for Rotors in Hover and Forward Flight. Journal of the American Helicopter Society, Vol. 34 (No. 4). Pitt, D. M., & Peters, D. A. (1981). Theoretical Prediction of Dynamic Inflow Derivatives. Vertica, Vol. 5, pp Pitt, D. M., & Peters, D. A.(1983, Sept ). Rotor Dynamic Inflow Derivatives and Time Constants from Various Inflow Models. In Proceedings of the Ninth European Rotorcraft Forum. Stresa, Italy. PLplot a Scientific Plotting Library. (26, July 12). net/. Porat, B.(1997). A Course in Digital Signal Processing. New York: John Wiley & Sons, Inc. Prasad, J. V. R., Zhao, J., & Peters, D. A. (21). Modeling of Rotor Dynamic Wake Distortion during Maneuvering Flight (No. AIAA ). Prasad, J. V. R., Zhao, J., & Peters, D. A.(22, Sept.). Helicopter Rotor Wake Distortion Models for Maneuvering Flight. In Proceedings of the 28the European Rotorcraft Forum. Bristol, UK. Ruijgrok, G. J. J. (199). Elements of Airplane Performance. Delft: Delft University Press. Takahashi, M. D. (199, Feb.). A Flight-Dynamic Helicopter Mathematical Model with a Single Flap-Lag-Torsion Main Rotor (No. NASA TM-12267). Tischler, M. B.(1991). System Identification Methods for Handling Qualities Evaluation. In Rotorcraft System Identification, LS 178. Neuilly sur Seine: AGARD. Turner, G., Bradley, R., & Brindley, G.(2, Sept.). Simulation of Pilot Control Activity for the Prediction of Workload Ratings in Helicopter/Ship Operations. In Proceedings of the 26th European Rotorcraft Forum. The Hague, Netherlands. van Aalst, R.(22). On the Question of Adequate Modelling of Steady-State Rotor Disc- Tilt for Helicopter Manoeuvring Flight. M.Sc. Thesis, Delft University of Technology, Delft. van der Vorst, J.(1998, Sept.). A Pilot Model for Helicopter Manoeuvres. In Proceedings of the 24th European Rotorcraft Forum. Marseilles, France. van Gool, P. C. A.(1997). Rotorcraft Responses to Atmospheric Turbulence. Ph.D. Thesis, Delft University of Technology, Delft. van Holten, T., & Melkert, J. A.(1994). Prestaties en Vliegeigenschappen van Hefschroefvliegtuigen (in Dutch). Delft: Delft University of Technology.

135 References 13 Van Hoydonck, W. R. M. (26, May 6). Literature Survey on the Simulation and Handling Qualities for Helicopters Operating near Ship Decks. be/tuinbels/afstuderen/litstudy final.pdf. van Kan, J.(2). Numerieke Wiskunde voor technici. Delft: Delft University Press. Visual Numerics - IMSL Library. (26, July 12). imsl/. Voorsluijs, G. M. (23). A Modular Generic Helicopter Model. M.Sc. Thesis, Delft University of Technology. Wheatley, J. B.(1935). An Aerodynamic Analysis of the Autogiro Rotor with a Comparison between Calculated and Experimental Results (NACA Report No. 487). Xin, H. (1999). Development and Validation of a Generalized Ground Effect Model for Lifting Rotors. Ph.D. Thesis, Georgia Institute of Technology, Georgia, Atlanta. Xin, H., & He, C. J.(22, June 11 13). A Combined Technique for Inverse Simulation Applied to Rotorcraft Shipboard Operations. In Proceedings of the 58th Annual Forum of the American Helicopter Society. Montreal, Canada. Xin, H., Prasad, J. V. R., & Peters, D. A.(1999, August 9 11). Dynamic Inflow Modeling for Simulation of a Helicopter Operating in Ground Effect. In Proceedings of the AIAA Modeling and Simulation Technology Conference. Portland, Oregon: AIAA. Xin, H., Prasad, J. V. R., Peters, D. A., Iboshi, N., & Nagashima, T. (2, May). Correlation of Experimental Measurements with a Finite-State Ground Effect Model. In Proceedings of the 56th Annual National Forum of the American Helicopter Society (pp ). Virginia Beach. Xin, H., Prasad, J. V. R., Peters, D. A., Itoga, N., Iboshi, N., & Nagashima, T. (1999, June). Ground Effect Aerodynamics of Lifting Rotors Hovering above Inclined Ground Plane. In Proceedings of the AIAA Applied Aerodynamics and CFD Conference. Norfolk, Virginia: AIAA. Ypma, F. A. K. (1996). On the Prediction of Blade Flapping Angles A Coupled Rotor Model Compared with Experimental Data. M.Sc. Thesis, Delft University of Technology, Delft. Zhao, J.(25). Dynamic Wake Distortion Model for Helicopter Maneuvring Flight. Ph.D. Thesis, School of Aerospace Engineering, Georgia Institute of Technology, Georgia, Atlanta. Zhao, J., Prasad, J. V. R., & Peters, D. A.(22, Aug.). Simplified Dynamic Wake Distortion Model for Helicopter Transitional Flight. In Proceedings of the AIAA Atmospheric Flight Mechanics Conference and Exhibit. Monterey, California. Zhao, J., Prasad, J. V. R., & Peters, D. A.(23, May). Investigation of Wake Curvature Dynamics for Helicopter Maneuvering Flight Simulation. In Proceedings of the 59th Annual Forum of the American Helicopter Society. Phoenix, Arizona.

136 14 References

137 Appendix A Main Rotor Mathematical Model A.1 Blade Element Velocity Components A.1.1 Assumptions and Restrictions The following assumptions and restrictions are used for the derivation of the velocity components at a blade section: - the rotor is articulated with centrally placed flapping hinges; - lead-lag hinges are not considered, so there is no in-plane motion of the blades; - pitch-flap coupling (δ 3 ) is introduced through the control system geometry; - the blades are rectangular and rigid in all directions and may have a root cutout; - a linear twist θ tw is applied; - the centre of gravity has three offsets with respect to the helicopter body fixed frame of reference; - the origin of the shaft axis also has three offsets with respect to the helicopter body fixed frame of reference; - the shaft of the rotor has a tilt angle s around the positive Y-axis of the body fixed frame of reference; r - the induced velocity is of the form v i (ψ) = v +v 1s R sin ψ+v 1c r R cos ψ and is defined positive in the positive z-direction of the tip path plane frame of reference. In order to calculate the forces and moments on a blade section, the velocity components of that blade section must be derived. The helicopter centre of gravity is allowed to have linear and angular velocities in the centre of gravity frame of reference 1. The linear velocity components are expressed as V cg = (u, v, w) { E cg } 1 Definitions of the reference frames used in this derivation can be found in Chapter 3. (A.1) 15

138 16 Main Rotor Mathematical Model where { E cg } is the unit vector in the centre of gravity frame of reference. The angular velocity of the centre of gravity is defined in a similar way ω cg = (p, q, r) { E cg } (A.2) The centre of gravity is allowed to have three arbitrary offsets (f x, f y, f z ) with respect to the body fixed frame of reference, { E bf }, r bf = (f x, f y, f z ) { E bf } (A.3) The respective axes of both reference frames are located in the same direction, so that the unit vectors stay the same { Ebf } { Ecg } The rotor system is defined with the reference frames as described hereafter. (A.4) A.1.2 Rotor System The rotor origin frame of reference {E ro } is obtained through a translation from the body fixed frame of reference { E bf } over r ro = (r x, r y, r z ) { E bf } (A.5) The shaft axis frame of reference {E sa } is obtained by a rotation s over the positive Y -axis of the rotor origin frame of reference. The transformation matrix that defines the relation between these two reference frames is given in Chapter 3. The actual rotor head, which is the origin of the hub plane frame of reference, is located at a distance h r in the positive z-direction of the shaft axis frame of reference. The linear velocity of this origin is the sum of the linear velocity of the centre of gravity and the velocity with respect to that origin due to the angular velocity of the centre of gravity, V hp = V cg + ω cg r cg hp which, when expressed in the centre of gravity frame of reference, becomes, u + q(r z f z + h r C s ) r(r y f y ) V hp = v + r(r x f x + h r S s ) p(r z f z + h r C s ) w + p(r y f y ) q(r x f x + h r S s ) T { Ecg } (A.6) (A.7) The angular velocity of the blade is the sum of the angular velocity of the centre of gravity of the helicopter, the clockwise rotor speed Ω and the angular motion of the blade due to flapping. The total angular velocity of the blade is then ω bl = (p, q, r) {E ca } + Ω(,, 1) {E ca } + β(, 1, ) {E ba } (A.8)

139 A.1 Blade Element Velocity Components 17 The relation between a unit vector in the blade axis system and a unit vector in the hub plane is defined by two successive matrix transformations, {E ba } = [β][ψ] { E hp } (A.9) The total angular velocity of the blade element (Eq. (A.8)) expressed in the hub plane equates to p C s r S s β T S ψ ω bl = q β { } C ψ Ehp (A.1) p S s + r C s + Ω The total kinematic velocity of an arbitrary point on the blade at a distance r b from the flapping hinge expressed in the hub plane frame of reference is the sum of the velocity of the control axis reference frame and the velocity relative to this reference frame V bl = V hp + ω bl r bl (A.11) with r bl defined as r bl = (r b,, ) {E ba } (A.12) The total velocity of the blade element needs to be expressed in the blade axis reference frame, [ ( C β C ψ C s u + q(rz f z ) r(r y f y ) ) T ( S s w + p(ry f y ) q(r x f x ) ) ] + qh r [ + C β S ψ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ] [ ( S β C s w + p(ry f y ) q(r x f x ) ) ( + S s u + q(rz f z ) r(r y f y ) )] [ C ψ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ] [ ( S ψ C s u + q(rz f z ) r(r y f y ) ) ( V ba = S s w + p(ry f y ) q(r x f x ) ) ] + qh r [ ] {E ba } (A.13) +r b C β Ω + r C s + p S s [ ] r b S β Cψ (p C s r S s ) q S ψ [ ( S β C ψ C s u + q(rx f x ) r(r y f y ) ) ( S s w + p(ry f y ) q(r x f x ) ) ] + qh r [ + S β S ψ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ] [ ( + C β C s w + p(ry f y ) q(r x f x ) ) ( + S s u + q(rz f z ) r(r y f y ) )] [ +r b q Cψ + S ψ (p C s r S s ) β ]

140 18 Main Rotor Mathematical Model V Xba X ba V Yba Yhp Ω β V Zba X pa ψ ψ ψ Y pa = Y ba X hp β β Z ba Z hp = Z pa Figure A.1: Rotor axes transformations For comparison with Ypma [1996, p. 95, Eq. A.1], the geometric offsets f x, f y, f z, r x, r y and r z and the tilt angle s should be set zero, which reduces the above set of equations to C β C ψ (u + qh r ) + C β S ψ (v ph r ) w S β V ba = S ψ (u + qh r ) + C ψ (v ph r ) + r b C β (Ω + r) r b S β (p C ψ q S ψ ) C ψ S β (u + qh r ) + S ψ S β (v ph r ) + r b (q C ψ + p S ψ β) + w C β T {E ba } (A.14) Except for the sign changes of the term involving h r and Ω, which are due to the different definition of rotor origin offset and opposite rotor rotational direction 2, the equations are identical. A.2 Blade Element Forces In this section, we zoom in on a single blade element along the span of the blade. 2 This derivation is for a clockwise rotating rotor, while Ypma derived her model for an anticlockwise rotating rotor.

141 A.2 Blade Element Forces 19 A.2.1 Assumptions For the derivation of the blade element forces, the following assumptions regarding aerodynamics are used: - the flow is assumed to be two dimensional, the radial component is neglected for the calculation of the local angle of attack and the total velocity; - angles are not assumed to be small; - all blade elements have a symmetrical profile, so that the profile pitching moment around the quarter-chord line can be neglected 3 ; - all control points are located on the quarter chord line of the blade, in the middle of their respective blade sections. It is assumed that the aerodynamic properties at that point are valid for the whole blade segment. A.2.2 Blade Segment Aerodynamics For the derivation of the forces at the blade element, the relative velocity of the air with respect to blade must be known. This is done by fixing the blade and looking along the X-axis of the blade frame of reference (Fig. A.2). The relative radial (U r ), tangential (U t ) and perpendicular (U p ) velocity components at any blade segment are [ ( U r = C β C ψ C s u + q(rz f z ) r(r y f y ) ) S s ( w + p(ry f y ) q(r x f x ) ) + qh r ] [ C β S ψ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ] [ ( + S β C s w + p(ry f y ) q(r x f x ) ) + S s ( u + q(rz f z ) r(r y f y ) )] ( U t = C ψ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ) ( ( S ψ C s u + q(rz f z ) r(r y f y ) ) S s ( w + p(ry f y ) q(r x f x ) ) + qh r ) ( ) ( ) + r b C β Ω + r C s + p S s rb S β Cψ (p C s r S s ) q S ψ [ ( U p = S β C ψ C s u + q(rx f x ) r(r y f y ) ) S s ( w + p(ry f y ) q(r x f x ) ) + qh r ] S β S ψ [ v + r(rx f x + h r S s ) p(r z f z + h r C s ) ] C β [ C s ( w + p(ry f y ) q(r x f x ) ) + S s ( u + q(rz f z ) r(r y f y ) )] r b [ q Cψ + S ψ (p C s r S s ) β ] (A.15) 3 This is only true at small angles of attack. The centre of pressure in the poststall regime moves to the midpoint of the airfoil when α = 9. With the maximum drag coefficient approximating the drag coefficient of a flat plate (C d = 2), the maximum moment coefficient is approximately.5.

142 11 Main Rotor Mathematical Model As already said, only the radial and perpendicular velocity components are used to df z dr dl ϕ θ U t α dd ϕ U p df y Figure A.2: Rotor blade sideview: definition of blade forces calculate the flight path angle ϕ, ( ) Up tan ϕ = U t (A.16) The blade pitch angle consists of two parts: a first-order Fourier series with the collective pitch angle θ, longitudinal and lateral cyclic pitch angles θ 1s and θ 1c and the blade twist θ tw, linearly varying with the rotor radius, θ = θ + θ 1c cos ψ + θ 1s sin ψ + θ tw r b R + θ pf (A.17) It should be noted that the blade pitch as used here is defined as a rotation around the pitch bearing of the blade. The longitudinal and lateral cyclic pitch angles are therefore not necessarily the same as the angles that define the orientation of the swashplate. The angle of attack of the blade section now follows as α = θ + ϕ and the local two-dimensional relative velocity equates to V = Up 2 + U t 2 (A.18) (A.19)

143 A.3 Derivation of the Blade Flapping Equation 111 Then, the dimensional local lift and drag forces are obtained from 1 dl = C l 2 ρv 2 cr b (A.2a) 1 dd = C d 2 ρv 2 cr b (A.2b) where c and r b are the local chord and the width of the section in question, respectively. For low angles of attack (between 14 and 14 ), the lift coefficient is assumed to vary proportional to the angle of attack whereas the drag coefficient follows a quadratic equation, C l = C lα α C d = C d + C d1 α + C d2 α 2 (A.21a) (A.21b) In high speed forward flight, the inboard part of the retreating blade may experience reversed and/or stalled flow. When the flow becomes fully separated, the resulting aerodynamic properties become independent of airfoil shape and are close to the values of a flat plate. The quasi-steady lift coefficient can be modeled using [Leishman, 22, pp ] C l = A sin 2(α α ) (A.22) where A = 1.1 for the NACA 12 airfoil. The drag coefficient in the poststall regime can be modeled with C d = D + E cos 2(α α ) (A.23) with D = and E = 1.5. The resulting shapes for the lift and drag coefficients over the full angle of attack regime are depicted in Fig. A.3. Decomposition of the elementary blade element forces of Eqs. (A.2) in the blade element system goes as follows, df y = dl sin ϕ + dd cos ϕ df z = dl cos ϕ + dd sin ϕ (A.24a) (A.24b) Note that here, the blade element forces are decomposed in the negative directions of the blade axis system as can be seen by comparing Figs. 3.2 and A.2. A.3 Derivation of the Blade Flapping Equation A.3.1 Derivation in the Rotating Frame of Reference For the derivation of the blade flapping equation, use is made of Lagrange s equation in a non-conservative force field, ( ) d T dt β T β + V β = Q β (A.25) where β is the blade flapping degree of freedom.

144 112 Main Rotor Mathematical Model Cl [-] cd [-] α [deg] Figure A.3: Lift and drag coefficient as a function of angle of attack. The rotor blades are straight and rigid with the mass concentrated at the quarter-chord line. The blade flapping hinge is also located at the quarter-chord line, so only the moment of inertia around the flapping hinge is important. This means that for the kinetic energy due to rotation (T ω = 1 2 I ω 2 ) only the angular velocities around the Y ba and Z ba axes make a contribution 4. Transformation of Eq. (A.1) to the blade axis system results in C β C ψ (p C s r S s ) + C β S ψ q S β (p S s + r C s + Ω) ω ba = S ψ (p C s r S s ) q C ψ + β S β ( C ψ (p C s r S s ) q S ψ ) + C β (p S s + r C s + Ω) T {E ba } (A.26) As already mentioned, only the second and third entry of the blade angular velocity are used to calculate the kinetic energy, 1 2 I ( β ω 2 + ω2 Yba Z ba) (A.27) 4 The kinetic energy due to translation of the rotor hub is not taken into account in this derivation. This assumption is only valid in the lower flight regime.

145 A.3 Derivation of the Blade Flapping Equation 113 The first term in Lagrange s equation equals ( ) d T ( ( ) dt β =I β β + Ω Cψ (p C s r S s ) q S ψ ) S ψ (ṗ C s ṙ S s ) q C ψ (A.28) The derivative of the kinetic energy with respect to the flapping angle β is somewhat more involved, 1 T [ ( I β β = S ) 2 β C β Cψ (p C s r S s ) q S ψ ( p S s + r C s + Ω ) 2 ] + (1 2 C 2 β ) [ Ω ( C ψ (p C s r S s ) q S ψ ) ( p 2 r 2 ) + C ψ S 2 s + rp C 2 s 2 ] q S ψ (r C s + p S s ) (A.29) The rotor blades of the Puma helicopter has flapping hinges, which means that the hub stiffness is significantly lower than the hub stiffness of a hingeless or bearingless rotor design, but it is not zero and therefore, it is taken into account here. V β = d ( ) 1 dβ 2 K ββ 2 = K β β (A.3) and the right-hand side of the equation, Q β = M a (A.31) The aerodynamic moment is obtained by integrating the perpendicular component of the aerodynamic force (Eq. (A.24)) on every blade element along the span of the blade, M a = R df Z rdr = R n df Z (i) r(i) i=1 (A.32) Rearranging the terms ultimately results in β = M a K β β I β Ω ( C ψ (p C s r S s ) q S ψ ) + S ψ (ṗ C s ṙ S s ) + q C ψ [ ( ) 2 ( + S β C β Cψ (p C s r S s ) q S ψ p S s + r C s + Ω ) ] 2 (2Cβ 2 1) [Ω ( ) C ψ (p C s r S s ) q S ψ + C ψ ( p 2 r 2 2 ) ] S 2 s + rp C 2 s q S ψ (r C s + p S s ) (A.33)

146 114 Main Rotor Mathematical Model This equation can considerably be simplified when the influence of the rotor shaft tilt s is completely neglected and a small angle approximation 5 for the blade flapping angle β is assumed β = M a K β β I β Ω (p C ψ q S ψ ) + β ( (p C ψ q S ψ ) 2 (r + Ω) 2) ( (p C ψ q S ψ )(r + Ω) ) (A.34) with exactly the same result as what Ypma (1996) derived during her thesis work with the exception of the sign changes of the terms involving Ω. A.3.2 Conversion from Rotating to the Non-Rotating Frame of Reference Due to the periodic behaviour of an N-bladed helicopter rotor rotating at a speed Ω, the flapping motion of the i-th blade with respect to the hub can be represented as an infinite Fourier series with fundamental period 2π as follows β i = β + β 1s sin ψ i + β 1c cos ψ i + β 2s sin 2ψ i + β 2c C 2ψi + ( 1) i β N/ = β + ( 1) i β N/2 + (β ns sin nψ + β nc cos nψ) n=1 (A.35) So instead of using the position and motion of the individual blades as degrees of freedom of the rotor system, one can study the behaviour of the rotor as a whole by transforming the motion of the individual blades to the non-rotating frame of reference. The usefulness of the Fourier series description of rotor motion is based on the fact that only the lowest few harmonics have significant magnitude, so that the complete periodic motion is described by a small set of numbers 6. For a four-bladed rotor, the coning angle β and lateral and longitudinal tilt angles β 1s and β 1c are used to represent the motion of the rotor in the nonrotating frame. The first reactionless mode β N/2 is not used. Then, Eq. (A.35) reduces to β i = β + β 1s sin ψ i + β 1c cos ψ i (A.36) The inverse transformations can easily be found to be β = 1 N β 1s = 2 N β 1c = 2 N N i=1 β i N β i sin ψ i i=1 N β i cos ψ i i=1 (A.37) 5 cos β 1 and sin β β. 6 Johnson, W., Helicopter Theory, Princeton, Princeton University Press, 198.

147 A.4 Rotor Aerodynamic Forces and Moments 115 The time derivatives of the rotor states in the non-rotating frame are β = 1 N β 1s = 2 N β 1c = 2 N N β i i=1 N ( ) βi sin ψ i Ω β 1c i=1 N ( ) βi cos ψ i + Ω β 1s i=1 (A.38) Using Eqs. (A.35) and (A.38), the angular accelerations of the three states in the nonrotating frame of reference can be calculated β = 1 N β 1s = 2 N β 1c = 2 N N β i i=1 N i=1 N i=1 ( β sin ψi ) 2 Ω β 1s + Ω 2 β 1s ( β cos ψi ) + 2 Ω β 1c + Ω 2 β 1c (A.39) A.4 Rotor Aerodynamic Forces and Moments A.4.1 Rotor Forces and Moments at the Hub Once the blade element forces are known, a transformation to the hub plane is done and all contributions from the individual elements are summed. The blade element forces from Eqs. (A.24) are transformed to the blade axis system with (df x, df y, df z ) ba = (, df y, df z ) be (A.4) and from the blade axis system via the projection axis system to the hub plane with (df x, df y, df z ) hp = (df x, df y, df z ) ba [β][ψ] { E hp } (A.41) The total aerodynamic forces at the centre of the hub are the found by summing all contributions of all blade elements of all blades. The thrust T hp then is the negative of F z hp, the in-plane H-force is the negative of the force contributions in the x-direction and the in-plane Y-force is found by summing the force contributions in the negative y-direction, F xhp = df x = H hp F yhp = df y = Y hp F zhp = df z = T hp (A.42) (A.43) (A.44)

148 116 Main Rotor Mathematical Model X pa X ca Y pa dy dq Y ca dh sin β df zba ψ df yba Figure A.4: Rotor hub top view: definition of rotor forces and moments Every single blade element has in the most general case three geometric offsets with respect to the centre of the hub plane frame of reference. These positions are calculated as follows (x, y, z) hp = (r b,, ) ba [β][ψ] { E hp } (A.45) with r b the dimensional position of the blade element along the span of the blade in question. The moment contribution at the centre of the hub can now be calculated for every blade element using Eqs. (A.45) and (A.41), dm xhp = df z y df y z dm yhp = df x z df z x dm zhp = df y x df x y (A.46) (A.47) (A.48) or M hp = x hp F hp (A.49)

149 A.5 Tip Path Plane Frame of Reference 117 The total aerodynamic moments in the hub plane are found by summing all contributions of every blade element of all blades, similar to (A.42), M xhp = dm xhp = M roll M yhp = dm yhp = M pitch M zhp = dm zhp = M torque (A.5a) (A.5b) (A.5c) The first and second terms, M roll and M pitch respectively, are needed in the differential equations of the Pitt-Peters dynamic inflow model. For that, they must be transformed to the tip path plane frame of reference, which is discussed in section A.5 The third term, M zhp, is the resultant torque of the rotor, mainly due to the drag force acting on the blade elements. The aerodynamic roll and pitch moments M roll and M pitch at the rotor hub are not transferred to the centre of gravity of the helicopter since a rotor with central flapping hinges does not have a resulting moment at the flapping hinges. However, this only holds true for a rotor system without central springs. The addition of which adds a moment for every blade which may have two resultant moments in the hub plane frame of reference. A.4.2 Main Rotor Forces and Moments at the Centre of Gravity The resulting forces and moments the centre of the hub plane are transformed to the centre of gravity of the helicopter as follows. Due to the tilt of the shaft axis with respect to the body fixed frame of reference, the forces must be multiplied with the transformation matrix [ s ]. The force contribution of the main rotor at the centre of gravity then becomes F cg = F hp [ s ] { E cg } (A.51) The same is done for the torque, M cg = ( M hp + F hp x hp cg )[ s ] { E cg } (A.52) where the intermediate rotor origin frame of reference {E ro } is skipped by directly using the offsets between the rotor hub and the centre of gravity. A.5 Tip Path Plane Frame of Reference This section discusses the tip path plane frame of reference, which is often utilised as the primary rotor frame of reference in helicopter performance calculations and momentum theory 7. Two transformation matrices are derived, first the exact transformation matrix and afterwards an approximate transformation matrix which uses the same concept as the Euler transformation matrices used to determine the orientation of an airplane.

150 118 Main Rotor Mathematical Model β 1s β 1c β 1c j tpp e 1 k tpp Y tpp β 1s Y hp Z tpp Z hp X hp Figure A.5: Tip path plane orientation wrt the hub plane reference frame A.5.1 Exact Transformation Matrix The orientation of the tip path plane frame of reference is defined with respect to the hub plane frame of reference (see Fig. A.5), through the angles β 1s and β 1c, which are the first-order coefficients of the following Fourier series [Leishman, 22] β n = β + (β nc cos nψ + β ns sin nψ) (A.53) n=1 Eq. (A.53) defines the blade flapping motion with respect to the rotor hub. From the assumption that the solution for the blade flapping motion is given by the first harmonics only, that is, β n = β + β 1c cos ψ + β 1s sin ψ (A.54) follows that the coefficients β, β 1s and β 1c can be expressed as a function of the individual blade flapping angles β n N b β = 1 β n N b n=1 β 1s = 2 N b β n sin(ψ n ) N b n=1 N b β 1c = 2 β n cos(ψ n ) N b n=1 (A.55) (A.56) (A.57) These angles transform the flapping motion of the individual blades from a rotating frame of reference to a non-rotating frame of reference 8. Using Fig. A.6, two vectors (e 1 and e 2 ) can be derived that lie in the tip path plane ] e 1 = [cos β 1c sin β 1c (A.58) ] e 2 = j tpp = [ cos β 1s sin β 1s (A.59) 7 The dynamic inflow model of Pitt and Peters is an application of momentum theory. 8 A so-called multiblade coordinate transformation (MCT) or Coleman transformation [Pavel, 21]

151 A.5 Tip Path Plane Frame of Reference 119 The second one, (e 2 ) will be used directly in the rotation matrix. The vector defining the normal to a plane is equal to the crossproduct of any two vectors that lie in that plane, so N tpp = e 1 e 2 (A.6) Due to the non-orthogonality of e 1 and e 2, Eq. (A.6) must be normalised to get the unit vector in the z-direction of the tip path plane frame of reference k tpp = N tpp N tpp (A.61) With two out of three unit vectors known, the third is found to be i tpp = j tpp k tpp which leads to { Etpp } = [β1s β 1c ] { E hp } where [β 1s β 1c ] = C β1c norm S β1s S β1c C β1s norm C 2 β 1s S β1c norm C β1s S β1s S β1c C β1s norm C β1c S β1s norm C β1c C β1s norm (A.62) (A.63) (A.64) and norm = cos 2 β 1s sin 2 β 1c + cos 2 β 1c (A.65) A.5.2 Euler Transformation Approximation In Takahashi [199], the transformation matrix from the hub plane to the tip path plane is obtained by rotating around the Y hp -axis over the angle β 1s to an intermediate frame i hp X hp j tpp Y β tpp 1s e 1 β 1c jhp Yhp k hp k hp Z hp Z hp Figure A.6: Tip path plane orientation wrt the hub plane reference frame

152 12 Main Rotor Mathematical Model of reference, using the following transformation matrix 1 [β 1s ] = C β1s S β1s (A.66) S β1s C β1s Then, a rotation around the intermediate Y -axis is performed with the following transformation matrix C β1c S β1c [β 1c ] = 1 (A.67) S β1c C β1c The complete rotation matrix [β 1s β 1c ] then becomes C β1c S β1c S β1s S β1c C β1s [β 1s β 1c ] = C β1s S β1s (A.68) S β1c C β1c S β1s C β1c C β1s Inspection of Eqs. (A.64) and (A.68) reveals that the error won t be very big when using the approximate solution as long as both β 1s and β 1c remain small.

153 Appendix B Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage In this appendix, the mathematical models of the helicopter tail plane surfaces, tail rotor and fuselage are discussed. B.1 Force and Moment Contributions of Empennage and Tail Rotor In this section, the contributions of the horizontal stabiliser, the vertical fin and the tail rotor to the forces in the helicopter equations of motion are listed 1. Interactional aerodynamics and aerodynamic pitching moments are not taken into account for these models. B.1.1 Horizontal Stabiliser The horizontal stabiliser of the Puma helicopter is located near the top of the vertical fin, completely on the left side of the butt-line of the helicopter. Therefore, its centre of pressure has three offsets with respect to the centre of the body fixed frame of reference { Ebf }. Adding to this the three arbitrary offsets of the centre of gravity, the local velocity components at the horizontal stabiliser centre of pressure are written as u hs = u + q(z hs z cg ) r(y hs y cg ) v hs = v + r(x hs x cg ) p(z hs z cg ) w hs = w + p(y hs y cg ) q(x hs x cg ) With these velocities, the angle of attack α hs (see Fig. B.1) is determined ( ) whs α hs = α hs + arctan u hs 1 These models are taken from Voorsluijs [23] (B.1a) (B.1b) (B.1c) (B.2) 121

154 122 Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage In Eq. (B.2), the geometric angle of attack (positive up) is denoted by α hs. For small angles of attack ( 2 α hs 2 ), the lift coefficient C Lhs is supposed to vary linearly with this angle, C Lhs = C Lα α hs the drag coefficient C Dhs C Dhs = C D hs + C 2 L hs is then calculated with (B.3) (B.4) whereas for angles beyond the above range, the same approximations as used for the main rotor airfoils (section A.2.2) is utilised here C Lhs = A sin 2(α hs + α hs ) C Dhs = D + E cos 2(α hs + α hs ) (B.5) (B.6) The distinction between small angle large angles may seem unnecessary for conventional forward flight, but in backward flight, the linear approximation would severely overpredict the magnitudes of the forces generated by the horizontal stabiliser. This, in turn, has a substantial effect on the instability of the helicopter (and hence the flight envelope where the helicopter can be trimmed). For the vertical fin, the same discrimination is made between small and large angles since in sideward flight, the vertical fin is subject to angles of sideslip of the order of 9 degrees. PSfrag The resulting replacements lift and drag forces are then L X bf u hs α α w hs Z bf D Figure B.1: Angle of attack α of the horizontal stabiliser 1 L = C Lhs 2 ρ(u2 hs + w2 hs )S hs (B.7) 1 D = C Dhs 2 ρ(u2 hs + w2 hs )S hs (B.8) At the centre of gravity, these lift and drag forces induce the following forces and moments F x = L sin α D cos α F y = F z = L cos α D sin α M x = F z (y hs y cg ) F y (z hs z cg ) M y = F x (z hs z cg ) F z (x hs x cg ) M z = F y (x hs x cg ) F x (y hs y cg ) (B.9a) (B.9b) (B.9c) (B.9d) (B.9e) (B.9f)

155 B.1 Force and Moment Contributions of Empennage and Tail Rotor 123 B.1.2 Vertical Fin The derivation of the forces and moments due to the vertical fin is similar to the force and moment derivation of the horizontal stabiliser. The local airspeeds at the centre of pressure of the vertical fin are X bf u vf β β L D v vf Y bf Figure B.2: Angle of sideslip β of the vertical fin u vf = u + q(z vf z cg ) r(y vf y cg ) v vf = v + r(x vf x cg ) p(z vf z cg ) w vf = w + p(y vf y cg ) q(x vf x cg ) The angle of sideslip β vf at the vertical fin (Fig. B.2) is ( ) vv F β vf = β vf + arctan u V F (B.1a) (B.1b) (B.1c) (B.11) β vf so that the lift C Lvf and drag coefficient C Dhs for small angles of sideslip can be written as C Lvf = C Lβ β vf (B.12) C Dhs = C D vf + C 2 L vf Again, the large angle approximation is written as (B.13) C Lvf = A sin 2(β vf β vf ) (B.14) C Dvf = D + E cos 2(β vf β vf ) (B.15)

156 124 Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage The resulting lift and drag forces then become 1 L = C Lvf 2 ρ(u2 vf + v2 vf )S vf (B.16) 1 D = C Dvf 2 ρ(u2 vf + v2 vf )S vf (B.17) Similar to Eq. (B.9), transforming these forces to the body fixed frame of reference, the force and moment contributions of the vertical fin at the centre of gravity are F x = L sin β D cos β F y = L cos β D sin β F z = M x = F z (y vf y cg ) F y (z vf z cg ) M y = F x (z vf z cg ) F z (x vf x cg ) M z = F y (x vf x cg ) F x (y vf y cg ) (B.18a) (B.18b) (B.18c) (B.18d) (B.18e) (B.18f) Without the distinction between small and large sideslip angles, trim values for the tail rotor collective pitch in sideward flight are beyond the normal range. This may be the primary reason why in the past it has been so difficult to perform sidestep manoeuvres [van Aalst, 22, p. 31]. B.1.3 Tail Rotor The tail rotor mathematical model is based on Pavel [21] and adapted here to work for the Puma and the Bo15. Both helicopters feature a push rotor, which accelerates air away from the vertical fin, as opposed to a pull rotor, the air of which impinges (part of) the vertical fin. So, under normal operating conditions, the induced flow of the tail rotor for the Puma is to the right and the tail rotor thrust is directed to the left (Fig. B.3a). For the Bo15, the exact opposite holds true, the induced flow is to the left and the tail rotor thrust points to the right (see Fig. B.3b). The local airspeeds at the tail rotor hub due to the linear and angular velocities of the centre of gravity are u tr = u + q(z tr z cg ) r(y tr y cg ) v tr = v + r(x tr x cg ) p(z tr z cg ) w tr = w + p(y tr y cg ) q(x tr x cg ) (B.19a) (B.19b) (B.19c) from which the dimensionless airspeeds parallel µ par and perpendicular µ per to the rotor disc are calculated as u 2 µ par = tr + wtr 2 Ω tr R tr µ per = v tr rotdir Ω tr R tr (B.2) (B.21) where rotdir equals 1 for the Puma and -1 for the Bo15 (as is clear from Fig. B.3).

157 B.1 Force and Moment Contributions of Empennage and Tail Rotor 125 X bf X bf Y bf Y bf Ω P uma Ω Bo15 T µ per µ per T (a) Puma tail rotor top view, sign conventions v i v i (b) Bo15 tail rotor top view, sign conventions Figure B.3: Sign conventions for the tail rotors of the Puma and the Bo15 The tail rotor thrust coefficient based on analytical blade element theory is ( ) ) C Tbe = 1 2 C 1 L α σ (θ 3 + µ2 par + µ per λ itr 2 2 (B.22) The tail rotor thrust T tr along the Y-axis of the body fixed frame of reference is then calculated with T tr = C Tbe ρπr 2 tr (Ω tr R tr ) 2 F tr The fin blockage factor F tr is taken from Padfield [2], (B.23) F tr = S vf πr 2 tr (B.24) In Eq. (B.22) the induced flow through the tail rotor disc is needed. The derivative of the induced inflow is added to the equations of state and calculated as follows λ i = C T be C TGl τ λtr where the Glauert thrust coefficient C TGl is defined as (B.25) C TGl = 2λ µ 2 par + (λ i µ per ) 2 (B.26) The forces and moments contributions of the tail rotor at the centre of gravity become F x = F y = T tr rotdir F z = M x = F z (y tr y cg ) F y (z tr z cg ) M y = F x (z tr z cg ) F z (x tr x cg ) M z = F y (x tr x cg ) F x (y tr y cg ) (B.27a) (B.27b) (B.27c) (B.27d) (B.27e) (B.27f)

158 126 Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage B.2 Force and Moment contributions of the Helicopter Fuselage B.2.1 Drag Model 1 This model is taken from Pavel [21] and assumes that the fuselage drag D fus is proportional with the square of the the local airspeed, D fus = 1 2 ρv 2 F = C fus V 2 (B.28) where F represents the fuselage drag-surface coefficient. The fuselage angle of attack α fus and angle of sideslip β fus are derived using Fig. B.4, ( ) wfus α = arctan (B.29) u fus ( v ) β = arcsin (B.3) V The components of the total drag force in the fuselage axes can be calculated in two ways which will both be given below. From Fig. B.4, the following relations can be derived, sin α = w u 2 + w 2 cos α = u u 2 + w 2 (B.31) sin β = v cos β = V The force components along the body axis follow as u 2 + w 2 V (B.32) v w u α Y bf β X bf V Z bf Figure B.4: Angle of attack α and sideslip β of the fuselage F x = D cos β cos α = C fus V 2 u 2 + w 2 F y = D sin β = C fus V 2 v V = C fusvv V u u 2 + w 2 = C fusuv (B.33a) (B.33b) F z = D cos β sin α = C fus V 2 u 2 + w 2 V w u 2 + w 2 = C fuswv (B.33c) which actually means that for this model, it is not necessary to calculate the fuselage angle of attack α and angle of sideslip β.

159 B.2 Force and Moment contributions of the Helicopter Fuselage 127 Results for a wide range of velocities is given in Fig. B.5. From this figure, it is clear that the total drag force does not depend on the direction of the relative wind. The model described in the paragraph hereafter does make a difference between different relative wind vectors. Figure B.5: Simple fuselage drag force model 1 B.2.2 Drag Model 2 This drag model is similar to the one discussed in the previous section but here, different values for the parasitic drag area are used for the three different drag force components. The fuselage frontal, side and top surface areas are approximated by ellipses 2 and based on the proportions between the three areas found, values for the fuselage sideward and downward parasitic drag are calculated. The results 3 are summarised in Table B.1. The calculation of the drag generated by the relative airspeed is then quite straightforward and some results can be seen in Fig. B.6. It is clear that with a contant relative wind, most drag is experienced along the Y-axis of the body fixed frame of reference. F x = 1 2 ρuv F x F y = 1 2 ρvv F y F z = 1 2 ρwv F z (B.34) (B.35) (B.36) 2 Values based on the three-view drawing in Padfield [2, p. 265]. 3 The value for F x was taken from Pavel [21] and used to calculate the values in the other directions.

160 128 Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage Frontal(x) Sideward (y) Downward (z) a [cm] b [cm] area [cm 2 ] π rel. area [-] F i [m 2 ] Table B.1: Fuselage parasitic drag area calculation Figure B.6: Simple fuselage drag force model 2 B.2.3 Moment contribution It is assumed that the fuselage only produces a pitching moment, the model is taken from van Aalst [22], M y = ρv 2 K fus V ol fus α fus (B.37) In this equation, K fus is a correlation factor and V ol fus is the volume of an equivalent body with the same projection as the helicopter in the horizontal plane, but only with circular cross-sections.

161 B.2 Force and Moment contributions of the Helicopter Fuselage 129 B.2.4 Problems There are some problems with the mathematical model of the fuselage. First of all, it assumes that only a drag force and a pitching moment are produced. This is obviously a serious simplification, since it might also produce a yawing and rolling moment. Furthermore, it is assumed that the forces (and moments) have their origin at the centre of gravity, while in reality, they would originate at the fuselage centre of pressure. Under the assumption that this last point is located below the centre of gravity, an extra pitch-down moment would be generated by the drag force. It might be better to use lookup tables generated from wind tunnel data, as can be found in Padfield [2].

162 13 Mathematical models of Tail Plane Surfaces, Tail Rotor and Fuselage

163 Appendix C Additional Trim Results C.1 Trim Results for the Bo15 C.1.1 Data Conversions In this section, trim results for the Bo15 are discussed. It is the only manned helicopter for which some flight test data obtained for the German Aerospace Center DLR is available at the Faculty of Aerospace Engineering. And therefore, it is the only helicopter that can be used to validate mathematical models in the time domain against flight test data. Some trim data can also be found in Pavel [1996] and van Aalst [22]. This data is presented in the following figures and compared with trim data generated with Helix. The data for the tail rotor collective pitch angle θ tr is given in percentages (pilot pedal input) and is first converted to the actual pitch angles of the tail rotor blades using θ tr = pedal min + (pedal max pedal min )pedal/1 (C.1) which was found in Voorsluijs [23]. The values for pedal min and pedal max are 18. and -6., respectively. The cyclic tilt angles and the cyclic pilot inputs adhere to the following sign convention [Leishman, 22] θ = θ + θ 1s sin ψ + θ 1c cos ψ β = β + β 1s sin ψ + β 1c cos ψ (C.2) (C.3) whereas in Pavel [1996] and van Aalst [22], the following convention is used for the aforementioned angles, θ = θ A 1 cos ψ B 1 sin ψ β = a a 1 cos ψ b 1 sin ψ (C.4) (C.5) The four harmonic angles a 1, b 1, A 1 and B 1 must be negated before any comparison can be done. The conversion from the no feathering plane to the hub plane is again done using (β 1c ) NF P = β 1c + θ 1s (C.6) (β 1s ) NF P = β 1s θ 1c (C.7) 131

164 132 Additional Trim Results C.1.2 Trim Results in Forward Flight The trim values of the pilot controls are shown in Fig. C.1. The simulation model of van Aalst [22] had a consistent 4 degree error in longitudinal cyclic pitch θ 1s over the complete velocity range. It was concluded that the inclusion of a downwash correction factor and a factor accounting for the rotor system rigidity could reduce this discrepancy. In Fig. C.1b, the trim results are shown for the longitudinal cyclic pitch. They coincide quite good with the flight test data. The inclusion of the equivalent spring stiffness and the rotor shaft tilt seem to have a positive effect on the accuracy of the longitudinal cyclic trim results. In hover, van Aalst predicted a main rotor collective pitch of approximately 8.5 degrees whereas the flight tests predict a value of more than 13 degrees. Just as was the case for the Puma helicopter, the addition of blade twist reduces the error in collective pitch with almost 4 degrees as is shown in Fig. C.1a. The discrepancies in lateral cyclic pitch (Fig. C.1c) are of the same order of magnitude as those presented in [Padfield et al., 1996], where a combination of Pitt-Peters theory and empirical fitting of the coefficients in the longitudinal variation of rotor inflow is used to improve the lateral cyclic pitch. It gives an indication of the powerful effect of nonuniform inflow on rotor flapping and the corresponding trim cyclic pitch. The pedal positions are fairly well predicted near hover, but as the velocity is increased, the discrepancy becomes quite large. The simple uniform momentum theory model cannot capture the complex interactional aerodynamics that are present near the tail rotor. The inclusion of a nonuniform dynamic inflow model could increase the fidelity of the tail rotor model. This should be accompanied with a more accurate representation of the flapping dynamics of the tail rotor blades. The roll and pitch angles of the fuselage in trimmed forward flight are displayed in Fig. C.2. They are compared with flight test data found in Pavel [1996]. The resulting fuselage pitch angle Θ compares well with the flight test data. The roll angle Φ is underestimated over the complete flight regime by a considerable amount. The results for the rotor disc tilt angles β 1s and β 1c and the rotor coning angle β are shown in Fig. C.3. Just as with the longitudinal cyclic pitch, the longitudinal tilt flight test data coincides with the trim calculations. The error in the lateral tilt angle is again of the same order of magnitude as the lateral cyclic pitch input. The prediction of the coning angle is fairly accurate for the complete flight regime, especially when compared to the model of van Aalst, which showed a constant error of.4 degrees. Overall, the trim results for the Bo15 generated with Helix are an improvement with respect to the data generated by van Aalst [22]. The improvements are mainly due to the addition of main rotor blade twist and an equivalent spring to model the hub stiffness. Nonuniform inflow seems to be a good addition to increase the accuracy of the lateral cyclic pitch and lateral tilt of the main rotor.

165 C.1 Trim Results for the Bo θ [deg] θ1s [deg] V [m/s] (a) Main rotor collective pitch V [m/s] (b) Main rotor longitudinal cyclic pitch 12 θ1c [deg] 2 1 θtr [%] V [m/s] (c) Main rotor lateral cyclic pitch V [m/s] (d) Main rotor longitudinal cyclic pitch Figure C.1: Top level trim variables for the Bo15 compared with DLR flight test data 5 Θ [deg] -5 Φ [deg] V [m/s] (a) Fuselage pitch angle V [m/s] (b) Fuselage roll angle Figure C.2: Fuselage attitude angles in trimmed forward flight C.1.3 Influence of Pitch-Flap Coupling According to Padfield [2], δ 3 equals -45 degrees. This means that when the blade flaps a certain angle β bl upward, the pitch angle θ is increased by ( ) 45 π θ = β bl tan δ 3 = β bl tan = β bl (C.8) 18

166 134 Additional Trim Results 3 β [deg] β1s [deg] PSfrag replacements V [m/s] (a) Main rotor coning angle V [m/s] (b) Main rotor lateral tilt β1c [deg] V [m/s] (c) Main rotor longitudinal tilt Figure C.3: Rotor trim variables for the Bo15 compared with flight test data A reduced pitch angle means less lift, so as a consequence, the (collective) pitch angle must be increased. So, in theory, the error in Fig. C.1a should be reduced somewhat by including pitch-flap coupling. In Fig. C.4, the collective pitch angle θ, the lateral cyclic pitch angle θ 1c and the lateral tilt angle β 1s are shown for forward flight. Included are DLR flight test data [Padfield et al., 1996] and the trim results with and without pitch-flap coupling. C.2 Trim Routine Accuracy The trim accuracies and magnitudes of the perturbations 1 for the three trim routines were determined experimentally. They are listed in Table C.1. One can see that the fuselage (top level) trim routine is the least accurate of the three. In theory, the accuracy of the trim results as a whole should therefore be at least equal to 1.e-7. But as already mentioned in section 5.3.2, the coupling between the derivatives of the angular velocity and the main rotor flapping states was not taken into account in the trim routine. By recalculating all derivatives after a trim solution is found, one can asses what influence this simplification has on the accuracy of the trim results. Fig. C.5 shows the magnitude of the six rigid body velocity derivatives u, v, ẇ, ṗ, q and 1 All perturbations are a multiple of the machine precision eps which is the distance from 1. to the next larger double precision number, 2 52 or approximately 2.224e-16.

167 C.2 Trim Routine Accuracy Baseline Baseline + pitch-flap coupling DLR flight test data 3 θ [deg] θ1c [deg] V [m/s] (b) Main rotor collective pitch PSfragangle replacements V [m/s] (c) Main rotor lateral cyclic pitch angle β1s [deg] V [m/s] (d) Main rotor lateral tilt angle Figure C.4: Influence of pitch-flap coupling on the trim variables fuselage trim main rotor trim tail rotor trim trim accuracy 1.e-7 1.e-8 1.e-9 perturbation 2.224e e e-12 Table C.1: Trim accuracy and perturbation magnitude of the trim routines ṙ as a function of trim step. As mentioned before, the tail rotor inflow was added to the top level trim routine, so its derivative λ tr is also shown here. The accuracy is met for all top level variables during the sixth iteration. However, after recalculating all derivatives with all couplings enabled, the second order derivatives of the flap angles do not meet the accuracy for the main rotor derivatives as displayed in Table C.1. In fact, they are the least accurate off all derivatives, with β 1c being greater than 2.e-6. This reduction in trim accuracy is not visually noticeable in the graphs of the trim results, but it could become a problem with a rotor where the influence of the linear velocity derivatives on the rotor flapping is also taken into account. The obvious solution is to retrim the main rotor including all coupling effects after the main trim routine has converged. This was added as an extra option to the trim algorithm. It ensures that all

168 136 Additional Trim Results trim accuracies as specified in Table C.1 are met. derivatives of top level trim variables u v ẇ ṗ q ṙ λ tr β β 1s β 1c top level trim step Figure C.5: Convergence of the top level trim variables and the magnitude of the second order derivatives of the flapping angles after recalculating all derivatives

169 Appendix D Stability and Control Derivatives This appendix contains the stability and control derivatives as a function of forward speed (V = to 6 m/s) for both the Puma and the Bo15. The data points generated with the mathematical model derived for this thesis are shown with white and black circles for the nonuniform and uniform inflow, respectively. All stability and control derivatives are shown in the same order as the graphs in [Padfield], on pages 269 to 275. The corresponding page number is given in the caption of the respective figures for the readers convenience. The dimensions of all stability and control derivatives are listed in Table D.1. D.1 Puma Helicopter The stability and control derivatives of the Puma helicopter with uniform and nonuniform inflow are given in Figs. D.1 to D.7. D.2 Bo15 Helicopter The stability and control derivatives of the Bo15 are depicted in Figs. D.8 to D.14. In contrast with the stability derivatives of the Puma helicopter, the longitudinal and lateral Derivative group example Dimension force/linear velocity X u,... [1/s] force/angular velocity X p,... [m/(rad s)] moment/linear velocity L u,... [rad/(m s)] moment/angular velocity L p,... [1/s] force/control X θ1s,... [m/(s 2 rad)] moment/control M θ,... [1/s 2 ] Table D.1: Dimensions of the stability and control derivatives 137

170 138 Stability and Control Derivatives Helix nonuniform inflow Helix uniform inflow Xu -.2 Xw.3 Xq Zu Zw Zq Mu Mw V [m/s] V [m/s] V [m/s] Figure D.1: Puma longitudinal stability derivatives as a function of forward speed (V = to 6 m/s), cf. Padfield [2, p. 269] Mq stability derivatives are influenced by the inflow model. This is caused by the different main rotor (hingeless for the Bo15 versus articulated for the Puma).

171 D.2 Bo15 Helicopter Yv Helix nonuniform inflow Helix uniform inflow Yp Yr Lv -.4 Lp -.9 Lr.1 Nv Np V [m/s] V [m/s] V [m/s] Figure D.2: Puma lateral stability derivatives between and 6 m/s, cf. Padfield [2, p. 27] Nr Xv Zv Helix nonuniform inflow Helix uniform inflow Xp Zp 1.5 Xr Zr Mv Mp V [m/s] V [m/s] V [m/s] Figure D.3: Puma lateral into longitudinal stability derivatives between and 6 m/s, cf. Padfield [2, p. 271] Mr

172 14 Stability and Control Derivatives Yu Lu Nu Helix nonuniform inflow Helix uniform inflow Yw Lw Nw Yq Lq V [m/s] V [m/s] V [m/s] Figure D.4: Puma longitudinal into lateral stability derivatives between and 6 m/s, cf. Padfield [2, p. 272] Nq Helix nonuniform inflow Helix uniform inflow Xθ Zθ Mθ Xθ1s Zθ1s Mθ1s Xθ1c V [m/s] V [m/s] V [m/s] Figure D.5: Puma main rotor longitudinal control derivatives between and 6 m/s, cf. Padfield [2, p. 273] Zθ1c Mθ1c

173 D.2 Bo15 Helicopter 141 Yθ Lθ Nθ Helix nonuniform inflow Helix uniform inflow Yθ1s Lθ1s Nθ1s Yθ1c V [m/s] V [m/s] V [m/s] Figure D.6: Puma main rotor lateral control derivatives between and 6 m/s, cf. Padfield [2, p. 274] Lθ1c Nθ1c Xθ tr Lθ tr Helix nonuniform inflow Helix uniform inflow Yθ tr Mθ tr V [m/s] V [m/s] V [m/s] Figure D.7: Puma tail rotor control derivatives between and 6 m/s, cf. Padfield [2, p. 275] Zθ tr Nθ tr

174 142 Stability and Control Derivatives Helix nonuniform inflow Helix uniform inflow Xu Zu Mu Xw Zw Mw Xq V [m/s] V [m/s] V [m/s] Figure D.8: Bo15 longitudinal stability derivatives as a function of forward speed (V = to 6 m/s), cf. Padfield [2, p. 269] Zq Mq Yv Lv Nv Helix nonuniform inflow Helix uniform inflow Yp Lp Np Yr Lr V [m/s] V [m/s] V [m/s] Figure D.9: Bo15 lateral stability derivatives between and 6 m/s, cf. Padfield [2, p. 27] Nr

175 D.2 Bo15 Helicopter Helix nonuniform inflow Helix uniform inflow Xv.2 Xp.14 Xr -.1 Zv Mv Zp Mp V [m/s] V [m/s] V [m/s] Figure D.1: Bo15 lateral into longitudinal stability derivatives between and 6 m/s, cf. Padfield [2, p. 271] Zr Mr Yu Lu Nu Helix nonuniform inflow Helix uniform inflow Yw Lw Nw Yq V [m/s] V [m/s] V [m/s] Figure D.11: Bo15 longitudinal into lateral stability derivatives between and 6 m/s, cf. Padfield [2, p. 272] Lq Nq

176 144 Stability and Control Derivatives Xθ Zθ Helix nonuniform inflow Helix uniform inflow Xθ1s Zθ1s Xθ1c Zθ1c Mθ Mθ1s V [m/s] V [m/s] V [m/s] Figure D.12: Bo15 main rotor longitudinal control derivatives between and 6 m/s, cf. Padfield [2, p. 273] Mθ1c Helix nonuniform inflow Helix uniform inflow Yθ Lθ Nθ Yθ1s Lθ1s Nθ1s Yθ1c V [m/s] V [m/s] V [m/s] Figure D.13: Bo15 main rotor lateral control derivatives between and 6 m/s, cf. Padfield [2, p. 274] Lθ1c Nθ1c

177 D.2 Bo15 Helicopter 145 Xθ tr Lθ tr Helix nonuniform inflow Helix uniform inflow Yθ tr Mθ tr V [m/s] V [m/s] V [m/s] Figure D.14: Bo15 tail rotor control derivatives between and 6 m/s, cf. Padfield [2, p. 275] Zθ tr Nθ tr

178 146 Stability and Control Derivatives

179 Appendix E Helicopter Parameters Part Parameter Description Inertia Puma Value Bo15 m helicopter mass [kg] I x moment of inertia about x-axis [kg m 2 ] I y moment of inertia about y-axis [kg m 2 ] I z moment of inertia about z-axis [kg m 2 ] J xz product of inertia [kg m 2 ] f dimless. x-offset of cg wrt rotor head [-] f 1 dimless. y-offset of cg wrt rotor head.. [-] h dimless. z-offset of cg wrt rotor head [-] Unit Main Rotor Geometric x h x-offset of rotor origin frame of reference wrt E bf [m] y h y-offset of rotor origin frame of reference wrt E bf.. [m] z h z-offset of rotor origin frame of reference wrt E bf [m] s main rotor shaft tilt (positive backwards) [deg] h r origin of hub plane axis system [m] rotdir flag, 1: CCW rotor (Puma), -1: CW rotor (Bo15) 1-1 [-] Ω main rotor rotational speed [rad/s] R main rotor radius [m] c main rotor blade chord [m] n bl main rotor number of blades 4 4 [-] n s number of blade segments [-] d type segment discretisation type: 1: equal annuli, 2: constant 1 1 [-] I bl rotor blade moment of inertia [kg m 2 ] m bl rotor blade mass [kg] K β equivalent spring constant [Nm/rad] θ tw rotor blade twist [deg] e flapping hinge offset.. [m] ee root cut-out [m] δ 3 pitch-flap coupling [deg] β p blade precone angle. 2. [deg] 147

180 148 Helicopter Parameters Part Parameter Description Main Rotor Main Rotor Aerodynamic Derived Tail Rotor Fuselage Horizontal Vertical Fin Stabiliser Puma Value Bo15 τ mr quasi-dynamic inflow time constant.14.1 [s] c lα small angle of attack lift gradient [1/rad] a zero-lift angle of attack.. [deg] α lb range for low angle of attack behaviour [deg] c d zero-thrust drag coefficient [-] c d1 coeff. of linear variation of drag with aoa [-] c d2 coeff. of quadratic variation of drag with aoa.4.4 [-] A large angle of attack lift scaling factor [-] B large angle of attack moment scaling factor [-] C large angle of attack moment scaling factor [-] D large angle of attack constant drag factor [-] E large angle of attack drag scaling factor [-] A rotor disc area `= πr [m 2 ] σ rotor solidity = n blc [-] πr γ blade Lock number = ρc l α cr 4 « [-] I bl λ 2 β flap frequency ratio = 1 + k «bl I bl Ω [-] R tr tail rotor blade radius [m] c tr tail rotor blade chord [m] Ω tr tail rotor rotational speed ratio (wrt main rotor speed) [-] x tr x-offset wrt E bf [m] y tr y-offset wrt E bf [m] z tr z-offset wrt E bf [m] c la blade lift gradient [1/rad] n bl number of blades 5 2 [-] τ tr quasi-dynamic inflow time constant [-] F x equivalent plate area to define fuselage drag [m 2 ] F y equivalent plate area as seen from aside [m 2 ] F z equivalent plate area as seen from above [m 2 ] k coefficient of fuselage pitching moment [m] vol equivalent volume of circular body [m 3 ] c la lift gradient [-] c d zero-lift drag coefficient.. [-] α horizontal stabiliser incidence angle [deg] S hs horizontal stabiliser surface area [m 2 ] x hs x-offset wrt E bf [m] y hs y-offset wrt E bf [m] z hs z-offset wrt E bf [m] c lb lift gradient [-] c d zero-lift drag coefficient.. [-] β vertical fin incidence angle [deg] S vf vertical fin surface area [m 2 ] x vf x-offset wrt E bf [m] y vf y-offset wrt E bf.. [m] z vf z-offset wrt E bf [m] Table E.1: Geometric, inertial and aerodynamic parameters of the Puma and Bo15 helicopters as used in the simulation program Unit

181 Appendix F System Output Filtering As was already discussed in Chapter 5 concerning the trim results, the exact values of the trimmed states and controls show small variations with the azimuth angle of the main rotor blades, which are due to small n bl /rev variations of hub forces and moments. This behaviour is found in more high-order helicopter flight dynamics models with a numerical blade element main rotor (e.g. [Kim et al., 1993] and [Xin & He, 22]). The frequency of this vibration is f v = Ω mr 2π n bl = [Hz] (F.1) 2π where Ω mr is the angular velocity of the main rotor and n bl is the number of blades. According to Tischler [1991], pilots operate in the frequency range between.1 and 5 Hz, which is far less than the above value. When mathematical pilot models (e.g. PID controllers) are developed for off-line time simulations, the fuselage states and state derivatives are fed back to the pilot model to stabilise and control the helicopter. If these states include the high frequency components originating at the rotor hub, the resulting control inputs for the helicopter will consist of a combination of a low frequency and a high frequency component, all generated by the pilot. To prevent this unrealistic behaviour, the high frequency component in the states is filtered out before sending them to the pilot using a seventh-order Butterworth low-pass filter. The transfer function of this filter can be written as [Porat, 1997] H(s) = H 1 (s)h 2 (s) (F.2) with H 1 (s) = H 2 (s) = 1 ( ) ( ( ) ) 2 (F.3) s ω + 1 s ω s ω ( ( ) ) ( 2 ( ) ) 2 (F.4) s ω s ω + 1 s ω s ω

182 15 System Output Filtering 1 Normalized filter gain Gn (ω) ω p 5 ω 1 ω s 15 Angular frequency ω [rad/s] Figure F.1: Frequency plot of normalised gains for Butterworth filters with orders of 1 to 8, also shown: cutoff frequency ω, pass band frequency ω p and stop band frequency ω s, the dashed line is frequency response curve of the 7th order filter ω is the cutoff frequency (also called the 3dB frequency). The frequency response of the Butterworth filter is maximally flat (no ripples) in the passband, and approaches zero in the stopband. The frequency plot of Fig. F.1 shows the normalised gains of the Butterworth filters with orders 1 to 8 (the above 7th order filter is shown with a dashed line). The value of the cutoff frequency is calculated with a procedure described in Porat [1997, p. 332]. The values of the pass band edge frequency ω p and stop band edge frequency ω s are 4 and rad/s, respectively. The pass band ripple δ p and stop band attenuation δ s are both set to 1%, i.e. a signal with a frequency of rad/s will have a amplitude that is smaller than 1 % of its original amplitude after passing through the filter. The resulting cutoff frequency ω equals 56 rad/sec. In Fig. F.2, a 5 second time simulation of the pitch response of the helicopter is shown starting from a trimmed condition 1 with the controls held fixed. The upper plot is the response without the Butterworth filter from Eq. (F.2), the middle shows the output with the filter applied and the lower plot is the difference between the two. Overall, the filter has the desired effect on the system output, the mathematical pilot will not notice and subsequently try to correct the vibrations due to the fluctuating rotor forces and moments. One disadvantage is that it creates a small time delay (<.6 s) in the signals, which might destabilise the simulation somewhat. The reason for this time delay is a result of the conversion to the time domain of the transfer function by Simulink. A seventh-order transfer function has an equivalent state-space model with 1 V = 4 m/s, h = m

183 151.5 q [deg/s].25.5 q [deg/s].25 q [deg/s] time [s] Figure F.2: Butterworth filter applied to pitch rate q: upper - original; middle - filtered; lower: difference between upper and middle seven states, which needs seven initial conditions. Then seven time steps are needed to propagate the signal through the state-space model and as a result, the output during the first six steps equals zero. An elliptic or Chebyshev filter with the above specifications would have a lower order and therefor a smaller time delay, but not enough time was available to design such a filter properly.

184 152 System Output Filtering

185 Appendix G Level 2 C-Mex Gateway S-Function to Fortran G.1 Introduction The helicopter model described in Chapter 3 and Appendices A and B, was initially implemented as Matlab M-code. This has some advantages and disadvantages. The biggest advantage is that the time to develop or implement a program in an interpreted language (such as Matlab or Python) is many times shorter than implementing a similar program in a compiled language (such as Fortran, C or Ada). The main drawback is the speed of execution of the resulting program. To trim the Puma helicopter in level flight between and 8 m/sec with increments of 5 m/sec, Matlab needs approximately 25 minutes on a Intel Pentium GHz, which is a fairly long time. The reason for this slow execution is that the part of the code that calculates the blade element forces of the main rotor, is contained in two nested for-loops. The outer loop iterates over the individual rotor blades and the inner loop iterates over the blade segments of every blade. It should execute many times faster when this code is vectorised. One can read in the Using Matlab manual [22, p. 117] that Matlab is a matrix language, which means it is designed for vector and matrix operations. You can often speed up your M-file code by using vectorizing algorithms that take advantage of this design. Vectorization means converting for and while loops to equivalent vector or matrix operations. It is however not possible to vectorise the aforementioned main rotor loops due to complexity. To overcome this obstacle, the Matlab soucre code including all trim and linearisation routines was manually converted to Fortran 95 and in the mean time, some practical improvements were added 1. The resulting stand-alone program is capable of trimming and linearising the helicopter in a fraction 2 of the time required by Matlab. 1 Or, to put it another way, you often don t really understand the problem until after the first time you implement a solution. The second time, maybe you know enough to do it right. So if you want to get it right, be ready to start over at least once. [Eric S. Raymond, The Cathedral and the Bazaar] 2 Approximately 1 times as fast, a trim run which takes 25 minutes in Matlab is completed in around 1.5 seconds in Fortran. 153

186 154 Level 2 C-Mex Gateway S-Function to Fortran In the second part of this thesis, an off-line simulation is made which is capable of landing a helicopter on a (moving) ship deck. The easiest way to accomplish this task is by designing the various controllers using Simulink. For this, a gateway must be programmed which enables Simulink to speak with the relevant Fortran routines, since using Matlab directly is again too slow to be of any practical use. To ensure that it is possible to construct a C-interface between Fortran routines and Simulink under Linux using the Intel Fortran compiler and GCC without wasting too much time, first some small examples were compiled and tested. This will be discussed in the following section. G.2 C-MEX Gateway for a Continuous-Time State-Space System There are two distinct strategies for executing Fortran code from within Simulink. One is from a Level 1 Fortran-Mex (F-Mex) S-function, the other is from a level 2 gateway S-function written in C. The original S-function interface was called the Level 1 API. As the capabilities of Simulink grew, the S-function API was rearchitectured into the more extensible Level 2 API. The Level 1 API is still available, but it is wiser to use the newer Level 2 API, since it allows S-functions to have all capabilities of a full Simulink model. G.2.1 Mixed-Language Programming and Compiler Compatibility The Intel Fortran compiler is not supported by the Matlab mex command, so using a Level 1 interface will not work. By using a Level 2 C-MEX gateway, it is possible to use Fortran compilers not directly supported by the mex command, if the object file formats of the utilised C and Fortran compilers are binary compatible. Fortran is well suited for numerical computations, whereas C is suitable for system programming tasks. A combination of the two languages in one program gives the programmer a combination of the best of both. There are however some pitfalls when calling a Fortran subroutine from a C program and vice versa. The following are the most important points one must take into account. Indexation In Fortran, array indices start at 1, while in C they start at ; hence a passed array fort arr[1 : 1] must be used in the C program as c arr[ : 99]. Array Storage Order Two dimensional arrays are stored in memory in a one dimensional way. Fortran stores arrays in column-major order while C stores them row-major. An array with three rows and two columns is stored in memory by Fortran in the following order: (1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (G.1) where the number pairs denote the indices of the array elements. The same array is stored by C as (1, 1) (1, 2) (2, 1) (2, 2) (3, 1) (3, 2) (G.2)

187 G.2 C-MEX Gateway for a Continuous-Time State-Space System 155 So before using an array which originates from a submodule in another language, it should be converted or else computations will generate wrong answers or won t work at all. Pointer Passing By default, Fortran passed arguments as pointers instead of values and it also expects pointers in the argument list when called from C. A Fortran routine which is called from C should pass a pointer to the first index of an array, and as another argument, a pointer to an integer which defines the total length of the array. These two arguments fully determine the location in memory where a Fortran routine can find the actual data. For passing two dimensional arrays around, the lengths of both dimensions should be passed around. Symbol Decoration Most Fortran compilers append an underscore ( ) to the names of subroutines and functions both in the definition and in calls. In C, the simplest solution then is to add an underscore to the function prototypes. C also makes a distinction between upper and lower case variable names, whereas Fortran doesn t. Best practice is to only use lower cases for variable and subroutine names which in the same time increases source code readability. G.2.2 Fortran Continuous-Time State-Space System In section G.2.3, the basic ingredients necessary to construct a Level 2 gateway between Simulink and Fortran will be explained, specifically focused on continuous-time state space systems. This section gives an example of such a system 3. The reason it is included here, is that Matlab doesn t come with an example of a Level 2 interface for this kind of dynamic system written in Fortran. In fact, there is only one example of a Level 2 interface for a Fortran subroutine, whereas there are more than 2 examples of Level 2 interfaces written in pure C. Sine Wave x = A*X + B*U Y = C*X + D*U Continuous-Time state space S-Function 1 Out Scope Figure G.1: Fortran continuous-time state space dynamic system The system under consideration (see Fig. G.1) has two continuous states, x 1 and x 2, two inputs u 1 and u 2 and two outputs y 1 and y 2. It can be written in the following form, x = A x + Bū ȳ = C x + Dū (G.3) 3 It is the same state space system as the one defined in the file $matlabroot/simulink/src/csfunc.c

188 156 Level 2 C-Mex Gateway S-Function to Fortran where A, B, C and D are the usual square system matrices, [ ] [ ] time [sec] output A = [-] B = 1 2 [ ] [ ] 2 3 C = D = (G.4) (G.5) The initial conditions x are zero and the driving functions are two sinusoidal inputs with amplitudes of 1 and 2 and a constant frequency of 1 rad/sec. Simulating this system for 1 seconds gives the output as depicted in Fig. G.2. Two small Fortran subroutines are used to calculate the left hand side of Eq. (G.3), the source code is listed in section G y1 y2 output [-] time [sec] Figure G.2: Simulation output of the state space system G.2.3 C-MEX S-function Contents A C-MEX file that defines an S-function block provides information about the model to Simulink during the simulation. As the simulation proceeds, Simulink, the ODE solver and the MEX-file interact to perform specific tasks. These tasks include defining initial conditions and computing derivatives and output. The general format of a C-MEX S-function gateway for a continuous-time dynamic system calling Fortran subroutines is shown below: #define S_FUNCTION_NAME your_sfunction_name_here #define S_FUNCTION_LEVEL 2 #include "simstruc.h" #define VARIABLE_STEP <define external subroutine prototype(s) here> static void mdlinitializesizes(simstruct *S) { } static void mdlinitializesampletimes(simstruct *S)

189 G.2 C-MEX Gateway for a Continuous-Time State-Space System 157 { } static void mdlinitializeconditions(simstruct *S) { } static void mdloutputs(simstruct *S, int_t tid) { <call external subroutine here> } static void mdlderivatives(simstruct *S) { <call external subroutine here> } <additional S-function routines/code> static void mdlterminate(simstruct *S) { } #ifdef MATLAB_MEX_FILE /* Is this file being compiled as a MEX-file? */ #include "simulink.c" /* MEX-file interface mechanism */ #else #include "cg_sfun.h" /* Code generation registration function */ #endif In the beginning of the file, the external function prototypes are defined. For the system introduced in the previous section, a function prototype for calculating the derivatives and one to calculate the outputs is needed, extern void der_calc_( double *x, double *u, double *dx, int *numstates); extern void outp_calc_( double *x, double *u, double *outp, int *numstates); The C-MEX S-function must contain the following model callbacks in order to work properly: - mdlinitializesizes - mdlinitializesampletimes - mdlinitializeconditions - mdlderivatives - mdloutput - mdlterminate The first three are called during the initialisation and could optionally contain a call to an external Fortran subroutine to load parameters into memory and/or calculate the initial conditions of the states of the system. The Fortran subroutine der_calc_ is called from within mdlderivatives which passes its output to the internal Simulink integrator to advance the system in time. The other subroutine (outp_calc_) is called from within mdloutput which sends the outputs y 1 and y 2 to the Scope (see Fig. G.1). The complete interface file for this dynamic system is given in section G.5.

190 158 Level 2 C-Mex Gateway S-Function to Fortran G.3 Helix C-MEX Gateway Interface G.3.1 Introduction Similar to the interface as described in section G.2, an interface for the helicopter dynamics routines was constructed. The most important requirement for this interface is that the existing subroutines and configuration files can be utilised without any modification. A second requirement is that only the strictly necessary variables are passed to the C-MEX interface, e.g. the subroutine derivative_calculation outputs the total forces and moments at the centre of gravity in addition to the state derivatives. These forces and moments are only needed for the linearisation routine and should be hidden from the C-MEX interface. The resulting interface consists of two files, a C-side and a Fortran-side. These will be discussed in the section hereafter. A visual representation of the interface is given in Fig. G.3, at the end of this appendix. The actual subroutines that constitute the helicopter dynamics model are accessed through a static library, helix_static.a. The resulting three files (helix_tsim.c, helix_simulink_interface.f9 and helix_static.a) are compiled into a binary MEX-file (helix_tsim.mexglx), which can be used in any Simulink model under Linux. G.3.2 Interface: Fortan-side The Fortran side of the gateway interface consists of one file, named helix gateway interface.f9. This file contains the following three subroutines, - helix_start ( states ) - helix_derivatives ( states,th_,th_1s,th_1c,th_tr,derivatives ) - helix_outputs ( states, outputs ) The first, helix_start, is a wrapper around the initialisation and trim routines. It has no input (the name of the program control file pcf is statically compiled in) and it outputs one vector containing the initial values of all states. helix_derivatives gets the states and controls as input from Simulink, from which it calculates the derivatives using the subroutine derivative_calculation. In this subroutine, the individual controls are put in a vector before being passed to derivative_calculation. By using an extra wrapper around the actual subroutine where the derivatives are calculated, the objective of hiding the for Simulink unneeded total forces and moments is met. The last one (helix_outputs) calculates the output as a function of the states without calling any underlying subroutines. G.3.3 Interface: C-side In the beginning of the file (helix_tsim.c), three function prototypes are defined for the subroutines described in the previous section. These are

191 G.4 Fortran Continuous-Time State-Space System 159 extern void helix_derivatives_( double *x, double *th_, double *th_1s, double *th_1c, double *th_tr, double *dx); extern void helix_start_( extern void helix_outputs_( double *x); double *x, double *outp); The first function (mdlinitializesizes) sets the number of states of the model and the number and width of the input and output ports of the simulink block. In the call to mdlinitializeconditions, the Fortran subroutine helix_start_ is called, which returns the values of the initial conditions of the states of the helicopter model. These values are then passed to the differential equation solver of Simulink, which initiates the time simulation. Using the values of state variables, the state derivatives are calculated with a call to the Fortran subroutine helix_derivatives_ from within the function mdlderivatives. When the integrator scheme has advanced the system in time, the outputs are calculated from the states in the function mdloutputs by calling helix_outputs_. These outputs are then passed to Simulink, for use in controllers. G.4 Fortran Continuous-Time State-Space System subroutine outp_calc(states,controls,outp,numstates) implicit none!==============!! declarations!!==============! integer, intent(in) :: numstates double precision, intent(in), dimension(numstates) :: states,controls double precision, intent(out), dimension(numstates) :: outp double precision, dimension(numstates,numstates) :: c_mat, d_mat! define matrices c_mat(1,:) = (/ +.D+, +2.D+/) c_mat(2,:) = (/ +1.D+, -5.D+/) d_mat(1,:) = (/ -3.D+, +.D+ /) d_mat(2,:) = (/ +1.D+, +.D+ /) outp = matmul(c_mat,states) + matmul(d_mat,controls) end subroutine outp_calc subroutine der_calc(states,controls,ders,numstates) implicit none!==============!! declarations!

192 16 Level 2 C-Mex Gateway S-Function to Fortran!==============! integer, intent(in) :: numstates double precision, intent(in), dimension(numstates) :: states,controls double precision, intent(out), dimension(numstates) :: ders double precision, dimension(numstates,numstates) :: a_mat, b_mat! define matrices a_mat(1,:) = (/ -.9D+, -.1D+ /) a_mat(2,:) = (/ +1.D+, +.D+ /) b_mat(1,:) = (/ +1.D+, -7.D+ /) b_mat(2,:) = (/ +.D+, -2.D+ /) ders = matmul(a_mat,states) + matmul(b_mat,controls) end subroutine der_calc G.5 C-MEX Gateway Interface /* * File: csfunf_gateway.c * * Abstract:: * * C->Fortran gateway TEMPLATE for a Level 2 S-function. * Copy, rename and then edit this file to call your Fortran * code in the solver mode you want, then build it. * * To build the mex file, first compile the Fortran file(s), * then include their object file names in the mex command. * For example, if your Fortran compiler is invoked with the * g77 command, a mex session looks like this at the command * prompt: * * >>!ifort -fpic -c ss_calc.f9 * >> mex csfunf_gateway.c ss_calc.o */ /* * You must specify the S_FUNCTION_NAME as the name of your S-function * (i.e. replace sfungate with the name of your S-function, which has * to match the name of the final mex file, e.g., if the S_FUNCTION_NAME * is my_sfuntmpl_gate_fortran, the mex filename will have to be * my_sfuntmpl_gate_fortran.dll on Windows and * my_sfuntmpl_gate_fortran.mexxxx on unix where XXX is the 3 letter * mex extension code for your platform). */ #define S_FUNCTION_LEVEL 2 #define S_FUNCTION_NAME csfunf_gateway /* * Need to include simstruc.h for the definition of the SimStruct and * its associated macro definitions. */ #include "simstruc.h" /* * Digital Fortran s external symbols are in capitals * on Windows platforms; preceding underscore is implicit */ #if defined (_MSC_VER)

193 G.5 C-MEX Gateway Interface 161 #define atmos_ ATMOS #endif /* * Some compilers don t use a trailing * underscore on Fortran external symbols */ #if defined( xlc ) defined( hpux) #define atmos_ atmos #endif /* * We have a continuous system, so we can make use of variable step solvers. */ #define VARIABLE_STEP /* * The interface (function prototype) for your the Fortran subroutine, * which are defined in the file ss_calc.f9. */ #ifdef VARIABLE_STEP extern void der_calc_( double *x, double *u, double *dx, int *numstates); extern void outp_calc_( double *x, double *u, double *outp, int *numstates); #endif /*====================* * S-function methods * *====================*/ /* Function: mdlinitializesizes =============================================== * Abstract: * The sizes information is used by Simulink to determine the S-function * block s characteristics (number of inputs, outputs, states, etc.). */ static void mdlinitializesizes(simstruct *S) { /* See sfuntmpl.doc for more details on the macros below */ sssetnumsfcnparams(s, ); /* Number of expected parameters */ if (ssgetnumsfcnparams(s)!= ssgetsfcnparamscount(s)) { /* Return if number of expected!= number of actual parameters */ return; } sssetnumcontstates(s, 2); /* how many continuous states? */ sssetnumdiscstates(s, ); if (!sssetnuminputports(s, 1)) return; sssetinputportwidth(s,, 2); /* * Set direct feedthrough flag (1=yes, =no). * A port has direct feedthrough if the input is used in either * the mdloutputs or mdlgettimeofnextvarhit functions. * See matlabroot/simulink/src/sfuntmpl_directfeed.txt. */ sssetinputportdirectfeedthrough(s,, 1);

194 162 Level 2 C-Mex Gateway S-Function to Fortran sssetinputportrequiredcontiguous(s,, 1); if (!sssetnumoutputports(s, 1)) return; sssetoutputportwidth(s,, 2); } sssetnumsampletimes(s, 1); sssetnumnonsampledzcs(s, ); sssetoptions(s, ); /* Function: mdlinitializesampletimes ========================================= * Abstract: * This function is used to specify the sample time(s) for your * S-function. You must register the same number of sample times as * specified in sssetnumsampletimes. */ static void mdlinitializesampletimes(simstruct *S) { #ifdef VARIABLE_STEP #else #endif } /* * For Fortran code with either no states at * all or with continuous states that you want * to support with variable time steps, use * a sample time like this: */ sssetsampletime(s,, CONTINUOUS_SAMPLE_TIME); sssetoffsettime(s,,.); /* * If the Fortran code implicitly steps time * at a fixed rate and you don t want to change * the code, you need to use a discrete (fixed * step) sample time, 1 second is chosen below. */ sssetsampletime(s,,.1); /* Choose the sample time here if discrete */ sssetoffsettime(s,,.); #define MDL_INITIALIZE_CONDITIONS /* Change to #undef to remove function */ #if defined(mdl_initialize_conditions) /* Function: mdlinitializeconditions ======================================== * Abstract: * In this function, you should initialize the continuous and discrete * states for your S-function block. The initial states are placed * in the state vector, ssgetcontstates(s) or ssgetrealdiscstates(s). * You can also perform any other initialization activities that your * S-function may require. Note, this routine will be called at the * start of simulation and if it is present in an enabled subsystem * configured to reset states, it will be call when the enabled subsystem * restarts execution to reset the states. */ static void mdlinitializeconditions(simstruct *S) { /* * #undef MDL_INITIALIZE_CONDITIONS if you don t have any * continuous states. */ double *x = ssgetcontstates(s); /* set the values of the states (x) to start with */

195 G.5 C-MEX Gateway Interface 163 x[] =.; x[1] =.; /* int lp; for (lp=;lp<2;lp++) { *x++=.; }*/ } #endif /* MDL_INITIALIZE_CONDITIONS */ #undef MDL_START /* Change to #undef to remove function */ #if defined(mdl_start) /* Function: mdlstart ======================================================= * Abstract: * This function is called once at start of model execution. If you * have states that should be initialized once, this is the place * to do it. */ static void mdlstart(simstruct *S) { } #endif /* MDL_START */ /* Function: mdloutputs ======================================================= * Abstract: * In this function, you compute the outputs of your S-function * block. The default datatype for signals in Simulink is double, * but you can use other intrinsic C datatypes or even custom * datatypes if you wish. See Simulink document "Writing S-functions" * for details on datatype topics. */ static void mdloutputs(simstruct *S, int_t tid) { #ifdef VARIABLE_STEP /* * For Variable Step Code WITH CONTINUOUS STATES * * For Fortran code that implements continuous states and uses * the mdlderivatives interface, call your Fortran code s output * routines from here. If it alters the states, you have to * reset the solver. Remember, in Simulink the continuous states * must be of type double, so be prepared to copy them to float * if your Fortran code uses REAL as the datatype for the states. * */ double *y = (double *) ssgetoutputportrealsignal(s,); double *x = (double *) ssgetcontstates(s); double *u = (double *) ssgetinputportsignal(s,); UNUSED_ARG(tid); /* not used in single tasking mode */ int numstates = ssgetnumcontstates(s); double fu[numstates], fx[numstates], fy[numstates]; /* set the input value */ fu[] = (double) u[]; fu[1] = (double) u[1]; fx[] = (double) x[]; fx[1] = (double) x[1];

196 164 Level 2 C-Mex Gateway S-Function to Fortran /* call the Fortran routine using pass-by-reference */ outp_calc_(&fx[], &fu[], &fy[], &numstates); /* get output from the Fortran routine*/ y[] = (double) fy[]; y[1] = (double) fy[1]; #endif } #define MDL_UPDATE /* Change to #undef to remove function */ #if defined(mdl_update) /* Function: mdlupdate ====================================================== * Abstract: * This function is called once for every major integration time step. * Discrete states are typically updated here, but this function is useful * for performing any tasks that should only take place once per * integration step. */ static void mdlupdate(simstruct *S, int_t tid) { } #endif /* MDL_UPDATE */ #define MDL_DERIVATIVES /* Change to #undef to remove function */ #if defined(mdl_derivatives) /* Function: mdlderivatives ================================================= * Abstract: * In this function, you compute the S-function block s derivatives. * The derivatives are placed in the derivative vector, ssgetdx(s). */ static void mdlderivatives(simstruct *S) { #ifdef VARIABLE_STEP /* * For Variable Step Code Only * * If your Fortran code needs to support continuous states * with variable timestep solvers, you need to call into * your Fortran routine (or perhaps one that shares a * common block but only calculates derivatives) here to * extract/calculate state derivatives WITHOUT ADVANCING TIME. */ double *dx = (double *) ssgetdx(s); double *x = (double *) ssgetcontstates(s); double *u = (double *) ssgetinputportsignal(s,); int numstates = ssgetnumcontstates(s); double fu[numstates], fx[numstates], fdx[numstates]; /* set the input value */ fu[] = (double) u[]; fu[1] = (double) u[1]; fx[] = (double) x[]; fx[1] = (double) x[1]; /* call the Fortran routine using pass-by-reference */ der_calc_(&fx[], &fu[], &fdx[], &numstates); /* get output from the Fortran routine*/ dx[] = (double) fdx[];

197 G.5 C-MEX Gateway Interface 165 dx[1] = (double) fdx[1]; #endif } #endif /* MDL_DERIVATIVES */ /* Function: mdlterminate ===================================================== * Abstract: * In this function, you should perform any actions that are necessary * at the termination of a simulation. For example, if memory was * allocated in mdlstart, this is the place to free it. */ static void mdlterminate(simstruct *S) { } /*=============================* * Required S-function trailer * *=============================*/ #ifdef MATLAB_MEX_FILE /* Is this file being compiled as a MEX-file? */ #include "simulink.c" /* MEX-file interface mechanism */ #else #include "cg_sfun.h" /* Code generation registration function */ #endif

198 166 Level 2 C-Mex Gateway S-Function to Fortran Simulink ODE-solver helix tsim.mexglx helix tsim.c mdlinitializeconditions mdlderivatives mdloutputs helix simulink interface.f9 helix start helix derivatives helix outputs Figure G.3: Schematic overview of the Fortran-Simulink gateway interface helix static.a process pcf load parameters mr discr st ct init guess initial trim full heli trim derivative calculation

199 Appendix H Helix User Guide H.1 Introduction This appendix contains some documentation for the adventurous persons that want to work with Helix. In the first section, the main programs and libraries used to develop Helix under Linux are given. Then, the installation, compilation and execution under Linux is explained, followed by a short explanation of the compilation under Windows in section H.4. In the fifth section, the contents of the program control file is explained. Section H.6 sheds some light on the structure of the source code of Helix. H.2 Prerequisites The program was developed on a computer running Fedora Core 4 [26] and is written in Fortran 9/95. The compiler used is the Intel Fortran Compiler [26] for Linux, which is free for personal, non-commercial use. The program needs direct access to three subroutines of LAPACK Linear Algebra PACKage [26], which are conveniently accessed through the Intel Math Kernel Library [26], also free for non-commercial use. Matlab and Simulink were used for the time simulations. The Simulink interface that was created between the Fortran code and Simulink needs a C compiler that is binary compatible with both the Fortran compiler and Simulink. In this case, GCC GNU Compiler Collection [26] was used. The PLplot a Scientific Plotting Library [26] was used to generate direct graphical output in Fortran. Table H.1 summarises the different programs with their respective versions. In Fedora, the Plplot library can be installed from the Extras repository. Once installed, compile and test the Fortran examples in the /usr/share/plplot5.6.1/examples/f77/ directory. To compile them with the Intel Fortran compiler, change the definition for the Fortran compiler from gfortran to ifort in the Makefile in that directory. When running one of the examples, Plplot may exit with an error about some TrueType fonts it could not find. They are not included in Fedora, but an rpm-file containing the necessary fonts can be found here: ftp://ftp.pbone.net/mirror/apt.unl.edu/apt/ 167

200 168 Helix User Guide Program/Library Name Version Intel Fortran Compiler for Linux 9. Intel Math Kernel Library 8..1 Plplot library GCC 4..2 Matlab 7.. Simulink 6. Table H.1: Versions of the programs and libraries used to develop Helix fedora/all/rpms.stable/freefont-ttf-.-.fdr noarch.rpm. Installing this file with rpm places all required fonts below /usr/share/fonts/freefont/, while Plplot needs them one directory higher. Copy or simlink them in /usr/share/fonts/ and you re done. H.3 Installation and Compilation on Linux H.3.1 Compiler Installation The installation of the compiler and library on Linux should not give any problems, the only thing that may need some attention is the initialisation of the environment variables which point to the installation directories of the compiler and libraries. The compiler is invoked on the command line as ifort yoursource.f9. Before this can be done, the environment variables should be set. In a subdirectory of the compiler (and library) installation directory (below /opt/intel/), two shell scripts can be found which, when executed, set the environment variables. These are (or should be) mklvars.sh and ifortvars.sh. To automate this, they should be copied to /etc/profile.d/ directory, which contains all kinds of shell scripts which are executed at start-up. The contents of these two scripts is displayed below for your reference. Contents of mklvars.sh: if [ -z "${INCLUDE}" ] then INCLUDE="/opt/intel/mkl/8..1/include"; export INCLUDE else INCLUDE="/opt/intel/mkl/8..1/include:$INCLUDE"; export INCLUDE fi if [ -z "${LD_LIBRARY_PATH}" ] then LD_LIBRARY_PATH="/opt/intel/mkl/8..1/lib/32"; export LD_LIBRARY_PATH else LD_LIBRARY_PATH="/opt/intel/mkl/8..1/lib/32:$LD_LIBRARY_PATH"; export LD_LIBRARY_PATH fi Contents of ifortvars.sh:

201 H.3 Installation and Compilation on Linux 169 #! /bin/sh if [ -z "${PATH}" ] then PATH="/opt/intel/fc/9./bin"; export PATH else PATH="/opt/intel/fc/9./bin:$PATH"; export PATH fi if [ -z "${LD_LIBRARY_PATH}" ] then LD_LIBRARY_PATH="/opt/intel/fc/9./lib"; export LD_LIBRARY_PATH else LD_LIBRARY_PATH="/opt/intel/fc/9./lib:$LD_LIBRARY_PATH"; export LD_LIBRARY_PATH fi if [ -z "${MANPATH}" ] then MANPATH="/opt/intel/fc/9./man":$(man -w); export MANPATH else MANPATH="/opt/intel/fc/9./man:${MANPATH}"; export MANPATH fi if [ -z "${INTEL_LICENSE_FILE}" ] then INTEL_LICENSE_FILE="/opt/intel/fc/9./licenses:/opt/intel/licenses:${HOME}/intel/licenses"; export INTEL_LICENSE_FILE else INTEL_LICENSE_FILE="${INTEL_LICENSE_FILE}:/opt/intel/fc/9./licenses:/opt/intel/licenses: ${HOME}/intel/licenses"; export INTEL_LICENSE_FILE fi H.3.2 Program Compilation, Execution and Command Line Options The program normally ships in a compressed tar archive, which should be unpacked to a location where you have write privileges (e.g. your home directory). All Fortran source code is located in the /src subdirectory. Navigate to this directory and compile the program by typing make at the command line. This should create an object file for every source code file (extension *.o), a static library (with extension *.ar) which contains the object files and a binary named helix. The program can then be invoked by typing./helix at the command line. You should now see (if you make the command window high enough) an ASCII representation of a certain helicopter. You are asked for the name of the program control file, it is located in the same directory as the source code and is conveniently named pcf. The program will now start doing stuff 1 depending on the options that are activated in the pcf-file. If in the pcf the proper flags to plot the results are set to 1, a list with output streams is shown from which one should be chosen. They are shown below: < 1> xwin X-Window (Xlib) < 2> tk Tcl/TK Window < 3> gcw Gnome Canvas Widget < 4> plmeta PLplot Native Meta-File < 5> ps PostScript File (monochrome) < 6> psc PostScript File (color) 1 It can trim and linearise a helicopter around one or more user-specifiable flight conditions. Sadly, making coffee is not (yet) implemented.

202 17 Helix User Guide < 7> xfig Fig file < 8> hp747 HP 747 Plotter File (HPGL Cartridge, Small Plotter) < 9> hp758 HP 758 Plotter File (Large Plotter) <1> lj_hpgl HP Laserjet III, HPGL emulation mode <11> pbm PDB (PPM) Driver <12> png PNG file <13> jpeg JPEG file <14> null Null device <15> tkwin New tk driver <16> mem User-supplied memory device <17> gif GIF file <18> wxwidgets wxwidgets DC <19> psttf PostScript File (monochrome) <2> psttfc PostScript File (color) From this list, options 1 to 7, 12, 13, 18, 2 and 21 are known to work. The third one, gcw (Gnome Canvas Widget) is the most flexible since it allows to view the plot results in a window with different tabs. From this window, it is also possible to save the results to a PostScript file. To make life easier for the user, a limited number of command line arguments are supported. These make it possible to start Helix in non-interactive mode. A list with the options, arguments and explanation is given in Table H.2. Flag Argument Explanation/Action -i pcf specify which program control file (pcf) to use -dev device specify which device Plplot should use to display the plots -h none print help message and quit -v none print version and quit Table H.2: Helix command line options H.3.3 Simulink Interface Compilation For controller design, an interface was written in C to link the Fortran program to Simulink. The resulting mex-file can be compiled with the following command, executed from the /src directory, mex -v -g helix_tsim.c helix_simulink_interface.o helix_static.a... /opt/intel/fc/9./lib/libguide.so /opt/intel/fc/9./lib/libirc.a... /opt/intel/fc/9./lib/libifcore.so.5 The C-interface is compiled with the Linux C-compiler, GCC. The above command will probably not work 2 if you didn t manually change the relevant portions of the mexopts.sh which should be located in $HOME/.matlab/R14/ (for Matlab release R14). 2 You also have to compile the Fortran source files first.

203 H.4 Installation and Compilation on Windows 171 The following code shows the relevant portion of the mexopts.sh for Linux for Matlab 7.. and up, # ;; glnx86) # RPATH="-Wl,--rpath-link,$TMW_ROOT/bin/$Arch" # gcc -v # gcc version 4..2 CC= gcc CFLAGS= -fpic -ansi -D_GNU_SOURCE -pthread -fexceptions CLIBS="$RPATH $MLIBS -lm" COPTIMFLAGS= -O -DNDEBUG CDEBUGFLAGS= -g # # g++ -v # gcc version 4..2 CXX= g++ CXXFLAGS= -fpic -ansi -D_GNU_SOURCE -pthread CXXLIBS="$RPATH $MLIBS -lm" CXXOPTIMFLAGS= -O -DNDEBUG CXXDEBUGFLAGS= -g # # g77 -fversion # GNU Fortran (GCC 3.2.3) (release) # NOTE: g77 is not thread safe # standard: g77, replace by ifort for Intel Fortran Compiler FC= ifort FFLAGS= -fpic -u -w95 -warn all FLIBS="$RPATH $MLIBS -lirc.a -lm" FOPTIMFLAGS= -O FDEBUGFLAGS= -g # LD="$COMPILER" LDEXTENSION=.mexglx LDFLAGS="-pthread -shared -Wl,--version-script,$TMW_ROOT/extern/lib/$Arch/$MAPFILE" LDOPTIMFLAGS= -O LDDEBUGFLAGS= -g # POSTLINK_CMDS= : H.4 Installation and Compilation on Windows First remove Windows from your computer, after all, it is of no use other than collecting viruses and Trojan horses. Then install Linux (or ask your local Linux guru 3 to do that for you) and follow the steps outlined in section H.3.1. Still here? Very well then, I might as well tell how to compile Helix on Windows, not? The following items are not supported with the Windows build: - graphical output with the Plplot library (no compiled version of Plplot was available), - eigenvalue and eigenvector calculations (no subroutine in double precision is available in the Visual Numerics - IMSL Library [26] to compute all of the eigenvalues and eigenvectors of a general matrix.) and 3 You may find one in the basement, wearing sandals and no deodorant ;)

204 172 Helix User Guide - the Simulink interface. The following therefore only applies to the stand-alone console application. The prerequisites for running Helix under Windows are the following, you need a Fortran 9/95 4 compiler and a (binary) compatible version of the IMSL library. Any library that implements the matrix inversion will do, but the one mentioned was the only one available at the time of writing. One of the subdirectories is named win32, which contains a workspace with project files to compile Helix. Settings for the two standard configurations Release and Debug are supplied. Change to the Release configuration and build all sources. The binary named win32 is placed in the subdirectory win32/debug or win32/release, depending on the selected configuration. To run the program, you will have to move the binaries to the \src directory, since moving pcf and the file(s) containing the helicopter data to the win32/debug or win32/release subdirectories doesn t seem to work. Relevant portions of code in the main file helix.f9 that call subroutines in the files plotfun.f9 and frequencyfun.f9 are enclosed in #ifdef statements, which effectively removes those parts from the program at compile time. H.5 Program Control File At the start of the program execution, the contents of the program control file (pcf) is read into memory. It contains approximately 6 flags and parameters that can be changed to control what must be done by the program. It is divided in the following nine major sections, - helicopter data file and helicopter name, - options related to the accuracy of the flight dynamics model and aerodynamics models - options related to wind and fuselage azimuth angle, - trim calculation options, - linearisation calculation options, - flight condition(s) to trim/linearise at, - eigenvalue and eigenvector calculations, - (Pl)plot options, - time simulation options. Each of these sections contains 1 or more options in the following two-line format 5,! ! variable_name [variable dimension or data type] short explanation variable_value Some of these sections will be discussed below in some more detail. 4 Compaq Visual Fortran version 6.6 was used. 5 In this report, explanations are continued on the second line if they do not fit on the page.

205 H.5 Program Control File 173 H.5.1 Flight Dynamics Model In this section of the program control file (see below), one can include or exclude the mathematical models of the fuselage and empennage. The inflow model to use for the main and tail rotor can also be set here. For the blade element aerodynamics model, one can choose between the default nbe_aero_simple and the Mach dependant model nbe_aero_mach. Note that combining nbe_aero_mach with the centre spring hub stiffness parameter k_re may cause the trim routine not to converge at high speeds. The exact reason for this is unknown.!=================! uniform inflow [int] uniform or nonuniform inflow! ! qd_unif_infl [int] quasi dynamic uniform inflow! ! wake_dist_dyn [int] include wake distortion dynamics! ! k_re [int] wake curvature parameter (1.D <= k_re <= 2.D) 2.D! ! with_fus [int] include fuselage forces and moments 1! ! with_hs [int] include horizontal stabiliser forces and moments 1! ! with_vf [int] include vertical fin forces and moments 1! ! with_kp [int] include centre spring stiffness in blade moment 1! ! tr_pp_uniform [int] model tail rotor w. Pitt-Peters uniform dynamic inflow, else w. "quasi-dynamic" model! ! nbe_aero_simple [int] use simple steady 2d nbe aero model (no mach correction) 1! ! nbe_aero_mach [int] use steady 2d nbe aero model with mach correction (from Prouty)! ! nbe_3d_velo [int] use 3d velocity calculation for blade elements H.5.2 Trim Options Most of the trim flags determine whether or not to save the calculated trim values for the states and controls (see below). All output files are written to the hard disk as Matlab readable m-files for which the filenames (and variable names) can be specified. With the second entry, one determines whether or not to save the trim data and the third line specifies the path and filename. The states and control can be saved separately in high precision for use as initial conditions in time simulations done with Simulink. Setting trim_usd to one changes the initial guess of the fuselage roll angle Φ to 18, which will trim the helicopter upside down (or not at all). The last one, post_mr_trim retrims the main rotor with all couplings enabled when the fuselage states have converged sufficiently, to ensure that they meet their accuracies.!=================! do_trim [int] do trim

206 174 Helix User Guide 1! ! trim_save_data [int] save trim data in (m-)file! ! trim_save_filename [char<=5] name of save file, if (suffix.eq. m or M ) then write in matlab m-file format../postpro/puma_trimdata.m! ! trim_save_controls [int] save controls in high precision for use as initial values by Simulink! ! ctrl file name [char<=2] name of file to save controls to./init_ctrls.m! ! ctrl var name [char<=2] name of variable(vector, length=4) to save control values in init_controls! ! save fus states [int] save average (fuselage) states in high precision for use as initial values for linear model! ! states file name [char<=2] name of file to save states to./init_stats.m! ! states var name [char<=2] name of variable vector to save states in init_states! ! trim_usd [int] trim up side down! ! post_mr_trim [int] retrim main rotor after complete trim, taking nonzero p_dot, q_dot and r_dot into account 1 H.5.3 Linearisation Options The linearisation options are quite similar to the trim options. lin_with_mr_trim will emulate a steady-state main rotor when linearizing the helicopter, which is best for time simulation that take longer than.1 second. Similarly, a steady-state tail rotor can be emulated with the lin_with_tr_trim. lin_with_psi adds one column and one row to the A-matrix and one row to the B-matrix. To investigate the effect of changing the state and control disturbances, one can change the values of the four lines starting with delta. The relevant portion is shown below.!=================! do_lin [int] linearise around trim condition(s) 1! ! lin_save_data [int] save linearisation data in m-file! ! lin_save_file [char<=5] name of save file, if (suffix.eq. m or M ) then write in matlab m-file format../postpro/puma_lindata.m! ! lin_with_mr_trim [int] linearise with or without main rotor trim (emulating steady state main rotor) 1! ! lin_with_tr_trim [int] linearise with or without tail rotor trim (emulating steady state tail rotor)! ! lin_with_psi [int] adds psi row and column to A and psi row to B matrix 1! ! with cross couple [int] add cross coupling submatrices 1! ! delta_linvel [3x m/s] linear velocity disturbances for linearisation (standard: u,v,w: ) ! ! delta_angvel [3x deg/s] angular velocity disturbances for linearisation (standard: p,q,r: 5 5 5)

207 H.5 Program Control File ! ! delta_euler [2x deg] euler disturbances for linearisation (standard: th,ph: 5 5) 6. 6.! ! delta_controls [4x deg] control disturbances for linearisation (standard: ) ! ! av_data [int] average linearisation data (mainly for use with multiple psi_1 values)! ! av_data file [char<=2] name of averaged linearisation data save file./av_data.m H.5.4 Flight Condition In essence, the flight condition to trim the helicopter at is set using 5 parameters. These are the speed range V, flight path angle range γ, flight path angle range χ, the altitude range h and the first blade azimuth range ψ 1. For each of these parameters, a lower value, upper value and the increment can be set as triplet of real numbers separated by a whitespace. With a set of five integers with values 1 or 3, one determines whether to trim a range of a certain parameter or just one value. The relevant section with some sensible defaults is shown below,!=================! flight condition [5x int] determines what to do with the following 5 entries, possible values: 1, 3 [int] ! ! speed range [3x m/s] lower, upper and increment ! ! gama range [3x deg] idem, flight path angle.. 7.! ! chi range [3x deg] idem, flight track angle ! ! alt range [3x m] idem, altitude ! ! psi_1 range [3x deg] idem, first blade azimuth position E.g. if the first integer is set to 3, then a trim loop will start at m/s and end at 8 m/s with increments of 2 m/s, with the values for the other parameters determined by the first value on their respective line (41 trim calculations in total). H.5.5 Plot Options The options in this part of the program control file only pertain to Linux, they are not available on Windows. The last option do_subplots splits the plot window for the trim results in 3 by 2 matrices. The first six subplots then show the results for the fuselage pitch and roll angle Θ and Φ and the four pilot controls (θ, θ 1s, θ 1c and θ tr ). The second window will show the trimmed main rotor states (four or six plots, depending on the number of states in the inflow model). These results are shown if plot_trimdata is set to 1. The third option, plot_lindata adds plots of the stability and control derivatives as a function of speed in the same order as given in Padfield [2, pp ], on

208 176 Helix User Guide pages 269 to 275. If the eigenvalues are calculated and plot_freqdata is set to one, then the coupled, uncoupled longitudinal and uncoupled lateral eigenvalues are shown in three subsequent windows/tabs.!=================! plot_data [int] plot data using libplplot 1! ! plot_trimdata [int] plot trim data (fuselage and main rotor states and pilot controls) 1! ! plot_lindata [int] plot linearisation data (state and control derivatives) 1! ! plot_freqdata [int] plot eigenvalues in complex plane 1! ! do_subplots [int] put them in subplots (3x2) 1 Note that if only one trim conditions is calculated, no plots will be made. Also, only trim ranges over the airspeed V and the first blade azimuth angle ψ 1 are supported. Other ranges may not work properly. Two output plot are shown below. If a the start of a run no device is specified on the command line, the user will be presented with the list of output devices given in section H.3.2. The following figures show what the output may look like when using the Gnome Canvas Widget driver 6. H.6 Source Code Structure This section is intended for those persons who want to modify the source code of Helix. The source files are fairly well documented, but they do not give a birds-eye view of the structure of the code as a whole, i.e. it doesn t give an overview of how the different files are related to each other. This section tries to do just that. The Fortran source files have the extension.f9 or.f9. Before compiling a file with the former extension, the Intel Fortran compiler will first preprocess the file. This may remove platform or library dependant code that is contained within #ifdef statements. The source files can be divided in three groups based on the contents. The first group contains all files that define subroutines, this is the biggest group. Module files are contained in a separate group and define variables that can be accessed from any subroutine, function or program that contains the appropriate include statement. It is similar to the frequently (ab)used common statement in Fortran 77. The file that contains the program constitutes the third and last group 7. The main file has the same name as the program, helix.f9. One can think of this file as the glue that connects all subroutines to something usefull. It starts with the initialisations, first processing the program control file pcf to determine what data file to read in and what exactly should be done later on. The main rotor blade element discretisation is calculated, after which the actual calculations start. 6 The command line looked like this: $./helix -i pcf -dev gcw 7 No functions are used in Helix.

209 H.6 Source Code Structure 177 Figure H.1: Sample graphical output from Plplot with Gnome Canvas Widget driver (actual plot area in inverted colours) The trim, linearisation and eigenvalue calculations are contained within five nested loops, iterating over the desired flight speed V, flight path angle γ, flight track angle χ, altitude h and first blade azimuth angle ψ 1. The values of the trim variables and the A and B matrix of the linearised model are shown in the console. Optionally, results can be written to files for later processing. On Linux, it is also possible to show the results directly in a window with the Plplot library. The two-page figure 8 at the end of this appendix (Fig. H.2) graphically shows the relations between all the subroutines in Helix. The conventions are as follows. Per file, all subroutines are grouped in rectangles with the name of the file in a smaller rectangular box. All subroutines have an ellipse drawn around their names. The internal subroutines are located inside the rectangle of the file in which they are defined and the external subroutines (from the LAPACK, IMSL and Plplot libraries) and standard Fortran subroutines are located outside these rectangles. The program has a diamond drawn around its name, and the module names are drawn inside a parallelogram. A short explanation of the meaning of the internal subroutines is given below. 8 It can be generated with the Python script src struct.py in the utils directory. This generates a file with the extension dot, that must be used as input for the dot program, which is a part of GraphViz.