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Houston, S. and Thomson, D. (2017) on the Modelling of Gyroplane Flight Dynamics

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Houston, S. and Thomson, D. (2017) on the Modelling of Gyroplane Flight Dynamics Houston, S. and Thomson, D. (2017) On the modelling of gyroplane flight dynamics. Progress in Aerospace Sciences, 88, pp. 43-58. (doi:10.1016/j.paerosci.2016.11.001) This is the author’s final accepted version. There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it. http://eprints.gla.ac.uk/131442/ Deposited on: 15 October 2016 Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk On the Modelling of Gyroplane Flight Dynamics Dr Stewart Houston Dr Douglas Thomson Aerospace Sciences Research Division School of Engineering University of Glasgow Glasgow G12 8QQ On the Modelling of Gyroplane Flight Dynamics Abstract The study of the gyroplane, with a few exceptions, is largely neglected in the literature which is indicative of a niche configuration limited to the sport and recreational market where resources are limited. However the contemporary needs of an informed population of owners and constructors, as well as the possibility of a wider application of such low-cost rotorcraft in other roles, suggests that an examination of the mathematical modelling requirements for the study of gyroplane flight mechanics is timely. Rotorcraft mathematical modelling has become stratified in three levels, each one defining the inclusion of various layers of complexity added to embrace specific modelling features as well as an attempt to improve fidelity. This paper examines the modelling of gyroplane flight mechanics in the context of this complexity, and shows that relatively simple formulations are adequate for capturing most aspects of gyroplane trim, stability and control characteristics. In particular the conventional 6 degree-of-freedom model structure is suitable for the synthesis of models from flight test data as well as being the framework for reducing the order of the higher levels of modelling. However, a high level of modelling can be required to mimic some aspects of behaviour observed in data gathered from flight experiments and even then can fail to capture other details. These limitations are addressed in the paper. It is concluded that the mathematical modelling of gyroplanes for the simulation and analysis of trim, stability and control presents no special difficulty and the conventional techniques, methods and formulations familiar to the rotary- wing community are directly applicable. Keywords: rotary-wing; gyroplane; flight dynamics; simulation 1 Nomenclature 2 퐴 linearised model system matrix; rotor disc area, m 퐴11, etc minors of 퐴 푎0 blade element lift coefficient at zero angle of attack 푎0 blade element lift curve slope, 1/rad ℎ푖푛푔푒 ℎ푖푛푔푒 2 푎푥 , 푎푧 hinge acceleration components, m/s 퐵 linearised model control matrix 퐵1, 퐵2 minors of 퐵 푏 number of blades 푑퐷 blade element drag, N 푑퐿 blade element lift, N 푑푄 blade element torque, N 2 퐼푓푙푎푝, 퐼푝푖푡푐ℎ, 퐼푙푎푔 blade moments of inertia, kgm 푖 imaginary operator 푳 dynamic inflow model static gain matrix; rotor moment vector, Nm 퐿푢, 푒푡푐 derivative - rolling moment with respect to 푢, 1/ms, etc 퐿푎푒푟표 aerodynamic rolling moment, Nm 푀푎푒푟표 aerodynamic pitching moment, Nm 푀푢, 푒푡푐 derivative - pitching moment with respect to 푢, 1/ms, etc 푏푙 푏푙 푏푙 푀푓푙푎푝, 푀푝푖푡푐ℎ, 푀푙푎푔 blade flap, feather and and lag moments, Nm 푚푏푙 blade mass, kg 푚̇ inflow, m/s 푁푢, 푒푡푐 derivative - yawing moment with respect to 푢, 1/ms, etc 푝, 푞, 푟 body axes angular velocity components, rad/s 푅 rotor radius, m 푅푒[ ], 퐼푚[ ] real and imaginary components of [ ] 푟 radial position on disc or blade, m 2 풓푐푔 position of centre of mass, body axes, m 풓ℎ푖푛푔푒 position of flap, lag and feather hinge, rotor axes, m 풓ℎ푢푏 position of rotor hub, body axes, m 2 푆 vorticity source, 1/s 푠 rotor solidity 푇 rotor thrust, N 푇푝푟표푝 propeller thrust, N 푇푎푒푟표 aerodynamic thrust moment, N 푇1, 푇2, 푇3 transformation matrices 푢, 푣, 푤 translational velocity components, body axes, m/s 푢(휔) Fourier-transformed control vector 푢푝푟표푏푒, 푣푝푟표푏푒, 푤푝푟표푏푒 translational velocity components at air data probe, body axes, m/s 풖 control vector 푢ℎ푢푏, 푣ℎ푢푏, 푤ℎ푢푏 rotor hub velocities in non-rotating reference frame, m/s 푉푓 airspeed, m/s 푣푖(푟, 휓) induced velocity at position (푟, 휓), m/s 푣푖푚 momentum induced velocity, m/s 푣푖0, 푣1푠 , 푣1푐 components of induced velocity, m/s 푣푚 wake mass flow velocity, m/s 푣푇 wake velocity, m/s 푿 rotor force vector, N 푎푒푟표 푖푛푒푟푡푖푎푙 푿푒푙푒푚, 푿푒푙푒푚, 푿푒푙푒푚 blade element contributions to rotor force - total, aerodynamic, inertial, N 푋푢, 푒푡푐 derivative – longitudinal force with respect to 푢, 1/s, etc 풙 state vector 푥(휔) Fourier-transformed state vector 푥푏푙푎푑푒 blade Ox axis 푥푅 horizontal distance of rotor hub ahead of c.g., m 3 푥푣푎푛푒, 푦푣푎푛푒, 푧푣푎푛푒 air data probe location, body axes, m 풙1, 풙2 subspace of linearised model state vector 푌푢, 푒푡푐 derivative – lateral force with respect to 푢, 1/s, etc 푏푙 푦푐푔 distance of blade centre of mass from hinge, m 푍푢, 푒푡푐 derivative – normal force with respect to 푢, 1/s, etc 푧푝푟표푝 vertical distance of propeller thrust line above c.g., m 푧푅 vertical distance of rotor hub above c.g., m 훼퐷 rotor disc angle of attack, rad 훼푣푎푛푒, 훽푣푎푛푒 angles of attack and sideslip at air data probe, rad 훽 blade flapping angle, rad 훽0, 훽1푠 , 훽1푐 disc coning, lateral and longitudinal tilt, rad Δ푓 frequency increment, rad/s Δ푡 time increment, s 훿 blade element drag coefficient 휃 pitch attitude; blade feather angle, rad 휂푐, 휂푠, 휂푟푢푑푑푒푟, 휂푝푟표푝 control angles - lateral and longitudinal rotor tilt, rudder and propeller, rad 휆푐표푛푒, 휆푎푑푣, 휆푟푒푔 stability roots of the coning, advancing and regressing flap modes 휈 flow velocity, m/s 3 휌 air density, kg/m 휙 roll attitude; inflow angle, rad [횻] time constant matrix, sec 휒 wake skew angle, rad 휓 azimuthal position on rotor hub (zero to rear); yaw angle, rad Ω rotorspeed, rad/s 휔 frequency, rad/s; flow vorticity, 1/s 푏푙 푏푙 푏푙 휔푥 , 휔푦 , 휔푧 blade angular velocities, rad/s 4 Introduction There is little application of rotorcraft mathematical modelling techniques to the study of gyroplane flight mechanics; indeed there is little on the study of autorotation other than examination of fundamental concepts found in rotorcraft textbooks and some references of historical significance. Prior to 1992 when the UK Civil Aviation Authority (CAA) started its programme of research into gyroplane airworthiness and flight safety, only two recorded applications of contemporary rotorcraft flight mechanics modelling techniques to the gyroplane problem are to be found [1, 2]. The first was conducted in response to a fatal accident in an attempt to understand the behaviour of the aircraft; the second was a postgraduate dissertation. Both used a disc, rather than individual blade, representations of the rotor and could therefore be classified as Level 1 models. This has benefits in terms of simplicity of formulation, understanding and interpretation but limitations in terms of the level of approximation necessary to achieve solutions. The CAA study required an examination of a number of issues associated with flight mechanics modelling of gyroplanes, [3 - 12] encompassing both theoretical modelling as well as derivation from flight test data. An analytical model was not specifically prepared for gyroplane analyses; rather, a generic rotorcraft simulation code was populated with configuration-specific data. In principle the only salient difference in gyroplane modelling is the inclusion of the rotorspeed variable as a degree of freedom, [4]. Latterly a flowfield code was incorporated, albeit still with rigid hinged blades, [13]. This allowed the study of aerodynamic interactions between the rotor, propeller and empennage, [12]. Analysis of steady flight test data included synthesis of inflow models from blade flapping measurements [14], providing some degree of validation of the commonly-used dynamic inflow model, [15], in the context of autorotation. The outcome of this work, conducted over a period of 16 years, was support of the CAA’s management of gyroplane airworthiness and flight safety [16]. This activity appeared to stimulate broader interest in gyroplane flight, either to support potential applications of the configuration, as reviews of historical data in a contemporary context or simply 5 for technical curiosity, [17-37]. There is much of general technical interest, in particular review items by McCormick [24], Leishman [36] and the splendid volume by Harris [37]. However studies specifically addressing flight dynamics modelling are rare, [29, 38, 39]. The former is primarily concerned with the behaviour of the rigid-body modes of motion, while the latter two address validation against flight test data, particularly in terms of trim and time response to control inputs. The objective of this Paper is to assess the mathematical modelling requirements for simulation of gyroplane flight mechanics for stability and control analyses. In particular, the appropriateness or otherwise of simplified models is addressed, especially with reference to those higher order dynamics found to be of significance in rotorcraft modelling such as those associated with inflow and flapping. The usefulness of the conventional 6 degree-of-freedom, linearised small-perturbation theory structure is a primary focus given its timeless and all-pervading role in understanding the behaviour of flight vehicles. The paper begins with a review of established methods of modelling rotary-wing aircraft. The focus of this initially is on the methods used in helicopter simulation as naturally, the vast majority of relevant research has been associated with this class of vehicle. The discussion is then broadened to include the gyroplane (or autogyro) a qualitative discussion being followed by a detailed description of the method used to develop the models used in this paper. A comprehensive nonlinear model is developed and this is the primary source for other linear reduced order models used in the analysis in this paper.
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