Masters thesis of engineering aerodynamic performance comparison of a conventional uav wing and a fishbac morphing wing

Aerodynamic Performance Comparison of a Conventional UAV Wing and a FishBAC Morphing Wing A thesis submitted in fulfilment of the requirements for the degree of Master of Engineering Arthur Wong Bachelor of Engineering (Aerospace Engineering) (Honours), RMIT University School of Engineering College of Science, Technology, Engineering and Maths RMIT University June 2021 Declaration I certify that except where due acknowledgement has been made, the work is that of the author alone; the work has not been submitted previously, in whole or in part, to qualify for any other academic award; the content of the thesis is the result of work which has been carried out since the official commencement date of the approved research program; any editorial work, paid or unpaid, carried out by a third party is acknowledged; and, ethics procedures and guidelines have been followed I acknowledge the support I have received for my research through the provision of an Australian Government Research Training Program Scholarship Signed Arthur Wong 08 June 2021 i Acknowledgments I would like to take this opportunity to thank all those who have supported me during this program I thank RMIT University for allowing me to continue my development in the masters by research program I would like to thank my senior supervisors Professor Cees Bil and Dr Matthew Marino for looking over my growth and providing guidance to me completing the program I thank the technical staff team particularly Gil Atkin and Paul Muscat for assisting me in getting from a concept and design to a prototype of the morphing wing and a compliant morphing skin providing guidance, advice, teaching composite lay-up techniques necessary to reach the end goal Additionally I thank Nhu Huynh, my long-time girlfriend for her endless encouragement and support during the program as well as my friends and family for their support ii Table of Contents Declaration i Introduction 2 Literature Review 2.1 Types of Morphing Wings 2.1.1 Planform Morphing 2.1.2 Out-of-Plane Morphing 11 2.1.3 Airfoil Adjustment 13 2.2 Morphing Wing Actuation 14 2.2.1 Internal Mechanisms 14 2.2.2 Piezoelectric Actuators 15 2.2.3 Shape Memory Alloys 15 2.3 Examples of Morphing Structures 15 2.3.1 Fish Bone Active Camber (FishBAC) 15 2.3.2 Zig-Zag Wingbox 18 2.3.3 GNAT Spar 19 2.4 Morphing Wings in Industry 20 2.5 Morphing Wing Concept Selection 21 2.6 Literature Review on Morphing Skins 22 2.6.1 Honeycomb and Honeycomb Variants 24 2.6.2 Corrugated structures 25 2.6.3 Flexible Matrix Composites (FMC) 26 2.6.4 Concept Selection 27 2.6.5 Further Investigation into Flexible Matrix Composites (FMC) 27 Motivations and Past Research 28 3.1 Wing Concept and Conceptual Design 28 3.1.1 Morphing Wing Actuation Method 32 3.2 Wing Design 34 Research Questions 35 4.1 Project Scope 35 Research Methodology 36 5.1 Airfoil Development 36 5.2 Simulations 37 5.2.1 XFLR5 37 iii 5.2.2 Tornado 45 5.2.3 Wind tunnel Testing 48 Building the Morphing Wing 49 6.1 Morphing Skin Concept 50 6.3 Evolution of the Morphing Skin 51 6.3.1 Morphing Skin Manufacturing Process 54 6.4 Summary of the Morphing Wing Design 55 Results 57 7.1 Flow Visualization 57 7.1.2 Summary of Flow Visualization Behaviour 57 7.2 Wind Tunnel Data Post Processing 59 7.3 Experimental Results and Discussion 60 7.3.1 Conventional T240 Wind Tunnel Results 61 7.3.2 Morphing Wing Wind Tunnel Results 66 7.3.3 Roll Results 76 7.3.4 Discussion of Results 82 7.3.5 Summary of Comparison – Conventional T240 vs Morphing Wing 84 Conclusions 90 8.1 Recommendations/Further research 91 References 92 APPENDIX A – Wind tunnel Calibration 100 APPENDIX B – Assembly of the Wing and Preparation of the Wing 109 APPENDIX C – Flow Visualization for Various Morphing Deflections 117 APPENDIX D – Experimental Results 137 APPENDIX E – Comparison between Wind Tunnel Test and Simulation 151 APPENDIX F – XFLR5 Convergence 159 APPENDIX G – Further Information on the Vortex Lattice Method 163 iv List of Figures Figure Principles of aircraft drag polar affected by airfoil camber variation in steady cruise flight [10] Figure Precedent T240 aircraft and its wing’s dimensions in plan view Figure Morphing wing dimensions in plan view Figure Makhonine Mak-10 aircraft [4] Figure Examples of variable sweep wings [4] Figure Span morphing wing via telescopic wing [28] 10 Figure Planform alteration types [2] 11 Figure Camber morphing concept visualization [2] 11 Figure Span-wise bending morphing concept [2] 12 Figure 10 Wing twisting concept seen in the 1899 Wright Kite [36] 13 Figure 11 Airfoil adjustment morphing concept visualization [2] 14 Figure 12 Airfoil adjustment via actuators inside the wing [37] 14 Figure 13 A SMA spring actuator recovering its original shape after heating [47] 15 Figure 14 FishBAC rib design [23] 16 Figure 15 FishBAC utilized as a morphing trailing edge and model parameters [49] 17 Figure 16 Top-view of the zig-zag wingbox concept [25] 18 Figure 17 Schematic of GNATSpar concept [24] 19 Figure 18 Rack and pinion actuation system for GNATSpar [6] 20 Figure 19 Flexsys' Flexfoil deflected [22] 21 Figure 20 Composite Cellular Material Morphing Wing [56] 21 Figure 21 FishBAC and corrugated morphing trailing edge concept [60] 25 Figure 22 FMC fibre orientation for a) span morphing and b) for camber morphing [52] 26 Figure 23 Three-view of the initial rib design that connects to the trailing edge [69] 28 Figure 24 Colour coded isometric view of morphing wing concept [69] 29 Figure 25 Revised rib design and its assembly [69] 29 Figure 26 Revised rib displacements [19] 30 Figure 27 Velcro strips on revised rib [69] 31 Figure 28 Step by step assembly of the wing [69] 31 Figure 29 Four view of the fuselage wingbox without the covering panel 32 Figure 30 Assembled fuselage wingbox 33 Figure 31 Proposed servo locations in the fuselage wingbox and morphing wing 33 Figure 32 CAD model of wing design (without stringers attached) [70] 34 Figure 33 Complete CAD model of 2nd wing design [70] 35 Figure 34 Construction of the T240 airfoil 36 Figure 35 Morphing wing airfoil construction 37 Figure 36 XFLR5 simulation process for wing aerodynamic analysis 38 Figure 37 T240 airfoil in XFLR5 39 Figure 38 T240 airfoil with flap deflections in XFLR5 39 Figure 39 2D analysis results of T240 airfoil with Flaps applied at various Reynolds numbers in XFLR5 42 Figure 40 XFLR5 Analysis for 𝑪𝑳 vs 𝜶 at various Reynolds numbers 45 Figure 41 Tornado simulation process 46 v Figure 42 TORNADO Analysis for 𝑪𝑳 vs 𝜶 at various Reynolds numbers 47 Figure 43 Schematic of the industrial wind tunnel at RMIT University 48 Figure 44 Electronic turntable aft of the contraction point in the wind tunnel 48 Figure 45 Isometric view of the Morphing Wing 50 Figure 46 2D Morphing Wing splines from XFLR5 50 Figure 47 Initial morphing skin design A to morphing skin design C 51 Figure 48 Morphing skin design D to morphing skin design E 52 Figure 49 Morphing skin F to Morphing skin G 53 Figure 50 Morphing skin H to the Final Skin 54 Figure 51 Compliant morphing skin demonstration 54 Figure 52 Exploded isometric view of the Morphing Wing 56 Figure 53 Conventional T240 Wing results for 𝑪𝑳 vs 𝜶 at various Reynolds numbers 63 Figure 54 Conventional T240 Wing Wind Tunnel results for 𝑪𝑫 vs 𝜶 at various Reynolds numbers 65 Figure 55 Morphing Wing results for 𝑪𝑳 vs 𝜶 at various Reynold numbers 68 Figure 56 Morphing Wing Experimental results for 𝑪𝑫 vs 𝜶 at various Reynold numbers 71 Figure 57 Wind tunnel results of 𝑳/𝑫 vs 𝑪𝑳 at various Reynolds numbers for Morphing Wing and Conventional Wing 74 Figure 58 Wind tunnel results of 𝑳/𝑫 vs 𝑪𝑳 at various Reynolds numbers for Morphing Wing and Conventional Wing with flaps 75 Figure 59 TORNADO and wind tunnel results for 𝑪𝒍 vs 𝜹 at various Reynolds number 78 Figure 60 𝑪𝑳 comparison for the Conventional T240 and Morphing wing at various Reynolds numbers 79 Figure 61 Difference in TORNADO and wind tunnel testing for 𝒑 vs 𝜹 at various Reynolds numbers 80 Figure 62 𝒑 comparison between Conventional T240 and Morphing Wing at various Reynolds numbers 81 Figure 63 𝒑 comparison between Conventional T240 and Morphing Wing at various Reynolds numbers 82 Figure 64 Calibration setup in the y-axis (drag axis) of the JR3 Load cell, measured at z= m above the load cell 100 Figure 65 Calibration curve for the Lift axis of the JR3 load cell – “Lift” Force output vs “Lift” Force input 102 Figure 66 Calibration curve for the phantom outputs of the JR3 load cell – “Drag” Force output vs “Rolling” Moment output 102 Figure 67 Calibration curve for the Drag axis of the JR3 load cell – “Drag” Force output vs “Drag” Force input 103 Figure 68 Calibration curve for the Drag axis of the JR3 load cell – “Drag” Force output vs “Yawing” Moment input 104 Figure 69 Calibration of the JR3 load cell by applying pure moments in the Yaw axis, at x= -0.3m via Tbeam 104 Figure 70 Calibration of the JR3 load cell by applying pure moments in the Yaw axis, at x= 0.3m via Tbeam 105 Figure 71 Pure Yaw moment configuration calibration in the Drag axis of the JR3 load cell 106 Figure 72 Pure Yaw moment configuration calibration in the Drag axis of the JR3 load cell 106 Figure 73 Using calibration data from Yaw moment calibration, calibration curve for the Drag axis of the JR3 load cell 107 vi Figure 74 Calibration of the JR3 load cell by applying pure moments in the Roll axis, at y= -0.3m via Tbeam 108 Figure 75 Using calibration data from Rolling moment calibration, calibration curve for the Lift axis of the JR3 load cell 108 Figure 76 Using calibration data from Rolling moment calibration, calibration curve for the Lift axis of the JR3 load cell 109 Figure 77 Layout of Leading edge and Spar to be bonded 110 Figure 78 Bonding Ribs to the Leading edge 111 Figure 79 Bonding Ribs to the Trailing edge 111 Figure 80 Bonding the thin Al sheet to the Wing 112 Figure 81 Bonding the thin Al sleeve to the Trailing edge 112 Figure 82 Bonding reinforcing L shape carbon fibre angles to the ribs 112 Figure 83 Assembled morphing wing minus the wingtip 113 Figure 84 Isometric view of Wing tip post modifications; removal of spar box and addition of thin Al strips 113 Figure 85 Bottom view of Wing tip post modifications; removal of spar box and addition of thin Al strips 114 Figure 86 Isometric view of the Bonding of the wingtip cover to the wingtip 114 Figure 87 Top view of the Bonding of the wingtip cover to the wingtip 115 Figure 88 Curing of the Epoxy resin applied to the foam components of the Morphing wing and wingtip 115 Figure 89 Morphing wing spray painted 116 Figure 90 Wingtip spray painted 116 Figure 91 Morphing wing - post cure of the spray paint 116 Figure 92 Flow visualization for 𝜹𝒎 = 0° 118 Figure 93 Flow visualization for 𝜹𝒎 = 2° 120 Figure 94 Flow visualization for 𝜹𝒎 = 3° 121 Figure 95 Flow visualization for 𝜹𝒎 = 5° 123 Figure 96 Flow visualization for 𝜹𝒎 = 10° 125 Figure 97 Flow visualization for 𝜹𝒎 = 15° 127 Figure 98 Flow visualization for 𝜹𝒎 = 20° 128 Figure 99 Flow visualization for 𝜹𝒎 = 25° 130 Figure 100 Flow visualization for 𝜹𝒎 = 30° 132 Figure 101 Flow visualization for 𝜹𝒎 = 35° 134 Figure 102 Flow visualization for 𝜹𝒎 = 40° 136 Figure 103 Conventional T240 Wing Experimental results for 𝑪𝑳 vs 𝜶 and 𝑪𝑫 vs 𝜶 at Re 202000 and Re 269000 138 Figure 104 Morphing Wing Experimental results for 𝑪𝑳 vs 𝜶 and 𝑪𝑫 vs 𝜶 at Re 202000 and Re 269000 140 Figure 105 Experimental results of 𝑳/𝑫 vs 𝑪𝑳 at various Reynolds numbers with error bars 143 Figure 106 Wind tunnel results of 𝑳/𝑫 vs 𝑪𝑳 at various Reynolds numbers for Morphing Wing and Conventional Wing with flaps and error bars 145 Figure 107 𝑳/𝑫 vs 𝑪𝑳 comparison of the ideal morphing deflection and conventional wing with flaps and error bars (using the agreeable data) 146 vii Figure 108 TORNADO and wind tunnel testing results for𝑪𝒍 vs 𝜹 and 𝒑 vs 𝜹 at Re 202000 and Re 269000 147 Figure 109 XFLR5 Analysis for 𝑪𝑳 vs 𝜶 at Re 202000 and Re 269000 148 Figure 110 TORNADO Analysis for 𝑪𝑳 vs 𝜶 at Re 202000 and Re 269000 148 Figure 111 𝑪𝑳 and 𝒑 comparison between Conventional T240 and Morphing Wing at Re 202000 and Re 269000 149 Figure 112 𝒑 comparison between Conventional T240 and Morphing Wing at various Reynolds Numbers 151 Figure 113 Morphing Wing performance comparisons at 𝜹𝒎 = 0° to 𝜹𝒎 = 40° at various Reynolds numbers 159 Figure 114 Lifting lines in both spanwise and chordwise directions superimposed onto a wing [33, 89] 163 Figure 115 Velocity (the direction is coming out of the paper) induced at point P by the infinitesimal segment of the lifting surface[33] 164 Figure 116 Single horseshoe to a system of horseshoe vortices (Vortex lattice) on a finite wing [33] 166 Figure 117 Nomenclature for calculating induced velocity by a finite length vortex segment [89] 167 Figure 118 A typical horseshoe vortex [89] 168 Figure 119 Vector elements for the calculation of induced velocities [89] 169 Figure 120 Nomenclature for tangency condition: (a) normal to element of mean camber surface, (b) section AA, (c) section BB [89] 171 Figure 121 Dihedral angle [89] 171 List of Tables Table Morphing Skin Concepts 22 Table Material combinations tested by Kirn [66] 27 Table Summation of flow visualisation behaviour 58 Table Difference in results for XFLR5 and TORNADO to experimental results 83 Table High-lift device comparison of the conventional T240 and the morphing wing at Re 168000 85 Table Comparison of the roll performance between the conventional T240 wing and the morphing wing at Re 337000 86 Table Comparison of conventional T240 and morphing wing in cruise condition at Re 337000 87 Table Experimental results of similar morphing concepts in literature [34, 49, 51, 76, 86] 88 Table Load results for the calibration of the JR3 load cell in the x-axis (Lift axis) 101 Table 10 Load results for the calibration of the JR3 load cell in the y-axis (Drag axis) 103 Table 11 Pure moment Yaw results for the calibration of the JR3 load cell in the y-axis (Drag axis) 104 Table 12 Pure moment Roll results for the calibration of the JR3 load cell in the x-axis (Lift axis) 107 Table 13 List of non-converged conditions in XFLR5 159 viii Nomenclature b c 𝐶𝐿 𝐶𝐿𝑙𝑜𝑐𝑎𝑙 Wingspan, in 𝑚 Chord, in 𝑚 Coefficient of Lift Section Lift coefficient 𝐶𝐷 Coefficient of Drag 𝐶𝐷0 Parasite Drag 𝐶𝐿𝛼 Lift-curve slope 𝑐𝑝 Coefficient of Pressure 𝐷 Drag, in N 𝐹 Fuel consumption, in 𝑘𝑔/ℎ 𝐼𝑥𝑥 Mass moment of Inertia about x-axis, in 𝑘𝑔/𝑚2 𝐶𝐷𝑖 Induced Drag 𝐶𝑙 Rolling moment coefficient 𝐶𝑙𝛿 Aileron effectiveness 𝑐𝜏 Specific fuel consumption 𝑒 Oswald efficiency factor 𝑓 Frequency, in ℎ𝑧 𝐿 Lift, in N L Roll Moment, in Nm 𝑀 Moment, in Nm 𝑝̇ Roll rate acceleration, in 𝑑𝑒𝑔/𝑠 𝐿/𝐷 Lift to Drag ratio 𝑝 Roll rate, in 𝑑𝑒𝑔/𝑠 𝑞 Dynamic pressure, 1/2𝜌𝑉 S Wing area, in 𝑚2 ix 236000 10 10 to 20 236000 15 4.5 to 5.5 236000 20 -0.5 to 236000 20 16.5 to 20 236000 25 -5 to 236000 25 7.5 to 20 236000 30 -5 to -4.5 236000 30 -1.5 to 236000 30 to 20 236000 35 -3.5 to 19.5 236000 40 -4 236000 40 to 236000 40 14 to 15 236000 40 16 to 16.5 236000 40 18.5 Conditions that didn’t converge in 2D XFLR5 Re 𝛿𝑚 269000 15 to 10.5 269000 20 -0.5 to 269000 20 11.5 to 20 269000 25 -5 to 20 269000 30 269000 30 16.5 to 20 269000 35 -3.5 to 269000 35 11.5 to 20 161 𝛼 269000 40 -0.5 to 269000 40 14 to 15.5 Conditions that didn’t converge in 2D XFLR5 Re 𝛿𝑚 337000 10 337000 10 337000 10 17 to 20 337000 15 -3 337000 20 -0.5 to 20 337000 25 -5 to 337000 25 to 20 337000 30 to 337000 30 8.5 to 20 337000 35 -5 to 337000 35 to 7.5 337000 35 17 to 20 337000 40 -4.5 337000 40 -2.5 to 0.5 337000 40 11 to 14.5 162 𝛼 APPENDIX G – Further Information on the Vortex Lattice Method Vortex lattice method is a simpler approach to the lifting surface method Hence the lifting surface method will be explored briefly in this section Prandtl’s classic lifting line theory yields reasonable results for straight wings with moderate to high aspect ratios Lifting line theory is not suitable for wings with low aspect ratio (straight wings), swept wings and delta wings [33] Lifting surface method covers for this weakness of lifting line theory Imagine several horseshoe vortices superimposed along a lifting line, now impose a series of further lifting lines at different chordwise positions (parallel to the 𝑦-axis), this is illustrated in Figure 114 Figure 114 Lifting lines in both spanwise and chordwise directions superimposed onto a wing [33, 89] A vortex sheet is formed where the vortex lines are parallel to the 𝑦 axis [33] The strength of the vortex sheet is 𝛾 (per unit length in x direction), where 𝛾 varies in 𝑦 direction Like a single lifting line, Γ varies along the vortex sheet Remember that each lifting line will generally have different strength, so 𝛾 varies also varies with 𝑥 Therefore 𝛾 is dependent on both 𝑥 and 𝑦 i.e 𝛾 = 𝛾(𝑥, 𝑦) Also remember that each lifting line has a 163 system of trailing vortices, therefore the lifting lines are crossed by superimposed trailing vortices parallel to the 𝑥-axis [33] The trailing vortices form another vortex sheet of strength 𝛿 (per unit length in the 𝑦 direction) From the leading edge to the trailing edge, additional superimposed trailing vortices are picked up each time a lifting line is crossed (since lifting lines run parallel to the 𝑥 axis and trailing vortices in 𝑦 axis) Trailing vortices are part of horseshoe vortex systems, where the leading edges/bound vortices make up various lifting lines [33] Since circulation about each lifting line varies in the 𝑦 direction, so does the strength of trailing vortices, therefore 𝛿 = 𝛿(𝑥, 𝑦), which is seen in Figure 114 There are now two vortex sheets, one running parallel in the 𝑥 axis and the other parallel in the 𝑦 axis Where the one running parallel to 𝑦 is 𝛾 (per unit length in 𝑥 direction) and the one running parallel to 𝑥 is 𝛿 (per unit length in 𝑦 direction) Which results in a lifting surface distributed over the entire wing seen in Figure 114 At any point of the lifting surface, the strength of the lifting surface is determined by both 𝛾 and 𝛿, where both are functions of 𝑥 and 𝑦 [33] 𝛾 = 𝛾(𝑥, 𝑦) is the spanwise vortex distribution and 𝛿 = 𝛿(𝑥, 𝑦) is the chordwise vortex strength distribution Downstream of the trailing edge there are no spanwise vortex distribution hence only chordwise vortices are present in the wake The strength of the wake vortex sheet is 𝛿𝑤 (per unit length in the 𝑦 direction) Remembering that no vortex lines are present in the wake, the strength of the trailing vortex is constant with 𝑥 Hence 𝛿𝑤 is solely a function of 𝑦 i.e 𝛿𝑤 (𝑦) is equal to its value at the trailing edge At point P (refer to Figure 114), both the lifting surface and wake vortex sheet induce a normal component of velocity, the normal velocity is 𝑤(𝑥, 𝑦) Because the wing planform needs to be a stream surface of flow, the sum of the induced 𝑤(𝑥, 𝑦) and normal component of the freestream velocity must be zero at all points on the wing (including point P), this is called the flow-tangency condition on the wing surface [33] Hence the main point of the lifting-surface theory is to find 𝛾 = 𝛾(𝑥, 𝑦) and 𝛿 = 𝛿(𝑥, 𝑦), so that flow-tangency condition is satisfied on all points of the wing Figure 115 Velocity (the direction is coming out of the paper) induced at point P by the infinitesimal segment of the lifting surface[33] An expression for induced normal velocity can be derived from Figure 115, considering point with coordinates (𝜉, 𝜂) The spanwise vortex strength is 𝛾(𝜉, 𝜂) Imagine a thin segment of the spanwise vortex sheet of incremental length 𝑑𝜉 in the 𝑥 direction, the strength of the segment is 𝛾 𝑑𝜉 and the 164 filament stretches in the 𝑦 (𝜂 direction in this case) Consider point P at (𝑥, 𝑦), which is at a distance of 𝑟 from point (𝜉, 𝜂) From Biot-Savart law, the incremental velocity induced at 𝑃 by a segment 𝑑𝜂 of this vortex filament of strength 𝛾 𝑑𝜉 |𝑑𝑉| = | Γ 𝑑𝑙∗𝑥 | 4𝜋 |𝑟|3 = 𝛾𝑑𝜉 (𝑑𝜂)𝑟 sin θ 𝑟3 4𝜋 (23) In Figure 115, following right-hand rule for strength 𝛾, note that |𝑑𝑉| is induced downward into the plane of the wing (in the negative z direction) Following usual sign convention that 𝑤 is positive in the upward direction (positive 𝑧 direction) Induced velocity 𝑤 can be defined as (𝑑𝑤)γ = −|𝑑𝑉| and sin 𝜃 = (𝑥 − 𝜉)/𝑟 now |𝑑𝑉| equation can be written as: 𝛾 (𝑥−𝜉)𝑑𝜉𝑑𝜂 𝑟3 (𝑑𝑤)γ = − 4𝜋 (24) Considering contribution of elemental chordwise vortex of strength 𝛿 𝑑𝜂 to induced velocity at point P then 𝛿 (𝑥−𝜂)𝑑𝜉𝑑𝜂 𝑟3 (𝑑𝑤)δ = − 4𝜋 (25) Note that 𝑟 = √(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2 Both equations must be integrated over the wing planform (designated point S) to obtain the velocity induced at point P by the entire lifting surface Note that velocity at point P by the complete wake is given by equation (24) but with 𝛿𝑤 instead of 𝛿 and integrated over the wake, region W in Figure 115 Note that normal velocity induced at point P by both lifting surface and wake is: 𝑤(𝑥, 𝑦) = − 4𝜋 ∬𝑠 (𝑥−𝜉)𝛾(𝜉,𝜂)+(𝑦−𝜂)𝛿(𝜉,𝜂) [(𝑥−𝜉)2 +(𝑦−𝜂)2 ]3/2 𝑑𝜉𝑑𝜂 − 4𝜋 (𝑦−𝜂)𝛿 (𝜉,𝜂) 𝑤 ∬𝑤 [(𝑥−𝜉)2 +(𝑦−𝜂) ]3/2 𝑑𝜉𝑑𝜂 (26) The problem of lifting-surface theory is to solve equation 26 for both 𝛾(𝜉, 𝜂) and 𝛿(𝜉, 𝜂) such that the sum of 𝑤(𝑥, 𝑦) and the normal component of the freestream is zero So that the flow is tangent to planform surface S Now for the vortex lattice method, the wing is represented by a surface with many horseshoe vortices superimposed onto it, where the horseshoe vortices have different strengths The velocities induced by all the horseshoe vortices at control point is calculated by the Biot-Savart law [33, 89] Performing a summation for all the control points on the wing leads to a set of linear algebraic equations for the horseshoe vortices strengths that satisfy the no flow boundary condition of the wing [89] The strength of the horseshoe vortices is related to the wing circulation and pressure differential between the upper and lower surfaces of the wing [33] Integrating the pressure differentials yield the total forces and moments on the wing[33] 165 Figure 116 Single horseshoe to a system of horseshoe vortices (Vortex lattice) on a finite wing [33] In the theoretical analysis, the vortex lattice panels are located on the mean camber surface of the wing and trailing vortices follow a curved path once they leave the wing In general, using linear straight-line trailing vortices extending downstream to infinity yields acceptable accuracy For a linear approach, trailing vortices are either aligned parallel to the free stream or parallel to the vehicle axis, both orientations provide similar accuracy within assumptions of linearized theory In general, the trailing vortices are aligned to be parallel to the vehicle axis, because the orientation’s influence coefficients of other vortices are simpler to solve It should be noted that geometric coefficients not change as angle of attack is changed Examining Figure 116, where the panel of size 𝑙 (in the length of the flow direction), note that the panels in this method are trapezoidal A horseshoe vortex (𝑎𝑏𝑐𝑑) of strength Γ𝑛 is placed so that the leading edge or the bound vortex of the horseshoe (𝑏𝑐) is 𝑙/4 from the front of the panel (25% from the front of the panel) [33, 89] A control point is placed at 𝑙 from the front of the panel (75% from the front of the panel) and on the centerline, the derivation of this location is shown in “aerodynamics for engineers” work [89] There are now many horseshoe vortices covering the entire wing such as 𝑎𝑏𝑐𝑑, 𝑎𝑒𝑓𝑑, 𝑎𝑔ℎ𝑑, 𝑎𝑖𝑗𝑑 and etc Remember each horseshoe vortex has it’s a different strength of Γ𝑛 , hence why a summation of all the horseshoe vortices must be conducted Velocity induced by general horseshoe vortex The induced velocity by a vortex filament of strength Γ𝑛 and length of 𝑑𝑙 is given by the Biot-Savart law [89]: ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ = Γ𝑛 (𝑑𝑙 X 𝑟) 𝑑𝑉 4𝜋𝑟 (27) From Figure 117, the magnitude of the induced velocity is: ⃗⃗⃗⃗⃗ = Γ𝑛 sin 𝜃𝑑𝑙 𝑑𝑉 4𝜋𝑟 166 (28) Figure 117 Nomenclature for calculating induced velocity by a finite length vortex segment [89] The flow field induced by the horseshoe vortex of three straight segments, using equation 27 to calculate the effect of each segment Where 𝐴𝐵 (Figure 117) be the segment where the vorticity vector is directed from point A to point B And point C be where the normal distance from the line 𝐴𝐵 is 𝑟𝑝 Integrating between point A and point B, calculates the magnitude of the induced velocity: 𝑉= 𝜃2 Γ𝑛 ∫ sin 𝜃 4𝜋𝑟𝑝 𝜃1 𝑑𝜃 = Γ𝑛 (cos 𝜃1 4𝜋𝑟𝑝 − cos 𝜃2 ) Considering that the vortex filament extends to infinity in both directions then 𝜃1 = and 𝜃2 = 𝜋 Γ𝑛 4𝜋𝑟𝑝 𝑉 = (29) (30) (Remember, this is for infinite span airfoils) For ease of reading let 𝑟⃗⃗⃗0 , ⃗⃗⃗ 𝑟1 , ⃗⃗⃗ 𝑟2 designate the vectors ⃗⃗⃗⃗⃗ 𝐴𝐵, ⃗⃗⃗⃗⃗ , 𝐵𝐶 ⃗⃗⃗⃗⃗ , respectively this is illustrated in Figure 117, hence 𝐴𝐶 𝑟𝑝 = |𝑟⃗⃗⃗1 x ⃗⃗⃗ 𝑟2 | 𝑟⃗⃗⃗0 ∙ 𝑟⃗⃗⃗1 𝑟⃗⃗⃗0 ∙ 𝑟⃗⃗⃗2 cos 𝜃1 = cos 𝜃2 = 𝑟0 𝑟1 𝑟0 𝑟2 𝑟0 Note: 𝑟⃗⃗⃗0 is vector, 𝑟0 is the magnitude of vector ⃗⃗⃗ 𝑟0 , |𝑟⃗⃗⃗1 x ⃗⃗⃗ 𝑟2 | is the magnitude of the vector cross product The direction of induced velocity is given by unit vector 𝑟⃗⃗⃗1 x ⃗⃗⃗ 𝑟2 |𝑟⃗⃗⃗1 x ⃗⃗⃗ 𝑟2 | Results in ⃗ = Γ𝑛 ⃗⃗⃗⃗𝑟1 x ⃗⃗⃗⃗𝑟2 [𝑟⃗⃗⃗0 ∙ (⃗⃗⃗⃗𝑟1 − 𝑟⃗⃗⃗⃗2 )] 𝑉 |𝑟 ⃗⃗⃗⃗ ⃗⃗⃗⃗ | 𝑟1 4𝜋 x 𝑟2 167 𝑟2 (31) This equation is for the calculation of the induced velocity by the horseshoe vortices in VLM [89] This equation can be used for either vortex orientations Using equation 31 to calculate the velocity that is induced at a general location (𝑥, 𝑦, 𝑧) by the horseshoe vortex, illustrated in Figure 118 Figure 118 A typical horseshoe vortex [89] Segment 𝐴𝐵 represents the bound vortex portion of the horseshoe system and is located on the ¼ chord line of the panel The trailing vortices are parallel to the 𝑥-axis The resultant induced velocity vector is calculated by considering the influence of each of the elements ⃗⃗⃗⃗⃗ For the bound vortex, segment 𝐴𝐵 ⃗⃗⃗⃗⃗ = (𝑥2𝑛 − 𝑥1𝑛 )𝑖̂ + (𝑦2𝑛 − 𝑦1𝑛 )𝑗̂ + (𝑧2𝑛 − 𝑧1𝑛 )𝑘̂ 𝑟0 = 𝐴𝐵 ⃗⃗⃗ 𝑟⃗⃗⃗1 = (𝑥 − 𝑥1𝑛 )𝑖̂ + (𝑦 − 𝑦1𝑛 )𝑗̂ + (𝑧 − 𝑧1𝑛 )𝑘̂ 𝑟⃗⃗⃗2 = (𝑥 − 𝑥2𝑛 )𝑖̂ + (𝑦 − 𝑦2𝑛 )𝑗̂ + (𝑧 − 𝑧2𝑛 )𝑘̂ 168 (32) (33) (34) Figure 119 Vector elements for the calculation of induced velocities [89] Using equation 31 to calculate the velocity induced at some point 𝐶 (x, y, z) by the vortex filament 𝐴𝐵 (shown in Figure 118 and Figure 119) then, Γ𝑛 ⃗⃗⃗⃗⃗⃗ {F𝑎𝑐1𝐴𝐵 }{F𝑎𝑐2𝐴𝐵 } 𝑉 𝐴𝐵 = 4𝜋 Where, {F𝑎𝑐1𝐴𝐵 } = And (35) 𝑟2 𝑟⃗⃗⃗1 x ⃗⃗⃗ |𝑟⃗⃗⃗1 x ⃗⃗⃗ 𝑟2 |2 {F𝑎𝑐1𝐴𝐵 } = {[(𝑦 − 𝑦1𝑛 )(𝑧 − 𝑧2𝑛 ) − (𝑦 − 𝑦2𝑛 )(𝑧 − 𝑧1𝑛 )]𝑖̂ − [(𝑥 − 𝑥1𝑛 )(𝑧 − 𝑧2𝑛 ) − (𝑥 − 𝑥2𝑛 )(𝑧 − 𝑧1𝑛 )]𝑗̂ + [(𝑥 − 𝑥1𝑛 )(𝑦 − 𝑦2𝑛 ) − (𝑥 − 𝑥2𝑛 )(𝑦 − 𝑦1𝑛 )]𝑘̂ } / {[(𝑦 − 𝑦1𝑛 )(𝑧 − 𝑧2𝑛 ) − (𝑦 − 𝑦2𝑛 )(𝑧 − 𝑧1𝑛 )]2 + [(𝑥 − 𝑥1𝑛 )(𝑧 − 𝑧2𝑛 ) − (𝑥 − 𝑥2𝑛 )(𝑧 − 𝑧1𝑛 )]2 + [(𝑥 − 𝑥1𝑛 )(𝑦 − 𝑦2𝑛 ) − (𝑥 − 𝑥2𝑛 )(𝑦 − 𝑦1𝑛 )]2 } {F𝑎𝑐2𝐴𝐵 } = (𝑟⃗⃗⃗0 ∙ {F𝑎𝑐2𝐴𝐵 } = { 𝑟⃗⃗⃗1 𝑟⃗⃗⃗2 − 𝑟⃗⃗⃗0 ∙ ) 𝑟1 𝑟2 [(𝑥2𝑛 − 𝑥1𝑛 )(𝑥 − 𝑥1𝑛 ) + (𝑦2𝑛 − 𝑦1𝑛 )(𝑦 − 𝑦1𝑛 ) + (𝑧2𝑛 − 𝑧1𝑛 )(𝑧 − 𝑧1𝑛 )] √(𝑥 − 𝑥1𝑛 )2 + (𝑦 − 𝑦1𝑛 )2 + (𝑧 − 𝑧1𝑛 )2 [(𝑥2𝑛 − 𝑥1𝑛 )(𝑥 − 𝑥2𝑛 ) + (𝑦2𝑛 − 𝑦1𝑛 )(𝑦 − 𝑦2𝑛 ) + (𝑧2𝑛 − 𝑧1𝑛 )(𝑧 − 𝑧2𝑛 )] − } √(𝑥 − 𝑥2𝑛 )2 + (𝑦 − 𝑦2𝑛 )2 + (𝑧 − 𝑧2𝑛 )2 To calculate the velocity induced by the filament that extends from point 𝐴 to infinity, first the velocity induced by the collinear, finite-length filament that extends from point 𝐴 to point 𝐷 (seen in Figure 119) needs to be determined Since 𝑟⃗⃗⃗0 is in the direction of the vorticity vector, ⃗⃗⃗⃗⃗ = (𝑥1𝑛 − 𝑥3𝑛 )𝑖̂ 𝑟⃗⃗⃗0 = 𝐷𝐴 169 (36) ⃗⃗⃗ 𝑟1 = (𝑥 − 𝑥3𝑛 )𝑖̂ + (𝑦 − 𝑦1𝑛 )𝑗̂ + (𝑧 − 𝑧1𝑛 )𝑘̂ (37) 𝑟⃗⃗⃗2 = (𝑥 − 𝑥1𝑛 )𝑖̂ + (𝑦 − 𝑦1𝑛 )𝑗̂ + (𝑧 − 𝑧1𝑛 )𝑘̂ (38) From Figure 119, the induced velocity is Where, And Γ𝑛 ⃗⃗⃗⃗⃗⃗ 𝑉 𝐴𝐷 = 4𝜋 {F𝑎𝑐1𝐴𝐷 }{F𝑎𝑐2𝐴𝐷 } {F𝑎𝑐1𝐴𝐷 } = (39) (𝑧 − 𝑧1𝑛 )𝑗̂ + (𝑦1𝑛 − 𝑦)𝑘̂ [(𝑧 − 𝑧1𝑛 )2 + (𝑦 − 𝑦1𝑛 )2 ](𝑥3𝑛 − 𝑥1𝑛 ) {F𝑎𝑐2𝐴𝐷 } = (𝑥3𝑛 𝑥3𝑛 − 𝑥 − 𝑥1𝑛 ) { √(𝑥 − 𝑥3𝑛 )2 + (𝑦 − 𝑦1𝑛 )2 + (𝑧 − 𝑧1𝑛 )2 𝑥 − 𝑥1𝑛 + } √(𝑥 − 𝑥1𝑛 ) + (𝑦 − 𝑦1𝑛 )2 + (𝑧 − 𝑧1𝑛 )2 Setting 𝑥3 go to infinity, the first term of {F𝑎𝑐2𝐴𝐷 } goes to 1.0 Therefore, velocity induced by the vortex filament which extends from point 𝐴 to inifinity in the positive direction that is parallel to the 𝑥-axis is given by: ̂ Γ𝑛 (𝑧−𝑧1𝑛 )𝑗̂ +(𝑦1𝑛 −𝑦)𝑘 ⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝐴∞ = 4𝜋 {[(𝑧−𝑧 )2 +(𝑦 −𝑦)2 ]} [1.0 + 1𝑛 1𝑛 𝑥−𝑥1𝑛 √(𝑥−𝑥1𝑛 )2 +(𝑦−𝑦1𝑛 )2 +(𝑧−𝑧1𝑛 )2 ] (40) Velocity induced by vortex filament that extends from point 𝐵 to infinity in positive direction that is parallel to the 𝑥 axis given by: ̂ Γ𝑛 (𝑧−𝑧2𝑛 )𝑗̂ +(𝑦2𝑛 −𝑦)𝑘 ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝐵∞ = − 4𝜋 {[(𝑧−𝑧 )2 +(𝑦 −𝑦)2 ]} [1.0 + 2𝑛 2𝑛 𝑥−𝑥2𝑛 √(𝑥−𝑥2𝑛 )2 +(𝑦−𝑦2𝑛 )2 +(𝑧−𝑧2𝑛 )2 ] (41) Total velocity induced at some point (𝑥, 𝑦, 𝑧) by horseshoe vortex representing one of the surface elements is the sum of components in equation 39,40,41 Let point (𝑥,𝑦,𝑧) be the control point of the 𝑚th panel, which is designated by coordinates (𝑥𝑚 , 𝑦𝑚 , 𝑧𝑚 ) The velocity induced at the 𝑚th control ⃗⃗⃗⃗⃗⃗⃗⃗ point by the vortex representing the 𝑛th panel is designated by 𝑉 𝑚,𝑛 Implementing this into 39,40,41 ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝑚,𝑛 = 𝐶𝑚,𝑛 Γ𝑛 (42) ⃗⃗⃗⃗⃗⃗⃗⃗ Where the influence coefficient 𝐶 𝑚,𝑛 is dependent on the geometry of the 𝑛th horseshoe vortex and the distance from the control point of 𝑚th panel Because the governing equation is linear, velocities induced by 2𝑁 vortices are added together for the total induced velocity at 𝑚rh control points: 2𝑁 ⃗⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝑚,𝑛 = ∑𝑛=1 𝐶𝑚,𝑛 Γ𝑛 We Have 2𝑁 of these equations, one for each of the control points Application of boundary conditions 170 (43) It is possible to determine the resultant induced velocity at any point in space if the strengths of the 2𝑁 horseshoe vortices are known Using The boundary condition of a streamline surface, the strength of the vortices Γ𝑛 can be determined, where Γ𝑛 represents the lifting flow field of the wing The boundary condition of a streamline surface, where resultant flow is tangent to the wing for each control point, where the control point lies ¾ chord and on the centerline of each panel If the trailing vortices are parallel to the vehicle axis [i.e., the x axis for equation 39,40,41 is the vehicle axis] then the induced velocity components can be evaluated Figure 120 Nomenclature for tangency condition: (a) normal to element of mean camber surface, (b) section AA, (c) section BB [89] From Figure 120 the tangency requirement yields the relation, −𝑢𝑚 sin 𝛿 cos 𝜙 − 𝜐𝑚 cos 𝛿 sin 𝜙 + 𝑤𝑚 cos 𝜙 cos δ + 𝑈∞ sin(𝛼 − 𝛿) cos 𝜙 = (44) Figure 121 Dihedral angle [89] Where 𝜙 is the dihedral angle (shown in Figure 121) and 𝛿 is slope of mean camber line at the control point hence, 𝑑𝑧 𝛿 = tan−1 (𝑑𝑥) 171 𝑚 (45) For wings where the slope of the mean camber line is small and at small angles of attack equation 44 can be simplified to, 𝑑𝑧 𝑤𝑚 − 𝜐𝑚 tan 𝜙 + 𝑈∞ [𝛼 − ( ) ] = 𝑑𝑥 (46) 𝑚 Noting that the approximation is consistent with the assumptions of linearized theory The circulation strengths Γ𝑛 are required to satisfy the tangent flow boundary conditions, by solving the simultaneous equations represented by equation 43 Relations for a planar wing For a planar wing, 𝑧1𝑛 = 𝑧2𝑛 = for all bound vortices and 𝑧𝑚 = for all control points hence Γ𝑛 ⃗⃗⃗⃗⃗⃗ 𝑉 𝐴𝐵 = (𝑥 4𝜋 ̂ (𝑥 −𝑥 )(𝑥 −𝑥 )+(𝑦 −𝑦 )(𝑦 −𝑦 ) 𝑘 [ 2𝑛 1𝑛 𝑚 2𝑛 2𝑛 1𝑛 𝑚 1𝑛 )(𝑦 )(𝑦 ) −𝑥 −𝑦 )+(𝑥 −𝑥 −𝑦 √(𝑥𝑚 −𝑥1𝑛 ) +(𝑦𝑚 −𝑦1𝑛 ) 𝑚 1𝑛 𝑚 2𝑛 𝑚 2𝑛 𝑚 1𝑛 (𝑥2𝑛 −𝑥1𝑛 )(𝑥𝑚 −𝑥2𝑛 )+(𝑦2𝑛 −𝑦1𝑛 )(𝑦𝑚 −𝑦2𝑛 ) √(𝑥𝑚 −𝑥2𝑛 )2 +(𝑦𝑚 −𝑦2𝑛 )2 ] (47) ̂ 𝑥𝑚 −𝑥1𝑛 𝑘 [1 + ] √(𝑥𝑚 −𝑥1𝑛 )2 +(𝑦𝑚 −𝑦1𝑛 )2 1𝑛 −𝑦𝑚 Γ𝑛 ⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝐴∞ = 4𝜋 𝑦 − (48) ̂ 𝑘 𝑥𝑚 −𝑥2𝑛 [1 + ] 4𝜋 𝑦2𝑛 −𝑦𝑚 √(𝑥𝑚 −𝑥2𝑛 )2 +(𝑦𝑚 −𝑦2𝑛 )2 Γ𝑛 ⃗⃗⃗⃗⃗⃗⃗⃗ 𝑉 𝐵∞ = − (49) For planar wings, the three components of the vortex representing 𝑛th panel induce a velocity at the control point of the 𝑚th panel in the 𝑧 direction, i.e downwash Combining the three components, Γ 𝑤𝑚,𝑛 = 4𝜋𝑛 {(𝑥 𝑚 −𝑥1𝑛 )(𝑦𝑚 −𝑦2𝑛 )−(𝑥𝑚 −𝑥2𝑛 )(𝑦𝑚 −𝑦1𝑛 ) (𝑥2𝑛 −𝑥1𝑛 )(𝑥𝑚 −𝑥2𝑛 )+(𝑦2𝑛 −𝑦1𝑛 )(𝑦𝑚 −𝑦2𝑛 ) √(𝑥𝑚 −𝑥2𝑛 )2 +(𝑦𝑚 −𝑦2𝑛 )2 +𝑦 1𝑛 −𝑦𝑚 (𝑥𝑚 −𝑥2𝑛 ) [ (𝑥2𝑛 −𝑥1𝑛 )(𝑥𝑚 −𝑥1𝑛 )+(𝑦2𝑛 −𝑦1𝑛 )(𝑦𝑚 −𝑦1𝑛 ) [1 + √(𝑥𝑚 −𝑥2𝑛 )2 +(𝑦𝑚 −𝑦2𝑛 )2 √(𝑥𝑚 −𝑥1𝑛 )2 +(𝑦𝑚 −𝑦1𝑛 )2 (𝑥𝑚 −𝑥1𝑛 ) √(𝑥𝑚 −𝑥1𝑛 )2 +(𝑦𝑚 −𝑦1𝑛 )2 ]]} ]−𝑦 2𝑛 −𝑦𝑚 − [1 + (50) Summing the contributions of all the vortices to the downwash at the control point of the 𝑚th panel, 𝑤𝑚 = ∑2𝑁 𝑛=1 𝑤𝑚,𝑛 (51) 𝑤𝑚 + 𝑈∞ sin 𝛼 = (52) Applying the tangency condition defined by equations 44 and 46, because a planar wing is used then (𝑑𝑧⁄𝑑𝑥 )𝑚 = at all locations and 𝜙 = Since the 𝑈∞ sin 𝛼 is the component of the freestream velocity that is perpendicular to the wing at any point on the wing Hence resultant flow is tangent to the wing if the total vortex-induced downwash at the control point of 𝑚th panel, which is determined using equation 51, balances the normal component of the freestream velocity For small angles of attack, 𝑤𝑚 = −𝑈∞ 172 Once the strength of the horseshoe vortices is determined by fulfilling the boundary condition that flow is tangent to the surface at each of the control points, the lift of the wing can be determined [89] For wings with no dihedral, lift is generated by the freestream velocity crossing the spanwise vortex filaments, since there are no sidewash or backwash velocities Since panels extend from the leading edge to the trailing edge then the lift (per unit span) acting on the 𝑛th panel is, 𝑙𝑛 = 𝜌∞ 𝑈∞ Γ𝑛 (53) Since flow is symmetric then total lift of the wing is, 0.5𝑏 (in terms of finite element panels) 𝐿 = ∫0 𝜌∞ 𝑈∞ Γ(𝑦) 𝑑𝑦 𝐿 = 2𝜌∞ 𝑈∞ ∑4𝑛=1 Γ𝑛 Δ𝑦𝑛 The section lift coefficient for the 𝑛th panel is 2Γ ∞ 𝑐𝑎𝑣 𝐶𝑙(𝑛𝑡ℎ) = 𝑙𝑛 𝐶𝑙 𝑐 𝑐𝑎𝑣 𝑙 ) 𝑞∞ 𝑐𝑎𝑣 𝑗 𝜌 𝑈 2𝑐 ∞ ∞ 𝑎𝑣 =𝑈 (54) (55) (56) The values of Γ for those bound vortex filaments at the spanwise location (chordwise strip) of interest, for a chordwise row, 𝐽𝑚𝑎𝑥 = ∑𝑗=1 ( (57) Where 𝑐𝑎𝑣 is average chord and is equal to 𝑆/𝑏, 𝑐 is local chord, and 𝑗 is the index for an elemental panel in the chordwise row By integrating the lift over the span, the total lift coefficient is determined 𝐶𝑙 𝑐 𝐶𝐿 = ∫0 𝑐𝑎𝑣 2𝑦 𝑑(𝑏) (58) Once section lift coefficient for the chordwise strip of the wing, the induced drag coefficient may be calculated with, Where 𝛼𝑖 , which induced incidence, 𝛼𝑖 = − For symmetrical loading then 𝛼𝑖 becomes, 𝛼𝑖 = − +0.5𝑏 𝐶𝐷𝑣 = ∫−0.5𝑏 𝐶𝑙 𝑐𝛼𝑖 𝑑𝑦 𝑆 +0.5𝑏 𝐶𝑙 𝑐 𝑑 ∫ 8𝜋 −0.5𝑏 (𝑦−𝜂)2 𝐶𝑐 0.5𝑏 [ 𝑙 ∫ 8𝜋 0.5𝑏 (𝑦−𝜂)2 𝐶𝑐 𝑙 + (𝑦+𝜂) 2] 𝑑 (59) (60) (61) Approximating the spanwise lift distribution across the strip by a parabolic function, at the 𝑚th chordwise strip with semi-width 𝑒𝑚 and centerline at 𝜂 = 𝑦𝑚 , 𝐶𝑙 𝑐 ) 𝐶𝐿 𝑐̅ 𝑚 ( = 𝑎𝑚 𝜂 + 𝑏𝑚 𝜂 + 𝑐𝑚 173 Solving for 𝑎𝑚 , 𝑏𝑚 and 𝑐𝑚 , 𝑦𝑚+1 = 𝑦𝑚 + (𝑒𝑚 + 𝑒𝑚+1 ) 𝑦𝑚−1 = 𝑦𝑚 − (𝑒𝑚 + 𝑒𝑚−1 ) Hence 𝐶𝑙 𝑐 ) 𝐶𝐿 𝑐̅ 𝑚 𝑐𝑚 = ( 𝑎𝑚 = − 𝑎𝑚 𝜂 − 𝑏𝑚 𝜂 (62) 𝐶𝑙 𝑐 𝐶𝑙 𝑐 𝐶𝑙 𝑐 − (𝑑𝑚𝑖 + 𝑑𝑚𝑜 ) ( ) + 𝑑𝑚𝑜 ( ) {𝑑𝑚𝑖 ( ) } 𝐶𝐿 𝑐̅ 𝑚+1 𝑑𝑚𝑖 𝑑𝑚𝑜 (𝑑𝑚𝑖 + 𝑑𝑚𝑜 ) 𝐶𝐿 𝑐̅ 𝑚 𝐶𝐿 𝑐̅ 𝑚−1 𝑏𝑚 = Where 𝐶𝑙 𝑐 𝐶𝑙 𝑐 ] {𝑑𝑚0 (2𝜂𝑚 − 𝑑𝑚𝑜 ) [( ) − ( ) 𝐶𝐿 𝑐̅ 𝑚 𝑑𝑚𝑖 𝑑𝑚𝑜 (𝑑𝑚𝑖 + 𝑑𝑚𝑜 ) 𝐶𝐿 𝑐̅ 𝑚−1 𝐶𝑙 𝑐 𝐶𝑙 𝑐 − ( ) ]} − 𝑑𝑚𝑖 (2𝜂𝑚 − 𝑑𝑚𝑖 ) [( ) 𝐶𝐿 𝑐̅ 𝑚 𝐶𝐿 𝑐̅ 𝑚+1 𝑑𝑚𝑖 = 𝑒𝑚 + 𝑒𝑚−1 For symmetrical load distribution 𝑑𝑚𝑜 = 𝑒𝑚 + 𝑒𝑚+1 ( 𝐶𝑙 𝑐 𝐶𝑙 𝑐 ) =( ) 𝐶𝐿 𝑐̅ 𝑚 𝐶𝐿 𝑐̅ 𝑚−1 𝑒𝑚−1 = 𝑒𝑚 At the root, ( 𝐶𝑙 𝑐 ) =0 𝐶𝐿 𝑐̅ 𝑚+1 𝑒𝑚+1 = At the tip, the numerical form for the induced incidence 𝛼𝑖 (𝑦) 𝐶𝐿 𝑐 𝑦 (𝑦𝑚 +𝑒𝑚 )𝑎𝑚 +𝑦 𝑏𝑚 +(𝑦𝑚 +𝑒𝑚 )𝑐𝑚 𝑦 (𝑦𝑚 −𝑒𝑚 )𝑎𝑚 +𝑦 𝑏𝑚 +(𝑦𝑚 −𝑒𝑚 )𝑐𝑚 ∑𝑁 { − + 𝑦 −(𝑦𝑚 +𝑒𝑚 )2 4𝜋 𝑚=1 𝑦 −(𝑦𝑚 −𝑒𝑚 )2 2 (𝑦−𝑒 )2 −𝑦 𝑦 −(𝑦 +𝑒 )2 1 𝑦𝑎𝑚 log [(𝑦+𝑒𝑚)2 𝑚2 ] + 𝑏𝑚 log [ 𝑚 𝑚)2 ] + 2𝑒𝑚 𝑎𝑚 } ( 63) 𝑦 −(𝑦𝑚 −𝑒𝑚 𝑚 −𝑦𝑚 = − Assuming that the product 𝐶𝑙 𝑐𝛼𝑖 has a parabolic variation across the strip, 𝐶𝑐 𝛼 [(𝐶 𝑙 𝑐̅) (𝐶 𝑖𝑐̅)] = 𝑎𝑛 𝑦 + 𝑏𝑛 𝑦 + 𝑐𝑛 𝐿 𝐿 𝑛 (64) Coefficients of 𝑎𝑛 , 𝑏𝑛 and 𝑐𝑛 are determined using the same approach as determining 𝑎𝑚 , 𝑏𝑚 and 𝑐𝑚 The numerical form of the induced drag coefficient, is then a generalization of Simpson’s rule: 174 𝐶𝐷𝑣 𝐶𝐿 = ∑𝑁 𝑒 {[𝑦𝑛 𝐴𝑅 𝑛=1 𝑛 175 + ( ) 𝑒𝑛 ] 𝑎𝑛 + 𝑦𝑛 𝑏𝑛 + 𝑐𝑛 } (65) ... hinges and actuators as seen in Vasista et al and Yang et al [41, 46] Variable geometry truss manipulators have also been used as a mechanism to achieve morphing [40] Linear actuators and pneumatic... has higher lift-drag ratio and can maintain it longer over a range of angles of attacks roughly a range of 9.05°, it does however plateau While flaps have a smaller range of maximum lift-drag... situations in flight Smart material-based actuators can limit some of the structural properties of a wing, such as usage of smart material-based actuators as the part of the skin to be compliant