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CHAPTER TWO - Static Aeroelasticity – Unswept Wing Structural Loads and Performance 21 2.1 Background

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CHAPTER TWO - Static Aeroelasticity – Unswept Wing Structural Loads and Performance 21 2.1 Background Static aeroelasticity – structural loads and performance CHAPTER TWO - Static Aeroelasticity – Unswept wing structural loads and performance 21 2.1 Background ........................................................................................................................... 21 2.1.2 Scope and purpose ....................................................................................................................... 21 2.1.2 The structures enterprise and its relation to aeroelasticity ............................................................ 22 2.1.3 The evolution of aircraft wing structures-form follows function ................................................ 24 2.2 Analytical modeling............................................................................................................... 30 2.2.1 The typical section, the flying door and Rayleigh-Ritz idealizations ................................................ 31 2.2.2 – Functional diagrams and operators – modeling the aeroelastic feedback process ....................... 33 2.3 Matrix structural analysis – stiffness matrices and strain energy .......................................... 34 2.4 An example - Construction of a structural stiffness matrix – the shear center concept ........ 38 2.5 Subsonic aerodynamics - fundamentals ................................................................................ 40 2.5.1 Reference points – the center of pressure..................................................................................... 44 2.5.2 A different reference point - the aerodynamic center................................................................... 45 2.6 Lifting surface flexibility and lift generation-the typical section model ................................. 46 A comment on the typical section model used in this text ..................................................................... 50 A sidebar – why the ¾ semi-span position? ........................................................................................... 51 2.7 Example problem – The effect of elastic axis offset on wing lift effectiveness and divergence .................................................................................................................................................... 52 2.8 Using the typical section model to understand early monoplane aircraft structural failures .................................................................................................................................................... 56 2.9 Load factor sensitivity to aeroelastic effects-fatal changes in response .............................. 58 2.10 A simple single degree of freedom (torsion only) aeroelastic model ................................ 58 2.10.1 Twist angle and load aeroelastic amplification, the feedback process .......................................... 61 2.10.2 Static stability – defining the critical dynamic pressure point .................................................... 64 2.11 Illustrative examples – doing the math ............................................................................... 66 2.11.1 A non-aeroelastic example – static stability of a magneto-elastic device ..................................... 66 2.11.2 Example – a flexible aircraft in steady turning flight ................................................................... 68 2.12 Linear analysis vs. nonlinear analysis-divergence with a nonlinear aerodynamic load ....... 71 2.13 The effect of Mach number on divergence-the match point ............................................... 75 2.14 Divergence of multi-degree of freedom systems................................................................. 78 2.14.1 Static aeroelastic stability of multi-degree-of-freedom systems .................................................. 81 2.14.2 Divergence eigenvalues ............................................................................................................ 82 2.14.3 Divergence eigenvectors .......................................................................................................... 84 2.14.4 Summary - divergence of multi-degree of freedom systems ..................................................... 85 2.15 Example - Two degree of freedom wing divergence ........................................................... 86 2.16 Example – the shear center, center of twist and divergence ............................................. 88 2.16.1 Locating the model shear center ............................................................................................... 90 Chapter 2 page 23 - Static aeroelasticity 2.16.2 Locating the center of twist ........................................................................................................ 91 2.16.3 Wing divergence ....................................................................................................................... 91 2.17 Example - Divergence of a three-degree-of-freedom system .............................................. 93 2.18 Example - Divergence of 2 DOF offset wing segment sections .......................................... 97 2.18.1 Solution using a different coordinate system ............................................................................. 100 2.18.2 Equation development using the Principle of Virtual Work ....................................................... 102 2.19 Torsional divergence of an unswept flexible wing - differential equation model .............. 105 2.19.1 The perturbation solution for differential equation divergence dynamic pressure ..................... 108 2.19.2 Example - Divergence of a wing with flexible wing/fuselage attachment .................................. 110 2.19.3 – Stiffness criteria – an energy approach ................................................................................... 114 Summary .................................................................................................................................. 115 2.20 Control surface effectiveness .......................................................................................... 117 Two-dimensional airfoil control surface aerodynamic coefficients ....................................................... 119 Effect of control surface size on center of pressure location ................................................................ 119 The reversal dynamic pressure and its relationship to divergence ........................................................ 120 2.20.1 Aileron effectiveness including rolling motion ......................................................................... 123 2.21 Two degree of freedom wing aileron reversal .................................................................. 127 The traditional measure of aileron effectiveness - generating airplane rolling moment ........................ 131 The effect of moving the outboard aileron to the inboard panel .......................................................... 132 Rolling moment with the inboard aileron ............................................................................................ 134 2.22 Artificial stabilization - wing divergence feedback control ................................................ 136 2.22.1 – A two dimensional example ................................................................................................... 136 2.22.2 Example - Active control of typical section divergence speed ................................................... 139 Control effectiveness - summary ......................................................................................................... 144 2.23 The relationship between divergence and dynamic instability-typical section torsional vibration with quasi-steady airloads......................................................................................... 145 2.24 Summary - lift effectiveness, divergence and aileron effectiveness .............................. 147 Torsional stiffness is fundamental to all aeroelastic problems .............................................................. 147 Boundary conditions are important..................................................................................................... 148 Divergence and control effectiveness are different but have the same origin ....................................... 148 2.25 Homework problems ..................................................................................................... 150 2.26 Chapter 2 References ....................................................................................................... 158 Static aeroelasticity Static aeroelasticity (4 th edition) - page 21 CHAPTER TWO Static Aeroelasticity – Unswept wing structural loads and performance 2.1 Background All structures deform when external loads are applied although the deflections may be barely discernible. In most cases, the external and internal loads do not depend on the structural deformation. From an analysis perspective this means we can compute the internal loads and the external deflections independently. Figure 2.1.1 – Static aeroelasticity encompasses problems involving the intersection between These structural analysis problems are called steady-state aerodynamic and structural statically determinate and include structural deformation interactions. stability problems such as column buckling. However, if the loads and structural deflection interact the structural analysis problem becomes very different, both physically
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