 | Tobias M. Flexible aircraft modelling for flight loads analysis of wake vortex encounters: Diss. … Dr.-Ing. / Institute of Aeroelasticity, Göttingen. - Köln: DLR, Bibliotheks- und Informationswesen, 2010. - xxi, 190 p.: ill., graph. - (Forschungsbericht; 2010-37). - Bibliogr.: p.137-146. - ISSN 1434-8454
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Nomenclature ................................................. xvii
List of Acronyms .............................................. xxi
1 Introduction ................................................. 1
1.1 Previous Work ........................................... 4
1.1.1 Unsteady Aerodynamic Models for Simulation ....... 4
1.1.2 Integral Flexible Aircraft Models ................ 6
1.1.3 Wake Vortex Encounters ........................... 7
1.2 Objectives and Document Structure ....................... 8
1.3 A Motivational Example .................................. 9
1.3.1 Aeroelastic System ............................... 9
1.3.2 Non-Linear Steady Aerodynamic Data .............. 11
1.3.3 Linear Unsteady Aerodynamic Model ............... 11
1.3.4 Static Analysis using Non-Linear Steady
Aerodynamic Data ................................ 14
1.3.5 Static and Dynamic Analysis using the Linear
Aerodynamic Model ............................... 14
1.3.6 Comparison of the System Response ............... 15
1.3.7 The Residual Model Method ....................... 16
2 Aerodynamic Models .......................................... 21
2.1 Quasi-steady Aerodynamic Database ...................... 21
2.1.1 Examples of Modelling Assumptions ............... 23
2.1.2 Quasi-Steady Approximation ...................... 24
2.1.3 Static Aeroelastic Considerations ............... 25
2.1.4 Mapping of Load Distributions ................... 26
2.1.5 Implementation .................................. 28
2.2 Unsteady Aerodynamic Model ............................. 31
2.2.1 Governing Equations ............................. 31
2.2.2 Unsteady Vortex Lattice Method .................. 34
2.2.3 Linearized Formulation .......................... 50
2.2.4 Steady and Quasi-Steady Formulations ............ 55
2.2.5 Incremental Formulation and Analogy to
Rational Function Approximation ................. 58
2.2.6 Harmonic Formulations ........................... 58
2.2.7 Numerical Examples .............................. 60
2.3 Steady Correction of the UVLM by External Data ......... 67
2.3.1 Formulation of the Correction Problem ........... 67
2.3.2 Application ..................................... 68
2.4 Reduction of Model Order by Balanced Truncation ........ 72
2.4.1 Balanced Truncation ............................. 72
2.4.2 Balanced Truncation of Lower Order Sub-Models ... 74
2.4.3 Application ..................................... 74
3 Structural Model ............................................ 85
3.1 Stiffness and Mass Model ............................... 85
3.2 Modal Decomposition .................................... 87
3.3 Non-Linear Equations of Motion ......................... 88
3.4 Equation of Loads ...................................... 91
3.5 Small Disturbance Equations ............................ 92
3.6 Transformation from Inertial to Body-Fixed
Coordinates ............................................ 93
4 Aeroelastic Model Integration ............................... 97
4.1 Aero-Structural Coupling ............................... 97
4.1.1 Methodology ..................................... 98
4.1.2 Principle of Virtual Work ...................... 102
4.1.3 Numerical Example .............................. 103
4.2 Model Integration ..................................... 104
4.2.1 Model Integration without Quasi-steady
Aeroelastic Corrections ........................ 107
4.2.2 Model Integration with Quasi-steady
Aeroelastic Corrections ........................ 107
4.2.3 Model Integration with Dynamic Incremental
Gust ........................................... 108
5 Application to Wake Vortex Encounter ....................... 109
5.1 Wake Vortex Model ..................................... 110
5.2 Aircraft Models ....................................... 112
5.3 Loads Model ........................................... 113
5.4 Simplified Autopilot .................................. 115
5.5 Definition of Encounter ............................... 116
5.6 Simulation and Data Processing ........................ 117
5.7 Results and Discussion ................................ 117
5.7.1 Anatomy of the Symmetrical and Asymmetrical
Encounter ...................................... 117
5.7.2 Influence of Encounter Type on Flight Loads .... 120
5.7.3 Influence of Aircraft Modeling on Flight
Loads .......................................... 124
6 Conclusion and Outlook ..................................... 133
6.1 Outlook ............................................... 135
Bibliography .................................................. 136
A Validation of the Vortex Lattice Method .................... 147
A.l Lifting Surfaces ...................................... 147
A.l.l Steady 3D Solution ............................. 147
A.1.2 Unsteady 2D Solutions .......................... 147
A.1.3 Unsteady 3D Solutions .......................... 152
A.2 Validation of Bodies .................................. 155
A.2.1 Steady 3D Solution ............................. 155
A.2.2 Unsteady 3D Solution ........................... 157
В Linearized Formulation of the Unsteady Vortex Lattice
Method ..................................................... 159
B.l Linearized Downwash ................................... 159
B.2 Linearized Loads of Lifting Surface Panels ............ 160
B.3 Linearized Loads of Body Panels ....................... 161
B.4 Discrete to Continuous Time Conversions of the Wake
State Equation ........................................ 164
С Aircraft Data .............................................. 167
D Envelopes for Wake Vortex Encounter ........................ 169
D.l Integrated Loads ...................................... 170
D.1.1 Wing Envelopes ................................. 170
D.1.2 Horizontal Tail Envelopes ...................... 173
D.1.3 Vertical Tail Envelopes ........................ 175
D.1.4 Fuselage Envelopes ............................. 177
D.2 Accelerations ......................................... 183
D.2.1 Wing Envelopes ................................. 183
D.2.2 Horizontal Tail Envelopes ...................... 185
D.2.3 Vertical Tail Envelopes ........................ 186
D.2.4 Fuselage Envelopes ............................. 187
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