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© Copyright 2016
Sang Wu

Contributions to Physics-Based Aeroservoelastic Uncertainty Analysis

Sang Wu
A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

University of Washington
2016

Reading Committee:
Eli Livne, Chair Mehran Mesbahi Dorothy Reed Frode Engelsen

Program Authorized to Offer Degree:
Aeronautics and Astronautics University of Washington

Abstract

Contributions to Physics-Based Aeroservoelastic Uncertainty Analysis
Sang Wu
Chair of the Supervisory Committee:
Professor Eli Livne
The William E. Boeing Department of Aeronautics and Astronautics

The thesis presents the development of a new fully-integrated, MATLAB based simulation capability for aeroservoelastic (ASE) uncertainty analysis that accounts for uncertainties in all disciplines as well as discipline interactions. This new capability allows probabilistic studies of complex configuration at a scope and with depth not known before. Several statistical tools and methods have been integrated into the capability to guide the tasks such as parameter prioritization, uncertainty reduction, and risk mitigation.
The first task of the thesis focuses on aeroservoelastic uncertainty assessment considering control component uncertainty. The simulation has shown that attention has to be paid, if notch filters are used in the aeroservoelastic loop, to the variability and uncertainties of the aircraft involved. The second task introduces two innovative methodologies to characterize the unsteady aerodynamic uncertainty. One is a physically based aerodynamic influence coefficients element by element correction uncertainty scheme and the other is an alternative approach focusing on rational function approximation matrix uncertainties to evaluate the relative impact of uncertainty in aerodynamic stiffness, damping, inertia, or lag terms. Finally, the capability has been applied to obtain the gust load response statistics accounting for uncertainties in both aircraft and gust profiles. The gust response analysis shows a significant increase of the critical

loads when system’s uncertainties are included.

The work is expected to make a contribution to the understanding of the propagation of uncertainty and the resulting reliability of realistic integrated aeroservoelastic system inherent in real modern aircraft. This work will motivate more research in the aeroservoelastic uncertainty area regarding both methods development in general and reliability studies of different configurations of interest. The mathematical derivations which serve as the foundation of the capability are also well documented.

TABLE OF CONTENTS

List of Figures................................................................................................................................. v List of Tables ................................................................................................................................. xi List of Symbols............................................................................................................................. xii Chapter 1. Introduction ................................................................................................................... 1
1.1 1.2
Background and Challenges ........................................................................................... 1 The Needs and Goals ...................................................................................................... 7

Chapter 2. Tool Development......................................................................................................... 9
2.1 2.2 2.3 2.4 2.5
The Matlab Based FEM Capability ................................................................................ 9 Unsteady Aerodynamics – The Doublet Lattice Method ............................................. 11 The Aeroservoelastic State-Space Model ..................................................................... 13 Aeroelastic System with Gust Excitation ..................................................................... 16 Probabilistic Aeroservoelastic Analysis ....................................................................... 18

Chapter 3. Probabilistic Aeroservoelastic Reliability Assessment Considering Control System Component Uncertainty................................................................................................................ 21

3.1 3.2 3.3
Overview....................................................................................................................... 21 Introduction................................................................................................................... 22 Model Derivation.......................................................................................................... 24
3.3.1 The Aeroservoelastic State-Space Model ................................................................. 24 3.3.2 Model of Electrohydraulic Servo Actuator............................................................... 24

i

  • 3.4
  • Procedure of Uncertainty Mitigation ............................................................................ 27

3.4.1 Parameter Prioritization with Statistical Sensitivity Analysis .................................. 27 3.4.2 Bayesian Update and Risk Reduction....................................................................... 29

  • 3.5
  • Numerical Aeroservoelastic Reliability Analysis......................................................... 32

3.5.1 The Nominal (Baseline) Case ................................................................................... 32 3.5.2 Uncertainty in the Actuator of the Outboard Elevon................................................ 39 3.5.3 Uncertainty in the Actuator of the Outboard Elevon and the Structure.................... 43 3.5.4 Added Unsteady Aerodynamic Uncertainty ............................................................. 44

  • 3.6
  • Application of Uncertainty Mitigation.......................................................................... 49

3.6.1 Parameter Prioritization ............................................................................................ 49 3.6.2 Risk Reduction.......................................................................................................... 50

  • 3.7
  • Summary....................................................................................................................... 53

Chapter 4. Alternative Aerodynamic Uncertainty Modeling Approaches for Flutter Reliability Analysis......................................................................................................................................... 56

4.1 4.2 4.3 4.4 4.5
Overview....................................................................................................................... 56 Introduction................................................................................................................... 56 AICs by Linear Panel Methods..................................................................................... 60 Accounting for the Aerodynamic Uncertainty.............................................................. 65 Numerical Example ...................................................................................................... 69
4.5.1 Nominal Flutter Analysis.......................................................................................... 71 4.5.2 Aeroelastic Simulation Subject to Aerodynamic Uncertainty.................................. 76 4.5.3 Wing Section Prioritization with Statistical Sensitivity Analysis ............................ 81 4.5.4 Aerodynamic Uncertainty via Rational Function Approximation............................ 86 ii
4.5.5 The Addition of Structural Uncertainty.................................................................... 89

  • 4.6
  • Conclusion .................................................................................................................... 92

Chapter 5. Probabilistic Gust Loads Analysis Accounting for Aeroservoelastic System Uncertainty.................................................................................................................................... 95

5.1 5.2 5.3
Overview....................................................................................................................... 95 Introduction................................................................................................................... 95 Model Derivation.......................................................................................................... 98
5.3.1 Aeroelastic System with Gust Excitation ................................................................. 98 5.3.2 Gust Input Uncertainty in the Time Domain ............................................................ 98 5.3.3 Gust Input Uncertainty in the Frequency Domain.................................................. 100

5.3.4 The Addition of a Gust “Filter” to the Math Model of the System ........................ 102

  • 5.4
  • Numerical Example of Gust Response with Uncertainty ........................................... 105

5.4.1 Discrete Gust Response with System Uncertainty.................................................. 106 5.4.2 Frequency Domain Gust Response Accounting for System Uncertainty............... 110 5.4.3 Time Domain Turbulence Response with System Uncertainty.............................. 113

  • 5.5
  • Summary..................................................................................................................... 115

Chapter 6. Conclusion................................................................................................................. 118 BIBLOIOGRAPHY.................................................................................................................... 121 Appendix A. The Finite Element Formulation ........................................................................... 130 Appendix B. Unsteady Aerodynamics – The Doublet Lattice Method...................................... 144 Appendix C. Derivation of Aeroservoelastic State-Space Model .............................................. 158

  • iii
  • iv

LIST OF FIGURES

Figure 1.1. Aeroelastic / Aeroservoelastic Triangle and Pyramid.................................................. 1 Figure 1.2. ASE systems Inputs / Outputs flow.............................................................................. 3 Figure 2.1. FEM model of the 15 deg backswept wing................................................................ 10 Figure 2.2. Sketch of the swept and tapered wing........................................................................ 12 Figure 2.3. Closed loop flow chart of ASE system....................................................................... 15 Figure 2.4. Block diagram of aeroelastic / aeroservoelastic system............................................. 17 Figure 2.5. Uncertainty propagation in integrated aeroservoelastic models................................. 19 Figure 3.1. Schematic representation of proportional position control actuator loop .................. 26 Figure 3.2. The Matlab Finite Element model of the AFA configuration .................................... 33 Figure 3.3. The Matlab Doublet Lattice Aerodynamic model of AFA ........................................ 33 Figure 3.4. Rigid body roll mode.................................................................................................. 33 Figure 3.5. First structural mode................................................................................................... 33 Figure 3.6. Second structural mode .............................................................................................. 34 Figure 3.7. Third structural mode ................................................................................................. 34 Figure 3.8. Control mode one ....................................................................................................... 34 Figure 3.9. Control mode two....................................................................................................... 34 Figure 3.10. Nominal roll rate control block for AFA.................................................................. 36

v
Figure 3.11. Root locus plot of the ASE system without notch filter........................................... 37 Figure 3.12. Roll rate control block with notch filter ................................................................... 37 Figure 3.13. Root locus plot of the ASE system with notch filter................................................ 37 Figure 3.14. Nyquist Plot of a) the closed loop AFA system b) third quadrant ........................... 38 Figure 3.15. Singular value of a) sensitivity function b) complimentary sensitivity function ..... 38 Figure 3.16. Flow chart of ‘trial and error’ procedure.................................................................. 41 Figure 3.17. Histogram of unstable frequency.............................................................................. 48 Figure 3.18. Histogram of unstable pole (damping coefficient)................................................... 48 Figure 3.19. Results of the analysis of system sensitivity to servo-actuator parameters.............. 52 Figure 3.20. Fifty new measurements of actuator fluid bulk modulus......................................... 52 Figure 3.21. Comparison of original and updated actuator fluid bulk modulus........................... 52 Figure 4.1. DLM mesh definition of upwash and load points ...................................................... 62 Figure 4.2. Structure FEM model ................................................................................................. 70 Figure 4.3. Aerodynamic panel model.......................................................................................... 70 Figure 4.4. Structure mode 1......................................................................................................... 70 Figure 4.5. Structure mode 2......................................................................................................... 70 Figure 4.6. Structure mode 3......................................................................................................... 70 Figure 4.7. Structure mode 4......................................................................................................... 70 Figure 4.8. Structure mode 5......................................................................................................... 71 vi

Figure 4.9. v-g and v- plots at M = 0.567 ................................................................................. 73

Figure 4.10. v-g and v- plots at M = 0.901 ............................................................................... 73

Figure 4.11. v-g and v- plots at M = 0.954 ............................................................................... 73 Figure 4.12. Mach number vs. flutter velocity index.................................................................... 74 Figure 4.13. Mach number vs flutter frequency ratio ................................................................... 74 Figure 4.14. Histogram of Vf index with M=0.678...................................................................... 77 Figure 4.15. Histogram of freq ratio with M=0.678 ..................................................................... 77 Figure 4.16. Histogram of Vf index with M=0.901...................................................................... 77 Figure 4.17. Histogram of freq ratio with M=0.901 ..................................................................... 77 Figure 4.18. Histogram of Vf index with M=0.954...................................................................... 78 Figure 4.19. Histogram of freq ratio with M=0.954 ..................................................................... 78 Figure 4.20. Mach number vs. flutter velocity index aero uncertainty included.......................... 80 Figure 4.21. Mach number vs. flutter frequency ratio aero uncertainty included ........................ 80 Figure 4.22. Mach number vs. flutter velocity index aero & experiment uncertainty included... 81 Figure 4.23. Mach number vs. flutter frequency ratio aero & experiment uncertainty included 81 Figure 4.24. Mach zone definition for the M=0.678 case............................................................. 84 Figure 4.25. Flutter index and frequency sensitivity to zone discretization for the M=0.678 case ....................................................................................................................................................... 85

Figure 4.26. Surface Mach zone definition for the M=0.901 case ............................................... 85 vii
Figure 4.27. Flutter index and frequency sensitivity to zone discretization for the M=0.901 case ....................................................................................................................................................... 85

Figure 4.28. Surface zone definition at the M=0.954 case ........................................................... 86 Figure 4.29. Flutter index and frequency sensitivity to zone discretization for the M=0.954 case ....................................................................................................................................................... 86

Figure 4.30. Histogram of Vf index with Roger approximation uncertainty................................ 88 Figure 4.31. Histogram of freq ratio with Roger approximation uncertainty............................... 88 Figure 4.32. Flutter index (left) and flutter frequency (right) sensitivity to Roger terms for the M=0.954 case................................................................................................................................ 88

Figure 4.33. Histogram of Vf index with aero + structural uncertainty at the detailed level ....... 91 Figure 4.34. Histogram of flutter frequency ratio with aero + structural uncertainty at the detailed level............................................................................................................................................... 91

Figure 4.35. Histogram of Vf index with aero + structural uncertainty at higher level................ 91 Figure 4.36. Histogram of frequency ratio with aero + structural uncertainty at high level ........ 91 Figure 4.37. Histogram of Vf index with damping uncertainty.................................................... 91 Figure 4.38. Histogram of freq ratio with damping uncertainty................................................... 91 Figure 4.39. Flutter index and frequency sensitivity to structural modal damping for the M=0.954 case................................................................................................................................................ 92

Figure 5.1. Excitation by the discrete gust 1-cosine profile ......................................................... 99

Figure 5.2. ‘1-cosine’ gust profiles with combination of gust velocity and gradient ................... 99

viii
Figure 5.3. PSD of the Dryden model with different scales L.................................................... 100 Figure 5.4. PSD of Dryden model with different rms value....................................................... 100 Figure 5.5. FE model of GTA aircraft ........................................................................................ 106 Figure 5.6. DLM aerodynamic model......................................................................................... 106 Figure 5.7. V-g plot of nominal GTA model.............................................................................. 106 Figure 5.8. Hist of max/min wing tip acceleration ..................................................................... 109 Figure 5.9. Hist of max/min wing root shear force..................................................................... 109 Figure 5.10. Sensitivity indices of max (left) and min (right) wing tip acceleration in pie chart110 Figure 5.11. Sensitivity indices of max (left) and min (right) wing root shear force in pie chart ..................................................................................................................................................... 110

Figure 5.12. Hist of rms of wing tip acceleration in the PSD-based gust response simulation case ..................................................................................................................................................... 112

Figure 5.13. Hist of rms of wing root shear force in the PSD-based gust response simulation case ..................................................................................................................................................... 112

Figure 5.14. Sensitivity indices of wing tip acceleration (left) and wing root shear force (right) in the PSD-based gust response simulations................................................................................... 112

Figure 5.15. Hist of max/min wing tip acceleration. PSD-equivalent time domain approach ... 114 Figure 5.16. Hist of max/min wing root shear force PSD-equivalent time domain approach.... 114 Figure 5.17. Sensitivity indices of max (left) and min (right) wing tip acceleration in pie chart115 PSD-equivalent time domain approach ...................................................................................... 115 ix
Figure 5.18. Sensitivity indices of max (left) and min (right) wing root shear force in pie chart ..................................................................................................................................................... 115

Figure A.1. The Linear Strain Triangle (LST) in local and global axes..................................... 132 Figure B.1. Kernel function coordinate definition...................................................................... 151 Figure B.2. DLM mesh definition of downwash and load points............................................... 152 Figure C.1. Closed loop flow chart of ASE system.................................................................... 158 Figure C.2. System actuator block.............................................................................................. 167

x

LIST OF TABLES

Table 2.1. Comparison of results from modal analysis ................................................................ 11 Table 2.2. Unsteady aerodynamics comparison ........................................................................... 13 Table 3.1. Nominal (baseline) values of actuator physical parameters ........................................ 38 Table 3.2. Monte Carlo simulation results for the actuator-only uncertainty case....................... 41 Table 3.3. Monte Carlo simulation results with structural and actuator variations ...................... 47 Table 3.4. Monte Carlo simulation results with unsteady aerodynamic uncertainty.................... 47 Table 3.5. Monte Carlo simulation results with variations of actuator, structural, and aerodynamic parameters. .............................................................................................................. 48

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    Redesigning Google Form Responses Sage Isabella Stanford University [email protected] Abstract 2 Related Work This project was motivated by personal struggles with Taking a simplistic design and updating it to better fit user Google Forms, a free service provided by Google for form needs is becoming a more common practice in the data creation, sharing, and reviewing, but with limited tools for visualization field. A popularly cited article renovates hotel exploratory data analysis. These struggles prompted the organization methods by researching what managers question: What interactions can be designed that improve physically tracked, updating those features to a digital data analysis that fit well with Google’s simple to use interface, and using the new platform to create insightful interface? This question was explored by reviewing prior interactions [Weaver, Fyfe, Robinson, Holdsworth, Peuquet, visualization research, collecting an online survey to gauge Maceachren 2006]. Redesigning Google Form Responses use and satisfaction, ideating on possible solutions, uses similar methodology to design a new interface that generating three video prototypes (chart filter, highlighting assists and not overwhelms current users. table, and drag to combine), interviewing current Google Forms users on prototypes, and collecting survey data on Several platforms already exist for form creation with prototypes. It was found that clicking on aspects of charts to varying features for data visualization. (See Table 1) filter the Summary of Responses and highlight entries in Google Sheets was intuitive and useful to most users. Table 1: Top Online Form Building Platforms Dragging charts to combine data was a conceptual leap, but useful to some. Name Public Price Analysis Features 1 Introduction Survey 1999 $0-$85 - Filter Monkey monthly - Crosstab There are hundreds of form platforms online, but arguably - Word Clouds/Text none as accessible as Google Forms.
  • Modeling Just Noticeable Differences in Charts

    Modeling Just Noticeable Differences in Charts

    To appear in IEEE Transactions on Visualization and Computer Graphics Modeling Just Noticeable Differences in Charts Min Lu, Joel Lanir, Chufeng Wang, Yucong Yao, Wen Zhang, Oliver Deussen, and Hui Huang Fig. 1. Four charts of the same dataset, but with different pairs below the Just Noticeable Difference (JND) threshold. From left to right: a bar chart with no pairs below JND, the difference between all pairs of bars in the graph is noticeable; with a different order, there is a pair of bars below JND (A-B); two pairs of indistinguishable fans are detected in the pie chart (A-B and C-F); three pairs of circles are detected as not distinguishable in the bubble chart (A-B, C-F and D-E). Abstract—One of the fundamental tasks in visualization is to compare two or more visual elements. However, it is often difficult to visually differentiate graphical elements encoding a small difference in value, such as the heights of similar bars in bar chart or angles of similar sections in pie chart. Perceptual laws can be used in order to model when and how we perceive this difference. In this work, we model the perception of Just Noticeable Differences (JNDs), the minimum difference in visual attributes that allow faithfully comparing similar elements, in charts. Specifically, we explore the relation between JNDs and two major visual variables: the intensity of visual elements and the distance between them, and study it in three charts: bar chart, pie chart and bubble chart. Through an empirical study, we identify main effects on JND for distance in bar charts, intensity in pie charts, and both distance and intensity in bubble charts.