Aeroelastic Optimization of Varying Sweep
Angle Wing
Research Thesis
In Partial Fulfillment of the
Requirements for the Degree of
Master of Science
In Aerospace Engineering
Shlomi Buchnik
Submitted to the Senate of
The Technion – Israel Institute of Technology
HESHVAN 5773
HAIFA
SEPTEMBER 2013
The Research Thesis was Done Under the
Supervision of Professor Moti Karpel in the
Faculty of Aerospace Engineering
The Generous Financial Help of MAFAT and
REFAEL Ltd is Gratefully Acknowledged
Table of Content
Abstract ………………………………………………………………………………1
List of Symbols ………………………………………………………………………2
1.
Introduction…………………………………………………………………...4
1.1
Literature Survey……………………………………………………………..6
1.2
The Purpose of the Research…………………………………………………9
2.
Research Approach…………………………………………………………..10
3.
Mathematical Model………………………………………………………...11
3.1
Aeroelastic Mathematical Model…………………………………………….11
3.2
Modal Coupling Mathematical Model………………………………………13
3.3
State Space Model……………………………………………………………19
3.4
Control System Model……………………………………………………….20
3.5
Stability and Closed Loop Performances…………………………………….24
3.6
Aeroelastic Effectiveness…………………………………………………….25
3.7
Model Matrices Derivatives………………………………………………….27
3.8
System's Performance Derivatives…………………………………………...29
3.8.1 Flutter Dynamic Pressure Derivative………………………………………...29
3.8.2
Stability Margin Derivatives………………………………………………...31
3.8.3 System Performance Derivatives…………………………………………….33
3.8.4 Structure Derivatives…………………………………………………………35
3.9
Optimization Process and Optimization Algorithms………………………...37
4.
Numerical Test Case…………………………………………………………40
4.1
General Description of the Test Case………………………………………...40
4.2
Test Case Aerodynamic Model……………………………………………...42
4.3
Aerodynamic Rational Approximation………………………………………43
4.4
Test Case Roll Control System………………………………………………45
4.5
Test Case Stability in Closed Loop…………………………………………..48
4.6
Test Case Performance in Regard To Design Parameters…………………...55
4.7
Test Case Optimization Process……………………………………………...68
4.7
Test Case Optimization Results……………………………………………..69
5.
Conclusions…………………………………………………………………..76
Table of Content (cont.)
Appendix
A.1.
Wing's Finite Element Model………………………………………………...77
A.2.
Body's Finite Element Model………………………………………………..80
A.3.
Modal Analysis Results………………………………………………………82
A.4.
Modal Coupling Method Verification……………………………………….92
A.5.
Structure Linear Change Method Verification……………………………...100
Bibliography………………………………………………………………………...109
List of Figures
Figure 1: Optimization Process……………………………………………………....39
Figure 2: Test Case Configuration at 0° Wing Deploy Angle……………………….40
Figure 3: Test Case Configuration at 20° Wing Deploy Angle……………………...41
Figure 4: Test Case Configuration at 40° Wing Deploy Angle……………………...41
Figure 5: Aerodynamic Panel Model and Structure Model………………………….42
Figure 6: First Bending Mode In Aerodynamic Model………………………………43
Figure 7: Rational Approximation Verification……………………………………...44
Figure 8: Roll Control Architecture………………………………………………….45
Figure 9: Servo Frequency Response………………………………………………...46
Figure 10: IMU Position……………………………………………………………..47
Figure 11: Q-G Plots at 0° Wing Deploy Angle……………………………………..49
Figure 12: Q-G Plots at 20° Wing Deploy Angle……………………………………50
Figure 13: Q-G Plots at 40° Wing Deploy Angle……………………………………51
Figure 14: Flutter Animation 1, 2……………………………………………………53
Figure 15: Flutter Animation 3, 4……………………………………………………53
Figure 16: Flutter Animation 5, 6……………………………………………………53
Figure 17: Flutter Animation Pilot's View 1…………………………………………54
Figure 18: Flutter Animation Pilot's View 2…………………………………………54
Figure 19: Flutter Animation Pilot's View 3…………………………………………54
Figure 20: Flutter Animation Pilot's View 4…………………………………………54
Figure 21: Flutter Animation Pilot's View 5…………………………………………54
Figure 22: Flutter Animation Pilot's View 6…………………………………………54
Figure 23: Performance Derivative by Wing Width at Section 1 0° Wing
Deploy Angle……………………………………………………………..56
Figure 24: Performance Derivative by Wing Width at Section 2 0° Wing
Deploy Angle……………………………………………………………..57
Figure 25: Performance Derivative by Wing Width at Section 3 0° Wing
Deploy Angle……………………………………………………………..58
Figure 26: Performance Derivative by Wing Width at Section 4 0° Wing
Deploy Angle……………………………………………………………..59
List of Figures (cont.)
Figure 27: Performance Derivative by Control Gains A' 0° degree Wing
Deploy Angle……………………………………………………………..60
Figure 28: Performance Derivative by Control Gains B' 0° degree Wing
Deploy Angle……………………………………………………………..61
Figure 29: Performance Derivative by Wing Width at Section 1 20° Wing
Deploy Angle……………………………………………………………..62
Figure 30: Performance Derivative by Wing Width at Section 2 20° Wing
Deploy Angle……………………………………………………………..63
Figure 31: Performance Derivative by Wing Width at Section 3 20° Wing
Deploy Angle……………………………………………………………..64
Figure 32: Performance Derivative by Wing Width at Section 4 20° Wing
Deploy Angle……………………………………………………………..65
Figure 33: Performance Derivative by Control Gains A' 20° Wing
Deploy Angle……………………………………………………………..66
Figure 34: Performance Derivative by Control Gains B' 20° Wing
Deploy Angle……………………………………………………………..67
Figure 35: Wing Structure and Dimensions………………………………………….77
Figure 36: Wing FE Model…………………………………………………………..78
Figure 37: Wing Section Profile……………………………………………………...78
Figure 38: Factious Mass Position and Node Constrain……………………………..79
Figure 39: Body FE Model…………………………………………………………...80
Figure 40: First Wing Elastic Mode………………………………………………….84
Figure 41: Second Wing Elastic Mode………………………………………………84
Figure 42: Third Wing Elastic Mode………………………………………………...84
Figure 43: Fourth Wing Elastic Mode……………………………………………….85
Figure 44: Fifth Wing Elastic Mode…………………………………………………85
Figure 45: Sixth Wing Elastic Mode………………………………………………...85
Figure 46: Seventh Wing Elastic Mode……………………………………………...86
Figure 47: Eighth Wing Elastic Mode……………………………………………….86
Figure 48: Ninth Wing Elastic Mode………………………………………………...86
Figure 49: Tenth Wing Elastic Mode………………………………………………...87
List of Figures (cont.)
Figure 50: First Body Elastic Mode………………………………………………….89
Figure 51: Second Body Elastic Mode……………………………………………….89
Figure 52: Third Body Elastic Mode…………………………………………………89
Figure 53: Fourth Body Elastic Mode………………………………………………..90
Figure 54: Fifth Body Elastic Mode………………………………………………….90
Figure 55: Sixth Body Elastic Mode…………………………………………………90
Figure 56: Seventh Body Elastic Mode………………………………………………91
Figure 57: Eighth Body Elastic Mode………………………………………………..91
Figure 58: Ninth Body Elastic Mode………………………………………………...91
Figure 59: Tenth Body Elastic Mode………………………………………………...92
Figure 60: Comparison of First Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………96
Figure 61: Comparison of Second Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………96
Figure 62: Comparison of Third Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………96
Figure 63: Comparison of Fourth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………97
Figure 64: Comparison of Fifth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………97
Figure 65: Comparison of Sixth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………97
Figure 66: Comparison of Seventh Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………98
Figure 67: Comparison of Eighth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………98
Figure 68: Comparison of Ninth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………98
Figure 69: Comparison of Tenth Mode Shape Full Calculation and Modal
Coupling..…………………………………………………………………99
List of Figures (cont.)
Figure 70: Comparison of Approximated Wing Structure Calculation to Full
Calculation A'..………………………………………………………….101
Figure 71: Comparison of Approximated Wing Structure Calculation to Full
Calculation B'..………………………………………………………….102
Figure 72: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 0° Wing Deploy Angle A'..……………………….103
Figure 73: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 0° Wing Deploy Angle B'..………………………..104
Figure 74: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 20° Wing Deploy Angle A'..………………………105
Figure 75: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 20° Wing Deploy Angle B'..………………………106
Figure 76: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 40° Wing Deploy Angle A'..………………………107
Figure 77: Comparison of Approximated Full Configuration Structure Calculation to
Full Calculation for 40° Wing Deploy Angle B'..………………………108
List of Tables
Table 1: Control System Gains..……………………………………………………..46
Table 2: Wing Material..……………………………………………………………..77
Table 3: Body Modal Analysis with Factious Mass..………………………………..88
Table 4: Comparison of Modal Coupling Technique to Full Calculation for 0° Wing
Deploy Angle..……………………………………………………………...93
Table 5: Comparison of Modal Coupling Technique to Full Calculation for 20° Wing
Deploy Angle..……………………………………………………………...94
Table 6: Comparison of Modal Coupling Technique to Full Calculation for 40° Wing
Deploy Angle..……………………………………………………………...95
I
Abstract
It is common in modern aerodynamic platform design to have multiple design
conditions in which the platform is expected to function properly. A way to have high
performance levels at very different flight conditions can be the use of a morphing
wing with variable sweep angle. Multidisciplinary design of the structure and the
control system for the platform at different flight conditions and different wing sweep
angles poses an engineering challenge. The different design goals such as low wing
inertia, sufficient platform performance, adequate control system gain and phase
margins and sufficient flutter margins, form an elaborate optimization problem with
contradictory requirements and constraints. This work proposes a novel automated
approach for solving the aeroservoelastic optimization problem by using the modal
approach with modal coupling techniques.
Aeroservoelastic optimization has been studied in a large number of studies. Usually
the solution approach is based on the modal approach. According to the modal
approach, which is well suited for aeroelastic problem, a limited low frequency set of
structural vibration modes are taken into account. By doing so the number of
problem's dof is reduced significantly. Reducing problem's dof enables rapid
optimization sessions even on ordinary of the shelf personal computers. The
optimization sessions includes derivatives calculations and design updates, which
demands extensive CPU time. The optimization sessions includes control system's
gains change and structure design parameters change.
This research goal is to develop a method for interdisciplinary optimization which
optimizes the control system's gains and structure design parameters for varying
sweep angle wing. The work is focused at optimizing predefined aerodynamic
platform while changing its control gains and wing's structure. As stated the
optimization is performed using the modal assumption, while larger number of
structure modes is used than normal aeroelastic analysis. The aerodynamic model of
the configuration is calculated using panel method with appropriate compression
correction, the aerodynamic matrices are approximated using minimum states method.
The control system is formalized with state space representation. Combining the
different models creates a full state space model of the configuration.
II
The cost function and the design constraints for the optimization process are derived
from roll-performance requirements, control stability margins, closed-loop flutter
velocities, aileron aeroelastic effectiveness and the wing moment of inertia about its
rotation axis that affects the sweep-angle rate. The main theoretical novelty is in the
definition of the equations of motion and the derivatives of the parameters that affect
the design criteria with respect to the design variable in the coupled modal system.
The application novelty is in the use of the models and derivatives for the
optimization of an important multi-disciplinary engineering problem while engaging
common industrial codes.
The optimization process starts with an initial platform design with high fidelity
structural FEM, control system architecture with initial gain values and detailed
aerodynamic panel model while the wing is modeled separately from the body. The
fictitious-mass modal coupling method is used to combine the models at various
sweep angles, which reduces the amount of FEM and sensitivity computations
needed. The unsteady generalized aerodynamic matrix is approximated using the
Minimum State method, the servo actuator and the control system are formulated
using state-space equations with which a complete vehicle state space representation
is formatted. An iterative optimization process is then conducted. During the process
the vehicle control gains and structure are modified for improving the predefined cost
function. At the end of the optimization process a full high fidelity calculation is
performed for verifying the updated design, and the optimization cycle is repeated if
necessary. The method enables a comprehensive view of the configuration's
performance and the effect of the different design variables.
In the research the optimization method is demonstrated on a test case. The test case
includes Wing – Body configuration, with varying wing sweep angle. Using the test
case modal coupling technique was validated and demonstrated, in addition different
structure assumptions were validated. The test case preliminary design suffers from
low stability margins and low flutter margins, after the optimization method is applied
and the design is updated. The platform enjoys proper stability margins without major
reduction in platform performances.