MULTIDISCPLINARY OPTIMIZATION AND SENSITIVITY ANALYSIS OF FLUTTER OF AN AIRCRAFT WING WITH UNDERWING STORE IN TRANSONIC REGIME

Transcription

1 MULTIDISCPLINARY OPTIMIZATION AND SENSITIVITY ANALYSIS OF FLUTTER OF AN AIRCRAFT WING WITH UNDERWING STORE IN TRANSONIC REGIME A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering By CHAKRADHAR BYREDDY B.Tech., Jawaharlal Nehru Technological University, India, Wright State University 1

2 WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES August 4, 2003 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDERMY SUPERVISION BY Chakradhar Byreddy ENTITLED Multidisciplinary optimization and sensitivity analysis of flutter of an aircraft wing with underwing store in transonic regime BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Engineering. Ramana V. Grandhi, Ph.D. Thesis Director Committee on Final Examination Richard J. Bethke, Ph.D. Department Chair Ramana V. Grandhi, Ph.D. Mitch Wolff, Ph.D. Ravi C. Penmetsa, Ph.D. Joseph F. Thomas, Jr., Ph.D. Dean, School of Graduate Studies 2

3 Abstract Chakradhar R., Byreddy. M.S.Egr., Department of Mechanical and Materials Engineering, Wright State University, Multidisciplinary Optimization and Sensitivity Analysis of Flutter of an Aircraft Wing with Underwing Store in Transonic Regime. The main objective of this research is to obtain a preliminary design of an aircraft wing with an underwing store for improved flutter performance in the transonic regime. Since the transonic flow regime is already highly non-linear, the presence of an underwing store creates store-induced non-linearities in addition to those non-linearites associated with the wing. All these non-linearities are captured using a computational tool called Computational Aeroelasticity Program Transonic Small Disturbance (CAP-TSD). In this work, a methodology is developed to incorporate the non-linearities into a multidisciplinary optimization algorithm. Generally, combining a non-linear analysis with optimization is a computationally expensive and difficult task. Therefore, the parameters that are insignificant in the analyses are identified and excluded. A wing with different store configurations is modeled using finite elements. Using CAP- TSD, several parametric studies on the flutter of various wing-underwing store configurations are conducted in the transonic regime. The sensitivity of flutter is analyzed for the following store parameters: (i) location of underwing store center of gravity with respect to aerodynamic root chord, (ii) location of underwing store along the span of the wing, and (iii) underwing clearance (pylon length). Analyses are also performed to identify the onset of Limit-Cycle Oscillations (LCO) for different configurations of the underwing store and flight regimes. In addition, an investigation of the effects of 3

4 underwing store aerodynamics on the onset of flutter is studied. These studies help in excluding the parameters that are considered insignificant for the optimization and in choosing the critical wing-underwing store configuration. A final study is conducted to find the significance of including the parameters associated with the CAP-TSD into the optimization. This is done by optimizing the structural parameters such as the thicknesses of skins, spars, ribs and cross-sectional areas of the posts associated with the wing using Automated STRuctural Optimization System (ASTROS), and then analyzing the flutter for this optimized structure using CAP-TSD. From this, the nature of the flutter sensitivities helps in understanding whether it is essential to conduct non-linear analysisbased optimization using CAP-TSD. 4

5 Contents Abstract... iii 1 Introduction Introduction Literature Review. 4 2 Analysis Methodology Governing Equations Analysis Procedure Computational Models Structural Modeling of Wing and Underwing Store Aerodynamic Modeling of Wing and Underwing Store using CAP-TSD Effect of Store Aerodynamics on Flutter Different Cases Analysis of Clean Wing Analysis of Wing with Store (Mass only) Analysis of Wing with Store (Store Aerodynamics) Optimization of Wing with Underwing Store for Improved Flutter Performance

6 5.1 Optimization Methodology Structural Analysis Linear Aerodynamic Analysis Unsteady Non-linear Aeroelastic Analysis Multidisciplinary Design Optimization - Formulation Optimization Results Summary Summary 46 Bibliography 48 Appendix A.. 52 A.1 Example 1: ICW without store aerodynamics. 52 A.2 Example 2: ICW with store aerodynamics.. 55 Appendix B.. 60 B.1 Example 1: ICW without store aerodynamics. 60 Appendix C.. 72 C.1 Example 1: ICW with store (All Constraints) 72 6

7 List of Figures 1 Flow chart for analysis methodology Modified Intermediate Complexity Wing with underwing store 17 3 Aerodynamic modeling of wing and underwing store using CAP-TSD 18 4 Flutter velocities for a clean wing Unsteady Pressure Distribution: Indicating presence of shocks at Mach 0.90 and 0.92 respectively Center of gravity representation for various store configurations Sensitivity of flutter velocity to underwing store with center of gravity (mass only) Sensitivity of flutter velocity to underwing store along the span of the wing (mass only) for store configuration Sensitivity of flutter velocity to underwing store with underwing clearance (mass only) for store configuration Sensitivity of flutter velocity to underwing store with location of span with respect to center of gravity of store (mass only) Sensitivity of flutter velocity to underwing store with underwing clearance with respect to center of gravity of store (mass only) Aerodynamic grid of wing with underwing store using CAP-TSD

8 13 Comparison of Flutter velocities for Linear (ASTROS), Linear (CAP-TSD) and Non-Linear (CAP-TSD) for store configuration Comparison of flutter velocities (knots) for M=0.9 and M=0.92 using Non-Linear (CAP-TSD) for store configuration Comparison of flutter for store configurations 1 and 2 with and without store aerodynamics Flow chart describing the design methodology Aerodynamic model of wing using DLM CAP-TSD grid of wing Flutter velocities of initial and optimized wing-underwing store configuration with frequency constraints Flutter velocities of initial and optimized wing-underwing store configuration with flutter constraint Flutter velocities of initial and optimized wing-underwing store configuration with both the flutter and frequency constraints Percentage change in flutter velocities of wing-underwing store configuration with various constraints Thickness (in inches) distribution of initial and optimum design of wing structure at Mach

9 List of Tables 1 Different store configurations Modal frequencies for different wing-store configurations. 25 9

10 NOMENCLATURE φ inviscid small disturbance velocity potential M free stream Mach number γ u i f i K B M F ρ ratio of specific heats time varying generalized displacements vertical components of the mode shapes structural stiffness structural damping structural mass external aerodynamic loads free stream density c r wing reference chord u free stream velocity p lifting pressure z i q mode shape dynamic pressure u i W nloc generalized velocities total structural weight of the wing and number of local design variables 10

11 ρ i mass density of the i th structural element V i volume of the i th structural element σ Tensile tensile stress σ Compressive compressive stress σ Shear shear stress ω natural frequency γ jreq required level of damping at the j th velocity γ ij calculated damping value for the i th mode at j th velocity. GFACT NF i value in order to scale the constraint non-linear flutter velocity (CAP-TSD) in Knots F i linear flutter velocity (ASTROS) in Knots 11

12 Acknowledgements There are many people who have helped and motivated me during my M.S.Egr degree program. My professors, family and friends have all helped in one way or another. I am very much obliged to all of them for their constant support and encouragement. In particular, my very special thanks goes to my advisor, Dr. Ramana V. Grandhi for his discerning guidance and also, for taking the time to understand me as an individual and motivating all the time. I would like to thank Dr. Ravi Penmetsa and Dr. Mitch Wolff for their time in reviewing the thesis and also, for being part of my final thesis defense committee. I would also like to thank Dr. Philip Beran, Dr. Frank Eastep, Dr. Narendra Khot and Dr. Brian Sanders from WPAFB, for their genuine interest in my research work and for various numerous discussions and meetings in this project. Furthermore, I would like to thank Dr. Nathan Klingbeil and Dr. Joseph Slater for their valuable suggestions. Finally and most importantly, I would like to thank my parents, brother, and my wife for their endless support during my course of study. 12

13 This thesis is dedicated to my father Late Mr. Ravindra Reddy, mother Mrs. Pushpa Reddy, and brother Mr. Sarath Kumar 13

14 Chapter 1 Introduction 1.1 Introduction Many fighter aircraft carry out their missions in the transonic regime, and the presence of external stores pose complex and dangerous problems in this regime. In transonic flow regimes, the effect of aerodynamic nonlinearities becomes significant due to the presence of shocks on the wing surface, and dynamic aeroelastic instabilities such as flutter and LCO are induced due to the presence of external stores. A computational method based on the inviscid Transonic Small Disturbance theory is used to predict the nonlinear unsteady aerodynamics associated with shock motions in the transonic flow region [1]. This method is used to solve the nonlinear governing equations in aeroelastic analysis, and provides an efficient but accurate alternative to linear methods such as the doublet lattice method (panel method). Previous literature helped in understanding the implications of an aircraft wing with external stores (stores considered as rigid bodies) on the static aeroelastic phenomena and unsteady pressure distributions in the transonic regime [2]. Also, some work has been performed on the LCO of an aircraft wing, but not considering the effect of the underwing store structural parameters on the dynamic aeroelastic phenomena in the transonic regime. The present work advances the ongoing research that is being performed at the Air Force Research Laboratory (AFRL) by investigating the effects of 14

15 dynamic aeroelastic phenomena taking place in flight vehicles carrying stores (missiles, launchers, fuel tanks, etc.) [3]. In the present work, different underwing store configurations were chosen so as to understand the influence of the structural parameters of store on the dynamic aeroelastic instabilities. Presence of underwing stores causes flutter and store induced LCO in the transonic regime, which can lead to several problems associated with target-locking system, roll maneuverability etc. Therefore, it plays an important role in the preliminary design stage. The research work has been divided into three phases. The first phase involves the validation of computation of flutter by conducting an analysis on a clean wing (i.e., one without store) using Automated STRuctural Optimization System (ASTROS) [4] and Computational Aeroelasticity Program Transonic Small Disturbance (CAP-TSD) (linear and nonlinear analysis) in the subsonic regime. The second phase of the work involves an investigation of the effect of variation in the store parameters such as the underwing store center of gravity, underwing store location along the span of the wing and underwing clearance in the transonic region. One of the core issues in the second phase of the work is the inclusion of store aerodynamics in the calculation of flutter velocities [5]. Therefore, parametric studies are conducted by considering the underwing store mass only and underwing store aerodynamics. The accuracy of computed flutter velocity is compared in both cases to understand the impact of inclusion of store aerodynamics. Thus, identifying whether the effect of store aerodynamics has to be included or neglected in the optimization algorithms (which are iterative in nature). These analyses also help in identifying the critical parameters that directly affect the flutter and LCO in the transonic region. By obtaining the sensitivities of these parameters to flutter and 15

16 LCO, least sensitive parameters can be ignored in the analysis, resulting in reduced computational time and costs. With the results obtained from the second phase, it is viable to incorporate nonlinear analysis into the preliminary design process, which is the third phase of the research work. Based on the information obtained from the above analyses, a multidisciplinary optimization methodology was developed to design a wing structure with external stores to delay the occurrence of dynamic aeroelastic phenomena such as flutter and LCO. Using conventional design methods, it is a difficult task to find a feasible design that satisfies the conditions of nonlinearities. Thus, this study helps in developing an automated methodology to incorporate the nonlinearities associated with the wing and store into the multidisciplinary design environment. This study helps in the preliminary design of aircraft structures for improved flutter in the transonic regime with the presence of stores. This research work involves optimization which, by automating the analysis for a particular store configuration and different flight conditions, helps to delay the occurrence of flutter of a fighter aircraft. In the optimization problem, a wing with a particular underwing store configuration is chosen and modeled [6]. This underwing store configuration has a center of gravity almost near the elastic axis of the wing. A slight variation in the store center of gravity with respect to the elastic axis has a significant effect on the dynamic aeroelastic phenomena. Also, the location of the underwing store along the wing span and the amount of underwing clearance have significant effects on flutter. Since the combination of nonlinear analysis with optimization is difficult and computationally expensive, many parameters that are insignificant in the analyses are 16

17 excluded. One such parameter is the addition of store aerodynamics in the optimization problem [6]. Plus a separate study is conducted to find the significance of including the parameters associated with the non-linear aerodynamic analysis of CAP-TSD into the optimization problem. This is conducted by optimizing the structural parameters associated with the wing using Automated STRuctural Optimization System (ASTROS), and then analyzing the flutter for this optimized structure using the higher fidelity aerodynamic tool CAP-TSD. Furthermore, the nature of flutter sensitivity with Mach number helps in understanding whether it is essential to conduct non-linear aerodynamic analysis- based optimization using CAP-TSD. The optimization of these critical parameters helps to increase the air vehicle life, performance, and flight envelope during the mission. The information obtained from this work using higher fidelity models helps in certification by analysis, thereby reducing cost and time which are prevalent in the flight tests. 1.2 Literature Review Many problems associated with fluid-structure interaction are quite complicated, particularly that of wing-store interaction. Different research approaches have been extensively studied and developed in order to understand the impact of structures and aerodynamics associated with the wing and store. High computing power led to the advent of various numerical methods to solve the aeroelastic problems for application to realistic aircraft configurations in the transonic regime such as CAP-TSD [7]. Several researchers in the late 1980 s and early 1990 s emphasized structural optimization for improved aeroelastic performance. The work during this period concentrated on 17

18 integrating the existing techniques of flutter analysis with other disciplines such as static strength, dynamics, static aeroelasticity, etc. The unsteady aerodynamics considered to calculate flutter was linear such as Doublet Lattice Method (DLM) for the subsonic regime, and strip theory, Mach box method etc. for the supersonic regime. Previous literature mainly concentrated on the subsonic and supersonic regimes for wings with tip/underwing stores. Also, most of the research work for wing-body configuration was carried out in either subsonic or supersonic flow regimes [8]. A comprehensive work analyzed by Cattarius [9] on the wing-underwing store of a parametric F-16 wing considered the effect of flutter of stores. In that work, the author developed an unsteady vortex method and used it in order to understand the effects of store aerodynamics in the subsonic regime. In the case of transonic regime, studies were conducted mainly on a clean wing. One such study includes the preliminary design of aircraft structures for improved control effectiveness (steady-state roll performance) in the transonic region for a clean wing [10]. Furthermore, Kolonay [11] developed a design methodology to include transonic flutter requirements for the preliminary structural design of a clean wing. The loads that arise in the transonic regime have profound influence on the aeroelastic stability of a fighter aircraft. Meijer [12] developed a methodology in order to determine these loads. Inclusion of these loads into the optimization procedure helps in better understanding the mechanisms that drive the dynamic aeroelastic instabilities such as flutter and LCO. In the case of a wing with external stores in the transonic regime, the influence of store aerodynamics on different wing configurations have been studied and compared with present flight flutter data [13]. The various factors that affect LCO have been 18

19 extensively studied [14] by considering the structural nonlinearities as well as aerodynamic nonlinearities of a wing with tip store. The current work involves a comprehensive parametric study of underwing store structural parameters on flutter and LCO by using various nonlinear analysis tools. Toth, et al. [15] conducted various studies on non-linear transonic flutter of F-16 aircraft wing with stores configuration clearance. This work studied various flutter prediction methods and, hypothesized the mechanisms that govern the onset of Limit- Cycle Oscillations (LCO). Research work that was performed by Beran [3] involved the investigation of dynamic aeroelastic instabilities in flight vehicles carrying stores (missiles, launchers, fuel tanks, etc.). They used Computational Fluid Dynamics (CFD) methods linked with optimization techniques to determine the aerodynamics aspect rather than the structural aspects. Therefore, in this research an extensive study on the structural aspects of the wing with underwing store for enhanced flutter performance was studied. In the present research a design methodology is developed by integrating various tools associated with nonlinear analysis for improved air vehicles with external stores in the transonic regime. This work can be advanced by including the effects of pylon stiffness [16], i.e., various types of attachments and also the flutter of store. The flutter of store itself might cause extensive fatigue to the pylon. Therefore, studies involving all these effects help in better understanding the mechanisms and physical significance that govern the onset of flutter and LCO due to the presence of underwing stores. Also, the optimization of critical store structural parameters helps in increasing the air vehicle life, performance and flight envelope during their mission. Thus, it helps in the study of the 19

20 preliminary design of aircraft structures with and without stores for improved flutter and LCO performance in the transonic regime. 20

21 Chapter 2 Analysis Methodology 2.1 Governing Equations The TSD theory is based on the assumption that in the transonic flow regime, there are small disturbances, or perturbations around, a thin wing. The TSD equation in conservation form is given as 2 2 ( Aφ t Bφx) + ( Eφx + Fφ x + Gφ y ) + ( φy + Hφxφ y ) + ( φz ) = 0 (2.1.1) t x y z where φ is the inviscid small disturbance velocity potential. It is the nonlinearity in φ that helps in modeling weak shock waves in the transonic regime. In the analyses, only two different forms of the TSD equation are used by choosing either the linear equation coefficients or the AMES coefficients. The coefficients A, B, E, are A = M (2.1.2) 2 2 B = M (2.1.3) 2 2 E = M (2.1.4) 2 1 where M the free stream Mach number. The coefficients F, G and H are called AMES coefficients, given as F = γ M (2.1.5) 2 1/ 2( + 1) 21

22 2 3) 2( / 1 = M G γ (2.1.6) 2 1) ( = M H γ (2.1.7) where γ is the ratio of specific heats. The value of γ used in these analyses is 1.4 (air) The nonlinear results are computed by using the AMES coefficients given by the equation 0 ) ( )) 1) ( (1 ( ) 3) ( 2 1 1) ( 2 1 ) ((1 ) 2 ( = z x y y x x x t z M y M M M x M M t φ φ γ φ φ γ φ γ φ φ φ (2.1.8) The linear results are computed by setting the coefficients given by the equation 0 ) ( ) ( ) ) ((1 ) 2 ( = z y x x t z y M x M M t φ φ φ φ φ (2.1.9) When the linear equation was used, the wing and store was modeled as a flat plate in order to produce results similar to other methods such as the doublet-lattice method. When the nonlinear equation was used, the wing was modeled using an appropriate airfoil such as NACA0004, (zero camber, symmetric and four percent thick) so that the nonlinear effects (such as moving shock waves) can be realistically captured. However the store is modeled as a flat plate, for inclusion of store aerodynamics. Coupling of the structural equations of motion with the unsteady aerodynamics of wing and store is implemented and only the vertical component of the mode shape is used for both the wing and store. 2.2 Analysis Procedure 22

23 The CAP-TSD code solves the unsteady transonic small disturbance equation using an implicit time accurate approximate factorization algorithm [17]. The unsteady aerodynamics is simultaneously integrated with the structural equation of motions in time. For this the vibration analysis is performed using ASTROS [18] and the displacements are splined on to the CAP-TSD grid of the wing using a Thin Infinite Plane Spline (IPS) [19]. This integration is represented by the structural response in time to some initial perturbations. The structure is modeled by a series of orthogonal mode shapes weighted with time varying coefficients called the generalized displacements. The generalized coordinate transformation represents the physical deformations of the structure. The modal deflections in the streamwise and spanwise directions are minute in comparison to the vertical modal displacements, and thus, neglected. Therefore, the position of the wing at any point in time is given as z ( x, y, t) = i ModeNumber i= 1 u f ( x, y) i i (2.2.1) where u i is the time varying generalized displacements and f i represents the vertical components of the mode shapes. The structural equations of motion in generalized coordinates are given as M u+ B u+ Ku = F (2.2.2) K- Structural Stiffness B- Structural Damping M- Structural Mass F- External aerodynamic loads 23

24 where c p 2 2 r F = ρ u z i ds 2 (2.2.3) 2 c s 2 r ρ u 2 ρ - Free stream density c r - Wing reference chord u - Free stream velocity p - Lifting pressure z i - Modal displacements described in equation 10 Equation is solved with equation by using an implicit time-marching aeroelastic solution procedure based on approximate factorization [20]. In the current work, the procedure for the assessment of flutter prediction is described using the flow chart in Figure 1. In this method, the flutter velocity is calculated by varying free stream 1 2 velocity and dynamic pressure ( q = ρ u ) while holding the density constant at a given 2 Mach number (which is called an unmatched analysis). To compute the point at which flutter first occurs for a given Mach number, several executions of the CAP-TSD code are required at different dynamic pressures. All CAP-TSD calculations include the effects of shock generated entropy and vorticity. A static aeroelastic analysis is performed at a given dynamic pressure (that is assumed to be near neutral stability) to create a steady flow field that reflects the wing thickness, camber and mean angle of attack. This steady flow field is essential for the proper computing of the free decay transients in the dynamic aeroelastic analysis. 24

25 Static aero analysis using CAP-TSD Increase number of iterations Use higher dynamic pressure Static solution converged? No Yes Dynamic aeroelastic analysis Yes Is the solution stable? No Flutter occurs: Determine flutter conditions Sensitivity analysis, Surrogate models & design optimization Figure 1 Flow chart of research methodology If the static aeroelastic solution is converged, then the dynamic aeroelastic analysis is performed by restarting the calculation from the converged static aeroelastic solution with some initial disturbance on the vertical velocity of the wing. If the solution is not converged, then the number of iterations is increased till the static solution converges. 25

26 After the dynamic analysis is run, the stability of the system (coefficient of lift) is determined. If the system is stable, the entire procedure is repeated by increasing the dynamic pressure; else the damping value is computed. The flutter dynamic pressure value is determined by linearly extrapolating the damping information using the logarithmic decrement method [21]. Moreover, further refinement of the damping can be obtained by additional aeroelastic analyses if improved accuracy of flutter velocity is desired. 26

27 Chapter 3 Computational Models 3.1 Structural modeling of Wing and Underwing store Figure 2 represents the structural model of wing and store. A wing model called the Intermediate Complexity Wing (ICW2001), which has characteristics of a conventional fighter aircraft, is chosen for the study of wing-store interaction. The ICW2001 is modeled using 199 elements. In this model, 80 membrane elements with bending capability are used to represent the wing skins, 70 shear panels represent the spars and ribs, and 49 rod elements represent the posts. The wing skin is modeled using QUAD4 and TRIA3 elements in ASTROS. SHEAR elements are used for modeling the spars and ribs. ROD elements are used for modeling the posts. The wing root is fully constrained by using single point constraints. The following are the characteristics of ICW2001: Configuration o Thickness of the wing section: A constant thickness of four percent is maintained through out the span in order to have a NACA0004 airfoil [22]. o Extension of wing tip: The span of the wing model is extended to 108 inches, in case of attachment of tip store and for further analysis for multiple store configurations. Structural mass of wing 27

28 o 98 pounds Non structural mass of wing o 327 pounds The ratio of structural mass to non-structural mass is 0.3 and angle of sweep is The lengths of the root and tip chord are 48 and 26 inches respectively. The material used for modeling the wing is Aluminum (AL-7050-T7451) with E = 10.3 x 10 6 psi. Since the skin of wing is modeled with bending capability, the store connections (pylon) are modeled as BAR elements, with high stiffness (E = 30 x 10 6 psi) to represent a rigid body. The pylon mass is lb. The connections are modeled in the form of V shape. The store is modeled with BAR elements. The material properties of the store are E = 10.3 x 10 6 psi and specific weight = 0.3 lb/in 3. Mass of store with non-structural mass is lb. The store configuration properties are presented in Table 1. The percentage of store mass to wing mass is 44 percent. Each configuration has a different center of gravity. The center of gravity is varied by changing the non structural mass distribution on the nodes of the store. We chose these configurations so as to understand the impact of position of various guidance systems, warheads, etc. on the underwing store. The total weight of the wing-underwing store configuration is 726 lbs. Finally, a preliminary design is modeled which represents the centerline symmetry of the entire aircraft wing with store. 28

29 Configuration Number Description of the store configuration Type of Configuration Properties of the Underwing store Configuration 1 Store center of gravity at 22 percent and underwing clearance at 7.7 percent of aerodynamic root chord respectively, and store at 67 percent of aerodynamic span Fore Weight = lbs Length = 80 inches Area Moments of Inertia = in 4 Torsion Constant = in 4 Configuration 2 Store center of gravity at 44 percent and underwing clearance at 7.7 percent of aerodynamic root chord respectively, and store at 67 percent of aerodynamic span Near Elastic Weight = lbs Length = 80 inches Area Moments of Inertia = in 4 Torsion Constant = in 4 Configuration 3 Store center of gravity at 66 percent and underwing clearance at 7.7 percent of aerodynamic root chord respectively, and store at 67 percent of aerodynamic span Aft Weight = lbs Length = 80 inches Area Moments of Inertia = in 4 Torsion Constant = in 4 Configuration 4 Store center of gravity at 44 percent and underwing clearance at 7.7 percent of aerodynamic root chord respectively, and store at 56 percent of aerodynamic span Span Weight = lbs Length = 80 inches Area Moments of Inertia = in 4 Torsion Constant = in 4 Configuration 5 Store center of gravity at 44 percent, and underwing clearance at 16.7 percent of aerodynamic root chord respectively, and store at 67 percent of aerodynamic span pylon Weight = lbs Length = 80 inches Area Moments of Inertia = in 4 Torsion Constant = in 4 Table 1: Different store configurations 29

30 Figure 2: Structural model of wing-underwing store configuration 3.2. Aerodynamic modeling of Wing and Underwing store using CAP-TSD Figure 3 represents the aerodynamic model of wing with store. The dimension for this computational grid is 90 x 30 x 60. The actual aerodynamic model of wing with root and tip chord is 90 and 48 inches respectively. This aerodynamic model is normalized to the computational grid which represents the physical region of the wing (schematically represented as a horizontal lifting surface in solid green lines). The dimension of the computational grid representing the wing is 50 x 15. Similarly the aerodynamic model of the store with the chord length 90 inches is normalized to the computational grid which represents the physical region of the store (second horizontal lifting surface in solid green lines). Even though the store planform in CAP-TSD grid is not rectangular it primarily helps in understanding the significance of store aerodynamics on the dynamic aeroelastic instabilities. The dimension of the computational grid representing the store is 56 x 2. 30

31 The transformation region is defined in order to relate the computational grid and physical region. In this modeling, the effect of the vertical lifting surface (pylon) is neglected. Physical Region Computational Grid Y Eta X Xi Figure 3: Aerodynamic modeling of wing and underwing store using CAP-TSD 31

32 Chapter 4 Effect of Store Aerodynamics on Flutter 4.1 Different Cases This section discusses the numerical results obtained for the intermediate complexity wing and also wing with underwing store. Prior to making numerous aeroelastic calculations, a convergence study was done to show that the time step ( t = 0.05 and 0.02 seconds) and grid dimensions (90 x and 120 x 60 x 90) had no effect on the CAP-TSD aeroelastic solutions. Also, each dynamic solution is computed by using the converged static solution with some form of initial condition for the generalized displacements or velocities for each structural mode. A small initial perturbation is created by giving a value of one to generalized velocities for each mode in all the dynamic aeroelastic calculations ( u i = 1.0, i = 1, 6) to initiate the motion of the wing. Also, the initial conditions on the generalized displacements of each mode are taken to be zero to reduce the numerical transients and corresponding instabilities. Sensitivity of 32

33 aeroelastic solutions to the initial conditions is also verified. The choice of using an initial generalized velocity of one is arbitrary and its effect on the structural response depends on how the mode shapes are scaled [23]. Finally, dynamic aeroelastic results are presented for the following cases. Case I: Clean wing Case II: Wing with underwing store (Mass only) and Case III: Wing with underwing store (Store aerodynamics included) Case I constitutes the first phase, and Case II and Case III constitutes the second phase of the research work. Several flutter points were computed with CAP-TSD using the linear (to compare with ASTROS) and nonlinear equations. For all sets of results, the flutter points were computed by holding the density constant and varying the velocity, as described above. No structural damping was used, and all the calculations were performed with wing root set at zero angle of attack Analysis of Clean Wing This work helps to validate the CAP-TSD analysis conducted on a clean wing. Figure 4 shows that the CAP-TSD linear results at low Mach numbers are in excellent agreement with ASTROS. The good agreement between CAP-TSD linear and ASTROS at low Mach numbers should be expected, since nonlinear aerodynamic effects there are insignificant. Figure 4 also shows the nonlinear results in comparison with linear results. Nonlinearities do not become significant until Mach 0.8, leading to the presence of a transonic dip at Mach number 0.9 due to the effect of shocks. The large differences between the results using linear and nonlinear equations in the transonic region are due to 33

34 the presence of shocks in the flow field, as shown in Figure 5 for Mach 0.90 and 0.92 respectively. Figure 4: Flutter velocities for a clean wing Figure 5: Unsteady Pressure Distribution - Indicating presence of shocks at Mach 0.90 and 0.92 respectively 34

35 4.1.2 Analysis of Wing with Store (Mass only) The various store configurations used have their center of gravity at 22, 44 and 66 percent of the aerodynamic chord, respectively, as shown in Figure 6. Figure 7 shows the sensitivity of flutter to location of store center of gravity. The results indicate that the flutter velocity of the aircraft wing with underwing store increases, a favorable change, if the underwing store center of gravity is forward of the elastic axis of the wing. The extent to which the store can be moved forward depends on the design constraints of the store parameters. Also, there is a shift of transonic dip from Mach number 0.90 to 0.92 due to the presence of store. Fore Near Elastic Aerodynamic Model Structural Model Aft Figure 6: Center of gravity representation for various store configurations 35

36 Figure 7: Sensitivity of flutter velocity to underwing store with center of gravity (mass only) Figure 8 shows the sensitivity of flutter with location of the store along the span of the wing. The results indicate that as the store is located near the aerodynamic root chord the flutter velocity increases, thereby indicating that underwing stores can be placed near the fuselage in order to delay the occurrence of flutter. Also from the Figure 8, it can be concluded that if heavy stores can be placed near to the aerodynamic root chord for delaying the occurrence of dynamic aeroelastic instabilities. 36

37 Figure 8: Sensitivity of flutter velocity to underwing store along the span of the wing (mass only) for store configuration 2 Figure 9 shows the sensitivity of flutter with underwing clearance. The results indicate that as the underwing clearance increases, the flutter velocity decreases, due to the structural deformations associated with the pylon. Even though the pylon is realistically stiff, these deformations are dominant over the aerodynamic interference effects between the store and pylon (inclusion of pylon aerodynamics using panel method does not have any significant change in flutter velocity but change in the structural properties of pylon has significant effect. Thus, the same can be attributed to the nonlinear region since it is mass only aerodynamics). Also, there is a significant decrease in the first bending and torsion modal frequencies (shown in Table 2) with the increase in the underwing 37

38 clearance, thereby indicating that the flutter velocity will be decreased with increase in underwing clearance. Mode Number Clean Wing Store Configuration 1 Store Configuration 2 Modal Frequencies (Hz) Store Configuration 3 Store Configuration 4 Store Configuration Table 2: Modal frequencies for different wing-store configurations Figure 9: Sensitivity of flutter velocity to underwing store with underwing clearance (mass only) for store configuration 2 38

39 Figure10 shows the sensitivity of flutter with location of store (for various store configurations) along the span, which indicates that the presence of store center of gravity at fore of the elastic axis increases the flutter velocity. Similarly, Fig 11 shows the sensitivity of flutter with underwing clearance (for various store configurations), which also supports the conclusion that as underwing clearance increases the flutter velocity decreases. There was no presence of LCO for different wing-store configurations at various Mach numbers with mass only aerodynamics. Figure 10: Sensitivity of flutter velocity to underwing store with location of span with respect to center of gravity of store (mass only) 39

40 Figure 11: Sensitivity of flutter velocity to underwing store with underwing clearance with respect to center of gravity of store (mass only) Analysis of Wing with Store (Store Aerodynamics) Figure 12 shows a three-dimensional view of the aerodynamic model using CAP-TSD. Here the underwing store is modeled as a horizontal lifting surface. Splining of displacements on to the aerodynamic grid of the underwing store was done by considering the relative displacements of the underwing store with the wing. The splined displacements of the wing enclosing the store region are extrapolated on to the grid of the underwing store, as shown in Figure 3. 40

41 Figure 12: Aerodynamic grid of wing with underwing store using CAP-TSD Figures 13 and 14 shows that the inclusion of store aerodynamics in the transonic regime does not have any significant effect in the case of wing-store configuration 2. Figure 13: Comparison of flutter velocities for Linear (ASTROS), Linear (CAP-TSD) and Non-Linear (CAP-TSD) for store configuration 2 41

42 Figure 14: Comparison of flutter velocities (knots) for M=0.9 and M=0.92 using Non-Linear (CAP-TSD) for store configuration 2 Figure 15 shows that for store configuration 2 at Mach 0.94, the flutter velocity was more for wing with store (aerodynamics) than with the wing and store (mass only), but the flutter velocity is same with store aerodynamics and store (mass only) for store configuration 1. Another interesting observation made was that the flutter velocity is more at Mach 0.94 than at Mach 0.90 with inclusion of store aerodynamics for the store configuration 2, but it is different in case of store configuration 1. Store configuration 1 shows greater sensitivity to store aerodynamics, reducing onset flutter speed by 10 percent (approximately) at Mach Thus, from the analyses we can conclude which wing-store configurations have significant impact on flutter with inclusion of store aerodynamics at various Mach numbers. Also, no LCO was found for different wingstore configurations at various Mach numbers with inclusion of store aerodynamics. 42

43 Figure 15: Comparison of flutter for store configurations 1 and 2 with and without store aerodynamics 43

44 Chapter 5 Optimization of Wing with Store for Improved Flutter Performance 5.1 Optimization Methodology The flow chart shown in Figure 16 describes the design methodology developed in this study. This approach constitutes in establishing whether non-linear aerodynamic analysisbased optimization is essential for improved flutter performance in the transonic regime. The design approach consists of the following phases when using the gradient-based optimization techniques: (5.1.1) structural analysis using ASTROS, (5.1.2) linear aerodynamic analysis (DLM) using ASTROS, (5.1.3) unsteady non-linear aeroelastic analysis using CAP-TSD, (5.1.4) multidisciplinary design optimization using ASTROS, and (5.1.5) comparison of optimized results using ASTROS and CAP-TSD. The details of these phases are discussed with a fighter aircraft wing structure and AIM store configuration. 44

45 Optimization based on Linear Aerodynamic Analysis ASTROS INPUT ASTROS OPTIMIZATION (1) Initial airframe structure and boundary conditions (2) Loading conditions (3) Design variables (thickness of skin, spars, ribs and cross-sectional area of posts) (4) Constraints (stress, multiple frequency and flutter constraints) (1) Stress analysis (2) Frequency analysis (eigen value analysis using Lancoz method) (3) Unsteady aerodynamics (flutter analysis using DLM) ASTROS OUTPUT (1) Weight of the optimized wing structure (2) Maximum stress values (3) Modal frequencies (mode1 and mode2) (4) Flutter velocity of the wing CAP-TSD INPUT (1) Vibrational modal data: generalized mass and frequencies (2) Dynamic pressure, Mach number and velocity (3) Computational grid and airfoil data Non-Linear Aeroelastic Analysis CAP-TSD ANALYSIS CAP-TSD OUTPUT (1) Perform static aeroelastic analysis until the convergence to create basic flow field (2) Perform dynamic aeroelastic analysis by initiating small disturbance on the velocity of the wing (3) Check the response (coefficient of lift) of the dynamic analysis (1) Calculate the damping of the response using Logarithmic decrement method (2) Calculate the flutter velocity by extrapolating the damping Figure 16: Flow chart describing computer simulation for optimization 45

46 5.1.1 Structural analysis The equation of motion for the structural analysis is written in the form [ M ]{ q} + [ B]{ q} + [ K]{ q} = { F} ( ) where [ M ], [ B ], [ K ], and { F } are the generalized mass, damping, stiffness and aerodynamic forces, respectively. {q} is a vector of independent generalized coordinates defined by where {u} { u} = [ Φ]{ q} ( ) is a vector of structural translations and rotations in physical coordinates and [Φ] is the modal transformation matrix. The finite element method is used to represent [ M ], [ B ], [ K ] of the wing and store. A frequency analysis (eigenvalues- Lancoz method) is conducted using ASTROS Linear aerodynamic analysis In Equation ( ) { F} is a vector of generalized aerodynamic forces { F } = [Φ] T {F} ( ) {F} is calculated using the Doublet-Lattice Method. DLM is based on linearized potential theory. In this method, the flow is uniform or steady and all lifting surfaces are parallel to the flow. In addition, the panels are divided into trapezoidal lifting elements, such that they are parallel to the free stream. The unknown lifting pressures are concentrated 46

47 uniformly across the one-quarter chord line of each panel. Thus, the aerodynamic model for the wing planform is defined by 72 panels (nine spanwise and eight chordwise panels in the model) as shown in Figure 17. The lengths of aerodynamic root and tip chords are 90 and 48, respectively. The aerodynamic loads for the pylon and store are excluded [6]. Finally, flutter analysis is conducted using the P-k method [24]. Figure 17: Aerodynamic model of wing using DLM Unsteady non-linear aeroelastic analysis In this analysis, {F}, the non-linear aerodynamic force vector is calculated using the transonic small disturbance theory and is defined by F = 2 2 c r p ρ u z i ds 2 ( ) 2 c s 2 r ρ u 2 47

48 where ρ is the free stream density, C r is the wing reference chord, u is the free stream velocity, p is the lifting pressure, and z i is the i th mode shape that defines the position of wing at any point in time. In this analysis, unsteady aerodynamics is integrated with the structural equations of motion. This integration is achieved by conducting modal analysis using ASTROS and splining the modal displacements onto the CAP-TSD grid of the wing using a thin infinite plate spline technique. The CAP-TSD grid of the wing (90x30x60 dimension) is shown in Figure 18. The modal displacements that are considered in this analysis are only vertical modal displacements (represented by z i ), since the spanwise and streamwise displacements are negligible. Y X Figure 18: CAP-TSD grid of wing The equation of motion ( ) is solved by using an implicit time-marching aeroelastic solution procedure based on approximate factorization. Since it is necessary to find the 48

49 flutter velocity at a given Mach number, flutter analysis is conducted using an unmatched analysis method. In this method, the flutter velocity is calculated by varying 2 the free stream velocity and the dynamic pressure ( q = / 2ρu ) while holding the density 1 constant for a given Mach number. The static aeroelastic analysis is performed to generate the basic flow field. After the static aeroelastic analysis is conducted, dynamic aeroelastic analysis is initiated by restarting from the converged static aeroelastic solution. A small initial disturbance is imposed on the vertical velocity of the wing to initiate the dynamic aeroelastic analysis. Once two or more transients have been computed, the dynamic pressure (hence, the flutter velocity) for neutral stability is determined by extrapolating the damping value. The damping information is calculated by using the logarithmic decrement method Multidisciplinary design optimization Formulation The multidisciplinary optimization problem has many possibilities for increasing the flutter boundary. For the given wing structure, one could select a store configuration from the hundreds of available stores for maximizing the dynamic stability. In addition, the location of store along the span, underwing clearance, store center of gravity, store mass, store mass moment of inertia etc., could be potential variables for a given wing. Looking from another perspective, where the preliminary design of aircraft wing structures is the main goal, the wing geometry could be designed for a critical store. In this case again, the layout of the wing could be fixed, and the sizing of the spars, ribs and skins can be done for maximizing the flutter limit with a minimum weight 49

50 penalty. Also, one could vary the planform of the wing by fixing the above stated sizing parameters. In this paper, the wing structure is sized for a given airfoil shape with preselected store configuration, span location and underwing clearance. In the optimization of the wing-store structure, the inclusion of store aerodynamics in the unsteady aeroelastic analysis is excluded, since the calculated flutter velocity using CAP-TSD with the addition of store aerodynamics and with mass only aerodynamics is about the same in the transonic regime [6]. In this study a preliminary design of wing structure for a given store configuration is developed, therefore the design parameters are associated with the wing, and they are the thicknesses of skins, ribs, spars and cross-sectional areas of the posts. The performance constraints are (i) Stress: Material properties of AL7050-T7451 plate (tensile stress limit = 6.4 x 10 4 psi, compressive stress limit = 6.2 x 10 4 psi and shear stress limit = 4.3 x 10 4 psi) are chosen as the constraints. The stress constraints help in designing the wing with store to withstand static loads, such as the forward center of pressure, after center of pressure, and wing twist conditions (ii) Frequency: In this case, multiple frequency constraints are considered. These are the first bending mode 6.00 Hz and the first torsional mode Hz. The frequency constraint plays a vital role since the flutter is coupled between the bending and torsional modes, and (iii) Flutter: This constraint is defined by satisfying the values on the modal damping for a series of velocities. A velocity of 700 knots was chosen as the limit at zero damping and a velocity of 650 knots at 0.1 damping for all the Mach numbers (subsonic and transonic regimes) using the DLM. The flutter constraint enhances the flutter velocity for the wing with store. Finally, the manufacturing constraints for the aircraft wing structure are considered 50

51 during optimization. These constraints are defined by the minimum gauge of the skin, spars, ribs and cross-sectional areas of the posts. The mathematical representation of the optimization of the wing-store structure problem is described below. Objective function: Minimize structural weight nloc W = ρiv i ( ) i= 1 where nloc is the number of elements. ρ i and V i are the mass density and volume, th respectively, of the i structural element participating in the design. Stress constraints: σ 6.4 x 10 4 psi, σ 6.2 x 10 4 psi, σ 4.3 x 10 4 psi ( ) Tensile Compressive Multiple frequency constraints: ω Hz and ω Hz ( ) Flutter constraint: Shear g j γ ij γ jreq = ( ) GFACT γ γ i = 1, 2 number of modes ij jreq j = 1, 2 number of velocities where g j is the damping constraint, γ jreq is the required level of damping at the j th velocity and γ ij is the calculated damping value for the i th mode at j th velocity. GFACT is a normalizing factor that converts the damping numbers consistent with the other constraints in the design task. A value of 0.1 for GFACT is used in this study for all the cases. 51

52 5.1.5 Optimization results The constraints are imposed, individually and simultaneously, for the wing-store structure problem using the methodology described above. The methodology helps in understanding the significance of these constraints on the objective function and to further increase the flutter velocity of the wing-store configuration. In the case of strength requirements, static loads are applied on the wing structure. These loads consists of forward center of pressure, after center of pressure, and wing twist conditions on the wing. When the strength and multiple frequency constraints are applied, there is adequate frequency separation between the bending and torsional modes at the optimum design than at the initial design. The initial design has bending and torsional frequencies of 5.5 and 15.5 Hz, respectively, whereas the optimum design has bending and torsional frequencies of 6.0 and 18.6 Hz, respectively. Using the simple minded frequency separation approach, it is expected that the flutter velocity at the optimum design is expected to be greater than the initial design in both the subsonic and transonic regimes. However this is not true since it is not the separation of frequencies that is important, but rather the rate at which the bending and torsional frequencies coalesce at zero damping, which is the flutter point. With the optimum design, there is a greater rate at which the two frequencies coalesce than with the initial design. Also the problem is formulated with stress constraints in addition to the frequency separation, hence the structure is resized to meet the strength requirements. The mass redistribution had negative impact on the flutter boundary in this particular case. The material distribution and aerodynamic 52

53 coupling caused the decrease in the flutter velocity. Figure 19 shows that the initial design has a higher flutter velocity than that of the optimum design in the subsonic regime as well as in the transonic regime. The weight of the optimized wing structure when these constraints are applied is 421 lbs. Figure 19: Flutter velocities of initial and optimized wing-underwing store configuration with frequency constraints In the case of both the strength and flutter constraints, there is adequate frequency separation between the bending and torsional modes at the optimum design than at the initial design. Also, at the optimum design, the point at which the two frequencies coalesce is higher than that of the initial design. The flutter constraint is applied specifically at a particular Mach number. Therefore, the optimum wing structure satisfies the flutter constraint at that Mach number. The flutter velocity and weight of the wing structure varies at different Mach numbers. Considering the case of the transonic dip 53

54 (drastic flow conditions) at Mach number 0.92 due to the presence of the store, it has been found that the optimum design has bending and torsional frequencies of 5.9 and 21.6 Hz, respectively. The weight of the optimized wing design at Mach 0.92 is 428 lbs. The flutter velocity using CAP-TSD at Mach 0.92 is found to be 591 knots. Figure 20 shows that the optimum design has higher flutter velocity in both the subsonic and transonic regimes than the initial design. Also, the difference in the weight of the wing between the optimum and initial designs is significant in the subsonic regime. In the case of the transonic regime, the difference in weight is minimal, but the difference in flutter velocity is very significant. This indicates that the flutter constraint is the active constraint in this optimization problem. Figure 20: Flutter velocities of initial and optimized wing-underwing store configuration with flutter constraint 54

55 When all the strength, frequencies and flutter constraints are applied, considering the case of the transonic dip, it has been found that the optimum design has bending and torsional frequencies of 6.0 and 21.6 Hz, respectively, satisfying the frequency constraints. Also, the optimum weight of the wing structure at Mach 0.92 is 429 lbs. The flutter velocity using CAP-TSD at the transonic dip is 592 knots. The results mentioned above further signify that the flutter constraint is the active constraint because the results, as shown in Figure 21, are so similar when flutter constraint is only applied. Figure 21: Flutter velocities of initial and optimized wing-underwing store configuration with both the flutter and frequency constraints Figure 22 shows the percentage change in flutter velocity for different Mach numbers with the initial design and the optimum design subjected to various constraints. The percentage change is defined as 55

56 NFi Fi NF i x 100 i = initial and optimum values ( ) where NF is the flutter velocity using transonic analysis (CAP-TSD) and F is the flutter i velocity using linear aerodynamic analysis (ASTROS). Consider the case of transonic dip at Mach 0.92, the initial difference of the flutter velocity is %, whereas, the optimized difference of the flutter velocity is %. Thereby, concluding that there is significant increase in the flutter velocity (91 knots) for 4 lb change in the weight of the wing structure at Mach i Figure 22: Percentage change in flutter velocities of wing-underwing store configuration with various constraints The increase in flutter velocity is achieved by varying the thickness distribution, as shown in Figure 23, of the elements of the wing structure. In the optimum design there is 56

57 an increase in the thickness of the skin along the leading edge of the wing structure. This is due to the fact that in the torsional mode there is a twist of the pylon and, it is the pylon that connects the skin of the structure along the leading edge of the wing. The increase in the thickness of the skin along the leading edge of the wing is to counteract the effect of torsional mode (dominant mode). Figure 23: Thickness (in inches) distribution of initial and optimum design of wing structure at Mach 0.92 Finally, the nature of percentage change in flutter velocity is the same in the transonic regime for all the constraints. This indicates that the flutter velocity can be predicted for any constraint in the non-linear region (transonic regime), if the flutter velocity is calculated for a constraint at a Mach number. Therefore, it can be concluded that it is not essential to conduct a non-linear aerodynamic analysis-based optimization for improving 57