Document
Aeroelastic Tailoring Study of an N+2 Low-boom Supersonic
Commercial Transport Aircraft
Chan-gi Pak NASA Armstrong Flight Research Center, Edwards, CA 93523-0273 The Lockheed Martin N+2 Low-boom Supersonic Commercial Transport (LSCT) aircraft was optimized in this study through the use of a multidisciplinary design optimization tool developed at the National Aeronautics and Space Administration Armstrong Flight Research Center. A total of 111 design variables were used in the first optimization run. Total structural weight was the objective function in this optimization run. Design requirements for strength, buckling, and flutter were selected as constraint functions during the first optimization run.
The MSC Nastran code was used to obtain the modal, strength, and buckling characteristics.
Flutter and trim analyses were based on ZAERO code, and landing and ground control loads were computed using an in-house code. The weight penalty to satisfy all the design requirements during the first optimization run was 31,367 lb, a 9.4% increase from the baseline configuration. The second optimization run was prepared and based on the big-bang big-crunch algorithm. Six composite ply angles for the second and fourth composite layers were selected as discrete design variables for the second optimization run. Composite ply angle changes can’t improve the weight configuration of the N+2 LSCT aircraft. However, this second optimization run can create more tolerance for the active and near active strength constraint values for future weight optimization runs.
Nomenclature AFRC = Armstrong Flight Research Center AIL1 = aileron #1 AIL2 = aileron #2 BBBC = Big-Bang Big-Crunch BF = body flap BLF = buckling load factor b = full span length CG = center of gravity C = pressure coefficient p c = chord length DLW = design landing weight DTOW = design take-off weight DV = design variable i i d = distance from CG to main landing gear 𝐶𝐺2𝑀𝐺 d = distance from CG to nose landing gear 𝐶𝐺2𝑁𝐺 d = distance from nose landing gear to main landing gear 𝑁𝐺2𝑀𝐺 E = vertical height of the CG of the airplane above the ground in the 1.0 g static condition EFEP = empty fuel empty payload EFFP = empty fuel full payload FE = finite element FFEP = full fuel empty payload FFFP = full fuel full payload FM = Z component of the main landing gear load under a level landing condition 𝐿𝑣 Senior Aerospace Engineer, Aerostructures Branch, P.O. Box 273, Edwards, California/Mailstop 48201A, Senior Member AIAA.
American Institute of Aeronautics and Astronautics FM = Z component of the main landing gear load under a static condition 𝑆𝑡 FN = Z component of the nose landing gear load under a level landing condition 𝐿𝑣 FN = Z component of the nose landing gear load under a static condition 𝑆𝑡 FX = X component of load vector FY = Y component of load vector FZ = Z component of load vector 𝐹(𝐗) = objective function f = dynamic response factor; 2.0 is to be used unless a lower factor is substantiated f = main landing gear ratio of level landing reactions to total weight of an aircraft 𝐿𝑀𝐺 f = nose landing gear ratio of level landing reactions to total weight of an aircraft 𝐿𝑁𝐺 GD = gear down (extended) GU = gear up (retracted) g = gravitational acceleration 𝑔 (𝐗) = inequality constraints (design requirements) 𝑗 HSCT = high speed civil transport L = left LMSW = Lockheed Martin Skunk Works LSCT = low-boom supersonic commercial transport M2W = DTOW-fuel burned to reach Mach = 2 MDAO = multi-disciplinary analysis and optimization MLG = main landing gear MS = margin of safety NASA = National Aeronautics and Space Administration NLG = nose landing gear Nx = acceleration in x-direction (fore and aft) Nz = acceleration in z-direction (up and down) ncon = number of constraints ndv = number of design variables O = object-oriented optimization P = roll rate Pdot = roll acceleration PI = performance index from the buckling post-processor module B PI = performance index from the flutter post-processor module F PI = performance index from the strength post-processor module S PI = performance index from the weight post-processor module W PLdB = perceived loudness in decibels Post = post-processor module Pre = pre-processor module Q = pitch rate Qdot = pitch acceleration R = right S = slope SL = sea level TEF = trailing-edge flap V = velocity V = equivalent speed e V-f = velocity versus frequency V = flutter speed F V-g = velocity versus damping V = limit speed L W = total weight T 𝑇 X = design variable vector, 𝐗 = ⌊𝑋 𝑋 … , 𝑋 ⌋ 1, 2, 𝑛 X = i-th design variable i 𝑋 = X coordinate of the CG location 𝐶𝐺 𝑋 = X coordinate of the ground contact point of the main landing gear 𝑀𝐺𝐶𝑃 American Institute of Aeronautics and Astronautics 𝑋 = X coordinate of the ground contact point of the nose landing gear 𝑁𝐺𝐶𝑃 𝑌 = Y coordinate of the CG location 𝐶𝐺 𝑍 = Z coordinate of the CG location 𝐶𝐺 𝑍 = Z coordinate of the ground contact point of the main landing gear 𝑀𝐺𝐶𝑃 𝑍 = Z coordinate of the ground contact point of the nose landing gear 𝑁𝐺𝐶𝑃 ZFW = zero fuel weight = angle of attack 𝜇 = coefficient of friction (CV) = constraints violated I. Introduction HE first supersonic flights of a commercial transport aircraft Tu-144 (Tupolev OKB, Moscow, Russia) and Concorde (British Aircraft Corporation, now British Aerospace, Westminster, London, United Kingdom) were in T 1968 and 1969, respectively. The dream of flying from New York to Sydney in four hours hasn’t been abandoned even with catastrophic failures leading to crashes of the Tu-144 and Concorde. The Tu-144 crashed during the Paris Air Show in 1973 and during delivery in 1978, and was then retired in 1983. At the Paris Air Show, the Tu-144 lost the left-hand (port) side of the whole wing during a descending maneuver. Concorde Air France Flight 4590 crashed during take-off in 2000 and the last Concorde flight was in 2003. Supersonic commercial transport aircraft designers are tasked to meet many requirements that include safety, sonic boom, and fuel efficiency issues. Outboard wing flutter and divergence associated with extensive outboard engine motion were one of the major issues during the 1,2 design of High Speed Civil Transport (HSCT) aircraft as shown in Fig. 1.
When an aircraft flies at supersonic speed, it creates a shock wave that imparts a thunder-like boom on the ground.
To mitigate the unacceptable boom magnitudes, the United States and other countries limited supersonic routes of commercial transport aircraft to only those over the ocean. This sonic boom needs to be reduced drastically to enable supersonic commercial transport aircraft operation over the land.
The National Aeronautics and Space Administration (NASA) and the major private aerospace companies in the United States; The Boeing Company (Chicago, Illinois), Lockheed Martin (Bethesda, Maryland), Gulfstream Aerospace Corporation (Savannah, Georgia), and Aerion Corporation (Reno, Nevada); have continued to conduct research into Low-boom Supersonic Commercial Transport (LSCT) aircraft concepts to reduce the level of sonic boom 3-5 on the ground within an acceptable range. Lockheed Martin Skunk Works (LMSW) has developed an N+2 LSCT 4,5 aircraft under a NASA contract. The second next generation is referred as N+2. An artist concept of this aircraft is shown in Fig. 2. Cruise Mach number of 1.7 and a range of over 5,000 nautical miles were selected to design this 80- passenger aircraft. This Lockheed Martin designed tri-jet aircraft achieved a sonic boom level of 79 PLdB at cruise speed. This sonic boom level was 6 dB, 20 dB, and 25 dB less than NASA’s N+2 goal, HSCT aircraft, and Concorde, respectively.
Based on the current outer mold-line configuration, LMSW developed a detailed internal structural layout and delivered an aeroelastically optimized finite element (FE) model in gear-up (retracted) and gear-down (extended) configurations. The Lockheed Martin baseline FE model was sized using MSC Nastran (MSC Software Corporation, Newport Beach, California) solution 200 (design optimization). One of the major difficulties in using MSC Nastran solution 200 for design optimization is that it is not easy to handle multiple structural models with multiple flight conditions in a single optimization run. In this study, the N+2 LSCT aircraft design at eight Mach numbers utilizes five and two fuel and payload conditions for landing gear-up and gear-down configurations, respectively. An object- oriented multidisciplinary design, analysis, and optimization (MDAO) tool that has been developed at the NASA Armstrong Flight Research Center (AFRC) (Edwards, California) will be used to perform an aeroelastic tailoring study with multiple structural configurations and Mach numbers in a single optimization run.
The primary objective of the current aeroelastic tailoring study is to develop a baseline FE model for the Lockheed Martin N+2 LSCT aircraft. The long term objective of this design optimization study is using a game-changing approach for a light-weight aircraft design procedure. In this game-changing approach, flutter of an aircraft will be passively suppressed (i.e. use aeroelastic constraints) up to the limit speed line, and then actively suppressed between limit speed line and 15% limit speed margin line instead of using passive flutter suppression technique all the way up 8,9 to 15% limit speed margin line as shown in Fig. 3. Therefore, simultaneous structural and control optimization for reducing the structural weight using the aeroelastic tailoring and flexible motion control will be achieved in a single optimization run. Not only will the structural design variables, but also the control law design variables, such as coefficients of polynomials in the transfer functions et cetera, will be simultaneously changed during the optimization American Institute of Aeronautics and Astronautics to satisfy the open- and closed-loop flutter, gain and phase margin of the aeroservoelastic system, buckling, and overall strain requirements.
In the case of the current baseline optimization study, flutter of an aircraft will be suppressed all the way up to the 15% limit speed line. Therefore, results of the current baseline optimization study can be compared with the optimization results from the game-changing optimization study in the future.
The pre-matured version of a FE model delivered from LMSW in June 2013 was selected as a demonstration model in this study. The object-oriented MDAO tool is used in this study with structural behavior constraints, such as strength, buckling, and flutter. Structural analyses are based on MSC Nastran solution 103 (modal analysis) and 105 (buckling and strength analyses). The ZAERO code (Zona Technology Inc., Scottsdale, Arizona) is used to obtain the aeroelastic characteristics of the N+2 LSCT aircraft in subsonic as well as supersonic speed regimes.
II. Object-Oriented Multidisciplinary Design, Analysis, and Optimization Tool Supporting the Aeronautics Research Mission Directorate guidelines, NASA AFRC has developed an object- 3 11 oriented optimization (O ) tool to leverage existing tools and practices, and to allow the easy integration and adoption of new state-of-the-art software.
Details of flow diagrams about pre-processor modules, discipline modules, and post-processor modules for flutter, buckling, and strength analyses used in this study are summarized in Fig. 4. These modules are script commands which mainly perform submission of the computing job, copying files, changing directories, saving files, and deleting files. Although these modules are developed mainly for MSC Nastran and ZAERO codes, they can be easily customized for other analytical tools. Some of these modules are discussed below.
A. Update Design Pre-processor Module (Pre: Update Design in Fig. 4) This module reads in a template file for the MSC Nastran input deck, design variable data created by O tool, and design variable to structural property relationship information, and then creates a new MSC Nastran input deck corresponding to the current design configuration. Design variables can be thicknesses and ply angles for composite plate/shell elements and area, area moment of inertia, and torsional constant for bar and beam elements, et cetera.
B. Modal Analysis Module (Discipline: Modal in Fig. 4) MSC Nastran solution 103 (modal analysis) is used to determine the modal characteristics (natural frequencies and mode shapes), the global mass matrix, total weight, center of gravity (CG) locations, and mass moment of inertia of a structural model. Total weight, CG locations, and mass moment of inertia in the MSC Nastran output file (f06 file) are used in weight post-processor, update ZAERO pre-processor, and landing and ground control loads pre-processor modules. Natural frequencies and mode shapes are used for flutter and trim analyses and the global mass matrix is needed for trim analyses.
C. Weight Post-processor Module (Post: Weight in Fig. 4) Two different weight computation modules were incorporated when the object-oriented MDAO tool was developed. The first module was based on the MSC Nastran output file (f06 file) from modal analysis. Total weight, CG locations, and mass moment of inertia are computed in this module. However, this module can be used only for light weight structural models due to the issue associated with the number of effective digits in the MSC Nastran output file. The second weight computation module was developed in this study to overcome this number of effective digits issue in the MSC Nastran output file. The second weight computation program reads in the MSC Nastran input deck and computes the total weight of a structural model.
In this study, weight computation is based on the design take-off weight (DTOW) condition. The DTOW is equivalent to full fuel full payload (FFFP) condition. Performance index from the weight post-processor module is the total weight as shown in Eq. (1): PI = 𝑊 (1) W 𝑇 D. Flutter Analysis and Flutter Post-processor Modules (Discipline: Flutter and Post: Flutter in Fig. 4) The ZAERO code with g-method solution technique and an in-house mode tracking code are used to determine the flutter speeds and frequencies. In the in-house mode tracking code, the flutter speeds are computed using the following definition together with the speed versus damping, V-g, and speed versus frequency, V-f, data obtained from ZAERO code. Definitions of flutter speed are illustrated in Fig. 5.
American Institute of Aeronautics and Astronautics 1) V ≤ V : When speed is lower than a limit speed V , zero structural damping is assumed to compute flutter L L speed.
2) V ≥ 1.15V : When speed is higher than 1.15V , flutter speed computation is based on three percent structural L L damping.
3) V < V < 1.15V : When speed is between V and 1.15V , linearly varying structural damping value is used L L L L to determine flutter speed.
In this study, a flutter speed V is designed to be higher than 1.15V , as shown in Eq. (2): F L V > 1.15V , (2) F L at a selected Mach number and altitude to have flutter free aircraft within flight envelope. Rewrite Eq. (2) as shown in Eq. (3): 1.15V − V < 0. (3) L F Dividing Eq. (3) by 1.15V gives the following design requirement shown in Eq. (4): L V F 1 − < 0. (4) 1.15V L Therefore, the performance index from the flutter post-processor module is defined in Eq. (5): V F PI ≡ 1 − (5) F 1.15V L where, V is the flutter speed obtained from the post-processor module.
F E. Update ZAERO Pre-processor, Trim Analysis, and Trim Loads Pre-processor Modules (Pre: Update ZAERO, Discipline: Trim, and Pre: Trim Loads in Fig. 4) The ZAERO trim analysis is used to compute a design load (inertia load + aerodynamic load) for various design configurations. During optimization, an input deck for ZAERO trim analysis is updated in the update ZAERO pre- processor module using total weight, CG locations, moment of inertias, and the global mass matrix computed from the modal analysis module.
Sometimes, computed design loads are not symmetric from symmetric trim analysis due to the numerical difficulties associated with a splining procedure. In this study, the trim loads pre-processor module reads in external loads computed from the trim analysis module, generates symmetric or anti-symmetric loads, and writes manipulated design loads for strength and buckling analyses. Trim flight conditions used in this study are summarized in Table 1.
F. Landing and Ground Control Loads Pre-processor Module (Pre: Landing & Ground Loads in Fig. 4) This pre-processor module computes landing, ground control, and emergency landing loads. Landing conditions used in the design procedure are as follows: Level landing Spin up landing Spring back landing Lateral drift landing Right gear landing Left gear landing Side load right (R) to left (L) Side load L to R Ground control loads are computed using the following conditions: Three-point braking roll Two-points braking roll Dynamic roll braking Turning condition Nose wheel yaw and steering 1 Nose wheel yaw and steering 2 American Institute of Aeronautics and Astronautics Nose wheel yaw and steering 3 Reverse braking 2g taxi Finally, emergency landing loads applied to three engine structures are computed based on the following conditions: 9g forward loading 1.5g rearward loading 3g sideway loading 6g downward loading Landing and ground control loads computations are based on equations in Tables 2 and 3, respectively.
G. Buckling and Strength Analyses and Strength Post-processor Modules (Discipline: Buckling and Strength and Post: Strength in Fig. 4) Based on design loads computed from trim, landing (regular as well as emergency), and ground control analyses, strength and buckling analyses are performed simultaneously using MSC Nastran solution 105. Once a landing gear configuration and a weight condition are selected then all kinds of different load subcases can be analyzed in a single MSC Nastran solution 105.
For the static safety of a structure, design load multiplied by safety factor applied to a structural element should be smaller than a corresponding failure load as shown in Eq. (6): Design Load × Safety Factor < Failure Load. (6) Rearranging above equation gives Eq. (7): Design Load × Safety Factor − Failure Load < 0. (7) Dividing Eq. (7) by “ Design Load × Safety Factor ” gives Eq. (8): Failure Load 1 − < 0. (8) Design Load ×Safety Factor Margin of safety (MS) is defined in Eq. (9): Failure Load MS ≡ − 1. (9) Design Load ×Safety Factor The minimum margin of safety from all of the structural elements under all of the different load subcases is selected as the performance index from strength post-processor. Therefore, one performance index, that is critical MS, is obtained from one MSC Nastran solution 105 run as shown in Eq. (10): PI ≡ −min(MS) (10) s A safety factor of 1.5 is used for all metal and composite materials in this study.
H. Buckling Post-processor Module (Post: Buckling in Fig. 4) The buckling load factor (BLF) is the factor of safety against buckling phenomena and possible BLF values with corresponding buckling status are summarized as follows: 0 ≤ BLF ≤ 1 : Buckling predicted BLF < 0 or BLF > 1 : Buckling not predicted Therefore, buckling will be predicted when the BLF value is within the following ranges in Eq. (11): 0 ≤ BLF ≤ 1 (11) Subtracting 1/2 from Eq. (11) gives Eq. (12): −1/2 ≤ BLF − 1/2 ≤ 1/2 (12) American Institute of Aeronautics and Astronautics Eq. (12) is equal to Eq. (13): 2 2 (BLF − 1/2) ≤ (1/2) (13) Therefore, if the opposite description in Eq. (14) is true, 2 2 (BLF − 1/2) > (1/2) , (14) then buckling is not predicted. Rewrite Eq. (14), as shown in Eq. (15): 2 2 (1/2) − (BLF − 1/2) < 0. (15) A performance index from buckling post-processor is defined using the positive minimum BLF value from all of the different load subcases, and computed from Eq. (16).
2 2 PI ≡ (1/2) − {positive min(BLF) − 1/2} (16) B III. Multidisciplinary Analysis of the Aircraft Model before Optimization In this section, modal, flutter, trim, landing and ground control, strength, and buckling analyses have been performed before starting optimization in order to have reference structural characteristics of the N+2 LSCT aircraft.
The MSC Nastran code is used to obtain the modal, strength, and buckling characteristics. Flutter and trim analyses are based on ZAERO code, and landing and ground control loads are computed using an in-house code.
A. Modal Analysis A structural FE model with gear-down configuration is shown in Fig. 6. The total number of grid points in the gear-up FE model is 55,635, and the total weight in the DTOW condition is 332,738 lbf.
Fifty and sixteen modes are computed for the flutter and trim analyses, respectively. Natural frequencies for the first ten elastic modes from gear-up with DTOW, full fuel empty payload (FFEP), DTOW minus fuel burned to reach Mach 2 weight (M2W), and zero fuel weight (ZFW) configurations; and gear-down with DTOW and design landing weight (DLW = ZFW+35% Fuel) configurations are summarized in Table 4. The ZFW is equivalent to empty fuel full payload (EFFP) condition. The first six flexible mode shapes obtained from gear-up with DTOW configuration are shown in Fig. 7.
B. Flutter Analysis The aerodynamic model of the N+2 LSCT aircraft based on ZAERO computation is shown in Fig. 8. This ZAERO aerodynamic model has 5,060 surface elements. The matched flutter analyses are performed at six Mach numbers of 0.66, 0.89, 1.41, 1.80, 2.00, and 2.30 using DTOW, FFEP, empty fuel empty payload (EFEP), and ZFW conditions.
The velocity versus damping, V-g, and velocity versus frequency, V-f, curves of the baseline model with the DTOW condition at a Mach number of 0.66 from the matched flutter analyses are given in Fig. 9. The primary flutter mode shape using DTOW condition at Mach 0.66 is given in Fig. 10. In this flutter mode shape, the outboard wing and V-tail are coupled through the flexibility of the aft inner wing section, and the center engine pitch motion is also involved in this first flutter mode shape. Flutter boundaries before optimization are summarized in Fig. 11. It should be noted in Fig. 11 that the fuel effect on flutter boundaries are larger than the payload effect. Flutter speeds at Mach 0.66 and 0.89 for full fuel conditions DTOW and FFEP are between V and 1.15V . Therefore, flutter design L L requirements are violated at these two Mach numbers with full fuel conditions.
C. Trim Analysis The control surfaces for trim analyses are displayed in Fig. 12. Trim analyses are also performed using the ZAERO code, and trim results are given in the appendix. Symmetric trim analyses are performed using the gear-up configuration with DTOW, ZFW, and M2W conditions; and gear-down configuration with DTOW and DLW conditions. Anti-symmetric trim analyses are also performed in the case of the gear-up configuration with the DTOW condition.
The ZAERO based aerodynamic model gives more realistic aerodynamic loads on the fuselage area compared to the MSC Nastran model (using the doublet lattice method) in reference 5. In this reference, the accuracy of the design aerodynamic load on the fuselage area was questionable since the fuselage of the N+2 LSCT aircraft had been American Institute of Aeronautics and Astronautics idealized using flat horizontal panels instead of using three dimensional body type elements as shown in Fig. 8.
Especially, since there are no applied horizontal design aerodynamic loads for the fuselage design in reference 5.
D. Landing and Ground Control Analyses Landing and ground control load vectors for twenty one load cases with DTOW and DLW conditions are computed in this study. Eight, nine, and four load cases as given in section II-F are from landing, ground control, and emergency landing conditions, respectively. In the case of landing analysis, trimmed aerodynamic loads from load cases 1500 and 1800 are added for weight conditions DTOW and DLW, respectively.
E. Buckling and Strength Analyses In this study, buckling and strength analyses are performed simultaneously using MSC Nastran solution 105. A total number of five buckling and strength analyses with gear-up (three analyses) and gear-down (two analyses) configurations are performed. These five different structural models are summarized in Table 5. Minimum buckling load factors before optimization from each analysis set are summarized in Table 6, and a buckling mode shape under the gear-up and DTOW condition is shown in Fig. 13. The gear-up configuration of the N+2 LSCT aircraft has buckling issues as shown in Table 6. On the other hand, the gear-down configuration is buckling free.
Minimum margin of safety values of the N+2 LSCT aircraft before optimization from five MSC Nastran solution 105 are summarized in Table 7. Negative margin of safety values are observed for all five different sets of the model, and that means strength design requirements are all violated with the ZAERO based aerodynamic model. Violation of margin of safety values probably means that the aerodynamic loads distribution computed using ZAERO trim analysis are different than the MSC Nastran generated aerodynamic loads.
IV. Multidisciplinary Design Optimization In this study, the optimization problem is stated as follows: 𝑇 Find design variables 𝐗 = ⌊𝑋 𝑋 … , 𝑋 ⌋ which minimizes Eq. (17): 1, 2, 𝑛 𝐹(𝐗) objective function (17) subjected to Eqs. (18) and (19): (𝐗) < 0.
𝑔 𝑗 = 1,2, … , 𝑛𝑐𝑜𝑛 inequality constraints (18) 𝑗 𝐿 𝑈 𝑋 ≤ 𝑋 ≤ 𝑋 𝑖 = 1,2, … , 𝑛𝑑𝑣 side constraints (19) 𝑖 𝑖 𝑖 Based on the analyses before optimization in section III, most of the flutter, buckling, and strength design requirements are violated under ZAERO based aerodynamics as shown in Table 8. The fuselage aerodynamic model is better with the ZAERO model, shown in Fig. 8, than the MSC Nastran based model. The ZAERO based aerodynamic loads are larger than MSC Nastran based loads, and therefore most of the design requirements are violated as shown in Table 8 under the before optimization column.
A. First Optimization Run (Sizing Optimization) One of the major issues with the gradient based optimization algorithms such as Automated Design Synthesis (ADS) or Design Optimization Tools (DOT) is that the starting configuration of an optimization run should be in the feasible domain. Otherwise, there is no guarantee that the optimizer program will eventually push the design from an infeasible domain to a feasible domain. In this study, composite ply thicknesses of the failed finite elements are manually stiffened to have a feasible design. A total number of 111 variables are selected and thickened to have a feasible design at a starting configuration. Objective function value and corresponding constraint function values at a starting configuration are also given in Table 8 under the iteration 1 column. The weight penalty for having an achievable (or feasible) design was 93,026 lb (a 28.0% increase from baseline).
These 111 variables are selected as design variables for the first optimization run. Structural components affected by these 111 design variables are shown in Fig. 14 using colored elements. In the baseline model, composite materials are based on the stacking of the nine plies, and cross sectional configuration is shown in Fig. 15. Design variable linking is defined based on the different structural components and summarized as follows: Wing, inner-wing, tail, and fuselage skins; three design variables per each composite laminate property st nd th rd o 1 ply thickness, 2 ply thickness = 4 ply thickness, and 3 ply thickness American Institute of Aeronautics and Astronautics Spars and ribs for wing, inner-wing, and tail as well as bulkheads and walls for fuselage; one design variable per each composite laminate property st nd rd th o 1 ply thickness = 2 ply thickness = 3 ply thickness = 4 ply thickness Spars and ribs for inner-wing; one design variable per each composite laminate property th o 5 ply thickness Objective as well as constraint functions and corresponding performance indices for the first optimization run are summarized in Table 9. After the first seven iterations, structural weight of the DTOW condition is reduced, and all the constraint functions 𝑔 (𝐗) satisfy required values, i.e. 𝑔 (𝐗) < 0. j = 1,2, … , 16 . The weight reduction was 𝑗 𝑗 61,659 lb, and therefore, weight penalty at the end of the first optimization run is 31,367 lb (a 9.4% increase from baseline). Change of the composite ply thickness design variables after iteration 7 are shown in Fig. 16. In this figure, color spectrums represent “composite laminate thickness after optimization” divided by “composite laminate thickness before optimization”.
The active and near active strength constraints at the end of the first optimization run are from buckling and strength analyses set number 4, 2, and 1 as shown in Table 8 under the iteration 7 column. Corresponding weight and gear configurations are gear up with DTOW and ZFW conditions and gear down with the DTOW condition. Strain distributions under these active and near active strength constraints are shown in Fig. 17.
The active constraint is from load case number 3013, and this case corresponds to nose wheel yaw and steering case number 1 as shown in Tables 3 and 5. The margin of safety value before optimization was -0.781 as shown in Table 7. This margin of safety value becomes 5.36e-6 (effectively zero) after the first optimization run. The active element is located at the second rib of the inner wing near the main landing gear bay area as shown in Fig. 17a.
Two near active strength constraints are load case numbers 1700 and 300 from the gear up with ZFW and DTOW conditions, respectively. These cases match with 2.7g gust and 2.5g pull up maneuver loading cases under Mach numbers of 0.89 and 0.48, respectively, as shown in Table 1. Strain distributions are shown in Figs. 17b and 17c. In these figures, active elements are located at the floor of the main landing gear bay. The margin of safety value for load case 1700 was -0.998 before optimization and becomes 0.061 after the first optimization run. On the other hand, load case 1400 for the near active strength constraint in Table 7 is switched to load case 300 after the first optimization run, and the margin of safety value is changed from -0.999 to 0.161.
The near active flutter constraints are from DTOW and FFEP weight conditions at Mach 0.89. The only area in the wing affected by design variables is near the trailing-edge of the wing tip section as shown in Fig. 14. By doubling the total thickness of this area, flutter boundaries are outside 1.15 V line. Adding weight near the trailing-edge of the L wing tip section is a kind of mass balancing effect on flutter boundaries which can be observed in reference 7.
B. Second Optimization Run (Aeroelastic Tailoring Optimization) Six design variables are selected for the second optimization run. From the active strength constraint case, the design variable for the active element area in Fig. 17a was the composite core thickness, and therefore, angles for the second and fourth plies in the area of zone 1 in Fig. 18 are selected as the first design variable for the second optimization. Right and left hand side as well as upper and lower ply angle design variables are linked in this study.
The second design variable is selected from zone 2 in Fig. 18. From the near active strength constraint, the active elements in Figs 17b and 17c are located at the floor of the main landing gear bay (near the junction of the floor and centerline wall). However, composite laminate thicknesses in this area were between 6 to 10 times thicker than the starting configuration as shown in Fig. 16, and therefore, angles for the second and fourth plies are selected as the second design variable, which are also linked, the same as the first design variable.
The third and fourth design variables are also selected near the main landing gear bay area as shown in Fig. 18, zone 3 and zone 4. As shown in Fig. 16, composite laminate thicknesses in these areas were between 2 to 6 times thicker compared to the starting configuration, and these two zones are connected to the floor of the main landing gear bay. Therefore angles for the second and fourth plies for zones 3 and 4 are also selected as design variables.
Finally, the fifth and sixth design variables are angles for the second and fourth plies for zone 5 and zone 6 in the aft fuselage skin area as shown in Fig. 18. In Fig. 16, composite laminate thicknesses changes in these two zones are almost 3 to 16 times thicker than the starting configuration, and therefore, design variables are selected in these two zones.
Objective and constraint functions for the second optimization run are also summarized in Table 9. Composite ply angle change cannot improve weight configuration. Therefore, the active and the near active strength constraint values; minimum margin of safety values from three strength analyses set numbers 1, 2, and 4; will be maximized to create more tolerance, as defined in Fig. 19 for the future weight optimization runs. Therefore, the linear combination of these three margin of safety values becomes the objective function for the second optimization run. Weighting factors for these three performance indices, 𝑔 (𝐗) , 𝑔 (𝐗) , and 𝑔 (𝐗) , for the objective function are 1.0, 0.5, and 0.5, 15 13 12 American Institute of Aeronautics and Astronautics respectively. Constraint functions for the second optimization run are the same as the first optimization run, and (𝐗) (𝐗) (𝐗) therefore performance indices 𝑔 , 𝑔 , and 𝑔 are both in objective and constraint functions.
15 13 12 Discrete variables are used for ply angles which are discretized every 5° from 0° to 90°. These design variables are related to the ±45° ply angles shown in Fig. 15; the second, fourth, sixth, and eighth plies. Plus 45° and minus 45° angles are also linked to reduce the total number of design variables for the second optimization run.
15-18 Discrete design variables together with the big-bang big-crunch (BBBC) algorithm are used in the second optimization run. Number of populations and BBBCs are 60 and 2, respectively. Objective as well as constraint functions and discrete design variable histories are given in Tables 10 and 11, respectively. In Table 10, the active and two near active strength constraint values of -5.63e-6, -0.061, and -0.161 become -0.159, -0.145, and -0.232, respectively. Therefore, the second optimization run creates more tolerance values from constraint boundaries for future weight optimization runs.
V. Conclusion The Lockheed Martin pre-matured N+2 LSCT aircraft is optimized in this study through the use of a multidisciplinary design optimization tool developed at the NASA AFRC. The baseline design of the pre-matured N+2 LSCT aircraft was infeasible when ZAERO based aeroelastic analyses were used. Most of the flutter, buckling, and strength design requirements were violated, and therefore, the composite thickness variables were changed manually to have a feasible starting configuration. The starting configuration of the optimization run should be an achievable design, and the weight penalty for this was 93,026 lb (a 28.0% increase from baseline).
A total of 111 design variables are used in the first optimization run. During the first optimization run, the weight reduction was 61,659 lb, and therefore, the weight penalty at the end of the first optimization run is 31,367 lb (a 9.4% increase from baseline). All the design requirements of the N+2 LSCT aircraft are satisfied at the end of the first optimization run. The active strength constraint at the end of the first optimization run was under the nose wheel yaw and steering case number 1. The minimum margin of safety under this load condition was 5.36e-6, effectively zero.
This minimum value is associated with the structural component located at the second rib of the inner wing near the main landing gear bay area. Two near active constraints are also due to the strength requirement. Minimum margin of safety values of 0.061 and 0.161 are observed at the floor of the main landing gear bay area. Corresponding load cases are the 2.7g gust load case at Mach 0.89 and an altitude of 20,000 ft and a 2.5g maneuver load case at sea level under Mach 0.48. The near active flutter constraint is related to the flutter boundary requirement at Mach number of 0.89 under DTOW and FFEP load conditions, and the design was improved due to the mass balancing effect.
The second optimization run is prepared and based on the six discrete design variables with the BBBC algorithm.
Angles for the second and fourth plies are selected as discrete design variables for the second optimization runs. Ply angle changes cannot improve the weight configuration of the N+2 LSCT aircraft. However, strength properties can be changed with different ply angles. Therefore, the second optimization run can create more tolerance for the active as well as near active strength constraint values for future weight optimization runs.
American Institute of Aeronautics and Astronautics Tables Table 1. Trim flight conditions.
Load Load Mach Landing c ase Maneuver Weight Altitude Trim v ariables f actor n umber g ear ID 100 Pull up 2.5g 0.66 DTOW Up SL BF( R=L ) 200 Push over - 1g 0.66 DTOW Up SL BF( R=L ) 300 Pull up 2.5g 0.48 DTOW Up SL BF( R=L ) 400 Pull up 2.5g 2.00 M2W Up 49 , 770ft BF=TEF( R=L ) 500 Push over - 1g 2.00 M2W Up 49 , 770ft BF( R=L ) 600 Pull up 2.5g 1.41 DTOW Up 49 , 770ft BF=TEF=AIL1=AIL2( R=L ) 700 Pull up 2.5g 0.66 ZFW Up SL BF( R=L ) 800 Push over - 1g 0.66 ZFW Up SL BF( R=L ) 900 Pull up 2.5g 2.00 ZFW Up 49 , 770ft BF=TEF( R=L ) 1000 Push over - 1g 2.00 ZFW Up 49 , 770ft BF( R=L ) 1100 Steady roll 0g 0.48 DTOW Up SL Load Case 2100+2300 1200 Abrupt roll 0g 0.48 DTOW Up SL Load Case 2200+2300 1300 Steady roll 1.67g 0.48 DTOW Up SL Load Case 2100+2400 1400 Abrupt roll 1.67g 0.48 DTOW Up SL Load Case 2200+2400 1500 Landing 1g 0.3092 DTOW Down SL BF( R=L ) 1600 Cruise 1g 1.80 DTOW Up 55 , 000ft BF=TEF( R=L ) 1700 Gust l oads 2.7g 0.89 ZFW Up 20 , 000ft BF( R=L ) 1800 Landing 1g 0.3092 DLW Down SL BF( R=L ) 2100 Steady roll 0g 0.48 DTOW Up SL AIL1=AIL2( R= - L ) 2200 Abrupt roll 0g 0.48 DTOW Up SL AIL1=AIL2( R= - L ) 2300 Pull up 0 g 0.48 DTOW Up SL BF( R=L ) 2400 Pull up 1.67g 0.48 DTOW Up SL BF( R=L ) American Institute of Aeronautics and Astronautics Table 2. Equations for landing load computations.
Load c ases Case n umber Load Right - MLG Left - MLG NLG FX 0 . 25 FM 0 . 25 FM 0 . 25 FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 300 1 (DTOW) Level + t rim load FY 0.0 0.0 0.0 & 4001 (DLW) FZ FM = f W FM FN = f W 𝐿𝑣 𝐿𝑀𝐺 𝑇 𝐿𝑣 𝐿𝑣 𝐿𝑁𝐺 𝑇 ( ) ( ) ( ) FX 0 . 8 × 0 . 8 FM 0 . 8 × 0 . 8 FM 0 . 8 × 0 . 8 FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 Spin up + t rim 3002 (DTOW) FY 0.0 0.0 0.0 load & 4002 (DLW) FZ 0 . 8FM 0 . 8FM 0 . 8FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 FX − ( 0 . 8 × 0 . 8 ) FM − ( 0 . 8 × 0 . 8 ) FM − ( 0 . 8 × 0 . 8 ) FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 Spring back + 3003 (DTOW) FY 0.0 0.0 0.0 t rim load & 4003 (DLW) FZ 0 . 8FM 0 . 8FM 0 . 8FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 FX ( ) ( ) 0 . 4 × 0 . 75 FM 0 . 4 × 0 . 75 FM 0 . 4 FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 Lateral drift + 3004 (DTOW) FY ( 0 . 25 × 0 . 75 ) FM ( 0 . 25 × 0 . 75 ) FM 0 . 25 FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 t rim load & 4004 (DLW) FZ 0 . 75FM 0 . 75FM FN 𝐿𝑣 𝐿𝑣 𝐿𝑣 FX 0 . 25 FM 0.0 0.0 𝐿𝑣 Right one gear + 3005 (DTOW) FY 0.0 0.0 0.0 t rim load & 4005 (DLW) FZ 0.0 0.0 FM 𝐿𝑣 FX 0.0 0 . 25 FM 0.0 𝐿𝑣 Left one gear + 3006 (DTOW) FY 0.0 0.0 0.0 t rim load & 4006 (DLW) FZ 0.0 FM 0.0 𝐿𝑣 FX 0.0 0.0 0.0 Side load RtoL + 3007 (DTOW) FY ( 0 . 8 × 0 . 5 ) FM ( 0 . 6 × 0 . 5 ) FM 0.0 𝐿𝑣 𝐿𝑣 t rim load & 4007 (DLW) FZ 0.0 0 . 5 FM 0 . 5 FM 𝐿𝑣 𝐿𝑣 FX 0.0 0.0 0.0 Side load LtoR + 3008 (DTOW) FY ( ) ( ) 0.0 − 0 . 6 × 0 . 5 FM − 0 . 8 × 0 . 5 FM 𝐿𝑣 𝐿𝑣 t rim load & 4008 (DLW) FZ 0 . 5 FM 0 . 5 FM 0.0 𝐿𝑣 𝐿𝑣 DTOW f =0.36; f =0.0639; Trim load case ID = 1500 𝐿𝑀𝐺 𝐿𝑁𝐺 DLW f =1.20; f =0.1477; Trim load case ID = 1800 𝐿𝑀𝐺 𝐿𝑁𝐺 American Institute of Aeronautics and Astronautics Table 3. Equations for ground control load computations.
Load c ases Case n umber Load Right - MLG Left - MLG NLG FX 0.0 0.0 0.0 FY 0.0 0.0 0.0 Static FM 𝑆𝑡 d W c ondition 𝐶𝐺 2 𝑀𝐺 𝑇 0 . 5d W FZ 𝐶𝐺 2 𝑁𝐺 𝑇 FM FN = 𝑆𝑡 𝑆𝑡 = d 𝑁𝐺 2 𝑀𝐺 d 𝑁𝐺 2 𝑀𝐺 FX 0 . 8FM 0 . 8FM 0.0 𝑆𝑡 𝑆𝑡 FY 0.0 0.0 0.0 3 - point 3009 (DTOW) 2 d FM + 2 ( 𝑍 − 𝑍 ) 0 . 8FM braked roll & 4009 (DLW) 𝐶𝐺 2 𝑀𝐺 𝑆𝑡 𝐶𝐺 𝑀𝐺𝐶𝑃 𝑆𝑡 FZ FM FM 𝑆𝑡 𝑆𝑡 d 𝐶𝐺 2 𝑁𝐺 FX 0 . 8FM 0 . 8FM 0.0 𝑆𝑡 𝑆𝑡 2 - point 3010 (DTOW) FY 0.0 0.0 0.0 braked roll & 4010 (DLW) FZ FM FM 0.0 𝑆𝑡 𝑆𝑡 FX 0 . 8FM 0 . 8FM 0.0 𝑆𝑡 𝑆𝑡 FY 0.0 0.0 0.0 W f d 𝜇𝐸 𝑇 𝐶𝐺 2 𝑁𝐺 [ d + ] 𝐶𝐺 2 𝑀𝐺 Dynamic 3011 (DTOW) d d + 𝜇𝐸 𝑁𝐺 2 𝑀𝐺 𝑁𝐺 2 𝑀𝐺 roll braking & 4011 (DLW) where, 𝐸 = { 𝑍 − 𝑍 − ( 𝑋 − 𝐶𝐺 𝑁𝐺𝐶𝑃 𝐶𝐺 FZ FM FM 𝑆𝑡 𝑆𝑡 𝑋 ) 𝑆 } 𝑁𝐺𝐶𝑃 𝑍 − 𝑍 𝑀𝐺𝐶𝑃 𝑁𝐺𝐶𝑃 and 𝑆 = ( 𝑋 − 𝑋 ) 𝐶𝐺 𝑁𝐺𝐶𝑃 𝑋 − 𝑋 𝑀𝐺𝐶𝑃 𝑁𝐺𝐶𝑃 FX 0.0 0.0 0 . 25FN 𝑆𝑡 Turning 3012 (DTOW) FY 0 . 5FM 0 . 5FM 0 . 5FN 𝑆𝑡 𝑆𝑡 𝑆𝑡 Condition & 4012 (DLW) FZ FM FM FN 𝑆𝑡 𝑆𝑡 𝑆𝑡 FX 0.0 0.0 0.0 Nose wheel 3013 (DTOW) yaw and FY 0.0 0.0 0 . 8FN 𝑆𝑡 & 4013 (DLW) steering (1) FZ FM FM FN 𝑆𝑡 𝑆𝑡 𝑆𝑡 FX 0 . 8FM 0.0 0.0 𝑆𝑡 Nose wheel FY 0.0 0.0 0.0 30 14 (DTOW) yaw and ( ) & 40 14 (DLW) 2 d FM + 𝑍 − 𝑍 0 . 8FM 𝐶𝐺 2 𝑀𝐺 𝑆𝑡 𝐶𝐺 𝑀𝐺𝐶𝑃 𝑆𝑡 steering (2) FZ FM FM 𝑆𝑡 𝑆𝑡 d 𝐶𝐺 2 𝑁𝐺 FX 0.0 0 . 8FM 0.0 𝑆𝑡 Nose wheel FY 0.0 0.0 0.0 30 15 (DTOW) yaw & ( ) & 40 15 (DLW) 2 d FM + 𝑍 − 𝑍 0 . 8FM 𝐶𝐺 2 𝑀𝐺 𝑆𝑡 𝐶𝐺 𝑀𝐺𝐶𝑃 𝑆𝑡 steering (3) FZ FM FM 𝑆𝑡 𝑆𝑡 d 𝐶𝐺 2 𝑁𝐺 FX − 0 . 55FM − 0 . 55FM 0.0 𝑆𝑡 𝑆𝑡 Reversed 30 16 (DTOW) FY 0.0 0.0 0.0 braking & 40 16 (DLW) FZ FM FM 0.0 𝑆𝑡 𝑆𝑡 FX 0.0 0.0 0.0 30 17 (DTOW) 2G t axi FY 0.0 0.0 0.0 & 40 17 (DLW) FZ 2FM 2FM 2FN 𝑆𝑡 𝑆𝑡 𝑆𝑡 𝜇 = 0 . 80 ; f = 2.00 d ≡ d + d 𝑁𝐺 2 𝑀𝐺 𝐶𝐺 2 𝑁𝐺 𝐶𝐺 2 𝑀𝐺 American Institute of Aeronautics and Astronautics Table 4. Natural frequencies of the N+2 LSCT aircraft with different fuel and payload conditions before optimization.
Natural f requency (Hz) Mode Notes Gear - up Gear - down n umber DTOW FFEP M2W ZFW DTOW DLW 7 2.049 2.055 2.071 2.266 2.048 2.158 Aft fuselage torsion 8 2.235 2.262 2.277 2.554 2.238 2.424 First symmetric fuselage bending 9 2.498 2.509 2.539 2.993 2.503 2.714 First symmetric wing bending 10 2.754 2.769 2.935 3.415 2.752 3.265 First anti - symmetric wing bending 11 3.060 3.069 3.115 3.731 3.057 3.403 Symmetric tail bending 12 3.562 3.608 3.689 4.044 3.574 3.945 Forward fuselage lateral bending 13 4.440 4.449 4.511 4.790 4.429 4.602 First anti - symmetric tail bending 14 4.456 4.537 4.555 5.532 4.437 5.142 Second symmetric wing bending 15 4.818 4.842 5.146 5.832 4.809 5.542 Second anti - symmetric wing bending 16 5.449 5.465 5.550 6.158 5.444 5.994 Symmetric aft inner wing bending Table 5. Buckling and strength analyses.
Analysis Gear Weight Load c ases s et c onfiguration c ondition 1 Up DTOW 100, 200, 300, 600, 1100, 1200, 1300, 1400, & 1600 2 Up ZFW 700, 800, 900, 1000, & 1700 3 Up M2W 400 & 500 4 Down DTOW 3001 ~ 30 17 + 3018 ~ 3021 (emergency) + 1500 (for landing) 5 Down DLW 4001 ~ 40 17 + 4018 ~ 4021 (emergency) + 1800 (for landing) Table 6. Minimum buckling load factor for baseline model before optimization from each analysis set.
Minimum Analysis Gear Weight Case Load c ase buckling load Buckling s et c onfiguration c ondition n umber f actor 1 Up DTOW 300 2.5G pull up; M=0.48 0.152 yes 2 Up ZFW 1700 2.7G gust loads; M=0.89 0.195 yes 3 Up M2W 400 2.5G pull up; M=2.00 0.151 yes 4 Down DTOW 3006 Left one gear landing 1.71 no 5 Down DLW 4006 Left one gear landing 1.52 no Table 7. Minimum margins of safety for baseline model before optimization from each analysis set.
Minimum Analysis Gear Weight Case Load c ase margin of Failure s et c onfiguration c ondition n umber s afety 1 Up DTOW 1400 1.67G abrupt roll; M=0.48 - 0.999 yes 2 Up ZFW 1700 2.7G gust loads; M=0.89 - 0.998 yes 3 Up M2W 400 2.5G pull up; M=2.00 - 0.997 yes 4 Down DTOW 3013 Nose wheel yaw & steering (1) - 0.781 yes 5 Down DLW 4003 Spring back landing - 0.657 yes American Institute of Aeronautics and Astronautics Table 8. Optimization histories of the first optimization run.
Performance Before Design c onfiguration Iteration 1 Iteration 7 i ndex o ptimization Objective Total w eight DTOW; GU 332738 425764 364105 f unction ( ) 𝑔 𝐗 DTOW; GU; M=0.66 0. 067 ( C V) - 0.362 - 0.342 𝑔 ( 𝐗 ) DTOW; GU; M=0.89 0. 048 ( C V) - 0.543 -0.096 𝑔 ( 𝐗 ) DTOW; GU; M=1.41 - 0.079 - 1.34 - 0.297 ( ) FFEP; GU; M=0.66 0.066 ( C V) - 0.365 - 0.337 𝑔 𝐗 Flutter ( ) 𝑔 𝐗 FFEP; GU; M=0.89 0. 034 ( C V) - 0.586 -0.094 𝑔 ( 𝐗 ) FFEP; GU; M=1.41 - 0.095 - 1.32 - 0.255 𝑔 ( 𝐗 ) DTOW; GU 0.152( C V) - 1.05 - 1.29 Constraint 𝑔 ( 𝐗 ) ZFW; GU 0.186( C V) - 2.36 - 3.09 f unctions ( ) M2W; GU 0.151( C V) - 1.28 - 1.91 𝑔 𝐗 𝑔 ( 𝐗 ) 𝑗 𝑔 ( 𝐗 ) DTOW; GD - 0.960 - 3.27 - 3.88 Buckling 𝑔 ( 𝐗 ) DLW; GD - 0.561 - 0.308 - 1.07 𝑔 ( 𝐗 ) DTOW; GU 0.999(CV) - 0.267 -0.161 ( ) ZFW; GU 0.998( C V) - 0.780 𝑔 𝐗 -0.061 ( ) 𝑔 𝐗 M2W; GU 0.997( C V) -0.179 - 0.537 𝑔 ( 𝐗 ) DTOW; GD 0.781( C V) - 0.751 -5.63e-6 Strength 𝑔 ( 𝐗 ) DLW; GD 0.657( C V) - 0.210 - 0.320 ( C V): Constraint v iolated GU: Gear u p (retracted) GD: Gear d own (extended) Iteration 2 through 6 not shown Table 9. Summary of objective and constraint functions for the first and second optimization runs.
Functions Performance indices Notes 2 2 Objective (first run) 𝐹 ( 𝐗 ) = ( P I ) = 𝑊 DTOW W 𝑇 Objective (second run) 𝐹 ( 𝐗 ) = − { 0 . 5 𝑔 ( 𝐗 ) + 0 . 5 𝑔 ( 𝐗 ) + 𝑔 ( 𝐗 ) } Safety factor = 1.5 12 13 15 V F ( ) Flutter constraint 𝑔 𝐗 = P I = 1 . − < 0 . j = 1 , 2 , … , 6 15% margin 𝑗 F 1 . 15 V L 2 2 { } 𝑔 ( 𝐗 ) = P I = ( 1 / 2 ) − positive min ( BLF ) − 1 / 2 < 0 .
𝑗 B Buckling constraint Safety factor = 1.5 j = 7 , 8 , … , 11 Strength constraint 𝑔 ( 𝐗 ) = P I = − min ( MS ) < 0 . j = 12 , 13 , … , 16 Safety factor = 1.5 𝑗 s American Institute of Aeronautics and Astronautics Table 10. Optimization histories of the second optimization run.
Performance i ndex Design c onfiguration Starting BBBC 1 BBBC 2 Objective − 𝑔 ( 𝐗 ) / 2 − 𝑔 ( 𝐗 ) / 2 − 𝑔 ( 𝐗 ) 0.111 0.337 0.348 12 13 15 f unction 𝑔 ( 𝐗 ) DTOW; GU; M=0.66 - 0.342 - 0.342 - 0.342 𝑔 ( 𝐗 ) DTOW; GU; M=0.89 -0.096 -0.096 -0.096 ( ) DTOW; GU; M=1.41 - 0.297 - 0.297 - 0.297 𝑔 𝐗 𝑔 ( 𝐗 ) FFEP; GU; M=0.66 - 0.337 - 0.337 - 0.337 Flutter 𝑔 ( 𝐗 ) FFEP; GU; M=0.89 -0.094 -0.094 -0.094 𝑔 ( 𝐗 ) FFEP; GU; M=1.41 - 0.255 - 0.255 - 0.255 ( ) DTOW; GU - 1.29 - 1.38 - 1.38 𝑔 𝐗 Constraint ( ) 𝑔 𝐗 ZFW; GU - 3.09 - 3.09 - 3.09 f unctions 𝑔 ( 𝐗 ) M2W; GU - 1.91 - 2.23 - 2.23 ( ) 𝑔 𝐗 𝑗 𝑔 ( 𝐗 ) DTOW; GD - 3.88 - 4.19 - 4.19 Buckling ( ) DLW; GD - 1.07 - 1.07 - 1.07 𝑔 𝐗 ( ) 𝑔 𝐗 DTOW; GU -0.161 -0.232 -0.232 𝑔 ( 𝐗 ) ZFW; GU -0.061 -0.145 -0.145 𝑔 ( 𝐗 ) M2W; GU - 0.537 - 0.542 - 0.542 𝑔 ( 𝐗 ) DTOW; GD -5.63e-6 -0.159 -0.159 Strength 15 ( ) DLW; GD - 0.320 - 0.419 - 0.419 𝑔 𝐗 ( C V): Constraint v iolated GU: Gear u p (retracted) GD: Gear d own (extended) Table 11. Design variable histories of the second optimization run.
Design v ariables Starting BBBC 1 BBBC 2 nd 1 (2 rib at inner - wing) 45° 15° 15° 2 (floor at main landing gear bay) 45° 65° 65° 3 (center wall at main landing gear bay) 45° 65° 55° 4 (aft bulkhead at main landing gear bay) 45° 65° 55° 5 (aft fuselage skin 1) 45° 30° 40° 6 (aft fuselage skin 2) 45° 50° 55° American Institute of Aeronautics and Astronautics Figures Figure 1. A high speed civil transport aircraft.
Figure 2. Artist’s concept of the Lockheed Martin N+2 LSCT aircraft.
American Institute of Aeronautics and Astronautics Figure 3. Aeroelastic stability envelope.
From O tool Pre: Update Design Post: Weight (gear-up;DTOW) Discipline: Modal Discipline: Flutter Post: Flutter To O tool (gear-up;DTOW) (gear-up;DTOW) (gear-up;DTOW) Discipline: Modal Discipline: Flutter Post: Flutter (gear-up;FFEP) (gear-up;FFEP) (gear-up;FFEP) Discipline: Trim Pre: Trim Loads Post: Strength (gear-up;DTOW;sym) (gear-up;DTOW;sym) Discipline: Buckling (gear-up;DTOW) Pre: Update ZAERO & Strength (gear-up;DTOW) Discipline: Trim (gear-up;DTOW) Post: Buckling Pre: Trim Loads (gear-up;DTOW;Asym) (gear-up;DTOW) (gear-up;DTOW; Asym) Post: Strength Discipline: Buckling (gear-up;ZFW) Discipline: Modal Discipline: Trim Pre: Update ZAERO Pre: Trim Loads & Strength (gear-up;ZFW) (gear-up;ZFW;sym) (gear-up;ZFW) (gear-up;ZFW;sym) (gear-up;ZFW) Post: Buckling (gear-up;ZFW) Post: Strength Discipline: Buckling (gear-up; M2W) Discipline: Modal Discipline: Trim Pre: Update ZAERO Pre: Trim Loads & Strength (gear-up;M2W) (gear-up; M2W;sym) (gear-up; M2W) (gear-up; M2W;sym) (gear-up; M2W) Post: Buckling (gear-up; M2W) Pre: Landing & Ground Loads (gear-down;DTOW) Post: Strength Discipline: Buckling (gear-down;DTOW) Discipline: Modal Discipline: Trim Pre: Update ZAERO Pre: Trim Loads & Strength (gear-down;DTOW) (gear-down;DTOW;sym) (gear-down;DTOW) (gear-down;DTOW;sym) (gear-down;DTOW) Post: Buckling (gear-down;DTOW) Pre: Landing & Ground Loads (gear-down;DLW) Post: Strength (gear-down;DLW) Discipline: Buckling Discipline: Modal Discipline: Trim Pre: Update ZAERO Pre: Trim Loads & Strength (gear-down;DLW) (gear-down;DLW;sym) (gear-down;DLW) (gear-down;DLW;sym) (gear-down;DLW) Post: Buckling (gear-down;DLW) Figure 4. Details of pre-processor, discipline, and post-processor modules.
American Institute of Aeronautics and Astronautics b 3 % a c 0 % Speed V Structural Damping V 1.15V L L Figure 5. Definitions of flutter speed in three different speed regimes.
Front View Top View Bottom View Side View Figure 6. Finite element model of the N+2 LSCT aircraft in a gear-down configuration.
American Institute of Aeronautics and Astronautics Figure 7. First six flexible mode shapes of the N+2 LSCT aircraft with a gear-up and DTOW weight configuration (before optimization).
American Institute of Aeronautics and Astronautics Top Down View Front View Flow Through Top View Bottom View Side View Figure 8. Aerodynamic model of the N+2 LSCT aircraft.
American Institute of Aeronautics and Astronautics Figure 9. V-g and V-f curves of the N+2 LSCT aircraft with a gear-up and DTOW configuration at Mach 0.66 before optimization.
American Institute of Aeronautics and Astronautics Figure 10. The primary flutter mode shape using the DTOW condition at Mach 0.66.
Figure 11. Flutter boundaries of the baseline N+2 LSCT aircraft before optimization.
American Institute of Aeronautics and Astronautics Trailing-Edge Flap (R) (L) Aileron #1 (R) Aileron #1 (L) Aileron #2 (R) Aileron #2 (L) (R) (L) Body Flap Figure 12. Control surfaces for trim analyses.
Figure 13. Buckling model shape under a 2.5g pull up maneuver at Mach 0.66 and sea level.
American Institute of Aeronautics and Astronautics Figure 14. Structural components affected by thickness design variables.
Figure 15. Symmetric stacking of nine plies into a composite laminate.
American Institute of Aeronautics and Astronautics Figure 16. Thickness change of structural elements after iteration 7.
Figure 17. Strain distribution after iteration 7.
American Institute of Aeronautics and Astronautics Bottom up Top down Zone 1 Forward Zone 1 view view Aft Zone 4 Aft Zone 2 Zone 3 Zone 5 Forward Zone 6 Aft Figure 18. Design variable zones for composite ply angles.
DV Infeasible domain Active constraint Tolerance Constraint Feasible boundary domain DV DV : Design Variable i i Figure 19. Definition of tolerance.
American Institute of Aeronautics and Astronautics Appendix Trim Results In Table A1 (steady roll maneuver, load cases 1100 and 1300), roll rate in the trim card is defined in non- dimensional terms as equation (A1): 3.1416 𝑟𝑎𝑑 30°/𝑠𝑒𝑐 × ( ) × 83.89𝑓𝑡 × 12 𝑖𝑛/𝑓𝑡 Pb 180° = = 0.0410 𝑟𝑎𝑑 (A1) 2𝑉 2 × (0.48 × 13391.7 𝑖𝑛/𝑠𝑒𝑐) In the case of an abrupt roll maneuver (load cases 1200 and 1400), roll acceleration is in units of rad/sec /g, since accelerations have been defined to be in units of g, and therefore equation (A2) is shown as: 3.1416 𝑟𝑎𝑑 1 𝑟𝑎𝑑 Pdot = 30°/𝑠𝑒𝑐 × ( ) × 0.002588 ( ) = 0.0014 . (A2) 180° 𝑔 𝑠𝑒𝑐 ×𝑔 Table A1. Trim results.
Load c ase 100 200 300 400 500 600 700 800 900 Trim a nalysis Symmetric Nx ( g ) - 0.007 - 0.003 - 0.005 0.001 0.002 - 0.004 - 0.016 - 0.006 0.003 Nz ( g ) 2.5 - 1.0 2.5 2.5 - 1.0 2.5 2.5 - 1.0 2.5 Pdot (rad/s /g) None None None None None None None None None Qdot (rad/s /g) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Pb/2V (rad) None None None None None None None None None Qc/2V (rad) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.75 - 2.50 14.37 8.32 - 2.81 16.90 4.10 - 1.04 5.07 °) 2.01 - 6.07 6.12 - 5.25 5.42 - 25.68 - 8.01 - 1.59 - 12.92 Body f lap °) Trailing - edge f lap °) - 5.25 - 25.68 - 12.92 Aileron #1 °) - 25.68 - 25.68 Aileron #2 °) Mach n umber 0.66 0.66 0.48 2.00 2.00 1.41 0.66 0.66 2.00 Altitude (ft) SL SL SL 49770 49770 49770 SL SL 49770 Weight c onfiguration DTOW DTOW DTOW M2W M2W DTOW ZFW ZFW ZFW Gear c onfiguration Up Up Up Up Up Up Up Up Up Load c ase 1000 1100 1200 1300 1400 1500 1600 1700 1800 Trim a nalysis Sym Asymmetric (sym + anti - sym ) Symmetric Nx ( g ) 0.004 - 0.002 - 0.002 - 0.004 - 0.004 - 0.002 0.002 - 0.022 - 0.001 Nz ( g ) - 1.0 0.0 0.0 1.67 1.67 1.0 1.0 2.7 1.0 Pdot (rad/s /g) None 0.0 0.0014 0.0 0.0014 None None None None Qdot (rad/s /g) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Pb/2V (rad) None 0.0410 0.0 0.0410 0.0 None None None None Qc/2V (rad) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 - 1.44 0.40 0.40 9.74 9.74 13.91 6.02 4.95 9.07 °) Body f lap °) 11.89 - 2.86 - 2.86 3.14 3.14 8.00 - 9.87 - 13.20 22.52 Trailing - edge f lap °) - 9.87 Aileron #1 °) 19.07 48.63 19.07 48.63 19.07 48.63 19.07 48.63 Aileron #2 °) Mach n umber 2.00 0.48 0.48 0.48 0.48 0.3092 1.80 0.89 0.3092 Altitude (ft) 49770 SL SL SL SL SL 55000 20000 SL We ight c onfiguration ZFW DTOW DTOW DTOW DTOW DTOW DTOW ZFW DLW Gear c onfiguration Up Up Up Up Up Down Up Up Down American Institute of Aeronautics and Astronautics References Bhatia, K. G., and Wertheimer, J., “Aeroelastic Challenges for a High Speed Civil Transport,” AIAA -93-1478-CP, 1993.
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