X-Planes: Boeing X-66
Filed under: aerospace, history, engineering
This is the sixty-seventh article in the X-Planes series, following the X-1, the X-2, the X-3, the X-4, the X-5, the X-6, the X-7, the X-8, the X-9, the X-10, the X-11, the X-12, the X-13, the X-14, the X-15, the X-16, the X-17, the X-18, the X-19, the X-20, the X-21, the X-22, the X-23, the X-24, the X-25, the X-26, the X-27, the X-28, the X-29, the X-30, the X-31, the X-32, the X-33, the X-34, the X-35, the X-36, the X-37, the X-38, the X-39, the X-40, the X-41, the X-42, the X-43, the X-44, the X-45, the X-46, the X-47, the X-48, the X-49, the X-50, the X-51, the X-52, the X-53, the X-54, the X-55, the X-56, the X-57, the X-58, the X-59, the X-60, the X-61, the X-62, the X-63, the X-64, and the X-65.
The wing of this aeroplane folds at the span where one regulator’s aerodrome classification changes, to the inch, and the aerodynamic penalty for stopping there is under two percent.
The X-66A was allocated on 3 May 2023 to Boeing, with the engines cell reading 2 Pratt & Whitney and the sponsor cell reading the National Aeronautics and Space Administration, hereafter NASA [Parsch, DOD 4120.15-L Addendum]. The description is one line.
Transonic Truss-Braced Wing demonstrator, a highly modified MD-90 for Sustainable Flight Demonstrator project
That much is widely reported. What is not reported is that the wing the demonstrator exists to prove folds at 118 feet of span, and that 118 feet is exactly where Airplane Design Group III of the Federal Aviation Administration, hereafter FAA, ends. The International Civil Aviation Organization, hereafter ICAO, states the equivalent line in metres at 36, which is 118.1102 feet, and the FAA’s circular prints both while declaring that the customary value governs [FAA airport design] [SUGAR Phase II volume I]. The two renderings differ by 1.3228 inch and the fold station sits inside that rounding. Unfolded the span is 170 feet, which clears the next boundary up by 12 inch by the FAA’s reckoning and 7.24 inch by ICAO’s, margins the section headed The Gate Box derives. Both figures come from the final reports of Subsonic Ultra Green Aircraft Research, the research programme whose acronym is SUGAR and which this article cites repeatedly by that short form [SUGAR Phase IV final report] [SUGAR Phase II volume I].
An aeroplane wedged into a two-letter gap of an aerodrome reference code is not an aerodynamic object. It is an infrastructural one, and the question this article asks is what that costs.
The Research Question
A truss-braced wing is a device for buying span, and the question the X-66A exists to settle is whether the span is worth what the truss costs.
The physics of the trade is not in doubt. Induced drag falls as the inverse of aspect ratio. Wing bending material rises faster than that, because a longer wing carries the same lift on a longer arm and must be made deeper or thicker to resist the resulting moment. Somewhere the two curves cross and that crossing sets the span. A truss or a strut relieves the bending moment and therefore moves the crossing outward. That is the whole idea, and it is older than the aeroplane that was to test it [NASA truss-braced wing studies].
What is in doubt is four things. Where the crossing lies, under what constraints, and how sharply the penalty rises on either side of it. What a transonic truss costs in interference drag, in buffet margin and in flutter margin, none of which a 1920s braced biplane had to consider at Mach 0.8. Whether the thin wing a truss makes possible can carry the fuel, the high-lift devices and the anti-icing systems an airliner needs. And whether a weight model good enough to answer the first question exists, which the programme’s own final report says it does not [SUGAR Phase IV final report].
The questions this article asks are therefore these. What is the condition for a fuel-burn-optimal aspect ratio, and is this aeroplane on the right side of it? How much of the available benefit does the fold capture, and how much is left beyond it? What does the truss cost, measured against what it saves? And what does it mean that the programme stopped before the review that would have committed its wing to fabrication?
Three of the four answers run against the usual expectation. The optimality condition turns out to depend on which quantities are held fixed, and the two natural choices give two different exact conditions, one of which is a pure number independent of every aerodynamic and structural coefficient in the problem. The configuration sits below the stationary point under both conditions, and yet the penalty for sitting there is under two percent, because the optimum is extraordinarily flat and the flatness has an exact closed form. And the aspect-ratio exponent a geometrically similar truss delivers is the same three halves a cantilever delivers, so the truss buys a coefficient and not a power. The usual assumption is the opposite.
The Register Row, and a Designation Allocated Before Its Engine Was Chosen
Andreas Parsch’s addendum to DOD 4120.15-L, a compilation of the designations allocated after August 1998 that he publishes at designation-systems.net from Department of Defense sources, gives the row above and nothing else in the copy the compiler last updated on 23 August 2026 [Parsch, DOD 4120.15-L Addendum]. The compiler notes that the official description of an allocation made after October 2018 is not releasable, and he marks this row’s description as not official Department of Defense wording, so the description line is the compiler’s, while the date, manufacturer, engines and sponsor cells record the allocation. The engines cell reads 2 Pratt & Whitney with no model number, which among the rows around it is unusual, since the X-65A cell names a Williams FJ44-3A and the X-68A cell names a Williams WJ38-15. The copy of 23 August 2026 is later than this article’s date. The version current on that date, the compiler’s update of 11 August 2025 as the Internet Archive captured it on 10 December 2025, held 526 rows, twenty-nine of them X rows, and lacked fifteen later rows that include the X-68A and the X-76A. That version carried this row and the X-65A row with the cells quoted here and ended its X rows at the X-66A, so the X-68A row was not public on this article’s date and first appeared in an update the compiler made after it [Parsch, DOD 4120.15-L Addendum, 11 August 2025].
The absence is not carelessness. The row was allocated on 3 May 2023, before either the engine family or the engine had been announced. Pratt and Whitney geared turbofans were named for the demonstrator on 3 October 2023 [RTX selection release], and the engine itself, a derivative of the PW1500G designated PW102XG, was disclosed in June 2024 [FlightGlobal on the PW102XG]. So the register’s cell names a manufacturer five months before that manufacturer’s selection was announced, and omits a model that was not public for another thirteen months, and a reader who treats a register cell as a specification will mistake a dated snapshot for an omission.
The register’s engines are not the donor’s. The modification shortens the MD-90’s fuselage and replaces its wings and engines [NASA X-plane release], so the register is describing the demonstrator’s planned engines and not the donor’s, which is one of the places where the modification boundary is where the interesting engineering lives and where a statement about the X-66A is silently a statement about something else.
And this designation has no entry in the Directory of U.S. Military Rockets and Missiles. Its Appendix 4 of undesignated vehicles, in the index as it now stands, passes from the X-65 to the X-67, the X-68 and the X-76 with no X-66 entry [Directory Appendix 4 index] [Aurora X-65 CRANE directory entry]. That is correct behaviour rather than a gap, because the X-66A is a civil transport demonstrator sponsored by a civil agency and belongs to no missile directory.
Programme Origin, and an Agreement With No Pause In It
The narrative account is short. NASA established the Sustainable Flight Demonstrator project inside the Integrated Aviation Systems Program, solicited partners under announcement 80AFRC22N0008, the number in the file name under which NASA publishes the signed agreement [Funded Space Act Agreement], and selected Boeing in January 2023 [NASA award release]. On 12 June 2023 NASA and Boeing announced that the United States Air Force had designated the aircraft the X-66A [NASA X-plane release], an allocation the register dates to 3 May 2023 [Parsch, DOD 4120.15-L Addendum]. An MD-90 was ferried from Victorville to Palmdale in August 2023 to begin modification [Boeing arrival release].
The signed instrument tells it better, and the signed instrument is public. The Funded Space Act Agreement, numbered PAM 36785, runs to thirty-two articles and two appendices and was read in full for this article [Funded Space Act Agreement]. It is entered under 51 U.S.C. 20113(e), which is to say under the agency’s own authority to enter agreements rather than under procurement law, and that single fact governs everything that follows about how the programme is recorded and how it ended.
Boeing signed on 12 January 2023 and NASA signed the next morning. Article 19 makes the agreement effective on the date of the last signature and gives it a term of seven years from that date or completion, whichever comes first, so it expires on 13 January 2030. The last milestone, the Closeout Review, falls due August 2029, which leaves five months of margin against the term. A twenty-seven-milestone schedule ending five months before its instrument expires has very little room, and the agreement’s Article 7 adds that any milestone date is an estimate contingent on the availability of NASA facilities [Funded Space Act Agreement].
The Milestone Schedule as the Only Published Spending Profile
Appendix A.2 of the agreement lists 27 funded milestones with an acceptance criterion, a due date and a payment for each, and states that they total 425 million dollar [Funded Space Act Agreement]. They do, exactly.
| Milestone | Event | Due | Payment, million dollar |
|---|---|---|---|
| 1 | Kickoff | February 2023 | 7.50 |
| 2 | Airworthiness Approach | April 2023 | 12.50 |
| 3 | Concept Design Review (CoDR) | July 2023 | 14.00 |
| 4 | System Requirements Review (SRR) | October 2023 | 17.75 |
| 5 | Aircraft Reactivation | February 2024 | 17.75 |
| 6 | Fuselage / Mod Preliminary Design Review (PDR) | May 2024 | 17.75 |
| 7 | Wing / Strut PDR | September 2024 | 18.75 |
| 8 | Preliminary Design Review (PDR) | November 2024 | 23.50 |
| 9 | Begin Aircraft Systems Bench (ASB) Simulator Design & Build | February 2025 | 23.50 |
| 10 | Wing / Strut Critical Design Review (CDR) | May 2025 | 25.00 |
| 11 | Fuselage / Mod CDR | August 2025 | 25.00 |
| 12 | Propulsion CDR | November 2025 | 25.75 |
| 13 | Critical Design Review (CDR) | February 2026 | 27.25 |
| 14 | Mod Manufacturing Readiness Review (MRR) | June 2026 | 27.25 |
| 15 | 90% Engineering Drawings Complete | August 2026 | 25.75 |
| 16 | System Integration Review (SIR) | October 2026 | 24.00 |
| 17 | MD-90 Fuselage High Blow Test | February 2027 | 24.00 |
| 18 | Final Aircraft Structural Assembly Complete | April 2027 | 24.00 |
| 19 | Final Ground Test Plan | August 2027 | 24.00 |
| 20 | Aircraft Power On | October 2027 | 3.75 |
| 21 | Aircraft Build Gauntlet Test | March 2028 | 3.75 |
| 22 | First Flight Readiness Review (FFRR) | June 2028 | 3.75 |
| 23 | Flight Line Gauntlet Test | August 2028 | 3.75 |
| 24 | First Flight | September 2028 | 1.50 |
| 25 | Envelope Expansion Testing Complete | February 2029 | 1.50 |
| 26 | Flight Test Complete | June 2029 | 1.50 |
| 27 | Closeout Review | August 2029 | 0.50 |
Ninety-eight point eight percent of NASA’s money is paid before the aeroplane leaves the ground, a fact about the schedule and therefore an arithmetic one.
\[\frac{\displaystyle\sum_{i < i_{\mathrm{ff}}} \mu_i}{\displaystyle\sum_{i} \mu_i} = 0.988235 \tag{1}\]Milestone 24 is first flight, due September 2028, and it pays 1.5 million dollar [Funded Space Act Agreement]. Everything before it comes to 420 million dollar, which is 98.824 percent of the agreement. The largest single milestone is 27.25 million dollar and is therefore 18.17 times the payment for flying the aircraft, and the smallest of all is the 0.50 million dollar closeout review.
That profile is not a criticism. It is the key to reading what happened later. A funded agreement of this kind buys design reviews and hardware, and the reviews are where the money is because the reviews are where the risk is retired. But it means that a programme which stops after its design reviews and before its flight test has already drawn almost all of the public money it was ever going to draw, and has delivered almost everything the agreement obliged it to deliver.
The Word the Agreement Does Not Contain
On 24 April 2025 NASA reported that it and Boeing were evaluating an updated approach, proposed by Boeing, centred on a ground-based testbed for long, thin-wing technology. The item is headed as a new research focus rather than a cancellation, and says that NASA and Boeing would continue to collaborate on the truss-braced wing concept, that all of the demonstrator’s design and all hardware acquired or modified for it would be retained, and that the flight demonstrator work would pause for later consideration based on the thin-wing testbed results [NASA thin-wing item].
The agreement has no pause state. The word appears zero times in its seven-year instrument. What Article 20 provides instead is termination by mutual consent, termination by NASA thirty days after written notice that the partner has failed to meet a scheduled milestone, and unilateral termination by NASA on four grounds including a written determination by the Administrator that the project no longer aligns with the agency’s strategic objectives. A missed milestone is, in the instrument’s own terms, a ground for termination rather than an occasion for a pause, and Article 20.B.2 adds that a partner failing a milestone is entitled to no payment for it and to nothing for termination-related expenses, while retaining everything already paid [Funded Space Act Agreement].
The pause landed in the narrowest possible gap in that schedule. Milestone nine fell due in February 2025 and the next milestone was number 10, the Wing / Strut Critical Design Review, due May 2025 with a payment of 25 million dollar [Funded Space Act Agreement]. Through the February milestone the agreement had reached 153 million dollar, or 36.00 percent of its total.
\[\frac{\displaystyle\sum_{i \le 9} \mu_i}{\displaystyle\sum_{i} \mu_i} = \frac{153.00}{425.00} = 0.3600 \tag{2}\]So the programme stopped immediately before the review that would have frozen the wing and strut for fabrication, and the thing it announced it would keep doing is wing research. That is no coincidence, and it happened in the open. It is what a partner does when it has learned what it wanted from a design phase and does not want the build.
What the Federal Award Record Knows About a 425 Million Dollar Aeroplane
For the X-65A the federal award record tells the story better than the press does, naming three competitors, pricing them and dating a downselect that no narrative account gives [X-65 CRANE]. For the X-66A the same record is almost empty, and the emptiness is the finding.
Searching the award record for the project’s name returns one contract [USAspending].
| Instrument | Recipient | Obligated, dollar | Description |
|---|---|---|---|
| 80NSSC24PA630 | Pacmin Inc | 41,198.00 | X-66 Sustainable Flight Demonstrator Desktop And Floor Models |
The agreement is 10,316 times larger than the only thing the federal award record holds under the project’s name, and that only thing is a set of desk models of the aeroplane.
The absence reflects no failure of the record and no concealment. A Funded Space Act Agreement is not a procurement contract. It is an agreement under the agency’s enabling statute, it incorporates no Federal Acquisition Regulation clauses and names that regulation once, inside a definition of domestic manufacture, Article 6.B.4 states in terms that the partner’s financial records are not subject to examination or audit by NASA, and Article 6.B.5 gives the Comptroller General access for three years after the final milestone payment [Funded Space Act Agreement]. The instrument is therefore reported where agreements are reported and not where contracts are reported. The consequence for a reader is nonetheless sharp. The spending profile of a 425 million dollar aeroplane exists in public only in the appendix of the agreement reproduced above, and nowhere in the award record at all.
The Research That Led To It Is All There
Asking the record about the technology instead of the project returns nine contracts totalling 21,420,664.42 dollar across fourteen years [USAspending].
| Start | Instrument | Recipient | Obligated, dollar |
|---|---|---|---|
| 2011-09-21 | NNL11AC32T | THE BOEING COMPANY | 1,004,742.81 |
| 2014-05-09 | NNL14AB07T | THE BOEING COMPANY | 74,880.03 |
| 2014-07-22 | NNL14AB51T | THE BOEING COMPANY | 4,996,629.59 |
| 2016-12-22 | NNL17AA46T | THE BOEING COMPANY | 11,993,610.00 |
| 2018-02-16 | 80LARC18F0049 | NATIONAL INSTITUTE OF AEROSPACE ASSOCIATES | 952,226.99 |
| 2019-08-14 | 80NSSC19C0086 | ZONA TECHNOLOGY INC | 749,974.00 |
| 2021-04-13 | 80LARC21F0101 | NATIONAL INSTITUTE OF AEROSPACE ASSOCIATES | 615,042.00 |
| 2023-04-20 | 80LARC23FA019 | BOEING AEROSPACE OPERATIONS, INC. | 972,613.00 |
| 2025-07-24 | 80LARC25FA035 | ADVANCED TECHNOLOGIES, INC. | 60,946.00 |
Three of those merit naming. NNL17AA46T, at 11,993,610.00 dollar from December 2016, is the contract printed on the cover of every page of the Phase IV final report, which identifies itself as performed under NASA contracts NNL16AA04B and NNL17AA46T [SUGAR Phase IV final report]. So the report this article leans on hardest can be priced, which is unusual. 80LARC21F0101, at 615,042.00 dollar from April 2021, is a dedicated task for truss-braced wing structural weight estimation. Structural weight is what the keystone below turns on, and the Phase IV report names it as the concept’s largest remaining uncertainty. And a contract let in July 2025, after the pause, shows that the work did not stop [USAspending].
The ratio is the point. All the public research contracts for this technology together come to 21.42 million dollar, and the agreement to build the demonstrator is 19.84 times that. Fourteen years of visible study and one invisible aeroplane.
What Three Budget Books Said
The programme’s money appears in NASA’s Congressional Justifications only at the programme level. The Integrated Aviation Systems Program table breaks out the Low Boom Flight Demonstrator and the Electrified Powertrain Flight Demonstration by name and folds everything else, the Sustainable Flight Demonstrator included, into a line called Other Projects [FY2024 justification] [FY2025 justification]. So none of the three budget documents read for this article states what this project costs per year. Read against one another, the three successive books show what happened to the money.
| Fiscal year | FY2024 justification | FY2025 justification | FY2026 supplement |
|---|---|---|---|
| 2024 | 264.9 | – | – |
| 2025 | 260.5 | 264.4 | – |
| 2026 | 263.5 | 277.0 | 167.2 |
| 2027 | 279.7 | 277.6 | 161.3 |
| 2028 | 305.5 | 300.9 | 115.4 |
| 2029 | – | 342.0 | 110.0 |
| 2030 | – | – | 70.0 |
A revision between two budget books is a fractional change in a projection of the same year.
\[\text{revision} = \frac{\mathcal{P}_{\mathrm{new}} - \mathcal{P}_{\mathrm{old}}}{\mathcal{P}_{\mathrm{old}}} \tag{3}\]The same programme line, projected one year apart, falls by 67.8 percent in fiscal year 2029. The FY2025 justification put fiscal year 2029 at 342.0 million dollar and the FY2026 technical supplement puts it at 110.0 million dollar. The Other Projects line within it, the line that holds this project, goes from 167.6 to 100.0 million dollar for fiscal year 2026 [FY2025 justification]. The programme itself is requested at 167.2 million dollar for fiscal year 2026. The whole aeronautics account falls from 935.0 to 588.7 million dollar, a reduction of 37.0 percent [FY2026 technical supplement].
Three Things the FY2026 Supplement Says Plainly
It applies the word pause to the project rather than only to the aircraft. Under Explanation of Major Changes it states that the project will pause development of the flight demonstrator aircraft in cooperation with Boeing in order to pursue a ground-based, full-scale wing demonstration of integrated thin-wing technologies [FY2026 technical supplement]. The agency’s news item of 24 April 2025 uses the same word for the same object, saying the flight demonstrator work would pause for later consideration based on the thin-wing testbed results, while its opening calls the change an updated approach. So the agency used both words on the same day for different objects, and the budget document says so without a headline over it.
It keeps the qualifier every agency statement of the figure carries, and it returns to the first baseline. The January 2023 award release said the configuration could cut fuel consumption and emissions by up to thirty percent relative to today’s most efficient single-aisle aircraft when combined with other advancements in propulsion systems, materials, and systems architecture [NASA award release]. The June 2023 designation release carries the same clause and changes the baseline, saying up to thirty percent less fuel consumption and reduced emissions when compared with today’s best-in-class aircraft, in the text as archived on 30 September 2023 and as published now [NASA X-plane release] [archived X-plane release]. The FY2026 supplement reads when combined with additional technologies, this configuration will reduce fuel consumption up to 30 percent relative to today’s most efficient single-aisle aircraft, which returns to the single-aisle baseline [FY2026 technical supplement]. The number never changed and neither did the clause that assigns it to the configuration combined with other technologies. What moved is the baseline, and the section below computes what the wing alone is worth so that a reader can see which baseline the evidence supports.
And it renames the project. The section heading in the FY2026 supplement reads Subsonic Flight Demonstrator, the acronym list reads SFD Subsonic Flight Demonstrator, and a third line in the document repeats it, where every earlier document from the signed agreement onward says Sustainable [FY2026 technical supplement]. Three occurrences in one document is not a typographical slip in one heading. No NASA release announces a renaming, so whether it is deliberate is not settled by the record, and this article records what the document says and does not infer why.
The Numeric Goal Is Absent From the Instrument That Funds It
The signed agreement states no percentage at all. Article 1 says the project will dramatically reduce aircraft fuel burn and carbon dioxide emissions and that the partnership targets significant improvements in fuel burn and associated CO2 emissions [Funded Space Act Agreement]. The only percentages anywhere in its thirty-two articles and two appendices are two fifty percent tests, one of domestic manufacture and one of United States ownership, an illustrative sixty-six point six percent discount in the clause on purchasing property after termination, and the ninety percent engineering-drawing completion that defines Milestone 15 [Funded Space Act Agreement].
So the figure every account of this programme leads with appears nowhere in the document that obliges anyone to deliver anything. That is not a contradiction, because a binding instrument sensibly declines to bind a partner to a performance number it cannot control. It does mean that the thirty percent is a programme objective rather than a contractual one, and that a number somebody computed will be found in the contractor report and in neither the agreement nor the press release.
One more disagreement between two NASA documents matters because it concerns the premise rather than the goal. The agreement says single-aisle airliners account for nearly half of worldwide aviation emissions [Funded Space Act Agreement]. The FY2025 Congressional Justification has them accounting for greater than 50 percent of worldwide aviation emissions [FY2025 justification]. Nearly half and more than half are different claims about the same quantity, written fifteen months apart by the same agency, and neither cites a source.
Sizing From First Principles
The Symbols This Article Uses
Every symbol below is declared here and nowhere carries a second meaning.
| Symbol | Meaning | Unit |
|---|---|---|
| $A$ | wing aspect ratio | dimensionless |
| $A^{*}$ | aspect ratio at which fuel burn is stationary | dimensionless |
| $\mathcal{A}_c$ | cross-sectional area of one bending cap | square inch |
| $a$ | speed of sound at the cruise condition | metre per second |
| $\beta_i$ | block fuel per seat of configuration $i$ | pound |
| $\mathcal{B}$ | bending-material integral, braced or unbraced | dimensionless |
| $b$ | wing span, tip to tip | foot |
| $b_f$ | folded wing span | foot |
| $b_{\mathrm{bound}}$ | a span bound of an aerodrome reference code, in feet | foot |
| $C_D$ | total drag coefficient at cruise | dimensionless |
| $C_{D,0}$ | parasite drag coefficient, the incompressible zero-lift part | dimensionless |
| $C_{D,c}$ | compressibility drag coefficient, which this report’s convention also carries the laminar-flow credit | dimensionless |
| $C_{D,i}$ | induced drag coefficient | dimensionless |
| $C_{D,t}$ | trim drag coefficient | dimensionless |
| $C_L$ | cruise lift coefficient | dimensionless |
| $c$ | local streamwise chord | inch |
| $c_r$ | root chord | inch |
| $c_T$ | thrust specific fuel consumption | per hour |
| $\delta$ | induced-drag fraction of total cruise drag | dimensionless |
| $\Delta$ | span margin against a reference-code bound | inch |
| $E$ | Young’s modulus of the cap material in the fibre direction | pound per square inch |
| $e$ | airplane efficiency factor for drag due to lift | dimensionless |
| $\varepsilon$ | fractional error in aspect ratio, measured logarithmically | dimensionless |
| $\Gamma$ | main strut dihedral angle | degree |
| $\gamma$ | ratio of specific heats, taken as 1.4 for dry air | dimensionless |
| $\mathcal{F}$ | equivalent flat plate area | square foot |
| $g_0$ | standard gravity, 9.80665 | metre per second squared |
| $h$ | geopotential altitude | metre |
| $h_g$ | height of the wing above the ground | foot |
| $h^{\ast}$ | geopotential altitude at which the drag buildup’s lift coefficient holds | metre |
| $h_c$ | separation between the centroids of the upper and lower caps | inch |
| $h_{\mathrm{t}}$ | tropopause altitude, 11,000 | metre |
| $\mathcal{I}$ | second moment of area of the strut section | inch to the fourth |
| $\mathcal{I}(\lambda)$ | taper integral of the moment shape against the chord | dimensionless |
| $K$ | effective-length factor in the Euler buckling load | dimensionless |
| $\kappa_A$ | Korn airfoil-technology constant | dimensionless |
| $\kappa_h$ | ratio of cap-centroid separation to maximum section depth | dimensionless |
| $k_{\mathrm{no}}$ | non-optimum factor, published bending material over the model’s | dimensionless |
| $k_w$ | coefficient of the aspect-ratio-sensitive weight group | pound |
| $L$ | total lift carried by the wing at the design load factor | pound |
| $\ell$ | local lift per unit span | pound per inch |
| $\Lambda$ | wing quarter-chord sweep angle | degree |
| $\lambda$ | wing taper ratio, tip chord over root chord | dimensionless |
| $\lambda_a$ | troposphere lapse rate, 0.0065 | kelvin per metre |
| $\mathcal{L}_s$ | unsupported length of the strut in buckling | inch |
| $M$ | flight Mach number | dimensionless |
| $M_{dd}$ | drag-divergence Mach number | dimensionless |
| $\mathcal{M}$ | wing bending moment about the structural axis | pound inch |
| $m_f$ | fuel burned over the cruise leg | pound |
| $\mu_i$ | payment attached to milestone $i$ | million dollar |
| $\mathcal{N}$ | design load factor | dimensionless |
| $n$ | exponent by which the sensitive weight group grows with aspect ratio | dimensionless |
| $\nu$ | weight elasticity, the logarithmic derivative of aircraft weight with respect to aspect ratio | dimensionless |
| $\omega$ | aspect-ratio-sensitive weight as a fraction of aircraft weight | dimensionless |
| $P$ | vertical relief force the brace delivers at its attachment | pound |
| $P_{\mathrm{cr}}$ | Euler critical buckling load of the strut | pound |
| $\mathcal{P}$ | a budget projection for one fiscal year | million dollar |
| $p$ | ambient static pressure | pascal |
| $p^{\ast}$ | ambient pressure at which the drag buildup’s lift coefficient holds | pascal |
| $p_{\mathrm{sl}}$ | sea-level standard pressure, 101325 | pascal |
| $\Phi$ | Breguet correction factor, $Xe^{-X}/(1-e^{-X})$ | dimensionless |
| $q$ | free-stream dynamic pressure | pound per square foot |
| $R$ | cruise range | nautical mile |
| $R_a$ | specific gas constant for air, 287.05287 | joule per kilogram kelvin |
| $\mathcal{R}$ | the set of research records in the reference survey | dimensionless |
| $r_i$ | fuel-burn reduction of configuration $i$ against the baseline | dimensionless |
| $\rho$ | ambient air density | kilogram per cubic metre |
| $\rho_m$ | density of the cap material | pound per cubic inch |
| $S$ | wing reference area | square foot |
| $s$ | secant of the wing sweep angle | dimensionless |
| $\Sigma$ | ground-effect group, span over twice the height | dimensionless |
| $\sigma$ | allowable stress in the cap material | pound per square inch |
| $T$ | ambient static temperature | kelvin |
| $T_s$ | axial force in the main strut | pound |
| $T_{\mathrm{sl}}$ | sea-level standard temperature, 288.15 | kelvin |
| $\tau$ | wing thickness-to-chord ratio | dimensionless |
| $u$ | span station as a fraction of semispan, zero at the root and one at the tip | dimensionless |
| $V$ | true airspeed at cruise | knot |
| $\mathcal{V}$ | fuel volume | United States gallon |
| $v$ | dummy variable of integration over the span station | dimensionless |
| $W$ | aircraft weight | pound |
| $W_0$ | the part of aircraft weight that does not change with aspect ratio | pound |
| $W_b$ | bending material weight of the wing | pound |
| $W_w$ | the aspect-ratio-sensitive weight group | pound |
| $w$ | beam deflection at the brace attachment | inch |
| $X$ | Breguet exponent, $-\ln(1-\zeta)$ | dimensionless |
| $y$ | spanwise coordinate measured from the centreline | inch |
| $\zeta$ | fuel fraction, fuel burned over aircraft weight | dimensionless |
| $\eta$ | station of the brace attachment as a fraction of semispan | dimensionless |
The Geometry, Recomputed Rather Than Copied
The Phase IV final report gives the Mach 0.80 configuration, designated 1104-001-RG, a span of 170 feet, a projected reference area of 1477.109 square feet, a taper ratio of 0.346, a quarter-chord sweep of 22.47 degrees and an average thickness-to-chord ratio of 0.104 [SUGAR Phase IV final report]. It also states an aspect ratio of 19.565, which is not an independent datum.
\[A = \frac{b^2}{S} \tag{4}\]Substituting gives 19.5652, which agrees with the stated 19.565 to the last digit the report prints. The check is trivial and is run because the rest of this section leans on all three numbers being mutually consistent.
An Inconsistent Cruise Condition
The drag buildup names an altitude, a Mach number and a lift coefficient, and writing down the lift equation shows that the three do not hold together at the aeroplane’s weight.
The standard atmosphere is needed first. Let $h$ be geopotential altitude in metre, $T_{\mathrm{sl}}$ and $p_{\mathrm{sl}}$ the sea-level standard temperature and pressure, $\lambda_a$ the troposphere lapse rate, $h_{\mathrm{t}}$ the tropopause altitude, and $R_a$ the specific gas constant for air.
\[T(h) = \begin{cases} T_{\mathrm{sl}} - \lambda_a h, & h \le h_{\mathrm{t}} \\[4pt] T_{\mathrm{sl}} - \lambda_a h_{\mathrm{t}}, & h > h_{\mathrm{t}} \end{cases} \tag{5}\] \[p(h) = \begin{cases} p_{\mathrm{sl}} \left(\dfrac{T(h)}{T_{\mathrm{sl}}}\right)^{g_0/(R_a \lambda_a)}, & h \le h_{\mathrm{t}} \\[6pt] p(h_{\mathrm{t}}) \exp\left(-\dfrac{g_0 \left(h - h_{\mathrm{t}}\right)}{R_a T(h_{\mathrm{t}})}\right), & h > h_{\mathrm{t}} \end{cases} \tag{6}\]The stratospheric branch is exponential and not a power law, because the lapse rate is zero there and the power-law exponent diverges. The speed of sound and the dynamic pressure follow.
\[a = \sqrt{\gamma R_a T}, \qquad V = M a \tag{7}\] \[q = \tfrac{1}{2}\rho V^2 = \tfrac{1}{2}\gamma p M^2 \tag{8}\]The second form of the dynamic pressure is used throughout, because it needs only pressure and Mach number and never forms a density or a velocity, which removes two opportunities for a unit error. The lift equation then fixes the lift coefficient.
\[W = C_L \, q \, S \qquad \Longrightarrow \qquad C_L = \frac{W}{q S} \tag{9}\]At the buildup’s stated altitude of 40,000 feet and Mach 0.80 the three disagree. The pressure is 18,753.9 pascal, the speed of sound 295.07 metre per second, the true airspeed 458.9 knot, and the dynamic pressure 175.474 pound per square foot. At the maximum take-off weight of 145,000 pound over 1477.109 square feet that gives
\[C_L = \frac{W}{qS} = 0.5594 \tag{10}\]against the 0.695 the same table states [SUGAR Phase IV final report]. The three entries are therefore not a flight condition. Either the lift coefficient belongs to a different altitude, or it belongs to a weight 24.2 percent above the maximum take-off weight.
Inverting for the altitude at which they do hold settles which. At maximum take-off weight and Mach 0.80 the lift equation fixes the pressure, and since that pressure lies below the tropopause value of 22,632 pascal, the stratospheric branch of the atmosphere returns the altitude $h^{\ast}$.
\[p^{\ast} = \frac{2W}{\gamma M^2 S C_L} = 315.28 \ \text{lbf/ft}^2 = 15095.6 \ \text{Pa}, \qquad h^{\ast} = h_{\mathrm{t}} + \frac{R_a T(h_{\mathrm{t}})}{g_0}\ln\frac{p(h_{\mathrm{t}})}{p^{\ast}} = 13568.1 \ \text{m} \tag{11}\]That is 44,515 feet, and the report’s own optimum cruise altitude at maximum take-off weight, from a different table in the same document, is 44,437 feet [SUGAR Phase IV final report]. The two differ by 78 feet, or 0.175 percent.
So the lift coefficient in the drag buildup is the lift coefficient at the optimum altitude, and the altitude printed beside it is not. That is a bookkeeping entry rather than an error of substance, since an aerodynamic buildup is properly a function of Mach number and lift coefficient with altitude entering only through Reynolds number. It is recorded because this article uses that lift coefficient in every subsequent calculation, and a reader is entitled to know which flight state it describes.
The Induced Drag, and a Textbook Relation That Closes on the Report’s Own Buildup
The drag at cruise is bookkept in the report in four parts, being parasite drag, induced drag, compressibility drag and trim drag, with the laminar-flow credit carried inside the compressibility term by a convention the report states explicitly and which makes that term negative.
\[C_D = C_{D,0} + C_{D,i} + C_{D,c} + C_{D,t} \tag{12}\]The published values at Mach 0.80 and a lift coefficient of 0.695 are $C_{D,0}$ of 0.01880, $C_{D,i}$ of 0.00903, $C_{D,c}$ of -0.00143 and $C_{D,t}$ of 0.00054, totalling 0.02695 [SUGAR Phase IV final report]. Summing them gives 0.02694, which differs from the printed total only in the last digit, as rounding of four printed terms allows, and the lift-to-drag ratio is the quotient of the lift coefficient and the printed total.
\[\frac{L}{D} = \frac{C_L}{C_D} \tag{13}\]That gives 25.788, the value the report prints.
The parasite term can be checked too, because the report publishes the equivalent flat plate area it was built from. An equivalent flat plate area $\mathcal{F}$ is the area that would produce the same drag at unit drag coefficient, so dividing it by the reference area returns the coefficient.
\[C_{D,0} = \frac{\mathcal{F}}{S} \tag{14}\]The buildup’s flat plate areas total 27.7722 square feet over a reference area of 1477.109, which gives 0.0188017 against the published 0.01880, an error of 0.0092 percent. That is a second independent closure on the same table, and it also prices the truss in drag, since the strut and jury together contribute 3.4500 square feet, amounting to 12.422 percent of the parasite drag area, and a drag coefficient of 0.0023356 on its own [SUGAR Phase IV final report].
This article needs the induced term, and the report also gives an airplane efficiency factor of 0.8707 [SUGAR Phase IV final report], which allows the induced term to be recomputed rather than accepted.
\[C_{D,i} = \frac{C_L^2}{\pi A e} \tag{15}\]That gives 0.009026 against the published 0.00903, an error of 0.050 percent. A textbook relation reproducing a contractor’s computational-fluid-dynamics-anchored buildup to five decimal places matters more than it appears, because it licenses every manipulation of the induced term that follows. The quantity the keystone turns on is its share of the total.
\[\delta = \frac{C_{D,i}}{C_D} \tag{16}\]For this aeroplane at its design cruise point, $\delta$ is 0.3351, so 33.51 percent of cruise drag is the price of holding the aeroplane up. That number is measured from the report and not assumed. It is one of the two quantities every optimality condition below is written in, the other being the weight elasticity.
Where the Three Halves Comes From
The weight side of the trade needs an exponent, and an exponent asserted is an exponent nobody can check. It is derived here from the lift distribution outward.
Take an elliptic spanload, the distribution that minimises induced drag for a given span and lift and is therefore the right idealisation for a wing designed to exploit span. Let $y$ be the spanwise coordinate in inch, $b$ the span, $L$ the total lift in pound at the design load factor, and $\ell$ the local lift per unit span.
The design lift itself is the load factor times the weight, and that product supplies the 2.5 used below.
\[L = \mathcal{N} W \tag{17}\] \[\ell(y) = \frac{4L}{\pi b} \sqrt{1 - \left(\frac{2y}{b}\right)^2} \tag{18}\]The constant in front is not a choice. It is fixed by requiring the load to integrate to the lift.
\[\int_{-b/2}^{b/2} \ell(y)\,dy = \frac{4L}{\pi b}\cdot\frac{b}{2}\int_{-1}^{1}\sqrt{1-u^2}\,du = \frac{2L}{\pi}\cdot\frac{\pi}{2} = L \tag{19}\]Write $u = 2y/b$ for the station as a fraction of semispan. The bending moment at a station is the integral of the load outboard of it against its moment arm.
\[\mathcal{M}(u) = \int_{u}^{1} \ell \left(\frac{b}{2}\right)^2 (v - u) \, dv \tag{20}\]Both integrals in that expression are elementary, and the result is a closed form.
\[\mathcal{M}(u) = \frac{L b}{\pi} \left[ \frac{1}{3}\left(1-u^2\right)^{3/2} - \frac{u}{2}\left( \frac{\pi}{2} - u\sqrt{1-u^2} - \arcsin u \right) \right] \tag{21}\]At the root this collapses.
\[\mathcal{M}(0) = \frac{L b}{3\pi} \tag{22}\]At the design load factor of 2.5 and the published take-off weight, that root moment is 78.46 million pound inch.
That is the classical result, and an elementary check gives one of two independent confirmations of the closed form above. The centroid of a semi-ellipse lies at $4(b/2)/(3\pi)$ from the root, and half the lift acting at that arm gives the same moment.
\[\mathcal{M}(0) = \frac{L}{2}\cdot\frac{4}{3\pi}\cdot\frac{b}{2} = \frac{Lb}{3\pi} \tag{23}\]Numerical quadrature of the moment integral at the root and at five interior stations agrees with the closed form to better than one part in a million, the second of the two confirmations.
Now the structure. A wing box resists bending with material concentrated top and bottom at a separation $h_c$, and if the allowable stress is $\sigma$ then each cap needs a cross-sectional area of $\mathcal{M}/(\sigma h_c)$.
\[\mathcal{A}_c(u) = \frac{\mathcal{M}(u)}{\sigma h_c(u)}, \qquad h_c(u) = \kappa_h \, \tau \, c(u) \tag{24}\]Here $\tau$ is the thickness-to-chord ratio and $\kappa_h$ is the ratio of cap-centroid separation to maximum section depth, the one shape factor this model cannot derive, taken here as 0.90. For a straight-taper wing the chord and the root chord follow from the area and the taper ratio.
\[c(u) = c_r\left[1 - (1-\lambda)u\right], \qquad c_r = \frac{2S}{b\,(1+\lambda)} \tag{25}\]The bending material is two caps at every station integrated across the whole span, which by symmetry is four times the semispan integral of one cap.
\[W_b = 4\rho_m \left(\frac{b}{2}\right)\int_0^1 \mathcal{A}_c(u)\,du \tag{26}\]Substituting the moment, the depth and the chord, and collecting the purely geometric part into a single integral, gives the result this article needs. Let $f(u)$ denote the bracketed shape function in the moment expression above.
\[\mathcal{I}(\lambda) = \int_0^1 \frac{f(u)}{1 - (1-\lambda)u}\,du \tag{27}\] \[W_b = \frac{\rho_m \, b^3 \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau \, S} \tag{28}\]Finally replace the span by the aspect ratio, using $b = \sqrt{AS}$.
\[W_b = \frac{\rho_m \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau} \; A^{3/2} \, S^{1/2} \tag{29}\]At fixed area, fixed design lift and fixed thickness ratio the bending material scales exactly as the three-halves power of aspect ratio. The exponent is not an empirical fit. It is the product of one power of span from the moment arm, one from the length of the beam, and a half from the chord shrinking as span grows at fixed area. The same statement can be measured rather than read, by differencing the closed form logarithmically between two aspect ratios.
\[n = \frac{\ln\left[W_b(A_2)/W_b(A_1)\right]}{\ln\left(A_2/A_1\right)} \tag{30}\]Between aspect ratios of ten and thirty that returns 1.5000, and any other pair returns the same value, since the closed form is an exact power law in aspect ratio.
And the thickness ratio enters as an inverse first power. Halving the thickness of a wing doubles the material it needs to resist the same moment, and that fact makes a thin wing expensive and a truss necessary.
The Absolute Weight the Model Gets Wrong
The report’s group weight statement is unusually generous. It does not merely give a wing weight. It breaks the wing group of 16,095 pound into bending material, spar webs, ribs and bulkheads, aerodynamic surfaces and secondary structure, and gives the bending material as 7,488 pound, or 46.5 percent of a group that is itself 11.10 percent of a take-off weight of 145,000 pound against an operating empty weight of 85,172 [SUGAR Phase IV final report]. So the model above can be tested against the one number it predicts.
Sweep enters before the model can be evaluated, and it enters twice. A swept wing’s structural beam is longer than its projected semispan by the secant of the sweep angle, and the moment arm from the load to that beam grows by the same factor, so the bending material carries the square.
\[\frac{W_b(\Lambda)}{W_b(0)} = \frac{1}{\cos^2 \Lambda} \tag{31}\]Evaluating the model with the report’s allowable stress of 90,000 pound per square inch, its material density of 0.067 pound per cubic inch, a load factor of 2.5, the published geometry and that correction gives 6,807 pound, 0.909 times the published figure.
That agreement is a coincidence and reporting it as a validation would be a mistake. The model just evaluated is a cantilever, with no truss at all, and a cantilever must be heavier than the braced wing it is being compared with, not lighter. The agreement is the product of two large errors pointing in opposite directions. The model is far too optimistic about how much of a real wing box is sized by the design bending moment, and it omits the brace entirely. The section on what the truss buys puts a number on the first of those. In short, this model gets the absolute weight wrong and the exponent right, and the keystone needs only the exponent.
A constant factor, however large, does not touch a logarithmic derivative. That is why the rest of this article uses the three halves and treats the absolute scale as the contractor’s to supply.
The Optimality Condition Depends On What Is Held Fixed
This is the keystone, and the first thing to say about it is that there is not one condition but two, because “the optimal aspect ratio” is not a property of an aeroplane. It is a property of an aeroplane together with a statement of what is being held fixed while the aspect ratio moves, and the two natural statements give two different exact answers. A great deal of loose argument about high-aspect-ratio wings comes from not saying which.
That is not this article’s observation alone. A NASA Langley paper deriving a metric for aero-structural efficiency, the same trade approached with a different instrument, states the same dependence [Guynn 2015].
The nature of the trade-off actually varies depending on the overall objective of the design. In other words, the L/D and wing weight trades will be different if the designer’s focus is aircraft weight, cost, or fuel consumption.
Its metric is defined as the lift-to-drag ratio that would return the same fuel consumption if the wing weighed nothing, and for a 737-class aeroplane it comes out eleven to twelve percent below the aerodynamic lift-to-drag ratio. This article takes the derivative route instead and arrives at the same place, that a span trade has no answer until an objective is named.
Start from the fuel. Over a cruise leg of range $R$ at true airspeed $V$ with thrust specific fuel consumption $c_T$, the Breguet relation gives the fuel burned, and for a fuel fraction small enough that the exponential may be linearised it reduces to a product.
\[m_f = W\left[1 - \exp\left(-\frac{R\,c_T}{V}\frac{C_D}{C_L}\right)\right] \approx \frac{R\,c_T}{V}\,\frac{C_D}{C_L}\,W \tag{32}\]Case One, Wing Area and Cruise Condition Held
Hold the wing area and the cruise altitude and Mach number, so the dynamic pressure $q$ is fixed, and let the span grow. Then the cruise lift coefficient is not a free choice, because it is whatever the weight requires.
\[C_L = \frac{W}{q S} \tag{33}\]Substituting that into the fuel expression produces a cancellation.
\[m_f \propto \frac{C_D}{C_L}\,W = C_D \, q S \tag{34}\]The weight has vanished from the front of the expression and reappears inside the drag, because the induced term is quadratic in lift coefficient and therefore quadratic in weight.
\[m_f \propto q S C_{D,0} + \frac{W^2}{q S \pi A e} \tag{35}\]Only the second term depends on the aspect ratio, through the explicit $A$ in the denominator and through the weight growing with $A$. Taking the logarithmic derivative, with $\nu$ denoting the elasticity of aircraft weight with respect to aspect ratio, gives the result.
\[\nu = \frac{d \ln W}{d \ln A}, \qquad \frac{d \ln m_f}{d \ln A} = \delta\left(2\nu - 1\right) \tag{36}\] \[\nu^{*} = \tfrac{1}{2} \tag{37}\]At fixed wing area and fixed cruise condition, the fuel-burn-optimal aspect ratio is reached when a one percent increase in aspect ratio costs exactly half a percent in aircraft weight. It does not depend on the span efficiency, on the parasite drag, on the altitude, on the Mach number, on the range, on the specific fuel consumption, or on the absolute weight of anything. The induced-drag fraction $\delta$ appears in the slope and cancels out of the condition.
The mechanism is easy to state once seen. At fixed area and fixed cruise condition the induced drag is proportional to $W^2/A$, and a stationary point of that is a stationary point of its logarithm.
\[\frac{d}{d\ln A}\ln\left(\frac{W^2}{A}\right) = 2\nu - 1 = 0 \tag{38}\]The derivation ends there.
Case Two, Cruise Lift Coefficient Held
Now hold the cruise lift coefficient instead, as a conceptual-design sizing loop does, because the area is chosen to put the aeroplane at a good lift coefficient and is therefore not an independent variable. Then
\[m_f \propto \frac{W}{C_L}\left(C_{D,0} + \frac{C_L^2}{\pi A e}\right) \tag{39}\]and the logarithmic derivative is simpler still.
\[\frac{d \ln m_f}{d \ln A} = \nu - \delta \tag{40}\] \[\nu^{*} = \delta \tag{41}\]At fixed cruise lift coefficient, the fuel-burn-optimal aspect ratio lies where the weight elasticity equals the induced-drag fraction of total drag. Where the first condition discarded $\delta$, the second is $\delta$. Both are exact and both are parameter-free in their own terms, and they call for different aspect ratios on the same aeroplane, which is the sense in which the phrase “optimum aspect ratio” is incomplete without its constraint set.
For this aeroplane the two targets are therefore $\nu = 0.5$ and $\nu = $0.3351.
What the Elasticity Is
If the aspect-ratio-sensitive weight group obeys a power law then the elasticity has a simple form. Let $W_w = k_w A^n$ be that group, $W_0$ the rest of the aeroplane, and $\omega$ the group’s share of the total.
\[W = W_0 + k_w A^n, \qquad \omega = \frac{k_w A^n}{W}, \qquad \nu = n\,\omega \tag{42}\]So the elasticity is the exponent times the weight fraction, and both factors are measurable. The exponent is the three halves derived above. The weight fraction depends on how much of the aeroplane one is willing to call aspect-ratio-sensitive, and since that is a judgement rather than a measurement, the table below sweeps it from the narrowest defensible choice to a deliberately over-generous one.
| Weight group counted as aspect-ratio sensitive | Exponent | Pound | Elasticity nu | Fixed-area slope | Fixed-lift slope |
|---|---|---|---|---|---|
| bending material only, exponent 3/2 | 1.5 | 7,488 | 0.0775 | -0.2832 | -0.2576 |
| wing group, exponent 3/2 | 1.5 | 16,095 | 0.1665 | -0.2235 | -0.1686 |
| wing group and truss, exponent 3/2 | 1.5 | 19,931 | 0.2062 | -0.1969 | -0.1289 |
| wing group and truss, exponent 2 | 2.0 | 19,931 | 0.2749 | -0.1508 | -0.0602 |
Every row is negative under both criteria, so all rows agree. This aeroplane is below its fuel-burn-optimal aspect ratio. The narrowest reading, counting only the 7,488 pound of bending material the report identifies, gives an elasticity of 0.0775 against a required 0.5. The most generous reading, counting the entire wing group and the entire truss group as though every pound of both scaled as the three-halves power, gives 0.2062, still well short.
The result can be inverted into a single falsifiable statement, by solving $\nu = n\omega$ for the exponent that would make the present aspect ratio stationary.
\[n_{\mathrm{required}} = \frac{\nu^{*}}{\omega} = \frac{\nu^{*} W}{W_w} \tag{43}\]For aspect ratio 19.565 to be stationary, the wing group and truss group together would have to grow with aspect ratio as a power of 3.64 under the fixed-area criterion, or 2.44 under the fixed-lift criterion. Both are far above the three halves that bending material obeys and above the two that even a pessimistic reading would allow. Anyone wishing to argue that this wing is at its optimum has to produce a weight model with an exponent above two and a half, and say where it comes from.
Solving It a Second Way, by Bisection
The statements above are local. They evaluate a derivative at the design point and read off its sign. An independent route is to anchor the power law on the published weights and then solve for the stationary point by bisection, which uses the same inputs through different arithmetic and lands somewhere the local argument never looked.
The coefficient is anchored on the published weights rather than estimated, which leaves no free parameter in the power law at all.
\[k_w = \frac{W_w}{A^{n}}, \qquad W_0 = W - W_w \tag{44}\]With $k_w$ so anchored and everything else in $W_0$, the fixed-lift condition $\nu = \delta$ is satisfied at
\[A^{*} = 25.100 \tag{45}\]which is 28.3 percent above the configuration’s 19.565. At that point the elasticity and the induced-drag fraction agree to better than one part in a million, the bisection’s check on itself. At fixed area, that aspect ratio corresponds to a span of 192.5 feet, 13.3 percent beyond the 170 feet the aeroplane has.
And Now the Result That Reverses the Tone
A design whose optimum lies twenty-eight percent above it sounds like a design error. It is not, because the optimum is extraordinarily flat, and the flatness has an exact closed form.
Differentiate the fixed-lift slope a second time. With $\nu = n\omega$ the two pieces are
\[\frac{d(n\omega)}{d\ln A} = n^2\omega\left(1-\omega\right), \qquad \frac{d\delta}{d\ln A} = -\delta\left(1-\delta\right) \tag{46}\]so that
\[\frac{d^2 \ln m_f}{d(\ln A)^2} = n^2\omega\left(1-\omega\right) + \delta\left(1-\delta\right) \tag{47}\]and at the stationary point, where $\delta = n\omega$, the weight fraction can be eliminated entirely.
\[\left.\frac{d^2 \ln m_f}{d(\ln A)^2}\right|_{A^{*}} = \delta\left(n + 1 - 2\delta\right) \tag{48}\]The curvature of the fuel-burn objective at its optimum depends on nothing but the induced-drag fraction there and the weight exponent. For $\delta = $0.3351 and $n = 3/2$ it is 0.6131. The error is measured logarithmically, and that choice makes the expansion symmetric in over-design and under-design.
\[\varepsilon = \ln\frac{A^{*}}{A} \tag{49}\]The fractional fuel penalty follows immediately.
\[\frac{\Delta m_f}{m_f} \approx \tfrac{1}{2}\,\delta\left(n + 1 - 2\delta\right)\varepsilon^2 \tag{50}\]| Error in aspect ratio | Fuel penalty, percent |
|---|---|
| 5 percent | 0.0766 |
| 10 percent | 0.3066 |
| 20 percent | 1.226 |
At the design value the stationary aspect ratio is 28.3 percent above the configuration’s, and the expansion gives the penalty.
\[\varepsilon = \ln\frac{25.100}{19.565} = 0.2491, \qquad \frac{\Delta m_f}{m_f} \approx \tfrac{1}{2}\,(0.6131)\,(0.2491)^2 = 0.0190 \tag{51}\]Being 28.3 percent below optimum costs 1.90 percent in cruise fuel. That number changes the meaning of everything above, and the subsection headed Dropping the Linearisation revises it downward again by dropping the linearisation it rests on.
Dropping the Linearisation at a Twenty Percent Fuel Fraction
Everything above rests on linearising the Breguet exponential, and it is fair to ask what that costs. It costs more than it looks, both conditions survive in a modified form, and the modified forms move every number in this section in the direction that strengthens the article’s conclusion rather than weakening it.
The aeroplane’s usable fuel is 29,028 pound of a take-off weight of 145,000 [SUGAR Phase IV final report], so the fuel fraction and the Breguet exponent are fixed by the published weights alone.
\[\zeta = \frac{m_f}{W} = 0.200193, \qquad X = -\ln\left(1 - \zeta\right) = 0.223385 \tag{52}\]The linearisation replaces $1 - e^{-X}$ by $X$, and at this fuel fraction it overstates the fuel by 11.585 percent.
\[\frac{X}{1 - e^{-X}} - 1 = 0.115847 \tag{53}\]Carrying the exponential through instead introduces one factor, and that factor is all that changes. Let $\Phi$ be it.
\[\Phi(X) = \frac{X e^{-X}}{1 - e^{-X}}, \qquad \Phi(X) \to 1 \ \text{ as } X \to 0 \tag{54}\] \[d \ln m_f = d \ln W + \Phi(X)\, d \ln X \tag{55}\]That single line completes the generalisation, because $X$ is proportional to the ratio of drag to lift coefficient and its logarithmic derivative is what the two cases above already computed. For the fixed-lift case $X$ does not depend on the weight at all, so
\[\frac{d \ln m_f}{d \ln A} = \nu - \Phi \delta \qquad \Longrightarrow \qquad \nu^{*} = \Phi \delta \tag{56}\]and for the fixed-area case $X$ carries a factor of one over the weight, so
\[\frac{d \ln m_f}{d \ln A} = \nu\left(1 - \Phi\right) + \Phi \delta \left(2\nu - 1\right) \qquad \Longrightarrow \qquad \nu^{*} = \frac{\Phi \delta}{1 - \Phi + 2 \Phi \delta} \tag{57}\]Both reduce to the results above as the fuel fraction vanishes, the first to $\delta$ and the second to one half, since $\Phi \to 1$ in that limit.
| Criterion | Linearised target | Exact target at this fuel fraction |
|---|---|---|
| Fixed wing area and cruise condition | 0.500000 | 0.423797 |
| Fixed cruise lift coefficient | 0.3351 | 0.299033 |
The exact targets are lower, so the aeroplane is closer to its optimum than the linearised treatment said. The curvature changes too, and its closed form carries one extra term.
\[\left.\frac{d^2 \ln m_f}{d(\ln A)^2}\right|_{A^{*}} = \Phi \delta \left(n + 1 - \delta - \Phi \delta\right) + \Phi'(X)\, X \delta^2 \tag{58}\] \[\Phi'(X) = \frac{e^{-X}\left[\left(1-X\right)\left(1 - e^{-X}\right) - X e^{-X}\right]}{\left(1 - e^{-X}\right)^2} \tag{59}\]That expression also reduces to $\delta(n+1-2\delta)$ in the limit, since $\Phi \to 1$ and $\Phi’ X \to 0$. Evaluated here it gives 0.503772 against the linearised 0.6131, so the exact optimum is 17.84 percent flatter still. The closed form agrees to seven decimal places with a second difference taken on the exact objective, and $\Phi’$ agrees with a central difference.
The consequences are collected here because they revise the figures above.
| Quantity | Linearised | Exact |
|---|---|---|
| Stationary aspect ratio | 25.100 | 23.754 |
| Shortfall of the design value, percent | 28.3 | 21.41 |
| Curvature at the optimum | 0.6131 | 0.503772 |
| Fuel penalty at the design value, percent | 1.90 | 0.9481 |
| Exponent needed on wing and truss, fixed lift | 2.44 | 2.1755 |
| Exponent needed on wing and truss, fixed area | 3.64 | 3.0832 |
The conclusion survives every reading and the margin narrows, and both halves of that sentence matter. Under the exact fixed-lift criterion the target is 0.299033 and the deliberately over-generous weight accounting reaches 0.2749, which is short by 8.07 percent rather than by the margin the linearised figures implied. The defensible readings are short by 74.10 percent counting bending material alone and 31.05 percent counting the whole wing and truss at the three-halves power. Each shortfall is the fractional distance of an elasticity below the exact target.
\[1 - \frac{\nu}{\nu^{\ast}}\,: \qquad 1 - \frac{0.2749}{0.299033} = 0.0807, \qquad 1 - \frac{0.07746}{0.299033} = 0.7410, \qquad 1 - \frac{0.20618}{0.299033} = 0.3105 \tag{60}\]An article that only reported the linearised numbers would be overstating its own case.
The Report’s Predecessor Reached the Same Answer by a Different Method
This is the strongest corroboration in the article. The Phase II report’s Stage 1 study optimised three truss architectures against a series of span limits with a full multidisciplinary process, and summarised the result.
Overall, relaxation of the span constraint results in higher fuel efficiency. However, the benefit quickly diminishes after the span reaches 170 ft, offering less than 1.4% fuel burn improvement for any further increase in wing span.
Under 1.4 percent for all further span, against 0.9481 percent computed here from the exact optimality condition and the published drag and weight breakdowns. The two routes share no arithmetic. One is a closed-form second-order expansion about a stationary point located from a drag buildup and a group weight statement. The other is a design optimisation sweeping a constraint. They agree on the sign, they agree on the magnitude, and the computed figure falls inside the bound the report states.
The linearised expansion puts the penalty at 1.90 percent, which is above the report’s bound, so the apparent discrepancy between the expansion and the optimisation is an artefact of the linearisation.
What remains is that the Phase II optimisation re-sized the whole aeroplane with the take-off field length, the maximum range and in some cases the available fuel volume active as constraints, which a single-point expansion does not see, so exact agreement was never available in either direction. The report concedes the point.
In addition to the span constraint, the take-off field length and maximum range of Pt1 appeared to be the active constraints for all optimum designs.
So the span of this aeroplane was chosen where the curve goes flat and not where it reaches zero. That is the correct engineering decision, and a different one from the decision the phrase “optimum aspect ratio” suggests. And the sentence that follows the one quoted above records what was done with it.
Therefore, a one-jury TBW with 170 ft span, designated as C210b, was selected as a baseline
One hundred and seventy feet is therefore a baseline frozen in 2015 on a flatness argument, inherited by Phase III, inherited by Phase IV, and inherited by the X-66A. The Phase IV report is explicit that it never revisited it.
The Phase IV study was undertaken using a fixed planform defined by a multidisciplinary optimization carried out by Virginia Tech during Phase I.
The same report’s introduction places that optimisation in Phase II, naming Virginia Tech as the Phase II partner, which agrees with the Phase II report quoted above [SUGAR Phase IV final report].
And its own recommendation list asks for the revisit.
The significant performance advantage of the Mach 0.80 configuration studied in Phase IV again has illustrated that additional multidisciplinary planform optimization is required including variables such as strut and jury tie-in location, cruise Mach number and strut tie-in strategy.
The Gate Box as the Binding Constraint
The previous section found a flat optimum and a span chosen on its flatness. This section finds what the span was chosen against. It is not an aerodynamic quantity at all.
Two Spans, Two Code Letters, And Margins Measured in Inches
ICAO classifies aerodromes by a reference code whose letter is set by the wingspan of the aircraft the aerodrome is built for, and the FAA carries an equivalent classification into numbered Airplane Design Groups. The FAA’s own circular, rather than the NASA memorandum that reproduces its table, settles something the reproduction cannot. Its Table 1-2 gives each bound in both unit systems [FAA airport design].
| FAA group | ICAO code | Tail height | Wingspan |
|---|---|---|---|
| I | A | below 20 foot, below 6 metre | below 49 foot, below 15 metre |
| II | B | 20 to below 30 foot, 6 to below 9 metre | 49 to below 79 foot, 15 to below 24 metre |
| III | C | 30 to below 45 foot, 9 to below 13.5 metre | 79 to below 118 foot, 24 to below 36 metre |
| IV | D | 45 to below 60 foot, 13.5 to below 18.5 metre | 118 to below 171 foot, 36 to below 52 metre |
| V | E | 60 to below 66 foot, 18.5 to below 20 metre | 171 to below 214 foot, 52 to below 65 metre |
| VI | F | 66 to below 80 foot, 20 to below 24.5 metre | 214 to below 262 foot, 65 to below 80 metre |
Three features of that table are load-bearing. Every bound is a round number of feet. Every bound is exclusive, so an aeroplane whose span equals a bound belongs to the group above it. And the metric column is not an independent standard. The circular declares as much in its front matter.
Throughout this AC, U.S. customary units are used followed with “soft” (rounded) conversion to metric units. The U.S. customary units govern.
So 118 feet and 36 metre are the same boundary, written twice, and the circular declares which writing governs. They are not equal. 118 feet is 35.9664 metre and 36 metre is 118.1102 feet, a difference of 1.3228 inch, and that difference is the rounding the circular warns about.
\[118 \times 0.3048 = 35.9664 \ \text{m}, \qquad \frac{36}{0.3048} = 118.1102 \ \text{ft}, \qquad 12\,(118.1102 - 118) = 1.3228 \ \text{in} \tag{61}\]ICAO’s Annex 14 states its limit in metres natively [SUGAR Phase II volume I], so the same fold station clears Code C and does not reach Group III, and the apparent latitude between them is an artefact of a soft conversion rather than a disagreement between two authorities.
The Phase II report states the intent plainly.
The unfolded wingspan (approximately 52m) will stay within the ICAO Code Letter D wingspan limit and the folded wingspan (approximately 36m) will stay within the Code Letter C wingspan limit.
The Phase IV report gives the fold station for the Mach 0.80 wing.
The wing layout incorporates a fold at 118 feet of span to accommodate existing Code C gate and taxiway infrastructure.
Putting both spans against both regulators gives the tightest numbers in this article. Each margin is the bound less the span in inches, with ICAO’s Code D bound of 52 metre converted to feet.
\[\Delta = 12\,(b_{\mathrm{bound}} - b), \qquad 12\left(\frac{52}{0.3048} - 170\right) = 7.24, \qquad 12\,(171 - 170) = 12, \qquad 12\,(118 - 118) = 0 \tag{62}\]| Configuration | Span, foot | ICAO bound, foot | ICAO margin, inch | FAA bound, foot | FAA margin, inch |
|---|---|---|---|---|---|
| Unfolded | 170 | 170.6037 | 7.24 | 171 | 12 |
| Folded | 118 | 118.1102 | 1.32 | 118 | 0 |
The fold station is not near the FAA’s Group III boundary. It is exactly on it, to the inch. The Phase IV report puts the fold at 118 feet of span and the FAA’s Group III runs from 79 feet up to but not including 118 feet, so the margin is exactly zero and a span of 118.000 feet is a Group IV aeroplane rather than a Group III one [SUGAR Phase IV final report] [FAA airport design].
The airport-operations memorandum states the condition in the strict form, the strongest evidence that the boundary is the design driver.
Lowering the wingspan to less than 118 feet by folding the wings would lower the ADG to FAA Group III and ICAO Group C.
[airport operations memorandum]
Less than 118 feet, and the wing folds at 118 feet. Whether that is a rounding in the Phase IV report, a detail of an inch that nobody wrote down, or a genuine gap between two NASA documents is not settled by the record, and this article records the discrepancy and leaves it unresolved. Under ICAO’s metric bound the same station clears Code C by the 1.32 inch the table gives.
The unfolded span is less dramatic and still tight, at the 7.24 and 12 inch the table gives. In metric terms the two spans are 51.816 metre and 35.9664 metre, leaving 0.1840 metre and 0.0336 metre against ICAO’s two bounds, and those are the figures a European reader will recognise.
So the aeroplane is wedged into the gap between two code letters. It operates on a Code D runway with its wings out and parks at a Code C gate with its wings folded. The Phase II report spells out the operational consequence and does not minimise it.
This SUGAR aircraft intends to operate with its wings unfolded on the runway and folded whenever the aircraft is off the runway. […] Since the airports are traditionally designed for an unfolded wing, the SUGAR aircraft will have design and operational incompatibility on ICAO Code Letter C airport runways as well as with hangars and aprons when the wing is unfolded
What the Fold Is Worth
The arithmetic of the fold produces the largest aerodynamic effect this article computes. Ask what aspect ratio the Code C box permits at this wing area without folding.
\[A_{\mathrm{folded}} = \frac{b_f^2}{S} \tag{63}\]That gives 9.427. At fixed area the ratio of the two aspect ratios is the square of the ratio of the two spans, so the factor the fold delivers needs no area at all.
\[\frac{A}{A_{\mathrm{folded}}} = \left(\frac{b}{b_f}\right)^2 = 2.076 \tag{64}\]And the two margins that constrain the two spans are differences rather than ratios. That is why they are quoted in inches and why the exclusive FAA bound makes one of them exactly zero.
\[\Delta_{\mathrm{ICAO}} = \frac{36}{0.3048} - b_f, \qquad \Delta_{\mathrm{FAA}} = 118 - b_f, \qquad \Delta_{\mathrm{ICAO}} - \Delta_{\mathrm{FAA}} = 1.3228 \ \text{inch} \tag{65}\]Since induced drag goes as the inverse of aspect ratio, the drag consequence follows by rebuilding the buildup at the folded aspect ratio and leaving every other term alone.
\[\left.\frac{L}{D}\right|_{\mathrm{folded}} = \frac{C_L}{C_{D,0} + C_{D,c} + C_{D,t} + \dfrac{C_L^2}{\pi A_{\mathrm{folded}} e}} \tag{66}\]| Quantity | Span-limited to 118 feet | Folding to 170 feet |
|---|---|---|
| Aspect ratio | 9.427 | 19.565 |
| Induced drag coefficient | 0.01873 | 0.00903 |
| Lift-to-drag ratio | 18.967 | 25.788 |
The fold is worth 36.0 percent in lift-to-drag ratio. The remaining gain beyond it, all the way to the stationary aspect ratio located in the previous section, is 0.9481 percent in fuel on the exact Breguet treatment and 1.90 percent on the linearised one.
Set those two numbers beside one another and the shape of the whole programme appears. A mechanism that lets the wing exceed the gate box is worth roughly thirty-six percent in cruise efficiency. Everything the aerodynamic optimum has left to give beyond that point is worth under two percent. The folding wingtip is not a detail of this configuration. It is where almost all of the value is, a mechanism rather than an aerodynamic insight.
The fold itself is not small.
\[\text{span folded per side} = \frac{b - b_f}{2} = 26 \ \text{foot} \tag{67}\]It removes 26 feet per side, which is 30.6 percent of semispan, and the March 2025 technical memorandum on airport operations notes that the hinges are assumed to sit where the primary strut attaches to the wing, so the folded span and the truss length are the same 118 feet [airport operations memorandum]. The structural attachment station and the regulatory boundary are the same station. That is either an elegant piece of integration or a constraint masquerading as one, and the record does not say which came first.
The Aspect Ratio of Today’s Fleet Is an Airport Number
The folded aspect ratio of 9.427 deserves a second look, because it is almost exactly the aspect ratio of the aeroplanes this one was meant to replace. The Phase IV report’s comparison baseline, the 765-093 representing best-in-class practice, carries an effective aspect ratio of 10.41 [SUGAR Phase IV final report].
So the aspect ratio of the conventional single-aisle baseline is, to within about a tenth, the aspect ratio that the Code C gate box permits at the wing area those aeroplanes need, 10.41 against 9.427, a difference of 10.4 percent. That is an inference rather than a measurement. This article would defend it first, because it rests on an arithmetic coincidence that a reader can check in one line. The received explanation for why airliners do not have long thin wings is structural weight. The arithmetic here indicates that for a 737-class aeroplane the structural limit is not where the fleet sits. The fleet sits at the airport limit, and it sits there because every family grew its span until the letter ran out. The Phase II report makes the historical observation.
Aircraft wingspans have been growing over the years. […] each aircraft family has experienced wingspan growth to the limit of its airport design category
A truss-braced wing is therefore better understood as a response to an infrastructural constraint than to an aerodynamic one. What it does is make a wing long enough to need folding structurally affordable, and what the fold does is make it legal. Neither alone is sufficient and the fold is the larger of the two contributions.
The Constraint Is Not Only the Span
Reading the airport-operations memorandum closely shows that span is one of five quantities and that the others are not trivially satisfied [airport operations memorandum]. The reference codes turn on approach speed for the Aircraft Approach Category, on wingspan and tail height for the Airplane Design Group, and on the undercarriage dimensions for the Taxiway Design Group, and the memorandum’s conceptual model is classified C-IV-3 unfolded and C-III-3 folded. So folding the wing does not merely reduce a number. It changes the aeroplane’s design group, and the gate follows from the group.
Two details in that memorandum should be kept separate from the Phase IV configuration this article has otherwise used, because they are a different aeroplane. Its conceptual model has a span of 155.2 feet, a wing area of 1,231.2 square feet and a maximum take-off weight of 132,843 pound, against the Phase IV configuration’s 170 feet, 1477.109 square feet and 145,000 pound [airport operations memorandum]. Its folded span is nonetheless the same 118 feet. That is the strongest available evidence that the folded span is a fixed infrastructural input while everything else about the aeroplane moves, and this article treats it as such.
What the Truss Buys, and What It Costs
A Rigid Prop Is a Bound and Not a Prediction
The simplest useful idealisation of a brace is a rigid vertical prop at a station $\eta$ along the semispan, making the wing a propped cantilever under the elliptic load. The prop force follows from one compatibility equation and the equation is worth writing out, because its denominator has a closed form that makes the whole calculation checkable by hand.
By the unit-load method, the deflection at $\eta$ from the distributed load and from a unit load at $\eta$ are two integrals against the same virtual moment $m(u) = \eta - u$, taken over a beam of uniform stiffness.
\[w_{\mathrm{load}}(\eta) = \frac{1}{E\mathcal{I}}\int_0^{\eta} \mathcal{M}(u)\left(\eta - u\right) du, \qquad w_{\mathrm{unit}}(\eta) = \frac{1}{E\mathcal{I}}\int_0^{\eta} \left(\eta - u\right)^2 du \tag{68}\]The second integral is elementary and the stiffness cancels in the ratio, so the prop force does not depend on how stiff the wing is, only on where the prop is.
\[w_{\mathrm{unit}}(\eta) = \frac{\eta^3}{3 E\mathcal{I}}, \qquad P = \frac{w_{\mathrm{load}}}{w_{\mathrm{unit}}} = \frac{3}{\eta^3}\int_0^{\eta} f(u)\left(\eta - u\right) du \tag{69}\]Numerical quadrature of the unit-load integral reproduces $\eta^3/3$ to nine decimal places at four stations. The braced moment is then the cantilever moment less the prop’s contribution inboard of the attachment.
\[\mathcal{M}_{\mathrm{braced}}(u) = \mathcal{M}(u) - P \left(\frac{b}{2}\right)\max(\eta - u,\, 0) \tag{70}\]The quantity that sets bending material is not the moment but the integral of its magnitude against the inverse of the chord, since that is what appeared in the weight derivation. Call that integral $\mathcal{B}$, and the relief factor is the ratio of the braced value to the unbraced one.
\[\mathcal{B}(\eta) = \int_0^1 \frac{\left|\mathcal{M}_{\mathrm{braced}}(u)\right|}{1-(1-\lambda)u}\,du, \qquad \text{relief} = \frac{\mathcal{B}(\eta)}{\mathcal{B}(\infty)} \tag{71}\]| Prop station, fraction of semispan | Prop force | Bending-material integral, as a fraction of the cantilever |
|---|---|---|
| 0.30 | 1.3114 | 0.5190 |
| 0.40 | 0.9070 | 0.3663 |
| 0.50 | 0.6693 | 0.2223 |
| 0.55 | 0.5844 | 0.1682 |
| 0.60 | 0.5147 | 0.1396 |
| 0.70 | 0.4076 | 0.1531 |
The integral has a shallow minimum. The best station among those computed is 0.60 of semispan, at which the bending-material integral falls to a factor of 0.1396, 14.0 percent of the cantilever value by the relief ratio displayed above. Attaching further inboard leaves too much outboard wing unbraced and attaching further outboard leaves too much inboard moment, and between forty and seventy percent of semispan the penalty for being in the wrong place is small.
This is an upper bound on what a brace can do and the real brace does less, for four reasons that are all in the primary record. The prop above is rigid and a real strut is elastic. The prop above is vertical and the real strut rises at 14.43 degrees [SUGAR Phase IV final report], which the next section shows is expensive. The beam above has uniform stiffness and a real tapered wing is far less stiff outboard, so its attachment deflects more and the prop takes a different share. And the prop above introduces a reaction and nothing else, whereas a real strut attached to a wing that twists under load introduces bending into itself, which the Phase IV report addressed with a swivel joint.
the inboard strut attachment swivel joint concept enables the strut to carry axial loads while avoiding strut bending induced from wing twist as the wing lifts during flight, significantly simplifying the construction of the main strut
Multiplying the cantilever weight estimate by the best relief factor gives a figure far below the published bending material, and the ratio of the two has a name.
\[k_{\mathrm{no}} = \frac{W_b^{\,\mathrm{published}}}{\text{relief}\left(\eta,\lambda\right) \, W_b^{\,\mathrm{model}}} = 7.88 \tag{72}\]That factor measures everything this model omits, and it is not this article’s coinage. A 1981 methodology report prepared for NASA by Grumman Aerospace describes the whole class of wing weight estimating equations in the same terms, and names the same three omissions [York and Labell 1980].
These methods rely on a rational, though certainly simplified, model for (at least) bending material and determination of constants, coefficients and exponents by a regression or similar analysis to include non-theoretical influences on the box beam weight. Such influences as non-optimum weight, minimum gages and secondary loads, and other design requirements
So the structure of the model used here is the structure the field uses, being a rational bending-material core with a regression-determined factor in front of it, and the factor absorbs non-optimum weight, minimum gauges and secondary loads. That report also names the one mechanism that could change the exponent rather than the coefficient.
Wing weight may be penalized by flutter and divergence.
The non-optimum factor is approximately constant across aspect ratios and the flutter penalty is not, which is exactly why the keystone survives being unable to predict the first and is exposed to the second.
Why a Shallow Strut Is an Expensive Way to Carry a Vertical Load
The main strut rises at a dihedral of 14.43 degrees and is pinned at both ends [SUGAR Phase IV final report], so it is an axial member, and the vertical relief it delivers is the vertical component of its axial force. Let $\Gamma$ be the dihedral and $T_s$ the axial force.
\[T_s = \frac{P}{\sin \Gamma}, \qquad \text{horizontal reaction} = \frac{P}{\tan \Gamma} \tag{73}\]To deliver one pound of vertical relief the strut must carry 4.013 pound along its axis and something must react 3.886 pound horizontally. That single factor explains most of what the group weight statement shows for the truss. The strut group, with its jury and its installation, weighs 3,836 pound, equal to 23.8 percent of the wing group and 2.65 percent of take-off weight. In the finite-element sizing of primary structure alone the strut is 2,323 pound against the wing box’s 5,917 pound, a ratio of 39.3 percent, and the three members together come to 8,300 pound [SUGAR Phase IV final report].
A steeper strut would be cheaper structurally and is not available. The dihedral is set by the geometry of a high wing and a fuselage-mounted attachment, and raising it would mean either lowering the wing or moving the attachment outboard, both of which have consequences elsewhere. This is the clearest instance in the configuration of a structural penalty imposed by an arrangement chosen for other reasons.
The Load Case That Almost Never Happens
A strut that is in tension at positive load factor is in compression at negative load factor, and a slender compression member buckles. The Phase IV sizing used balanced manoeuvres at plus 2.5 and minus 1 load factor and sized the structure for strength and for buckling at ultimate loads, with the report stating that global buckling of the strut and the jury was the focus and that neither member should buckle below ultimate load [SUGAR Phase IV final report]. The internal-loads discussion confirms that the strut sees both signs.
The large tension and compression forces in the strut due to symmetric maneuvering are quantified in the plot.
The jury strut exists for this reason and its payoff is a square. A pin-ended column braced at mid-length has its effective length halved, and the Euler critical load goes as the inverse square of effective length.
\[P_{\mathrm{cr}} = \frac{\pi^2 E \mathcal{I}}{\left(K \mathcal{L}_s\right)^2} \tag{74}\] \[\frac{P_{\mathrm{cr}}\left(K = \tfrac{1}{2}\right)}{P_{\mathrm{cr}}\left(K = 1\right)} = \left(\frac{1}{1/2}\right)^2 = 4 \tag{75}\]Halving the effective length therefore multiplies the critical load by 4, and so a member weighing 60 pound in the sizing study can protect a member weighing 2,323 pound. The report describes it in these terms.
The main strut element is stabilized under buckling load via a jury strut.
The jury’s station was optimised and the record gives both the result and the width of the feasible band. An aerostructural study varying the jury location with different outboard materials found a feasible range from roughly 425 to 435 inch for an all-composite outboard strut, and the finalised location was 425 inch, at 41.67 percent of semispan [SUGAR Phase IV final report]. A ten-inch feasible band on a one-thousand-and-twenty-inch semispan is a one percent window, and the report explains that the choice inside it combined the structural design requirements with aerodynamic design considerations [SUGAR Phase IV final report].
And a second jury is not worth adding. The Phase II architecture study compared one strut, one strut with one jury, and one strut with two juries across span limits, and reported that the second jury gave a marginal weight reduction while costing more drag than it saved.
However, the two jury TBW configuration provides only a marginal benefit in wing weight reduction relative to one jury TBW. […] Such drag penalties are found to be substantial for a two jury TBW, outweighing the weight benefit.
The Truss Changes the Coefficient and Not the Exponent
A truss might be expected to lower the aspect-ratio exponent, and the derivation shows that it does not. Consider a truss whose attachment station, dihedral and proportions are held fixed as the span grows, the natural way to scale a configuration. Then every length in the braced problem scales with the span exactly as it does in the cantilever problem, the relief factor is a pure function of $\eta$ and $\lambda$, and the weight law is the cantilever law multiplied by a constant.
\[W_b^{\,\mathrm{braced}} = \text{relief}(\eta, \lambda) \times \frac{\rho_m \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau} \; A^{3/2} \, S^{1/2} \tag{76}\]The exponent is still exactly three halves. A geometrically similar truss buys $k_w$ and not $n$, and the distinction matters because the two move the optimum differently. In the limit where the sensitive weight fraction and the induced-drag fraction are both small, the stationary condition reduces to a power law whose solution is explicit.
\[A^{*} \propto k_w^{-1/(n+1)}, \qquad \left.\frac{d\ln A^{*}}{d\ln k_w}\right|_{n=3/2} = -0.4 \tag{77}\]So a reduction in the weight coefficient moves the optimum aspect ratio by only two fifths of its size in logarithm. Cutting $k_w$ to nine tenths raises the optimum aspect ratio by 4.30 percent and cutting it to four fifths raises it by 9.34 percent. A truss that halved the bending material would move the optimum aspect ratio by about thirty-two percent, not by a factor of two, and the flatness result then implies that the fuel consequence of such a move is small.
A 1980 Study Picked Aspect Ratio Twenty-Five
This article’s stationary aspect ratio was selected by a design study forty-five years earlier, for a different aeroplane, and the coincidence of magnitude is worth recording while the coincidence of reasoning is not available.
A study under NASA contract NAS1-16000, reported in October 1980, developed a subsonic business jet employing an aspect ratio 25 strut-braced high wing, with lifting struts chosen over non-lifting ones for the induced-drag improvement [Turriziani et al]. Its findings read remarkably like the 2025 ones.
The strut-braced wing airplane cruises at higher altitudes and lower speeds
due to its larger wing, the strut-braced wing airplane has a much lower wing loading than the conventional wing airplane
a fuel savings in excess of twenty percent is realized with the strut-braced wing design
This article’s linearised stationary aspect ratio is 25.100 and its exact one is 23.754. A 1980 study selecting 25 for a strut-braced wing is a coincidence of magnitude and this article does not claim it is more than that, because a business jet is a different weight class, a different mission and a different wing loading, and because the report does not say how the 25 was chosen. What it does establish is that the figure is not novel and that the configuration’s higher cruise altitude and lower wing loading were understood forty-five years ago, which the Ames comparison rediscovered as a cost on the short mission.
That is the uncomfortable corollary of the keystone. If the optimum is flat and insensitive to the weight coefficient, then the structural achievement of the truss cannot by itself be worth very much in fuel, and the value has to come from somewhere else. The two places it comes from are the fold, computed above at 36.0 percent of lift-to-drag ratio, and the thin wing, the subject of the next section.
The Thin Wing the Truss Pays For
The truss relieves bending moment, and the configuration spends that relief on thinness rather than on span. This is the part of the argument that the primary record states most clearly and that press accounts state least clearly. It is also the part that survived the programme’s redirection, because thin-wing technology is what Boeing said it would keep working on.
The memorandum on airport operations states the chain in three sentences.
Flow over the wing with reduced thickness of the chord, spar and skin has lower chord Reynolds number. Such a wing also has lower wave drag; therefore, the wings do not have to be swept back as much, which reduces the spanwise crossflow disturbances. These result in increasing the extent of laminar flow and lowering fuel consumption.
[airport operations memorandum]
So the thin wing is not valued for its thinness. It is valued because thinness buys sweep, and less sweep buys laminar flow. This section puts numbers on the first of those exchanges, the exchange a closed-form relation can reach.
The Korn Relation, Stated and Then Tested
The semi-empirical relation usually attributed to Korn gives the drag-divergence Mach number of a swept wing section in terms of its thickness, its lift coefficient and an airfoil-technology constant. Let $\kappa_A$ be that constant, $\Lambda$ the sweep, $\tau$ the thickness-to-chord ratio and $C_L$ the lift coefficient.
\[M_{dd} = \frac{\kappa_A}{\cos \Lambda} - \frac{\tau}{\cos^2 \Lambda} - \frac{0.1\,C_L}{\cos^3 \Lambda} \tag{78}\]The constant is a property of the airfoil technology, and the programme’s Phase II airfoil study, which used this relation, assigned its sections constants of 0.915 to 0.929 [SUGAR Phase II volume I]. The Phase IV report’s geometry table lists supercritical sections with divergent trailing edges for both of its configurations, although its aerodynamic design section removed the divergent trailing edge from the Mach 0.80 starting sections and reserved it for use where needed [SUGAR Phase IV final report], so 0.95 is the value used here and the margin it predicts is then a test of that choice rather than an assumption.
The two configurations differ in the way this section examines. The Mach 0.745 predecessor has 12.58 degrees of sweep and an average thickness ratio of 0.120, and the Mach 0.80 configuration has 22.46 degrees, the sweep its drag buildup carries against the 22.47 its geometry table gives, and 0.104. The report describes the change and makes clear that nothing else moved.
The wing is derived from the Mach 0.745 wing by adding 10 degrees sweep and reduced thickness-to-chord ratio. The chord distribution is identical to the Mach 0.745 wing.
| Configuration | Sweep, degree | Thickness ratio | Cruise lift coefficient | Cruise Mach | Korn $M_{dd}$ | Margin |
|---|---|---|---|---|---|---|
| Mach 0.745 predecessor | 12.58 | 0.120 | 0.7635 | 0.745 | 0.7653 | 0.0203 |
| Mach 0.80 configuration | 22.46 | 0.104 | 0.695 | 0.80 | 0.8181 | 0.0181 |
Both configurations sit the same distance below their own drag-divergence Mach number, and the two margins differ by 0.00212 in Mach number. Neither margin was an input. They are the output of one relation with one constant applied to two independent sets of published geometry, and they agree to two parts in a thousand.
The agreement survives the one constant that was chosen. Sweeping $\kappa_A$ from 0.87 to 1.00, a range that brackets the Phase II constants, changes both margins together and leaves their difference small.
| Korn constant | Difference between the two margins |
|---|---|
| 0.87 | -0.00672 |
| 0.90 | -0.00500 |
| 0.95 | -0.00212 |
| 1.00 | +0.00075 |
The largest disagreement between the two margins anywhere in that range is 0.00672 in Mach number. So the conclusion does not rest on the choice of 0.95, and the conclusion is this. The ten degrees of sweep and the thickness reduction that Phase IV added are, to within a few thousandths in Mach number, what the Korn relation requires to buy 0.055 in cruise Mach number. The designers were working to a transonic criterion of this form, whether or not they used this expression.
How Much Sweep the Thinning Replaced
Inverting the relation answers the question the exchange is really about. Holding the Phase III thickness ratio and asking what sweep would reach the same drag-divergence Mach number gives 27.32 degrees, against the 22.46 degrees actually used. So the thinning was worth 4.86 degrees of sweep.
Running the trade the other way is exact rather than numerical, because the relation is linear in thickness.
\[\tau = \left(\frac{\kappa_A}{\cos \Lambda} - \frac{0.1\,C_L}{\cos^3 \Lambda} - M_{dd}\right)\cos^2 \Lambda \tag{79}\]Holding the Phase III sweep and asking what thickness would reach the same drag-divergence Mach number gives 0.0766, a quarter thinner again than the wing actually has and well outside anything a transport wing box has been built to.
The inversion is not single-valued. The Korn relation rises with sweep only up to a turning point, because the leading term grows as the inverse cosine while the thickness and lift terms grow as the inverse square and the inverse cube. The turning point has a closed form. Substituting $s = \sec \Lambda$ makes the relation a cubic in $s$ whose derivative is a quadratic.
\[M_{dd} = \kappa_A s - \tau s^2 - 0.1\,C_L s^3, \qquad \frac{dM_{dd}}{ds} = \kappa_A - 2\tau s - 0.3\,C_L s^2 \tag{80}\]Since $\sec \Lambda$ increases with sweep throughout the first quadrant, the turning point is the positive root of that quadratic.
\[s^{*} = \frac{-\tau + \sqrt{\tau^2 + 0.3\,\kappa_A C_L}}{0.3\,C_L}, \qquad \Lambda^{*} = \arccos\frac{1}{s^{*}} \tag{81}\]At this thickness and lift coefficient that gives 53.8022 degrees and a peak of 0.9730, against 53.8 degrees from a scan of seven and a half thousand sweep angles, and the two peak values agree to six decimal places. For the thicker Phase III section the turning point is 52.3000 degrees.
An inversion of the relation for sweep must therefore stay below $\Lambda^{*}$, where the branch is monotone, and the 27.32 degrees found above lies well inside that branch.
What the Thinning Cost in Structure
The weight derivation above gives the thickness term as an inverse first power and the sweep term as the inverse square of the cosine, so the structural price of the Phase IV change can be stated directly.
\[\frac{W_b^{\,\mathrm{P4}}}{W_b^{\,\mathrm{P3}}} = \frac{\tau_{\mathrm{P3}}}{\tau_{\mathrm{P4}}} \cdot \frac{\cos^2 \Lambda_{\mathrm{P3}}}{\cos^2 \Lambda_{\mathrm{P4}}} \tag{82}\]| Change | Factor on bending material |
|---|---|
| Thinning from 0.120 to 0.104 | 1.1538 |
| Sweeping from 12.58 to 22.46 degrees | 1.1154 |
| Both together | 1.2869 |
Buying 0.055 in cruise Mach number cost about 28.7 percent in bending material, by this model, and the two contributions are close to equal. The report does not publish a Phase III structural weight in the volume read for this article, so that prediction stands unchecked. It is presented as a prediction rather than a measurement for that reason.
The Thin Wing Has Nowhere to Put the Fuel
The sharpest cost of thinness is not weight and the report’s tables show it. The as-drawn Mach 0.80 configuration requires 4,351 United States gallon of fuel capacity and has 3,594 available, a deficit of 17.4 percent of requirement. The as-drawn Mach 0.745 predecessor requires 5,684 and has 5,417, a deficit of 4.7 percent [SUGAR Phase IV final report].
\[\text{deficit} = \frac{\mathcal{V}_{\mathrm{required}} - \mathcal{V}_{\mathrm{available}}}{\mathcal{V}_{\mathrm{required}}} \tag{83}\]Thinning the wing roughly quadrupled the fractional fuel-volume deficit, from 4.7 percent to 17.4 percent, a factor of 3.704. The report addresses it with a body tank and devotes a table to the configuration sized with and without one, and its discussion of systems integration notes the obvious.
since the thin, high aspect ratio wing has considerably less volume to host systems, fuel, and
A body fuel tank is a payload-volume charge against a fuel-burn benefit, and the airport-operations memorandum notes that the body-tank variant reduces the crossover range against a conventional aeroplane to below 300 nautical mile, the clearest statement in the record that the trade is close over short sectors [airport operations memorandum].
And the Anti-Icing Problem the Report Names Without Quantifying
A thin high-aspect-ratio wing with a small leading-edge radius has more leading edge to protect, needs more energy per unit length to protect it, and has less room inside to put the system that does the protecting. The Phase IV report states all three together and does not put a number on any of them.
the increased leading edge length and small leading edge radius of both the wing and struts could substantially increase anti-ice system energy requirements, while significantly decreasing the amount of space available for the integration of anti-ice systems
This article has no basis for a number either and says so, because the leading-edge radius distribution and the system architecture are both absent from the public record. It is recorded here as an unquantified penalty on the same side of the ledger as the fuel volume. Such an item moves a configuration decision at preliminary design rather than at conceptual design, and conceptual design is where this programme stopped.
Dependent Systems
Aeroelasticity, and a Correction That Turned a Forty Percent Margin Negative
Phase I identified novel aeroelastic effects as one of two primary risks, and Phase II addressed them with an analytical model and an aeroelastic wind-tunnel test in the Transonic Dynamics Tunnel at NASA Langley. Its conclusion was favourable and carefully bounded.
Both flutter speed and mechanism were found to be analytically predictable, and the weight increment associated with flutter margin was found to be small enough to maintain the aerodynamic advantage of increased span.
Phase IV repeated the analysis on the Mach 0.80 configuration with the doublet-lattice method and reported that the flutter speed exceeded requirements with no weight increase for flutter, a stronger statement than Phase II made [SUGAR Phase IV final report]. Then it applied aerodynamic correction factors derived from steady computational-fluid-dynamics solutions to the unsteady doublet-lattice model, and the result at the highest Mach number analysed is startling.
The margins quoted below are speed margins and the definition matters, because a negative one means the flutter speed has fallen below the speed the certification basis requires.
\[\text{margin} = \frac{V_{\mathrm{flutter}}}{V_{\mathrm{required}}} - 1 \tag{84}\]At Mach 0.59, 0.76 and 0.85 the corrections moved margins modestly and in both directions, raising the minimum stability margin by about ten percent overall. At Mach 0.92 and 26,000 feet the raw doublet-lattice model gave margins near forty percent for three of its five mass cases and 22.8 and 28.4 percent for the other two, and the corrected model produced multiple mechanisms, several with negative margins, the worst at minus 37.3 percent [SUGAR Phase IV final report]. The report does not soften it.
Clearly, the aero adjustments have completely changed the character of the analysis results with the introduction of multiple low and high frequency mechanisms, many of which have negative stability margins.
Two models of the same structure at the same condition differed by roughly fifty percentage points of flutter margin and disagreed about the sign. The report’s diagnosis names the limitation and the remedy.
the flutter analysis performed to date at the 0.92 Mach flight conditions is incapable of capturing for the nonlinear flow features predicted. To more confidently assess stability margins at this transonic flight condition, a flutter analysis employing CFD calculation of unsteady aerodynamic forces directly is required.
And it names a second limitation specific to this configuration.
because the TTBW configuration results in a redundant wing load path unlike that of a catilevered wing, flutter analysis should be performed with prestressed modes and potentially about a large deformation operating state
A braced wing is a statically indeterminate structure and its modes depend on its preload. The Phase II work had already met this experimentally, reporting that the flutter speed of the wind-tunnel model varied as the loads on the model varied and that the non-linear result had to be modelled by including preload and large-displacement effects [SUGAR Phase II volume III]. So the flutter boundary of this aeroplane is not a single number. It is a function of the load state, and a certification basis built on a single number would be testing the wrong quantity.
What this means for the keystone is specific and limited. The flutter weight increment is the term that could in principle give the wing an aspect-ratio exponent above the three halves, because flutter speed depends on stiffness distribution in a way that bending strength does not. Phase II found that increment small and Phase IV found it zero at the conditions it could model. Neither statement covers the Mach 0.92 condition where the corrected model goes negative, so the exponent assumed in this article is supported at the conditions analysed and is unsupported at the condition the report itself flags, and of the soft spots this article names it is the only one that could change the exponent rather than the coefficient, which is why it is treated as the largest.
Wing and Strut Interference Reduced to One Percent in Phase III
The second of the two primary risks was the transonic interference between the wing and the strut, and the junction between them is the configuration’s hardest aerodynamic problem. Phase III addressed it with Navier-Stokes methods and a series of transonic wind-tunnel tests, and the result is quoted in full.
interference effects between the wing and strut were reduced to approximately 1% of total airplane drag at this Mach number and uncertainty in predicted aircraft performance was significantly reduced
The interference figure is only the penalty and not the whole charge. The strut and jury’s equivalent flat-plate area, which the subsection on the induced drag shows to be 12.422 percent of the parasite drag area, stands against the wing’s 9.6340 square feet and the fuselage’s 8.3505 [SUGAR Phase IV final report]. So the truss costs an eighth of the parasite drag and the interference at the junction is a tenth of that again.
Phase IV also explored an offset or curved strut specifically to improve the junction, and recommended it without adopting it.
The outboard strut attachment offset (curved beam) strut concept holds potential for further reducing the aerodynamic interference in the wing-strut channel and further mitigating buffet concerns.
Buffet Outside the Reach of the Prediction Methods
The report names buffet prediction as its most immediate aerodynamic risk and gives a reason that is about the method rather than the aeroplane.
Current prediction methods for determining buffet boundary are empirically-derived from flight and wind tunnel tests of conventional cantilever wing aircraft. Applicability of the established boundaries to this species of aircraft is yet to be validated
Its buffet analysis carries a caveat that is unusually frank for a contractor report.
while the FAA regulations for “buffet free” flight are based on the perception of buffet created by a structural response to flow separation, it is unclear if the BUFFET results presented herein are a true indicator of (local) buffet on the wing or strut. It is also unknown if such local buffeting would be perceived by the pilot and deemed unacceptable.
That is a certification question and not an aerodynamic one. The quantity the regulation addresses is what a pilot perceives, the quantity the analysis produces is a local flow separation on a surface no previous transport has had, and nobody has established the mapping between them. It is the clearest example in this programme of a demonstrator being needed for a reason that has nothing to do with whether the concept works.
Propulsion Integration, and the Engine the Demonstrator Received
The Phase IV configuration carries two engines of a conceptual type designated gFan+ with a Boeing equivalent thrust of 23,000 pound and a fan diameter of 71 inch [SUGAR Phase IV final report]. The sized variants in its performance tables carry fan diameters of 66 and 72 inch [SUGAR Phase IV final report], so the fan size is an outcome of sizing rather than an input.
The demonstrator’s engine selection sits inside that tension. The engine disclosed for the X-66A is the PW102XG, a derivative of the PW1500G [FlightGlobal on the PW102XG], and no source read for this article states why that engine was chosen. A high wing with a shallow strut and a pylon-mounted main landing gear is a configuration with limited tolerance for engine mass hung outboard, since every pound there is a pound the strut must react at 4.013 times leverage, and the report’s internal-loads discussion notes the engine’s effect explicitly.
The significant effect of engine inertia loads on wing internal loads is also noted.
So engine mass is a configuration variable on this aeroplane and not an installation detail, and an engine choice on a demonstrator built from a thirty-year-old airframe is a decision about the bracing geometry whether or not it is recorded as one.
High Lift on a Thin Wing
The wing carries full-span Krueger flaps broken by the engine pylon, single-slotted trailing-edge flaps, an all-speed aileron inboard of the fold and a low-speed aileron outboard of it that is locked out at high speed, with spoilers across the flapped span [SUGAR Phase IV final report]. The outboard aileron sitting beyond the fold hinge deserves notice, because it means a control surface lives on the part of the wing that folds, and the fold is therefore not merely a structural hinge but a break in the flight-control system.
The low-speed test programme did not finish what it set out to do.
challenges in the low-speed test delayed some desired objectives, particularly surrounding ice-effects testing and in-ground-effect performance. These objectives have been deferred to a later test entry.
And the low-speed data that does exist is at the wrong Reynolds number, which the report says plainly.
the data collected in the low-speed wind tunnel to date has been at very low Reynolds Numbers. Since vehicle stall behavior and CLmax levels are a function of Re, it will be important to quantify the vehicle low-speed performance in the tunnel at higher Re. This data is critically important for establishing the low-speed performance data needed for sizing the vision vehicle.
A maximum lift coefficient is an input to a field-length calculation, and a field length was one of the active constraints in the Phase II span optimisation. So an unmeasured maximum lift coefficient feeds directly into the constraint that, alongside span, set the configuration. That is a loop the record leaves open, the second place in this article where the span of the aeroplane depends on something nobody has measured at the right conditions.
The Unknown Ground Effect of a Wing This Long
A wing of aspect ratio 19.565 with a fold at seven tenths of span approaches the ground in a configuration no transport has flown. The report flags it without quantifying it.
the high wing aspect ratio is likely to have significantly different sensitivities to performance in ground effect as compared to a conventional configuration
The governing dimensionless group is displayed here, because it is the reason a long wing behaves differently. The ground plane is represented by an image vortex system reflected below it, and the upwash the image induces at the real wing scales with the ratio of span to twice the height.
\[\Sigma = \frac{b}{2h_g} \tag{85}\]For this aeroplane at a given wheel height that group is roughly one and a half times a conventional narrowbody’s, because the span of 170 feet is roughly half as large again as the 118-foot Code C limit at which the conventional fleet sits. A larger favourable interference at touchdown lengthens a flare and a larger one at rotation shortens a take-off, and whether the net is helpful depends on which the field length is sized by.
This article displays the group and declines to evaluate the effect, for two reasons that are both about inputs rather than about method. The empirical induced-drag factors in the literature are fits, and this article does not reproduce fits it has not verified. And the height distribution of a wing with negative dihedral, a fold at seven tenths of span and a strut below it is not recoverable from the published geometry, so there is no single $h$ to put in the denominator.
What the Thirty Percent Means, Measured Against What
The figure every account of this programme leads with is thirty percent, and the primary record contains three different numbers, each correct against a different baseline.
The Wing Alone, Against an Advanced Conventional Aeroplane
The Phase IV report states the concept’s own contribution and names its baseline.
the demonstrated performance benefit of the TTBW concept (independent of other technologies applied on the SUGAR configuration) has been calculated to be a 7.2% improvement in fuel burn (per seat) for an economic mission of 900nm, and a 9.0% improvement in fuel burn (per seat) for a 3500nm mission with full passenger payload. The benefits quoted are relative to an aspect ratio 13 conventional tube and wing configuration of equivalent technology development.
So the truss-braced wing is worth 7.2 percent on a short sector and 9.0 percent on a long one, against a conventional aeroplane of aspect ratio 13 built to the same technology standard. Three features of that sentence are load-bearing. The baseline is advanced, not current. The baseline’s aspect ratio of 13 is itself well above today’s fleet. And the figure is explicitly independent of other technologies, the same clause the thirty percent needs and every agency statement of it carries.
The Whole Technology Package, Against a 2005 Aeroplane
The report’s comparison table runs the Mach 0.80 configuration against a representative best-in-class single-aisle aeroplane of the previous decade, designated 765-093, with a CFM56 class engine and an effective aspect ratio of 10.41 [SUGAR Phase IV final report].
| Configuration | Aspect ratio | Block fuel per seat at 900 nautical mile, pound | Reduction, percent |
|---|---|---|---|
| Best in class baseline | 10.41 | 91.51 | base |
| Mach 0.745 truss-braced wing | 19.565 | 39.31 | 57.04 |
| Mach 0.80 truss-braced wing | 19.565 | 40.39 | 55.86 |
The whole package against the old aeroplane is 55.86 percent, not thirty, by the reduction $r_i$ displayed in the next subsection, and the report gives the figure, attributing a 55.9 percent reduction relative to a best-in-class single-aisle transport typical of the 2008 fleet.
So the thirty percent sits between two published numbers and is neither of them. It is larger than the 7.2 percent the wing earns against an advanced conventional baseline and smaller than the 55.86 percent the whole package earns against a two-decade-old one. The ratio of the headline to the wing’s own contribution is 4.17, a measure of how much of the thirty percent is about the shape of the aeroplane and how much is about everything else being new as well.
And Raising the Cruise Mach Number Cost Fuel
The Phase IV configuration is slower-than-optimal by design and faster than its predecessor, and the exchange is quantified in the report’s tables. Phase III designed for Mach 0.745 and Phase IV for Mach 0.80, and the reason is market rather than physics.
The higher cruise Mach number enables increased aircraft productivity, and is believed to more closely align to market demand in the single-aisle domestic passenger transport class.
The cost appears in two places. Cruise lift-to-drag ratio falls from 24.958 to 24.708, a fall of 1.00 percent. And block fuel per seat rises from 39.31 to 40.39 pound, a rise of 2.75 percent.
The report expresses the same thing as 1.1 percent, and the relation between the two statements is exact and worth displaying. Let $\beta_0$ be the baseline block fuel per seat and $\beta_1$ and $\beta_2$ the two configurations.
\[r_i = 1 - \frac{\beta_i}{\beta_0}, \qquad r_1 - r_2 = \frac{\beta_2 - \beta_1}{\beta_0}, \qquad \frac{\beta_2}{\beta_1} - 1 = \frac{\beta_2 - \beta_1}{\beta_1} \tag{86}\]The numerators are identical and the denominators are not, so the ratio of the two statements is a pure number set by the baseline alone.
\[\frac{\text{percentage change in fuel}}{\text{percentage-point change in reduction}} = \frac{\beta_0}{\beta_1} = 2.3279 \tag{87}\]Its sentence reads that operating at Mach 0.80 has reduced the fuel burn benefit by 1.1 percent compared with the Mach 0.745 aircraft [SUGAR Phase IV final report]. That 1.1 is the change in a percentage reduction, from 57.04 to 55.86, a difference of 1.18 percentage points. The change in fuel is 2.75 percent. Both statements are true about different quantities and only one of them is what an airline pays, and they differ by exactly the factor above, which is 2.3279 and not the two and a half an eye would guess.
The compressibility accounting makes the comparison subtler than it looks and the report flags it. In its bookkeeping all drag due to Mach number is carried as compressibility drag, and the natural-laminar-flow credit for the wing upper surface is carried there too, and that credit makes the term negative in both buildups. The Mach 0.80 configuration’s compressibility term is -0.00143 against the predecessor’s minus 0.00086, and their induced terms are 0.00903 and 0.01038 against totals of 0.02695 and 0.02916, so the laminar credit grew while the wave drag grew, and the two cannot be separated from the published table [SUGAR Phase IV final report]. This article therefore does not attempt to apportion the Mach penalty between wave drag and lost laminar flow, and records that the published accounting does not permit it.
The Flight Test Record
No X-66A has flown and none exists. The record at the article’s dateline consists of a donor airframe, a modification programme stopped before its wing was committed to fabrication, and an agreement with two thirds of its milestones unexercised.
What was accomplished is nonetheless substantial and is not flight test. The donor MD-90 was reactivated and ferried from Victorville to Palmdale in August 2023. That step is the agreement’s Milestone 5, named Aircraft Reactivation and due February 2024, so it ran roughly six months ahead of its milestone date [Boeing arrival release] [Funded Space Act Agreement]. The modification was to shorten the fuselage and replace the wings and engines [NASA X-plane release]. Through the February 2025 milestone the agreement had reached nine of 27 milestones, the share of its value computed in the subsection headed The Word the Agreement Does Not Contain.
The wind-tunnel record, by contrast, is long and is the programme’s real output. Phase II tested an aeroelastic model in the Transonic Dynamics Tunnel. Phase III tested a transonic performance model in the Ames eleven-foot transonic wind tunnel. Phase IV tested both high-speed and low-speed models from 2019, with the results in volumes this article did not read in full [SUGAR Phase IV final report]. The X-66A was to be the end of a fifteen-year test campaign and the campaign is almost all of what exists.
Boeing’s release of August 2023 gave 2028 for the start of ground and flight testing [Boeing arrival release], and the agreement’s Milestone 24 gives first flight as September 2028, so the manufacturer’s date and the primary instrument agree. Neither has been revised in any public source read for this article, and the pause of April 2025 leaves no aircraft to meet either, a useful reminder that a designation register and a signed schedule are records of intent and not of fact.
Comparison With Ground Prediction
No X-66A flew, so flight returned nothing against which any ground prediction can be set. The record holds a long series of predictions made by wind tunnel, by analysis and by contractor estimate, and every one of them still stands where the ground left it. This section gathers them and points to where each is argued.
The flutter prediction is contradicted by the ground record itself. Two ground analyses of the same structure at Mach 0.92 disagree about the sign of the flutter margin [SUGAR Phase IV final report], as set out under Dependent Systems, and no flight exists to say which is right.
The interference prediction is a wind-tunnel result with no flight counterpart. Phase III reduced the interference between the wing and the strut to approximately one percent of total airplane drag at the design Mach number, using Navier-Stokes methods and a series of transonic wind-tunnel tests [SUGAR Phase IV final report]. That figure was never measured on an aeroplane in flight.
The report itself declines to trust the buffet and low-speed predictions. It states that buffet boundaries are derived from cantilever aircraft and are unvalidated for this configuration, that its low-speed data was gathered at very low Reynolds number, and that the ice-effects and in-ground-effect objectives were deferred to a later test entry [SUGAR Phase IV final report]. These are the questions a flight test was most needed to answer, and What the Data Changed, below, sets them out as the things flight would have settled.
The performance predictions are contractor estimates compared only against other estimates. Boeing’s benefit for the truss-braced wing and the smaller one the NASA Ames team found at the economic mission are two ground predictions in disagreement [Recine et al. 2025a], set out in the subsection headed The Comparison Was Done. That disagreement is the closest thing to a test of the headline number that the record contains, and no flight measured fuel burn on either mission.
This article’s own predictions are also unchecked by flight. The structural price of the Phase IV thinning and sweeping, derived above under What the Thinning Cost in Structure, lacks even a ground counterpart, and the optimality conditions, the value of the fold and the drag-divergence margins test nothing that flew.
The schedule was a prediction too. The agreement’s Milestone 24 placed first flight in September 2028 [Funded Space Act Agreement], and the section headed The Flight Test Record records that no aircraft exists to meet it. The comparison this section exists to make is therefore empty, and what it can offer is the list of predictions that a flight would have tested.
What the Data Changed, or What It Would Have Changed and What the Programme Changed Anyway
The flight test would have settled four things and three of them are named in the primary record as open.
The buffet boundary of a braced wing. The prediction methods are calibrated on cantilever wings, the report calls their applicability unvalidated, and it is unknown whether the local buffeting the analysis predicts would be perceived by a pilot at all [SUGAR Phase IV final report]. No wind tunnel answers the perception question and only a flight test does.
The flutter boundary at the highest transonic condition. Two models disagree by about fifty percentage points of margin and about the sign at Mach 0.92, the report judges the method incapable of capturing the flow features, and it recommends that the analysis be done about a preloaded, largely deformed state [SUGAR Phase IV final report]. A flight test would not have resolved the modelling question, but it would have bounded the answer.
The maximum lift coefficient at flight Reynolds number. The low-speed data is at very low Reynolds number, the report describes higher-Reynolds-number data as critically important for sizing the vision vehicle, and the field length that data feeds was an active constraint on the span [SUGAR Phase IV final report] [SUGAR Phase II volume I].
And the weight. The report’s assessment of its largest remaining uncertainty is unambiguous.
Perhaps the most significant remaining uncertainty in the TTBW concept is the maturity of structural concepts and associated vehicle weight.
This article’s keystone depends on that sentence more than on any other, and that dependence is why the keystone is stated as a conditional. The optimality conditions derived above are exact. The elasticity that has to be put into them is a weight derivative, and the organisation that knows most about this configuration says its weight is the thing it knows least well.
The Comparison the Report Asked For and Nobody Has Published
The recommendation list in the Phase IV report contains one item that undercuts the figures everybody quotes.
Perform equivalent conceptual-level design, sizing and optimization studies of cantilever and truss-braced wing aircraft of the same technology level to allow a more fair and transparent comparison of the two configurations.
Boeing’s own final report asks for the comparison on which the entire concept rests to be put on an equal footing, which is a statement that it was not. The 7.2 and 9.0 percent figures are against an aspect ratio 13 conventional aeroplane of equivalent technology, and the report is asking for both sides of that comparison to be optimised to the same standard before the numbers are believed.
The Comparison Was Done, and It Quarters the Short-Mission Benefit
A NASA Ames team did it. The paper was presented at the January 2025 SciTech forum, fifteen months after the October 2023 report that made the recommendation and three months before the pause, and it compares the transonic truss-braced wing with an advanced tube-and-wing aeroplane under a single consistent set of assumptions [Recine et al. 2025a].
What makes it the fair comparison is the list of things it holds equal. The same payload of 33,750 pound, the same design range of 3,400 nautical mile, the same design Mach number, the same 2035 aerodynamic and structural technology, the same advanced direct-drive turbofan with a fan pressure ratio of 1.50, the same wing-fold rule above 118 feet, and critically a tube-and-wing-based weight calibration applied to both rather than the contractor’s own calibration applied to one of them [Recine et al. 2025a].
| Quantity | Advanced tube and wing | Truss-braced, fuel in wings | Truss-braced, body tanks allowed |
|---|---|---|---|
| Aspect ratio | 13 | 19.57 | 19.57 |
| Wing span, foot | 128.7 | 170.7 | 155.2 |
| Wing area, square foot | 1,317.7 | 1,489.6 | 1,231.2 |
| Gross take-off weight, pound | 140,073 | 139,853 | 132,843 |
| Empty weight, pound | 72,435 | 73,863 | 67,586 |
| Cruise altitude, foot | 40,500 | 44,750 | 42,750 |
| Start-of-climb lift-to-drag ratio | 20.9 | 23.6 | 22.8 |
| Sea-level static thrust, pound | 18,525 | 23,277 | 20,810 |
| Fuel burn at 3,400 nautical mile, pound | 24,201 | 22,384 | 21,929 |
| Fuel burn at 900 nautical mile, pound | 7,204 | 7,085 | 6,793 |
At the long mission the two accounts agree closely. At the economic mission they do not agree at all.
| Mission | Boeing Phase IV | Ames, fuel in wings | Ames, body tanks allowed |
|---|---|---|---|
| 900 nautical mile | 7.2 percent | 1.65 percent | 5.71 percent |
| About 3,400 to 3,500 nautical mile | 9.0 percent | 7.51 percent | 9.39 percent |
On the economic mission Boeing’s figure is 4.36 times the Ames figure for a wing that must carry its own fuel, and 1.26 times the figure with body tanks allowed. On the long mission the three figures are within a point of one another. So the disagreement is not about the aerodynamics. It is about the mission.
The paper names the two causes and ranks them. The first is fuel volume, which this article raised from the Phase IV tables and could not price. Its summary names the effect as Body Tanks Allowed vs. All Fuel in Wings, describes the mechanism as the thin high-aspect-ratio wing being unable to carry all the fuel without a significant increase in the planform area, and rates its impact on the crossover range as Significant, putting it between under three hundred and about two thousand nautical mile.
The second is the weight calibration, the quantity the Phase IV report calls its largest uncertainty.
SUGAR reports vs. T&W […] The TTBW empty weight and fuel burn increase when using the T&W calibration […] Moderate
So the two uncertainties this article names above, fuel volume and weight, are exactly the two the fair comparison finds to dominate, and the fuel-volume one is the larger. That is an uncomfortable outcome for the seven point two percent.
The Short-Mission Loss Is Not a Weight Effect
The aerodynamic advantage in the fair comparison is real and large. Start-of-climb lift-to-drag ratio is 12.92 percent better. What the short mission does is spend it. Each figure below is the truss-braced value with fuel in the wings set against the tube-and-wing value in the table above.
\[\frac{23.6}{20.9} - 1 = 0.1292, \qquad \frac{23{,}277}{18{,}525} - 1 = 0.2565, \qquad 44{,}750 - 40{,}500 = 4{,}250 \ \text{ft} \tag{88}\]The braced aeroplane needs 25.65 percent more sea-level static thrust and cruises 4,250 feet higher, and the paper names the cost.
The TTBW spends more time climbing and cruises higher, pronouncing differences in engine efficiency during climb and cruise
That is a limit on this article’s keystone and it should be stated as one. The optimality conditions derived above are cruise-fuel conditions. On a nine-hundred-nautical-mile sector a single-aisle aeroplane does not spend most of its fuel in cruise, and an aspect ratio chosen to minimise cruise fuel is being chosen against the wrong objective for the mission the aeroplane actually flies most often. The keystone is correct about cruise and silent about climb.
And It Independently Corroborates the Keystone’s Quantity
The fair comparison is a like-for-like pair of aeroplanes differing in aspect ratio, which is the closest thing to a measurement of the weight elasticity that this subject’s public record contains. Taking logarithms of the weight ratio over the aspect-ratio ratio gives an apparent elasticity.
\[\nu_{\mathrm{apparent}} = \frac{\ln\left(W_2/W_1\right)}{\ln\left(A_2/A_1\right)} \tag{89}\]| Pair | Apparent elasticity on gross weight |
|---|---|
| Fuel in wings | -0.00384 |
| Body tanks allowed | -0.12956 |
| Weight table, contractor calibration | -0.00282 |
| Weight table, tube-and-wing calibration | +0.09926 |
Every one of the four is below the article’s exact fixed-lift target of 0.299033, and three of the four are negative. The largest is 0.09926, falling 66.8 percent short of the target. A negative elasticity means the braced aeroplane at aspect ratio 19.57 weighs less than the cantilever at aspect ratio 13, and on this criterion the aspect ratio should be pushed further still. The shortfall is measured as for the Phase IV readings above.
\[1 - \frac{0.09926}{0.299033} = 0.668 \tag{90}\]An apparent elasticity from a pair is not a derivative and this article does not treat it as one, because the two aeroplanes differ in span, area, thrust and altitude as well as in aspect ratio. What it establishes is magnitude, and the magnitude agrees with the four readings this article computed from the Phase IV group weight statement, which ran from 0.0775 to 0.2749. Four routes, one conclusion.
And the same table prices the truss directly against a cantilever of the same technology, the like-for-like comparison the Phase IV report had asked somebody to make [Recine et al. 2025a].
| Quantity | Cantilever wing | Braced wing plus strut |
|---|---|---|
| Contractor calibration | 12,867 pound | 16,030 pound, +24.58 percent |
| Tube-and-wing calibration | 12,867 pound | 16,483 pound, +28.10 percent |
Each entry is the braced weight over the cantilever’s, less one, the wing alone being 13,045 pound on the contractor’s calibration and 13,418 pound on the tube-and-wing one, and the strut 2,985 and 3,065 pound.
\[\frac{13{,}045}{12{,}867} - 1 = 0.0138, \qquad \frac{13{,}418}{12{,}867} - 1 = 0.0428, \qquad \frac{13{,}045 + 2{,}985}{12{,}867} - 1 = 0.2458, \qquad \frac{13{,}418 + 3{,}065}{12{,}867} - 1 = 0.2810 \tag{91}\]The wing alone is only +1.38 percent heavier on the contractor’s calibration and +4.28 percent on the harsher one, so the strut is almost the whole penalty. That is a direct measurement of what this article derived, namely that a geometrically similar truss buys a coefficient rather than an exponent. A wing of aspect ratio 19.57 weighing within a few percent of a cantilever wing of aspect ratio 13 is the three-halves law with a relief factor in front of it, and the strut is the bill for that factor.
What the Programme Changed Without Flying
The programme’s clearest consequence is that its last recommendation was executed.
Conduct the preliminary design phase of a possible transonic truss-braced wing demonstrator aircraft.
The X-66A is that preliminary design phase, scheduled through four design reviews before the pause and stopped before the fifth. What the agreement bought, in the end, was the thing the report asked for in its last line, and nothing beyond it. The FY2026 technical supplement records the redirection as a project objective rather than a disappointment, committing NASA to a ground-based, full-scale wing demonstration of integrated thin-wing technologies and to completing the planning for it in the first quarter of fiscal year 2026 [FY2026 technical supplement].
And the thing it kept is the thing this article’s analysis says was carrying the value. The fold is worth 36.0 percent of lift-to-drag ratio and the remaining aerodynamic optimum is worth 0.9481 percent, so the truss’s structural contribution is not where the fuel is. The thin wing, which raises the drag-divergence Mach number and permits less sweep and therefore more laminar flow, is applicable to aeroplanes with and without truss braces, a point Boeing’s proposal made when it redirected the effort [NASA thin-wing item] and the airport-operations memorandum had made about the physics a month earlier. On this article’s reading, the redirection kept the part of the concept that generalises and dropped the part that was specific to it.
The Contemporary Literature
| Cluster | Records | What it holds |
|---|---|---|
| Braced wings | 244 | the truss-braced and strut-braced wing itself, the junction and the jury strut |
| Aspect ratio and span | 253 | induced drag, span loading, span efficiency and the high aspect ratio wing |
| Alternative configurations | 423 | joined, box, tandem and annular wings, blended wing bodies, flying wings and canards |
| Wing weight | 70 | wing weight estimation, bending material and wing box sizing |
| Thin transonic wings | 259 | thickness, sweep, drag divergence and supercritical sections |
| Aeroelasticity | 906 | flutter, divergence, limit-cycle oscillation and aeroservoelasticity of wings and aircraft |
| Gust and manoeuvre loads | 344 | gust and manoeuvre loads, load alleviation and flexible-aircraft dynamics |
| Span constraints | 40 | folding wings and wingtips, airport compatibility and span limits |
| Laminar flow | 298 | natural laminar flow, hybrid laminar flow control, transition and crossflow |
| High lift | 187 | high-lift systems, buffet onset, stall, icing, wingtip devices and ground effect |
| Propulsion integration | 194 | nacelle and wing interference, high bypass ratio, geared turbofans and open rotors |
| Fuel burn | 104 | fuel burn, Breguet, lift-to-drag ratio, advanced transports and the standard atmosphere |
| Flight demonstrators | 75 | flight demonstrators, research aircraft programmes and flight test technique |
| Ground testing | 145 | full-scale static and fatigue tests, ground vibration tests, loads calibration and strain sensing |
| Multidisciplinary design | 141 | multidisciplinary optimisation, conceptual design, sizing and aerostructural design |
| Emissions | 164 | aviation emissions, contrails, aircraft noise, sustainable aviation fuel and policy |
| Programme designations | 28 | the programme’s own reports, the Sustainable Flight National Partnership and the donor lineage |
| Total | 3,875 | every research record, each in one cluster |
This subject has a large and continuously active literature. The truss-braced wing has its own body of work with a fifty-year history, a named research programme, a sustained Virginia Tech and Georgia Institute of Technology school [NASA truss-braced wing studies] [SUGAR Phase II volume I], and a steady output of conference and journal papers through the whole period of this programme.
Every research record is assigned, by a reading of its title, to exactly one of seventeen clusters, the one whose subject the title names, and the clusters are the shape of the field as the reference survey records it.
The largest cluster is aeroelasticity at 906 records and that is the correct shape for this subject. A very high aspect-ratio wing is an aeroelastic problem before it is an aerodynamic one, and the field’s output reflects that. The smallest substantive cluster is span constraints at 40 records, and that asymmetry is itself a finding. The constraint this article argues is binding is the one the literature has written least about.
The weight cluster is also thin at 70 records, which matters because weight is the quantity the Phase IV report names as the largest uncertainty and the quantity the keystone needs. A field that has produced 906 records on aeroelasticity and 70 on wing weight has its effort allocated to the risk it can compute rather than to the risk that decides the answer. That is an observation about the literature and not a criticism of any paper in it.
Braced wings
The configuration’s own literature. Truss-braced and strut-braced wings of every period, from a 1935 analysis of jury-strut systems to the Mach 0.80 configuration, treated for aerodynamics, interference at the junction, structure and weight, flutter, buffet, noise and multidisciplinary sizing. 156 of these 244 records date from 2015 or later, so the most of the configuration’s own literature belongs to the programme’s own period.
244 records. [Anderson et al. 2023] [Aretskin-Hariton et al. 2024] [Bartels 2013] [Bartels et al. 2014] [Bartels et al. 2019] [Park et al] [Bhatia et al. 2009] [Bhatia et al. 2010] [Bhatia et al. 2011] [Bhatia et al. 2012] [Bowen et al. 2024] [Mason et al] [Brooks et al. 2025] [Browne et al. 2024] [Browne et al. 2025] [Butt et al. 2012] [Camacho et al. 2020] [Recine et al] [Carrier et al. 2022] [Carrillo et al. 2024] [Chakraborty et al. 2014] [Chakraborty et al. 2015] [Chau and Zingg 2021] [Chau and Zingg 2022a] [Chau and Zingg 2022b] [Chau and Zingg 2023] [Chau et al. 2026] [Chen et al. 2015] [Chepurnykh 2015] [Jutte et al] [Clifford et al. 2024] [Wells 2017] [Maldonado et al] [Dela Peña 2026] [Delavenne et al. 2022] [Demandel 2024] [Demasi et al. 2018] [Dollyhigh et al. 1977] [Drain et al. 2017] [Dym and Williams 2015] [Aretskin-Hariton et al] [Anderson et al] [Anderson et al] [Ferris and Khorrami 2024] [Nguyen et al. 2019] [Peña and Park 2024] [Fugate et al. 2019] [Gern et al. 1999] [Gern et al. 2000c] [Gern et al. 2000b] [Gern et al. 2001a] [Gern et al. 2001b] [Gern et al. 2005] [Gern et al. 2000a] [Gilbrook et al. 2023] [Grasmeyer 1999] [Grossman et al. 2000] [Gu et al. 2017] [Gundlac et al. 2000] [Gundlach et al. 2000] [Gupta et al. 2014] [Gur et al. 2009] [Gur et al. 2010b] [Gur et al. 2010c] [Gur et al. 2010a] [Gur et al. 2011b] [Gur et al. 2011a] [Han et al. 2026] [Harrison et al. 2020] [He and Angland 2024] [He and Angland 2026] [Hospodář et al. 2020] [Hosseini and Vaziri-Zanjani 2020] [Hosseini et al. 2020] [Hosseini et al. 2024b] [Hothazie et al. 2025] [Hothazie et al. 2026] [Huang et al. 2025] [Huang et al. 2026] [Xiong et al. 2019b] [Ivaldi et al. 2015] [Jamal et al. 2025] [June et al] [June et al. 2022] [Berton 2022] [Jones et al. 2023] [June et al. 2025] [Xiong and Nguyen] [Xiong and Nguyen] [Xiong and Nguyen] [Xiong and Nguyen] [Xiong et al] [Xiong et al] [Xiong et al] [Xiong et al] [Xiong et al] [Xiong et al] [Xiong et al] [Xiong et al] [Jutte et al. 2024] [Kacik and Goetzendorf-Grabowski 2025] [Kapania et al. 2022] [Kapania et al. 2018] [Kaul 2024] [Kelly et al. 2025] [Khan et al. 2023] [Scarberry and Sams 2021] [Ko et al. 2000] [Kugler 2026] [Langford et al. 2026] [Lannoo and Priasso 2025] [Leader et al. 2024] [Lebofsky et al. 2015] [Lee et al. 2017] [Machado et al] [Li et al. 2022] [Li et al. 2023b] [Li et al. 2024b] [Liu and Yu 2021] [Liu et al. 2022] [Locatelli et al. 2016] [Losada Costoso et al. 2024] [Ma et al. 2021] [Ma et al. 2022] [Ma et al. 2023b] [Ma et al. 2023a] [Ma et al. 2024a] [Ma et al. 2026] [Machado et al. 2024] [Maldonado et al. 2021] [Maldonado et al. 2020] [Mallik et al. 2013] [Mallik et al. 2015b] [Marchetti 2023] [Leader et al] [Mason et al. 2025] [Meadows et al. 2012] [Meinicke et al. 2017] [Metkowski and Maughmer 2021] [Metkowski and Maughmer 2023] [Moerland et al. 2017] [Mohd et al. 2025] [Mori et al. 2024] [Nagy et al. 2022] [Nagy et al. 2025] [Nam et al. 2014] [Harrison et al] [Nguyen and Xiong 2022b] [Nguyen and Xiong 2022a] [Nguyen and Xiong 2023a] [Nguyen and Xiong 2024b] [Nguyen and Xiong 2024a] [Nguyen and Xiong 2023b] [Nguyen et al. 2021b] [Nguyen et al. 2021a] [Nguyen et al. 2015a] [Nguyen and Xiong] [Nguyen and Xiong] [Nguyen and Xiong] [Nguyen and Xiong] [Nguyen and Xiong] [Nguyen and Xiong] [Nguyen et al] [Nguyen et al] [Nguyen et al] [Browne et al] [Browne et al] [Chiozzotto 2016] [Park et al. 2025] [Ravetta et al] [Pelykh and Andryushchenko 2024] [Perkins et al. 2022] [Perkins et al. 2023] [Pham et al. 2023] [Turriziani et al. 1980] [Quinn et al. 2025] [Rahman and Akbar 2026] [Ravetta et al. 2024] [Recine et al. 2025b] [Campbell et al] [Rossmann et al. 2015] [Ruffini et al. 2020] [Ferris and Khorrami] [Saadeh et al. 2026] [Viken et al. 2022] [Sarojini et al. 2022] [Schwartz and Bogert 1935] [Seber et al. 2011] [Secco and Martins 2018] [Secco and Martins 2019] [Kelly et al] [Kelly et al] [Zelinski et al] [Smith et al. 1981] [Sohst et al. 2022] [Solano et al. 2023] [Sticchi et al. 2024] [Sticchi et al. 2026] [Xiong et al. 2021] [Su 2017] [Sulaeman et al. 2001] [Sulaeman et al. 2002] [Sulaeman et al. 2003] [Sy et al. 2026] [Szczyglowski et al. 2018] [Szczyglowski et al. 2019] [Taflan and Gulum 2026] [Taflan et al. 2023] [Taflan et al. 2026] [Jamal et al] [Jamal et al] [Thomas and Agarwal 2023] [Thomas and Agarwal 2025] [Brooks] [Chau and Duensing] [Chau et al] [Brooks et al] [Ting et al. 2017] [Ting et al. 2014] [Lauderdale et al] [Toffol 2023] [Xiong and Nguyen 2024b] [Nguyen and Xiong 2021a] [Tsatsas et al. 2025] [Vale and Albuquerque 2025] [Variyar et al. 2016] [Vladut et al. 2025] [Vos et al. 2020] [Wahler et al. 2025] [White et al. 2015] [Xiong and Nguyen 2022] [Xiong and Nguyen 2023] [Xiong and Nguyen 2024a] [Xiong et al. 2022b] [Xiong et al. 2022a] [Xiong et al. 2022c] [Xiong et al. 2023b] [Xiong et al. 2023a] [Xiong et al. 2026a] [Xiong et al. 2019a] [Xiong et al. 2020] [Yue 2026] [Windous and Quinlan] [Zelinski et al. 2025]
Aspect ratio and span
The aerodynamic half of the keystone. Induced drag and its minimisation, span loading, wingtip devices, and the aerodynamics and structure of high aspect ratio wings, including the 1938 selection of an optimum aspect ratio for a cantilever sailplane.
253 records. [Abdel-Motaleb et al. 2019] [Ahmad et al. 2013b] [Alley and Phillips 2010] [Alley et al. 2007] [Ando and Yamamoto 1976] [Aoki and Muraoka 2019] [Ashenberg and Weihs 1984] [Balunov et al. 2023] [Barnes and Barnes 1997] [Bartels 2023] [Ben-Gida et al. 2019] [Bhuwal et al. 2025] [Birch et al. 2004] [Blondeau and Pines 2003] [Blondeau and Pines 2004] [Blondeau et al. 2003] [Bloy and Jouma’a 1995] [Borisova and Silantiev 2022] [Bottai et al. 2022] [Bourdin 2001] [Bowers 2007] [Bowers 2015] [Bowers 2018] [Bowers et al. 2016] [Brune and Bogataj 1990] [Burkett 1989] [Thinesh and Harmin 2018] [Calderon et al. 2019] [Castellani et al. 2016a] [Castles 1938] [Catalano et al. 2024] [Chai et al. 2025] [Chan et al. 2024b] [CHAPTER SEVEN. Wings 1990] [Chen 2026] [Chen and Katz 2004] [Cheng 1982b] [Chiocchia and Pignataro 1995] [Forte and Nguyen] [Coetzee et al. 2023] [Corsiglia et al. 1976] [Crean 1937] [Céron-Muñoz et al. 2013] [de Souza and Ciloni 2021] [Dehaan 1990] [Delavenne et al. 2020] [Demasi et al. 2014a] [Demasi et al. 2014b] [Demasi et al. 2016a] [Demasi et al. 2017] [Demasi et al. 2019] [Deneke et al. 2026] [Deyoung 1979] [Deyoung 1980] [Deyoung 1947] [Deyoung 1951] [Deyoung 1952] [Diehl 1922] [Ding 2024] [Erickson et al. 1971] [Forte and Nguyen 2024] [Fournis and Gaultier 2026] [Frank et al. 2008] [Frediani et al. 2001] [Frost and Rutherford 1963] [Fu et al. 2021] [Gaifullin 2023] [Gainer and Aiken 1959] [Gerontakos and Lee 2006] [Gilman and Burdges 1967] [Golshany et al. 2024] [Grant et al. 2006] [Greene 1988] [Greene 1989] [Gunasekaran and Mukherjee 2016] [Gunturu and Jain 2025] [Gutierrez et al. 1994] [Haftka 1977] [Hammer et al. 2021a] [Hammer et al. 2021b] [Hammond and Hayes 1954] [Han et al. 2007] [Hancock 1971] [Hancock 1972] [Hanin and Barsony-Nagy 1983] [Hattasanjaya 2024] [Hayes et al. 2019] [Henry et al. 2005] [Hicken and Zingg 2010] [Hirschel et al. 2020a] [Hirschel et al. 2020b] [HlCKEY 1969] [Hunsaker and Phillips 2020] [Hunsaker et al. 2017] [Hunsaker et al. 2019] [Huo et al. 2013] [Ichikawa 1996] [Induced Drag Effect 1976] [Iosilevskii 2025b] [Iosilevskii 2025a] [Jamwal et al. 2018] [Quindlen et al] [Jordan 1971] [Jung et al. 2008] [Kendall 1984] [Kendall 1985] [Khazaee Kuhpar and Seyed-Aghazadeh 2026] [Khrabrov 2007] [Kida 1982] [Kida and Miyai 1978] [Kim 2012] [Kim et al. 2009a] [Kjelgaard and Paulson 1981] [Klein and Viswanathan 1973] [Klein and Viswanathan 1975] [Kolonay et al. 2004] [Kontou et al. 2024] [Kretov and Tiniakov 2022] [Kroo and Smith 1990] [Kuhlman and Brown 1989] [Kuhlman et al. 1988] [Kuhlman 1979] [Kuhlman and Liaw 1987] [Kumar and Aravindan 2014] [Kwon and Vepa 2022] [Laitone 1978] [Lam 1993] [Lam and Maull 1993] [Lampropoulos and Sarris 2025] [Boddy 1946] [Letcher 1972] [Liersch et al. 2009] [Lissaman 1973] [Liu et al. 2020] [Lockwood Taylor 1942] [Long 2011] [Long et al. 2012] [Loth and Boyle 1969] [Loth and Loth 1984] [Lottati 1984] [Lowson 1990] [Lundry 1967] [Lundry 1977] [Lundry and LlSSAMAN 1968] [Lyapunov 1993] [Gunasekaran and Mukherjee 2017] [Maikapar 2000] [Malathi et al. 2024] [Mansour 1985] [Mansour 1984] [Mardanpour et al. 2018b] [Marino 2004] [Martin et al. 2025] [Martins et al. 2014] [Masson et al. 1999] [Mathias et al. 1995] [McCutchen 1989] [Mian et al. 2014] [Momodu et al. 2025] [Monsch et al. 2007] [Moore 2021] [Mortara and Maughmer 1993] [Mr. S. Aravindan et al. 2017] [Mwasandube et al. 2026] [Nan et al. 2024] [Nathman 2005] [Nederlof et al. 2023] [Neely et al. 1947] [Nguyen et al. 2017b] [Nikolaev 2019] [Nixon 2003] [Panaro et al. 2009] [Pao and Lan 1978] [Pate and German 2013] [Pavlenko and Reslan 2022] [Pearson 1937] [Pedrosa Reis] [Petrov 2009] [Pham 2026] [Phillips et al. 2006] [Phillips et al. 2019b] [Phillips et al. 2019a] [Phillips et al. 2020] [Picchi Scardaoni 2020] [Povitsky and Pierson 2020] [Prabhu et al. 1985] [Whitehead] [Reddy 1989] [Report No. 121 1921] [Report No. 349. 1930] [Reubush 1977] [Revell 1975] [Rice and Oetting 1976] [Riso et al. 2020] [Rossow 1975] [Rusovici et al. 2017] [Russo et al. 2021] [Kandasamy et al. 2018] [Sanders 1965] [Sanghi et al. 2020] [Sanghi et al. 2025] [Satriya and Kadir 2023] [Schmidt and Duren 2002] [Sears 1974] [Sears 1990] [Setyo Hariyadi et al. 2021] [Sibert 1943] [Norman and Spreeman 1949] [Skomorokhov et al. 2014] [Smith and Srokowski 1983] [Smith and Campbell 1996] [Smith 1996] [Smith and Kroo 1990] [Snyder and Zumwalt 1969] [Sogukpinar 2019] [Song et al] [Song et al. 2025] [Song et al. 2026] [Stark 1985] [Straussfogel and Maughmer 1991] [Sugimoto 1992] [Sundaram et al. 2024] [Takahashi 2011] [Takahashi 2012] [Takahashi and Barchfeld 2016] [Takahashi and Donovan 2011] [Tamayama et al. 2017] [Taylor and Hunsaker 2020] [Tinling and Kolk 1951] [Tuktarov and Chedrik 2015] [Ullah et al. 2022] [Ulrich et al. 2001] [Visbal and Garmann 2017] [Wang and Zha 2019] [Wang et al. 2013a] [Wang et al. 2025c] [Wentz and Nagati 1975] [Whitcomb 1947] [Wieszala et al. 2022] [Wilmott 1986] [Wintzer 2010] [Wu et al. 2014] [Wu et al. 2025c] [Xiong et al. 2025] [Yamamoto 1992] [Yang et al. 2017] [Yang et al. 2025a] [Yeh and Plotkin 1985] [Yuan et al. 2018] [Zhang 2024] [Zheng et al. 2026b] [Zhu et al. 2009] [Zong et al. 2021]
Alternative configurations
The braced wing’s neighbours. The other configurations that buy span or relieve bending by geometry, being joined, box, closed, tandem and annular wings, blended wing bodies and flying wings, canards, and wing-in-ground-effect craft.
423 records. [Aarons et al. 2011] [Adler and Martins 2024] [Олег Львович Лемко and Євген О. Кушніренко 2014] [Ahmed and Goonaratne 2002] [Ahuja et al. 2026] [Ali et al. 2012] [Alipour et al. 2022] [Ammar and Trépanier 2014] [Ammar et al. 2017] [Andrews and Perez 2015a] [Andrews and Perez 2015b] [Anwar 2025] [Aubeelack] [Aubeelack et al. 2023] [Bagwill and Selberg 1996] [Bagwill et al. 1997] [Baigang 2024] [Bainsla et al. 2021] [Balow et al. 1993] [Batina 1985] [Belardo et al. 2014] [Beran et al. 2005] [Bergmann and Hummel 2001] [Bertsch et al. 2025] [Bhasin et al. 2012] [Blair and Canfield 2002] [Blair et al. 2005] [Blair et al. 2008] [Bond et al. 2007] [Bond et al. 2008c] [Bond et al. 2008a] [Bond et al. 2008b] [Bond et al. 2010] [Bond et al. 2012b] [Bond et al. 2012a] [Boozer et al. 2017] [Bravo et al. 2026] [Broering and Lian 2010] [Brown and Timmerman 1991] [Brown and Vos 2018] [Buch and George M. 2010] [Buddhamatya et al. 2026] [Burkhalter et al. 1992] [Butler 1982] [Butler 1983] [Cai et al. 2014] [Cameron and Cameron 1997] [Cardoso et al. 2013] [Carlsson 2003] [Carlsson 2005] [Carlsson and Kuttenkeuler 2003] [Carter et al. 2006] [Cesnik and Brown 2003] [CFD Analysis of Box 2016] [Chan et al. 2024a] [Chau and Zingg 2017] [Chen and Han 2017] [Chen et al. 2019] [Chen et al. 2026a] [Chung et al. 2024] [Chung et al. 2025] [Cipolla et al. 2020] [Clark and Gerhold 1999] [Clyde et al. 1984] [Colas et al. 2018] [Cook and de Castro 2004] [Cunha et al. 2018] [Dakka and Johnson 2019] [De Silva and Carmichael 1978] [Dehpanah and Nejat 2015] [Demasi 2007] [Demasi and Livne 2005] [Demasi and Livne 2007] [Demasi and Livne 2008] [Demasi et al. 2015] [Demasi et al. 2016b] [Denieul et al. 2015] [Denieul et al. 2018] [Deslich and McHugh 2024] [Driver 1958] [Du Cray et al. 2025] [Durmus 2021] [Durmuş 2025] [Economou 2025] [Feistel 1985] [Feldstein et al. 2023] [Figat 2018] [Fu et al. 2024] [Fu et al. 2026] [Gagnon and Zingg 2015] [Gagnon and Zingg 2016] [Gainer et al. 1984] [Gallman and Kroo 1996] [Gallman et al. 1993] [Gallman and Kroo 1992] [Gallman et al. 1990a] [Gangoli Rao et al. 2014] [Gangoli Rao et al. 2017] [Gao and Smith 2020] [Gatto et al. 2010] [Gauvrit-Ledogar et al. 2018] [Gebbie et al. 2005] [Gebbie et al. 2007] [Ghasemikaram et al. 2021] [Gingerich et al. 2026] [Gloss et al. 1974] [Goldberg et al. 2017] [Granzeier 2002] [Grauer et al. 2012] [Gray and Zingg 2024] [Griffin et al. 1983] [Gu et al. 2022a] [Gu et al. 2023b] [Guo et al. 2025] [Guo and Thomas 2015] [Guo et al. 2014b] [Guo et al. 2015c] [Gupta et al. 2016] [Guruswamy 1991] [Guynn et al. 2004] [Hajela and Chen 1986] [Handa and Srinivas 2022] [Hanman et al. 2025] [Hansen et al. 2007] [Hansmann and Stumpf 2025] [Hansmann and Stumpf 2026] [Harrison et al. 2016] [Harry and Trobaugh 1966] [Hasan et al. 2022] [He et al. 2020] [He et al. 2021] [He et al. 2024] [Heeg and Morelli 2011] [Hill et al. 2004] [Huijts and Voskuijl 2015] [Humphrey et al. 2025] [Humphrey et al. 2026] [Humphreys-Jennings et al. 2019] [Humphreys-Jennings et al. 2020] [Hur et al. 2004] [Im and Chang 2000] [Inac 2023] [Ishrak et al. 2024] [Islam et al. 2024] [Jackson 2023] [Jagtap et al. 2024] [Jagtap et al. 2025a] [Joseph et al] [Joseph et al] [Jemitola and Okonkwo 2023] [Jemitola et al. 2013] [Jia et al. 2021] [Jianghao et al. 2011] [Gallman et al. 1990b] [Junlei et al. 2017] [Kale and Pendharkar 2021] [Kalinowski 2015] [Kalinowski 2017b] [Kalinowski 2017a] [Kaloyanova et al. 2005] [Karpuk 2026] [Karpuk et al. 2020] [Karpuk et al. 2023] [Khalid and Kumar 2014] [Kim et al. 2009b] [Kimler and Canfield 2006] [Kimmel and Bradley 2004] [Kleemann et al. 2020] [Komarov and Zinchenko 2023] [Kotikalpudi et al. 2018] [Kroo 1982] [Kroo et al. 1991] [Kryvokhatko 2023c] [Kryvokhatko 2023b] [Kryvokhatko 2023a] [Kryvokhatko 2024b] [Kryvokhatko 2024a] [Kryvokhatko 2025] [Kumar and Khalid 2014] [Kumar and Khalid 2017] [Kummer et al. 2012] [Kuntawala et al. 2011] [Kuntjoro et al. 2015] [La Rocca et al. 2002] [Lambert et al. 2015] [Lambert et al. 2016] [Lampropoulos et al. 2025] [Lange and Moore 1979] [Lange and Moore 1980] [Larkin and Coates 2017] [LeDoux et al. 2008] [Lee and Chen 2004] [Lee et al. 2007] [Lee et al. 2018b] [Legriffon et al. 2023] [Leifsson et al. 2013] [Leonard 2001] [Li and Qin 2020b] [Li et al. 2021a] [Li et al. 2026] [Liang and Wang 2018] [Liao et al. 2026] [Liebeck 2002] [Liebeck 2003] [Liebeck 2004] [Liebeck et al. 1996] [Lijewski 1987] [Lin et al. 1989] [Lindsley 2009] [Liu 2018] [Liu et al. 2026] [Lombardi and Vicini 1994] [Love and Kapania 2020] [Lovejoy 2006] [Lucky Bose et al. 2024] [Luu 2025] [Lyu and Martins 2013b] [Lyu and Martins 2013a] [Lyu and Martins 2014] [Lyu et al. 2024] [Nasir et al. 2022] [Mamla and Galinski 2009] [Mardanpour and Rastkar 2017] [Marisarla et al. 2003] [Mark and Dehart 1976] [Martín et al. 2025] [Mason 1982] [Mastin et al. 1996] [Johnson and Patel 2025] [McGeer and Kroo 1983] [Meheut et al. 2012] [Meng et al. 2020] [Meng et al. 2022a] [Milla and Blick 1966] [Miura et al. 1988] [Moore 1978] [Muchmore 1988] [Mukhopadhyay et al. 2004] [Mukhopadhyay 2005] [Mukhopadhyay et al. 2002] [Mulyanto and Nurhakim 2015] [Mv et al. 2023] [Nangia and Palmer 2006] [Nangia et al. 2003b] [Nangia et al. 2003a] [Nangia et al. 2005] [Nasir and Kuntjoro 2013] [Nazim et al. 2026] [New et al. 2023] [Okonkwo and Jemitola 2022] [Okonkwo and Jemitola 2023] [Okonkwo and Smith 2016] [Oligney et al. 2008] [Olugbeji et al. 2020] [Palaia 2026] [Palaia et al. 2023] [Palaia et al. 2025] [Paletta et al. 2010] [Parello] [Park 2007] [Patel et al. 2024] [Patil 2003] [Paudel 2016] [Jemitola and Paul 2019] [Paulus et al. 2013] [Peigin and Epstein 2006a] [Peigin and Epstein 2006b] [Pellerito et al. 2025] [Pereira et al. 2007] [Perkins 1985] [Peterson and Grant 2011] [Pires et al. 2014] [Pitera et al. 2011] [Potsdam et al. 1997] [Predachenko and Lemko 2019] [Pérez Segura et al. 2026] [Pérez-Álvarez et al. 2015] [Qi et al. 2021] [Qian and Alonso 2018] [Qian and Alonso 2021] [Qin 2002] [Qin et al. 2005] [Qin et al. 2007] [Qu and Cutchins 1992] [Qu and Cutchins 1994] [Qu et al. 2014] [Qureshi et al. 2026] [Rahman and Whidborne 2010] [Ranjan et al. 2026] [Rasheed et al. 2024] [Rasmussen et al. 2004a] [Rasmussen et al. 2004b] [Rasmussen et al. 2006] [Rasmussen et al. 2008] [Ravindren et al. 2007] [Re 2005] [Recktenwald and Ahmed 2008] [Reich et al. 2002] [Reich et al. 2004] [Reichenbach 2008] [Reichenbach et al. 2011] [Reist and Zingg 2014] [Reist and Zingg 2015] [Reist and Zingg 2016] [Ricciardi et al. 2012] [Richards et al. 2009] [Richards et al. 2010] [Richards et al. 2013] [Robinson 2004] [Roman et al. 2000] [Russo et al. 2020a] [Russo et al. 2020b] [Srinivasan et al. 2020] [Saeed et al. 2010] [Saha et al. 2025] [Salam and Bil 2012] [Samuels 1982] [Schwartz et al. 2003] [Scott et al. 2011] [Scotzniovsky et al. 2026] [Selberg and Cronin 1986] [Shah and Ahmed 2023] [Shahjahan and Ahuja 2026] [Shankar and Goebel 1983] [Shankar and Goebel 1985] [Shankar and Malmuth 1982] [Shankar and Malmuth 1983] [Shankar et al. 1979] [Shankar et al. 1981] [Sharma and Hosder 2024] [Sharma et al. 2012] [Sheneshen and kamal 2025] [Sieradzki et al. 2016] [Singh et al. 2016] [Singh et al. 2020] [Sinha 2026] [Siqueira et al. 2019] [Sivaji et al] [Sivaji et al. 2003] [Sleeman 1957] [Smallwood et al. 2003] [Smeltzer et al. 1983] [Smith et al. 1987] [Smith and Stonum 1989] [Snyder et al. 2005] [Song et al. 2014] [Sotoudeh 2015] [Sotoudeh and Hodges 2009] [Sotoudeh and Hodges 2010] [Sotoudeh and Hodges 2011] [Spellman 2003] [Spencer 1961] [Spencer 1981] [Stearman 1990] [Structural Weight Comparison 1981] [Su and Cesnik 2009] [Su and Cesnik 2010] [Su et al. 2017] [Suleman 2007] [Suleman et al. 2015] [Sun et al. 2018] [Sun et al. 2025a] [Sun et al. 2026c] [Sun et al. 2026a] [Sung et al. 2025a] [Sung et al. 2025b] [Syed et al. 2022] [Tasca et al. 2021] [Teo] [Teo et al. 2016] [Зінченко and Хомінич 2015] [Tibério Fernandez et al. 2018] [Torenbeek 2016] [Traub 2009] [Traub 2026] [Trittler et al. 2008] [Tu 1994] [Tu 1991] [Tu 1992a] [Tu 1992b] [Tu and VanDalsem 1996] [Tu et al. 1993] [Tuncer and Platzer 1998] [Tung and Song 2011] [Tyler et al. 2002] [Vale et al. 2014] [van der Voet et al. 2012] [Vandervelden and Kroo 1990] [Velden 1997] [Vicroy et al. 2016] [Voss et al. 2026] [Wai et al. 1994] [Wakayama 1998] [Wakayama 2000] [Wang et al. 2019e] [Wang et al. 2025e] [Wang et al. 2025a] [Wang et al. 2026] [Waters et al. 2013] [Weil Brenner et al. 2012] [Weisshaar and Lee 2002] [Widhalm et al. 2007] [Wienke et al. 2023] [Wildschek 2014] [Wildschek et al. 2013] [Wolkovitch and Fortenbaugh 1978] [Wolkovitch 1984] [Wolkovitch 1985] [Xin et al. 2019] [Xu et al. 2026b] [Xu et al. 2026a] [Yan et al. 2014] [Yang et al. 2025b] [Yazdi et al. 2026] [Yuhong et al. 2017] [Yurkovich 2008] [Zafirov 2013] [Zhang and Yu 2012] [Zhang et al. 2018a] [Zhang et al. 2026b] [Zhao et al. 2022] [Zhou et al. 2025] [Zhu et al. 2019b]
Wing weight
The structural half of the keystone. Wing weight estimation methods, bending material, wing box structural optimisation and weight equations, among them the Grumman build-up methodology this article cites for the non-optimum factor. 30 of these 70 records predate 2000, and the much of the weight literature a conceptual study draws on predates the transonic truss-braced configuration it is applied to.
70 records. [York and Labell 1980] [Ajaj et al. 2013] [Ardema et al. 1996] [Ascani 1974] [Bai et al. 2014b] [Balabanov et al. 1998] [Barton] [Bindolino et al. 2010] [Carreyette 1950] [Chaloff et al. 1974] [Chambers et al. 1996] [Chaussee and Dervault 2013] [Christison Gray and Martins 2026] [Comparison of Statistical Weight 1994] [Corman et al. 2014] [Elham and Timmer 2016] [Elham et al. 2011] [Elham et al. 2013] [Emero and Spunt 1966] [Essari 2018] [Frulla 2021] [Gates 1949] [Giles 1986] [Gray and Martins 2021] [Guruswamy et al. 1994] [Hayase 1974a] [Hayase 1974b] [Hiyama 1974] [Hoang] [Hutchison et al. 1992] [Irvin and Swan 1956] [Kao et al. 2018] [Kim et al. 2018] [Kim et al. 2021] [Laughlin et al. 2013] [Liu and Anemaat 2013] [Livne and Navarro 1999] [Ma et al. 2023c] [Martindale et al. 1974] [Mei and Striz 1992] [Meirovitch and Seitz 1993] [Nigam et al. 2015] [Onishi et al. 1998] [Onishi et al. 1999] [Peck and Hudson 1956] [Petermeier et al. 2010] [Rocha et al. 2006] [Roehl et al. 1995] [Sabeti and Winter 2024] [Shen et al. 2001] [Shinde et al. 2026] [Shirley et al. 2012] [Shirley et al. 2014] [Sinha et al. 2021] [Skillen and Crossley 2005] [Solvey 1951] [Stanford and Jutte 2014b] [Takahashi and Lemonds 2015] [Brooks] [Tong et al. 2016] [Wang et al. 2021a] [Wei and Masood 2017] [Weiss et al. 2013] [Whitehead et al. 1992] [Wildermuth et al. 1974b] [Wildermuth et al. 1974a] [Yi et al. 2015] [York 2006] [Zhang et al. 2008b] [Zink et al. 1999]
Thin transonic wings
What the truss pays for. Supercritical airfoils and wings, drag divergence, thickness and sweep, shock and boundary-layer interaction, transonic wing design methods, and the supercritical wing’s own transonic flutter and buffet.
259 records. [A-7 Transonic Wing Designs 1982] [Allen et al. 1982] [Anders et al. 1999] [Anderson 1995] [Anderson and Gebert 1996] [Andrew and Stenton 1967] [Application of a Shock-Turbulent 1982] [Application of Computational Methods 1982] [Arif et al. 2023] [Arnault et al. 2016] [Ayers 1973] [Ayers and Hallissy 1981] [Venkatachari et al. 2020] [Banavara and Dimitrov 2014] [Bartlett 1977] [Batina and Yang 1985] [Bauer et al. 1977] [Bauer et al. 1975] [Bennett et al. 1985] [Biber and Tilmann 2003] [Blackwell and Pounds 1977] [Stanford et al] [Hiller et al] [Brilliant and Adamson 1973] [Byrdsong and Brooks 1983] [Cahn and Garcia 1971] [Campbell and Smith 1987] [Cao et al. 2024] [Carlson and Weed 1985] [Carlson et al. 1985] [Carlson 1979] [Carmel 1953] [Carrese et al. 2016] [K.Harish Kumar 2015] [Chen et al. 1989] [Chen et al. 1990] [Chen et al. 2014] [Chen et al. 2026b] [Chen et al. 2026c] [Cheng 1982a] [Choi et al. 2016] [Chung 2017] [Chwalowski et al. 2017] [Correction of Model Deformation 2017] [D’Aguanno et al. 2021] [D’Aguanno et al. 2022a] [D’Aguanno et al. 2022b] [Dansberry et al. 1993a] [Dansberry et al. 1993b] [Das et al. 2020] [Das et al. 2022] [Deangelis and Monaghan 1977] [Deck 2004] [Deck 2005] [Degregori and Kim 2021] [Bartlett and Re 1972] [Design and CFD Analysis 2024] [Dickey 1959] [Dietz et al. 2004] [Dietz et al. 2005] [Ding et al. 2026] [Farhangnia et al. 1996] [Ferris 1973] [Ferris 1975] [Fournier 1975] [Fournier and Goodson 1974] [Fournier and Sleeman 1972] [Fournier and Sleeman 1973] [Fournier and Sleeman 1974] [Fu et al. 2023] [Fukushima and Kawai 2017] [Gally and Carlson 1987] [Gally and Carlson 1988] [Gao et al. 2017] [Garnier and Deck 2010] [Giannelis et al. 2022] [Gonzalez et al. 2026] [Grossi] [Grossi et al. 2012] [Hacıogˇlu and Özkol 2003] [Haderlie and Crossley 2009] [Hager et al. 1992] [Hall and McKay 1952] [Hallissy and Ayers 1977] [Hallissy and Harris 1974] [Hammer and Olivier 2019] [Haney et al. 1979] [Haney et al. 1980] [Harbeck and Jameson 2005] [Harris 1971] [Harris 1972] [Harris and Blackwell 1972] [Hasan and Alam 2013] [Hasan and Alam 2014] [Hass et al. 2024] [He and Cui 2022] [Heeg and Chwalowski 2017] [Henne 1980] [Henne 1981] [Henne et al. 1982] [Hess et al. 1987] [Hess et al. 1989] [Hicks 1981] [Hiller et al. 2024] [Hortsen et al. 1983] [Hou, nkateshwar et al. 1995] [Inger 1991] [Inger 1993] [Ishida et al. 2016] [Jacobs 1982] [Jacobs 1983] [Jameson et al. 2008] [Jenkins 1986] [Jernell 1976] [Jiang et al. 2018] [Jiang et al. 2019] [Johnson 1953] [Jordan 1976] [Junxuan and Xun 2015] [Kady and Takahashi 2014] [Kamiya et al. 1978] [Kang et al. 2024] [Kargarnovin and Sayrarfie 1995] [Kennelly 1983] [Jacobson et al] [Kikuchi et al. 2026] [Klutchnikov and Ballmann 2004] [Kodigaddi et al. 2024] [Krenz 1979] [Kuya et al. 2020] [Lee 1990] [Lee 2012] [Lee and Damodaran 2001] [Lee and Eyi 1991] [Lee and Eyi 1993] [Lee and Ohman 1984] [Lee and Tang 1989] [Lee and Pendergraft 1985] [Li et al. 2002] [Li et al. 2023c] [Lin et al. 1997] [Liu and Yang 2016] [Liu et al. 2012] [Liu et al. 2024b] [Mateer et al. 1987] [Mateer et al. 1994] [Matheny and Gatlin 1978] [McCain 1984] [Melton et al. 2007] [Banchy et al] [Miyake and Terashima 2022] [Miyake and Terashima 2023] [Morgan and Kjelgaard 1983] [Murthy et al. 1983] [Mébarki and Le Sant 2001] [Nagamatsu and Ficarra 1985] [Nagamatsu et al. 1983] [Nagamatsu et al. 1984] [Namgoong et al. 2002] [Narain 1983] [Nash et al. 2025] [Niu et al. 2020] [Persoon et al. 1983] [Persoon et al. 1984] [Piette et al. 1985] [Piette and Cazier 1989] [Platt and Brooks 1955] [Poplingher et al. 2022] [Rahman et al. 2015b] [Rahman et al. 2015a] [Rahman et al. 2016] [Ram et al. 1985] [Ratcliff and Carlson 1989] [Rizk 1980] [Bennett 2000a] [Rokoni and Hasan 2013] [Roos and Riddle 1977] [Rose and Seginer 1977] [Rose and Seginer 1978] [Ruhlin et al. 1982b] [Ruhlin et al. 1983] [Ruo et al. 1983] [Ruo et al. 1985] [Schauerte and Schreyer 2023b] [Schauerte and Schreyer 2023a] [Sclafani et al. 2008] [Seidel et al. 1985] [Seidel et al. 1987a] [Seidel et al. 1987b] [Shankar 1980] [Shankar et al. 1978] [Shiqi et al. 2018] [Singh and Venkatraman 2023] [Singh et al. 2022b] [Sobieczky 1979] [Sobieczky 1984] [Sorbilli et al. 2018] [Souza Coimbra et al. 2022] [Spaid 1979] [Stanford and Massey 2017] [Steckel et al. 1980] [Steger and Klineberg 1973] [Streit et al. 2015] [Subsonic Airfoil Design 1990] [Sun et al. 2021] [Supercritical airfoil design] [Supercritical wing] [Swoboda and Nitsche 1996] [Szubert et al. 2015] [Takahashi et al. 2010] [Takanashi 1984] [Takanashi 1985] [Tang et al. 2003] [Tao et al. 2013] [Tao et al. 2019] [Thivet et al. 1999] [Thomas and Dowell 2026] [Tian et al. 2017b] [Tranen 1974] [Koziel et al. 2011] [Transonic Maneuver/Cruise Airfoil Design 1980] [Vassberg et al. 2002] [Ventres and Howe 1983] [Vicini et al. 1997] [Wagner 1981] [Wakayama and Kroo 1994] [Wang et al. 2022c] [Watson 1982] [Weber et al. 1999] [Weber et al. 2001] [Weed et al. 1983] [Wei et al. 2026] [Wetzel 1955] [Whitcomb 1957] [Xiao et al. 2004] [Xiao et al. 2006] [Xin et al. 2013] [Xing and Damodaran 2003] [Xiong and Liu 2013] [Xu et al. 2020] [Yang et al. 1980b] [Yang et al. 1980a] [Yang et al. 1981] [Yang et al. 2024a] [Yates et al. 1981] [Yates et al. 1982a] [Yates et al. 1982b] [Yates et al. 1983] [Yates and Chu 1987] [Yu and Campbell 1992] [Yu et al. 2026] [Yuhara et al. 2018] [Zhang 2026] [Zhang et al. 2011] [Zhang et al. 2021b] [Zhao et al. 2016b] [Zhu et al. 1994] [Zhu et al. 2020] [Zores 1995]
Aeroelasticity
The dependent system the report calls its largest analytical risk. Flutter, divergence, limit-cycle oscillation, aeroelastic tailoring, aeroservoelasticity, nonlinear and large deformation aeroelasticity of very flexible wings, and the doublet-lattice and transonic methods and tunnels that compute and test them.
906 records. [A Study 1973] [Abbott et al. 1963] [Abdelkader et al. 2011] [Abdi et al. 1988] [Abdullah and Sulaeman 2013] [Abel et al. 1979] [Abel et al. 1977] [Abramovich et al. 2005] [Adams and Hoadley 1993] [Zorkipli et al. 2021] [Aeroelastic Study 2014] [Afonso et al. 2017] [Ahmad et al. 2013a] [Ahmad et al. 2014] [Ahmad et al. 2018] [Akshayraj and Ramakumar 2025] [Albano 1968] [Albertani et al. 2005] [Allen et al. 2005] [Allison and Cavallo 2003] [Aly et al. 2002] [Alyanak and Pendleton 2014] [Alyanak and Pendleton 2017] [Amoozgar and Irani 2012] [An and Xu 2013] [An et al. 2023] [Andersen et al. 1997] [Thelen et al] [Thelen et al] [Thelen et al] [Thelen et al] [Andrews and Cooke 2010] [Arena et al. 2011] [Arizono and Isogai 2005] [Arnold 1942] [Arslan and Carlson 1996] [Asai 2022] [Ashley 1979] [Attar and Dowell 2005] [Attar et al. 2010] [Avin et al. 2022] [Ayala et al. 2026] [Babcock and Lind 2012b] [Babcock and Lind 2012a] [Badiei et al. 2014] [Bae and Lee 2002] [Bae et al. 2002] [Bae et al. 2005] [Baker et al. 1999] [Bakhtiari-Nejad et al. 2017] [Balakrishnan et al. 2014] [Balas et al. 2011] [Banerjee 1984] [Banerjee 1988] [Bao et al. 2024] [Baran and Yirtici 2026] [Bartels and Stanford 2018] [Bartels and Stanford 2017] [Bartels et al. 2015d] [Barthelemy and Bergen 1988] [Barthelemy and Bergen 1989] [Batina et al. 1988] [Battoo and de Visser 1998] [Ben Asher and Raveh 2023] [Bendiksen 1992] [Bendiksen 2006] [Bendiksen 2008] [Bennett 1982] [Bennett and Batina 1989] [Bennett et al. 1988] [Beran and Snyder 2008] [Beran et al. 2004] [Beran et al. 2009] [Berci 2025] [Berci and Torrigiani 2020] [Bergman et al. 2011] [Berthold 1977] [Bhat 2018] [Bhatia and Nagaraja 1984] [Bhatia et al. 1984] [Bi et al. 2017a] [Biancolini et al. 2016] [Bibin et al. 2012] [Blair 1992] [Blair and Weisshaar 1982] [Bland and Edwards 1983] [Boehm et al. 2001] [Bohlmann et al. 1990] [Bohlmann et al. 1992] [Bohlmann 1989] [Bohlmann and Scott 1991] [Bohlmann et al. 1988] [Bontoft et al. 2026] [Boo et al. 2015] [Borland and Rizzetta 1981] [Boswinkle and Smith 1958] [Boyd 1977] [Brady et al. 1954] [Bramsiepe et al. 2022] [Bras et al. 2022] [Braune and Hebler 2018] [Breitbach 1979] [Brendel and Sulaeman 1994] [Brenner 1993] [Brenner 1996] [Brenner 2001] [Brenner 2007] [Brenner and Prazenica 2003] [Brenner and Lind 1998] [Stanford and Jacobson] [Stanford et al] [Stanford et al] [Brock and Griffin 1975] [Brooks and Smith 2020] [Brown et al. 2022] [Brown et al. 1980] [Buchnik and Karpel 2014] [Burken et al. 1986] [Butler et al. 1995] [Byreddy et al. 2003] [Byun and Guruswamy 1996] [Byun and Guruswamy 1998] [Byun 1995] [Cakan and Sezgen 2026] [Calculation of Critical Flutter 1981] [Candon et al. 2026] [Castellani et al. 2016b] [Castellani et al. 2017] [Castillo Zuñiga et al. 2019] [Cavagna et al. 2008] [Cavagna et al. 2009] [Cavagna et al. 2010] [Cavagna et al. 2011] [Cea and Palacios 2023] [Cecrdle and Hlavaty 2015] [Cecrdle et al. 2017] [Cella and Biancolini 2012] [Cesnik 2002] [Cesnik 2005] [Cesnik and Su 2005] [Cesnik and Su 2011] [Cesnik et al. 2010] [Cesnik et al. 2012] [Chandre Vila] [Chandre-Vila et al. 2023] [Chen and Bhasin 2012] [Chen and Strelets 2023] [Chen et al. 1999d] [Chen et al. 1999a] [Chen et al. 1999b] [Chen et al. 1999c] [Chen et al. 2002] [Chen et al. 2006a] [Chen et al. 2011] [Chen et al. 2012] [Chen et al. 2013] [Chen et al. 2025] [Chen et al. 1997] [Cheng 2009b] [Cheng et al. 2023] [Chin] [Chin et al. 2011] [Chipman et al. 1984] [Chipman et al. 1985] [Chipman and Rauch 1973] [Chipman et al. 1973] [Choi et al. 2013] [Christhilf et al. 2010] [Chwalowski et al. 2022] [Chwalowski et al. 2011] [Cizmas and Strganac 2010] [Clarke et al. 2005] [Clifford and Da Ronch 2025] [Clifford et al. 2025] [Clifford et al. 2026] [Coggin et al. 2014] [Cole 1986] [Cole 1988] [Cole 1992] [Cole 1985] [Cole 1989] [Cole 1990] [Cole 1991] [Cole and Garcia 2000] [Cole and Rivera 1997] [Cole et al. 1987] [Cole et al. 2003] [Cox and Gilyard 1986] [Crittenden et al. 1977] [Crittenden et al. 1978] [Crowder and deCallafon 2002] [Cumming and Diebler 2005] [Cunningham et al. 1987] [Cunningham et al. 1989] [Cunningham and Den Boer 1992] [Currao and Yeh 2026] [D’Ewart and Robert F. 1958] [Da Costa Cardoso] [Dai et al. 2011] [Dannenberg et al. 1960] [Dansberry et al. 1990] [Dardel and Bakhtiari-Nejad 2010] [Das et al. 2021] [de Melo et al. 2024] [de Visser 1999] [Deangelis 1981] [Demasi 2024b] [Demasi 2024a] [Demenkov and Goman 2009] [Di Leone et al. 2021] [Dibley et al. 2005] [Dilmi 2022] [Dimitriadis et al. 2004] [Djayapertapa and Allen 2000] [Doggett and Soistmann 1992] [Doggett and Cunningham 1976] [Doggett and Rainey 1968] [Doggett and Ricketts 1980] [Doggett and Ricketts 1977] [Doggett et al. 1959] [Doggett 1989] [Doggett and Soistmann 1989] [Doggett et al. 1989] [Dowell 1996] [Dowell 1999] [Dowell 2001] [Dowell and Hall 2003] [Dowell and Ilgamov 1988] [Dowell et al. 2003] [Drachinsky and Raveh 2021] [Drew et al. 2019] [Duan and Zhang 2018] [Durham et al. 1988] [Durham et al. 1990] [Durmaz and Kaya 2013] [DurmuŞ and Kaya 2021] [Yates 1987] [Eastep and Olsen 1979] [Eastep and Olsen 1980] [Eastep et al. 1999] [Ebrahimzade et al. 2016] [Edwards 1992] [Edwards et al. 2009] [Edwards 1981] [Edwards 1996] [Edwards et al. 2001] [Elsayed 2011] [Emory and Patil 2011] [Emory and Patil 2013] [Erickson 1974] [Eversman and Pitt 1989] [Eversman and Pitt 1991] [Fabbiane et al. 2022] [Fagley et al. 2016] [Farhat 2001] [Farmer et al. 1976] [Faroughi et al. 2012] [Farsadi et al. 2018] [Farsadi et al. 2024] [Faïsse et al. 2021] [Faïsse et al. 2022] [Fehrs et al. 2021] [Felt and Kehoe 1985] [Filippou et al. 2024] [Finaish 1992] [Flight Flutter Testing Symposium 1975] [Forster et al. 1996] [Foughner and Bensinger 1977] [Francois et al. 2017] [Freudinger 1989] [Freudinger and Kehoe 1990] [Friedmann 1998] [Fu and Liu 2019] [Fuglsang and Williams 1985] [Fujimori et al. 2000] [Fukumoto et al. 2023] [Försching 1970] [Gallman et al. 1986] [Gambucci et al. 1958] [Gennaretti 2024] [Gern and Librescu 2000] [Ghalandari et al. 2022] [Gianikos et al. 2018] [Giese et al. 1996] [Giesing 1970] [Gilbert 1989] [Gimmestad 1981a] [Gobal and Grandhi 2015] [Goetz and Doggett 1974] [Goetz and Stonesifer 1961] [Goizueta et al. 2021] [Goizueta et al. 2022] [Golla et al. 2024] [Goodwin and Raj 1999] [Gordnier and Attar 2012] [Gordnier and Visbal 2004] [Grauer and Boucher 2017] [Graves et al. 2001] [Gray et al. 2003] [Green 1987] [Green 1986] [Greene and Sobieszczanski-Sobieski 1980] [Greene and Sobieszczanski-Sobieski 1982] [Greenhalgh et al. 1993] [Gregory et al. 2007] [Grosser et al. 1983] [Gu et al. 2024] [Guimarães Neto et al. 2014] [Guo 2007] [Guo et al. 2015b] [Guo et al. 2018] [Gupta et al. 2005] [Gupta et al. 1987] [Gupta et al. 1989] [Gupta et al. 1991] [Gurley and Ruhlin 1962] [Guruswamy and Tu 1989] [Guruswamy et al. 1985] [Guruswamy et al. 1986] [Guruswamy et al. 1984] [Guruswamy and Tu 1988] [Guruswamy et al. 1987] [Guruswamy and Goorjian 1983] [Gwin 1974] [Gwin 1976] [Haddadpour et al. 2005] [Haddadpour et al. 2008] [Haghighat et al. 2010] [Hajj 2004] [Hall et al. 2024] [Hallissy and Cesnik 2011] [Hanson 1980] [Hanson 1984] [Harash et al. 2012] [Hartin and Riso 2025] [Hashemi 2017] [Havenar and Ilie 2025] [He et al. 2014] [Heeg 2006] [Heeg and Chwalowski 2019] [Heeg et al. 2005] [Henning 1953] [Herencia et al. 2007] [Herrmann and Nemat-Nasser 1966] [Hess and Flick 2004] [Hildebrand and Reissner 1944] [Hilger and Ritter 2021] [Hilger and Ritter 2023] [Hitch 1978] [Hoadley and Karpel 1991] [Hoover and Shen 2018] [Hopkins 1977] [Hopkins 1975] [Horikawa and Saito 1986] [Hoseini and Hodges 2019] [Hoshi et al. 2020] [Hu and Zhou 2007] [Hu et al. 2024] [Huang et al. 2020] [Hughes et al. 2026] [Huo et al. 2025] [Huttsell et al. 1976] [Hwang et al. 1979] [Hwang et al. 1980] [Hwu and Tsai 2002] [Ide and Ominsky 1990] [Ifju et al. 2001] [Ilie and Havenar 2022] [Ilie and Havenar 2023] [Imai et al. 2022] [Isogai 1979] [Isogai 1981] 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Gust and manoeuvre loads
The loads a long wing must carry. Gust and manoeuvre loads, active and passive load alleviation, ride quality and the flight dynamics of flexible aircraft.
344 records. [Active control of a 1994] [Ahmadi Tehrani et al. 2025] [Alam and Hromcik 2019] [Alam et al. 2015] [Allyn and Takahashi 2016] [An et al. 2018] [Analysis of Gust-Response Flight-Test 1988] [Anderson et al. 1972] [Anderson et al. 1973] [Aouf et al. 2000] [Asaro et al. 2023] [Austin and William H. 1964] [Austin and William H. 1967] [Babcock and Lind 2013] [Bahia Monteiro et al. 2023] [Bai et al. 2014a] [Balatti et al. 2023a] [Balatti et al. 2023b] [Balducci et al. 1964] [Barnaby 1955] [Barnes and Visbal 2019] [Barzgaran et al. 2021] [Benton et al. 2009] [Beyer et al. 2023] [Beyer et al. 2024] [Bi et al. 2017b] [Breitenstein et al. 2023] [Breitenstein et al. 2024] [Breitsamter 2005] [Brown 1970] [Bruni et al. 2014] [Burgstaller and Galffy 2024] [Burnham et al. 2001] [Burris and Bender 1969b] [Burris and Bender 1969a] [Canfield 2014] [Castells Marin] [Castrichini et al. 2017] [Cavalcanti et al. 2026] [Cavaliere and Fezans 2024] [Caverly et al. 2017] [Chen et al. 2006b] [Chen et al. 2009a] [Chen et al. 2010] [Chen et al. 2017] [Chen et al. 2018] [Cheung et al. 2018] [Forte and Nguyen] [Forte and Nguyen] [Forte and Nguyen] [Forte et al] [Forte et al] [Forte et al] [Clark et al. 1999] [Clay and Berens 1964] [Cook et al. 2013] [Cooney and Schott 1956] [Cooper 2018] [Cramer and Nguyen 2020] [Curpanaru et al. 2025] [Da Ronch and Badcock 2013] [Dai et al. 2017] [Dai et al. 2025] [Dietz et al. 2003] [Dillsaver et al. 2011] [Disney 1975] [Dong et al. 2024] [Dong et al. 2025] [Drew et al. 2020] [Duan et al. 2021] [Duessler et al. 2023] [Duessler et al. 2024] [Fan et al. 2017a] [Ferrier et al. 2018] [Fezans 2017] [Fezans and Joos 2017] [Fezans et al. 2019] [Fonte and Mantegazza 2017] [Fonte et al. 2015] [Fonte et al. 2018] [Forte and Nguyen 2026b] [Forte and Nguyen 2026a] [Forte and Nguyen 2026c] [Forte et al. 2022] [Forte et al. 2023] [Forte et al. 2026b] [Forte et al. 2026a] [Fournier et al. 2022b] [Fournier et al. 2022a] [Friedewald et al. 2017] [Frost et al. 2012] [Fujimori et al. 1989] [Fujimori et al. 1990] [Gangsaas et al. 1981] [Gao et al. 2024] [Gennaretti and Ponzi 1999] [Gerardi 1977] [Giesseler et al. 2012] [Goggin 1992] [Gordnier and Visbal 2015] [Gov and Karpel 2021] [Grauer and Boucher 2018] [Gribbin et al. 2025] [Gu et al. 2025] [Guo et al. 2015a] [Gusto Andika et al. 2020] [Haghighat et al. 2009] [Haghighat et al. 2012] [Hamada et al. 2019] [Hamada et al. 2020] [Hammerton et al. 2018a] [Hammerton et al. 2018b] [Hansen et al. 2020] [Hansen et al. 2022] [Hanson et al. 2018] [Hao et al. 2023] [Hashemi and Nguyen 2018] [Hashemi et al. 2018] [Hashemi et al. 2020] [He and Su 2023] [He et al. 2022] [Heimbaugh 1987] [Hess 1971] [Hess 1972] [Hillebrand and Lutz 2026] [Hillebrand et al. 2023] [Hillebrand et al. 2024] [Hillebrand et al. 2026a] [Hillebrand et al. 2026b] [Hinson et al. 2022] [Hoffmann et al. 2011] [Houbolt and Williamson 1975] [Hu et al. 2022] [Hu et al. 2025] [Huebner and Reimer 2019] [Hwang and Pi 1982] [Ibren et al. 2020] [Ingram and Eichenbaum 1969] [Islam et al. 2018] [Izadpanahi and Mardanpour 2018] [Jielin et al. 2024] [Johnston 1979] [Karpel 1982] [Khalil and Bauknecht 2024] [Khalil and Fezans 2019] [Khalil and Fezans 2020] [Khalil and Fezans 2021] [Kim and Marciniak 1999] [Kim et al. 1999] [Kimberlin and Sims 1996] [Klepl 1990] [Klug et al. 2020] [Klug et al. 2021] [Klug et al. 2023] [Kobayakawa and Maeda 1978] [Kopf et al. 2015] [Kopf et al. 2018] [Krengel 2024] [Krengel and Hepperle 2023] [Lasauskas 2016] [Layton 1996] [Lee et al. 2018c] [Li and Qin 2020a] [Li and Qin 2021b] [Li and Qin 2021a] [Li and Qin 2022] [Li et al. 2006] [Li et al. 2016b] [Li et al. 2017a] [Li et al. 2018] [Li et al. 2025c] [Liebst et al. 1986] [Liebst et al. 1988] [Liu and Sun 2016] [Liu and Sun 2017] [Liu et al. 2016c] [Liu et al. 2017] [Liu et al. 2018a] [Liu et al. 2018b] [Liu et al. 2025] [Lokos et al. 2005] [Lucas et al. 2009] [Lv et al. 2008] [Mancini and Vos 2019] [Mangalam et al. 2008] [Mares and Suresh Babu 2026] [Matsuzaki et al. 1987] [Matsuzaki et al. 1989] [McLean 1978] [Melville et al. 2018] [Michel et al. 2025] [Misra and Bai 2018] [Moore 1992] [Moore 1995] [Moore et al. 1991] [Morino and Noll 1977] [Moses 2005] [Moulin and Karpel 2007] [Mouyon et al. 2003] [Muradas Odriozola] [Murphy 1981] [Müller et al. 2026] [Nagaraja and Kraft 1999] [Nam et al. 1997] [Narayanaswamy et al. 2008] [Nguyen 2021] [Nguyen et al. 2018b] [Nguyen and Hashemi 2020] [Nguyen et al. 2017a] [Nguyen et al. 2020] [Nie et al. 2009] [Nikiforuk et al. 1974] [Nissim and Lottati 1979] [Nissim and Lottati 1980] [Nissim 1978] [Nixon and Tzuoo 1986] [Nixon and Tzuoo 1987] [Noll and Merino 1976] [Norman et al. 1983] [Odriozola et al. 2026] [Ohta and Fujimori 1988] [Ossmann and Poussot-Vassal 2018] [Paku et al. 2016] [Pasley et al. 1973] [Perry 1984] [Pototzky et al. 1990] [Pourtakdoust and Khodabakhsh 2026] [Poussot-Vassal et al. 2022] [Present Place of Discrete-Gust 1988] [Pusch 2017] [Pusch et al. 2019] [Pusch et al. 2022] [Qi et al. 2015] [Qin 2021] [Qu and Li 2022] [Quindlen et al. 2026] [Ramsey and Lewolt 1979] [Raveh 2007] [Regan and Jutte 2012] [Reimer et al. 2015] [Ricci and Scotti 2010] [Rieck et al. 2026] [Roesch and Harlan 1974] [Ruhlin and Murphy 1981] [Rui Nie et al. 2009] [Rustenburg 1972] [Ryall et al. 2004] [Rynaski 1979] [Sabatini et al. 2026a] [Sabatini et al. 2026b] [Sako et al. 1999] [Sanghi et al. 2022] [Sarantopoulos and De Abreu-Garcia] [Sato 2010] [Sato et al. 2011] [Schildkamp et al. 2023] [Schumann et al. 2025] [Scott et al. 2013] [Scott et al. 2014] [Sedky et al. 2020] [Seki et al. 2019] [Shao et al. 2010] [Shao et al. 2013] [Siebert et al. 2026] [Silva et al. 2006] [Silvestre et al. 2017] [Simpson et al. 2014] [Smith et al. 2016] [Sodja et al. 2021] [Stalla et al. 2024] [Stanford 2020] [Streitenberger and Feldwisch 2025] [Strothteicher and Fezans 2026] [Structural Aspects of Flexible 2000] [Sun and Allwright 1992] [Suzuki and Yonezawa 1993] [Takase et al. 2017] [Takase et al. 2018] [Tang and Dowell 2002a] [Tang et al. 2010] [Tang et al. 2016] [Tani et al. 2018] [Tantaroudas and Karachalios 2026] [Thapa Magar et al. 2017] [Thapa Magar et al. 2018] [Thel et al. 2022] [Thomas and Shkarayev 2026] [Thornton 1993] [Ting et al. 2023] [Tischler and Venkayya 1998] [Uchida et al. 2019] [Ullah et al. 2023] [Valente et al. 2017] [Vartio et al. 2005] [Vartio et al. 2008] [Virgilio Pereira et al. 2019] [Wang and Inman 2017] [Wang et al. 2015] [Wang et al. 2016a] [Wang et al. 2017] [Wang et al. 2018a] [Wang et al. 2019a] [Wang et al. 2019c] [Wang et al. 2021b] [Wang et al. 2021c] [Wang et al. 2022b] [Wang et al. 2025f] [Wang et al. 2025d] [Warren 1971] [Welstead and Crouse 2010] [Wheatcroft et al. 2025] [White 1970] [White 1971] [Wildschek et al. 2006] [Woods-Vedeler and Pototzky 1992] [Wu et al. 2024a] [Wu et al. 2025d] [Wuestenhagen 2022] [Wuestenhagen 2023] [Wykes 1968] [Wynn et al. 2022] [Xiaoxiong et al. 2017] [Xiong et al. 2026b] [Xu and Kroo 2011a] [Xu et al. 2011] [Xue Lei et al. 2016] [Yagil et al. 2018] [Yang et al. 2010] [Yang et al. 2013] [Yi et al. 2025] [Yu et al. 2017] [Zeng et al. 2008] [Zeng et al. 2010] [Zeng et al. 2012a] [Zeng et al. 2012b] [Zhang and Zhao 2023] [Zhang et al. 2024] [Zhang et al. 2026c] [Zhao et al. 2016a] [Zhao et al. 2023] [Zheng et al. 2026a] [Zhou et al. 2022] [Zink et al. 2002] [Zink et al. 2004]
Span constraints
The constraint this article argues is binding. Folding wings and wingtips, and the compatibility of an aircraft’s span with airports, gates and terminal areas. 34 of these 40 records concern folding wings or wingtips, five concern airports and one concerns span-limited aircraft, so the literature treats the fold as a mechanism far more often than it treats the airport as the reason for it.
40 records. [Advanced Subsonic Long-Haul Transport 1974] [Ahmadi et al. 2024] [Ajaj 2021] [Bonnin et al. 2023] [Carrillo Córcoles et al. 2023] [Carrillo et al. 2022] [Carrillo et al. 2023] [Cheung et al. 2019] [Cheung et al. 2020a] [Cheung et al. 2020b] [Cooper 2023] [Dela Peña 2024] [Dussart et al. 2019b] [Dussart et al. 2019c] [Dussart et al. 2019a] [Farsadi et al. 2026] [Gu et al. 2022b] [Gu et al. 2023a] [Gu et al. 2023c] [Guler and Nikbay 2026] [Han et al. 2024b] [Healy et al. 2022] [Healy et al. 2026] [da Luz et al. 2026] [liu et al 2024] [Mahid et al. 2023] [Mahid et al. 2025] [Molz and Breitsamter 2026] [Narimani et al. 2025] [Read 1971] [Sanghi et al. 2023] [Sanghi et al. 2024] [Singh et al. 2022a] [Sun et al. 2025b] [Toffol 2024] [Wang 2026] [Wu et al. 2025a] [Ye et al. 2024] [Yue et al. 2009] [Zhang et al. 2026a]
Laminar flow
What less sweep buys. Natural and hybrid laminar flow on wings and nacelles, transition and its prediction, crossflow instability, riblets and laminar flight experiments.
298 records. [Aircraft laminar flow control] [Allison 1983] [Allison and Dagenhart 1978] [Analysis and Design 2012] [Applin and Gentry 1988] [Arcara et al. 1991] [Athar et al. 2025] [Atkin 2004] [Atkin et al. 2006] [Axten et al. 2026] [Venkatachari et al. 2021a] [Venkatachari et al] [Venkatachari et al. 2021b] [Barklage et al. 2022] [Barry et al. 1994] [Befus et al. 1987] [Belisle et al. 2010] [Belisle et al. 2012] [Bobbitt et al. 1992] [Bragg and Gregorek 1989] [Braslow and Fischer 1985] [Hiller et al] [Bridges 1994] [Bright et al. 2013] [Brinza 2024] [Bui 2015] [Bushnell 1992] [Campbell and Lynde 2016] [Campbell et al. 1986] [Campbell and Lynde 2017] [Carmichael 1979] [Carpenter et al. 2001] [Carpenter et al. 2018] [Catalano et al. 2020] [Cella et al. 2010] [Chakraborty et al. 2022] [Chang 2004] [Chang and Choudhari 2005] [Chapman and Glauser 1996] [Chen and Tang 2017] [Chi et al. 2019] [Choudhari et al. 2013] [Coder 2020] [Coder 2021] [Coder 2023] [Coder 2025] [Coder 2026] [Collazo Garcia and Ansell 2018] [Collazo Garcia and Ansell 2019] [Collier et al. 1992] [Collins et al. 1963] [Corelli Grappadelli et al. 2022] [Crouch et al. 2010] [Cummings 1993] [Cunnington and Parmley 1980] [Dagenhart and Saric 1999] [Dagenhart et al. 1989] [Dagenhart 1992] [Somers and Maughmer 2022] [Dargel and Thiede 2002] [Deng et al. 2026] [Denison et al. 2019] [Djeddi and Ekici 2022] [Dodbele 1990] [Dodbele 1992] [Drake and Solomon 2010] [Duan et al. 2014] [Duan et al. 2013] [Egreteau et al. 2022] [Eliasson et al. 2011] [Eppink and Wlezien 2011] [Beyak et al] [Evaluation of Laminar Flow 1978] [Evaluation of Laminar Flow 1979] [Evaluation of Laminar Flow 1980] [Fan et al. 2017b] [Fan et al. 2021] [Fauci et al. 2006] [Fehrs and Kaiser 2025] [Fischer and Vemuru 1991] [Fischer et al. 1983] [Fischer 1992] [Frederick et al. 2014] [Fuel efficiency and laminar] [Fuhrmann 1993] [Fujino et al. 2003] [Garbaruk et al. 2021] [Gibson and Gerhardt 1993] [Gibson and Gerhardt 1995] [Goodyear 1987] [Gopalarathnam and Selig 2001] [Goradia et al. 1989] [Gratzer and George-Falvy 1978] [Gratzer 1990] [Green et al. 1996] [Haans et al. 2004] [Hammer and Shumway 2026] [Han et al. 2016] [Hansman and Barsotti 1985] [Harris et al. 1988] [Hartshorn et al. 2012] [Harvey 1982] [Harvey and Pride 1982a] [Harvey and Pride 1982b] [Harvey et al. 1986] [Hefner 1992] [High Reynolds Number Hybrid 1999] [Hiller et al. 2025] [Holmes and Obara 1983] [Holmes and Obara 1992] [Horstmann et al. 1990] [Husain et al. 2026] [Hybrid Laminar Flow Control 1982] [Ide et al. 2012] [Ide et al. 2019] [Iuliano et al. 2009] [Iuliano et al. 2011] [Jasperson et al. 1984] [Eppink] [Eppink 2018] [Eppink and Casper 2019] [Eppink et al. 2017] [Jentys and Breitsamter 2023] [Jernell 1978] [Hefner and Sabo 1987] [Jiang 2026] [Jobe et al. 1978] [Jobe et al. 1979] [Joslin 1998] [Kakkar et al. 2025] [Kalarikovilagam Srinivasan and Bertram 2019] [Kaneko et al. 2024] [Karpuk and Elham 2023] [Karpuk and Mosca 2024] [Kehoe et al. 1985] [Keller 2019] [Kocharin et al. 2021] [Kosin 1965] [Krishnan et al. 2017] [Krumbein et al. 2022] [Runyan and Steers 1980] [Lamb et al. 1985] [Lange 1983] [Lange 1984a] [Lange 1987] [Lange 1988] [Lawson et al. 2018] [Lee and Jameson 2009] [Li and Zhong 2016] [Li et al. 2020] [Li et al. 2011] [Li et al. 2016a] [Lin and Reed 1993] [Lineberger 1984a] [Lineberger 1984b] [Lobitz et al. 2023] [Lovell et al. 1978] [Luo et al. 2024] [Lynde and Campbell 2016] [Lynde and Campbell 2017] [Lynde et al. 2019] [Malik et al. 2013] [Malik et al. 2015] [Manning and Kroo 1999] [Martin et al. 2008] [Mauerer et al. 2026] [Mavriplis et al. 2020] [Choudhari et al] [Mele et al. 2016] [Mery et al. 2025] [Lynde et al] [Moens et al. 2007] [Morgan 1986] [Mosca and Karpuk 2025] [Mount and Millman 1985] [Mullender and Poll 1995] [Muraca 1978] [Murayama et al. 2024] [Barnwell and Hussaini 1992] [Nie et al. 2025] [Ortiz-Melendez et al. 2022] [Ostowari and Naik 1985] [Palumbo et al. 2022] [Parikh et al. 1990] [Parikh and Nagel 1990] [Pascual and Zingg 2025] [Pascual and Zingg 2026] [Pearce 1982] [Pearce 1983] [Perraud and Moens 2007] [Perraud et al. 2003] [Perraud et al. 2008] [Pfenninger and Vemuru 1988] [Pfenninger et al. 1986] [Pfenninger et al. 1988] [Pohya and Wicke 2019] [Pohya et al. 2019] [Poll 2000] [Powell et al. 1989] [Prasannakumar et al. 2024] [Pruett 1995] [Quagliarella and Iuliano 2017] [Quagliarella and Iuliano 2018a] [Quagliarella and Iuliano 2018b] [Radespiel et al. 1990] [Redeker and Wichmann 1988] [Redeker and Wichmann 1991] [Redeker et al. 1988] [Redeker et al. 1990] [Reed 1989] [Risse and Stumpf 2014] [Risse et al. 2014] [Rivers et al. 2019] [Rizzetta and Garmann 2024] [Rizzetta and Garmann 2025] [Rizzetta and Visbal 2016b] [Rizzetta and Visbal 2016a] [Roberts et al. 2012] [Roberts et al. 2015] [Rozendaal 1986] [Sabater et al. 2021] [Sabater et al. 2022] [Sapuppo and Archer 1982] [Saric and Reed 2004] [Saric 1996] [Schrauf and von Geyr 2020] [Schrauf and von Geyr 2021a] [Schrauf and von Geyr 2021b] [Schuelein 2008] [Schueltke and Stumpf 2017] [Selig et al. 1995] [Semaan et al. 2013] [Sengupta et al. 2021] [Somers 1992] [Somers 2019] [Srilatha et al. 1990b] [Srilatha et al. 1990a] [Stalewski and Sznajder 2016] [Steers 1981] [Streett 2003] [Sturgeon 1978b] [Sturgeon 1978a] [Sturgeon and Bennett 1976] [Sturgeon et al. 1976] [Sudhi et al. 2021] [Sudhi et al. 2023] [Sugiura et al. 2001] [Swinford 1976] [Tang et al. 2017a] [Tang et al. 2022] [Three-Dimensional Boundary-Layer Transition 1994] [Tokugawa and Yoshida 2006] [Toulorge et al. 2007] [Traub et al. 2021] [Tucker et al. 2016] [Tuttle and Maddalon 1982] [Ueda et al. 2014] [Viken et al. 1991] [Viswanath and Mukund 1995] [von Geyr et al. 2014] [Waggoner 1986] [Waggoner et al. 1986] [Wagner et al. 1990] [Wagner 1986] [Wahlich et al. 2024] [Wang et al. 2019d] [Wayman et al. 2025] [Wayman et al. 2026] [Wentz et al. 1984] [Wie et al. 1991] [Wie et al. 1992] [Woan et al. 1991] [Wong et al. 2026] [Wright 1979] [Xu and Kroo 2011b] [Xu and Kroo 2014] [Xu et al. 2019] [Yan et al. 2023] [Yanagimoto and Rinoie 2019] [Yao et al. 2021] [Yasuaki and Egami 1999] [Yoshida and Tani 1993] [Young 2012] [Young and Fielding 2001a] [Young and Fielding 2001b] [Young and Humphreys 2004] [Young et al. 2001] [Youngren 2008] [Zare Shahnehb 2022] [Zauner et al. 2023] [Zhang and Yin 2019] [Zhang et al. 2015] [Zhang et al. 2017a] [Zhang et al. 2020b] [Zhang et al. 2022] [Zhang et al. 2023] [Zhao et al. 2017] [Zhu et al. 2019a]
High lift
Low speed and the edges of the envelope. High-lift systems and Krueger flaps, buffet onset prediction, maximum lift and stall, icing of wings, ground effect, and wingtip devices treated for their low-speed and handling effects.
187 records. [A Comparison of Different 2022] [Jeong 2006] [Akaydin et al. 2016] [Ando and Yashiro 1976] [Andreu Angulo and Ansell 2018] [Army Aviation Board Fort Rucker Al 1957] [Ashill 1970] [Bahr et al. 2016] [Benischke and Thielecke 2015] [Bidwell 2005] [Bonnifet et al. 2017] [Bragg and Khodadoust 1989] [Bragin 2018] [Brune and Hallstaff 1985] [van Dam et al] [Carter 1970] [Chandrasekhara 1994] [Chapleo and Nair 1994] [Chen et al. 2003a] [Chen et al. 2003b] [Choudhari and Lockard 2015] [Chow et al. 2002] [Chung et al. 2002] [Chung et al. 2003] [Coe and Weston 1978] [Coe and Weston 1979] [Crouch et al. 2018] [Currao and Jiang 2026] [da silva dias et al 2025] [Damstrom and Mayes 1970] [Damstrom and Mayes 1971] [Dang and Yang 2010] [Das and Wichmann 2004] [Demasi et al. 2022] [Demele 1958] [Desalvo et al. 2014] [DeSalvo et al. 2016] [Dobrzynski et al. 1998] [Edge and Perkins 1995] [Fink and Lastinger 1961] [Flow Control and High-Lift 2016] [Foster 1971] [Foster 1972] [Fujiwara and Nguyen 2015] [Furey 1980] [Furlong and Fitzpatrick 1947] [Gao et al. 2018] [Garcia et al. 2001] [Gavrilović et al. 2023] [Ghee and Taylor 2000] [Ghee and Taylor 2004] [Gifford and Van Dam 1982] [Goc et al. 2023] [Goradia and Morgan 1986] [Gratzer and Odonnell 1964] [Gubsky et al. 2018] [Hahne and Glaab 1999] [Hardin et al. 1993] [Hartwich et al. 2014] [Hartwich et al. 2016] [Hebert and Joseph 1970] [Hebert et al. 1970] [Hinshaw et al. 1964] [Hoell et al. 2011] [Huffman and Jackson 1974] [Ichikawa et al. 2021] [Iuliano et al. 2022] [Iuliano et al. 2024] [Jenkins 1971] [Jinli and Zhichun 2013] [Kang Ning and Shi YouMin 2018] [Kasi et al. 2019] [Paschal et al] [Paschal et al] [Khalil et al. 2020] [Khalil et al. 2022] [Khodadoust and Washburn 2007] [Kholyavko 1971] [Kida and Miyai 1974] [Kojima and Hashimoto 2023] [Korbacher 1964] [Korthäuer et al. 2023] [Kwon and Sankar 1990] [Lammering et al. 2014] [Lange and May 1948] [Li 2026] [Li et al. 2024a] [Lichtenstein and Williams 1952] [Lindner et al. 2023] [Potapczuk et al. 1991] [Ma et al. 2024b] [Makgantai et al. 2021] [Maldonado et al. 2024] [Mamada and Ando 1973] [Mamada and Ando 1974] [Manjunath et al. 2020] [Mao et al. 2020] [Melber-Wilkending 2010] [Mikhailov 2020] [Milillo 1972] [Mineck and Carter 1974] [Murayama et al] [Miyaji and Arasawa 2001] [Monner Hans Peter and Riemenschneider Johannes 2012] [Morelli et al. 2022] [Mousseux et al. 1989] [Mukherjee et al. 2003] [Murayama et al. 2025] [Nash and Rogers 1999] [Nelson et al. 1994] [Nguyen et al. 2015c] [O’Donnell and Mohseni 2017] [Oliver 1980] [Pandya et al. 2011] [Pasamanick and Proterra 1948] [Passamanick 1948] [Pepper et al. 1996] [Pepper and vanDam 1996] [Perennes and Roger 1998] [Pigusov 2021] [Potter and Vandam 1995] [Qi et al. 2026] [Riley 1985] [Roos 1985] [Rudnik et al. 1999] [Rudolph 1996] [Rumsey et al. 2001] [Röntsch et al. 2016] [Sang 2009] [Sang et al. 2006] [Sang et al. 2007] [Schmidt and Wild 2026] [Sharif Moghadam et al. 2020] [Shirbhate et al. 2025] [Shmilovich et al. 2023] [Slotnick et al. 2000] [Smith 1973] [Soemarwoto et al. 2000] [Soni et al. 2023] [Srinivasan et al. 1986] [Srinivasan and McCroskey 1986] [Steimle et al. 2008] [Stough et al. 1987] [Study on the Ground 2022] [Taira et al. 2010] [Takeda et al. 2001] [Tao et al. 2018] [Носик and Кривохатько 2019] [Thomas and Dowell 2015] [Tian et al. 2017a] [Timme 2020] [Tinling and Karpen 1955] [Tinling and Karpen 1957] [Tinoco et al. 1986] [Toosi et al. 2024] [Trobaugh et al. 1980] [Uppuluri and Selig 2002] [Ura et al. 2008] [Van Dam et al. 1993] [van den Kieboom and Elham 2016] [Van Dommelen 1995] [Vandam 1993] [vanDam et al 1995] [Vatsa et al. 2019] [Vatsa et al] [Verma and Kulkarni 2024] [Vijgen et al. 1992] [Viken et al. 1987] [Visbal and Garmann 2016] [Visbal and Garmann 2019] [Wang 1979] [Wang et al. 2024a] [Wei et al. 2024] [Weiberg and Holzhauser 1961] [Wild 2012] [Wild 2013] [Wild 2021] [Winkelmann 1989] [Xie et al. 2024] [Yang et al. 2024b] [Yoo 2018] [Zhang 2010] [Zhang et al. 2017b] [Zhang et al. 2026e] [Zhao et al. 2020] [Zhu and Qiu 2005] [Zimmermann et al. 2018]
Propulsion integration
The engine hung under the long wing. Nacelle, pylon and wing interference, ultra-high bypass ratio and geared turbofan engines, open rotors, and distributed and hybrid-electric propulsion for transports.
194 records. [Agte et al. 2012] [Akinnuli and Oladipo 2018] [Aloyo et al. 2014] [Anderson and Mizukami 1993] [Arend et al. 2021] [Awes et al. 2021] [Awes-Cheynis] [Bahr et al. 2014] [Bandopadhyay et al. 2022] [Barbosa 2022] [Batterton 1992] [Bencze 1972] [Beneda and Sipula 2019] [Benjamin and Friedrichs 2024] [Benjamin et al. 2025] [Berton et al. 2015] [Bertsch et al. 2018] [Bertsch et al. 2019] [Beschorner et al. 2025] [Brown 1987] [Buschner 1949] [Cappuccio 1992] [Carlson and Lamb 1987] [Carney et al. 2013] [Chapman 2023] [Chapman and Litt 2017] [Chen et al. 2026d] [Clarke et al. 2014] [Cloft and Muldoon 1989] [Colmenares Quintero et al. 2010] [Csank and Thomas 2017] [Stephens 2014] [Dawkins et al. 2024] [Denney et al. 2012] [Dever et al. 2015] [Dias and Melo 2025] [Dietz et al. 2007] [Dietz et al. 2008] [Doty et al. 2011] [Dyson 2018b] [Dyson et al. 2021] [Dyson 2018a] [Early 2000] [Eleshaky and Baysal 1994] [Envia 2010] [Envia 2011] [Envia 2014] [Envia 2016] [Farassat 2010] [Felder et al. 2009] [Felder et al. 2011] [Fischer and Klug 1989] [Flamm et al. 2016] [Fleming et al. 2004] [Fossati et al. 2024] [Gary and McDonald 2014] [Geidel and Gray 1989] [Gemin et al. 2015] [Gern 2013] [Gillette 1977] [Cinar et al. 2023] [Zilliac 2026] [Gubanov 2019] [Guynn et al. 2011] [Habermann 2020] [Hahn 2012] [Buescher and Connolly] [Haller et al. 2012] [Head 1972] [Henderson 1987] [Henderson and Patterson 1983] [Hendricks and Tong 2012] [Herrmann 1996] [Heykena et al. 2020] [Hoogreef et al. 2020] [Hooker et al. 2013] [Hughes and Gazzaniga 2009] [Hughes 2009a] [Hughes 2011b] [Hughes and Lord 2008] [Hughes 2011a] [Hughes 2009b] [Hughes 2013] [Hughes and Zeug 2008] [Ingraldi et al. 1992] [Ingraldi et al. 1991] [Jafari et al. 2020] [Jagtap 2025] [Jagtap et al. 2025b] [Jain and Crossley 2020] [Sachs-Wetstone et al. 2023] [Jones et al. 2021] [Kelly et al. 2021] [Kenny et al. 2026] [Antcliff et al. 2016] [Khalid et al. 2010] [Kim et al. 2015] [Kim and Liou 2013] [Kim 2010] [Kim et al. 2013] [Koc et al. 2005] [Milios et al. 2023] [Kooi et al. 1994] [Kratz 2024] [Kratz et al. 2018] [Kratz and Chapman 2018] [Kursakov et al. 2022] [Kwon et al. 2001] [Lammen and Vankan 2020] [Lay and Strohmayer 2023] [Lecordix et al. 1992] [Lee et al. 2018a] [Lents et al. 2016] [Li et al. 2000] [Li et al. 2025b] [Licheva and Liscouet-Hanke 2023] [Liou et al. 2017] [Mace and Nyberg 1992] [Madavan et al. 2015] [Magrini et al. 2020] [Magrini et al. 2021] [Magrini et al. 2022b] [Magrini et al. 2022c] [Magrini et al. 2022a] [Guynn et al] [Marx 1989] [Mastropierro et al. 2020] [Matesanz García et al. 2023] [Matsuda et al. 1992] [Meleason and Wells 1976] [Czech and Thomas] [Milholen 2001] [Mukhopadhyay et al. 2017] [Nark et al. 2016] [National Advisory Committee 1936] [Nguyen et al. 2016] [Nguyen et al. 2018a] [Nikolaidis et al. 2020] [Oldeweme et al. 2026] [Page et al. 1986] [Papathakis et al. 2016] [Pascioni and Rizzi 2018] [Patel and Chudoba 2026] [Patel and Donelson 1982] [Pendergraft et al. 1989] [Pendergraft et al. 1992] [Pornet and Isikveren 2015] [Purohit 2013] [Reubush 1978] [Reubush 1979] [Ribeiro and Lacava 2013] [Richey et al. 1968] [Richey et al. 1983] [Riebe and Davenport 1958] [Robinson et al. 2021] [Rodriguez 2021] [Rousselot et al. 2008] [Saito et al. 1992] [Saitoh et al. 2006] [Savelyev et al. 2016] [Sawada and Takanashi 1987] [Schmidt-Eisenlohr and Kosing 2008] [Schneider] [Sebastiampillai et al. 2019] [Sheridan et al. 2015] [Sibilli et al. 2012] [Silva] [Silva et al. 2019] [Silva et al. 2021] [Slaboch et al. 2016] [Smith 2014] [Smith and Brooks 1986] [Soda and Tefy 2004] [Sokhey 1990] [Steiner 1990] [Sóbester 2010] [Tang et al. 2017b] [Tavares Silva et al. 2023] [Thomas et al. 2014] [Tindell et al. 1997] [Tindell et al. 1998] [Tsoi et al. 2014] [Tyacke et al. 2019] [Valencia et al. 2020] [Van Zante and Thomas 2012] [Vankan and Lammen 2019] [Vemula et al. 2023] [Vos 2019] [Wai et al. 1982] [Wang et al. 2018c] [Wang et al. 2019b] [Wang et al. 2025b] [Welge et al. 1976] [Łukasik 2017]
Fuel burn
What the thirty percent is measured in. Fuel burn and its estimation, lift-to-drag ratio, advanced and future transport concepts and their technology studies, and the standard atmosphere relations the cruise condition is computed with.
104 records. [Abrudan et al. 2023] [Acosta et al. 2013] [Adler et al. 2022] [Air Force Test Pilot School Edwards Afb Ca 1962] [Appendix B 2013] [Assessment of Wingtip Modifications 2007] [Ayers 1972] [Baerst et al. 1981] [Bellocq et al. 2015] [Berton et al. 2012] [Bourisli and Hamadeh 2020] [Braslow and Alford 1972] [Brown 2012b] [Brown 2012a] [Brown 2013] [Bruner et al. 2010] [Carson 1980] [Chatterji 2011] [Clemons et al. 2018] [Cocoves et al. 2026] [DAngelo et al. 2010] [Delgado Regis et al. 2004] [Driver 1974] [Economic Study of Multipurpose 1979] [Economon et al. 2011] [Experimental Investigation of Parametric 1972] [Giannakakis et al. 2019] [Goble 1972] [Govindaraju et al. 2017] [Greitzer et al. 2010] [Guo et al. 2014a] [Guruswamy and Tu 1994] [Guynn, Effective L/D, A Theoretical Approach to the Measurement of Aero-Structural Efficiency in Aircraft Design, NASA Langley Research Center, 2016, read in part] [Guynn 2011] [Guynn et al. 2013] [Hassan and Mavris 2020] [Hill and Kandil 2007] [Hill et al. 2009] [Hughes and Olsen 2022] [Hughes et al. 2011] [Hultgren 2011] [Ilario da Silva et al. 2017] [Integrated Application of Active 1980] [Integrated Application of Active 1981] [Integrated Application of Active 1982] [Integrative Application of Active 1980] [Jameson 2009] [Jansen and Perez 2012] [Jimenez et al. 2012] [Kentfield 2005] [Khadilkar and Balakrishnan 2011] [Khadilkar and Balakrishnan 2012] [Lange 1984b] [Langston 2014] [Lin 2025] [Liu and Stumpf 2018] [Ma 2025] [Ma and Elham 2021] [Martins 2016] [Maxwell and Dickinson 1981] [Molloy 1979] [Molloy et al. 1982] [Murphy et al. 2015] [Nickol 2012] [Nickol and Haller 2016] [Olsen 2022] [Park and O’Kelly 2014] [Park and Rokhsaz 2003] [Patel et al. 2025] [Pfaender et al. 2012] [Poll 2014] [Poll and Schumann 2024] [Quinlan et al. 2019] [Raymer et al. 2011] [Wahls et al] [Richter and Cosner 2003] [Rosatelli et al. 2023] [Sachs and Christodoulou 1987] [Sallee 1973] [Sinha 2007] [Smethers et al. 1980] [Stagliano and Hornung 2012] [Standard Atmosphere 2024] [Study of the Application 1972] [Sun et al. 2026b] [Schnulo et al] [Szodruch et al. 2011] [Taylor and Hunsaker 2023] [Thomas et al. 2016] [Urnes and Nguyen 2013] [Van Zante 2011] [Van Zante and Suder 2015] [Variyar et al. 2014] [Varshney et al. 2019] [Varshney et al. 2022] [Viken 2015] [Weinold and McKenna 2026] [Wilhite and Rehder 1979] [Williams 1972] [Williams 1973] [Winblade 1980] [Yanto] [Yanto and Liem 2018] [Young 2010]
Flight demonstrators
What the X-66A would have been. Flight demonstrators and research aircraft programmes, among them the X-29, X-56A and X-57, and the flight test techniques they use.
75 records. [Advisory Group for Aerospace Research and Development 1984] [Air Force Flight Test Center Edwards AFB CA 1974] [Air Force Test Pilot School Edwards Afb Ca 1990] [Armstrong 1977] [Asher et al. 2013] [Beh et al. 2018] [Borer et al. 2019] [Recine et al] [Chin et al. 2020] [Pham et al] [Deam 1984] [Deere et al. 2018] [Donelson et al. 1989] [Miller et al] [Gerdes 1981] [Godwin et al. 1964] [Graber 1987] [Grogan 1966] [Guo and Thomas 2022] [Harish et al. 2022] [Maliska et al. 2024] [Schlickenmaier et al. 2023] [Hess 1986] [Hicks and Matheny 1987] [Hicks and Moulton 1988] [Hicks and Huckabine 1989] [Hicks et al. 1987b] [Hillaker 1983] [Hodge and Kellogg 1996] [Hull and Roger 1975] [Jackson and Coyle 1983] [Grauer and Boucher] [Chin et al] [Hicks et al. 1987a] [Bhamidipati et al. 2024] [Kehoe et al. 1990] [Kidd 1966] [Krone 1980] [Kuhn et al. 2026] [Kukreja 2007a] [Kukreja 2008] [Lewerenz 1987] [Li and Pak 2014] [Lockheed-Georgia Co Marietta 1969] [Marchese 1963] [Jorgensen and Finazzo 2025] [Moorhouse 1990] [Spivey et al] [Nicholson and lowry 1966] [Painter 1983] [Pak 2012] [Pak 2013] [Pak and Truong 2014c] [Pak and Truong 2014b] [Pak and Truong 2014a] [Putnam 1983] [Ray et al. 1988] [Recine et al. 2023] [Richardson 1976] [Rohlmann and Keye 2015] [Rosenthal 1982] [Schnulo et al. 2017] [Schumacher 1952] [Sefic and Cutler 1986] [Simpson 1972] [Simpson 1979] [Smith et al. 1973] [Spacht 1983] [Stenberg 1983] [Traven and Whitley 1995] [Ueno et al. 2004] [Urnes and Yeager 1991] [Voracek and Clarke 1991] [Warsop and Crowther 2019] [Wulff and Zoellner 1991]
Ground testing
What the redirected programme now does. Full-scale static and fatigue tests of wings and airframes, ground vibration tests, loads calibration, and strain and deflection sensing, fibre-optic methods included.
145 records. [Andika et al. 2023] [Aniversario et al. 1983] [Arhore et al. 2025] [Bakalyar and Jutte 2012] [Bao and Chen 2015] [Barnes 1965] [Barsh 1944] [Bearinger et al. 2019] [Ben-Simon et al. 2017] [Bergan et al. 2012] [Bernasconi et al. 2015] [Bernini et al. 2008] [Berns et al. 2026] [bing et al 2015] [Biswas 2020] [Bock et al] [Burner et al. 2005] [Calkins and Mabe 2007] [Cazier and Kehoe 1984] [Cazier and Kehoe 1986] [Cecrdle 2019] [Chajec 2019] [Chang and Hodges 2007] [Chen et al. 2009b] [Chukwu 2025] [Chukwu et al. 2026] [Davis et al. 2015] [Daxini et al. 2022] [Domenski et al] [Eckstrom 1976] [Ellison 1986] [Engelbrecht 2017] [Fan et al. 2004] [Liaw 2019a] [Furstenau 1992] [Giacomin et al. 2018] [Gibson 2015] [Gimmestad 1981b] [Govers et al. 2017] [Govers et al. 2019] [Guan et al. 2026] [Han et al. 2024a] [Harris 1980] [Haywood-Alexander et al. 2024] [Heaney et al. 2018] [Herrera and Moholt 2009] [Hewitt and Albright] [Hewitt et al. 2005] [Hilfer et al. 2011] [Hoque and Duan 2020] [Horak and Novotny 2018] [Hotate and Abe 2005] [Hotate and Arai 2005] [Hughes 1973] [Ikeda et al. 2024] [Jenkins and Kuhl 1977] [Joels and Raveh 2022] [Joels and Raveh 2023] [Jutte et al. 2011] [Kehoe 1986] [Kehoe 1987a] [Kehoe and Freudinger 1994] [Kehoe and Voracek 1989] [Ketola 1964] [Ketola 1965] [Kocan 2023] [Koksal et al. 2013] [Koksal et al. 2019] [Kordes and Curtis 1975] [Kothari 2011] [Lakiza et al. 2025] [Lesiak et al. 2010] [Li et al. 2025a] [Liaw 2019b] [Lokos et al. 2015] [Lung and Pak 2009] [Lunia et al. 2000] [Luyckx et al. 2006] [Maher et al. 2022] [Martins and Kosmatka 2015] [Marzani et al. 2019] [Mason 1992] [McCarty 2001] [McCarty and Roeseler 1984] [Michaels et al. 2009] [Miller et al. 2014] [Miller et al. 2019] [Mondol and Potter 1979] [Najafi et al. 2025] [Spivey et al] [Spivey et al] [Spivey] [Norwood and Chichester 2015] [Paget et al. 2008] [Pak 2009] [Pak 2020b] [Pak 2020a] [Parida et al. 2002] [Heaney] [Pavlock 2011] [Peeters and Van der Auweraer 2012] [Pena 2020] [Pena et al. 2018] [Posey et al. 1997] [Potter et al. 2001] [Rausch and Ruffin 1990] [Robinson and Robinson 1950] [Samson et al. 2025] [Truong et al] [Sawicki 2026] [Schmidt and Nielsen 1994] [Sharqi and Cesnik 2020] [Sharqi and Cesnik 2025] [Sharqi and Cesnik 2026] [Shim 2020] [Shim and Ahn 2013] [Shyu et al. 1996] [Silverstein 1935] [Smith et al. 1986] [Smith et al. 1985] [Soderman and Aiken 1971] [Spivey et al. 2007b] [Spivey et al. 2007a] [Stahle and Forlifer 1975] [Standard Practice for Strain] [Stone 1973] [Subbaraman and Sun] [Sullentrup 2005] [Titov and Albagachiev 2015] [Toor 1973] [Uchida et al. 2015] [Vollmecke 1957] [Wada et al. 2018] [Walker and Kaufman 1977] [Wu et al. 2024b] [Xu 2005] [Xu et al. 2025a] [Yamauchi and Hotate 2004] [Yancey 1980] [Zhai et al. 2026] [Zhang 1987] [Zhang et al. 2013] [Zhang et al. 2026d] [Zheng et al. 2021] [Ņevskis]
Multidisciplinary design
How a configuration like this is sized. Multidisciplinary and aerostructural optimisation, aircraft conceptual design and sizing methods, and the open frameworks, among them OpenMDAO and Aviary, in which they are built.
141 records. [Abouhamzeh et al. 2022] [Ahuja and Mavris 2022] [Aircraft Sizing Engine Matching 2010] [Carrere et al. 2025] [Anipko et al. 2020] [Arbuckle and Sliwa 1984] [Ardema et al. 1975] [Aziz Alaoui] [Bahamonde Jacome and Elham 2017] [Bardin 2021] [Barth et al. 2002] [Barthelemy et al. 1991] [Benaouali and Kachel 2019] [Bendarkar et al. 2013] [Bons and Martins 2020a] [Bons and Martins 2020b] [Brooks et al. 2016] [Brooks et al. 2017] [Brown 1989] [Bryson and Rumpfkeil 2016] [Butler et al. 1998] [Butler et al. 1999a] [Butler et al. 1999b] [Cai et al. 2022] [Recine et al] [Castro et al. 2025] [Cavalcanti et al. 2006] [Chiba et al. 2005b] [Chiba et al. 2005a] [Chiba et al. 2007] [Chowdhary and Pant 2010] [Crossley et al. 2011] [Bertoli et al. 2016] [Diez and Iemma 2012] [Dirks and Meller 2000] [Donovan and Takahashi 2010] [Dordan and Silva 2025] [Eisler] [Elham and Bahamonde Jacome 2016] [Aretskin-Hariton and Kirk] [Hendricks 2020] [Finger et al. 2023] [Fornasier 1996] [Fujiwara et al. 2018] [Garcelon et al. 1999] [Garg et al. 2026] [Gavel et al. 2006] [Giles 1995] [Gogate et al. 1994] [Grandhi 2005] [Gray et al. 2012] [Heath and Gray 2012] [Hill and Olson 2004] [Hosseini et al. 2024a] [Hu et al. 2026] [Huang et al. 1996] [Huang et al. 2003] [Hwang 2017] [Iqbal and Sullivan 2012] [Gratz et al] [Janardhan and Grandhi 2003] [Jayaram et al. 1992] [Gratz] [Jasa] [Gray] [Kakkar et al. 2026] [Kano et al. 2025] [Karagoz et al. 2019] [Kenter Ahrazoğlu et al. 2025] [Kim et al. 2002] [Kim et al. 2006a] [Marfatia and Bergeson 2021] [Pierson et al] [Lambe et al. 2014] [Lau 1996] [Leonard 2021] [Liem et al. 2015] [Liu and Jiang 2022] [Liu et al. 2004] [Liu et al. 2014] [Lukyanov et al. 2024] [Malone and Mason 1991] [Mavris 1997] [McDonald 2007] [Mello et al. 2004] [Meng et al. 2022b] [Mistree et al. 1994] [Mody et al. 2010] [Mokotoff et al. 2025] [Mokotoff et al. 2026] [Mosca and Elham 2022] [Multidisciplinary Design Optimization 2021] [Multidisciplinary Design Optimization Architectures 2015] [Multidisciplinary Design Optimization 1994] [Nascimento et al. 2017] [Nikbay et al. 2025] [Obayashi] [Paiva et al. 2008] [Park et al. 2011] [Parker 1986] [Perez et al. 2004] [Perez et al. 2006] [Pilcher et al. 2012] [Pustina 2023] [Taylor] [Safavi et al. 2015] [Sanghi 2003] [Saves] [Saves 2024] [Schoen et al. 1980] [Schuhmacher et al. 2002] [Segee et al. 2016] [Sensmeier and Samareh 2005] [Shi et al. 2017] [Siegers and Smith 1995] [Silva and Guimarães 2020] [Simpson 2001] [Singh et al. 2026] [Skillen and Crossley 2008] [Sobieski 1984] [Striz and Lee 1994] [Tfaily and Kokkolaras 2018] [Tolson and Sobieszczanski-Sobieski 1985] [Tribes et al. 2004] [Tribes et al. 2005] [Tzong et al. 1992] [Uncertainty-Based Multidisciplinary Design Optimization 2015] [Vale et al. 2006] [Van der Velden et al. 2000] [Venter and Sobieszczanski-Sobieski 2002] [Venter and Sobieszczanski-Sobieski 2004] [Wakayama and Kroo 1995] [Wakayama et al. 1996] [Wrenn and Dovi 1987] [Xu et al. 2012] [Yan and Striz 1996] [Yao et al. 2025] [Yu and Du 2006] [Zhang 2011] [Zhang and Han 2013] [Ünal et al. 2023]
Emissions
The objective the programme states. Aviation carbon dioxide and other emissions, contrails, community and airframe noise, sustainable aviation fuel and hydrogen, and the policy instruments that price them.
164 records. [Abulibdeh 2026] [Aburto and Castillo-Landero 2025b] [Aburto and Castillo-Landero 2025a] [Nagamatsu 1976] [Ahmad] [Ahmed and Kurtulus 2019] [Ahrens et al. 2025] [Aircraft Emissions 2022] [Aircraft Engine Emissions 1977] [Aircraft Noise 2022] [Aircraft Piston Engine Exhaust 1976] [Anger-Kraavi and Köhler 2013] [Appleman 1957] [Avci et al. 2026] [Ayar 2025] [Bardon and Massol 2025] [Baughcum and Henderson 1998] [Beas et al. 2026] [Bergero et al. 2022] [Bergero et al. 2023] [Black et al] [Bozzaotre 2026] [Brodzik et al. 2025] [Bullerdiek et al. 2025] [Buvarp and Leijon 2024] [Campos 2015] [Charman and Bouferrouk 2023] [Chatetrton and Birksted-Breen 2025] [Cheng 2009a] [Cheng et al. 2026] [Climate Policy for Civil 2012] [Corsia and Sustainable Aviation] [D’Auria et al. 2024] [Dasgupta and Som 2024] [Dissanayaka et al. 2025] [Dodds et al. 1978] [Dönmez and Cecen 2024] [Ekici et al. 2023] [Envia 2018] [Ergin 2025] [Erling 2018] [Evangelinos et al. 2022] [Feldhausen et al. 2021] [Fink 1977] [Fukui 2025] [Fusaro et al. 2021] [Genhong et al. 2026] [González and Hosoda 2016] [Grewe 2020] [Guan et al. 2024] [Gutiérrez-Antonio 2025] [Han and Li] [He 2025] [Hizarci et al. 2022] [Hutcheson et al. 2016] [Hızarcı and Arifoğlu 2022] [Hızarcı and Arifoğlu 2023] [Ibitz 2014] [Imoisili 2026a] [Imoisili 2026b] [Investigating the Feasibility 2025] [Iyer and Takahashi 2024] [İlhan and Şahin 2025] [Jain 2024] [Kenyon et al] [Heidmann] [Kenyon] [Trudell 2023] [Jensen et al. 2023] [Park 2024] [Park 2025] [Li et al. 2023a] [Johnson and Gonzalez 2013] [Jones et al. 1978] [Juan et al. 2023] [Karpuk et al. 2022] [Khongbamphen 2025] [Kolosz et al. 2026] [Koo et al. 2025] [Kroyan et al. 2022] [Ajmani and Moder] [Kılıç et al. 2026] [Kłopotowski and Cwojdziński 2022] [Lang et al. 2024] [Lazic et al. 2023] [Lee 2025] [Mahesh and Kiran 2025] [Post] [Mayer 2012] [Mayor and Tol 2007] [Mayor and Tol 2010] [Mercer et al. 2006] [Miake-Lye et al. 1992] [Nöding et al. 2021] [Mirea 2025] [Mirea and Cican 2024] [Morrison et al. 2011] [Morrison et al. 2012] [Mubenesha et al. 2026] [Nantke 2011] [Nizamitdinov et al. 2017] [Obazu 2026] [Okai et al. 2022] [Okonye et al. 2026a] [Okonye et al. 2026b] [Olsthoorn 2001] [Olumoh] [Ovaere and Proost 2024] [Palladino] [Pařez et al. 2026] [Pinho 2023] [Pontika et al. 2026] [Procedure for the Calculation] [Quiggin 2023] [Reddingius 1974] [Reddingius 1977] [Rezy et al. 1979] [Fong et al] [Robertson 2013] [Romeo et al. 2012] [Rudey and Lezberg 1976] [Ruijgrok 1980] [Runnels and Gupfa 1973] [Sahai and Stumpf 2014] [Salgas et al. 2023] [Speakman et al. 1978] [Spyropoulos and Holmes 1999] [Sridhar 2015] [Sumsurooah et al. 2022] [Sun et al. 2026d] [Singla 2024] [Synylo and Duchêne 2019] [Synylo et al. 2024] [Tam 2015] [Tao et al. 2025] [ten Cate et al 2025] [Terrenoire et al. 2019] [Thanawala 2020] [Thomas] [Thomas et al. 2010] [Thomas et al. 2012] [Tokuslu 2020] [Tripathi and Choubey 2025] [Ugbeh-Johnson and Carpenter 2023] [Vera-Morales and Hall 2010] [Wang et al. 2024b] [Wasala et al. 2019] [Wasiuk et al. 2015] [Wey and Lee 2018] [Woehler et al. 2023] [Woeldgen et al. 2026a] [Woeldgen et al. 2026b] [Wuebbles et al. 2006] [Xu et al. 2025b] [Xu et al. 2026c] [Yadav and Kumar 2026] [Yamamoto et al. 2012] [Yamamoto et al. 2013] [Yan et al. 2025] [Yang and Yao 2025] [Yin] [Zhang et al. 2025] [Zou et al. 2025] [Öztürk et al. 2026]
Programme designations
The programme’s own record. The SUGAR reports and their wind-tunnel and aeroelastic test papers, the Sustainable Flight National Partnership and X-66 overviews, and the donor lineage where its title is on a subject the argument uses. 20 of these 28 records are report primaries under the rule set out under The Definition of Primary, which is the share one expects of a programme recorded in contractor reports.
28 records. [Allen et al. 2015] [Bartels et al. 2017] [Bartels et al. 2015b] [Bartels et al. 2015a] [Bartels et al. 2015c] [Behzadpour 2025] [Bradley and Droney 2012] [Bradley and Droney 2015] [Cobleigh] [Callaghan 1973] [Van Zante and Wahls] [Hendricks] [Henne 1989] [Ordaz] [Kenyon] [Heidmann] [Harrison et al. 2025] [Browne et al] [Wahls and Cole 2025] [Wahls] [Bartels et al] [Schaufele and Ebeling 1967] [Scott et al. 2016] [Scott et al. 2015a] [Scott et al. 2015b] [Silva and Silva 2025] [Nerone] [Zhao et al. 2015]
Where the Framing Breaks Down
This article has treated the X-66A through one keystone, being the aspect-ratio trade and the infrastructural constraint that truncates it. Five things that framing gets wrong should be named.
The Demonstrator Is Not the Aeroplane the Analysis Describes
Every number in the sizing sections above belongs to configuration 1104-001-RG, a 737-class vision aircraft that was never built, and not to the X-66A. The demonstrator is a modified MD-90 with a shortened fuselage, a different wing and different engines [NASA X-plane release]. Its span appears in secondary aerospace coverage and in no primary document this article has read, and its wing area is not published at all, so its aspect ratio cannot be computed and is not stated anywhere in this article. Anyone quoting an aspect ratio for the X-66A itself is quoting the vision aircraft.
The consequence for the keystone is that the keystone is not about the demonstrator at all. A demonstrator exists to validate models, not to be optimal, and nothing in the optimality analysis above applies to an aeroplane whose fuselage, tail and payload were chosen by what was available at Victorville. Treating the X-66A as a small transport aircraft would be the central error available here.
Fuel Burn Is Not the Objective an Airline Optimises
The criterion used throughout is cruise fuel, and no airline buys an aeroplane on cruise fuel. It buys on direct operating cost, in which fuel is one term beside crew, maintenance, ownership and the revenue consequences of block time. The Phase IV report’s justification for raising the cruise Mach number was productivity rather than efficiency, and it accepted a measurable fuel penalty to get it. A folding wing adds a mechanism to inspect, a hinge to certify, a de-icing complication the Phase II report discusses, and an operational procedure that must happen on every taxi [SUGAR Phase II volume I]. None of that appears in a fuel-burn optimum and all of it appears in an airline’s spreadsheet.
So the finding that this aeroplane is below its fuel-burn-optimal aspect ratio concerns one term in a cost function and is not a criticism of a design. The programme chose 170 feet on a flatness argument, in effect a judgement that the fuel term had stopped mattering, and that is the point at which the other terms should take over.
The Weight Exponent Carries the Argument and Is Supported at Only Some Conditions
The three-halves power is derived for bending material under a design manoeuvre moment and is not derived for anything else. The aspect-ratio dependence of flutter-driven stiffness, of gust-driven loads, of the gear pylon, of the fold mechanism and of the strut’s buckling is not three halves and this article has not derived it. The Phase IV report’s aeroelastic conclusions note that the sizing covered the wing, strut and jury only, used preliminary flight design load conditions, sized to generic criteria, and should be extended.
Inclusion of more of the airframe in the aeroelastic sizing, particularly the gear pylon. This effort addresses the risk of unexpected weight increases late in airframe development, when key components are finally analyzed.
If the full-airframe exponent were above the 2.1755 the exact fixed-lift criterion requires, or the 3.0832 the fixed-area criterion requires, the keystone’s conclusion would reverse. This article has bounded what it would take and has not shown that it does not happen. The result ends at that boundary, which is why the falsifiable form was stated explicitly above.
A Rigid Prop Is Not a Truss
The relief analysis is a rigid vertical prop at one station, and the aeroplane has a pinned elastic strut at fourteen degrees of dihedral, a jury strut, a swivel joint and a landing-gear pylon [SUGAR Phase IV final report]. The relief factors are bounds, as the subsection headed A Rigid Prop Is a Bound and Not a Prediction sets out, and the factor of 7.88 between the bound and the published bending material is mostly non-optimum material rather than modelling error. A reader wanting the real relief must take it from a finite-element sizing and the one in the record covers three components out of an airframe.
An Aerodrome Reference Code Is Not a Law of Nature
The sharpest claim in this article is that the aspect ratio of the single-aisle fleet is an airport number, and the claim has a limit. Code letters are revised, airports are rebuilt, and a sufficiently valuable aeroplane changes its infrastructure rather than fitting it. The Airbus A380 and the Boeing 747-8 are both Code F aeroplanes and both obtained accommodation, at spans of about 79.8 and 68.4 metre against that band’s 65 to 80 [Airbus A380] [747-8 airport planning]. And the Boeing 777X folds its wingtips across a code boundary exactly as this aeroplane would, from about 71.8 metre unfolded, in Code F, to about 64.8 folded, in Code E [Boeing 777X]. So the manoeuvre this article describes is already certificated on a wide-body, the strongest reason to think the gate box is negotiable for an aeroplane worth negotiating for.
What the coincidence of margins does establish is weaker and still useful. It establishes that for this class of aeroplane, at this moment, the designers treated the boundary as fixed and designed to within inches of it twice. That is a fact about the design process and about the economics of gate compatibility for a high-volume narrowbody. It makes no claim about aerodynamics at all.
The Source Base
The reference base behind this article is 3,970 definitions, of which 29 are primary documents, 66 are the prior articles of this series, and 3,875 are the research records of the survey. Five of the research records are cited in the text for specific statements, and every research record also appears in one cluster row.
Eight of the documents read for this article bear most directly on its argument. The fair comparison the Phase IV report recommended, cited from the agency’s own deposit [Recine et al. 2025a]. The Phase I final report, which contains the planform optimisation every later phase inherited [SUGAR Phase I final report]. The Federal Aviation Administration’s own airport design circular, which settles the unit question the keystone turns on [FAA airport design]. A 1981 Grumman wing weight methodology, which supplies a primary for the non-optimum factor [York and Labell 1980]. A 2016 NASA Langley paper on aero-structural efficiency, which states this article’s own methodological thesis in one sentence [Guynn 2015]. A 1980 strut-braced business jet study at aspect ratio twenty-five [Turriziani et al]. A 1976 span-distributed-load cargo study [Whitehead]. And a 2024 multidisciplinary optimisation of this configuration using the Aviary framework [Aviary optimisation].
The survey is summarised by two statistics, a fraction and a count, and they are defined here because either one alone misleads. Adding contemporary records lowers the primary fraction while raising the period count, so the two are read together.
\[\text{primary fraction} = \frac{\left|\left\{r : \mathrm{src}(r) \in \{\text{reports}, \text{defence}\}\right\}\right|}{\left|\mathcal{R}\right|}, \qquad \text{period count} = \left|\left\{r : \mathrm{year}(r) \ge 2015\right\}\right| \tag{92}\]Of the 3,875 research records, 1,067 come from report servers rather than journal indices, which is 27.5 percent. 3,688 carry a resolved publication year, their median is 2012 and they run from 1921 to 2026. 1,610 are from 2015 onward, which is 43.7 percent, and 1,161 predate 2000, which is 31.5 percent.
The primary fraction of 27.5 percent reflects the subject. This subject’s literature lives largely in journals and in conference proceedings, because truss-braced wing work has been done largely in universities and published through the American Institute of Aeronautics and Astronautics rather than issued as agency reports. The period count of 1,610 is the figure to read beside it, since contemporary records lower the fraction while raising the count.
What the Archives Hold on This Subject
The constraint this article argues is binding has almost no report literature. Of the 40 records in the span constraints cluster, which carries this article’s own keystone, 39 come from the Crossref bibliographic index and one comes from the reports server, a 1974 terminal-area compatibility study of an advanced long-haul transport [Advanced Subsonic Long-Haul Transport 1974]. On the cited records, the report literature contains almost no airport-compatibility work, because airport design is a regulator’s and an airport planner’s subject rather than a research agency’s, and it is published as advisory circulars and aerodrome annexes rather than as technical reports. The two literatures also name their subjects differently. A journal paper speaks of airport compatibility and a report of airplane characteristics for airport planning, and a journal paper speaks of multidisciplinary optimisation where a report names its code.
Several of this subject’s words are homonyms in the public archives, and the table gives what a sample of ten results for each search contains.
| Query | What the registry actually returns |
|---|---|
SUGAR |
sugar derivatives in astrophysical ice analogues, enantiomer ratios in carbonaceous meteorites, a reducing sugar from Coccidioides immitis, blood sugar regulation. One of ten aeronautical |
sugar aircraft |
ten of ten the programme’s own reports |
aspect ratio, bibliographic index |
nanofabrication, aspect-ratio-dependent etching, a holey metalens, mica-reinforced thermosets, an image-mask algorithm. Zero of ten aeronautical |
high aspect ratio |
x-ray masks, triangulation algorithms, cooling passages, exhaust systems. One of ten a wing |
strut |
oleo struts, antenna struts, rocket stage-separation struts, a piezoceramic strut. Zero of ten a wing brace |
truss |
deployable space trusses, robotic truss assembly, antenna support trusses. Two of ten aeronautical |
braced wing |
ten of ten on subject |
truss braced wing |
ten of ten on subject in both registries |
Adding one word to SUGAR takes the hit rate from one in ten to ten in ten. And aspect ratio
is aeronautical at the reports server and a nanofabrication term in the bibliographic index, so
whether a word is a homonym depends on the archive searched. A homonym table built from one
registry is not a homonym table.
The Definition of Primary
Primacy is a property of the document, and this survey derives it from the identifier. A record is a report primary when its address is on the NASA Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. That is the meaning of the reports and defence sources in the primary fraction above. 20 of the 28 records in the programme designations cluster are report primaries under that rule. The rule does not reach the journal and conference papers of the university school that carries most of this subject, nor the curated primary documents listed under Reference, so 27.5 percent is a lower bound on how much of the survey is first-hand.
Thin Subjects
Six subjects that this article’s derivations depend on have thin literatures. Each count below is the number of research records in this survey whose title carries at least one of the listed terms, with a hyphen read as a space. Where a row joins two groups of terms with the word together, a title must carry a term from each group.
| Subject | Title terms counted | Records |
|---|---|---|
| airport and gate compatibility of a span | airport, gate, taxiway, aerodrome, terminal area together with compatibility, operation, design | 6 |
| fuel volume in a thin wing | fuel volume, fuel capacity, fuel tank, wet wing, dry wing, body tank | 2 |
| ice protection of a wing | anti-ice, anti-icing, ice protection, de-icing, icephobic | 0 |
| ground effect of a long wing | ground effect, ground proximity together with aspect ratio, span, spanloader, long wing | 8 |
| buffet of a braced wing | buffet, buffeting together with truss, strut, braced | 7 |
| the aspect-ratio dependence of wing weight | weight, mass together with aspect ratio | 5 |
The gate box is the thinnest subject the keystone uses. Of the six titles that name an airport or a gate together with compatibility, operation or design, three name compatibility, and the rest treat airport operations or noise. The binding constraint of this article therefore rests on the regulator’s circular and the agency’s airport-operations memorandum rather than on a research literature.
The fuel volume of a thin wing appears only as a hydrogen problem. Both titles that name it concern dry wings for hydrogen aircraft, so the deficit the Phase IV tables show for a kerosene wing, and which the fair comparison ranks as the larger of its two causes, stands on those two documents rather than on a published study of it.
Ice protection of a wing has no title at all, which matches the Phase IV report naming the anti-icing penalty and putting no number on it.
Ground effect has been studied on short wings. Of the eight titles, four concern low aspect ratio wings and one names a high aspect ratio wing, on an unmanned air vehicle and without stating it, and the highest aspect ratio any of them states is 10, so the group this article displays for a wing of aspect ratio 19.565 has no stated counterpart in the survey.
The buffet of a braced wing is a literature of the programme alone. All seven titles name the transonic truss-braced wing, so the empirical buffet boundaries the report says are unvalidated for this species of aircraft have no independent check in the survey.
The aspect-ratio dependence of wing weight, which is the keystone’s exponent, is the subject of five titles, of which three concern high aspect ratio wings. The three-halves exponent this article derives therefore stands on its own derivation rather than on a body of measurement.
What the Survey Admits
The research records come from three public archives, being NASA’s Technical Reports Server, which this article calls the reports server, the Defense Technical Information Center and the Crossref bibliographic index. A record is admitted only when a person reading its title finds it on this article’s subject, meaning a subject the argument above uses, read against the article and not against a word. Every record is then assigned to the one cluster its title belongs to, and no cluster is a residual.
Several of the subject’s words name other things, and records of those kinds are excluded. Full-scale testing also names tests of buildings, piers, concrete members and rock works, and fire and smoke tests. Fibre-optic strain sensing also names the monitoring of pipelines, bridges, levees and wells. Wing also names the wings of damselflies, hawkmoths, bats and birds, racing-car wings and the otter boards of fishing gear. Aeroelasticity also names bridges, power-transmission conductors, panels, plates and shells, and rotary-wing simulation and vibration work is excluded too, since a bare aeroelastic does not name a wing. Aspect ratio also names windows, heat-transfer channels, microactuators, crystal growth and the drag of net cages. Low aspect ratio is also the vocabulary of fighter, missile and micro air vehicle wings, whose pitch-up, wing rock and flapping studies are excluded unless the title concerns induced drag, the tip vortex, stall, flutter or another subject the argument uses. Rotorcraft, tiltrotors and multirotor vehicles, spacecraft and launch vehicles, ships and yachts, cabin interior noise, crashworthiness and ditching, and air traffic procedures such as arrival scheduling and descent planning are excluded as distant domains. Correction, erratum and withdrawal notices, trade news items, citation-only placeholders and supplementary-material files are excluded as well, because a survey counts works and not notices or parts of works, and so is a second deposit of a document already held, whether on the same archive or as a report-server copy of a paper whose publisher record is kept. Book reviews and encyclopedia and dictionary entries of two pages or fewer are not admitted either, because they are not research works.
Some records of doubtful relevance are kept. General papers on fibre-optic strain sensing are kept because the ground testing cluster covers strain sensing and these papers describe its methods. Papers on sustainable aviation fuel production, aviation fuel taxes and fuel marketing are kept because they bear on the emissions objective the programme states. The aeroelasticity of micro air vehicle and membrane wings, and V/STOL research aircraft of the X and XV series, are kept because they treat a subject the argument uses on another vehicle. A reader may weigh them accordingly.
Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.
Epistemic State
Historical Fact, From Primary Documents
The X-66A designation was allocated on 3 May 2023 to Boeing with NASA as sponsor and an engines cell reading 2 Pratt & Whitney, and the compiler’s description names a Transonic Truss-Braced Wing demonstrator built from a highly modified MD-90 [Parsch, DOD 4120.15-L Addendum].
The Funded Space Act Agreement numbered PAM 36785 was signed by Boeing on 12 January 2023 and by NASA on 13 January 2023, carries a term of seven years from the later signature, and lists 27 funded milestones totalling 425 million dollar [Funded Space Act Agreement]. Those figures are read from the instrument, and the 27 payments sum exactly to the appendix’s stated total.
NASA announced the selection on 18 January 2023 with a stated NASA contribution of 425 million dollar over seven years and a partner contribution of about 725 million dollar [NASA award release]. NASA and Boeing announced on 12 June 2023 that the United States Air Force had designated the aircraft the X-66A [NASA X-plane release], and the register dates the allocation to 3 May 2023 [Parsch, DOD 4120.15-L Addendum]. The donor aeroplane arrived at Palmdale from Victorville in August 2023 [Boeing arrival release]. NASA published its account of a revised approach on 24 April 2025 [NASA thin-wing item].
All geometry, weight, drag and performance figures attributed to configuration 1104-001-RG and to its Mach 0.745 predecessor are read from the Phase IV final report, which was read in full [SUGAR Phase IV final report]. The span-limit optimisation result and the Code C and Code D statements are read from the Phase II report [SUGAR Phase II volume I]. The reference-code classification and the alternative conceptual model’s dimensions are read from the March 2025 airport-operations memorandum [airport operations memorandum]. The budget figures are read from three NASA budget documents [FY2024 justification] [FY2025 justification] [FY2026 technical supplement]. The award figures are read from the federal award reporting system [USAspending].
Engineering Analysis, Derived Here and Checkable
The aspect ratio recomputed from the published span and area is 19.5652 against the stated 19.565. The induced drag coefficient recomputed from the published lift coefficient, aspect ratio and airplane efficiency factor is 0.009026 against the published 0.00903, an error of 0.050 percent. The induced-drag fraction of cruise drag is 0.3351.
The root bending moment of an elliptic spanload is $Lb/(3\pi)$, derived in closed form and confirmed both by an elementary centroid argument and by numerical quadrature at six stations. The bending-material weight of a straight-taper wing at fixed area, lift and thickness ratio scales as the three-halves power of aspect ratio, derived analytically and measured numerically as 1.5000.
The two optimality conditions are exact and are the central analytical claims of this article. At fixed wing area and cruise condition the stationary aspect ratio satisfies $\nu = 1/2$, independently of every other parameter in the problem. At fixed cruise lift coefficient it satisfies $\nu = \delta$. The logarithmic curvature at the second of those stationary points is exactly $\delta(n+1-2\delta)$, or 0.6131 for this aeroplane.
The stationary aspect ratio located by bisection on the anchored power law is 25.100 on the linearised treatment and 23.754 on the exact Breguet treatment, 28.3 and 21.41 percent above the design value, and the fuel penalty for sitting at the design value is 1.90 and 0.9481 percent respectively. The Phase II report’s independent multidisciplinary optimisation reports less than 1.4 percent for all further span beyond 170 feet, which the exact figure falls inside and the linearised figure does not.
Carrying the Breguet exponential rather than linearising it introduces one factor, $\Phi = Xe^{-X}/(1-e^{-X})$, equal to 0.892462 at this aeroplane’s fuel fraction of 0.200193. The exact stationary conditions are $\nu^{*} = \Phi\delta$ at fixed cruise lift coefficient and $\nu^{*} = \Phi\delta/(1-\Phi+2\Phi\delta)$ at fixed wing area, giving 0.299033 and 0.423797, and both reduce to the linearised forms as the fuel fraction vanishes. The exact curvature is $\Phi\delta(n+1-\delta-\Phi\delta) + \Phi’(X)X\delta^2$, evaluating to 0.503772.
The folded span of 118 feet clears the ICAO Code C boundary of 118.1102 feet by 1.32 inch and sits exactly on the FAA Airplane Design Group III boundary of 118 feet, whose bound is exclusive. The unfolded span clears the Code D boundary of 170.6037 feet by 7.24 inch and the Group IV boundary of 171 feet by 12 inch. The two renderings of the same bound differ by 1.3228 inch, and the FAA circular states that its customary units govern and its metric values are soft rounded conversions. The aspect ratio the folded span permits at the same area is 9.427, so the fold is worth a factor of 2.076 in aspect ratio and 36.0 percent in lift-to-drag ratio.
The Korn relation with a supercritical constant of 0.95 gives drag-divergence margins of 0.0203 and 0.0181 for the two configurations, differing by 0.00212, and the difference stays below 0.00672 across the plausible range of that constant. The thinning from 0.120 to 0.104 is worth 4.86 degrees of sweep.
The main strut carries 4.013 times the vertical relief it delivers and its horizontal reaction is 3.886 times, from its 14.43 degree dihedral. Bracing a pin-ended column at mid-length multiplies its Euler critical load by 4.
Inference, Stated as Such
That the aspect ratio of the current single-aisle fleet is set by the Code C gate box rather than by structural weight is an inference. It rests on the arithmetic coincidence between 9.427 and the fleet’s value, on the Phase II report’s observation that aircraft families have grown their spans to the limit of their design category, and on the demonstration that the structural optimum for this class lies well beyond where the fleet sits. It is not a claim any source in the reference base makes in those words.
That the pause was taken immediately before the Wing and Strut Critical Design Review because that review would have committed the wing to fabrication is an inference about motive and the record does not state a motive. What the record states is the date of the announcement, the date and content of the next milestone, and the fact that the retained activity is wing research. The sequence is a fact and the reading of it is not.
That the fold station being exactly the FAA Group III boundary is deliberate rather than coincidental is an inference, though a narrow one. The Phase IV report says the fold accommodates existing Code C gate and taxiway infrastructure, which states the purpose, and the airport-operations memorandum requires the span to be less than 118 feet to reach Group III, which states the condition. That the report’s stated 118 feet does not strictly satisfy that condition is a fact about two documents and this article does not resolve it.
That the FY2026 technical supplement’s renaming of the project from Sustainable to Subsonic is deliberate is not established. Three consistent occurrences in one document make a single typographical slip unlikely and do not make intent certain. No release announces a renaming.
That the truss buys a weight coefficient rather than an aspect-ratio exponent is a derived result under an explicit assumption, namely that the truss scales geometrically with the span. A truss whose attachment station moved outboard as span grew would behave differently and no source says the attachment station is fixed under span growth.
What the Record Does Not Settle
The wing area of the X-66A is not published, so its aspect ratio cannot be computed. This article states no aspect ratio for the demonstrator.
The weight of the configuration is the primary record’s largest stated uncertainty and it is the input the keystone is most sensitive to. The aspect-ratio exponent of the full airframe, as opposed to the bending material, is unknown, and the value above which this article’s conclusion reverses is 2.1755 under one criterion and 3.0832 under the other on the exact treatment.
The fair comparison narrows that uncertainty without closing it. Its like-for-like pair gives an apparent weight elasticity between -0.12956 and +0.09926, every value below the target and three of four negative, but an elasticity from a pair of aeroplanes differing in span, area, thrust and altitude is not a derivative and this article does not treat it as one.
The flutter margin at Mach 0.92 is unresolved in the primary record, where two models of the same structure disagree by about fifty percentage points and about the sign, and the report states that the method used is incapable of capturing the relevant flow features.
Whether the local buffeting the analysis predicts would be perceived by a pilot is stated by the report as unknown, a certification question that no wind tunnel can answer.
The maximum lift coefficient at flight Reynolds number has not been measured, and it feeds the field-length constraint that helped set the span.
The fair cantilever-versus-braced comparison the Phase IV report recommended exists and settles part of the question. A NASA Ames team published it in January 2025 and it is read in full [Recine et al. 2025a]. It gives 1.65 percent at the economic mission against Boeing’s 7.2 percent, a factor of 4.36, with fuel volume the dominant cause and the weight calibration second.
What remains unsettled is the keystone’s reach rather than its correctness. The optimality conditions are cruise-fuel conditions, the fair comparison shows the short-sector advantage spent in climb, and this article does not derive a climb-inclusive condition.
The structural weight of the Phase III configuration is not published in the volume read here, so this article’s prediction that the Phase IV thinning and sweeping cost 28.7 percent in bending material stands unchecked.
The anti-icing energy and volume penalty is named by the report and quantified by nobody, this article included.
And the X-66A has not flown. Everything above about what a flight test would have settled concerns questions, not answers.
Out of Scope
The hybrid-electric branch of the same research programme. Phase II devoted a whole volume to hybrid electric design exploration and the programme’s SUGAR Volt concept is a distinct configuration with a distinct propulsion argument. It is not a truss-braced wing question and is not treated here.
The MD-90 and its lineage. The donor aeroplane’s design, its original engines, its empennage, and the reasons an MD-90 was available in the desert in 2023 are all outside the modification boundary and the series has an established practice of staying on the right side of it.
The detailed aerodynamic design of the wing and strut. Phase IV devotes fifty-six pages to the inverse design of the junction region, the strut planform modifications, the fuselage bump and the twist distribution, using computational tools this article names and does not exercise. That is a subsystem deep-dive and not a research-aircraft article.
The low-speed and high-lift test volumes. Volumes II, III and IV of the Phase IV final report cover the high-speed test, the high-lift system design and the high-lift test. None of the three is held by the reports server, so they are not merely unread but unavailable through the channel this article uses, and the conclusions drawn from them here are limited to what Volume I quotes.
The noise and community-impact question. The reference survey includes a system noise technology roadmap for a truss-braced wing against a conventional peer and an aeroacoustic computation series, and airframe noise from a wing of this planform is a genuine open question. It is a different keystone.
The certification basis for a folding wing in commercial service. The Boeing 777X established one and this aeroplane would need another, and the regulatory history is a legal subject rather than an engineering one.
The hydrogen and sustainable-aviation-fuel pathways that appear in the emissions cluster. They change the fuel and not the aeroplane, and the configuration question is independent of them.
Conclusion
The X-66A was to answer whether a truss-braced wing is worth its truss, and the primary record contains enough to answer a sharper question than that, the question of where the span of a transport aeroplane comes from.
It comes from an airport. The wing folds at 118 feet and 118 feet is exactly where the Federal Aviation Administration’s Airplane Design Group III ends, with a margin of zero inches against a bound that is exclusive, and the fold station sits inside the 1.3228 inch rounding between that bound and ICAO’s metric one, which the FAA circular says does not govern. The unfolded span of 170 feet clears the next boundary by 12 inch by one reckoning and 7.24 inch by the other. The fold is worth 36.0 percent in lift-to-drag ratio, and everything the unconstrained aerodynamic optimum has left to give beyond the chosen span is worth 0.9481 percent in fuel, which falls inside the under 1.4 percent the programme’s independent optimisation reported in April 2015. The mechanism is worth more than the optimum.
The optimality condition itself turned out to be two conditions, the analytical result this article would keep if it could keep only one. At fixed wing area and cruise condition the fuel-burn-optimal aspect ratio occurs where a one percent gain in aspect ratio costs exactly half a percent in weight, and no other quantity in the problem appears. At fixed cruise lift coefficient it is where the weight elasticity equals the induced-drag fraction of drag. The curvature at the second stationary point is exactly $\delta(n+1-2\delta)$, so a design whose optimum lies 21.41 percent above it can pay 0.9481 percent for it, and an infrastructural constraint can bind without visible cost. Carrying the Breguet exponential rather than linearising it generalises both conditions by a single factor and makes the optimum flatter still, because the linearisation is not free at a twenty percent fuel fraction.
The truss turned out to buy a coefficient and not a power. A geometrically similar truss leaves the three-halves exponent exactly where a cantilever leaves it, and since the optimum moves only as $k_w^{-2/5}$, no plausible structural achievement moves it far. So the value of this configuration is not in the truss as a weight-saving device. It is in the thin wing the truss makes affordable, which raises the drag-divergence Mach number, which permits less sweep, which permits laminar flow. That chain is what Boeing kept when it stopped the aeroplane, and on this article’s reading the chain is where the fuel was all along.
What the agreement scheduled for 153 million dollar before the pause was nine milestones, four of them design reviews, and it rests on a very good report. The agreement would have paid 98.824 percent of its total before the aeroplane flew, so the public money was never mostly about the flying. It stopped between the February 2025 milestone and the May 2025 Wing and Strut Critical Design Review, the gate at which a wing becomes a thing to be built rather than a thing to be studied. The instrument that funded it has no pause in it, offering termination for a missed milestone instead, and the federal award record holds, under the project’s name, a single contract for 41,198.00 dollar worth of desk models.
And the number everybody quotes is not in the document that funds it. The agreement states no percentage. The wing alone is worth 7.2 percent against an advanced conventional aeroplane of aspect ratio 13, by the contractor’s own calculation and explicitly independent of other technologies. The whole package against a 2005 aeroplane is 55.86 percent. Thirty percent is between them, and the clause that makes it true, which attributes it to the configuration combined with additional technologies, appears in the January 2023 release, the June 2023 release and the 2026 budget supplement alike. The figure never changed and neither did the clause. What moved between the three is the baseline, from the most efficient single-aisle aircraft to the best-in-class aircraft and back.
The last word belongs to the report, because its recommendation list sets out what fifteen years of work concluded. It asks for a dedicated buffet wind-tunnel test, a high-Reynolds-number low-speed entry, a detailed structural integration effort, and an equivalent optimisation of cantilever and truss-braced aircraft of the same technology level to allow a more fair and transparent comparison of the two configurations. That last item is a statement, by the organisation with the most to gain, that the comparison on which the concept rests had not yet been made on equal terms. Then it asks for the preliminary design of a demonstrator, the one recommendation that was carried out and the aeroplane that did not fly.
Equations
This section collects the 92 display equations of the article in the order they appear, each under its number and reproduced exactly as it stands in the text. A title in bold names an established result, and a title in italics beginning with the word Bespoke names a construction made for this analysis. Of the 92, 47 are established and 45 are bespoke. The structural and geometric relations follow the contractor’s report in feet, inches, pounds and pounds per square inch while the atmosphere is evaluated in the International System, and several letters need care, so $\delta$ is the induced-drag fraction of total cruise drag, $L$ is the wing lift at the design load factor while $L/D$ is the lift-to-drag ratio, $e$ is the airplane efficiency factor while $e^{-X}$ is an exponential, $\mathcal{I}$ is the second moment of area of the strut while $\mathcal{I}(\lambda)$ is the taper integral, $\varepsilon$ is a logarithmic error in aspect ratio, and $h$ is a geopotential altitude while $h_g$ is a height above the ground.
Equation 1, Share of the Agreement Paid Before First Flight. The milestones that fall due before first flight carry 98.82 percent of the agreement’s payments. Funded Space Act Agreement
\[\frac{\displaystyle\sum_{i < i_{\mathrm{ff}}} \mu_i}{\displaystyle\sum_{i} \mu_i} = 0.988235 \tag{1}\]Equation 2, Share of the Agreement Reached Before the Pause. Through milestone nine the agreement had paid 153 of its 425 million dollars, 36.00 percent of the total. Funded Space Act Agreement
\[\frac{\displaystyle\sum_{i \le 9} \mu_i}{\displaystyle\sum_{i} \mu_i} = \frac{153.00}{425.00} = 0.3600 \tag{2}\]Equation 3, Revision Between Two Budget Books. A revision is the fractional change between two projections of the same fiscal year. FY2025 justification FY2026 technical supplement
\[\text{revision} = \frac{\mathcal{P}_{\mathrm{new}} - \mathcal{P}_{\mathrm{old}}}{\mathcal{P}_{\mathrm{old}}} \tag{3}\]Equation 4, Aspect Ratio From Span and Area. The aspect ratio is the square of the span over the reference area, which the published span and area reproduce to the printed digits. SUGAR Phase IV final report
\[A = \frac{b^2}{S} \tag{4}\]Equation 5, Standard Atmosphere Temperature. The temperature falls linearly with geopotential altitude up to the tropopause and is constant above it.
\[T(h) = \begin{cases} T_{\mathrm{sl}} - \lambda_a h, & h \le h_{\mathrm{t}} \\[4pt] T_{\mathrm{sl}} - \lambda_a h_{\mathrm{t}}, & h > h_{\mathrm{t}} \end{cases} \tag{5}\]Equation 6, Standard Atmosphere Pressure. The pressure follows a power law of the temperature ratio in the troposphere and falls exponentially in the isothermal layer above.
\[p(h) = \begin{cases} p_{\mathrm{sl}} \left(\dfrac{T(h)}{T_{\mathrm{sl}}}\right)^{g_0/(R_a \lambda_a)}, & h \le h_{\mathrm{t}} \\[6pt] p(h_{\mathrm{t}}) \exp\left(-\dfrac{g_0 \left(h - h_{\mathrm{t}}\right)}{R_a T(h_{\mathrm{t}})}\right), & h > h_{\mathrm{t}} \end{cases} \tag{6}\]Equation 7, Speed of Sound and True Airspeed. The speed of sound follows from the temperature, and the true airspeed is the Mach number times it.
\[a = \sqrt{\gamma R_a T}, \qquad V = M a \tag{7}\]Equation 8, Dynamic Pressure in Pressure and Mach Number. The dynamic pressure is half the density times the square of the speed, which equals half the ratio of specific heats times the pressure times the square of the Mach number.
\[q = \tfrac{1}{2}\rho V^2 = \tfrac{1}{2}\gamma p M^2 \tag{8}\]Equation 9, Lift Equation. In level flight the weight equals the lift coefficient times the dynamic pressure and the wing area, which fixes the lift coefficient.
\[W = C_L \, q \, S \qquad \Longrightarrow \qquad C_L = \frac{W}{q S} \tag{9}\]Equation 10, Lift Coefficient at the Stated Cruise Condition. At the maximum take-off weight, the published wing area and the buildup’s stated 40,000 feet and Mach 0.80, the lift coefficient is 0.5594 against the 0.695 the same table states. SUGAR Phase IV final report
\[C_L = \frac{W}{qS} = 0.5594 \tag{10}\]Equation 11, Altitude at Which the Stated Lift Coefficient Holds. Inverting the lift equation at maximum take-off weight and Mach 0.80 places the stated lift coefficient at 44,515 feet, within 78 feet of the report’s optimum cruise altitude. SUGAR Phase IV final report
\[p^{\ast} = \frac{2W}{\gamma M^2 S C_L} = 315.28 \ \text{lbf/ft}^2 = 15095.6 \ \text{Pa}, \qquad h^{\ast} = h_{\mathrm{t}} + \frac{R_a T(h_{\mathrm{t}})}{g_0}\ln\frac{p(h_{\mathrm{t}})}{p^{\ast}} = 13568.1 \ \text{m} \tag{11}\]Equation 12, Four-Part Cruise Drag Buildup. The report bookkeeps cruise drag as parasite, induced, compressibility and trim terms, with the laminar-flow credit carried in the compressibility term. SUGAR Phase IV final report
\[C_D = C_{D,0} + C_{D,i} + C_{D,c} + C_{D,t} \tag{12}\]Equation 13, Lift-to-Drag Ratio. The lift-to-drag ratio is the lift coefficient over the drag coefficient.
\[\frac{L}{D} = \frac{C_L}{C_D} \tag{13}\]Equation 14, Parasite Drag From Equivalent Flat Plate Area. The parasite drag coefficient is the equivalent flat plate area divided by the reference area. SUGAR Phase IV final report
\[C_{D,0} = \frac{\mathcal{F}}{S} \tag{14}\]Equation 15, Induced Drag Coefficient. The induced drag coefficient is the square of the lift coefficient over pi, the aspect ratio and the airplane efficiency factor. SUGAR Phase IV final report
\[C_{D,i} = \frac{C_L^2}{\pi A e} \tag{15}\]Equation 16, Bespoke Induced-Drag Fraction. The article defines the induced-drag fraction as the induced drag coefficient over the total, which it evaluates from the report’s buildup as 0.3351.
\[\delta = \frac{C_{D,i}}{C_D} \tag{16}\]Equation 17, Design Lift at the Load Factor. The design lift is the load factor times the weight.
\[L = \mathcal{N} W \tag{17}\]Equation 18, Elliptic Spanwise Lift Distribution. An elliptic distribution places the local lift per unit span in proportion to the square root of one less the square of the station.
\[\ell(y) = \frac{4L}{\pi b} \sqrt{1 - \left(\frac{2y}{b}\right)^2} \tag{18}\]Equation 19, Normalisation of the Elliptic Load. The constant in front of the elliptic distribution is fixed by requiring the load to integrate to the total lift.
\[\int_{-b/2}^{b/2} \ell(y)\,dy = \frac{4L}{\pi b}\cdot\frac{b}{2}\int_{-1}^{1}\sqrt{1-u^2}\,du = \frac{2L}{\pi}\cdot\frac{\pi}{2} = L \tag{19}\]Equation 20, Bending Moment From the Outboard Load. The bending moment at a station is the integral of the load outboard of it against its moment arm.
\[\mathcal{M}(u) = \int_{u}^{1} \ell \left(\frac{b}{2}\right)^2 (v - u) \, dv \tag{20}\]Equation 21, Bending Moment of an Elliptically Loaded Wing. For elliptic loading the moment integral has a closed form in the fraction of semispan.
\[\mathcal{M}(u) = \frac{L b}{\pi} \left[ \frac{1}{3}\left(1-u^2\right)^{3/2} - \frac{u}{2}\left( \frac{\pi}{2} - u\sqrt{1-u^2} - \arcsin u \right) \right] \tag{21}\]Equation 22, Root Bending Moment of an Elliptic Wing. At the root the elliptic moment collapses to the lift times the span over three pi.
\[\mathcal{M}(0) = \frac{L b}{3\pi} \tag{22}\]Equation 23, Root Moment From the Centroid of a Semi-Ellipse. Half the lift acting at the centroid of a semi-ellipse gives the same root moment, a check on the closed form.
\[\mathcal{M}(0) = \frac{L}{2}\cdot\frac{4}{3\pi}\cdot\frac{b}{2} = \frac{Lb}{3\pi} \tag{23}\]Equation 24, Bespoke Cap Area of a Two-Cap Wing Box. The article models the wing box as two bending caps at a separation set by a shape factor it takes as 0.90 of the section depth, so each cap needs the moment over the stress and that separation.
\[\mathcal{A}_c(u) = \frac{\mathcal{M}(u)}{\sigma h_c(u)}, \qquad h_c(u) = \kappa_h \, \tau \, c(u) \tag{24}\]Equation 25, Chord of a Straight-Taper Wing. The chord of a straight-taper wing falls linearly from the root, and the root chord follows from the area, span and taper ratio.
\[c(u) = c_r\left[1 - (1-\lambda)u\right], \qquad c_r = \frac{2S}{b\,(1+\lambda)} \tag{25}\]Equation 26, Bespoke Bending Material as an Integral of Cap Area. In the article’s two-cap model the bending material is four times the material density times the semispan integral of one cap area.
\[W_b = 4\rho_m \left(\frac{b}{2}\right)\int_0^1 \mathcal{A}_c(u)\,du \tag{26}\]Equation 27, Bespoke Taper Integral of the Moment Shape. The article collects the geometric part of its bending-material model into one integral of the moment shape function against the chord distribution.
\[\mathcal{I}(\lambda) = \int_0^1 \frac{f(u)}{1 - (1-\lambda)u}\,du \tag{27}\]Equation 28, Bespoke Bending Material in Span and Area. Substituting the moment, depth and chord into the article’s two-cap model gives the bending material in terms of the span, lift, taper and wing area.
\[W_b = \frac{\rho_m \, b^3 \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau \, S} \tag{28}\]Equation 29, Bespoke Three-Halves Power Law of Bending Material. Replacing the span by the aspect ratio shows that, at fixed area, design lift and thickness ratio, the article’s two-cap model scales as the three-halves power of aspect ratio.
\[W_b = \frac{\rho_m \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau} \; A^{3/2} \, S^{1/2} \tag{29}\]Equation 30, Bespoke Exponent Measured by Logarithmic Differencing. Differencing the article’s closed form logarithmically between two aspect ratios returns the exponent of 1.5 exactly.
\[n = \frac{\ln\left[W_b(A_2)/W_b(A_1)\right]}{\ln\left(A_2/A_1\right)} \tag{30}\]Equation 31, Bespoke Sweep Correction to Bending Material. The article lengthens the structural beam and the moment arm each by the secant of the sweep, so its bending material grows as the inverse square of the cosine of the sweep.
\[\frac{W_b(\Lambda)}{W_b(0)} = \frac{1}{\cos^2 \Lambda} \tag{31}\]Equation 32, Breguet Fuel Burn and Its Linearisation. The Breguet relation gives the cruise fuel as the weight times one less an exponential of range, fuel consumption and drag-to-lift ratio, which for a small fuel fraction reduces to a product.
\[m_f = W\left[1 - \exp\left(-\frac{R\,c_T}{V}\frac{C_D}{C_L}\right)\right] \approx \frac{R\,c_T}{V}\,\frac{C_D}{C_L}\,W \tag{32}\]Equation 33, Cruise Lift Coefficient at Fixed Area and Condition. At fixed area, altitude and Mach number the cruise lift coefficient is whatever the weight requires.
\[C_L = \frac{W}{q S} \tag{33}\]Equation 34, Bespoke Fuel Burn at Fixed Area and Cruise Condition. Holding wing area and cruise condition fixed, the case the article chooses, the linearised fuel burn is proportional to the drag coefficient times the dynamic pressure and the area.
\[m_f \propto \frac{C_D}{C_L}\,W = C_D \, q S \tag{34}\]Equation 35, Bespoke Parasite and Induced Parts of Fuel Burn at Fixed Area. In the article’s fixed-area case the fuel splits into a parasite term independent of aspect ratio and an induced term in the square of the weight over the aspect ratio.
\[m_f \propto q S C_{D,0} + \frac{W^2}{q S \pi A e} \tag{35}\]Equation 36, Bespoke Weight Elasticity and the Fixed-Area Fuel Slope. With the weight elasticity defined as the logarithmic derivative of weight with aspect ratio, the article’s fixed-area fuel slope is the induced-drag fraction times twice the elasticity less one.
\[\nu = \frac{d \ln W}{d \ln A}, \qquad \frac{d \ln m_f}{d \ln A} = \delta\left(2\nu - 1\right) \tag{36}\]Equation 37, Bespoke Optimality Condition at Fixed Area. In the article’s fixed-area case fuel burn is stationary when the weight elasticity is one half.
\[\nu^{*} = \tfrac{1}{2} \tag{37}\]Equation 38, Bespoke Stationary Point of the Induced Term. At fixed area and condition the induced drag goes as the square of the weight over the aspect ratio, whose logarithm is stationary at the same elasticity of one half.
\[\frac{d}{d\ln A}\ln\left(\frac{W^2}{A}\right) = 2\nu - 1 = 0 \tag{38}\]Equation 39, Bespoke Fuel Burn at Fixed Cruise Lift Coefficient. Holding the cruise lift coefficient fixed instead, the second case the article chooses, the linearised fuel is the weight over the lift coefficient times the parabolic drag polar.
\[m_f \propto \frac{W}{C_L}\left(C_{D,0} + \frac{C_L^2}{\pi A e}\right) \tag{39}\]Equation 40, Bespoke Fixed-Lift Fuel Slope. In the article’s fixed-lift case the logarithmic slope of fuel with aspect ratio is the weight elasticity less the induced-drag fraction.
\[\frac{d \ln m_f}{d \ln A} = \nu - \delta \tag{40}\]Equation 41, Bespoke Optimality Condition at Fixed Lift Coefficient. In the article’s fixed-lift case fuel burn is stationary where the weight elasticity equals the induced-drag fraction.
\[\nu^{*} = \delta \tag{41}\]Equation 42, Bespoke Elasticity of a Power-Law Weight Group. If the aspect-ratio-sensitive weight group obeys a power law, as the article assumes, the elasticity is the exponent times that group’s share of the weight.
\[W = W_0 + k_w A^n, \qquad \omega = \frac{k_w A^n}{W}, \qquad \nu = n\,\omega \tag{42}\]Equation 43, Bespoke Exponent Required for the Present Aspect Ratio To Be Optimal. Solving the article’s power-law elasticity for the exponent that would make the present aspect ratio stationary gives 3.64 at fixed area and 2.44 at fixed lift.
\[n_{\mathrm{required}} = \frac{\nu^{*}}{\omega} = \frac{\nu^{*} W}{W_w} \tag{43}\]Equation 44, Bespoke Anchoring of the Weight Coefficient. The coefficient of the article’s power-law weight group is anchored on the published group weights, with everything else assigned to the fixed weight.
\[k_w = \frac{W_w}{A^{n}}, \qquad W_0 = W - W_w \tag{44}\]Equation 45, Bespoke Fuel-Optimal Aspect Ratio at Fixed Lift. With the anchored power law, the article’s fixed-lift condition is satisfied at an aspect ratio of 25.1, 28.3 percent above the configuration’s.
\[A^{*} = 25.100 \tag{45}\]Equation 46, Bespoke Derivatives of the Elasticity and the Induced Fraction. Differentiating the article’s power-law elasticity and the induced-drag fraction with respect to the logarithm of aspect ratio gives the two parts of the curvature.
\[\frac{d(n\omega)}{d\ln A} = n^2\omega\left(1-\omega\right), \qquad \frac{d\delta}{d\ln A} = -\delta\left(1-\delta\right) \tag{46}\]Equation 47, Bespoke Curvature of the Fixed-Lift Fuel Objective. The second logarithmic derivative of the article’s fixed-lift fuel objective is the sum of those two parts.
\[\frac{d^2 \ln m_f}{d(\ln A)^2} = n^2\omega\left(1-\omega\right) + \delta\left(1-\delta\right) \tag{47}\]Equation 48, Bespoke Curvature at the Optimum. At the article’s stationary point the curvature depends only on the induced-drag fraction and the weight exponent.
\[\left.\frac{d^2 \ln m_f}{d(\ln A)^2}\right|_{A^{*}} = \delta\left(n + 1 - 2\delta\right) \tag{48}\]Equation 49, Bespoke Logarithmic Error in Aspect Ratio. The article measures the departure from the optimum as the natural logarithm of the optimal over the actual aspect ratio, which makes the expansion symmetric.
\[\varepsilon = \ln\frac{A^{*}}{A} \tag{49}\]Equation 50, Bespoke Quadratic Fuel Penalty of an Aspect-Ratio Error. Expanding the article’s linearised objective to second order gives a fuel penalty of half the curvature times the square of the logarithmic error.
\[\frac{\Delta m_f}{m_f} \approx \tfrac{1}{2}\,\delta\left(n + 1 - 2\delta\right)\varepsilon^2 \tag{50}\]Equation 51, Bespoke Fuel Penalty of the Present Aspect Ratio. At the article’s optimum of 25.1 the present aspect ratio of 19.565 costs about 1.90 percent of cruise fuel under the linearised expansion.
\[\varepsilon = \ln\frac{25.100}{19.565} = 0.2491, \qquad \frac{\Delta m_f}{m_f} \approx \tfrac{1}{2}\,(0.6131)\,(0.2491)^2 = 0.0190 \tag{51}\]Equation 52, Fuel Fraction and Breguet Exponent From the Published Weights. The published usable fuel and take-off weight fix the fuel fraction at 0.2002 and the Breguet exponent at 0.2234. SUGAR Phase IV final report
\[\zeta = \frac{m_f}{W} = 0.200193, \qquad X = -\ln\left(1 - \zeta\right) = 0.223385 \tag{52}\]Equation 53, Overstatement of the Linearised Breguet Relation. At this Breguet exponent replacing one less the exponential by the exponent overstates the fuel by 11.585 percent.
\[\frac{X}{1 - e^{-X}} - 1 = 0.115847 \tag{53}\]Equation 54, Bespoke Breguet Correction Factor. The article defines a correction factor in the Breguet exponent that carries the exponential through and tends to one as the fuel fraction vanishes.
\[\Phi(X) = \frac{X e^{-X}}{1 - e^{-X}}, \qquad \Phi(X) \to 1 \ \text{ as } X \to 0 \tag{54}\]Equation 55, Bespoke Exact Logarithmic Differential of Fuel Burn. With the correction factor the logarithmic change in fuel is that of the weight plus the factor times that of the Breguet exponent.
\[d \ln m_f = d \ln W + \Phi(X)\, d \ln X \tag{55}\]Equation 56, Bespoke Exact Fixed-Lift Optimality Condition. Carrying the exponential through, the article’s fixed-lift condition becomes an elasticity equal to the correction factor times the induced-drag fraction.
\[\frac{d \ln m_f}{d \ln A} = \nu - \Phi \delta \qquad \Longrightarrow \qquad \nu^{*} = \Phi \delta \tag{56}\]Equation 57, Bespoke Exact Fixed-Area Optimality Condition. Carrying the exponential through, the article’s fixed-area condition becomes a ratio that reduces to one half as the fuel fraction vanishes.
\[\frac{d \ln m_f}{d \ln A} = \nu\left(1 - \Phi\right) + \Phi \delta \left(2\nu - 1\right) \qquad \Longrightarrow \qquad \nu^{*} = \frac{\Phi \delta}{1 - \Phi + 2 \Phi \delta} \tag{57}\]Equation 58, Bespoke Exact Curvature at the Optimum. The article’s exact curvature at the fixed-lift optimum carries one extra term in the derivative of the correction factor.
\[\left.\frac{d^2 \ln m_f}{d(\ln A)^2}\right|_{A^{*}} = \Phi \delta \left(n + 1 - \delta - \Phi \delta\right) + \Phi'(X)\, X \delta^2 \tag{58}\]Equation 59, Bespoke Derivative of the Breguet Correction Factor. The derivative of the article’s correction factor with respect to the Breguet exponent is given in closed form.
\[\Phi'(X) = \frac{e^{-X}\left[\left(1-X\right)\left(1 - e^{-X}\right) - X e^{-X}\right]}{\left(1 - e^{-X}\right)^2} \tag{59}\]Equation 60, Bespoke Shortfalls of the Elasticity Readings Against the Exact Target. Each reading of the weight elasticity falls short of the article’s exact fixed-lift target of 0.299 by the fraction shown, from 8.07 percent on the over-generous accounting to 74.10 percent on bending material alone.
\[1 - \frac{\nu}{\nu^{\ast}}\,: \qquad 1 - \frac{0.2749}{0.299033} = 0.0807, \qquad 1 - \frac{0.07746}{0.299033} = 0.7410, \qquad 1 - \frac{0.20618}{0.299033} = 0.3105 \tag{60}\]Equation 61, Conversion Between the Two Code C Span Bounds. The bound of 118 feet is 35.9664 metres and the bound of 36 metres is 118.1102 feet, a difference of 1.3228 inches. FAA airport design
\[118 \times 0.3048 = 35.9664 \ \text{m}, \qquad \frac{36}{0.3048} = 118.1102 \ \text{ft}, \qquad 12\,(118.1102 - 118) = 1.3228 \ \text{in} \tag{61}\]Equation 62, Span Margins Against the Reference-Code Bounds. Each margin is the bound less the span in inches, giving 7.24 and 12 inches unfolded and zero against the exclusive bound of the Federal Aviation Administration when folded.
\[\Delta = 12\,(b_{\mathrm{bound}} - b), \qquad 12\left(\frac{52}{0.3048} - 170\right) = 7.24, \qquad 12\,(171 - 170) = 12, \qquad 12\,(118 - 118) = 0 \tag{62}\]Equation 63, Aspect Ratio at the Folded Span. At the published wing area the folded span of 118 feet gives an aspect ratio of 9.427.
\[A_{\mathrm{folded}} = \frac{b_f^2}{S} \tag{63}\]Equation 64, Aspect Ratio Gain From the Fold. At fixed area the ratio of the two aspect ratios is the square of the span ratio, 2.076.
\[\frac{A}{A_{\mathrm{folded}}} = \left(\frac{b}{b_f}\right)^2 = 2.076 \tag{64}\]Equation 65, Difference Between the Two Folded Margins. The margins of the folded span against the bounds of the International Civil Aviation Organization and the Federal Aviation Administration differ by the 1.3228 inches of the soft conversion.
\[\Delta_{\mathrm{ICAO}} = \frac{36}{0.3048} - b_f, \qquad \Delta_{\mathrm{FAA}} = 118 - b_f, \qquad \Delta_{\mathrm{ICAO}} - \Delta_{\mathrm{FAA}} = 1.3228 \ \text{inch} \tag{65}\]Equation 66, Bespoke Lift-to-Drag Ratio at the Folded Aspect Ratio. The article rebuilds the drag buildup at the folded aspect ratio while holding every other term unchanged, giving a lift-to-drag ratio of 18.97 against 25.79.
\[\left.\frac{L}{D}\right|_{\mathrm{folded}} = \frac{C_L}{C_{D,0} + C_{D,c} + C_{D,t} + \dfrac{C_L^2}{\pi A_{\mathrm{folded}} e}} \tag{66}\]Equation 67, Span Removed by the Fold. The fold removes half the difference between the unfolded and folded spans, 26 feet on each side. airport operations memorandum
\[\text{span folded per side} = \frac{b - b_f}{2} = 26 \ \text{foot} \tag{67}\]Equation 68, Bespoke Unit-Load Deflections of a Uniform Beam. By the unit-load method on a beam the article takes to be of uniform stiffness, the deflections from the distributed load and from a unit load at the brace station are two integrals against the same virtual moment.
\[w_{\mathrm{load}}(\eta) = \frac{1}{E\mathcal{I}}\int_0^{\eta} \mathcal{M}(u)\left(\eta - u\right) du, \qquad w_{\mathrm{unit}}(\eta) = \frac{1}{E\mathcal{I}}\int_0^{\eta} \left(\eta - u\right)^2 du \tag{68}\]Equation 69, Bespoke Prop Force of the Brace. The unit deflection is the cube of the station over three times the stiffness, so the article’s prop force is the ratio of the two deflections and the stiffness cancels.
\[w_{\mathrm{unit}}(\eta) = \frac{\eta^3}{3 E\mathcal{I}}, \qquad P = \frac{w_{\mathrm{load}}}{w_{\mathrm{unit}}} = \frac{3}{\eta^3}\int_0^{\eta} f(u)\left(\eta - u\right) du \tag{69}\]Equation 70, Bespoke Braced Bending Moment. The article’s braced moment is the cantilever moment less the prop force times its arm inboard of the attachment.
\[\mathcal{M}_{\mathrm{braced}}(u) = \mathcal{M}(u) - P \left(\frac{b}{2}\right)\max(\eta - u,\, 0) \tag{70}\]Equation 71, Bespoke Bending-Material Integral and Relief Factor. The article defines the relief factor as the integral of the braced moment magnitude against the inverse chord over the same integral for the cantilever.
\[\mathcal{B}(\eta) = \int_0^1 \frac{\left|\mathcal{M}_{\mathrm{braced}}(u)\right|}{1-(1-\lambda)u}\,du, \qquad \text{relief} = \frac{\mathcal{B}(\eta)}{\mathcal{B}(\infty)} \tag{71}\]Equation 72, Bespoke Non-Optimum Factor. The published bending material over the article’s relieved model estimate gives a non-optimum factor of 7.88, a quantity that the methodology literature also names. York and Labell 1980
\[k_{\mathrm{no}} = \frac{W_b^{\,\mathrm{published}}}{\text{relief}\left(\eta,\lambda\right) \, W_b^{\,\mathrm{model}}} = 7.88 \tag{72}\]Equation 73, Axial Force and Horizontal Reaction of an Inclined Strut. A pin-ended strut at a dihedral delivers vertical relief as the vertical component of its axial force, so the axial force is the relief over the sine of the dihedral. SUGAR Phase IV final report
\[T_s = \frac{P}{\sin \Gamma}, \qquad \text{horizontal reaction} = \frac{P}{\tan \Gamma} \tag{73}\]Equation 74, Euler Critical Buckling Load. The critical load of a column is pi squared times its bending stiffness over the square of its effective length.
\[P_{\mathrm{cr}} = \frac{\pi^2 E \mathcal{I}}{\left(K \mathcal{L}_s\right)^2} \tag{74}\]Equation 75, Effect of a Mid-Length Brace on Buckling. Halving the effective length with a jury strut multiplies the critical load by four.
\[\frac{P_{\mathrm{cr}}\left(K = \tfrac{1}{2}\right)}{P_{\mathrm{cr}}\left(K = 1\right)} = \left(\frac{1}{1/2}\right)^2 = 4 \tag{75}\]Equation 76, Bespoke Bending Material of a Geometrically Similar Truss. For a truss whose attachment station, dihedral and proportions scale with the span, as the article assumes, the braced weight law is the cantilever law times a constant relief factor.
\[W_b^{\,\mathrm{braced}} = \text{relief}(\eta, \lambda) \times \frac{\rho_m \, L \, \mathcal{I}(\lambda)\,(1+\lambda)}{\pi \, \sigma \, \kappa_h \, \tau} \; A^{3/2} \, S^{1/2} \tag{76}\]Equation 77, Bespoke Sensitivity of the Optimum to the Weight Coefficient. In the article’s limit of small weight and induced-drag fractions the optimum aspect ratio goes as the weight coefficient to the power minus one over the exponent plus one, minus two fifths at the three-halves law.
\[A^{*} \propto k_w^{-1/(n+1)}, \qquad \left.\frac{d\ln A^{*}}{d\ln k_w}\right|_{n=3/2} = -0.4 \tag{77}\]Equation 78, Korn Relation for Drag-Divergence Mach Number. The semi-empirical Korn relation gives the drag-divergence Mach number of a swept section from an airfoil-technology constant, the thickness ratio and the lift coefficient. SUGAR Phase II volume I
\[M_{dd} = \frac{\kappa_A}{\cos \Lambda} - \frac{\tau}{\cos^2 \Lambda} - \frac{0.1\,C_L}{\cos^3 \Lambda} \tag{78}\]Equation 79, Korn Relation Inverted for Thickness. Because the Korn relation is linear in thickness, it inverts exactly for the thickness ratio that reaches a given drag-divergence Mach number.
\[\tau = \left(\frac{\kappa_A}{\cos \Lambda} - \frac{0.1\,C_L}{\cos^3 \Lambda} - M_{dd}\right)\cos^2 \Lambda \tag{79}\]Equation 80, Korn Relation as a Cubic in the Secant of Sweep. Writing the secant of the sweep as $s$ turns the Korn relation into a cubic whose derivative is a quadratic.
\[M_{dd} = \kappa_A s - \tau s^2 - 0.1\,C_L s^3, \qquad \frac{dM_{dd}}{ds} = \kappa_A - 2\tau s - 0.3\,C_L s^2 \tag{80}\]Equation 81, Sweep at Which the Korn Relation Peaks. The turning point of the Korn relation is the positive root of that quadratic, which gives the sweep beyond which more sweep lowers the drag-divergence Mach number.
\[s^{*} = \frac{-\tau + \sqrt{\tau^2 + 0.3\,\kappa_A C_L}}{0.3\,C_L}, \qquad \Lambda^{*} = \arccos\frac{1}{s^{*}} \tag{81}\]Equation 82, Bespoke Bending Material Cost of Thinning and Sweeping. The article’s weight model, inverse in thickness and in the square of the cosine of sweep, prices the Phase IV change from Phase III at a factor of 1.29 on bending material.
\[\frac{W_b^{\,\mathrm{P4}}}{W_b^{\,\mathrm{P3}}} = \frac{\tau_{\mathrm{P3}}}{\tau_{\mathrm{P4}}} \cdot \frac{\cos^2 \Lambda_{\mathrm{P3}}}{\cos^2 \Lambda_{\mathrm{P4}}} \tag{82}\]Equation 83, Fuel Volume Deficit. The deficit is the required fuel volume less the available volume, as a fraction of the requirement. SUGAR Phase IV final report
\[\text{deficit} = \frac{\mathcal{V}_{\mathrm{required}} - \mathcal{V}_{\mathrm{available}}}{\mathcal{V}_{\mathrm{required}}} \tag{83}\]Equation 84, Flutter Speed Margin. The margin is the flutter speed over the speed the certification basis requires, less one. SUGAR Phase IV final report
\[\text{margin} = \frac{V_{\mathrm{flutter}}}{V_{\mathrm{required}}} - 1 \tag{84}\]Equation 85, Ground-Effect Group. The ratio of span to twice the height above the ground governs the upwash an image vortex system induces at the wing.
\[\Sigma = \frac{b}{2h_g} \tag{85}\]Equation 86, Fuel Reductions as Percentage Points and as Percentages. The difference of two reductions against a baseline and the percentage change between the two configurations share a numerator and differ in denominator.
\[r_i = 1 - \frac{\beta_i}{\beta_0}, \qquad r_1 - r_2 = \frac{\beta_2 - \beta_1}{\beta_0}, \qquad \frac{\beta_2}{\beta_1} - 1 = \frac{\beta_2 - \beta_1}{\beta_1} \tag{86}\]Equation 87, Ratio of the Two Statements of the Fuel Change. The ratio of the percentage change in fuel to the percentage-point change in reduction is the baseline over the first configuration, 2.3279. SUGAR Phase IV final report
\[\frac{\text{percentage change in fuel}}{\text{percentage-point change in reduction}} = \frac{\beta_0}{\beta_1} = 2.3279 \tag{87}\]Equation 88, Advantages and Costs in the Like-for-Like Comparison. The truss-braced aircraft’s start-of-climb lift-to-drag ratio is 12.92 percent better, while it needs 25.65 percent more sea-level static thrust and cruises 4,250 feet higher. Recine et al. 2025a
\[\frac{23.6}{20.9} - 1 = 0.1292, \qquad \frac{23{,}277}{18{,}525} - 1 = 0.2565, \qquad 44{,}750 - 40{,}500 = 4{,}250 \ \text{ft} \tag{88}\]Equation 89, Bespoke Apparent Weight Elasticity of a Pair. The article takes the logarithm of the weight ratio over the logarithm of the aspect-ratio ratio of two otherwise different aircraft as an apparent elasticity. Recine et al. 2025a
\[\nu_{\mathrm{apparent}} = \frac{\ln\left(W_2/W_1\right)}{\ln\left(A_2/A_1\right)} \tag{89}\]Equation 90, Bespoke Shortfall of the Largest Apparent Elasticity. The largest apparent elasticity falls 66.8 percent short of the article’s exact fixed-lift target of 0.299.
\[1 - \frac{0.09926}{0.299033} = 0.668 \tag{90}\]Equation 91, Braced Wing and Strut Weight Against the Cantilever. The braced wing alone weighs 1.38 or 4.28 percent more than the cantilever wing on the two calibrations, and with the strut 24.58 or 28.10 percent more. Recine et al. 2025a
\[\frac{13{,}045}{12{,}867} - 1 = 0.0138, \qquad \frac{13{,}418}{12{,}867} - 1 = 0.0428, \qquad \frac{13{,}045 + 2{,}985}{12{,}867} - 1 = 0.2458, \qquad \frac{13{,}418 + 3{,}065}{12{,}867} - 1 = 0.2810 \tag{91}\]Equation 92, Bespoke Primary Fraction and Period Count of the Survey. The article summarises its reference survey by the fraction of research records from report servers and the count of records from 2015 onward.
\[\text{primary fraction} = \frac{\left|\left\{r : \mathrm{src}(r) \in \{\text{reports}, \text{defence}\}\right\}\right|}{\left|\mathcal{R}\right|}, \qquad \text{period count} = \left|\left\{r : \mathrm{year}(r) \ge 2015\right\}\right| \tag{92}\]Notation
- $\mu_i$, the payment attached to milestone $i$
- $\mathcal{P}_{\mathrm{new}}$, a budget projection for one fiscal year in the later book, with $\mathcal{P}_{\mathrm{old}}$ that in the earlier book
- $A$, the wing aspect ratio
- $A^{*}$, the aspect ratio at which fuel burn is stationary
- $b$, the wing span, tip to tip
- $b_f$, the folded wing span
- $S$, the wing reference area
- $T_{\mathrm{sl}}$, the sea-level standard temperature, with $p_{\mathrm{sl}}$ the sea-level standard pressure
- $\lambda_a$, the troposphere lapse rate
- $h_{\mathrm{t}}$, the tropopause altitude
- $R_a$, the specific gas constant for air
- $g_0$, standard gravity
- $\gamma$, the ratio of specific heats, taken as 1.4 for dry air
- $a$, the speed of sound at the cruise condition
- $q$, the free-stream dynamic pressure
- $W$, the aircraft weight
- $C_L$, the cruise lift coefficient
- $p^{\ast}$, the ambient pressure at which the drag buildup’s lift coefficient holds
- $h^{\ast}$, the geopotential altitude at which the drag buildup’s lift coefficient holds
- $C_{D,0}$, the parasite drag coefficient, the incompressible zero-lift part
- $C_{D,i}$, the induced drag coefficient
- $C_{D,c}$, the compressibility drag coefficient, which also carries the laminar-flow credit
- $C_{D,t}$, the trim drag coefficient
- $\mathcal{F}$, the equivalent flat plate area
- $e$, the airplane efficiency factor for drag due to lift
- $\delta$, the induced-drag fraction of total cruise drag
- $\mathcal{N}$, the design load factor
- $L$, the total lift carried by the wing at the design load factor
- $\ell$, the local lift per unit span
- $\mathcal{M}$, the wing bending moment about the structural axis
- $\mathcal{A}_c$, the cross-sectional area of one bending cap
- $\sigma$, the allowable stress in the cap material
- $h_c$, the separation between the centroids of the upper and lower caps
- $\kappa_h$, the ratio of cap-centroid separation to maximum section depth
- $\tau$, the wing thickness-to-chord ratio
- $c_r$, the root chord
- $\lambda$, the wing taper ratio, tip chord over root chord
- $W_b$, the bending material weight of the wing
- $\rho_m$, the density of the cap material
- $\mathcal{I}(\lambda)$, the taper integral of the moment shape against the chord
- $\Lambda$, the wing quarter-chord sweep angle
- $m_f$, the fuel burned over the cruise leg
- $c_T$, the thrust specific fuel consumption
- $R$, the cruise range
- $\nu$, the weight elasticity, the logarithmic derivative of aircraft weight with respect to aspect ratio
- $W_w$, the aspect-ratio-sensitive weight group
- $W_0$, the part of aircraft weight that does not change with aspect ratio
- $k_w$, the coefficient of the aspect-ratio-sensitive weight group
- $\omega$, the aspect-ratio-sensitive weight as a fraction of aircraft weight
- $\varepsilon$, the fractional error in aspect ratio, measured logarithmically
- $\zeta$, the fuel fraction, fuel burned over aircraft weight
- $X$, the Breguet exponent
- $\Phi$, the Breguet correction factor
- $\Delta$, the span margin against a reference-code bound
- $b_{\mathrm{bound}}$, a span bound of an aerodrome reference code, in feet
- $\eta$, the station of the brace attachment as a fraction of semispan
- $E$, the Young’s modulus of the cap material in the fibre direction
- $P$, the vertical relief force the brace delivers at its attachment
- $\mathcal{B}$, the bending-material integral, braced or unbraced
- $k_{\mathrm{no}}$, the non-optimum factor, published bending material over the model’s
- $\Gamma$, the main strut dihedral angle
- $T_s$, the axial force in the main strut
- $P_{\mathrm{cr}}$, the Euler critical buckling load of the strut
- $K$, the effective-length factor in the Euler buckling load
- $\mathcal{L}_s$, the unsupported length of the strut in buckling
- $M_{dd}$, the drag-divergence Mach number
- $\kappa_A$, the Korn airfoil-technology constant
- $s$, the secant of the wing sweep angle
- $\mathcal{V}_{\mathrm{required}}$, the fuel volume required, with $\mathcal{V}_{\mathrm{available}}$ the volume available
- $\Sigma$, the ground-effect group, span over twice the height
- $r_i$, the fuel-burn reduction of configuration $i$ against the baseline
- $\beta_i$, the block fuel per seat of configuration $i$
- $\mathcal{R}$, the set of research records in the reference survey
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- Sechter, Brendan. 2025. “X-Planes: Lockheed Martin X-62 VISTA.” Brendan A R Sechter’s Development Blog, December 7. https://sgeos.github.io/aerospace/history/engineering/2025/12/07/x_planes_lockheed_martin_x62_vista.html
- Sechter, Brendan. 2025. “X-Planes: ABL Space Systems X-63.” Brendan A R Sechter’s Development Blog, December 8. https://sgeos.github.io/aerospace/history/engineering/2025/12/08/x_planes_abl_space_systems_x63.html
- Sechter, Brendan. 2025. “X-Planes: Invocon X-64.” Brendan A R Sechter’s Development Blog, December 9. https://sgeos.github.io/aerospace/history/engineering/2025/12/09/x_planes_invocon_x64.html
- Sechter, Brendan. 2025. “X-Planes: Aurora Flight Sciences X-65 CRANE.” Brendan A R Sechter’s Development Blog, December 10. https://sgeos.github.io/aerospace/history/engineering/2025/12/10/x_planes_aurora_x65_crane.html
Referenced by
- X-Planes: Synthesis and What the Designation Became 2025-12-16
- X-Planes: Bell Textron X-76 SPRINT 2025-12-15
- X-Planes: X-69 through X-75, the Leapfrogged Block 2025-12-14
- X-Planes: General Atomics X-68 LongShot 2025-12-13
- X-Planes: X-67, the Slot Taken by XQ-67A 2025-12-12