On a Variational Principle in Lifting-Line Theory

Flax, Alexander H.

doi:10.2514/8.1732

Cited by 7 publications

(2 citation statements)

References 2 publications

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“…On the other hand, if the wingspan is high enough, the wing-structure weight required for the stress-limited design will be less than that required for the deflection-limited design, and the wing design will be deflection limited. For the case of fixed wing loading, combining Equations (71) and 73, the wingstructure weight that results when the wingspan is the same for both the stress-limited and deflection-limited designs is obtained from the relation…”

Section: Resultsmentioning

confidence: 99%

Minimising induced drag with weight distribution, lift distribution, wingspan, and wing-structure weight

2020

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ABSTRACTBecause the wing-structure weight required to support the critical wing section bending moments is a function of wingspan, net weight, weight distribution, and lift distribution, there exists an optimum wingspan and wing-structure weight for any fixed net weight, weight distribution, and lift distribution, which minimises the induced drag in steady level flight. Analytic solutions for the optimum wingspan and wing-structure weight are presented for rectangular wings with four different sets of design constraints. These design constraints are fixed lift distribution and net weight combined with 1) fixed maximum stress and wing loading, 2) fixed maximum deflection and wing loading, 3) fixed maximum stress and stall speed, and 4) fixed maximum deflection and stall speed. For each of these analytic solutions, the optimum wing-structure weight is found to depend only on the net weight, independent of the arbitrary fixed lift distribution. Analytic solutions for optimum weight and lift distributions are also presented for the same four sets of design constraints. Depending on the design constraints, the optimum lift distribution can differ significantly from the elliptic lift distribution. Solutions for two example wing designs are presented, which demonstrate how the induced drag varies with lift distribution, wingspan, and wing-structure weight in the design space near the optimum solution. Although the analytic solutions presented here are restricted to rectangular wings, these solutions provide excellent test cases for verifying numerical algorithms used for more general multidisciplinary analysis and optimisation.

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Section: Resultsmentioning

confidence: 99%

Minimising induced drag with weight distribution, lift distribution, wingspan, and wing-structure weight

2020

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“…31,32 For unswept wings of aspect ratio greater than about 4, lifting-line theory agrees well with experimental data and computational fluid dynamics solutions, and lifting-line solutions are widely accepted. 33–76 Nickel 75,76 found a solution for the optimum lift distribution to create a rolling moment while minimizing induced drag using lifting-line theory. However, his work neglected the influence of rolling rate.…”

Section: Introductionmentioning

confidence: 99%

Analytic and computational analysis of wing twist to minimize induced drag during roll

Hunsaker

Montgomery

Joo

2019

Proceedings of the Institution of Mechanical Engineers, Part G:

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Geometric and/or aerodynamic wing twist can be used to produce a lift distribution that results in a rolling moment. A decomposed Fourier-series solution to Prandtl’s lifting-line theory is used to develop analytic spanwise antisymmetric twist distributions for roll control that minimize induced drag on wings of arbitrary planform in pure rolling motion. Roll initiation, steady rolling rate, and the transition between the two are each considered. It is shown that if these antisymmetric twist distributions are used, the induced drag is proportional to the square of the rolling moment, and the induced drag during a steady rolling rate is equal to that on the wing at the same lift coefficient with no rolling rate or antisymmetric twist distribution. Results also show that if these antisymmetric twist distributions are used on straight, tapered wings without symmetric twist, any rolling maneuver for which the rolling rate and rolling moment have the same sign will always produce a yawing moment in the opposite direction. Computational results are also included, which were obtained using a gradient-based optimization algorithm in combination with a modern numerical lifting-line algorithm to find the optimum twist solutions. The resulting twist, induced drag, and yawing moment solutions compare favorably with the analytic solutions developed in the text. The solutions presented here can be used to inform the design of morphing aircraft.

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