Geometrically Nonlinear Aeroelastic Scaling for Very Flexible Aircraft Zhiqiang Wan * Beihang University, Beijing 100191, People’s Republic of China and Carlos E. S. Cesnik † University of Michigan, Ann Arbor, Michigan 48109-2140 DOI: 10.2514/1.J052855 High-aspect-ratio wings present in very flexible aircraft can undergo large deformations, which results in significant changes in natural frequencies as well as in static and dynamic aeroelastic response. This geometric nonlinear behavior becomes an integral part of any aeroelastic analysis to be conducted in such class of vehicles. Aeroelastic scaling is an important way to study the aeroelastic behavior of aircraft, and it is an integral part in risk mitigation for aircraft development. However, the current aeroelastic scaling methodologies have focused on geometrically linear structures. This paper demonstrates a methodology for geometrically nonlinear aeroelastic scaling of very flexible aircraft. The known linear scaling factors and similarity rules are extended to address geometrically nonlinear aeroelastic scaling. A high-aspect-ratio flying wing in free flight is taken as an example to verify the new scaling procedure, and numerical studies are conducted using the University of Michigan’s Nonlinear Aeroelastic Simulation Toolbox. Numerical results support the new approach for aeroelastic scaling of very flexible aircraft. Nomenclature  B = strain-displacement operator B L = linear strain-displacement differential operator B NL fug = nonlinear strain-displacement operator C h = generalized (structural) damping matrix D = material matrix dV = domain volume EA = beam extensional stiffness EI y = out-of-plane bending stiffness EI z = in-plane bending stiffness GJ = beam torsional stiffness i = imaginary number K = structural stiffness matrix K h = generalized structural stiffness matrix K L = linear stiffness matrix due to small deformations K NL = nonlinear stiffness matrix due to large geometric deformation K T = tangential stiffness matrix in global coordinate system K σ = geometric stiffness matrix due to prestress k b = length scaling factor k EA = extensional stiffness scaling factor k EI = bending stiffness scaling factor k GJ = torsional stiffness scaling factor k K = stiffness scaling factor k L = lift scaling factor k M = moment scaling factor k m = mass scaling factor k Re = Reynolds-number scaling factor k t = time scaling factor k V = speed scaling factor k ρ = air density scaling factor k ω = frequency scaling factor k μ = air dynamic viscosity scaling factor M = structural mass matrix M h = generalized structural mass matrix P = vector of additional applied loads Q = matrix of aerodynamic influence coefficients Q g = matrix of generalized unsteady aerodynamic forces due to gust Q h = matrix of generalized unsteady aerodynamic forces due to vehicle motion Q x = matrix of unit aerodynamic loads R = external forces u = structural deformation u h = vector of generalized structural degrees of freedom u x = vector of aerodynamic trim parameters V = uniform flow speed w g = vertical gust speed w max = maximum vertical gust speed x g = gust wavelength δu = virtual displacement δε = virtual strain ε 0 = initial strain λ = eigenvalue ρ = air density σ = stress σ 0 = initial stress φ = eigenvector ψ = summation of the internal and external forces ω = circular frequency I. Introduction A EROELASTIC scaled models for wind-tunnel testing or flight test play a key role in studying the aeroelastic characteristics of full-size aircraft, in which the aeroelastic scaling laws are the key elements [1]. The scaled models are also widely used in research studies such as active control of aeroelastic response, flutter Presented as Paper 2013-1894 at the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, 8–11 April 2013; received 7 June 2013; revision received 15 April 2014; accepted for publication 15 April 2014; published online 17 July 2014. Copyright © 2014 by Zhiqiang Wan and Carlos E. S. Cesnik. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-385X/ 14 and $10.00 in correspondence with the CCC. *Associate Professor, School of Aeronautic Science and Engineering; Visiting Professor, Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2140; wzq@buaa.edu.cn. Senior Member AIAA. † Professor, Department of Aerospace Engineering, 1320 Beal Avenue — 3024 FXB; cesnik@umich.edu. Fellow AIAA (Corresponding Author). 2251 AIAA JOURNAL Vol. 52, No. 10, October 2014 Downloaded by University of Michigan - Duderstadt Center on December 13, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.J052855