An Extended Modal Approach for Nonlinear Aeroelastic Simulations of Highly Flexible Aircraft Structures

Kurzfassung

Some classes of aircraft are characterized by highly flexible wings undergoing large structural deformations in steady and maneuvering flight. In most cases this flexibility is the result of a high aspect ratio, which is in turn forced by dedicated design criteria such as the reduction of the induced drag. Prominent examples for such configurations are High Altitude Long Endurance (HALE) aircraft as well as modern, high performance sailplanes of the Open Class. The design and the analysis of highly flexible aircraft put high demands on the methods and tools employed. Multidisciplinary analysis taking into account aerodynamics, flight mechanics, and structural dynamics is indispensable where nonlinearities due to large rigid-body motions and large structural deflections are inherent in each of these disciplines. For the structural part, only few methods exist to date for the calculation of general aircraft structures subjected to large deformations. Commercial finite element solvers are mostly limited to clamped structures in their nonlinear solution capabilities, i.e. concurrent rigid-body motions are inadmissible. On the other hand, sophisticated methods incorporating nonlinear rigid-body as well as nonlinear elastic motions have been developed mainly for beam-type structures but not for complex and three-dimensional models. In aeroelasticity, the modal approach is a well-established and elegant method for simulating and analyzing the dynamic behavior of aircraft structures. However, its applicability is limited to small structural deformations and an extension into the nonlinear regime with respect to large geometric deflections would be desirable. The goal of this thesis is thus the extension of the modal approach towards large geometric deformations of highly flexible aircraft structures. The extensions include stiffness terms of higher order as well as higher-order modal components for the calculation of the nonlinear displacement field. It is shown that the method is applicable to different kinds of structural models composed of beam and shell elements with anisotropic material properties. Furthermore, an integrated set of equations based on these extensions is presented that allows the nonlinear time-domain simulation of the free-flying elastic aircraft in steady and unsteady maneuvering flight with large structural deformations. In the first part of the thesis, the two extensions are presented in detail. The linear relationship between load and structural displacements given in the classical modal approach is extended by generalized stiffness terms that depend quadratically and cubically on the generalized coordinates. The higher-order stiffness terms are derived by a series expansion of the strain energy of the structure which is formulated as a nonlinear function of deformation. The linear transformation between modal and Euclidean space by the eigenvectors of the structure is extended by higher (second-, third-, and fourth-) order mode components to approximate the geometrically nonlinear displacement field. The higher-order stiffness and mode components of the structure are determined in a preprocessing step where a series of nonlinear static solutions are generated using a commercial finite element solver. Higher-order polynomials are fitted to the solutions, the polynomial coefficients then correspond to the higher-order components. The second part introduces the extensions described above into the governing equations of motion of the free-flying elastic aircraft. The governing equations are derived using Lagrange's equations of the second kind where particular attention is paid to admit as few assumptions as possible. One widely applied assumption in aeroelastic analysis of free-flying aircraft is the mean axes condition to inertially decouple rigid-body and elastic degrees of freedom. This assumption requires the frequencies of the typical rigid-body and elastic modes of the aircraft to be largely separated and is limited to small structural deformations. By contrast, flight dynamics of highly flexible aircraft is characterized by strong inertial coupling between flight mechanic and elastic degrees of freedom due to low structural frequencies. Consequently, the governing equations derived in this thesis consider inertial coupling between rigid-body and elastic motions. The third part presents applications of the method to three different test cases with increasing complexity. First, the basics of the method are outlined by a simple beam structural model in static structural response. The second test case is a slender wing box represented by a full 3D finite element model with anisotropic materials. The third test case is the very flexible X-HALE unmanned aerial vehicle from the University of Michigan. Static structural and aeroelastic responses as well as free-flight maneuver simulations with unsteady excitation of the aircraft by gust and tail inputs demonstrate further capabilities and the limits of the method.