Recent investigations pertaining to high aspect ratio wings have demonstrated the influence of geometric nonlinearities on structural and aeroelastic response when large deflections occur [1,2]. While utilization of nonlinear analyses techniques is beneficial for more realistic predictions of large deflection behaviour, it is accompanied with the drawback of high computational costs since finite element (FE) solvers are based on iterative predictor-corrector models. Nonlinear reduced order modelling can be an effective tool for conducting efficient analyses in such cases. In the proposed study we aim to exploit the recent developments in the Koiter-Newton (K-N) model reduction technique [3] for nonlinear dynamic response analysis of a high aspect ratio wing and thus, demonstrate the achievable reduction in computational costs in comparison to full FE simulations. The K-N reduction is a FE-based formulation which describes a system of nonlinear governing equations comprising quadratic and cubic stiffness terms. The higher order stiffness terms are evaluated as derivatives of the in-plane strain energy. To ensure that the effect of large rotations is accounted for, the reduced order model (ROM) is updated at fixed load intervals. Linear eigenmodes of the deformed structure are used to formulate the reduction subspace at the different load steps. The test structure chosen in this work is a variant of the Pazy wing [4] which is an experimental benchmark wing designed for nonlinear aeroelastic studies. The Pazy wing variant is based on the dimensions of the original design with minor modifications in the inner geometry. The FE model is constructed entirely using shell elements with 21,712 grid points and 130,272 degrees of freedom. For the initial validation, we conduct a nonlinear static analysis with a concentrated follower force, and compare it to FE solution from MSC Nastran. It is seen that by using just a single degree of freedom ROM, the nonlinear static solution is reproducible within a 2 % error margin for up to 40 % tip deflections. Subsequently, a nonlinear dynamic response analysis is conducted where the wing is subjected to large amplitude transient loads. The preliminary studies show a reduction in simulation time of over 95 % without significant loss in solution accuracy. References [1] Riso, Cristina, and Carlos E. Cesnik. "Low-Order Geometrically Nonlinear Aeroelastic Modeling and Analysis of the Pazy Wing Experiment." AIAA SciTech 2022 Forum. 2022. [2] Hilger, Jonathan, and Markus Raimund Ritter. "Nonlinear aeroelastic simulations and stability analysis of the Pazy wing aeroelastic benchmark." Aerospace 8.10 (2021): 308. [3] Sinha, Kautuk, et al. "Koiter–Newton Based Model Reduction for Large Deflection Analysis of Wing Structures." AIAA Journal (2023): 1-10. [4] Avin, Or, et al. "Experimental aeroelastic benchmark of a very flexible wing." AIAA Journal 60.3 (2022): 1745-1768. @en

Wing structures subjected to large deflections are prone to nonlinear load-deflection behavior. Geometric nonlinearities can arise due to the accompanying large rotations and in-plane deflections that manifest in the form of stiffening effects in the nonlinear static response. To account for these nonlinearities, reduced-order modeling techniques in combination with nonlinear finite element formulations have been previously proposed. However, these methods often have a limited range of validity due to linear eigenmode-based formulations with assumptions of small rotations. In this paper, a large deflection analysis framework based on the Koiter–Newton model reduction technique is presented. It is demonstrated that the reduced model in its basic form is ineffective for large deflection analysis. To resolve this, an incremental updating procedure is used for the reduced-order model that incorporates the necessary nonlinear effects. The model updating enables the computation of nonlinear response for a large range of deflections.

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The article proposes a method developed for model order reduction in a Finite Element (FE) framework that is capable of computing higher order stiffness tensors. In the method, a truncated third order asymptotic expansion is used for transformation of an FE model to a reduced system. The basis matrix in the formulation of the reduced-order model (ROM) is derived from linear mode shapes of the structure. The governing equations are derived using Hamilton's principle and the method is applied to geometrically nonlinear vibration problems to test its effectiveness. An initial validation of the numerical formulation is obtained by comparison of results from time domain nonlinear vibration analyses of a rectangular plate using Abaqus. Subsequently, a stiffened plate is modeled to test a more complex structure and a continuation algorithm is used in combination with the ROM to compute nonlinear frequency response curves. The validation of the stiffened plate has been performed through comparisons with the results of nonlinear vibration experiments. The experiments are conducted with Polytec Laser Doppler Vibrometer and PAK MK-II measurement systems for large amplitude vibrations to validate the nonlinear vibration analyses.

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