A Hybrid Reduced-Order Model for the Aeroelastic Analysis of Flexible Subsonic Wings—A Parametric Assessment

Abstract

A hybrid reduced-order model for the aeroelastic analysis of flexible subsonic wings with arbitrary planform is presented within a generalised quasi-analytical formulation, where a slender beam is considered as the linear structural dynamics model. A modified strip theory is proposed for modelling the unsteady aerodynamics of the wing in incompressible flow, where thin aerofoil theory is corrected by a higher-fidelity model in order to account for three-dimensional effects on both distribution and deficiency of the sectional air load. Given a unit angle of attack, approximate expressions for the lift decay and build-up are then adopted within a linear framework, where the two effects are separately calculated and later combined. Finally, a modal approach is employed to write the generalised equations of motion in state-space form. Numerical results were obtained and critically discussed for the aeroelastic stability analysis of a uniform rectangular wing, with respect to the relevant aerodynamic and structural parameters. The proposed hybrid model provides sound theoretical insights and is well suited as an efficient parametric reduced-order aeroelastic tool for the preliminary multidisciplinary design and optimisation of flexible wings in the subsonic regime.

On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Comparative Assessment

“…However, SU and SQU models were demonstrated to be quite robust in providing reliable flutter analyses when the elastic axis lay ahead of the aerofoil's control point and flutter occurs before static divergence, as is typical in practice [8,9]. As for the computational cost, lower-fidelity semi-analytical solutions are almost instantaneous and demand minimal pre-and post-processing (if any at all), while providing an enhanced theoretical understanding; therefore, they enable efficient multidisciplinary exploration of a large design variable space for innovative aircraft concepts and configurations [12,13]. Note that further numerical savings may be obtained by exploiting efficient tuning techniques and surrogate models [106,107] with the effective synthesis of problem complexity.…”

“…Theodorsen's (complex transfer) function can then be considered as the Fourier transform of the (unit) unsteady airload due to impulsive plunging of the aerofoil at its control point [5] in the reduced-frequency domain: it introduces a lag in the airload build-up because of the counteracting vorticity that is shed in the wake from infinitesimal variations of the bound circulation according to Kelvin's and Helmholtz's theorems [7]. Theodorsen's function generalises for the swept wing and subsonic compressible flow [45][46][47] by considering the effective freestream relative to the aerodynamic axis and Prandtl-Glauert's compressibility factor [10,32]; for finite wings [34,48], it is modified by unsteady downwash effects (which reduce the sectional effective angle of attack and the related airload as well as its build-up rate [49,50]) within modified strip theory [12]. Finally, the total "unsteady" airload acting on the oscillating aerofoil reads [4,16]:…”

“…with aerodynamic inertia, damping and stiffness contributions to the aeroelastic response of the typical section. Within the framework of flutter analysis using tuned strip theory [12] (i.e., scaling the circulatory contributions by the ratio between wing's and aerofoil's lift coefficient derivatives with respect to the angle of attack in order to account for the overall steady effect of the three-dimensional downwash [11]), it is worth stressing that this lowerfidelity aerodynamic approximation did compare favourably with higher-fidelity tools coupling a beam-based finite element model with doublet lattice or boundary element aerodynamics [87], as the latter provides a faster development of the indicial airload than Wagner's model and approaches a quasi-steady behaviour while decreasing the wing's aspect ratio (i.e., increasing unsteady downwash effects) [49,50].…”

“…In particular, such formulations grant a clear and complete overview of the problem at a conceptual level, which is essential for any engineering application, providing full control in terms of both results and their underlying assumptions. Although inherently limited, analytical approaches offer a wealth of qualitative information and quantitative details, as well as rigorous validation, for both educational and practical purposes [12,13]. As far as the p-k method is adopted for flutter analyses in the reduced frequency domain, all presented aerodynamic approximations lead to similar negligible computational cost with modern computers; however, unsteady and quasi-unsteady aerodynamics become more expensive as soon as added aerodynamic states are defined to write the unsteady airload in the time domain [18] within a state-space formulation (typically required to study the stability of nonlinear aeroelastic equations) where the p method can then be directly adopted for flutter analyses in Laplace's domain [6].…”

“…Although inherently limited in applicability [9,10], explicit airload expressions and theoretical solutions grant a clear overview of the underlying phenomena and offer rigor-ous validation as well as fundamental insights for both educational and practical scopes in both academic and industrial activities, with a convenient trade-off between reliability and affordability [11][12][13]. Focusing on two-dimensional, incompressible, potential flow without loss of generality for the fundamental purpose of this conceptual work, the unsteady aerodynamic model is first reviewed by adopting a vorticity-based approach [14,15], showing its formal equivalence with Thodorsen's and Wagner's models [16,17].…”