Numerical Continuation of Limit Cycle Oscillations and Bifurcations in High-Aspect-Ratio Wings

This paper applies numerical continuation techniques to a nonlinear aeroelastic model of a highly flexible, high-aspect-ratio wing. Using continuation, it is shown that subcritical limit cycle oscillations, which are highly undesirable phenomena previously observed in numerical and experimental studies, can exist due to geometric nonlinearity alone, without need for nonlinear or even unsteady aerodynamics. A fully nonlinear, reduced-order beam model is combined with strip theory and one-parameter continuation is used to directly obtain equilibria and periodic solutions for varying airspeeds. The two-parameter continuation of specific bifurcations (i.e., Hopf points and periodic folds) reveals the sensitivity of these complex dynamics to variations in out-of-plane, in-plane and torsional stiffness and a ‘wash out’ stiffness coupling parameter. Overall, this paper demonstrates the applicability of continuation to nonlinear aeroelastic analysis and shows that complex dynamical phenomena, which cannot be obtained by linear methods or numerical integration, readily exist in this type of system due to geometric nonlinearity.

[1]  Mark H Lowenberg,et al.  The Dynamical Systems Toolbox: Integrating AUTO into MATLAB , 2010 .

[2]  J. E. Cooper,et al.  On the geometrically exact low-order modelling of a flexible beam: formulation and numerical tests , 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  S. J. Price,et al.  NONLINEAR AEROELASTIC ANALYSIS OF AIRFOILS : BIFURCATION AND CHAOS , 1999 .

[4]  Bernd Krauskopf,et al.  Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  Chetan Nichkawde,et al.  Nonlinear aeroelastic analysis of high aspect-ratio wings using the method of numerical continuation , 2006 .

[6]  Earl H. Dowell,et al.  Experimental and Theoretical Study on Aeroelastic Response of High-Aspect-Ratio Wings , 2001 .

[7]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[8]  Carlos E. S. Cesnik,et al.  Limit-cycle oscillations in high-aspect-ratio wings , 2002 .

[9]  D. Gee,et al.  Numerical Continuation Applied to Panel Flutter , 2000 .

[10]  Carlos E. S. Cesnik,et al.  Nonlinear Aeroelasticity and Flight Dynamics of High-Altitude Long-Endurance Aircraft , 2001 .

[11]  R. Bhat.  Nonlinear Aeroelasticity , 2018, Principles of Aeroelasticity.

[12]  Christopher K. Droney,et al.  Subsonic Ultra Green Aircraft Research Phase II: N+4 Advanced Concept Development , 2012 .

[13]  Bret Stanford,et al.  Direct flutter and limit cycle computations of highly flexible wings for efficient analysis and optimization , 2013 .

[14]  Grigorios Dimitriadis.  Bifurcation Analysis of Aircraft with Structural Nonlinearity and Freeplay Using Numerical Continuation , 2008 .

[15]  Earl H. Dowell,et al.  Limit-Cycle Hysteresis Response for a High-Aspect-Ratio Wing Model , 2002 .

[16]  Jonathan E. Cooper,et al.  Aeroelastic Tailoring of a Representative Wing-Box Using Tow-Steered Composites , 2017 .

[17]  Frank Schilder,et al.  Recipes for Continuation , 2013, Computational science and engineering.