Continuation of Higher-Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.

[1]  Earl H. Dowell,et al.  Multi-Harmonic Analysis of Dry Friction Damped Systems Using an Incremental Harmonic Balance Method , 1985 .

[2]  Earl H. Dowell,et al.  Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface , 2005 .

[3]  F. H. Ling,et al.  Fast galerkin method and its application to determine periodic solutions of non-linear oscillators , 1987 .

[4]  Q. Cao,et al.  Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method , 2003 .

[5]  S. T. Noah,et al.  Stability and Bifurcation Analysis of Oscillators With Piecewise-Linear Characteristics: A General Approach , 1991 .

[6]  S. Lau,et al.  Solution Diagram Of Non-Linear Dynamic Systems By The IHB Method , 1993 .

[7]  Lee,et al.  [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Aeroelastic Studies on a Folding Wing Configuration , 2005 .

[8]  Jian Xu,et al.  An efficient method for switching branches of period-doubling bifurcations of strongly non-linear autonomous oscillators with many degrees of freedom , 2003 .

[9]  Zhichun Yang,et al.  Analysis of limit cycle flutter of an airfoil in incompressible flow , 1988 .

[10]  Construction of chaotic regions , 1989 .

[11]  Jeffrey P. Thomas,et al.  Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique , 2002 .

[12]  Kwok-wai Chung,et al.  Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces , 2005 .

[13]  Jeffrey P. Thomas,et al.  Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations , 2001 .

[14]  C. G. Broyden.  A Class of Methods for Solving Nonlinear Simultaneous Equations , 1965 .

[15]  Earl H. Dowell,et al.  Higher Order Harmonic Balance Analysis for Limit Cycle Oscillations in an Airfoil with Cubic Restoring Forces , 2005 .

[16]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[17]  Soon-Yi Wu,et al.  Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems , 1983 .

[18]  Y. Cheung,et al.  A Variable Parameter Incrementation Method for Dynamic Instability of Linear and Nonlinear Elastic Systems , 1982 .

[19]  Andreas Griewank,et al.  The Calculation of Hopf Points by a Direct Method , 1983 .

[20]  A. Ferri.  On the Equivalence of the Incremental Harmonic Balance Method and the Harmonic Balance-Newton Raphson Method , 1986 .

[21]  Jeffrey P. Thomas,et al.  Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach , 2002 .

[22]  A. Leung,et al.  A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators , 1995 .

[23]  Dirk Roose,et al.  A Direct Method for the Computation of Hopf Bifurcation Points , 1985 .

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

[25]  A. Leung,et al.  Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method , 1995 .

[26]  S. Narayanan,et al.  Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method , 1999 .

[27]  D. Ewins,et al.  The Harmonic Balance Method with arc-length continuation in rotor/stator contact problems , 2001 .

[28]  Toeplitz Jacobian Matrix for Nonlinear Periodic Vibration , 1995 .

[29]  Jonathan E. Cooper,et al.  Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem , 2007 .

[30]  Jonathan E. Cooper,et al.  Application of Higher-Order Harmonic Balance to Non-Linear Aeroelastic Systems , 2006 .

[31]  J. Griffin,et al.  An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems , 1989 .

[32]  Aldo A. Ferri,et al.  Frequency domain solutions to multi-degree-of-freedom, dry friction damped systems under periodic excitation , 1988 .

[33]  S. Lau,et al.  Nonlinear Vibrations of Piecewise-Linear Systems by Incremental Harmonic Balance Method , 1992 .

[34]  Earl H. Dowell,et al.  The Secondary Bifurcation of an Aeroelastic Airfoil Motion: Effect of High Harmonics , 2004 .

[35]  S. Narayanan,et al.  Non-linear dynamics of a two-dimensional airfoil by incremental harmonic balance method , 1999 .

[36]  Higher Approximation of Steady Oscillations in Nonlinear Systems with Single Degree of Freedom : Suggested Multi-Harmonic Balance Method , 1981 .