Non-linear dynamics of wind turbine wings

Abstract The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli–Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω . Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω 2 / ω 1 ≃ 2 , internal resonances between these modes have been studied. It is demonstrated that for ω / ω 1 ≃ 0.66 , 1.33 , 1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω / ω 1 ≃ 1.0 and ω / ω 2 ≃ 1.0 partly caused by the additive load term.

[1]  M. Raffel,et al.  Experimental and numerical investigations of dynamic stall on a pitching airfoil , 1996 .

[2]  S. Ariaratnam,et al.  DYNAMIC STABILITY OF A COLUMN UNDER RANDOM LOADING , 1967 .

[3]  Santosha K. Dwivedy,et al.  Nonlinear Response of a Parametrically Excited System Using Higher-Order Method of Multiple Scales , 1999 .

[4]  Michael S. Selig,et al.  The effect of rotation on the boundary layer of a wind turbine blade , 2000 .

[5]  J. B. Yang,et al.  Dynamic modelling and control of a rotating Euler–Bernoulli beam , 2004 .

[6]  J. G. Leishman,et al.  A Semi-Empirical Model for Dynamic Stall , 1989 .

[7]  Raouf A. Ibrahim,et al.  Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading-I. Single internal resonance , 1996 .

[8]  Poul Henning Kirkegaard,et al.  Super and combinatorial harmonic response of flexible elastic cables with small sag , 2002 .

[9]  Yu-Kweng Michael Lin,et al.  Probabilistic Structural Dynamics: Advanced Theory and Applications , 1967 .

[10]  Steen Krenk,et al.  Dynamic Stall Model for Wind Turbine Airfoils , 2007 .

[11]  R. Khas'minskii,et al.  Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems , 1967 .

[12]  G. Tagata,et al.  Non-linear string random vibration , 1989 .

[13]  R. N. Iyengar,et al.  Internal resonance and non-linear response of a cable under periodic excitation , 1991 .

[14]  A. H. Nayfeh,et al.  The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation , 1983 .

[15]  Nader Jalili,et al.  Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion , 1998 .

[16]  R. Langley.  An investigation of multiple solutions yielded by the equivalent linearization method , 1988 .

[17]  G. Rega,et al.  Experimental Investigation of the Nonlinear Response of a Hanging Cable. Part II: Global Analysis , 1997 .

[18]  S. I. Al-Noury,et al.  Large-amplitude vibrations of parabolic cables , 1985 .

[19]  W. Xie,et al.  On the Monte Carlo Simulation of Moment Lyapunov Exponents , 2006 .

[20]  Søren Nielsen,et al.  Nonlinear parametric instability of wind turbine wings , 2007 .

[21]  J. Gordon Leishman,et al.  Principles of Helicopter Aerodynamics , 2000 .

[22]  R. Alaggio,et al.  Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions , 1995 .

[23]  A. Preumont.  Vibration Control of Active Structures , 1997 .

[24]  Z. Ge,et al.  NON-LINEAR DYNAMIC ANALYSIS AND CONTROL OF CHAOS FOR A TWO-DEGREES-OF-FREEDOM RIGID BODY WITH VIBRATING SUPPORT , 2001 .

[25]  Earl H. Dowell,et al.  Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades , 1974 .

[26]  Dynamic Response of Non-Linear Systems to Random Trains of Non-Overlapping Pulses , 2002 .

[27]  Giuseppe Rega,et al.  Analysis of finite oscillations of elastic cables under internal/external resonance conditions , 1994 .

[28]  R. T. Jones.  The unsteady lift of a wing of finite aspect ratio , 1940 .

[29]  Jon Juel Thomsen,et al.  Vibrations and Stability , 2003 .

[30]  T. R. Kane,et al.  Dynamics of a cantilever beam attached to a moving base , 1987 .

[31]  Raouf A. Ibrahim,et al.  Stochastic excitation of suspended cables involving three simultaneous internal resonances using Monte Carlo simulation , 1999 .

[32]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[33]  Yuan-Cheng Fung,et al.  An introduction to the theory of aeroelasticity , 1955 .

[34]  M. Selig,et al.  A 3-D stall-delay model for horizontal axis wind turbine performance prediction , 1998 .

[35]  John A. Ekaterinaris,et al.  Evaluation of turbulence models for unsteady flows of an oscillating airfoil , 1995 .

[36]  J. C. Samuels.  The Dynamics of Impulsively and Randomly Varying Systems , 1963 .

[37]  Santosha K. Dwivedy,et al.  DYNAMICS OF A SLENDER BEAM WITH AN ATTACHED MASS UNDER COMBINATION PARAMETRIC AND INTERNAL RESONANCES, PART II: PERIODIC AND CHAOTIC RESPONSES , 1999 .

[38]  S. Krenk,et al.  A triangulation procedure for elastic cross sections with moderate wall thickness , 1986 .

[39]  Zheng-Ming Ge,et al.  Non-linear dynamics and control of chaos for a tachometer , 2002 .

[40]  J. Leishman.  Validation of approximate indicial aerodynamic functions for two-dimensional subsonic flow , 1988 .

[41]  J. R. Connell,et al.  Three-Dimensional Wind Simulation , 1998 .

[42]  Christophe Pierre,et al.  Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems , 1995 .

[43]  S. Hanagud,et al.  Problem of the dynamics of a cantilevered beam attached to a moving base , 1989 .

[44]  Noel C. Perkins,et al.  Three-dimensional oscillations of suspended cables involving simultaneous internal resonances , 1995, Nonlinear Dynamics.

[45]  Carl W. Helstrom,et al.  Two notes on a Markoff envelope process (Corresp.) , 1959, IRE Transactions on Information Theory.

[46]  J. Anderson,et al.  Fundamentals of Aerodynamics , 1984 .

[47]  J. K. Hammond,et al.  The instantaneous Lyapunov exponent and its application to chaotic dynamical systems , 1998 .

[48]  A. H. Nayfeh,et al.  The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance , 1983 .

[49]  I. H. Abbott,et al.  Theory of Wing Sections , 1959 .

[50]  G. Tagata,et al.  Analysis of a randomly excited non-linear stretched string , 1978 .

[51]  Søren Nielsen,et al.  Stochastic stability of mechanical systems under renewal jump process parametric excitation , 2005 .

[52]  G. V. Anand,et al.  NON-LINEAR RESONANCE IN STRINGS UNDER NARROW BAND RANDOM EXCITATION, PART I: PLANAR RESPONSE AND STABILITY , 1983 .

[53]  S. Ariaratnam,et al.  Lyapunov exponents and stochastic stability of coupled linear systems under white noise excitation , 1991 .

[54]  T. S. Beddoes,et al.  A Generalised Model for Airfoil Unsteady Aerodynamic Behaviour and Dynamic Stall Using the Indicial Method , 1986 .

[55]  S. Berman.  Stationary and Related Stochastic Processes , 1967 .

[56]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[57]  Liviu Librescu,et al.  Effects of pretwist and presetting on coupled bending vibrations of rotating thin-walled composite beams , 2003 .

[58]  R. Ibrahim,et al.  Multiple Internal Resonance in Suspended Cables under Random In-Plane Loading , 1997 .

[59]  Steen Krenk,et al.  Whirling motion of a shallow cable with viscous dampers , 2003 .

[60]  Søren Nielsen,et al.  Non-linear stochastic response of a shallow cable , 2006 .

[61]  C. W. S. To,et al.  Lyapunov Exponents and Information Dimensions of Multi-Degree-of-Freedom Systems Under Deterministic and Stationary Random Excitations , 1996 .

[62]  S. Krenk.  The torsion-extension coupling in pretwisted elastic beams , 1983 .

[63]  H. M. Irvine,et al.  The linear theory of free vibrations of a suspended cable , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[64]  Franklin D. Harris,et al.  Prediction Of Inflight Stalled Airloads From Oscillating Airfoil Data , 1969 .

[65]  A Linear Theory for Pretwisted Elastic Beams , 1983 .

[66]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[67]  Franklin D. Harris.  Preliminary Study of Radial Flow Effects on Rotor Blades , 1966 .

[68]  Westland Helicopters,et al.  Onset of Leading Edge Seperation Effects Under Dynamic Conditions and Low Mach Number , 1978 .

[69]  P. K. Chaviaropoulos,et al.  Investigating Three-Dimensional and Rotational Effects on Wind Turbine Blades by Means of a Quasi-3D Navier-Stokes Solver , 2000 .

[70]  R. A. McD. Galbraith,et al.  Measurements of the dynamic stall vortex convection speed , 1992, The Aeronautical Journal (1968).

[71]  L. Arnold,et al.  A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems , 1984 .

[72]  T. Vrouwenvelder,et al.  NUMERICAL MODELLING OF RAIN-WIND-INDUCED VIBRATION: ERASMUS BRIDGE, ROTTERDAM , 1998 .

[73]  Dongpu Cao,et al.  A stochastic approach to cable dynamics with moving rivulets , 2003 .

[74]  Santosha K. Dwivedy,et al.  SIMULTANEOUS COMBINATION, PRINCIPAL PARAMETRIC AND INTERNAL RESONANCES IN A SLENDER BEAM WITH A LUMPED MASS: THREE-MODE INTERACTIONS , 2001 .

[75]  M. Akbari,et al.  Simulation of dynamic stall for a NACA 0012 airfoil using a vortex method , 2003 .

[76]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[77]  Santosha K. Dwivedy,et al.  DYNAMICS OF A SLENDER BEAM WITH AN ATTACHED MASS UNDER COMBINATION PARAMETRIC AND INTERNAL RESONANCES PART I: STEADY STATE RESPONSE , 1999 .

[78]  Jaan Liiva,et al.  Unsteady aerodynamic and stall effects on helicopter rotor blade airfoil sections. , 1969 .

[79]  H. Madsen,et al.  A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations , 2004 .

[80]  L. Trilling,et al.  Incompressible Aerodynamics , 1961, Nature.

[81]  Santosha K. Dwivedy,et al.  Nonlinear Dynamics of a Cantilever Beam Carrying an Attached Mass with 1:3:9 Internal Resonances , 2003 .

[82]  Sundaram Suresh,et al.  Lift coefficient prediction at high angle of attack using recurrent neural network , 2003 .