Aeroelastic Equations Of Motion Research Articles

A New Dynamic Stall Approach for Investigating Bifurcation and Chaos in Aeroelastic Response of a Blade Section with Flap Free-Play Section

This paper investigates the effects of the unsteady nonlinear aerodynamic, plunge/pitch cubic nonlinearities, flap free-play nonlinearity, and coupled nonlinear aeroelasticity on the dynamics of the three-dimensional blade section. The dynamic stall model is developed based on the unsteady Wagner aerodynamics. Coupling the developed nonlinear aerodynamic model and nonlinear elasticity model results in the nonlinear aeroelastic model. The nonlinear aeroelastic equation of motion is converted into a state-space form. The resulting nonlinear state-space equation of motion is simulated by a standard Runge–Kutta algorithm in MATLAB. The proposed model is validated against test data of distinct two- and three-degrees-of-freedom studies and is compared to the ONERA model. Bifurcation diagrams show that there is distinct airspeed, in which the system experiences limit cycle oscillations (LCOs) or chaos. Both hysteresis air loads and structural nonlinearity make the system unstable at airspeed less than linear flutter speed. The nonlinearity of the structure causes supercritical pitchfork bifurcation. Elastic-aerodynamic nonlinearity interaction causes sub-supercritical bifurcation at the lower airspeed and chaotic motion at a higher airspeed. Furthermore, the effects of the initial condition on the response of the nonlinear aero-servo-elastic system are investigated by the Lyapunov exponent method.

Proper orthogonal decomposition method for analysis of nonlinear panel flutter with thermal effects in supersonic flow

The proper orthogonal decomposition (POD) method for analysis of nonlinear panel flutter subjected to supersonic flow is presented. Optimal POD modes are extracted from a chaotic Galerkin mode responses. The aeroelastic equations of motion are constructed using von Karman plate theory, first-order piston theory and quasi-steady thermal stress theory. A simply-supported plate with thermal loads from a uniformly distributed temperature is considered. Many types of panel behaviors, including stable flat, dynamically stable buckled, limit cycle oscillation, nonharmonic periodic motion, quasi-periodic motion and chaotic motion are observed. Our primary focus is on chaos and the route to chaos. It is found that a sudden transition from the buckled state to chaos occurs. Time history, phase portrait, Poincaré map, bifurcation diagram and Lyapunov exponent are employed to study chaos. The POD chaotic results obtained are compared with the traditional Galerkin solutions. It is shown that the POD method can obtain accurate chaotic solutions, using fewer modes and less computational effort than the Galerkin mode approach; additionally, the POD method converges faster in the analysis of chaotic transients. Effects of length-to-width ratios and thermal loads are presented. It is found that a smaller width for fixed length will produce more stable flutter response, while the thermal loads degrade the flutter boundary and result in a more complex evolution of dynamic motions. The numerical simulations show that the robustness of the POD modes depends on the dynamic pressure but not on temperature.

Aeroelastic behavior of long-span suspension bridges under arbitrary wind profiles

The aeroelastic behavior of suspension bridges under generic temporal and space wind distributions is studied employing a parametric one-dimensional structural model coupled with a strip-based indicial representation of the unsteady aerodynamic loads. The aeroelastic response to spatially nonuniform gusts is also investigated. The equations of motion linearized about the prestressed aerostatic equilibrium are expressed in terms of incremental kinematic variables. The aerodynamic loading characteristics of the deck cross sections of the Runyang Suspension Bridge (over the Yangtze river in China) are evaluated by means of computational fluid dynamics simulations performed to extract the aeroelastic derivatives from which the indicial aerodynamic representation is obtained. The aeroelastic equations of motion are then reduced to the state-space ordinary differential form by using the Faedo–Galerkin approach. The reduced-order bridge dynamics and the description of the aerodynamic loading via the added aerodynamic states are solved simultaneously by using a time-integration numerical scheme. Dependence of the flutter condition and the transient response to gusts is shown through systematic parametric studies.

Nonlinear Aeroelastic Behavior of Slender Wings Considering a Static Stall Model Based on Wagner Function

The aeroelastic behavior of high aspect ratio wings in an incompressible flow is investigated. The nonlinear nonplanar bending-bending-twisting motions of beam theory is used for the structural equations assuming large deformations with small strains, small Poisson effects, inextensional beam theory, and linear elastic material characteristics by neglecting warping and shear deformation. An Unsteady nonlinear aerodynamic static stall model based on the Wagner function is introduced and then is used for determination of aerodynamic loading of the wing. In this aerodynamic model, the static lift curve vs. angle of attack is approximated by a piece-wise curve and for each linear part of this curve a corrected Wagner theory is used. Combining these two types of formulation yields fully nonlinear integro-differentials aeroelastic equations of motion. The governing equations will be solved to predict the nonlinear aeroelastic response of a wing in the stall and post stall regions using Galerkin's method and a numerical method without the need of adding any aerodynamic state-space variables and their corresponding equations. The obtained equations are solved for some test cases and the obtained results are compared with the results given in the literature. Also a study is done to show effects of nonlinear aerodynamic static stall model on the limit cycle oscillations.

Nonlinear Aeroelastic Behavior of Slender Wings Considering a Static Stall Model Based on Wagner Function

The aeroelastic behavior of high aspect ratio wings in an incompressible flow is investigated. The nonlinear nonplanar bending-bending-twisting motions of beam theory is used for the structural equations assuming large deformations with small strains, small Poisson effects, inextensional beam theory, and linear elastic material characteristics by neglecting warping and shear deformation. An Unsteady nonlinear aerodynamic static stall model based on the Wagner function is introduced and then is used for determination of aerodynamic loading of the wing. In this aerodynamic model, the static lift curve vs. angle of attack is approximated by a piece-wise curve and for each linear part of this curve a corrected Wagner theory is used. Combining these two types of formulation yields fully nonlinear integro-differentials aeroelastic equations of motion. The governing equations will be solved to predict the nonlinear aeroelastic response of a wing in the stall and post stall regions using Galerkin's method and a numerical method without the need of adding any aerodynamic state-space variables and their corresponding equations. The obtained equations are solved for some test cases and the obtained results are compared with the results given in the literature. Also a study is done to show effects of nonlinear aerodynamic static stall model on the limit cycle oscillations.

Improved Mu Method with Mixed Perturbation to Dynamic Pressure

Mu method is a flutter solution technique based on frequency domain mu-analysis, which directly applies perturbation to dynamic pressure in aeroelastic equation of motion. In a previous study, the dynamic pressure perturbation was modeled as a purely real uncertainty, which could not guarantee the continuity of mu. To improve the continuity of mu, a small amount of complex perturbation is added to the dynamic pressure in addition to the real perturbation. Thus the mu method is improved with this mixed real/complex perturbation. Formulations and algorithm of the improved method are presented and validated with sample test cases. It is demonstrated that the improved mu method is more feasible and conservative than the original mu method due to the mixed perturbation.

Stability analysis of pitch‐regulated, variable‐speed wind turbines in closed loop operation using a linear eigenvalue approach

AbstractA multi‐body aeroelastic design code based on the implementation of the combined aeroelastic beam element is extended to cover closed loop operation conditions of wind turbines. The equations of a controller for variable generator speed and pitch‐controlled operation in high wind speeds are combined with the aeroelastic equations of motion for the complete wind turbine, in order to provide a compound aeroservoelastic system of equations. The control equations comprise linear differential equations for the pitch and generator torque actuators, the control feedback elements (proportional–integral control) and the various filters acting on the feedback signals. In its non‐linear form, the dynamic equations are integrated in time to provide the reference state, while upon linearization of the system and transformation in the non‐rotating frame, the linear stability equations are derived. Stability results for a multi‐MW wind turbine show that the coupling of the controller dynamics with the aeroelastic dynamics of the machine is important and must be taken into account in view of defining the controller parameters. Copyright © 2008 John Wiley & Sons, Ltd.

Viscosity Influence on Flutter Boundary and Limit Cycle Oscillation in Transonic Regime

Viscosity has a great influence on nonlinearity especially in the transonic regime: however only preliminary research into the viscous effect on the aeroelstic response was conducted. Nonlinearity of the flowfield assumed to be the main source of flutter behaviours. This paper reports the results of intensive numerical simulations carried out on 2D aerofoil sections to investigate the effect of viscosity on flutter boundary and especially on limit cycle oscillation (LCO) characteristics. Unsteady Euler and Navier-Stokes flow solvers are developed for use in aeroelastic analysis with the moving grid method, while the aeroelastic response is calculated with the aeroelastic equations of motion in two degrees of freedom. The flow solvers are verified with simulations of pitching aerofoil. Two aeroelastic test cases ranging from subsonic to transonic flow regimes (M∞=0.3∼0.92, Rec = 12.3×106) were simulated in depth to obtain flutter boundaries as well as characteristics of LCO. The results indicate that the LCO occurs in the subsonic flow regime as well as in the transonic flow regime with far greater intensity caused by viscous effect. It has been shown that the weak divergence is found to be a transonic aeroelastic phenomenon.

Aeroelastic response and stability analysis of composite rotor blades in forward flight

Abstract The aeroelastic phenomena of a composite rotor blade are investigated using a finite element method. The aeroelastic equations of motion of the rotor blade are formulated using a large deflection-type beam model that has no artificial restrictions on the magnitudes of displacements and rotations due to the degree of nonlinearity. The sectional elastic constants of a composite box beam including warping deformations are determined from the refined cross-sectional finite element method. The nonlinear, periodic blade steady response is obtained using a time finite element method on full finite element equation. The aeroelastic stability of a rotor blade is investigated by linearizing the dynamics of the rotor blade about the nonlinear equilibrium position. Numerical results of the steady equilibrium deflections and the aeroelastic modal damping are presented for various configurations of composite rotor blades and are compared with those obtained from a previously published modal analysis using a moderate deflection-type beam theory.

FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a two-dimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a fourth order Runge-Kutta scheme. Results for soft and hard springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit-cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analysed using analytical techniques developed for nonlinear dynamical systems. The Hopf bifurcation point is determined analytically and the amplitude of the limit-cycle oscillation in post-Hopf bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit-cycle oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit-cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom.

Unified aeroelastic flutter theory for very low aspect ratio panels

A theory unifying the flutter analyses of orthotropic and isotropic panels having very low aspect ratios and exposed to an inviscid potential flow on the upper surfaces is developed. The analysis considers an infinitely long panel of finite width, simply supported on the side edges, and resting on a spring foundation. Three-dimensional linearized aerodynamic and classical plate theories are used. Traveling wave modes are utilized as solutions to the panel's aeroelastic equation of motion. The resulting aerodynamic integral is evaluated approximately using both numerical and analytical methods. The structural and aerodynamic generic variables, which are instrumental in establishing this unified theory, are derived using affine transformations. Subsonic as well as supersonic flutter and divergence boundaries are determined. The effects of air to panel mass ratios and midplane forces are examined. Viscous structural damping is found to be destabilizing. There is a general agreement between the results of this analysis and those obtained for isotropic panels by previous investigators.