Nonlinear Aeroelastic System Research Articles

Classical and Fuzzy Sliding Mode Control for a Nonlinear Aeroelastic System with Unsteady Aerodynamic Model

Classical and Fuzzy Sliding Mode Control for a Nonlinear Aeroelastic System with Unsteady Aerodynamic Model

Modeling and Design of a Piezoelectric Nonlinear Aeroelastic Energy Harvester

In the present era, the possibility to generate electrical energy from an operational environment is a critical factor for an aerospace industry to drive microelectronic components. In this research, a solution for the energy harvesting mechanism based on fluid-structure interaction (FSI) is investigated. The possibility to harvest energy from post-critical aeroelastic behavior, known as Limit Cycle Oscillations (LCOs) through piezoelectric transduction is presented. A typical condition for energy harvesting, which requires a strong interaction between the external energy and the components where the harvester is embedded. The LCOs arise after the flutter speed in nonlinear aeroelastic systems, and these oscillations are utilized for harvesting phenomenon. The analytical model is developed for both the FSI and electromechanical behavior of the piezoelectric harvester. In particular, the importance of the aerodynamic model for determining the performance of the harvester is stressed. Two different piezoelectric materials, i.e. Lead zirconate titanate (PZT-5A) and Barium titanate (BaTiO3) is used in the designed harvester. The presented model is suitable to harvest energy and to drive wireless sensors. The maximum power output obtained by the designed piezoelectric aeroelastic energy harvester (PAEH) is found to be 5.5 mW for 0.1 mΩ of resistance.

A novel grey decision-DE optimized internal model controller for vibration control of nonlinear uncertain aeroelastic blade system

In this paper, a novel grey decision-differential evolution (GD-DE) optimized internal model controller is proposed for the control of nonlinear aeroelastic blade system (ABS) under time-varying uncertainties and control saturations. Two-port internal model control structure is presented for saturation compensation. The filter parameter of the controller is optimally tuned by proposed GD-DE algorithm, which is designed based on the grey decision incidence theory and differential evolution algorithm. The superiority of GD-DE optimized tuning technique is verified, compared to conventional IMC tuning method and other evolutionary algorithm based techniques. The robustness is analysed by comprehensive simulations from time-invariant uncertainties to large periodic or random time-varying uncertainties. The realistic condition with input/output disturbances is also involved. Simulation results show that the proposed controller outperforms existing adaptive IMC controller and other advanced controllers with greatly improved dynamic performance, stronger robustness and better saturation compensation on various conditions.

Optimal control of nonlinear aeroelastic system with non‐semi‐simple eigenvalues at Hopf bifurcation points

AbstractThis study discusses an efficient method of the Hopf bifurcation control for nonlinear aeroelastic system. The nonlinear aeroelastic system whose linear part has multiple non‐semi‐simple eigenvalues at critical point gives rise to Hopf bifurcations. The method of the multiple scales and the well‐known linear quadratic regulator method are used to deal with the optimal control of the nonlinear system at Hopf bifurcation points. The modal optimal control equation and modal Riccati equation of the nonlinear system are developed to simplify the computations. The conventional Potter's algorithm is extended to solve modal Riccati equation for the modal Riccati matrix of the Hopf bifurcation control. The first‐order approximation solutions are developed, which include the gain vectors and inputs. By the way of optimal control, the admissible control input and trajectory of the linear part of the nonlinear aeroelastic system are obtained to minimize the performance measure. Then, we set the appropriate first‐order gain vector to adjust the convergence speed of this nonlinear system.

Data-Driven Forecasting of Postflutter Responses of Geometrically Nonlinear Wings

Ensuring adequate flutter margins is a critical step in aircraft design. However, in the presence of nonlinear effects, subcritical limit-cycle oscillations can arise even before flutter occurs. When nonlinear effects are anticipated, postflutter analyses have to be integrated into aircraft design in addition to flutter computations for preventing this undesirable subcritical behavior. Therefore, there is a need for computationally fast postflutter analysis methods suitable for design applications. This paper investigates a data-driven method to forecast the bifurcation behavior of a geometrically nonlinear wing, demonstrating its suitability to analyze realistic nonlinear aeroelastic systems. The method efficiently forecasts postflutter responses using only a few system transient computations in the preflutter regime. First, the accuracy of forecasted bifurcation diagrams is verified against time-marching results. Next, wing design parameters are varied to investigate their impact on postflutter responses and show the method suitability to parametric studies.

Aeroelastic responses of airfoil under dynamic stall forced to oscillate by cyclic pitch input

The helicopter blades are involved in a nonlinear aeroelastic system in which unsteady dynamic stall flow and flexible structure are coupled. This problem can be reduced to a 2D aeroelastic problem in which airfoil is forced to oscillate through cyclic pitch input. In this paper, a modified dynamic stall model which focuses on revising the modeling of aerodynamic loads in reattachment stage is adopted in the numerical simulation to predict the aeroelastic responses of the above-mentioned system. The results are validated against experiments, after which parametric studies are performed to investigate the complex aeroelastic responses. In single Degree-Of-Freedom (DOF) system, secondary resonances happen in specific ratios of natural frequency to forcing frequency, which are found in both experiments and simulation. Moreover, as free-stream velocity varies, the aerodynamic stiffness has an apparent influence on the pitching natural frequency of system with low structural frequency. The varying pitching natural frequency brings about different aeroelastic responses including 1/2 subharmonic resonances and quenching phenomenon in single-DOF system as well as internal resonances and combination resonances in 2-DOF system. The aeroelastic system with low structural frequency is further investigated through multiple scale method, which provides an illustration for the mechanism of those phenomena found in the aeroelastic simulation adopting a semi-empirical aerodynamic model.

Nonlinear aeroelastic behavior of an airfoil with free-play in transonic flow

An investigation has been made into the nonlinear aeroelastic behavior of an airfoil system with free-play nonlinear stiffness in transonic flow. Computational Fluid Dynamics (CFD) and Reduced Order Model (ROM) based on Euler and Navier-Stokes equations are implemented to calculate unsteady aerodynamic forces. Results show that the nonlinear aeroelastic system experiences various bifurcations with increasing Mach number. Regular subcritical bifurcations are observed in low Mach number region. Subsequently, complex Limit Cycle Oscillations (LCOs) and even non-periodic motions appear at specific airspeed regions. When the Mach number is increased above the freeze Mach number, regular subcritical bifurcations occur again. Comparisons with inviscid solutions are used to identify and elaborate the effect of viscosity with the help of aeroelastic analysis techniques, including root locus, Single Degree of Freedom (SDOF) flutter and aerodynamic influence coefficient (AIC). For low Mach numbers in the transonic regime, the viscosity has little effect on the linear flutter characteristic because of limited influence on AIC, but a remarkable impact on the nonlinear dynamic behavior due to the sensitivity of the nonlinear structure. As the Mach number increases, the viscosity becomes significantly important due to the existence of shock-boundary layer interaction. It affects the unstable mechanism of linear flutter, impacts the aerodynamic center and hence the snap-through phenomenon, influences the AIC and consequently the nonlinear aeroelastic response. When the Mach number is increased further, the shock wave dominates the air flow and the viscosity is of minor importance.

Reduced order nonlinear aeroelasticity of swept composite wings using compressible indicial unsteady aerodynamics

Nonlinear dynamic aeroelasticity of composite wings in compressible flows is investigated. To provide a reasonable model for the problem, the composite wing is modeled as a thin walled beam (TWB) with circumferentially asymmetric stiffness layup configuration. The structural model considers nonlinear strain displacement relations and a number of non-classical effects, such as transverse shear and warping inhibition. Geometrically nonlinear terms of up to third order are retained in the formulation. Unsteady aerodynamic loads are calculated according to a compressible model, described by indicial function approximations in the time domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to derive the final explicit form of the coupled fluid-structure equations of motion. The final nonlinear governing aeroelastic system of equations is solved using the eigenvectors of the linear structural equations of motion to approximate the spatial variation of the corresponding degrees of freedom in the Ritz solution method. Direct time integrations of the nonlinear equations of motion representing the full aeroelastic system are conducted using the well-known Runge–Kutta method. A comprehensive insight is provided over the effect of parameters such as the lamination fiber angle and the sweep angle on the stability margins and the limit cycle oscillation behavior of the system. Integration of the interpolation method employed for the evaluation of compressible indicial functions at any Mach number in the subsonic compressible range to the derivation process of the third order nonlinear aeroelastic system of equations based on TWB theory is done for the first time. Results show that flutter speeds obtained by the incompressible unsteady aerodynamics are not conservative and as the backward sweep angle of the wing is increased, post-flutter aeroelastic response of the wing becomes more well-behaved.

Parameter Identification of Nonlinear Aeroelastic System with Time-Delayed Feedback Control

In this paper, an enhanced response sensitivity approach is proposed for parameter identification of nonlinear time-delay aeroelastic systems. Two common aeroelastic models are investigated, namely, the aeroelastic system under the unsteady Theodorsen aerodynamic forces and the airfoil-store system under the steady incompressible flow. The work of this paper is mainly threefold. First, the motion equations of two aeroelastic models including time-delay signals are derived, and the response sensitivity analysis with respect to the system parameters is conducted. Then, parameter identification is formulated as a nonlinear least-squares problem, and the enhanced response sensitivity approach is developed to get the solution. Finally, numerical examples are studied to verify the effectiveness of the proposed method. The results reveal that the proposed approach can give accurate, robust, and efficient identification of the parameters for time-delay aeroelastic systems.

Nonlinear Aeroelastic Modeling of a Folding Wing Structure

The nonlinear aeroelastic response of a folding wing with the hinge free-play nonlinearity is investigated. A nonlinear structural dynamic model is established based on the modal synthesis of free-substructures by modification of the connected relationships, and is unnecessary to re-establish it if the stiffness changes. The nonlinear aeroelastic equation is obtained by rational function approximation of the unsteady aerodynamic force, and solved by Runge-Kutta method to predict the aeroelastic response of the folding wing. The results show that the free-play at hinges changes the aeroelastic characteristic of the folding wing, leading to the occurrence of the limit cycle oscillation within a certain velocity range. Moreover, the sensitivity to the free-play is different for the inboard and outboard hinges. If the free-play is controlled within a certain value, the motion of nonlinear aeroelastic system of the folding wing can be considered linear approximately.

Hypersonic flutter and flutter suppression system of a wind tunnel model

The aeroelastic behavior of a thin flat rudder model was numerically simulated and experimentally investigated in a hypersonic wind tunnel. In particular, a flutter suppression system taking advantage of collision within small gaps was proposed and a novel system for the flutter simulation of the whole nonlinear aeroelastic system including the flutter suppression system was developed. First, the critical flutter dynamic pressure of the rudder without the flutter suppression system was calculated with different methods. Then, the whole nonlinear aeroelastic system, including the flutter suppression system, was simulated to design the gap size. Finally, the flutter suppression system was experimentally validated in a hypersonic wind tunnel operating at Mach number 5. The typical phenomenon of Limit Cycle Oscillation (LCO) was observed, avoiding the structural failure of the model and the consistency between numerical and experimental results was demonstrated. The proposed suppression system can improve the design and reusability of test models of hypersonic flutter experiments.

Uncertainty Quantification Using Generalized Polynomial Chaos Expansion for Nonlinear Dynamical Systems With Mixed State and Parameter Uncertainties

This paper develops a framework for propagation of uncertainties, governed by different probability distribution functions in a stochastic dynamical system. More specifically, it deals with nonlinear dynamical systems, wherein both the initial state and parametric uncertainty have been taken into consideration and their effects studied in the model response. A sampling-based nonintrusive approach using pseudospectral stochastic collocation is employed to obtain the coefficients required for the generalized polynomial chaos (gPC) expansion in this framework. The samples are generated based on the distribution of the uncertainties, which are basically the cubature nodes to solve expectation integrals. A mixture of one-dimensional Gaussian quadrature techniques in a sparse grid framework is used to produce the required samples to obtain the integrals. The familiar problem of degeneracy with high-order gPC expansions is illustrated and insights into mitigation of such behavior are presented. To illustrate the efficacy of the proposed approach, numerical examples of dynamic systems with state and parametric uncertainties are considered which include the simple linear harmonic oscillator system and a two-degree-of-freedom nonlinear aeroelastic system.

Bifurcation behavior and chaotic dynamics of a three-degree-of-freedom aeroelastic system

In this paper, we combined analytic method with numerical method to investigate complex dynamic behavior in a three-degree-of-freedom nonlinear aeroelastic system of an airfoil with external store. Center manifold and normal form theories are performed to derive the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves. Stable and unstable Hopf bifurcations appear when the damping coefficient of the plunge and the positive parameter Q, which is associated with the flow velocity and the pitch movement, are perturbed a little at the linear critical values. Under certain conditions, quasi-periodic motions on 2-D and 3-D torus may occur. Furthermore, with the aid of Poincare map, the largest Lyapunov exponent and frequency spectra, chaotic dynamics arise when 2-D tori become unstable and are violated. These rich results contribute to show the various characteristics of structural nonlinearities and avoid aerodynamic flutter of the airfoil. In the end, the numerical solutions achieved by fourth order Runge–Kutta method agree with the analytic results.

Nonlinear Aeroelastic System Identification Based on Neural Network

This paper focuses on the nonlinear aeroelastic system identification method based on an artificial neural network (ANN) that uses time-delay and feedback elements. A typical two-dimensional wing section with control surface is modelled to illustrate the proposed identification algorithm. The response of the system, which applies a sine-chirp input signal on the control surface, is computed by time-marching-integration. A time-delay recurrent neural network (TDRNN) is employed and trained to predict the pitch angle of the system. The chirp and sine excitation signals are used to verify the identified system. Estimation results of the trained neural network are compared with numerical simulation values. Two types of structural nonlinearity are studied, cubic-spring and friction. The results indicate that the TDRNN can approach the nonlinear aeroelastic system exactly.

Geometrically nonlinear aeroelastic behavior of pretwisted composite wings modeled as thin walled beams

Geometrically nonlinear aeroelastic behavior of pretwisted composite high aspect ratio wings, structurally modeled as thin walled beams (TWB) is studied. The structural equations of motion are obtained for the circumferentially asymmetric stiffness (CAS) configuration TWB based on the kinematic relations governing the thin walled beams, including the nonlinear strain–displacement relations and utilizing the principles of analytical dynamics. Unsteady aerodynamic loads in the incompressible flow regime are expressed using Wagner’s function in time-domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to come up with the final coupled fluid–structure equations of motion. The governing equations of the aeroelastic system is solved, for the TWB with CAS composite layup, by means of a Ritz based solution methodology utilizing the mode shapes of the linear structural system to approximate the spatial variation of degrees of freedom in the thin walled beam. Time response of the nonlinear aeroelastic system is obtained via the Runge–Kutta direct integration algorithm. Effects of the fiber angle and pretwist angle of the CAS layup configuration on the nonlinear aeroelastic stability margins and limit cycle oscillation behavior are studied in depth.

Identification of Volterra kernels for improved predictions of nonlinear aeroelastic vibration responses and flutter

Aeroelastic structural systems are intrinsically nonlinear and accurate predictions of dynamic responses of nonlinear aeroelastic systems have become of paramount importance since these directly affect the accuracy and reliability of subsequent stability analyses. Such nonlinear systems can be generally represented with Volterra series whose kernels have been found to be effective in their dynamic characterizations. This paper examines how first- and second-order Volterra kernels of nonlinear aeroelastic systems can be accurately identified and then incorporated into the theoretical models of aeroelastic analyses such as predictions of dynamic response and onset of aeroelastic flutter. A novel identification method based on correlation analysis to extract frequency components has been developed which can be applied to general nonlinear aeroelastic systems to obtain accurately the required Volterra transfer functions. The method is very accurate and extremely robust against measurement noise contaminations in both input and output signals, due to the correlation scheme which effectively filters uncorrelated signal components. Detailed aeroelastic behavior of a representative pitch-plunge airfoil dynamic model with nonlinear pitch stiffness has been examined. Its Volterra transfer functions are then identified which are found to be close to their exact analytical counterparts, though interaction between kernels becomes apparent as input level increases. Once inverse Fourier transformed, these identified Volterra kernels are then included in the modeling of the dynamics of aeroelastic systems for vibration response and flutter. Extensive numerical simulation results have demonstrated that the proposed method is very accurate and resilient to measurement errors when applied to the identification of second-order Volterra kernels, and the improvement in predictions of vibration response and flutter become significant when the contributions of these second-order Volterra kernels are included in the overall aeroelastic system dynamics. The identification and subsequent inclusion of second-order Volterra kernels into system dynamics model offer improved design capabilities of nonlinear aeroelastic structural systems.

Efficient Nonlinear Reduced-Order Model for Computational Fluid Dynamics-Based Aeroelastic Analysis

In this work, an efficient nonlinear state-space reduced-order model (ROM) is introduced. The nonlinear state-space ROM consists of both linear terms and nonlinear terms, the former of which is obtained based on Volterra series theory and eigensystem realization algorithm method, whereas the latter is determined by an output error-minimization procedure. Two approaches are proposed to determine the training signal for identification of the nonlinear terms. Aeroelastic analysis of a weakened wall-mounted AGARD 445.6 wing is conducted and it is demonstrated that 1) the nonlinear state-space ROM can get more accurate unsteady aeroelastic response than traditional linear Volterra-based ROM with computational fluid dynamics coupling with computational structural dynamic as a reference method for comparison; 2) the nonlinear state-space ROM performs well in range of subsonic, transonic, and supersonic; and 3) the nonlinear state-space ROM can better characterize the nonlinear aeroelastic system than traditional linear Volterra-based ROM without the need of identifying second-order Volterra kernels, which thus can be easily applied to solve engineering problems for its concise form.

A combined Multiple Time Scales and Harmonic Balance approach for the transient and steady-state response of nonlinear aeroelastic systems

The majority of methods for calculating the dynamic response of nonlinear aeroelastic systems considers only steady-state periodic behaviour. An exception is the Multiple Time Scales method, which can estimate both the transient and steady-state solutions of such systems; nevertheless, this approach is only accurate close to the Hopf bifurcation. This paper proposes a novel combined approach whereby the transient response is obtained from the Multiple Time Scales method and the asymptotic periodic behaviour is corrected using the Harmonic Balance method. This consistent and efficient framework mutually empowers both techniques and accounts for large parameter variations around the critical condition. The effect of cubic aero-structural nonlinearities on the dynamic response of a generic aeroelastic system is then investigated. Both the Multiple Time Scales and Harmonic Balance methods are adopted and perfect agreement of the explicit results is demonstrated, albeit near the system instability. In contrast, the proposed combined solution is valid for a wider range of perturbations, is analytical and has negligible computational cost while retaining accuracy. The role of key parameters and terms on the core mechanism of the dynamic behaviour is rigorously identified and discussed, from both physical and mathematical points of view. Galloping is finally considered as the simplest but complete application to a fundamental yet practical problem, featuring full conceptual complexity while exploiting the solid synthesis capability of the newly proposed analytical approach. Excellent agreement was found in all cases with results from the numerical integration of the nonlinear equations of motion in the time domain.

Dynamic assessment of nonlinear typical section aeroviscoelastic systems using fractional derivative-based viscoelastic model

Nonlinear aeroelastic systems are prone to the appearance of limit cycle oscillations, bifurcations, and chaos. Such problems are of increasing concern in aircraft design since there is the need to control nonlinear instabilities and improve safety margins, at the same time as aircraft are subjected to increasingly critical operational conditions. On the other hand, in spite of the fact that viscoelastic materials have already been successfully used for the attenuation of undesired vibrations in several types of mechanical systems, a small number of research works have addressed the feasibility of exploring the viscoelastic effect to improve the behavior of nonlinear aeroelastic systems. In this context, the objective of this work is to assess the influence of viscoelastic materials on the aeroelastic features of a three-degrees-of-freedom typical section with hardening structural nonlinearities. The equations of motion are derived accounting for the presence of viscoelastic materials introduced in the resilient elements associated to each degree-of-freedom. A constitutive law based on fractional derivatives is adopted, which allows the modeling of temperature-dependent viscoelastic behavior in time and frequency domains. The unsteady aerodynamic loading is calculated based on the classical linear potential theory for arbitrary airfoil motion. The aeroelastic behavior is investigated through time domain simulations, and subsequent frequency transformations, from which bifurcations are identified from diagrams of limit cycle oscillations amplitudes versus airspeed. The influence of the viscoelastic effect on the aeroelastic behavior, for different values of temperature, is also investigated. The numerical simulations show that viscoelastic damping can increase the flutter speed and reduce the amplitudes of limit cycle oscillations. These results prove the potential that viscoelastic materials have to increase aircraft components safety margins regarding aeroelastic stability.

Quasi-periodic aeroelastic response analysis of an airfoil with external store by incremental harmonic balance method

This paper presents an investigation on the nonlinear aeroelastic system of an airfoil with external store by incremental harmonic balance (IHB) method. Besides solving limit cycle (LC) solutions, the IHB method is implemented to obtain quasi-periodic (QP) solutions by introducing multiple irreducible time scales. Steady state responses such as LC and QP oscillations obtained by the presented method are verified by numerical examples. One pair of Floquet multipliers for LC solutions first leave and then enter a unit cycle at complex conjugate values, indicating the existence of a secondary Hopf bifurcation and its reversal one. Along with the fundamental frequency of LC oscillation, an additional frequency arises at the secondary bifurcation, and finally disappears at the reversal bifurcation. The appearance and disappearance of the irreducible frequency cause the steady state responses changing from LC to QP and back to LC oscillation.