Bifurcation analysis of an aeroelastic system with concentrated nonlinearities

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system??s instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed.     

An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system

We perform an analytical and experimental investigation into the dynamics of an aeroelastic system consisting of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The experimental results show that the onset of flutter takes place at a speed smaller than the one predicted by a quasi-steady aerodynamic approximation. On the other hand, the unsteady representation of the aerodynamic loads accurately predicts the experimental value. The linear analysis details the difference in both formulation and provides an explanation for this difference. Nonlinear analysis is then performed to identify the nonlinear coefficients of the pitch spring. The normal form of the Hopf bifurcation is then derived to characterize the type of instability. It is demonstrated that the instability of the considered aeroelastic system is supercritical as observed in the experiments.     

Dimensionless modeling and nonlinear analysis of a coupled pitch–plunge–leadlag airfoil-based piezoaeroelastic energy harvesting system

In this paper, an airfoil-based piezoaeroelastic energy harvesting system is proposed with an additional supporting device to harvest the mechanical energy from the leadlag motion. A dimensionless dynamic model is built considering the large-effective-angle-of-attack vibrations causing (1) the nonlinear coupling between the pitch–plunge–leadlag motions, (2) the inertia nonlinearity, and (3) the aerodynamic nonlinearity modeled by the ONERA dynamic stall model. Cubic hardening stiffness is introduced in the pitch degree of freedom for persistent vibrations with acceptable amplitude beyond the flutter boundary. The nonlinear aeroelastic response and the average power output are numerically studied. Limit cycle oscillations are observed and, as the flow velocity exceeds a secondary critical speed, the system experiences complex vibrations. The power output from the leadlag motion is smaller than that from the plunge motion, whereas the gap is narrowed with increasing flow velocity. Case studies are performed toward the effects of several dimensionless system parameters, including the nonlinear torsional stiffness, airfoil mass eccentricity, airfoil radius of gyration, mass of the supporting devices, and load resistances in the external circuits. The optimal parameter values for the power outputs from the plunge and leadlag motions are, respectively, obtained. Beyond the secondary critical speed, it is shown that the variations of the power outputs with those parameters become irregular with fluctuations and multiple local maximums. The bifurcation analysis shows the mutual transitions between the limit cycle oscillations, multi-periodic vibrations, and possible chaos. The influences of these complex vibrations on the power outputs are discussed.     

Effects of multiple structural nonlinearities on limit cycle oscillation of missile control fin

Aeroelastic analyses are performed for a 2-D typical section model with multiple nonlinearities. The differences between a system with multiple nonlinearities in its pitch and plunge spring and a system with a single nonlinearity in its pitch are thoroughly investigated. The unsteady supersonic aerodynamic forces are calculated by the doublet point method (DPM). The iterative V-g method is used for a multiple-nonlinear aeroelastic analysis in the frequency domain and the freeplay nonlinearity is linearized using a describing function method. In the time domain, the DPM unsteady aerodynamic forces, which are based on a function of the reduced frequency, are approximated by the minimum state approximation method. Consequently, multiple structural nonlinearities in the 2-D typical wing section model are influenced by the pitch to plunge frequency ratio. This result is important in that it demonstrates that the flutter speed is closely connected with the frequency ratio, considering that both pitch and plunge nonlinearities result in a higher flutter speed boundary than a conventional aeroelastic system with only one pitch nonlinearity. Furthermore, the gap size of the freeplay affects the amplitude of the limit cycle oscillation (LCO) to gap size ratio.     

Output Feedback Modular Adaptive Control of a Nonlinear Prototypical Wing Section

The paper presents nonlinear adaptive control systems for the control of limit cycle oscillations of a prototypical wing section with structural nonlinearities using only output feedback. The chosen model describes the plunge and pitch motion of a wing. The model includes plunge and pitch nonlinearities, and has a single control surface for the purpose of control. Using a canonical representation of the aeroelastic system, a modular output feedback adaptive control system consisting of an input-to-state stabilizing controller and a passive identifier (an observer and adaptation law) is derived. In the closed-loop system, asymptotic stabilization of the pitch and plunge motion is accomplished. Simulation results show that the control system is effective in regulating the state vector to the origin in spite of large parameter uncertainties.     

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.     

Response analysis of a pitch–plunge airfoil with structural and aerodynamic nonlinearities subjected to randomly fluctuating flows

This study focuses on numerically investigating the response dynamics of a pitch–plunge airfoil with structural nonlinearity under dynamic stall conditions. The aeroelastic responses are investigated for both deterministic and randomly time varying flow conditions. To that end, a pitch–plunge airfoil under dynamic stall condition is considered and the nonlinear aerodynamic loads are computed using a Leishman–Beddoes formulation. It is shown that the presence of structural nonlinearities can give rise to a variety of dynamical responses in the pre-flutter regime. Next, a response analysis under the presence of a randomly fluctuating wind is carried out. It is demonstrated that the route to flutter occurs via a regime of pre-flutter oscillations called intermittency. Finally, the manifestation of these stochastic responses is characterized by invoking stochastic bifurcation concepts. The route to flutter via intermittency is presented in terms of topological changes occurring in the joint-probability density function of the state variables.     

Thrust augmentation of flapping airfoils in low Reynolds number flow using a flexible membrane

The unsteady aerodynamic thrust and aeroelastic response of a two-dimensional membrane airfoil under prescribed harmonic motion are investigated computationally with a high-order Navier–Stokes solver coupled to a nonlinear membrane structural model. The effects of membrane prestress and elasticity are examined parametrically for selected plunge and pitch–plunge motions at a chord-based Reynolds number of 2500. The importance of inertial membrane loads resulting from the prescribed flapping is also assessed for pure plunging motions. This study compares the period-averaged aerodynamic loads of flexible versus rigid membrane airfoils and highlights the vortex structures and salient fluid–membrane interactions that enable more efficient flapping thrust production in low Reynolds number flows.     

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports

Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems. This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks.     

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.     

CHARACTERIZATION OF THE BEHAVIOUR OF A SIMPLE AEROSERVOELASTIC SYSTEM WITH CONTROL NONLINEARITIES

The characterization of the behaviour of nonlinear aeroelastic systems has become a very important research topic in the Aerospace Industry. However, most work carried to-date has concentrated upon systems containing structural or aerodynamic nonlinearities. The purpose of this paper is to study the stability of a simple aeroservoelastic system with nonlinearities in the control system and power control unit. The work considers both structural and control law nonlinearities and assesses the stability of the system response using bifurcation diagrams. It is shown that simple feedback systems designed to increase the stability of the linearized system also stabilize the nonlinear system, although their effects can be less pronounced. Additionally, a nonlinear control law designed to limit the control surface pitch response was found to increase the flutter speed considerably by forcing the system to undergo limit cycle oscillations instead of fluttering. Finally, friction was found to affect the damping of the system but not its stability, as long as the amplitude of the frictional force is low enough not to cause stoppages in the motion.     

Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity

We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum value of the load resistance exists, at which the level of the harvested power is maximized.     

Assessment of added mass effects on flutter boundaries using the Leishman–Beddoes dynamic stall model

We consider the dynamics of a typical airfoil section both in forced and free oscillations and investigate the importance of the added mass terms, i.e. the second derivatives in time of the pitch angle and plunge displacement. The structural behaviour is modelled by linear springs in pitch and plunge and the aerodynamic loading represented by our interpretation of the state-space version of the Leishman–Beddoes semi-empirical model. The added mass terms are often neglected since this leads to an explicit system of ODEs amenable for solution using standard ODE solvers. We analyse the effect of neglecting the added mass terms in forced oscillations about a set of mean angles of incidence by comparing the solutions obtained with the explicit and implicit systems of ODEs and conclude that their differences amount to a time lag that increases at a constant rate with increases of the reduced frequency. To determine the effect of the added mass terms in free oscillations, we introduce a spring offset angle to obtain static equilibrium positions at various degrees of incidence. We analyse the stability of the explicit and implicit aeroelastic systems about those positions and compare the locations of the respective flutter points calculated as Hopf bifurcation points. For low values of the spring offset angle, added mass effects are significant for low values of the mass ratio, or the ratio of natural frequencies, of the aeroelastic system. For high values of the spring offset angle, corresponding to stall flutter, we observe that their effect is greater for large values of the mass ratio.     

Design of piezoaeroelastic energy harvesters

We design a piezoaeroelastic energy harvester consisting of a rigid airfoil that is constrained to pitch and plunge and supported by linear and nonlinear torsional and flexural springs with a piezoelectric coupling attached to the plunge degree of freedom. We choose the linear springs to produce the minimum flutter speed and then implement a linear velocity feedback to reduce the flutter speed to any desired value and hence produce limit-cycle oscillations at low wind speeds. Then, we use the center-manifold theorem to derive the normal form of the Hopf bifurcation near the flutter onset, which, in turn, is used to choose the nonlinear spring coefficients that produce supercritical Hopf bifurcations and increase the amplitudes of the ensuing limit cycles and hence the harvested power. For given gains and hence reduced flutter speeds, the harvested power is observed to increase, achieve a maximum, and then decrease as the wind speed increases. Furthermore, the response undergoes a secondary supercritical Hopf bifurcation, resulting in either a quasiperiodic motion or a periodic motion with a large period. As the wind speed is increased further, the response becomes eventually chaotic. These complex responses may result in a reduction in the generated power. To overcome this adverse effect, we propose to adjust the gains to increase the flutter speed and hence push the secondary Hopf bifurcation to higher wind speeds.     

Nonlinear dynamic behaviors of a rotor-labyrinth seal system

The nonlinear model of rotor-labyrinth seal system is established using Muszynska’s nonlinear seal forces. We deal with dynamic behaviors of the unbalanced rotor-seal system with sliding bearing based on the adopted model and Newmark integration method. The influence of the labyrinth seal one the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare’ maps. Various phenomena in the rotor-seal system, such as periodic motion, double-periodic motion, quasi-periodic motion and Hopf bifurcation are investigated and the stability is judged by Floquet theory and bifurcation theorem. The influence of parameters on the critical instability speed of the rotor-seal system is also included.     

Transient mode localization in coupled strongly nonlinear exactly solvable oscillators

Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.     

Influence of pitching angle of incidence on the dynamic stall behavior of a symmetric airfoil

This study presents the influence of pitch angle of an airfoil on its near-field vortex structure as well as the aerodynamic loads during a dynamic stall process. Dynamic stall behavior in a sinusoidally pitching airfoil is usually analyzed at low to medium reduced frequencies and with the maximum angle of attack of the airfoil not exceeding 25°. In this work, we study dynamic stall of a symmetric airfoil at medium to high reduced frequencies even as the maximum angle of attack goes from 25° to 45°. The evolution and growth of the laminar separation bubble, also known as a dynamic stall vortex, at the leading edge and the trailing edge are studied as the pitch cycle goes from the minimum to the maximum angle of attack. The effect of reduced frequencies on the vortex structure as well as the aerodynamic load coefficients is investigated. The reduced frequency is shown to be a bifurcation parameter triggering period doubling behavior. However, the bifurcation pattern is dependent on the variation of the pitch angle of incidence of the airfoil.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.