Bifurcation and chaos of airfoil with multiple strong nonlinearities

The bifurcation and chaos phenomena of two-dimensional airfoils with multiple strong nonlinearities are investigated.First,the strongly nonlinear square and cubic plunging and pitching stiffness terms are considered in the airfoil motion equations,and the fourth-order Runge-Kutta simulation method is used to obtain the numerical solutions to the equations.Then,a post-processing program is developed to calculate the physical parameters such as the amplitude and the frequency based on the discrete numerical solutions.With these parameters,the transition of the airfoil motion from balance,period,and period-doubling bifurcations to chaos is emphatically analyzed.Finally,the critical points of the period-doubling bifurcations and chaos are predicted using the Feigenbaum constant and the first two bifurcation critical values.It is shown that the numerical simulation method with post-processing and the prediction procedure are capable of simulating and predicting the bifurcation and chaos of airfoils with multiple strong nonlinearities.     

Nonlinear aeroelastic analysis of an airfoil-store system with a freeplay by precise integration method

The aeroelastic system of an airfoil-store configuration with a pitch freeplay is investigated using the precise integration method (PIM). According to the piecewise feature, the system is divided into three linear sub-systems. The sub-systems are separated by switching points related to the freeplay nonlinearity. The PIM is then employed to solve the sub-systems one by one. During the solution procedures, one challenge arises when determining the vibration state passing the switching points. A predictor-corrector algorithm is proposed based on the PIM to tackle this computational obstacle. Compared with exact solutions, the PIM can provide solutions to the precision in the order of magnitude of 10⁻¹². Given the same step length, the PIM results are much more accurate than those of the Runge–Kutta (RK) method. Moreover, the RK method might falsely track limit cycle oscillations (LCOs), bifurcation charts or chaotic attractors; even the step length is chosen much smaller than that for the PIM. Bifurcations and LCOs are obtained and analyzed by the PIM in detail. Interestingly, it is found that multiple LCOs and chaotic attractors can exist simultaneously. With this magnitude of precision and efficiency, the PIM could become a solution technique with excellent potential for piecewise nonlinear aeroelastic systems.     

Equivalent damping of aeroelastic system of an airfoil with cubic stiffness

This paper presents a mathematical study on the subsonic aerodynamics acting on an airfoil with a cubic stiffness. One portion of aerodynamics is assumed as and replaced by an equivalent damping. Using the harmonic balance method, an equivalent system is deduced and studied by a numerical integration method. Numerical examples show the validity and feasibility of the proposed mathematical treatment of the aerodynamics. It reveals that the unsteady aerodynamics acting on the airfoil can not only affect the pitch stiffness but also result in additional damping.     

Nonlinear aeroelastic analysis of a damped elastica-aerofoil system

Nonlinear Dynamics - This work formulates a comprehensive model of a nonlinear aeroelastic system developed for the analysis of complex aeroelastic phenomena related to structural and aerodynamic...     

Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem

A global stability and bifurcation analysis of the transverse galloping of a square section beam in a normal steady flow has been implemented. The model is an ordinary differential equation with polynomial damping nonlinearity. Six methods are used to predict bifurcation, the amplitudes and periods of the ensuing Limit Cycle Oscillations: (i) Cell Mapping, (ii) Harmonic Balance, (iii) Higher Order Harmonic Balance, (iv) Centre Manifold linearization, (v) Normal Form and (vi) numerical continuation. The resulting stability predictions are compared with each other and with results obtained from numerical integration. The advantages and disadvantages of each technique are discussed. It is shown that, despite the simplicity of the system, only two of the methods succeed in predicting its full response spectrum. These are Higher Order Harmonic Balance and numerical continuation.     

Analysis of an electromechanical energy harvester system with geometric and ferroresonant nonlinearities

Enhancing the performance of vibrating energy harvesting systems has been the backbone of several research contributions for the last few years, and it is considered in this paper. Specifically, an electromechanical energy harvester is analyzed, and the effects of geometric and ferroresonant nonlinearities on the electric power are discussed. The geometric nonlinearity includes the small- and high-order terms in Euler internal force while the ferroresonant nonlinearity is included by assuming different levels of saturation in the circuit. Our results reveal regions in the parameter space where nonlinear stiffness is better than linear stiffness and vice versa. Similarly, increasing the saturation parameter can be used to enhance the electric power.     

Bifurcation of non-negative solutions for an elliptic system

In the paper, we consider a nonlinear elliptic system coming from the predator-prey model with diffusion. Predator growth-rate is treated as bifurcation parameter. The range of parameter is found for which there exists nontrivial solution via the theory of bifurcation from infinity, local bifurcation and global bifurcation.     

Bifurcation analysis of an arch structure with parametric and forced excitation

The subharmonic bifurcation and universal unfolding problems are discussed for an arch structure with parametric and forced excitation in this paper. The amplitude–frequency curve and some dynamical behavior have been shown for this class of problems by Liu et al. Here, by means of singularity theory, in the case of strict 1:2 internal resonance, the bifurcation behavior of the amplitude with respect to a parameter (which is related to the amplitude of the live load imposed on the arch structures) is studied. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, 20 forms of two parameter unfoldings with some constraints are studied. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper present some new dynamic buckling patterns and abundant bifurcation phenomena.     

Bifurcation analysis for nonlinear multi-degree-of-freedom rotor system with liquid-film lubricated bearings

The oil-film oscillation in a large rotating machinery is a complex high-dimensional nonlinear problem. In this paper, a high pressure rotor of an aero engine with a pair of liquid-film lubricated bearings is modeled as a twenty-two-degree-of-freedom nonlinear system by the Lagrange method. This high-dimensional nonlinear system can be reduced to a two-degree-of-freedom system preserving the oil-film oscillation property by introducing the modified proper orthogonal decomposition (POD) method. The efficiency of the method is shown by numerical simulations for both the original and reduced systems. The Chen-Longford (C-L) method is introduced to get the dynamical behaviors of the reduced system that reflect the natural property of the oil-film oscillation.     

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.     

Bifurcation analysis of a linear Hamiltonian system with two kinds of impulsive control

In this paper, the dynamical behavior of a linear Hamiltonian system under two kinds of impulsive control is discussed by means of both theoretical and numerical ways. The existence and stability of the periodic solution are investigated. Moreover, the conditions of existence for a Neimark?CSacker bifurcation are derived by using a discrete map. Numerical results for phase portraits, periodic solutions, and bifurcation diagrams are in good agreement with the theoretical analysis.     

Bifurcation and stability analysis of an anaerobic digestion model

This paper presents the dynamic behaviour of the anaerobic digestion process, based on a simplified model. The hydraulic, biological and physicochemical processes such as those which underpin anaerobic digestion have more than one stable stationary solution and they compete with each other. Further, the attractive domains of the stable solutions vary with the key parameters. Thus, some initial transient process moving toward one stable solution could suddenly move towards another solution, at which a so-call catastrophe takes places (e.g. washout). The paper systematically analyses the stationary solutions with their associated stability, which provides insight and guidance for anaerobic digestion reactor design, operation and control.     

Bifurcation of multi-freedom gear system with spalling defect

This study focuses on the bifurcation characteristics of the four degree-of-freedom gear system with local spalling defect to explore the spalling nonlinear dynamic mechanism. The dynamic model of the gear system with spalling defect, time-variant mesh stiffness, and nonlinear clearance is established to investigate the effect of spalling defect on mesh stiffness and dynamic bifurcation. The primary resonance and internal resonance responses of the spalling model are analyzed by the averaging method, and the bifurcation characteristics with the evolvement of spall and internal excitation are studied by employing the singularity theory for the two-state variable system, which reveal the different bifurcation characteristics caused by the spalling defect. The results obtained herein can provide a theoretical basis to spalling fault diagnosis of gearbox.     

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

Nonlinear dynamic behaviors of an aeroelastic airfoil with free-play in transonic air flow are studied. The aeroelastic response is obtained by using time-marching approach with computational fluid dynamics (CFD) and reduced order model (ROM) techniques. Several standardized tests of transonic flutter are presented to validate numerical approaches. It is found that in time-marching approach with CFD technique, the time-step size has a significant effect on the calculated aeroelastic response, especially for cases considering both structural and aerodynamic nonlinearities. The nonlinear dynamic behavior for the present model in transonic air flow is greatly different from that in subsonic regime where only simple harmonic oscillations are observed. Major features of the responses in transonic air flow at different flow speeds can be summarized as follows. The aeroelastic responses with the amplitude near the free-play are dominated by single degree of freedom flutter mechanism, and snap-though phenomenon can be observed when the air speed is low. The bifurcation diagram can be captured by using ROM technique, and it is observed that the route to chaos for the present model is via period-doubling, which is essentially caused by the free-play nonlinearity. When the flow speed approaches the linear flutter speed, the aeroelastic system vibrates with large amplitude, which is dominated by the aerodynamic nonlinearity. Effects of boundary layer and airfoil profile on the nonlinear responses of the aeroelastic system are also discussed.     

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.     

Characteristic nonlinear system identification of local attachments with clearance nonlinearities

Nonlinear Dynamics - In this paper, a SEIR epidemic model for the COVID-19 is built according to some general control strategies, such as hospital, quarantine and external input. Based on the data...