Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance

In this paper, we investigate nonlinear dynamical responses of two-degree-of-freedom airfoil (TDOFA) models driven by harmonic excitation under uncertain disturbance. Firstly, based on the deterministic airfoil models under the harmonic excitation, we introduce stochastic TDOFA models with the uncertain disturbance as Gaussian white noise. Subsequently, we consider the amplitude–frequency characteristic of deterministic airfoil models by the averaging method, and also the stochastic averaging method is applied to obtain the mean-square response of given stochastic TDOFA systems analytically. Then, we carry out numerical simulations to verify the effectiveness of the obtained analytic solution and the influence of harmonic force on the system response is studied. Finally, stochastic jump and bifurcation can be found through the random responses of system, and probability density function and time history diagrams can be obtained via Monte Carlo simulations directly to observe the stochastic jump and bifurcation. The results show that noise can induce the occurrence of stochastic jump and bifurcation, which will have a significant impact on the safety of aircraft.     

Bifurcation and Chaos Analysis of Stochastic Duffing System Under Harmonic Excitations

The Chebyshev polynomial approximation is applied to the dynamic response problem of a stochastic Duffing system with bounded random parameters, subject to harmonic excitations. The stochastic Duffing system is first reduced into an equivalent deterministic non-linear one for substitution. Then basic non-linear phenomena, such as stochastic saddle-node bifurcation, stochastic symmetry-breaking bifurcation, stochastic period-doubling bifurcation, coexistence of different kinds of steady-state stochastic responses, and stochastic chaos, are studied by numerical simulations. The main feature of stochastic chaos is explored. The suggested method provides a new approach to stochastic dynamic response problems of some dissipative stochastic systems with polynomial non-linearity.     

Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation

The Laguerre polynomial approximation method is applied to study the stochastic period-doubling bifurcation of a double-well stochastic Duffing system with a random parameter of exponential probability density function subjected to a harmonic excitation. First, the stochastic Duffing system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then nonlinear dynamical behavior about stochastic period-doubling bifurcation can be fully explored. Numerical simulations show that similar to the conventional period-doubling phenomenon in the deterministic Duffing system, stochastic period-doubling bifurcation may also occur in the stochastic Duffing system, but with its own stochastic modifications. Also, unlike the deterministic case, in the stochastic case the intensity of the random parameter should also be taken as a new bifurcation parameter in addition to the conventional bifurcation parameters, i.e. the amplitude and the frequency of harmonic excitation.     

Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations

The stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping of order α (0<α<1) under combined harmonic and white noise excitations are studied. First, the system state is approximately represented by two-dimensional time-homogeneous diffusive Markov process of amplitude and phase difference using the stochastic averaging method. Then, the method of reduced Fokker–Plank–Kolmogorov (FPK) equation is used to predict the stationary response of the original system. The phenomenon of stochastic jump and bifurcation as the fractional orders' change is examined.     

Jump and bifurcation of Duffing oscillator under narrow-band excitation

The jump and bifurcation of Duffing oscillator with hardening spring subject to narrow-band random excitation are systematically and comprehensively examined. It is shown that, in a certain domain of the space of the oscillator and excitation parameters, there are two types of more probable motions in the stationary response of the Duffing oscillator and jumps may occur. The jump is a transition of the response from one more probable motion to another or vise versa. Outside the domain the stationary response is either nearly Gaussian or like a diffused limit cycle. As the parameters change across the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation. It is also shown that, for a set of specified parameters, the statistics are unique and they are independent of initial condition. It is pointed out that some previous results and interpretations on this problem are incorrect. The project supported by National Natural Science Foundation of China     

含噪双稳杜芬振子矩方程的分岔与随机共振

研究了含噪声的双稳杜芬振子矩方程的分岔与随机共振的关系，并根据它们的关系, 从另 一个角度揭示了随机共振发生的机制. 首先在It?方程的基础上，导出了双稳杜芬振子在白噪声和弱周期信号作用下的矩方程，其次以噪声强度 为分岔参数分析了矩方程的分岔特性，再次分析了矩方程的分岔与双稳杜芬振子随机共振 之间的关系，最后根据该对应关系从另一种观点提出了双稳杜芬振子随机共振的机制，该 机制是由于以噪声强度为分岔参数的矩方程发生了分岔，而分岔使得原系统响应均值的能量分布发生了转移，使能 量向频率等于输入信号频率的分量处集中，使得弱信号得到了放大，随机共振发生了.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Bifurcation Control of Parametrically Excited Duffing System by a Combined Linear-Plus-Nonlinear Feedback Control

For a parametrically excited Duffing system we propose a bifurcation control method in order to stabilize the trivial steady state in the frequency response and in order to eliminate jump in the force response, by employing a combined linear-plus-nonlinear feedback control. Because the bifurcation of the system is characterized by its modulation equations, we first determine the order of the feedback gain so that the feedback modifies the modulation equations. By theoretically analyzing the modified modulation equations, we show that the unstable region of the trivial steady state can be shifted and the nonlinear character can be changed, by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the suppression of the discontinuous bifurcation due to the change of the nonlinear character allows the elimination of the jump in the quasistationary force response. Furthermore, by performing numerical simulations, and by comparing the responses of the uncontrolled system and the controlled one, we clarify that the proposed bifurcation control is available for the stabilization of the trivial steady state in the frequency response and for the reduction of the jump in the nonstationary force response.     

Nonlinear dynamics of a $$\varvec{\phi ^6}-$$ modified Duffing oscillator: resonant oscillations and transition to chaos

The nonlinear dynamics of a hybrid Rayleigh–Van der Pol–Duffing oscillator includes pure and impure quadratic damping are investigated. The multiple timescales method is used to study exhaustively various resonances states. It is noticed that the system presents nine resonances states. The frequency response curves of quintic, third and second superharmonic, and subharmonic resonances states are obtained. Bistability, hysteresis, and jump phenomenon are also obtained. It is pointed out that these resonance phenomena are strongly related to the nonlinear cubic and quadratic damping and to the external force. The numerical simulations are used to make bifurcation sequences displayed by the model for each type of oscillatory. It is noticed that the pure quadratic, impure cubic damping, and external excitation affect regular and chaotic states.     

谐和与随机噪声联合作用下非线性系统的响应

研究了Duffing振子在谐和与随噪声联合激励下的响应和稳应性问题。用谐波平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统晌应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象。数值模拟表明本文提出的方法是有效的。     

Response of Duffing system with delayed feedback control under bounded noise excitation

This paper presents a procedure for predicting the response of Duffing system with time-delayed feedback control under bounded noise excitation by using stochastic averaging method. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged It? stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker?CPlank?CKolmogorov equation associated with the averaged It? equations. It is shown that the time delay in feedback control will deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise

A global analysis of stochastic bifurcation in a special kind of Duffing system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell flow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. Through numerical simulations the evolution of the Poincare cell maps of the random flow against the variation of noise intensity is explored systematically. Our study reveals that as a powerful tool for global analysis, the generalized cell mapping method using digraph is applicable not only to deterministic bifurcation, but also to stochastic bifurcation as well. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.     

Response of nonlinear oscillator under narrow-band random excitation

IntroductionThestudyoftheresponseofnonlinearsystemstonarrow_bandrandomexcitationofconsiderableimportance.Forexample ,theexcitationofsecondarysystemwouldbeanarrow_bandrandomprocessiftheprimarysystemcouldbemodeledasasingle_degree_of_freedomsystemwithlightdampingsubjecttowide_bandexcitation .Inthetheoryofnonlinearrandomvibration ,mostresultsobtainedsofarareattributedtotheresponseofnonlinearoscillatorstowide_bandrandomexcitation .Incomparison ,resultsontheeffectofnarrow_bandexcitationonnonlinearos…     

Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method

The smooth and discontinuous oscillator with fractional derivative damping under combined harmonic and random excitations is investigated in this paper. The short memory principle is introduced so that the evolution process of the oscillator with fractional derivative damping can be described by the Markov chain. Then the stochastic generalized cell mapping method is used to obtain the steady-state probability density functions of the response. The stochastic response and bifurcation of the oscillator with fractional derivative damping are discussed in detail. We found that both the smoothness parameter, the noise intensity, the amplitude and frequency of the harmonic force can induce the occurrence of stochastic P-bifurcation in the system. Monte Carlo simulation verifies the effectiveness of the method we adopt in the paper.     

Quasi-harmonic analysis of the behaviour of a hardening Duffing oscillator subjected to filtered white noise

The behaviour of a hardening Duffing oscillator subjected to narrow band random excitation is examined. The influence of possible jumps, between competing states, on the probability distribution of the response amplitude is addressed. A quasi-harmonic approximation of system behaviour is adopted which is capable of reproducing the observed concave shape of probability functions and compares well with predictions obtained via stochastic averaging techniques and with digital simulations.     

Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state

Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with time-delay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application.

     

A functional series expansion method for response analysis of non-linear systems subjected to random excitations

A Wiener-Hermite functional series expansion method for response analysis of nonlinear systems under random excitations is presented. The formal procedure for the derivation of the deterministic equations governing the Wiener-Hermite kernel functions is described. A singleterm expansion is used to analyze the non-stationary responses of a damped Duffing oscillator subjected to several modulated white noises. An iterative procedure for evaluating the kernel functions and the mean-square responses of the Duffing oscillator is developed. For several values of non-linearity strength and different damping coefficients the non-stationary variances and autocorrelations of the response are obtained. The results are compared with those found by the digital simulations and the existing methods.     

An approach to the problem of closure in non-linear stochastic mechanics

An approach combining the method of moment equations and the statistical linearization technique is proposed for analysis of the response of non-linear mechanical systems to random excitation. The adaptive statistical linearization procedure is developed for obtaining a more accurate mean square of responses. For these, a Duffing oscillator and an oscillator with cubic non-linear damping subject to white noise excitation are considered. It is shown that the adaptive statistical linearization proposed yields good accurate results for both weak and strong non-linear stochastic systems.Presented at the First European Solid Mechanics Conference, September 9–13, 1991. Munich, Germany     

Energy Based Stochastic Estimation of Nonlinear Oscillators with Parametric Random Excitation

A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.     

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.