Marginal stability boundaries for some driven, damped, non-linear oscillators

A procedure is developed for averaging the differential equations for certain non-linear oscillators which are damped and externally driven. The procedure makes possible the obtaining of marginal stability boundaries for bifurcations in parameter space and is useful for systems with unperturbed solutions involving Jacobi elliptic functions. Specific cases of a driven, damped pendulum, an anharmonie oscillator, a Duffing oscillator, and a non-linear Helmholtz oscillator are examined.     

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth ( α > 0) and discontinuous ( α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.     

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 and Chaos in the Duffing Oscillator with a PID Controller

We discuss in this paper the bifurcation, stability and chaos of the non-linear Duffing oscillator with a PID controller. Hopf bifurcation can occur and we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system and so is fractal basin boundary for soft spring system. The Melnikov method is used to obtain the criterion of fractal basin boundary.     

Duffing 系统的主-亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.      

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     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

Reconstructing the transient,dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

Analysis of Duffing oscillator with time-delayed fractional-order PID controller

The free vibration of Duffing oscillator with time-delayed fractional-order Proportional-Integral-Derivative (FOPID) controller based on displacement feedback is studied. The second-order approximate analytical solution is obtained by KBM asymptotic method. The effects of the parameters in FOPID controller on the dynamical properties are characterized by some equivalent parameters. The correctness of the approximate analytical results is verified by the numerical results. The effects of the time-delayed FOPID controller with displacement feedback on control performances of Duffing oscillator are analyzed in detail by time response, and the stability conditions of zero solution and periodic motions are also presented. Finally, the control performances on Duffing oscillator with large damping are further analyzed. And the results show that one could take the advantage of time delay, when the parameters of time-delayed FOPID controller are chosen reasonably.     

A nonlinear oscillatory system subjected to driving forces of elliptic type

The aim of this paper is to show how Jacobi elliptic functions in combination with the averaging and the harmonic balance methods can be applied to obtain the approximate solution of two coupled, ordinary differential equations having a spring with cubic nonlinearity and subjected to driving forces of elliptic type. By an appropriate choice of the system parameter values, it is possible to show that our derived solution represents the exact steady-state solution of the undamped Duffing equation with driving force of elliptic type. At the end of this work, we also demonstrate the validity of our derived solution by comparing the amplitude–time response curves with those of the numerical integration solutions.     

Effects of a crack on the stability of a non-linear rotor system

The stability of a rotor system presenting a transverse breathing crack is studied by considering the effects of crack depth, crack location and the shaft's rotational speed. The harmonic balance method, in combination with a path-following continuation procedure, is used to calculate the periodic response of a non-linear model of a cracked rotor system. The stability of the rotor's periodic movements is studied in the frequency domain by introducing the effects of a perturbation on the periodic solution for the cracked rotor system.It is shown that the areas of instability increase considerably when the crack deepens, and that the crack's position and depth are the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. The effects of some other system parameters (including the disk position and the stiffness of the supports) on the dynamic stability of the non-linear periodic response of the cracked rotor system are also investigated.     

Modified Parker-Sochacki method for solving nonlinear oscillators

A new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.     

Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator

An electronic model of Duffing oscillator with a characteristic memristive nonlinear element is proposed instead of the classical cubic nonlinearity. The memristive Duffing oscillator circuit system is mathematically modeled, and the stability analysis presents the evolution of the proposed system. The dynamical behavior of this circuit is investigated through numerical simulations, statistical analysis, and real-time hardware experiments, which have been carried out under the external periodic force. The chaotic dynamics of the circuit is studied by means of phase diagram. It is found that the proposed circuit system shows complex behaviors, like bifurcations and chaos, three tori, transient chaos, and intermittency for a certain range of circuit parameters. The observed phenomena and scenario are illustrated in detail through experimental and numerical studies of memristive Duffing oscillator circuit. The existence of regular and chaotic behaviors is also verified by using 0–1 test measurements. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio. The numerically observed results are confirmed from the laboratory experiment.     

Nonlinear dynamics and performance analysis of modified snap-through vibration energy harvester with time-varying potential function

Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications. Enhancing the performance of a vibration energy harvester(VEH) incorporating nonlinear techniques, for example, the snap-through VEH with geometric non-linearity, has gained attention in recent years. A conventional snap-through VEH is a bi-stable system with a time-invariant potential function, which was investigated extensively in the past. In this work, a modified snap-through VEH wit...     

Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y-axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

分数阶达芬振子的超谐与亚谐联合共振

研究了含分数阶微分项的达芬(Duffing)振子的超谐与亚谐联合共振.采用平均法得到了系统的一阶近似解析解,提出了超、亚谐联合共振时等效线性阻尼和等效线性刚度的概念.建立了联合共振定常解幅频曲线的解析表达式,并对联合共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,证明了求解过程及近似解析解的正确性.然后,将等效线性阻尼和等效线性刚度的概念与传统整数阶系统进行比较,证明分数阶微分项不仅起着阻尼的作用同时还起着刚度的作用.最后,通过数值仿真研究了不同的分数阶微分项系数和阶次对联合共振幅频曲线多值性和跳跃现象的影响,并与单一频率下超谐共振或亚谐共振进行了对比.研究发现,分数阶微分项系数与阶次不仅影响着系统的响应幅值、共振频率,同时还对系统的周期解个数、发生区域面积、发生先后等有重要影响.并且,在不同的基本参数下该系统分别表现出单独超谐共振、单独亚谐共振以及超谐共振和亚谐共振同时存在的现象.这些结果对系统动力学特性的研究具有重要意义.     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.