Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback

We investigate the primary resonance of an externally excited van der Pol oscillator under state feedback control with a time delay. By means of the asymptotic perturbation method, two slow-flow equations on the amplitude and phase of the oscillator are obtained and external excitation-response and frequency-response curves are shown. We discuss how vibration control and high amplitude response suppression can be performed with appropriate time delay and feedback gains. Moreover, energy considerations are used in order to investigate existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-period modulated motion for the van der Pol oscillator. We demonstrate that appropriate choices for the feedback gains and the time delay can exclude the possibility of modulated motion and reduce the amplitude peak of the primary resonance. Analytical results are verified with numerical simulations.     

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.     

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.     

Applications of the integral equation method to delay differential equations

In this paper, we compare two approaches for determining the amplitude equations; namely, the integral equation method and the method of multiple scales. To describe and compare the methods, we consider three examples: the parametric resonance of a Van der Pol oscillator under state feedback control with a time delay, the primary resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay, and the primary resonance together with 1:1 internal resonance of a two degree-of-freedom model. Using the integral equation method and the method of multiple scales, the amplitude equations are obtained. The stability of the periodic solution is examined by using the Floquet theorem together with the Routh–Hurwitz criterion (without time delay) and the Nyquist criterion (with time delay). By comparison with the solution obtained by the numerical integration, we find that the accuracy of the integral equation method is much better.     

Fast parametrically excited van der Pol oscillator with time delay state feedback

We investigate the effect of a fast vertical parametric excitation on self-excited vibrations in a delayed van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic in the vicinity of the trivial equilibrium. Then, we apply the multiple scales method on this slow dynamic to derive a second-order slow flow system describing the modulation of slow dynamic. In particular we analyze the slow flow to obtain the effect of a fast excitation on the regions in parameter space where self-excited vibrations can be eliminated. We have shown that in the case where the time delay and the feedback gains are imposed, fast vertical parametric excitation can be an alternative to suppress undesirable self-excited vibrations in a delayed van der Pol oscillator.     

Vibration control for parametrically excited Liénard systems

We apply a new vibration control method for time delay non-linear oscillators to the principal resonance of a parametrically excited Liénard system under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase. Their fixed points correspond to limit cycles for the Liénard system. Vibration control and high-amplitude response suppression can be performed with appropriate time delay and feedback gains. Using energy considerations, we investigate existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a two-period quasi-periodic modulated motion for the starting system and in order to reduce the amplitude peak of the parametric resonance and to exclude the existence of two-period quasi-periodic motion, we find the appropriate choices for the feedback gains and the time delay.     

Energy harvesting from quasi-periodic vibrations using electromagnetic coupling with delay

In this paper, we study quasi-periodic vibrational energy harvesting in a delayed self-excited oscillator with a delayed electromagnetic coupling. The energy harvester system consists in a delayed van der Pol oscillator with delay amplitude modulation coupled to a delayed electromagnetic coupling mechanism. It is assumed that time delay is inherently present in the mechanical subsystem of the harvester, while it is introduced in the electrical circuit to control and optimize the output power of the system. A double-step perturbation method is performed near a delay parametric resonance to approximate the quasi-periodic solutions of the harvester which are used to extract the quasi-periodic vibration-based power. The influence of the time delay introduced in the electromagnetic subsystem on the performance of the quasi-periodic vibration-based energy harvesting is examined. In particular, it is shown that for appropriate values of amplitudes and frequency of time delay the maximum output power of the harvester is not necessarily accompanied by the maximum amplitude of system response.     

Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay

This paper investigates quasi-periodic vibration-based energy harvesting in a delayed nonlinear MEMS device consisting of a delayed Mathieu–van der Pol–Duffing type oscillator coupled to a delayed piezoelectric coupling mechanism. We use the multiple scales method to approximate the quasi-periodic response and the related power output near the principal parametric resonance. The effect of time delay on the energy harvesting performance is studied. It is shown that for appropriate combination of time delay parameters, there exists an optimum range of excitation frequency beyond the resonance where quasi-periodic vibration-based energy harvesting is maximum. Numerical simulations are performed to confirm the analytical predictions.     

Parametric Excitation for Two Internally Resonant van der Pol Oscillators

The response of a system of two nonlinearly coupled van der Poloscillators to a principal parametric excitation in the presence ofone-to-one internal resonance is investigated. The asymptoticperturbation method is applied to derive the slow flow equationsgoverning the modulation of the amplitudes and the phases of the twooscillators. These equations are used to determine steady-stateresponses, corresponding to a periodic motion for the starting system(synchronisation), and parametric excitation-response andfrequency-response curves. Energy considerations are used to studyexistence and characteristics of limit cycles of the slow flowequations. A limit cycle corresponds to a two-period amplitude- andphase-modulated motion for the van der Pol oscillators. Two-periodmodulated motion is also possible for very low values of the parametricexcitation and an approximate analytic solution is constructed for thiscase. If the parametric excitation increases, the oscillation period ofthe modulations becomes infinite and an infinite-period bifurcationsoccur. Analytical results are checked with numerical simulations.     

Dynamics of two delay coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol oscillators with delayed position and velocity coupling is studied by the method of averaging together with truncation of Taylor expansions. According to the slow-flow equations, the dynamics of 1:1 internal resonance is more complex than that of non-1:1 internal resonance. For 1:1 internal resonance, the stability and the number of periodic solutions vary with different time delay for given coupling coefficients. The condition necessary for saddle-node and Hopf bifurcations for symmetric modes, namely in-phase and out-of-phase modes, are determined. The numerical results, obtained from direct integration of the original equation, are found to be in good agreement with analytical predictions.     

Arbitrary amplitude periodic solutions for parametrically excited systems with time delay

Periodic solutions for parametrically excited system under state feedback control with a time delay are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the parametric resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the starting system. In the system without control, periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.     

Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control

Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones.     

Nonlinear dynamics of self-, parametric,and externally excited oscillator with time delay: van der Pol versus Rayleigh models

The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.

     

Study of mixed-mode oscillations in a parametrically excited van der Pol system

This paper investigates the effects of slowly varying parametric excitation on the dynamics of van der Pol system. Periodic bifurcation delay behaviors are exhibited when the parametric excitation slowly passes through Hopf bifurcation value of the controlled van der Pol system. The first bifurcation delay behavior relies on initial conditions, while the bifurcation delay behaviors that follow the first one are immune to initial conditions. These bifurcation delay behaviors result in a hysteresis loop between the spiking attractor and the rest state, which is responsible for the generation of mixed-mode oscillations. Then an approximate calculation for the number of spikes in each cluster of repetitive spiking of mixed-mode oscillations is explored based on bifurcation delay behaviors. Theoretical results agree well with numerical simulations.     

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.     

Oscillators with a power-form restoring force and fractional derivative damping: Application of averaging

In this paper free oscillators with a power-form restoring force and with a fractional derivative damping term are considered. An analytical approach based on the averaging method is adjusted to derive analytical expressions for the amplitude and phase of oscillations. Effects of the fractional-order derivative on the amplitude and frequency of oscillations are discussed in several examples, including a generalized van der Pol oscillator, purely nonlinear oscillators and a linear oscillator.     

时滞反馈控制对弹簧摆动力学行为的影响

利用解析和数值方法,以弹簧摆为对象讨论了线性的时滞位移反馈控制对一类平方非线性系统动力学行为的影响。根据多尺度法得到了1:2内共振情况下一次近似解的慢变方程,基于此讨论了反馈控制参数对零解的稳定性和周期解振幅的影响。结果表明:耦合的反馈项在平均方程中并不出现。根据罗斯-霍尔维茨判据发现,没有反馈控制时该系统的零解总是不稳定的,而通过调整反馈增益或反馈时滞就可以很容易地使零解稳定。反馈时滞对周期解振幅的影响呈现周期性,反馈增益或时滞发生变化时,周期解振幅的变化会表现出鞍结分岔现象;同时基于MATLAB软件的数值计算结果验证了该理论分析的正确性。     

Bifurcations in a parametrically excited non-linear oscillator

A non-linear parametrically excited oscillator, that includes van der Pol as well as Duffing type non-linearities, is studied for its small non-linear motions using the method of averaging. The averaged equations, which form a dynamical system on the plane and depend on the linear damping and the detuning, are analyzed for their constant and periodic solutions. Bendixon's criterion is used to deduce the existence and the non-existence of limit cycle solutions for various values of the parameters. Then, using local bifurcation theory for “saddle-node”, pitchfork and “Hopf” bifurcations and some results from one and two parameter unfoldings of degenerate singularities, a partial bifurcation set is constructed. Since constant and periodic solutions of the averaged system correspond, respectively, to the periodic solutions and almost periodic or amplitude modulated motions of the original oscillator, the bifurcation set indicates some ways in which periodic solutions can become “entrained” or can break the entrainment for almost periodic oscillations.     

Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation

The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.     

水平地震时斜面水坝受到的非线性动水压力的全过程的解析解

本文给出在非线性Ｒａｙｌｅｉｇｈ阻尼作用下，由水平地震激发的水库内动水压力变化全过程的解析解：在激发初期其幅值较小，可用小参数法使控制方程线性化，从而求得各阶渐近控制方程的解析解；当动水压力的幅值增大到小参数展开不适用时，用ｖａｎ ｄｅｒｐｏｌ法求得非线性控制方程的渐近解，从而显示了动水压力的幅值从有序变化到发生突然跳跃的多值性的非线性本质。