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.     

The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback

We investigate the parametric resonance of a van der Pol oscillator under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow-flow equations on the amplitude and phase ofthe oscillator. Their fixed points correspond to a periodic motion forthe starting system and we show parametric excitation-response andfrequency-response curves. We analyze the effect of time delay andfeedback gains from the viewpoint of vibration control and use energyconsiderations to study the existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-periodmodulated motion for the van der Pol oscillator. Analytical results areverified with numerical simulations. In order to exclude the possibilityof quasi-periodic motion and to reduce the amplitude peak of theparametric resonance, we find the appropriate choices for the feedbackgains and the time delay.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

The Response of a Rayleigh–Liénard Oscillator to a Fundamental Resonance

A Rayleigh–Liénard oscillator excited by a fundamentalresonance is investigated by using an asymptotic perturbation method based on Fourier expansion and time rescaling. Two first-order nonlinear ordinarydifferential equations governing the modulation of the amplitude andthe phase of solutions are derived. These equations are used todetermine steady-state responses and their stability. Excitationamplitude-response and frequency-response curves are shown and checkedby numerical integration. Dulac's criterion, the Poincaré–Bendixsontheorem, and energy considerations are used in order to study the existenceand characteristics of limit cycles of the two modulation equations. Alimit cycle corresponds to a modulated motion for the Rayleigh–Liénardoscillator. For small excitation amplitude, the analytical results arein excellent agreement with the numerical solutions. In certain caseswhen the excitation amplitude is very low, an approximate analyticsolution corresponding to a modulated motion can be obtained andnumerically checked. Moreover, if the excitation amplitude is increased,an infinite-period bifurcation occurs because the modulation periodlengthens and becomes infinite, while the modulation amplitude remainsfinite and suddenly the attractor settles down into a periodic motion.     

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.     

Existence and nonexistence of periodic solutions for the Liénard and related equations

The existence of periodic solutions of Liénard type equations is converted into the existence of special fixed point problems with auxiliary conditions. A general method for exact calculation of the limit cycles is given, and the corresponding numerical iterative procedure is carried out in significant cases, with comparison to standard Runge-Kutta numerical integration. On the basis of the general theory, criteria for the existence of limit cycles are given and tested in particular cases.     

Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback

The paper presents analytical and numerical studies of the primary resonance and the 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay. By using the method of multiple scales, the first order approximations of the resonances are derived and the effect of time delay on the resonances is analyzed. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. In order to numerically solve the problem of history dependence prior to the start of excitation, the concepts of the Poincaré section and fixed points are generalized. Then, a modified shooting scheme associated with the path following technique is proposed to locate the periodic motion of the delayed system. The numerical results show the efficacy of the first order approximations of the resonances.     

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.     

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.

     

Three-Period Quasi-Periodic Solutions in the Self-Excited Quasi-Periodic Mathieu Oscillator

In this work, we investigate three-period quasi-periodic (QP) oscillations in the vicinity of 2:2:1 resonance in a self-excited QP Mathieu equation using perturbation method. Two successive averaging are performed to reduce the original QP equation to an autonomous amplitude and phase system describing the modulation of the slow flow dynamic. Approximation of three-period QP solution is obtained via the study of limit cycle of the reduced autonomous system. The efficiency of the method is illustrated by comparison between analytical approximations and numerical integration. The double reduction procedure, applied in previous works to construct two-period QP solution, can be implemented to approximate excplicit analytical three-period QP solutions.     

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.     

基于压电纤维复合材料的旋转叶片主动控制

旋转叶片结构的振动失效占据了航空发动机整机故障的相当比重. 发展针对旋转叶片结构的减振技术对于减轻叶片重量, 提升叶片性能, 延长叶片寿命具有重要意义. 通过引入压电纤维复合材料(macro fiber composite, MFC)传感器和作动器, 研究预变形旋转叶片2:1内共振的主动控制. 建立考虑时滞效应的旋转叶片比例微分闭环控制系统运动方程. 通过摄动分析推导出受控叶片的演化方程, 并结合延拓法揭示速度增益、位移增益、时滞量等系统参数对受控系统稳态响应及稳定性的影响规律. 理论研究结果与数值结果得到相互验证. 研究发现时滞量对系统稳定性影响显著, 当时滞超过某临界值时, 演化方程原有的平衡点失稳, 闭环受控系统将缓慢进入一个大振幅的周期运动, 从而丧失控制效果. 位移增益存在一个范围使得系统出现多值稳态响应, 进而破坏了增益平面内系统稳定区和非稳定区域的直线边界. 不恰当的速度增益和位移增益会给受控系统引入新的共振. 研究结果为叶片结构的减振提供了理论基础.      

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.     

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.     

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

We investigate the resonant double Hopf bifurcation in a diffusive complex Ginzburg–Landau model with delayed feedback and phase shift. The conditions for the existence of resonant double Hopf bifurcation are obtained by analyzing the roots’ distribution of the characteristic equation, and a general formula to determine the bifurcation point is given. For the cases of 1:2 and 1:3 resonance, we choose time delay, feedback strength and phase shift as bifurcation parameters and derive the normal forms which are proved to be the same as those in non-resonant cases. The impact of cubic terms on the unfolding types is discussed after obtaining the normal form till 3rd order. By fixing phase shift, we find that varying time delay and feedback strength simultaneously can induce the coexistence of two different periodic solutions, the existence of quasi-periodic solutions and strange attractors. Also, the effects on the existence of transient quasi-periodic solution exerted by the phase shift are illustrated.

     

Phase space structure of spinning disks

Using a Hamiltonian formalism and a sequence of canonical transformations, we show that the ordinary differential equations associated with the forced oscillations of rotating circular disks admit the first integral of motion. This reduces the phase space dimension of the governing equations from five to three. The phase space flows of the reduced system are then visualized using Poincaré maps. Our results show that single mode oscillations of rotating disks are subject to chaotic behavior through the emergence of higher-order resonant islands that surround fundamental periodic cycles. We extend our new formalism to imperfect disks and construct adiabatic invariants near to and far from resonances. For low-speed imperfect disks, we find a new kind of bifurcations of the phase space flows as the system parameters vary. We study the effect of structural damping using Hamilton's principle for non-conservative systems and reveal the existence of asymptotically stable limit cycles for the damped system near the 1:1 resonance. We show that a low-speed disk is eventually flattened due to damping effect.     

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.     

Vibration control by nonlocal feedback and jerk dynamics

Jerk dynamics is used for a new method for the suppression of self-excited vibrations in nonlinear oscillators. Two cases are considered, the van der Pol equation and nonlinear oscillator with quadratic and cubic nonlinearities. A nonlocal control force is introduced in such a way to obtain a third order nonlinear differential equation (jerk dynamics). Using the asymptotic perturbation method, two slow flow equations on the amplitude and phase of the response are obtained, and subsequently the performance of the control strategy is investigated. The feedback gains are connected with the stability and response of the system under control. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the self-excitations. Numerical simulation confirms the validity of the new method.     

Control of the nonlinear oscillator bifurcation under a superharmonic resonance

A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddle-node bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddle-node bifurcations, while the nonlinear feedback gain can eliminate saddle-node bifurcations. Feasible ways of further research of saddle-node bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steady-state response.     

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.