2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator

The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance. Numerical simulations are performed to confirm the analytical results.     

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.     

Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

Nonlinear Dynamics - We analyze the effect of a fast harmonic excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1...     

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.     

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.     

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.     

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.     

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.     

切削加工中刀具颤振的迟滞反馈控制

从金属切削加工的实际情况出发,分析了刀具在切削加工中产生颤振的原因．从力学的角度解释了刀具径向非线性颤振的机理,说明这是一种自激振荡,可以通过施加线性迟滞反馈力加以控制．这种反馈力是位移信号的迟滞函数,可以用于抑制振幅,增加刀具的平稳性．数值仿真验证了理论结果．     

Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

     

反馈时滞对van der Pol振子张弛振荡的影响

研究反馈控制环节时滞对van derPol振子张弛振荡的影响. 首先, 通过稳定性切换分析, 得到了系统的慢变流形的稳定性和分岔点分布图, 结果表明, 当时滞大于某临界值时, 系统慢变流形的结构发生本质的变化.其次, 基于几何奇异摄动理论, 分析了慢变流形附近解轨线的形状, 发现时滞反馈会引起张弛振荡中的慢速运动过程中存在微幅振荡, 其中微幅振荡来自于内部层引起的振荡和Hopf分岔产生的振荡两个方面; 同时, 时滞对张弛振荡的周期也具有显著的影响. 实例分析表明理论分析结果与数值结果相吻合.      

Piezoelectric energy harvesting from concurrent vortex-induced vibrations and base excitations

We investigate the potential of using a piezoelectric energy harvester to concurrently harness energy from base excitations and vortex-induced vibrations. The harvester consists of a multilayered piezoelectric cantilever beam with a circular cylinder tip mass attached to its free end which is placed in a uniform air flow and subjected to direct harmonic excitations. We model the fluctuating lift coefficient by a van der Pol wake oscillator. The Euler–Lagrange principle and the Galerkin procedure are used to derive a nonlinear distributed-parameter model for a harvester under a combination of vibratory base excitations and vortex-induced vibrations. Linear and nonlinear analyses are performed to investigate the effects of the electrical load resistance, wind speed, and base acceleration on the coupled frequency, electromechanical damping, and performance of the harvester. It is demonstrated that, when the wind speed is in the pre- or post-synchronization regions, its associated electromechanical damping is increased and hence a reduction in the harvested power is obtained. When the wind speed is in the lock-in or synchronization region, the results show that there is a significant improvement in the level of the harvested power which can attain 150 % compared to using two separate harvesters. The results also show that an increase of the base acceleration results in a reduction in the vortex-induced vibrations effects, an increase of the difference between the resonant excitation frequency and the pull-out frequency, and a significant effects associated with the quenching phenomenon.     

Canard Explosion in Delay Differential Equations

We analyze canard explosions in delay differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow–fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of ‘classical’ canard explosions ends as a Bogdanov–Takens bifurcation occurs at the folds points of the S-shaped critical manifold.     

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.     

随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

Parametric excitations and lock-in of flexible hydrofoils in two-phase flows

This work introduces a reduced-order method to study the parametric excitations and lock-in of flexible hydrofoils caused by unsteady two-phase (cavitating) flow. The reduced-order method is based on a 1-DOF structural model coupled with a van der Pol wake oscillator with empirically derived relations for the variation in lift, cavity-length, and cavity-shedding frequency as a function of a non-dimensional cavitation parameter. The results are compared with several available data from both numerical simulations and experimental measurements. The frequency content of both the predicted and measured vibrations suggested that, in addition to the primary cavity-shedding frequency and the hydrofoil natural frequencies, unsteady two-phase flows may excite additional modulated frequencies due to time-varying fluid-added mass effects. The results show that these frequency modulations might cause the flexible hydrofoil to undergo higher-order resonances, as well as parametric resonances. While the maximum deformations for the primary and higher-order resonances were observed to damp out, parametric resonances might persist even with realistic fluid damping coefficients (4–12%). It was observed that with higher effective foil flexibility, the cavity-shedding frequencies may be significantly modified from the rigid foil trends, and may instead lock-in with the system natural frequencies.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

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.

     

含外激励van der Pol-Mathieu方程的非线性动力学特性分析

本文针对含有自激励, 参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析, 发现其准周期运动的频谱中含有均匀边频带这一新的特性. 首先, 采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应, 得到了其非线性频率响应曲线; 再利用Floquet理论对周期解进行稳定性分析, 得到了两种类型的分岔及它们的位置. 然后, 基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的, 另一个基频事先是未知的), 推导了相应的两时间尺度IHB法, 精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值, 尤其在临界点附近处的准周期运动响应. 得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合. 最后, 研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.      

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.