STOCHASTIC AVERAGING OF STRONGLY NON-LINEAR OSCILLATORS UNDER BOUNDED NOISE EXCITATION

A stochastic averaging method for strongly non-linear oscillators under external and/or parametric excitation of bounded noise is proposed by using the so-called generalized harmonics functions. The method is then applied to study the primary resonance of Duffing oscillator with hardening spring under external excitation of bounded noise. The stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase. Subsequently, the method is applied to study the dynamical instability and parametric resonance of Duffing oscillator with hardening spring under parametric excitation of bounded noise. The primary unstable region is delineated by evaluating the Lyapunov exponent of linearized system, and the response and jump of non-linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase.     

Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars

Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation （unlike the Duffing oscillator, where they exist only in the fundamental wave equation）. In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken： （i） for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained; （ii） for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd- （or even-） order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.     

Predicting optimal drive sweep rates for autoresonance in Duffing-type oscillators: A beat method using Teager-Kaiser instantaneous frequency

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. But in oscillators with nonlinear restoring forces such as the pendulum, Duffing and Duffing-Van der Pol oscillator, the resonant frequency changes as the amplitude changes, so a constant frequency drive results in a beat oscillation instead of sustained resonance. Duffing-type nonlinear oscillators can be driven into sustained resonance, called autoresonance, when the drive frequency is swept in time to match the changing resonant frequency of the oscillator. We find that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specifically, a least squares estimate of the Teager-Kaiser instantaneous frequency versus time for the beat response to a stationary drive provides a near-optimal estimate of the nonstationary drive linear sweep rate needed to sustain resonance in the pendulum, Duffing and Duffing-Van der Pol oscillators. We confirm these predictions with model-based numerical simulations. An advantage of the beat method of estimating optimal drive sweep rates for maximal autoresonant response is that no model is required so experimentally generated beat oscillation data can be used for systems where no model is available.     

Fokker–Planck equation analysis of randomly excited nonlinear energy harvester

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.     

Performance measures for single-degree-of-freedom energy harvesters under stochastic excitation

We develop performance criteria for the objective comparison of different classes of single-degree-of-freedom oscillators under stochastic excitation. For each family of oscillators, these objective criteria take into account the maximum possible energy harvested for a given response level, which is a quantity that is directly connected to the size of the harvesting configuration. We prove that the derived criteria are invariant with respect to magnitude or temporal rescaling of the input spectrum and they depend only on the relative distribution of energy across different harmonics of the excitation. We then compare three different classes of linear and nonlinear oscillators and using stochastic analysis methods we illustrate that in all cases of excitation spectra (monochromatic, broadband, white-noise) the optimal performance of all designs cannot exceed the performance of the linear design. Subsequently, we study the robustness of this optimal performance to small perturbations of the input spectrum and illustrate the advantages of nonlinear designs relative to linear ones.     

广义Duffing扰动振子随机共振机理的渐近解

利用非线性方法研究了一类广义Duffing扰动方程.首先求得了典型的Duffing方程的解.然后利用泛函广义变分迭代原理得到了广义Duffing扰动振子随机共振机理的近似解,并论述了解的一致有效性.     

Analysis of nonlinear oscillators using Volterra series in the frequency domain

The Volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems, both in the time and the frequency domain. The Volterra series is associated with the so-called weakly nonlinear systems, but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid Volterra series representation for a given nonlinear differential equation. Barrett (1965) [1] proposed a time domain criterion to prove that the Volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity. In this paper this time-domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation. A common and severe nonlinear phenomenon called jump, a behavior associated with the Duffing oscillator and the multi-valued properties of the response solution, is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators, to predict the onset point of the jump, and the Volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques.     

Response of uni-modal duffing-type harvesters to random forced excitations

Linear energy harvesters have a narrow frequency bandwidth and hence operate efficiently only when the excitation frequency is very close to the fundamental frequency of the harvester. Consequently, small variations of the excitation frequency around the harvester's fundamental frequency drops its small energy output even further making the energy harvesting process inefficient. To extend the harvester's bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. From a steady-state perspective, this hardening-type nonlinearity can extend the coupling between the excitation and the harvester to a wider range of frequencies. In this effort, we investigate the response of such harvesters, which can be modeled as a uni-modal duffing-type oscillator, to White Gaussian and Colored excitations. For White excitations, we solve the Fokker-Plank-Kolmogorov equation for the exact joint probability density function of the response. We show that the expected value of the output power is not even a function of the nonlinearity. As such, under White excitations, nonlinearities in the stiffness do not provide any enhancement over the typical linear harvesters. We also demonstrate that nonlinearities in the damping and inertia may be used to enhance the expected value of the output power. For Colored excitations, we use the Van Kampen expansion and long-time numerical integration to investigate the influence of the nonlinearity on the expected value of the output power. We demonstrate that, regardless of the bandwidth or the center frequency of the excitation, the expected value of the output power decreases with the nonlinearity. With such findings, we conclude that energy harvesters modeled as uni-modal duffing-type oscillators are not good candidates for harvesting energy under forced random excitations. Using a linear transformation, results can be extended to the base excitation case.     

Energy harvesting from the nonlinear oscillations of magnetic levitation

This paper investigates the design and analysis of a novel energy harvesting device that uses magnetic levitation to produce an oscillator with a tunable resonance. The governing equations for the mechanical and electrical domains are derived to show the designed system reduces to the form of a Duffing oscillator under both static and dynamic loads. Thus, nonlinear analyses are required to investigate the energy harvesting potential of this prototypical nonlinear system. Theoretical investigations are followed by a series of experimental tests that validate the response predictions. The motivating hypothesis for the current work was that nonlinear phenomenon could be exploited to improve the effectiveness of energy harvesting devices.     

Energy harvesting in the nonlinear two-masses piezoelastic system driven by harmonic excitations

We examine the energy harvesting system consisted of two different masses (magnets) attached to piezoelastic oscillators, coupled by the electric circuit, and driven by harmonic excitations. The nonlinearity of the system is achieved by variable distance between vibrating magnetic masses and the magnets attached directly to the harvester. We also introduce the mistuning parameter which describes the disproportion of vibrating masses (their ratio). In our work we examine the dependence of output power (in terms of mean squared voltage) generated on electric load on excitation frequencies for different values of mistuning parameter and additionally for different values of system nonlinearity parameter. We compare obtained results with the dia- grams presenting relative displacements of these oscillators (in terms of standard deviation) vs. excitation frequencies. In the second part of this paper we present the phase boundary lines (phase portraits) for selected values of applied frequency to show the complicated behavior of the oscillators in the nonlinear regime when the mistuning appears.     

Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises

This study presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. First the Zhuravlev non-smooth coordinate transformation is adopted to convert a vibro-impact oscillator into an oscillator without barriers. The stationary PDF of the converted oscillator is governed by the Fokker–Planck (FP) equation. The FP equation is solved by the exponential-polynomial closure (EPC) method. Illustrative examples are presented with vibro-impact Duffing oscillators under external and parametric Gaussian white noises to show the effectiveness of the solution procedure. The parametric excitation is acting in displacement and the constraint is a unilateral zero-offset barrier. The restitution coefficient of impacts is taken as 0.90. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. This difference may be due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity.     

A feedback model of millimeter wave harmonic oscillators and its application

In this paper a feedback model of second harmonic oscillators is developed. By using describing functions of nonlinearity of active devices, the performances of second harmonic oscillators are studied. Frequency dependence of I–V characteristics of active element are taken into account. The ratio of maximum output power of second harmonic to fundamental is given. The maximum harmonic locking bandwidth of injected harmonic oscillator is derived. The theoretical prediction is compared with experimental results.     

Steady-State Responses of NRD-Guide LSE/LSM Mode Oscillators in MM-Wave Bands

The analysis methods for the steady-state responses of the mm-wave band NRD-guide negative impedance oscillators based on nonlinear microwave autonomous circuits harmonic balanced method are presented in the paper in details. Firstly, the large-signal nonlinear lumped equivalent circuits of the Gunn diode are studied in mm-wave bands. Then, the performances of two kinds of NRD-guide autonomous circuits, the probe-exciting LSM ₁₁ ^o-mode oscillator and the LSE ₁₁ ^o-mode oscillator, are analyzed by the way of extracting the large-signal dynamic harmonic admittance (conductance) of the Gunn diode or directly configuring the harmonic balance equations for the oscillator. The input impedance of the exciting probe in the oscillator and the performances of the load-pulling and the local stability of the NRD-guide oscillator are also involved.     

Linearizability conditions for the Rayleigh-like oscillators

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.     

Three superintegrable two-dimensional oscillators: Superintegrability,nonlinearity, and curvature

The superintegrability of three different two-dimensional oscillators is studied: (i) a nonlinear oscillator dependent on a parameter λ (two-dimensional version of the oscillator of Lakshmanan and Mathews), (ii) a nonlinear oscillator related to the Riccati equation, and (iii) the standard harmonic oscillator on constant curvature spaces. They can be considered as nonlinear deformations, or curvature-dependent versions, of the linear harmonic oscillator. The text was submitted by the authors in English.     

Virial Theorem for a Class of Quantum Nonlinear Harmonic Oscillators

In this paper,the Virial Theorem based on a class of quantum nonlinear harmonic oscillators is presented.This relationship has to do with parameter λ and ∂/∂λ,where the λ is a real number.When λ=0,the nonlinear harmonic oscillator naturally reduces to the usual quantum linear harmonic oscillator,and the Virial Theorem also reduces to the usual Virial Theorem.     

脉冲激励下环形耦合Duffing振子间的瞬态同步突变现象

研究非周期信号激励下Duffing振子动力学行为变化特征时,发现处于倍周期分岔的环形耦合Duffing振子系统,在一定的参数条件下,脉冲信号能引起其中一个振子与其他振子运动轨迹间出现短暂失同步的现象即瞬态同步突变现象.利用这种现象可以快速检测出强噪声背景中的微弱脉冲信号,从而扩展了现有的Duffing振子对非周期信号的检测范围及应用领域. 关键词： 瞬态同步突变 微弱信号检测 脉冲信号 Duffing振子     

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.     

Advantages of plasma harmonic generation using high pulse repetition rate lasers: Review of recent studies

The studies of coherence properties of the harmonics generating in laser-produced plasmas, the analysis of the optical nonlinearities of deoxyribonucleic acid components, the resonance enhancement of harmonic in the cases of excitation of indium plasma by multi- and few-cycle pulses, and the application of nanoparticle-based emitters of harmonics using high-pulse repetition rate lasers are reviewed. The analysis of various aspects of plasma harmonic generation at the conditions of optimal excitation of the ablated targets irradiating by 1 kHz lasers is presented. The growth of plasma harmonic conversion efficiency, single harmonic emission, nonlinear spectroscopy of complex organic components, as well as high coherency of harmonic radiation show the advantages of using plasma harmonic technique for optimization of the sources of coherent extreme ultraviolet radiation and for the material science studies. These studies allowed a significant growth of the average power of harmonics compared with the case of 10 Hz lasers.