A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations

A new stochastic averaging method for predicting the response of vibro-impact (VI) systems to random perturbations is proposed. First, the free VI system (without damping and random perturbation) is analyzed. The impact condition for the displacement is transformed to that for the system energy. Thus, the motion of the free VI systems is divided into periodic motion without impact and quasi-periodic motion with impact according to the level of system energy. The energy loss during each impact is found to be related to the restitution factor and the energy level before impact. Under the assumption of lightly damping and weakly random perturbation, the system energy is a slowly varying process and an averaged Itô stochastic differential equation for system energy can be derived. The drift and diffusion coefficients of the averaged Itô equation for system energy without impact are the functions of the damping and the random excitations, and those for system energy with impact are the functions of the damping, the random excitations and the impact energy loss. Finally, the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equation is derived and solved to yield the stationary probability density of system energy. Numerical results for a nonlinear VI oscillator are obtained to illustrate the proposed stochastic averaging method. Monte-Carlo simulation (MCS) is also conducted to show that the proposed stochastic averaging method is quite effective.     

Stochastic resonance in a time-delayed mono-stable system with correlated multiplicative and additive white noise

This paper considers the stochastic resonance for a time-delayed mono-stable system, driven by correlated multiplicative and additive white noise. It finds that the output signal-to-noise ratio (SNR) varies non-monotonically with the delayed times. The SNR varies non-monotonically with the increase of the intensities of the multiplicative and additive noise, with the increase of the correlation strength between the two noises, as well as with the system parameter.     

Moment Lyapunov exponents of a two-dimensional system under both harmonic and white noise parametric excitations

The moment Lyapunov exponents and the Lyapunov exponents of a 2D system under both harmonic and white noise excitations are studied. The moment Lyapunov exponents and the Lyapunov exponents are important characteristics determining the moment and almost-sure stability of a stochastic dynamical system. The eigenvalue problem governing the moment Lyapunov exponent is established. A singular perturbation method is applied to solve the eigenvalue problem to obtain second-order, weak noise expansions of the moment Lyapunov exponents. The influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated.     

Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation

Recent years have shown increasing interest of researchers in energy harvesting systems designed to generate electrical energy from ambient energy sources, such as mechanical excitations. In a lot of cases excitation patterns of such systems exhibit random rather than deterministic behaviour with broad-band frequency spectra. In this paper, we study the efficiency of vibration energy harvesting systems with stochastic ambient excitations by solving corresponding Fokker-Planck equations. In the system under consideration, mechanical energy is transformed by a piezoelectric transducer in the presence of mechanical potential functions which are governed by magnetic fields applied to the device. Depending on the magnet positions and orientations the vibrating piezo beam system is subject to characteristic potential functions, including single and double well shapes. Considering random excitation, the probability density function (pdf) of the state variables can be calculated by solving the corresponding Fokker-Planck equation. For this purpose, the pdf is expanded into orthogonal polynomials specially adapted to the problem and the residual is minimized by a Galerkin procedure. The power output has been estimated as a function of basic potential function parameters determining the characteristic pdf shape.     

Space and energy characteristics of a circular antenna array with radial reflectors under mode excitations

On the basis of an electrodynamic analysis, we study a multimode circular antenna array with radial reflectors. We show that it is possible to attenuate the mutual coupling between the radiators and, thereby, improve the energy properties of the array. Samara Branch Research Institute of Radio, Samara, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 11, pp. 1085–1091, September 1999.     

Potential of a piezoelectric energy harvester from sea waves

A sea wave energy harvester from the longitudinal wave motion of water particles is developed. The harvester consisting of a cantilever substrate attached by piezoelectric patches and a proof mass is used to collect electrical energy owing to the electromechanical coupling effect of the piezoelectric patches from the longitudinal wave motion. To describe the energy harvesting process, a mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and classical elastic beam model. Results show that the mean value of the generated power increases with the increase in the ratio of the width to the thickness of the cantilever, the wave height, the sea depth (which equals to the cantilever height in this study), the ratio of the proof mass to the cantilever mass, and the ratio of the sea depth to the wave length. A value of the power up to 55 W can be realized for a practical sea wave with the values of the sea depth, wave height and wave length to be 3 m, 2 m, and 15 m, respectively. The collected power harvesting with respect to different categories of the sea waves are provided. Our simulations also show the generated electric power can be further increased by an increase in dimensions of the harvester considering the scale effect. This research develops a new technique for energy harvesting from sea waves by piezoelectric energy harvesters.     

窄带随机激励双稳压电悬臂梁响应机制与能量采集研究

具有中心频率的窄带随机振动是一种典型的环境振动,其振动特征与环境的变化密切相关.本文以双稳压电悬臂梁能量采集系统为研究对象,分析系统在不同磁铁间距下的等效线性固有频率特性,以带通滤波器输出一定带宽的窄带随机激励模拟环境振动,研究系统的响应和能量采集特征.研究表明,对于一定带宽的窄带随机激励,一方面系统始终存在一个固定的磁铁间距使其输出达到峰值,另一方面当激励中心频率在一定范围内变化时,系统还分别存在另外两个或一个不同磁铁间距也能使系统输出达到峰值,而且该峰值特性是系统在其等效线性固有频率处诱导双稳或单稳“共振”形成的.研究结果可为具有窄带随机激励特征的振动能量采集提供一定的理论和技术支持.     

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.     

Investigations of a nonlinear energy harvester with a bistable potential well

This paper investigates a nonlinear energy harvester that uses magnetic interactions to create an inertial generator with a bistable potential well. The motivating hypothesis for this work was that nonlinear behavior could be used to improve the performance of an energy harvester by broadening its frequency response. Theoretical investigations study the harvester's response when directly powering an electrical load. Both theoretical and experimental tests show that the potential well escape phenomenon can be used to broaden the frequency response of an energy harvester.     

Sustained high-frequency energy harvesting through a strongly nonlinear electromechanical system under single and repeated impulsive excitations

This work investigates a vibration-based energy harvesting system composed of two oscillators coupled with essential (nonlinearizable) stiffness nonlinearity and subject to impulsive loading of the mechanical component. The oscillators in the system consist of one grounded, weakly damped linear oscillator mass (primary system), which is coupled to a second light-weight, weakly damped oscillating mass attachment (the harvesting element) through a piezoelastic cable. Due to geometric/kinematic mechanical effects the piezoelastic cable generates a nonlinearizable cubic stiffness nonlinearity, whereas electromechanical coupling simply sees a resistive load. Under single and repeated impulsive inputs the transient damped dynamics of this system exhibit transient resonance captures (TRCs) causing high-frequency ‘bursts’ or instabilities in the response of the harvesting element. In turn, these high-frequency dynamic instabilities result in strong and sustained energy transfers from the directly excited primary system to the lightweight harvester, which, through the piezoelastic element, are harvested by the electrical component of the system or, in the present case, dissipated across a resistive element in the circuit. The primary goal of this work is to demonstrate the efficacy of employing this type of high-frequency dynamic instability to achieve enhanced nonlinear vibration energy harvesting under impulsive excitations.     

Performance of a cantilever piezoelectric energy harvester impacting a bump stop

Piezoelectric cantilever beam energy harvesters are commonly used to convert ambient vibration into electrical energy. In practical applications, energy harvesters are subjected to large shocks which can shorten the service life by causing mechanical failure. In this work, a bump stop is introduced into the design of a piezoelectric cantilever beam energy harvester to limit the maximum displacement of the cantilever and prevent excessively high bending stresses developing as a result of shocks. In addition to limiting the maximum displacement of the beam, it is inevitable that the deflected shape of the beam and the electrical output are modified. A theoretical model for a piezoelectric cantilever beam harvester impacting against a stop is derived, which aims to develop an understanding of the vibration characteristics of the cantilever and quantify how the electrical output of the harvester is affected by the stop. An experiment is set up to measure the dynamics and the electrical output of a bimorph energy harvester and to validate the theoretical model. Numerical simulation results are presented for energy harvesters with different initial gaps and different stop locations, and it is found that the reduction in maximum bending stress is at the expense of the electrical power of the harvester.     

Modeling of a nanoscale flexoelectric energy harvester with surface effects

This work presents the modeling of a beam energy harvester scavenging energy from ambient vibration based on the phenomenon of flexoelectricity. By considering surface elasticity, residual surface stress, surface piezoelectricity and bulk flexoelectricity, a modified Euler-Bernoulli beam model for the energy harvester is developed. After deriving the requisite energy expressions, the extended Hamilton's principle and the assumed-modes method are employed to obtain the discrete electromechanical Euler-Lagrange's equations. Then, the expressions of the steady-state electromechanical responses are given for harmonic base excitation. Numerical simulations are conducted to show the output voltage and the output power of the flexoelectric energy harvesters with different materials and sizes. Particular emphasis is given to the surface effects on the performance of the energy harvesters. It is found that the surface effects are sensitive to the beam geometries and the surface material constants, and the effect of residual surface stress is more significant than that of the surface elasticity and the surface piezoelectricity. The axial deformation of the beam is also considered in the model to account for the electromechanical coupling due to piezoelectricity, and results indicate that piezoelectricity will diminish the output electrical quantities for the case investigated. This work could lead to the development of flexoelectric energy harvesters that can make the micro- and nanoscale sensor systems autonomous.     

Relative performance of a vibratory energy harvester in mono- and bi-stable potentials

Motivated by the need for broadband vibratory energy harvesting, many research studies have recently proposed energy harvesters with nonlinear characteristics. Based on the shape of their potential function, such devices are classified as either mono- or bi-stable energy harvesters. This paper aims to investigate the relative performance of these two classes under similar excitations and electric loading conditions. To achieve this goal, an energy harvester consisting of a clamped-clamped piezoelectric beam bi-morph is considered. The shape of the harvester's potential function is altered by applying a static compressive axial load at one end of the beam. This permits operation in the mono-stable (pre-buckling) and bi-stable (post-buckling) configurations. For the purpose of performance comparison, the axial load is used to tune the harvester's oscillation frequencies around the static equilibria such that they have equal values in the mono- and bi-stable configurations. The harvester is subjected to harmonic base excitations of different magnitudes and a slowly varying frequency spanning a wide band around the tuned oscillation frequency. The output voltage measured across a purely resistive load is compared over the frequency range considered. Two cases are discussed; the first compares the performance when the bi-stable harvester has deep potential wells, while the second treats a bi-stable harvester with shallow wells. Both numerical and experimental results demonstrate the essential role that the potential shape plays in conjunction with the base acceleration to determine whether the bi-stable harvester can outperform the mono-stable one and for what range of frequencies. Results also illustrate that, for a bi-stable harvester with shallow potential wells, super-harmonic resonances can activate the inter-well dynamics even for a small base acceleration, thereby producing large voltages in the low frequency range.     

Laser cooling and optical detection of excitations in a LC electrical circuit

We explore a method for laser cooling and optical detection of excitations in a room temperature LC electrical circuit. Our approach uses a nanomechanical oscillator as a transducer between optical and electronic excitations. An experimentally feasible system with the oscillator capacitively coupled to the LC and at the same time interacting with light via an optomechanical force is shown to provide strong electromechanical coupling. Conditions for improved sensitivity and quantum limited readout of electrical signals with such an "optical loud speaker" are outlined.     

Nonlinear dynamics of a bistable piezoelectric-composite energy harvester for broadband application

The continuing need for reduced power requirements for small electronic components, such as wireless sensor networks, has prompted renewed interest in recent years for energy harvesting technologies capable of capturing energy from ambient vibrations. A particular focus has been placed on piezoelectric materials and devices due to the simplicity of the mechanical to electrical energy conversion and their high strain energy densities compared to electrostatic and electromagnetic equivalents. In this paper an arrangement of piezoelectric layers attached to a bistable asymmetric laminate is investigated experimentally to understand the dynamic response of the structure and power generation characteristics. The inherent bistability of the underlying structure is exploited for energy harvesting since a transition from one stable configuration to another, or “snap-through”, is used to repeatedly strain the surface bonded piezoelectric and generate electrical energy. This approach has been shown to exhibit high levels of power extraction over a wide range of vibrational frequencies. Using high speed digital image correlation, a variety of dynamic modes of oscillation are identified in the harvester. The sensitivity of such modes to changes in vibration frequency and amplitude are investigated. Power outputs are measured for repeatable snap-through events of the device and are correlated with the measured modes of oscillation. The typical power generated is approximately 3.2?mW, comparing well with the needs of typical wireless senor node applications.     

Invariable optical properties of phosphor-free white light-emitting diode under electrical stress

This paper reports that a dual-wavelength white light-emitting diode is fabricated by using a metal-organic chemical vapor deposition method. Through a 200-hours’ current stress, the reverse leakage current of this light-emitting diode increases with the aging time, but the optical properties remained unchanged despite the enhanced reverse leakage current. Transmission electron microscopy and cathodeluminescence images show that indium atoms were assembled in and around V-shape pits with various compositions, which can be ascribed to the emitted white light. Evolution of cathodeluminescence intensities under electron irradiation is also performed. Combining cathodeluminescence intensi- ties under electron irradiation and above results, the increase of leakage channels and crystalline quality degradation are realized. Although leakage channels increase with aging, potential fluctuation caused by indium aggregation can effectively avoid the impact of leakage channels. Indium aggregation can be attributed to the mechanism of preventing optical degradation in phosphor-free white light-emitting diode.     

Power flow and energy distribution in conservatively or non-conservatively coupled oscillators under correlative excitations

As a foundation of SEA for consevatively or non-conservativelycoupled systems under correlative excitations,power flow and energy distribu-tion in two coupled osillators under proportionally correlative excitations arestudied mainly in this paper.Theoretical relationship among power flow andtime averaged vibational energy of the osillators,as well as power balance equa-tions and energy ratio or them,are derived.Further along,power flow,energydistribution and power balance equations of the systems in conservative ornon-conservative coupling and under proportional correlative excitations arediscussed in detail,with comparing with characteristics of those innon-correlative excitations.Numerical resu1ts supporting the basic theory aregiven,too.Results show that the power flow and energy distribution of theosillators under correlative excitations,either conservative or non-conservativecoupling,are much different to those in the condition of non-correlativeexcitations.     

The refractive nonlinearities of InAs/GaAs quantum dots above-bandgap energy

The third-order optical nonlinear refractive properties of InAs/GaAs quantum dots grown by molecular beam epitaxy have been measured using the reflection Z-scan technique at above-bandgap energy. The nonlinear refractive index and nonlinear absorption index of the InAs/GaAs quantum dots were determined for wavelengths from 740 to 777 nm. The measured results are compared with the nonlinear refractive response of several typical III-V group semiconductor materials. The corresponding mechanisms responsible for the large nonlinear response are discussed.     

相关激励形式下保守和非保守耦合振子的能量分布与功率流

作为相关输入形式下保守或非保守耦合系统统计能量分析的基础,本文主要研究耦合振子在比例相关输入条件下的能量分布与功率流.在推导了相关输入条件下耦合振子间的功率流与振子能量的基本关系式和功率平衡方程式及振子能量比的表达式之后,本文分别详细讨论了保守耦合振子和非保守耦合振子系统的能量分布和功率流问题,并与互不相关输入时耦合振子的能量分布与功率流的特点进行了比较,对于各种耦合阻尼情况下的振子能量分布与功率流也进行了数值计算,研究结果表明,相关输入下耦合振子的能量分布与功率流的特征与互不相关输入时的特征有着显著的差别。