Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Topological entropy and geometric entropy and their application to the horizontal visibility graph for financial time series

Seismic wave control is very important both in civil and mechanical engineering. Common passive methods for isolating a building or a device include base isolators and tuned mass dampers. In the present paper, a time-varying controllable spring is considered as a vibration isolator for a linear mechanical system. The controller works as follows: When the seismic movement is active, the velocity of the moving mass is monitored as the reference velocity. When such reference velocity is positive, the stiffness is reduced; when it is negative, the stiffness is increased. Numerical investigations show that the controller is capable of filtering seismic excitation close to the natural frequency of the controlled system and reducing the total seismic energy transfer up to 5 times. The role played by the gravity in the active vibration filtering is pointed out by showing that no filtering action can be observed in gravity-free simulations. Moreover, control effectiveness has been proven for a measured seismic signal, showing its robustness in presence of noise.     

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

Nonlinear energy harvesting with dual resonant zones based on rotating system

An electromagnetic nonlinear energy harvester(NEH)based on a rotating system is proposed,of which the host system rotates at a constant speed and vibrates harmonically in the vertical direction.This kind of device exhibits several resonant phenomena due to the combinations of the rotating and the vibration frequencies of the host system as well as the cubic nonlinearity of the NEH.The governing equation of motion for the NEH is derived,and the dynamic responses and output power are investigated with the multiple scale method under the 1:1 primary and 2:1 superharmonic resonant conditions.The effects of system parameters including the nondimensional external frequency,the rotating speed,and the nonlinear stiffness on the responses of free vibration for the system are studied.The results of the primary resonance show that the responses exhibit not only the resonant characteristics but also the nonlinear dynamic characteristics such as the saddle-node(SN)bifurcation.The coexistence of multiple solutions and the varying trends of responses are verified with the direct numerical simulation.Moreover,the effects of system parameters on the average output power are investigated.The results of the analyses on the two resonant conditions indicate that the large power can be harvested in two resonant frequency bands.The effect of resonance on the output power is dominant for the 2:1 superharmonic resonance.Moreover,the results also show that introducing the nonlinearity can increase the value of the output power in large frequency bands and induce the occurence of new frequency bands to harvest the large power.The efficiency of the harvested power could be improved by the combined effects of the resonance as well as the nonlinearity of the NEH device.Suitable parameter conditions could help optimize the power harvesting in design.     

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.     

Nonlinear resonances in infinitely long 1D continua on a tensionless substrate

In this work, we investigate the primary nonlinear resonance response of a one-dimensional continuous system, which can be regarded as a model for semi-infinite cables resting on an elastic substrate reacting in compression only, and subjected to a constant distributed load and to a small harmonic displacement applied to the finite boundary. By introducing a straightforward small amplitude expansion characterized by a smallness parameter ε and by performing a Fourier analysis, we first determine the frequencies of the oscillations of the system about the static solution at all orders. We find that, at each order, there exists a critical (cutoff) frequency, above which the solution behaves as a traveling wave toward infinity, while it decays exponentially below it. We then examine the resonance response of the system when an external harmonic excitation is applied at the finite boundary. To this aim, we scale the external excitation with the third power of ε and perform a Multiple-Time-Scale analysis, whose third-order consistency conditions give the differential equations which govern the behavior of the amplitude on the long time scale. In this way, we determine the third-order bending of the resonance curves, whose hardening or softening behavior depends upon the frequency of the chosen primary resonance.     

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.     

磁力耦合道路能量收集设计与动力学分析

通过在交通环境布置无线传感器等小型机电系统, 实现交通状况监测、交通系统管控、交通设施健康状态监测等, 可以使交通系统更加安全、有序、高效地运行. 但是, 如何为这些广泛分布的小型机电系统供能？本文提出了一种磁力耦合道路能量收集设计, 用以收集车辆滚压能量并转换成电能. 通过磁力耦合进行无接触能量传递, 减小了装置受到的冲击并使得装置具有良好密封性, 从而提升装置的鲁棒性. 通过升频齿轮机构、棘轮机构将车辆滚压激励转换为高速单向旋转, 并且通过换向齿轮机构能够继续收集复位弹性势能, 提高了收集装置的输出功率. 基于磁力耦合道路能量收集系统的工作原理建立了机电耦合动力学模型. 数值仿真探究了减速带限位距离和复位弹簧刚度等关键设计参数对能量采集系统动力学和电学性能的影响. 能量采集系统在车速为50 km/h时最大输出电压为76.28 V, 最大功率为59.94 W. 磁力耦合道路能量收集装置可以成为未来智慧交通系统的重要组成部分, 俘获交通环境能量为交通环境中小型机电系统提供可持续的绿色无碳电力.      

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.     

New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems

We present new analytical approach to the problem of energy pumping in strongly non- homogeneous nonlinear two-degree-of-freedom (2DOF) systems with single anchor spring under condition of initial impact. Energy pumping is a passive, almost irreversible transfer of mechanical energy from the main substructure of the system to the light auxiliary attachment. The mechanism of energy pumping in the system under consideration is a resonance capture. The approach is based on application of Laplace transformation to the principal asymptotic approximation of the equations of motion in complex form and using the power expansion of the solution in terms of time. Obtained temporal dependence of the system energetic characteristics gives a tool for estimation of energy pumping efficiency. In particular, we show that the system without an anchor spring in attachment is more efficient than the system with such a spring. Numerical simulations confirm the analytical results.     

Study of the Damping Mechanisms of Loose Spring Skirts on Vibrating Pipes

Damping of the loose spring skirt (LSS) of a vibrating pipe under operating conditions was studied experimentally and numerically. Motion images made using a high speed charge-coupled device (CCD) camera as well as acceleration signals were analyzed in order to correlate the physical states of the LSS pipe system with the vibration transmissibility (TM) through the pipe. Experiments on the pipe undergoing harmonic excitation demonstrate that the damping of the LSS is caused by the same mechanism as with a conventional impact damper. Measurements of the acceleration TM as a function of excitation amplitude verified that the momentum exchange between the pipe and the auxiliary spring is the cause of the damping produced by the LSS pipe. Another finding of this research is that the experimental representation of the resonance shift phenomenon is related to two main operating conditions, namely: 1) the driving frequency and 2) the excitation amplitude. This research demonstrated the occurrence of the resonance shift phenomenon through experimental mapping of the system response at increasing excitation amplitudes using frequency sweep tests. Simulations of the LSS pipe system based on principal coordinate and impact damping analysis were performed in order to investigate and provide better explanations for LSS pipe system damping mechanisms. As a result of this research study, a new damping mechanism is proposed which is different from previous damping models. This research study has also addressed more practical boundary and excitation conditions for LSS pipe systems.     

Effect of the system imperfections on the dynamic response of a high-static-low-dynamic stiffness vibration isolator

The dynamic response of a high-static-low-dynamic stiffness (HSLDS) isolator formed by parallelly connecting a negative stiffness corrector which uses compressed Euler beams to a linear isolator is investigated in this study. Considering stiffness and load imperfections, the resonance frequency and response of the proposed isolator are obtained by employing harmonic balance method. The HSLDS isolator with quasi-zero stiffness characteristics can offer the lowest resonance frequency provided that there is only stiffness or load imperfection. If load imperfection always exists, there is no need to make the stiffness to zero since it cannot provide the lowest resonance frequency any longer. The reason for this unusual phenomenon is given. The dynamic response will exhibit softening, hardening, and softening-to-hardening characteristics, depending on the combined effect of load imperfection, stiffness imperfection, and excitation amplitude. In general, load imperfection makes the response exhibit softening characteristic and increasing stiffness imperfection will weak this effect. Increasing the excitation level will make the isolator undergo complex switch between different stiffness characteristics.     

Design and evaluation of a low-power mobile shaker for vibration tests on heavy wheeled vehicles

In the field of mobile agricultural and off-road machinery, there is a growing interest to examine the dynamic behaviour of the vehicle under construction with a view to improve passenger comfort and machine performance. As many of these machine constructors are small sized, specialised companies are required to execute vibration tests and to evaluate vibration levels. Because most of these machines are heavy constructions with exceptional proportions, it is obvious to perform vibration tests on the factory floor by means of a low power mobile excitation device such as an electro-hydraulic shaker. This paper shows that an air spring, parallel to the actuator of an electro-hydraulic shaker, results in considerable power savings. The behaviour of the shaker is discussed at its resonance frequencies. For small band excitation signals, extra power savings can be achieved by tuning the spring stiffness and by using the air spring at constant pressure. Experiments on a one degree of freedom vertical shaker show good agreement with the theory and demonstrate that the different mode shapes of an agricultural tractor can be sufficiently excited with a nominal hydraulic power of 500 W.     

Integration of a nonlinear energy sink and a piezoelectric energy harvester

A mechanical-piezoelectric system is explored to reduce vibration and to harvest energy. The system consists of a piezoelectric device and a nonlinear energy sink(NES), which is a nonlinear oscillator without linear stiffness. The NES-piezoelectric system is attached to a 2-degree-of-freedom primary system subjected to a shock load. This mechanical-piezoelectric system is investigated based on the concepts of the percentages of energy transition and energy transition measure. The strong target energy transfer occurs for some certain transient excitation amplitude and NES nonlinear stiffness. The plots of wavelet transforms are used to indicate that the nonlinear beats initiate energy transitions between the NES-piezoelectric system and the primary system in the transient vibration, and a 1:1 transient resonance capture occurs between two subsystems.The investigation demonstrates that the integrated NES-piezoelectric mechanism can reduce vibration and harvest some vibration energy.     

Theoretical and Experimental Study on the Parametrically Excited Vibration of Mass-Loaded String

The parametrically excited lateral vibration of a mass-loaded string is investigated in this paper. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher-order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieu's equation with cubic nonlinearity. Based on the stability criterion for Mathieu's equation, the critical conditions inducing parametric resonance are clarified. Theoretical analysis shows that when the natural frequency f _s of the string lateral vibration and the vertical excitation frequency f satisfy f _s= (n/2)f, n= 1, 2, 3, ..., parametric resonance occurs in the case of no damping. For a damped system, parametric resonance most likely occurs when f is close to 2f _s, and depends on the damping of the system and the vertical excitation. The critical excitation has been derived at different frequencies. If the natural frequency of the mass vertical vibration happens to be twice that of the string lateral vibration, the parametric resonance may occur due to a small disturbance. Numerical simulations show that the lateral vibration of the string does not increase infinitely at parametric resonance because the parametric excitation is self-tuned due to the coupling between the vertical and lateral vibrations. Finally, the theoretical results are supported by some experimental work.     

外激励作用下弹性支撑浅拱的非线性动力学分析

研究了外激励下两端采用转动弹簧约束的铰支浅拱在发生1:1内共振时的非线性动力学行为。通过引入基本假定和无量纲化变量得到浅拱的动力学控制方程, 将阻尼项、外荷载项和非线性项去掉后,所得线性方程及对应边界条件即可确定考虑转动弹簧影响的频率和模态, 发现转动约束取不同刚度值时系统存在模态交叉与模态转向两种内共振形式。对动力方程进行Galerkin全离散, 并采用多尺度法对内共振进行了摄动分析, 得到了极坐标和直角坐标两种形式的平均方程, 其中平均方程系数与转动弹簧刚度一一对应。最低两阶模态之间1:1内共振的数值研究结果表明: 外激励能激发内共振模态的非线性相互作用, 参数处于某一范围时系统存在周期解、准周期解和混沌解窗口, 且通过(逆)倍周期分岔方式进入混沌。     

Non-linear interactions in imperfect beams at veering

Non-linear interactions in a hinged-hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force- and frequency-response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.     

Non-linear dynamics of a thermomechanical pseudoelastic oscillator excited by non-ideal energy sources

This paper discusses the non-linear dynamical response of a shape-memory non-ideal oscillator. The non-ideal excitation originates from a DC electric motor with limited power supply driving an unbalanced rotating mass. The restoring force provided by the shape-memory device is described by a thermomechanical model capable of accounting for the hysteretic behavior via the evolution of a suitable internal variable. The non-linear dynamic response of the system is investigated with the voltage as control parameter. Numerical simulations show the occurrence of regular and quasi-periodic motions, which are investigated via bifurcation diagrams and phase plane portraits. The 0–1 test is used for quantitative characterization of chaotic responses. The computation of basins of attraction points out the strong dependence of the response on small changes of initial conditions, along with meaningful modifications of competing basins with variations of the control parameter. Finally, variations of the mechanical and thermal parameters of the pseudoelastic oscillator are considered, with the aim to evaluating the effects produced by the non-ideal excitation source on the non-linear dynamics of the shape memory device.     

Internal resonances in non-linear vibrations of a laminated circular cylindrical shell

Internal resonances in geometrically non-linear forced vibrations of laminated circular cylindrical shells are investigated by using the Amabili?CReddy higher-order shear deformation theory. A harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retain dissipation. Numerical results are obtained by using the pseudo-arc length continuation method and bifurcation analysis. A one-to-one-to-two internal resonance is identified, giving rise to pitchfork and Neimark?CSacher bifurcations of the non-linear response. A threshold level in the excitation has been observed in order to activate the internal resonance.     

Influence of spring dynamics and friction on a spring-actuated cam system

Summary  The paper investigates two important aspects, friction and spring motion, of the dynamics of a spring-actuated cam system. The characteristics of the friction on the camshaft are analyzed using the nonlinear pendulum experiment, while the parameters of the friction model are estimated using the optimization technique. The analysis reveals that the friction of the camshaft depends on stick–slip, Stribeck effect and viscous damping. Spring elements are found to have much influence on the dynamic characteristics. Hence, they are modeled as four-degrees-of-freedom lumped masses with equivalent springs. The lumped masses and equivalent springs are obtained to match the static stiffness and natural frequency of the actual spring. The appropriateness of the derived friction and spring model are verified by its application to a vacuum circuit-breaker mechanism of the cam-follower type. Received 23 March 2000; accepted for publication 21 November 2000