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... Xu et al. [27] introduced the stochastic averaging of energy envelope for Duffing-type vibration energy harvesters, and discussed the effects of the system parameters on the mean square output voltage and power. Kumar et al. [28] used the finite element method to solve the FPK equation of the associated bistable energy harvester, and analyzed the effects of the system parameters on the mean square output voltage and power. Jin et al. [29] introduced the generalized harmonic transformation to decouple the electromechanical equations, and applied the equivalent nonlinearization technique to derive a semi-analytical solution of the corresponding nonlinear vibration energy harvesters subjected to Gaussian white noise excitation. ...
... Its exact solution is difficult to calculate, even for exact stationary probability densities. Therefore some approximate methods for solving the FPK equation of the coupled electromechanical system have been reported which include the statistical linearization techniques [14,[17][18][19], the moment differential equations method [22,26], the Galerkin method [23], the finite element method [24,25,28], the stochastic averaging of energy envelope [27,32,33], the equivalent nonlinearization technique [29] and the cell mapping method [31]. Recently, Er [36] first proposed the SSS method as a scheme to reduce the high-dimensional FPK equation and demonstrated it is an effective method for Gaussian white noise [37] and Poisson impulses excitations [38,39]. ...
... Thus, obtaining its exact solution is not a simple problem. As a result, some researchers proposed the statistical linearization method [20], Galerkin method [23] and finite element method [25,28] to approximate the response statistics of the bistable system. In addition, some studies presented the Monte Carlo numerical simulation [13][14][15]26] to calculate the response of the system. ...
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This paper aims to investigate the statistical characteristics of strongly nonlinear vibratory energy harvesters under Gaussian white noise excitation. The high-dimensional Fokker–Planck–Kolmogorov (FPK) equation of the coupled electromechanical system is reduced to a low-dimensional equation by using the state-space-split method. The conditional moment given by the equivalent linearization method is employed to decouple the FPK equations of coupled system, and then obtained an equivalent nonlinear uncoupled subsystem. The exact stationary solution of the reduced FPK equation of the subsystem is established. The mean output power is derived by the second order conditional moment from the associated approximate probability density function of mechanical subsystem. The procedure is applied to mono- and bi-stable energy harvesters. Effectiveness of the probability density function of the proposed approach is examined via comparison with equivalent linearization method and Monte Carlo simulation. The effects of the system parameters on the mean-square displacement and the mean output power are discussed. The approximate analytical outcomes are qualitatively and quantitatively supported by the numerical simulations.
... To evaluate the performance of the VEHs under noise, it is important to develop analytical approaches for solving the mean output power. Recently, some analytical techniques have been proposed to study the response of nonlinear energy harvester under Gaussian white noise excitation [24][25][26][27][28][29][30][31]. For example, Daqaq [24] presented the voltage response statistics by using the method of moment and demonstrated that the time constant ratio of the energy harvester plays a key role in developing the performance of VEHs under Gaussian white noise. ...
... He et al. [28] employed the statistical linearization techniques and a finite element method of FPK equation to investigate the mean steadystate output of the energy harvester. Kumar et al. [29] used the finite element method to solve the FPK equation of the associated bistable energy harvester and analyzed the effects of the system parameters on the mean-square output voltage and power. Jin et al. [30,31] introduced the generalized harmonic transformation to decouple the electromechanical equations and applied the approximate analytical technique to derive a semi-analytical solution of the corresponding nonlinear VEHs subjected to Gaussian white noise excitation. ...
... The mean-square displacement E(x 2 ) and the mean output power E(P) are important for the miniaturization of device and the energy harvesting. To understand the above theoretical results (29) and (31), we study the E(x 2 ) and E(P) for different noise intensities and system parameters in Figs. 3 and 4. Figure 3 shows the E(x 2 ) and E(P) increase as the multiplicative noise intensity D increases. However, the cross-correlation λ has no obvious influence on the E(x 2 ) and E(P) (see Figs. 3a, b). ...
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Energy harvesting of a monostable duffing-type harvester with piezoelectric coupling under correlated multiplicative and additive white noise is investigated in this paper. The generalized harmonic transformation is applied to decouple the electromechanical equations, which leads to an uncoupled equivalent nonlinear system. Using the stochastic averaging method, an analytical solution of random response for vibration energy harvesters (VEHs) is obtained. The effects of the system parameters on the mean-square displacement, the mean output power and the power spectral density are explored. It is found that the correlated noise can improve the performance of the nonlinear VEHs. The curve of the mean output power first increases with increasing the ratio of time constant, reaches a maximum and then decreases. This phenomenon is of great significance to energy harvesting. Finally, the theoretical results are well verified through the numerical simulations.
... Depending on the system properties, excitation and initial conditions, the response of the Duffing oscillator to harmonic excitation can be periodic, chaotic or quasi-periodic. The Duffing oscillator has been used as a classical prototype model for modelling the behaviour of stiffening springs, beam buckling, nonlinear electronic circuits [13,99,100]. The equation of motion of (a) (b) ...
... The FP equation corresponding to Eq. (17) is [100] op ot ...
... The solution for the stationary jpdf of Eq. (17) admits a closed form solution in separable form as [100] ...
Article
Dynamics of nonlinear oscillators with discontinuous nonlinearities subjected to harmonic and random excitations is investigated. Impact, dry friction and Hertzian type compliant contact nonlinearities are considered. Stochastic bifurcations like the P-bifurcation and D-bifurcation are discussed. P and D bifurcations are characterized respectively by the joint probability density functions (jpdf) of the response and the largest Lyapunov exponent. The jpdf is obtained by the solution of the corresponding Fokker–Planck equation by the finite element and path integral methods. The results are verified by Monte Carlo simulation methods. Adaptive time step integration procedure (ATSP) is adopted which accurately determines the point of discontinuity. A bisection method and a Brownian tree approach are used in this process and direct the solution along the correct Brownian path. Numerical results are also obtained using non-smooth coordinate transformations like the Zhuvarlev and Ivanov transformations converting the discontinuous systems to equivalent smooth systems and compared with the results of the ATSP. The Filippov convex transformation is used in the case of the dry friction nonlinearity in the integration near the discontinuity. The results are discussed with respect to some examples like the Duffing and Van der Pol oscillators with impact and dry friction.
... Xu et al. [13] introduced the stochastic averaging of energy envelope for Duffing-type VEHs. Kumar et al. [14] used the finite element method to solve the Fokker-Planck-Kolmogorov (FPK) equation of the associated bistable energy harvester. Jin et al. [15] introduced the equivalent nonlinearization technique to derive a semi-analytical solution of the corresponding nonlinear VEHs. ...
... By substituting the solution of Eq. (19) into Eq. (13), the path integration PDFs can been calculated by using the initial distribution (14) and Eq. (12). ...
... The system parameters are selected as ζ = 0.2, δ = 0.5, κ = 0.5, α = 0.5, λ = 1, and D = 0.01. Therefore, the path integration PDFs can be rendered by using the initial distribution (14) and Eq. (12). Figure 4 depicts the PDFs and logarithmic PDFs of the displacement, velocity, and electricity for the quadratic-cubic VEH, and Fig. 5 shows the joint PDFs of displacement and velocity obtained with the path integration method and the MCS results of the systems (20) and (21). ...
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A transition Fokker-Planck-Kolmogorov (FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters (VEHs) under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function (PDF) from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the Gauss-Legendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation (MCS).
... Xu et al. [22] proposed a novel decoupling technique to develop a stochastic averaging of energy envelope for Duffing-type vibration-based energy harvesters, and discussed the effects of the system parameters on the mean square output voltage and power. Kumar et al. [23] used the finite element method to solve the FPK equation of the associated bistable energy harvester, and analyzed the effects of the system parameters on the mean square output voltage and power. Jin et al. [24] introduced the generalized harmonic transformation to decouple the electromechanical equations, and applied the equivalent nonlinearization technique to derive a semi-analytical solution of the corresponding nonlinear vibration energy harvesters subjected to Gaussian white noise excitation. ...
... To date all known exact stationary solutions of the FPK equation have been obtained only for degraded cases of the decoupled electromechanical system [14][15][16]. Therefore some approximate methods for solving the FPK equation of the coupled electromechanical system have been reported which included the statistical linearization techniques [17,20,21], the moment differential equations method [19], the finite element method [20,21,23], the stochastic averaging of energy envelope [22], the equivalent nonlinearization technique [24]. ...
... For specified parameters and initial conditions, Eqs. (33) can be numerically integrated via the Stochastic Communication Toolbox in Matlab through the Euler-Maruyama algorithm [23] with a time step of 0.001 and 50 sample trajectories. The stationary probability density of the Eq. ...
... In the last decade, it has been found that bistable structures produce noticeable improvement in the efficiency of mechanical energy absorption compared with traditional monostable structures [14]. They have also proved to be promising in the design of broadband vibration energy harvesters as self-powered sources for portable devices or wireless sensor network sys-tems, and the reader is referred to [15][16][17][18][19] for more details. For other emerging topics, e.g., unidirectional wave propagation characteristics, or applications, e.g., for morphing designs, see [20][21][22][23][24][25][26] and a more comprehensive review in [27]. ...
... It is noted that in this case the inherent continuity of the physical response is neglected and cannot be recovered from the PDF. The forward Kolmogorov equations, i.e., Fokker-Planck equations (FPE) [4,[28][29][30][31], are commonly used and can be seen in some investigations involving stochastic dynamics of bistable structures [11,18,19]. ...
... Consequently, time averaging of a single sample path with remarkably reduced amount of computation replaces necessary ensemble averaging, but the statistics obtained may be unreliable since the response process of a structure may be not ergodic, and the statistical evolutionary regularities of the physical response are not revealed by this approach. On the other hand, when FPEs are adopted [19], the finite element method (FEM) or finite differential method (FDM) is usually needed to solve these equations numerically since their analytical solutions can rarely been found [4,[28][29][30][31][34][35][36]. In this paper, a complete MCS using ensemble averaging and a FPE analysis using FEM are implemented to capture the evolution process of the physical response PDF. ...
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This paper investigates some basic issues on the stochastic dynamic analysis and assessment of bistable structures from an applications perspective, illustrated with a classical spring–mass–rod structure. A complete Lagrangian-description-based Monte Carlo simulation and an Eulerian-description-based Fokker–Planck equation analysis are implemented, respectively, to capture the evolution process of the physical response probability density function, with special focus on the dynamics under the statistical steady state condition. A comparison of these two methods outlines their capabilities. As a representative example, quantitative counting and statistical analysis of the number and amplitudes of snapping-through of the structure indicate that physical quantities for structural assessment may show certain statistical regularities under the statistical steady state condition, which can be utilized efficiently to reduce the efforts of structural assessment without loss of precision.
... One can use two types of arrangements of magnets in these devices. In the first case, one uses the magnetic attraction between harvester's beam and two permanent magnets located symmetrically with respect to the beam's equilibrium position [7][8][9][10][11][12]. In the second case, one uses the piezoelectric cantilever with magnetic tip mass and the repulsion permanent magnet mounted in the beam's equilibrium position [5,10,[13][14][15][16][17]. ...
... In the second case, one uses the piezoelectric cantilever with magnetic tip mass and the repulsion permanent magnet mounted in the beam's equilibrium position [5,10,[13][14][15][16][17]. e energy harvesting device can be adapted to the vibrating system which in real life vibrates in some spectrum of frequencies and additionally, the vibrations have the stochastic origin [7,9,11,13,14,18,19]. Vocca et al. [13] introduced the stochastic term in the equation of motion of the moving magnet. ...
... 11: e effective frequency range of the oscillators versus the load resistance R. ...
Article
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In this paper, we examine the influence of the background’s stochastic excitations on an output power generated by using an energy harvester. The harvester is composed of two magnets attached to a piezoelastic oscillators separated by a distance Δ from the static magnets fastened directly to the device. We also introduce the parameter α which describes the mass ratio of moving magnets. We examine the output power for different excitation frequencies, different values of α , and different amplitudes δ0 of the stochastic force. We also analyze the influence of δ0 and Δ on the effective output power (EOP), the mean value of output power averaged over the considered frequencies, produced by using the harvester. We have observed that increasing δ0 causes the growth of generated mean power, especially in the low-frequency regime, while the maximum power near the resonance frequency remains unchanged. The EOP also grows with increasing δ0 for all examined values of α . The environment’s stochastic behavior improves slightly the harvester’s efficiency as compared to the purely harmonic case. Analyzing the dependence of EOP on Δ , we observed the maximum which appears at values of Δ corresponding to the situation when the system starts to work in the unsynchronized regime.
... We illustrate the method by applying it to the Lorenz-63 system [8] with additive stochastic forcing. Although a phenomenological FPE has been applied for a quantum system without the addition of stochastic forcing [9], we follow previous work [10][11][12][13][14] and add small additive white noise to wash out fractal structure below the lattice scale. ...
... The Lorenz-63 attractor is a three dimensional, chaotic system that was originally derived by applying a severe Galerkin approximation to the equations of motion (EOMs) for Rayleigh-Benard convection with stress-free boundary conditions [8]. We study an extension with additive stochastic forcing [10][11][12][13][14] that obeys Eq. 3. Such additive white noise can model fast or unresolved physical processes that are not explicitly described. ...
Article
We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz-63 attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator using linear algebra. Two variants are also studied: A self-adjoint construction of the linear operator, and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. Comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.
... He and Daqaq [38,39] employed the finite element method of the FPK equation to investigate how the shape of the potential energy function influences the mean steady-state approximate output power. Kumar et al. [23,38,40], the stationary mean square output electric current [16,18,20] and the stationary mean power [18,20,23,38,39,40,41] of energy harvesters will be obtained by using the stationary probability density of the FPK equation. The stationary mean square output voltage (electric current) or the stationary mean power can be used to assess the performance of energy harvesters under random excitation. ...
... He and Daqaq [38,39] employed the finite element method of the FPK equation to investigate how the shape of the potential energy function influences the mean steady-state approximate output power. Kumar et al. [23,38,40], the stationary mean square output electric current [16,18,20] and the stationary mean power [18,20,23,38,39,40,41] of energy harvesters will be obtained by using the stationary probability density of the FPK equation. The stationary mean square output voltage (electric current) or the stationary mean power can be used to assess the performance of energy harvesters under random excitation. ...
Article
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A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.
... Daqaq [16] investigated theoretically a BEH under Gaussian white and exponentially correlated noise and pointed out that bistable harvester is not always preferable to monostable harvester. Kumar et al. [17] proved Daqaq's conclusion through finite element method solving the Fokker-Planck-Kolmogorov (FPK) equation and gave the joint probability distribution functions (PDF) of response. ...
... Similar observations were reported by Kumar et. al. [17] using different values of system parameters. Looking at the effect various viscous damping (γ = 0.05, γ = 0.08) on the joint PDF of displacement (x) and velocity (y) P st (x, y) for D = 0.5κ v = 0.1, κ c = 1, a = 1.5 and b = 0.1 shown in Figs. ...
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In the present study, the dynamic characterization of a flexible spinning shaft with constant eccentricity driven by a non-ideal energy source (DC motor) with external and internal damping has been focused. It is well established that the structural response of a vibratory system to which a non-ideal drive is connected may act as an energy sink under certain conditions such that a part of the energy supplied by the source is spent to vibrate the structure rather than to increase the drive speed. This phenomenon is formally known as the Sommerfeld effect. The Sommerfeld effect characterized by jump phenomena is studied through the steady-state amplitude obtained by instantaneous power balance method and further verified through numerical simulation. Finally, power balance equation is transformed into a characteristic equation through which jump phenomena and Sommerfeld effect are predicted numerically for the first and third modes using root loci method. The break-in and breakaway points of the root loci represent the values of the supply voltage at which jumps in shaft speed and flexural vibration amplitudes take place during coast-up and coast-down operation.
... However, the FPK equation is a partial differential equation (PDE), and it cannot be solved analytically except for special classes of systems [10,11]. In the absence of exact analytical solutions for the FPK equation, researchers try to find approximate analytical solutions for it [12][13][14][15] or intend to solve the corresponding PDE numerically [16,17]. Contrary to the FPK equation, other analytical techniques are approximate methods and are different in terms of simplicity and accuracy. ...
... Monte Carlo simulation of the stochastic differential equation, if executed for long enough times, exposes the statistical properties of the system and may reveal the behavior of the system if repeated for a large set of values for the parameters appeared in the governing equations [20][21][22][23]. Another approach is to solve the FPK equation numerically, which also requires to be repeated for many different parameter values [17,19,24]. In addition to numerical methods, some novel techniques have been proposed to predict the behavior of the bistable systems under random excitation. ...
Article
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There are many physical phenomena in engineering applications that are mostly modeled as stochastic differential equations. Sensor noise and environmental disturbance are two main sources of randomness. Determination of a closed-form analytical solution for the dynamical systems under random excitation is significantly useful for further investigations of these systems. Linear systems with stochastic excitation obey simple classified rules, leading to straightforward procedures for derivation of their analytical solutions. However, it is not the case for nonlinear systems with stochastic excitation, where no comprehensive method has been developed to be applicable to all such systems. This article brings in a novel method for statistical analysis of stochastic nonlinear systems, especially the ones with multiple equilibria, for which the traditional methods such as statistical linearization give incorrect approximations. The proposed method is mainly based on the moment closure method and assumes the joint probability distribution function of the state vector, as a linear combination of several normal distribution functions. The method represents a solution for essentially nonlinear systems, which are far from being linearizable. Duffing oscillator with negative linear stiffness is discussed as a case study to illustrate the advantage of the proposed method compared to the traditional ones. Such nonlinear systems especially arise in energy harvesting applications, when linear harvester designs do not fulfill performance requirements.
... The effects of piezoelectric nonlinearity have been examined in energy harvesting in Refs. [11,12]. With the increase in the nonlinearity, the operation effectiveness decreases. ...
... The simulations have shown that the chaotic behavior of piezoelastic configuration increases the power generation [13,14]. The piezomagnetoelastic configuration has also been examined in Gaussian white noise excitation [12]. Generally, the piezoelectric energy harvesters are made with piezoceramic layers integrated on a beam for energy harvesting from ambient vibration. ...
Article
Full-text available
Typically, two configurations are used for energy harvesting, with different advantages: piezoelastic and piezomagnetoelastic. Best performance of piezoelastic configuration is limited to its narrow bandwidth around the resonance frequency. If the excitation frequency slightly deviates from the resonance frequency, the power-out is severely reduced. To overcome this, the piezomagnetoelastic has been introduced. This configuration can be used in non-resonant frequency for the generation of high power. This paper investigates the effects of frequency on the two mentioned configurations. The results of the study indicate that at high frequency, piezomagnetoelastic operation is better than the piezoelastic; but at low frequencies, this configuration has a weakness.
... Due to the fact that the influence of stochastic bifurcation becomes crucial in the choices made by the system in the course of its evolution between the numerous basins of attraction, or dissipative structures, to which bifurcations give rise [1] , its study in the field of energy harvesting could give rise to a better harvesters understanding and control. A particularly simple demonstration of this feature can be achieved by finding the so-called probability structure of system's response based on the Fokker-Plank-Kolmogorov equation (FPK) [7] . ...
... Although as now, energy harvesting from nonlinear oscillators subjected to random excitation has been the subject of investigation by a number of researchers [16][17][18][19] , the study of stochastic pbifurcation in harvester models just started. Recently Kumar et al. [7] , demonstrated the agreement between their results from the Fokker-Planck approach and Monte Carlo simulation; and a good agreement when compared their simulation with analytic results obtained by Daqaq [20,21] . ...
Article
In this paper, a sandwiched buckled beam with axial compressive force under Gaussian white noise is considered as a piezoelectric energy harvester. A stochastic averaging method is proposed to analytically predict the system's response, the stability and the estimation of system's reliability. By using the generalized harmonic transformation, the Itô differential equations with respect to the mechanical and electrical amplitude are derived through this technique. From these differential equations, we construct the Fokker–Plank–Kolmogorov equation for the electrical and mechanical subsystem where the solution of each equation in the stationary state is a probability density. The mean first passage time (MFPT) is numerically provided in order to study the attractor stability(stable equilibrium point observed in the effective potential) which give rise to the noise-enhanced stability(NES) phenomenon. The mean square response and voltage are obtained for different white noise intensities and others system parameters. The effects of linear damping and noise intensity on the mean square voltage are investigated. We notice that harvested energy can be enhanced by suitable choice of noise intensity and others system parameters. In additional, by combining the random signal with harmonic excitation, the stochastic resonance(SR) phenomenon is observed via the mean residence time(TMR) which give rise to the large amplitude of vibrations and consequently, an optimization of harvested energy. The agreements between the analytical method and those obtained numerically validate the effectiveness of analytical investigations.
... Thus mean power output of PE and EM element of nonlinear hybrid energy harvester are illustrated as: (23) and (24), the output characteristics of our nonlinear hybrid piezoelectric and electromagnetic energy harvester under colored excitation can be presented. In order to validate the proposed theoretical method, Monte Carlo simulation [39,43] was used to analyze the responses of the designed nonlinear hybrid piezoelectric and electromagnetic energy harvester under colored excitation. ...
... Besides, reductions of mean output powers obtained in the other two cases are similar. In conclusion, from [32,[43][44][45], researchers arrived that for Duffing-type PE or EM energy harvester under band-limited excitation tuning the natural frequency at a larger or lower frequencies can obtain higher output power according to the value of bandwidth and nonlinearity. Same results have been found for nonlinear hybrid PE and EM energy harvester excited by colored noise with large bandwidth. ...
Article
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It is well known that when excited by a stochastic base acceleration, the power absorbed by a nonlinear Duffing-type piezoelectric (PE) or electromagnetic (EM) energy harvester might not be increased effectively compared to a linear one. When intentionally introducing nonlinear magnetic forces into a doubly-clamped hybrid PE and EM energy harvester subjected to narrow-band (colored) excitation however, the power output could be improved to a much higher value. Also, in comparison with the typical nonlinear PE or EM generator, the influence of load and excitation parameters on the performance of a nonlinear hybrid energy harvester under colored excitation has been proven rather different as well. These results are derived analytically by solving the Fokker-Planck (FP) equation, and numerically by Monte Carlo (MC) simulations for validation. Besides, for a nonlinear hybrid configuration excited by colored noise approaching white Gaussian excitation, theoretical output characteristics are discussed and compared with results from a reported theory for white Gaussian excited case, which again verifies the feasibility of the theoretical analysis.
... We illustrate this method by applying it to the Lorenz system [7] with additive stochastic forcing. Although a phenomenological FPE has been applied for a quantum system without the addition of stochastic forcing [8], we follow previous work [9][10][11][12][13] and add small additive white noise to wash out fractal structure below the lattice scale. ...
... The Lorenz attractor is a three-dimensional chaotic system that was originally derived by applying a severe Galerkin approximation to the equations of motion (EOMs) for Rayleigh-Bénard convection with stress-free boundary conditions [7]. We study an extension with additive stochastic forcing [9][10][11][12][13] that obeys Eq. (3). Such additive white noise can model fast or unresolved physical processes that are not explicitly described, ...
Article
We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.
... Daqaq [16] investigated theoretically a BEH under Gaussian white and exponentially correlated noise and pointed out that bistable harvester is not always preferable to monostable harvester. Kumar et al. [17] proved Daqaq's conclusion through finite element method solving the Fokker-Planck-Kolmogorov (FPK) equation and gave the joint probability distribution functions (PDF) of response. ...
... Similar observations were reported by Kumar et. al. [17] using different values of system parameters. Looking at the effect various viscous damping (γ = 0.05, γ = 0.08) on the joint PDF of displacement (x) and velocity (y) P st (x, y) for D = 0.5κ v = 0.1, κ c = 1, a = 1.5 and b = 0.1 shown in Figs. ...
Article
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In this paper, the system parameters of a nonlinear bistable energy harvester excited by Gaussian white noise was investigated. Using Fokker–Planck–Kolmogorov equation, the probability distribution functions of displacement and velocity of the oscillator are obtained. The effect of various system parameters on the probability distributions of displacement and velocity of the oscillator and the mean square of the output voltage are investigated when the time constant of the piezoelectric circuit takes a larger value to achieve maximum voltage gain. The maximum peak values of the joint probability distribution of displacement and velocity of the oscillator decrease with the larger values of noise strength. The effect of parameters of bistable potential function on mean square of output voltage was also investigated. The system equations are numerically solved and mean square value of output voltage is numerically estimated and is seen to be increased as noise intensity increases and decreased as viscous damping increases. The result also shows that better power output can be achieved when the time constant takes larger value.
... The effects of piezoelectric nonlinearity have been examined in energy harvesting in Refs. [11,12]. With the increase in the nonlinearity, the operation effectiveness decreases. ...
... The simulations have shown that the chaotic behavior of piezoelastic configuration increases the power generation [13,14]. The piezomagnetoelastic configuration has also been examined in Gaussian white noise excitation [12]. Generally, the piezoelectric energy harvesters are made with piezoceramic layers integrated on a beam for energy harvesting from ambient vibration. ...
Article
Full-text available
Typically two configurations are used for energy harvesting with different advantages: piezoelastic and piezomagnetoelastic. Best performance of the piezoelastic configuration is when the excitation frequency is close to the resonance frequency. If the input frequency slightly deviates from the natural frequency, the generated power is severely decreased. To tackle the problem, the piezomagnetoelastic configuration has been introduced. This configuration can be used near the non-resonant frequencies. This paper examines the effects of frequency and damping in the two above-mentioned configurations. The results of the study indicate that with increasing the damping, the harvested energy decreases. Also the results show that at higher frequencies, piezomagnetoelastic operation is better than the piezoelastic one; but at low frequencies, piezoelastic configuration is the better option.
... and equivalent linearization method, Green deduced output characteristics of nonlinear EM energy harvester based magnetic spring, and got that natural frequency of energy harvester can be regulated with no change of structural mass and stiffness by means of Monte Carlo simulation and experimental test (Green et al. 2012;Green et al. 2013). In addition, Kumar establish state equation of coupled electromechanical characteristics by FPK equation, and analyzed output voltage of nonlinear energy harvester in different acceleration spectral density (Kumar et al. 2014). Meimukhin analyzed output power of nonlinear energy harvester based soft spring structure under white noise excitation, and derived that nonlinear structures with negative stiffness can be used to enhance the conversion, and bistable oscillators performance considerably better than their linear counterpart under band-limited excitation (Meimukhin et al. 2013). ...
Article
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For the designed nonlinear hybrid piezoelectric (PE)–electromagnetic (EM) energy harvester, electromechanical coupling state equations are established at stochastic excitation, and vibration response, output mean power, voltage and current are derived by statistical linearization method. Then, effects of nonlinear strength, load resistance and excitation spectral density on vibration response and electric output of nonlinear hybrid energy harvester are studied by theoretical analysis, simulation and experimental test. It is obtained that mean power of nonlinear hybrid energy harvester increases linearly with acceleration spectral density; the bigger nonlinear strength, the bigger output power of energy harvester and the lower resonant frequency are; besides, mean amplitude of nonlinear hybrid energy harvester reaches the minimum at PE optimal load, but it increases with EM load increasing. Compared with linear hybrid energy harvester, the resonant frequency of nonlinear energy harvester can be decreased by 57%, while output power can be increased by 72%.
... The PDF is an important characteristic quantity for analyzing the transition phenomenon of noise-induced systems. Analytical and numerical methods are available in the literature to calculate the joint and marginal PDFs up to a four-dimensional system [31][32][33] . In this study, the Monte Carlo simulation is used to obtain the joint and marginal PDFs and then explore the effect of Gaussian white noise intensity on the response of the modified snap-through hybrid VEH. Figure 14 shows the joint PDFs for three values of noise intensity. ...
Article
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Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications. Enhancing the performance of a vibration energy harvester (VEH) incorporating nonlinear techniques, for example, the snap-through VEH with geometric non-linearity, has gained attention in recent years. A conventional snap-through VEH is a bi-stable system with a time-invariant potential function, which was investigated extensively in the past. In this work, a modified snap-through VEH with a time-varying potential function subject to harmonic and random base excitations is investigated. Modified snap-through VEHs, such as the one considered in this study, are used in wave energy harvesters. However, the studies on their dynamics and energy harvesting under harmonic and random excitations are limited. The dynamics of the modified snap-through VEH is represented by a system of differential algebraic equations (DAEs), and the numerical schemes are proposed for its solutions. Under a harmonic excitation, the system exhibits periodic and chaotic motions, and the energy harvesting is superior compared with the conventional counterpart. The dynamics under a random excitation is investigated by the moment differential method and the numerical scheme based on the modified Euler-Maruyama method. The Fokker-Planck equation representing the dynamics is derived, and the marginal and joint probability density functions (PDFs) are obtained by the Monte Carlo simulation. The study shows that the modified snap-through oscillator based VEH performs better under both harmonic and random excitations. The dynamics of the system under stochastic resonance (SR) is investigated, and performance enhancement is observed. The results from this study will help in the development of adaptive VEH techniques in the future.
... Lefeuvre (Lefeuvre et al., 2007) were one of the first to investigate an energy harvesting system subjected to a random excitation. This same system was also studied by Kumar (Kumar et al., 2014), where a Gaussian white noise was considered and an investigation of the effects of the system parameters on the mean square voltage was studied. De Paula (De Paula et al., 2015) presented a numerically and experimentally investigation of the piezomagnetoelastic system when it is subjected to a Gaussian white noise and presented the influence of nonlinearities in the system. ...
Conference Paper
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The renewable energy is in the focus of many researches in the last decades, and the use of piezoelectric material can be used to obtain one source of this renewable energy. In this case, energy harvesting explores mainly the source of ambient motion and the piezoelectric material convert mechanical energy, present in the ambient motion, into electrical energy. In the work, we present a nonlinear bistable piezomagnetoelastic structure subjected to harmonic and random base excitation. At first, harmonic excitation is of concern and then, the system subjected to random excitation is analyzed. The goal of the numerical analysis is to present an investigation of the best electrical output response of the system given harmonic and random excitations.
... However, as opposed to the large amount of research on harmonic excitation (Tang et al., 2010), limited research has been conducted for non-harmonic or random excitations of resonant harvesters. Random vibration energy harvesting has been studied with lumped-parameter (Adhikari et al., 2009;Barton et al., 2010;Blystad et al., 2010;Ferrari et al., 2009;Scruggs, 2009;Tvedt et al., 2010) and distributed-parameter (Zhao and Erturk, 2013a) modeling, while nonlinear structures with higher complexities, such as Duffing oscillators, have been modeled and analyzed by others (Daqaq, 2010(Daqaq, , 2011Kumar et al., 2014;Litak et al., 2010;Ramlan et al., 2010;Zhao and Erturk, 2013b) in parallel. ...
Article
Vibrational energy harvesting using piezoelectric cantilever beams has received significant attention over the past decade. When compared to piezoelectric cantilever-based harvesters, piezopatch energy harvesters integrated on plate-like thin structures can be a more efficient and compact option to supply electrical power for wireless structural health and condition monitoring systems. In this article, electroelastic modeling, analytical and numerical solutions, and experimental validations of piezopatch-based energy harvesting from stationary random vibrations of thin plates are presented. Electroelastic models for the series and parallel connected multiple piezopatches are given based on a distributed-parameter modeling approach for a thin host plate excited by a transverse point force. The analytical and numerical solutions for the mean power output and the mean-square shunted vibration response are then derived. The experimental measurements are carried out by employing a fully clamped thin plate with three piezopatches connected in series. It is shown that the analytical and numerical model predictions for the mean power output and the mean-square velocity response are in very good agreement with the experimental measurements. The electroelastic modeling framework and solution methods presented in this work can be used for design, performance analysis, and optimization of piezoelectric energy harvesting from stationary random vibration of thin plates.
... Assume that excitation ηðtÞ is a Gaussian white noise with intensity D. This excitation is considered as a good model of ambient broadband excitations (see e.g. [16][17][18][19][20][21][22][23]. The processes ηðtÞ and telegraphic process ξðtÞ are assumed to be independent. ...
... Lefeuvre (Lefeuvre et al., 2007) were one of the first to investigate an energy harvesting system subjected to a random excitation. The system was also studied by Kumar (Kumar et al., 2014), a Gaussian white noise was considered, and an investigation of the effects of the system parameters on the mean square voltage was studied. De Paula (De Paula et al., 2015) presented a numerically and experimentally investigation of the piezomagnetoelastic system when it is subjected to a Gaussian white noise and presented the influence of nonlinearities in the system. ...
Conference Paper
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Many alternative energy sources have been investigated in the last decades, energy harvesting from the environment is of these possibilities and have been explored recently by using piezoelectric material. This vibration-based energy harvesting using piezoelectric elements is possible by exploring the direct effect, where the piezoelectric material is able convert mechanical in to electrical energy, and can be very useful for applications in powering small electronic devices. Analyses involving oscillators subjected to harmonic excitations have been well studied, demonstrating that nonlinearities can enhance the harvested energy, mostly the nonlinearities that lead to bistable oscillators. This work addresses the influence of nonlinearities in energy harvesting from a piezomagnetoelastic structure subjected to random and harmonic vibrations. The energy harvesting system is a magnetoelastic structure that consists of a ferromagnetic cantilevered beam with two permanent magnets, one located in the free end of the beam and the other at a vertical distance d from the beam free end, subjected to harmonic and random base excitation. In order to use this device as a piezoelectric power generator, two piezoceramic layers are attached to the root of the cantilever and a bimorph generator is obtained. Nonlinear equations of motion that describe the electromechanical system are given along with theoretical simulations. The numerical analysis presents a comparison between the average electrical power provided from nonlinear system and the Power Spectral Density (PSD) of the electrical power due to two different excitations, at first a harmonic excitation after random excitation. The goal of the proposed analysis is to establish how the average electrical power and PSD value evaluated the system.
... For the linear harvester, the response and output power to single harmonic, period and random excitations are analytically derived by applying the method of the undetermined coefficients [51,52]. For the nonlinear energy harvester, the deterministic response can be determined through method of multi scale [53], while the random response is evaluated by Monte-Carlo simulation [54], the moment method [55], the finite element method [56], the Galerkin procedure [57], the equivalent linearization technique [58], the stochastic averaging [59] and the equivalent non-linearization technique [60]. ...
Article
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As rapid development in wearable/implantable electronic devices benefit human life in daily health monitoring and disease treatment medically, all kinds of flexible and/or stretchable electronic devices are booming, together with which is the demanding of energy supply with similar mechanical property. Due to its ability in converting mechanical energy lying in human body into electric energy, energy harvesters based on piezoelectric materials are promising for applications in wearable/ implantable device’s energy supply in a renewable, clean and life-long way. Here the mechanics of traditional piezoelectrics in energy harvesting is reviewed, including why piezoelectricity is the choice for minor energy harvesting to power the implantable/wearable electronics and how. Different kinds of up to date flexible piezoelectric devices for energy harvesting are introduced, such as nanogenerators based on ZnO and thin and conformal energy harvester based on PZT. A detailed theoretical model of the flexible thin film energy harvester based on PZT nanoribbons is summarized, together with the in vivo demonstration of energy harvesting by integrating it with swine heart. Then the initial researches on stretchable energy harvesters based on piezoelectric material in wavy or serpentine configuration are introduced as well. © 2015, Science China Press and Springer-Verlag Berlin Heidelberg.
... It was pointed out that the bistable harvester was not preferable to the monostable harvester always. Kummar [20] used the finite element method to solve the FP equation and gave the joint probability density functions of response as well as the output voltage. His analyses proved Daqaq's conclusion. ...
... The PDF solution is governed by the FP equation. The form of the FP equation is too complicated to be solved exactly for the coupled equations of an energy harvester, even for a stationary PDF solution [27]. ...
Article
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This paper proposes a solution procedure to formulate an approximate joint probability density function (PDF) of a Duffing-type energy harvester system under Gaussian white noise. The joint PDF solution of displacement, velocity, and an electrical variable is governed by the Fokker-Planck (FP) equation. First, the FP equation is reduced to a lower-dimensional FP equation only about displacement and velocity by a state-space-split (SSS) method. The stationary joint PDF of displacement and velocity can be solved exactly. Then, the joint PDF of displacement, velocity, and the electrical variable can be approximated by the product of the obtained exact PDF and the conditional Gaussian PDF of the electrical variable. A parametric study is further conducted to show the effectiveness of the proposed solution procedure. The study considers weak nonlinearity, strong nonlinearity, high excitation level, and a bistable oscillator. Comparison with the simulated results shows that the proposed solution procedure is effective in obtaining the joint PDF of the energy harvester system in the examined examples.
... . In other words, this finding supports results in the literature indicating that bistable harvesters outperform, in general, monostable harvesters (e.g., Refs. [5,22,30,32,33,50], and [51]). ...
Article
A methodology based on the Wiener path integral technique (WPI) is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation data demonstrate the satisfactory performance of the methodology.
... This observation was similar to the one mentioned in Zhao and Erturk (2013) regarding the bi-stable harvester. Kumar et al. (2014) derived the FP equations for the bi-stable Duffing oscillator (piezo-magneto-elastic configuration) subjected to the Gaussian white noise excitation. These equations were solved to obtain the joint and marginal PDFs of the response and voltage, using the FE method that involves the Galerkin Projection approach. ...
Article
Small and micro-scale energy harvesting is an essential and viable option for the powering of portable and maintenance free electronic devices, wireless sensor nodes, and similar applications. In this regard, piezoelectric harvesters have presented promising outcomes. This article provides a sequential, comprehensive, and informative survey of potential well based models and studies related to piezoelectric harvesters (PEH). Piezoelectric materials used for energy harvesting are discussed briefly, following which a non-dimensional generalized model is derived to set the discussion on a common platform. Dynamics of various potential well configurations are presented using the generalized model before discussing specific models and related studies. The survey is classified into symmetric and asymmetric potential well categories. Under the symmetric head, lumped and distributed parameter linear models and tuning methods for improving the broadband response are discussed. Subsequently, studies related to nonlinear mono-stable, bi-stable, and tri-stable potentials showing interwell, multi-periodic and chaotic oscillations with improved broadband response are discussed. The asymmetric section studies the influence of asymmetries on the performance of the mono-stable, bi-stable, and tri-stable configurations. Few other configurations outside the cantilever type PEH were mentioned, realizing the widespread research in this field. Important observations and future challenges for performance improvement are also discussed.
... It was found, however, that the optimal parameters of regenerative series TMD for vibration control and energy harvesting were very close to each other when white noise input was considered (Zuo and Wen 2013). Meanwhile, the energy harvesting performance of linear or nonlinear harvesters was analyzed comprehensively by considering stochastic excitations (Hawes and Langley 2017;Joo and Sapsis 2014;Kumar et al. 2014;Langley 2014Langley , 2015. To date, however, the ambiguity still exists in the answer regarding the consistency between vibration control and energy harvesting for a structure-EMDEH system. ...
Article
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With the extensive studies on energy harvesting via electromagnetic devices in civil or mechanical structures, a novel dual-function device, termed electromagnetic damping and energy harvesting (EMDEH) device, has been proposed in recent years, whose design represents a dual-objective optimization problem by considering both vibration control and energy harvesting performance. However, an argument on whether or not vibration control and energy harvesting is contradictory has arisen, based mainly on the intuition that the former tries to suppress vibration magnitude while the latter benefits from large vibration amplitude. This paper clarifies this question through a theoretical analysis of coupled structure-EMDEH systems under stochastic excitations. The closed-form solutions of the damping power, as well as the output power of the EMDEH device when attached to single-degree-of-freedom (SDOF) and multidegree-of-freedom (MDOF) structures, are given based on random vibration theory. A numerical analysis of a 20-story steel structure installed with EMDEH devices is conducted to validate the theoretical predictions. The consistency between vibration control and energy harvesting in randomly excited structures with EMDEH devices in the presence of inherent structural damping is demonstrated for the first time.
... However only two and three dimensional systems were solved using the proposed methodology. Recently Kumar et al. (2014) and He and Daqaq (2014) have used the Bubnov-Galerkin FE formulation to study three dimensional vibratory energy harvester under additive Gaussian white excitations. They have used the FE method to solve the associated 3D FP equation to obtain the approximate pdfs of the response as well as the voltage generated from the piezoelectric patches. ...
... For other types of SDEs, finding an approximate analytical solution is nontrivial and the procedure is case dependent. One other method is to solve the FP equation numerically for example with the FE method [32], but numeric solution of the FP equation is challenging and massive, especially for high-dimensional state spaces [33,34]. ...
Article
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Statistical analysis of stochastic dynamical systems is of considerable importance for engineers as well as scientists. Engineering applications require approximate statistical methods with a trade-off between accuracy and simplicity. Most exact and approximate methods available in the literature to study stochastic differential equations (SDEs) are best suited for linear or lightly nonlinear systems. When a system is highly nonlinear, e.g., a system with multiple equilibria, the accuracy of conventional methods degrades. This problem is addressed in this article, and a novel method is introduced for statistical analysis of special types of essentially nonlinear SDEs. In particular, second-order dynamical systems with nonlinear stiffness and additive random excitations are considered. The proposed approximate method can estimate second-order moments of the state vector (namely position and velocity), not only in the case of white noise excitation, but also when the excitation is a correlated noise. To illustrate the efficiency, a second-order dynamical system with bistable Duffing-type nonlinearity is considered as the case study. Results of the proposed method are compared with the Gaussian moment closure approximation for two types of colored noise excitations, one with first-order dynamics and the other with second-order dynamics. In the absence of exact closed-form solutions, Monte Carlo simulations are considered as the reference ideal solution. Results indicate that the proposed method gives proper approximations for the mean square value of position, for which the Gaussian moment closure method cannot provide good estimations. On the other hand, both methods provide acceptable estimations for the mean square value of velocity in terms of accuracy. Such nonlinear SDEs especially arise in energy-harvesting applications, when the ambient vibration can be modeled as a wideband random excitation. In such conditions, linear energy harvesters are no longer optimal designs, but nonlinear broadband harvesting techniques are hoped to show much better performance.
... Hence, this large-amplitude inter-well dynamic behavior may significantly improve the electromechanical transduction, leading to high-efficiency energy harvesting [11]. The bistable energy harvester's interwell dynamics for energy harvesting performance are extensively investigated [8,[12][13][14][15][16][17][18][19][20][21][22][23]. From those studies, it was discovered that the effective way for highefficiency energy harvesting is to improve the feature of the inter-well dynamic response by enhancing its stability, bandwidth and response amplitude [11]. ...
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This paper proposes two new coupled lever-bistable nonlinear energy harvesters to enhance the inter-well dynamic response for improvement of vibration energy harvesting. For the first harvester, the oscillator mass and lever-supporting mass are on different sides of the lever pivot; for the second, both the masses are on the same side. The fundamental-periodic inter-well dynamics of both the lever-bistable energy harvesters are analytically, numerically and experimentally investigated in this study. Their variation trends are firstly studied analytically with respect to different lever parameters, which are also mathematically interpreted afterward. Subsequently, experiments are conducted to validate the theoretically predicted variation trends. The analytical and experimental results show that both the lever-bistable energy harvesters with appropriate lever parameters can significantly outperform the conventional bistable energy harvester. Finally, through numerical investigations, this paper reveals that different initial conditions of the lever-bistable energy harvesters can lead to other different types of inter-well responses apart from the fundamental-periodic responses, e.g., subharmonic response. The numerical results show that the fundamental-periodic inter-well response is more beneficial to energy harvesting than other inter-well responses. The basin-of-attraction map corresponding to each type of inter-well response is drawn to describe the distributions of each type’s initial condition. Based on the basin-of-attraction maps, both the lever-bistable energy harvesters’ occurring probabilities of exhibiting fundamental-periodic inter-well response are evaluated, which can quantitatively illustrate the lever structural benefit to stabilization of this favorable inter-well response.
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The shallow cylindrical structure is suitable to develop broadband vibration energy harvesters due to the property of the inherent mechanical bistability. In this letter, the optimum design of the bistable cylindrical shell for broadband energy harvesting application is investigated from the structural point of view. The output power is evaluated by the concept of the harvestable power, which balances the frequency of snap through and the referred output energy associated with each snap through. The non-dimensional harvestable power is analytically expressed as the function of the non-dimensional curvature parameter and one constructed parameter. The universal dependence of the optimal curvature parameter and the associated optimal harvestable power on the constructed parameter is derived, which can be well approximated by the linear relation in double logarithmic coordinate.
Article
The nonlinear beam with vertical combined excitations is proposed as an energy harvester. The system is modelled as a cantilever beam with included a tip mass and piezoelectric patches which transduce the bending strains induced by both, the harmonic and as an additive stochastic forces. The excitation affects in vertical directions by kinematic forcing into electrical charge. The main goal is to analyse the dynamics of the electro-mechanical beam system and the influence of the mixed excitation forces into an effectiveness of the energy harvesting. Overcoming the potential barrier by the beam system is also analysed, where large output amplitudes occur. Such region of the vibration affects more power generation as well as voltage spikes, which is crucial in terms of load resistors sensitivities. By increasing the additive noise level with fixed harmonic force we observe the transition from single well oscillations to inter-well stochastic jumps, and the output power is measured during different system behaviours.
Article
A path integration procedure based on Gauss-Legendre integration scheme is developed to analyze probabilistic solution of nonlinear vibration energy harvesters (VEH) in this paper. First, traditional energy harvesters are briefly introduced and their non-dimensional governing and moment equations are given. These moment equations could be solved through the Runge-Kutta and Gaussian closure method. Then, the path integration method is expanded to three-dimensional situation, solving the probability density function (PDF) of VEH. Three illustrative examples are considered to evaluate the effectiveness of this method. The effectiveness of nonlinearity of traditional monostable VEH and a bistable VEH are further studied too. At the same time, Equivalent linearization method(EQL) and Monte Carlo simulation are employed too. The results indicate that three-dimensional path integration method can give satisfactory results for the global PDF, especially for the tail PDF, and they have better agreement with the simulation results than those of the EQL. In addition, the different degrees of hardening and softening behaviors of the PDFs occur when the nonlinearity coefficient increases and the bistable type is considered.
Article
The piezomagnetoelastic energy harvester system subjected to harmonic and Poisson white noise excitations is studied by using the generalized cell mapping method. The transient and stationary probability density functions (PDFs) of response based on the global viewpoint are obtained by the matrix analysis method. Monte Carlo simulation results verify the accuracy of this method. It can be observed that evolutionary direction of transient and stationary PDFs is in accordance with the unstable manifold for this system, and a stochastic P-bifurcation occurs as the intensity of Poisson white noise increases. This study presents an efficient numerical tool to solve the stochastic response of a three-dimensional dynamical system and provides a new idea to analyze the energy harvester system.
Article
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The nonlinear beam with vertical combined excitations is proposed as an energy harvester. The nonlinearities are included both, in the beam model and also in the electrical subsystem. The system is modelled as a cantilever beam with included a tip mass and piezoelectric patches which convert the bending strains induced by both, the harmonic and the additive stochastic forces. The excitation affects in vertical directions by kinematic forcing into electrical charge. The first main goal is to analyse the dynamics of the electro-mechanical beam system and the influence of the mixed excitation forces into an effectiveness of the energy harvesting. Overcoming the potential barrier by the beam system is also analysed, where large output amplitudes occur. Such region of the vibration affects more power generation, which is crucial in terms of load resistors sensitivities. By increasing the additive noise level with fixed harmonic force it is observed the transition from single well oscillations to inter-well stochastic jumps. The second mail goal is analysing the influence of the piezoelectric nonlinear characteristic and compare the results to the linear piezoelectric cases. The output power is measured during different system behaviours provided by different piezoelectric characteristic as well as introduced stochastic components by modulated tip mass of the system.
Article
The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.
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The use of nonlinear dynamic phenomena for frequency bandwidth enhancement in vibration-based energy harvesting has received growing attention over the last few years. Various designs have been studied to create Duffing oscillators usually by introducing magnetoelastic coupling. In such devices, magnetic forces are typically coupled with elastic beams involving ferromagnetic components to achieve a nonlinear restoring force of the monostable or bistable Duffing type. Other than the increased volume and structural complexity due to additional magnets and discrete components, these magnetoelastic piezoelectric energy harvesters are not suitable to use in various compact applications and in systems that are sensitive to magnetic fields. The M-shaped structural configuration studied in this work overcomes these issues due to the asymmetric stiffness nonlinearity created by a simple structural configuration composed of a bent spring steel beam with piezoelectric patches. The electroelastic dynamics of the Mshaped broadband piezoelectric energy harvester is governed by various interacting nonlinearities, such as the deliberately introduced stiffness nonlinearity of hardening type resulting from the substrate geometry, inherent elastic nonlinearities of softening type as well as hysteretic losses associated with piezoelectric patches, and other dissipative effects due to large velocities experienced in response to base excitation. A recently developed nonlinear non-conservative electroelastic modeling framework for the piezoelectric patches is combined with geometric nonlinearities of the M-shaped energy harvester to establish a nonlinear dissipative model of the electromechanically coupled system. Energy harvesting experiments for a set of resistors and base excitation levels are then performed to experimentally characterize the bandwidth enhancement using the M-shaped broadband piezoelectric energy harvester.
Article
This paper develops a new solution procedure for obtaining the joint probability density function (PDF) of non-linear energy harvesters under Poisson impulses using the state-space-split method and the exponential-polynomial closure method. In the beginning, a generic electromechanical energy harvester is introduced and its governing equations are transformed into non-dimensional ones. According to the non-dimensional governing equations, a Duffing-type oscillator with piezoelectric conversion mechanism is further considered in this study. The proposed solution procedure includes three steps. First the joint PDF of this system is governed by the generalized Fokker-Plank-Kolmogorov (FPK) equation. The state-space-split method is used to reduce this generalized FPK equation to a low-dimensional one only about displacement and velocity. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation. Finally, the joint PDF of displacement, velocity and voltage can be approximated by the product of the obtained PDF and the conditional Gaussian PDF of voltage. Four cases are investigated considering the effects of non-linearity coefficient, impulse arrival rate and a bistable Duffing oscillator. The PDFs of these state variables and the harvested power are compared with the simulation results, respectively. The good agreement between the obtained PDFs and the simulated results is achieved. Compared with the results of the equivalent linearization method, the strong non-linearity in displacement and lower impulse rate lead the tail PDF of the harvested power to exhibiting hardening and softening behaviors, respectively. For the bistable Duffing oscillator, the PDF of the harvested power differs significantly from the result of the equivalent linearization method in the tail region.
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Vibration energy harvesting provides prominent potential on leveraging the converted energy to realize self-powered electronics. To satisfy the demand of electronics and rechargeable batteries for DC voltages in practical applications, the high-performing, nonlinear bistable energy harvesters are considered to be interfaced with standard rectifying electrical circuits to extract DC power from environment-like, random base accelerations. To lead to an effective and efficient set of design guidelines for system development, this research proposes a theoretical method to characterize the stationary stochastic dynamic responses and the energy harvesting performance under white noise accelerations. Considering that the bistable harvesters possess multiple vibration regimes which induce drastically different energy harvesting performances, a novel state-probability estimation approach is presented based on the theoretically predicted probability density function (PDF) of system energy to statistically classify the stationary probability of the stochastic vibrations being in the snap-through or intrawell states. Via investigating the effects of the base acceleration strength and system parameters on the dynamic responses and harvested DC power, it is revealed that high energy harvesting performance would be achieved via low system damping and suitable system parameters, such as moderate coupling constant and load resistance. The theoretical predictions are compared with numerical simulations and experimental results to validate the effectiveness of the proposed theoretical method.
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In this paper, we propose a penta-stable energy harvest (PEH) and demonstrate that it can reach a high efficiency in harvesting vibration energy. The PEH is composed of a cantilever beam with a tip magnet and a pedestal with four fixed magnets. To realize the best penta-stability, the four magnets are mounted on two inclined planes and oriented with their opposite poles facing the tip magnet such that the ensuing attractive forces create five stable equilibrium positions. Compared with bi-stable energy harvester (BEH), the potential energy of PEH owns shallower potential wells and farther outermost potential wells, implying that it is easy to realize snap-through and large amplitudes. The validation experiments are carried out, and the results prove that the PEH can realize snap-through between potential wells more easily and generate higher output powers than the BEH even for a weak random excitation. Therefore, the proposed PEH provides a promising alternative in practical applications.
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In this paper, recent published techniques for vibration energy harvesting with piezoelectric materials have been summarized. The various techniques described are classified in terms of linear and nonlinear vibration energy harvesters, harvesting electrical circuits, large scale vibration energy harvesting concept. The focus is on linear vibration energy harvester but with multiple resonant mode models. The paper concluded with an overview of vibration energy harvesting techniques that aim to maximize the extracted power and the future utilization of the vibration energy harvester. © 2014 Praise Worthy Prize S.r.l. - All rights reserved.
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In this paper, an electromechanical energy harvesting system exhibiting the fractional properties and subjected to the harmonic excitation is investigated. The main objective of this paper is to discuss the system performance with parametric coupling and fractional derivative. The dynamic of the system is presented, plotting bifurcation diagram, poincaré map, power spectral density and phase portrait. These results are confirmed by using test. The harmonic balance method is used with the goal to provide the analytical response of the electromechanical system. The numerical simulation validates the results obtained by this analytical technique. In addition, replacing the harmonic by the random excitation, the impact of noise intensity, the fractional order derivatives κ and the amplitude of the parametric coupling γ is investigated in detail. It points out from these results that for the best choice of D, κ and γ, the output power can be improved.
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Typically two configurations are used for energy harvesting with different advantages: piezoelastic and piezomagnetoelastic. Best performance of the piezoelastic configuration is when the excitation frequency is close to the resonance frequency. If the input frequency slightly deviates from the natural frequency, the generated power is severely decreased. To tackle the problem, the piezomagnetoelastic configuration has been introduced. This configuration can be used near the non-resonant frequencies. This paper examines the effects of frequency and damping in the two above-mentioned configurations. The results of the study indicate that with increasing the damping, the harvested energy decreases. Also the results show that at higher frequencies, piezomagnetoelastic operation is better than the piezoelastic one; but at low frequencies, piezoelastic configuration is the better option.
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Vibration to electricity energy conversion strategies are discussed by using nonlinear stochastic dynamics. General principles for the exploitation of nonlinear oscillators in energy harvesting that provide useful leads for the realization of micropower generators of practical interest are presented.
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The concept of capturing the normally lost energy surrounding a system and converting it into electrical energy that can be used to extend the lifetime of that system’s power supply or possibly provide an endless supply of energy to an electronic device has captivated many researchers and has brought forth a growing amount of attention to power harvesting. One of the most common methods of obtaining the energy surrounding a system is to use piezoelectric materials. Piezoelectric materials have a crystalline structure that provides a unique ability to convert an applied electrical potential into a mechanical strain or vice versa, or convert an applied strain into an electrical current. The latter of these two properties allows the material to function as a power harvesting medium. In most cases the piezoelectric material is strained through the ambient vibration around the structure, thus allowing a frequently unused energy source to be utilized for the purpose of powering small electronic systems. However, the amount of energy generated by these piezoelectric materials is far smaller than that needed by most electronic devices. For this reason, the methods of accumulating and storing the energy generated, until sufficient power has been captured, is the key to developing completely self-powered systems. This article quantifies the amount of energy generated by a piezoelectric plate and investigates two methods of accumulating the energy thus produced. The first method uses a capacitor, which in early research has been the most common method of storing the energy generated and the second utilizes rechargeable nickel metal hydride batteries. The advantages of each method are discussed and the rechargeable battery is found to have more desirable qualities for power harvesting than the capacitor. Additionally, this manuscript represents, for the first time, the fact that the power output by a piezoelectric material is capable of recharging a discharged battery. Through the excitation of a piezoelectric plate, it is demonstrated that a 40 mAh battery can be charged in less than half an hour at resonance and in only a few hours with a random signal similar to that of a typical vibrating piece of machinery.
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Energy harvesting for the purpose of powering low power electronic sensor systems has received explosive attention in the last few years. Most works using deterministic approaches focusing on using the piezoelectric effect to harvest ambient vibration energy have concentrated on cantilever beams at resonance using harmonic excitation. Here, using a stochastic approach, we focus on using a stack configuration and harvesting broadband vibration energy, a more practically available ambient source. It is assumed that the ambient base excitation is stationary Gaussian white noise, which has a constant power-spectral density across the frequency range considered. The mean power acquired from a piezoelectric vibration-based energy harvester subjected to random base excitation is derived using the theory of random vibrations. Two cases, namely the harvesting circuit with and without an inductor, have been considered. Exact closed-form expressions involving non-dimensional parameters of the electromechanical system have been given and illustrated using numerical examples.
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This paper presents an analysis of piezomagnetoelastic energy harvesters under broadband random ambient excitations for the purpose of powering low-power electronic sensor systems. Their nonlinear behavior as a result of vibration in a magnetic field makes piezomagnetoelastic energy harvesters different from classical piezoelastic energy harvesters. An equivalent linearization-based analytical approach is developed for the analysis of harvested power. A closed-form approximate expression for the ensemble average of the harvested power is derived and validated against numerical Monte Carlo simulation results. Our results show that it is possible to optimally design the system such that the mean harvested power is maximized for a given strength of the input broadband random ambient excitation.
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The process of acquiring the energy surrounding a system and converting it into usable electrical energy is termed power harvesting. In the last few years, there has been a surge of research in the area of power harvesting. This increase in research has been brought on by the modern advances in wireless technology and low-power electronics such as microelectromechanical systems. The advances have allowed numerous doors to open for power harvesting systems in practical real-world applications. The use of piezoelectric materials to capitalize on the ambient vibrations surrounding a system is one method that has seen a dramatic rise in use for power harvesting. Piezoelectric materials have a crystalline structure that provides them with the ability to transform mechanical strain energy into electrical charge and, vice versa, to convert an applied electrical potential into mechanical strain. This property provides these materials with the ability to absorb mechanical energy from their surroundings, usually ambient vibration, and transform it into electrical energy that can be used to power other devices. While piezoelectric materials are the major method of harvesting energy, other methods do exist; for example, one of the conventional methods is the use of electromagnetic devices. In this paper we discuss the research that has been performed In the area of power harvesting and the future goals that must be achieved for power harvesting systems to find their way into everyday use.
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We model and experimentally validate a nonlinear energy harvester capable of bidirectional hysteresis. In particular, both hardening and softening response within the quadratic potential field of a power generating piezoelectric beam (with a permanent magnet end mass) is invoked by tuning nonlinear magnetic interactions. Not only is this technique shown to increase the bandwidth of the device but experimental results additionally verify the capability to outperform linear resonance. Engaging this nonlinear phenomenon is ideally suited to efficiently harvest energy from ambient excitations with slowly varying frequencies.
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Recently, the idea of using nonlinearity to enhance the performance of vibration-based energy harvesters has been investigated. Nonlinear energy harvesting devices have been shown to be capable of operating over wider frequency ranges delivering more power than their linear counterparts, rendering them more suitable for real applications. In this paper, we propose to exploit the rich nonlinear behavior of a bistable composite plate with bonded piezoelectric patches for broadband nonlinear energy harvesting. The response of the structure is experimentally investigated revealing different large amplitude oscillations. Substantially large power is extracted over a wide frequency range achieving broadband nonlinear energy harvesting.
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A new mechanism is presented for the local amplification and possible global dynamo maintenance of non-axisymmetric large-scale magnetic fields in disk galaxies. Shear in a galactic wind or large-scale flow of ionised gas with components axial and radial to the disk plane may regenerate large-scale magnetic fields. Numerical results are presented from kinematic mathematical models based on a local (thin disk) approximation and an exact three-dimensional formulation. The one-dimensional thin-disk model illustrates the possibility of exponential amplification and the resulting local axial spatial structure of large-scale galactic magnetic fields. Three-dimensional results support the possibility of global wind dynamo action.
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We present a new architecture for wideband vibration-based micro-power generators (MPGs). It replaces a linear oscillator with a piecewise-linear oscillator as the energy harvesting element of the MPG. A prototype of an electromagnetic MPG designed accordingly is analyzed analytically, numerically and experimentally. We find that the new architecture increases the bandwidth of the MPG during a frequency up-sweep, while maintaining the same bandwidth in a down-sweep. Closed-form expressions for the response of the new MPG as well as the up-sweep bandwidth are presented and validated experimentally. Simulations show that under random-frequency excitations, the new MPG collects more energy than the traditional MPG.
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The field of power harvesting has experienced significant growth over the past few years due to the ever-increasing desire to produce portable and wireless electronics with extended lifespans. Current portable and wireless devices must be designed to include electrochemical batteries as the power source. The use of batteries can be troublesome due to their limited lifespan, thus necessitating their periodic replacement. In the case of wireless sensors that are to be placed in remote locations, the sensor must be easily accessible or of a disposable nature to allow the device to function over extended periods of time. Energy scavenging devices are designed to capture the ambient energy surrounding the electronics and convert it into usable electrical energy. The concept of power harvesting works towards developing self-powered devices that do not require replaceable power supplies. A number of sources of harvestable ambient energy exist, including waste heat, vibration, electromagnetic waves, wind, flowing water, and solar energy. While each of these sources of energy can be effectively used to power remote sensors, the structural and biological communities have placed an emphasis on scavenging vibrational energy with piezoelectric materials. This article will review recent literature in the field of power harvesting and present the current state of power harvesting in its drive to create completely self-powered devices.
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A dramatic consumption reduction of integrated circuits related to the development of mobile electronic devices has been reached over the past years, enabling the use of ambient energy instead of batteries. The focus is here on the transformation of ambient mechanical vibrations into electrical energy. This paper compares the performances of a vibration-powered electrical generator using PZT piezoelectric ceramics associated to two different power conditioning circuits. A new approach of the piezoelectric power conversion based on a nonlinear voltage processing is presented and implemented using a particular circuit. Theoretical predictions and experimental results show that the new technique may increase the power harvested by a factor up to 4 compared to the Standard technique. The power optimization problem is in particular examined in the case of broadband, random vibrations.
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This letter considers a nonlinear piezomagnetoelastic energy harvester driven by stationary Gaussian white noise. The increase in the energy generated by this device has been demonstrated for harmonic excitation with slowly varying frequency in simulation and validated by experiment. This paper considers the simulated response of this validated model to random base excitation and shows that the system exhibits a stochastic resonance. If the variance of the excitation were known then the device may be optimized to maximize the power harvested, even under random excitation.
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This paper reviews the state-of-the art in vibration energy harvesting for wireless, self-powered microsystems. Vibration-powered generators are typically, although not exclusively, inertial spring and mass systems. The characteristic equations for inertial-based generators are presented, along with the specific damping equations that relate to the three main transduction mechanisms employed to extract energy from the system. These transduction mechanisms are: piezoelectric, electromagnetic and electrostatic. Piezoelectric generators employ active materials that generate a charge when mechanically stressed. A comprehensive review of existing piezoelectric generators is presented, including impact coupled, resonant and human-based devices. Electromagnetic generators employ electromagnetic induction arising from the relative motion between a magnetic flux gradient and a conductor. Electromagnetic generators presented in the literature are reviewed including large scale discrete devices and wafer-scale integrated versions. Electrostatic generators utilize the relative movement between electrically isolated charged capacitor plates to generate energy. The work done against the electrostatic force between the plates provides the harvested energy. Electrostatic-based generators are reviewed under the classifications of in-plane overlap varying, in-plane gap closing and out-of-plane gap closing; the Coulomb force parametric generator and electret-based generators are also covered. The coupling factor of each transduction mechanism is discussed and all the devices presented in the literature are summarized in tables classified by transduction type; conclusions are drawn as to the suitability of the various techniques.
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In this paper, a method for the design and optimization of an electret-based vibration-to-electric microconverter is presented, using a nonlinear dynamical model of the device. The dynamics of the converter is analyzed in detail, showing the importance of properly accounting for the nonlinearity in the optimization process. A procedure to determine a set of optimization parameters is finally presented.
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This work is concerned with the performance of a single degree of freedom electromagnetic energy harvester when subjected to a broadband white noise base acceleration. First, using the Fokker–Planck–Kolmogorov equation, it is shown that Duffing-type nonlinearities can be used to reduce the size of energy harvesting devices without affecting their power output. This is then verified using the technique of Equivalent Linearisation. Second, it is shown analytically that the optimum load resistance of the device is different to that which is dictated by the principle of impedance matching. This result is then verified experimentally.
Book
1. Probability and Statistics.- 2. Probability and Stochastic Processes.- 3. Ito Stochastic Calculus.- 4. Stochastic Differential Equations.- 5. Stochastic Taylor Expansions.- 6. Modelling with Stochastic Differential Equations.- 7. Applications of Stochastic Differential Equations.- 8. Time Discrete Approximation of Deterministic Differential Equations.- 9. Introduction to Stochastic Time Discrete Approximation.- 10. Strong Taylor Approximations.- 11. Explicit Strong Approximations.- 12. Implicit Strong Approximations.- 13. Selected Applications of Strong Approximations.- 14. Weak Taylor Approximations.- 15. Explicit and Implicit Weak Approximations.- 16. Variance Reduction Methods.- 17. Selected Applications of Weak Approximations.- Solutions of Exercises.- Bibliographical Notes.
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Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.
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A new approach for an efficient numerical implementation of the path integral (PI) method based on non-Gaussian transition probability density function (PDF) and the Gauss–Legendre integration scheme is developed. This modified PI method is used to solve the Fokker–Planck (FP) equation and to study the nature of the stochastic and chaotic response of the nonlinear systems. The steady state PDF, periodicity, jump phenomenon, noise induced changes in joint PDF of the states are studied by the modified PI method. A computationally efficient higher order, finite difference (FD) technique is derived for the solution of higher-dimensional FP equation. A two degree of freedom nonlinear system having Coulomb damping with a variable friction coefficient subjected to Gaussian white noise excitation is considered as an example which can represent a bladed disk assembly of turbo-machinery blades. Effects of normal force and viscous damping on the mean square response are investigated.
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This letter introduces a piezomagnetoelastic device for substantial enhancement of piezoelectric power generation in vibration energy harvesting. Electromechanical equations describing the nonlinear system are given along with theoretical simulations. Experimental performance of the piezomagnetoelastic generator exhibits qualitative agreement with the theory, yielding large-amplitude periodic oscillations for excitations over a frequency range. Comparisons are presented against the conventional case without magnetic buckling and superiority of the piezomagnetoelastic structure as a broadband electric generator is proven. The piezomagnetoelastic generator results in a 200% increase in the open-circuit voltage amplitude (hence promising an 800% increase in the power amplitude).
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This book addresses random vibration of mechanical and structural systems commonly encountered in aerospace, mechanical, and civil engineering. Techniques are examined for determining probabilistic characteristics of the response of dynamic systems subjected to random loads or inputs and for calculating probabilities related to system performance or reliability. Emphasis is given to applications.
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The following problem is considered in this paper: Let xt be a solution to the stochastic differential equation: dxt = m[xt, t] dt+ σ[xt, t] dyt where yt is the Brownian motion process. Let xt(n) be the solution to the ordinary differential equation which is obtained from the stochastic differential equation by replacing yt with yt(n) where yt(n) is a continuous piecewise linear approximation to the Brownian motion and yt(n) converges to yt as n → ∞. If xt is the solution to the stochastic differential equation (in the sense of Ito) does the sequence of the solutions xt(n) converge to xt? It is shown that the answer is in general negative. It is however, shown that xt(n) converges in the mean to the solution of another stochastic differential equation which is: .RésuméL'auteur étudie le problème suivant: xt est une solution de l'équation différentielle stochastique dxt = m(xt, t)dt + σ (xt, t)dyt où yt représente le mouvement Brownien. Soit xt(n), une solution de l'équation différentielle ordinaire, obtenue en partant de l'équation différentielle stochastique en remplacant yt par yt(n), oü yt(n) est une approximation linéaire continue du mouvement Brownien, convergeant vers yt lorsque n → ∞.Si xt est la solution de l'équation différentielle stochastique (prise dans le sens de Ito), la séquence des solutions xt(n) peut elle converger vers xt ? L'auteur montre que la réponse à cette question est en général négative. Il montre, cependant, que xt(n) converge, en moyenne, vers la solution d'une autre équation différentielle stochastique qui s'exprime par .ZusammenfassungDieser Beitrag befasst sich mit dem folgenden Problem: xt sei eine Lösung der stochastischeri Differentialgleichung: dxt = m(xt, t) dt + σ(xt, t) dyt, wo yt der Brownsche Bewegungsprozess ist. xt(n) sei die Lösung der gewöhnlichen Differentialgleichung, und zwar erhält man diese von der stochastischen Differentialgleichung, indem man yt durch yt(n) ersetzt, wo yt(n) sich in kontinuierlichen Stücken und linear der Brownschen Bewegung annähert, und wo yt(n) auf yt konvergiert, wenn n gegen ∞ geht. Es erhebt sich die Frage: Konvergiert die Lösungsfolge xt(n) auf xt, ween xt die Lösung der stochastischen Differentialgleichung ist ? Es wird gezeigt, dass die Antwort im allgemeinen negativ ist. Es wird jedoch nachgewiesen, dass xt(n) im Mittelwert auf die Lösung einer anderen stochastischen Differentialgleichung konvergiert, nämlich .SumàrioIn questa memoria viene considerato il seguente problema: Sia xt una soluzione dell'equazione differenziale stocastica: dxt = m(xt, t) dt + σ (xt, t)dyt in cui yt è il processo di movimento Browniano. Sia xt(n) la soluzione della equazione differenziale ordinaria ottenuta dall'equazione differenziale stocastica sostituendo yt come n → ∞. Se xt é la soluzione dell'equazione differenziale stocastica (nel senso di Ito), la sequenza delle soluzioni xt(n) convergerà su xt ? E' indicato come la risposta in generale sia negativa; è però dimostrato che xt(n) converge in media sulla soluzione di un'altra equazione differenziale stocastica che è: .РефератB paбoтe paccмaтpивaeтcя cлeдyющaя зaдaчa: Пycь xt. являeтcя peщeниeм cтoчacтичecкoгo диффepeнциaльнoгo ypaвнeния dxt = m(xt, t)dt + σ(xt, t)dyt, гдe yt oбoзнaчaeт пpoцecc движeния Бpayнa. Пycть xt(N) являeтcя peщним oбынoeeннoo диффepeнциaльнoгo ypaвнeния, пoлyчeннoгo из cтoчacтичecкoгo ypaвнeния пyтeм зaмeны yt нa yt(n), гдe yt(n) нeпpepывнaя, кycoчнo линeйнaя, aпpoгcимaцня движeкия Бpayнa и, чтo yt(n) cтpeмитcя к yt для n → ∞. Ecли xt являeтcя peщeниeм cтoчacтичecкoгo диффepeнциaльнoгo ypaвнeния (в cмыcлe Итo) имeeм вoпpoc: cчoдитcя пocдeлoвaтeльнocть peщeний xt(n)kxt? Дoкaзывaeтcя, чтo в oбщeм cлyчae, oтвeт бyдeт oмpццaмe ьным, нo дoкaзывaeтcя тaкжe, чтo xt(n) cчoдитcя в cpeнeм к peщeнию ∂pyoo cтoчacтичecкoгo ypaвнeния, для кoтopoгo имeeм; .
Article
The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.
Article
The vast reduction in the size and power consumption of sensors and CMOS circuitry has led to a focused research effort on the on-board power sources which can replace the batteries. The concern with batteries has been that they must always be charged before use. Similarly, the sensors and data acquisition components in distributed networks require centralized energy sources for their operation. In some applications such as sensors for structural health monitoring in remote locations, geographically inaccessible temperature or humidity sensors, the battery charging or replacement operations can be tedious and expensive. Logically, the emphasis in such cases has been on developing the on-site generators that can transform any available form of energy at the location into electrical energy. Piezoelectric energy harvesting has emerged as one of the prime methods for transforming mechanical energy into electric energy. This review article provides a comprehensive coverage of the recent developments in the area of piezoelectric energy harvesting using low profile transducers and provides the results for various energy harvesting prototype devices. A brief discussion is also presented on the selection of the piezoelectric materials for on and off resonance applications. Analytical models reported in literature to describe the efficiency and power magnitude of the energy harvesting process are analyzed.
Article
Simple analytical models have proved very useful in understanding vibration energy harvesters driven by a sinusoidal acceleration. Corresponding analyses for broadband excitations have been absent. In this paper, we present new closed-form results on the output power, proof mass displacement, and optimal load of linear resonant energy harvesters driven by broadband vibrations. Output power dependence on signal bandwidth is also considered. The results are compared with those that are already well established for a sinusoidal acceleration. We formulate a stochastic description of more general energy-harvester models and show that the influence of elastic mechanical stoppers on the output power is dependent on the electrical load for large amplitude vibrations.
Article
In this theoretical study, the response of an inductive power generator with a bistable symmetric potential to stationary random environmental excitations is investigated. Both white and Ornstein–Uhlenbeck-type excitations are considered. In the white noise limit, the stationary Fokker–Plank–Kolmagorov equation is solved for the exact joint probability density function (PDF) of the response. The PDF is then used to obtain analytical expressions for the response statistics. It is shown that the expected value of the generator's output power is independent of the potential shape leading to the conclusion that under white noise excitations, bistabilities in the potential do not provide any enhancement over the traditional linear resonant generators which have a single-well potential. In the case of Ornstein–Uhlenbeck (exponentially correlated) noise, an approximate expression for the mean power of the generator which depends on the potential shape, the generator's design parameters and the noise bandwidth and intensity is obtained. It is shown that there exists an optimal potential shape which maximizes the output power. This optimal shape guarantees an optimal escapement frequency between the potential wells which remains constant even as the noise intensity is varied.
Article
New approaches for numerical implementation of the path integration (PI) method are described. In essence the PI method is a stepwise calculation of the joint probability density function (PDF) of a set of state space variables describing a white noise excited nonlinear dynamic system. The basic idea behind the proposed procedure is to apply a splines interpolation method to the logarithm of the calculated PDF to obtain an accurate representation of the PDF over the whole domain and not only at the chosen grid points. This exploits the fact that the logarithm of the PDF shows a more polynomial behaviour than the PDF itself, and therefore is better adapted to a splines representation. It is demonstrated that the proposed techniques may lead to significantly improved performance in calculating the response statistics of large classes of nonlinear oscillators excited by white or coloured noise when compared to other available implementations of the PI method. An advantage of the new approaches is that they allow time-variant dynamic systems to be analysed without significant increase in computer time. Numerical results for both 2D and 3D problems are presented.
Article
Future MEMS devices will harvest energy from their environment. One can envisage an autonomous condition monitoring vibration sensor being powered by that same vibration, and transmitting data over a wireless link; inaccessible or hostile environments are obvious areas of application. The base excitation of an elastically mounted magnetic seismic mass moving past a coil, considered previously by several authors, is analysed in detail. The amplitude of the seismic mass is limited in any practical device and this, together with the magnitude and frequency of the excitation define the maximum power that can be extracted from the environment. The overall damping coefficient (part of which is mechanical) is associated with the harvesting and dissipation of energy and also the transfer of energy from the vibrating base into the system. It is shown that net energy flow from the base through the damper is positive (negative) for , but is zero when ω=ωn. The mechanical part of the damper cannot contribute more power than it dissipates and is neutral, at best, when ω/ωn→∞. Maximum power is delivered to an electrical load when its resistance is equal to the sum of the coil internal resistance and the electrical analogue of the mechanical damping coefficient, which differs from what has been claimed. A highly damped system has the advantage of harvesting energy over a wider band of excitation frequencies on either side of the natural frequency, is smaller, but will harvest marginally less power. One possible strategy for variable amplitude excitation is proposed.
Article
Advances in low power VLSI design, along with the potentially low duty cycle of wireless sensor nodes open up the possibility of powering small wireless computing devices from scavenged ambient power. A broad review of potential power scavenging technologies and conventional energy sources is first presented. Low-level vibrations occurring in common household and office environments as a potential power source are studied in depth. The goal of this paper is not to suggest that the conversion of vibrations is the best or most versatile method to scavenge ambient power, but to study its potential as a viable power source for applications where vibrations are present. Different conversion mechanisms are investigated and evaluated leading to specific optimized designs for both capacitive MicroElectroMechancial Systems (MEMS) and piezoelectric converters. Simulations show that the potential power density from piezoelectric conversion is significantly higher. Experiments using an off-the-shelf PZT piezoelectric bimorph verify the accuracy of the models for piezoelectric converters. A power density of 70 μW/cm3 has been demonstrated with the PZT bimorph. Simulations show that an optimized design would be capable of 250 μW/cm3 from a vibration source with an acceleration amplitude of 2.5 m/s2 at 120 Hz.
High fidelity numerical solutions of the Fokker–Planck equation
  • S F Wojtkiewicz
  • L A Bergman
  • B F Spencer
  • Jr
S.F. Wojtkiewicz, L.A. Bergman, B.F. Spencer Jr., High fidelity numerical solutions of the Fokker–Planck equation, in: ICOSSAR 97, The 7th International Conference on Structural Safety and Reliability, Kyoto, Japan 1997, 24–28.
On the relation between ordinary and stochastic differential equation
  • E Wong
  • M Zakai
E. Wong, M. Zakai, On the relation between ordinary and stochastic differential equation, International Journal of Engineering Science 3 (1965) 213–229.