Stochastic Averaging Of Energy Envelope Research Articles

Stationary Response of a Kind of Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative Damping

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of nonlinear stochastic systems with stochastic variable-mass and fractional derivative, damping is investigated in this paper. Firstly, an approximately equivalent system of the studied nonlinear stochastic system is presented according to the Taylor expansion technique. Then, based on stochastic averaging of energy envelope, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced, which gives an approximated analytical solution of stationary response. Finally, a nonlinear oscillator with variable mass and fractional derivative damping is proposed in numerical simulations. The approximated analytical solution is compared with Monte Carlo numerical solution, which could verify the effectiveness of the obtained results.

Stabilization of electrostatic MEMS resonators using a stochastic optimal control

Noise-induced motions are a significant source of uncertainty in the response of electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and dramatic loss of stability. In this manuscript, we present an analysis for the stabilization of an electrostatically actuated MEMS resonator in the presence of noise processes using a stochastic optimal control scheme. A stochastic lumped-mass model that accounts for the uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude of the resonator is presented. The Ito^ equations, describing the averaged modulations of the resonator's total energy and phase difference, are obtained via the stochastic averaging of energy envelope. The stochastic optimal control law aiming at minimization the pull-in probability of the resonator is determined via the stochastic dynamic programming equations associated with the Ito^ equations. We show that the resulting stochastic optimal feedback control, with a careful selection of its control gain, can be used to suppress the noise-activated pull-in instability of the disturbed resonator. The impact of the time delay inherent in the feedback control input on the control effectiveness is investigated. In addition, the control scheme also shows superior performance in counteracting the deterministic pull-in instability of the resonator operating in the dynamic pull-in frequency band. Good agreement between the theoretical results and numerical simulation is demonstrated.

Response of a vibro-impact energy harvesting system with bilateral rigid stoppers under Gaussian white noise

In this paper, responses of the stochastic energy harvesting systems with bilateral rigid stoppers are studied, in which the Newton’s impact model is adopted to describe the impact force between the proof mass and stoppers. To the best knowledge of the authors, the dynamical characteristics of the vibro-impact energy harvesting system have been scarcely investigated by utilizing the stochastic averaging of energy envelope. First, such theoretical method is employed to study the mono-stable energy harvesting system with bilateral rigid impact. After obtaining the equivalent system and considering the energy loss due to impact in every quasi period, the probability density functions are theoretically derived, and the effectiveness of the theoretical method is validated by numerical simulations. Then, the time history of response for the stochastic bi-stable system with bilateral rigid stoppers is given to analyze the effects of the noise intensity and the impact distance on the system. In addition, effects of the coefficients on the mean square voltage in the mono-stable and bi-stable systems are discussed to help to enhance the energy harvesting performance, and some interesting phenomena are observed.

Stochastic dynamics of a piezoelectric energy harvester with correlated colored noises from rotational environment

The energy harvesting from rotational automotive tire is studied through an electromechanical-coupled system which is a harmonically driven bi-stable potential with correlated additive and multiplicative colored noises. By integrating the voltage equation and introducing generalized harmonic transformation, the equivalent uncoupled system based on energy-dependent frequency is derived. The improved stochastic averaging of energy envelope is then carried out to obtain the stationary probability density (SPD) of the equivalent uncoupled system. The effects of correlated colored noises and other system parameters on SPD and mean output power are discussed theoretically and numerically. Besides, signal-to-noise ratio (SNR) is obtained analytically to measure stochastic resonance (SR). It is found that the curve of SNR has a single peak versus noise intensity and system parameters. That is, the appropriate choice of system parameters, such as noise intensity, electromechanical coupling coefficient and time constant ratio, can enhance power conversion efficiency greatly and produce more mean output power from the viewpoint of parameter-optimized SR. Especially, with a large periodic rotational angular velocity, the RMS voltage can be significantly enhanced by increasing the magnetic end mass gravity. Finally, the proposed stochastic averaging is well verified by numerical simulations.

Response Evaluation and Optimal Control for Stochastically Excited Vibro-Impact System with Hertzdamp Contact

This manuscript devotes to procedures to evaluate random responses and explore stochastic optimal control for a vibro-impact system, in which the inelastic impact is modeled as Hertzdamp contact model. The dimension of governing equation is halved with respect to the total energy of uncontrolled/controlled system by conducting the stochastic averaging of energy envelope. The stationary probability density and then the response statistics of the uncontrolled system are much easier to solve due to a low-dimensional Fokker-Planck-Kolmogorov equation. The original optimal control problem with respect to states of system is converted to that with respect to system total energy. The latter is quite simpler than the former although it only can be regarded as the suboptimal control of the original system. The optimal control with damping property is determined by adopting the stochastic maximum principle and solving simultaneously a set of forward-backward stochastic differential equations. The accuracy of the proposed procedure for evaluation of the random response is verified by comparing with Monte-Carlo simulations and the efficiency of the optimal control strategy is also illustrated. The consistence between the analytical result from energy dissipative balance technique and that from stochastic averaging of energy envelope is discussed, which means the rigorous restriction on stochastic averaging, i.e., light damping and weak excitation, can be removed. The optimal control strategy from stochastic maximum principle and that from stochastic dynamic programming principle holds close connection based on our related discussion.

Stationary response of stochastic viscoelastic system with the right unilateral nonzero offset barrier impacts**Project supported by the National Natural Science Foundation of China (Grant Nos. 11872305 and 11872307) and the Excellent Doctorate Cultivating Foundation of Northwestern Polytechnical University, China.

The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact (VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope (SAEE) is adopted to seek the stationary probability density functions (PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo (MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.

Approximately analytical procedure to evaluate random stick-slip vibration of Duffing system including dry friction

As a mechanical system is placed on a belt moving with a constant speed, the friction between the mass block and the belt induces the stick-slip vibration. If the mechanical system is imposed by random external excitations at the same time, it will suffer from more complex random stick-slip vibration which is quite difficult to predict analytically. This manuscript tries to establish an approximately analytical procedure to evaluate the response statistics of the random stick-slip vibration of a Duffing system which is influenced by friction, viscous damping and Gaussian white noise external excitation. The first integral of the degenerated nonlinear system is first introduced, and then the averaged Itô stochastic differential equation with respect to the first integral is derived through the stochastic averaging of energy envelope. The drift and diffusion coefficients are piecewise functions of the first integral and possess dramatically different forms for the slip and stick stages. The stationary probability density of the first integral is derived by solving the Fokker-Plank-Kolmogorov equation associated with the averaged equation. The stationary probability densities and response statistics of the displacement and velocity are finally obtained. By comparing with Monte-Carlo simulations, the effectiveness and precision of the approximately analytical procedure are verified. The influences of belt speed, excitation intensity, viscous damping, dynamic friction coefficient and nonlinear stiffness on response statistics are discussed in detail. It is found that the analytical procedure is valid so long as the portion of stick stage is not too heavy in the stick-slip motion. The belt speed and excitation intensity have significant influences on the random response, and the dramatically different random behavior occurs as these parameters vary.

Trajectory control and random vibration control of nonlinear system with dielectric elastomer actuator

Dielectric elastomer actuators have gained extensive attention in scientific and industrial communities with the rapid development of soft robot technology. There still remain some questions on the control aspect of nonlinear system with dielectric elastomer actuator. The first is whether the soft actuator can successfully drive the primary structure to track an arbitrary prescribed trajectory. The second is how to suppress the random vibration around the equilibrium position when the primary structure is disturbed by external excitation. This article seeks the answers for these two questions. By directly solving the governing equation of motion, an open-loop control technique is designed to track a prescribed trajectory. The effectiveness of the trajectory tracking technique is investigated and the limitation is illustrated by the influence of inertia of the primary structure. Based on the stochastic averaging of energy envelope and stochastic dynamic programming principle, a clipped control strategy is proposed by slightly adjusting the voltage in real time to suppress the random vibration around the equilibrium position. The good effectiveness and high robustness of the clipped control strategy are verified numerically. This work may provide some guidelines for the control aspect of nonlinear systems with dielectric elastomer actuators.

Reliability‐based control algorithms for nonlinear hysteretic systems based on enhanced stochastic averaging of energy envelope

SummaryCurrent reliability‐based control techniques have been successfully applied to linear systems; however, incorporation of stochastic nonlinear behavior of systems in such control designs remains a challenge. This paper presents two reliability‐based control algorithms that minimize failure probabilities of nonlinear hysteretic systems subjected to stochastic excitations. The proposed methods include constrained reliability‐based control (CRC) and unconstrained reliability‐based control (URC) algorithms. Accurate probabilistic estimates of nonlinear system responses to stochastic excitations are derived analytically using enhanced stochastic averaging of energy envelope proposed previously by the authors. Convolving these demand estimates with capacity models yields the reliability of nonlinear systems in the control design process. The CRC design employs the first‐level and second‐level optimizations sequentially where the first‐level optimization solves the Hamilton–Jacobi–Bellman equation and the second‐level optimization searches for optimal objective function parameters to minimize the probability of failure. In the URC design, a single optimization minimizes the probability of failure by directly searching for the optimal control gain. Application of the proposed control algorithms to a building on nonlinear foundation has shown noticeable improvements in system performance under various stochastic excitations. The URC design appears to be the most optimal method as it reduced the probability of slight damage to 8.7%, compared with 11.6% and 19.2% for the case of CRC and a stochastic linear quadratic regulator, respectively. Under the Kobe ground motion, the normalized peak drift displacement with respect to stochastic linear quadratic regulator is reduced to 0.78 and 0.81 for the URC and CRC cases, respectively, at comparable control force levels. Copyright © 2017 John Wiley & Sons, Ltd.

Random response of dielectric elastomer balloon to electrical or mechanical perturbation

Random response of dielectric elastomer balloon disturbed by electrical or mechanical fluctuation is analytically investigated in this article. The stochastic differential equation governing the oscillating behavior around the stable equilibrium position is first derived by introducing the translation transformation. The stationary joint probability density about the disturbing stretch ratio and its rate of change is analytically established by adopting the stochastic averaging of energy envelope; the statistics quantities, such as the mean value and the standard deviation of the disturbing stretch ratio, are then subsequently calculated. Two special cases, the first case with only voltage fluctuation and the second one with only pressure variation, are discussed in detail, and the random response properties are summarized. The accuracy of the analytical solution is verified by comparing with the Monte Carlo simulation for the perturbation with weak intensity, and the valid ranges for the mean voltage and pressure are illustrated. This work provides an effective technique to evaluate the detection precision of new type of sensors based on the dielectric elastomer.

Stationary response analysis of vibro-impact system with a unilateral nonzero offset barrier and viscoelastic damping under random excitations

The stationary response analysis of stochastic vibro-impact (VI) dynamical system with a unilateral nonzero offset barrier and viscoelastic damping under additive and multiplicative random excitations is explored. First, the conversion of the original stochastic VI system coupled with viscoelastic damping into an equivalent stochastic VI system without viscoelastic damping terms is performed by adding the equivalent stiffness and damping. Then, the equivalent stochastic VI system is studied according to the level of system energy instead of non-smooth transformation of the previous literature; thus, description on effects of impacts draws support from the loss of the system energy. The energy loss at each impact is found to be related to the restitution factor and the energy level before impact. Furthermore, the probability density functions of the equivalent VI system based on assumption of the weak damping and random perturbation is produced by the application of the stochastic averaging of energy envelope. The effectiveness of the suggested method is verified by comparing the analytical results with those from Monte Carlo simulations.

Stochastic D-bifurcation for a damped sine-Gordon equation with noise

We investigate the stochastic bifurcation of a damped sine-Gordon equation with Dirichlet boundary conditions under the influence of multiplicative Gaussian white noise. Introducing a slow time scale, we derive the amplitude equations near the trivial solution by multiscale analysis. And the stationary probability density functions are formulated analytically using the stochastic averaging of energy envelope. The numerical calculations show that the system undergoes a stochastic D-bifurcation of energy envelope from a delta measure to new stationary measures when the control parameter crosses a critical point.

First-passage failure of preisach hysteretic systems

The first-passage failure of a single-degree-of-freedom hysteretic system with nonlocal memory is investigated. The hysteretic behavior is described through a Preisach model with excitation selected as Gaussian white noise. First, the equivalent nonlinear non-hysteretic system with amplitude-dependent damping and stiffness coefficients is derived through generalized harmonic balance technique. Then, equivalent damping and stiffness coefficients are expressed as functions of system energy by using the relation of amplitude to system energy. The stochastic averaging of energy envelope is adopted to accept the averaged Itô stochastic differential equation with respect to system energy. The establishing and solving of the associated backward Kolmogorov equation yields the reliability function and probability density of first-passage time. The effects of system parameters on first-passage failure are investigated concisely and validated through Monte Carlo simulation.

Non-linear stochastic optimal control of acceleration parametrically excited systems

Acceleration parametrical excitations have not been taken into account due to the lack of physical significance in macroscopic structures. The explosive development of microtechnology and nanotechnology, however, motivates the investigation of the acceleration parametrically excited systems. The adsorption and desorption effects dramatically change the mass of nano-sized structures, which significantly reduces the precision of nanoscale sensors or can be reasonably utilised to detect molecular mass. This manuscript proposes a non-linear stochastic optimal control strategy for stochastic systems with acceleration parametric excitation based on stochastic averaging of energy envelope and stochastic dynamic programming principle. System acceleration is approximately expressed as a function of system displacement in a short time range under the conditions of light damping and weak excitations, and the acceleration parametrically excited system is shown to be equivalent to a constructed system with an additional displacement parametric excitation term. Then, the controlled system is converted into a partially averaged Itô equation with respect to the total system energy through stochastic averaging of energy envelope, and the optimal control strategy for the averaged system is derived from solving the associated dynamic programming equation. Numerical results for a controlled Duffing oscillator indicate the efficacy of the proposed control strategy.

Stationary response of bilinear hysteretic system driven by Poisson white noise

Stationary response of single-degree-of-freedom (SDOF) bilinear hysteretic system driven by Poisson white noise is investigated via stochastic averaging of energy envelope in this paper. The averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for SDOF bilinear hysteretic system driven by Poisson white noise is derived and the approximate stationary solutions of the averaged GFPK equation are obtain by using a modified exponential polynomial closure method. The effectiveness and accuracy of the approximate solution are assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and the effect of non-Gaussianity of the excitation process on stationary probability densities of total energy and displacement of bilinear hysteretic system is predicted successfully via stochastic averaging of energy envelope.

Stochastic analysis of dynamical system with double-well potential

The method of quasi-conservative averaging, or the stochastic averaging of energy envelope, is extended to a system with a double-well potential subjected to both the external and parametric non-white noise excitations. Instead of using the power spectral densities, the correlation functions of the excitation processes are used directly so that the averaging procedure can be performed for different types of motions appearing in the system with a double-well potential. The obtained Markov process of the energy envelope allows to carry out the dynamic analysis of the system, such as to assess the asymptotic behaviors at the boundaries, to find the probability density of the response, and to investigate the transition of the system motion between the two wells, as done in the present paper. Furthermore, the proposed procedure can be applied not only to systems with double-well potentials, but also to those with potentials of more complicated shapes.

Response probability density functions of Duffing–Van der Pol vibro-impact system under correlated Gaussian white noise excitations

This paper aims to investigate the stationary probability density functions (PDFs) of a Duffing–Van der Pol vibro-impact system excited by correlated Gaussian white noise. With the help of non-smooth transformation, the stationary PDFs are formulated analytically by the stochastic averaging of energy envelope. The analytical results are verified by numerical simulation results. Stochastic bifurcations for different parameters are considered, and several special PDF forms are observed in this paper. The first form is the shape of the PDF of total energy can be similar to a crater, which has a minimum and a maximum. The second one is the shape of the joint PDF with three peaks, that is to say, the section of joint PDF has three maximum and two minimum. In addition, the influence of the degree of the correlation of Gaussian white noises is explored.

Stochastic averaging of energy envelope of Preisach hysteretic systems

A new stochastic averaging technique for analyzing the response of a single-degree-of-freedom Preisach hysteretic system with nonlocal memory under stationary Gaussian stochastic excitation is proposed. An equivalent nonhysteretic nonlinear system with amplitude-envelope-dependent damping and stiffness is firstly obtained from the given system by using the generalized harmonic balance technique. The relationship between the amplitude envelope and the energy envelope is then established, and the equivalent damping and stiffness coefficients are expressed as functions of the energy envelope. The available range of the yielding force of the system is extended and also the strong nonlinear stiffness of the system is incorporated so as to improve the response prediction. Finally, an averaged Itô stochastic differential equation for the energy envelope of the system as one-dimensional diffusion process is derived by using the stochastic averaging method of energy envelope, and the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is solved to obtain stationary probability densities of the energy envelope and amplitude envelope. The approximate solutions are validated by using the Monte Carlo simulation.

The Peak Factor for Gust Loading: A Review and Some New Proposals

Abstract. Wind forces on structures that can be considered stiff are usually calculated by using the so-called gust factor G that magnifies the effects of the statical part of the wind speed U. In the expression of G a peak factor g is introduced to account for the maxima of the response dynamical displacement Xd(t),which is a zero mean stationary Gaussian process. The peak factor is derived by assuming that the upcrossings of a given level are a Poisson process, which is deemed very conservative by several authors. Thus, other ways for computing g are proposed herein, preserving its classical definition. They are: (1) the use of the envelope of the response process; (2) the solution of the backward Kolmogorov equation; (3) the use of some approximate formulae such as those by Preumont, Lutes et al., and Vanmarcke. The theoretical models are applied to the response of a linear SDOF oscillator for two values of the ratio of critical damping. In the last part of the paper a nonlinear response, that of a Duffing oscillator, is considered and the problem of the peak factor for this nonlinear case is attacked by using the stochastic averaging of energy envelope. The results of the various approaches are compared with those obtained by numerical simulation.

Stochastic Averaging of Quasi-Integrable Hamiltonian Systems

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.