Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu system

The first-passage statistics of Duffing-Rayleigh- Mathieu system under wide-band colored noise excitations is studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging. The diffusion process method for first-passage problem is used and the corresponding backward Kolmogorov equation and Pontryagin equation are constructed and solved to yield the conditional reliability function and mean first-passage time with suitable initial and boundary conditions. The analytical results are confirmed by Monte Carlo simulation.

Studies on Stochastic Parametric Roll of Ship with Stochastic Averaging Method

The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed.Random sea surface is treated as a narrow-band stochastic process,and the stochastic parametric excitation is studied based on the effective wave theory.The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function.By using the stochastic averaging method,the differential equation of motion is transformed into Ito’s stochastic differential equation.The steady-state probability density function of roll motion is obtained,and the results are validated with the numerical simulation and model test.

Stochastic averaging of quasi integrable and resonant Hamiltonian systems excited by fractional Gaussian noise with Hurst index 1/2

A stochastic averaging method of quasi integrable and resonant Hamiltonian systems under excitation of fractional Gaussian noise （fGn） with the Hurst index 1/2 〈 H 〈 1 is proposed. First, the definition and the basic property of fGn and related fractional Brownian motion （iBm） are briefly introduced. Then, the averaged fractional stochastic differential equations （SDEs） for the first integrals and combinations of angle variables of the associated Hamiltonian systems are derived. The stationary probability density and statistics of the original systems are then obtained approximately by simulating the averaged SDEs numerically. An example is worked out to illustrate the proposed stochastic averaging method. It is shown that the results obtained by using the proposed stochastic averaging method and those from digital simulation of original system agree well.

Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system

A stochastic wheelset model with a nonlinear wheel-rail contact relationship is established to investigate the stochastic stability and stochastic bifurcation of the wheelset system with the consideration of the stochastic parametric excitations of equivalent conicity and suspension stiffness.The wheelset is systematized into a onedimensional(1D)diffusion process by using the stochastic average method,the behavior of the singular boundary is analyzed to determine the hunting stability condition of the wheelset system,and the critical speed of stochastic bifurcation is obtained.The stationary probability density and joint probability density are derived theoretically.Based on the topological structure change of the probability density function,the stochastic Hopf bifurcation form and bifurcation condition of the wheelset system are determined.The effects of stochastic factors on the hunting stability and bifurcation characteristics are analyzed,and the simulation results verify the correctness of the theoretical analysis.The results reveal that the boundary behavior of the diffusion process determines the hunting stability of the stochastic wheelset system,and the left boundary characteristic value cL=1 is the critical state of hunting stability.Besides,stochastic D-bifurcation and P-bifurcation will appear in the wheelset system,and the critical speeds of the two kinds of stochastic bifurcation decrease with the increase in the stochastic parametric excitation intensity.

Performance enhancement of a viscoelastic bistable energy harvester using time-delayed feedback control

This paper focuses on the stochastic analysis of a viscoelastic bistable energy harvesting system under colored noise and harmonic excitation, and adopts the time-delayed feedback control to improve its harvesting efficiency. Firstly, to obtain the dimensionless governing equation of the system, the original bistable system is approximated as a system without viscoelastic term by using the stochastic averaging method of energy envelope, and then is further decoupled to derive an equivalent system. The credibility of the proposed method is validated by contrasting the consistency between the numerical and the analytical results of the equivalent system under different noise conditions. The influence of system parameters on average output power is analyzed, and the control effect of the time-delayed feedback control on system performance is compared. The output performance of the system is improved with the occurrence of stochastic resonance(SR). Therefore, the signal-to-noise ratio expression for measuring SR is derived, and the dependence of its SR behavior on different parameters is explored.

A NEW STOCHASTIC OPTIMAL CONTROL STRATEGY FOR HYSTERETIC MR DAMPERS

A new stochastic optimal control strategy for randomly excited quasi-integrable Hamiltonian systems using magneto-rheological (MR) dampers is proposed. The dynamic be- havior of an MR damper is characterized by the Bouc-Wen hysteretic model. The control force produced by the MR damper is separated into a passive part incorporated in the uncontrolled system and a semi-active part to be determined. The system combining the Bouc-Wen hysteretic force is converted into an equivalent non-hysteretic nonlinear stochastic control system. Then It?o stochastic di?erential equations are derived from the equivalent system by using the stochastic averaging method. A dynamical programming equation for the controlled di?usion processes is established based on the stochastic dynamical programming principle. The non-clipping nonlin- ear optimal control law is obtained for a certain performance index by minimizing the dynamical programming equation. Finally, an example is given to illustrate the application and e?ectiveness of the proposed control strategy.

Stochastic response of fractional-order van der Pol oscillator

We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. Through stochastic averaging, an averaged Ito equation is deduced. We obtained the Fokker–Planck–Kolmogorov equation connected to the averaged Ito equation and solved it to yield the approximate stationary response of the system. The analytical solution is confirmed by using Monte Carlo simulation.

STOCHASTIC OPTIMAL CONTROL OF STRONGLY NONLINEAR SYSTEMS UNDER WIDE-BAND RANDOM EXCITATION WITH ACTUATOR SATURATION

A bounded optimal control strategy for strongly non-linear systems under non-white wide-band random excitation with actuator saturation is proposed. First, the stochastic averaging method is introduced for controlled strongly non-linear systems under wide-band random excitation using generalized harmonic functions. Then, the dynamical programming equation for the saturated control problem is formulated from the partially averaged Itō equation based on the dynamical programming principle. The optimal control consisting of the unbounded optimal control and the bounded bang-bang control is determined by solving the dynamical programming equation. Finally, the response of the optimally controlled system is predicted by solving the reduced Fokker-Planck-Kolmogorov （FPK） equation associated with the completed averaged Itō equation. An example is given to illustrate the proposed control strategy. Numerical results show that the proposed control strategy has high control effectiveness and efficiency and the chattering is reduced significantly comparing with the bang-bang control strategy.

STOCHASTIC OPTIMAL CONTROL OF HYSTERETIC SYSTEMS UNDER EXTERNALLY AND PARAMETRICALLY RANDOM EXCITATIONS

A stochastic optimal control method for nonlinear hysteretic systems under externally and/or parametrically random excitations is presented and illustrated with an example of hysteretic column system. A hysteretic system subject to random excitation is first replaced by a nonlinear non-hysteretic stochastic system. An It$\hat {\rm o}$ stochastic differential equation for the total energy of the system as a one-dimensional controlled diffusion process is derived by using the stochastic averaging method of energy envelope. A dynamical programming equation is then established based on the stochastic dynamical programming principle and solved to yield the optimal control force. Finally, the responses of uncontrolled and controlled systems are evaluated to determine the control efficacy. It is shown by numerical results that the proposed stochastic optimal control method is more effective and efficient than other optimal control methods.

Optimal control strategies for stochastically excited quasi partially integrable Hamiltonian systems

In this paper two different control strategies designed to alleviate the response of quasi partially integrable Hamiltonian systems subjected to stochastic excitation are proposed. First, by using the stochastic averaging method for quasi partially integrable Hamiltonian systems, an n-DOF controlled quasi partially integrable Hamiltonian system with stochastic excitation is converted into a set of partially averaged It^↑o stochastic differential equations. Then, the dynamical programming equation associated with the partially averaged It^↑o equations is formulated by applying the stochastic dynamical programming principle. In the first control strategy, the optimal control law is derived from the dynamical programming equation and the control constraints without solving the dynamical programming equation. In the second control strategy, the optimal control law is obtained by solving the dynamical programming equation. Finally, both the responses of controlled and uncontrolled systems are predicted through solving the Fokker-Plank-Kolmogorov equation associated with fully averaged It^↑o equations. An example is worked out to illustrate the application and effectiveness of the two proposed control strategies.

Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation

This paper aims to investigate the stochastic response of the van der Pol （VDP） oscillator with two kinds of fractional derivatives under Gaussian white noise excitation. First, the fractional VDP oscillator is replaced by an equivalent VDP oscillator without fractional derivative terms by using the generalized harmonic balance technique. Then, the stochastic averaging method is applied to the equivalent VDP oscillator to obtain the analytical solution. Finally, the analytical solutions are validated by numerical results from the Monte Carlo simulation of the original fractional VDP oscillator. The numerical results not only demonstrate the accuracy of the proposed approach but also show that the fractional order, the fractional coefficient and the intensity of Gaussian white noise play important roles in the responses of the fractional VDP oscillator. An interesting phenomenon we found is that the effects of the fractional order of two kinds of fractional derivative items on the fractional stochastic systems are totally contrary.

Stochastic bifurcations of generalized Duffing–van der Pol system with fractional derivative under colored noise

The stochastic bifurcation of a generalized Duffing–van der Pol system with fractional derivative under color noise excitation is studied. Firstly, fractional derivative in a form of generalized integral with time-delay is approximated by a set of periodic functions. Based on this work, the stochastic averaging method is applied to obtain the FPK equation and the stationary probability density of the amplitude. After that, the critical parameter conditions of stochastic P-bifurcation are obtained based on the singularity theory. Different types of stationary probability densities of the amplitude are also obtained. The study finds that the change of noise intensity, fractional order, and correlation time will lead to the stochastic bifurcation.

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS

A strategy is proposed based on the stochastic averaging method for quasi non- integrable Hamiltonian systems and the stochastic dynamical programming principle.The pro- posed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation.By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional av- eraged It stochastic differential equation.By using the stochastic dynamical programming princi- ple the dynamical programming equation for minimizing the response of the system is formulated. The optimal control law is derived from the dynamical programming equation and the bounded control constraints.The response of optimally controlled systems is predicted through solving the FPK equation associated with It stochastic differential equation.An example is worked out in detail to illustrate the application of the control strategy proposed.

STOCHASTIC PROPAGATION OF SEMI-ELLIPTICAL SURFACE CRACK

The semi-elliptical surface crack growth of structural components with uncertain material resistance under random loading is studied by using the stochastic averaging principle.The FPK equation governing the transition probability density function of crack lengths is derived.The analytical solution of the FPK equation for the case of that the equations for the crack growth in the surface and depth directions are uncoupled is obtained.The effects of the parameters of the stress process and of the material property on the behavior of semi-elliptical fatigue crack growth of the components with deterministic resistance to crack growth in the stationary Gaussian stress process are examined.The comparison of the analytical result with digital simulation shows the effectiveness of the present method.

STOCHASTIC MODELLING OF FATIGUE CRACK GROWTH UNDER CONSTANT AMPLITUDE LOADING

Two stochastic models of fatigue crack growth under constant amplitude cyclic loading are proposed and studied by using the stochastic averaging method and total probability theorem to account for high frequency component and low frequency component, respectively, of the irregular nature of fatigue crack growth observed in Virkler's experiment. Particular attention is paid to the prediction performance of the models under a change in the initial crack length. It is shown that the models proposed in the present paper yields better agreement with experimental data than other models available in literature.

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first- passage time （MFPT） of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kraraers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

Time-delayed feedback control optimization for quasi linear systems under random excitations

A strategy for time-delayed feedback control optimization of quasi linear systems with random excitation is proposed. First, the stochastic averaging method is used to reduce the dimension of the state space and to derive the stationary response of the system. Secondly, the control law is assumed to be velocity feedback control with time delay and the unknown control gains are determined by the performance indices. The response of the controlled system is predicted through solving the Fokker-Plank-Kolmogorov equation associated with the averaged Ito equation. Finally, numerical examples are used to illustrate the proposed control method, and the numerical results are confirmed by Monte Carlo simulation .

FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM

Studies on first-passage failure are extended to the multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.

Response of a Duffing-Rayleigh system with a fractional derivative under Gaussian white noise excitation

In this paper,we consider the response analysis of a Duffing-Rayleigh system with fractional derivative under Gaussian white noise excitation.A stochastic averaging procedure for this system is developed by using the generalized harmonic functions.First,the system state is approximated by a diffusive Markov process.Then,the stationary probability densities are derived from the averaged Ito stochastic differential equation of the system.The accuracy of the analytical results is validated by the results from the Monte Carlo simulation of the original system.Moreover,the effects of different system parameters and noise intensity on the response of the system are also discussed.

Optimization for vibro-impact nonlinear energy sink under randomexcitation

As a promising vibration control device, the vibro-impact nonlinear energy sink (VI-NES) gathered extensively attention in recent years. However, general optimization procedures have not been available forthe design of VI-NES subjected to random excitations. To this end, this paper constitutes a research effortto address this gap. Specifically, the approximate analytical solution of the system stochastic responseis obtained in conjunction with non-smooth conversion and stochastic averaging methodology. Takingadvantages of this approximate solution, the variance of the system is defined and easily minimized tocalculate the optimal parameters for VI-NES. In addition, the results computed by this way fairly correlatewith direct numeric simulations.