Method Of Stochastic Averaging Research Articles

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

Cantilever model with curvature nonlinearity and longitudinal inertia excited by lateral basal moments being Gaussian white noise

A slender in-extensional cantilever model excited by this transverse pedestal moments with the form of the Gaussian white noise was studied. The purpose of this paper was to explore how the geometrical nonlinearity influence the responses of the model excited by the Gaussian white noise. The curvature nonlinearity and inertia nonlinearity problems were addressed emphatically. The standard stochastic averaging method was applied to simplify the system into the Ito functions of amplitude and phase. Stationary probability density function (PDF) of amplitude, the joint PDF of the displacement and velocity together with the reliability function and the probability density of the first passage failure time were studied theoretically and numerically. The theoretical approximations coincide with the digital results very well when noise density is not too large. When the noise is stronger, a prediction-correction method was proposed to improve the predicting performance. Numerical simulations verified the effectiveness of this approach.

Stochastic Stability of Gyroscopic Systems Under Bounded Noise Excitation

Dynamic stochastic stability of a two-degree-of-freedom gyroscopic system under bounded noise parametric excitation is studied in this paper through moment Lyapunov exponent and the largest Lyapunov exponent. A rotating shaft subject to stochastically fluctuating thrust is taken as a typical example. To obtain these two exponents, the gyroscopic differential equation of motion is first decoupled into Itô stochastic differential equations by using the method of stochastic averaging. Then mathematical transformations are used in these Itô equation to obtain a partial differential eigenvalue problem governing moment Lyapunov exponents, the slope of which at the origin is equal to the largest Lyapunov exponent. Depending upon the numerical relationship between the natural frequency and the excitation frequencies, the gyroscopic system may fall into four types of parametric resonance, i.e. no resonance, subharmonic resonance, combination additive resonance, and combination differential resonance. The effects of noise and frequency detuning parameters on the parametric resonance are investigated. The results pave the way to utilize or control the vibration of gyroscopic systems under stochastic excitation.

Simulation of AntiMatter–Matter Interactions in Geant4

Cross sections of antiproton and antinucleus interactions with nuclei are calculated using stochastic averaging method. A new implementation of the Quark-Gluon-String Model (QGSM) is proposed for simulation of multi-particle production in antinucleus-nucleus collisions. A combination of the cross sections and the new implementation of QGSM allows experimental data on antiproton and antinucleus interactions with nuclei to be described. The combination is included in the well-known Geant4 simulation toolkit.

A comparative study of the stochastic averaging method and the path integration method for nonlinear ship roll motion in random beam seas

In this paper, the energy-based stochastic averaging method and the path integration method are applied to study the stochastic response of the nonlinear roll motion in random beam seas. Specifically, the Markov diffusion theory is applied to describe the random roll motion such that the probabilistic properties of the ship roll motion are governed by the Fokker–Planck (FP) equation. The stochastic averaging method focuses on the roll energy envelope process and reduces the difficulty in calculating the stochastic response via a dimensional reduction of the original system. In contrast, the path integration method is based on the Markov property of the original dynamical system and provides approximate solutions to the FP equation by linking the explicitly known local solutions. Although the stochastic averaging technique is well established in the study of the stochastic responses of various dynamical systems, its accuracy for determining the high-level roll response has not yet been a focus of study. This paper aims to provide a comparative study for the performance of the two above-mentioned methods; additionally, the advantages and shortcomings of using each method for studying the nonlinear roll motion in beam seas are demonstrated by practical calculations.

Bifurcation Regulations Governed by Delay Self-Control Feedback in a Stochastic Birhythmic System

We aim to investigate bifurcation behaviors in a stochastic birhythmic van der Pol (BVDP) system subjected to delay self-control feedback. First, the harmonic approximation is adopted to drive the delay self-control feedback to state variables without delay. Then, Fokker–Planck–Kolmogorov (FPK) equation and stationary probability density function (SPDF) for amplitude are obtained by applying stochastic averaging method. Finally, dynamical scenarios of the change of delay self-control feedback as well as noise that markedly influence bifurcation performance are observed. It is found that: the big feedback strength and delay will suppress the large amplitude limit cycle (LC) while the relatively big noise strength facilitates the large amplitude LC, which imply the proposed regulation strategies are feasible. Interestingly enough, the inner LC is never destroyed due to noise. Furthermore, the validity of analytical results was verified by Monte Carlo simulation of the dynamics.

Probabilistic response and analysis for a vibro-impact system driven by real noise

In this paper, a new stochastic averaging method is proposed to derive the stationary probabilistic response of vibro-impact system subject to real noise excitation, and the effects of real noise and the restitution coefficient on the response are discussed in detail. According to the size relationship between the system energy and the potential energy of the impact position $$\varDelta $$ , the motions of the unperturbed vibro-impact system are divided into two types: the free motion with no energy loss and the impact motion with energy loss. Based on those relationships, a new method that combines a transformation and the Fourier expansion scheme by introducing, the mean drift and diffusion coefficients of the averaged Ito stochastic differential equation for the two types of motions are obtained, which are depended on the bandwidth and the noise density of real noise. The probability density function of stationary response of vibro-impact system is derived through solving the associated Fokker–Planck–Kolmogorov equation. Finally, the effects of the restitution coefficient and the noise intensity of real noise in different bandwidths are detailedly analyzed through an example. The main results on probabilistic response analysis of vibro-impact Duffing system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo simulation method is also conducted to show the effectiveness of the proposed method.

Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise

A new quasi-conservative stochastic averaging method is proposed to analyze the Probabilistic response of nonlinear vibration energy harvesting (VEH) system driven by exponentially correlated Gaussian colored noise. By introducing a method combining a transformation and the residual phase, the nonlinear vibration electromechanical coupling system is equivalent to a single degree of freedom system, which contains the energy-dependent frequency functions. Then the corresponding drift and diffusion coefficients of the averaged Ito^ stochastic differential equation for the equivalent nonlinear system are derived, which are dependent on the correlation time of Gaussian colored noise. The probability density function (PDF) of stationary responses is derived through solving the associated Fokker–Plank–Kolmogorov (FPK) equation. Finally, the mean-square electric voltage and mean output power are analytically obtained through the relation between the electric voltage and the vibration displacement, and the output power has a linear square relationship with the electric voltage, respectively. The main results on probabilistic response of VEH system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo (MC) simulation method is also conducted to show that the proposed method is quite effective.

Higher-order stochastic averaging for a SDOF fractional viscoelastic system under bounded noise excitation

The moment Lyapunov exponents and stochastic stability of a single-degree-of-freedom (SDOF) fractional viscoelastic system under bounded noise excitation are studied by using the method of higher-order stochastic averaging. A realistic example of such a system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The viscoelastic material is assumed to follow a fractional Kelvin–Voigt constitutive relation. The method of higher-order stochastic averaging is used to approximate the fractional stochastic differential equation of motion, and then moment Lyapunov exponents are determined for the system with small damping and weak random fluctuation. The approximate results are confirmed by Monte-Carlo simulations. It is found that convergence of moment Lyapunov exponents depends on the width of power spectral density of the bounded noise process. For this viscoelastic structure, second-order averaging analysis is adequate for stability analysis. The effects of various parameters on the stochastic stability of the system are discussed and possible explanations are explored.

Comparative Study of the Path Integration Method and the Stochastic Averaging Method for Nonlinear Roll Motion in Random Beam Seas

In this work, the path integration method and the energy-based stochastic averaging method are introduced in order to study the stochastic responses of ship roll motion in random beam seas. Based on the Markov property of the dynamical system, the path integration (PI) method provides approximate solution to the governing Fokker-Planck (FP) equation. On the other hand, the stochastic averaging method leads to a dimension reduction of the original system, but the essential behavior is retained. Then numerical solution or even analytical solution of the low-dimensional FP equation can be obtained. Since the principles of the above two stochastic methods are different, a comparative study is valuable and noticeable. The accuracy and performance of each method are evaluated with the assistance of Monte Carlo simulation (MCS).

Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments

The approximate transient response of a nonlinear vibro-impact system under Gaussian white noise excitation is investigated by the methods of stochastic averaging and the Mellin transform. The Zhuravlev nonsmooth transformation is utilized to convert the nonlinear vibro-impact system into an equivalent system without velocity jumps by introducing an impulsive damping term. The Ito stochastic differential equation with respect to the amplitude response and the related Fokker–Plank–Kolmogorov (FPK) equation governing the amplitude response probability density of the vibro-impact system are derived with the stochastic averaging method. The Mellin transform is introduced to solve the FPK equation. The differential relations of complex fractional moments are obtained. The probability density function for transient response of this system constructed by solving a set of differential equations yields complex fractional moments. Two illustrative examples are examined to evaluate the effectiveness of the proposed solution procedure. The effects of restitution factors are investigated on the transient probability density distribution of the vibro-impact systems. At the same time, the convergence and the error analyses for different restitution factors are illustrated. Influences of the truncate term on the convergence and the error are further illustrated. The results obtained from the proposed procedure agree well with the results from Monte Carlo simulations.

Phase Noise Reduction in an MEMS Oscillator Using a Nonlinearly Enhanced Synchronization Domain

We investigate the phase dynamics of a closed-loop MEMS-based oscillator and demonstrate how one can exploit nonlinear behavior to improve oscillator phase noise characteristics by synchronizing the oscillator with a weak harmonic drive. Analytical predictions are based on an oscillator model that incorporates a resonator element with a weak cubic (i.e., Duffing type) nonlinearity, weak coupling to a clean external harmonic drive, and both thermal and frequency noise terms. The method of stochastic averaging is used to derive an expression for the rate of phase diffusion induced by the noises, and the results predict a remarkable phase noise reduction of three order of magnitude when the oscillator is synchronized to the weak external field. The predictions are experimentally demonstrated using a closed-loop oscillator with a double-anchored double-ended-tuning-fork MEMS resonator with coupling to a small external sinusoidal signal from a signal generator, demonstrating a 30 dB/Hz drop in phase noise. The results show how one can generate a clean high-power signal from a relative noisy oscillator by synchronizing it with a low-power external signal. The results also confirm recent studies showing that the parameter range of synchronization is significantly expanded when the resonator operates in its nonlinear regime. [2016-0067]

Stochastic averaging of energy harvesting systems

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.

First-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations

In this paper, first-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations is studied theoretically. By using stochastic averaging method, the equations of motion of the original internally resonant Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established under appropriate boundary and (or) initial conditions. An example is given to show the accuracy of the theoretical method. Numerical solutions of high-dimensional backward Kolmogorov and Pontryagin equation are obtained by finite difference. All theoretical results are verified by Monte Carlo simulation.

Dynamic stability of a viscoelastic rotating shaft under parametric random excitation

This paper deals with dynamic stability of a viscoelastic rotating shaft subjected to a parametric random axial compressive thrust, by using moment Lyapunov exponents and the largest Lyapunov exponents as indicators. The equation of motion for the shaft is derived, which is a system of gyroscopic stochastic differential equations. The method of stochastic averaging is used to decouple the governing equations into Itô equations, from which the moment Lyapunov exponent is obtained by using mathematical transformations only. The largest Lyapunov exponent is obtained through its relation with moment Lyapunov exponents. The effects of various parameters on the stochastic dynamic stability are discussed. The approximate analytical results are confirmed by Monte Carlo simulation.

Phase Noise Reduction and Optimal Operating Conditions for a Pair of Synchronized Oscillators

We investigate the dynamics of a pair of noisy coupled oscillators and derive operating conditions that minimize phase noise. The generic model employed allows for general nonlinear dynamics of the resonant elements of the oscillators, in terms of their amplitude-dependent frequencies, general asymmetric coupling, and both additive and multiplicative noise sources. The model is analyzed using the method of stochastic averaging and the results suggest an optimal operating point for minimizing phase noise in either of the oscillators by varying the operating amplitudes and the nature and strength of the coupling. We also show that the system with asymmetric coupling has the ability to reduce the phase noise of the noisier oscillator beyond the expected result for synchronized oscillators. The analytical predictions, which are validated by Monte-Carlo simulations, show how one can exploit nonlinear behavior to improve phase noise characteristics using a pair of coupled oscillators. The results also point to a means of designing and optimizing larger sets of coupled oscillators to improve phase noise performance.

Stochastic Perturbations of Periodic Orbits with Sliding

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.

Stochastic stability of SDOF linear viscoelastic system under wideband noise excitation

The stochastic moment stability and almost-sure stability of a single-degree-of-freedom (SDOF) viscoelastic system subject to parametric fluctuation is investigated by using the method of higher-order stochastic averaging. The stochastic parametric excitation is modeled as a wideband noise, which is taken as Gaussian white noise and real noise. The viscoelastic material is assumed to follow ordinary Maxwell linear constitutive relation. For small damping and weak stochastic fluctuation, analytical expressions are derived for the moment Lyapunov exponent and the Lyapunov exponent, which indicate moment stability and almost-sure stability respectively. The effects of various system and loading parameters on the stochastic stability are discussed. Both analytical and simulation results show that higher-order stochastic averaging improves the accuracy compared with the first-order stochastic averaging. However, results of the third-order averaging are almost overridden by those of second-order averaging and the third-order averaging involves far more calculation. It is advisable to consider a balance between accuracy achievement and calculation endeavor when using higher-order stochastic averaging.

Moment Lyapunov exponents of the Parametrical Hill's equation under the excitation of two correlated wideband noises

The Lyapunov exponent and moment Lyapunov exponents of Hill\' s equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged Ito differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.