Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments

In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem and related concepts, the solution of the Kolmogorov–Feller equation is obtained in a very easy way by solving a set of linear differential equations. Results are compared with those of Monte Carlo simulation showing the robustness of the solution pursued in terms of CFMs. Further, a very elegant strategy, to give a relationship between the CFMs evaluated for different value of real part, is introduced as well.     

Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation

The stochastic averaging method for strongly non-linear oscillators with lightly fractional derivative damping of order α (0<α≤1) subject to bounded noise excitations is proposed by using the generalized harmonic function. The system state is approximated by a two-dimensional time-homogeneous diffusion Markov process of amplitude and phase difference using the proposed stochastic averaging method. The approximate stationary probability density of response is obtained by solving the reduced Fokker–Planck–Kolmogorov (FPK) equation using the finite difference method and successive over relaxation method. A Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary resonance, the stochastic jump of the Duffing oscillator with fractional derivative damping and its P-bifurcation as the system parameters change are examined for the first time using the stationary probability density of amplitude.     

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree-of -freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set...     

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.     

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations is studied. The motion equation of the system is reduced to a set of averaged Itô stochastic differential equations by stochastic averaging in the case of resonance. Then, the backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function and the conditional probability density and mean first-passage time are obtained by solving the backward Kolmogorov equation and Pontryagin equation with suitable initial and boundary conditions. The procedure is applied to Duffing–van der Pol system in resonant case and the analytical results are verified by Monte Carlo simulation.     

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

The stationary response of Duffing oscillator with hardening stiffness and fractional derivative under Gaussian white noise excitation is studied. First, the term associated with fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, and the original system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the stochastic averaging method of energy envelope is applied to the equivalent nonlinear stochastic system to yield the averaged Itô equation of energy envelope, from which the corresponding Fokker–Planck–Kolmogorov (FPK) equation is established and solved to obtain the stationary probability densities of the energy envelope and the amplitude envelope. The accuracy of the analytical results is validated by those from the Monte Carlo simulation of original system.     

Dynamic stability and response of fluttered beams subjected to random follower forces

This paper presents a study of non-linear response of a fluttered, cantilevered beam subjected to a random follower force at the free end. The random follower force is characterized as the sum of a post-critical static force and a stationary process with a zero mean. First, the Ritz-Galerkin method is applied to yield a set of discretized system equations. The system equations are then partially uncoupled by a special modal analysis based on normal modes of the corresponding linear, autonomous system at the onset of fluttering. Next, the stochastic averaging method is utilized to get Ito's differential equation governing the amplitude of the fluttered mode. Finally, the probability density function for the amplitude of the fluttered mode is obtained by solving the FPK equation. Numerical results show that the probability density function for the amplitude of the fluttered mode is determined by the sample behavior of the beam near the trivial equilibrium configuration.     

First-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations

A procedure for studying the first-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations is proposed. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is worked out in detail to illustrate the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed. The validity of the proposed method is confirmed by digital simulation.     

粘滞和粘弹性阻尼器减震结构的随机响应特性

利用积分型本构关系,针对带支撑任意线性粘滞和粘弹性阻尼器单自由度耗能结构,建立了微分和积分混合地震响应方程;基于随机平均分析法,推导出耗能结构振幅与相位瞬态联合概率密度函数、位移与速度瞬态联合概率密度函数、位移与速度瞬态响应方差、振幅动力可靠性、振幅首超时间任意阶统计矩的一般解析解;给出了带支撑广义Maxwell阻尼器和广义微分模型阻尼器耗能结构上述各种随机响应特性,从而建立了带支撑任意线性粘滞和粘弹性阻尼器单自由度减震结构的各种随机响应特性分析的统一解析解法。     

FIRST-PASSAGE FAILURE OF PREISACH HYSTERETIC SYSTEMS

The first-passage failure of a single-degree-of-freedom hysteretic system with non- local 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 sys- tem 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 aver- aging of energy envelope is adopted to accept the averaged It5 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.     

First passage of stochastic fractional derivative systems with power-form restoring force

In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.     

Cross-correlation and cross-power spectral density representation by complex spectral moments

A new approach to provide a complete characterization of normal multivariate stochastic vector processes is presented in this paper. Such proposed method is based on the evaluation of the complex spectral moments of the processes. These quantities are strictly related to the Mellin transform and they are the generalization of the integer-order spectral moments introduced by Vanmarcke.The knowledge of the complex spectral moments permits to obtain the power spectral densities and their cross counterpart by a complex series expansions. Moreover, with just the aid of some mathematical properties the complex fractional moments permit to obtain also the correlation and cross-correlation functions, providing a complete characterization of the multivariate stochastic vector processes.Some numerical applications are reported in order to show the capabilities of this method. In particular, the examples regard two dimensional linear oscillators forced by Gaussian white noise, the characterization of the wind velocity field, and the stochastic response analysis of vibro-impact system under Gaussian white noise.     

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.     

Responses of Duffing oscillator with fractional damping and random phase

This paper aims to investigate dynamic responses of stochastic Duffing oscillator with fractional-order damping term, where random excitation is modeled as a harmonic function with random phase. Combining with Lindstedt–Poincaré (L–P) method and the multiple-scale approach, we propose a new technique to theoretically derive the second-order approximate solution of the stochastic fractional Duffing oscillator. Later, the frequency–amplitude response equation in deterministic case and the first- and second-order steady-state moments for the steady state in stochastic case are presented analytically. We also carry out numerical simulations to verify the effectiveness of the proposed method with good agreement. Stochastic jump and bifurcation can be found in the figures of random responses, and then we apply Monte Carlo simulations directly to obtain the probability density functions and time response diagrams to find the stochastic jump and bifurcation. The results intuitively show that the intensity of the noise can lead to stochastic jump and bifurcation.     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

Stochastic averaging for nonlinear vibration energy harvesting system

A stochastic averaging technique for the nonlinear vibration energy harvesting system to Gaussian white noise excitation is developed to analytically evaluate the mean-square electric voltage and mean output power. By introducing the generalized harmonic transformation, the influence of the external circuit on the mechanical system is equivalent to a quasi-linear stiffness and a quasi-linear damping with energy-dependent coefficients, and then the equivalent nonlinear system with respect to the mechanical states is completely established. The Itô stochastic differential equation with respect to the mechanical energy of the equivalent nonlinear system is derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the stationary probability density of the mechanical states, and then the mean-square electric voltage and mean output power are analytically obtained through the approximate relation between the electric quantity and the mechanical states. The agreements between the analytical results and those from the moment method and from Monte Carlo simulations validate the effectiveness of the proposed technique.     

Amplitude stochastic response of Rayleigh beams to randomly moving loads

We consider the problem of the nonlinear response of a Rayleigh beam to the passage of a train of forces moving with stochastic velocity. The Fourier transform and the theory of residues are used to estimate the mean square amplitude of the beam, while the stochastic averaging method gives the stationary probability density function of the oscillations amplitude. The analysis shows that the effect of the load random velocities is highly nonlinear, leading to a nonmonotonic behavior of the mean amplitude versus the intensity of the stochastic term and of the load weight. The analytic approach is also checked with numerical simulations. The effect of loads number on the system response is numerically investigated.     

Transient stochastic response of quasi-partially integrable Hamiltonian systems

The approximate transient response of multi-degree-of-freedom (MDOF) quasi-partially integrable Hamiltonian systems under Gaussian white noise excitation is investigated. First, the averaged Itô equations for first integrals and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the transient probability density of first integrals of the system are derived by applying the stochastic averaging method for quasi-partially integrable Hamiltonian systems. Then, the approximate solution of the transient probability density of first integrals of the system is obtained from solving the FPK equation by applying the Galerkin method. The approximate transient solution is expressed as a series in terms of properly selected base functions with time-dependent coefficients. The transient probability densities of displacements and velocities can be derived from that of first integrals. One example is given to illustrate the application of the proposed procedure. It is shown that the results for the example obtained by using the proposed procedure agree well with those from Monte Carlo simulation of the original system.     

First-Passage Time of Duffing Oscillator under Combined Harmonic and White-Noise Excitations

The first-passage time of Duffing oscillator under combined harmonic andwhite-noise excitations is studied. The equation of motion of the system is firstreduced to a set of averaged Itô stochastic differential equations by using thestochastic averaging method. Then, a backward Kolmogorov equation governing theconditional reliability function and a set of generalized Pontryagin equationsgoverning the conditional moments of first-passage time are established. Finally, theconditional reliability function, and the conditional probability density and momentsof first-passage time are obtained by solving the backward Kolmogorov equation andgeneralized Pontryagin equations with suitable initial and boundary conditions.Numerical results for two resonant cases with several sets of parameter values areobtained and the analytical results are verified by using those from digital simulation.     

A moment-based approach for nonlinear stochastic tracking control

This paper describes a new stochastic control methodology for nonlinear affine systems subject to bounded parametric and functional uncertainties. The primary objective of this method is to control the statistical nature of the state of a nonlinear system to designed (attainable) statistical properties (e.g., moments). This methodology involves a constrained optimization problem for obtaining the undetermined control parameters, where the norm of the error between the desired and actual stationary moments of state responses is minimized subject to constraints on moments corresponding to a stationary distribution. To overcome the difficulties in solving the associated Fokker–Planck equation, generally experienced in nonlinear stochastic control and filtering problems, an approximation using the direct quadrature method of moments is proposed. In this approach, the state probability density function is expressed in terms of a finite collection of Dirac delta functions, and the partial differential equation can be converted to a set of ordinary differential equations. In addition to the above mentioned advantages, the state process can be non-Gaussian. The effectiveness of the method is demonstrated in an example including robustness with respect to predefined uncertainties and able to achieve specified stationary moments of the state probability density function.