An exact closed-form solution for linear multi-degree-of-freedom systems under Gaussian white noise via the Wiener path integral technique

The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi-degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions. In this regard, the herein derived analytical results are also compared with readily available in the literature closed-form exact solutions obtained by alternative stochastic dynamics techniques. In addition to the mathematical merit of the derived exact solution, the closed-form joint response transition PDF can also serve as a benchmark for assessing the performance of alternative numerical solution methodologies.     

Efficient solution of the first passage problem by Path Integration for normal and Poissonian white noise

In this paper the first passage problem is examined for linear and nonlinear systems driven by Poissonian and normal white noise input. The problem is handled step-by-step accounting for the Markov properties of the response process and then by Chapman–Kolmogorov equation. The final formulation consists just of a sequence of matrix–vector multiplications giving the reliability density function at any time instant. Comparison with Monte Carlo simulation reveals the excellent accuracy of the proposed method.     

Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

Equivalent linearization of MDOF systems under external Poisson white noise excitation

Equivalent linearization (EQL) techniques are developed and evaluated for multidimensional systems under external Poisson white noise excitation. Especially, a simulation strategy for the calculation of the linearization coefficients is proposed. The methods are illustrated by several examples that have been treated under Gaussian white noise excitation in the literature. It is shown that EQL for MDOF systems under Poisson white noise excitation is able to deal with problems of nearly the same dimension as under Gaussian white noise excitation.     

The Wong-Zakai theorem for dynamical systems with parametric Poisson white noise excitation

The theorem of Wong and Zakai is applied to obtain a mathematical model for systems under random impulse excitation, in case that the excitation process tends to be delta-correlated. By means of the Wong-Zakai transformation, a class of reducible non-linear stochastic integro-differential equations is identified. Exact stationary probability density functions for reducible stochastic integro-differential equations are calculated.     

Equivalent linearization for systems driven by Lévy white noise

The equivalent linearization method is extended to the case of nonlinear systems driven by Lévy white noise. The classical objective function used for the determination of the equivalent linear system cannot be applied because the Lévy process generating the Lévy white noise has no variance and higher order moments. An alternative objective function based on the characteristic function of the state of the linear system is used for solution. Two simple examples are presented to illustrate the proposed extension of the equivalent linearization method. The examples show that the proposed equivalent linearization method provides approximations of quality similar to the classical equivalent linearization method.     

Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method

In this paper the response in terms of probability density function of nonlinear systems under combined normal and Poisson white noise is considered. The problem is handled via a Path Integral Solution (PIS) that may be considered as a step-by-step solution technique in terms of probability density function. A nonlinear system under normal white noise, Poissonian white noise and under the superposition of normal and Poisson white noise is performed through PIS. The spectral counterpart of the PIS, ruling the evolution of the characteristic functions is also derived. It is shown that at the limit when the time step becomes an infinitesimal quantity an equation ruling the evolution of the probability density function of the response process of the nonlinear system in the presence of both normal and Poisson White Noise is provided.     

Path integral solution for non-linear system enforced by Poisson White Noise

In this paper the response in terms of probability density function of non-linear systems under Poisson White Noise is considered. The problem is handled via path integral (PI) solution that may be considered as a step-by-step solution technique in terms of probability density function. First the extension of the PI to the case of Poisson White Noise is derived, then it is shown that at the limit when the time step becomes an infinitesimal quantity the Kolmogorov–Feller (K–F) equation is fully restored enforcing the validity of the approximations made in obtaining the conditional probability appearing in the Chapman Kolmogorov equation (starting point of the PI). Spectral counterpart of the PI, ruling the evolution of the characteristic function is also derived. It is also shown that using appropriately the PI for Poisson White Noise also the case of Normal White Noise be easily derived.     

Stochastic averaging of quasi-linear systems driven by Poisson white noise

The averaged generalized Itô and Fokker–Planck–Kolmogorov (FPK) equations for single-degree-of-freedom (SDOF) quasi-linear systems driven by Poisson white noise are derived and the approximate stationary solutions of the averaged generalized FPK equations are obtained by using the perturbation method for four typical quasi-linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and system with power law damping. The effectiveness and accuracy of the perturbation 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 is not negligible for predicting the probability densities of total energy and displacement of quasi-linear systems in most cases.     

Delay-independent stability of moments of a linear oscillator with delayed state feedback and parametric white noise

The stability of a linear oscillator with delayed state feedback driven by parametric Gaussian white noise is studied in this paper. The first and second order moment equations of the system response are derived by using moment method and Itô differential rule. Based on the moment equations, the delay-independent stable conditions of both moments are proposed: For the first order moment, the sufficient and necessary condition that guarantee delay-independent stability is identified to that of the deterministic system; for the second order moment, the sufficient condition that ensure delay-independent stability depends on noise intensity. The theoretical results are also illustrated with numerical simulations.     

Path integral solution handled by Fast Gauss Transform

The path integral solution method is an effective tool for evaluating the response of non-linear systems under Normal White Noise, in terms of probability density function (PDF).     

Characteristic function equations for the state of dynamic systems with Gaussian, Poisson, and Lévy white noise

The Itô formula for semimartingales is applied to develop equations for the characteristic function of the state of linear and non-linear dynamic systems with Gaussian, Poisson, and Lévy white noise, viewed as the formal derivatives of Brownian, compound Poisson, and Lévy processes, respectively. These equations can be obtained if the drift and diffusion coefficient of a dynamic system are polynomials of the system state and the driving noise is Gaussian or Poisson. It was not possible to derive equations for the characteristic function for the state of systems driven by Lévy white noise. Numerical results are presented for dynamic systems with real-valued states driven by Gaussian, Poisson, and Lévy white noise processes.     

Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.     

Probabilistic response of linear structures equipped with nonlinear damper devices (PIS method)

Passive control introducing energy absorbing devices into the structure has received considerable attention in recent years. Unfortunately the constitutive law of viscous fluid dampers is highly nonlinear, and even supposing that the structure behaves linearly, the whole system has inherent nonlinear properties. Usually the analysis is performed by a stochastic linearization technique (SLT) determining a linear system equivalent to the nonlinear one, in a statistical sense. In this paper the effect of the non-Gaussianity of the response due to the inherent nonlinearity of the damper device will be studied in detail via the Path Integral Solution (PIS) method. A systematic study is conducted showing that for a very wide range of parameters the SLT gives satisfactory results in terms of variance of displacement and velocity but not in terms of joint Probability Density Function (PDF). It has also been shown that at steady state the two processes, displacement and velocity, may be considered as independent ones.     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

泊松白噪声激励下斜拉索的面内随机振动

目前针对斜拉索非线性随机振动的研究已广泛开展,但仅限于高斯随机激励情形。然而,现实中大部分的随机扰动都是非高斯的。若使用高斯激励模型将产生较大误差。假设拉索所受非高斯激励为泊松白噪声,研究了泊松白噪声激励下斜拉索面内随机振动。推导了受泊松白噪声激励的斜拉索面内振动的随机微分方程,建立了支配系统平稳响应概率密度函数的广义FPK方程。提出迭代加权残值法求解了四阶广义FPK方程,得到了系统响应概率密度函数的近似稳态闭合解。考察了垂跨比、阻尼系数以及脉冲到达率对拉索面内随机振动响应的影响。结果表明:拉索的响应随着垂跨比的增大,响应呈现不对称现象愈加明显;随阻尼比增加,系统响应得到显著抑制;当脉冲到达率增大,拉索的响应也随之增大,并逐渐接近于高斯白噪声激励的情形。另外,获得的理论结果与蒙特卡罗模拟的结果吻合地非常好。     

Multiplicative cases from additive cases: Extension of Kolmogorov–Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined.As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completely defined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformed system in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the proper solution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, we must pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that is known). Moreover, introducing this invertible nonlinear transformation into the Itô’s rule for the original system driven by external input, one can derive the Itô’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of the Wong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively. Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found.     

Probabilistic response of an elastic perfectly plastic oscillator under Gaussian white noise

The response of an elastic perfectly plastic oscillator under zero mean Gaussian white noise excitation is studied in this paper. Considering the works of previous studies, a closed form expression of the mean maximum of the plastic drift is given assuming that the plastic process is equivalent to a Brownian motion. In order to better describe the plastic drift a probabilistic model is proposed for the yield increments which occur in clumps. To estimate the input parameters of this model, three methods, based on numerical computations of some relevant integrals, are presented. Alternatively, these parameters can be estimated, more conveniently, according to the results obtained more recently in the literature with the Slepian model approach. The results of numerical simulations show a quite satisfactory agreement with theoretical predictions.     

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.