Alberto Di Matteo

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

THE TLCD PASSIVE CONTROL: NUMERICAL INVESTIGATIONS VS EXPERIMENTAL RESULTS

Very recently the tuned liquid column damper (TLCD) is receiving an increasing interest from researchers concerned with vibration control, to be considered an alternative device with respect to the tuned mass damper (TMD), since the former has low cost, easy adjustment, flexible installation.However, in recent studies the authors [1] have pointed out that for TMD the analytical formulation provides results that are in good agreement with the experimental ones, while for TLCD it has been deducted that the analytical formulation needs further investigation.In fact using the classical formulation of the problem, numerical results are very different from the experimental results obtained by the authors using the facilities at the experimental dynamic laboratory of University of Palermo.In particular it has been shown that the total liquid length should be corrected in an effective one, but in a different way from what has been done in literature, where only the variation of section of the vessel has been taken into account. On the other hand, from experimental investigations it is seen that the liquid moves more in the central area of the tube and less in the area in contact with the side walls. This aspect plays a fundamental role for capturing the real performance of TLCD. In fact, being the TLCD a special type of auxiliary damping device which relies on the inertia of liquid column in a U-tube to counteract the forces acting on the structure, then it is necessary to identify the effective moving liquid mass. To aim at this, in this paper the authors differentiate the total liquid mass into a liquid dead mass and a liquid dynamic mass, then introducing these values into a properly modified mathematical formulation numerical results match the experimental ones for all tests.© 2012 ASME

The moment equation closure method revisited through the use of complex fractional moments

In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones the probability density function response of nonlinear systems may be written in discretized form in terms of complex fractional moment not requiring a closure scheme. Excitations such as ground motion, wind turbulence, sea waves, surface roughness, blasts and impacts loads being stochastic processes induce that structural responses are stochastic processes too. Thus, the analyst is concerned with the problem of the response statistical characterization. However, to consider a model closer to reality a nonlinear system has to be considered, then a complete statistical characterization of the response may be performed by solving the Fokker–Planck– Kolmogorov (FPK) equation, a partial differential equation whose solution is the joint probability density function (PDF) of the response variables (Lin and Cai, 1995). Unfortunately, the FPK equation admits analytical solution in very few cases, for this reason we resort to numerical methods. Among the numerical approaches, more attractive, from a computational point of view, is the moment equation (ME) approach, in which the response statistical characterization is given by the response moments or by other quantities related to the former such as cumulants or quasimoments (Stratanovich, 1997; Ibrahim, 1985). This method consists of writing differential equations for the response statistical moments of any order. However, when dealing with nonlinear systems, a serious problem arises in the ME approach, the entire system is hierarchic in the sense that the equations for the moments of a fixed order, say K, contain moments of order higher than K. In this way, the ME form an infinite hierarchy. In order to overcome this difficulty, the so-called closure methods were born. The key idea is to express the response PDF as a Edgeworth or Gram-Charlier series, truncating it at a certain term. The coefficients of the above mentioned series can be written as functions of the response central moments or of the response cumulants or of the response quasimoments. Thus, neglecting the terms beyond a 12 th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 given order is equivalent to make central moments or cumulants or quasi-moments zero, which makes the ME solvable. The moments of order larger than K are expressed in terms of moments of order equal or lower to K by means of the relationships that are obtained by putting the above cited quantities equal to zero. Recently in Di Paola (2014) it has been introduced the complex fractional moments (CFM) through which the FPK equation has been converted, returning a simple method to perform a PDF response function; in Di Matteo et al. (2014) and in Alotta and Di Paola (2015) the method has been applied successfully also for the case of Kolmogorov-Feller equation (Poissonian white noises) and for fractional FPK equation (αstable white noises), respectively. In this paper it will be explored the useful tool of the complex fractional moments to overcome this moment closure procedure. 1. SERIES FORM OF PDF THROUGH COMPLEX FRACTIONAL MOMENTS Starting from the equation of motion of a nonlinear half oscillator in the form, consider the scalar stochastic process ( ) X t satisfying the stochastic differential equation ( ) ( ) ( ) 0 , 0 X f X t W t X X  = +   =  ɺ (1) Where ( ) W t is a Gaussian white noise with zero mean and correlation function ( ) ( ) ( ) ( ) 0 [ ] 2 τ π δ δ + = = E W t W t S t q t with 0 S being the spectral density of ( ) W t . Moreover ( ) , f X t is a nonlinear function of the process ( ) X t and 0 X is the random variable with assigned PDF in zero ( ) ( ) ,0 X p x p x = . The equation ruling the evolution of the PDF of the response process ( ) X t is the so-called Fokker-Planck (FP) equation that may be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2

PATH INTEGRAL METHOD FOR FIRST-PASSAGE PROBABILITY DETERMINATION OF NONLINEAR SYSTEMS UNDER LEVY WHITE NOISE

In this paper the problem of the first-passage probabilities determination of nonlinear systems under alpha-stable Lévy white noises is addressed. Based on the properties of alpha-stable random variables and processes, the Path Integral method is extended to deal with nonlinear systems driven by Lévy white noises with a generic value of the stability index alpha. Furthermore, the determination of reliability functions and first-passage time probability density functions is handled step-by-step through a modification of the Path Integral technique. Comparison with pertinent Monte Carlo simulation reveals the excellent accuracy of the proposed method.

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

In this paper the probabilistic characterization of a nonlinear system enforced by parametric Poissonian white noise in terms of complex fractional moments is presented. In fact the initial system driven by a parametric input could be transformed into a system with an external type of excitation through an invertible nonlinear transformation. It is shown that by using Mellin transform theorem and related concepts, the solution of the Kolmogorov-Feller equation for the system with external input may be obtained in a very easy way.