Pol D. Spanos

Stochastic averaging: An approximate method of solving random vibration problems

Abstract Results obtained by applying the method of stochastic averaging to random vibration problems are discussed. This method is applicable to a variety of problems involving the response of lightly damped systems to broad-band random excitations. Solutions pertaining to both linear and non-linear vibrations are reviewed, and it is shown that the technique enables, in the case of parametric excitation, stability criteria to be established. Some results which have been obtained relating to the first-passage reliability problems are also surveyed. Various applications of the theory to engineering problems are outlined.

Polynomial Chaos in Stochastic Finite Elements

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials

Computational stochastic mechanics

Random processes and fields random vibrations (I and II) control and optimization earthquake engineering Monte Carlo methods stochastic finite elements stochastic boundary elements and continua applied random vibrations safety and reliability damage and fatigue concrete and geotechnical applications.

A stochastic Galerkin expansion for nonlinear random vibration analysis

Abstract An approach is developed for the numerical solution of random vibration problems. It is based on treating random variables as functions in a certain Hilbert space. Stochastic processes are described as curves defined in this space, and concepts from deterministic approximation theory are applied to represent the solution as a series involving a known basis of stochastic processes, and a set of unknown coefficients which are deterministic functions of time. Then, a Galerkin projection procedure is utilized to derive a set of ordinary differential equations which can be solved numerically to determine the coefficients in the series. The versatility of the proposed approach is demonstrated by its application to a nonlinear vibration problem involving the probability density of a non-Markovian oscillator response.

Recursive Simulation of Stationary Multivariate Random Processes—Part II

Stability and invertibility aspects of the AR to ARMA procedures developed in Part I in connection with simulation of multivariate random processes are addressed. A general criterion is proved for this purpose. Furthermore, several properties regarding the matching of the correlations at various time lags of the target and the simulated processes are shown. Finally, the reliability and efficiency of the discussed procedures are demonstrated by application to spectra encountered in earthquake engineering, offshore engineering, and wind engineering.

Dynamic analysis of stacked rigid blocks

The dynamic behavior of a structural model of two stacked rigid blocks subjected to ground excitation is examined. Assuming no sliding, the rocking response of the system standing free on a rigid foundation is investigated. The derivation of the equations of motion accounts for the consecutive transition from one pattern of motion to another, each being governed by a set of highly nonlinear differential equations. The system behavior is described in terms of four possible patterns of response and impact between either the two blocks or the base block and the ground. The equations governing the rocking response of the system to horizontal and vertical ground accelerations are derived for each pattern, and an impact model is developed by conservation of angular momentum considerations. Numerical results are obtained by developing an ad hoc computational scheme that is capable of determining the response of the system under an arbitrary base excitation. This feature is demonstrated by using accelerograms from the Northridge, CA, 1994, earthquake. It is hoped that the two-blocks model used herein can facilitate the development of more sophisticated multi-block structural models.

Nonlinear Stochastic Drill-String Vibrations

ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC2000-251AbstractA study of downhole vibrations of drill-strings bottom-hole assemblies is undertaken. The lateral behaviorof the system is of interest. The nonlinear nature of the problem is addressed by considering a lateral clear-ance between the drill-string and the borehole that induces a stiffening of the system when exceeded. Thestochastic input force is defined by its power spectral density and it is applied laterally to the bit. The methodof statistical linearization is used, nd expressions for computing the equivalent linear system of the bottom-hole assembly are presented. The adopted procedure involves a prefiltering of the bit excitation to derive adynamic system under white-noise and colored white-noise excitations. Then, the Lyapunov equation for thecovariance of the linearized system is solved. Further, a Monte-Carlo simulation is conducted by means ofan auto-regressive moving-average digital filter, and the equations of motion are integrated by the Newmarkmethod. Numerical results pertaining to data obtained by measurement-while-drilling tools are presented.The study facilitates the assessment of the appropriateness of the method of statistical linearization for “realworld” problems encountered even in conservative industrial applications such as drilling.

Spectral techniques for stochastic finite elements

SummaryA formulation for the stochastic finite element method is presented which is a natural extension of the deterministic finite element method. Discretization of the random dimension is achieved via two spectral expansions. One of them is used to represent the coefficients of the differential, equation which model the random material properties, the other is used to represent the random solution process. The method relies on viewing the random aspect of the problem as an added dimension, and on treating random variables and processes as functions defined over that dimension. The versatility of the method is demonstrated by discussing, as well, some non-traditional problems of stochastic mechanics.

Oil and gas well drilling: A vibrations perspective

The process of oil well rotary drilling is discussed from a vibrations perspective. Furthermore, a synopsis of the equipment used and of the procedures followed is included. Performance modeling and monitoring are discussed, along with data acquisition and utilization. Axial, torsional, and lateral vibrations are reviewed. Typical operational difficulties such as sticking, buckling, and fatiguing of strings are also reviewed. Finally, emerging techniques of analysis such as stochastic treatment, and optimization are discussed. To enhance the utility of the paper, references readily available in the form of books or archival journal publications are primarily cited. Furthermore, mathematical rigor and phenomenological completeness are sporadically moderated to conform with paper length limitations.

ARMA Monte Carlo simulation in probabilistic structural analysis

Autoregressive moving average (ARMA) systems for sysnthesizing realizations of stochastic processes are discussed in context with the technique of Monte Carlo simulation. Strictly autoregressive (AR) or strictly moving average (MA) systems are considered as special cases of the ARMA systems. Their applicability in wind, ocean, and earthquake engineering is briefly reviewed