C.S. Manohar

Progress in Structural Dynamics With Stochastic Parameter Variations: 1987-1998

This paper is an update of an earlier paper by Ibrahim (1987) and is aimed at reviewing the papers published during the last decade in the area of vibration of structures with parameter uncertainties. Analytical, computational, and experimental studies conducted on probabilistic modeling of structural uncertainties and free and forced vibration of stochastically defined systems are discussed. The review also covers developments in the areas of statistical modeling of high frequency vibrations and behavior of statistically disordered periodic systems.

An improved response surface method for the determination of failure probability and importance measures

The problem of response surface modeling of limit surface lying within two hyper spheres of prescribed radii is considered. The relevance of this problem in structural reliability analysis involving performance functions with multiple design points and/or multiple regions that make significant contributions to failure probability is discussed. The paper also proposes global measures of sensitivity of failure probability with respect to the basic random variables. The performance of the proposed improvements is examined by comparing simulation based results with results from the proposed procedure with reference to two specific structural reliability analysis problems.

A time-domain approach for damage detection in beam structures using vibration data with a moving oscillator as an excitation source

The problem of detecting local/distributed change of stiffness in bridge structures using ambient vibration data is considered. The vibration induced by a vehicle moving on the bridge is taken to be the excitation source. A validated finite element model for the bridge structure in its undamaged state is assumed to be available. Alterations to be made to this initial model, to reflect the changes in bridge behaviour due to occurrence of damage, are determined using a time-domain approach. The study takes into account complicating features arising out of dynamic interactions between vehicle and the bridge, bridge deck unevenness, spatial incompleteness of measured data and presence of measurement noise. The inclusion of vehicle inertia, stiffness and damping characteristics into the analysis makes the system time variant, which, in turn, necessitates treatment of the damage detection problem in time domain. The efficacy of the procedures developed is demonstrated by considering detection of localized/distributed damages in a beam-moving oscillator model using synthetically generated vibration data.

DYNAMIC ANALYSIS OF FRAMED STRUCTURES WITH STATISTICAL UNCERTAINTIES

The forced harmonic vibration analysis of portal frames consisting of viscously damped beams with spatial stochastic variation of mass and stiffness properties is considered. The analysis is based on the assembly of element stochastic dynamic stiffness matrices. The solution involves inversion of the global dynamic stiffness matrix, which, in this case, turns out to be a complex-valued symmetric random matrix. Three alternative approximate procedures, namely, random eigenfunction expansion method, complex Neumann expansion method and combined analytical and simulation method are used to invert the matrix. The performance of these approximate procedures is evaluated using Monte Carlo simulation results.

Dynamic stiffness of randomly parametered beams

A finite element-based methodology is developed for the determination of the dynamic stiffness matrix of Euler-Bernoulli beams with randomly varying flexural and axial rigidity, mass density and foundation elastic modulus. The finite element approximation made employs frequency dependent shape functions and the analysis avoids eigenfunction expansion which, not only eliminates modal truncation errors, but also, restricts the number of random variables entering the formulations. Application of the proposed method is illustrated by considering two problems of wide interest in engineering mechanics, namely, vibration of beams on random elastic foundation and the problem of seismic wave amplification through randomly inhomogeneous soil layers. Satisfactory agreement between analytical solutions and a limited amount of digital simulation results is also demonstrated.

A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems

The problem of estimating parameters of nonlinear dynamical systems based on incomplete noisy measurements is considered within the framework of Bayesian filtering using Monte Carlo simulations. The measurement noise and unmodelled dynamics are represented through additive and/or multiplicative Gaussian white noise processes. Truncated Ito–Taylor expansions are used to discretize these equations leading to discrete maps containing a set of multiple stochastic integrals. These integrals, in general, constitute a set of non-Gaussian random variables. The system parameters to be determined are declared as additional state variables. The parameter identification problem is solved through a new sequential importance sampling filter. This involves Ito–Taylor expansions of nonlinear terms in the measurement equation and the development of an ideal proposal density function while accounting for the non-Gaussian terms appearing in the governing equations. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter.

Monte Carlo filters for identification of nonlinear structural dynamical systems

The problem of identification of parameters of nonlinear structures using dynamic state estimation techniques is considered. The process equations are derived based on principles of mechanics and are augmented by mathematical models that relate a set of noisy observations to state variables of the system. The set of structural parameters to be identified is declared as an additional set of state variables. Both the process equation and the measurement equations are taken to be nonlinear in the state variables and contaminated by additive and (or) multiplicative Gaussian white noise processes. The problem of determining the posterior probability density function of the state variables conditioned on all available information is considered. The utility of three recursive Monte Carlo simulation-based filters, namely, a probability density function-based Monte Carlo filter, a Bayesian bootstrap filter and a filter based on sequential importance sampling, to solve this problem is explored. The state equations are discretized using certain variations of stochastic Taylor expansions enabling the incorporation of a class of non-smooth functions within the process equations. Illustrative examples on identification of the nonlinear stiffness parameter of a Duffing oscillator and the friction parameter in a Coulomb oscillator are presented.

Rocking response of rectangular rigid blocks under random noise base excitations

The response of a rigid rectangular block resting on a rigid foundation and acted upon simultaneously by a horizontal and a vertical random white-noise excitation is considered. In the equation of motion, the energy dissipation is modeled through a viscous damping term. Under the assumption that the body does not topple, the steady-state joint probability density function of the rotation and the rotational velocity is obtained using the Fokker-Planck equation approach. Closed form solution is obtained for a specific combination of system parameters. A more general but approximate solution to the joint probability density function based on the method of equivalent non-linearization is also presented. Further, the problem of overturning of the block is approached in the framework of the diffusion methods for first passage failure studies. The overturning of the block is deemed incipient when the response trajectories in the phase plane cross the separatrix of the conservative unforced system. Expressions for the moments of first passage time are obtained via a series solution to the governing generalized Pontriagin-Vitt equations. Numerical results illustra- tive of the theoretical solutions are presented and their validity is examined through limited amount of digital simulations.

Improved Response Surface Method for Time-Variant Reliability Analysis of Nonlinear Random Structures Under Non-Stationary Excitations

The problem of time-variant reliability analysis of randomly driven linear/nonlinear vibrating structures is studied. The excitations are considered to be non-stationary Gaussian processes. The structure properties are modeled as non-Gaussian random variables. The structural responses are therefore non-Gaussian processes, the distributions of which are not generally available in an explicit form. The limit state is formulated in terms of the extreme value distribution of the response random process. Developing these extreme value distributions analytically is not easy, which makes failure probability estimations difficult. An alternative procedure, based on a newly developed improved response surface method, is used for computing exceedance probabilities. This involves fitting a global response surface which approximates the limit surface in regions which make significant contributions to the failure probability. Subsequent Monte Carlo simulations on the fitted response surface yield estimates of failure probabilities. The method is integrated with professional finite element software which permits reliability analysis of large structures with complexities that include material and geometric nonlinear behavior. Three numerical examples are presented to demonstrate the method.

CRITICAL CROSS POWER SPECTRAL DENSITY FUNCTIONS AND THE HIGHEST RESPONSE OF MULTI-SUPPORTED STRUCTURES SUBJECTED TO MULTI-COMPONENT EARTHQUAKE EXCITATIONS

The highest response of multi-supported structures subjected to partially specified multi-component earthquake support motions is considered. The seismic inputs are modelled as incompletely specified vector Gaussian random processes with known autospectral density functions but unknown cross spectral densities and these unknown functions are determined such that the steady state response variance of a given linear system is maximized. The resulting cross power spectral density functions are shown to be dependent on the system properties, autospectra of excitation and the response variable chosen for maximization. It emerges that the highest system response is associated neither with fully correlated support motions, nor with independent motions, but, instead, specific forms of cross power spectral density functions are shown to exist which produce bounds on the response of a given structure. Application of the proposed results is demonstrated by examples on a ground based extended structure, namely, a 1578 m long, three span, suspension cable bridge and a secondary system, namely, an idealized piping structure of a nuclear power plant.