Loren D. Lutes

Spectral characteristics of nonstationary random processes — a critical review

Abstract This article analyzes the approaches to defining “spectral characteristics” derived from the spectral functions of nonstationary random processes. The processes considered are those for which an evolutionary power spectrum as designated by Priestley can be defined. Two basic approaches to defining spectral characteristics are reviewed. The first, characterized as geometric, leads to Vanmarkes spectral moments, which have proven to be very useful characteristics for stationary processes. However, these moments may be infinite for nonstationary processes, which creates problems for applications. The second approach, viewed as nongeometric, is based on Di Paolas pre-envelope covariances. The advantages and deficiencies of both approaches are discussed. It is also shown that the nongeometric spectral characteristics can be directly defined from the frequency domain as integrals of the one-sided auto- and cross-spectra of the evolutionary process and its derivatives. These nongeometric spectral characteristics are then used in defining parameters that characterize the central frequency and the bandwidth of evolutionary processes. To this end, the probability distributions of the process envelope are analyzed. It is demonstrated that suitable central frequencies and bandwidth factors can be defined from the probability density functions of the derivatives of the envelope and the phase.

Spectral characteristics of nonstationary random processes—response of a simple oscillator

Abstract A nongeometric approach for defining spectral characteristics of evolutionary (nonstationary) processes is applied to the transient stochastic response of a simple oscillator with modulated white noise excitation. The first three nonstationary spectral characteristics of the response are considered. Although the nongeometric spectral characteristics can generally be defined as integrals of the one-sided evolutionary auto- and cross-spectra of the nonstationary process and its derivatives, the study exploits their time-domain definitions as the variances and the covariance of the response and an “auxiliary” process. The approach is applied to three particular excitation functions which have simple analytical forms. Each of these functions has been frequently studied in the past, and/or has potential for use in modeling a variety of practical engineering problems. Approximate expressions for the first three spectral characteristics of the response are developed by neglecting small oscillatory terms. Based on these expressions, approximations are derived for the central frequency and the bandwidth factor of the response.

Response variability for a structure with soil-structure interactions and uncertain soil properties

Abstract A non-classical modal analysis based formulation is used to quantify the effect of uncertain soil-foundation properties on structural response in seismically excited soil–structure interacting (SSI) systems. This formulation allows the response of the interacting system to be represented as a superposition of the responses of uncoupled modal equations which include an additional random vector to represent the system uncertainties. The method is implemented by an effective Gaussian quadrature integration method to find the frequency-domain stochastic response properties of the SSI systems. Numerical examples of MDOF SSI systems illustrate that the main effect of uncertain soil properties on such SSI systems is to alter the magnitudes of modal response near the system resonant frequencies, rather than to shift the resonant frequencies. When the uncertainty of soil-foundation properties is not negligible, there may be significant variations of the transfer functions for modal response and significant uncertainty about the spectral density of structural response may occur near the system’s resonant frequencies.

Response variability of an SSI system with uncertain structural and soil properties

Abstract A non-classical modal-analysis-based formulation including an additional random vector to represent the system uncertainties is used to quantify the effect of uncertain soil–foundation and superstructure properties on structural response for seismically excited soil–structure interacting (SSI) systems. It is shown that the pertinent equations are essentially the same as those for only soil parameter uncertainty and very similar to those for only structural parameter uncertainty. The method is implemented here by using an effective Gaussian quadrature method to find accurate frequency-domain stochastic response properties of the SSI systems. Numerical examples of multiple-degree-of-freedom SSI systems with quite large parameter uncertainty illustrate that the unconditional spectral density of the structural response is often very similar to that in the absence of uncertainty. However, deterministic variations of parameters may cause large changes in the response spectral density. In particular, when the parameter uncertainty is not negligible, there may be significant uncertainty about the spectral density of structural response near the systems resonant frequencies. It is shown that the effects of uncertainty about multiple parameters can generally be adequately and efficiently approximated by applying a simple first-order second-moment approximation to the results for single-parameter uncertainty.

Stochastic cumulant analysis of MDOF systems with polynomial-type nonlinearities

A new methodology is presented for formulating the equations governing the evolution of the response cumulants of MDOF nonlinear dynamic systems subjected to external delta-correlated processes. The system nonlinearities are represented by polynomial terms involving the system variables. Kronecker algebra and matrix calculus are used as efficient mathematical tools to organize the information and present the cumulant equations in a compact form. Appropriate recursive relationships are developed to relate the joint cumulants involving Kronecker powers of the response vector variables to the joint cumulants involving the original response vector variables. It is found that the differential equations governing the system of cumulants has a form similar to the state space form of equations for the original dynamic system. The state and excitation matrices describing the system of cumulants are obtained, respectively, from the state and excitation matrices describing the original dynamic system. Cumulant-neglect closure techniques, in which the joint cumulants of the response variables with order higher than a specified order are neglected, can be directly incorporated and efficiently used in the analysis to truncate the infinite hierarchy of the resulting system of cumulant equations. Examples are presented to illustrate the use of the formulation, as well as to investigate convergence and accuracy issues related to the higher-order cumulant-neglect closure scheme.

Stochastic response moments for linear systems

Abstract State space moment analysis is developed as a practical tool for investigating the response of a linear system subjected to stochastic excitation. General formulations are presented to show that the method can be used to evaluate response moments, or cumulants, of any order for both stationary and nonstationary response. The limitation is that the excitation of the linear system must be a generalized white noise called a delta-correlated process. This generalization of the Lyapunov method for finding response covariances gives a comparable matrix method for finding the higher order moments which are often important in predicting failure due to first-passage or fatigue. The technique used here involves rewriting the mth order tensor of mth order cumulants into a minimum length vector, making use of all inherent symmetry, in order to minimize the size of the resulting matrix. Easily implemented algorithms are presented for finding the terms in this matrix. General relationships are also given relating the eigenvalues and eigenvectors of this matrix for mth order cumulants to those of a much smaller matrix. This eigen solution is needed for evaluating nonstationary response cumulants, and the given relationships provide a particularly efficient method for evaluating the eigenvalues. The method is illustrated by evaluating the 35 fourth cumulants of nonstationary response for a class of two-degree-of-freedom oscillators.

Model identification and control of a tuned liquid damper

Experimental studies have been conducted to investigate the feasibility of using active control of the tuned liquid damper for reducing the vibration of large civil structures. The results indicate that a simple mechanism can be designed to actively control the tuning of the system by adjusting the length of the liquid tank with rotatable baffles driven by stepping motors. The tank can be permanently installed and no powerful actuators would be required to regulate its motion. As a result, the system can be built inexpensively and can be tested frequently to ensure its reliability. A control strategy based on detection of frequency content and two-state control is simple and can be easily implemented in a microprocessor. Preliminary results of the damper model identification are provided, allowing computer simulations and the design of advanced control strategies for fine tuning of the liquid damper in future studies. The performance of this novel damper is verified by computer simulations and experiments using a physical model.

Trispectrum for the response of a non-linear oscillator

Abstract Simulation is used to obtain information about non-Gaussian aspects of the absolute response acceleration of a bi-linear hysteretic oscillator with an excitation which is Gaussian white noise. Attention is focused on the frequency content of the fourth cumulant of the response. This frequency content is studied by consideration of the trispectrum and also by the simplified technique of looking at the coefficient of excess for the response of a narrowband linear system mounted on the non-linear oscillator. Attempts are also made to model the non-Gaussian response of the non-linear oscillator by a filtered delta correlated (FDC) process, but it is shown that no process of this type can exhibit some of the significant characteristics of the non-linear response. In particular, the trispectrum of the non-linear response appears to be more narrowband than the power spectral density, and also it sometimes does not have the same sign at every point in the three-dimensional frequency space, and this behavior is distinctly different from that of any FDC process. Modifications of the FDC model are suggested in order to obtain improved approximations of the non-linear response.

Approximate analysis of higher cumulants for multi-degree-of-freedom random vibration

Abstract A computationally efficient methodology is presented for calculating the higher-order cumulants of the response of a linear system subjected to stationary or non- stationary stochastic excitation. The technique of state space analysis is used to derive, from the original linear system, a new system of ordinary differential equations governing the evolution of cumulants of various response variables. Complex modal analysis is employed to uncouple the new system and obtain the approximate cumulants in a reduced space of modal coordinates. An approximate methodology is developed to obtain simplified analytical solutions for some of the contributing modes by neglecting secondary dynamical effects in the corresponding modal equations. Valuable insight into the important factors affecting the reliability of such approximations is provided. The derivation and storage of the large matrix describing the system of cumulants is avoided. Instead, one uses only the eigenvalues and eigenvectors of the matrix describing the system of cumulants and of its transpose, and in the approximate analysis only the first few of these eigensolutions are usually needed. These eigensolutions are efficiently obtained from the eigensolutions of two much smaller matrices. The approximate methodology reduces drastically the computer storage and the computational effort required to solve for the response cumulants of any order. The performance and accuracy of the methododology are illustrated by examples.

Efficiency and accuracy in simulation of random fields

A direct method for the conditional simulation of a stationary, Gaussian scalar random field is compared with an alternative formulation which uses frequency domain probability density functions. In both cases the random field is described by given correlation or spectral density functions, and no restrictions are placed on these functions, except that they must be positive definite. Efficient implementation techniques are investigated for both general methods. The major computational effort in the most efficient implementations of both procedures is in the solution of linear algebraic equations in which the coefficients are spectral densities. The direct method is shown to be significantly more efficient than existing methods for applying the probability density function technique. However, a new implementation method for the latter technique is also presented, and it equals the efficiency of the direct method. Problems of numerical accuracy due to ill-conditioned matrices are shown not to be severe except when using an inherently problematic form for the spectral density. Numerical examples demonstrate that either method can simulate highly coherent time histories.