Wilfred D. Iwan

A generalization of the concept of equivalent linearization

Abstract A method is presented whereby a non-linear second order dynamical system is replaced by a linear system in such a way that an average of the difference between the two systems is minimized. Provided the averaging operator possesses certain properties, it is shown that the replacement is unique and can be accomplished in a straightforward manner. The parameters of the replacement linear system are expressed in terms of averages of functions of the linearized solution.

Equivalent linearization for systems subjected to non-stationary random excitation

Abstract The method of eauivalent linearization is applied to the general problem of the response of non-linear discrete systems to non-stationary random excitation. Conditions for minimum equation difference are determined which do not depend explicitly on lime but only on the instantaneous statistics of the response process. Using the equivalent linear parameters, a deterministic non-linear ordinary differential equation for the covariance matrix is derived. An example is given of a damped Duffing oscillator subjected to modulated white noise.

On the existence and uniqueness of solutions generated by equivalent linearization

Abstract The existence and uniqueness of approximate solutions generated by the generalized method of equivalent linearization is considered. For the stationary analysis of systems with harmonic or Gaussian random excitation, it is shown that even though the equivalent linear system may not be unique, a simple element-by-element substitute system exists. Furthermore, this system is at least as good as any other similarly defined substitute system.

On the dynamic response of non-linear systems with parameter uncertainties

This paper presents a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties. Consideration is given to systems with polynomial non-linearity subjected to deterministic excitation. The uncertain parameters are modeled as time-independent random variables. The set of orthogonal polynomials associated with the probability density function is used as the solution basis, and the response variables are expanded in terms of a finite sum of these polynomials. A set of deterministic non-linear differential equations is derived using the weighted residual method. The discrete-time solutions to the equation set are evaluated numerically using a step-by-step time-integration scheme and the response statistics are determined. Application of the proposed method is illustrated through the analysis of non-linear single-degree-of-freedom structural systems exhibiting uncertain stiffnesses. Both hardening and softening stiffness characteristics are examined. The accuracy of the results is validated by direct numerical integration.

On defining equivalent systems for certain ordinary non-linear differential equations

Abstract This paper deals with a method for generating approximate periodic solutions for systems of ordinary non-linear differential equations. The method is based on the idea of finding a system which is equivalent to the original system in the sense of minimum mean square difference. The equivalent system may be either linear of nonlinear. The relation of the presented method to some of the more well known techniques of analysis is discussed and examples of application are given.

Nonlinear system identification based on modelling of restoring force behaviour

This paper introduces a system identification algorithm based upon modelling of the restoring force behaviour of the structure. This algorithm is more efficient than traditional algorithms based upon matching the time history of response of the structure, since error evaluation does not require the solution of any differential equations. The effectiveness of this system identification approach, in coordination with a model for deteriorating structures is demonstrated by an example using actual earthquake data from the Bank of California building which was damaged during the 1971 San Fernando earthquake.

The transient and steady-state response of a hereditary system

Abstract The distinctive features of the response of a system with limited slip and trigonometric excitation are discussed. Both the steady-state and transient behavior of this hereditary system are predicted using a first order approximation technique, and the predictions are compared with the results of “exact” numerical solutions.

Some observations on two piecewise-linear dynamic systems with induced hysteretic damping

Significant hysteretic damping can be introduced into dynamical systems for vibration control purpose by means of semi-active control implementation. The dynamic behavior of two such models with a single-degree-of-freedom are examined in this paper. Control performance is found to be not only dependent on the stiffness ratio between the primary spring and auxiliary spring, but also on the forcing frequency of the harmonic excitation. Some problems associated with the equivalent linearization of these types of dynamical systems are also discussed.

Nonstationary equivalent linearization of nonlinear continuous systems

Abstract This paper presents a technique for obtaining the nonstationary stochastic response of a nonlinear continuous system. The method of equivalent linearization is generalized to continuous systems subjected to nonstationary random excitation. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. A numerical implementation is also described. In this procedure, the linear replacement system is discretized by the finite element method. Application to systems satisfying the one-dimensional wave equation with a constitutive nonlinearity is discussed. Results are presented for nonlinear stress-strain laws of a strain-hardening type.

Preface to the Special Issue on the Great Sumatra Earthquakes and Indian Ocean Tsunamis of 26 December 2004 and 28 March 2005

At 7:58 A.M. local time on Sunday, 26 December 2004, a great earthquake occurred in the Indian Ocean approximately 250 km west of Sumatra, Indonesia. With a moment magnitude of 9.1–9.3, this was the second largest instrumentally recorded earthquake in history. The earthquake had an average source duration of about 500 seconds, the longest ever recorded, and a rupture length of about 1,300 km, the largest ever determined instrumentally. This earthquake generated one of the most devastating tsunamis in recorded history.