Equivalent Linearization Method Research Articles

Nonlinear Stresses and Deflections of Beams Subjected to Random Time Dependent Uniform Pressure

The nonlinear mean-square multimode response of beams subjected to uniform pressure uncorrelated in time is investigated. The method of equivalent linearization is used to obtain mean-square stresses and displacements in beams with arbitrary end conditions. Calculations are carried out for beams with both ends either simply-supported or clamped, for the case of white noise excitation. Although the maximum displacement can be obtained with the use of only a single-degree-of-freedom model, it is necessary to consider as many as 100 modal functions for accurate determination of the stresses. The maximum mean-square deflection of the clamped beam is found to be somewhat less than the simply-supported beam, whereas the maximum mean-square stresses are as much as twice as large.

TEST FAIRNESS: A COMMENT ON FAIRNESS IN STATISTICAL ANALYSIS

ABSTRACTAn argument is presented to suggest that the analysis of covariance may in some circumstances be an unfair method to use in the study of the question of test fairness. As an alternative, the use of equipercentile methods or equivalent linear methods may be preferred in these circumstances.

Stationary mode of a nonlinear elastically hereditary oscillator

The steady-state forced oscillations of a single-mass system subject to an external pure harmonic force are considered. The role of restoring force is played by a nonlinear function, which takes account of hereditary effects as a sum of multiple integrals in accordance with the Volterra theory. The problem is solved by the method of equivalent linearization, the discussion being confined to a triple integral of the hereditary type. The influence of the hereditary nonlinearity on the system dynamic characteristics, namely, its amplitude, phase, dynamic rigidity, hysteresis loop area, and Q, is investigated. In particular, the reciprocal of the Q, which can be taken as a measure of the internal friction, is shown to be independent of the amplitude of oscillation and to be the same as that obtained by linear theory. The other dynamic characteristics prove sensitive to the nonlinear properties. The exponential rational fractions proposed by Yu. N. Rabotnov are used as concrete hereditary functions.

Nondeterministic Analysis of Offshore Structures

The development of a rational procedure for predicting the dynamic response of offshore structures to random sea waves has been presented in earlier papers. However, the fluid-structure interaction effects in a nondeterministic method of analysis has not been fully considered. The inclusion of these effects in the general method of analysis is presented herein. The general theory presented, combines known methods of analysis, such as wave force definitions, structural response techniques, equivalent linearization methods, and extreme value statistics.

Response Of Multiple Horizontal Soil LayersUsing Higher Order Frequency ResponseFunctions

For several years the Equivalent Linear Method was the most used method to solve dynamic problems of stratified soil deposits. This is valid for soils showing small or medium nonlinearities. It is an iterative procedure without a mathematical approach associated to it. The proposed mathematical method is based on the use of the First and Higher Order Frequency Response functions as required by the Volterra series. This method is based in the determination of the functions that relates the input motion to the response. In this study, up to the third order frequency response functions (kernel) are developed. The use of these expressions in some numerical examples and a comparison with the Equivalent Linear Method will be presented. Also, a sensitivity analysis is done to show how the series is affected with the quantity of terms in it. It will be demonstrated that at low or medium shear strain, three terms in the series are acceptable to obtain the result produced by the Equivalent Linear Method.

On the zero shift problem of a vertical pendulum

The effect of the vibration transmitted through pendulum support on the accuracy of a pendulum used as a vertical reference is examined. Based upon the method of equivalent linearization of Kryloff and Bogoliuboff together with a perturbation scheme, the average angular displacement of a vertical pendulum is obtained in terms of the functional averages involving support excitations. It is shown that the method leads to general and simple formulas for estimating the zero shift and, in the case of random support excitations, second-moment properties. The analysis is valid for both heavily damped and weakly damped systems. The zero shifts due to several types of deterministic and random excitations are computed. Methods of minimizing the zero shift are considered for these cases.

The Buckling of a Column With Random Initial Deflections

We consider the buckling of an ensemble of infinitely long columns, with initial deflections, resting on nonlinear elastic foundations. The initial deflections are assumed to be Gaussian stationary random functions of known autocorrelation, and the problem is solved by the method of equivalent linearization. We find that each column in the ensemble has the same buckling load that depends only on the autocorrelation of the initial deflection functions. Results are presented for columns whose initial deflection functions have an exponential-cosine autocorrelation.

An analysis of ground motions during the 1957 San Francisco earthquake

ABSTRACT The seismic responses of soil deposits are calculated at four sites where recordings were obtained during the 1957 San Francisco earthquake. The analyses are made utilizing an equivalent linear lumped-mass method of analysis incorporating the soil characteristics at the four sites and estimates of the rock motions at each site based on the record obtained on rock during this earthquake. Comparisons of the maximum amplitudes of ground motions, the time history of ground accelerations, velocity and acceleration spectra and spectral intensities for both computed and recorded ground accelerations indicate a reasonably high degree of agreement. The influence of the amplitude of base rock accelerations on the response of a soil deposit is also discussed and illustrated.

Response of the Bilinear Hysteretic System to Stationary Random Excitation

Time-average statistics of the response of the bilinear hysteretic system to an excitation with approximately white-power spectral density and approximately Gaussian probability distribution are determined, using electronic-analog techniques. Results are presented for the mean-squared amplitude, the power spectral density, and the probability distribution of the response. The applicability of the Krylov-Bogoliubov method of equivalent linearization to this problem is investigated by comparing predicted and experimentally measured values of the mean-squared level of response.

Contraction Maps and Equivalent Linearization*

This study is primarily concerned with the question: If the method of equivalent linearization indicates the existence of a periodic solution, is there actually a periodic solution near the approximation of equivalent linearization? To answer this question, we use a modification of the contraction mapping fixed point theorem. We discuss applications to differential equations and difference-differential equations (with forcing functions). Also, we show that our use of contraction maps is not applicable (without modification) to autonomous systems because the mapping evaluated in the neighborhood of a periodic solution to an autonomous system is not a contraction in a space of periodic functions.

Analysis of the Shimmy Phenomenon

The shimmy phenomenon has been treated as a non-linear problem which is solved using the method of equivalent linearization and also with the aid of an analogue computer. Three non-linear elements are considered: the tyre characteristics, the Coulomb friction in the king-pin bearings, and clearance in the wheel bearings. The theory is substantiated by experiments with a test vehicle and a mechanical model. Two different kinds of shimmy appear to exist, one originating from the tyre elasticity and the other from the gyroscopic coupling. Results showing the effect of changing a number of important parameters on the shimmy phenomenon are presented.

Equivalent Linearization Techniques

The method of equivalent linearization of Kryloff and Bogoliubov is generalized to the case of nonlinear dynamic systems with random excitation. The method is applied to a variety of problems, and the results are compared with exact solutions of the Fokker-Planck equation for those cases where the Fokker-Planck technique may be applied. Alternate approaches to the problem are discussed, including the characteristic function method of Rice.

Comment on "Response of Nonlinear Flat Panel to Periodic and Randomly Varying Loadings"

L1 presents an interesting analysis of the dynamic response of a flat plate when the nonlinear effects of the membrane forces are significant. Unfortunately, the author has neglected to reference several previous publications presenting results pertinent to this problem, some of which are listed here. Also, the author's treatment of the random-excitation case deserves some comment. The equivalent linearization technique used seems to be of questionable value in this case, since an exact solution for the mean-square response (as well as other statistical measures) of the randomly excited cubic system is available.' It is that the exact solution is limited to the case of white-noise excitation; however, the limitations of the equivalent linearization method imply much the same restriction. That is, the true linear frequency and the equivalent linear frequency cannot be far separated, and the spectrum of the excitation normally may be considered constant over this restricted frequency range. In any case, it is not clear from the article why the author feels that the results of this approximate technique are valid for the case of nonwhite excitation. The statements concerning the effective transfer function also may be questioned. Not all people who are working on the problem of nonlinear panel response would agree that physical occurrence of the upper branch of the nonlinear response curve is as unusual as the author implies. Certainly, if the response of a panel does lie on the upper branch, the effect of damping may not be considered unimportant.

Approximate Power Spectral Density of the Response of a Nonlinear System to Stationary Random Excitation

An approximation is obtained for the power spectral density of the response of a hard-spring resonator to a class of random inputs by 2 methods of equivalent linearization. Comparison of the theoretical spectral densities is made with those obtained experimentally from an analog computer with white-noise excitation.

Equivalent Linearization Techniques

The method of equivalent linearization of Kryloff and Bogliubov is generalized to the case of nonlinear dynamic systems with random excitation. The method is applied to a variety of problems, and the results compared with exact solutions of the Fokker-Planck equation for those cases where the Fokker-Planck technique may be applied. Alternate approaches to the problem are discussed, including the characteristic function method of Rice.

Analysis of a Nonlinear First-Order System with a White Noise Input

A general technique for computing the power spectrum of nonlinear systems is presented, and the method is applied to the Brownian motion of a particle with idealized Coulomb damping. In this method the transition probability is obtained by solving the appropriate Fokker-Planck equation. Then the autocorrelation function is calculated by an appropriate integration, and finally the power spectrum is found by means of the Wiener-Khintchine relation. Comparison is made with the method of equivalent linearization, which is applicable to a wide class of nonlinear problems.

Simultaneous Oscillations in Oscillators

An oscillator with two degrees of freedom can, under certain conditions, oscillate simultaneously at two different frequencies. The ratio of the two frequencies may be rational (synchronous oscillations) or irrational (asynchronous oscillations). The conditions necessary for 'simultaneous oscillations are discussed for these two cases. The problem is essentially non-linear and cannot be solved by the methods of linear analysis. The method of equivalent linearization, developed by Kryloff and Bogoliuboff, is shown to give a good correlation with experimental results.

The Method of Equivalent Linearization

The Method of Equivalent Linearization

A Method of Equivalent Linearization for Non-Linear Oscillatory Systems with Large Non-Linearity

This paper introduces a method, based partially on existing non-linear theory and partially on experimental results, for transforming a non-linear oscillatory system into an equivalent linear one, regardless of the amount of non-linearity in the original system.

Resistance of iron to repeated blows

Cross-ties are employed as passive devices for the mitigation of stay-cable vibrations, exhibited on cable-stayed bridges under wind and wind-rain excitation. Large-amplitude oscillation can result in damage to the cables or perceived discomfort to bridge users. The “cable-cross-ties system” derived by connecting two or more stays by transverse cross-ties is often referred to as an “in-plane cable network”. Linear modeling of network dynamics has been available for some time. This framework, however, cannot be used to detect incipient failure in the restrainers due to slackening or snapping. A new model is proposed in this paper to analyze the effects of a complete loss in the pre-tensioning force imparted to the cross-ties, which leads to the “unilateral” free-vibration response of the network (i.e., a cross-tie with linear-elastic internal force in tension and partially inactive in compression).Deterministic free vibrations of a three-cable network are investigated by using the “equivalent linearization method”. A performance coefficient is introduced to monitor the relative reduction in the average (apparent) stiffness of the connector during free vibration response (“mode by mode”), exhibiting unilateral behavior. A reduction of fifty percent in the apparent stiffness was observed in the cross-tie when the pre-tensioning force is small in comparison with the tension force in the stay. This coefficient may be used as a damage indicator for the selection of the initial pre-tensioning force in the cross-ties needed to avoid slackening.