Gaussian Equivalent Linearization Research Articles

Approximation of the Statistical Characteristics of Piecewise Linear Systems with Asymmetric Damping and Stiffness under Stationary Random Excitation

In this paper, the dynamic response of piecewise linear systems with asymmetric damping and stiffness for random excitation is studied. In order to approximate the statistical characteristics for each significant output of piecewise linear system, a method based on transmissibility factors is applied. A stochastic linear system with the same transmissibility factor is attached, and the statistical parameters of the studied output corresponding to random excitation having rational spectral densities are determined by solving the associated Lyapunov equation. Using the attached linear systems for root mean square and for standard deviation of displacement, the shift of the sprung mass average position in a dynamic regime, due to damping or stiffness asymmetry, can be predicted with a good accuracy for stationary random input. The obtained results are compared with those determined by the Gaussian equivalent linearization method and by the numerical integration of asymmetric piecewise linear system equations. It is shown that the piecewise linear systems with asymmetrical damping and stiffness characteristics can provide a better vibration isolation (lower force transmissibility) than the linear system.

Performance analysis of global local mean square error criterion of stochastic linearization for nonlinear oscillator

The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of stochastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi-degree of freedom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two-degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL).

Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization

The analysis of large-scale structures subject to transient random loads, coherent in space and time, is a classic problem encountered in earthquake and wind engineering. The simulation-based framework is usually seen as the most convenient approach for both linear and nonlinear dynamics. However, the generation of statistically consistent samples of an excitation field remains a heavy computational task. In light of this, perturbation techniques are applied to develop and improve evolutionary spectral analysis.Advantageously performed in a standard modal basis, this evolutionary spectral analysis for linear structures requires the computation of the modal impulse response matrix. However, this matrix has no general closed-form expression in the presence of modal coupling. We propose therefore to model it by an asymptotic approximation, obtained by the inverse Fourier transform of an asymptotic expansion of the modal transfer matrix of the structure. This latter expansion considers the modal coupling as a perturbation of a main decoupled system. This strategy leads to an expansion known in a closed-form. Finally, the semi-group property allows the use of an efficient recurrence relation to approximate the modal evolutionary transfer matrix, i.e. the evolutionary extension of the transfer matrix.The asymptotic expansion-based method and the recurrence relation are then applied to nonlinear transient dynamics by using Gaussian equivalent linearization. This extension is formalized by a multiple timescales approach, allowing to consider a linearized structure, namely a time variant system, as piecewise linear time invariant depending on a statistical timescale. The proposed developments are finally illustrated on realistic civil engineering applications.

An asymptotic expansion-based method for a spectral approach in equivalent statistical linearization

Equivalent linearization consists in replacing a nonlinear system with an equivalent linear one whose parameters are tuned with regard to the minimization of a suitable function. In particular, the Gaussian equivalent linearization expresses the properties of an equivalent linear system in terms of the mean vector and the covariance matrix of the responses, which are the unknowns of the optimization problem in a spectral approach. Even though the system has been linearized, the resulting set of equations is nonlinear. The computational effort in this method pertains to the solution of a possibly large set of nonlinear algebraic equations involving integrals and inversions of full matrices. This work proposes to develop and apply an asymptotic expansion-based method to facilitate and to improve the statistical linearization for large nonlinear structures. The proposed developments demonstrate that for slightly to moderately coupled nonlinear systems, the equivalent linearization can be applied with an appropriate modal approach and eventually seen as a convergent series initiated with the stochastic response of a main decoupled linear system. With this method, the computational effort is attractively reduced, the conditioning of the set of nonlinear algebraic equations is improved and inversion of full transfer matrices and repeated integrations are avoided. The paper gives a formal description of the method and illustrates its implementation and performances with the computation of stationary responses of nonlinear structures subject to coherent random excitation fields.

Non-Gaussian Stochastic Equivalent Linearization Method for Inelastic Nonlinear Systems with Softening Behaviour, under Seismic Ground Motions

A non-Gaussian stochastic equivalent linearization (NSEL) method for estimating the non-Gaussian response of inelastic non-linear structural systems subjected to seismic ground motions represented as nonstationary random processes is presented. Based on a model that represents the time evolution of the joint probability density function (PDF) of the structural response, mathematical expressions of equivalent linearization coefficients are derived. The displacement and velocity are assumed jointly Gaussian and the marginal PDF of the hysteretic component of the displacement is modeled by a mixed PDF which is Gaussian when the structural behavior is linear and turns into a bimodal PDF when the structural behavior is hysteretic. The proposed NSEL method is applied to calculate the response of hysteretic single-degree-of-freedom systems with different vibration periods and different design displacement ductility values. The results corresponding to the proposed method are compared with those calculated by means of Monte Carlo simulation, as well as by a Gaussian equivalent linearization method. It is verified that the NSEL approach proposed herein leads to maximum structural response standard deviations similar to those obtained with Monte Carlo technique. In addition, a brief discussion about the extension of the method to muti-degree-of-freedom systems is presented.

Time Dependent Gaussian Equivalent Linearization of Duffing Oscillator Using Continuous Wavelet Transform

Time Dependent Gaussian Equivalent Linearization of Duffing Oscillator Using Continuous Wavelet Transform

Evaluation of Non-Gaussian Response Characteristics of an Asymmetric Nonlinear System Subjected to Nonwhite Excitation Using Higher Order Statistical Values

Non-Gaussian response characteristics of an asymmetric nonlinear system subjected to nonwhite random excitation are investigated. Applying a previously developed analytical method, which contains moment equations method and non- Gaussian equivalent linearization technique, stationary responses are numerically computed. Non-Gaussian indicators i.e. skewness, kurtosis and equivalent linear coefficients which are defined by the 3rd and 4th order moments are considered to evaluate the non-Gaussianity of the responses. Skewness and kurtosis of the response in a family of the asymmetric parameter of the system and the dominant frequency of the excitation are plotted onto a diagram as the map of non- Gaussianity. The non-Gaussian map indicates the variation of the shape of probability density function. The equivalent linear coefficients are illustrated in the same way. These numerical results demonstrate the effects of the asymmetric properties of the system and the dominant frequencies of the excitation upon the non-Gaussian response characteristics.

An improved criterion of Gaussian equivalent linearization for analysis of non-linear stochastic systems

An improved criterion of Gaussian equivalent linearization for analysis of non-linear stochastic systems

Approximate analysis of some two-degree-of-freedom nonlinear random systems by an extension of Gaussian equivalent linearization

The paper presents the analysis of some two-degree-of-freedom nonlinear systems under random excitation using Local Mean Square Error Criterion which is an extension of Gaussian Equivalent Linearization. The results obtained shows that the new technique can be very efficiently used not only for simple-degree-of-freedom systems as presented in the previous papers, but also for multi-degree-of-freedom ones. The solution accuracy obtained by the proposed technique is much more improved than that using the traditional linearization. The conclusions in the paper point up the significance of this technique.

Linearization of randomly excited nonlinear systems by using local mean square error criterion

The paper presents the analysis of three non-linear systems under random excitation by using Local Mean Square Error Criterion which is an extension of Gaussian Equivalent Linearization proposed by Caughey. The obtained results shows that the new technique allows to get much more accurate solutions than .that using Caughey Criterion. The paper leads out some new conclusions which have not been found yet by the previous researches. The new conclusions more clarify the significance of this technique.

A technique for obtaining approximate solutions in Gaussian equivalent linearization

Within the scope of Gaussian equivalent linearization, a new mean square criterion of error sample function for determining the coefficients of the linearized equivalent equation is proposed to treat stationary response of nonlinear systems under zero mean Gaussian random excitation. Application to the Duffing and Van der Pol oscillators subjected to white noise is presented that shows significant improvement over corresponding accuracy of the classical Gaussian equivalent linearization for both weak and strong nonlinearity.

An equivalent linearization for the drag force in the Morison's equation using Hermite polynomial error sample function

Within the scope Gaussian equivalent linearization (GEL), a new mean square criterion based on Hermite polynomial error sample functions for determining the coefficients of the equivalent linearization is proposed to linearize non-linear functions of the zero mean Gaussian random process. Application to the Morison's equation for wave forces is presented that shows significant improvements over the corresponding accuracy of the classical GEL.

Probabilistic criterion for Gausian equivalent linearization

Within the scope of Gaussian equivalent linearization, a new probabilistic criterion for determining the coefficients of the linearized equivalent equation is proposed to treat stationary response of non-linear systems under zero mean Gaussian ro.nd.om excitation. Application to the Duffing oscillate subjected to white noise is presented that shows significant improvement over corresponding accuracy of the classical Gaussian equivalent linearization for both weak and strong non-linear.