Analysis of nonlinear sliding structures by modified stochastic linearization methods

1994 ◽  
Vol 5 (3) ◽  
pp. 299-312 ◽  
Author(s):  
Ruichong Zhang ◽  
Isaac Elishakoff ◽  
Masanobu Shinozuka
Keyword(s):  
Linearization Methods ◽  
Stochastic Linearization ◽  
Sliding Structures

Stochastic linearization: the theory

1998 ◽  
Vol 35 (3) ◽  
pp. 718-730 ◽  
Author(s):  
Pierre Bernard ◽  
Liming Wu
Keyword(s):  
Large Deviation ◽  
Random Excitation ◽  
Kolmogorov Equation ◽  
Parabolic Partial Differential Equations ◽  
Statistical Linearization ◽  
Linearization Methods ◽  
Large Deviation Principles ◽  
Kullback Information ◽  
Stochastic Linearization ◽  
Global Linearization

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

Stochastic linearization: the theory

1998 ◽  
Vol 35 (03) ◽  
pp. 718-730 ◽  
Author(s):  
Pierre Bernard ◽  
Liming Wu
Keyword(s):  
Large Deviation ◽  
Random Excitation ◽  
Kolmogorov Equation ◽  
Parabolic Partial Differential Equations ◽  
Statistical Linearization ◽  
Linearization Methods ◽  
Large Deviation Principles ◽  
Kullback Information ◽  
Stochastic Linearization ◽  
Global Linearization

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

A new modified stochastic linearization technique to analyze structures with nonlinear fluid viscous dampers

2021 ◽  
pp. 107754632110195
Author(s):  
Ghasem Asadpour ◽  
Payam Asadi ◽  
Iman Hajirasouliha
Keyword(s):  
Probability Density ◽  
Ground Motions ◽  
Soft Soil ◽  
Probability Density Functions ◽  
Degree Of Freedom ◽  
Density Functions ◽  
Viscous Dampers ◽  
Linearization Technique ◽  
Linearization Methods ◽  
Stochastic Linearization

Nonlinear viscous dampers can efficiently improve the seismic performance of structures by dissipating large amounts of earthquake-induced energy. In common practice, the spectral analysis of structures with nonlinear viscous dampers is generally conducted based on an estimated equivalent damping ratio. To this end, the stochastic linearization technique can be used as an effective probabilistic approach to take into account the evolutionary characteristics of the input earthquake excitation. This study aims to present optimal non-Gaussian probability density functions to improve the accuracy of the stochastic linearization technique for nonlinear viscous dampers in both firm and soft soil-based structures. It is shown that by using the optimum probability density functions, the computational error of the stochastic linearization technique for a single-degree-of-freedom structure under simulated ground motions, with a range of peak ground accelerations between 0.1 and 0.6 g, is reduced by up to 70%. The efficiency of the proposed probability density functions is then demonstrated for multi-degree-of-freedom structures, by estimating the roof displacements of a six-story steel frame with nonlinear viscous dampers under a set of natural ground motions using different linearization methods. The comparison of the stochastic linearization technique estimated responses with the exact values confirms that using the proposed probability density functions leads to considerably lower errors in both firm and soft soil-based structures compared with the other linearization techniques.

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

2019 ◽  
Author(s):  
Nguyen Cao Thang ◽  
Luu Xuan Hung
Keyword(s):  
Performance Analysis ◽  
Mean Square Error ◽  
Nonlinear Oscillators ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Mean Square ◽  
Error Criterion ◽  
Stochastic Linearization ◽  
Local Mean ◽  
Mean Square Error Criterion

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).

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

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