An Analytical Scheme for Stochastic Dynamic Systems Containing Fractional Derivatives

1999 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
White Noise ◽  
Expansion Method ◽  
Stochastic Dynamic ◽  
Stochastic Response ◽  
Analytical Technique ◽  
Analytical Scheme ◽  
Damped System ◽  
Duhamel Integral ◽  
General Response ◽  
Stochastic Dynamic Systems

Abstract This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method proposed by Suarez and Shokooh is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions. The general response contains two parts, namely zero state and zero input. For a stochastic analysis the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise.

Stochastic Analysis of a 1-D System With Fractional Damping of Order 1/2

2002 ◽  
Vol 124 (3) ◽  
pp. 454-460 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
White Noise ◽  
Stochastic Analysis ◽  
Damping Ratio ◽  
Expansion Method ◽  
Stochastic Dynamic ◽  
Stochastic Response ◽  
Analytical Technique ◽  
Damped System ◽  
Duhamel Integral ◽  
General Response

This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions for the response of the system. The general response contains two parts, namely zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics, namely the variance and covariance responses of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise. Results show that stochastic response of the fractionally damped system oscillates even when the damping ratio is greater than its critical value.

Analytical Solution for Stochastic Response of a Fractionally Damped Beam

2004 ◽  
Vol 126 (4) ◽  
pp. 561-566 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Stochastic Analysis ◽  
Normal Modes ◽  
Continuous Beam ◽  
Stochastic Response ◽  
Analytical Technique ◽  
System A ◽  
Laplace Transform Technique ◽  
Duhamel Integral ◽  
Fractional Green’S Function ◽  
Response Expression

This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations, each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green’s function and a Duhamel integral-type expression for the system’s response. The response expression contains two parts, namely, zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed-form stochastic response expressions are obtained for white noise for two cases, and numerical results are presented for one of the cases. The approach can be extended to all those systems for which the existence of normal modes is guaranteed.

A Partition Expansion Method for Nonlinear Response Analysis of Stochastic Dynamic Systems With Local Nonlinearity

2013 ◽  
Vol 8 (3) ◽  
Author(s):  
Changqing Bai ◽  
Hongyan Zhang
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Dynamic Response ◽  
Dynamic Systems ◽  
Expansion Method ◽  
Direct Monte Carlo Simulation ◽  
Stochastic Dynamic ◽  
Integration Algorithm ◽  
Local Nonlinearity ◽  
Stochastic Dynamic Systems

This paper focuses on the problem of nonlinear dynamic response variability resulting from stochastic system properties and random loads. An efficient and accurate method, which can be employed to analyze the dynamic responses of random finite element systems with local nonlinearity, is presented in this paper. This method, dubbed as the partition expansion method, is based on the partitioned time integration algorithm in conjunction with the Neumann expansion technique within the framework of the Monte Carlo simulation. Two numerical examples involving structural and mechanical stochastic vibration problems are employed to illustrate the advantage of the proposed method with respect to accuracy and efficiency. By comparing the results obtained by the direct Monte Carlo simulation, the dynamic response statistics can be accurately determined using the proposed method with four order expansion while the computational efforts are significantly reduced. The comparison of computing time indicates that the proposed method is efficient and practical for analyzing the statistical quantities of stochastic dynamic systems with local nonlinearity.

A New Reliability Method Combining Kriging and Probability Density Evolution Method

2017 ◽  
Vol 17 (10) ◽  
pp. 1750113 ◽  
Author(s):  
Zhongming Jiang ◽  
Jie Li
Keyword(s):  
Probability Density ◽  
New Method ◽  
Stochastic Dynamic ◽  
Kriging Method ◽  
Density Evolution ◽  
Stochastic Response ◽  
Probability Density Evolution Method ◽  
Numerical Precision ◽  
Stochastic Dynamic Systems ◽  
Probability Density Evolution

Stochastic dynamic analysis of structures with random parameters continues to be an open question in the field of civil engineering. As a newly developed method, the probability density evolution method (PDEM) can provide the probability density function (PDF) of the dynamic responses of highly nonlinear structures. In this paper, a new method based on PDEM and the kriging surrogate model, named the K-PDEM, is proposed to study the stochastic response of a structure. Being an exact interpolation method, the Gaussian process regression or the so-called kriging method is capable of producing highly accurate results. Unlike the traditional PDEM numerical method whose numerical precision is strongly influenced by the number of representative points, the K-PDEM employs the kriging method at each instant to generate additional time histories. Then, the PDEM, which is capable of capturing the instantaneous PDF of a dynamic response and its evolution, is employed in nonlinear stochastic dynamic systems. Because of the decoupling properties of the K-PDEM, the numerical precision of the result is improved by the enrichment of the generalized density evolution equations without increasing the computation time. The result shows that the new method is capable of calculating the stochastic response of structures with efficiency and accuracy.

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