Some Closure Problems in Stochastic Dynamics of Solids (Process, Random Vibration, Cumulant-Neglect, Crack Propagation, Earthquake)
Wu, Wen-Fang
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https://hdl.handle.net/2142/70633
Description
Title
Some Closure Problems in Stochastic Dynamics of Solids (Process, Random Vibration, Cumulant-Neglect, Crack Propagation, Earthquake)
Author(s)
Wu, Wen-Fang
Issue Date
1985
Department of Study
Aeronautical and Astronautical Engineering
Discipline
Aeronautical and Astronautical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Applied Mechanics
Abstract
In the studying of a nonlinear system under random excitation, it is usually difficult to determine the probability density of the system response. On the other hand, the equations for the statistical moments can be formulated rather easily, except that these equations form an infinite hierarchy rendering an exact solution impossible. A closure procedure is then needed to truncate the infinite hierarchy and to solve for some of the moments approximately. Several closure schemes are discussed in the context, particularly the cumulant-neglect closure. Emphasis is put on the applications of these closure schemes to several stochastic dynamics problems rather than theoretical development of these schemes.
Three nonlinear oscillators under random external or/and parametric excitations are explored first. These specially chosen oscillatory systems have very different characteristics and their stationary exact solutions are known. Hence the accuracy of the approximate solutions obtained by using different closure schemes can be determined by comparing with respective exact solutions in order to gain confidence for further applications.
In most dynamic analyses of structural response to earthquake excitation only the horizontal ground motion is considered and the soil mass supporting the structure is treated as being rigid. In the present thesis the effects of gravity, vertical direction earthquake and soil compliance are all taken into consideration. Thus the influences of these different additional factors can be evaluated.
The Gaussian closure scheme is applied to a fatigue crack propagation analysis to compute the probability distribution of the crack size at any given time instant and the probability distribution of the random time at which a given crack size is reached. The mathematical model for fatigue crack propagation used in the analysis is a modified Paris-Erdogen model recently proposed by Lin and Yang. The results are compared with available experimental data and results obtained by using other methods.
Cumulant-neglect closure, which are extensively used in this thesis, is compared with Gaussian closure and the equivalent linearization method. The comparison is worthwhile since the equivalent linearization method is perhaps the most widely used tool in stochastic analyses of nonlinear systems. Examples are given in the text.
The final chapter summarizes the main results and concludes with a discussion of some unresolved questions.
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