Fractional Stochastic Dynamics in Structural Stability Analysis
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Date
2013-08-30T20:05:43Z
Authors
Deng, Jian
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
The objective of this thesis is to develop a novel methodology of fractional
stochastic dynamics to study stochastic stability of viscoelastic
systems under stochastic loadings.
Numerous structures in civil engineering are driven by dynamic forces, such as
seismic and wind loads, which can be described satisfactorily only by using
probabilistic models, such as white noise processes, real noise processes, or
bounded noise processes. Viscoelastic materials exhibit time-dependent stress
relaxation and creep; it has been shown that fractional calculus provide a
unique and powerful mathematical tool to model such a hereditary property.
Investigation of stochastic stability of viscoelastic systems with fractional
calculus frequently leads to a parametrized family of fractional stochastic
differential equations of motion. Parametric excitation may cause parametric
resonance or instability, which is more dangerous than ordinary resonance as it
is characterized by exponential growth of the response amplitudes even in the
presence of damping.
The Lyapunov exponents and moment Lyapunov exponents provide not only the
information about stability or instability of stochastic systems, but also how
rapidly the response grows or diminishes with time. Lyapunov exponents
characterizes sample stability or instability. However, this sample stability
cannot assure the moment stability. Hence, to obtain a complete picture of the
dynamic stability, it is important to study both the top Lyapunov exponent and
the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the
accurate values of theses two exponents. One has to resort to numerical and
approximate approaches.
The main contributions of this thesis are: (1) A new numerical simulation
method is proposed to determine moment Lyapunov exponents of fractional
stochastic systems, in which three steps are involved: discretization of
fractional derivatives, numerical solution of the fractional equation, and an
algorithm for calculating Lyapunov exponents from small data sets. (2)
Higher-order stochastic averaging method is developed and applied to
investigate stochastic stability of fractional viscoelastic
single-degree-of-freedom structures under white noise, real noise, or bounded
noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and
gyroscopic viscoelastic systems under random excitation, the Stratonovich
equations of motion are set up, and then decoupled into four-dimensional Ito
stochastic differential equations, by making use of the method of stochastic
averaging for the non-viscoelastic terms and the method of Larionov for
viscoelastic terms. An elegant scheme for formulating the eigenvalue problems
is presented by using Khasminskii and Wedig’s mathematical transformations from
the decoupled Ito equations. Moment Lyapunov exponents are approximately
determined by solving the eigenvalue problems through Fourier series expansion.
Stability boundaries, critical excitations, and stability index are obtained.
The effects of various parameters on the stochastic stability of the system are
discussed. Parametric resonances are studied in detail. Approximate analytical
results are confirmed by numerical simulations.
Description
Keywords
stochastic dynamics, dynamic stability of structures, fractional dynamics, random vibration, Lyapunov exponents, moment Lyapunov exponents, viscoelastic structures, stochastic differential equations, fractional differential equations, stochastic averaging, coupled system, gyroscopic system