14,587 research outputs found
A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise
We develop a moment equation closure minimization method for the inexpensive
approximation of the steady state statistical structure of nonlinear systems
whose potential functions have bimodal shapes and which are subjected to
correlated excitations. Our approach relies on the derivation of moment
equations that describe the dynamics governing the two-time statistics. These
are combined with a non-Gaussian pdf representation for the joint
response-excitation statistics that has i) single time statistical structure
consistent with the analytical solutions of the Fokker-Planck equation, and ii)
two-time statistical structure with Gaussian characteristics. Through the
adopted pdf representation, we derive a closure scheme which we formulate in
terms of a consistency condition involving the second order statistics of the
response, the closure constraint. A similar condition, the dynamics constraint,
is also derived directly through the moment equations. These two constraints
are formulated as a low-dimensional minimization problem with respect to
unknown parameters of the representation, the minimization of which imposes an
interplay between the dynamics and the adopted closure. The new method allows
for the semi-analytical representation of the two-time, non-Gaussian structure
of the solution as well as the joint statistical structure of the
response-excitation over different time instants. We demonstrate its
effectiveness through the application on bistable nonlinear
single-degree-of-freedom energy harvesters with mechanical and electromagnetic
damping, and we show that the results compare favorably with direct Monte-Carlo
Simulations
A Method for the Combination of Stochastic Time Varying Load Effects
The problem of evaluating the probability that a structure becomes unsafe under a
combination of loads, over a given time period, is addressed. The loads and load effects
are modeled as either pulse (static problem) processes with random occurrence time, intensity and a specified shape or intermittent continuous (dynamic problem) processes which
are zero mean Gaussian processes superimposed 'on a pulse process. The load coincidence
method is extended to problems with both nonlinear limit states and dynamic responses,
including the case of correlated dynamic responses. The technique of linearization of a
nonlinear limit state commonly used in a time-invariant problem is investigated for timevarying
combination problems, with emphasis on selecting the linearization point. Results
are compared with other methods, namely the method based on upcrossing rate, simpler
combination rules such as Square Root of Sum of Squares and Turkstra's rule. Correlated
effects among dynamic loads are examined to see how results differ from correlated static
loads and to demonstrate which types of load dependencies are most important, i.e., affect'
the exceedance probabilities the most.
Application of the load coincidence method to code development is briefly discussed.National Science Foundation Grants CME 79-18053 and CEE 82-0759
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
A mathematical framework for critical transitions: normal forms, variance and applications
Critical transitions occur in a wide variety of applications including
mathematical biology, climate change, human physiology and economics. Therefore
it is highly desirable to find early-warning signs. We show that it is possible
to classify critical transitions by using bifurcation theory and normal forms
in the singular limit. Based on this elementary classification, we analyze
stochastic fluctuations and calculate scaling laws of the variance of
stochastic sample paths near critical transitions for fast subsystem
bifurcations up to codimension two. The theory is applied to several models:
the Stommel-Cessi box model for the thermohaline circulation from geoscience,
an epidemic-spreading model on an adaptive network, an activator-inhibitor
switch from systems biology, a predator-prey system from ecology and to the
Euler buckling problem from classical mechanics. For the Stommel-Cessi model we
compare different detrending techniques to calculate early-warning signs. In
the epidemics model we show that link densities could be better variables for
prediction than population densities. The activator-inhibitor switch
demonstrates effects in three time-scale systems and points out that excitable
cells and molecular units have information for subthreshold prediction. In the
predator-prey model explosive population growth near a codimension two
bifurcation is investigated and we show that early-warnings from normal forms
can be misleading in this context. In the biomechanical model we demonstrate
that early-warning signs for buckling depend crucially on the control strategy
near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio
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