Random dynamical systems: addressing uncertainty, nonlinearity and predictability

We present new solutions for the dynamics of a pendulum energy converter which is vertically excited at its suspension point. Thereby, we deal with a random excitation by a non‐white Gaussian stochastic process. We formulate the pendulum energy converter as a weakly perturbed Hamiltonian system. The random process across the energy levels of the Hamiltonian system is then approximated by a Markov process, which is obtained by stochastic averaging. This procedure leads to analytical results for the energy of the pendulum motion, which are used for analyzing the required probability of reaching higher energy states of the pendulum energy converter in order to maximize the harvested energy.

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“…These measures are given by

0true Σ_{l} (x_{l}) = \int_{x_{l}}^{x} ({}, \int_{η}^{x} μ (ξ) normald ξ) s (η) normald η, N_{l} (x_{l}) = \int_{x_{l}}^{x} ({}, \int_{η}^{x} s (ξ) normald ξ) μ (η) normald η .

Since the system has an exit boundary from libration to rotation at

H = 2 α

, a stationary density for would involve additional conditions, cf. . If we assume a reflecting boundary at

H_{r}

(i.e.…”

Section: Probability Density and First Passage Probabilitymentioning

confidence: 99%

Pendulum energy converter excited by random loads

et al. 2017

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“…Using approximation techniques such as stochastic averaging, as described for example in [2,4,8,9], many random nonlinear problems of higher dimension can be approximated by a lower dimensional system of Itô stochastic differential equations. The averaged system will then have values on a lower dimensional subspace Γ, which can be further divided into N spaces Γ i , i = 1, .., N , as is described in more detail in [6]. These subspaces are glued together at π j (γ j ), which are mappings of homoclinic or heteroclinic manifolds γ j .…”

Section: Introductionmentioning

confidence: 99%

“…The gluing conditions at vertex O roughly mean that the stochastic process will continue in the subspace Γ i with likelihood σ 2 i /(σ 2 1 + ... + σ 2 N ), cf. [6].…”

Section: Introductionmentioning

confidence: 99%

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Reduction of nonlinear dynamical systems excited by random loads

Dostal

2019

Proc Appl Math and Mech

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The presented research deals with methods for the study of complex interactions between noise and nonlinearities in dynamical systems. Practical applications of this research are beginning to appear across the spectrum of mechanics; for example vibration absorbers, ship dynamics, energy harvesting, and variable speed machining processes. In the presented approach perturbed Hamiltonian systems are considered, which are damped and excited by an absolutely regular non-white Gaussian process. Analytical and semi-analytical solutions to the corresponding nonlinear stochastic differential equations (SDE) are determined. The approach is based on a limit theorem by Khashminskii, from which a class of methods has been derived known as stochastic averaging. In the presented approach Homoclinic orbits, which divide the phase space of the corresponding Hamiltonian flow into different regions, are mapped to a graph in the one degree of freedom case. For two degree of freedom systems this procedure results in a mapping to a book, and so on for higher degrees of freedom. From the drift and diffusion of the resulting averaged process, probability density functions and first passage times are obtained.

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