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Distribution of Stochastic Impulses Acting on an Oscillator as a Function of Its Motion

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In previous papers formulas have been derived describing distribution of a random variable whose values are positions of an oscillator at the moment t, which, in the interval [0, t], underwent the influence of stochastic impulses with a given distribution. In this paper we present reasoning leading to an opposite inference thanks to which, knowing the course of the oscillator, we can find the approximation of distribution of stochastic impulses acting on it. It turns out that in the case of an oscillator with damping the stochastic process ξ_{t} of its deviations at the moment t is a stationary and ergodic process for large t. Thanks to this, time average of almost every trajectory of the process, which is the n-th power of ξ_{t} is very close to the mean value of ξ_{t}^{n} in space for sufficiently large t. Thus, having a course of a real oscillator and theoretical formulae for the characteristic function ξ_{t} we are able to calculate the approximate distribution of stochastic impulses forcing the oscillator.
  • Faculty of Mathematics and Computer Science, Jagiellonian University, Gołębia 24, 31-007 Cracow, Poland
  • Department of Mechanic and Vibroacoustics, University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland
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