Some Applications of the Theory of Continuous Markoff Processes to Random Oscillation Problems

Citation

Dienes, John Kalman (1961) Some Applications of the Theory of Continuous Markoff Processes to Random Oscillation Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R51Y-W418. https://resolver.caltech.edu/CaltechETD:etd-04102006-132153

Abstract

Many random problems of engineering interest can be looked upon as examples of continuous Markoff processes. Such processes are completely determined if a certain function, the transition probability, is prescribed. It is shown that all of the functions of interest in random problems can be derived from the transition probability.

Some of the concepts of probability theory and of spectral analysis are reviewed and using these results, the Gaussian white noise function is defined. A new derivation of the Fokker-Planck equation is given which emphasizes the role of the Gaussian white input in the analysis of Markoff processes. The transition probability is the fundamental solution of this equation.

It is then shown that the autocorrelation is closely related to the mean motion of a system and can be calculated from the transition probability. This relation can be used, in principle at least, to determine the autocorrelation of nonlinear systems. The Method of Equivalent Linearization for random problems and the First Passage Problem are discussed briefly.

These methods are used to solve a number of problems. A discussion of linear systems is presented, and by a similar treatment the solution to a problem in random parametric excitation is given. Next, the first probability density of a class of nonlinear problems is discussed. Finally, the power spectra for two nonlinear systems are calculated.

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