Atkinson, John David (1967) Spectral density of first order piecewise linear system excited by white noise. California Institute of Technology . (Unpublished) http://resolver.caltech.edu/CaltechEERL:1967.DYNL.1967.002

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Abstract
[Contains mathematical notation that does not convert: see report for the correct formula.] The FokkerPlanck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form m . . . where f and the . . . are piecewise linear functions (not necessarily continuous), and the . . . are stationary Gaussian white noise. For such systems, it is shown how the Laplace transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32. 11. This method is applied to 4 subclasses: (1) . . . (forcing function excitation); (2) . . . (parametric excitation); (3) . . .; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed. Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steadystate density (the limit of the transition density as . . .).
Item Type:  Report or Paper (Technical Report) 

Additional Information:  PhD, 1967 
Group:  Dynamics Laboratory 
Record Number:  CaltechEERL:1967.DYNL.1967.002 
Persistent URL:  http://resolver.caltech.edu/CaltechEERL:1967.DYNL.1967.002 
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ID Code:  26517 
Collection:  CaltechEERL 
Deposited By:  Imported from CaltechEERL 
Deposited On:  11 Jul 2002 
Last Modified:  26 Dec 2012 13:59 
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