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# Spectral Density of First Order Piecewise Linear Systems Excited by White Noise

## Citation

Atkinson, John David (1967) Spectral Density of First Order Piecewise Linear Systems Excited by White Noise. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V6YX-7864. https://resolver.caltech.edu/CaltechTHESIS:11022015-090737162

## Abstract

The Fokker-Planck (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 ẋ + f(x) = m/Ʃ/j = 1 hj(x)nj(t) where f and the hj are piecewise linear functions (not necessarily continuous), and the nj 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) m = 1, h1 = const. (forcing function excitation); (2) m = 1, h1 = f (parametric excitation); (3) m = 2, h1 = const., h2 = f, n1 and n2 correlated; (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 steady-state density (the limit of the transition density as t → ∞).

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Applied Mechanics and Mathematics)
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Minor Option:Mathematics
Thesis Availability:Public (worldwide access)
• Caughey, Thomas Kirk
Thesis Committee:
• Unknown, Unknown
Defense Date:2 May 1967
Funders:
Funding AgencyGrant Number
University of SydneyUNSPECIFIED
Ford FoundationUNSPECIFIED
Inland Steel-Ryerson FoundationUNSPECIFIED
CaltechUNSPECIFIED
Record Number:CaltechTHESIS:11022015-090737162
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:11022015-090737162
DOI:10.7907/V6YX-7864
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9263
Collection:CaltechTHESIS
Deposited By:INVALID USER
Deposited On:02 Nov 2015 18:21