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/V6YX7864. https://resolver.caltech.edu/CaltechTHESIS:11022015090737162
Abstract
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 ẋ + f(x) = m/Ʃ/j = 1 h_{j}(x)n_{j}(t) where f and the h_{j} are piecewise linear functions (not necessarily continuous), and the n_{j} are stationary Gaussian white noise. For such systems, it is shown how the Laplacetransformed 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, h_{1} = const. (forcing function excitation); (2) m = 1, h_{1} = f (parametric excitation); (3) m = 2, h_{1} = const., h_{2} = f, n_{1} and n_{2} 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 steadystate 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)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  2 May 1967  
Funders: 
 
Record Number:  CaltechTHESIS:11022015090737162  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:11022015090737162  
DOI:  10.7907/V6YX7864  
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  
Last Modified:  11 Mar 2024 22:29 
Thesis Files

PDF
 Final Version
See Usage Policy. 34MB 
Repository Staff Only: item control page