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Stability and related problems in randomly excited systems

Citation

Gray, Augustine H. (1964) Stability and related problems in randomly excited systems. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09192002-104048

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Introductions to parts are included in .pdf document.

Part 1. THE CONTINUOUS MARKOV PROCESS

Since the first treatments of Brownian Motion as an example of a continuous Markov Process, the applications of Markov Processes in physical situations have extended over a wide range which includes such extremes as barometric pressure distributions and structural responses to earthquakes.

In this part, the notion of a continuous Markov Process is presented and described in terms of a transition probability and a Fokker-Planck Equation. Two uniqueness theorems are presented here, as well as a heuristic discussion of the large time behavior of such a process.

Part 2. THE MARKOV PROCESS AS GENERATED BY DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS

In Part 1, the Markov Process was treated from the point of view of the Fokker-Planck Equation alone, and no discussion was presented treating the process itself. In this part, it will be demonstrated that a system of differential equations can define a Markov Process, and the Fokker-Planck Equation for such a process will be derived.

Particular emphasis is given to a discussion of the differing results of various authors in the case of "parametric white noise." These differing results have led to a controversy concerning the coefficients A[subscript k] of the Fokker-Planck Equation, Eq. 1.14, even in some of the simplest examples.

Among the examples given will be the general linear differential equation with "parametric white noise," and an "equivalent" linear differential equation with no parametric white noise will be derived.

Part 3. SOME SUFFICIENT STABILITY CONDITIONS FOR LINEAR SYSTEMS WITH RANDOM (NON-WHITE) COEFFICIENTS

As pointed out in Part 2, the stability and instability of linear systems whose coefficients are sums of constants and Gaussian White Noise can be determined by Laplace transforms applied to appropriate moment equations. This will lead explicitly to stability boundaries for the various moments, and is most often useful in determining "mean square stability." Unfortunately, such a procedure cannot be extended to cover non-white parameters. At present, there is no general method that may be used to show instability when the coefficients are random but not white, and only conservative sufficient conditions for various forms of stability can be obtained.

Some attempts at determining stability boundaries have been published by Chelpanov (11) and Samuels (12). In the former paper, correlation times of the random parameters were assumed to be much smaller than the natural times of the system, thus, as described in Part 2 of this thesis, making the random signals essentially white. The latter paper is unfortunately erroneous in parts, and its results are questionable.

In a later paper, F. Kozin (7) treated sufficient stability conditions by utilizing an ergodic property of the random terms and by using the Gronwall-Bellman Lemma (14). Less conservative conditions have been obtained by T. K. Caughey* [*Communicated verbally to the author] by using an appropriate Lyaponof function and the same ergodic property.

Caughey's Lyaponof function was quadratic in form, and suggested the possibility of using a general quadratic Lyaponov function. Herein is presented an approach for obtaining sufficient conditions for stability, utilizing a general quadratic Lyaponov function.

Part 4. FIRST PASSAGE TIMES IN A SECOND ORDER SYSTEM

Of major interest in vibrational systems with random excitation is the mean and mean square time for the system to get from one state to another. A more general problem is that of determining the probability distribution of the elapsed time in getting from one state to another.

In principle, this problem can be solved for a stationary Markov Process. Using the notation of Part 1 (Sections 1.2.0 and 1.2.1), if one defines T(z/x,t) as the probability density for the time of passage, t, to get from the point x to the point z in n dimensional phase space, and if P[subscript T](z/x,t) represents the transition probability for the variable z, then Eq. 1.4 of Part 1 states that [...]. By using Laplace Transform techniques, the convolution equation indicated by Eq. 4.1 can be readily solved, giving T(z/x,t) as the inverse Laplace Transform of a ratio of Laplace Transform of P[subscript T](z/x,t) and P[subscript T}(z/z,t). In general, the calculations necessary are far too difficult to perform, except in the simple case of a one-dimensional Markov Process, generated by the linear differential equation [...], where n(t) is white noise.

A similar problem, that of the evaluation of the frequency with which a variable y(t) crosses a given value z, has been worked out by Rice (16), for the case where y(t) is a stationary random variable. In particular, when y(t) is a Gaussian variable with zero mean, then the mean number of times per unit time that y is equal to z is given by [...], where [...].

Consider the resonant system defined by [...], where n(t) is white noise, with an autocorrelation function given by [...]. Such a system is used as an approximation of a structural response to an earthquake.* [*See Rosenblueth and Bustamante (17).] A desired result is that of the "probability of failure" which is the probability that in a given interval of length T, [...] will have exceeded a fixed value of X at least once. Another desired result if the "mean time to failure," or the mean time to get from one value of [...] to the fixed level X. Neither of these desired results fall into the type of problems just discusses.

Though this problem does not appear complex on the surface, there is no known technique for solving it. Messrs. Rosenblueth and Bustamante (17) have utilized approximations and boundary value techniques to obtain approximate solutions for the probability of failure. Their approach will be justified here by the obtaining of the same results in another manner, and further the mean and mean square times to failure will be calculated. It will be assumed that the system is highly resonant, so that b < < w, and that the fixed level X will be such that [...]. This latter condition will be satisfied if X is much larger than the steady state standard deviation of y, for the [...].

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Engineering and Applied Science
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Caughey, Thomas Kirk
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1964
Record Number:CaltechETD:etd-09192002-104048
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-09192002-104048
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3633
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:01 Oct 2002
Last Modified:26 Dec 2012 03:01

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