Response Of A Nonlinear System To Random Vibration

This is a systematic presentation of several classes of analytical techniques in non-linear random vibration. The book also includes a concise treatment of Markovian and non-Markovian solutions of non-linear differential equations.

This self-contained volume explains the general method of statistical linearization and its use in solving random vibration problems. Numerous examples show advanced undergraduate and graduate students many practical applications. 1990 edition.

Several aspects of the problem of random vibration in nonlinear systems are discussed in terms of a particular set-up spring system. Exact solutions for the mean square response and the expected frequency of zero crossings are obtained by means of the Fokker-Planck equation. These are compared with approximate solutions obtained from equivalent linearization techniques. The distribution of response peaks is then studied and a probability density is derived for the peaks on a relative frequency basis. The probability density of the response envelope is also obtained and the relationship between the two distributions is discussed.

The topic of Random Vibrations is the behavior of structural and mechanical systems when they are subjected to unpredictable, or random, vibrations. These vibrations may arise from natural phenomena such as earthquakes or wind, or from human-controlled causes such as the stresses placed on aircraft at takeoff and landing. Study and mastery of this topic enables engineers to design and maintain structures capable of withstanding random vibrations, thereby protecting human life. Random Vibrations will lead readers in a user-friendly fashion to a thorough understanding of vibrations of linear and nonlinear systems that undergo stochastic-random-excitation. Provides over 150 worked out example problems and, along with over 225 exercises, illustrates concepts with true-to-life engineering design problems Offers intuitive explanations of concepts within a context of mathematical rigor and relatively advanced analysis techniques. Essential for self-study by practicing engineers, and for instruction in the classroom.

This second edition of the book, Nonlinear Random Vibration: Analytical Techniques and Applications, expands on the original edition with additional detailed steps in various places in the text. It is a first systematic presentation on the subject. Its features include: • a concise treatment of Markovian and non- Markovian solutions of nonlinear stochastic differential equations, • exact solutions of Fokker-Planck-Kolmogorov equations, • methods of statistical linearization, • statistical nonlinearization techniques, • methods of stochastic averaging, • truncated hierarchy techniques, and • an appendix on probability theory. A special feature is its incorporation of detailed steps in many examples of engineering applications. Targeted audience: Graduates, research scientists and engineers in mechanical, aerospace, civil and environmental (earthquake, wind and transportation), automobile, naval, architectural, and mining engineering.

This second edition of the book, Nonlinear Random Vibration: Analytical Techniques and Applications, expands on the original edition with additional detailed steps in various places in the text. It is a first systematic presentation on the subject. Its features include: a concise treatment of Markovian and non- Markovian solutions

When a linear vibratory system is excited by a stationary random process the most useful statistical description of the response is given by either the autocorrelation function or the spectral density. The exact calculation of these statistics for a nonlinear oscillator remains an unsolved problem although a general approach has been outlined. This paper describes three approximate procedures which apply to systems which are only slightly nonlinear. The procedures are perturbation, equivalent linearization, and a new heuristic method based on simply averaging the deteministic properties of the nonlinear system. This latter procedure is an extension of the heuristic method recently introduced for obtaining the expected frequency of the stationary response of a nonlinear oscillator. It is shown that for vibratory systems with small stiffness nonlinearities all three procedures give autocorrelations (and spectra) which are identical to first order in the nonlinearity parameter. (Author).

Introduction to Random Vibrations presents a brief review of probability theory, a concise treatment of random variables and random processes, and a comprehensive exposition of the theory of random vibrations.

Two procedures are discussed for studying the response of nonlinear systems to random excitation. The Fokker-Planck method is potentially applicable to a wide class of problems but the integrations required have only been effected in a few special cases. The equivalent linearization method is easy to carry out but is limited to small nonlinearities. It is demonstrated that to first order in the nonlinearity the equivalent linearization estimate of mean square stationary response is identical with that provided by the Fokker-Planck equation. A new equivalent linearization procedure for estimating the expected frequency of zero crossings in the random response is described and shown to give the same result as the Fokker-Planck equation. (Author).