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**Random vibration and statistical linearization.**
*(English)*
Zbl 0715.73100

Chichester etc.: John Wiley & Sons Ltd. xiii, 407 p. £37.50 (1990).

This is a book for engineers about an approximation method for the design of structures under random actions such as wind, waves, earthquakes. This method, statistical linearization, consists in changing the mechanical equation of motion of the structure, which is usually nonlinear, into a linear one which is chosen in such a way that it minimizes the difference to the nonlinear term in the sense of quadratic mean. This method is not rigorous by the fact that this quadratic mean is taken for the probability which governs the linearized response, the only one which is computable, instead of the true response of the structure.

It would be a mistake to believe that this point disqualifies the method, on the contrary it is a very convenient and useful tool during the first step of the design process (predimensioning) provided that the user be well warned about its field of application. It is precisely what the authors are doing quite clearly in the first chapter discussing the place of the method among the available procedures for dealing with nonlinear responses of structures to stochastic inputs.

The second chapter is a presentation of the mechanical arguments yielding the equations of the motion of structures. An interesting and well organized survey of the different types of nonlinearities is proposed with a discussion of the most convenient mathematical representations including a detailed treatment of hysteresis. The two following chapters give an elementary account of probability theory and second order stationary processes.

The explanation of the statistical linearization method begins in chapter 5 by the case of systems of single degree of freedom. Numerous examples are given corresponding to the mechanical classification of nonlinearities precedingly stated. The multivariate case, the most usual for engineers, is then detailed with an emphasis on effective procedures for getting numerically the linearized equations. The case where the coefficients of the linearized equation depend on time allows to reach nonstationary systems. A whole chapter is devoted to the case of systems with hysteretic nonlinearity. The book ends with an analysis of the accuracy of the method obtained by comparison with exact analytical solutions of the nonlinear equations when it is possible or with results of Monte Carlo simulations. This is quite enlightening on the size of the errors which remain small, generally.

The point of relaxing the Gaussian hypotheses and extending the method by other closure techniques or by using explicitely solvable nonlinear equations, or by direct optimization on paths of the response, is the subject of chapter 9. It were be worth, when discussing the closure methods by expansion on Hermite polynomials or by truncating the sequence of cumulants to quote that the positivity of the density is not preserved in general by these procedures yielding sometimes nonpositive expectation of positive quantities.

This book, which is easy to read and well written, is not only a reference book for engineers but a quite motivating reading for mathematicians interested by improvement of the correctness and research of effective bounds in the asymptotic expansions related to this method or its extensions.

It would be a mistake to believe that this point disqualifies the method, on the contrary it is a very convenient and useful tool during the first step of the design process (predimensioning) provided that the user be well warned about its field of application. It is precisely what the authors are doing quite clearly in the first chapter discussing the place of the method among the available procedures for dealing with nonlinear responses of structures to stochastic inputs.

The second chapter is a presentation of the mechanical arguments yielding the equations of the motion of structures. An interesting and well organized survey of the different types of nonlinearities is proposed with a discussion of the most convenient mathematical representations including a detailed treatment of hysteresis. The two following chapters give an elementary account of probability theory and second order stationary processes.

The explanation of the statistical linearization method begins in chapter 5 by the case of systems of single degree of freedom. Numerous examples are given corresponding to the mechanical classification of nonlinearities precedingly stated. The multivariate case, the most usual for engineers, is then detailed with an emphasis on effective procedures for getting numerically the linearized equations. The case where the coefficients of the linearized equation depend on time allows to reach nonstationary systems. A whole chapter is devoted to the case of systems with hysteretic nonlinearity. The book ends with an analysis of the accuracy of the method obtained by comparison with exact analytical solutions of the nonlinear equations when it is possible or with results of Monte Carlo simulations. This is quite enlightening on the size of the errors which remain small, generally.

The point of relaxing the Gaussian hypotheses and extending the method by other closure techniques or by using explicitely solvable nonlinear equations, or by direct optimization on paths of the response, is the subject of chapter 9. It were be worth, when discussing the closure methods by expansion on Hermite polynomials or by truncating the sequence of cumulants to quote that the positivity of the density is not preserved in general by these procedures yielding sometimes nonpositive expectation of positive quantities.

This book, which is easy to read and well written, is not only a reference book for engineers but a quite motivating reading for mathematicians interested by improvement of the correctness and research of effective bounds in the asymptotic expansions related to this method or its extensions.

Reviewer: N.Bouleau

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P99 | Optimization problems in solid mechanics |

74H50 | Random vibrations in dynamical problems in solid mechanics |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |