By Josef L. Zeman

ISBN-10: 3211811311

ISBN-13: 9783211811313

ISBN-10: 3709127408

ISBN-13: 9783709127407

**Read Online or Download Approximate Analysis of Stochastic Processes in Mechanics: Course Held at the Department of General Mechanics October 1971 PDF**

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**Extra resources for Approximate Analysis of Stochastic Processes in Mechanics: Course Held at the Department of General Mechanics October 1971**

**Sample text**

As a first approximation one may use the corresponding quantities of the response of the corresponding linear system. The calculations are usually quite simple, particularly in cases where Ut(t) and U2 (t) are Gaussian, but often lead to very rough approximations only. , are the centered inputs, may lead to differ- ent results and the second even to contradictions ( ~~) • Better approximations are obtained by using the stochastic properties of the response of the equivalent linear systems, given by Eqs.

In what follows, only systems are considered where the functions als a~ and b~k are or have been a:Qproximated by polynomi- in the components of the state vector. Under this proposition the expectations on the right-hand side of Eq. 27) may pressed in terms of moments ClC~h···,~n {Zt(t), ... 28) results. For nonlinear problems the expression on the right-hand side in general will contain also moments whose order is greater than the order of the moment on the left-hand side. Consider systems of such differential equations obtained for all different combinations r~ such that r!.

Z and, with the results of Sect. JE{Z 6} - E{Z 3 }E{Z 3 } /flz. Since the input process U(t) is Gaussian with zero mean the response of the corresponding linear system is also Gaussian with vanishing mean, and using this v 0 = v~ = v = v* = 0 follows for this version. That this solution is the proE er one can only be decided by other means. But if one starts from these results, the responses of the equivalent linear systems have then also vanishing means and are, of course, Gaussian, the parameters mediately as v1 and \1~ follow for all different versions im- so Chap.

### Approximate Analysis of Stochastic Processes in Mechanics: Course Held at the Department of General Mechanics October 1971 by Josef L. Zeman

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