By Josef L. Zeman

ISBN-10: 3211811311

ISBN-13: 9783211811313

ISBN-10: 3709127408

ISBN-13: 9783709127407

**Read 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**

In this case the covariance function of the output process of this equivalent linear system is an upper bound of that of the original system. t/· e:xp [:-xlt1 -t:d]. Mean function and covariance function of the output process Z(t) of the simple nonlinearity 26 Chap. 2 - Memoryless Systems can be determined easily usingthe results of Sect. 1. r/ = at 2 + + a 22 + a 32 • The covariance functions of the output processes of the nonlinearity and the two equivalent linear systems for the "two cases mu = 0 and mu ...

For instance note that for a Gaussian input U2( t) the process (1) sian but Z (t) (0) Z ( t) is also Gaus- is not anymore. 15b) + tRz(o)z(i)(t 1,tz) + lRzlt> 1tOl(t 1 ,t 2 ) + O(t2). In order to determine the correlation function of Z(t) one thus m needs not only the autocorrelation functions of the Z (t) but also their crosscorrelation functions. +O. Special caution is appropriate in cases where the impulse response function corresponding to N(D,~ is not absolutely integrable over [ 0, oo], see, for instance, Example 2, Cases c and d.

22) where V1 , ... ,Vk stand, symbolically, for Z,U 1 , and the various derivatives of the two that may occur in the expression forni_), According to what has been sa1d in Sect. 22a) Insertion of this approximation into Eq. 21) renders the differential equation of the equivalent linear system: N(D,t)Y(t) + v(t) + vv(t)V~(t) = M(D,t)U 2(t) . In this equation the summation convention is used with (4. 2. 23) L ranging Chap. 4 - Approximation Methods 46 from 1 through k , and Z(t) has been replaced by Y(t) to indicate that the solution of this equation is an approximation of the solution of Eq.

### 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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