By Josef L. Zeman
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Extra resources for Approximate Analysis of Stochastic Processes in Mechanics: Course Held at the Department of General Mechanics October 1971
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convention is used with (4. 2. 23) L ranging Chap. 4 - Approximation Methods 46 from 1 through k , and Z(t) has been replaced by Y(t) to indicate that the solution of this equation is an approximation of the solution of Eq.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