In the case of a polynomial expansion the coefficients G imn and D imnp are not

In the case of a polynomial expansion the

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In the case of a polynomial expansion the coefficients G imn and D imnp are not equal to zero. Hence the moment equations form an infinite hierarchy. This hierarchy will be closed with the help of the cumulant neglect closure technique. In the present case the joint cumulants above the fourth order will be neglected, which results in the following relationships for the joint central moments of the 5th and 6th order, [9] 10 ijklm ij klm s   (15) 15 10 30 ijklmn ij klmn ijk lmn ij kl mn s s s      (16) The symbol   s indicates a symmetry operation producing the arithmetic mean of all terms similar to those indicated, obtained by permuting all free indices. In Eq. (1) only the third and fourth equation are non-linear. For the constitutive equation the following expression is obtained   2 1 1 2 1 2 3 2 2 2 3 1 2 3 4 , , , c e c c c c c c x z x c c c c c c c c c g x z k x z x E k x z x a b x b z c x c x z c z d x d x z d x z d z (17) were c x x x and c z z z   . The 10 coefficients in (17) are determined from the following system of linear equations similar to (14) 20 11 02 30 21 12 03 1 20 11 30 21 12 40 31 22 13 1 02 21 12 03 31 22 13 04 40 31 22 50 41 32 23 22 13 41 32 23 14 04 32 23 14 05 60 51 42 33 42 33 24 24 15 06 1 0 0 . a b symm 2 2 1 2 2 3 3 1 2 2 3 2 4 3 0 c c c c c c c c c c c c x g z g b x g c x z g c z g c x g d d x z g d x z g d z g (18) where i j c c ij x z and , c c g g x z . For i + j 4 these are obtained from the differential equations (13). The expectations on the right hand side, , E x z x and ij , i + j = 5, 6 are performed with the help of an approximate joint p.d.f. , c c c c x z f x z . This is assumed in the form of a truncated bivariate Gram-Charlier expansion 463
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Chapter 37 2 2 2 2 3 3 2 2 2 3 , 2 3 3 6 3 3 1 2 , exp 2 1 2 1 1 1 , , , , ... ! ! 2! ! ! ! ! c c c c x z x z N N ij ij kl ij i k j l i j i j k l i j k l f x z H H i j i j k l                  19) where 2 3 , c c x z x z (20) 23 20 02 , , x z x z   (21) ij ij i x zx   (22) ij is the joint j + i th order cumulant of x and z , and 2 3 , , ij H is the bivariate standardized Hermite polynomial of order ( i , j ). The Gram-Charlier expansion (19) is truncated at order N = 4. Then ij , i + j 4 can be calibrated from the joint central moments of x and z available. An expansion similar to (17) applies to the fourth equation of (1). From (3) it follows that the only non-zero coefficients become 1 1 2 2 , , , 1 x z z x a E xz b b c   (23) In case of equivalent linearization b 1 and b 2 are the only non-zero coefficients in both expansions. 4. Numerical Example Let the random variable R be Rayleigh distributed, i.e.
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  • Summer '20
  • Probability theory, hysteretic systems

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