By Cho W.S. To
It is a systematic presentation of a number of periods of analytical options in non-linear random vibration. The ebook additionally features a concise therapy of Markovian and non-Markovian strategies of non-linear differential equations.
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Additional resources for Nonlinear Random Vibration: Analytical Techniques and Applications (Advances in Engineering Series)
25), and the remaining symbols have their usual meaning. To proceed further one can express the quantities of interest of the above oscillator as x = q, and dx/dt = p such that the equation of motion can be re- Markovian and Non-Markovian Solutions 17 written as two first order stochastic differential equations (I-2) The solution process in Eq. (I-2) is NMNR due to the fact that > is not a white noise. By applying Eq. 28], one can show that (I-4) and the approximate equations, to first order in J, for the second moments are (I-5) Equations (I-3) and (I-5) can be solved in closed form or by so me numerical integration algorithm, such as the fourth order Runge-Kutta (RK4) scheme.
Equation (IV-9) was independently presented in Refs. 11], with different notations. Example V. (8) and f(8) are arbitrary functions, and 8 is the total energy (V-2) Note that Eq. (V-1) is similar to Eq. (I-1) above except for the RHS. Applying the same symbols as in the method presented above, the two Itô stochastic differential equations for Eq. (V-1) are (V-3) and (V-4) where B(t) or written simply as B is a unit Wiener process. The corresponding reduced FPK equation becomes (V-5) Exact Solutions of Fokker-Planck-Kolmogorov Equations 33 The first and second derivate moments are divided into two parts as those in the procedure described above except that A2 (2) = - g(x1 ) is chosen in accordance with Eq.
A possible example of application isin the analysis of a vibration isolator that uses an elastomer, such as neoprene, as the spring element. 23]. The joint stationary probability density function of equation (II-1) can be obtained by replacing g(x1 ) = S2 x + gx3 with g(x1 ) = [2k0 x0 /(Bm)] tan[Bx1 /(2x0 )] so that (II-2) where S2 = k0 /m and C is the normalization constant. Writing F0 2 = BS/($S2 ) and performing the integration in Eq. (II-2), (II-3) The above probability density function can be factored by marginal distributions as indicated in the last example due to the solutions given in Eqs.