By Chen T.Q.
This paintings offers the development of an asymptotic method for fixing the Liouville equation, that's an analogue of the Enskog-Chapman process for the Boltzmann equation. as the assumption of molecular chaos has now not been brought, the macroscopic variables outlined via the mathematics technique of the corresponding microscopic variables are random as a rule. consequently, it's handy for describing the turbulence phenomena. The asymptotic method for the Liouville equation unearths a time period exhibiting the interplay among the temperature and the speed of the fluid flows, for you to be misplaced less than the idea of molecular chaos.
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For na T small, i. e. P. Thrall [THR 64] showed that na T constitutes an upper limit of the probability of crossing a positive (or negative) threshold in time T without using the assumption of independence of the up-crossings. An arbitrary correlation between successive extrema tends to decrease the probability that such peaks exist during a given time period [GRA 66] [THR 64]. 6. 6 shows the variations of with n*0 T, for various values of P. N. Yang and M. Shinozuka [YAN 71] express the same results in the form where P(N) is the probability that the first excursion above threshold a takes place in the N first half-cycles, corresponding to a duration T: 4.
11. Coefficients a in assumptions 2° and 3° in relation to a. 6. 1. H. Crandall If the threshold level a of response is sufficiently large, one can consider that, in a given time, there are as many crossings of this threshold by the envelope R(t) with positive slope than the maxima of the envelope. That means that the envelope does not have any peak below a [CRA 70]. For an arbitrary threshold b > a, the peak distribution of the envelope, as in assumption n°2, is therefore dictated by the form P = prob(peak of envelope where m ^ is the mean number by unit time of up-crossings of the threshold b by the envelope R(t), m0 is the expected frequency of R(t).
Threshold level a is sufficiently high and the threshold excursions are so rare that they can be regarded as statistically independent. 2. The maxima of the response can be supposed to be independent. 3. The threshold up crossings of the envelope of maxima are independent. 4. The maxima of the envelope of the peaks are independent. 5. The amplitudes of the peaks follow a Markov process. 6. The response peaks are divided into groups for each of which the envelope of the peaks varies slowly. 2. Definitions Consider a response random signal u ( t ) , whose derivative is ii(t), and let it be / x (co being the natural pulsation of the single degreeplaced in a diagram M , u(t) 0 co0 of-freedom system subjected to vibration).