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The knowledge of the relations between the distribution of a random vibratory process and that of its envelope is required in many engineering applications. Under the assumption that the vibratory process is of narrow band type and the phase is uniformly distributed over the interval $(0, 2\pi)$, the integral transform giving the relation between the two distributions in question may be derived considering that the distribution of the envelope is known and the distribution function of the vibratory process is to be estimated. The aim of the paper is to summarize some most useful types of distribution functions which are important in the technical practice. Analytical expressions for the distributions of the corresponding vibratory processes are given for ten one-parametric distributions and for distributions with threeshold values, all related to the envelope processes. Approximate analytical description using the Gram-Charlier series may be used for cases where the analytical solution is inaccessible. This procedure is shown for three two-parametric distributions and for the generalized three- and four-parametric gamma-distributions.
[1] Abramowitz A., Stegun I.: Handbook of mathematical functions with formulas, graphs and mathematical tables. Nat. Bureau Stand., Washington, 1964. Zbl 0171.38503
[2] Барк Л. С., Большее Л. H., Кузнецов П. И., Черенков А. П.: Таблицы распределения Релея-Райса. Изд. Выч. центра АН СССР, Москва, 1964. Zbl 1117.65300
[3] Бункин Ф. В.: О свойствах огибаюшей стационарного случайного процесса. Радиотех. и электроника, Том 5 (1960), вып. 9, стр. 1555-1556. MR 0134806 | Zbl 1004.90500
[4] Crandall S. H., Mark W. D.: Random vibrations in mechanical systems. Academic Press, New York, 1963.
[5] Drexler J., Kropáč O.: Contribution to random vibration of one class of non-linear parametrically excited two-mass systems. Proceed. Fifth Conf. Dynamics of Machines, Liblice, 1968, pp. 101-109.
[6] Drexler J., Kropáč O.: One class of non-linear stochastic differential equations characterized by random excitation. Proceed. 12th Internat. Congress Applied Mechanics, Stanford Univ. Aug. 1968, Springer-Verlag, Berlin, 1969, pp. 179-191. MR 0359461
[7] Dwight H. B.: Tables of integrals and other mathematical data. Macmillan, New York, 1961 (4th edit.). MR 0129577
[8] Градштейн И. С., Рыжык И. М.: Таблицы интегралов, сумм, рядов и произведений. Физматгиз, Москва, 1963 (4-е изд.). Zbl 1145.93303
[9] Jahnke-Emde-Lösch: Tafeln höherer Funktionen. Teubner, Stuttgart, 1960 (6th edit.)
[10] Kendall M. G., Stuart A.: The advanced theory of statistics, Vol. I, Distribution theory. Griffin, London, 1963 (2nd edit.).
[11] Kropáč О.: On treatment of samples of random processes given by their envelopes. Acta technica ČSAV, Vol. 15 (1970), No. 5, pp. 594-610. MR 0277079
[12] Kropáč O.: On some qualitative characteristics of one class of non-linear parametrically excited stochastic differential equations. Journ. Sound Vibrat., Vol. 14 (3971), No. 2, pp. 241-249. DOI 10.1016/0022-460X(71)90387-7
[13] Rice S. O.: Mathematical analysis of random noise. Bell System Techn. Journ., Vol. 23 (1944), pp. 282-332, Vol. 24 (1945), pp. 46-156. DOI 10.1002/j.1538-7305.1944.tb00874.x | MR 0011918 | Zbl 0063.06485
[14] Рытое C. M.: Введение в статистическую радиофизику. Изд. Наука-ФМЛ, Москва, 1966. Zbl 1155.78304
[15] Tables of error functions and of its twenty derivatives. Harvard Univ. Press, Cambridge (Mass.), 1952.
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