Analysis of the trajectories of non-differentiable random processes

The calculations based on the analysis of random processes in mechanical systems are currently being increasingly used. This is especially true when the behavior of such systems is analyzed over time. The article deals with the analysis of the trajectories of differentiable and non-differentiable random processes. The objective of this analysis is to define the expected number of zeros, extrema, inflection points of the trajectory, and the probability distribution of the maxima and absolute maxima of random processes. The spectral densities of nondifferentiable random processes are represented as a product of two complex conjugate functions (quasi-spectra), with white noises being excluded. This makes it possible to solve all problems stated for simple processes. The spectra of complex processes running in systems with two or more degrees of freedom are represented in the form of delta-functions of their natural frequencies. Simple formulas are obtained for determining the parameters of the trajectories. A beam loaded by white noise forces is considered as an example.

References

[1] Vibratsii v tekhnike. Spravochnik. T. 1. Kolebaniia lineinykh sistem [Vibration technology. Handbook. Vol. 1. Oscillations of linear systems]. Ed. Bolotin V.V. Moscow, Mashinostroenie publ., 1999. 504 p.

[2] Kuper Dzh., Makgillem K. Veroiatnostnye metody analiza signalov i system [Probabilistic methods for analyzing signals and systems]. Moscow, Mir publ., 1989. 301 p.

[3] Bolotin V.V. Sluchainye kolebaniia uprugikh system [Random vibrations of elastic systems]. Moscow, Nauka publ., 1979. 335 p.

[4] Li W., Qi-Man Shao. Stochastic Processes: Theory and Methods. Handbook of Statistics, 2001, vol. 19, pp. 533–598.

[5] Gusev A.S. Soprotivlenie ustalosti i zhivuchest’ konstruktsii pri sluchainykh nagruzkakh [Fatigue resistance and survivability of structures under random loads]. Moscow, Mashinostroenie publ., 1989. 245 p.

[6] Ayache A., Linde W. Approximation of Gaussian random fields: General results and optimal wavelet representation of the Levy fractional motion. Journal of Theoretical Probability, 2008, vol. 21, issue 1, pp. 69–96.

[7] Gusev A.S. Veroiatnostnye metody v mekhanike mashin i konstruktsii [Probabilistic methods in the mechanics of machines and structures]. Moscow, Bauman Press, 2009. 223 p.

[8] Dandekar D.P., Hall C.A., Chhabildas L.C., Reinhart W.D. Shock response of a glass-fiberreinforced polymer composite. Composite Structures, 2003, vol. 61, issue 1–2, pp. 51–59.

[9] Gusev A.S., Karunin A.L., Kramskoi N.A., Starodubtseva S.A. Nadezhnost’ mekhanicheskikh sistem i konstruktsii pri sluchainykh vozdeistviiakh [Reliability of mechanical systems and structures under random actions]. Moscow, MGTU MAMI publ., 2001. 282 p.