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Dynamic systems with random initial state

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Summary

A perturbation scheme is described to treat time variable cryptodeterministic systems. According to Moyal's degree of randomness criteria the method provides a complete stochastic characterization of the system response. Certain digital computational features, when the perturbation scheme is not applicable, are also outlined. For an assumed random initial state, the results are then applied to describe the transient flapping oscillations of a helicopter blade which in forward flight has periodically varying aerodynamic damping and spring parameters.

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References

  1. R. Syski,Stochastic Differential Equations, Modern Nonlinear Equations, McGraw-Hill, New York, pp. 346–456, 1967.

    Google Scholar 

  2. W. E. Boyce,Random Eigenvalue Problems, Probabilistic Methods in Applied Mathematics, Academic Press, New York, pp. 1–72, 1968.

    Google Scholar 

  3. K. von Hasselmann, über zufallserregte Schwingungssysteme,ZAMM, 42 (1962) 465–476.

    Google Scholar 

  4. H. L. Van Trees,Detection, Estimation and Modulation Theory, John Wiley, New York, pp. 527–534, 1968.

    Google Scholar 

  5. G. H. Gaonkar and K. H. Hohenemser, Flapping Response of Lifting Rotor Blades to Atmospheric Turbulence,Journal of Aircraft, 6 (1969) 496–503.

    Google Scholar 

  6. G. H. Gaonkar and K. H. Hohenemser, Stochastic Properties of Turbulence Excited Rotor Blade Vibrations,AIAA Journal, 9 (1971) 419–424.

    Google Scholar 

  7. B. D. O. Anderson, J. B. Moore and G. L. Sonny, Spectral Factorization of Time Varying Covariance Functions,IEEE Transaction on Information Theory, IT-15, 5, (1969) 550–557.

    Google Scholar 

  8. J. D. Collins and W. T. Thomson, The Eigenvalue Problem for Structural Systems with Statistical Properties,AIAA Journal, 7 (1969) 642–648.

    Google Scholar 

  9. M. Hoshiya, Dynamic Eigenvalue Analysis of Stochastic Structural Systems,Ph.D. dissertation, Stanford University, 1969.

  10. R. L. Barnoski and J. R. Maurer, Mean-Square Response of Simple Mechanical Systems to Nonstationary Random Excitation,Journal of Applied Mechanics, Paper No. 69-APM-25. (Also seeJournal of Applied Mechanics, p. 250, March 1970.)

  11. A. A. Sveshnikov,Applied Methods of the Theory of Random Functions, Pergamon Press, New York, pp. 127–139, 1966.

    Google Scholar 

  12. W. E. Boyce, A “dishonest” approach to certain Stochastic Eigenvalue Problems,SIAM Journal of Applied Mathematics, 15 (1967) 143–152.

    Google Scholar 

  13. J. M. Richerdson, Application of Truncated Hierarchy Techniques,Proceedings of Symposia in Applied Mathematics, AMS, Providence, Rhode Island, pp. 290–302, 1964.

    Google Scholar 

  14. C. W. Haines, Hierarchy Methods for Random Vibrations of Elastic Strings and Beams,Journal of Engineering Mathematics (1967) 293–305.

  15. G. Adomian, Stochastic Green's Functions,Proceedings of Symposia in Applied Mathematics, AMS, Providence, Rhode Island, pp. 1–39, 1964.

    Google Scholar 

  16. J. E. Moyal, Stochastic Processes and Statistical Physics,Journal of Royal Statistical Society, Ser. B, 11 (1949) 150–210.

    Google Scholar 

  17. D. A. Edwards and J. E. Moyal, Stochastic Differential Equations,Proceedings of the Cambridge Philosophical Society, 51 (1955) 663–677.

    Google Scholar 

  18. R. Bellman,Perturbation Methods in Mathematics, Physics, and Engineering. Holt, Rinehart and Winston, New York, pp. 1–16, 1966.

    Google Scholar 

  19. V. S. Pugachev,Theory of Random Functions and its Application to Control Problems, Pergamon Press, New York, pp. 120–130, 1965.

    Google Scholar 

  20. A. Blanc-Lapierre and R. Fortet,Theory of Random Functions, Gordon and Breach, New York, pp. 66, 1965.

    Google Scholar 

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Gaonkar, G.H. Dynamic systems with random initial state. J Eng Math 5, 171–178 (1971). https://doi.org/10.1007/BF01535409

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  • DOI: https://doi.org/10.1007/BF01535409

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