On The Probability of Response of a Linear Oscillator to a Random Pulse Train
Dynamic response of a linear oscillator to a Poisson-distributed train of general pulses is considered. The complete expansion for the one-dimensional probability density function of the response is presented in explicit form. The coefficients of skewness and of excess are evaluated for the steady-state response to a stationary train of square pulses and their behaviour is analyzed. The truncated serics is used to examine approximately the probability density function of the stationary response. The effect of the pulse duration and of the expected rate of pulses occurrence on the approximate probability density is discussed. Positive skewness and the departure of the response probability density from the Gaussian behaviour arc explained.
References
Y.K. LIN, Non-stationary excitation in linear systems treated as sequences of random pulses, J. Acoustical Soc. of Am, 38, 453-460, 1965.
J.B. ROBERTS, The response of linear vibratory systems to random impulses, J. Sound and Vibration, 2, 375-390, 1965.
S.K. SRINIVASAN, R. SUBRAMANIAN, S. KUMARASWAMY, Response of linear vibratory systems to non-stationary stochastic impulses, J. Sound and Vibration, 6, 169-179, 1967.
Y.K. LIN, Probabilistic theory of structural dynamics, McGraw-Hill Book Company, New York 1967.
E. PARZEN, Stochastic processes, Holden-Day, San Francisco 1962.
I.I. GIKHMAN, A V. SKOROKHOD, Stochastic Differential Equations, Ergebnisse der Mathematik, 72, Springer, New York 1972.
A. RENGER, Equation for probability density of vibratory systems subjected to continuous and discrete stochastic excitation, Zeitschrift fiir Angewandte Mathematik und Mechanik, 59, 1-13, 1979 (in German).
A. TYLIKOWSKI, A method of investigation of linear systems subjected to Poissonian impulse excitation, In: Proc. of the XI Symposium "Vibration in Physical Systems", Poznań - Błażejewko, 1984 (in Polish).
A. TYLIKOWSKI, Nonstationary response of linear discrete systems to Poissonian impulse sequences, Facta Universitatis Yugoslavia (to be published).
A. TYLIKOWSKI, W. MAROWSKI, Vibration of a non-linear single-degree-of-freedom system due to Poissonian impulse excitation, Int. J. of Non-linear Mechanics, 21, 229-238, 1986.
E. ORSINGHER, F. BATIAGLIA, Probability distributions and level crossings of shot noise models, Stochastics, 8, 45-61, 1982.
E.N. GILBERT, H.O. POLLAK, Amplitude distribution of shot noise, The Bell System Techn. J., 39, 33J.-350, 1960.
R. RACICOT, F. MOSES, Filtered Poisson process for random vibration problems. J. Engineering Mechanics Division, Proc. of ASC E, 98, 159-176, 1972.
R.A. JANSEEN, R.F. LAMBERT, Numerical calculation of some response statistics for a linear oscillator under impulsive-noise excitation, J. Acoustical Soc. of Am., 41, 827-835, 1967.
J.B. ROBERTS, On the response of a simple oscillator to random impulses, J. Sound and Vibration, 4, 51-61, 1966.
A. TYLIKOWSKI, Vibrations of the harmonic oscillator produced by a sequence of random collisions, Publ. of the Inst. of Fundamentals of Machines Construction, No. 13, 101-112. Warsaw Technical University, 1982 (in Polish).
Z. KOTULSKI, K. SOBCZYK, Linear systems and normality, J. Statistical Physics, 24, 359-373, 1981.
M. OCHI, Non-Gaussian random processes in ocean engineering, Probabilistic Engineering Mechanics, 1, 28-39, 1986.
J.B. ROBERTS, Distribution of the response of linear systems to Poisson distributed random pulses, J. Sound and Vibration, 28, 93-103, 1973.
J. KAWCZYŃSKI, Stochastic point processes; properties and applications, M. Sc. thesis, Warsaw Technical University, 1979 (in Polish).
M. GŁADYSZ, P. ŚNIADY, Random vibration of a discrete system due to a Poissonian stream of forces, Arch. Inż. Ląd., 30, 1984 (in Polish).
R. IWANKIEWICZ, P. ŚNIADY, Vibration of a beam under a random stream of moving forces, J. Structural Mechanics, 12, 13-26, 1984.
M.S. LONGUET-HIGGINS, The effect of non-linearities on statistical distributions in the theory of sea waves, J. Fluid Mechanics, 17, 459-480, 1963.
H. CRAMER, Random variables and probability distributions, Cambridge, 1970.
D.L. WALLACE, Asymptotic approximations to distributions, Annals of Mathematical Statistics, 29, 635-654, 1958.
R.N. BHATTACHARYA, J.K. GHOSH, On the validity of the formal Edgeworth expansion, Annals of Statistics, 6, 434-451, 1978.
A. PAPOULIS, High density shot noise and Gaussianity, J. Applied Probability, 8, 118-127, 1971.