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Non-linear time series models for non-linear random vibrations

Published online by Cambridge University Press:  14 July 2016

T. Ozaki
T. Ozaki*
Affiliation:
Institute of Statistical Mathematics, Tokyo
*
∗Postal address: Institute of Statistical Mathematics, 4–6–7 Minami-Azabu, Minato-Ku, Tokyo, Japan.
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Abstract

Non-linear time series models for non-linear vibrations are presented. Some typical behaviour of non-linear vibrations generated from Duffing's equation or van der Pol's equation are explained through the models.

Keywords

NON-LINEAR TIME SERIESNON-LINEAR VIBRATIONSDUFFING'S EQUATIONVAN DER POL EQUATIONAMPLITUDE-DEPENDENT FREQUENCYJUMP PHENOMENALIMIT CYCLE
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 84 - 93
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

This work was supported in part by the U.K. Science Research Council at the University of Manchester Institute of Science and Technology, and in part by a grant from the Ministry of Education, Culture and Science, Japan, at the Institute of Statistical Mathematics, Tokyo.

References

Akaike, H. (1974) Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Ann. Inst. Statist. Math. 26, 363–387.Google Scholar
Akaike, H. (1978) On newer statistical approaches to parameter estimation and structure determination. Proc. IFAC World Conference, Helsinki.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco.Google Scholar
Caravani, P., Watson, M. L. and Thomson, W. T. (1977) Recursive least-squares time domain identification of structural parameters. J. Appl. Mech. 44, 135–140.Google Scholar
Duffing, G. (1918) Erwungen Schwingungen bei Veränderlicher Eigenfrequenz. F. Vieweg u. Sohn, Braunschweig.Google Scholar
Gersch, W., Nielsen, N. N. and Akaike, H. (1973) Maximum likelihood estimation of structural parameters from random vibration data. J. Sound and Vibration 31, 295–308.Google Scholar
Haggan, V. and Ozaki, T. (1978) Amplitude dependent AR model fitting for nonlinear random vibrations. Technical Report No. 103, Department of Mathematics (Statistics), UMIST.Google Scholar
Lennox, W. C. and Kuak, Y. C. (1976) Narrow band excitation of a nonlinear oscillator. J. Appl. Mech. 43, 340–344.CrossRefGoogle Scholar
Ozaki, T. (1979) Statistical analysis of Duffing's process through non-linear time series models. Research Memo No. 151, The Institute of Statistical Mathematics, Tokyo.Google Scholar
Ozaki, T. and Oda, H. (1977) Non-linear time series model identification by Akaike's Information Criterion. Proc. IFAC Workshop on Information and Systems, Compiègne.Google Scholar
Pandit, S. M. and Wu, S. M. (1975) Unique estimates of the parameters of a continuous statinary stochastic process. Biometrika 62, 497–501.Google Scholar
Poincaré, H. (1881) Mémoire sur les courbes définies par une équation différentielle. J. Math. (3) 7, 375–422. (Oeuvres I, 3–84.) Google Scholar
Van Der Pol, B. (1927) Forced oscillations in a circuit with non-linear resistance. Phil. Mag. (7) 3, 65–80.Google Scholar
Rayleigh, Lord (1894) The Theory of Sound. Macmillan, London. Reprinted (1945) Dover, New York.Google Scholar
Stoker, J. J. (1950) Non-Linear Vibrations in Mechanical and Electrical Systems. Interscience, New York.Google Scholar