Abstract
In this paper, we present a method for computing upper and lower bounds of the natural frequencies of a structure with parameters which are unknown, except for the fact that they belong to given intervals. These parameters are uncertain, yet they are not treated as being random, since no information is available on their probabilistic characteristics. The set of possible states of the system is described by interval matrices. By solving the generalized interval eigenvalue problem, the bounds on the natural frequencies of the structure with interval parameters are evaluated. Numerical results show that the proposed method is extremely effective.
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References
Ibrahim, R.,Structural Dynamics with Parameter Uncertainties, Applied Mechanics Reviews, Vol. 40, pp. 309–328, 1987.
Kozin, F.,On the Probability Densities of the Output of Some Random Systems, Journal of Applied Mechanics, Vol. 28, pp. 161–164, 1961.
Elishakoff, I., andSpencer, B. F., Jr.,Reliability of an Uncertain Structure, Journal of Sound and Vibration, Vol. 114, pp. 399–404, 1987.
Thomson, W. T.,Parameter Uncertainty in Dynamic Systems, Shock and Vibration Digest, Vol. 7, pp. 3–9, 1975.
Chen, S. H.,Vibration Theory of Structures with Random Parameters, Jilin Science and Technology Press, Changchun, China, 1992 (in Chinese).
Ananta Ramu, S., andGanesan, R.,Parametric Instability of Stochastic Columns, International Journal of Solids and Structures, Vol. 30, pp. 1339–1354, 1993.
Ananta Ramu, S., andGanesan, R.,A Galerkin Finite-Element Technique for Stochastic Field Problems, Computer Methods in Applied Mechanics and Engineering, Vol. 105, pp. 315–331, 1993.
Shinozuka, M.,Structural Response Variability Journal of Engineering Mechanics, Vol. 113, pp. 825–842, 1987.
Moore, R.,Methods and Applications of Interval Analysis, SIAM Publications, Philadelphia, Pennsylvania, 1979.
Alefeld, G., andHerzberger, J.,Introduction to Interval Computations, Academic Press, New York, New York, 1983.
Deif, A.,Advanced Matrix Theory for Scientists and Engineers, Abacus Press, Tunbridge Wells, England, 1991.
Elishakoff, I.,Some Questions in Eigenvalue Problems in Engineering, Numerical Treatment of Eigenvalue Problems, Edited by J. Albrecht, L. Collatz, P. Hagedorn, and W. Welte, Birkhauser Publishers, Basel, Switzerland, pp. 71–107, 1991.
Shi, Z. C., andGao, G. B.,A Necessary and Sufficient Condition for the Positive Definiteness of Interval Symmetric Matrices, International Journal of Control, Vol. 41, pp. 325–328, 1986.
Bialas, S.,A Necessary and Sufficient Condition for the Stability of Interval Matrices, International Journal of Control, Vol. 37, pp. 717–722, 1983.
Schweppe, F. C.,Uncertain Dynamical Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.
Leitmann, G.,On One Approach to the Control of Uncertain Systems, Journal of Dynamic Systems, Measurement, and Control, Vol. 115, pp. 373–380, 1993.
Ben-Haim, Y., andElishakoff, I.,Convex Models of Uncertainty in Applied Mechanics, Elsevier Science Publishers, Amsterdam, The Netherlands, 1990.
Elishakoff, I., andBen-Haim, Y.,Dynamics of a Thin Cylindrical Shell under Impact with Limited Deterministic Information on Initial Imperfection, Structural Safety, Vol. 8, pp. 103–112, 1990.
Elishakoff, I.,Convex Versus Probabilistic Modeling of Uncertainty in Structural Dynamics, Structural Dynamics: Recent Advances, Edited by M. Petyt, M. F. Wolfe, and C. Mei, Elsevier Applied Science Publishers, London, England, pp. 3–21, 1991.
Elishakoff, I.,Essay on Uncertainties in Elastic and Viscoelastic Structures: From A. M. Freudenthal's Criticisms to Modern Convex Modeling, Technical Report, College of Engineering, Florida Atlantic University, 1993.
Elishakoff, I., Li, Y. W., andStarnes, J. H., Jr.,A Deterministic Method to Predict the Effect of Unknown-but-Bounded Elastic Moduli in the Buckling of Composite Structures, Computer Methods in Applied Mechanics and Engineering, Vol. 111, pp. 155–167, 1994.
Elishakoff, I., Elisseeff, P., andGlegg, S. A. L.,Nonprobabilistic, Convex-Theoretic Modeling of Scatter in Material Properties, AIAA Journal, Vol. 32, pp. 843–849, 1994.
Elishakoff, I., Lin, Y. K., andZhu, L. P.,Probabilistic and Convex Modeling of Acoustically Excited Structures, Elsevier Science Publishers, Amsterdam, The Netherlands, 1994.
Hollot, C., andBartlett, A.,On the Eigenvalues of Interval Matrices, Technical Report, Department of Electrical Engineering and Computer Engineering, University of Massachusetts, 1987.
Qiu, Z. P., Chen, S. H., andNa, J. X.,The Rayleigh Quotient Method for Computing Eigenvalue Bounds of Vibrational Systems with Interval Parameters, Acta Mechanica Sinica, Vol. 6, pp. 309–318, 1993 (English Edition).
Hu, H. C.,The Theory of Natural Vibration of MDOF Structures, Science Press of China, Beijing, pp. 1–80, 1987 (in Chinese).
Chen, S. H.,Matrix Perturbation Theory in Structural Dynamics, International Academic Publishers, Beijing, P.R. China, 1993.
Lancaster, P.,Theory of Matrices, Academic Press, New York, New York, pp. 289–290 and 350–380, 1985.
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Communicated by G. Leitmann
The research reported in this paper has been supported by the PRC National Natural Science Foundation and by the USA National Science Foundation Grant MSM-9215698 (Program Director Dr. K. P. Chong).
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Qiu, Z.P., Chen, S.H. & Elishakoff, I. Natural frequencies of structures with uncertain but nonrandom parameters. J Optim Theory Appl 86, 669–683 (1995). https://doi.org/10.1007/BF02192164
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DOI: https://doi.org/10.1007/BF02192164