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IssuesArchive of Issues2008-1pp.13-17

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V. G. Grigoriev, "On paradoxes of the formal approach to determining the eigenvalues of one-dimensional boundary value problems," Mech. Solids. 43 (1), 13-17 (2008)
Year 2008 Volume 43 Number 1 Pages 13-17
DOI 10.3103/S0025654408010020
Title On paradoxes of the formal approach to determining the eigenvalues of one-dimensional boundary value problems
Author(s) V. G. Grigoriev (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993, Russia, valgrigg@rambler.ru)
Abstract The conventional approach to determining the eigenvalues of a one-dimensional boundary value problem consists in writing out the solution of the differential equation in general form containing indeterminate coefficients and constructing a system of homogeneous linear algebraic equations for these coefficients on the basis of the expressions for the boundary conditions. The eigenvalue is determined from the condition that the determinant of the system thus constructed is zero.

In the classical problems (of string, rod, etc. vibrations), this method, as a rule, does not cause any difficulties, although several examples in which the zero value of the frequency satisfying the characteristic equation thus constructed is not an eigenfrequency were constructed and investigated, for example, in [1, p. 220]. We show that in some cases more complicated than the classical ones a similar situation can lead to paradoxical conclusions and erroneous results.
References
1.  Ya. G. Panovko and I. I. Gubanova, Stability and Oscillations of Elastic Systems: Modern Concepts, Paradoxes, and Errors (Nauka, Moscow, 1979) [in Russian].
2.  V. I. Feodosev, "Oscillations and Stability in a Tube Containing a Flowing Liquid," Inzh. Sb., No. 10, 169-170 (1951).
3.  S. V. Arinchev, Theory of Oscillations of Nonconservative Systems (with examples on compact disks) (Izdvo MGTU im. Baumana, Moscow, 2002) [in Russian].
Received 15 June 2005
Link to Fulltext http://www.springerlink.com/content/n7401247v7733173
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