Abstract
A beam of circular cross-section, made of viscoelastic material of Kelvin–Voigt type, is considered. The beam is symmetric with respect to its center, the length and volume of the beam are fixed and its ends are simply supported. The radius of the cross-section is a cubic function of co-ordinate. The beam interacts with a foundation of Winkler, Pasternak or Hetényi type and is axially loaded by a non-conservative force P(t) = P0 + P1 cos #t. Only the first instability region is taken into account. The shape of the beam is optimal if the critical value of P1 is maximal. A few numerical examples are presented on graphs.References
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8. A. Gajewski, Optimization of a compressed column under dynamical stability constraints, Third World Congress of Structural and Multidisciplinary Optimization, Buffalo, New York 1999.
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11. A.D. Kerr, Elastic and viscoelastic foundation models, Journal of Applied Mechanics, 31, 491–498, 1964.
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14. A.S. Volmir, Stability of Deformable Systems [in Russian]. Moscow: Nauka, 1967
2. A.S. Foryś, Two-parameter optimization of an axially loaded beam on a foundation, Journal of Sound and Vibration, 199, 801–812, 1997.
3. A. Gajewski and A.S. Foryś Optimal structural design of a nonconservative viscoelastic system with respect to dynamic stability, Euromech Colloquium 190, Hamburg-Harburg, 1984.
4. A.S. Foryś and A. Gajewski, Parametric optimization of a viscoelastic rod with respect to its dynamic stability [in Polish], Engineering Transactions, 35, 297–308, 1987.
5. A. Gajewski and M. Życzkowski, Optimal structural design under stability constraints, Kluwer Academic Publishers, Dordrecht/Boston /London 1988.
6. A. Foryś, Optimization of mechanical systems in conditions of parametric resonance and in autoparametric resonances [in Polish], Monograph 199, Cracow University of Technology, Krakow 1996.
7. A. Gajewski, Optimization of a compressed column under dynamical stability constraints, XVIII Symposium – Vibration in Physical Systems, Poznań – Błażejewko 1998.
8. A. Gajewski, Optimization of a compressed column under dynamical stability constraints, Third World Congress of Structural and Multidisciplinary Optimization, Buffalo, New York 1999.
9. A.P. Seyranian, O.G. Privalova, The Lagrange problem on an optimal column: old and new results, Struct. Multidisc. Optim., 25, 393–410, 2003.
10. A.A. Mailybaev, H. Yabuno and H. Kaneko, Optimal shapes of parametrically excited beams, Struct. Multidisc. Optim., 27, 435–445, 2004.
11. A.D. Kerr, Elastic and viscoelastic foundation models, Journal of Applied Mechanics, 31, 491–498, 1964.
12. R.S. Engel, Dynamic stability of an axially loaded beam on an elastic foundation with damping, Journal of Sound and Vibration, 146, 463–477, 1991.
13. V.V. Bolotin, Dynamic Stability of Elastic Systems [in Russian], Moscow: Izd. Teor. Lit., 1956.
14. A.S. Volmir, Stability of Deformable Systems [in Russian]. Moscow: Nauka, 1967
