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On optimal design of vibrating structures

  • R. M. Brach
Technical Note

Abstract

Optimal design problems of linearly elastic vibrating structural members have been formulated in two ways. One is to minimize the total mass holding the frequency fixed; the other is to maximize the fundamental frequency holding the total mass fixed. Generally, these two formulations are equivalent and lead to the same solution. It is shown in this work that the equivalence is lost when the design variable (the specific stiffness) appears linearly in Rayleigh's quotient and when there is no nonstructural mass. The maximum-frequency formulation then is a normal Lagrange problem, whereas the minimum-mass problem is abnormal. The lack of recognition of this can lead to incorrect conclusions, particularly concerning existence of solutions. It is shown that existence depends directly on the boundary conditions and, when a sandwich beam has a free end, a solution to the maximum-frequency problem does not exist.

Keywords

Sandwich Beam Optimal Design Problem Mass Constraint Specific Stiffness Fundamental Natural Frequency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Prager, W., andTaylor, J. E.,Problems of Optimal Structural Design, Journal of Applied Mechanics, Vol. 35, series E, No. 1, 1968.Google Scholar
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    Taylor, J. E.,Minimum Mass Bar for Axial Vibration at Specified Natural Frequency, AIAA Journal, Vol. 5, No. 10, 1967.Google Scholar
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    Brach, R. M.,On the Extremal Fundamental Frequencies of Vibrating Beams, International Journal of Solids and Structures, Vol. 4, pp. 667–674, 1968.MATHCrossRefGoogle Scholar
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    Karihaloo, B. L., andNiordson, F. I.,Optimum Design of Vibrating Beams under Axial Compression, Danish Center for Applied Mathematics and Mechanics, Report No. 29, 1972.Google Scholar
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    Vepa, K.,Optimally Stable Structural Forms, University of Waterloo, Canada, Department of Civil Engineering, Ph.D. Thesis, 1972.Google Scholar
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    Pars, L. A.,An Introduction to the Calculus of Variations, Heinemann, London, 1965.Google Scholar

Copyright information

© Plenum Publishing Corporation 1973

Authors and Affiliations

  • R. M. Brach
    • 1
  1. 1.Department of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre Dame

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