On the existence of solutions to optimization problems with eigenvalue constraints
Author:
Kosla Vepa
Journal:
Quart. Appl. Math. 31 (1973), 329-341
MSC:
Primary 73.49
DOI:
https://doi.org/10.1090/qam/428893
MathSciNet review:
428893
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The optimum tapering of Bernoulli—Euler beams, i.e. the shape for which a given total mass yields the highest possible value of the first fundamental frequency of harmonic transverse small oscillations, is determined. The question of the existence of a solution to the optimization problem is considered. It is shown that, irrespective of the relationship between the flexural rigidity and linear mass density of the cantilever beam, the necessary conditions for optimality lead to a contradiction. This result is in partial disagreement with that obtained by earlier investigators. By imposing additional constraints on the optimization variable, a numerical solution for the case of the cantilever beam is obtained, using the formulation of the maximum principle of Pontryagin.
L. S. Pontryagin, et al., The mathematical theory of optimal processes, English, tr., Interscience Publishers, N. Y., 1962
R. M. Brach, On the extremal fundamental frequencies of vibrating beams, Int. J. Solids Structures 4, 667–674 (1968)
B. L. Karihaloo and F. I. Niordson, Optimum design of vibrating cantilevers, Rpt. No. 15, Dept. of Solid Mechanics, The Technical University of Denmark.
T. A. Weisshaar, An application of control theory methods to the optimization of structures having dynamic or aeroelastic constraints, AFOSR, 70-2862 TR, SUDAAR 412, Dept. of Aeronautics and Astronautics, Stanford University, 1970
- Frithiof I. Niordson, On the optimal design of a vibrating beam, Quart. Appl. Math. 23 (1965), 47–53. MR 175392, DOI https://doi.org/10.1090/S0033-569X-1965-0175392-8
W. Prager, Optimality criteria derived from classical extremum principles in An introduction to structural optimization, Study No. 1, Solid Mechanics Division, University of Waterloo (1968)
- Gilbert Ames Bliss, The Problem of Lagrange in the Calculus of Variations, Amer. J. Math. 52 (1930), no. 4, 673–744. MR 1506783, DOI https://doi.org/10.2307/2370714
R. G. Gottleib, Rapid convergence to optimum solutions using a Min-H strategy, AIAA J. 5, 322–329 (1969)
- J. Albrecht and L. Collatz (eds.), Numerische Behandlung von Differentialgleichungen. Band 3, Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], vol. 56, Birkhäuser Verlag, Basel, 1981 (German). MR 784038
L. S. Pontryagin, et al., The mathematical theory of optimal processes, English, tr., Interscience Publishers, N. Y., 1962
R. M. Brach, On the extremal fundamental frequencies of vibrating beams, Int. J. Solids Structures 4, 667–674 (1968)
B. L. Karihaloo and F. I. Niordson, Optimum design of vibrating cantilevers, Rpt. No. 15, Dept. of Solid Mechanics, The Technical University of Denmark.
T. A. Weisshaar, An application of control theory methods to the optimization of structures having dynamic or aeroelastic constraints, AFOSR, 70-2862 TR, SUDAAR 412, Dept. of Aeronautics and Astronautics, Stanford University, 1970
F. I. Niordson, The optimal design of a vibrating beam, Quart. Appl. Math. 23, 47–53 (1965)
W. Prager, Optimality criteria derived from classical extremum principles in An introduction to structural optimization, Study No. 1, Solid Mechanics Division, University of Waterloo (1968)
G. A. Bliss, The problem of Lagrange in the calculus of variations, Am. J. Math. 52, 673–744 (1930)
R. G. Gottleib, Rapid convergence to optimum solutions using a Min-H strategy, AIAA J. 5, 322–329 (1969)
L. Collatz, The numerical treatment of differential equations, Springer-Verlag, Berlin, 1960
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73.49
Retrieve articles in all journals
with MSC:
73.49
Additional Information
Article copyright:
© Copyright 1973
American Mathematical Society
