On the optimal design of a vibrating beam
Author:
Frithiof I. Niordson
Journal:
Quart. Appl. Math. 23 (1965), 47-53
MSC:
Primary 73.49
DOI:
https://doi.org/10.1090/qam/175392
MathSciNet review:
175392
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Abstract: This paper deals with the problem of finding the best possible tapering of a simply supported beam, which for a given volume would have the highest possible value of the natural frequency for the lowest mode of lateral vibrations. The differential equation governing the deflection is given and a procedure to solve this non-linear equation by successive iterations is presented.
J. L. de Lagrange, Sur la figure des colonnes, Miscellanea Taurinensia 5 (1770–1773) 123
T. Clausen, Über die Form archilektonischer Säulen, Bulletin physico-math. et astron. 1 (1849–1853) 279
- Joseph B. Keller, The shape of the strongest column, Arch. Rational Mech. Anal. 5 (1960), 275–285 (1960). MR 128160, DOI https://doi.org/10.1007/BF00252909
- I. Tadjbakhsh and J. B. Keller, Strongest columns and isoperimetric inequalities for eigenvalues, Trans. ASME Ser. E. J. Appl. Mech. 29 (1962), 159–164. MR 137381
- Paul R. Beesack, Isoperimetric inequalities for the nonhomogeneous clamped rod and plate, J. Math. Mech. 8 (1959), 471–482. MR 0107410, DOI https://doi.org/10.1512/iumj.1959.8.58034
- Binyamin Schwarz, On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density, J. Math. Mech. 10 (1961), 401–422. MR 0125271
- Binyamin Schwarz, Some results on the frequencies of nonhomogeneous rods, J. Math. Anal. Appl. 5 (1962), 169–175. MR 148986, DOI https://doi.org/10.1016/S0022-247X%2862%2980001-8
- Binyamin Schwarz, Bounds for the principal frequency of nonuniformly loaded strings, Israel J. Math. 1 (1963), 11–21. MR 155038, DOI https://doi.org/10.1007/BF02759796
J. L. de Lagrange, Sur la figure des colonnes, Miscellanea Taurinensia 5 (1770–1773) 123
T. Clausen, Über die Form archilektonischer Säulen, Bulletin physico-math. et astron. 1 (1849–1853) 279
J. B. Keller, The shape of the strongest column, Archive Ratl. Mech. Anal. 5 (1960) 275
I. Tadjbakhsh and J. B. Keller, Strongest columns and isoperimetric inequalities for eigenvalues, Journal Appl. Mech. 29 (1962) 159
P. R. Beesack, Isoperimetric inequalities for the nonhomogeneous clamped rod and plate, Journal Math. Mech. 8 (1959) 471
B. Schwarz, On the extrema of frequencies of nonhomogeneous strings with equimeasurable density, Journal Math. Mech. 10 (1961) 401
B. Schwarz, Some results on the frequencies of nonhomogeneous rods, Journal Math. Anal, and Appl. 5 (1962) 169
B. Schwarz, Bounds for the principal frequency of nonuniformly loaded strings, Israel Journal Math. 1 (1963) 11
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Article copyright:
© Copyright 1965
American Mathematical Society