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On the free motions of a uniform elastic helicoidal bar

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Sommario

Si presenta un'analisi completa e rigorosa al fine di disaccoppiare un sistema di quattro equazioni vettoriali ordinarie (a coefficienti costanti) che governano l'equilibrio dinamico tridimensionale libero di un elemento d'arco di una barra eliocoidale elastica. Attraverso questo procedimento di disaccoppiamento il sistema differenziale equivalente risulta, rispetto ad uno degli spostamenti generalizzati, del dodicesimo ordine. Sotto l'ipotesi che l'integrale generale di questa equazione sia dato si formula, per il tipo più generale di risposta, l'equazione trascendente che determina le frequenze naturali del sistema. Infine, come applicazione del metodo e sotto alcune ipotesi restrittive, si determina l'integrale generale dell'equazione differenziale dei moti armonici di una barra elicoidale elastica per mezzo di funzioni elementari.

Summary

A thorough and rigorous analysis for the decoupling of a first-order partial differential system of four (3×1)-vectorial equations (with constant coefficients), governing the free three-dimensional dynamic equilibrium of an arc element of an elastic circular helicoidal bar, is presented. Through this decoupling procedure the equivalent to the system differential equation, with respect to one of the generalized displacements, results of the twelfth-order. Under the assumption that the general integral of this equation is given and for the most general case of response, the transcendental equation for the determination of the natural frequencies of the system is formulated. Finally, as an application on the method and under some restrictions the general integral of the differential equation of harmonic motions of an elastic helicoidal bar is determined in the form of elementary functions.

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References

  1. Nowacki, W.,Dynamics of Elastic Structures, Chapman and Hall, London, 1963.

    Google Scholar 

  2. Timoshenko, S., Young D. H. andWeaver W.,Vibrations Problems in Engineering, Wiley, New York, London, 1974.

    Google Scholar 

  3. Biggs I.M.,Introduction to Structural Dynamics, McGraw Hill Book Co., New York, 1975.

    Google Scholar 

  4. Herrmann G., Forced Motions of Timoshenko Beams,ASME Journal of Applied Mechanics, Vol. 22, pp. 53–56.

  5. Cowper G.R., The Shear Coefficient in Timoshenko's Beam Theory,ASME Journal of Applied Mechanics, Vol. 33, 1966, pp. 335–340.

    MATH  Google Scholar 

  6. Panayotounakos D.E. andTheocaris P.S., The Dynamically Loaded Circular Beam on Elastic Foundation,ASME Journal of Applied Mechanics, Vol. 47, 1980, pp. 139–144.

    Google Scholar 

  7. Ball, R.E., Dynamic Analysis of Rings by Finite Difference,Jour. of the Eng. Mech. Div., Vol. 93, 1967, pp. 1–10.

    Google Scholar 

  8. Culver G.C., Natural Frequencies of Horizontally Curved Beams,Jour. of the Struc. Div., Vol. 93, 1967, pp. 189–203.

    Google Scholar 

  9. Panayotounakos D.E. andTheocaris P.S., A Closed Form Solution for the Static Analysis of Skew-Curved Beams,Jour. de Méchanique Appl., Vol. 4, 1980, pp. 83–102.

    Google Scholar 

  10. Panayotounakos D.E. andTheocaris P.S., Forced Harmonic Vibration of a Cantilever Skew-Curved Beam Considered as a Discrete System by the Flexibility Method,SM ARCHIVE, Vol. 7, 1982, pp. 379–409.

    Google Scholar 

  11. Theocaris P.S., andPanayotounakos D.E., The Differential Equations of Uniform Skew-Curved Bars Due to Harmonic Motions,Jour. of Sound and Vibration, Vol. 89, 1983, pp. 195–211.

    ADS  Google Scholar 

  12. Theocaris P.S. andPanayotounakos D.E., A Theorem on the Decoupling of Higher Order Linear Ordinary Differential Systems with Variable Coefficients,Indus. Mathem., Vol. 33, 1983, pp. 33–49.

    MathSciNet  Google Scholar 

  13. Pereira C.A.L., Free Vibration of Circular Arches,Master of Science Thesis, Rice University, Houston, Texas, 1968.

    Google Scholar 

  14. Wung S.J., Vibration of Hinged Circular Arches,Master of Science Thesis Rice University, Houston, Texas, 1967.

    Google Scholar 

  15. Korn G.A. andKorn T.M.,Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Co., N.Y., 1968.

    Google Scholar 

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, National Technical University of Athens, Athens, Greence

    D. E. Panayotounakos (Lecturer)

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  1. D. E. Panayotounakos
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Panayotounakos, D.E. On the free motions of a uniform elastic helicoidal bar. Meccanica 20, 151–159 (1985). https://doi.org/10.1007/BF02337634

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  • DOI: https://doi.org/10.1007/BF02337634

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