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The Normal Functions of Beam Vibration in Series Solutions of Static Problems

Published online by Cambridge University Press:  28 July 2016

R. E. D. Bishop
R. E. D. Bishop*
Affiliation:
Engineering Laboratory, Cambridge
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Extract

The normal functions of beam vibration may be used in series to solve statical problems of beam flexure and the recent appearance of tables of these functions has rendered this method practicable. An outline is given of the procedure.

The simple equation of free flexural vibration of beams is

1

where v is the displacement, EI the flexural rigidity andAp the mass per unit length. The separation of variables and the application of appropriate boundary conditions at the ends x = 0 and x = l, gives the following normal functions ϕ(x).

Type
Technical Notes
Information
The Aeronautical Journal , Volume 57 , Issue 512 , August 1953 , pp. 527 - 529
Copyright
Copyright © Royal Aeronautical Society 1953

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References

1. Duncan, W. J. (1950). Normalized Orthogonal Deflexion Functions for Beams. R. & M. 2281, 1950.Google Scholar
2. Feloar, R. P. (1950). Formulas for Integrals Containing Characteristic Functions of a Vibrating Beam. Circular No. 14, Bureau of Engng. Research, Univ. of Texas, 1950.Google Scholar
3. Rayleigh, Lord (1894). Theory of Sound. Art. 92. 2nd Edition. Macmillan, London, 1894.Google Scholar
4. Sokolnikoff, I. S. (1939). Advanced Calculus. Chap. 4. McGraw–Hill, 1939.Google Scholar
5. Von Kármán, Th. (1938). On the Use of Orthogonal Functions in Structural Problems. Timoshenko Anniversary Volume, p. 114. Macmillan, New York, 1938.Google Scholar
6. Young, Dana and Felgar, R. P. (1949). Tables of Characteristic Functions Representing Normal Nodes of Vibration of a Beam. Publication No. 4913, Univ. of Texas, 1949.Google Scholar