Numerical Evaluation of Uniform Beam Modes
Publication: Journal of Engineering Mechanics
Volume 129, Issue 12
Abstract
The equation for calculating the normal modes of a uniform beam under transverse free vibration involves the hyperbolic sine and cosine functions. These functions are exponential growing without bound. Tables for the natural frequencies and the corresponding normal modes are available for the numerical evaluation up to the 16th mode. For modes higher than the 16th, the accuracy of the numerical evaluation will be lost due to the round-off errors in the floating-point math imposed by the digital computers. Also, it is found that the functions of beam modes commonly presented in the structural dynamics books are not suitable for numerical evaluation. In this paper, these functions are rearranged and expressed in a different form. With these new equations, one can calculate the normal modes accurately up to at least the 100th mode. Mike’s Arbitrary Precision Math, an arbitrary precision math library, is used in the paper to verify the accuracy.
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References
Chang, T., and Craig, R. R., Jr. (1969). “Normal modes of uniform beams.” J. Eng. Mech. Div., EM4(95), 1027–1031.
Felgar, R. P., Jr. (1950). “Formulas for integrals containing characteristic functions of a vibrating beam.” Circular No. 14, Bureau of Engineering Research, The Univ. of Texas, Austin, Tex.
Gerald, C. F. (1978). Applied numerical analysis, 2nd Ed. Addison-Wesley, Reading, Mass.
Tang, Yu. (1996). “Laterally excited flexible tanks with nonuniform density liquid.” J. Eng. Mech., 122(10), 948–956.
Young, D., and Felgar, R. P., Jr. (1949). “Tables of characteristic functions representing normal modes of vibration of a beam.” Engineering Research Series No. 44, Bureau of Engineering Research, Austin, Tex.
Ring, M. C.(2001). “MAPM, A portable arbitrary precision math library in C.” C/C++ User J., 19(11), 28–38.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 129 • Issue 12 • December 2003
Pages: 1475 - 1477
Copyright
Copyright © 2003 American Society of Civil Engineers.
History
Received: Sep 19, 2002
Accepted: Apr 23, 2003
Published online: Nov 14, 2003
Published in print: Dec 2003
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