Nonlinear Dynamics of Unidirectional, Fiber-Reinforced Tori
Publication: Journal of Engineering Mechanics
Volume 122, Issue 3
Abstract
The symbolic manipulator Mathematica is used to model the nonlinear dynamic behavior of closed, elastic toroidal shells. Transverse shears are neglected and the nonlinearities are of the Von Kármán type. Two fiber-reinforcing schemes are considered: reinforcement with fibers along the major direction of the torus, and reinforcement with fibers along the minor direction of the torus. These schemes result in orthotropic material characteristics. Differential geometry is used to derive the nonlinear kinematic relationships, and a combination of the Rayleigh-Ritz technique and the method of harmonic balance is used to approximate the nonlinear natural frequencies of the tori. Numerical examples show that the linear natural frequency increases as the fiber volume fraction increases for any radii ratio. On the other hand, the nonlinear analysis of some reinforcing schemes shows a competition between the geometric and material parameters of the tori. This competition has a significant effect on the qualitative behavior of the torus and demarks the borders separating shell-like behavior and ring like behavior.
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References
1.
Boresi, A. P., and Chong, K. P. (1987). Elasticity in engineering mechanics . Elsevier Science Publishing Co., Inc., New York, N.Y.
2.
Clark, R. A.(1950). “On the theory of thin elastic toroidal shells.”J. Mathematics and Physics, 29(3), 146–178.
3.
Dowell, E. H., and Ventres, C. S.(1968). “Modal equation for the nonlinear flexural vibrations of a cylindrical shell.”Int. J. of Solids and Struct., 4(10), 975–991.
4.
Evensen, D. A.(1963). “Some observations on the nonlinear vibration of thin cylindrical shells.”AIAA J., 1(12), 2857–2858.
5.
Gibson, R. F. (1994). Principles of composite materials mechanics . McGraw-Hill Book Co., New York, N.Y.
6.
Hayashi, C. (1964). Nonlinear oscillations in physical systems . McGraw-Hill Book Co., New York, N.Y.
7.
Nakajima, Y., and Padovan, J.(1987). “Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structures—III: Impact impact/contact simulations.”Comp. and Struct., 27(2), 275–286.
8.
Noor, A. K., and Peters, J. M.(1988). “Stress and vibration analyses of anisotropic shells of revolution.”Int. J. Numer. Methods in Engrg., 26(5), 1145–1167.
9.
Padovan, J.(1975). “Traveling waves vibrations and buckling of rotating anisotropic shells of revolution by finite elements.”Int. J. Solids and Struct., 1, 1367–1380.
10.
Raouf, R. A.(1994). “Tailoring the dynamic characteristics of composite panels using fiber orientation.”Composite Struct., 29(3), 259–267.
11.
Reddy, J. N. (1984). Energy and variational methods in applied mechanics, Wiley-Interscience, New York, N.Y.
12.
Redekop, D.(1994). “Dynamic response of a toroidal shell panel to blast loading.”Comp. and Struct., 51(3), 235–239.
13.
Simo, J. C., Rifai, M. S., and Fox, D. D.(1992). “On a stress result and geometrically exact shell model. Part VI: Conserving algorithms for nonlinear dynamics.”Int. J. for Numer. Methods in Engrg., 34(1), 117–164.
14.
Simo, J. C., and Tarnow, N.(1994). “A new energy and momentum conserving algorithm for the nonlinear dynamics of shells.”Int. J. for Numer. Methods in Engrg., 37, 2527–2549.
15.
Synge, J. L., and Schild, A. (1969). Tensor calculus . Dover Publications, New York, N.Y.
16.
Timoshenko, S., and Woinowski-Krieger, S. (1959). Theory of plates and shells . McGraw-Hill Book Co., New York, N.Y.
17.
Wolfram, S. (1991). Mathematica, a system for doing mathematics by computer, Advanced Books Program, Addison-Wesley Publishing Co., Inc. Reading, Mass.
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Copyright © 1996 American Society of Civil Engineers.
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Published online: Mar 1, 1996
Published in print: Mar 1996
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