Simple and Effective Equilibrium Models for Vibration Analysis of Curved Rods
Publication: Journal of Engineering Mechanics
Volume 122, Issue 4
Abstract
New finite-element models for curved beam vibration analysis are derived from classical complementary variational principles of elastodynamics. The use of a spline approximation of the axis line (as previously introduced by the writers in the static case) allows for the a priori satisfaction of the dynamic differential equilibrium equations in a simple and effective way. More precisely, starting from the Hellinger-Reissner principle and making use of a linear interpolation of displacements and momentum fields, a very simple hybrid-mixed model is obtained that can be easily linked with general-purpose finite element packages. Alternatively, fully equilibrated models are derived from the complementary energy principle assuming as unknowns either the momentum or the stress resultant fields; in both cases highly accurate finite element models are obtained for which upper and lower bounds on eigenvalue estimates are readily available. Several examples are worked out that are capable of showing the efficiency and the wide spectrum of applicability of the proposed method. The comparison with two general-purpose finite element packages of large diffusion let us assess the high level of performance of the complementary energy models for curved elements.
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References
1.
ABAQUS/Standard user's manual—Version 5.4. (1994). Vols. I and II, Hibbit, Karlsson, Sorensen, Inc.
2.
Alessandri, C., Benedetti, A., and Tralli, A. (1989). “A new F.E. hybrid model for elasto-plastic curved beam analysis.”Proc., Int. Conf. on Computational Plasticity II, D. R. J. Owen, E. Hinton, and E. Oñate, eds., Pineridge Press, U.K., 699–710.
3.
Argyris, J., and Sharpf, D.(1969). “Some general considerations on the natural mode technique. Part I: small displacements.”Areonautical J., 73(2), 218–226.
4.
Ashwell, D. G., and Ghallagher, R. H. (1976). Finite elements for thin shells and curved members . J. Wiley, London, England.
5.
Babu, R., and Prathap, G.(1986). “A liner thick curved beam element.”Int. J. Numer. Meth. Engrg., 23(10), 1313–1328.
6.
Bathe, K. J., Wilson, E. L., and Peterson, F. E. (1974). “SAP IV—A structural analysis program for static and dynamic response of linear systems.”EERC Rep. 73-11, Univ. of California, Berkeley.
7.
Benedetti, A., and Tralli, A.(1989). “A new hybrid F.E. model for arbitrarily curved beam. Part I: Linear analysis.”Comp. and Struct., 33(6), 1437–1449.
8.
Colautti, M. P. (1968). “Sulle vibrazioni trasversali di una biella cuneiforme appoggiata agli estremi.”Rend. Acc. dei Lincei, 2, serie VIII, vol. XLIV, 158–171 (in Italian).
9.
Davis, R., Henshell, R. D., and Warburton, G. B.(1972). “Constant curvature beam finite elements for in plane vibration.”J. Sound and Vibration, 25(4), 561–576.
10.
Den Hartog, J. P.(1928). “The lowest natural frequency of circular arcs.”Philosophical Magazine, Series 7, 5, 400–408.
11.
De Veubecke, F. (1965). “Displacement and equilibrium models in F.E. methods.”Stress analysis, O. C. Zienkiewicz and G. S. Holister, eds., J. Wiley, London, England.
12.
Elias, Z. M. (1972). “Dynamic analysis of frame structures by the force method.”Advances in computational methods in structural mechanics and design, J. T. Oden et al., eds., University of Alabama Press, Tuscaloosa, Ala.
13.
Fichera, G. (1978). Numerical and quantitative analysis . Pitman, London, England.
14.
Fujii, F., and Gong, S. X.(1988). “Field transfer matrix for nonlinear curved beams.”J. Struct. Engrg., ASCE, 114(3), 675–692.
15.
Ghallagher, R. H. (1975). Finite element analysis fundamentals . Prentice-Hall, Englewood Cliffs, N.J.
16.
Grammel, R. (1939). “Ein neues Verfahren zur Lösung technischer Eigenwertprobleme.”Ingenieur Archiv, 10, 35–46 (in German).
17.
Hoppe, R.(1871). “The bending vibrations of a circular ring.”Crelles J. Math., 73(1), 158–170.
18.
Kuhl, W.(1942). “Measurements to the theories of resonant vibrations of circular rings of arbitrary wall thickness.”Akustische Zeitschrift, 7(1), 125–152.
19.
Ladeveze, P., and Pelle, J. P. (1989). Accuracy in finite element computation for eigenfrequencies.”Int. J. Numer. Meth. Engrg., 28(11), 1929–1949.
20.
Lamb, H.(1888). “On the flexure and vibration of a curved bar.”Proc., London Math. Soc., 19(2), 365–376.
21.
Lee, P. G., and Sin, H. C.(1994). “Locking free curved beam element based on curvature.”Int. J. Numer. Meth. Engrg., 37(6), 989–1007.
22.
Love, A. E. H. (1944). A treatise on the mathematical Theory of Elasticity. 4th Ed., Dover Publications, New York, N.Y.
23.
Morris, D. L.(1968). “Curved beam stiffness coefficients.”J. Struct. Div., ASCE, 94(5), 1165–1174.
24.
Pian, T. H., Chen, P., and Kang, D. (1983). “A new formulation of hybrid/mixed finite element.”Comp. and Struct., 16(1–4), 81–87.
25.
Ping, L. H.(1969). “Stiffness matrix of a curved beam element.”AIAA J., 7(1), 57–63.
26.
Rao, S. S., and Sundarajan, V.(1969). “In plane flexural vibrations of circular rings.”J. Appl. Mech., 36, 620–625.
27.
Reddy, B. D., and Volpi, M. B.(1992). “Mixed finite element methods for the circular arch problem.”Comp. Meth. Appl. Mech. and Engrg., 97(1), 125–145.
28.
Saleeb, A. F., and Chang, T. Y.(1987). “On the hybrid formulation of C° curved beam.”Comp. Meth. Appl. Mech. and Engrg., 60(3), 95–121.
29.
Seidel, B. S., and Erdelyi, E. A.(1964). “On the vibration of a thick ring in its own plane.”J. Engrg. for Industry, series B, 86(2), 240–244.
30.
Tabarrok, B.(1971). “A variational principle for the dynamic analysis of continua by the hybrid F.E. method.”Int. J. Solids and Struct., 7, 251–268.
31.
Tabarrok, B. (1984). “Complementary variational principles in elastodynamics.”Comp. and Struct., 19(1–2), 239–246.
32.
Tabarrok, B., Farshad, M., and Yi, H.(1988a). “Finite element formulation of spatially curved and twisted rods.”Comp. Meth. Appl. Mech. and Engrg., 70(2), 275–299.
33.
Tezcan, S. S., and Ovunc, B.(1965). “Analysis of plane and space frameworks with curved members.” IABSE Bull., 25(3), 339–352.
34.
Toupin, R. A.(1952). “A variational principle for the mesh-type analysis of a mechanical system.”J. Appl. Mech., 74(1), 151–152.
35.
Washizu, K.(1966). “Note on the principle of stationary complementary energy applied to free vibration of an elastic body.”Int. J. Solids and Struct., 2(1), 27–35.
36.
Washizu, K. (1982). Variational methods in elasticity and plasticity, 3rd Ed., Pergamon Press, Oxford, England.
37.
Wolf, J. Jr.(1971). “Natural frequencies of circular arches.”J. Struct. Div., ASCE, 97(9), 2337–2349.
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Copyright © 1996 American Society of Civil Engineers.
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Published online: Apr 1, 1996
Published in print: Apr 1996
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