Summary
In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
Similar content being viewed by others
References
Archer, R.R.: Small vibrations of thin incomplete circular rings. Internat. J. Mechanical Sciences1, 45–56 (1960)
Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems III. Eigenvalue problems. Numer. Math.12, 120–133 (1968)
Ciarlet, P.G.: The finite element method for elliptic problems, Amsterdam: North-Holland 1978
Collatz, L.: The numerical treatment of differential equations. Berlin Heidelberg New York: Springer 1960
Dawe, D.J.: Numerical studies using circular arch finite elements. Computers & Structures4, 729–740 (1974)
Den Hartog, J.P.: The lowest natural frequency of circular arcs. Philosophical Magazine5 (Series 7), 400–408 (1928)
Gellert, M., Laursen, M.E.: Formulation and convergence of a mixed finite element method applied to elastic arches of arbitrary geometry and loading. Comput. Methods Appl. Mech. Engrg.7, 285–302 (1976)
Ishihara, K.: A mixed finite element method for the biharmonic eigenvalue problems of plate bending. Publ. Res. Inst. Math. Sci.14, 399–414 (1978)
Ishihara, K.: On curved finite element and straight beam element approximations for vibration problems of circular arch structures. J. Math. Kyoto Univ.20, 753–782 (1980)
Kikuchi, F.: On the validity of the finite element analysis of circular arches represented by an assemblage of beam elements. Comput. Methods Appl. Mech. Engrg.5, 253–276 (1975)
Krieg, R.D., Key, S.W.: Transient shell response by numerical time integration. Internat. J. Numer. Methods Engrg.7, 273–286 (1973)
Moan, T.: A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems. Comput. Methods Appl. Mech. Engrg.3, 17–30 (1971)
Sabir, A.B., Ashwell, D.G.: A comparison of curved beam finite elements when used in vibration problems. J. Sound Vibration18, 555–563 (1971)
Schultz, M.H.: Spline analysis. London: Prentice-Hall 1973
Strang, G., Fix, G.: An analysis of the finite element method. New York: Prentice-Hall 1973
Surana, K.S.: Lumped mass matrices with non-zero inertia for general shell and axisymmetric shell elements. Internat. J. Numer. Methods Engrg.12, 1635–1650 (1978)
Timoshenko, S., Woinowsky-Krieger, S., Theory of plates and shells. New York: McGraw-Hill 1959
Volterra, E., Morell, J.D.: A note on the lowest natural frequency of elastic arcs. Trans. ASME Ser. E. J. Appl. Mech.27, 744–746 (1960)
Wilkinson, J.H.: The algebraic eigenvalue problem. Oxford: Oxford University Press 1965
Yosida, K.: Functional analysis. Berlin Heidelberg New York: Springer 1968
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ishihara, K. A finite element lumped mass scheme for solving eigenvalue problems of circular arches. Numer. Math. 36, 267–290 (1980). https://doi.org/10.1007/BF01396655
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01396655