# Motion equation of a finite dynamic elastic plane lineal element plane lineal element

Américo G Hossne

## Keywords

Hamilton principle, elastic dynamic planar element, four barplanar mechanism, lagrangian, mass matrix, rigid matrix and gyroscopic matrix.

## Abstract

A lineal finite element with constant traverse section, it can adopt any orientation in the plane, and their ends or nodes tie it to the rest of the elements. The kinetic energy (T ) and potential (V ) of a dynamic elastic element are the basement in the implementation of the Hamilton principle for the definition of a finite element. The definition of the kinetic energy and potential is the first step for the preliminary variational formulation to the enunciation for finite elements that it is used to solve, say, the problems of mechanisms that move in the plane using the Hamilton equation. The general objective consisted on defining the equation of the movement of a finite lineal dynamic elastic plane element using the equation of Hamilton, starting from the lagrangiana (T − V ) obtained with the use of a polynomial of fifth and first degree, with eight degrees of freedom, four in each node that represented the deformations: axial (u(x)), traverse (w(x)), slope ((dw(x)/dx)) and bend ((d2w(x)/dx2)). The deformation due to traverse shearing, insignificant with respect to flexional and axial deformations, the rotational inertia and the frictional forces in the nodes, were underrated with the purpose of producing a friendly element. The specific objectives were to take place: (a) the translational mass matrix [MD], (b) the translational gyroscopic matrix [AD], (c) the translational total rigidity matrix [KD], and (d) the deformation vector (S). As a result the movement equation of a finite lineal dynamic elastic plane element was forged [MD]( ¨ S) − 2¨[AD]( ˙S ) + {[K] − ˙2[MD] − ¨[AD]}(S) = (Q) . On concluded that the equation obtained variationally with the application of the Hamilton Principle is the state–of–the–art pattern, and that the procedure can be used when it is required to increase the number of the pattern freedom degrees.

PACS. 45.20.Jj, 47.10.Df

MSC: 37Jxx

## References

[1] K. H. Huebner, E. A. Thornton and T. G. Byrom. The finite element method for engineers, 3rd Edition, ISBN 0471547425. John Wiley, 1995.

[2] S. Kobayashi, S–I Oh and T. Altan.Metal forming and the finite element method, ISBN 0195044029. Oxford University Press, 1989.

[3] S. V. Patankar. Numerical heat transfer and fluid flow, ISBN 0891165223, Taylor & Francis, 1980.

[4] M. Gams, M. Saje, I. Srpeie and I. Planinc. Strain–base finite element for the dynamic analysis of elastic planar beams, http://www.scix.net/Files/ aceptedpapers/Accepted/Gams.pdf, University of Ljubljana, Faculty of Civil and Geodetic Engineering, SI-1115 Ljubljana, Jamova 2, Slovenia, 2007.

[5] E. Reissner. On one–dimensional finite–strain beam theory: the plane problem. Journal of Applied Mathematics and Physics (ZAMP), 23(5), 795–804 (1972).

[6] A. Midha. Dynamics of high speed linkages with elastic members. Doctoral dissertation, University of Minnesota, Minnesota, 1977.

[7] A. Midha, A. G. Erdman, G. N. Sandor and A. G. Frohrib. An alternate computationally efficient and conservative method for kineto–elastodynamic analysis of mechanisms. Proc. 4th OSU appl. Mech. Conf. Chicago, Illinois, (1975).

[8] R. M. Alexander and K. L. Lawrence. An experimental investigation of the dynamic response of an elastic mechanism. Journal of Engineering for Industry, ISSN 0022–1817, 96(1), 268–274 (1974).

[9] U. Oktay. Finite element method–basic concepts and applications . Intext Educational Publishers, New York, 1973.

[10] I. A. Iman. General method for kineto–elastodynamic analysis and design of high speed mechanisms. Doctoral dissertation, Rensselaer Polytechnic Institute, New York, 1973.

[11] A. G. Erdman. A general method for kineto–elastodynamic analysis and synthesis of mechanisms. Doctoral dissertation, Rensselaer Polytechnic Institute, New York, 1972.

[12] R. C. Winfrey. Dynamics of mechanisms with elastic links. Doctor dissertation, UCLA, 1969.

[13] J. S. Przemieniecki. Theory of matrix structural analysis, ISBN 0070509042. McGraw–Hill, New York, 1968.

[14] C. H. Walter and F. R. Moshe. Dynamics of structures, ISBN 013222075X. Prentice–Hal, Englewood cliffs, New Jersey, 1964.

[15] J. P. Sadler. A lumped parameter approach to kineto–elastodynamic analysis of mechanisms. Doctoral dissertation, Rensselaer Polytechnic Institute, New York, 1972.

[16] A. Midha, A. G. Erdman and D. A. Frohrib. Finite element approach to mathematical modeling of high–speed elastic linkages. Mechanism and Machine Theory, ISSN 0094–114X, 13(6), 603-618 (1978).

[17] R. Avilés, G. Ajuria, V. G´omez–Garay and S. Navalpotro. Comparison among nonlinear optimization methods for the static equilibrium analysis of multibody systems with rigid and elastic elements. Mechanisms and Machine Theory, ISSN 0094–114X, 35(8), 1151–1168 (2000).

[18] Z. E. Boutaghou and A. G. Erdman. A design methodology for system parameters synthesis of elastic planar linkages. Journal of Mechanical Design, ISSN 1050-0472, 114(4), 542–546 (1993).

[19] W. L. Cleghorn, R. G. Fenton and B. Tabarrock. Optimum design of high-speed flexible mechanismsCalcul optimum de mecanismes flexibles de grande vitesse. Mechanisms and Machine Theory, ISSN 0094–114X, 16(4), 394–406 (1981).

[20] L. Saggere and S. Kota. Synthesis of Planar, compliant four–bar mechanisms for compliant–segment motion generation. Journal of Mechanical Design, ISSN 1050-0472, 123(4), 535–541 (2001).

[21] A. J. Hossne. Lagrangiano de un elemento finito plano elástico dinámico con ocho grados de libertad. Ingeniería, ISSN 1665–529X, 11(1), 25–36 (2007)

[22] D. Bel y M. Doblaré. Formulación de elementos finitos Lagrangianos y Hamiltonianos Bond–Graph para la simulación dinámica de sistemas multidisciplinares. Universidad de Zaragoza, IV Congreso de Métodos Numéricos en Ingeniería. Sevilla, 1–21 (1999).

[23] Dare A. Wells. Schaum’s Outline of Lagrangian Dynamics: With a Treatment of Euler’s Equations of Motion, Hamilton’s Equations and Hamilton’s Principle, ISBN 978–0070692589, Schaum’s Outline, 1967.

[24] O. Bruls and G. Kerschen. Flexible multibody systems with finite elements. ULg–Department of Aerospace and Mechanical, 1–55 (2007).

[25] E. Bansch, L. Tobiska and N. Walkington. Mini–Workshop: Interface problems in Computational fluid dynamics. Mathematisches Forschungsinstitut Oberwolfach, 2(1), 465–502 (2005).