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

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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

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