The eigen modes of continuous bars, strings and Euler beams constrained only at their 2 ends have an important qualitative property called oscillation property. Appreciate discrete models of bars, strings and Euler beams are anticipated to mirror the oscillation property in discrete form. With help of an algebraic approach involving tri- and penta-diagonal matrix, the discrete eigen modes of bars, strings and beams obtained through finite difference method were proved to have the oscillation property invariably, regardless of grid and mass distribution. In this paper, we prove that if the modes of finite element model equal to the modes of analytic models at nodes, then the modes of finite element model own oscillation property. Furthermore, the oscillation property does also correctly for the eigen modes of bars and strings discretized by 2-nodes finite elements with lumped mass matrix. However, Euler beam meshed with the 2-nodes cubic Hermitian elements, the stiffness matrixes are no longer tri- or penta-diagonal and the algebraic approach does not work. In this paper, the discrete oscillation property of bars, strings and Euler beam meshed with finite elements is discussed by a new approach, i.e., checking an equivalent condition of the oscillation property. We proved in this paper that eigen modes of FE discrete Euler beam with lumped mass has oscillation property invariably, if stiffness matrices are derived by flexibility approach. The 2-nodes cubic Hermitian element may lead to failure of oscillation property, if the beam section varies severely in elements. Numerical examples supported our conclusion.
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