Application of the Finite Element Method to Nonconservative Stability Problems with Damping.
WATERVLIET ARSENAL N Y
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The method of finite elements in conjunction with an adjoint variational principle is applied to nonconservative problems in the theory of elastic stability. The effects of internal and external damping are included by assuming a Kelvin-Voigt material and a dissipative force proportional to the velocity, respectively. Because boundary value problems in the theory of nonconservative elastic stability are nonself-adjoint, by virtue of the fact that nonconservative forces do not possess potentials, no complete functional exists for the classical form of Hamiltons principle. A boundary value problem that is the adjoint of the original problem is introduced, and in terms of the original and adjoint variables a complete functional is derived for an extended or generalized version of Hamiltons principle. This principle contains generalized expressions for the kinetic, potential, and dissipation energies that are bilinear forms in the original and adjoint variables. A CANTILEVERED BEAM SUBJECTED TO A CONCENTRATED FOLLOWERS FORCE WITH DAMPING EFFECTS IS ANALYZED. The finite element formulation emanates from this generalized version of Hamiltons principle. The beam is divided into several segments elements. The displacement function in each element is approximated by a cubic polynomial for both theoriginal and the adjoint problems, with the coefficinets to be determined. These approximate expressions are used in the complete functional of the generalized Hamiltons principle. When the boundary and continuity conditions are imposed on the displacement functions, this principle leads to a eigenvalue matrix equation. Author, modified-PL
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