Dynamic modeling of beams with non-material, deformation-dependent boundary conditions

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In conventional problems of structural mechanics, both kinematic boundary conditions and external forces are prescribed at fixed material points that are known in advance. If, however, a structure may move relative to its supports, the position of the imposed constraint relations generally changes in the course of motion. A class of problems which inherently exhibits this particular type of non-material boundary conditions is that of axially moving continua. Despite varying in time, the positions of the supports relative to the material points of the body have usually assumed to be known a priori throughout the deformation process in previous investigations. This requirement is abandoned in the present paper, where the dynamic behavior of a structure is studied, which may move freely relative to one of its supports. As a consequence, the position of such a non-material boundary relative to the structure does not only change in time but also depends on the current state of deformation of the body. The variational formulation of the equilibrium relations of a slender beam that may undergo large deformations is presented. To this end, a theory based on Reissner's geometrically exact relations for the plane deformation of beams is adopted, in which shear deformation is neglected for the sake of brevity. Before a finite element scheme is developed, a deformation-dependent transformation of the beam's material coordinate is introduced, by which the varying positions of the constraint relations are mapped onto fixed points with respect to the new non-material coordinate. By means of this transformation, additional convective terms emerge from the virtual work of the inertia forces, whose symmetry properties turn out to be different from what has previously been presented in the literature. In order to obtain approximate solutions, a finite element discretization utilizing absolute nodal displacements as coordinates is subsequently used in characteristic numerical examples, which give an insight into the complex dynamic behavior of problems of this type. On the one hand, the free vibrations of a statically pre-deformed beam are investigated, on the other hand, an extended version of the sliding beam problem is studied.