Abstract
This paper presents the 3D dynamic formulations for a flexible beam sliding through a revolute-prismatic joint. Considering the geometric nonlinearity, the configuration space of the 3D flexible beam is a nonlinear differentiable manifold (R3×SO(3)). Moreover, the beam manipulated by the revolute-prismatic joint can undergo large overall motion and slide through the joint. Because of the difficulty mentioned above, most studies on these problems focus on 2D cases or are tackled under a small deformation assumption. In this paper, the rotation matrices are parameterized using rotational vectors to describe accurately the spatial configuration of flexible beams. For convenience, to describe the finite deformation of the beams, the material frame is fixed on the revolute-prismatic joint but will change over time. The corotational method is introduced to take the geometric nonlinearity (small strain and large rotation) of the beam into account. In the corotational frame, the strain energy and kinetic energy of the elements are derived with the same shape functions, which are used to describe the local displacements, to maintain the element-independent framework. Then a ‘standard element’ can be embedded within this framework. In order to consider the shear deformation, the flexible beam is discretized using a fixed number of variable-domain interdependent interpolation elements. Rotary inertia is also considered in this paper. The nonlinear equations of motion are derived by using the extended Hamilton's principle and solved by using the Hilber-Hughes-Taylor method and the Newton-Raphson iteration method. Four examples are presented to demonstrate the validity, accuracy and versatility of the present dynamic formulation.
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