Rational Finite Elements and Flexible Body Dynamics

The goal of this study is to develop the dynamic differential equations of the first finite element based on the rational absolute nodal coordinate formulation (RANCF) and to demonstrate its use in the nonlinear dynamic and vibration analysis of flexible bodies that undergo large displacements, including large deformations and finite rotations. New RANCF elements, which correctly describe rigid body displacements, will allow representing complex geometric shapes that cannot be described exactly using nonrational finite elements. Developing such rational finite elements will facilitate the integration of computer aided design and analysis and will allow for developing analysis models that are consistent with the actual geometry. In order to demonstrate the feasibility of developing RANCF finite elements, an Euler–Bernoulli beam element, called in this investigation as the cable element, is used. The relationship between the nonrational absolute nodal coordinate formulation (ANCF) finite elements and the nonrational Bezier curves is discussed briefly first in order to shed light on the transformation between the control points used in the Bezier curve representation and the ANCF gradient coordinates. Using similar procedure and coordinate transformation, the RANCF finite elements can be systematically derived from the computer aided design geometric description. The relationships between the rational Bezier and the RANCF interpolation functions are obtained and used to demonstrate that the new RANCF finite elements are capable of describing arbitrary large deformations and finite rotations. By assuming the weights of the Bezier curve representation to be constant, the RANCF finite elements lead to a constant mass matrix, and as a consequence, the Coriolis and centrifugal inertia force vectors are identically equal to zero. The assumption of constant weights can be used to ensure accurate representation of the geometry in the reference configuration and also allows for the use of the same rational interpolating polynomials to describe both the original geometry and the deformation. A large strain theory is used to formulate the nonlinear elastic forces of the new RANCF cable element. Numerical examples are presented in order to demonstrate the use of the RANCF cable element in the analysis of flexible bodies that experience large deformations and finite rotations. The results obtained are compared with the results obtained using the nonrational ANCF cable element.