In the present paper a coupled problem of the dynamic response of a beam-like structure under moving body is analyzed. The beam is modelled using the kinematically non-linear beam theory and rotations are parameterized with the rotational quaternions. Mass is modelled by a particle. A particle is moving along the axis of the three dimensional beam. The equation of motion of the particle is added to the system of the beam dynamic equations and an additional unknown representing the coordinate of the curvilinear path of the particle is introduced. The specially designed finite-element formulation of the three-dimensional beam based on the weak form of consistency conditions has been employed. Such an approach has been proven to be most suitable, since only the boundary conditions are affected by the contact forces. Numerical studies have shown that the uncoupled models might highly underestimate the response of the structure compared to the present coupled analysis.
COBISS.SI-ID: 5899105
The rotational quaternions represent a unique four dimensional parametrization of rotations in the three dimensional Euclidean space. In the present paper they are used as the basic rotational parameters in formulating the finite-element approach of geometrically exact beam-like structures. The classical concept of parameterizing the rotation matrix by the rotational vector is completely abandoned so that the only rotational parameters are the rotational quaternions representing both rotations and rotational strains in the beam. The space discretization based on the collocation method is used and the adjustment of the Newmark time integration algorithm to the quaternion parameterizations of rotation is presented. We provide a general procedure for consistent adaptation of integrators on SO(3) group onto the sphere of unit quaternions.
COBISS.SI-ID: 5825377
Because the rotational quaternions in contrast to standard rotational vector do not introduce any singularity into representation of rotation their use in the present finite-element quaternion-based beam theory is an advantage. This makes possible to apply any standard time integration methods. The main advantage of the proposed dynamics formulation is probably its simple implementation into the standard Runge-Kutta procedures. Thus several benefits of the Runge-Kutta methods such as the local error control and the adaptive time steps are automatically incorporated in the present procedure.
COBISS.SI-ID: 5751393