Study On Modeling Of Large Deformation Beams In Multibody System Based On Geometrically Exact Beam Theory

According to the precision requirements of the dynamics simulation, joints and bodies of a multibody system should be properly modeled. Among these, the beam is not only one of the most used flexible components, but also one of the most popular research objects in multibody system dynamics. Geometric nonlinearity has become increasely prominent in flexible beams in multibody system. The phenomena of dynamic stiffening occur in high-speed spinning flexible beams. Beam elements based on linear theory could account for the effects of dynamic stiffening by adding some dynamic stiffening terms. However, the magnitude of displacement and rotation of these elements is limited. Geometrically exact beams are modeled with only one assumption of rigid cross-section. Strain measures defined in the theory are objective and suitable for arbitrary magnitude of displacements and rotations. Hence geometrically exact beams can be used to analyze beams with large displacements and rotations. Traditional geometrically exact spatial beam elements widely adopte independent interpolations of displacement and rotation, which causes the problem of shear locking when modeling extremely slender beams. Beams in flexible multibody system are slender in general. Euler-Bernoulli beam theory is suitable for the modeling of shear-free slender beams with the basic assumption of cross-section keeping perpendicular to the current neutral axes.This constraint of rotation and displacement makes interpolation of rotation very challenging. The process of obtaining internal nodal forces involves interpolations of rotation, resulting in some numerical difficulties. Improper interpolation leads to a non-objectivity of the interpolated strain measures. Some beam elements directly interpolate the strain measures, in which strain measures are invariant to rigid body motions. However, the integration of rotation from the interpolated rotational strains cannot be given in a closed form for three dimensional beams.In allusion to the problems in modeling of geometric nonlinear beams mentioned above, based on the internal virtual power equations for geometrically exact beams and the relationships between tangents of the beam centreline and curvatures, a geometric nonlinear spatial beam element with global nodal coordinates was presented. First, Hermitian interpolation of the beam centerline was used for calculating nodal curvatures. Then, internal curvatures of the beam were re-interpolated. The nodal parameters of the presented element are the same with those of the element based on the geometrically exact beam model, but the internal forces can be obtained without angular interpolations. In this element, objectivity of strain measures could be persevered, and also the internal force, as a function of inertia nodal coordinates, is integrable which has an advantage in numerical simulation. The proposed beam element can be degenerated into linear beam element under the condition of small deformation.This spatial Euler-Bernoulli beam element is suitable for geometric nonlinear analysis and automatically counts dynamic stiffening.Meanwhile, a new geometric nonlinear spatial beam element formulation exactly fulfilling the constraint of cross sectional rotation and displacement of beam centerline to vanish shear strains is developed according to the geometrically exact beam theory. A novel interpolation strategy is presented. Beam cross sections remain perpendicular to the tangent vector of the beam centerline in Euler-Bernoulli beams. Hence the normal vector of a cross section, which is also the base vector of the local frame, is coinciding with the unit tangential vector of the beam centerline. Therefore, the rotation of the base vector from the initial orientation onto the local frame can be described via 3 quaternion equations consisting 4 variables which have a general solution. By using a beam centerline interpolation, the continuity requirements are guaranteed. Rotational quaternions resolved from the equation automatically fulfill the constraint between cross section and beam centerline. Adopting quaternion instead of rotational vector as nodal variables avoids the traditionally encountered singularity problem. Strain measures of the beam element are objective and could be consistently derived from displacements and rotations.Besides, in many multibody softwares based on the traditional theories of dynamics of multibody systems, parameters such as the mass, the center of mass and the moments of inertia of beam's cross-sections and rigid bodies usually have to be given artificially. The requirement results in many difficulties in practice applications, because these parameters are not easy to obtain, especially when bodies are complicated in shape. Meanwhile as the multibody system dynamics spread, both the theory and business software tend to reduce the operations of the user in order to minimize the rate of invalid datas. For all this, a new methodology is presented which enables to overcome the aforementioned difficulties. The main idea is to divide a body into many pieces with regular shape called rigid elements. By using rigid elements, the dynamic equations of rigid multibody system could be automatically integrated. Two kinds of element, the rigid beam element and the rigid tetrahedron element were constructed in this paper, with the consideration of their ability to represent any solid body. The triangular element was proposed for irregular sections. A software with mesh generator was developed based on the methodology which can obtain the inertia parameters of bodies with any shape automatically.Rigid elements and two kinds of large deformation geometrically exact Euler-Bernoulli beam elements are proposed in this manuscript for the study of dynamics modeling of multibody systems containing complex shaped rigid bodies and geometric nonlinear beams.