Comparison Of The Absolute Nodal Coordinate And Geometrically Exact Formulations For Beams

As a large number of lightweight flexible components are widely used in huge mechanical systems,the modeling method for large deformation flexible multibody systems has received increasing attention in recent years.Different modeling methods have significant influence on the computational precision and efficiency of engineering problems.Recently,two formulations have emerged which have the potential of handling largedeformation flexible multibody systems with overall motion: the Absolute Nodal Coordinate Formulation(ANCF)proposed by Shabana and the Geometrically Exact Beam Formulation(GEBF)firstly proposed by Reissner and later extended by Simo and Vu-Quoc by means of the large rotation vector approach.The Absolute Node Coordinates Formulation employs the nodal slope vector to describe the three-dimension rotation for flexible components,which has the advantages of being intuitive and easy to understand.However,the shortcomings of this method include: containing much more nodal degrees of freedom,suffering from low computational efficiency,suffering from the Poisson and shear locking problem such that the convergence is poor in the process of simulation.Geometrically exact beam formulation directly employs the rotation parameter with less degree of freedom as the generalized coordinate.And,this approach adopts the assumption of flat section to make that the whole beam's shape can be completely determined by displacement of the beam axis and rotation of the corresponding section.Geometrically exact beam elements have the advantages of less nodal degree of freedom,high computational efficiency and accuracy.However,the interpolation and updating algorithms for rotation parameter are very complicated due to their nonlinearities.This dissertation presents a detailed comparison between these two modeling methods,and the main contents are as follows:1)The basic theory of Absolute Node Coordinate Formulation is reviewed briefly,and the calculation formulations for the elastic force,stiffness matrix and mass matrix are presented for the fully parameterized beam element and the slope deficient beam element.The equation of motion for multibody systems is established based on the Lagrange equations of the first kind,and the time integration algorithm is also introduced.2)The basic theory of geometrically exact beam elements is revisited firstly.The rotation of the beam cross-section is described by means of the rotational tensor.The calculation formulation for the angular velocity vector and curvature vector are given for the beam element.Then,the equilibrium equation and the geometrical equation with strain-displacement relationship are presented as well as physical equation for the geometrically exact beam formulation.3)The Cartesian rotation vector parameterization technique is exploited to describe the section rotation in geometrically exact beam elements.Three types of geometrically exact beam models are presented.Then,the linearization algorithm for the strain,the elastic force and the inertial force are studied.The updating and interpolation of the vectorial parameterization of rotation is deduced in details.4)The source program of the two-dimensional and three-dimensional geometrically exact beam elements is accomplished by MATLAB language.And a series of classical statics examples verify the correctness and effectiveness of the three-dimensional geometrically exact beam element.Finally,the computational accuracy and efficiency among three different beam elements including ANCF fully parameterized element,slope deficient element,GEF beam element are investigated in detail through several statics and dynamics numerical examples.