Research On Dynamic Modeling And New Element Development Based On The Absolute Nodal Coordinate Formulation

With the development of the modern high-speed,lightweight and precision mechanical systems,the disadvantages of traditional modeling and analysis methods used for the flexible multi-body system dynamics based on the assumption of small deformation are increasingly exposed since they cannot accurately describe the mutual coupling between large overall rigid motion and large deformation.The absolute node coordinate formulation(ANCF)is based on an absolute coordinate system.The absolute position gradient vectors are used instead of the infinitesimal rotation angles to represent the rotation.There is no restriction on the magnitude of deformation,and the derived mass matrix is constant.The system equation of motion has the advantages of non-incremental form,zero Coriolis force and zero centrifugal acceleration.Therefore,the ANCF method has inherent advantages in solving large-rotation and large-displacement dynamic problems.At the same time,the use of gradient coordinates makes it possible to describe complex geometries.Additionally,the ANCF finite element mesh is compatible with the non-uniform rational B-splines method widely used in computer-aided design software.Therefore,the ANCF method facilitates the integration of computer aided design(CAD)and analysis(CAA)system.Thus,the tedious transformation from the geometric model to the finite element model before the traditional finite element analysis is avoided.As a consequence,the error introduced in the conversion process is eliminated.Due to these advantages,the ANCF method has been a hot topic after it was proposed.Compared with the library of the conventional finite element,that of the ANCF finite element is not complete.Taking into account of such conditions,this paper focuses on the development of the new ANCF elements,the improvement of the existing ANCF finite elements and the application of ANCF finite element in the parabolic leaf spring dynamic analysis.All the work aims to facilitate the application of the ANCF finite element in the integration of CAD and CAA system.Based on the area coordinates,one ANCF full cubic and two incomplete cubic triangle elements are developed.The Bézier triangle basis functions are used as the element interpolation polynomials.The corresponding non-independent area coordinates are eliminated at each vertex and the position vector of the triangle mass center is adopted as the nodal coordinate vector.The independent area gradients with clear physical interpretation are introduced.By so doing,a concise shape function is deduced and the full cubic triangular element is obtained.As for the assembly of the element,the Cartesian coordinate parameter is used to guarantee the continuity of the strain at the nodes.Based on the full cubic element,the three-node incomplete cubic triangular element is developed using the position constraint method.Then,the polynomial completeness of the resulting incomplete cubic triangular element is discussed.The development method of the incomplete cubic triangular element with the quadratic completeness is proposed.The incomplete cubic triangular element is improved to achieve the quadratic accuracy.Numerical examples including the patch test are used to verify the accuracy and convergence of the developed elements.The linear transformation between the ANCF triangle element coordinates and those of the Bézier triangle control points is proposed.The general method of constructing both the straight and curved configuration using triangular element is presented,which facilitates its application in the integration of the CAD and CAA system.Using two sets of parameter,i.e.,the volume coordinate and Cartesian coordinate parameter set,three ANCF tetrahedral elements are developed.Adopting all the terms of the full cubic polynomial as the basis polynomials of the interpolation,the derivation of the shape function is simplified by introducing the concept of the independent volume coordinate gradient and the full cubic tetrahedron element is developed.With the algebraic constraint method,the incomplete cubic tetrahedral elements with four vertices as nodes are derived based on the full cubic tetrahedral element.From the perspective of polynomial completeness,the incomplete cubic elements obtained are investigated and the criterion of selecting the appropriate polynomial basis is proposed to guarantee the quadratic accuracy of the incomplete cubic tetrahedron elements.Patch test and other numerical examples are used to verify the accuracy and convergence of the proposed element.The linear transformation relationship between tetrahedral element nodes and Bézier tetrahedron control points is proposed,which promotes the application of ANCF tetrahedral element in the integration of the CAD and CAA system.Although the ANCF element has many advantages,compared with conventional finite element used in the structural analysis and multi-body dynamics analysis based on the floating frame of reference(FFR),it adopts position gradient as nodal coordinate,the number of nodal degrees of freedom is much larger than that of the conventional finite element.As a result,the computational efficiency is weaken.In structural analysis and small deformation dynamic problems,the advantage of the ANCF finite element is not obvious.Accounting for the conditions above,the rotation matrix represented by the finite rotation angles is introduced and the gradient coordinates are expressed with the finite rotation angles.Thus,the number of nodal coordinates is reduced.Introducing the small deformation assumption and linearizing the velocity transformation matrix,the obtained ANCF/FFR element retains the valuable properties of the ANCF element,i.e.,the constant mass,zero Coriolis and centrifugal acceleration terms.Additionally,it takes into account of the initially curved configuration through the constant geometry coefficients.It has the ability to describe complex geometries and is compatible with the geometric description methods used in the CAD system.Compared with the ANCF element,the reduced-order ANCF/FFR element has the same number of the nodal coordinates as that of the conventional finite element.Accordingly,the calculation efficiency is improved.Based on the above ideas,ANCF/FFR planar beam element and triangular element are developed.The eigenvalue analyses are used to verify the validity of the new elements.The geometric model is reconstructed by extracting the leaf spring geometric configuration using three-dimensional scanning technology.A novel interpolation mode of the leaf spring profile curve is proposed.Based on the compatibility between the geometrical description method in CAD system and the ANCF finite element mesh,the geometric model is automatically transformed into the finite element model.A procedure to automatically create the geometric and the further finite element model from an actual physical model is proposed.Different ANCF elements are used to simulate the multi-body suspension system with leaf springs.According to the comparison on the performance of different elements in the simulation of the leaf spring system,a parabolic leaf spring is constructed using the high-order beam element for simulation.The obtained result is consistent with that given by the commercial finite element software ANSYS.The influence of the pre-stress and friction between leaves on the leaf spring is discussed.A simplified suspension model is constructed and simulations are performed under different working conditions including the unilateral bump negotitation.The newly proposed strain split method is tested under different conditions to evaluate its effect of alleviating the locking problems.