Element Coordinate System Research Articles

DEVELOPMENT OF SIMPLE MODELS FOR THE ELASTIC FORCES IN THE ABSOLUTE NODAL CO-ORDINATE FORMULATION

The objective of this study is to develop simple and accurate elastic force models that can be used in the absolute nodal co-ordinate formulation for the analysis of two-dimensional beams. These force models which account for the coupling between bending and axial deformations are derived using a continuum mechanics approach, without the need for introducing a local element co-ordinate system. Four new different force models that include different degrees of complexity are presented. It is shown that the vector of the elastic forces can be significantly simplified as compared to the elastic force model developed for the absolute nodal co-ordinate formulation using a local element frame [1]. Despite the simplicity of the new models, they account for elastic non-linearity in the strain–displacement relationship. Therefore, they lead to more accurate results as compared to the more complex models developed using the local frame method which does not account for the non-linearities in the strain–displacement relationships. Numerical results are presented in order to demonstrate the use of the new models and test their performances in the analysis of large deformations of flexible multibody systems.

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory

The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.

Exact Modeling of the Spatial Rigid Body Inertia Using the Finite Element Method

In the classical finite element literature beams and plates are not considered as isoparametric elements since infinitesimal rotations are used as nodal coordinates. As a consequence, exact modeling of an arbitrary rigid body displacement cannot be obtained, and rigid body motion does not lead to zero strain. In order to circumvent this problem in flexible multibody simulations, an intermediate element coordinate system, which has an origin rigidly attached to the origin of the deformable body coordinate system and has axes which are parallel to the axes of the element coordinate system in the undeformed configuration was introduced. Using this intermediate element coordinate system and the fact that conventional beam and plate shape functions can describe an arbitrary rigid body translation, an exact modeling of the rigid body inertia can be obtained. The large rigid body translation and rotational displacements can be described using a set of reference coordinates that define the location of the origin and the orientation of the deformable body coordinate system. On the other hand, as demonstrated in this investigation, the incremental finite element formulations do not lead to exact modeling of the spatial rigid body mass moments and products of inertia when the structures move as rigid bodies, and such formulations do not lead to the correct rigid body equations of motion. The correct equations of motion, however, can be obtained if the coordinates are defined in terms of global slopes. Using this new definition of the element coordinates, an absolute nodal coordinate formulation that leads to a constant mass matrix for the element can be developed. Using this formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, beam and plate elements can be treated as isoparametric elements.

Finite Element Incremental Approach and Exact Rigid Body Inertia

In the dynamics of multibody systems that consist of interconnected rigid and deformable bodies, it is desirable to have a formulation that preserves the exactness of the rigid body inertia. As demonstrated in this paper, the incremental finite element approach, which is often used to solve large rotation problems, does not lead to the exact inertia of simple structures when they rotate as rigid bodies. Nonetheless, the exact inertia properties, such as the mass moments of inertia and the moments of mass, of the rigid bodies can be obtained using the finite element shape functions that describe large rigid body translations by introducing an intermediate element coordinate system. The results of application of the parallel axis theorem can be obtained using the finite element shape functions by simply changing the element nodal coordinates. As demonstrated in this investigation, the exact rigid body inertia properties in case of rigid body rotations can be obtained using the shape function if the nodal coordinates are defined using trigonometric functions. The analysis presented in this paper also demonstrates that a simple expression for the kinetic energy can be obtained for flexible bodies that undergo large displacements without the need for interpolation of large rotation coordinates.

EFFECT OF MEMBER INITIAL CURVATURE ON A FLEXIBLE MECHANISM RESPONSE

A co-rotational finite element formulation of slender curved beam element is presented to investigate the effect of member initial curvature on the dynamic behaviour of planar flexible mechanisms. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of curved beam. The nodal co-ordinates, incremental displacements and rotations, velocities, accelerations and the equations of motion of the system are defined in terms of a fixed global co-ordinate system, while the total strains in the beam element are measured in element co-ordinates which are constructed at the current configuration of the beam element. The element equations are constructed first in the element co-ordinate system and then transformed to the global co-ordinate system by using standard procedures. Both the deformation nodal forces and the inertia nodal forces of the beam element are systematically derived by consistent linearization of the non-linear beam theory in the element co-ordinates. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the non-linear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the flexible mechanisms.

Dynamic Stiffness Matrix for a Beam Element with Shear Deformation

A method for calculating the dynamic transfer and stiffness matrices for a straight Timoshenko shear beam is presented. The method is applicable to beams with arbitrarily shaped cross sections and places no restrictions on the orientation of the element coordinate system axes in the plane of the cross section. These new matrices are needed because, for a Timoshenko beam with an arbitrarily shaped cross section, deflections due to shear in the two perpendicular planes are coupled even when the coordinate axes are chosen to be parallel to the principal axes of inertia.

Dynamic Stiffness Matrix for a Beam Element with Shear Deformation

A method for calculating the dynamic transfer and stiffness matrices for a straight Timoshenko shear beam is presented. The method is applicable to beams with arbitrarily shaped cross sections and places no restrictions on the orientation of the element coordinate system axes in the plane of the cross section. These new matrices are needed because, for a Timoshenko beam with an arbitrarily shaped cross section, deflections due to shear in the two perpendicular planes are coupled even when the coordinate axes are chosen to be parallel to the principal axes of inertia.

A Motion Process for Large Displacement Analysis of Spatial Frames

A motion process of a spatial beam element which can retain the continuity of spatial frames at rigid joints is poroposed to remove the restriction of small rotations between two successive increments for large displacement analysis of spatial frames using a corotational formulation combined with incremental-iterative methods. Two different strategies are proposed to determine the current configuration. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system which is determined by the locations of end nodes and the orientations of end cross sections of the beam element. The element deformations are determined by rotations of end cross sections relative to the element coordinate system. An incremental-iterative method based on the Newton-Raphson method combined with are length control is adopted. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

Dynamic analysis of planar flexible mechanisms by co-rotational formulation

A co-rotational formulation of beam element and numerical procedure for the dynamic analysis of planar flexible mechanisms is presented. The nodal coordinates, velocities, accelerations, incremental displacements and rotations, and equations of motion of the system are defined in terms of fixed global coordinates, while the total strains in the beam element are measured in an element coordinate system which rotates and translates with the element but does not deform with the element. The element equations are constructed using the small deflection beam theory with the inclusion of the effect of axial force first in the element coordinate system, and then transformed to the global coordinate system using a standard procedure. In the proposed approach the resulting equations of motion are the same as those typically arising in nonlinear strctural dynamics. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for numerical studies. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

Nonlinear Finite Element Formulation for the Large Displacement Analysis of Plates

In this investigation a nonlinear total Lagrangian finite element formulation is developed for the dynamic analysis of plates that undergo large rigid body displacements. In this formulation shape functions are required to include rigid body modes that describe only large translational displacements. This does not represent any limitation on the technique presented in this study, since most of commonly used shape functions satisfy this requirement. For each finite plate element an intermediate element coordinate system, whose axes are initially parallel to the axes of the element coordinate system, is introduced. This intermediate element coordinate system, which has an origin which is rigidly attached to the origin of the deformable body, is used for the convenience of describing the configuration of the element with respect to the deformable body coordinate system in the undeformed state. The nonlinear dynamic equations developed in this investigation for the large rigid body displacement and small elastic deformation analysis of the rectangular plates are expressed in terms of a unique set of time invariant element matrices that depend on the assumed displacement field. The invariants of motion of the deformable body discretized using the plate elements are obtained by assembling the invariants of its elements using a standard finite element procedure.

Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part II—System Equations

The first step in the derivation of the equations of motion for general elastic mechanism systems was described in Part I of this work. The equations were derived at the elemental level using Lagrange’s equation and the generalized coordinates were both the rigid body degrees of freedom, and the elastic degrees of freedom of element ‘e’. Each rigid body degree of freedom gave rise to a scalar equation of motion, and the elastic degrees of freedom of element e gave rise to a vector equation of motion. Since both the rigid body degrees of freedom and elastic degrees of freedom are considered as generalized coordinates, the equations derived take into account the mutual dependence between the rigid body and elastic motions. This is important for mechanisms that are built using lightweight and flexible members and which operate at high speeds. A schematic diagram of how the equations of motion are obtained in this work is shown in Fig. 1 in Part I. The transformation step in the figure refers to the rotational transformation of the nodal elastic displacements (which were measured in the element coordinate system), so that they are measured in terms of the reference coordinate system. This transformation is necessary in order to ensure compatibility of the displacement, velocity and acceleration of the degrees of freedom that are common to two or more links during the assembly of the equations of motion. This final set of equations after assembly are obtained in closed form, and, given external torques and forces, can be solved for the rigid body and elastic response simultaneously taking into account the mutual dependence between the two responses.

Nonlinear dynamic analysis of elastic frames

A co-rotational formulation and numerical procedure for the dynamic analysis of in-plane frames is presented. The nodal coordinates, incremental displacements and rotations, velocities, acceler-ations, and equations of motion of the system are defined in terms of fixed global coordinates, whereas the total strains in the beam element are measured in an element coordinate system which rotates and translates with the element but does not deform with the element. The element equations are constructed using the small deflection beam theory with the inclusion of the effect of axial force first in the element coordinate system, and then transformed to the global coordinate system by standard procedure. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is used here for the solution of the nonlinear dynamic equilibrium equations. To improve convergence properties of the equilibrium iterations, the previous convergent axial force is used to calculate the geometric stiffness matrix. Numerical examples show the accuracy and efficiency of the present method.

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n- cycle iteration scheme is introduced. The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations. An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.

NONLINEAR ANALYSIS OF THIN-WALLED STRUCTURES BY A COUPLED FINITE ELEMENT METHOD

This paper presents a finite element method which enables to analyze a large displacement behavior of elasto-plastic thin-walled structures which fail by overall, local or interactive instability. An incremental equilibrium equation for a thin-walled beam-column is derived in a stiffness matrix form by using a moving element coordinate system and an incremental variational principle. So called multipoint constraints technique is used to connect plate elements with a beam element at the coupling nodal point in the cross section. Numerical results of the present method are compared with the exact solutions and experimental results. Validity and efficiency of the present method are confirmed.

NONLINEAR ANALYSIS OF THIN-WALLED FRAMES AND MEMBERS WITH ARBITRARY OPEN CROSS SECTIONS

An incremental equilibrium equation for a beam-cloumn with arbitrary open cross sections is derived in a stiffness matrix form by using a moving element coordinate system and an incremental variational principle. A segmentation method for the cross section is used to analyze the development of plastic zones in the cross section and the effect of initial residual stresses arbitrarily distributed in the cross section. Yielding of the material is judged as a bi-axial stress problem under normal and shear stresses by using the flow theory of plasticity associated with von Mises yield criterion. Validity and efficiency of the present method are shown by illustrative examples.

Scale-Free Refined Element Technique

Scale-free finite elements are elements whose stiffness properties depend only on relative dimensions and material constants. These elements are used in conjunction with standard finite-element techniques to formulate refined elements that are applicable to problems of different dimensions. This generalization of the refinedelement method is illustrated with the case of a square shear panel containing a circular hole. Eigenvalue techniques are employed to show that the stiffness properties of this refined element are improved when it is modeled with a larger number of elements while retaining the same number of edge nodes. The accuracy of the resulting stress concentration factor remains essentially constant as the hole is located at different positions in a stress field. Nomenclature E = Young's modulus of elasticity t = uniform thickness v = Poisson's ratio x,y = element coordinate system xityf = nodal location of the /th node a = ratio of lengths (alb for a rectangle, (x2-x3)/x2 for a triangle) |8 = ratio of lengths ((x2 —x3) /y3 for a triangle)

Geopotential coefficient recovery from very long arcs of resonant orbits

Many very long (up to 1100 days) maneuver-free orbital arcs of resonant 12- and 24-hour satellites exist. These orbits are extremely sensitive to low-degree and low-order terms of the geopotential, To utilize these long orbital arcs for geodesy we integrate numerically orbits and variational equations in a mean element coordinate system. Step sizes of 1–10 days are obtained. All gravitational perturbations and drag and radiation-pressure effects are included. Tests with published gravity models on these long arcs indicate that considerable improvement in low-order coefficients, even beyond the second degree, is possible and desirable. These tests as a whole show that the low-order portions of 1969 Smithsonian Astrophysical Observatory gravity fields are superior to those derived in 1966, all without the use of this sensitive resonant satellite data. New geopotential results from combined 12- and 24-hour satellite data have also been obtained. A resonance-corrected Smithsonian 1969 field is presented that should have an accurate set of low-order and low-degree coefficients. These coefficients agree quite well with 1969 Smithsonian solutions.