悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.     

An accurate polynomial displacement function for unite ring elements

A finite element solution is developed for a circular ring or curved beam. A polynomial displacement function is used. The solution gives high accuracy for an element which subtends an angle as great as 90°. Both inextensional and extensional deformations are considered. The loads may be either concentrated or distributed. It is shown that failures which have occurred in earlier attempts to use polynomial displacement functions for curved elements are due not to the neglect of rigid body motions as frequently stated in the literature, but to the neglect of coupling between normal and tangential displacements.     

Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

Geometrically exact physics-based modeling and computer animation of highly flexible 1D mechanical systems

This paper presents a geometrically exact beam theory and a corresponding displacement-based finite-element model for modeling, analysis and natural-looking animation of highly flexible beam components of multibody systems undergoing huge static/dynamic rigid-elastic deformations. The beam theory fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, concepts of local displacements and orthogonal virtual rotations, and three Euler angles to exactly describe the coordinate transformation between the undeformed and deformed configurations. To demonstrate the accuracy and capability of this nonlinear beam element, nonlinear static and dynamic analysis of two highly flexible beams are performed, including the twisting a circular ring into three small rings and the spinup of a flexible helicopter rotor blade (Graphical abstract). These numerical results reveal that the proposed nonlinear beam element is accurate and versatile for modeling, analysis and 3D rendering and animation of multibody systems with highly flexible beam components.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Timoshenko beam finite element with four dof and convergence of O(h)

In situations where transverse shear deformations and rotary inertia in beams are important, elements based on the Timoshenko beam theory are useful. Among the two-noded, four DOF elements derived from the minimum total potential energy principle, the HTK. element proposed by Hughes et al. using linear displacement functions for both w and θ and the T1CC4 element proposed by Tessler et al. using quadratic displacement function for w and linear displacement function for θ are well known in the literature. The convergence of the HTK element in the thin beam situation has been too poor due to shear locking but by using selective integration this element can be shown to be equivalent to the T1CC4 element which has a rate of convergence of O(h²). In this paper a five DOF element with w and θ at the end nodes and θ at the middle node and based on the cubic displacement function for w and the quadratic displacement function for θ is first developed. Statically condensing the middle rotational DOF, the well-known (4 × 4) stiffness matrix using the φ-factor defined as φ = 12EI/kGAL² and hitherto obtained only through a flexibility approach or closed-form solution of the governing equations of the Timoshenko beam theory is derived. This element based on cubic displacement function for w has rate of convergence of O(h⁴), is completely free of shear locking and performs equally well in thin as well as thick beam situations.     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

Effect of thermal deformations of an optical pick-up base on the optical properties of DVD optical system

An analytical method is presented to analyze the effect of thermal deformations of an optical pick-up base on the optical properties of DVD optical system. To measure the amount of thermal deformations of an optical pick-up base, finite element analysis and holographic interferometry were used. First, thermal deformations of an aluminum pick-up base was analyzed in thermal environments using finite element analysis; finite element analysis was carried out without the initial surface stress condition. The measurement of thermal deformations by holographic interferometry was carried out to verify finite element analysis results. However, since the finite element analysis results were deviated from those by experiment, the effect of the initial surface stress condition was considered; finite element analysis was carried out with the initial surface residual stress condition, which was obtained from X-ray diffraction measurement. The finite element analysis results with the initial surface stress condition agreed well with the experimental results by holographic interferometry. Finally, to analyze the effect of thermal deformations of the pick-up base on the optical properties of DVD optical system, the deformation of optical path was analyzed. However, the drastic changes of beam spot, beam intensity profile, modulation transfer function curve and wavefront aberration were not observed.     

Non-linear analysis of structures by the finite element method as a supplement to a linear analysis

The discrete elements of finite dimensions which replace the structural continuum in the finite element method can always be chosen sufficiently small that the linear relations between element deformations and element stresses remain valid to the same degree of approximation as is considered acceptable in the linear theory of elasticity. This observation formed the basis for the treatment of geometrical nonlinearities by Argyris and his co-workers in their natural mode technique [1]and [2].Here we give an alternative development of the theory. The element deformations, linearly related to nodal displacements and rotations in a local coordinate system, are expressed as analytic functions of the nodal coordinates in the global system. Then, for structures with an initially linear behaviour, the stability and postbuckling analysis is developed on the basis of the general theory founded by Koiter [3].The theory is illustrated by the example of frame-structures. The location of the nodal points is defined in terms of the displacement vector, while the orientation of an orthogonal triad attached to each nodal point is described by means of modified angular coordinates of Euler. The accuracy of the analysis is demonstrated for a problem solved analytically by Koiter [5]and verified experimentally by Roorda [4].     

Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object's surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C ¹ continuity. The surface pose is estimated based on the model's translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data.     

Plastic instability of beam structures using co-rotational elements

In a previous paper [Comput. Methods Appl. Mech. Engrg. 191 (2002) 1755], the authors have presented a 3D co-rotational elastic beam element including warping effects. This formulation is now further developed in order to incorporate elasto-plastic deformations. The element possesses seven degrees of freedom at each node and can be used to model beams with arbitrary cross-sections. Thus, within the present approach, the centroid and shear center of the cross-section are not necessarily coincident. The main purpose of this element is to model elasto-plastic instability problems. In this context, two methods of branch-switching are tested and discussed. In the first one, the bifurcation point is isolated by successive bisections and the branch-switching is operated by using the eigenvector associated to the negative eigenvalue. In the second one, introduced by Petryk, an energy approach is used to select automatically the stable post-bifurcation path. Six examples, including large displacement and stability problems, are used in order to assess the performances of the element.     

Geometrically non-linear formulation for two dimensional curved beam elements

This paper presents a geometrically non-linear formulation using a total lagrangian approach for the two dimensional curved beam elements. The beam element is derived using linear, paralinear and cubic-linear plane stress elements. The basic element geometry is constructed using the coordinates of the nodes on the element center line (η = 0) and the nodal point normals. The element displacement field is described using two translations of the node on the center line and a rotation about the axes normal to the plane containing the center line of the element. The existing beam element formulations are restricted to small nodal rotations between two successive load increments. The element formulation presented here removes such a restriction. This is accomplished by retaining non-linear nodal rotation terms in the definition of the displacement field and the consistent derivation of the element properties. The formulation presented here is very general and yet can be made specific by selecting appropriate non-linear functions representing the effects of nodal rotations. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the behavior and the accuracy of the two dimensional beam elements for geometrically non-linear applications. In all cases comparisons made with theory and/or other published data show that the beam elements product accurate results and permit large load increments with good convergence characteristics.     

Modular finite element approach to structural crashworthiness prediction

A modular approach is formulated for purposes of analyzing vehicle crashworthiness, and to provide the flexibility to model different parts of a vehicle with different levels of sophistication, depending on the type of information one is seeking. The idea is to approximate a vehicle structure by a number of modules, each suitable for modeling particular portions of the structure. Basic elements are used for this purpose and include a beam element which allows large three dimensional rotations and plastic hinge action, a special spring element which can only take either tension or compression, a finite size rigid body element and a general elastic element using modal approximations. Development of such elements is described together with an example of the simulation of the impact of a rail car locomotive into the rear of a caboose.     

Three-dimensional finite-element analysis of laminated composites

A three-dimensional finite-element analysis treating the mechanical response of thick laminated composite plates in bending is presented. An isoparametric solid element with a cubic displacement expansion in planform and a linear variation through the thickness is used to model each layer of the laminate. The degrees-of-freedom of the element are retained at its boundaries so that interconnections between lamina with different fiber orientations can be made at their interfaces. An incore version of the conjugate gradient technique, which does not have bandwidth restrictions, is used to minimize the total potential energy of the system with the number of iterations to convergence being about one-fifth the total global degrees-of-freedom. Because a three-dimensional analysis is used, the effects of thickness-stretching, transverse shear, extension, and bending deformations are obtained. Comparisons with three-dimensional elasticity solutions are in excellent agreement and show the necessity of having individual elements for each layer when they have different fiber orientations and when the plates are thick.     

Incremental variational principles and finite element models for nonlinear problems

The classical variational principles are formulated for nonlinear problems by considering incremental deformations of a continuum. Associated finite element models are derived, adopting the terminology introduced by Pian for classification of linear finite element models. It is demonstrated how the classical incremental variational principles can be modified by relaxing the continuity requirements between adjoining elements. Nonlinear hybrid finite element models arising from such modifications are discussed.     

On a thin shell element for non-linear analysis, based on the isoparametric concept

This paper discusses a finite element method model for the large displacement, moderate strain analysis of thin shells. The model is based on an ‘adapted’ reference configuration for a displaced element, separating the displacements into rigid body displacements and strain-producing deformations. A strategy is developed, making use of the isoparametric concept for both the choice of reference configuration and in the element formulation. This makes the use of arbitrarily shaped elements possible. The model is shown to give accurate results for a range of relevant problems. Some problems in the general application of this type of model are discussed.