Showing papers on "Direct stiffness method published in 2001"

Stiffness estimation of a tripod-based parallel kinematic machine

Abstract: Presents a simple yet comprehensive approach that enables the stiffness of a tripod-based parallel kinematic machine to be quickly estimated. The approach arises from the basic idea for the determination of the equivalent stiffness of a group of serially connected linear springs and can be implemented in two steps. In the first step, the machine structure is decomposed into two substructures associated with the machine frame and parallel mechanism. The stiffness models of these two substructures are formulated by means of the virtual work principle. This is followed by the second step that enables the stiffness model of the machine structure as a whole to be achieved via linear superposition. The three-dimensional representations of the machine stiffness within the usable workspace are depicted and the contributions of different component rigidities to the machine stiffness are discussed. The results are compared with those obtained through experiments.

On obtaining machine tool stiffness by CAE techniques

Abstract: In this paper, a single module method and a newly developed hybrid modeling method for analyzing the stiffness of machine tools are introduced in detail. Techniques include building suitable finite element models, determining equivalent loads, simulating the interface between two modules, considering boundary constraints, and interpreting results. By taking a detailed finite element mesh for one of the five modules (the headstock, the column, the table, the saddle and the bed), together with simplified meshes for the other four modules, a hybrid finite element model is assembled. The elastic modulli of the four simplified meshes are kept several orders higher than that of the detailed one. Therefore, the calculated stiffness of the hybrid model is essentially the stiffness of the softer module with the detailed mesh. The stiffness of the five modules can be obtained one after another in the same manner. By supporting the hybrid model only at the middle of the short edge on the bottom surface of the bed, the machine tool can be properly constrained, and its stiffness can be estimated correctly. The controversial issue as to how to simulate properly the boundary condition of the casters under the bed will not occur in this method. A cumbersome procedure to transform the external loads into the equivalent forces as required in SMM is also avoided. There is no local effect due to unevenly distributed nodal forces. It is shown that the hybrid modeling method is better than the single module method in accuracy and efficiency.

Finite element modelling of thick plates on two‐parameter elastic foundation

Abstract: This paper is intended to give some information about how to build a model necessary for bending analysis of rectangular and circular plates resting on a two-parameter elastic foundation, subjected to combined loading and permitting various types of boundary conditions. The formulation of the problem takes into account the shear deformation of the plate and the surrounding interaction effect outside the plate. The numerical model based on an 18-node zero-thickness isoparametric interface element interacting with a thick Reissner–Mindlin plate element with three degrees of freedom at each of the nine nodes, which enforce C0 continuity requirements for the displacements and rotations of the midsurface, is proposed. Stiffness matrices of a special interface element are superimposed on the global stiffness matrix to represent the stiffening elastic foundation under and beyond the plate. Some numerical examples are given to illustrate the advantages of the method presented. Copyright © 2001 John Wiley & Sons, Ltd.

On the stiffness and stability of Gough-Stewart platforms

Abstract: This paper deals with a class of parallel mechanisms, called Gough-Stewart platforms. For these mechanisms, pre-loaded by driving forces, the stiffness matrix is derived and its basic properties are established. It is shown when the stiffness matrix becomes asymmetric, how the parametric imbalance may influence the system stability, and how the center of stiffness depends on the force pre-loading. Next, some necessary and some sufficient conditions for the stability are established in an analytical form by transforming the stiffness matrix to the center of stiffness.

Design sensitivities using high-order tetrahedral vector elements

Abstract: Obtaining design sensitivities for electromagnetic quantities by finite element analysis requires explicit expressions for the derivatives of the local stiffness matrix with respect to design parameters. By expressing the stiffness matrix for a vector tetrahedron in a certain way, the required derivatives are easily evaluated in terms of universal matrices. The method is validated by comparison with exact values or with derivatives obtained by a finite-difference formula.

Introductory Finite Element Method

Abstract: INTRODUCTION Basic Concept Process of Discretization Principles and Laws Cause and Effect Review Assignments STEPS IN THE FINITE ELEMENT METHOD Introduction General Idea Introduction to Variational Calculus Summary ONE-DIMENSIONAL STRESS DEFORMATION Introduction Explanation of Global and Local Coordinates Local and Global Coordinate System for the One-Dimensional Problem Interpolation Functions Relation Between Local and Global Coordinates Requirements for Approximation Functions Stress-Strain Relation Principle of Minimum Potential Energy Expansion of Terms Integration Direct Stiffness Method Boundary Conditions Strains and Stresses Formulation by Galerkin's Method Computer Implementation Other Procedures for Formulation Complementary Energy Approach Mixed Approach Bounds Advantages of the Finite Element Method ONE-DIMENSIONAL FLOW Theory and Formulation Problems Bibliography ONE-DIMENSIONAL TIME-DEPENDENT FLOW: Introduction to Uncoupled and Coupled Problems Uncoupled Case Time-Dependent Problems One-Dimensional Consolidation Computer Code FINITE ELEMENT CODES: ONE AND TWO-DIMENSIONAL PROBLEMS One-Dimensional Code Philosophy of Codes Stages Explanation of Major Symbols and Arrays User's Guide for Code DFT/C1DFE Two-Dimensional Code User's Guide for Plane-2D Sample Problems for Plane-2D User's Guide for Field-2D Sample Problems for Field-2D BEAM BENDING AND BEAM COLUMN Introduction Beam-Column Other Procedures for Formulation ONE-DIMENSIONAL MASS TRANSPORT Introduction Finite Element Formulation References Bibliography ONE-DIMENSIONAL STRESS WAVE PROPAGATION Introduction Finite Element Formulation Convection Parameter ux Bibliography TWO AND THREE DIMENSIONAL FORMULATIONS Introduction Two-Dimensional Formulation Three-Dimensional Formulation ONE-DIMENSIONAL STRESS WAVE PROPAGATION Introduction Finite Element Formulation Boundary and Initial Conditions Boundary Conditions TWO-AND THREE-DIMENSIONAL FORMULATIONS Introduction Two-Dimensional Formulation Triangular Element Quadrilateral Element Three-Dimensional Formulation Tetrahedron Element Brick Element TORSION Introduction Finite Element Formulation (Displacement Approach) Comparison of Numerical Predictions and Closed Form Solutions Stress Approach Review and Comments Hybrid Approach Mixed Approach Static Condensation OTHER FIELD PROBLEMS: POTENTIAL, THERMAL, FLUID, AND ELECTRICAL FLOW Introduction Potential Flow Finite Element Formulation Stream Function Formulation Thermal or Heat Flow Problem Seepage Electromagnetic Problems Computer Code TWO-DIMENSIONAL STRESS-DEFORMATION ANALYSIS Introduction Plane Deformations Finite Element Formulation Computer Code MULTICOMPONENT SYSTEMS: BUILDING FRAME AND FOUNDATION Introduction Various Components Computer Code Transformation of Coordinates APPENDIX 1: Various Numerical Procedures: Solution to Beam Bending Problem APPENDIX 2: Solution of Simultaneous Equations APPENDIX 3 Computer Codes Each chapter also contains sections of problems and references

Note on the normal form of a spatial stiffness matrix

Abstract: There has been some recent interest in the problem of designing compliance mechanisms with a given spatial stiffness matrix. A key result that has proven useful in the design of such mechanisms is Loncaric's normal form. When a spatial stiffness matrix is described in an appropriate coordinate frame, it will have a particularly simple structure. In this form the 3/spl times/3 off-diagonal blocks of the stiffness matrix are diagonal. It has been shown that generically, a spatial stiffness matrix can be written in normal form. For example, it is fairly well known that this is possible for any positive definite spatial stiffness matrix. In this article, it is shown that any symmetric positive semi-definite matrix can be written in normal form. As an application this result is used to design a compact parallel compliance mechanism with a prescribed positive semi-definite spatial stiffness matrix.

Stiffness Modeling of Reinforced Concrete Beam-Columns for Frame Analysis

Abstract: In defining member stiffness coefficients, simple distinctions between beams or columns can be misleading, especially for frames governed by earthquake or wind loads where column compression loads may be small relative to column strengths. In this paper, factors influencing beam-column stiffness in frame analysis are reviewed and simple formulas are proposed to determine effective flexural and shear stiffness coefficients of beam-columns as a function of the applied axial compression. The proposed stiffness coefficients represent conditions at incipient yield, and are applicable for linear analyses and the linear preyield region of nonlinear analyses. Proposed stiffness coefficients are compared to test data and alternative recommendations from several sources. More test data is needed for refinement of available models, particularly in the area of shear stiffness.

Pseudoforce method for nonlinear analysis and reanalysis of structural systems

Abstract: This paper develops a new solver to enhance the computational efficiency of finite-element pro- grams for the nonlinear analysis and reanalysis of structural systems. The proposed solver does not require the reassembly of the global stiffness matrix and can be easily implemented in present-day finite-element packages. It is particularly well suited to those situations where a limited number of members are changed at each step of an iterative optimization algorithm or reliability analysis. It is also applicable to a nonlinear analysis where the plastic zone spreads throughout the structure due to incremental loading. This solver is based on an extension of the Sherman-Morrison-Woodburg formula and is applicable to a variety of structural systems including 2D and 3D trusses, frames, grids, plates, and shells. The solver defines the response of the modified structure as the difference between the response of the original structure to a set of applied loads and the response of the original structure to a set of pseudoforces. The proposed algorithm requires O(mn) operations, as compared with traditional solvers that need O(m 2 n) operations, where m = bandwidth of the global stiffness matrix and n = number of degrees of freedom. Thus, the pseudoforce method provides a dramatic improvement of computational efficiency for structural redesign and optimization problems, since it can perform a nonlinear incremental analysis no harder than the inversion of the global stiffness matrix. The proposed method's efficiency and accuracy are demonstrated in this paper through the nonlinear analysis of an example bridge and a frame redesign problem.

Dynamic Analysis of Rotating Beams with Nonuniform Cross Sections Using the Dynamic Stiffness Method

Abstract: This study develops dynamic analysis based on the dynamic stiffness method for a rotating beam of nonuniform cross-section. To deal with nonuniform beams, coefficients related to material and geometric properties in the equation of motion are expressed in a polynomial form. A dynamic stiffness matrix is accordingly formulated in terms of power series. The dynamic response of the rotating beam is calculated by performing modal analysis. It is demonstrated that the present method provides an alternative to the finite element method in dealing with nonuniform rotating beams.

Customizing the mass and geometric stiffness of plane beam elements by Fourier methods

Abstract: Teaches by example the application of finite element templates in constructing high performance elements. The example discusses the improvement of the mass and geometric stiffness matrices of a Bernoulli‐Euler plane beam. This process interweaves classical techniques (Fourier analysis and weighted orthogonal polynomials) with newer tools (finite element templates and computer algebra systems). Templates are parameterized algebraic forms that uniquely characterize an element population by a “genetic signature” defined by the set of free parameters. Specific elements are obtained by assigning numeric values to the parameters. This freedom of choice can be used to design “custom” elements. For this example weighted orthogonal polynomials are used to construct templates for the beam material stiffness, geometric stiffness and mass matrices. Fourier analysis carried out through symbolic computation searches for template signatures of mass and geometric stiffness that deliver matrices with desirable properties when used in conjunction with the well‐known Hermitian beam material stiffness. For mass‐stiffness combinations, three objectives are noted: high accuracy for vibration analysis, wide separation of acoustic and optical branches, and low sensitivity to mesh distortion and boundary conditions. Only the first objective is examined in detail.

A Classification of Spatial Stiffness Based on the Degree of Translational–Rotational Coupling

Abstract: Previously, we have shown that, to realize an arbitrary spatial stiffness matrix, spring components that couple the translational and rotational behavior along/about an axis are required. We showed that, three such coupled components and three uncoupled components are sufficient to realize any full-rank spatial stiffness matrix and that, for some spatial stiffness matrices, three coupled components are necessary. In this paper, we show how to identify the minimum number of components that provide the translational-rotational coupling required to realize an arbitrarily specified spatial stiffness matrix. We establish a classification of spatial stiffness matrices based on this number which we refer to as the degree of translational-rotational coupling (DTRC). We show that the DTRC of a stiffness matrix is uniquely determined by the spatial stiffness mapping and is obtained by evaluating the eigenstiffnesses of the spatial stiffness matrix. The topological properties of each class are identified. In addition, the relationships between the DTRC and other properties identified in previous investigations of spatial stiffness behavior are discussed.

A Flexible 2-Level Neumann-Neumann Method for Structural Analysis Problems

Abstract: We discuss a new approach for the construction of the second-level Neumann-Neumann coarse space. Our method is based on an inexpensive and parallel analysis of the lower part spectrum of each subdomain stiffness matrix. We show that the method is flexible enough to converge fast on nonstandard decompositions and various types of finite elements used in structural analysis packages.

Aggregation Multilevel Iterative Solver for Analysis of Large-Scale Finite Element Problems of Structural Mechanics: Linear Statics and Natural Vibrations

Abstract: The preconditioned conjugate gradient method with aggregation multilevel preconditioning is employed for solution of large-scale finite element linear static and natural vibration problems. The aggregation approach, proposed by Prof. V.E.Bulgakov, is applied to create the aggregation multilevel preconditioning. Formulation of the restriction-prolongation operators and preparation of the coarse level matrix is based on clear mechanical interpretation. Application of the element-by-element (EBE) procedure makes evaluation of the coarse level matrix almost as fast as the assembly of the global stiffness matrix. Use of the shift improves properties of the aggregation multilevel preconditioning for solution of natural vibration problems. The PCG algorithm with acceleration by shifts is developed further on. Next, the implicit approach to solution of the additional equation set with the preconditioning ”shifted” respectively is applied. There is a correlational relationship between the well-known inverse iteration procedure with the shift and the PCG method accelerated by shifts. Efficiency of the approach proposed is illustrated by large-scale examples. These methods are incorporated in the Robot Millennium software (RoboBAT Software Company, Krakow, Poland, http://robot-structures.com/us/).