Showing papers on "Direct stiffness method published in 2002"

Stable recursive algorithm for elastic wave propagation in layered anisotropic media: stiffness matrix method.

Abstract: An efficient recursive algorithm, the stiffness matrix method, has been developed for wave propagation in multilayered generally anisotropic media. This algorithm has the computational efficiency and simplicity of the standard transfer matrix method and is unconditionally computationally stable for high frequency and layer thickness. In this algorithm, the stiffness (compliance) matrix is calculated for each layer and recursively applied to generate a stiffness (compliance) matrix for a layered system. Next, reflection and transmission coefficients are calculated for layered media bounded by liquid or solid semispaces. The results show that the method is stable for arbitrary number and thickness of layers and the computation time is proportional to the number of layers. It is shown both numerically and analytically that for a thick structure the solution approaches the solution for a semispace. This algorithm is easily adaptable to laminates with periodicity, such as multiangle lay-up composites. The repetition and symmetry of the unit cell are naturally incorporated in the recursive scheme. As an example the angle beam time domain pulse reflections from fluid-loaded multilayered composites have been computed and compared with experiment. Based on this method, characteristic equations for Lamb waves and Floquet waves in periodic media have also been determined.

Dramatically stiffer elastic composite materials due to a negative stiffness phase

Abstract: Composite materials of extremely high stiffness can be produced by employing one phase of negative stiffness. Negative stiffness entails a reversal of the usual codirectional relationship between force and displacement in deformed objects. Negative stiffness structures and materials are possible, but unstable by themselves. We argue here that composites made with a small volume fraction of negative stiffness inclusions can be stable and can have overall stiffness far higher than that of either constituent. This high composite stiffness is demonstrated via several exact solutions within linearized and also fully nonlinear elasticity, and via the overall modulus tensor estimate of a variational principle valid in this case. We provide an initial discussion of stability, and adduce experimental results which show extreme composite behavior in selected viscoelastic systems under sub-resonant sinusoidal load. Viscoelasticity is known to expand the space of stability in some cases. We have not yet proved that purely elastic composite materials of the types proposed and analyzed in this paper will be stable under static load. The concept of negative stiffness inclusions is buttressed by recent experimental studies illustrating related phenomena within the elasticity and viscoelasticity contexts.

Kinematic response functions and dynamic stiffnesses of bridge embankments

Abstract: Recognizing that soil-structure interaction affects appreciably the earthquake response of highway overcrossings, this paper compares approximate analytical solutions and finite element results to conclude on a simple procedure that allows for the estimation of the kinematic response functions and dynamic stiffnesses of approach embankments. It is shown that the shear-wedge model yields realistic estimates for the amplification functions of typical embankments and reveals the appropriate levels of dynamic strains which are subsequently used to estimate the stiffness and damping coefficients of embankments. The shear-wedge model is extended to a two-dimensional model in order to calculate the transverse static stiffness of an approach embankment loaded at one end. The formulation leads to a sound closed-form expression for the critical length, , that is the ratio of the transverse static stiffness of an approach embankment and the transverse static stiffness of a unit-width wedge. It is shown through two case studies that the transverse dynamic stiffness (“spring” and “dashpot”) of the approach embankment can be estimated with confidence by multiplying the dynamic stiffness of the unit-width wedge with the critical length, . The paper concludes that the values obtained for the transverse kinematic response function and dynamic stiffness can also be used with confidence to represent the longitudinal kinematic response function and dynamic stiffness respectively.

A Geometrical Approach to the Study of the Cartesian Stiffness Matrix

Abstract: The stiffness of a rigid body subject to conservative forces and moments is described by a tensor, whose components are best described by a 6X6 Cartesian stiffness matrix. We derive an expression that is independent of the parameterization of the motion of the rigid body using methods of differential geometry. The components of the tensor with respect to a basis of twists are given by evaluating the tensor on a pair of basis twists. We show that this tensor depends on the choice of an affine connection on the Lie group, SE (3). In addition, we show that the definition of the Cartesian stiffness matrix used in the literature [1,2] implicitly assumes an asymmetric connection and this results in an asymmetric stiffness matrix in a general loaded configuration. We prove that by choosing a symmetric connection we always obtain a symmetric Cartesian stiffness matrix. Finally, we derive stiffness matrices for different connections and illustrate the calculations using numerical examples.

Stiffness analysis of a Stewart platform-based parallel kinematic machine

Abstract: Stiffness is one of the important considerations in the design of Stewart platform based parallel kinematic machine (PKM). In most stiffness models of PKMs, the machine frames are thought of rigid bodies. In this paper, an approach is presented to establish the stiffness model of a Stewart platform-based PKM, considering the deformation of the frame. The deformation of legs and the frame is considered as kinematic parameter errors of Stewart platform. According to the differential error model, the machine structure is decomposed into two subsystems: the parallel links subsystem and the machine frame subsystem. The stiffness matrix of each subsystem is established respectively, assuming that the other one is a rigid structure. By linear superposition of the two subsystems, the stiffness model of the machine structure is obtained. A finite element analysis (FEA) model is used to simulate the physical structure. The FEA results are compared to those derived from a mathematical model, and the comparison shows the validity of this approach.

Formulation of Cracked Beam Element for Structural Analysis

Abstract: In this paper, a finite element for a cracked prismatic beam is developed. This element may be used in any matrix structural analysis. This paper details the derivation of the interpolation functions for a cracked Timoshenko beam finite element. These shape functions for rotational and translational displacements are also used to develop the consistent mass matrix for the cracked beam element. The crack effect on the stiffness matrix, as well as on the consistent mass matrix, is investigated and graphically represented.

Spatial stiffness realization with parallel springs using geometric parameters

Abstract: This paper investigates the synthesis of a spatial stiffness matrix using simple line springs. A new algorithm is developed, which enables the selection of constituent springs based on their positions and directions. The constraining space of the line springs is then investigated. It is shown that an isotropic stiffness matrix, in general, can be split into the sum of two rank-3 stiffness matrices. The three line springs of the first matrix can be selected to pass through any arbitrary points in space, while the three line springs of the second stiffness matrix lie on a quadric surface, which is usually a hyperboloid of one sheet.

Geometrical approach to the conservative congruence transformation (CCT) for robotic stiffness control

Abstract: In this paper, the conservative congruence transformation (CCT) for robot stiffness control is investigated by using geometrical methods. With the strategy of changing basis, it indicates that the formulation of stiffness matrix depends on the choice of coordinates. Thus, we show that the CCT can directly represent the spatial mapping relationship in robotic stiffness control. The CCT theory suggests a generalized transformation relationship in stiffness control and establishes the complete formulation of the 6/spl times/6 Cartesian stiffness matrix in the presence of external loads.

The buckling mode extracted from the LDLT‐decomposed large‐order stiffness matrix

Abstract: The present study proposes an innovated eigenanalysis-free idea to extract the buckling mode only from the LDLT-decomposed stiffness matrix in large-scale bifurcation analysis. The computational cost for extracting the critical eigenvector is negligible, because the decomposition of the stiffness matrix will continually be repeated during path-tracing to solve the stiffness equations. A numerical example is computed to illustrate that the proposed idea is tough enough even for multiple bifurcation. Copyright © 2002 John Wiley & Sons, Ltd.

Recursive Stiffness Matrix Method for Wave Propagation in Stratified Media

Abstract: A stable recursive stiffness matrix method is described for wave propagation in a layered elastic medium. The 4 × 4 global stiffness matrix for a stratified medium is recursively calculated from the layer stiffness matrices. The layer stiffness matrix has only six different elements, which are given in closed form. For general buried sources, a simple formulation has been developed to calculate the response for arbitrary positions of the receivers. Based on this method, characteristic equations for surface and guided modes in stratified media have also been obtained. The algorithm is unconditionally stable and retains the simplicity and flexibility of the transfer matrix method. Manuscript received 8 June 2001.

On the stiffness and stiffness control of redundant manipulators

Abstract: An analysis of the stiffness of redundant manipulators is undertaken in this paper. First, the matrix of the force-dependent stiffness is derived and its basic properties are analyzed. In particular, in the planar case the stability conditions for the force dependent stiffness (and gravity-dependent stiffness) are obtained in the analytical form. Next, dual properties of the stiffness and compliance are exploited to establish a decomposition of the joint stiffness and compliance in the form similar to the decomposition of the joint velocities and torques. Finally, a minimal, nonredundant parameterization of the joint stiffness and compliance is commented.

Stiffness control and transformation for robotic systems with coordinate and non-coordinate bases

Abstract: In this paper, the application of the conservative congruence transformation (CCT) to the stiffness mapping between non-coordinate basis and coordinate basis systems is studied and presented. Through the stiffness transformation between the 2 degree-of-freedom cylindrical and joint spaces, we illustrate that the CCT can be applied either directly or indirectly to the stiffness transformation between any two systems with either coordinate basis or noncoordinate basis. It is found that the same stiffness control for a conservative system will render a symmetric stiffness matrix with respect to a coordinate basis, but an asymmetric matrix with respect to a non-coordinate basis. The direct and indirect CCT methods are presented, with the latter requiring an intermediate coordinate system with a generalized coordinate basis. The relationships of the effective K/sub g/ matrices between the direct and indirect CCT methods are found and validated.

Dynamic stiffness matrix of a long rubber bush mounting

Abstract: The complete blocked dynamic stiffness matrix of along rubber bush mounting of particular interest for noise abatement is examined by an analytical model, where influences of audible frequencies, material properties, bush mounting length, and radius, are investigated The model is based on the dispersion relation for an infinite, thick-walled cylinder with arbitrary boundary conditions at the radial inner and outer surfaces; yielding the sought stiffness matrix, including axial, torsional, radial, and tilting stiffness A nearly incompressible material model is adopted, being elastic in dilatation while displaying viscoelasticity in deviation The applied deviatoric Mittag-Leffler relaxation function is based on a fractional standard linear solid, the main advantage being the minimum number of parameters required to successfully model the rubber properties over a broad structure-borne sound frequency domain The dynamic stiffness components display a strong frequency dependence at audible frequencies, resulting in acoustical resonance phenomena, such as stiffness peaks and troughs The low-frequency stiffness asymptotes of the presented model are shown to agree with those of static theories

Theory and Method for Structural Intelligent Health Diagnosis

Abstract: The paper systematically reviews and summarizes the problems and achievements, which are found in dynamic parameter based structural health diagnosis in the past two decades. Some popular structural health diagnosis methods which are model based and model free are expanded and improved. A series of novel structural health diagnosis formula are proposed. This paper discovers that ① and (0, where, and are the global stiffness matrix, modal eigenvalue, and modal shape of structure, respectively.② With respect of damaged structure, the event of emerging=0 and=0 at the same time is impossible. So it is certain that structural damage can be expressed completely by the change of . This discovery provides answers for the first time a longtime for the debate in the world whether the structural inner dynamical feature is sufficient to be the basis of structural damage recognition and localization. And our answer is positive.③ Strain type dynamic parameters are usually more sensitive to structural damages than displacement type. ④It is feasible to design a mixed type dynamic parameter based structure intelligent health diagnosis system. This system is based on the integration of the generalized genetic algorithm and artificial neural network, which use self learning function of artificial neural network and the global optimization capability of the generalized genetic algorithm. What is more, the knowledge bank is capable to realize self revolution.

Diagonal spatial stiffness matrices

Abstract: In this work we study in detail the conditions under which the stiffness matrix of a spatial system can be transformed into block-diagonal and diagonal form. That is the existence of a coordinate frame in which the stiffness matrix takes on these simple forms. The consequences of a block-diagonal or diagonal stiffness matrix for the invariants of the system, principal screws, von Mises' invariants and so forth, are also studied.

Wave propagation in the soil: theoretical background and application to traffic induced vibrations

Abstract: This paper reviews the direct stiffness method for the calculation of harmonic and transient wave propagation in horizontally layered media. Developed in the early eighties by Kausel and Roesset, this method has become a standard tool for the computation of the Green’s functions in layered media. As it is based on a superposition of plane harmonic waves in the frequency-wavenumber domain, it is very useful as a didactical tool for elastic wave propagation. The Green’s functions are subsequently used in a boundary element formulation to compute the dynamic impedance of the soil in a subdomain formulation for dynamic soil-structure interaction. This is illustrated for the case of road traffic induced vibrations, where the invariance of the problem domain in the longitudinal direction is exploited by working in the wavenumber domain. A numerical example demonstrates the influence of the soil stratification on free field vibrations during the passage of a two-axle truck on a traffic plateau.

Implementation of a new algorithm for evaluating 3D fracture analysis

Abstract: This paper presents a new method for calculating the elastic stress intensity factors for 3D structural components under complex loads and with complex geometry. To this end, a hybrid element formulation was developed where the hybrid element has its stiffness matrix corrected to exactly reflect the existence of the true 3D crack geometry within the element. To obtain the stiffness matrix of the hybrid element, this approach involves the combined use of the redundant force method together with the displacement field results arising from the finite element alternating technique (FEAT). This matrix was subsequently used together with a standard commercial finite element package to obtain the force field of the nodes. Finally, these forces were used to determine the stress intensity factors along the crack front. This procedure was validated by comparison with results available in the literature.

Elastic buckling analysis of 3-D trusses and frames with thin-walled members

Abstract: A finite element method is presented for the determination of the elastic buckling load of three-dimensional trusses and frames with rigid joints. The beam element stiffness matrix is constructed on the basis of the exact solution of the governing equations describing the coupled flexural-torsional buckling behaviour of a three-dimensional beam with an open thin-walled section in the framework of a small deformation theory. Large deformation effects are taken into account approximately through consideration of P−Δ effects. The structural stiffness matrix is obtained by an appropriate superposition of the various element stiffness matrices. The axial force distribution in the members is obtained iteratively for every value of the externally applied loading and the vanishing of the determinant of the structural stiffness matrix is the criterion used to numerically determine the elastic buckling load of the structure. The effect of initial member imperfections is also included in the formulation. Comparisons of accuracy and efficiency of the present exact finite element method against the conventional approximate finite element method are made. Cases where the axial force distribution determination can be done without iterations are also identified. The effect of neglecting the warping stiffness of some mono-symmetric sections is also investigated. Numerical examples involving simple and complex three-dimensional trusses and frames are presented to illustrate the method and demonstrate its merits.

Decomposition-Based Assembly Synthesis Based on Structural Stiffness Considerations

Abstract: This paper presents a method for systematically decomposes product geometry into a set of components considering the structural stiffness of the end product. A structure is represented a graph of its topology, and the optimal decomposition is obtained by combining FEM analyses with a Genetic Algorithm. As a case study, the side frame of a passenger car is decomposed for the minimum distortion of the front door panel geometry, where spot-welded joints are modeled as torsional springs. First, the rates of the torsional springs are treated as constant values obtained in the literature. Second, they are treated as design variables within realistic bounds. By allowing the change in the joint rates, it is demonstrated that the optimal decomposition can achieve the smaller distortion with less amount of joint stiffness (hence less welding spots), than the optimal decomposition with the typical joint rates available in the literature.

An efficient stable recursive algorithm for elastic wave propagation in layered anisotropic media

Abstract: An efficient recursive algorithm, the stiffness matrix method, has been developed for wave propagation in multilayered generally anisotropic media. This algorithm has the computational efficiency and simplicity of the standard transfer matrix method and is unconditionally computationally stable for high frequency and layer thickness. In this algorithm the stiffness (compliance) matrix is calculated for each layer and recursively applied to generate a stiffness (compliance) matrix for the layered system. Next, reflection and transmission coefficients are calculated for the layered media bounded by liquid or solid semispaces. Results show that the method is stable for arbitrary number and thickness of layers and the computation time is proportional to the number of layers.