Spatial Stiffness Matrix Research Articles

Vehicle-Bridge Coupling Vibration Analysis for Simply Supported Girders of High-Speed Railway Bridges Based on the Cross-Sectional Decentralized Centre of Mass and Shear

Taking the simply supported box girder bridge of high-speed railway as an example, the effect of cross-sectional decentralized centre of mass and shear on the spatial beam element stiffness matrix was theoretically derived. Based on the vehicle-bridge coupling vibration analysis method of the railway bridge, an analysis program of vehicle-bridge coupling vibration for the high-speed railway was compiled, and its reliability was verified through an example analysis. On this basis, considering the cross-sectional decentralized centre of mass and shear, the influence factors of vehicle-bridge coupling vibration response were studied, which included the offset distance of the beam section’s mass and shear centre, offset distance of track centreline, vehicle weight, and vehicle speed. The results show that the additional items of the spatial beam element stiffness matrix are generated by the torsion effect when the cross-sectional decentralized centre of mass and shear is considered, and it will affect the lateral and vertical stiffness of the element. The cross-sectional decentralized centre of mass and shear has a significant effect on the lateral dynamic response of the bridge’s mid-span, but the influence on the vertical response of the bridge and the dynamic response of the car body is small. The main influence factors of the lateral dynamic response of the bridge are the vertical offset distance of the beam section’s centre of mass and shear, the lateral offset distance of the track centreline, and the vehicle weight.

From shear centre to eigenwrenches

In the field of mechanism design, the behaviour of compliant shell mechanisms is commonly characterised via the eigenscrew decomposition of their spatial stiffness matrix. Recent developments have included the design of compliant shell mechanisms made with anisotropic materials. Conceptually, these compliant mechanisms are very similar to morphing composite structures that are typically designed using structural mechanics approaches. Eigenscrew decomposition could, therefore, provide additional insight to designers. To bridge the gap between the two communities, we present the equivalence of eigenscrews (eigenwrench and eigentwist) and familiar structural concepts such as shear centre and centre of twist for the special case of a cantilevered beam. It is hoped that this explicit link will help bridge these disparate fields, and encourage cross-fertilisation of ideas.

The best-approximate realization of a spatial stiffness matrix with simple springs connected in parallel

This paper presents an approximation algorithm to simplify the physical implementation of the non-isotropic stiffness matrices. The algorithm is inspired by the best approximation theorem in Matrix theory, by which the original stiffness matrix is projected onto a special linear space, thus an approximate isotropic matrix is derived. Accordingly, the desired force–deflection characteristics can be realized by purely simple springs (linear/torsional springs) connected in parallel. In mathematics, the approximation only changes the diagonal entries of the off-diagonal blocks of the original stiffness matrix. Moreover, the physical properties of the approximate isotropic matrix are identical to that of the original stiffness matrix in many aspects, including the center of stiffness, the wrench compliance axes as well as the translational and rotational stiffness value. The approximation is unique in both mathematics and physics. In order to illustrate the effectiveness of the proposed method, a numerical example is studied with comparison to some traditional algorithms of stiffness decomposition. The advantages of the approximation are well verified through the example.

The Principal Axes Decomposition of Spatial Stiffness Matrices

This paper presents an alternative decomposition of spatial stiffness matrices based on the concept of compliant axes. According to the congruence transformation of spatial stiffness, the coordinate-invariant aspects, which are referred to as the central principal components of the $6\times 6$ symmetric positive semidefinite matrices, can be derived uniquely. The proposed decomposition is free from the eigenvalue problems of the $6\times 6$ stiffness matrices so that both Plucker's ray and axis coordinates can be utilized to characterize the elastic system's force–deflection behavior. Hence, an arbitrary spatial stiffness matrix can be uniquely decomposed into two sets of orthogonal spring wrenches with finite and infinite pitches, respectively. The decomposed wrenches with finite pitches correspond to the stiffness’ wrench-compliant axes, along which linear deformations produce only wrenches parallel to them. As a result, three torsional and three screw springs are required, at the most, to realize a given spatial stiffness. Using the principal axes decomposition, some physical appreciations, such as the center of stiffness, the wrench-compliant axes, and the correspondence of compliance and stiffness, can be derived to reveal the inherent structure of spatial stiffness in an intuitive manner. In order to verify the effectiveness of the proposed method, two numerical examples are intensively studied with comparison to the eigenscrew decomposition. In addition, a potential application of the proposed stiffness decomposition method is also provided for the structural compliance modeling of flexible links in robot manipulators.

Surface Effect Preparatory Research of Vibrational Rotary Forging

As lack of a efficiency theoretical approximate solution for the surface effect mechanism analyzed during continuing and local vibrational plastic deformation process, according to vibrational rotary forging deforming characters, a three-dimension visco-elasticity plasticity constitutive equation was built by Kelvin model and Liewei-Mises model. After this constitutive equation was to be matrixing and to be introduced to tetrahedral solid elements, a visco-elasticity spatial stiffness matrix and a visco-plasticity spatial stiffness matrix were respectively derived. According to experimental parameters given, some simulation experimental projects were designed, a FEM models were established, and some FEM experiments were done on condition that amplitude and vibrational frequency were respectively changed, then different normal load-time curves were obtained. It is shown from experiment results that surface effect will appear during vibrational rotary forging forming process with little amplitude, the surface effect is related to the amplitude and the frequency, and this effect generation should be in special conditions, then preparatory research of surface effect mechanism of continuing and local vibrational plastic deforming process.

The conditions for mode decoupling of a linear vibration system with diagonalizable stiffness matrices via screw theory

In this article, a new geometric approach to the conditions for mode decoupling of an elastically suspended rigid body with a diagonalizable stiffness matrix is presented. The necessary and sufficient condition for diagonalization of a spatial stiffness matrix by a proper co-ordinate transformation is derived using the form of the stiffness matrix expressed by uniquely determined three orthogonal lines and three orthogonal torsional springs. Three orthogonal planes are defined by use of the axes of the line springs. Both the spatial displacements lying on or perpendicular to each of the planes and the spatial forces induced due to the displacements are decoupled into two linearly independent three-systems of screws. From this property and the geometric relations between those planes obtained from the stiffness and the principal planes of inertia, the conditions for mode decoupling of the vibration system are derived and the possible locations of the axes of the decoupled vibration modes are identified.

Hexahedral mesh smoothing using a direct method

This paper develops a generalisation of the mesh smoothing direct method proposed by Balendran, to three dimensions by applying its methodology to smoothed hexahedral meshes. A geometric criteria is used where irregular quadrilaterals are converted to regular quadrilaterals on each face of the hexahedron using a local stiffness matrix. This matrix is then converted to a spatial stiffness matrix with reference to absolute coordinates in order to assemble the geometrical stiffness matrix as a global stiffness matrix. The fixing of boundary conditions is analysed and results obtained in a series of hexahedral meshes, used in finite difference method analysis, are illustrated. To evaluate the quality of the obtained results, a degree of uniformity is calculated and a comparative study of the improvement obtained in each zone of the hexahedral mesh is conducted. The proposed method permits the marking out of the portions of mesh in which a Laplacian smoothening fails to improve the mesh quality, therefore, requiring the use of other methods in which inserted mesh points are taken into account.

Spatial stiffness realization with parallel springs using geometric parameters

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.

Compliance Planning For Dextrous Assembly Tasks Using Multi-Fingered Robot Hands

Abstract This paper deals with compliance planning for effective peg-in-hole and peg-outhole tasks using multi-fingered robot hands without inter-finger coupling. We first observe the fact that some of coupling stiffness elements of the spatial stiffness matrix cannot be planned arbitrary. Then, we analyze the conditions of the specified stiffness matrix in the operational space to successfully and more effectively achieve the given peg-in/out-hole tasks. It is clearly shown that the location of compliance center on the peg and the coupling stiffness element between the translational and the rotational direction play important roles for successful peg-in/out-hole tasks. Thus, the operational compliance characteristics should be carefully specified and the location of compliance center should be planned on-line for effective assembly and disassembly tasks. Simulation results are included to verify the feasibility of the analytic results.

THE GEOMETRICAL MODE AND FREQUENCY ANALYSES OF A VIBRATING SYSTEM WITH PLANES OF SYMMETRY

The geometrical mode and frequency analyses of a vibrating system can be performed via the theory of screw. From a screw theoretical standpoint, a vibration mode can be geometrically interpreted as a pure rotation about the center of vibration in a plane and as a twisting motion on a screw in a three-dimensional space. This paper presents a method to diagonalize a spatial stiffness matrix by use of a parallel axis congruence transformation when the stiffness matrix satisfies some conditions. It also describes that the diagonalized stiffness matrix can have the planes of symmetry depending on the location of the center of elasticity. For a system with the planes of symmetry, the vibration modes can be expressed by the axes of vibration. Analytical solutions for the axes of vibrations have been derived. The set of axes of vibrations yields the modal matrix and the response at the mass center is expressed by the reciprocal product between the axes of vibration and the applied wrench. A numerical example of an application to the vibrational analysis of an optical disc drive has been presented.

Minimal realizations of spatial stiffnesses with parallel or serial mechanisms having concurrent axes

AbstractThis article presents a new method for the synthesis of an arbitrary spatial elastic behavior with an elastic mechanism. The mechanisms considered are parallel and serial mechanisms with concurrent axes. We show that any full‐rank spatial stiffness matrix can be realized using a parallel mechanism with all spring axes intersecting at a unique point. It is shown that this intersection point must be the center of stiffness. We also show that any full‐rank spatial compliance matrix can be realized using a serial mechanism with all joint axes intersecting at a unique point. This point is shown to be the center of compliance. Synthesis procedures for mechanisms with these properties are provided. The realizations are shown to be minimal in the sense that both the number of screw components and the total number of components are minimum. © 2001 John Wiley & Sons, Inc.

Note on the normal form of a spatial stiffness matrix

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.

The eigenscrew decomposition of spatial stiffness matrices

A manipulator system is modeled as a kinematically unconstrained rigid body suspended by elastic devices. The structure of spatial stiffness is investigated by evaluating the stiffness matrix "primitives"-the rank-1 matrices that compose a spatial stiffness matrix. Although the decomposition of a rank-2 or higher stiffness matrix into the sum of rank-1 matrices is not unique, one property of the set of matrices is conserved. This property, defined as the stiffness-coupling index, identifies how the translational and rotational components of the stiffness are related. Here, we investigate the stiffness-coupling index of the rank-1 matrices that compose a spatial stiffness matrix. We develop a matrix decomposition that yields a set of rank-1 stiffness matrices that identifies the bounds on the stiffness-coupling index for any decomposition. This decomposition, referred to as the eigenscrew decomposition, is shown to be invariant in coordinate transformation. With this decomposition, we provide some physical insight into the behavior associated with a general spatial stiffness matrix.

The Duality in Spatial Stiffness and Compliance as Realized in Parallel and Serial Elastic Mechanisms

Spatial elastic behavior is characterized by a 6×6 positive definite matrix, the spatial stiffness matrix, or its inverse, the spatial compliance matrix. Previously, the structure of a spatial stiffness matrix and its realization using a parallel elastic system have been addressed. This paper extends those results to the analysis and realization of a spatial compliance matrix using a serial mechanism and identifies the duality in spatial stiffness and compliance associated with parallel and serial elastic mechanisms. We show that, a spatial compliance matrix can be decomposed into a set of rank-1 compliance matrices, each of which can be realized with an elastic joint in a serial mechanism. To realize a general spatial compliance, the serial mechanism must contain joints that couple the translational and rotational motion along/about an axis. The structure of a spatial compliance matrix can be uniquely interpreted by a 6-joint serial elastic mechanism whose geometry is obtained from the eigenscrew decomposition of the compliance matrix. The results obtained from the analysis of spatial compliant behavior and its realization in a serial mechanism are compared with those obtained for spatial stiffness behavior and its realization in a parallel mechanism.

Minimal realization of an arbitrary spatial stiffness matrix with a parallel connection of simple and complex springs

Presents a method for determining a minimal realization of an arbitrary spatial stiffness matrix K through the use of a mechanism constructed of a parallel connection of springs. The springs are of two types: simple and screw. The term simple spring refers to a purely translational or purely rotational passive spring, while a screw spring couples translational and rotational components. Any spatial stiffness matrix can be realized with a parallel connection of such springs. However, as the names suggest, simple springs are much easier to implement than screw springs. Thus, to reduce the compliance mechanism's complexity, it is desirable to minimize the number of screw springs. The article presents a method for determining minimal realizations for any symmetric positive definite or semidefinite spatial stiffness matrix. These realizations are minimal in the sense that they minimize both the number of screw springs and the total number of springs.

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

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.

Minimal realization of a spatial stiffness matrix with simple springs connected in parallel

This article presents a method for determining a minimal realization of an arbitrarily specified stiffness dominated spatial impedance through the use of a mechanism constructed of passive springs. This mechanism consists of simple springs connected in parallel where the term simple spring refers to a purely translational or purely rotational passive spring. Not every stiffness matrix K can be realized with a parallel connection of simple springs. The characterizing condition is that the upper right 3/spl times/3 submatrix of K has zero trace. Using this condition, the author shows how one can always synthesize any realizable spatial stiffness matrix with r parallel simple springs, where r is the rank of K. These results are also applicable to realizing a spatial damping matrix.

Achieving an Arbitrary Spatial Stiffness with Springs Connected in Parallel

In this paper, the synthesis of an arbitrary spatial stiffness matrix is addressed. We have previously shown that an arbitrary stiffness matrix cannot be achieved with conventional translational springs and rotational springs (simple springs) connected in parallel regardless of the number of springs used or the geometry of their connection. To achieve an arbitrary spatial stiffness matrix with springs connected in parallel, elastic devices that couple translational and rotational components are required. Devices having these characteristics are defined here as screw springs. The designs of two such devices are illustrated. We show that there exist some stiffness matrices that require 3 screw springs for their realization and that no more than 3 screw springs are required for the realization of full-rank spatial stiffness matrices. In addition, we present two procedures for the synthesis of an arbitrary spatial stiffness matrix. With one procedure, any rank-m positive semidefinite matrix is realized with m springs of which all may be screw springs. With the other procedure, any positive definite matrix is realized with 6 springs of which no more than 3 are screw springs.