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

1999 ◽  
Vol 123 (3) ◽  
pp. 353-358 ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Stiffness Matrix ◽  
Full Rank ◽  
Rotational Behavior ◽  
Topological Properties ◽  
Number Of Components ◽  
Stiffness Matrices ◽  
Minimum Number ◽  
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.

The Degree of Translational-Rotational Coupling of a Spatial Stiffness

1999 ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Stiffness Matrix ◽  
Full Rank ◽  
Rotational Behavior ◽  
Number Of Components ◽  
Stiffness Matrices ◽  
Minimum Number ◽  
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, 3 such coupled components and 3 uncoupled components are sufficient to realize any full-rank spatial stiffness matrix and that for some spatial stiffness matrices 3 coupled components are necessary. In this paper, we show how to identify the minimum number of components that provide translational-rotational coupling required to realize an arbitrarily specified spatial stiffness matrix. This number is defined 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. In addition, the relationships between the DTRC and other properties identified in previous investigations of spatial stiffness behavior are discussed.

Achieving an Arbitrary Spatial Stiffness with Springs Connected in Parallel

1998 ◽  
Vol 120 (4) ◽  
pp. 520-526 ◽  
Author(s):  
S. Huang ◽  
J. M. Schimmels
Keyword(s):  
Stiffness Matrix ◽  
Full Rank ◽  
Positive Definite Matrix ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
The Other ◽  
Positive Semidefinite Matrix ◽  
Stiffness Matrices ◽  
Rotational Components ◽  
Semidefinite Matrix

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.

scholarly journals Diagonal scaling of stiffness matrices in the Galerkin boundary element method

2000 ◽  
Vol 42 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mark Ainsworth ◽  
Bill McLean ◽  
Thanh Tran
Keyword(s):  
Integral Equation ◽  
Stiffness Matrix ◽  
Numerical Experiments ◽  
Degrees Of Freedom ◽  
Boundary Integral ◽  
Piecewise Constant ◽  
Stiffness Matrices ◽  
Diagonal Scaling ◽  
Mesh Ratio ◽  
Galerkin Boundary Element Method

AbstractA boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

Elastoplastic Analysis of Layered Metal Matrix Composite Cylinders—Part I: Theory

1996 ◽  
Vol 118 (1) ◽  
pp. 13-20 ◽  
Author(s):  
R. S. Salzar ◽  
M.-J. Pindera ◽  
F. W. Barton
Keyword(s):  
Metal Matrix Composite ◽  
Matrix Composite ◽  
Stiffness Matrix ◽  
Metal Matrix ◽  
Plastic Response ◽  
Solution Strategy ◽  
Composite Tubes ◽  
Elastic Plastic ◽  
Stiffness Matrices ◽  
Local Stiffness

An exact elastic-plastic analytical solution for an arbitrarily laminated metal matrix composite tube subjected to axisymmetric thermo-mechanical and torsional loading is presented. First, exact solutions for transversely isotropic and monoclinic (off-axis) elastoplastic cylindrical shells are developed which are then reformulated in terms of the interfacial displacements as the fundamental unknowns by constructing a local stiffness matrix for the shell. Assembly of the local stiffness matrices into a global stiffness matrix in a particular manner ensures satisfaction of interfacial traction and displacement continuity conditions, as well as the external boundary conditions. Due to the lack of a general macroscopic constitutive theory for the elastic-plastic response of unidirectional metal matrix composites, the micromechanics method of cells model is employed to calculate the effective elastic-plastic properties of the individual layers used in determining the elements of the local and thus global stiffness matrices. The resulting system of equations is then solved using Mendelson’s iterative method of successive elastic solutions developed for elastoplastic boundary-value problems. Part I of the paper outlines the aforementioned solution strategy. In Part II (Salzar et al., 1996) this solution strategy is first validated by comparison with available closed-form solutions as well as with results obtained using the finite-element approach. Subsequently, examples are presented that illustrate the utility of the developed solution methodology in predicting the elastic-plastic response of arbitrarily laminated metal matrix composite tubes. In particular, optimization of the response of composite tubes under internal pressure is considered through the use of functionally graded architectures.

Sign in / Sign up

Export Citation Format

Share Document