In this article, we described the equations of motion a rigid solid by matrix formulation. The matrices contained in our movement description are homogeneous to same unit. Inertial characteristics met 4x4 positive definite symmetric called "tensor generalized Poinsot." This consists 3x3 "inertia tensor Poinsot", coordinates center mass multiplied total body and body. formulated as gender skew m...

A ®nite element model for investigating damage evolution in brittle matrix composites was developed. This modeling is based on an axisymmetric unit cell composed of a ®ber and its surrounding matrix. The unit cell was discretized into linearly elastic elements for the ®ber and the matrix and cohesive elements which allow cracking in the matrix, ®ber-matrix interface, and ®ber. The cohesive elem...

We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain certain results for a perturbat...

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G), a question which was previously answered when G is a tree. In this paper, a number of new techniques are ...

Decomposition of matrix is a vital part of many scientific and engineering applications. It is a technique that breaks down a square numeric matrix into two different square matrices and is a basis for efficiently solving a system of equations, which in turn is the basis for inverting a matrix. An inverting matrix is a part of many important algorithms. Matrix factorizations have wide applicati...

For a given linear matrix function A1−B1XB 1 , where X is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3. As applications, we give necessary and sufficient conditions for the triple matrix equat...

We give in this paper some closed-form formulas for the maximal and minimal values of the rank and inertia of the Hermitian matrix expression A − BX ± (BX)∗ with respect to a variable matrix X. As applications, we derive the extremal values of the ranks/inertias of the matrices X and X ± X∗, where X is a (Hermitian) solution to the matrix equation AXB = C, respectively, and give necessary and s...

This paper describes new factorization algorithms that exploit branch-induced sparsity in the joint-space inertia matrix (JSIM) of a kinematic tree. It also presents new formulae that show how the cost of calculating and factorizing the JSIM vary with the topology of the tree. These formulae show that the cost of calculating forward dynamics for a branched tree can be considerably less than the...