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1

Performance analysis of Gough Stewart platform with 6 limbs

Chandrashekhar A., Satish Babu G.

Mechanics and Mechanical Engineering

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2018

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Vol. 22, nr 1

133--141

EN

This paper concentrates on widespread study of parallel manipulator. It focuses on optimal designing of manipulator which has a large number of application fields. Optimal design is an important criterion to improve the accuracy of a robot. Through optimal design a robot can achieve isotropic configurations where the condition number of its jacobian matrix equals one. In this we are also concentrating on transmission index and stiffness index along with their plots, which can affect the kinetostatic performance of the robot. In this the singularity of Gough Stewart platform is also studied.

2

Improvement of the finite element method equations conditioning for the magnetic field-circuital problems

Gołębiowski M.

Archives of Electrical Engineering

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2017

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Vol. 66, nr 2

325--337

EN

The presented systems with magnetically coupled windings are solved with the finite element method. If the issue of voltage supply is analyzed, a system of linear equations with a partially skew-symmetric sparse matrix is obtained. Iterative methods used to solve a system of equations are particularly effective for symmetric matrices. Resultant equations can be reduced to this symmetrical form by using the method known from the literature [1]. The ratio of the maximum to the minimum eigenvalue of the main matrix of this circuit, which is the condition number, is however very high. This means that the problem is ill-conditioned and leads to a very long iterative solution process. The method presented in the article allows for a direct solution of a system of equations on its part, corresponding to high eigenvalues of the system matrix. The remaining part of the system of equations is solved by iterative methods. This part has much better condition number, and therefore the computational process is fast. The proposed iterative process depends on multiplication of a sparse matrix by vectors. It is not necessary (and possible) to store the entire matrix. This is especially important for larger sizes of a matrix.

3

Numerical solutions of symmetric saddle point problem by direct methods

Smoktunowicz A., Okulicka-Dłużewska F.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

437--451

EN

Numerical stability of two main direct methods for solving the symmetric saddle point problem are analyzed. The first one is a generalization of Golub’s method for the augmented system formulation (ASF) and uses the Householder QR decomposition. The second method is supported by the singular value decomposition (SVD). Numerical comparison of some direct methods are given.

4

Improvement of stability conditions and uniqueness of penalty approach in contact modeling in sheet metal forming

Bednarek T., Kowalczyk P., Marczewski A., Sosnowski W.

Computer Methods in Materials Science

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2011

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Vol. 11, No. 2

411-417

EN

The main objective of this paper is to improve stability conditions, uniqueness and convergence of the flow approach algorithm with viscoplastic and plastic material models. In this paper, the problem of convergence and uniqueness ofthe problem of non-linear simulation of sheet metal forming processes modeled using rigid-viscoplastic material model is considered. In the numerical simulation of the deformation process MFP2D and MFP3D Finite Element programs were used. The simplicity of the algorithm is the main advantage of these codes, the Direct Differentiation method and optimization modules can be implemented in the source code. The numerical instability caused by high values of the condition number of the main system of equations is the main disadvantage of the codes. The penalty approach contact model used in the program makes the stiffness matrix condition number worse.

PL

Celem niniejszej pracy jest poprawa warunków stabilności, jednoznaczności i zbieżność zadania lepkoplastycznego płynięcia materiału tłoczonej blachy poprzez modyfikację procedury kontaktu z zastosowaniem zmiennego współczynnika kary. W niniejszej pracy rozważany jest problem zbieżności i jednoznaczności rozwiązania nieliniowego problemu symulacji procesów tłoczenia blach modelowanych z wykorzystaniem sztywno-lepkoplastycznego modelu materiałowego. W symulacji numerycznej procesu deformacji zastosowano programy MFP2D i MFP3D Metody Elementów Skończonych. Zaletą kodów numerycznych MFP2D i MFP3D jest prostota zastosowanego algorytmu analizy, pozwalająca w konsekwencji na implementację modułów analizy wrażliwości metodami ścisłymi i optymalizacji procesu tłoczenia blach. Wadą przyjętego modelu materiałowego jest niestabilność numeryczna zadania, wywołana wysokim wskaźnikiem uwarunkowania układu równań. Dodatkowo model kontaktu zawierający funkcję kary pogarsza wskaźnik uwarunkowania macierzy sztywności.

5

An inverse structured perturbation problem for the linear system ATAx = b

Szularz M., Smoktunowicz A., Pawelec E.

Demonstratio Mathematica

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2010

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Vol. 43, nr 4

755-763

EN

The paper deals with the following inverse perturbation problem for the linear system ATAx = b: assuming that there exist two (possibly different) perturbations E1 and E2 of A so that (A + E2)T (A + E1)y = b, we ask whether there is a single perturbation F of A so that (A + F)T (A + F)y = b. We consider only small relative normwise perturbations of A. It is shown that if yT b >0 and (...) is small, then our problem has a solution. Some practical upper and lower error bounds for the structured backward error are also given.

6

Numerical stability of the Richardson second order method

Smoktunowicz A., Wróbel I.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

255--263

EN

In this paper we study numerical properties of the Richardson second order method (RS) for solving a linear system Ax = b, where A € Rnxn is infinitysymmetric and positive definite. We consider the standard model of floating point arithmetic (cf. [6], [7], [11]). We prove that the RS-algorithm is numerically stable. This means that the algorithm computes approximations xk to the exact solution x* = A-1b such that the error limfk||xk - x*ll2 ls of order eMcond(A), where eM is the machine precision and cond(A) = ||A || 2 ||A-1|| denotes the condition number of the matrix A.

7

Optimization and Pole Assignment in Control System Design

Chu E. K.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 5

1035-1053

EN

Some elementary optimization techniques, together with some not so well-known robustness measures and condition numbers, will be utilized in pole assignment. In particular, "Method 0" by Kautsky et al. (1985) for optimal selection of vectors is shown to be convergent to a local minimum, with respect to the condition number 1/2 ||X||_F^2 - ln |det X|. This contrasts with the misconception by Kautsky et al. that the method diverges, or the recent discovery by Yang and Tits (1995) that the method converges to stationary points.

8

Analysis of an Isope-Type Dual Algorithm for Optimizing Control and Nonlinear Optimization

Tadej W., Tatjewski P.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 2

429-457

EN

First results concerning important theoretical properties of the dual ISOPE (Integrated System Optimization and Parameter Estimation) algorithm are presented. The algorithm applies to on-line set-point optimization in control structures with uncertainty in process models and disturbance estimates, as well as to difficult nonlinear constrained optimization problems. Properties of the conditioned (dualized) set of problem constraints are investigated, showing its structure and feasibility properties important for applications. Convergence conditions for a simplified version of the algorithm are derived, indicating a practically important threshold value of the right-hand side of the conditioning constraint. Results of simulations are given confirming the theoretical results and illustrating properties of the algorithms.