scholarly journals Use of instrumental variables methods in trajectory measurements’ problems

2021 ◽  
Vol 2131 (5) ◽  
pp. 052044
Author(s):  
V Krylov ◽  
A Tolstikov
Keyword(s):  
Numerical Experiment ◽  
Information Content ◽  
Instrumental Variables ◽  
Geometric Constraints ◽  
Base Stations ◽  
Algebraic Equations ◽  
System Matrix ◽  
The Matrix ◽  
Order Of Magnitude ◽  
Basic Functions

Abstract Possibilities of theinstrumental variables method using to improve the matrix properties of algebraic equations system are shown in the article. The problem of trajectory measurements from a network of ground-based, no-demand measuring stations using GLONASS navigation satellites is considered. The need to increase information content of trajectory measurements is especially important for trajectory measurements in the southern hemisphere of the Earth with a small number of measuring stations. The numerical experiment results with the simulation modeling use of trajectory measurements of the navigation satellite movement from two and three request-free measuring stations with geometric constraints are presented. The possibility of the measurements’ information content increasing from such base stations using the instrumental variables method is shown. Comparison of an indicator of the trajectory measurements informativenessis made according to the degree of the matrix conditionality of the algebraic equations system being solved. Based on the numerical experiment results, it isconsidered thepossibility to increase the information content of trajectory measurements from a network of non-query measuring stations with geometric constraints, with the instrumental variables involvement in the method processing. A quantitative estimate of the increase in information content was obtained in the form of a decrease in the condition number of the system matrix being solved by more than one order of magnitude. In the future, the research is predicted on the possibility of the trajectory measurements informativeness increasing for various basic functions used as instrumental variables, and their effect on the geometric constraints reducing an uneven network of measuring stations.

scholarly journals IMPROVING THE INFORMATIVITY OF TRAJECTORY MEASUREMENTS FOR GLONASS EPHEMERIS AND BALLISTIC SUPPORT

2021 ◽  
Vol 8 ◽  
pp. 192-198
Author(s):  
Vladimir S. Krylov ◽  
Alexander S. Tolstikov
Keyword(s):  
Numerical Experiment ◽  
Information Content ◽  
Instrumental Variables ◽  
Quantitative Estimation ◽  
Geometric Constraints ◽  
Base Stations ◽  
Algebraic Equations ◽  
System Matrix ◽  
The Matrix ◽  
Order Of Magnitude

Possibilities of the instrumental variables method using to improve the matrix properties of algebraic equations system are shown in the article. The problem of trajectory measurements from a network of ground-based, no-demand measuring stations using GLONASS navigation satellites is considered. The need to increase information content of trajectory measurements is especially important for trajectory measurements in the southern hemisphere of the Earth with a small number of measuring stations. The numerical experiment results of trajectory measurements of the navigation satellite movement from two and three request-free measuring stations with geometric constraints with use of simulation modeling are presented. The possibility of the measurements’ information content increasing from such base stations using the instrumental variables method is shown. Comparison of an indicator of the trajectory measurements informativity is made according to the degree of the matrix conditionality of the algebraic equations system being solved. Based on the results of the numerical experiment, it is concluded that it is possible to increase the information content of trajectory measurements from a network of non-query measuring stations with geometric constraints, involving the method of instrumental variables in processing. A quantitative estimation of the increase in information content was obtained in the form of a decrease in the condition number of the system matrix being solved by more than one order of magnitude. In the future, the research is predicted on the possibility of the trajectory measurements informativeness increasing for various basic functions used as instrumental variables, and their effect on the geometric constraints reducing an uneven network of measuring stations.

scholarly journals EXISTING APPROACHES TO DETERMINING THE MOTION PARAMETERS OF ARTIFICIAL EARTH SATELLITES

2020 ◽  
Vol 8 (2) ◽  
pp. 24-30
Author(s):  
Vladimir C. Krylov ◽  
Andrey C. Tomilov ◽  
Alexander C. Tolstikov
Keyword(s):  
Long Range ◽  
Instrumental Variables ◽  
Limited Area ◽  
Algebraic Equations ◽  
Indirect Measurements ◽  
Artificial Earth Satellites ◽  
Modern Methods ◽  
The Matrix ◽  
Artificial Earth ◽  
Motion Parameters

The article presents existing approaches to determining the motion parameters of artificial earth satellites. For ephemeris-ballistic support, this problem is reduced to solving a system of algebraic equations. In conditions of insufficient results of trajectory measurements from unquestioned measuring stations located on a limited area of the Earth's surface, and taking into account factors that affect the accuracy of pseudo-long-range measurements, the system of algebraic equations has an imperfectly defined right side. To improve the accuracy of trajectory measurements, it is proposed to use modern methods of processing the results of indirect measurements. The most effective method for improving the conditionality of the matrix of the specified system is the method of "instrumental variables".

scholarly journals Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

2021 ◽  
Vol 5 (1) ◽  
pp. 8
Author(s):  
Cundi Han ◽  
Yiming Chen ◽  
Da-Yan Liu ◽  
Driss Boutat
Keyword(s):  
Numerical Analysis ◽  
Fractional Order ◽  
Governing Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Matrix Product ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
The Matrix ◽  
The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

scholarly journals A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures

2021 ◽  
Vol 11 (12) ◽  
pp. 5570
Author(s):  
Binbin Wang ◽  
Jingze Liu ◽  
Zhifu Cao ◽  
Dahai Zhang ◽  
Dong Jiang
Keyword(s):  
Structural Characteristics ◽  
Complex Structure ◽  
Complex Structures ◽  
System Matrix ◽  
Substructure Method ◽  
Residual Structure ◽  
The Matrix ◽  
Multi Level ◽  
Fixed Interface ◽  
Selection Of

Based on the fixed interface component mode synthesis, a multiple and multi-level substructure method for the modeling of complex structures is proposed in this paper. Firstly, the residual structure is selected according to the structural characteristics of the assembled complex structure. Secondly, according to the assembly relationship, the parts assembled with the residual structure are divided into a group of substructures, which are named the first-level substructure, the parts assembled with the first-level substructure are divided into a second-level substructure, and consequently the multi-level substructure model is established. Next, the substructures are dynamically condensed and assembled on the boundary of the residual structure. Finally, the substructure system matrix, which is replicated from the matrix of repeated physical geometry, is obtained by preserving the main modes and the constrained modes and the system matrix of the last level of the substructure is assembled to the upper level of the substructure, one level up, until it is assembled in the residual structure. In this paper, an assembly structure with three panels and a gear box is adopted to verify the method by simulation and a rotor is used to experimentally verify the method. The results show that the proposed multiple and multi-level substructure modeling method is not unique to the selection of residual structures, and different classification methods do not affect the calculation accuracy. The selection of 50% external nodes can further improve the analysis efficiency while ensuring the calculation accuracy.

scholarly journals Approximation on Manifold

2021 ◽  
Vol 20 ◽  
pp. 62-73
Author(s):  
Yu.K. Dem’yanovich
Keyword(s):  
Multidimensional Case ◽  
Element Approximation ◽  
Algebraic Equations ◽  
Approximation Of Functions ◽  
Effective Evaluation ◽  
Linear Algebraic Equations ◽  
Basic Functions ◽  
Direct Approximation ◽  
Differential Manifold ◽  
Direct Use

The purpose of this work is to obtain an effective evaluation of the speed of convergence for multidimensional approximations of the functions define on the differential manifold. Two approaches to approximation of functions, which are given on the manifold, are considered. The firs approach is the direct use of the approximation relations for the discussed manifold. The second approach is related to using the atlas of the manifold to utilise a well-designed approximation apparatus on the plane (finit element approximation, etc.). The firs approach is characterized by the independent construction and direct solution of the approximation relations. In this case the approximation relations are considered as a system of linear algebraic equations (with respect to the unknowns basic functions ωj (ζ)). This approach is called direct approximation construction. In the second approach, an approximation on a manifold is induced by the approximations in tangent spaces, for example, the Courant or the Zlamal or the Argyris fla approximations. Here we discuss the Courant fla approximations. In complex cases (in the multidimensional case or for increased requirements of smoothness) the second approach is more convenient. Both approaches require no processes cutting the manifold into a finit number of parts and then gluing the approximations obtained on each of the mentioned parts. This paper contains two examples of Courant type approximations. These approximations illustrate the both approaches mentioned above.

scholarly journals A Simple Algorithm for Finding a Non-negative Basic Solution of a System of Linear Algebraic Equations

2021 ◽  
Vol 28 (3) ◽  
pp. 234-237
Author(s):  
Gleb D. Stepanov
Keyword(s):  
Simplex Method ◽  
Gaussian Elimination ◽  
Simple Algorithm ◽  
Basic Solution ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Linear Programming Problems ◽  
Two Stages ◽  
Computational Resources

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

Interply Behavior Exhibited in Compliant Filamentary Composite Laminates

1982 ◽  
Vol 55 (4) ◽  
pp. 1078-1094 ◽  
Author(s):  
J. L. Turner ◽  
J. L. Ford
Keyword(s):  
Shear Strain ◽  
Composite Laminates ◽  
Matrix Material ◽  
Matrix Composites ◽  
Efficient Manner ◽  
Strain Behavior ◽  
The Matrix ◽  
Order Of Magnitude ◽  
Extensional Strain ◽  
Interlaminar Shear

Abstract Cord-rubber composite systems allow a visualization of interply shear strain effects because of the compliant nature of the matrix material. A technique termed the pin test was developed to aid this visualization of interply shear strain. The pin test performed on both flat pads and radial tires shows that interlaminar shear strain behavior in both types of specimens is similar, most of the shear strain being confined to a region approximately 10 interly rubber thicknesses from the edge. The observed shear strain is approximately an order of magnitude greater than the applied extensional strain. A simplified mathematical model, called the Kelsey strip, for describing such behavior for a two-ply (±θ) cord-rubber strip has been formulated and demonstrated to be qualitatively correct. Furthermore, this model is capable of predicting trends in both compliant and rigid matrix composites and allows for simplified idealizations. A finite-element code for dealing with such interply effects in a simple but efficient manner predicts qualitatively correct results.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

A Linearized Two-Dimensional Theory for High-Speed Hydrofoils Near the Free Surface

1966 ◽  
Vol 10 (01) ◽  
pp. 25-48
Author(s):  
Richard P. Bernicker
Keyword(s):  
Direct Problem ◽  
High Speed ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Pressure Loading ◽  
Dimensional Theory ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sectional Shape ◽  
Infinite Set

A linearized two-dimensional theory is presented for high-speed hydrofoils near the free surface. The "direct" problem (hydrofoil shape specified) is attacked by replacing the actual foil with vortex and source sheets. The resulting integral equation for the strength of the singularity distribution is recast into an infinite set of linear algebraic equations relating the unknown constants in a Glauert-type vorticity expansion to the boundary condition on the foil. The solution is achieved using a matrix inversion technique and it is found that the matrix relating the known and unknown constants is a function of depth of submergence alone. Inversion of this matrix at each depth allows the vorticity constants to be calculated for any arbitrary foil section by matrix multiplication. The inverted matrices have been calculated for several depth-to-chord ratios and are presented herein. Several examples for specific camber and thickness distributions are given, and results indicate significant effects in the force characteristics at depths less than one chord. In particular, thickness effects cause a loss of lift at shallow submergences which may be an appreciable percentage of the total design lift. The second part treats the "indirect" problem of designing a hydrofoil sectional shape at a given depth to achieve a specified pressure loading. Similar to the "direct" problem treated in the first part, integral equations are derived for the camber and thickness functions by replacing the actual foil by vortex and source sheets. The solution is obtained by recasting these equations into an infinite set of linear algebraic equations relating the constants in a series expansion of the foil geometry to the known pressure boundary conditions. The matrix relating the known and unknown constants is, again, a function of the depth of submergence alone, and inversion techniques allow the sectional shape to be determined for arbitrary design pressure distributions. Several examples indicate the procedure and results are presented for the change in sectional shape for a given pressure loading as the depth of submergence of the foil is decreased.

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