An Improved Method for Calculating Free Vibrations in Systems of Several Degrees of Freedom

1938 ◽  
Vol 5 (2) ◽  
pp. A61-A66
Author(s):  
Winston M. Dudley
Keyword(s):  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Matrix Theory ◽  
Free Vibrations ◽  
Initial Disturbance ◽  
Improved Method ◽  
Matrix Methods ◽  
Computing Machine ◽  
The Matrix ◽  
Time Required

Abstract In 1934 two English investigators (1) published a method for calculating the various modes and frequencies of vibration of a system having several degrees of freedom. Their method, which is based on matrices, greatly shortens the time spent in obtaining numerical solutions in many important problems, notably those with immovable foundations. In this paper is presented a new theorem which (a) makes possible a further reduction of nearly one half in the time required, so that solutions up to 20 deg or more of freedom are now practical and (b) makes it then possible to determine the motion of the system after any initial disturbance in a few minutes, instead of several hours as required by older methods. It is useful in the latter respect whether the modes have been determined by matrix methods, or not. Although the paper gives simpler proofs than any previously published, knowledge of the matrix theory is not required in using the method. Problems are analyzed by a tabular process, in which an ordinary computing machine helps greatly. Comments based on computing experience are given. A simple numerical example has been given elsewhere (1).

scholarly journals MENENTUKAN DETERMINAN SUATU MATRIKS DENGAN METODE CHIO

2016 ◽  
Vol 6 (1:) ◽  
pp. 33-40
Author(s):  
Asep Saepudin
Keyword(s):  
Mathematical Model ◽  
Matrix Element ◽  
Matrix Theory ◽  
Human Life ◽  
Equation System ◽  
Inverse Matrix ◽  
Matrix Methods ◽  
Linear Equation System ◽  
The Matrix ◽  
Mathematical Sciences

Matrix theory is a branch of linear algebra that discussed in the mathematical sciences. Mathematical sciences play an important role in human life, it is necessary to solve problems that can not be solved directly. Thus, the problem can be transformed into the form of a mathematical model. One is the SPL (Linear Equation System). Various methods can be used to solve it. But for the SPL with a large variable can be solved by matrix methods, namely the inverse matrix. In the inverse matrix of the determinants involved. If the search value that ordo major determinant of the matrix (𝑛×𝑛), it would require an effective method. One is the method of Chio. Chio method can be applied to all square matrixas long as the element is 𝑎11 not equal to zero (𝑎11≠0). Chio method of calculating the determinant of the matrix by decomposing determinant will look into sub-determinant of degree two (2×2) using the matrix element row 1 and column 1 as pointof departure. The decomposition is performed using the following sized matrix:   Keywords: Matrix, Matrix Determinant, Chio method.

scholarly journals Partial Component Consensus of Discrete-Time Multiagent Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Wenjun Hu ◽  
Gang Zhang ◽  
Zhongjun Ma ◽  
Binbin Wu
Keyword(s):  
Multiagent Systems ◽  
Discrete Time ◽  
Matrix Theory ◽  
Pinning Control ◽  
Directed Network ◽  
Strong Function ◽  
Partial Stability ◽  
Partial Component ◽  
The Matrix ◽  
Error System

The multiagent system has the advantages of simple structure, strong function, and cost saving, which has received wide attention from different fields. Consensus is the most basic problem in multiagent systems. In this paper, firstly, the problem of partial component consensus in the first-order linear discrete-time multiagent systems with the directed network topology is discussed. Via designing an appropriate pinning control protocol, the corresponding error system is analyzed by using the matrix theory and the partial stability theory. Secondly, a sufficient condition is given to realize partial component consensus in multiagent systems. Finally, the numerical simulations are given to illustrate the theoretical results.

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 MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

A NUMERICAL PROOF THAT CERTAIN CELLS FOLLOW THE TRAILS OF THE DIFFUSIONS OF SOME CHEMICALS IN THE EXTRACELLULAR MATRIX

2021 ◽  
pp. 2150027
Author(s):  
İREM ÇAY ◽  
SERDAL PAMUK
Keyword(s):  
Mathematical Model ◽  
Extracellular Matrix ◽  
Tumor Angiogenesis ◽  
Numerical Solutions ◽  
Extra Cellular Matrix ◽  
The Matrix ◽  
Good Agreement ◽  
Cellular Matrix

In this work, we obtain the numerical solutions of a 2D mathematical model of tumor angiogenesis originally presented in [Pamuk S, ÇAY İ, Sazci A, A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix, Math Biosci 306:32–48, 2018] to numerically prove that the certain cells, the endothelials (EC), pericytes (PC) and macrophages (MC) follow the trails of the diffusions of some chemicals in the extracellular matrix (ECM) which is, in fact, inhomogeneous. This leads to branching, the sprouting of a new neovessel from an existing vessel. Therefore, anastomosis occurs between these sprouts. In our figures we do see these branching and anastomosis, which show the fact that the cells diffuse according to the structure of the ECM. As a result, one sees that our results are in good agreement with the biological facts about the movements of certain cells in the Matrix.

scholarly journals Application of microbial risk assessment on a residentially-operated Bio-toilet

2006 ◽  
Vol 4 (4) ◽  
pp. 479-486 ◽  
Author(s):  
Naoko Nakagawa ◽  
Hana Oe ◽  
Masahiro Otaki ◽  
Katsuyoshi Ishizaki
Keyword(s):  
Risk Assessment ◽  
Water Content ◽  
Risk Level ◽  
Microbial Risk Assessment ◽  
Microbial Risk ◽  
Infectious Risk ◽  
The Matrix ◽  
Pathogenic Viruses ◽  
Scale Experiment ◽  
Time Required

The Sustainable Sanitation System is a new wastewater treatment system that incorporates a non-flushing toilet (Bio-toilet) that converts excreta into a reusable resource (as fertilizer or humus for organic agriculture) and reduces the pollution load to environments of the rivers, the lakes, and the sea. However, the risk of exposure to pathogens should be considered, because excrement is stored in the Bio-toilet. The aim of the present work is to analyze the health risk of dealing with the matrix (excreta and urine mixed with sawdust) of the Bio-toilet. Therefore, the fate of pathogenic viruses was investigated using coliphages as a virus index, and the modeling of the die-off rate in matrix was introduced. Then the microbial risk assessment was applied to a Bio-toilet that was actually used in a residential house; the infection risks of rotavirus and enterovirus as reference pathogens were calculated. According to the lab-scale experiment using coliphages for investing the die-off rate of viruses in the Bio-toilet, Qβ had a higher die-off, which was greatly influenced by the water content and temperature. On the other hand, T4 showed a lower rate and was independent of water content. Therefore, these two phages' data were used as critical examples, such as viruses having high or low possibilities of remaining in the Bio-toilet during the risk assessment analysis. As the result of the risk assessment, the storage time required for an acceptable infectious risk level has wide variations in both rotavirus and enterovirus cases depending on the phage that was used. These were 0–260 days' and 0–160 days' difference, respectively.

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