Matrix dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics in Simulink

The dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics are based on the systems of algebraic and differential equations that describe a change in the basic technological parameters. The main difficulty in using such models in MathWorks Simulink™ computer simulation systems is the representation of ordinary differential equations (ODE) and partial differential equations (PDE) that describe the dynamics of a process as a MathWorks Simulink™ block set. The study was aimed at developing an approach to the synthesis of matrix dynamic models of elements of technological systems with perfect mixing and plug-flow hydrodynamics that allows for transition from PDE to an ODE system on the basis of matrix representation of discretization of coordinate derivatives. The process of synthesis of the dynamic matrix mathematical model was considered by the example of a sugar syrup cooler, the quality indicator of the finished product are selected as sucrose crystals and their portion in the total volume of caramel mass. Taking into account the dependence of syrup viscosity on temperature, thermal effects as a result of the process of crystallization of sucrose from syrup, design features of a typical caramel machine made it possible to clarify the dynamics of the process of syrup cooling. The model developed with this approach allows to obtain real-time estimates of temperatures at the outlet of the cooler, which makes it possible to study the dynamics of the technological process and synthesize the control system. The presented approach allows to implement mathematical models of ideal reactors in Simulink system and to move to matrix ordinary differential equations, which makes it possible to convert them into Simulink blocks. The approach is also applicable to other models of ideal reactors, which allows to form libraries of typical ideal reactors of Simulink system for synthesis of heat and mass exchange equipment. The proposed approach significantly simplifies the study and modernization of the current and the development of new technological equipment, as well as the synthesis of algorithms for controlling the processes therein.

Download Full-text

Glycemia monitoring

Nonlinear Analysis Modelling and Control ◽

10.15388/na.1998.2.0.15282 ◽

1998 ◽

Vol 2 ◽

pp. 23-30

Author(s):

Igor Basov ◽

Donatas Švitra

Keyword(s):

Diabetes Mellitus ◽

Mathematical Model ◽

Numerical Methods ◽

Differential Equations ◽

Blood Sugar ◽

Self Regulation ◽

Physiological System ◽

Non Linear ◽

The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

Download Full-text

Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere

Molecules ◽

10.3390/molecules25112694 ◽

2020 ◽

Vol 25 (11) ◽

pp. 2694

Author(s):

Amir Abbas ◽

Muhammad Ashraf ◽

Yu-Ming Chu ◽

Saqib Zia ◽

Ilyas Khan ◽

...

Keyword(s):

Mathematical Model ◽

Partial Differential Equations ◽

Differential Equations ◽

Mixed Convection ◽

Computational Study ◽

Convection Flow ◽

Mixed Convection Flow ◽

Partial Differential ◽

Primitive Form ◽

The Mathematical Model

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

Download Full-text

Study on the Dynamic Models of Systems Biology from the View of Engineering

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.220-223.952 ◽

2012 ◽

Vol 220-223 ◽

pp. 952-957

Author(s):

Chen Liu ◽

Xiao Yan Liu

Keyword(s):

Mathematical Model ◽

Systems Biology ◽

Dynamic Process ◽

Biological System ◽

Dynamic Models ◽

Model Parameters ◽

State Variables ◽

Biochemical Pathway ◽

The Right ◽

The Mathematical Model

From the view of engineering, based on expatiating the features of systems biology, the paper discusses the workflows and the research emphasis of systems biology. It also explains how to model and analyze the dynamic process of signal transmitting network for a biological system by an example. Based on the complexity and uncertainty of the mathematical model, the right methods are chosen to realize the effective estimation of state variables and model parameters for the biochemical pathway.

Download Full-text

Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

Bulletin of Kalashnikov ISTU ◽

10.22213/2413-1172-2021-4-17-24 ◽

2021 ◽

Vol 24 (4) ◽

pp. 17-24

Author(s):

V.M. Churkin ◽

T.Yu. Churkina ◽

A.M. Girin

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Solid Body ◽

Equations Of Motion ◽

Vertical Plane ◽

Spatial Motion ◽

Mathematical Task ◽

Spatial Movement ◽

Elastic Thread ◽

The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

Download Full-text

Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

Journal of Physics Conference Series ◽

10.1088/1742-6596/2092/1/012012 ◽

2021 ◽

Vol 2092 (1) ◽

pp. 012012

Author(s):

O Krivorotko ◽

D Andornaya

Keyword(s):

Mathematical Model ◽

Partial Differential Equations ◽

Differential Equations ◽

Sensitivity Matrix ◽

Partial Differential ◽

Identifiability Analysis ◽

Practical Identifiability ◽

Eigenvalue Method ◽

Simulation Results ◽

The Mathematical Model

Abstract A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

Download Full-text

Determination of technological parameters of soil compacting process

WAYS TO IMPROVE CONSTRUCTION EFFICIENCY ◽

10.32347/2707-501x.2020.45.3-15 ◽

2020 ◽

pp. 3-15

Author(s):

V. A. Basarab

Keyword(s):

Experimental Data ◽

Mathematical Model ◽

Differential Equations ◽

Fourier Analysis ◽

Soil Properties ◽

Soil Compaction ◽

Technological Parameters ◽

Compaction Process ◽

Properties Of Soil

The article is devoted to determine the technological parameters of soil compaction by rollers according to soil properties in construction. The compaction of the soil occurs due to certain force loads on the soil. Nowadays, there are enough methods to determine the technological parameters of soil compaction process, however, there are few research methods that consider the system of soil-machine-technological parameters. The urgency of the soil compaction process, which ultimately determines the quality of construction products, was emphasized. The basic physical and mechanical properties of soil and technological parameters of the compaction process were given. Analysis of discrete mathematical model of interaction between roller and soil was carried out in order to determine the technological parameters of soil compaction process in the field of construction technologies. A rheological model to describe the soil properties was used. The differential equations of the interaction between roller and soil have been made. A numerical calculation method to solve the differential equations has been proposed. The basic technological parameters of soil compaction process by rollers have been determined - the thickness of soil layer to be compacted, the number of the roller passes, and the speed of the roller movement. The methodology of theoretical and experimental researches of interaction between roller and soil has been proposed in order to clarify the mathematical model and to find the technological parameters of the compaction process. Experimental research methods are based on the determination of the stress-strain state of soil and on the Fourier analysis of the experimental data. Fourier analysis of the experimental data makes it possible to analyze the qualitative energy spectrum of the interaction between roller and soil to determine the rational amplitude-frequency characteristics of the compaction process. Experimental studies allow obtaining such rheological properties of soil as elasticity, viscosity, coefficient of elastic resistance, etc. The main recommendations for the choice of compaction method depending on the type of soil have been proposed.

Download Full-text

Mathematical Model of Forest Fire Soil-thrower Movement

KnE Life Sciences ◽

10.18502/kls.v5i1.6168 ◽

2020 ◽

Author(s):

Olga Dornyak ◽

Ivan Bartenev ◽

Mikhail Drapalyuk ◽

Dmitry Stupnikov ◽

Sergey Malyukov ◽

...

Keyword(s):

Mathematical Model ◽

Forest Fire ◽

Forest Fires ◽

Degrees Of Freedom ◽

Translational Motion ◽

Steady Motion ◽

Technological Parameters ◽

Lagrange Equations ◽

Milling Tool ◽

The Mathematical Model

The design of a forest fire soil-thrower made to prevent and eliminate ground forest fires is presented. A mathematical model of machine movement has been developed, which enables to study the laws of the interaction process of the design with the soil. It is accepted that the machine has two degrees of freedom. The mathematical model has been obtained using the Lagrange equations of the second kind. The design and technological parameters of the forest fire soil-throwing machine, affecting the efficiency of its work, including mass and width of the grip of the ripper casing, mass, radius and frequency of rotation of the milling tool, the number and geometric parameters of the blades are taken into account. Mathematical model enables to determine the effect of these parameters on the characteristics of the movement of ripper casing and milling working body. A mathematical model is needed to synchronize the translational motion of the unit and the rotational motion of the rotor. Formulas have been obtained for the steady motion of the forest fire soil-thrower, that determine the hauling power of tractor and torque that ensures the operation of milling tools.

Download Full-text

Mathematical Model with Feedback Loop in Company Control

SHS Web of Conferences ◽

10.1051/shsconf/202111401020 ◽

2021 ◽

Vol 114 ◽

pp. 01020

Author(s):

Arkadiy Plotnikov ◽

Tatyana Goryacheva ◽

Flyura Kazakova ◽

Ekaterina Zakharchenko

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Feedback Loop ◽

Structural Model ◽

Production Costs ◽

Operator Form ◽

Single Industry ◽

Dynamic Relationships ◽

The Mathematical Model ◽

Further Development

This article is devoted to the further development of feedback loop models. They are used in the management of a single-industry firm. The development of the studying consists in the mathematical modeling of transients and assessing their impact on economic indicators. To solve this problem, the following was done in the article. A structural model of the functioning of a single-industry firm (SIF) is presented. It is built in accordance with the theory of automatic control (TAC) and includes elements: a management unit, a production unit, a sales unit, information nodes, as well as a feedback loop (FL). Equations and relationships describing the logic of the functioning of the company as a production system are presented. They allowed us to derive dynamic relationships and differential equations that reflect feedback loops on revenue and production costs. The system of expressions in operator form is presented, which describes the contour of the FL SIF. It has the form of a system of differential equations. It forms the basis of the mathematical model of SIF in the control system. This model made it possible to obtain a graphical interpretation of transients with closed and open FL based on the use of the Mathcad editor. Transients are fluctuations in sales volumes and production costs in the presence of disturbing influences. This is the scientific result and determines the novelty of the article.

Download Full-text

THE NONLINEAR MODEL OF BEHAVIOR OF TWO COMPETITIVE FIRMS

BULLETIN OF CHERNIVTSI INSTITUTE OF TRADE AND ECONOMICS ◽

10.34025/2310-8185-2021-1.81.08 ◽

2021 ◽

Vol I (81) ◽

pp. 115-128

Author(s):

Bohdan Drin ◽

◽

Iryna Drin ◽

Svitlana Drin ◽

◽

...

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Mathematical Description ◽

Information Technologies ◽

Dynamic Models ◽

Nonlocal Problem ◽

Variable Coefficients ◽

Economic Behavior ◽

Algebraic Equations ◽

Mathematical Methods

The practical task of economics lies in applying the methods of substantiating its decisions. For economics, the main method is the modeling of economic phenomena and processes and, above all, mathematical modeling, which has been stipulated by the presence of stable MATHEMATICAL METHODS, MODELS AND INFORMATION TECHNOLOGIES IN ECONOMY Issue I (81), 2021 117 quantitative patterns and the possibility of a formalized description of many economic processes. The economic-mathematical model contains a system of equations of linear and nonlinear units that promote a mathematical description of economic processes and phenomena, consists of a set of variables and parameters and serves to study these processes and control them. Dynamic models of the economy describe it in development, as well as provide a detailed description of technological methods of production. Mathematical description of dynamic models is carried out with the use of a system of differential equations (in models with continuous time), difference equations (in models with discrete time), as well as systems of algebraic equations. It is important that the investigation of various economic issues has led to the development of the mathematical apparatus. In linear algebra, productive matrices are caused by the studies of intersectoral balance, whereas mathematical programming arose in the course of researching the optimal plan for the distribution of limited resources. In a similar way, there emerged the theory of economic indices and econometrics, the theory of production functions and the theory of consumption, the theory of general economic balance and social welfare, the theory of optimal economic growth. The paper under studies deals with the dynamic economic behavior of two competing objects, whose mathematical model is a nonlinear nonlocal problem for a system of ordinary differential equations with variable coefficients and argument deviation. The dynamic mathematical model is based on the assumption that the volume of output of both firms is determined by such factors on which output depends linearly. The model under discussion includes nonlinear factors, which describe the level of distrust of the competitors and depend on the time of observations and production volumes in previous moments, because the latter significantly affect the production activities of the firm. Such mathematical models are called time-delayed models.

Download Full-text