Mathematical model of the system “manual vibration shock shaker – fruit branch”

Purpose. Development of mathematical model of oscillating system “manual vibration shock shaker – fruit branch” for the purpose of theoretical substantiation of the parameters of the shaker. Methods. The basic positions of mathematics, theoretical mechanics, mathematical modeling, program development and numerical calculations on the PC using methods of constructing mathematical models of functioning of agricultural machines are used. Results. The paper proposes a mathematical system model “manual vibration shock shaker – fruit branch” of six differential equations describing the motion of five separate masses (the mass of branch and four masses of individual shaker strings) and differential equations of the transverse and rotational motion of the system as whole. The mathematical system model determines the regularity of the motion of all masses, as well as the reactions of the viscals of the oscillatory system to the impact and after the impact that is generated in the shock mechanism. The proposed nonlinear, complex system of differential equations solves the numerical Runge-Kutta method of the fourth order of accuracy. On the basis of the calculated data the theoretical regularities of change of movement, speed and acceleration of a branch in the place of capture are received, which confirm that in the case of interaction of the cups of the shock mechanism there is blow that is accompanied by an increase in the acceleration of the branch, which is 4–5 times greater than the acceleration of the vibration mode of operation. Conclusions 1. The mathematical model of oscillating system “manual vibration shock shaker – fruit branch” is proposed in the form of system of six differential equations that allows to theoretically substantiate the basic modes of work of the manual shaker in the vibration shock mode to provide the agrotechnical necessary extraction completeness. 2. The received theoretical regularities of change of displacement, speed and acceleration of branch at the place of capture confirm the effectiveness of the vibration shock mode of the shaker. Due to the vibration-shock mode, the acceleration of the branch at the point of transmission of disturbing forces is 4–5 times higher than the acceleration of the vibrational operation mode. Keywords: manual shakes, vibration shocking process, oscillation oscillators, mathematical model, fruit branch, harvesting.

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Research of kinetics of change in fractional composition of wood raw materials being shredded

Известия СПбЛТА ◽

10.21266/2079-4304.2020.230.126-140 ◽

2020 ◽

Author(s):

Ю.Н. Власов ◽

Е.В. Нестерова ◽

Е.Г. Хитров

Keyword(s):

Experimental Data ◽

Mathematical Model ◽

Differential Equations ◽

Fractional Composition ◽

Raw Materials ◽

Calculated Data ◽

General Transformation ◽

Differential Model ◽

Integral Differential Equations ◽

Kinetics Of

В технике при изучении кинетики измельчения материалов пользуются математическими моделями в виде интегро-дифференциальных уравнений, решение которых трудоемко и не всегда приводит к получению наглядных результатов. Цель настоящей статьи разработать математическую модель, раскрывающую кинетику изменения фракционного состава измельчаемых древесных материалов, позволяющую на практике проводить оценку фракционного состава обрабатываемого сырья во времени. Методы исследования математический анализ, численные методы решения дифференциальных уравнений и обработки расчетных данных. Измельчение рассмотрено как многостадийный процесс, при котором фракции материала (узкие классы) под воздействием рабочего органа машины-измельчителя претерпевают превращения, происходящие как последовательно, так и параллельно, причем скорости превращений и доли вновь образованных узких классов материала определяются исходными размерами измельчаемых фракций и параметрами рабочего органа измельчителя. Предложена система дифференциальных уравнений, описывающая в общем превращения узких классов при измельчении, причем коэффициенты уравнений позволяют учесть произвольный вид функций скоростей измельчения фракций и выхода продуктов измельчения. Предложенная система является альтернативой интегро-дифференциальному уравнению балансовой модели измельчения. Выполнена оценка значений параметров математической модели на примере измельчения коры. По результатам сопоставления результатов моделирования с экспериментальными данными, полученными предыдущими исследователями, установлено, что предложенная дифференциальная модель изменения фракционного состава материала при принятых предпосылках к расчету ее параметров качественно и количественно описывает экспериментальных данные с высокой точностью. In techniques at study of kinetics of shredding of materials use mathematical models in the form of the integral-differential equations, which solution is laborious and not always leads to reception of evident results. The purpose of this article is to develop a mathematical model, which reveals the kinetics of change in fractional composition of wood materials being shredded, allowing in practice to evaluate the fractional composition of the processed raw materials in time. Methods of research include mathematical analysis, numerical methods for solving differential equations and processing of calculated data. Shredding is considered as multistage process at which fractions of a material (narrow classes) under the influence of a working body of the shredder machine undergo transformations occurring both consistently and in parallel, and rates of transformations and a share of again formed narrow classes of the material are defined by initial sizes of shredded fractions and parameters of the working body. The system of the differential equations describing in the general transformation of narrow classes at grinding is offered, and factors of the equations allow to consider any kind of functions of speeds of grinding of fractions and the output of shredding products. The proposed system is an alternative to the integral-differential equation of the balance shredding model. The estimation of values of parameters of the mathematical model on an example of bark shredding is carried out. By results of comparison of results of modeling with the experimental data received by previous researchers it is established that the offered differential model of change of fractional composition of the material at the accepted preconditions to calculation of its parameters qualitatively and quantitatively describes the experimental data with high accuracy.

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Determining the loose medium movement parameters in a centrifugal continuous mixer using a discrete element method

Eastern-European Journal of Enterprise Technologies ◽

10.15587/1729-4061.2021.232636 ◽

2021 ◽

Vol 3 (7 (111)) ◽

pp. 59-67

Author(s):

Volodymyr Statsenko ◽

Oleksandr Burmistenkov ◽

Tetiana Bila ◽

Svitlana Demishonkova

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Discrete Element Method ◽

Pearson Correlation ◽

Discrete Element ◽

System Of Differential Equations ◽

Continuous Action ◽

Centrifugal Mixer ◽

The Impact ◽

Element Method

The processes to form the compositions of loose materials in centrifugal mixers of continuous action have been considered. Based on the method of discrete elements, a mathematical model of the movement of particles in the rotor of the centrifugal mixer was built, taking into consideration their geometric and physical-mechanical parameters. To assess the extent of influence of these parameters on the nature of particle movement, a well-known mathematical model in the form of a system of differential equations was used, which was built on the basis of classical laws of mechanics. The process of mixing particles of two loose materials under different initial conditions of movement was modeled. The trajectories of individual particles along the bottom and side wall of the rotor were calculated. The results of the research reported here have established that the model built on the basis of the discrete element method makes it possible to improve the accuracy of determining the parameters of the movement of loose materials in the mixing zone. Calculations that involved this method show that the length of the particle trajectory is 2.9, and the movement time is 9 times greater than those calculated by the system of differential equations. The built and known mathematical models demonstrated the same nature of the distribution of components in the mixer. The value of the Pearson correlation coefficient between the calculated values of the coefficients of variation is 0.758. The best homogeneity is achieved by separating the flows of the mixture components and reducing the distance between their centers. The experimental study was carried out using a centrifugal mixer of continuous action with a conical rotor. Particle trajectories were constructed; it was established that the shape of the trajectory built by a discrete element method is closer to the experimental one. The results reported in this paper make it possible to predict the impact of the structural and technological parameters of the mixers of continuous action on the uniformity of the mixture

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Modelling Blood and Pulmonary Pressure for Solving a Performance Optimal Problem for Sportsmen

ISRN Applied Mathematics ◽

10.5402/2012/470143 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16

Author(s):

Jean Marie Ntaganda

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Comparative Study ◽

Model Parameters ◽

Stability Conditions ◽

Nonlinear Ordinary Differential Equations ◽

Respiratory Parameters ◽

Blood Pressures ◽

The Impact ◽

Found Model

This paper aims at designing a three-compartmental mathematical model for determining the impact and response of blood pressures on cardiovascular and respiratory parameters. Three nonlinear ordinary differential equations are derived from three compartments. Stability conditions are established and inverse techniques are proposed for identifying model parameters. To test the efficiency of the found model, a validation is achieved based on an existing mathematical model through a comparative study.

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A mathematical investigation into the uptake kinetics of nanoparticles in vitro

PLoS ONE ◽

10.1371/journal.pone.0254208 ◽

2021 ◽

Vol 16 (7) ◽

pp. e0254208

Author(s):

Hannah West ◽

Fiona Roberts ◽

Paul Sweeney ◽

Simon Walker-Samuel ◽

Joseph Leedale ◽

...

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Mixed Culture ◽

Uptake Kinetics ◽

Multiphase System ◽

Chemotherapy Drugs ◽

Culture Model ◽

The Impact ◽

Kinetics Of

Nanoparticles have the potential to increase the efficacy of anticancer drugs whilst reducing off-target side effects. However, there remain uncertainties regarding the cellular uptake kinetics of nanoparticles which could have implications for nanoparticle design and delivery. Polymersomes are nanoparticle candidates for cancer therapy which encapsulate chemotherapy drugs. Here we develop a mathematical model to simulate the uptake of polymersomes via endocytosis, a process by which polymersomes bind to the cell surface before becoming internalised by the cell where they then break down, releasing their contents which could include chemotherapy drugs. We focus on two in vitro configurations relevant to the testing and development of cancer therapies: a well-mixed culture model and a tumour spheroid setup. Our mathematical model of the well-mixed culture model comprises a set of coupled ordinary differential equations for the unbound and bound polymersomes and associated binding dynamics. Using a singular perturbation analysis we identify an optimal number of ligands on the polymersome surface which maximises internalised polymersomes and thus intracellular chemotherapy drug concentration. In our mathematical model of the spheroid, a multiphase system of partial differential equations is developed to describe the spatial and temporal distribution of bound and unbound polymersomes via advection and diffusion, alongside oxygen, tumour growth, cell proliferation and viability. Consistent with experimental observations, the model predicts the evolution of oxygen gradients leading to a necrotic core. We investigate the impact of two different internalisation functions on spheroid growth, a constant and a bond dependent function. It was found that the constant function yields faster uptake and therefore chemotherapy delivery. We also show how various parameters, such as spheroid permeability, lead to travelling wave or steady-state solutions.

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ВИКОРИСТАННЯ МОДЕЛІ НЕЛІНІЙНОЇ ДИНАМІКИ ДЛЯ АНАЛІЗУ ВПЛИВУ СИСТЕМИ УПРАВЛІННЯ ЗНАННЯМИ НА РЕАЛІЗАЦІЮ ВИСОКОТЕХНОЛОГІЧНИХ ПРОЕКТІВ

Aerospace technic and technology ◽

10.32620/aktt.2019.3.09 ◽

2019 ◽

pp. 76-82

Author(s):

Василь Михайлович Вартанян ◽

Дар'я Олександрівна Штейнбрехер

Keyword(s):

Nonlinear Dynamics ◽

Mathematical Model ◽

Knowledge Management ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Management System ◽

Knowledge Management System ◽

Project Implementation ◽

The Impact ◽

The Mathematical Model

The article determines that at the present stage of the project management development in information-oriented society, the decision-making process on the efficiency of the knowledge management system assessment is complicated, due to the lack of approaches that determine the impact of the system on the project implementation. The article presents the main results of the analysis of the current state of the problem of modeling the knowledge management system of high-tech projects, which helped to establish that one of the factors of the introduction of an effective system of knowledge preservation is the ability to assess the impact of the system on the project environment.In order to solve the problem, the mathematical model of nonlinear dynamics for the analysis of the influence of knowledge management system on the project based on the Bulirsch–Stoer method was proposed, it is possible to evaluate the influence of elements of the knowledge management system on the projected stages of the project implementation and to calculate the duration of the project taking into account their influence. The mathematical model of nonlinear dynamics for the analysis of the influence of the knowledge management system on the Bulirsch–Stoer method is given to evaluate the influence of elements of the knowledge management system on the projected implementation stages and to calculate the duration of the project, taking into account their impact. Bulirsch–Stoer algorithm is a method for the numerical solution of ordinary differential equations which combines three powerful ideas: Richardson extrapolation, the use of rational function extrapolation in Richardson-type applications, and the modified midpoint method, to obtain numerical solutions to ordinary differential equations (ODEs) with high accuracy and comparatively little computational effort. The step-by-step tracking of the knowledge management system impact on the project development will allow the project manager to predict both its successful completion and the risks of deviation from the scheduled time due to the loss of critical knowledge, which largely stems from the successful implementation of the project.Further research will be aimed at developing a model that allows us to assess the profitability of the system in the design environment, based on the results of the proposed mathematical model.

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Installation Characteristics of Variable Cycle Engine Based on Inlet Flow Matching

International Journal of Turbo and Jet Engines ◽

10.1515/tjj-2018-0031 ◽

2019 ◽

Vol 0 (0) ◽

Cited By ~ 2

Author(s):

Haoying Chen ◽

Haibo Zhang ◽

Yong Wang ◽

Qiangang Zhen

Keyword(s):

Mathematical Model ◽

Fuel Consumption ◽

Geometry Optimization ◽

Engine Performance ◽

System Model ◽

Variable Geometry ◽

Inlet System ◽

Supersonic Inlet ◽

Analysis Scheme ◽

The Impact

Abstract As per few investigation in installed performance for variable cycle engines, an analysis scheme is proposed on the basis of integrating variable cycle engine and supersonic inlet system model. An integrated mathematical model, containing the inlet and the variable cycle engine is built, realizing the simulation of influences on the installation performance by varying geometry components. The impact on engine performance of variable geometric regulation was analyzed and concluded respectively. The experimental results show that the overflow resistance of the variable cycle engine with variable geometry optimization is reduced at subsonic cruise stage, and the installed fuel consumption is reduced, which significantly improves the installation performance.

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Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines

Open Physics ◽

10.1515/phys-2019-0021 ◽

2019 ◽

Vol 17 (1) ◽

pp. 206-213

Author(s):

Gafar G. Ismayilov ◽

Vugar M. Fataliyev ◽

Elman Kh. Iskenderov

Keyword(s):

Experimental Data ◽

Mathematical Model ◽

Oil And Gas ◽

Flow Characteristics ◽

Calculated Data ◽

Hydrodynamic Condition ◽

Bubble Point ◽

Bubble Point Pressure ◽

Point Pressure ◽

The Impact

Abstract The conventional equations for describing the flow characteristics of the mixtures merely consider fluid that is homogenic, if it is above the bubble point conditions but ignore that a system containing sub-micron sized gas or vapor bubbles distributed throughout the volume of the liquid, which can exhibit unexpected heterogenic and complex phase properties. In this paper, a new mathematical model for the flowing gas-liquid mixture is presented, which has been proposed considering the colloidal feature of the system above the saturation or bubble point pressure. This approach is more in line with the actual dynamic performance of the oil and gas mixture export pipeline. Experimental data, simulations and field case studies validate the new proposed mathematical model of flow characteristics in pipeline. The obtained results confirmed that the calculated data are in good agreement with the experimental data. Based on Azerbaijan oil-gas-condensate field “Guneshli” data, this new model was used for calculating the condition in which the transformation of the flow characteristics from stable into instable is occurred. It has been discovered that the flow becomes unstable at a pressure about 30% higher than Bubble Point Pressure, which causes pulsation effect in the pipeline structure. However, homogenic behavior should be observed in this hydrodynamic condition. Also, the model provides a guideline on how to optimize the flow rate by adjusting the pipeline parameters to minimize the flow resistance, liquid slugging and hydraulic hammering effects, which cause instable operation.

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

Foods and raw materials ◽

10.21603/2308-4057-2018-2-483-492 ◽

2018 ◽

Vol 6 (2) ◽

pp. 483-492 ◽

Cited By ~ 2

Author(s):

Anatoly Khvostov ◽

Viktor Ryazhskikh ◽

Gazibeg Magomedov ◽

...

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Dynamic Models ◽

Plug Flow ◽

Calculated Data ◽

Technological Systems ◽

Technological Parameters ◽

Perfect Mixing ◽

The Mathematical Model ◽

Simulation Systems

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. A sugar syrup cooler was chosen as an object of modeling. The mathematical model of the cooler is formalized by a set of perfect reactors. The simulation results showed that the mathematical model adequately describes the main regularities of the process, the deviation of the calculated data from the regulations did not exceed 10%. 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.

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METHOD FOR DETERMINING ENERGY LOSSES IN THE TRANSMISSION OF VEHICLES

The collection of scientific works of the National Academy of the National Guard of Ukraine ◽

10.33405/2409-7470/2020/2/36/223551 ◽

2020 ◽

Vol 2 (36) ◽

pp. 35-43

Author(s):

I. Shasha ◽

Keyword(s):

Mathematical Model ◽

Fuel Consumption ◽

Physical And Mechanical Properties ◽

Oscillatory System ◽

Transmission Efficiency ◽

Operating Conditions ◽

Transmission Losses ◽

Armored Vehicles ◽

The Moment ◽

The Impact

Generally the transmission can be considered as a mechanical system formed by a set of rigid links, the position and speed of which are determined by the problem of the law of motion of the engine crankshaft. The transmission of the car consists of a large number of flywheels which are interconnected by shafts, clutches and other elastic elements with different angular stiffness. These assembly units form a rather complex torsional oscillatory system with scattered parameters. Loads acting on the transmission of the car are due to environmental influences (macro- and micro-profile of the bearing surface, physical and mechanical properties of the soil, etc.) and fluctuations of the car itself, and are random. To determine the actual loads in its elements and the degree of non-uniformity of their motion, as well as finding the laws of this motion, it is necessary to take into account their elasticity and the impact of these processes on fuel consumption. The method of determining losses in the transmission of the car on the basis of the energy approach taking into account the operating conditions of armored vehicles of the NGU is proposed. The method should be based on a mathematical model of fuel consumption as the main source of energy, which takes into account the variety of operating conditions and follows from the theory of the car. Mechanical losses in the engine can be expressed through the magnitude of the piston stroke and crankshaft speed. The average effective pressure is determined by the force lost in the transmission and the force applied to the wheels of the car. The total moment of resistance of transmission given to driving wheels is presented in the form of two components: the moment caused by friction losses in knots transferring loadings (gearboxes, a transfer case, reducers of leading axles, cardan transfers), and the moment taking into account expenses energy for oil spraying in units (hydraulic losses). The total transmission losses consist of two main components: idling losses and torque transmission losses. Losses in transmission separately on units are defined by means of the special stands equipped with braking and loading devices. An important advantage of the method is the ability to obtain the equation of engine power balance and determine the efficiency of the transmission as a system with distributed parameters. Further research should be aimed at improving the mathematical model of fuel consumption, taking into account the calculated values of the transmission efficiency depending on the operating conditions of armored vehicles of the NGU.

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The Impact of Light Polarisation on Light Field of Scenes with Multiple Reflections

Light & Engineering ◽

10.33383/2019-023 ◽

2020 ◽

pp. 108-115 ◽

Cited By ~ 1

Author(s):

Vladimir P. Budak ◽

Anton V. Grimaylo

Keyword(s):

Mathematical Model ◽

Light Field ◽

Global Illumination ◽

Multiple Reflections ◽

Software Environment ◽

The Monte Carlo Method ◽

Mueller Matrices ◽

Volume Integration ◽

The Impact

The article describes the role of polarisation in calculation of multiple reflections. A mathematical model of multiple reflections based on the Stokes vector for beam description and Mueller matrices for description of surface properties is presented. On the basis of this model, the global illumination equation is generalised for the polarisation case and is resolved into volume integration. This allows us to obtain an expression for the Monte Carlo method local estimates and to use them for evaluation of light distribution in the scene with consideration of polarisation. The obtained mathematical model was implemented in the software environment using the example of a scene with its surfaces having both diffuse and regular components of reflection. The results presented in the article show that the calculation difference may reach 30 % when polarisation is taken into consideration as compared to standard modelling.

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