Mathematical modeling classification. Mathematical modeling

1. Economic and mathematical models are classified for different reasons.

By purpose, they are divided into:

Theoretical and analytical - in the study of general properties and patterns;

Applied - in solving specific economic problems (models of economic analysis, forecasting, management).

Economic and mathematical models can be used in the study of various aspects of production and its individual parts.

According to the substantive issues studied by economic processes, economic and mathematical models are divided into:

Models of production in general and its subsystems - industries, regions, etc.;

Complexes of models of production, consumption, formation and distribution of income, labor resources, pricing, financial relations, etc.

In accordance with the general classification of mathematical models, they are divided into:

functional;

Structural;

Structural and functional.

The use of structural models in research at the economic level is justified by the interconnection of subsystems. Typical in this case are models of intersectoral relations.

Functional models are widely used in the field of economic regulation. Typical in this case are models of consumer behavior in terms of commodity-money relations.

One and the same object can be presented in the form of both structural and functional models at the same time. So, for example, a structural model is used to plan a separate industry system, and a functional model is used at the economic level.

2. Differences between descriptive and normative models are revealed when considering their structure and nature of use.

Descriptive models provide an answer to the question: “How does this happen?” or “How is this most likely to develop further?”, that is, they explain the observed facts or predict the likelihood of any facts.

The purpose of the descriptive approach is the empirical identification of various dependencies in the economy. This may be the establishment of statistical patterns of economic behavior of social groups, the study of probable ways of development of any processes under unchanged conditions or without external influences, and other studies. An example here is a consumer demand model built on the basis of statistical data processing.

Normative models are recognized to answer the question: "How should it be?", that is, they imply purposeful activity. A typical example is the optimal planning model.

The economic-mathematical model can be both descriptive and normative. Thus, the intersectoral balance model is descriptive if it is used to analyze the proportions of the past period, and normative when calculating balanced options for the development of the economy.

3. Signs of descriptive and normative models are combined if the normative model of a complex structure combines separate blocks that are private descriptive models. Thus, the intersectoral model may include consumer demand functions that reflect the behavior of consumers when income changes.

The descriptive approach is widely used in simulation modeling.

By the nature of the discovery of cause-and-effect relationships, there are rigidly deterministic models and models that include elements of randomness and uncertainty. It is necessary to distinguish between uncertainty based on the law of probability theory and uncertainty that goes beyond the application of this law. The second type of uncertainty causes big problems in modeling.

4. According to the ways of reflecting the time factor, economic and mathematical models are divided into:

static;

Dynamic.

In static models, all the laws of the economy refer to one moment or period of time.

Dynamic models characterize changes over time.

According to the length of the time period, models of short-term (up to a year), medium-term (up to 5 years), long-term (5 years or more) forecasting and planning are distinguished. The flow of time in economic and mathematical models can change either continuously or discretely.

Models of economic phenomena differ in the form of mathematical dependencies.

The class of linear models is most convenient for analysis and calculations. But there are the following dependencies in the economy, which are non-linear:

Efficiency of resource use while increasing production;

Change in demand and consumption of the population with an increase in production;

Change in demand and consumption of the population with income growth, etc.

According to the ratio of exogenous and endogenous variables included in the model, they can be divided into open and closed.

A model must contain at least one endogenous variable, so there are no absolutely open models. Models that do not include exogenous variables (closed) are extremely rare - their construction requires complete abstraction from the "environment", that is, a serious coarsening of real economic systems that always have external connections.

Basically, the models differ in the degree of openness (closedness).

For business-level models, division into is important. aggregated and detailed.

Depending on whether economic models include spatial factors and conditions or do not include, spatial and point models are distinguished.

With the growth of achievements in economic and mathematical research, the problem of classifying the applied models becomes more complicated. Along with the emergence of new types of models (especially mixed types) and new grounds for their classification, the process of integrating models of different types into more complex model constructions is taking place.

Consider the concept: “Models. Classification of models ”from a scientific point of view.

Classification

Currently, there is a division of them into separate groups. Depending on the intended purpose, the following classification of economic and mathematical models is implied:

• theoretical and analytical types associated with studies of general characteristics and patterns;
• applied models aimed at solving certain economic problems. These include models of forecasting, economic analysis, management.

The classification of economic and mathematical models is also related to the scope of their practical application.

Depending on the content of the problem, such models are divided into groups:

• production models in general;
• individual options for regions, subsystems, industries;
• complexes of models of consumption, production, distribution and formation of labor resources, incomes, financial ties.

The classification of models of these groups implies the allocation of structural subsystems.

When conducting research at the economic level, structural models are explained by the relationship of individual subsystems. Models of intersectoral systems can be distinguished as common options.

Functional options are used for the economic regulation of commodity-money relations. One and the same object can be presented in the form of functional, structural forms at the same time.

The use of structural models in research at the economic level is justified by the interconnection of subsystems. Typical in this case are models of intersectoral relations.

Functional models are widely used in the field of economic regulation. Typical in this case are models of consumer behavior in terms of commodity-money relations.

Differences between models

Let's analyze different models. The classification of models currently used in the economy involves the allocation of normative and descriptive options. Using descriptive models, it is possible to explain the analyzed facts, to predict the possibility of the existence of certain facts.

The purpose of the descriptive campaign

It involves the empirical identification of various dependencies in the modern economy. For example, the statistical regularities of various social groups are established, the probable ways of development of certain processes are studied under constant conditions or without external influences. Based on the results obtained in the course of a sociological survey, it is possible to build a model of consumer demand.

Regulatory Models

With their help, it is possible to assume purposeful activity. An example is the optimal scheduling model.

It can be both normative and descriptive. If the model is used in the analysis of the proportions of the past period, it is descriptive. When calculating with its help the optimal ways of economic development, it is normative.

Model features

The classification of models involves taking into account individual functions that help clarify controversial issues. The descriptive approach has found its maximum distribution in simulation modeling.

Depending on the nature of the discovery of causal relationships, there is a classification of models into options, including individual elements of uncertainty and randomness, as well as rigidly deterministic models. It is important to distinguish between uncertainty, which is based on the theory of probability, and uncertainty, which goes beyond the limits of the law.

Division of models according to the ways of reflecting the time factor

It is supposed to classify models according to this factor into dynamic and static types. Static models involve consideration of all regularities in a certain period of time. Dynamic options are characterized by changes over time. Depending on the duration of use, the classification of models into the following options is allowed:

• short-term, the duration of which does not exceed one year;
• medium-term, calculated for a period of one to five years;
• long-term, calculated for a period of more than five years.

Depending on the specifics of the project, it is allowed to make changes in the process of using the model.

According to the form of mathematical dependencies

The basis for the classification of models is the form of mathematical dependencies chosen for work. They mainly use the class of linear models for calculations and analysis. Consider the economic types of models. The classification of models of this type helps to study the change in consumption and demand of the population in the event of an increase in their material income. In addition, with the help, changes in the needs of the population are analyzed in the event of an increase in production, and the effectiveness of the use of resources in a particular situation is assessed.

Depending on the ratio of endogenous and exogenous variables that are included in the model, the classification of models of these types into closed and open systems is applied.

Any model must include at least one endogenous variable, and therefore it is very problematic to find completely open systems. Models that do not include exogenous variables (closed variants) are also practically uncommon. In order to create such a variant, it is necessary to completely abstract from the environment, to allow serious coarsening of the real economic system that has external ties.

As the achievements of mathematical and economic research increase, the classification of models and systems becomes much more complicated. Currently, mixed types are used, as well as complex model designs. A unified classification of information models has not yet been established. At the same time, about ten parameters can be noted, according to which the types of models are aligned.

Model types

A monographic or verbal model involves a description of a process or phenomenon. Often we are talking about rules, a law, a theorem, or a combination of several parameters.

The graphic model is drawn up in the form of a drawing, a geographical map, a picture. For example, the relationship between consumer demand and product costs can be represented using coordinate axes. The graph clearly demonstrates the relationship between the two quantities.

Real or physical models are created for objects that do not yet exist in reality.

Degree of object aggregation

There is a classification of information models on this basis into:

• local, with the help of which the analysis and forecast of certain indicators of the development of the industry are carried out;
• on microeconomic, intended for a serious analysis of the structure of production;
• macroeconomic, based on the study of the economy.

There is also a separate classification of management models for macroeconomic types. They are divided into one-, two-, multi-sector options.

Depending on the purpose of creation and use, the following options are distinguished:

• deterministic, having uniquely understandable results;
• stochastic, which involve probabilistic outcomes.

In the modern economy, balance models are distinguished, which reflect the requirement of matching the resource base and their application. They are written in the form of square chess matrices.

There are also econometric types, for the evaluation of which methods of mathematical statistics are used. Such models express the development of the main indicators of the created economic system through a long trend (trend). They are in demand in the analysis and forecasting of certain economic situations associated with real statistical information.

Optimization models make it possible to choose the optimal variant of production, consumption or distribution of resources from a variety of alternative (possible) options. The use of limited resources in such a situation will be the most effective means to achieve the goal.

Assume the participation in the project not only of an expert, but also of specialized software, computers. The resulting expert database is intended to solve one or more tasks by simulating human activity.

Network models are a set of operations and events interconnected in time. Most often, such a model is intended for the implementation of work in such a sequence as to achieve the minimum time for the project.

Depending on the chosen type of mathematical apparatus, models are distinguished:

• matrix;
• correlation-regressive;
• network;
• inventory management;
• mass service.

Stages of economic and mathematical modeling

This process is purposeful, it is subject to a certain logical program of actions. Among the main stages of creating such a model are:

• formulation of the economic problem and its qualitative analysis;
• development of a mathematical model;
• preparation of initial information;
• numerical solution;
• analysis of the obtained results, their use.

When posing an economic problem, it is necessary to clearly formulate the essence of the problem, note the important features and parameters of the object being modeled, analyze the relationship of individual elements in order to explain the development and behavior of the object under consideration.

When developing a mathematical model, the relationship between equations, inequalities, and functions is revealed. First of all, the type of model is determined, the possibility of its application in a specific problem is analyzed, and a specific list of parameters and variables is formed. When considering complex objects, multidimensional models are built so that each characterizes individual aspects of the object.

Conclusion

Currently, there is no separate concept of the model. The classification of models is conditional, but this does not reduce their relevance.

Calculations of the stress-strain state of the beam

Performed by student gr. 6-Sm-1 Melnikov R.V.

Head Semenov A.A.

St. Petersburg

Introduction………………………………………………………………………………………………………….2

1. Classification of mathematical models…………………………………………………3

2. Ritz method………………………………………………………………………………………..……5

3. Calculations of the stress-strain state of the beam………….…….7

3.1. Calculation of a linear-elastic problem for a steel beam…………….…….7

3.2. Calculation of a nonlinear elastic problem for a steel beam……….….…..9

3.3. Calculation of a linear-elastic problem for a concrete beam………….…….12

3.4. Calculation of the creep problem for a concrete beam………………………….13

Conclusion………………………………………………………………………………..………………….15

List of used literature……………………………………………………………….16

Introduction

With the advent of electronic computers, a new method was developed for the theoretical study of complex processes, i.e. study of natural science problems by means of computational mathematics.

The essence of a computational experiment is to compile a mathematical model of the process or phenomenon under study, which is some mathematical equations, then a computational algorithm is developed to solve these equations, a computer program is compiled and specific options for the state of the object are calculated when the parameters included in the equation change. That. the basis for studying various objects is the construction of a mathematical model of their functioning.

The purpose of the course work is the development of mathematical models of deformation of elements of building structures, the construction of a methodology for studying the stress-strain state of steel and concrete beams.

Classification of mathematical models

A mathematical model is a mathematical representation of reality, one of the variants of a model as a system, the study of which allows obtaining information about some other system.

The process of building and studying mathematical models is called mathematical modeling.

Mathematical models can be classified according to several main features.

1. Static and dynamic models

A model is called static if the value of the output depends on the value of the input at the same time. In dynamic models, the output value may depend on the entire past input process. For dynamic models, the subject of study is the change in the object under study in time.

2. Deterministic and probabilistic models.

If the mathematical model includes random variables that obey statistical laws, then it is called probabilistic or stochastic. A mathematical model that does not contain random components is called deterministic.

3. Discrete and continuous models.

Values ​​can be of two types - discrete, i.e., taking individual values, allowing natural numbering, and continuous, taking all values ​​from a certain interval. A mixed case is also possible, for example, when a value behaves as a discrete one on one interval, and continuous on another. Similarly, mathematical models can be either discrete, or continuous, or mixed. It is necessary to take into account the possibility of using either discrete or continuous devices in the construction of a mathematical model and the method of its study.

4. Linear and non-linear models.

The linear dependence of one quantity on another is the proportionality of their increments, i.e. the dependence of the form y=ax+b, from which we get △y=a△x. Similarly, the concept of a linear model is also defined. If the model is considered as a converter, for which each input corresponds to some output. Then the model is called linear if the principle of superposition is satisfied in it, i.e. when adding the inputs, the outputs are added, when multiplying the input by any number, the output is multiplied by the same number. Using the principle of superposition, it is not difficult, having found a solution in any case, to construct a solution in a more general situation. Therefore, the qualitative properties of the general case can be judged by the properties of the particular one - the difference between the two solutions is only quantitative.

One of the most important properties of mathematical models is their universality. Its essence lies in the fact that the same mathematical models can describe processes that are completely different in nature, i.e. the same techniques and methods for constructing and studying mathematical models are suitable for various problems.

However, the solution of such problems requires the integration of a complex system of partial differential equations and is associated with significant mathematical difficulties. Therefore, when solving a direct problem, approximate methods are often used, for example, direct methods of variational problems (the Ritz method), as well as the finite element method.

Ritz Method

The Ritz method is a direct method for finding an approximate solution to boundary value problems in the calculus of variations.

The method provides for the choice of a test function, which should minimize a certain functional, in the form of superpositions of known functions that satisfy the boundary conditions. In this case, the problem is reduced to finding unknown superposition coefficients. The spatial operator in the operator equation that describes the boundary value problem must be linear, symmetric, and positive definite.

The Ritz method allows one to find unknown displacement functions from the condition of the minimum of the total strain energy functional.

Consider the energy functional:

It is required to find the minimum of the functional (3.1), i.e., to find the functions u(x, y), v(x, y) , w(x, y) , given in some area D= {0 ≤ x≤ a; 0 ≤ y≤ b) satisfying some homogeneous boundary conditions on the boundary Γ , under which functional (1) has a minimum value. We will look for an approximate solution of the problem in the form:

u(x,y)=u N = ,

v(x,y)=v N = ,

w(x,y)=w N= .

To avoid two indices, we represent the moves as:

Here U(I), V(I), W(I) are unknown numerical parameters; X 1(I), X 2(I), X 3(I) are known approximating functions of the variable x, satisfying at x= 0, x= a given boundary conditions; Y 1(I), Y 2(I), Y 3(I) are known approximating functions of the variable y, satisfying at y= 0, y= b given boundary conditions. Functions X 1(I) − X 3(I) , Y 1(I) − Y 3(I) are called basis functions.

Substituting (2) into (1) and integrating from known functions, we reduce functional (1) to the function:

J=J(U(I),V(I),W(I)) (3)

parameters U(I), V(I), W(I), I=1,…,N.

For the function (3.3) to have a minimum, its partial derivatives with respect to the variables U(l),V(l),W(l), l=1,.., N must vanish:

System (4) is a system of linear algebraic equations, which can be solved using the Gauss method. Found parameter values U(I), V(I), W(I) are substituted into expansions (2) and we obtain an approximate solution of the problem posed. The existence of a minimum of the functionals of the total strain energy of the elements of building structures (rod, plate, shell) has been proven.

A mathematical model is a simplification of a real situation and is an abstract, formally described object, the study of which is possible by various mathematical methods.

Consider classification of mathematical models.

Mathematical models are divided into:

1. Depending on the nature of the displayed object properties:

· functional;

· structural.

Functional mathematical models designed to display information, physical, temporal processes occurring in operating equipment, in the course of technological processes, etc.

Thus, functional models- display the processes of the object functioning. They usually take the form of a system of equations.

Structural models- can take the form of matrices, graphs, lists of vectors and express the relative position of elements in space. These models are usually used in cases where the problems of structural synthesis can be set and solved, abstracting from the physical processes in the object. They reflect the structural properties of the designed object.

To obtain a static representation of the modeled object, a group of methods can be used, called schematic models - these are methods of analysis, including a graphical representation of the operation of the system. For example, flow charts, diagrams, multifunctional operation diagrams, and flowcharts.

2. According to the methods of obtaining functional mathematical models:

· theoretical;

· formal;

· empirical.

Theoretical obtained on the basis of the study of physical laws. The structure of the equations and the parameters of the models have a certain physical interpretation.

Formal are obtained based on the manifestation of the properties of the modeled object in the external environment, i.e. considering the object as a cybernetic "black box".

The theoretical approach makes it possible to obtain more universal models, valid for wider ranges of external parameters.

Formal - are more accurate at the point in the parameter space at which the measurements were taken.

Empirical mathematical models are created as a result of experiments (studying the external manifestations of the properties of an object by measuring its parameters at the input and output) and processing their results using mathematical statistics methods.

3. Depending on the linearity and nonlinearity of the equations:

· linear;

· non-linear.

4. Depending on the set of domains and values ​​of model variables, there are:

· continuous

· discrete (domains of definition and values ​​are continuous);

· continuous-discrete (the domain of definition is continuous, and the domain of values ​​is discrete). These models are sometimes called quantized;

· discrete-continuous (the domain of definition is discrete, and the domain of values ​​is continuous). These models are called discrete;

· digital (domains of definition and values ​​are discrete)

5. According to the form of links between output, internal and external parameters:

· algorithmic;

· analytical;

· numerical.

algorithmic are called models presented in the form of algorithms that describe a sequence of unambiguously interpreted operations performed to obtain the desired result.

Algorithmic mathematical models express the relationship between the output parameters and the input and internal parameters in the form of an algorithm.

Analytical mathematical models such a formalized description of an object (phenomenon, process) is called, which is an explicit mathematical expression of output parameters as functions of input and internal parameters.

Analytical modeling is based on an indirect description of the object being modeled using a set of mathematical formulas. The analytical description language contains the following main groups of semantic elements: criterion (criteria), unknowns, data, mathematical operations, restrictions. The most significant characteristic of analytical models is that the model is not structurally similar to the modeled object. Structural similarity here means a one-to-one correspondence of the elements and links of the model to the elements and links of the modeled object. Analytical models include models built on the basis of the apparatus of mathematical programming, correlation, regression analysis. An analytical model is always a construct that can be analyzed and solved mathematically. So, if the apparatus of mathematical programming is used, then the model basically consists of an objective function and a system of restrictions on variables. The objective function, as a rule, expresses the characteristic of the object (system) that needs to be calculated or optimized. In particular, it can be the performance of the technological system. Variables express the technical characteristics of the object (system), including variable ones, restrictions - their permissible limit values.

Analytical models are an effective tool for solving problems of optimizing processes occurring in technological systems, as well as optimizing and calculating the characteristics of technological systems themselves.

An important point is the dimension of a particular analytical model. Often for real technological systems (automated lines, flexible production systems), the dimension of their analytical models is so large that obtaining the optimal solution turns out to be very difficult from a computational point of view. To improve the computational efficiency in this case, various techniques are used. One of them is related to the division of a problem of high dimension into subproblems of smaller dimension so that autonomous solutions of subproblems in a certain sequence give a solution to the main problem. In this case, problems arise in organizing the interaction of subtasks, which are not always simple. Another technique involves reducing the accuracy of calculations, due to which it is possible to reduce the time for solving the problem.

The analytical model can be investigated by the following methods:

· analytical, when they seek to obtain in a general form the dependencies for the desired characteristics;

· numerical, when they seek to obtain numerical results with specific initial data;

· qualitative, when, having solutions in an explicit form, you can find some properties of the solution (estimate the stability of the solution).

However, analytical modeling gives good results in the case of fairly simple systems. In the case of complex systems, either a significant simplification of the original model is required in order to study at least the general properties of the system. This allows you to get approximate results, and to determine more accurate estimates, use other methods, for example, simulation modeling.

Numerical model is characterized by a dependence of such a form that allows only solutions obtained by numerical methods for specific initial conditions and quantitative parameters of the models.

6. Depending on whether the model equations take into account the inertia of processes in the object or do not take into account:

· dynamic or inertial models(written as differential or integro-differential equations or systems of equations) ;

· static or non-inertial models(written as algebraic equations or systems of algebraic equations).

7. Depending on the presence or absence of uncertainties and the type of uncertainties, the models are:

· deterministic e (no uncertainties);

· stochastic (there are uncertainties in the form of random variables or processes described by statistical methods in the form of laws or distribution functionals, as well as numerical characteristics);

· fuzzy (to describe uncertainties, the apparatus of the theory of fuzzy sets is used);

· combined (there are uncertainties of both kinds).

In the general case, the type of mathematical model depends not only on the nature of the real object, but also on the tasks for which it is created, and the required accuracy of their solution.

The main types of models presented in Figure 2.5.

Consider another classification of mathematical models. This classification is based on the concept of controllability of the control object. We will conditionally divide all MMs into four groups.1. Forecast models (computational models without control). They can be divided into static and dynamic.The main purpose of these models: knowing the initial state and information about the behavior at the boundary, to make a prediction about the behavior of the system in time and space. Such models can also be stochastic. As a rule, forecasting models are described by algebraic, transcendental, differential, integral, integro-differential equations and inequalities. Examples are models of heat distribution, electric field, chemical kinetics, hydrodynamics, aerodynamics, etc. 2. Optimization models. These models can also be divided into static and dynamic. Static models are used at the design level of various technological systems. Dynamic - both at the design level and, mainly, for the optimal control of various processes - technological, economic, etc. There are two directions in optimization problems. The first one includes deterministic tasks. All input information in them is completely definable. The second direction refers to stochastic processes. In these tasks, some parameters are random or contain an element of uncertainty. Many optimization problems for automatic devices, for example, contain parameters in the form of random noise with some probabilistic characteristics. Methods for finding the extremum of a function of many variables with various restrictions are often called methods of mathematical programming. Mathematical programming problems are one of the important optimization problems. In mathematical programming, the following main sections are distinguished.· Linear programming . The objective function is linear, and the set on which the extremum of the objective function is sought is given by a system of linear equalities and inequalities.· Nonlinear programming . The objective function is non-linear and non-linear constraints.· Convex programming . The objective function is a convex and convex set on which the extremal problem is solved.· Quadratic programming . The objective function is quadratic and the constraints are linear.· Multi-extremal problems. Problems in which the objective function has several local extrema. Such tasks seem to be very problematic.· Integer programming. In such problems, integer conditions are imposed on variables.

Rice. 4.8. Classification of mathematical models

As a rule, methods of classical analysis for finding the extremum of a function of several variables are not applicable to mathematical programming problems. Models of optimal control theory are one of the most important in optimization models. The mathematical theory of optimal control is one of the theories that have important practical applications, mainly for the optimal control of processes. There are three types of mathematical models of optimal control theory.· Discrete models of optimal control. Traditionally, such models are called dynamic programming models, since the main method for solving such problems is the Bellman dynamic programming method.· Continuous models of optimal control of systems with lumped parameters (described by equations in ordinary derivatives).· Continuous models of optimal control for systems with distributed parameters (described by partial differential equations).3. Cybernetic models (game). Cybernetic models are used to analyze conflict situations. It is assumed that the dynamic process is determined by several subjects who have several control parameters at their disposal. A whole group of subjects with their own interests is associated with the cybernetic system. 4. Simulation . The above types of models do not cover a large number of different situations, such that can be fully formalized. To study such processes, it is necessary to include a functioning “biological” link, a person, into the mathematical model. In such situations, simulation is used, as well as methods of expertise and information procedures.

Modeling, general concepts

The task of modeling is the study of complex objects or processes on their physical or mathematical models. The purpose of modeling is to find the optimal (best according to some criteria) technical solution. Modeling types:

Ø physical;

Ø mathematical;

Ø graphic (geometric).

When modeling, the most important properties of the system under study are replaced by strict, but simplified in relation to the original natural phenomenon, scientific formulations - models. The model provides the ability to accurately describe and predict the behavior of the system, but only in a strictly limited area of ​​application - so far, those initial simplifications on the basis of which the model was built are valid.

For example, when simulating the flight of a satellite around the Earth, its walls can be considered absolutely solid, and when simulating a collision of the same satellite with a micrometeorite, even superhard iron can be described with very high accuracy as an ideal incompressible fluid. This is the paradoxical feature of modeling - its accuracy, brought to life by fundamentally inaccurate, by its very essence approximate, suitable only in a certain area of ​​phenomena, models of a real system.

The functioning processes and structure of the system can be described by means of mathematical modeling. Mathematical modeling is the process of creating a mathematical model and acting on it in order to obtain information about a real system. A mathematical model is a set of mathematical objects and relationships between them that adequately reflects the most important properties of the system. Mathematical objects - numbers, variables, matrices, etc. Connections between mathematical objects - equations, inequalities, etc. Any scientific and technical calculations are specialized types of mathematical modeling.

A system is a set of elements that are naturally connected with each other, forming a single integrity, indicating the links between them and the purpose of functioning. The properties of a system differ from the sum of the properties of its elements. Examples: Machine ¹ å (parts + assemblies); Human ¹ å (brain + liver + spine).

Classification of mathematical models

According to the method of analysis, mathematical models are divided into analytical, algorithmic and simulation.

Analytical models can be:

1) qualitative, when the nature of the dependence of the output parameters on the input parameters, the very existence of the solution, etc. are determined. For example, will the cutting force increase or decrease with increasing speed, is it possible to move at a speed greater than the speed of light, etc. The construction of such a model is a necessary step in the study of a complex system.

2) counting (analytical) models are explicit mathematical relationships between the input, internal and output characteristics of the system. Such models are always preferable, since they are the most effective in analyzing the laws of system functioning, optimization, etc. Unfortunately, it is not always possible to obtain them and only with a significant simplification of the system under study. In addition to computational (analytical) models built on the basis of understanding the processes occurring in the system, these can also be models built on the basis of the analysis of the results of experiments with a "black box". An example is the dependence of cutting force on speed, feed and depth of cut.

3) numerical, when the numerical values ​​of the output parameters are obtained for the given input values. An example is finite element calculations. Numerical models are universal, but they give only partial results, from which it is difficult to draw generalized conclusions.

The algorithmic model is presented in the form of a calculation algorithm. Unlike analytical models, the calculation progress depends on intermediate results.

Simulation modeling is based on a direct description of the object being modeled. When constructing a simulation model, the laws of functioning of each element separately and the relationship between them are described. Unlike the analytical one, it is characterized by a structural similarity between the object and the model. Most often, simulation modeling is used in the study of complex random processes. For example, blanks are fed to the input of an automatic line (AL) model, the dimensions of which have a random spread. At the same time, the processing model on each AL machine is sensitive to the actual dimensions of the workpiece. After the virtual "processing" of hundreds of thousands of blanks, it is possible to find the combination of circumstances in which the AL will stop and avoid it even during the design.

According to the nature of functioning and the type of system parameters, mathematical models are also divided into

continuous and discrete;

static and dynamic;

deterministic and stochastic (probabilistic).

In continuous systems, the parameters change gradually, in discrete systems - abruptly, impulsively. For example, in the model of a turning tool, wear is constantly increasing, and breakage (chipping of the insert) occurs instantly - discretely.

In static models, all parameters included in the model have constant values, and the calculated parameters at the output of the system change simultaneously with changes in the parameters at the input. Such models describe systems with rapidly decaying transients.

Dynamic models take into account the inertia of the system. As a result, the change in the output parameter lags behind the change in the input. Such models more accurately describe the real system, but are more difficult to implement.

The output of deterministic systems is uniquely determined by their input and current state. Possible random changes in system parameters or input parameters are neglected. In stochastic systems, on the contrary, the probabilistic nature of the change in the parameters of the system, which take random values ​​in accordance with some distribution law, is taken into account.