1. Basic equations of dynamics
The following approaches to the development of mathematical models of technological objects can be distinguished: theoretical (analytical), experimental and statistical, methods for constructing fuzzy models and combined methods. Let us give an explanation of these methods.
Analytical methods compiling a mathematical description of technological objects usually refers to methods for deriving static and dynamic equations based on a theoretical analysis of the physical and chemical processes occurring in the object under study, as well as on the basis of specified design parameters of the equipment and characteristics of the processed substances. When deriving these equations, the fundamental laws of conservation of matter and energy are used, as well as the kinetic laws of the processes of mass and heat transfer, and chemical transformations.
To compile mathematical models based on a theoretical approach, it is not necessary to conduct experiments on the object, therefore such methods are suitable for finding static and dynamic characteristics newly designed objects, the processes of which are quite well studied. The disadvantages of such methods for constructing models include the difficulty of obtaining and solving a system of equations with sufficient full description object.
Deterministic models of oil refining processes are developed on the basis of theoretical ideas about the structure of the described system and the patterns of functioning of its individual subsystems, i.e. based on theoretical methods. Having even the most extensive experimental data about the system, it is impossible to describe its operation using the means of a deterministic model if this information is not generalized and its formalization is not given, i.e. are presented in the form of a closed system of mathematical dependencies that reflect, with varying reliability, the mechanism of the processes under study. In this case, you should use the available experimental data to build a statistical model of the system.
The stages of developing a deterministic model are presented in Fig. 4.
Formulation of the problem
Formulation of the mathematical model
Analytical method selected?
Selection of calculation parameters
body process
Experimental
Solving test problems definition
model constants
No
Control tests Adequacy check Adjustment
experiments on natural models
Object No. Yes
Optimization Process optimization with target definition
model using the function model and constraint
Process control with Management model
using the model
Fig.4. Stages of developing a deterministic model
Despite significant differences in the content of specific tasks for modeling various oil refining processes, the construction of a model includes a certain sequence of interrelated stages, the implementation of which allows one to successfully overcome emerging difficulties.
The first stage of the work is the formulation of the problem (block 1), including the formulation of the task based on the analysis of the initial data about the system and its knowledge, assessment of the resources allocated for building the model (personnel, finance, technical means, time, etc.) in comparison with the expected scientific, technical and socio-economic effect.
The formulation of the problem is completed by establishing the class of the model being developed and the corresponding requirements for its accuracy and sensitivity, speed, operating conditions, subsequent adjustments, etc.
The next stage of the work (block 2) is the formulation of a model based on an understanding of the essence of the described process, divided, in the interests of its formalization, into the elementary components of the phenomenon (heat exchange, hydrodynamics, chemical reactions, phase transformations, etc.) and, according to the accepted level of detail, into aggregates (macro level), zones, blocks (micro level), cells. At the same time, it becomes clear which phenomena are necessary or inappropriate to neglect, and to what extent the interconnection of the phenomena under consideration must be taken into account. Each of the identified phenomena is associated with a certain physical law (balance equation) and the initial and boundary conditions for its occurrence are established. Recording these relationships using mathematical symbols is the next stage (block 3), which consists of a mathematical description of the process being studied, forming its initial mathematical model.
Depending on the physical nature of the processes in the system and the nature of the problem being solved, the mathematical model may include mass and energy balance equations for all selected subsystems (blocks) of the model, kinetics equations chemical reactions and phase transitions and transfer of matter, momentum, energy, etc., as well as theoretical and (or) empirical relationships between various model parameters and restrictions on the conditions of the process. Due to the implicit nature of the dependence of the output parameters Y from input variables X in the resulting model, it is necessary to select a convenient method and develop an algorithm for solving the problem (block 4) formulated in block 3. To implement the adopted algorithm, analytical and numerical tools are used. In the latter case, it is necessary to compose and debug a computer program (block 5), select the parameters of the computational process (block 6) and carry out a control calculation (block 8). An analytical expression (formula) or a program entered into a computer represents new uniform a model that can be used to study or describe the process if the adequacy of the model to the full-scale object is established (block 11).
To check the adequacy, it is necessary to collect experimental data (block 10) on the values of those factors and parameters that are part of the model. However, the adequacy of the model can be verified only if some constants contained in the mathematical model of the process are known (from tabular data and reference books) or additionally experimentally determined (block 9).
A negative result of checking the adequacy of a model indicates its insufficient accuracy and may be the result of a whole set of different reasons. In particular, it may be necessary to rework the program in order to implement a new algorithm that does not produce such a large error, as well as adjusting the mathematical model or making changes to physical model, if it becomes clear that neglect of any factors is the cause of failure. Any adjustment to the model (block 12) will, of course, require repeating all the operations contained in the underlying blocks.
A positive result of checking the adequacy of the model opens up the possibility of studying the process by conducting a series of calculations on the model (block 13), i.e. exploitation of the received information model. Consistent adjustment of the information model in order to increase its accuracy by taking into account the mutual influence of factors and parameters, introducing them into the model additional factors and refinement of various “tuning” coefficients allows us to obtain a model with increased accuracy, which can be a tool for a more in-depth study of the object. Finally, establishing the objective function (block 15) using theoretical analysis or experiments and including an optimizing mathematical apparatus in the model (block 14) to ensure the targeted evolution of the system to the optimum region makes it possible to build an optimization model of the process. Adaptation of the resulting model to solve the problem of controlling the production process in real time (block 16) when automatic control means are included in the system completes the work on creating a mathematical control model.
The analytical method consists in drawing up a mathematical description of the object, in which equations of statics and dynamics are found based on fundamental laws that describe the physical and chemical processes occurring in the object under study, taking into account the design of the equipment and the characteristics of the substances being processed. For example: the laws of conservation of matter and energy, as well as the kinetic laws of the processes of chemical transformations, heat and mass transfer. The analytical method is used in the design of new technological objects, the physical and chemical processes of which have been sufficiently well studied.
Advantages:
Does not require experiments on a real object;
Allows you to determine a mathematical description at the control system design stage;
Allows you to take into account all the main features of the dynamics of the control object - nonlinearity, nonstationarity, distributed parameters, etc.;
Provides a universal mathematical description suitable for a wide class of similar control objects.
Flaws:
The difficulty of obtaining a sufficiently accurate mathematical model that takes into account all the features of a real object;
Verifying the adequacy of the model and the real process requires conducting full-scale experiments;
Many mathematical models have a number of parameters that are difficult to estimate in numerical terms
The experimental method consists of determining the characteristics of a real object by conducting a special experiment on it. The method is simple, low-labor, and allows one to fairly accurately determine the properties of a particular object.
Experimental methods for determining dynamic characteristics are divided into:
methods for determining the time characteristics of a control object;
methods for determining the frequency characteristics of the control object.
Temporary methods for determining dynamic characteristics are divided, in turn, into active and passive. Active methods involve sending test testing signals to the input of the object (stepped or rectangular pulses, periodic binary signal).
Advantages:
sufficiently high accuracy of obtaining a mathematical description;
relatively short duration of the experiment.
In passive methods, no test signals are sent to the input of the object, but only the natural movement of the object is recorded in the process of its normal functioning. The resulting data arrays about input and output signals are processed by statistical methods.
Flaws:
low accuracy of the resulting mathematical description (since deviations from the normal operating mode are small);
the need to accumulate large amounts of data in order to increase accuracy (thousands of points);
if the experiment is carried out on an object covered by a control system, then the effect of correlation (relationship) between the input and output signals of the object through the regulator is observed. This relationship reduces the accuracy of the mathematical description.
With the experimental method, it is impossible to identify functional connections between the properties of the processed and obtained substances, the operating parameters of the technological process and the design characteristics of the object. This drawback does not allow the results obtained by the experimental method to be extended to other objects of the same type.
The most effective is the experimental-analytical method, when, using the analytically obtained structure of an object, its parameters are determined during full-scale experiments. Being a combination of analytical and experimental methods, this method takes into account their advantages and disadvantages.
Smoothing of experimental data, methods
When processing experimental data, approximation and interpolation are used. If the data is recorded with an error, then it is necessary to use approximation - smoothing the data of a curve that does not generally pass through the experimental points, but tracks the dependence, eliminating possible mistakes caused by measurement error.
If the data error is small, then interpolation is used, i.e. calculate a smoothing curve passing through each experimental point.
One of best methods approximation is a method (method) of least squares, which was developed through the efforts of Legendre and Gauss more than 150 years ago.
The least squares method allows you to obtain the best functional dependence for a set of available points (the best means that the sum of squared deviations is minimal).
If you connect the points y1, y2, ..., n in succession with a broken line, it is not a graphical representation of the function y = f (x), since when repeating this series of experiments we will get a broken line different from the first. This means that the measured values of y will deviate from the true curve y = f(x) due to statistical scatter. The task is to approximate the experimental data with a smooth (not broken) curve that would be as close as possible to the true dependence y = f(x).
Regression analysis is used to obtain dependencies in processes in which parameters depend on many factors. Often there is a relationship between the variables x and y, but not a well-defined one. In the simplest case, one value x corresponds to several values (set) y. In such cases, the relationship is called regression.
Statistical dependencies are described by mathematical models of the process. The model should be as simple and adequate as possible.
The task of regression analysis is to establish the regression equation, i.e. type of curve between random variables, and assessment of the closeness of the connection between them, the reliability and adequacy of the measurement results.
To preliminarily determine the presence of such a connection between x and y, points are plotted on graphs and a so-called correlation field is constructed. The correlation field characterizes the type of connection between x and y. Based on the shape of the field, one can roughly judge the shape of the graph characterizing a rectilinear or curvilinear relationship.
If you average the points on the correlation field, you can get a broken line called the experimental regression dependence. The presence of a broken line is explained by measurement errors, insufficient quantity measurements, the physical essence of the phenomenon under study, etc.
“EXPERIMENTAL-ANALYTICAL METHOD FOR DETERMINING THE CHARACTERISTICS OF A QUASI-HOMOGENEOUS MATERIAL BY ELASTOPLASTIC ANALYSIS OF EXPERIMENTAL DATA A. A. Shvab Institute of Hydrodynamics named after. ..."
Vestn. Myself. state tech. un-ta. Ser. Phys.-math. Sciences. 2012. No. 2 (27). pp. 65–71
UDC 539.58:539.215
EXPERIMENTAL AND ANALYTICAL METHOD
DEFINITIONS OF CHARACTERISTICS OF QUASI-HOMOGENEOUS
MATERIAL ON ELASTOPLASTIC ANALYSIS
EXPERIMENTAL DATA
A. A. Shvab
Institute of Hydrodynamics named after. M. A. Lavrentieva SB RAS,
630090, Russia, Novosibirsk, Academician Lavrentiev Ave., 15.
Email: [email protected] The possibility of estimating the mechanical characteristics of a material based on solving non-classical elastoplastic problems for a plane with a hole is being studied. The proposed experimental and analytical method for determining the characteristics of a material is based on an analysis of the displacements of the contour of a circular hole and the size of the zones of inelastic deformation around it. It is shown that, depending on the specification of the experimental data, three problems can be solved to assess the mechanical characteristics of the material. One of these problems is considered in relation to rock mechanics. An analysis of the solution to this problem is carried out and the framework of its applicability is given. It is shown that such an analysis can be used to determine the characteristics of both homogeneous and quasi-homogeneous materials.
Key words: experimental-analytical method, material characteristics, elastoplastic problem, plane with a circular hole, rock mechanics.
The work examines the possibility of assessing the mechanical characteristics of a material based on solving non-classical elastoplastic problems using full-scale measurements at existing facilities. Such a statement of the problem implies the development of experimental and analytical methods for determining any mechanical characteristics and their values for objects or their models using some experimental information. The emergence of this approach was associated with the lack of necessary reliable information for the correct formulation of the problem of mechanics of a deformed solid. Thus, in rock mechanics, when calculating the stress-strain state near mine workings or in underground structures, there is often no data on the behavior of the material under a complex stress state. The reason for the latter, in particular, may relate to the heterogeneity of the geomaterials being studied, i.e., materials containing cracks, inclusions and cavities. The difficulty of studying such materials using classical methods lies in the fact that the sizes of inhomogeneities can be comparable to the sizes of the samples. Therefore, experimental data have a large scatter and depend on the nature of the inhomogeneities of a particular sample. A similar problem, namely a large scatter, arises, for example, when determining the mechanical characteristics of coarse concrete. This is due to the lack of a pattern in the distribution of the constituent elements of concrete, on the one hand, and to the dimensions of the standard Albert Aleksandrovich Schwab (Doctor of Physical and Mathematical Sciences, Associate Professor), leading scientific
– – –
sample (cube 150-150 mm) on the other. If the linear measurement base is increased by two or more orders of magnitude compared to the size of the inhomogeneities, then a model of a quasi-homogeneous medium can be used to describe the behavior of the material during deformation. To determine its parameters it is necessary, or, as already noted, to increase linear dimensions sample by two or more orders of magnitude compared to the size of inhomogeneities, or formulate the problem of the strength of the entire object and carry out appropriate field measurements in order to determine the mechanical characteristics of a quasi-homogeneous material. It is when solving such problems that it makes sense to use experimental and analytical methods.
In this work, the characteristics of the material are assessed based on solving inverse elastoplastic problems for a plane with a circular hole by measuring displacements on the contour of the hole and determining the size of the plastic zone around it. Note that on the basis of calculated data and experimental measurements, it is possible to carry out an analysis that allows us to assess the correspondence of various plasticity conditions to the actual behavior of the material.
Within the framework of the theory of plasticity, such a problem, when on part of the surface the load and displacement vectors are simultaneously specified, and on another part of it the conditions are not defined, is formulated as non-classical. Solving such an inverse problem for a plane with a circular hole, when the displacements of the contour and the load on it are known, makes it possible to find the field of stresses and strains in the plastic region and, in addition, to restore the elastoplastic boundary. Knowing the displacement and load at the elastoplastic boundary, it is possible to formulate a similar problem for the elastic region, which makes it possible to restore the stress field outside the hole. To determine the elastic-plastic characteristics of a material, it is necessary Additional Information. In this case, the dimensions of the inelastic deformation zones near the hole are used.
In this work, the ideal plasticity model is used to describe the behavior of the material: when stresses reach a critical value, the relationships between stresses and strains are inelastic.
Let us formulate the boundary conditions on the hole contour (r = 1):
– – –
where u, v are the tangential and tangent components of the displacement vector.
Here and in what follows, the values of r, u and v refer to the hole radius. Under the condition of Tresca plasticity, the stress distribution in the plastic region is described by the relations
– – –
In this case, it is possible to determine the size r of the region of inelastic deformations and the magnitude values.
Problem 2. On the contour of a circular hole (r = 1), conditions (12) and the value r are known.
In this case, from relations (10), (11) one of the material constants can be estimated.
Problem 3. Let an additional quantity be given to the known data of Problem 2.
In this case, the characteristics of the material can be clarified.
On the basis of the given experimental-analytical method, problem 2 was considered. For this purpose, a comparison of calculated and experimental data was carried out. The basis was taken as the displacement (convergence) of the excavation contour, the resistance of the support and the sizes r of the zones of inelastic deformations around the excavations in the Kuznetsk coal basin in the Moshchny, Gorely and IV Internal seams.
Essentially, the convergence of the excavation contour corresponds to the value u0, and the resistance of the support corresponds to the value P. When comparative analysis The goal was not to discuss the quantitative agreement of the calculations with the experimental data, but their qualitative agreement, taking into account the possible scatter of field measurements. It should be noted that the data on movements on the excavation contour and the sizes of the corresponding inelastic deformation zones have a certain scatter. Besides, mechanical characteristics arrays determined from experiments on samples also have scatter. Thus, for the Moschny formation, the value of E varies from 1100 to 3100 MPa, the value of s from 10 to 20 MPa, the value was based on the Experimental-analytical method for determining characteristics...
equal to 0.3. Therefore, all calculations were carried out at different meanings experimental data.
For the Moshchny formation, the table shows the corresponding calculation results for the Treska plasticity condition at 25 G/s 80. From the table data it follows that at 50 G/s 60 there is a satisfactory agreement between the calculated r and experimental rexp values in a fairly wide range of changes in the value of u0, and at G/s = 80 the calculated values of r are clearly overestimated. Therefore, when using the Tresca condition at a value of s = 10 MPa, it is advisable to select the elastic modulus E in the range from 1300 to 1600 MPa.
– – –
In the figure, the area of the entire square corresponds to the possible values of s and G found from experiments on the samples. As a result of the analysis, it was found that only the values of s and G that are in the shaded area (approximately 26% of the total area) correspond to the real behavior of the array.
Since the value of u0 took values from 0.01 to 0.1, i.e., was quite large, the question naturally arises about the legitimacy of using the proposed relationships obtained from the theory of small deformations. To do this, calculations were carried out taking into account changes in the geometry of the contour under the assumption that the speed of displacement of the contour points is small. The results obtained are practically no different from those given above.
The table shows that the spread of G/s values significantly affects the calculation of the value. Therefore, a quantitative assessment of the value is possible, on the one hand, with making the right choice conditions of plasticity, and on the other hand, with a more accurate determination of the values of E and s. If, due to a lack of experimental data, such an analysis is impossible, then based on data on the convergence of the excavation contour, only the nature of the change in value can be assessed. In fact, the increase in u0 from 0.033 to 0.1 is caused by an increase in stress in the formation mass by 1.53–1.74 times, i.e.
the growth coefficient of the value can be determined with an accuracy of 26%.
The advantage of this approach to estimating magnitude is that it belongs to macrostrain methods for estimating stresses.
Sh v a b A. A.
On the one hand, as noted in, factors such as uneven resistance of the support, the difference in the shape of the excavation from the circular one have little effect on the shape of the zone of inelastic deformations. On the other hand, the anisotropy of rocks can significantly influence both the nature of destruction and the formation of an inelastic zone. Obviously, for the general case of anisotropy, the analysis performed is unacceptable, but it can be used to describe the behavior of transversally isotropic rocks with an isotropy plane perpendicular to the Oz axis.
Summarizing the above, we can note the following:
1) under the condition of Tresca plasticity, taking into account the scatter in the experimental values of the shear modulus G and the yield strength s, the proposed experimental-analytical method makes it possible to satisfactorily describe the experiment at 50 G/s 60;
2) the considered method makes it possible to estimate the stress growth factor in the medium with an error of up to 26%;
3) the considered method, based on solving non-classical problems of mechanics, allows one to evaluate the elastic-plastic characteristics of the material for both homogeneous and quasi-homogeneous media;
4) in relation to rock mechanics, the considered method is a macrodeformation method.
BIBLIOGRAPHICAL LIST
1. Turchaninov I. A., Markov G. A., Ivanov V. I., Kozyrev A. A. Tectonic stresses in the earth’s crust and the stability of mine workings. L.: Nauka, 1978. 256 p.
2. Shemyakin E.I. On the pattern of inelastic deformation of rocks in the vicinity of development workings / In: Rock pressure in capital and development workings. Novosibirsk: IGD SB AN USSR, 1975. P. 3–17].
5. Litvinsky G. G. Patterns of influence of non-axisymmetric factors on the formation of a zone of inelastic deformations in mine workings / In the collection: Fastening, maintenance and protection of mining workings. Novosibirsk: SO AN USSR, 1979. pp. 22–27.
Received by the editor 23/V/2011;
in final version 10/IV/2012.
Experimental analytical method determine the characteristics...
MSC: 74L10; 74C05, 74G75
EXPERIMENTAL ANALYTICAL METHOD FOR
QUASI-HOMOGENEOUS MATERIAL CHARACTERISTICS
DETERMINATION BASED ON ELASTO-PLASTIC ANALYSIS
OF EXPERIMENTAL DATA
A. A. Shvab M. A. Lavrentyev Institute of Hydrodynamics, Siberian Branch of RAS, 15, Lavrentyeva pr., Novosibirsk, 630090, Russia.Email: [email protected] The possibility of material mechanical characteristics estimation based on solving of the elasto-plastic problems for plane with a hole is studied. The proposed experimentalanalytical method for the material characteristics determination depends on the analysis of circular hole contour displacement and the sizes of inelastic strains zones near it.
It is shown, that three problems can be solved for the material mechanical characteristics estimation according to the assignment of experimental data. One of such problems is considered relating to the rock mechanics. The analysis of this problem solution is made and the scope of its applicability is noted. The validity of similar analysis using for the characteristics determination of both homogeneous and quasihomogeneous material is presented.
Key words: experimental analytical method, characteristics of material, elasto-plastic problem, plane with a circular hole, rock mechanics.
– – –
Albert A. Schwab (Dr. Sci. (Phys. & Math.)), Leading Research Scientist, Dept. of Solid
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Physical processes can be studied by analytical or experimental methods.
Analytical methods make it possible to study processes based on mathematical models, which can be presented in the form of functions, equations, systems of equations, mainly differential or integral. Usually, a rough model is created at the beginning, which is then refined after research. This model makes it possible to study the physical essence of the phenomenon quite fully.
However, they have significant disadvantages. In order to find a particular solution from the entire class that is inherent only to a given process, it is necessary to set uniqueness conditions. Often, incorrect acceptance of boundary conditions leads to a distortion of the physical essence of the phenomenon, and finding an analytical expression that most realistically reflects this phenomenon is either completely impossible or extremely difficult.
Experimental methods make it possible to deeply study processes within the accuracy of the experimental technique, especially those parameters that are of greatest interest. However, the results of a particular experiment cannot be extended to another process, even one that is very similar in nature. In addition, it is difficult to establish from experience which parameters have a decisive influence on the course of the process, and how the process will proceed if various parameters change simultaneously. Experimental methods make it possible to establish only partial dependencies between individual variables in strictly defined intervals. Using these dependencies outside of these intervals can lead to serious errors.
Thus, both analytical and experimental methods have their advantages and disadvantages. Therefore, the combination is extremely fruitful positive aspects these research methods. This principle is the basis for the methods of combining analytical and experimental research, which, in turn, are based on the methods of analogy, similarity and dimensions.
Method of analogy. The analogy method is used when different physical phenomena are described by the same differential equations.
Let's look at the essence of the analogy method using an example. Heat flow depends on the temperature difference (Fourier's law):
Where λ - coefficient of thermal conductivity.
Mass transfer or transfer of a substance (gas, steam, moisture, dust) is determined by a difference in the concentration of the substance WITH(Fick's law):
– mass transfer coefficient.
The transfer of electricity through a conductor with linear resistance is determined by a voltage drop (Ohm's law):
Where ρ – electrical conductivity coefficient.
Three different physical phenomena have identical mathematical expressions. Thus, they can be studied by analogy. Moreover, depending on what is accepted as the original and the model, there may be different kinds modeling. So, if the heat flow q Since they are studied on a model with fluid movement, the modeling is called hydraulic; if it is studied on an electrical model, the simulation is called electrical.
The identity of mathematical expressions does not mean that the processes are absolutely similar. In order to study the process of the original using a model, it is necessary to comply with the criteria of analogy. Compare directly q t and q e, thermal conductivity coefficients λ and electrical conductivity ρ , temperature T and voltage U it makes no sense. To eliminate this incomparability, both equations must be presented in dimensionless quantities. Each variable P should be represented as a product of constant dimension P n to a dimensionless variable P b:
P= P p∙ P b. (4.25)
With (4.25) in mind, we write down expressions for q t and q e in the following form:
Let us substitute the values of the transformed variables into equations (4.22) and (4.24), resulting in:
; 
Both equations are written in dimensionless form and can be compared. The equations will be identical if
This equality is called the criterion of analogy. Using criteria, the parameters of the model are determined based on the original equation of the object.
Currently, electrical modeling is widely used. With its help, you can study various physical processes (oscillations, filtration, mass transfer, heat transfer, stress distribution). This simulation is universal, easy to use, and does not require bulky equipment. For electrical modeling, analogue ones are used. computing machines(AVM). By which, as we have already said, we mean a certain combination of various electrical elements in which processes occur that are described by mathematical dependencies similar to the dependencies for the object under study (original). A significant disadvantage of AVM is its relatively low accuracy and lack of versatility, since for each task it is necessary to have its own circuit, and therefore another machine.
To solve problems, other methods of electrical modeling are also used: the method of continuum, electrical grids, electromechanical analogy, electrohydrodynamic analogy, etc. Planar problems are modeled using electrically conductive paper, volumetric problems are modeled using electrolytic baths.
Dimensional method. In a number of cases, processes occur that cannot be directly described by differential equations. The relationship between variable quantities in such cases can be established experimentally. In order to limit the experiment and find the connection between the main characteristics of the process, it is effective to use the dimensional analysis method.
Dimensional analysis is a method of establishing the relationship between the physical parameters of the phenomenon being studied. It is based on the study of the dimensions of these quantities.
Physical characteristic measurement Q means comparing it with another parameter q of the same nature, that is, you need to determine how many times Q more than q. In this case q is a unit of measurement.
Units of measurement constitute a system of units, e.g. International system SI. The system includes units of measurement that are independent of one another, they are called basic or primary units. In the SI system these are: mass (kilogram), length (meter), time (second), current (ampere), temperature (degree Kelvin), luminous intensity (candela).
Units of measurement of other quantities are called derivatives or secondary. They are expressed using basic units. The formula that establishes the relationship between basic and derived units is called dimension. For example, the speed dimension V is
Where L – symbol length, and T– time.
These symbols represent independent units of the unit system ( T measured in seconds, minutes, hours, etc., L in meters, centimeters, etc.). The dimension is derived using the equation, which in the case of speed has next view:
from which follows the dimension formula for speed. Dimensional analysis is based on the following rule: the dimension of a physical quantity is the product of basic units of measurement raised to the appropriate power.
In mechanics, as a rule, three basic units of measurement are used: mass, length and time. Thus, in accordance with the above rule, we can write:
(4.28)
Where N– designation of the derived unit of measurement;
L, M, T– designations of basic (length, mass, time) units;
l, m, t– unknown indicators that can be represented by integers or fractions, positive or negative.
There are quantities whose dimension consists of basic units to a power equal to zero. These are the so-called dimensionless quantities. For example, the rock loosening coefficient is the ratio of two volumes, from which
therefore, the loosening coefficient is a dimensionless quantity.
If during the experiment it is established that the quantity being determined can depend on several other quantities, then in this case it is possible to create a dimensional equation in which the symbol of the quantity being studied is located on the left side, and the product of other quantities is on the right. The symbols on the right side have their own unknown exponents. To finally obtain the relationship between physical quantities, it is necessary to determine the corresponding exponents.
For example, you need to determine the time t, spent by a body having mass m, when moving straight along the path l under constant force f. Therefore, time depends on length, mass and force. In this case, the dimensional equation will be written as follows:
The left side of the equation can be represented as . If the physical quantities of the phenomenon being studied are chosen correctly, then the dimensions on the left and right sides of the equation should be equal. Then the system of equations of exponents will be written:
Then x=y=1/2 and z = –1/2.
This means that time depends on the path as , on the mass as and on the force as . However, it is impossible to obtain a final solution to the problem using dimensional analysis. You can only establish a general form of dependence:
Where k– dimensionless proportionality coefficient, which is determined by experiment.
In this way, the type of formula and experimental conditions are found. It is only necessary to determine the relationship between two quantities: and A, Where A= .
If the dimensions of the left and right sides of the equation are equal, this means that the formula in question is analytical and calculations can be performed in any system of units. On the contrary, if an empirical formula is used, it is necessary to know the dimensions of all terms of this formula.
Using dimensional analysis, we can answer the question: have we lost the main parameters that influence this process? In other words, is the equation found complete or not?
Suppose that in the previous example the body heats up when moving and therefore time also depends on temperature WITH.
Then the dimensional equation will be written:
Where is it easy to find that, i.e. the process being studied does not depend on temperature and equation (4.29) is complete. Our assumption is wrong.
Thus, dimensional analysis allows:
– find dimensionless relationships (similarity criteria) to facilitate experimental studies;
– select the parameters influencing the phenomenon under study in order to find an analytical solution to the problem;
– check the correctness of analytical formulas.
The dimensional analysis method is very often used in research and in more complex cases than the example discussed. It allows you to obtain functional dependencies in a criterion form. Let it be known in general view function F for any complex process
(4.30)
Values have a specific unit dimension. The dimensional method involves choosing from a number k three basic units of measurement independent from each other. The rest ( k–3) the quantities included in the functional dependence (4.30) are chosen so that they are represented in the function F as dimensionless, i.e. in similarity criteria. Conversions are made using the basic, selected units of measurement. In this case, function (4.30) takes the form:
Three ones means the first three numbers are a ratio n 1 , n 2 and n 3 to correspondingly equal values A, V, With. Expression (4.30) is analyzed according to the dimensions of the quantities. As a result, the numerical values of the exponents are established X…X 3 , at…at 3 , z…z 3 and determine the similarity criteria.
A clear example The use of the dimensional analysis method in the development of analytical and experimental methods is the calculation method of Yu.Z. Zaslavsky, which makes it possible to determine the parameters of the support of a single mine.
LECTURE 8
Similarity theory. Similarity theory is the doctrine of the similarity of physical phenomena. Its use is most effective in the case when it is impossible to find dependencies between variables based on the solution of differential equations. In this case, using the data of the preliminary experiment, an equation is created using the similarity method, the solution of which can be extended beyond the experiment. This method of theoretical study of phenomena and processes is possible only on the basis of combination with experimental data.
Similarity theory establishes criteria for the similarity of various physical phenomena and, using these criteria, explores the properties of the phenomena. Similarity criteria represent dimensionless ratios of dimensional physical quantities, defining the phenomena being studied.
The use of similarity theory gives important practical results. With the help of this theory, a preliminary theoretical analysis of the problem is carried out and a system of quantities characterizing phenomena and processes is selected. It is the basis for planning experiments and processing research results. Together with physical laws, differential equations and experiment, similarity theory allows us to obtain quantitative characteristics the phenomenon being studied.
Formulating a problem and establishing an experimental plan based on the theory of similarity is greatly simplified due to the functional relationship between the set of quantities that determine the phenomenon or behavior of the system. As a rule, in this case we are not talking about studying separately the influence of each parameter on the phenomenon. It is very important that results can be achieved with just one experiment on such systems.
The properties of similar phenomena and criteria for the similarity of the phenomena being studied are characterized by three similarity theorems.
First similarity theorem. The first theorem, established by J. Bertrand in 1848, is based on general concept Newton's dynamic similarity and his second law of mechanics. This theorem is formulated as follows: for similar phenomena, you can find a certain set of parameters, called similarity criteria, that are equal to each other.
Let's look at an example. Let two bodies having masses m 1 and m 2, move with accelerations accordingly A 1 and A 2 under the influence of forces f 1 and f 2. The equations of motion are:
By propagating the result to n similar systems, we obtain the similarity criterion:
(4.31)
It was agreed upon to denote the similarity criterion by the symbol P, then the result of the above example will be written:
Thus, in such phenomena, the ratio of parameters (similarity criteria) are equal to each other and for these phenomena it is true. The converse statement also makes sense. If the similarity criteria are equal, then the phenomena are similar.
The found equation (4.32) is called Newton's dynamic similarity criterion, it is similar to expression (4.29) obtained using the dimensional analysis method, and is a special case of the thermodynamic similarity criterion based on the law of conservation of energy.
When studying a complex phenomenon, several different processes may develop. The similarity of each of these processes is ensured by the similarity of the phenomenon as a whole. From a practical point of view, it is very important that similarity criteria can be transformed into criteria of another type using division or multiplication by a constant k. For example, if there are two criteria P 1 and P 2, the following expressions are valid:
If similar phenomena are considered in time and space, we are talking about the criterion of complete similarity. In this case, the description of the process is most complex; it allows not only the numerical value of the parameter (the impact force of the blast wave at a point 100 m from the explosion site), but also the development, change of the parameter in question over time (for example, an increase in impact force, speed process attenuation, etc.).
If such phenomena are considered only in space or time, they are characterized by criteria of incomplete similarity.
Most often, approximate similarity is used, in which parameters that influence this process to a small extent are not considered. As a result, the research results will be approximate. The degree of this approximation is determined by comparison with practical results. In this case we are talking about criteria of approximate similarity.
Second similarity theorem ( P – theorem). It was formulated at the beginning of the 20th century by scientists A. Federman and W. Buckingham as follows: each complete equation of a physical process can be presented in the form of () criteria (dimensionless dependencies), where m is the number of parameters, and k is the number of independent units of measurement.
Such an equation can be solved with respect to any criterion and can be presented in the form of a criterion equation:
. (4.34)
Thanks to P- theorem, it is possible to reduce the number of variable dimensional quantities to () dimensionless quantities, which simplifies data analysis, experimental planning and processing of its results.
Typically, in mechanics, three quantities are taken as the basic units: length, time and mass. Then, when studying a phenomenon that is characterized by five parameters (including a dimensionless constant), it is enough to obtain the relationship between the two criteria.
Let's consider an example of reducing quantities to dimensionless form, usually used in the mechanics of underground structures. The stressed deformed state of the rocks around the excavation is predetermined by the weight of the overlying strata γH, Where γ – volumetric weight of rocks, N– the depth of the excavation from the surface; strength characteristics of rocks R; support resistance q; displacements of the excavation contour U; the size of the workings r; deformation modulus E.
In general, the dependence can be written as follows:
In accordance with P- theorem system of P parameters and one determined quantity should give dimensionless combinations. In our case, time is not taken into account, therefore, we get four dimensionless combinations.
from which we can create a simpler dependence:
Third similarity theorem. This theorem was formulated by Acad. V.L. Kirpichev in 1930 as follows: a necessary and sufficient condition for similarity is the proportionality of similar parameters that form part of the condition of unambiguity, and the equality of similarity criteria for the phenomenon being studied.
Two physical phenomena are similar if they are described by the same system of differential equations and have similar (boundary) conditions of uniqueness, and their defining criteria of similarity are numerically equal.
The conditions of unambiguity are the conditions by which a specific phenomenon is distinguished from the entire set of phenomena of the same type. The similarity of unambiguity conditions is established in accordance with the following criteria:
– similarity of geometric parameters of systems;
– proportionality of physical constants that are of fundamental importance for the process being studied;
– similarity of initial conditions of systems;
– similarity of the boundary conditions of the systems throughout the entire period under consideration;
– equality of criteria that are of primary importance for the process being studied.
The similarity of two systems will be ensured if their similar parameters are proportional and the similarity criteria determined using P- theorems from the complete equation of the process.
There are two types of problems in similarity theory: direct and inverse. The direct task is to determine the similarity for known equations. The inverse problem is to establish an equation that describes the similarity of similar phenomena. Solving the problem comes down to determining similarity criteria and dimensionless proportionality coefficients.
The problem of finding the process equation using P- The theorem is solved in the following order:
– determine by one method or another all the parameters that influence this process. One of the parameters is written as a function of other parameters:
(4.35)
– assume that equation (4.35) is complete and homogeneous with respect to dimension;
– choose a system of units of measurement. In this system, independent parameters are selected. The number of independent parameters is equal to k;
– compose a matrix of dimensions of the selected parameters and calculate the determinant of this matrix. If the parameters are independent, then the determinant will not be equal to zero;
– find combinations of criteria using the dimensional analysis method, their number in the general case is equal to k–1;
– determine proportionality coefficients between criteria using experiment.
Mechanical similarity criteria. In mining science, mechanical similarity criteria are most widely used. It is believed that other physical phenomena (thermal, electrical, magnetic, etc.) do not affect the process being studied. To obtain the necessary criteria and constant similarities, Newton's law of dynamic similarity and the method of dimensional analysis are used.
The basic units are length, mass and time. All other characteristics of the process under consideration will depend on these three basic units. Therefore, mechanical similarity sets criteria for length (geometric similarity), time (kinematic similarity) and mass (dynamic similarity).
Geometric similarity two similar systems will occur if all dimensions of the model are changed in C l times in relation to a system having real dimensions. In other words, the ratio of distances in real life and on a model between any pair of similar points is a constant value, called geometric scale :
. (4.36)
The ratio of the areas of similar figures is equal to the square of the proportionality coefficient, the ratio of volumes is .
Kinematic similarity condition will take place if similar particles of systems, moving along geometrically similar trajectories, travel geometrically similar distances over time intervals t n in kind and t m for models that differ in proportionality coefficient:
(4.37)
Dynamic similarity condition will take place if, in addition to conditions (4.36) and (4.37), the masses of similar particles of similar systems also differ from each other by the coefficient of proportionality:
. (4.38)
Odds C l , Ct, And Cm called similarity coefficients.
Experimental method
Structural-analytical method
It is known that natural science owes its development to the use of experiment. An experiment differs from simple observation in that a researcher, studying a phenomenon, can arbitrarily change the conditions under which it occurs, and, observing the results of such an intervention, draw conclusions about the patterns of the phenomenon being studied. For example, an experimenter can study the rate of reaction in response to signals of different intensities given to him. Or, let’s say, study the actions of a subject who needs to find a way out of a maze different levels difficulties. At the same time, the experimenter observes and records what techniques, means and forms of behavior the subject uses when getting out of the proposed labyrinths. Further analysis of the results obtained, in which the experimenter traces the structural structure of the techniques used by the subject, is called the method of structural analysis.
In the examples given, we were talking about direct experiments in which the researcher, actively changing the conditions of the subjects’ activities, observed their behavior. Typically, such studies are carried out in so-called laboratory conditions. Hence the experiment received the name laboratory. They often use special equipment, the experiment is clearly planned, and the subject is included in the experiment voluntarily and knows that he is being studied.
All psychophysics, psychophysiology, as well as many studies of general psychology (memory, attention, thinking) are carried out in laboratory conditions. These experiments are not questionable when their purpose is to study externally observable reactions or behaviors. But is it possible to experimentally study mental phenomena themselves: perceptions, experiences, imagination, thinking? After all, they are inaccessible to direct observation, and to conduct an experiment it is necessary to change the conditions for these processes. Indeed, this is impossible directly, but it is possible indirectly if we obtain the consent of the subject for such an experiment and with his help, relying on his introspection (subjective method), we change the conditions for the occurrence of mental processes in his consciousness.
Experimental genetic method
Along with the structural-analytical method, the experimental-genetic method is widely used in psychology, which has especially great importance for child (genetic) psychology. With its help, the experimenter can investigate the origin and development of certain mental processes in a child, study which stages are included in it, what factors determine it. The answer to these questions can be obtained by tracing and comparing how the same tasks are performed at successive stages of child development. This approach is called genetic (or cross-sectional) cross-sectioning in psychology. Another modification of the experimental genetic method is a longitudinal study, i.e. long-term and systematic study of the same subjects, allowing us to determine age-related and individual variability of phases life cycle person.
Longitudinal research is often conducted under the conditions of a natural experiment, which was proposed in 1910 by A.F. Lazursky. Its meaning is to eliminate the stress that a person experiences when he knows that he is being experimented on, and to transfer the research to ordinary, natural conditions (a lesson, an interview, a game, homework, etc.).
An example of a natural experiment is a study of memorization productivity depending on the setting for the duration of retention of material in memory. In a lesson in two classes, students are introduced to the material that needs to be studied. The first class is told that they will be surveyed the next day, and the second class is told that the survey will be in a week. In fact, both classes were surveyed two weeks later. This natural experiment revealed the benefits of setting the mindset to retain material in memory for a long time.
In developmental and educational psychology, a combination of structural-analytical and experimental-genetic methods is often used. For example, in order to identify how this or that mental activity is formed, the subject is placed in various experimental conditions, asked to solve certain problems. In some cases, he is required to make an independent decision, in others he is provided with various kinds of hints. The experimenter, observing the activities of the subjects, determines the conditions under which the subject can optimally master this activity. At the same time, using the techniques of the experimental genetic method, it turns out to be possible to experimentally form complex mental processes and deeply explore their structure. In educational psychology, this approach is called a formative experiment.
Experimental genetic methods were widely used in the works of J. Piaget, L.S. Vygotsky, P.P. Blonsky, S.L. Rubinshteina, A.V. Zaporozhets, P.Ya. Galperina, A.N. Leontyev. A classic example of the use of the genetic method is the study of L.S. Vygotsky’s egocentric speech of the child, that is, speech addressed to oneself, regulating and controlling the child’s practical activities. L.S. Vygotsky showed that genetically egocentric speech goes back to external (communicative) speech. The child addresses himself out loud in the same way as one of the parents or raising adults addressed him. However, every year the child’s egocentric speech becomes more and more abbreviated and therefore incomprehensible to others, and by the beginning school age stops completely. The Swiss psychologist J. Piaget believed that by this age egocentric speech simply dies out, but L.S. Vygotsky showed that it does not disappear, but goes into the internal plane, becomes internal speech, which plays an important role in self-management of one’s behavior. Internal pronunciation, “speech to oneself,” retains the structure of external speech, but is devoid of phonation, i.e. pronouncing sounds. It forms the basis of our thinking when we pronounce to ourselves the conditions or process of solving a problem.