DISTRIBUTION OF TEMPERATURE IN A SPATIALLY ONE- DIMENSIONAL OBJECT AS A RESULT OF THE ACTIVE POINT SOURCE

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1 DISTRIBUTION OF TEMPERATURE IN A SPATIALLY ONE- DIMENSIONAL OBJECT AS A RESULT OF THE ACTIVE POINT SOURCE Yury Iyushin and Anton Mokeev Saint-Petersburg Mining University, Vasiievsky Isand, 1 st ine, Saint-Petersburg, Russia E-Mai: orov83@mai.ru ABSTRACT This artice presents the concept of systems with distributed parameters and investigates the method of distributed controer synthesis and a homogeneous contro object. This artice anayzes main methods of transferring heat energy. On the basis of the heat-transfer equation, the function of initia heating has been obtained, and the process has been mathematicay simuated, and the resuts obtained have been anayzed. The practica resuts of this research make it possibe to draw a concusion about the possibiity of buiding a siicon-carbide heating eement made in the shape of an isotropic rod. Keywords: function grinna, mathematica mode, portioned systems, anaysis. INTRODUCTION In continuous automatic contro systems, information is represented as signas described by continuous functions. Aong with continuous methods of signa transmission and conversion, discrete signas are widey used in which signas are quantized. Quantization or samping consists in the representation of a continuous signa by its discrete vaues. Depending on the type of quantization, the automatic contro systems with umped parameters are divided into discrete (reay), discrete (puse), discrete in eve and time (reay-puse) in eve. The impementation of the input effect in systems with distributed parameters is carried out by discretizing it over spatia coordinates. For exampe, the reaization of the heat fux fied is carried out by means of a sectiona heater, and the number of sections can be arbitrariy arge. Thus, to the specified types of discretization it is necessary to add discretization with respect to spatia coordinates, both the input action and the sensors for observing. The fied of appication of discrete systems is very diverse. There are two main categories of such systems. Systems that are discrete in their physica nature, i.e. information in them exists ony at discrete instants of time. Exampes of this category of discrete systems are the radar detection and tracking systems of the target. There are numerous physica and bioogica phenomena, processes in socia and economic systems, the dynamics of which can adequatey be described ony by discrete modes. Systems in which information exists continuousy but is intentionay quantized to obtain some new properties as compared to continuous systems. These properties can be: ease of impementation, increased reiabiity, increased accuracy, smaer overa dimensions and cost. Rapid progress in computer technoogy, the widespread use of microprocessors in contro systems further increase interest in discrete systems. MATERIALS AND METHODS Anaysis of the management system: as a spatia objec we consider a homogeneous cyindrica rod. We wi assume that the controing effect is the heat fux generated by the sources, reaized as sections of a sectiona heater, distributed aong the boundary of the atera surface of the cyinder. The sources are switched on using reay eements. The zero temperature is maintained at the ends of the rod. Let us set the probem of temperature stabiization at a eve of a certain vaue (Iyushin, Pervukhin, Afanasieva, Kavdiev, Koesnichenko, 1). RESULTS AND DISCUSSIONS Consider a cyindrica rod of radius R ength, presented in the picture 1. The mathematica mode of the process of heat propagation has the form (Iyushin, Pervukhin, Afanasieva, Kavdiev, Koesnichenko, 15): ät ät ä T 1 ät a är r är ä T äx T зад r R (1) T ( r, temperature fied of a cyindrica rod a preset factor R, given numbers r spatia coordinates t time. Figure-1. Cyindrica rod. 138

2 form: The boundary conditions of equation (1) have the T(, r, T (, r, () T( R, u( (3) ät(, ). () är The function of the output is the function (, R * * * T x,, R specified number ( R R). Condition () indicates that the ends of the rod have a temperature equa to zero. Condition () is a symmetry condition for the therma fieds. The input effect is distributed aong the boundary, which refects the condition (3). Suppose that the rod is thin enough that at any point in time the temperature at a points of the cross section coud be considered the same. In other words, we assume that the cyinder is spatiay one-dimensiona. In this case, we keep the boundary condition (3), assuming that the boundary is not ony the ends of the rod, but aso its atera surface. We introduce this boundary condition into the right-hand side of the basic equation. The necessity of the boundary condition () disappears. We add a zero initia condition. With a the assumptions made, the mathematica mode of the process of heat propagation wi take the form (Gakin, 17): дt a u ( x, t ) дt д T x дx Boundary and initia conditions: t (5) T(, T(, T ( ) (9) The coefficients and the input functions take on the vaues [3, ]: A A1 1 C a B1 C1, f ( x ) ( t ), g, g. 1 Then the genera soution of the boundary vaue probem in integra form takes the form: t T( G(, ) ) ( )dd ( x (, ) t (1) The obtained mode describes the process of heat propagation in the rod, under assumptions that aow the system to be considered inear and the contro process to be continuous. To stabiize the temperature, it is necessary to consider a cosed contro system, represented in the form of the foowing structura scheme (see Figure-). The reguator of such a system can be reaized as a noninear discrete agorithm. This agorithm must perform an impact on the deviation of temperature from a given vaue at certain points at a certain time. Before any negative deviations occur T( T, it is necessary to heat the rod to a temperature exceeding the vaue зад T зад aong the entire ength of the rod. That is, it is necessary to form, so-caed, the function of initia heating. This function can be formed as a resut of the initia incusion of a sources (Koesnikov, 1). T(, T (, (6) T ( ). (7) The contro effect u ( is created by sources that are switched on by means of reay eements. The output variabe of the reay eement has a rectanguar shape. When the number of sources tends to infinity, it wi take the form of an impuse created by a point source. Assuming that the action of each source occurs during an infinitesima time interva, it can be assumed that the contro action is created by instantaneous point sources and is represented as a product of deta functions (Chernyshev, 9, Chernyshev, 1). u ( ( x ) ( t ). The mathematica mode takes the form: дt a ( x ) ( t ) дt ä T äx x t. (8) Figure-. Structura diagram of a cosed-oop contro system. Сircuit eements: «Задающее устройство» - Master device «Регулятор» - Reguator «Контроллер» - Controer «Объект» - Object «Измерительное устройство» - Measuring device In the mathematica mode (8) - (9) it is necessary to change the initia conditions. As a resu the mathematica mode takes the form: 139

3 дt a ( x ) ( t ) дt ä T äx x t (11) T(, T (, (1) T ( ) ( x ) ( (13) Then, the genera soution in integra form wi be expressed as: T( G(, ( d G(, ) ( ) ( )dd x (, ) t. ) t (1) The mathematica mode (11) - (13) obtained and the integra representation of the soution (1) can ony be of a basic nature. The rea reguatory system under consideration is of a discrete nature. Sources and sensors are ocated at specific fixed points, their number is imited. The time for switching on the contro actions is determined in accordance with the software agorithm, when the output vaue reaches the set vaue at the sensor setting point (Koesnikov, Zarembo, 1). Let us anayze the infuence of the components of the series at n. At the initia instant of time, when, is obtained: t, na exp t 1 Then the temperature at the points of the rod wi be expressed by the formua: n n T( ) sin xsin T( ) n 1 Let the instantaneous heat source affect the point, then the formua takes the form: n1 n n sin sin x The ampitude of each component of the series wi ook ike: n A n sin. For the first six components of the series, for x, we get: T1 ( x) sin sin x sin x 1 T1 T ( x) sin sin x sin x T A 1 A T3 ( x) sin sin x sin x A 3 1 T3 T ( x) sin sin x A T T5 ( x) sin sin x sin x A 5 1 T T6 ( x) sin sin x sin x A 6 T6 It is known that G (, ), at any vaue x,,. When t and a arge number of terms of the series n negative vaues of the function wi disappear, and the range of vaues of x for which the function is positive wi narrow, approaching from both sides to the point of appication of the action, in this case to the point. In this case, the vaue of the function at the point wi tend to infinity (see Figure-3). The expression n 1 n n sin xsin is a spatia deta function in the form of a Fourier series, i.e. n n ( x ) sin xsin n 1 1

4 t t f ( ), åñëè f ( ( t ) dt, åñëè t t, t, t, As appied to the function f ( two variabes - a scaar spatia coordinate x and time t, vector function, simuating the impuse action appied at the time t,t at the point,, described by ( x ) ( t ). The integra representation takes the form: functions with 6 Figure-3. Graphs with n. n and function beongs to the cass of generaized functions. Apparatus functions are widey used in the study of modes of contro objects with distributed parameters. The argument of functions ( x ) is the spatia coordinate, considered as a scaar quantity in a spatiay one-dimensiona probem or as a vector coordinate for mutidimensiona spatia domains. The function ( x ) is zero within the entire spatia domain occupied by the objec except for the point, at which ( x ) it assumes a vaue equa to infinity (Peshivtseva, Rapopor 1)., ïðè x (x- )=., ïðè x For functions f (x), distributed over a segment,, we have the equaity: f ( x) ( x ) dx f ( ) Simiary, we introduce the concept of a time function ( t ), with the purpose of describing the impuse actions that are imited in energy consumption of a sufficienty short duration, but of very great magnitude, on objects of a different physica nature. In the imi infinitey arge impacts, concentrated at fixed instants of time. Deta function ( t ) is equa to zero for a t, except for the vaue of t, at which it turns to infinity, and ( t ) dt 1. If f ( any bounded and continuous function on the interva t, so t f ( x ) ( t ) dxdt f (, ) t (. Thus, assuming that the input of the object under consideration received a singe impuse disturbance appied at the point at the time, at the output we get: t T ( G(, ) ( ) ( ) dd G(, ) For a fixed time t, the ampitude of the components of the Fourier series wi have the form: A n sin n na exp t When n, vaue A n wi tend to zero, i.e. the effect of each subsequent harmonic of the series decreases. We define with a given accuracy, starting with which number n a subsequent components of the series can be negected. Let is A, it is known that n sin 1, then n sin, exp na exp t n t a eventuay: n,, n na t, na t n, 11

5 n a t n. For exampe, for vaues 1,, 1 t 1,, 1 is obtained: n 8, 7757 a,, that is, to determine the vaue of the function with accuracy,1 it suffices to take 8 components of the series. The ampitude decreases aso when t. It is possibe to determine from which time vaue t components of the n n series with, can be negected. t na n. Let's define as an exampe, starting from which moment of time, with an accuracy of.1, the vaue of the function wi be determined by ony one first harmonic. 1 t n 1978,,1 t. Dependence of temperature distribution in the rod on time, as a resut of the action of the source at the point, can be represented in the form of a graph (see Figure-). ( t1 t... t5 ). systems with distributed parameters, it is required to create a new apparatus on the basis of non-traditiona for the cassica theory of contro of mathematica means. Based on the theory of impuse response functions, research methods and methods for parametric optimization of discrete distributed systems have been deveoped, which aowed obtaining anaytica dependencies between a given error and samping parameters that make it possibe to justify the choice of the discretization step in practica impementation. ACKNOWLEDGEMENTS We shoud note that persons without whom these studies coud not have been carried out. The first is Pershin Ivan Mitrofanovich (Doctor of Technica Dciences, Professor, actua member of the Academy of Natura Sciences, Honored Worker of Higher Professiona Education of the Russian Federation), Koesnikov Anatoy Arkadievich (Honored Worker of Science and Technoogy of the Russian Federation, Doctor of Technica Sciences, Professor, member of the Academy of Sciences and the Academy of eectrica-technica sciences and Motion Contro, Corresponding Member of the Russian Academy of Natura Sciences, Soros Professor (four) in the exact sciences fied), and Chernyshev Aexander Borisovich (Doctor of Technica Sciences, Professor) who aid the foundation for the synthesis of distributed controers on the basis of the Green's function. REFERENCES Chernyshev A. 9. Adaptation of Absoute Stabiity Frequency Criterion to Systems with Distributed Parameters. Mekhatronika, avtomatizatsiya, upravenie. pp Chernyshev A. 1. Interpretation of Absoute Stabiity Criterion for Noninear Distributed Systems. Avtomatizatsiya. Sovremennye tekhnoogii. pp Gakin A.F. 17. Compex Use of Heat-Exchange Tunnes. Zapiski Gornogo instituta, : 9-1. Gakin A.F. 17. Compex Use of Heat-Exchange Tunnes. Journa of Mining Institute. : 9-1. Figure-. Spatio-tempora dependence of temperature distribution. CONCLUSIONS Thus, it can be observed that the task of impementing contro systems for objects with distributed parameters is consideraby more compicated than for systems with umped parameters. The number of contro actions of such systems can incude space-time contros, described by functions of severa arguments - time and spatia coordinates. For the anaysis and synthesis of Iyushin Y., Pervukhin D., Afanasieva O., Kavdiev A., Koesnichenko S. 1. Designing of Distributed Contro System with Puse Contro. Midde-East Journa of Scientific Research. 1(3): Iyushin Y., Pervukhin D., Afanasieva O., Kavdiev A., Koesnichenko S. 15. The Methods of the Synthesis of the Noninear Reguators for the Distributed One- Dimension Contro Objects. Modern Appied Science. 9(): -61. Koesnikov A. 1. Noninear Osciations Contro. Energy Invariants. Journa of Computer and Systems Sciences Internationa. 8():

6 Koesnikov A., Zarembo Ya., Zarembo V. 1. Discharge of a Copper-Magnesium Gavanic Ce in the Presence of a Weak Eectromagnetic Fied // Russian Journa of Physica Chemistry A. 81(7): Peshivtseva Y., Rapoport E. 1. The Successive Parameterization Method of Contro Actions in Boundary Vaue Optima Contro Probems for Distributed Parameter Systems. Journa of Computer and Systems Sciences Internationa. 8(3):