Wyniki wyszukiwania - BazTech

1

On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially “dominated” nonlinearity and singular weight

Baraket Sami, Mahdaoui Safia, Ouni Taieb

Opuscula Mathematica

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2023

|

Vol. 43, no. 1

5--18

EN

Let Ω be a bounded domain in R4 with smooth boundary and let x1, x2, . . . , xm be m-points in Ω. We are concerned with the problem [formula], where the principal term is the bi-Laplacian operator, H(x, u, Dku) is a functional which grows with respect to Du at most like |Du|q, 1 ≤ q ≤ 4, f : Ω → [0,+∞[ is a smooth function satisfying f(pi) > 0 for any i = 1, . . . , n, αi are positives numbers and g : R → [0,+∞[ satisfy |g(u)| ≤ ceu. In this paper, we give sufficient conditions for existence of a family of positive weak solutions (uρ) ρ>0 in Ω under Navier boundary conditions u = Δu = 0 on ∂Ω. The solutions we constructed are singular as the parameters ρ tends to 0, when the set of concentration S = {x1, . . . , xm} ⊂ Ω and the set Λ := {p1, . . . , pn} ⊂ Ω are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.

2

Detection and identification of nonlinear contact dynamics at workpiece clamping positions

Feng Qi, Maier Walther, Braun Steffen, Möhring Hans-Christian

Journal of Machine Engineering

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2023

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Vol. 23, No. 1

114--122

EN

All mechanical systems behave nonlinearly to a certain extent since there are always reasons for nonlinearities, such as friction and slip effects, in the actual structures. It is important to detect and identify the nonlinearity due to friction and contact in order to investigate their effect on the global behavior of the workpiece-fixture system. That is a prerequisite for modeling the dynamic contact behavior at the interface between the workpiece and clamping elements. In this research, the workpiece-fixture system was excited with a shaker using the swept sine signal. The nonlinearities could be detected by comparing and analyzing the frequency responses of the structures in Bode plots. However, the nonlinearities behaved differently at various frequencies within the observation range. Different mechanisms such as nonlinear stiffness and damping, micro-slip friction, are responsible for that. Then the nonlinear contact behavior at the clamping positions was successfully identified by means of the Hilbert transform. In addition, the clamping force directly influenced the nonlinear stiffness of the workpiece-fixture system.

3

The visual research of changes in the geometry of a rivet joint for material model effect for simulation riveted joints made of EN AW 5251

Lubas Monika

Technologia i Automatyzacja Montażu

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2022

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nr 4

56--64

EN

The paper presents the results of a numerical analysis of a single-lap joint with a blind rivet and its geometrical verification by inside views from the experiment. The research aimed to determine how the results of numerical analyses (FEM) were influenced by the method of modeling the material model and how it relates to the experimental results. As part of the analyses, a discrete riveted model and material model: linear and nonlinear were constructed. The analyses took into account various load cases (500, 800, and 900 N) to better illustrate the relationship between the numerical and experimental results. A new methodology of visualizing changes in a riveted joint's geometry was used to validate the results. The technology of making riveted joint cross-sections was developed during a static tensile test. Samples of a single lap joint with blind rivets made of aluminum sheets were subjected to a shear load. Deformations were "frozen" during the test, and sections were prepared. The microscope photos allowed for the development of a method for visualizing the deformation of the hole and rivet. The numerical results (for various loads and various material configurations) were compared with the experimental results of geometric parameters (i.e. angles between sheets or other dimensions) on the riveted joint cross-sections. The obtained results help to understand the mechanism of failure of the blind rivet under load and the complex state of loads in various stages of deformation.

PL

W pracy przedstawiono wyniki analizy numerycznej połączenia zakładkowego z nitem zrywalnym oraz weryfikację geometrii przekrojów połączeń z eksperymentu. Badania miały na celu określenie, jaki wpływ na wyniki analiz numerycznych (MES) ma sposób modelowania modelu materiałowego i jaki ma on związek z wynikami eksperymentalnymi. W ramach analiz skonstruowano model dyskretny połączenia nitowego oraz model materiałowy: liniowy i nieliniowy. W analizach uwzględniono różne przypadki obciążeń (500, 800 i 900 N), aby lepiej zilustrować związek między wynikami numerycznymi a wynikami eksperymentalnymi. Do walidacji wyników zastosowano nową metodologię wizualizacji zmian w geometrii połączenia nitowego. Technologia wykonywania przekrojów połączeń nitowych została opracowana podczas statycznej próby rozciągania. Próbki pojedynczego połączenia zakładkowego z nitami zrywalnymi i arkuszami aluminiowymi poddano testom ścinającym. Odkształcenia zostały „zamrożone” podczas eksperymentu, następnie przygotowano przekroje połączeń. Zdjęcia mikroskopowe pozwoliły na opracowanie metody wizualizacji deformacji otworu i nitu. Wyniki numeryczne (dla różnych obciążeń i różnych konfiguracji materiałów) porównano z wynikami eksperymentalnymi parametrów geometrycznych (tj. kąta między blachami, kąta obrotu nita itd.) na przekrojach połączeń nitowych. Uzyskane wyniki pozwalają zrozumieć mechanizm niszczenia nitu zrywalnego pod obciążeniem oraz złożony stan obciążeń w różnych stadiach odkształcenia.

4

Disassembly-free metrological control of analog-to-digital converter parameters

Bubela Tetiana, Kochan Roman, Więcław Łukasz, Yatsuk Vasyl, Kuts Victor, Yatsuk Jurij

Metrology and Measurement Systems

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2022

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Vol. 29, nr 4

669--684

EN

The authors update the issue disassembly-free control and correction of all components of the error of measuring channels with multi-bit analog-to-digital converters (ADCs). The main disadvantages of existing methods for automatic control of the parameters of multi-bit ADCs, in particular their nonlinearity, are identified. Methods for minimizing instrumental errors and errors caused by limited internal resistances of closed switches, input and output resistances of active elements are investigated. The structures of devices for determining the multiplicative and nonlinear components of the error of multi-bit ADCs based on resistive dividers built on single-nominal resistors are proposed and analyzed. The authors propose a method for the correction of additive, multiplicative and nonlinear components of the error at each of the specified points of the conversion range during non-disassembly control of the ADC with both types of inputs. The possibility of non-disassembly control, as well as correction of multiplicative and nonlinear components of the error of multi-bit ADCs in the entire range of conversion during their on-site control is proven. ADC error correction procedures are proposed. These procedures are practically invariant to the non-informative parameters of active structures with resistive dividers composed of single-nominal resistors. In the article the prospects of practical implementation of the method of error correction during non-dismantling control of ADC parameters using the possibilities provided by modern microelectronic components are shown. The ways to minimize errors are proposed and the requirements to the choice of element parameters for the implementation of the proposed technical solutions are given. It is proved that the proposed structure can be used for non-disassembly control of multiplicative and nonlinear components of the error of precision instrumentation amplifiers.

5

Badanie mostków termometrycznych przy wykorzystaniu kalibratora RBC

Kur Katarzyna, Burcon Emil, Kozicki Marek, Kowal Aleksandra, Dobosz Justyna, Manuszkiewicz Henryk

Metrologia i Probiernictwo : biuletyn Głównego Urzędu Miar

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2022

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nr 1 (28)

17--20

PL

Międzynarodowa Skala Temperatury z 1990 roku definiuje punkty stałe oraz liczbowe wartości ich temperatury. Dla zakresu temperatury, od punktu potrójnego wodoru w równowadze do punktu krzepnięcia srebra, jako termometry interpolacyjne stosuje się platynowe czujniki termometru rezystancyjnego (SPRT). W tego typu czujnikach do pomiaru temperatury wykorzystuje się zmianę rezystancji czujnika zależną od zmiany temperatury. Wartości temperatury definicyjnych punktów stałych MST-90 określane są z błędem do 0,1 mK. Osiągnięcie takiej dokładności pomiaru temperatury wymaga pomiaru rezystancji termometru platynowego z błędem nie większym niż 10 µΩ w całym zakresie jego stosowalności. Dokładność realizacji skali temperatury w dużej mierze zależy od dokładności pomiaru rezystancji. W pracy przedstawione są wyniki związane z wyznaczeniem błędów pomiaru mostków termometrycznych.

EN

The International Temperature Scale of 1990 defines temperature fixed points and their numerical values. For the temperature range from the triple point of equilibrium hydrogen to the freezing point of silver, standard platinum resistance thermometers (SPRT) are used as interpolation instruments. In this type of thermometers, the temperature measurement is based on the change of sensor resistance which is dependent of the change of temperature. The temperature values of the ITS-90 fixed points are specified with the error of 0,1 mK. Achieving such an accuracy requires the error of thermometer resistance measurement not worse than 10 µΩ in its entire temperature range. The accuracy of the temperature scale realization depends largely on the accuracy of resistance measurement. This paper presents the results of determination of the measure ment errors of thermometry Bridges.

6

Nonlinear vibration analysis of beam and plate with closed crack: a review

Tewelde Samrawit A., Krawczuk Marek

Acta Mechanica et Automatica

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2022

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Vol. 16, no. 3

274--285

EN

The effect of nonlinearity is high sensitivity in damage detection, especially for closed cracks and delamination. This review illustrates the results of several researchers dealing with nonlinear effects caused by the closure of cracks in the structure, i.e., beam and plate structures. Early detection of damage is an important aspect for the structure and, therefore, continuous progress is being made in developing new and effective methods that use nonlinear effects for early detection of damage and barely visible cracks, i.e., closed cracks and delamination, as well as for the determination of crack size and location. After analysing various methods, the merits, drawbacks and prospects of a number of nonlinear vibration methods for structural damage detection are discussed, and recommendations are made for future researchers.

7

The magnetic field calculation in electromechanical systems with saturated ferromagnetic structural elements

Zhyltsov Andrii, Kondratenko Ihor, Lyktei Viktoriia, Usenko Serhii

Przegląd Elektrotechniczny

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2021

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R. 97, nr 1

69--72

EN

A method for calculating the characteristics of the magnetic field in electromechanical systems with thin bridges based on the reduction of the initial-boundary value problem for Maxwell's equations to an equivalent system of integro-differential equations with its subsequent numerical solution is developed.

PL

Opracowano metodę obliczania charakterystyk pola magnetycznego w układach elektromechanicznych z cienkimi mostkami ferromagnetycznymi, polegającą na redukcji problemu początkowo-brzegowego dla równań Maxwella do równoważnego układu równań całkowo-różniczkowych z jego późniejszym rozwiązaniem numerycznym.

8

Investigation of Different Transfer Functions for Optical Limiting Amplifier used in a 2R Burst Mode Optical Regenerator

Deodhar Yash, Reddy Jeeru Jaya Sankar, Kakade Priyanka Desai, Kakade Rohan

International Journal of Electronics and Telecommunications

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2021

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Vol. 67, No. 3

409--416

EN

The major difference between a continuous mode optical regenerator (CMOR) and a burst mode optical regenerator (BMOR) is that a BMOR is capable of handling large variations in the input power which makes it useful in optical packet switched and optical burst switched networks. This is due to the optical limiting amplifier (OLA) present in the BMOR. Using computer modelling, the impact of using different OLA non-linear transfer functions on the output bit error rate of a system consisting of a cascade of 2R BMORs has been investigated. The effect of amplified spontaneous emission (ASE) noise introduced in the inter-regenerator links has also been taken into consideration. Also, a brief review of existing OLA designs is presented.

9

Ultrahigh nonlinear polarization maintaining dispersion compensating photonic crystal fiber

Ghunawat Ashish Kumar, Cherif Rim, Singh Ghanshyam

Optica Applicata

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2019

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Vol. 49, nr 1

51--63

EN

This work describes the suitability of a nano-dimension slot with low index material in the core region to achieve a highly nonlinear, polarization maintaining and dispersion compensating photonic crystal fiber. Our design is composed of a spiral shaped photonic crystal fiber with an elliptical slot made of silicon nanocrystals in the core region. The simulated results show that high nonlinear coefficients at the 1.55 μm wavelength for quasi-TE mode and quasi-TM mode are found to be equal to 1348 and 638 W–1m–1, respectively. The proposed design offers high birefringence up to 0.2503 and large negative dispersion value –1228 ps/nm/km. The proposed fiber has immense potential for realization of all-optical signal processing devices/networks and sensing applications while maintaining its polarization.

10

Model development of two-paramater half tone reproduction in inking units of printing machines

Durnyak B., Petriaszwili G., Lutskiv M., Kurka P.

Przegląd Papierniczy

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2018

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R. 74, nr 5

324--328

EN

A model of two-parameters halftone reproduction and method of adjustment process synthesis, aimed to compensate nonlinearities of screening and printing processes in consecutive inking units have been developed; the results of imitation modelling of linear screen scale reproduction have been offered in this article.

PL

Opracowano model odtwarzania półtonów reprodukcji o dwóch parametrach i metodę syntezy procesów przygotowawczych, mającą na celu kompensację nieliniowości procesów rastrowania i drukowania w ciągłych systemach farbowo-drukujących; w artykule przedstawiono wyniki modelowania imitacyjnego reprodukcji liniowej skali rastrowej.

11

Projecting procedure for metamaterial cylindrical waveguide

Reichel B., Kuszner M.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2017

|

Vol. 21, No 2

165--172

EN

In this paper a new way of derivation of an evolution equation for short pulses in a dielectric waveguide including one model of a metamaterial waveguide is shown. This derivation model relies upon projecting to an orthogonal basis. In our case such orthogonal basis for cylindrical waveguides is chosen as Bessel functions.

12

Nonlinearity and instability of the architecture language

Fomenko O., Danylov S.

Przestrzeń i Forma

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2017

|

nr 30

137--154

EN

The article deals with the issue of nonlinearity and instability of the architecture language which is manifested in modern architectural trends: architecture as performance, show, attraction. The research apparatus of the article includes elements of the systematic approach, the theory of architectural composition, methods and means of a synergetic approach in architecture.

PL

Adnotacja. W artykule rozpatrzona została kwestia nieliniowość i niestabilności architektonicznego języka, przejawiającego się we współczesnych architektonicznych tendencjach – architektura jako działanie, show, atrakcja. Naukowo-badawczy aparat artykułu zawiera elementy podejścia systemowego, teorii architektonicznej kompozycji, metodyki i środków synergetycznego podejścia w architekturze.

13

Dynamic response of a base-isolated concrete rectangular liquid-storage structure under large amplitude sloshing

Cheng X., Li D., Li P., Zhang X., Li G.

Archives of Civil Engineering

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2017

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Vol. 63, nr 1

33--45

EN

Considering concrete nonlinearity, the wave height limit between small and large amplitude sloshing is defined based on the Bernoulli equation. Based on Navier-Stokes equations, the mathematical model of large amplitude sloshing is established for a Concrete Rectangle Liquid-Storage Structure (CRLSS). The results show that the seismic response of a CRLSS increases with the increase of seismic intensity. Under different seismic fortification intensities, the change in trend of wave height, wallboard displacement, and stress are the same, but the amplitudes are not. The areas of stress concentration appear mainly at the connections between the wallboards, and the connections between the wallboard and the bottom.

PL

Gdy amplituda chlupotania cieczy jest zbliżona do częstotliwości drgań struktury magazynowania substancji ciekłych (CRLSS), wówczas osiągamy rezonans i możemy zaobserwować silne zjawisko nieliniowe. Szkoda jest znacznie większa niż chlupotanie o małej amplitudzie. Obecnie brak jest odpowiedniego raportu na temat badań dynamicznej odpowiedzi struktury magazynowania substancji ciekłych z wykorzystaniem betonu izolacyjnego (CRLSS) z chlupotaniem o dużej amplitudzie, a wpływ materiałów betonowych nie jest brany pod uwagę. W związku z tym, w niniejszej pracy, w oparciu o równanie Bernoulliego, otrzymano ograniczone wysokości fali o dużej amplitudzie chlupotania oraz małej amplitudzie chlupotania. Na podstawie równań Naviera-Stokesa ustanowiono matematyczny model chlupotania o dużej amplitudzie i zbadano odpowiedź sejsmiczną CRLSS podczas chlupotania o dużej amplitudzie. Rozważając równanie Bernoulliego i zadowalający stan, chlupotanie substancji ciekłej jest liniowe, a nieliniowy kwadratowy człon jest lekceważony. W stałym i płynnym interfejsie, struktura magazynowania substancji ciekłych spełnia warunki ciągłości przemieszczania i równowagi siły oddziaływań. Właściwość mechaniczna gumowego zabezpieczenia izolacyjnego została opisana w oparciu o konstytutywną relacje modelu Mooney-Rivlin.

14

Information about dynamics of the sea surface as a means to improve safety of the unmanned vessel at sea

Przyborski M.

Polish Maritime Research

|

2016

|

nr 4

3--7

EN

One of the fundamental states of the sea surface is its heave. Despite of years of the intense scientific inquiry, no clear understanding of the influence of this aspect on the dynamics of the sea environment has emerged. The separation of two nearby fluid elements which one may observed for example as a free floating of small objects on the sea surface (rescuers on the rough sea or small research vessels) is caused by the interaction of different components. On the other hand one may say that the heave of the sea is also a summary interaction of a few components describing the dynamics of the sea. Therefore it is the most important aspect, which influenced the dispersion phenomenon. This observation has important consequences for many different problems as for example conducting Search and Rescue missions and using unmanned ships. We would like to present results of our experiment focused on finding the answer to question about nature of the heave of the sea and its influence on safety of Unmanned Surface Vessels (USV).

15

Analysis of the stochastic technical stability of engineering structures on an example of a moving car

Kisilowski J., Zalewski J.

Journal of Theoretical and Applied Mechanics

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2016

|

Vol. 54 nr 4

1157--1167

EN

The article presents analysis of the stochastic technical stability of mathematical models (described by the second-order ordinary differential equations) of technical objects, on the example of a moving motor vehicle with the disturbed location of the center of mass, as a result of a non-uniform load. After conducting simulation of motion of a sport vehicle mathematical model, using the MSC Adams/Car software, executing a maneuver of double lane change without completely returning to the original road lane, interpretation of the results in accordance with the given definition has been attempted. Specifically, the aspect of determining the probability of the solution remaining in a certain area of feasible solutions has been highlighted, as well as determination of the magnitude of this probability for stable motion. In addition, attention has been paid to the selection of the upper limit of probability, for which, under certain conditions, stable motion can be determined.

16

Neuronowe modelowanie nieliniowości aktorów

Rykała Ł., Muszyńska M.

Przegląd Mechaniczny

|

2015

|

nr 1

30--36

PL

W artykule zaproponowano nową rozszerzoną strukturę sieci neuronowej. Struktura ta umożliwia aproksymację przedziałowo ciągłych funkcji z nieciągłościami, które mogą pojawiać się w sterowaniu aktorami. Nowa rozszerzona struktura sieci zbudowana jest z neuronów mających sigmoidalne funkcje aktywacji oraz dodatkowe neurony z funkcją aktywacji nazwaną „funkcją aproksymującą skoki funkcyjne”. Sieć ta pojawiająca się w sprzężeniu zwrotnym pętli układów sterowania znacząco zwiększa wydajność całego procesu, kompensując szkodliwe efekty tarcia. W artykule ponadto przedstawiono modele statycznego i dynamicznego tarcia z uwzględnieniem tzw. efektu Stribecka.

EN

The article proposes a new augmented structure of the neural network. This structure enables the approximation of piecewise continuous functions with discontinuities that may occur in the control process of actors. New augmented network structure is built of neurons with sigmoidal activation functions and additional neurons with activation function called “function approximating function jumps”. This network appears in the feedback loop control systems and it significantly increases the efficiency of the entire process compensating for the harmful effects of friction. In addition, the article presents models of static and dynamic friction, taking into account the Stribeck effect.

17

Kwadraturowy oscylator harmoniczny rzędu trzeciego stabilizowany elementem nieliniowym

Topór-Kamiński L., Kumor A., Guzik J., Pilśniak A., Holajn P.

Prace Naukowe Politechniki Śląskiej. Elektryka

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2015

|

z. 1

119--137

PL

W pracy opisana jest struktura realizacji oscylatora rzędu trzeciego w postaci układu połączonych w zamkniętą pętlę: wzmacniacza elektronicznego, dolnoprzepustowego układu selektywnego w postaci układu aktywnego, integratora oraz ogranicznika napięcia jako układu nieliniowego. Jako elementy aktywne zaproponowano zastosowanie klasycznych napięciowych wzmacniaczy operacyjnych. Przeprowadzono analizę pracy tego oscylatora metodą funkcji opisującej. Podano zależności określające pulsację i amplitudę wytwarzanych w układzie oscylacji. Opisano praktyczne jego realizacje z filtrem dolnoprzepustowym RLC i RC plus pojemność rzędu drugiego D. Przeprowadzono symulacje komputerowe oscylatora z układu RLD dla idealnych i rzeczywistych zastosowanych wzmacniaczy operacyjnych.

EN

In this paper a structure of the third order oscillator built with an operational amplifier, a low pass filter, an integrator and a non-linear circuit as a voltage limiter is presented. All is connected in a closed loop circuit. Properties of the oscillator are tested through simulations, in which the describing function method is used. The equations defining the amplitude and the pulsation of oscillation are given. Also implementation of the oscillator containing the low pass filter RLC and RC and the second order capacity D is shown. Finally the computer simulations of the oscillator RLD for ideal and real operational amplifiers are presented.

18

Higher Order Nonlinearity of Niho Functions

Sun G., Wu C.

Fundamenta Informaticae

|

2015

|

Vol. 137, nr 3

403--412

EN

The r-th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. In this paper we investigate the lower bound of the higher-order nonlinearity of Niho Boolean functions f(x) = tr(λxd) over F2n, where... [formula]

19

Badanie nieliniowości w precyzyjnym pomiarze przedziału czasu dla wybranych typów częstościomierzy-czasomierzy

Urban A., Czubla A., Osmyk R., Szterk P., Krehlik P., Śliwczyński Ł.

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej

|

2014

|

Nr 38

81--84

PL

Celem niniejszego artykułu jest zaprezentowanie metody i wyników badań nad nieliniowością częstościomierzyczasomierzy w pomiarach precyzyjnych przedziałów czasu, prowadzonych w GUM przy współpracy z AGH. Dokonano porównań wyników pomiarów dla częstościomierzy-czasomierzy SR620 i HP 53132A zarówno stałych i zmiennych przedziałów czasu. Potwierdzono przydatność dwukanałowego generatora/dzielnika typu Dual Channel 1/100 PPS Generator oraz komparatora częstotliwości typu A7-MX.

EN

In this paper we present the method and the last results of verification of nonlinearities in time interval measurements with the usage of Time Interval Counters (TIC), performed at Central Office of Measures (GUM) with cooperation of AGH University of Science and Technology. We consider the non-ideal operation of TIC manifested by fluctuations of differences between the current indication of TIC and the real value of the measured time interval. We observed instabilities of indications of TIC, even if the measured time interval is keeping constant. For verification of TIC operation at continuously changed time intervals, it was set up a special measurement system that performs simultaneous measurements of the phasetime changes between two standard frequency signals (with the usage of A7-MX standard frequency comparator) and time intervals between two pulse signals (with the usage of TIC under test). Due to applying a special, developed by AGH, dual channel 1/100 pps generator, the measured 1 pps (1 pulse per second) and 10 MHz signals are standardized and the phasetime between 1 pps and 10 MHz signals are kept in pairs constant. Differences between the indications of TIC and the proper measurement results for frequency allow to verify internal nonlinearities of TIC in time interval measurements. Here, we show and shortly discuss the results obtained for SR620 and 53132A universal counters. The obtained range of fluctuations of indications of SR6200 with relation to the measured time interval approaches about 190 ps, and for 53132A – about 450 ps, whereas the observed short-term noise of measurement for SR620 is about 10 ps, and for 53132A – about 100 ps typically. It was confirmed the complex and random influence of many factors on operation of TIC. Our investigation will be continued with the usage of 100 pps (100 pulses per second) signals and with automatic switching of the measured pulse signals in pairs.

20

Geometrically nonlinear free transversal vibrations of thin-walled elongated panels with arbitrary generatrix

Marchuk M., Goriachko T., Pakosh V.

Vibrations in Physical Systems

|

2014

|

Vol. 26

153--160

EN

The algorithm for finding a finite number of the first values of natural frequencies and forms of geometrically nonlinear free transverse vibrations of thin-walled elongated panels with arbitrary generatrix is proposed and verified. Under normal coordinate quadratic the approximation of displacements is used. Along the tangential coordinates used one-dimensional finite elements. The discrete variation problem is built. For its solving the perturbation method is applied. The numerical results are compared with previously obtained by other authors.