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1

Asymptotic analysis for coupled parabolic problem with Dirichlet-Fourier boundary conditions in a thin domain

Dilmi Mohamed

Journal of Mathematics and Applications

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2023

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Vol. 46

163--185

EN

This paper concerns the asymptotic behaviour of the initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) posed in a thin domain with Dirichlet-Fourier boundary conditions. We first prove the existence and uniqueness of the solution to the problem for fixed ε >0 by the Galerkin method. Then, we give the characterization of the limiting behaviour of these solution as the thinness tends to zero.

2

Free flexural vibration of a sandwich beam on an elastic foundation with variable properties

Magnucki K., Wstawska I., Kedzia P.

Archives of Mechanics

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2023

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Vol. 75, nr 6

619--646

EN

Free flexural vibration of a simply supported sandwich beam on an elastic foundation is the main purpose of the presented investigation. An analytical model of multi-layered beam on elastic foundation has been prepared. The authors submitted an original beam-foundation interaction model which based on variable parameters of the foundation and their influence on the beam response. This explanation leads to the possibility of continuous characterization of the beam-foundation interplay. A nonlinear mathematical function for symmetrical properties of the foundation has been adopted. The frequency equation as a function of geometric and mechanical properties of the beam and the parameters of the elastic foundation was derived using the Galerkin method. The analytical investigation has been divided into two parts: the analysis of elastic foundation with constant and variable properties. The unconventional shape function and the function of deflection have been introduced and employed. Moreover, the finite element analysis has been performed. Sample analytical and numerical calculations have been performed, demonstrating a good concurrence between both models. The difference between analytical and numerical values of the fundamental natural frequency did not exceed 0:5%.

3

Numerical study for a second order Fredholm integro-differential equation by applying Galerkin-Chebyshev-wavelets method

Henka Youcef, Lemita Samir, Aissaoui Mohamed-Zine

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

28--39

EN

In the present paper, we apply the Galerkin method using Chebyshev wavelets to approximate the exact solution for a second order Fredholm integro-differential equation with initial conditions. This numerical method gives us a nonlinear algebraic system that would be solved using the Picard successive approximations technique. Furthermore, we show the validity and the ability of the proposed method through some illustrative examples.

4

Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations

Slimani Ali, Bouzettouta Lamine, Guesmia Amar

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

558--575

EN

Keller-Segel chemotaxis model is described by a system of nonlinear partial differential equations: a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller-Segel model coupled with Boussinesq equations. The main objective of this work is to study the global existence and uniqueness and boundedness of the weak solution for the problem, which is carried out by the Galerkin method.

5

Stability of transverse vibration of drillstring conveying drilling fluid

Tang Song, Liang Zheng, Zhao Guang-Hui

Journal of Theoretical and Applied Mechanics

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2020

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Vol. 58 nr 4

1061--1074

EN

In this paper, an analytical model is proposed to predict lateral vibration of a drillstring in a vertical well. The effect of parameters, such as stiffness of spring, average weight on bit (WOB), amplitude of fluctuating WOB and so on, on the dynamic stability of the drillstring is discussed. It is found that the interaction between the drillstring and drilling mud has a great influence on drillstring buckling and dynamics. For constant drilling pressure, the mud flow rate stabilizes the drillstring under certain conditions, but the fluctuating WOB drives the drillstring parametric resonance.

6

Inertia effects in a multilobe conical bearing lubricated with a couple stress fluid

Walicka A., Walicki E., Jurczak P.

International Journal of Applied Mechanics and Engineering

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2019

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Vol. 24, no. 2

439--451

EN

In this paper, a multilobe conical bearing is analyzed. A lubricant modelled by a couple stress fluid flows In the bearing clearance . The Galerkin method is used to determine the mechanical parameters of multilobe journal bearings. An example of a two-lobe conical bearing is discussed in detail. The inertia of the flowing lubricant is taken into account in the analysis. It has been found that the increase of the couple stress generates an increase the pressure in the clearance.

7

Investigation of free vibration and buckling of Timoshenko nano-beam based on a general form of eringen theory using conformable fractional derivative and Galerkin method

Mohammadi Farogh Souf, Rahimi Zaher, Sumelka Wojciech, Yang Xiao-Jun

Engineering Transactions

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2019

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Vol. 67, iss. 3

347--367

EN

The purpose of this paper is to study the free vibration and buckling of a Timoshenko nano-beam using the general form of the Eringen theory generalized based on the fractional derivatives. In this paper, using the conformable fractional derivative (CFD) definition the generalized form of the Eringen nonlocal theory (ENT) is used to consider the effects of integer and noninteger stress gradients in the constitutive relation and also to consider small-scale effect in the vibration of a Timoshenko nano-beam. The governing equation is solved by the Galerkin method. Free vibration and buckling of a Timoshenko simply supported (S) nano-beam is investigated, and the influence of the fractional and nonlocal parameters is shown on the frequency ratio and buckling ratio. In this sense, the obtained formulation allows for an easier mapping of experimental results on nano-beams. The new theory (fractional parameter) makes the modeling more flexible. The model can conclude all of the integer and non-integer operators and is not limited to the special operators such as ENT. In other words, it allows to use more sophisticated/flexible mathematics to model physical phenomena.

8

Nonlinear analysis of high Q radio frequency energy harvesting networks

Merz Ch., Kupris G.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2018

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T. 8, nr 2

83--86

EN

The paper focuses on nonlinear analysis of high and low Q RF energy harvesting circuits. The analysis is made mathematically and by large signal simulation via Keysight Advanced Design System. The mathematical analysis of the nonlinear harvesting circuits is done by using the Galerkin method and the simulations are performed using the harmonic balance method, which is a special version of the Galerkin method.

PL

Artykuł poświęcony jest nieliniowej analizie obwodów zbierających energię o wysokim i niskim współczynniku dobroci Q, w zakresie częstotliwości radiowych. Z użyciem Keysight Advanced Design System wykonano analizę wielkosygnałową. Przeprowadzono również matematyczną analizę nieliniowych obwodów zbierających energię z użyciem metody Galerkina oraz symulacje za pomocą metody balansu harmonicznego, która jest wersją metody Galerkina.

9

Galerkin method for bending analysis of beams on non-homogeneous foundation

Musa A., E., S.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 3

61--72

EN

In this study, a mathematical formulation for static bending analysis of a beam on a non-homogenous foundation is presented. The proposed method offers an accurate procedure for analysis and design of a beam resting on a varying soil bed. The Winkler foundation model is used and presented using discontinuous functions to account for the sudden change in the soil stiffness coefficient. The solution of the governing differential equation is then obtained using the Galerkin method with the help of approximation functions that satisfy the boundary conditions. A systematic approach for setting the approximation functions for different support and soil conditions is suggested. The accuracy of the proposed method is verified through two numerical examples, and they showed an excellent agreement with the finite element method (FEM) and available literature results.

10

Nonlinear Free Vibration Analysis of Micro-beams Resting on Viscoelastic Foundation Based on the Modified Couple Stress Theory

Jam J. E., Noorabadi M., Namdaran N.

Archive of Mechanical Engineering

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2017

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Vol. LXIV, nr 2

239--256

EN

In this paper, nonlinear free vibration analysis of micro-beams resting on the viscoelastic foundation is investigated by the use of the modified couple stress theory, which is able to capture the size effects for structures in micron and sub-micron scales. To this aim, the governing equation of motion and the boundary conditions are derived using the Euler–Bernoulli beam and the Hamilton’s principle. The Galerkin method is employed to solve the governing nonlinear differential equation and obtain the frequency-amplitude algebraic equation. Finally, the effects of different parameters, such as the mode number, aspect ratio of length to height, the normalized length scale parameter and foundation parameters on the natural frequency-amplitude curves of doubly simply supported beams are studied.

11

Homotopy analysis of a forced nonlinear beam model with quadratic and cubic nonlinearities

Shahlaei-Far S., Nabarrete A., Balthazar J. M.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 4

1219--1230

EN

This study investigates forced nonlinear vibrations of a simply supported Euler-Bernoulli beam on a nonlinear elastic foundation with quadratic and cubic nonlinearities. Applying the homotopy analysis method (HAM) to the spatially discretized governing equation, we derive novel analytical solutions and discuss their convergence to present nonlinear frequency responses with varying contributions of the nonlinearity coefficients. A comparison with numerical solutions is conducted and nonlinear time responses and phase planes are compared to the results from linear beam theory. The study demonstrates that apart from nonlinear problems of free vibrations, HAM is equally capable of solving strongly nonlinear problems of forced vibrations.

12

Investigation of mathematical models for vibrations of one dimensional environments with considering nonlinear resistance forces

Pukach P., Sokhan P., Stolyarchuk R.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2016

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Vol. 5, No 1

97--102

EN

In this paper we consider important classes of one dimensional environments, bending stiffness of which can be neglected. It is impossible to apply approximate analytical method of solution of mathematical models of dynamic processes. So justification of existence and uniqueness of solutions, carried out a qualitative their evaluation, based on numerical analysis are considering in this paper. Also the features of dynamic processes of some of examined class of systems are analyzed. Methods of qualitative study of oscillations for restricted and unrestricted bodies under the influence of the resistance forces, described in this paper are based on the general principles of the theory of nonlinear boundary value problems – Galerkin method and the method of monotonicity. Scientific novelty consists in generalization these methods of studying for nonlinear problems at new classes of oscillating systems, justification of solution correctness for specified mathematical models that have practical application in real engineering vibration systems.

13

Modal analysis of viscous flow and reduced order models

Nowak M., Stankiewicz W., Morzyński M.

Vibrations in Physical Systems

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2014

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Vol. 26

223--228

EN

Phenomena occurring in the flows are very complex. Their interpretation, as well as an effective impact on them in the flow control is often only possible with the use of modal analysis and low-dimensional models. In this paper, the selected modal decomposition techniques, namely Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) and global stability analysis, are briefly introduced. The design of Reduced Order models basing on Galerkin projection is presented on the example of the flow past a bluff body. Finally, the issues of widening of the application of the models are addressed.

14

Numerical Analysis of the Problem of Flow Past a Cylindrical Body Applying the R-Functions Method and The Galerkin Method

Lamtyugova S.N., Sidorov M. V.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2014

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Vol. 3, No 3

43--50

EN

The article considers the stationary problem of viscous incompressible fluid flow past a cylindrical body. For solving the problem it is proposed a numerical method, based on the joint use of R-functions method and the Galerkin method. The computational experiment has been conducted for the task of flow past square cylinder for different Reynolds numbers.

15

Mathematical Modeling and Numerical Analysis of Nonstationary Plane-Parallel Flows of Viscous Incompressible Fluid by R-Functions and Galerkin Method

Artyukh A., Sidorov M.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2014

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Vol. 3, No 3

3-11

EN

This paper is dedicated to nonstationary plane-parallel flows of viscous incompressible fluid in finite simply connected domains. Theorem of the solution uniqueness is presented. The method of successive approximation, the Galerkin method and the R-functions method are used to obtain the numerical solution, which was tested on the problem with known solution.

16

Analysis of disturbance torque influence on critical state in rotational systems

Chiliński B.

Transport Problems

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2013

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T. 8, z. 4

137--146

EN

Currently most of existing means of transport contains different types of rotational systems. In many cases the dynamics of such rotors substantially can influence exploitation of the whole vehicle. Moreover, in order to minimize mass of the whole object modern construction materials are applied. This causes that the dynamic phenomena may be fundamental of exploitation. The paper presents preliminary analysis of disturbance torque influence on critical state in rotational system. The consideration assumed simple physical object in the form of heavy disk embedded on weightless, elastic shaft. The shaft was supported on two bearings. In particular chapters of paper, path leading from proposition of physical model, by solution of it, to qualitative conclusions about considered object and torque disturbances influence of motion of this system, was presented. In introduction, outline of considered problem and potential opportunities of it, were demonstrated. In the next chapter, physical and mathematical model of the analysed object, was described. Next and also the last but one chapter gives a detailed discussion of mathematical model in the form of nonlinear ordinary differential equations proposed earlier. The first part of the chapter presents the possibility to solve such a problem, then it shows the simplifications which are used. Furthermore, the influence of used simplifications on the shape of analysed problem was demonstrated. Additionally, the possibility of equations solution presented in the paper was discussed. Moreover, the series of interesting properties of analysed system of equations has been shown based on founded approximate solutions. The whole paper was summarized with plans for future work and synthetic conclusions concerning the innovative control method of critical states.

PL

Aktualnie większość istniejących środków transportu zawiera różnego typu układy wirujące. W wielu przypadkach dynamika takich wirników w istotny sposób wpływa na eksploatacje całego pojazdu. Ponadto w celu zminimalizowania masy całego obiektu stosuje się nowoczesne materiały konstrukcyjne. To powoduje, że zjawiska dynamiczne, mogą mieć podstawowe znaczenie eksploatacyjne. Artykuł przedstawia wstępną analizę wpływu zaburzenia momentu skręcającego na stany krytyczne układu wirującego. Do rozważań przyjęto prosty obiekt fizyczny w postaci ciężkiego krążka osadzonego na nieważkim podatnym wale podpartym w dwóch łożyskach. W poszczególnych rozdziałach artykułu przedstawiono drogę prowadzącą od zaproponowania modelu fizycznego, przez jego rozwiązanie do jakościowych wniosków o rozpatrywanym obiekcie i wpływie zaburzenia momentu skręcającego na jego ruch. We wstępie przedstawiono zarys rozpatrywanego problemu oraz potencjalne możliwości wykorzystania opracowanego zagadnienia. W kolejnym rozdziale zaprezentowano model fizyczny oraz matematyczny dla obiektu będącego podstawą niniejszego artykułu. Następny i zarazem przedostatni rozdział w sposób szczegółowy przedstawia dyskusję zaproponowanego wcześniej modelu matematycznego w postaci układu nieliniowych równań różniczkowych zwyczajnych. Na początku rozdziału przedstawiono rozwiązywalność takiego zagadnienia, następnie opisano zastosowane uproszczenia. W dalszym ciągu zademonstrowano wpływ zastosowanych uproszczeń na kształt analizowanych równań. Dodatkowo podjęto dyskusję o możliwości rozwiązania równań przedstawionych w pracy. Ponadto na podstawie znalezionych przybliżonych rozwiązań wykazano szereg ciekawych własności analizowanego układu równań. Całość została podsumowana celami dalszej pracy oraz syntetycznymi wnioskami, dotyczącymi propozycji innowacyjnej metody sterowania stanami krytycznymi.

17

An analytical approach to large amplitude vibration and post-buckling of functionally graded beams rest on non-linear elastic foundation

Yaghoobi H., Torabi M.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 1

39--52

EN

In this paper, non-linear vibration and post-buckling analysis of beams made of functionally graded materials (FGMs) rest on a non-linear elastic foundation subjected to an axial force are studied. Based on Euler-Bernoulli beam theory and von-Karman geometric non-linearity, the partial differential equation (PDE) of motion is derived.Then, this PDE problem is simplified into an ordinary differential equation problem by using the Galerkin method. Finally, the governing equation is analytically solved using the variational iteration metod (VIM). The results from the VIM solution are compared and shown to be in excellent agreement with the available solutions from the open literature. Some new results for the non-linear natural frequencies and buckling load of functionally graded (FG) beams, such as effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogenity are presented for future references.

PL

W pracy przedstawiono analizę drgań nieliniowych i zjawisk następujących po wyboczeniu w belkach wykonanych z funkcjonalnych materiałów gradientowych (FGMs), spoczywających na nieliniowo sprężystym podłożu i jednocześnie poddanych osiowemu ściskaniu. Na podstawie teorii Eulera-Bernoulliego oraz przy uwzględnieniu geometrycznej nieliniowości von Karmana wyprowadzono cząstkowe równanie różniczkowe ruchu takich układów. Równanie to sprowadzono do postaci różniczkowej zwyczajnej za pomocą metody Galerkina. Na koniec, rozwiązano je analitycznie poprzez zastosowanie iteracyjnej metody wariacyjnej (VIM), a uzyskane rozwiązanie porównano z innymi, już istniejącymi i znanymi w literaturze, stwierdzając doskonałą zgodność. Otrzymano również nowe rezultaty w postaci określenia wpływu amplitudy drgań, sprężystości podłoża, wartości siły osiowej, rodzaju podparcia brzegów oraz niejednorodności materiału na częstości własne i obciążenie krytyczne belek gradientowych.

18

Qualitative methods for research of transversal vibrations of semi-infinite cable under the action of nonlinear resistance forces

Pukach P., Kuzio I., Sokil M.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2013

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Vol. 2, No 1

43--48

EN

The aim of paper is to study the solution of the problem of nonlinear transverse vibrations of elastic elongated body under the force of resistance in unbounded domain. Such problems have applications in various technical systems - vibration of pipelines, railways, long bridges, electric lines, optical fibers. Unboundedness of the area creates more fundamental difficulties in the study of the problem. For the considered models of nonlinear oscillations have no general analytical techniques for determining the dynamic characteristics of the oscillatory process. Therefore it is sugges ted to use qualitative methods of the theory of nonlinear boundary value problems to obtain correct problem solution conditions (existence and uniqueness of the solution). In the paper conditions of the correctness of the solution of mathematical model for these nonlinear systems (sufficient conditions of the existence and uniqueness in the class of locally integrable functions) are obtained. Methods of qualitative study of semi-infinite cable vibrations under the forces of resistance based on general principles of the theory of nonlinear boundary value problems - method of monotony and Galerkin method. Scientific novelty of the work lies in particular in the generalization of methods of studying nonlinear problems on a new class of oscillatory systems In unbounded domains, justifying the correctness of the solution with specified mathematical model, which has practical applications in real engineering oscillatory systems. The technique allows not only for proving the correctness of the model solution, but also has an opportunity in its study to apply various approximate methods.

19

Influence of some speed parameters on the dynamics of nonlinear flexural vibrations of a drill column

Pukach P., Kuzio I., Nytrebych Z.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2013

|

Vol. 2, No 4

61-66

EN

We investigate the influence of the motion of fluid flushing the cutter of a well drilling column, and the angular rotational velocity upon dynamic characteristics of its flexural vibrations. We take into account the nonlinear elastic features of column material. As a base of the research we took the Galerkin method and the Van der Pol method. Combining those two methods made possible to obtain the relations describing the main parameters of the dynamical process In both nonresonance and resonance case.

20

Dobór parametrów wzbudnika piezoelektrycznego ze względu na żądane własności dynamiczne projektowanego układu mechatronicznego

Buchacz A., Płaczek B.

Modelowanie Inżynierskie

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2012

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T. 13, nr 44

37-47

PL

W pracy przedstawiono proces wyznaczenia charakterystyki dynamicznej drgającego układu mechatronicznego oraz analizę wpływu na nią parametrów stosowanego wzbudnika .Szukaną charakterystyką opisano zależność amplitudy drgań swobodnego końca belki wspornikowej od parametrów napięcia elektrycznego, doprowadzonego do zacisków wzbudnika piezoelektrycznego. Określono możliwości sterowania własnościami dynamicznymi układu poprzez dobór jego parametrów. Rozpatrywanym układem jest belka wspornikowa z naklejonym na jej powierzchni wzbudnikiem piezoelektrycznym zasilanym z zewnętrznego źródła napięcia prądu elektrycznego. Układ modelowany jest jako jednowymiarowy, drgający giętnie układ mechatroniczny, którego analizę przeprowadzono, stosując przybliżoną metodę Galerkina.

EN

The paper presents a process of mechatronic system's dynamic characteristic determining and analysis of influence of a piezoelectric actuator's parameters on the obtained results. The considered system is a flexural vibrating cantilever beam with piezoelectric actuator glued to the beam's surface and supplied by an external voltage source. The considered system was described by developed mathematical model and approximate Galerkin method was used to analyze it. The characteristic that describes relation between amplitude of the system's vibration and electric voltage applied to the piezoelectric actuator was calculated. An eccentric tension of a glue layer between the actuator and beam's surface was considered and mechatronic system was modelled as a combined beam.