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  1. 25 Vibrations in Physical Systems
  2. 16 Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
  3. 8 Modelling in Engineering
  4. 8 Modelowanie Inżynierskie
  5. 5 Scientific Letters of Rzeszow University of Technology. Mechanics
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  1. 14 Szmidla J.
  2. 12 Uzny S.
  3. 10 Tomski L.
  4. 7 Jędrysiak J.
  5. 6 Posiadała B.
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  1. 1 2021
  2. 2 2020
  3. 7 2019
  4. 4 2018
  5. 9 2017
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1
Content available Free vibrations of microstructured functionally graded plate band with clamped edges
Kaźmierczak-Sobińska Magda, Jędrysiak Jarosław
Vibrations in Physical Systems
|
2021
|
Vol. 32, nr 2
art. no. 2021216
EN
In this paper there are presented free vibrations of thin functionally graded plate band. This kind of plates has tolerance-periodic microstructure on the microlevel in planes parallel to the plate midplane. Dynamic problems of plates of this kind are described by partial differential equations with highly oscillating, tolerance-periodic, non-continuous coefficients. Thus, there are proposed here two models describing these plates by equations with smooth, slowly-varying coefficients. As an example there are analyses of free vibration frequencies for thin functionally graded plate band clamped on both edges. Using the known Ritz method the frequencies are obtained in the framework of proposed two models - the tolerance model and the asymptotic model.
2
Content available Liquid vibrations in cylindrical tanks with and without baffles under lateral and longitudinal excitations
Strelnikova E., Kriutchenko D., Gnitko V., Tonkonozhenko A.
International Journal of Applied Mechanics and Engineering
|
2020
|
Vol. 25, no. 3
117--132
EN
The paper is devoted to issues of estimating free surface elevations in rigid cylindrical fluid-filled tanks dunder external loadings. The possibility of baffles installation is provided. The liquid vibrations caused by lateral and longitudinal harmonic loadings are under consideration. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations are considered as basic functions for the analysis of forced and parametric vibrations. The modes of the free liquid vibrations in baffled and un-baffled cylindrical tanks are received by using single-domain and multi-domain boundary element methods. Effects of baffle installation are studied. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations is obtained. The numerical simulation of free surface elevations at different loadings and baffle configurations is accomplished. Beat phenomena effects are considered under lateral harmonic excitations. The phenomenon of parametric resonance is examined under longitudinal harmonic excitations.
3
Content available Effect of kerr foundation and in-plane forces on free vibration of fgm nanobeams with diverse distribution of porosity
Jankowski Piotr
Acta Mechanica et Automatica
|
2020
|
Vol. 14, no. 3
135--143
EN
In the present paper, the effect of diverse distribution of functionally graded porous material and Kerr elastic foundation on natu-ral vibrations of nanobeams subjected to in-plane forces is investigated based on the nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. All the displacements gradients are assumed to be small, then the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded (FG) porous material are expressed by nonlocal and length scale parameters and power-law variation of material pa-rameters in conjunction with cosine functions. It created possibility to investigate an effect of functionally graded materials with diverse dis-tribution of porosity and volume of voids on mechanics of structures in nano scale. The Hamilton’s variational principle is utilized to derive governing equations of motion of the FG porous nanobeam. Analytical solution to formulated boundary value problem is obtained in closed-form by using Navier solution technique. Validation of obtained results and parametric study are presented in tabular and graphical form. Influence of axial tensile/compressive forces and three different types of porosity distribution as well as stiffness of Kerr foundation on natural frequencies of functionally graded nanobeam is comprehensively studied.
4
Content available Nonlinear vibrations of the partially tensioned system taking into account the asymmetry of the stiffness of supports
Uzny Sebastian, Osadnik Michał
Vibrations in Physical Systems
|
2019
|
Vol. 30, nr 2
art. no. 2019217
EN
The nonlinear vibrations of the partially tensioned slender column are presented in this paper. The considered system is subjected to Euler’s load, which direction of action is consistent with the non-deformed axis of the column. The magnitude of the external load is variable and its application point is located at different heights between the upper and lower supports. In addition, the longitudinal displacement and rotation of both of the system ends are limited by the discrete elements in the form of translational and rotational springs. This nonlinear system is based on the screw drive used in the vertical lift platform for disabled people or cargo lift equipped with an engine room located in the lower part of the frame. The boundary problem of free vibrations of the mentioned system has been formulated on the basis of Bernoulli - Euler theory and due to nonlinear expressions the solution of the problem was conducted with small parameter method. The results of numerical simulations are concern on linear and nonlinear component of vibrations in relation to the location of external load application and influence of asymmetric value of supports stiffness on the free vibration frequency.
5
Content available Free vibrations of the column taking into account compressive and thermal load
Uzny Sebastian, Kutrowski Łukasz, Skrzypczak Tomasz
Vibrations in Physical Systems
|
2019
|
Vol. 30, nr 2
art. no. 2019211
EN
Free vibrations of slender systems are the subject of many scientific and research works. In this work, the boundary problem of free vibrations of a compressed column, which is additionally heat loaded, is considered. The issue of heat flow in the column is solved using the Finite Element Method. Averaged distribution of material properties is obtained in individual segments of the column in subsequent heating times. The mathematical model of free vibrations takes into account the thermal expansion of the material and the effect of changing the Young's modulus resulting from the effect of heat load. The boundary problem of the free vibrations of the considered system is limited to the linear range (the linear component of natural frequency is considered). The influence of the heat source exposure time on the course of characteristic curves (on the plane: load – natural frequency) is determined. The results are presented for various column diameters.
6
Content available Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials
Szybiński Józef, Ruta Piotr
Studia Geotechnica et Mechanica
|
2019
|
Vol. 41, nr 1
1--12
EN
This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).
7
Content available Shape optimization of a column under generalized load with a force directed towards positive pole
Szmidla Janusz, Jurczyńska Anna
Quality Production Improvement - QPI
|
2019
|
Vol. 1, Iss. 1
326--333
EN
In this paper, the issue of shape optimization of a column subjected to the generalized load with a force directed towards the positive pole (L. Tomski’s load, specific load) was considered. Based on the Hamilton’s principle, the differential equations of movement and boundary conditions describing the system were formulated. Taking into account a kinetic criterion of stability loss and a condition of constant total volume, the scope of changes in natural frequency as a function of an external load was determined with selected geometrical and physical parameters of the loading structure. On the basis of obtained results, values of geometrical parameters of individual column segments were determined, at which the maximum critical load value was obtained. In order to find the maximum critical force, which is a function of many variables, the simulated annealing algorithm was used.
8
Analiza drgań własnych osłony wału turbiny napędowej silnika lotniczego
Maciejowska Kaja, Noga Stanisław
Przegląd Mechaniczny
|
2019
|
nr 11
27--32
PL
W artykule omówiono drgania własne osłony wału turbiny napędowej silnika lotniczego PZL-10W. Zasadniczym celem pracy było zidentyfikowanie modelu modalnego osłony, składającego się z częstotliwości własnych, współczynników tłumienia i postaci drgań. Wykonano eksperyment modalny osłony nieuszkodzonej oraz pękniętej w eksploatacji w celu zbadania zmiany właściwości dynamicznych na skutek uszkodzenia. Ustalono zbieżność otrzymanych wyników na podstawie przyjętych miar niedopasowania. Wyznaczono błąd względny częstotliwości własnych oraz podobieństwo postaci drgań na podstawie wskaźnika MAC.
EN
In the paper the natural vibrations of the PZL-10W aircraft engine turbine shaft shield are discussed. The goal was to identify a shield modal model consisting of natural frequencies, damping factors and forms of vibration. A modal experiment of undamaged and cracked shield in operation was performed to examine the change in dynamic properties due to damage. The convergence of the results obtained was determined on the basis of adopted mismatch measures. The relative error of natural frequencies and the similarity of the vibrations were determined on the basis of the MAC indicator.
9
Koncepcja wykorzystania modułów tensegrity w konstrukcjach wież stalowych
Małyszko Leszek, Rutkiewicz Andrzej
Inżynieria i Budownictwo
|
2019
|
R. 75, nr 7-8
331--334
PL
Przedstawiono wyniki własnych badań właściwości statycznych i dynamicznych podstawowych modułów tensegrity, które mogą być wykorzystane w konstrukcji wież stalowych. Złożoność odpowiedzi na obciążenie statyczne zilustrowano za pomocą nieliniowego modelu modułu o trzech niewiadomych, obciążonego pionowo, z uwzględnieniem dużych przemieszczeń. Na podstawie eksperymentalnej analizy modalnej zbadano zależność podstawowej częstotliwości drgań własnych od stopnia sprężenia modułu.
EN
Results of research on static and dynamic tests carried out on tensegrity modules, which can be utilized in steel tower construction, were presented. Complexity of static response was illustrated by a nonlinear model of an axially loaded module, taking into account large displacements. Based on the experimental modal analysis tests, the dependence of the first eigenfrequency in relation of self-stress magnitude was found.
10
Content available Linear and geometrically nonlinear free and forced vibrations of multi-cracked beams
Chajdi Mohcine, Adri Ahmed, El Bikri Khalid, Benamar Rhali
Diagnostyka
|
2019
|
Vol. 20, No. 1
111--125
EN
The linear and geometrically nonlinear free and forced vibrations of Euler-Bernoulli beams with multicracks are investigated using the crack equivalent rotational spring model and the beam transfer matrix method. The Newton Raphson solution of the transcendental frequency equation corresponding to the linear case leads to the cracked beam linear frequencies and mode shapes. Considering the nonlinear case, the beam transverse displacement is expanded as a series of the linear modes calculated before. Using the discretised expressions for the total strain and kinetic energies and Hamilton’s principle, the nonlinear amplitude equation is obtained and solved using the so-called second formulation, developed previously for similar nonlinear structural dynamic problems, to obtain the multi-cracked beam backbone curves and the corresponding amplitude dependent nonlinear mode shapes. Considering the forced vibration case, the nonlinear frequency response functions obtained numerically near to the fundamental nonlinear mode using a single mode approach show the effects of the number of cracks, their locations and depths, and the level of the concentric harmonic force. The inverse problem is explored using the frequency contour plot method to identify crack parameters, such as the crack locations and depths. Satisfactory comparisons are made with previous analytical results.
11
Content available remote On the modeling of periodic sandwich structures with the use of the broken line hypothesis
Marczak J.
Mechanics and Mechanical Engineering
|
2018
|
Vol. 22, nr 3
761--773
EN
In this paper a dynamic analysis of sandwich plate with a certain periodic microstructure is considered. The initial system of governing equations is derived basing on the classic broken line hypothesis. As a result of transformations one can obtain a system of three differential equations of motion with periodic, highly oscillating and non-continuous coefficients. In order to derive a system of equations with constant coefficients tolerance averaging technique is applied. Eventually, in the calculation example a free vibration analysis of certain periodic plate strip is performed with the use of both the derived model and a FEM model. It can be observed that the consistency of obtained results is highly dependent on the calculation assumptions.
12
Content available remote Kładki dla pieszych. Cz.3. Parametry analiz dynamicznych
Pańtak M.
Builder
|
2018
|
R.22, nr 1
46--49
13
Content available A Fundamental Study on the Free Vibration of Geometrical Nonlinear Cantlever Beam using an Exact Solution and Experimental Investigation
Jamal-Omidi M., Shayanmehr M., Sazesh S.
Archive of Mechanical Engineering
|
2018
|
Vol. LXV, nr 1
65--82
EN
Two fundamental challenges in investigation of nonlinear behavior of cantilever beam are the reliability of developed theory in facing with the reality and selecting the proper assumptions for solving the theory-provided equation. In this study, one of the most applicable theory and assumption for analyzing the nonlinear behavior of the cantilever beam is examined analytically and experimentally. The theory is concerned with the slender inextensible cantilever beam with large deformation nonlinearity, and the assumption is using the first-mode discretization in dealing with the partial differential equation provided by the theory. In the analytical study, firstly the equation of motion is derived based on the theory of large deformable inextensible beam. Then, the partial differential equation of motion is discretized using the Galerkin method via the assumption of the first mode. An exact solution to the obtained nonlinear ordinary differential equation is developed, because the available semi analytical and approximated methods, due to their limitations, are not always sufficiently reliable. Finally, an experiment set-up is developed to measure the nonlinear frequency of oscillations of an aluminum beam within a domain of initial displacement. The results show that the proposed analytical method has excellent convergence with experimental data.
14
Content available remote Application of the homotopy analysis method for determining the free vibrations of beam
Gromysz K., Słota D.
Architecture Civil Engineering Environment
|
2018
|
Vol. 11, no. 1
61--71
EN
In this paper we present the application of the homotopy analysis method for determining the free vibrations of the simply supported beam. The linear and nonlinear cases are considered. The homotopy analysis gives the possibility to search for the solution of a wide range of problems described by means of the operator equations. The numerical examples are presented to confirm the exactness and fast convergence of the introduced method. The presented computational examples confirm the precision and the fast convergence of investigated method. In the linear case we knew the exact solutions, so we could compare with them the solutions obtained with the aid of homotopy analysis method. Differences between the obtained solutions were slight. However, the advantage of the examined method is that we receive here the approximate solution in the form of continuous function which can be used then in a further analysis or to perform various simulations.
PL
W artykule przedstawiono homotopijną metodę analizy równań opisujących drgania swobodne układu o jednym stopniu swobody opisane równaniami liniowymi oraz nieliniowymi. Homotopijna metoda analizy daje możliwość poszukiwania rozwiązania szerokiego zakresu problemów opisanych za pomocą równań operatorowych. W artykule przedstawiono przykłady liczbowe w celu potwierdzenia dokładności i szybkiej zbieżności metody. W przypadku liniowym znane było dokładne rozwiązanie, dzięki czemu można było porównać je z rozwiązaniami uzyskanymi za pomocą analizy homotopijnej. Różnice między otrzymanymi roztworami były niewielkie. Zaletą badanej metody jest jednak to, że otrzymano przybliżone rozwiązania w postaci ciągłych funkcji, które można wykorzystać w dalszych analizach, w tym do interpretacji wyników badań doświadczalnych.
15
Content available remote Ruchomy oscylator na moście składanym o pośrednich podporach pływających
Szcześniak W., Ataman M.
Przegląd Komunikacyjny
|
2017
|
R. 72, nr 9
30--33
PL
W artykule omówiono drgania trzyprzęsłowego mostu pontonowego wymuszone przejazdem ze stałą prędkością oscylatora jednomasowego. Przeanalizowano zarówno drgania wymuszone jak i swobodne, kiedy oscylator jest już poza mostem. Rozważano przypadek drgań tłumionych oraz drgania bez uwzględnienia tłumienia. Układ trzech równań opisujących ruch układu most pontonowy-oscylator jednomasowy rozwiązano numerycznie. Wybrane wyniki rozwiązania przedstawiono graficznie na rysunkach.
EN
The paper deals with vibration analysis of three span floating bridge under moving oscillator. Velocity of the oscillator is constant. Forced and free vibrations of pontoon bridge are discussed. Three equations of motion of the system for each span are solved numerically. Solution of damped vibrations and solution for undamped vibrations are presented. Some results are shown in the Figures.
16
Content available Studies of the system: beam - variable mass element modeling overhead crane
Kutrowski Ł., Miara M.
Prace Naukowe Akademii im. Jana Długosza w Częstochowie. Technika, Informatyka, Inżynieria Bezpieczeństwa
|
2017
|
T. 5
165--176
EN
This paper presents the free vibration problem of beam with additionally attached mass element. The considered system can be treated as simplified to twodimensional model of the bridge crane. In this model the displacement of the hoist along the girder is analyzed and the longitudinal motion of the bridge crane does not occur. Based on the Hamilton’s principle, the boundary problem for this system has been formulated. On the basis of numerical simulations the influence of the position of the mass element (representing the hoist and mass of the load) on the free vibration frequencies of the system has been presented. The accepted mathematical model has been verified by the experimental studies.
PL
W niniejszej pracy rozważano problem drgań swobodnych belki z dodatkowo zamocowanym elementem masowym. Układ ten może modelować konstrukcję suwnicy pomostowej, przy założeniu pewnych uproszczeń. Sprowadzono model obiektu rzeczywistego do zagadnienia dwuwymiarowego, w którym rozważa się możliwość przemieszczania wciągnika wzdłuż dźwigara, a ruch wzdłużny suwnicy nie występuje. Sformułowano zagadnienie brzegowe odnośnie do drgań układu na podstawie zasady Hamiltona. Przedstawiono symulacje numeryczne określające wpływ położenia elementu masowego (który reprezentuje układ wciągnika wraz z ładunkiem) na częstości drgań własnych układu. Przeprowadzono badania eksperymentalne odnośnie do drgań rozważanego układu, pozwalające na weryfikację przyjętego modelu matematycznego.
17
Content available The effect of torsional rigidity between elements on free vibrations of a telescopic hydraulic cylinder subjected to Euler’s load
Uzny S., Kutrowski Ł.
Journal of Applied Mathematics and Computational Mechanics
|
2017
|
Vol. 16, nr 3
117--122
EN
The boundary problem of free vibrations of a hydraulic telescopic cylinder, subjected to Euler’s load was presented in this work. The task was formulated on the basis of the Hamilton’s principle. The computational model, formulated by Tomski, was taken into account during analysis. Numerical calculations concern free vibrations of multi-stage hydraulic telescopic cylinder. Results were presented in the form of characteristic curves on the plane: external load-natural vibration frequency. Non-dimensional parameters of the structure were defined in reference to the characteristic parameters of the piston rod. During numerical simulations, an influence of a non-dimensional stiffness parameter between following elements on free vibrations of a hydraulic telescopic cylinder was determined for a different number of cylinder stages.
18
Content available Low frequency sloshing analysis of cylindrical containers with flat and conical baffles
Gnitko V., Naumemko Y., Strelnikova E.
International Journal of Applied Mechanics and Engineering
|
2017
|
Vol. 22, no. 4
867--881
EN
This paper presents an analysis of low-frequency liquid vibrations in rigid partially filled containers with baffles. The liquid is supposed to be an ideal and incompressible one and its flow is irrotational. A compound shell of revolution is considered as the container model. For evaluating the velocity potential the system of singular boundary integral equations has been obtained. The single-domain and multi-domain reduced boundary element methods have been used for its numerical solution. The numerical simulation is performed to validate the proposed method and to estimate the sloshing frequencies and modes of fluid-filled cylindrical shells with baffles in the forms of circular plates and truncated cones. Both axisymmetric and non-axisymmetric modes of liquid vibrations in baffled and un-baffled tanks have been considered. The proposed method makes it possible to determine a suitable place with a proper height for installing baffles in tanks by using the numerical experiment.
19
Content available Wyznaczanie częstości drgań własnych ortogonalnych ram portalowych o węzłach podatnych
Basiński W.
Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture
|
2017
|
z. 64, nr 1
53--68
PL
W pracy przedstawiono metodę diagnostycznego szacowania rzeczywistej sztywności obrotowej węzłów podatnych w portalowych ramach stalowych na podstawie analizy pierwszej częstości drgań swobodnych. Analizę przeprowadzono dla ram z ryglem podatnie zamocowanym w słupach oraz ram ze słupami podatnie zamocowanymi słupami w fundamentach. Sporządzono wykresy zależności współczynnika λ2(k) częstości drgań własnych ram w funkcji unormowanego parametru k sztywności obrotowej węzłów przy stałym stosunku sztywności słupa do rygla EJs/EJr. Wyznaczono pasmo częstości drgań ram portalowych z ryglem podatnie zamocowanym w słupach przy zmiennym stosunku sztywności słupa do rygla EJs/EJr. Wzięto pod uwagę ciągłą zmianę sztywności obrotowej węzłów podatnych począwszy od węzła przegubowego aż do sztywnego zamocowania. Zamieszczono tablicę parametrów dynamicznych λ2(k) w funkcji unormowanych parametrów sztywności węzłów k1/k2.
EN
The study shows the diagnostic method of estimation of the real rotational stiffness of semirigid joints in the portal steel frames on the basis of the measurement of the frequency of first-order free vibrations to estimate the real distribution of bending moments. The analysis was carried out for frames with a semirigid mounted beam in the columns and frames with a semirigid mounted columns in the foundation. Graphs of the coefficient λ2(k) frequency vibrations of the frame as a function of the normalized parameter k rotational stiffness joints was done at a constant ratio of the stiffness of the column to beam EJs/EJr. Zone of frequencies of vibration-story frame with a variable ratio of the stiffness of the column to beam EJs/EJr was determined. Taken for continuously changing the rotational stiffness semirigid joints from the pinned joints to a rigid mounting. Contains an array of dynamic parameters λ2(k) as a function of normalized parameters stiffness connections k1/k2.
20
Content available Analiza drgań poprzecznych smukłej kolumny o zmiennym przekroju poprzecznym przy obciążeniu uogólnionym z siłą skierowaną do bieguna dodatniego
Szmidla J., Jurczyńska A.
Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture
|
2017
|
z. 64, nr 2/I
191--202
PL
W pracy zawarto wyniki rozważań teoretycznych oraz analizę numeryczną zagadnienia drgań swobodnych smukłego układu o zmiennym przekroju poprzecznym poddanego działaniu wybranego przypadku obciążenia swoistego. Analizowane obciążenie uogólnione z siłą skierowaną do bieguna dodatniego realizowane jest poprzez strukturę zbudowaną z głowic z zarysie kołowym: wywołującą i przejmującą obciążenie. W celu zamodelowania niepryzmatyczności kolumny, układ podzielono na n pryzmatycznych segmentów o równej długości i grubości oraz zmiennej szerokości opisanej za pomocą funkcji liniowej oraz wielomianu drugiego stopnia, przy zachowaniu warunku stałej objętości sumarycznej. W oparciu o model fizyczny układu zdefiniowano zależności określające energię mechaniczną struktury. Problem sformułowano na podstawie zasady Hamiltona (metoda drgań, kinetyczne kryterium utraty stateczności). Biorąc pod uwagę geometryczne warunki brzegowe oraz geometryczne warunki ciągłości wyznaczono różniczkowe równania ruchu poszczególnych segmentów kolumny oraz brakujące do opisu układu naturalne warunki brzegowe i naturalne warunki ciągłości. W oparciu o tak zdefiniowany model matematyczny opracowano autorskie algorytmy obliczeniowe umożliwiające badania numeryczne drgań poprzecznych układu. W ramach przeprowadzonych obliczeń określono zakres zmian częstości drgań własnych w funkcji obciążenia zewnętrznego. Dyskusji poddano wpływ zmiennych parametrów geometrycznych kolumny na wartość częstości drgań oraz typ układu, uwzględniając parametry określające kształt kolumny oraz geometrię struktury realizującej obciążenie.
EN
The results of the theoretical considerations and numerical analysis of the issue of the free vibration of the slender system of the variable cross-section under selected case of the specific load were included in this work. Analyzed generalized load with a force directed towards the positive pole is realized by the structure built of heads of the circular outlines: loading and receiving heads. In order to model the variable cross-section of the column, the system was divided into n prismatic segments of the equal length and thickness and the variable width described by the linear function and the polynomial of degree 2, fulfilling the condition of the constant total volume. On the basis of the physical model of the system, the mechanical energy of the structure was defined. The issue of the free vibration was formulated taking into account the Hamilton’s principle (energetic method, kinetic criterion of the stability loss). Taking into consideration the geometric boundary conditions and the geometric continuity conditions, the differential equations of motion of particular segments of the column as well as the natural boundary condition and the natural continuity conditions were determined. On the basis of so-defined mathematical model, the computation algorithms enabling numerical examination of the transverse vibration of the column were developed. Within the scope of the carried-out calculations, the range of the changes in the frequency of the free vibration as a function of the external load was determined. An influence of the variable geometric parameters of the column on the value of the natural frequency and the type of the system was discussed, including the parameters describing the shape of the column as well as the geometry of the loading structure.
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