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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

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2017

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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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Free vibrations of non-prismatic slender system subjected to the follower force directed towards the positive pole

Szmidla J., Jurczyńska A.

Vibrations in Physical Systems

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2016

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Vol. 27

365--368

EN

The paper contains the results of theoretical and numerical studies within the scope of kinetic criterion of stability loss of slender non-prismatic column subjected to the follower force directed towards the positive pole (the case of specific load). Shape of the system approximation by a linear function and polynomial of degree 2 was considered. On the basis of the Bernoulli – Euler’s theory, the mechanical energy was defined. The differential equations of motion and natural boundary conditions were determined according to the Hamilton’s principle. The issue of free vibrations was solved using the small parameter method. Within the range of numerical calculations, the changes in the eigenvalues were presented as a function of external load with variable geometrical parameters of the system, including parameters resulting from the shape approximation and parameters of loading structure.

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Stateczność niepryzmatycznej kolumny smukłej poddanej obciążeniu siłą śledzącą skierowaną do bieguna dodatniego

Szmidla J., Jurczyńska A.

Modelowanie Inżynierskie

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2016

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T. 29, nr 60

67--73

PL

W pracy zaprezentowano wyniki badań teoretycznych i numerycznych odnośnie do stateczności smukłego układu niepryzmatycznego reprezentowanego przez kolumnę o zmiennym wzdłuż osi układu przekroju poprzecznym. Rozważany układ poddany jest obciążeniu siłą śledzącą skierowaną do bieguna dodatniego (przypadek obciążenia swoistego). Na podstawie modelu fizycznego analizowanej kolumny opracowano model matematyczny. Na podstawie teorii Bernoullego – Eulera określono energię mechaniczną rozważanej struktury. Zagadnienie brzegowe sformułowano przy wykorzystaniu zasady minimum energii potencjalnej (statyczne kryterium utraty stateczności). Na podstawie otrzymanych różniczkowych równań przemieszczeń oraz ich rozwiązań, przy uwzględnieniu warunków brzegowych i warunków ciągłości, opracowano programy obliczeniowe umożliwiające analizę numeryczną.

EN

In this work, results of theoretical and numerical research on the stability of a slender non-prismatic system represented by the column of a cross-section variable along the axis of the system are presented. Considered system is subjected to the follower force directed towards the positive pole (specific load). On the basis of the physical model of examined system, the mathematical model was developed. The mechanical energy of considered structure was determined based on the Bernoulli – Euler’s theory. The boundary problem was formulated using the minimum potential energy method (the static criterion of loss of the stability). Subsequently, on the basis of obtained differential equations of displacement describing the system and their solutions, with taking into considerations the geometrical and natural boundary conditions, the computing programmes were prepared.

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Local and Global Instability of the Slender Geometrically Nonlinear SystemWith Non-Prismatic Element Subjected to the Euler’s Load

Szmidla J., Skrzypczak T., Jurczyńska A.

Machine Dynamics Research

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2016

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Vol. 40, No. 4

79--88

EN

A theoretical considerations and numerical calculations concerning the issue of the stability of the geometrically nonlinear system with non-prismatic element are presented in this work. The analysed columns were subjected to the Euler’s load. On the basis of the minimum potential energy principle as well as the small parameter method, the differential equations of displacements were formulated and its solutions were obtained. The assumption that the approximation of the non-prismatic rod satisfies the condition of constant total volume resulting from the value of the coefficient of flexural stiffness distribution has been made. The results of the carried out numerical simulations refer to the local and global stability loss. It has been proved that taking into consideration in the geometrically nonlinear system appropriate shaped rod of variable cross-section causes an increase in the value of bifurcation load and in a consequence an „exit” from the area of the local instability (loss of rectilinear form of static equilibrium).

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Free Vibrations of the Slender Non-Prismatic Column Under the Specific Load Regarding the Joint Flexibility

Szmidla J., Jurczyńska A.

Machine Dynamics Research

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2016

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Vol. 40, No. 3

79--89

EN

The issue of the free vibrations of the non-prismatic rod subjected to the selected case of the specific load has been studied. In the carried out simulations, a flexibility of a constructional joints modelled by the translational and rotational springs at the point of mounting or on the free end of analysed system was taken into account. The shape of rod was approximated by linear function and by polynomial of degree 2, under the condition of constant volume of the column. After prior definition of total mechanical energy, a differential equations of motion as well as a boundary conditions were formulated on the basis of the Hamilton’s principle. The results of the numerical calculations refer to an influence of variable cross-section of the rod, joint flexibility and a geometry of a loading structure on the value of the frequency of free vibrations due to the external load (a characteristic curves) and on the critical load.

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Drgania swobodne wstępnie sprężonej kolumny geometrycznie nieliniowej poddanej obciążeniu siłą skierowaną do bieguna dodatniego miejscowo spoczywającej na podłożu typu Winklera

Szmidla J., Jurczyńska A.

Modelowanie Inżynierskie

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2015

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T. 23, nr 54

87--93

PL

Praca zawiera wyniki badań teoretycznych i numerycznych w zakresie kinetycznego kryterium stateczności miejscowo spoczywającej na podłożu sprężystym Winklera wstępnie sprężonej kolumny geometrycznie nieliniowej. Analizowany układ obciążono siłą skierowaną do bieguna dodatniego. Wykorzystując teorię Bernoullego – Eulera, sformułowano energię mechaniczną układu. Na podstawie zasady Hamiltona wyznaczono różniczkowe równania ruchu oraz naturalne warunki brzegowe. Zagadnienie drgań swobodnych rozwiązano za pomocą metody perturbacyjnej (metody małego parametru). W ramach obliczeń numerycznych wyznaczono zmiany wartości własnych w funkcji obciążenia zewnętrznego przy zadanych parametrach geometrycznych i fizycznych układu, w tym sztywności podłoża oraz parametru wstępnego sprężenia.

EN

The results of theoretical and numerical analysis of the kinetic criterion of stability of prestressed geometrically nonlinear column locally lying on Winkler elastic foundation are presented in this paper. Considered system is subjected to a force directed towards a positive pole. According to the Bernoulli - Euler’s theory total mechanical energy of structure is defined. Taking into account the Hamilton’s principle, the differential equations of motion and the natural boundary conditions are formulated. The issue of free vibrations is solved by using perturbation method (small parameter method). In the range of numerical analysis, the eigenvalues as a function of external load are determined at various values of geometrical and physical parameters including stiffness of elastic base and prestressing parameter.

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The Tapered Column Shape Optimisation in a Plane Perpendicular to the Buckling Plane Subjected to a Load by the Follower Force Directed to the Positive Pole

Szmidla J., Jurczyńska A.

Machine Dynamics Research

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2015

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Vol. 39, No. 2

33-44

EN

The results of theoretical and numerical research on shape optimisation of cantilever column subjected to a load by the follower force directed towards the positive pole are presented in this paper. Deflection line and cross section area equations of considered column and appropriate boundary problem were derived on the basis of static criterion of stability assuming constant volume of the rod. Having regard to the column’s taper in the plane perpendicular to the buckling plane, an extra conditions due to allowable stresses of structure were taken into account. Optimal shape and the critical load value of system meeting the compressive stressrequirement were determined at various values of radius of loading head.

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An influence of the Winkler elastic foundation of a geometrically nonlinear column loaded by force directed towards the positive pole on a type of system

Jurczyńska A., Szmidla J.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 4

79--91

EN

Depending on the mutual relation between external load, tendon’s length parameter resulting from the direction of loading force and the free vibration frequency parameter, the slope of characteristic curves of a considered column subjected to force directed towards the positive pole can take the negative, zero and positive value. The purpose of this paper is to determine the criterion that allows for classification of an analysed structure to divergent or divergent pseudo-fluttering type of system. On the basis of obtained formulas, the ranges of parameters describing the Winkler elastic foundation for which the considered system may be classified as one of the abovementioned types were determined.

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Free Vibrations and Stability of Geometrically Non-Linear Column Loaded by a Force Directed Towards a Positive Pole Partially Lying on Winkler Elastic Foundation

Szmidla J., Jurczyńska A.

Machine Dynamics Research

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2014

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Vol. 38, No. 2

59-68

EN

The results of theoretical and numerical research on the stability and free vibrations of a geometricaly non-linear column loaded by a force directed towards a positive pole partially lying on Winkler elastic foundation are presented in the paper. Equations of motion and boundary conditions of considered structures were formulated on the basis of the principle of minimum potential energy and the Hamilton’s principle and then solved using the small parameter perturbation method. For non-linear system and linear column respectively, the value of bifurcation and critical load was obtained for various values of geometrical and physical parameters of considered structures. The ranges of local and global instability were determined. Application of Winkler’s elastic base causes an increase in the value of the bifurcation load and this is the method which allows the geometrically non-linear system to exit from the region of the local instability. The courses of natural vibration frequencies in relation to the external load and the influence of the physical and geometrical parameters on its value are determined.