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1
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.
2
Content available 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.
3
Content available Free vibrations of geometrically nonlinear column locally resting on the winkler elastic foundation under the specific load
Szmidla J., Cieślińska-Gąsior I.
Vibrations in Physical Systems
|
2016
|
Vol. 27
355--362
EN
The results of theoretical and numerical studies concerning continuous system subjected to the follower force directed towards the positive pole, locally resting on Winkler elastic foundation are presented in this paper. The load by follower force directed towards the positive pole is guaranteed by loading structure built of loading and receiving heads made of elements of circular outlines. Abovementioned heads are real constructions, used in experimental research of continuous systems. Taking into account total mechanical energy of the system, the Hamilton’s principle and the small parameter method, the differential equations of motion and boundary conditions of the considered column were determined. On the basis of a solution of the issue of dynamics of the system, an appropriate formulas were formulated and then the trajectory of curves on the plane frequency of free vibrations – the value of external load were calculated taking into considerations physical and geometrical parameters of the structure, including parameter of loading head and parameters describing Winkler elastic foundation.
4
Wpływ parametrów podłoża Winklera na stateczność modelu kolumny geometrycznie nieliniowej przy obciążeniu swoistym
Szmidla J., Cieślińska-Gąsior I.
Modelowanie Inżynierskie
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2016
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T. 28, nr 59
69--75
PL
W pracy zaprezentowano wyniki badań teoretycznych i numerycznych kolumny geometrycznie nieliniowej poddanej działaniu obciążenia siłą śledzącą skierowaną do bieguna dodatniego, spoczywającej miejscowo na podłożu sprężystym Winklera. Jeden z prętów układu wsparto na pewnej długości podłożem sprężystym Winklera. Obciążenie siłą śledzącą skierowaną do bieguna dodatniego realizowane jest poprzez głowice: wymuszającą i przyjmującą obciążenie, zbudowane z elementów o zarysie kołowym. W pracy wyznaczono wartości obciążenia bifurkacyjnego kolumny przy wybranych parametrach geometrycznych i fizycznych układu w tym usytuowania i rozmiaru podłoża sprężystego Winklera względem długości kolumny. Uzyskane wyniki symulacji numerycznych zostały porównane z rezultatami obliczeń układu liniowego miejscowo spoczywającego na podłożu Winklera (układ odniesienia). Biorąc pod uwagę zmienne parametry głowicy realizującej obciążenie, wyznaczono krzywe obrazujące zjawisko lokalnej i globalnej utraty stateczności rozważanego układu przy zadanej asymetrii sztywności na zginanie. Celem nadrzędnym pracy jest określenie optymalnych parametrów opisujących podłoże sprężyste (rozmiar i położenie), przy których uzyskuje się maksymalne wartości obciążenia bifurkacyjnego, przy zadanej jego sprężystości.
EN
Results of theoretical and numerical research of geometrically nonlinear column subjected to the follower force directed towards the positive pole locally resting on the Winkler elastic foundation are presented in this paper. One of the members of the system is supported on a certain length on the Winkler elastic base. The follower force directed towards the positive pole is implemented through loading and receiving heads built of elements of a circular outlines. In this work, the values of bifurcation load for selected geometrical and physical parameters of structure, including location and size of elastic foundation in relation to the length of the column are determined. Obtained results of the numerical calculations are compared with the results corresponding to the linear system locally resting on Winkler elastic foundation (comparative system). Taking into account variable parameters of loading head, the curves presenting phenomenon of the local and global loss of stability of considered column for given asymmetry of flexural stiffness are determined. The main goal of the work is to obtain the optimal parameters describing the elastic base (length and location), for which the maximum values of the bifurcation load are achieved for given elasticity.
5
Stateczność niepryzmatycznej kolumny smukłej poddanej obciążeniu siłą śledzącą skierowaną do bieguna dodatniego
Szmidla J., Jurczyńska A.
Modelowanie Inżynierskie
|
2016
|
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.
6
Content available remote 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
|
2016
|
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).
7
Content available remote Free Vibrations of the Slender Non-Prismatic Column Under the Specific Load Regarding the Joint Flexibility
Szmidla J., Jurczyńska A.
Machine Dynamics Research
|
2016
|
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.
8
Content available Free vibrations of column locally resting on the Winkler elastic foundation under the specific load
Szmidla J., Cieślińska-Gąsior I.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
Vol. 15, nr 1
181-190
EN
The results of theoretical and numerical research into free vibrations of column subjected to the follower force directed towards the positive pole locally resting on Winkler elastic foundation were presented in this work. Differential equations of motion and boundary conditions of considered system were determined taking into account total mechanical energy and Hamilton’s principle. On the basis of solution of boundary problem, the courses of free vibrations frequency curves as a function of external load were obtained taking into consideration parameters of length and location of Winkler elastic base and also parameters describing loading head geometry.
9
Content available An influence of the geometrical nonlinearity of a frame column of flat frame type Γ subjected to the follower force directed towards the positive pole on its stability
Szmidla J., Wiktorowicz J.
Journal of Applied Mathematics and Computational Mechanics
|
2016
|
Vol. 15, nr 3
155--165
EN
The results of the theoretical and numerical research for the purpose of designation of local and global instability areas of a rectangular frame with a geometrically nonlinear frame column subjected to the follower force directed towards the positive pole are presented in this work. Taking into consideration the total potential energy of an analysed system, the equations describing transverse and longitudinal movements and the boundary conditions necessary for the problem solution are formulated. On the basis of the static criterion of the loss of stability, the influences of the variable flexural stiffness of the system and the geometrical and physical parameters of the flat frame on the value of the bifurcation load are obtained. The results of numerical computations are compared with appropriate values of critical load of the comparative system.
10
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
|
2015
|
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.
11
Content available remote 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
|
2015
|
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.
12
Content available 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
|
2015
|
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.
13
Content available The vibrations and the stability of a flat frame type gamma realizing the Euler’s load taking into account the vulnerability of the structural node connecting the pole and the bolt of the system
Szmidla J., Wiktorowicz J.
Vibrations in Physical Systems
|
2014
|
Vol. 26
297--304
EN
The paper presents the results of theoretical research and numerical calculations of the vibration and the stability of a twin rod flat frame subjected to the Euler's load. Considering the total mechanical energy of the system and using the kinematic stability criterion (Hamilton's principle) is determined by the equations of motion and boundary conditions considered system. The results of numerical calculations are presented at selected geometrical and physical parameters in the system for selected values of the rotary spring stiffness modeling the structural rigidity of the node connecting bolt with the column of frame.
14
Content available Impact of pre-stress on stability and vibration of geometric nonlinear column at a load force directed to the positive pole
Szmidla J., Cieślińska-Gąsior I.
Vibrations in Physical Systems
|
2014
|
Vol. 26
289--296
EN
The subject of the results of theoretical and numerical studies measures to designate the effect of pre-stress on the vibration of geometrically non-linear column exposed to the load force directed to the positive pole. Considering the total mechanical energy of the system the equation of motion and the boundary conditions necessary to solve the boundary problem were determined. Based on the kinetic stability criterion the range of values of internal compression forces was determined in which the growth of critical load of the column above the local loss of stability was achieved. In the research the geometrically non-linear system was analysed with variable asymmetry of bending stiffness and at selected values of geometrical parameters of the head realizing the load.
15
Content available remote The Stability of Geometrically Nonlinear Column Loaded by a Generalized Force Directed to Positive Pole at Stepped Variable of Flexural Rigidity
Szmidla J., Cieślińska-Gąsior I., Wiktorowicz J.
Machine Dynamics Research
|
2014
|
Vol. 38, No. 1
65-72
EN
The results of theoretical and numerical research that aims at determining areas of local and global instability of geometrically non-linear column loaded by a generalized force directed towards a positive pole at stepped variable of flexural rigidity are presented in the paper. Taking into account total potential energy of column, equations of transverse displacements and boundary conditions necessary to solve the issue were obtained. On the basis of the static criterion of stability an influence of variable flexural stiffness and geometrical and physical parameters of system on the value of the bifurcation load of considered structure was determined. The results of numerical calculations were referred to value of critical load of relevant linear column (a comparative column).
16
Content available remote 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
|
2014
|
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.
17
Stateczność i drgania swobodne układu dyskretnego poddanego działaniu obciążenia swoistego przy uwzględnieniu tłumienia w węzłach konstrukcyjnych
Szmidla J., Cieślińska I.
Modelowanie Inżynierskie
|
2013
|
T. 16, nr 47
179--184
PL
W pracy zaprezentowano badania teoretyczne mające na celu opis matematyczny układu dyskretnego poddanego działaniu obciążenia siłą śledzącą skierowaną do bieguna dodatniego. W węzłach konstrukcyjnych badanego układu zamodelowano zewnętrzne tłumienie wiskotyczne. Rozpatrywana kolumna jest układem dyskretnym, w przypadku, którego uwzględnia się skończoną sztywność występujących węzłów konstrukcyjnych, modelowanych za pomocą sztywności sprężyn rotacyjnych. Obciążenie siłą śledzącą skierowaną do bieguna dodatniego realizowane jest poprzez głowice wymuszające i przyjmujące obciążenie, zbudowane z elementów o zarysie kołowym. W pracy wyznaczono wartości obciążenia krytycznego oraz przebieg zmian częstości drgań własnych kolumny, w funkcji obciążenia zewnętrznego, dla zadanych stałych fizycznych, przy stałej i zmiennej geometrii układu w tym głowicy realizującej obciążenie.
EN
This article concerns the theoretical studies of mathematical description of the discreet system subjected to the load force directed to the positive pole and the tracking force directed to the positive pole. The external viscous damping was modeled in the structural nodes of the studied system. The column considered is a discreet system in case of which the finite stiffness of structural nodes occurring, modeled by rotational spring stiffness, is concerned. The load tracking force directed to the positive pole is realized by forcing heads and the host load, composed of the elements of a circular outline. The heads are real structures, used in experimental studies of continuous systems. The study determined the values of the critical load a vibration frequency changes of their own columns, as a function of an external load for the selected physical constants with a fixed and variable geometry system.
18
Content available remote Optimal Shape of a Uniformly Convergent Column Subjected to a Load by the Follower Force Directed Towards the Positive Pole with Regard to the Condition of Permissible Stresses
Tomski L., Szmidla J., Yatsenko D.
Mechanics and Mechanical Engineering
|
2013
|
Vol. 17, nr 2
187--196
EN
The results of theoretical research and numerical computations into the optimization of the shape of a cantilever column subjected to a load by the follower force directed towards the positive pole are presented in this paper. The equation of the bending line and the cross–section of the considered column were derived and an adequate boundary problem was formulated on the basis of the static criterion of stability. The additional condition of permissible stresses of an optimized column was assumed with regard to one of the three classes of the columns. The value of the critical load and optimum shape of the system, which fulfils the condition of compressive strength of the column rod for the chosen case of a load, were determined.
19
Content available remote The free vibrations of a column with an optimum shape with regard to the value of the critical load subjected to a generalized load by a force directed towards the positive pole
Szmidla J.
Vibrations in Physical Systems
|
2012
|
Vol. 25
387--392
EN
The results of numerical computations and theoretical research into the free vibrations of a column subjected to the generalized load by a force directed towards the positive pole are presented in this paper. The total mechanical energy of the column was formulated by taking into account the physical model of the system and the constructional solution of the loading head. The equations of motion and the boundary conditions of the considered system are determined. The curve courses of changes in the eigenvalues in the plane: load – natural frequency are shown on the basis of the solution to the boundary problem, which is obtained by considering the kinetic criterion of the stability. The changes in natural frequencies were determined for the chosen values of the geometrical parameters of the loading head. The accepted distribution of the bending rigidity along the length of the column corresponds to the systems with maximum values of the critical load at the assumed optimization condition of constant volume of the structure.
20
Content available remote The free vibrations and optimization of the shape of a column subjected to a load by a follower force directed towards the positive pole applying the variational method
Szmidla J., Yatsenko D.
Vibrations in Physical Systems
|
2012
|
Vol. 25
393--398
EN
The results of numerical computations and theoretical research into optimization of the shape and the free vibrations of a cantilever column subjected to a load by the follower force directed towards the positive pole are presented in this paper. The systems of equations: of motion and of cross-sectional area of the considered column were determined on the basis of the total mechanical energy and Hamilton’s principle. Taking into account the formulated boundary conditions, optimized shapes of the systems were determined for the chosen values of geometrical parameter of a head subjected to a load. The values of natural frequency of the considered system were obtained taking into account distribution of flexural rigidity along the column length while the values of critical load of optimized column were obtained on the basis of curves of changes in the basic natural frequency in relation to external load. The results of numerical computations were presented in the case of one from three classes of the systems – versatile uniformly convergence column for the assumed criterion of constant volume of the system.
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