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Influence of the specific load on stability and vibrations of the multimember column with internal cracks

Sokół Krzysztof, Kuliński Krzysztof

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 4

art. no. e146408

EN

Slender systems are mostly studied when Euler’s load or follower load is considered. The use of those types of external loads results in well-known divergence or flutter shape of the characteristic curve. In this study, one takes into account the specific load which allows one to obtain an interesting divergence – pseudo flutter shape of characteristic curves on the external load–vibration frequency plane. The curves can change inclination angle as well as one can observe the change in vibration modes along them. The shape of those curves depends not only on the parameters of the slender system but also on loading heads that induce the specific load. In this study, one considers the slender multimember system in which cracks are present and weaken the host structure. The results of theoretical as well as numerical simulations are focused on the influence of the parameters of the loading heads on vibrations, stability, and loading capacity of the investigated system as well as on the possibility of partial reduction of unwanted crack effect.

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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.

4

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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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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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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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

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2012

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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.

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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

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2012

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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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Vibrations and stability of columns loaded by four-side surfaces of circular cylinders

Tomski L., Podgórska-Brzdękiewicz I.

Journal of Theoretical and Applied Mechanics

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2006

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Vol. 44 nr 4

907-927

EN

The paper presents a new type of specific load realised through components making a side surface of four circular cylinders having pair-wise equal radii. This is a conservative load representing a follower force directed towards the positive pole. Theoretical considerations are made related to the determination of boundary conditions with installation of needle rolling bearings or rigid elements in the receiving head. The course of the natural vibration frequency against the external load is determined. The results of theoretical research are compared to results of experimental examinations. The rigidity of the substitute rotational spring is defined, considering "the rigidity impact" of the free end of the column.

PL

W pracy prezentuje się nowy typ obciążenia swoistego zrealizowanego poprzez elementy tworzące pobocznicę czterech walców kołowych o promieniach parami równych. Jest to obciążenie, które reprezentuje siłę śledzącą skierowaną do bieguna dodatniego. Przeprowadza się rozważania teoretyczne dotyczące sformułowania warunków brzegowych przy zastosowaniu w głowicy przejmującej obciążenie łożysk tocznych igiełkowych lub elementów sztywnych. Określa się przebieg częstości drgań własnych w funkcji obciążenia zewnętrznego. Wyniki badań teoretycznych porównuje się z wynikami badań eksperymentalnych. Wyznacza się sztywność zastępczej sprężyny rotacyjnej uwzględniającej "usztywnienie" nieutwierdzonego końca kolumny.