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1

Derivation of equations of motion and supplementary natural boundary conditions for a slender single column subjected to Euler’s load using Hamilton’s principle

Kuliński Krzysztof

Zeszyty Naukowe Politechniki Częstochowskiej. Budownictwo

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2023

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Z. 29 (179)

61--66

EN

In this paper the derivation process of motion equations and boundary conditions for a slender mechanical system on the basis of Hamilton’s principle is presented. In order to apply the Hamilton’s principle, first of all it is necessary to define appropriate general variables that describe the motion of the considered system. In the case of slender mechanical systems, natural coordinates are usually used, which are well suited to the system geometry and its motional characteristics. Based on Hamilton's principle, an appropriate action functional is constructed, which is the Lagrangian integral covering the appropriate general variables and time. The Lagrangian describes the well-known difference between the kinetic and potential energy of a system. A step by step derivation of a motion equation and supplementary natural boundary conditions in regard to an example of a slender clamped-free column subjected to Euler’s load is discussed. The obtained equation with a set of geometrical and natural boundary conditions gives the possibility to thoroughly analyse both analytically or numerically system dynamics and/or static response. Despite that the discussed method is time consuming and requires advanced mathematical techniques, it makes it possible to obtain exact or approximate motion equations even for complex problems, what can be difficult or even impossible to achieve using other known methods.

PL

Przedstawiono proces wyprowadzania rownań ruchu i naturalnych warunkow brzegowych smukłego układu mechanicznego na podstawie zasady Hamiltona. Aby zastosować zasadę Hamiltona, należy przede wszystkim zdefiniować odpowiednie zmienne ogolne opisujące ruch rozpatrywanego układu. W przypadku smukłych układow mechanicznych stosuje się zwykle tzw. wspołrzędne naturalne, ktore są dobrze dopasowane do geometrii układu i jego charakterystyk ruchu. Wykorzystując zasadę Hamiltona, konstruuje się odpowiedni funkcjonał ruchu, ktorym jest całka Lagrange’a smukłego układu po odpowiednich zmiennych ogolnych i czasie. Lagranżjan opisuje dobrze znaną rożnicę między energią kinetyczną i potencjalną układu. W pracy przedstawiono proces krok po kroku wyprowadzania rownania ruchu i uzupełniających naturalnych warunkow brzegowych w odniesieniu do smukłego słupa o jednym końcu utwierdzonym, a drugim wolnym, ktory to poddany jest obciążeniu Eulera. Otrzymane rownanie wraz ze zbiorem geometrycznych i naturalnych warunków brzegowych daje możliwość dokładnej analizy analitycznej lub numerycznej odpowiedzi dynamicznej i/lub statycznej układu. Pomimo tego, że omawiana metoda jest czasochłonna i wymaga zaawansowanych technik matematycznych, pozwala ona na uzyskanie dokładnych lub przybliżonych rownań ruchu nawet dla złożonych problemow, co przy innych znanych metodach może być trudne lub wręcz niemożliwe do osiągnięcia.

2

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

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2019

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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.

3

Free vibrations of the column taking into account compressive and thermal load

Uzny Sebastian, Kutrowski Łukasz, Skrzypczak Tomasz

Vibrations in Physical Systems

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2019

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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.

4

Strength analysis of a telescopic hydraulic cylinder elastically mounted on both ends

Uzny Sebastian, Kutrowski Łukasz

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 1

89--96

EN

Corrosion or contamination of flexible joints of a telescopic hydraulic cylinder may cause an increase of movement resistance in these places. In this work the influence of mounting rigidity on the strength of a telescopic hydraulic cylinder is under consideration. Buckling criterion and the strength of cylinder barrels (material effort) were included due to analysis. Boundary value problem concerning the stability of the system was formulated on the basis of the static stability criterion. Lame’s theory for thick pipes were used for determination of destructive load from the viewpoint of the material effort. Numerical simulations were performed. The results specifying the influence of mounting rigidity on stability and strength of cylinder barrels were presented by using non-dimensional parameters.

5

Nonlinear component of free vibration frequency of a partially tensioned system with consideration of rotational stiffness of support

Uzny Sebastian, Osadnik Michał

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 3

89--96

EN

The nonlinear vibrations of a slender system subjected to Euler’s load which is partially tensioned is discussed in this paper. The longitudinal displacement and rotation on both of the system ends are limited by the discrete elements in the form of translational and rotational springs. The results of numerical simulations concern the first vibration frequency (linear and non-linear components) in relation to the location and magnitude of external load application and different rotational spring stiffness. This nonlinear system is based on the screw drive used in the newly designed vertical platform lifts.