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EN
In this work a sewing needle of an industrial commercial sewing machine is investigated from the different points of view of geometrical and critical elastic statistical balance. The dimensions of the sewing needle are l = 60 mm, and ϕ = 2.5 mm and 1.45 mm diameter for the two stepped needle. The needle is treated as a fixed – free end column, with two sections: Ι0 (Ι1) = 1.9615 (Ε – 12) m4 and (Ι2) = 2.0516 (Ε – 13) m4. The upper partial lenght a = 31.2 mm, while the lower partial lenght b = 28.2 mm. The value of the critical load is 110 N, while the Euler load Ρ e = 1112 N, then γ = 3.1796. Another approach was used for a line diagram of the truncated cone of the sewing needle, where the critical load Ρ cr = 57 N, the Euler load Ρ e = 118 N, and γ = 3.13. For the line diagram of a sewing needle with a constant cross-section ϕ = 1.965 mm, l = 7.3185 (Ε – 13) m4, (formula) & η = 2.467. Then Ρ cr = 103 N, Ρ e = 413 N & (formula). For a sewing needle a constant cross-section and the penetration load is uniformly distributed, the geometric characterictic is found to be ῆ = 7.839, γ = 1.22, Ρ cr = Ρ cr(ql)cr = 328 N & q = 5472 N/m (≈6 N/mm). The present work aims to open as yet untouched areas for more and intensive academic and experimental studies, in the field of how penetration forces affect critical loads. The work is the first step in the evaluation of the mechanics of the industrial sewing needle.
PL
W pracy badano igłę do szycia stosowaną w przemysłowej maszynie do szycia. Do badań użyto dwustopniowej igły do szycia o wymiarach: dł. = 60 mm, ϕ = 2.5 mm i średnicy 1.45 mm. Igłę potraktowano jako kolumnę z nieruchomym i wolnym końcem, z dwiema sekcjami: Ι0 (Ι1) = 1.9615 (E – 12) m4 oraz (I2) = 2.0516 (E – 13) m4. Górna długość częściowa wynosiła: a = 31.2 mm, podczas gdy dolna długość częściowa to: b = 28.2 mm. Celem pracy było otwarcie jeszcze nietkniętych obszarów dla bardziej intensywnych badań akademickich i eksperymentalnych w zakresie wpływu sił penetracji na obciążenia krytyczne.
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.
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.
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.
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.
PL
Uszkodzenia siłowników hydraulicznych mogą być spowodowane wieloma czynnikami. Można wśród niech wyróżnić utratę stateczności oraz wytężenie materiału cylindrów, wywołane działaniem wysokiego ciśnienia wewnątrz układu. W niniejszej pracy w celu rozpatrywania pierwszego z czynników sformułowano zagadnienie brzegowe stateczności teleskopowego siłownika hydraulicznego przy uwzględnieniu obciążenia Eulera. Do określenia wytrzymałości wytężeniowej poszczególnych członów siłownika wykorzystano teorię Lamégo w przypadku rur grubościennych. Zestawiając ze sobą uzyskane wyniki, określono zakresy parametru grubości cylindrów oraz parametru grubości elementów uszczelniających, przy których do zniszczenia układu dochodzi na skutek utraty stateczności konstrukcji lub wytężenia materiału cylindrów.
EN
Damage of the hydraulic cylinders can be caused by many factors. Loos of stability and material effort, caused by the pressure inside the cylinder barrels can be distinguished. For analysis of the first factor, boundary value problem of the stability of a hydraulic telescopic cylinder, subjected to Euler’s load was formulated. For determination of the strength of individual cylinders, the Lamé’s theory for thick pipes was used. By comparing obtained results, the ranges of the cylinder thickness parameter and the sealing elements thickness parameter were determined, in which the destruction of the system occurs as a result of the loss of stability of the structure and the effort of the cylinder material.
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.
EN
The boundary problem concerning the free vibrations of a geometrically non-linear slender system subjected to a load by a force, unchanging in its direction (Euler’s force), is formulated and solved in the work. The considered column is a pipe with a focally mounted rod. A layer made of viscoelastic material is placed between the pipe and the rod. The Kelvin-Voigt model of this layer is taken into account to formulate the boundary problem. Numerical calculations connected to the free vibrations of the considered system are carried out on the basis of the boundary problem. Numerical simulations are assigned to different values of the parameters of the system for which the rigidity and damping parameters of the viscoelastic layer are ranked.
EN
In this paper, the results of numerical studies on vibration and instability of a geometrically nonlinear column subjected to Euler's load are presented. The system is loaded by external axial force P applied at the free end of the column. Additional concentrated mass is localised on one of column's rods. The Hamilton principle was used to formulate the boundary problem. Due to the geometrical nonlinearity, the solution of the problem was performed by means of the small parameter method. The main purpose of this paper is to investigate an influence of the concentrated mass on divergence instability and critical loading and natural vibration frequency.
PL
W pracy rozważa się drgania smukłego układu geometrycznie nieliniowego wokół krzywoliniowej postaci równowagi statycznej. Układ poddano działaniu ściskającej siły zewnętrznej Eulera. Na podstawie zagadnienia brzegowego przeprowadzono obliczenia numeryczne dotyczące liniowej i nieliniowej składowej częstości drgań własnych, które wykonano przy różnych wartościach parametrów układu.
EN
Boundary value problem concerning to free vibrations of geometrically nonlinear slender system subjected to compressed external Euler force was formulated and solved in this work. On account nonlinearity the small parameter method has been used. In this work curvilinear form of static equilibrium is considered. On the basis of boundary value problem the calculations concerning to linear and nonlinear components of free vibrations frequency have been carried out. Numerical calculations was carried out at different values of system parameters.
PL
W pracy sformułowano i rozwiązano zagadnienie brzegowe dotyczące drgań własnych smukłego układu geometrycznie nieliniowego poddanego działaniu ściskającej siły zewnętrznej Eulera. Ze względu na nieliniowość wykorzystano metodę małego parametru. Rozważaniom poddano prostoliniową postać równowagi statycznej. Na podstawie zagadnienia brzegowego przeprowadzono obliczenia numeryczne dotyczące liniowej i nieliniowej składowej częstości drgań własnych. Obliczenia numeryczne wykonano przy różnych wartościach parametrów układu.
EN
Boundary value problem concerning to free vibrations of geometrically nonlinear slender system subjected to compressed external Euler force was formulated and solved in this work. On account of nonlinearity the small parameter method has been used. In this work rectilinear form of static equilibrium is considered. On the basis of boundary value problem the calculations concerning to linear and nonlinear components of free vibrations frequency have been carried out. Numerical calculations was carried out at different values of system parameters.
EN
In this paper, the results of numerical studies on the local and global instability and vibration of a geometrically nonlinear column subjected to Euler's load are presented. The Hamilton principle was used to formulate the boundary problem. Due to the geometrical nonlinearity, the solution of the problem was performed by means of the perturbation method. The magnitude of the bifurcation load of the nonlinear column, the local and global instability regions and characteristic curves have also been presented.
PL
W artykule przeanalizowano wpływ rodzaju elementu pośredniczącego (łożysko toczne, sztywny element walcowy) pomiędzy głowicą wymuszającą i przejmującą obciążenie na stateczność i drgania kolumn poddanych obciążeniu Eulera. Zastosowanie elementu sztywnego walcowego, w głowicach realizujących obciążenie w porównaniu z elementem tocznym, powoduje usztywnienie węzła obrotowego pomiędzy dwoma współdziałającymi elementami struktury obciążającej układ.
EN
In this work has been analysed the influence of the adapter type (rolling bearing, stiff cylindrical element) between coactive and load taking heads on stability and vibrations of Euler loaded columns. Using the stiff cylindrical element in load charging heads in comparison to rolling element makes the rotating loop stiff between two cooperating structure loading elements.
PL
W niniejszej pracy przeprowadzono badania teoretyczne i numeryczne dotyczące stateczności i drgań własnych smukłego układu w postaci siłownika hydraulicznego, który poddano obciążeniu Eulera. Rozpatrywano sześć możliwości zamocowania siłownika. Na podstawie zagadnienia brzegowego przeprowadzono szereg symulacji numerycznych. Wyznaczono obciążenie krytyczne oraz krzywe charakterystyczne na płaszczyźnie obciążenie - częstość drgań własnych w zależności od parametrów układu.
EN
Theoretical and numerical research concerned to stability and free vibrations of slender system inform of hydraulic cylinder which was subjected to Euler load were made in this work. Six possibility offbced of hydraulic cylinder were considered in the paper. On the basis of the boundary problem a lot of numerical simulations were carried out. Critical load and characteristic curves on the plane load - natural frequency were determined in dependence on parameters of considered system. Coefficient of asymmetry flexural rigidity of piston rod and cylinder, coefficient of overlap and external load were used in the numerical calculations. Moreover limiting value of coefficient of asymmetry flexural rigidity piston rod and cylinder was determined in the work. Limiting value of coefficient of asymmetry flexural rigidity separated two regions. Hydraulic cylinder breaks down in result of lost of stability in first region . Hydraulic cylinder breaks down in result of effort of cylinder material in second region. Stress of part of cylinder which was filled by liquid was calculated on the basis of the Lame theory.
EN
In this paper, we present the shape optimization method applied to slender's systems (columns) subjected to Euler's loads. The method used to describe the problem, consist in dividing column into elements. These elements are described by their length (the same for every element) and variable diameter. Total potential energy is determined for two systems varying in loading method. Equation of motion and boundary conditions are determined by taking into account energetic method (the minimum of potential energy). Solution of boundary value problem leads to setting transcendental equation for the value of critical load. It is assumed that volume and total length of column are constant. Basing on this assumption, the shape's optimization comes down to matching diameters of particular system's elements. The maximum value of critical load is obtained for each set of diameters. The modified simulated annealing algorithm is used to finding maximum of critical force, which is described by function of several variables. Applied in the algorithm neighbourhood generator changes its behaviour with temperature function. Taking into account the method of column's mounting, the results of numeric calculations for chosen values of structures mounting's elasticity coefficient are presented.
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