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Free vibrations of the partially tensioned geometrically non-linear system subjected to Euler’s load

Uzny S., Sokół K., Osadnik M.

Vibrations in Physical Systems

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2016

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Vol. 27

399--406

EN

In this study the fixed-fixed column subjected to axial Euler’s load has been investigated. The load is placed between the fixed ends of the structure and its location can be changed along column’s length. The boundary problem of free vibrations of the mentioned system has been formulated on the basis of Bernoulli – Euler theory and taking into account non-linear axial deformation relationship. Due to non-linear expressions the solution of the problem was done with small parameter method. In the paper the change of the first vibration frequency in relation to location and magnitude of the loading force was obtained. The relationship between natural vibration frequency and the amplitude is also discussed.

2

Modelling and Simulation Research of the Gripper Manipulator

Uzny S., Sokół K.

Machine Dynamics Research

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2015

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Vol. 39, No. 2

5-14

EN

In this paper the boundary problem of instability of single slender system with consideration of Timoshenko theory is presented. The investigated structure is loaded by Euler’s force (the most common type of loading); additionally the different boundary conditions are taken into account. Simulated type of external load is characterized by constant line of action regardless to deflection of the system. In order to achieve more general form of the investigated system the springs limiting the rotations and displacement of both ends are used. Boundary problem is formulated on the basis of the minimum total potential energy. The results of numerical simulations obtained with Timoshenko and Bernoulli-Euler theories are compared. The simulations are done at different magnitudes of slenderness factor as well as translational and rotational springs stiffness. On the basis of the obtained results the difference in critical forces calculated on the basis of both theories can be easily presented.

3

Free vibrations of column subjected to Euler's load with consideration of Timoshenko's theory

Uzny S., Sokół K.

Vibrations in Physical Systems

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2014

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Vol. 26

319--326

EN

In this paper the single-rod cantilever column subjected to compressive Euler's load is investigated. The boundary problem has been formulated on the basis of Hamilton's principle and Timoshenko's theory. Numerical simulations of characteristic curves have been plotted on the plane external load-vibration frequency for different magnitudes of slenderness factor of the system. The results of numerical calculations of Timoshenko's beam are compared to the ones obtained from mathematical Bernoulli-Euler's model. The comparison of the results of characteristic curves calculated by means of Timoshenko's theory and Bernoulli-Euler's model are done for first three vibration frequencies.

4

Wpływ parametrów podłoża na drgania własne konstrukcji belkowych

Obara P.

Logistyka

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2014

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nr 6

8033--8042

PL

W pracy wyznaczone zostały częstotliwości drgań własnych belek pryzmatycznych przy użyciu trzech, stosowanych w mechanice konstrukcji, teorii: Bernoulliego-Eulera, Timoshenki oraz Bresse-Timoshenki. Tym samym określono wpływ odkształcalności postaciowej i bezwładności obrotowej na częstotliwości drgań. W rozważaniach uwzględniona została współpraca belki z podłożem sprężystym o dwóch charakterystykach sprężystości: pionowej – kw i poziomej – ku. Celem pracy było określenie, w jakim stopniu parametry podłoża sprężystego wpływają na częstotliwości drgań własnych. Rozpatrzono trzy sposoby podparcia belki. Pierwszy przypadek to belka swobodnie podparta, dla której wyznaczone zostały wzory analityczne na dwa pasma częstości drgań. Dwa pozostałe – to belka obustronnie utwierdzona i swobodnie leżąca na podłożu, dla tych przypadków wyznaczono równania warunkowe. Przedstawione formuły umożliwiają wyznaczenie dowolnej częstości drgań dla dowolnych charakterystyk materiałowych i geometrycznych belki oraz podłoża sprężystego i mogą być stosowane w praktyce inżynierskiej.

EN

The paper presents the dynamic analysis of uniform beam. The Bernoulli-Euler Timoshenko and Bresse-Timoshenko theories were used. Thus, the effect of transverse shear deformation and rotatory inertia on natural frequency was determined. The foundation stiffness parameters (vertical and horizontal) were taken into account. The aim of the study was to determine the extent to which the elastic parameters of the foundation affect the natural frequency. Three ways to support the beam were considered. The first case is a simply supported beam for which analytical formulas in two frequency bands of vibration were determined. Two others –clamped-clamped beam and beam freely lying on the foundation. For these cases conditional equations were determined. The obtained formulas allow to define any vibration frequency for any material and geometrical characteristics of the beam and the elastic foundation. These formulas can be used in engineering practice.