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Large amplitude forced vibrations of restrained beams resting on elastic point supports

Fakhreddine Hatim, Adri Ahmed, Chajdi Mohcine, Rifai Said, Benamar Rhali

Engineering Transactions

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2021

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Vol. 69, iss. 3

225--242

EN

The present paper concerns the study of geometrically non-linear forced vibrations of beams resting on two different types of springs: rotational and translational. Assuming that the motion is harmonic, the displacement is extended as a series of spatial functions determined by solving the linear problem. Hamilton’s principle and spectral analysis are used to reduce the problem to a non-linear algebraic system solved using a previously developed approximate method. The effects of the nature of the added springs and their location on the non-linear behaviour of the beam are examined. A multimode approach is used in the forced case to obtain results over a wide range of vibration amplitudes. This leads to examining the non-linear forced dynamic response for different positions of each spring and different levels of excitations. Following a parametric study, the non-linear forced mode shapes and their associated bending moments are presented for different levels of excitations and for different vibration amplitudes to give an estimation of the stress distribution over the beam length.

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A multimode approach to geometrically non-linear forced vibration of beams carrying point masses

Fakhreddine Hatim, Adri Ahmed, Chajdi Mohcine, Rifai Said, Benamar Rhali

Diagnostyka

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2020

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Vol. 21, No. 4

23--33

EN

The present work deals with the geometrically non-linear forced vibrations of beams carrying a concentric mass under different end conditions. Considering the axial strain energy and expanding the transverse displacement in the form of a finite series of spatial functions, the application of Hamilton's principle reduces the vibration problem to a non-linear algebraic system solved by an approximate method developed previously. In order to validate the approach, comparisons are made of the present solutions with those previously obtained by the finite element method. Focus is made here on the analysis of the non-linear stress distribution in the beam with an attached mass. The non-linear forced deflection shapes and their corresponding curvatures are presented for different magnitudes of the attached mass, different excitation levels and different vibration amplitudes.

3

Droppers’ stiffness influence on dynamic interaction between the pantograph and railway catenary

Bryja Danuta, Hyliński (Popiołek) Adam

Problemy Kolejnictwa

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2019

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Z. 183

89--98

EN

Droppers connecting the contact wire and messenger wire of the railway catenary are characterized by zero or negligible compressive stiffness, hence they become slack under compression that is similar to bars’ buckling. The paper presents a numerical analysis of the influence of droppers slackening phenomenon on the dynamic interaction between the pantograph and catenary. The analysis is based on a simulation method presented by the authors in previous papers, in which the catenary is modelled as a complex cable system. In this paper, the simulation method is modified by introducing the residual compressive stiffness of droppers that is assumed as a given percent of tensile stiffness. Modification leads to geometrically non-linear equations of motion of the pantograph-catenary system. Two different algorithms for solving the problem of non-linearity are proposed, in both of them the Newmark numerical integration method is applied. Results of dynamic response simulations performed for different values of residual compressive stiffness of droppers are presented and compared. It is shown that the contact wire does not cooperate with the messenger wire in a large area around the pantograph when the compressive stiffness of droppers is assumed zero. As a result, the pantograph moving at high speed induces severe vibrations of the catenary. It is also shown that droppers should be designed to have the residual compressive stiffness equal to at least one percent of their tensile stiffness. This is sufficient to ensure an appropriate cooperation between messenger wire and contact wire, which is demonstrated by simulation results fulfilling requirements given in the standard PN-EN 50318: 2002.