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A discrete model for geometrically nonlinear free and forced vibrations of stepped and continuously segmented Euler-Bernoulli AFG beams (SAFGB) carrying point masses

Moukhliss Anass, Rahmouni Abdellatif, Bouksour Othman, Benamar Rhali

Diagnostyka

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2022

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Vol. 23, No. 4

art. no. 2022401

EN

A discrete model is applied to handle the geometrically nonlinear free and forced vibrations of beams consisting of several different segments whose mechanical characteristics vary in the length direction and contain multiple point masses located at different positions. The beam is presented by an N degree of freedom system (N-Dof). An approach based on Hamilton's principle and spectral analysis is applied, leading to a nonlinear algebraic system. A change of basis from the displacement basis to the modal basis has been performed. The mechanical behavior of the N-Dof system is described in terms of the mass tensor mij, the linear stiffness tensor kij, and the nonlinear stiffness tensor bijkl. The nonlinear vibration frequencies as functions of the amplitude of the associated vibrations in the free and forced cases are predicted using the single mode approach. Once the formulation is established, several applications are considered in this study. Different parameters control the frequency-amplitude dependence curve: the laws that describe the variation of the mechanical properties along the beam length, the number of added masses, the magnitude of excitation force, and so on. Comparisons are made to show the reliability and applicability of this model to non-uniform and non-homogeneous beams in free and forced cases.

2

Drgania poprzeczne belek o skokowo zmiennym przekroju spoczywających na podłożu Winklera

Kuliński K., Przybylski J.

Modelowanie Inżynierskie

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2015

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T. 23, nr 54

33--40

PL

W niniejszej pracy dyskutuje się zagadnienie wpływu zmian parametrów strukturalnych belki z zamocowanymi końcami i usytuowanej na podłożu Winklera na częstość jej poprzecznych drgań własnych. Zaburzenie jednorodności struktury belki wynika z naklejenia na jej górnej i dolnej powierzchni dwóch warstw materiału piezoceramicznego. Do badań przyjęto pięć różnych sposobów podparcia końców belek, które uniemożliwiają ich przemieszczenie wzdłużne. Badania numeryczne podzielono na dwie części: pierwsza dotyczy wpływu zmiany geometrii układu na częstość jego poprzecznych drgań własnych. W drugiej części zademonstrowano zakres możliwej modyfikacji częstości pod wpływem pola elektrycznego przyłożonego do piezosegmentu w celu wygenerowania rozciągającej i ściskającej siły rezydualnej.

EN

In this paper the influence of structural parameters of the beam with two ends fixed resting on Winkler foundation on its natural vibration frequencies has been discussed. Structural inhomogeneity is the result of two surface-bonded piezoceramics, the first one for the top and the second one for the bottom surface, respectively. There have been adopted five different supports of beam ends which prevent their longitudinal displacements. Numerical analysis has been divided onto two parts. The first part concerns the influence of the system's geometry on its natural frequency, whereas in the second part a modification of the frequency resulting from the electric field applied to the piezosegment which generates a compressing or stretching residual axial force has been investigated.

3

The dynamic stability of a simply supported stepped beam with additional discrete elements

Sochacki W.

Vibrations in Physical Systems

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2012

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Vol. 25

367--372

EN

The dynamic stability of a simply supported stepped beam with additional discrete elements was investigated in the paper. These elements are a rotational spring and a rotary inertia, both of which are connected to the beam. The discrete elements can be mounted at any chosen position along the beam length. The influence of step changes in the cross-section of the beam on its dynamic stability was also investigated in the paper. The problem of dynamic stability was solved by applying the mode summation method. Applying an orthogonal condition of eigenfunctions, the dynamic of the system was described with the use of the Mathieu equation. The obtained equation allowed the dynamic stability of the tested system to be analysed. The considered beam was treated as Euler-Bernoulli beam.