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Use of the higher-order plate theory of I. N. Vekua type in problems of dynamics of heterogeneous plane waveguides

Egorova O. V., Rabinskiy L. N., Zhavoronok S. I.

Archives of Mechanics

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2020

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Vol. 72, nr 1

3--25

EN

The dynamics of elastic plane waveguides is studied on the basis of the extended formulation of the plate theory of Nth order. The plate model is based on the Lagrangian formalism of analytical dynamics combined with the dimensional reduction approach and the biorthogonal expansion of the spatial distribution of the displacement. The boundary conditions shifted from the faces onto the base plane are interpreted as constraints for the variational formulation of two-dimensional plate models. The normal wave dispersion in plates is modelled, the convergence of the approximate solutions is studied using the known exact solution for a plane layer as a reference. The proposed plate theory is used to analyse the normal wave dispersion in power graded waveguides of both symmetric and asymmetric structures, the locking phase frequencies for various power indices are computed.

2

Displacement analysis of the human knee joint based on the spatial kinematic model by using vector method

Knapczyk J., Góra-Maniowska M.

Acta Mechanica et Automatica

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2017

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Vol. 11, no. 4

322--327

EN

Kinematic model of the human knee joint, considered as one-degree-of-freedom spatial parallel mechanism, is used to analyse the spatial displacement of the femur with respect to the tibia. The articular surfaces of femoral and tibia condyles are modelled, based on selected references, as spherical and planar surfaces. The condyles are contacted in two points and are guided by three ligaments modelled as binary links with constant lengths. In particular, the mechanism position problem is solved by using the vector method. The obtained kinematic characteristics are adequate to the experimental results presented in the literature. Additionally, the screw displacements of relative motion in the knee joint model are determined.

3

Równania więzów pomiędzy układem wieloczłonowym i modelem złożonym z elementów skończonych

Lipiński K.

Modelowanie Inżynierskie

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2010

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T. 8, nr 39

127-134

PL

W pracy zaprezentowano równania punktowego kontaktu pomiędzy układem wieloczłonowym i strukturą odkształcalną. Układ wieloczłonowy zbudowano z nieodkształcalnych członów połączonych parami kinematycznymi. Część odkształcalną zbudowano z elementów skończonych. Zarysowano sposób wyznaczania równań modelu wieloczłonowego i modelu odkształcalnego. Wyznaczono równania więzów. Układ algebraiczno różniczkowy przekształcono do postaci różniczkowej. Wykorzystano do tego dedykowaną wersję metody podziału zmiennych. Wyeliminowano współrzędne zależne i mnożniki Lagrange'a. Zagadnienie zilustrowano przykładem numerycznym.

EN

In the paper, constraint equations are presented for a punctual contact between a multibody system and an elastic structure. The multibody system is composed of rigid bodies, joined by kinematic pairs. The elastic structure is built of deformable finite elements. Dynamic equations of the both (the multibody and the finite elements) models are presented. The main zoom is set on the constraint equations. Then the differentially-algebraic system of equations is transformed into a differential set of equations, by used of a dedicated version of the coordinate partitioning technique. Dependent coordinates, as well as the Lagrange' multipliers are eliminated. The method is illustrated with a numerical example.