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Stabilization of a 1-D transmission problem for the Rayleigh beam and string with localized frictional damping

Hong Gimyong, Hong Hakho

Journal of Applied Analysis

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2023

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Vol. 29, nr 1

77--90

EN

We are concerned with the stability of a 1-D coupled Rayleigh beam-string transmission system.We obtain the polynomial decay rate t−1 or the exponential decay rate for the given transmission system whether the frictional damping is only effective in the beam part or the string part, respectively. This paper generalizes the recent result in [Y.-F. Li, Z.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett. 78 (2018), 51-58]. The main ingredient of the proof is some careful analysis for the Rayleigh beam and string transmission system.

2

Free vibrations spectrum of periodically inhomogeneous Rayleigh beams using the tolerance averaging technique

Świątek Marcin, Domagalski Łukasz, Jędrysiak Jarosław

Journal of Theoretical and Applied Mechanics

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2019

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Vol. 57 nr 1

141--154

EN

In this paper, linear-elastic Rayleigh beams with a periodic structure are considered. Dynamics of such beams is described by partial differential equations with non-continuous highly oscillating coefficients. The analysis of dynamic problems using the aforementioned equations is very often problematic to perform. Thus, other simplified models of Rayleigh beams are proposed. Some of these models are based on the concept of the effective stiffness. Among them, one can distinguish the theory of asymptotic homogenization. However, in these models, the size of the mesostructure parameter (the size of a periodicity cell) is often neglected. Therefore, a non-asymptotic averaged model of the periodic beam is introduced, called the tolerance model, which is derived by applying the tolerance averaging technique (TA). The obtained tolerance model equations have constant coefficients, and in contrast to other averaged models, some of them depend on the size of the periodicity cell.

3

Linear vibrations of periodic Timoshenko and Rayleigh beams

Świątek M., Jędrysiak J., Domagalski Ł.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018035

EN

Elastic periodic structures with variable material and geometrical properties exhibit dynamic characteristics that are investigated in this contribution. The paper is devoted to analysis of geometrically linear vibrations of Rayleigh and Timoshenko beams with cross-sections and material properties periodically varying along the longitudinal axis. The period of inhomogeneity is assumed to be sufficiently small when compared to the beam length. Equations of motion in both beam theories under consideration have highly-oscillating coefficients. In order to derive the averaged model equations with constant coefficients for vibrations, the tolerance averaging approach is applied. The method of averaging differential operators with rapidly varying coefficients is applied to obtain averaged governing equations with constant coefficients. An assumed tolerance and indiscernibility relations and the definition of slowly varying function found the applied technique. Numerical results from the tolerance Rayleigh and Timoshenko beam model equations are compared.

4

Application of the difference equation method to the vibrations analysis of infinite Rayleigh beams by the isogeometric approach

Rakowski J., Wielentejczyk P.

Archives of Civil and Mechanical Engineering

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2015

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Vol. 15, no. 4

1108--1117

EN

An efficient method of vibration investigations of infinite Rayleigh beams applied to the isogeometric analysis called NURBS (non-uniform rational B-splines) is proposed. The research objective is to examine the influence of rotational inertia effects on the dynamic behaviour of the discrete systems approximated by NURBS and compare obtained results with the finite element method (FEM) and exact ones. In NURBS methodology transverse displacements are approximated by quadratic, cubic and quartic B-splines basis functions. For all types of approximations stiffness and consistent mass matrices with rotational inertia effects are found. The equilibrium conditions for an arbitrary interior element are expressed in the form of one difference equation equivalent to the infinite set of equations derived by numerical NURBS formulation for this dynamic problem. The convergence of these equations to the exact differential equations of motion is presented. Assuming the wavy nature of vibration propagation phenomenon the analytical dispersive equations are obtained for various orders of B-spline basis functions. The parametrical analysis of rotational inertia effects on the wave propagation is carried out. The influence of the adopted discretization and the mass distribution is taken into account, as well. The analytical NURBS results are compared with the FEM and exact ones.