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Modal analysis of frame structures with semi-rigid joints and viscoelastic connections modeled by fractional derivatives

Pawlak Z. M., Łasecka-Plura M., Zak-Sawiak M.

Archives of Mechanics

2023

Vol. 75, nr 3

337--363

The paper presents two methods of solving the problem of the dynamics of a frame structure with viscoelastic bonds in nodes. In the first approach, known from the literature, two-node beam elements with three degrees of freedom in each node were used. Exact shape functions were adopted to obtain a stiffness matrix, a consistent mass matrix and a damping matrix for the beam element. These matrices were then modified by introducing rotational viscoelastic constraints at the boundary nodes. In the second approach, a new method of modelling viscoelastic bonds in frame structures was proposed. It consists in removing rigid bonds between elements along selected degrees of freedom and replacing them with a new, additional element with viscoelastic properties. This approach allows the use of any rheological model to describe viscoelastic bonds (i.e. an additional element) without the need to create a new modified finite element. In this work, an advanced rheological model, i.e. the fractional Kelvin model, was used to describe rotational viscoelastic bonds. The use of fractional derivatives to describe the damping properties reduces the number of parameters needed in the model, but leads to a non-linear eigenproblem. In order to solve the eigenvalue problem, the continuation method was used, and the dynamic characteristics of the structure were determined on the basis of the calculated eigenvalues. Selected structures with viscoelastic nodes were analyzed and the obtained results confirm the effectiveness of the proposed approach.