Keywords: structures with viscoelastic dampers, dynamic characteristics, nonlinear eigenvalue problems, continuation method.

Viscoelastic (VE) dampers are often used to mitigate excessive vibrations of structures. The properties of VE materials or fluids used for manufacturing VE dampers depend on the temperature and excitation frequencies of the forces, acting on dampers. Many rheological models, including the classical rheological models and rheological models with fractional derivatives, are proposed for a correct description of the dynamical behaviour of VE dampers.

The dynamic characteristics of structures with VE dampers, such as natural frequencies, non-dimensional damping ratios and modes of vibrations, can be obtained as a result of solution to linear or nonlinear eigenvalue problems. A solution to the linear eigenvalue problem is obtained when the state-space approach is used whereas the nonlinear eigenvalue problem must be solved if equations of motion are written in a traditional form. The dimensions of the linear eigenvalue problem are more than twice as great as the dimension of the nonlinear eigenvalue problem. However, the methods of solution for the nonlinear eigenvalue problem obtained for VE systems or structures are less common. Some methods are presented in the papers [1,2,3,4,5] where the Biot model [1], the complex stiffness model [2,3,4] and the rheological models with fractional derivatives [5] are used to describe VE materials or dampers.

Planar frames with VE dampers modelled using the generalized Kelvin or Maxwell models are considered in this paper. The dynamic characteristics of this type of structure are of primary interest. Nonlinear eigenvalue problems resulting from the equations of motion written as a set of second-order differential equations are derived. The continuation method is proposed to solve the above-mentioned eigenvalue problem. The normalization condition for the eigenvector is adopted as the constrained equation and the artificial main parameter is introduced in the continuation method. The method does not require any assumptions about the viscoelasticity of dampers. Complex solutions to the eigenvalue problem considered are determined in a systematic way. The effectiveness and accuracy of the method are justified based on the results of calculations made, taken from the literature and for typical planar frames with VE dampers.

1

S. Adhikari, "A reduced second-order approach for linear viscoelastic oscillators", Journal of Applied Mechanics, 77, 041003-1-7, 2010. doi:10.1115/1.4000913

2

E.M. Daya, M. Potier-Ferry, "A numerical method for nonlinear eigenvalue problems and application to vibrations of viscoelastic structures", Computers and Structures, 79, 533-541, 2001. doi:10.1016/S0045-7949(00)00151-6

3

F. Cortes, M.J. Elejabarrieta, "An approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix", Journal of Sound and Vibration, 296, 166-182, 2006. doi:10.1016/j.jsv.2006.02.016

4

U. Miller, S. Bograd, A. Schmidt, L. Gaul, "Eigenpath following for systems with symmetric complex-valued stiffness matrices", Shock and Vibration, 17, 397-405, 2010.

5

R. Lewandowski, Z. Pawlak, "Dynamic analysis of frames with viscoelastic dampers modelled by rheological models with fractional derivatives", Journal of Sound and Vibration, 330, 923-936, 2011.

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