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Passive damping systems are energy dissipating devices that are used in building constructions to reduce excessive vibration caused by wind or earthquakes. Recently, the use of inerters, devices using the inertia of the rotating mass, as a passive damping system has become more and more popular. Inerter, regardless of the construction, generates a resistance force that is proportional to the relative acceleration at the terminal ends of the device. The purpose of this study is to evaluate the effectiveness of the viscoelastic (VE) dampers in combination with the inerter. For the derived equations of motion of structures with VE dampers and inerters, the Laplace transformation is applied, which leads to a nonlinear eigenproblem. The solution to the eigenproblem is obtained using the continuation method, also known as the homotopy or as the path following method. It was found that for some cases of inerter connection with ve damper, there are some additional solutions in relation to the degrees of freedom of the structure. Furthermore, the dynamic characteristics of the structure associated with the above-mentioned additional eigenvalue are strongly dependent on the equivalent inerter mass. Damping efficiency is tested by determining changes in the system's dynamic characteristics: natural frequencies, non-dimensional damping ratios and displacement transfer functions. The presented approach allows determining the appropriate parameters of the inerter and VE damper and their appropriate location on the structures that correspond to the most effective damping. The obtained numerical results confirm the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This study concerns the structural systems in which additional damping was introduced by means of viscoelastic (VE) dampers modeled with the fractional derivative models. The fractional derivative models have an ability to correctly describe the behavior of VE materials in a wide range of frequency, using a small number of model parameters. However, the governing equation of motion includes fractional derivatives together with ordinary ones. In the proposed approach, after applying the Laplace transform and the inverse transform to the equations of motion, the solution obtained for the system with fractional dampers is equivalent to the modal solution used in the case of proportional damping. In order to validate the proposed approach, the maximum response of the structural system equipped with dampers is determined in the time domain. Thus, the equations of motion with fractional derivatives derived for the considered system are numerically integrated. Moreover, the paper extends the response spectrum concept to structures equipped with fractional viscoelastic dampers. [ABSTRACT FROM AUTHOR]

In paper the design sensitivity analysis of dynamic characteristics of structures with viscoelastic (VE) dampers is considered. The dampers are modeled using the classical rheological model and the rheological model with fractional derivatives. The design sensitivity is analyzed by using direct differentiation method and adjoint variables method. The formulae enabling the calculation of sensitivity of the first and the second order with respect to chosen design parameters have been derived. Sensitivity analysis of eigenvalues, eigenvectors and dynamic characteristics such as natural frequencies and non-dimensional damping ratios are presented. The correctness of the presented formulae is illustrated in examples. [ABSTRACT FROM AUTHOR]

Highlights: [•] The dynamic characteristics of structures with viscoelastic dampers are determined. [•] The fractional, rheological model of damper is applied. [•] The proposed approach enables the simultaneous use of a few models of damper. [•] The equations of motion are solved using the continuation method. [•] The problem of existence of real eigenvalues is also discussed. [ABSTRACT FROM AUTHOR]

• The subspace iteration method is used to solve nonlinear eigenvalue problem. • The continuation method is used to solve the nonlinear Hermitian eigenproblem. • The method presented allows determination of only a part of eigenvalues. • The method was tested for structures with viscoelastic elements. • The models of VE elements are described with the help of fractional derivatives. The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the subspace iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solving the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]