Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models

Abstract

The visco-elastic dampers can be economically designed for response reduction of dynamical systems. Recent developments in fractional calculus have affected the modeling of visco-elastic materials. The fractional models for visco-elastic dampers, which are parsimonious, require fewer parameters in comparison with other models. In this paper, we use the visco-elastic dampers for control of structural responses. The visco-elastic damper utilizes three important parameters including damping coefficient, stiffness, and fractional order. The dynamic model of the building with visco-elastic dampers can be acquired by a system of fractional differential equations, which includes both fractional and ordinary derivatives. We apply spline collocation methods for obtaining the numerical solution of this complex system. The collocation method is implemented in a graded mesh. The discrete data of earthquake are smoothed by spline interpolation of higher degree. The introduced method is used to simulate the response of a structure with active and passive controllers. A comparison of using different dampers is considered. Also, $H_{\infty }$ norm of the transform function is used for response analysis. The results of simulations for four-story and 10-story buildings show significant reductions in structural responses. Moreover, our analysis presents that the large variety of values can be used for parameters of the visco-elastic damper to provide flexibility in designing appropriate visco-elastic-dampers.