The fractional order elastic-viscoelastic equations of motion: Formulationand solution methods

The equations of motion of structures with elastic and viscoelastic materia ls in the time domain are derived. The development is consistent with the f inite element formulation and leads to a system of equations where the matr ices are symmetric, real, and composed of constant coefficients. A four-par ameter fractional derivative model is used to model the frequency dependenc e of the linear viscoelastic material since experimental data can be fitted successfully over a wide frequency range. The resulting equations of motio n are known as the elastic-viscoelastic equations of motion. Numerical proc edures for solving the elastic-viscoelastic equations of motion in the time domain are developed, and a procedure based on the central-difference meth od that incorporates fractional derivatives is presented. The numerical sta bility of the procedure is presented, and criteria for selecting the size o f the time step are given. The closed-form, steady state solution of a single degree of freedom system is obtained in the frequency domain and is utilized to validate the result s obtained by using the numerical procedures. The proper selection of the s tiffness for viscoelastic dampers placed in elastic structural systems is d iscussed in order to ensure that the damper is effective in reducing dynami c amplification of the structure. The dynamic response of a multi-degree of freedom structure, obtained by using the numerical procedures, is used to demonstrate the effectiveness of the dampers in reducing the structural res ponse to dynamic loading.