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The Fractional Order Elastic-Viscoelastic Equations of Motion: Formulation and Solution Methods

The equations of motion of structures with elastic and viscoelastic materials in the time domain are derived. The development is consistent with the finite element formulation and leads to a system of equations where the matrices are symmetric, real, and composed of constant coefficients. A four-parameter fractional derivative model is used to model the frequency dependence of the linear viscoelastic material since experimental data can be fitted successfully over a wide frequency range. The resulting equations of motion are known as the elastic-viscoelastic equations of motion. Numerical procedures for solving the elastic-viscoelastic equations of motion in the time domain are developed, and a procedure based on the central-difference method that incorporates fractional derivatives is presented. The numerical stability of the procedure is presented, and criteria for selecting the size of 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 results obtained by using the numerical procedures. The proper selection of the stiffness for viscoelastic dampers placed in elastic structural systems is discussed in order to ensure that the damper is effective in reducing dynamic 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 response to dynamic loading.