A Comparison of Numerical Methods Applied to a Fractional Model of Damping Materials

Abstract

The use of fractional derivatives in the constitutive equations of systems with damping materials provides a powerful tool for modeling these systems because the model does not exhibit many of the short comings of those based on integer-order derivatives. The resulting equations of motion possess closed-form solutions only for single-degree-of-freedom systems and only for a small number of loadings. For practical applications, therefore, the equations of motion must be solved using numerical methods. This paper presents two numerical schemes to solve single-degree- and multi-degree-of-freedom systems with fractional damp ing subjected to a number of commonly used loading conditions. The techniques employed are based on the central difference method and the average acceleration method. Whenever possible, the numerical results are compared with the analytical solutions. The results of the two numerical methods are essentially identical, with the exact solutions for zero initial conditions, but differ for nonzero conditions and large damping. For small damping, the average method has the advantage of its simpler formulation, especially with regard to the starting values. For arbitrary damping, however, the central difference method, in view of its robustness, is the preferred method.

Numerical solution of fractionally damped beam by homotopy perturbation method

“…Similar observations may be made by keeping the order of the fractional derivative constant and varying the damping ratios as shown in Figures (3) and (4). It can clearly be seen that increasing the value of the damping ratios decreases the oscillations.…”

“…The eigenvector expansion method is successfully implemented in [17] to find the solution of dynamic systems containing fractional derivatives. Various numerical methods are applied in [4,18,20,21,27] to find the responses of a fractionally damped system. Recently, the homotopy perturbation method has been found to be a powerful tool for analysing this type of system involving fractional derivatives.…”

Formulation and solutions of fractional continuously variable order mass–spring–damper systems controlled by viscoelastic and viscous–viscoelastic dampers

“…Say we formulate the viscous-viscoelastic damping by q = 1=2½1 + (x(t)=L) 2 and viscoelastic-viscous damping by q = 1=2½1 À (x(t)=L) 2 , given in equation (21). Here, x(t) denotes the displacement that is defined for a guide x j j L. The guide is the length where the continuously variable damping force exists, and the motion of the mass is within the guide's limit.…”

“…The exact solution of fractional order of 1/2 was obtained by Elshehawey et al 12 The Green function approach for finding solution of dynamic system was studied by Agrawal, 13 which was followed by the Mittag-Leffler function proposed by Miller. 14 By using fractional Green function and Laplace transform, Hong et al 15 have obtained the solution of single-degree-of-freedom mass-spring system of order 0\a\1. The analytical solution of fractional systems mass-spring and spring-damper system formed by using Mittag-Leffler function was analysed by Gomez-Aguilar et al 16 The fractional Maxwell model for viscous-damper model and its analytical solution was proposed by Makris and Constantinou 17 and Choudhury et al 18 Other methods such as Fourier transform [19][20][21] and Laplace transform [21][22][23][24] have been proposed by researchers to find the solution of fractional damper systems. Recently, Saha Ray and Bera 25 used the Adomian decomposition method to determine the analytical solution of dynamic system of order one-half and proclaim that the acquired solutions coincided with the solutions obtained through eigenvector expansion method given by Suarez and Shokooh.…”

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

“…However, most of them are limited to the deterministic cases (see, e.g., [30], [52], [61], [62], [67], [71], [72], [85], [79], [77], [88]). Since stochastic perturbations are ubiquitous, it is necessary to investigate the fractionally damped systems subject to random excitations.…”