Hiroshi Nasuno

A nonlinear fractional derivative model of impulse motion for viscoelastic materials

Generally, force can be described as a function of displacement in the mechanical model. A nonlinear fractional derivative model with respect to displacement is proposed to describe force for a viscoelastic material based on the measured data of impulsive motion. In the model, the nonlinearity is assumed to appear in the term of the fractional derivative. Three types of nonlinearity in the fractional derivative term are considered as candidates for a suitable model for reproducing the impulsive responses of the measured data. The first one is the case where the nonlinearity appears in the coefficient of the fractional derivative and the second in the fractionally differentiated term. The third one is the case where the nonlinearity appears as the combination of the above two types. The equation of motion and the initial conditions are derived by employing the above nonlinear models for head-on collisions of a rigid body onto the viscoelastic material. The property of the impulsive responses for the system that is derived above is characterized by the time when the acceleration shows its maximum. The symmetry property of increasing and decreasing acceleration response about the time of maximum acceleration is also considered. The second-type nonlinearity in the model seems to be adequate for reproducing the measured response.

Power Time Numerical Integration Algorithm for Nonlinear Fractional Differential Equations

A numerical integration algorithm to solve single-degree-of-freedom nonlinear fractional differential equations (NFDEs) is proposed, by introducing the power time concept and by means of the Newmark-β method one step scheme. An NFDE involving fractional derivatives of the displacement and the displacement squared is solved numerically, using the proposed power time algorithm. The error analysis of the algorithm, and the contribution of the displacement squared fractional derivative term to the solution of NFDE, are also given.

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.Copyright

Numerical Integration Algorithm for Fractional Differential Equation by means of Power Time

This paper is concerned with the development of an efficient algorithm for the numerical solution of the fractional differential equation (FDE). The numerical integration of the FDE requires significant computational cost, because the fractional convolution integral included in the fractional derivative, requires O(N2) operations for N points calculation. The kernel of the fractional integral has singularity and consequently excessive small time-step near the singularity is needed to secure the high precision in the numerical calculation. This difficulty is solved by means of a new computational procedure for fractional derivative by introducing the variable trasformation from the physical time to the power time which is newly defined in this paper. The proposed algorithm is used to solved the nonlinear FDE. Computational results are compared with those by the former method (Nasuno and Shimizu, JSME (C), 2006). The proposed method shows remarkably higher performance than the former one.

A NUMERICAL ALGORITHM FOR DIFFERENTIAL EQUATIONS WITH NONLINEAR FRACTIONAL DERIVATIVES

Abstract A numerical integration algorithm to solve single-degree of freedom (1-DOF) nonlinear fractional differential equations (NFDEs) is developed by the use of the one stepscheme; Newmark- β method. The NFDE involving the fractional derivatives of the displacement and the displacement squared is numerically solved by the proposed numerical integration algorithm. In this paper, the derivation of the numerical integration algorithm, the error analysis of the algorithm, and the contribution of the displacement squared fractional derivative term on the solution of the NFDE are given.