A nonlinear fractional derivative model of impulse motion for viscoelastic materials

Abstract

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.