3. Ahmed Al-Jamel, Mohammad F Al-Jamal, and Ahmed El-Karamany, A memory-dependent derivative model for damping in oscillatory systems, Journal of Vibration and Control 1-9, 2016.

Classically, damping force is described as a function of velocity in the linear theory of mechanical models. In this work, a memory-dependent derivative model with respect to displacement is proposed to describe damping in various oscillatory systems of complex dissipation mechanisms where memory effects could not be ignored. A memory- dependent derivative is characterized by its time-delay ? and kernel function K(x, t) which can be chosen freely. Thus, it is superior to the fractional derivative in that it provides more access into memory effects and thus bet- ter physical meaning. To elucidate this, an equation of motion is proposed based on the prototype mass- spring model. The analytical solution is then attempted by the Laplace transform method. Due to the complexity of finding the inverse Laplace transform, a numerical inversion treatment is carried out using the fixed Talbot method and also compared with the finite difference discretization to validate the method. The calculations show that the response function is sensitive to different choices of ? and K(x, t). It is found that this proposed model supports the existence of memory-dependence in the structure of the material. The interesting case of resonance where the response function is classically increased rapidly is found to be weakened by an appropriate choice of ? and K(x, t)