• Variable-order calculus;
  • fractional calculus;
  • memory effects;
  • viscoelasticity


The complex dynamics of a variable viscoelasticity oscillator is studied using the novel concept of Variable-Order (VO) Calculus. The damping force in the oscillator varies continuously between the elastic and viscous regimes depending on the position of the mass. The oscillator considered here is composed of a linear spring of stiffness k that inputs a restitutive force Fk = -k x, a VO damper of order q(x(t)) that generates a damping force Fq = -cq ��q(x(t))x, and a mass m. A modified Runge-Kutta method is used in conjunction with a trapezoidal numerical integration technique to yield a second-order accurate method for the solution of the resulting VO Differential Equation (VODE). The VO oscillator is also modelled using a Constant Order (CO) formulation where a number of CO fractional order differentials are weighted to simulate the VO behavior. The CO formulation asymptotically approaches the VO results when a relatively large number of weights is used. For the viscoelastic range of 0 ≤ q ≤ 1, the dynamics of the oscillator is well approximated by the CO formulation when 5 or more fractional terms are included (e.g., 0, 1/4, 1/2, 3/4, and 1).