Dynamics With Variable-Order Operators

Abstract

In this work, the novel concept of Variable-Order (VO) Calculus is explored. VO Calculus extends the notion of Constant-Order (CO) Calculus by allowing the order of differentiation or integration to be a function of pertinent variables. VO Calculus is a powerful tool to model history-dependent processes and may allow solutions of CO partial differential equations to be reduced to simple VO expressions. Such expressions are extremely useful for parametric studies of complex systems such as fuel cells. Following a review of fractional calculus, the behavior of proposed Variable-Order Differential Operators (VODOs) is analyzed and it is shown that these operators must interpolate between integer-order derivatives. A VODO that returns the associated CO derivative at any instant in time is identified. The selected VODO is used to formulate a VO Differential Equation (VODE) of motion for a variable viscoelasticity oscillator. The damping force in the oscillator varies continuously between the elastic and viscous regimes depending on the position of the mass. The oscillator is composed of a linear spring of stiffness k that inputs a restitutive force Fk = -kx(t), a VO damper of order q(x(t)) that generates a damping force Fq = -cq1Ji(x(t)) x(t), and a mass m. A Runge-Kutta method is used in conjunction with a product-trapezoidal numerical integration technique to yield a second-order accurate method for the solution of the VODE. The VO oscillator also is modeled using a CO formulation where a number of CO fractional derivatives 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).