This work demonstrates the practicality of using variable order (VO) derivative
operators for modeling the dynamics of complex systems. First we review the various
candidate VO integral and derivative operator definitions proposed in the literature.
We select a definition that is appropriate for physical modeling based on the following
criteria: the VO operator must be able to return all intermediate values between 0
and 1 that correspond to the argument of the order of differentiation in addition to
the integer order derivatives, and the derivative of a true constant function should be
0. Then we apply the chosen operator to 3 different problems: a stationary analysis
of viscoelastic oscillators, the formulation of a Lagrangian equation of motion for a
sedimenting particle in a viscous fluid, and the development of a constitutive equation
for viscoelastic materials.
In the first problem we obtain an analytical solution for the order of the operator
and connect the meaning of functional order to the dynamic properties of a viscoelastic
oscillator. We replace the multi-term differential equation for the viscoelastic
oscillator with a single-term VO equation. We determine that the order of differentiation
for a single operator describing all dynamic elements in the stationary equation
of motion (mass, damping and spring) is equal to the normalized phase shift. The normalization
constant is found by taking the difference between the order of the inertial
term (2) and the order of the spring term (0) and dividing this difference by the angular
phase shift between acceleration and position in radians (π), so that the normalization
constant is simply 2/π.
For the second problem we focus on the transient equation of motion for a
spherical particle sedimenting in a quiescent viscous liquid. In particular, we examine
the various force terms in the equation of motion and propose a new form for the
history drag acting on the particle at finite Reynolds numbers. This new form equates
the history drag to the VO derivative of the velocity of the particle. Using numerical
results from a finite element simulation of the particle we solve for order of the derivative
q and evaluate how the order changes over time. Based on these results we propose a
simple form for q and obtain a correlation for the history drag acting on the particle that
is in good agreement with the numerical data for terminal Reynolds numbers ranging
from 2.5 to 20.
In the final problem we present a simple constitutive equation for linear viscoelastic
materials strained at constant strain rates. We propose a relationship in which
the stress is related to the q(t) derivative of strain, where q(t) in this case is a function
of normalized time. This order function is postulated to be proportional to the rate
of change of disorder within the material. From a statistical mechanics based theory,
we find that q(t) is proportional to tInt. Using experimental data for an epoxy resin
and carbon/epoxy composite undergoing compression, we determine the final form for
the constitutive equation that models the linear viscoelastic deformation in time. The
resulting dimensionless constitutive equation agrees well all the normalized data.