This paper investigates the equilibrium of fractional derivative and 2nd derivative, which occurs if the original function is damped (damping of a power-law viscoelastic solid with viscosities η of 0 ≤ η ≤ 1), where the fractional derivative corresponds to a force applied to the solid (e.g. an impact force), and the second derivative corresponds to acceleration of the solid’s centre of mass, and therefore to the inertial force. Consequently, the equilibrium satisfies the principle of the force equilibrium. Further-more, the paper provides a new definition of under- and overdamping that is not exclusively disjunctive, i.e. not either under- or over-damped as in a linear Voigt model, but rather exhibits damping phases co-existing consecutively as time progresses, separated not by critical damping, but rather by a transition phase. The three damping phases of a power-law viscoelastic solid—underdamping, transition and overdamping—are characterized by: underdamping—centre of mass oscillation about zero line; transition—centre of mass reciprocation without crossing the zero line; overdamping—power decay. The innovation of this new definition is critical for designing non-linear visco-elastic power-law dampers and fine-tuning the ratio of under- and overdamping, considering that three phases—underdamping, transition, and overdamping—co-exist consecutively if 0 < η < 0.401; two phases—transition and overdamping—co-exist consecutively if 0.401 < η < 0.578; and one phase— overdamping—exists exclusively if 0.578 < η < 1.
The equilibrium of time derivatives of the function of x is denoted as
where d is the differential operator; and n1 and n2 are the orders of differentiation, where n1 ≠ n2, and n1 and n2 are not necessarily integers. These kinds of differential equations are common in dynamics, e.g. in a mass-spring system, where the spring force (0th derivative) is in equilibrium with the inertial force (2nd derivative).
The term equilibrium should not be confused with the term equality of derivatives, such as
The latter equalities define the so-called cyclodifferential functions [
1) if
2) if
3) if
4) if
5) if
The aforementioned equilibrium, however, yields solutions different from the traditional cyclodifferential functions:
1) if
2) if
3) if
It is therefore of interest to investigate other equilibria involving fractional derivatives, e.g. with orders of n1 = 0.5 and n2 = 2, or, in general, 0 < n1 < 1 and n2 = 2.
These kinds of (extraordinary) differential equations characterize the behaviour of power-law visco-elastic solids. The constitutive equation of a power-law visco-elastic solid, derived from stress relaxation, is
[
Substituting the function
Equation (1) corresponds to Scott Blair’s [
In horizontal oscillations or impact, i.e. without gravitational force, two forces are in equilibrium: the inertial force and the applied or impact force at the contact between free-body diagram and the environment (
The aim of this paper is to analyse this equilibrium and to evaluate the properties of deflection; velocity; and forces, as well their relationship to fractal derivatives.
The force equilibrium of horizontal oscillations or impact is defined as:
[
Solving for
Let
then
Substituting Equation (7) for
Equation (8) is the fractional (ηth) derivative of Equation (5) times kΓ(1 − η).
The first derivative of Equation (7) yields the velocity of the solid’s COM, whose x0 is zero
The second derivative of Equation (7) yields the acceleration of the solid’s COM, which, times m, corresponds to the inertial force acting on the COM
and
If η = 0, then the equation of horizontal oscillation or impact becomes the equilibrium of the function of x (times k) and its second derivative (times m):
The analytical solution after applying the inverse Laplace transform equals the constitutive equation of undamped oscillation, i.e. the behaviour of a Hookean spring:
If η = 1, then the equation of horizontal oscillations or impact becomes the equilibrium of first derivative of x and second derivative of x:
the analytical solution of which is the integral of e−Ct plus an integration constant for satisfying the initial condition of x0 = 0:
Considering that C → ∞ (i.e. Γ0) when η → 1, Equation (15) reduces to x = 0, i.e. the behaviour of an incompressible, iso-volumetric fluid, and therefore of a Newtonian damper. Consequently, for the equation of a horizontal oscillation or impact, 0 ≤ η ≤ 1, as there is a solution for η = 1. Furthermore, Equation (4), the force equilibrium of horizontal oscillations or impact, reflects the principle of Scott Blair’s rheological element [
If η = 0.5, then the equation of horizontal oscillation or impact becomes the equilibrium of semi-derivative and second derivative of x:
Analytical solutions of Equation (16) and the more generalized Equation (7) are found when using the Mittag-Leffler Equation. Having different powers of s in the denominator of the transformed equations, e.g. 2 and 0.5 in Equation (16), and 2 and η in Equation (7), leads to Equation (17) with generalized powers of s, namely α and β,
the inverse Laplace transform of which is
[
so that β = 2, β − α = η, α = 2 − η. Note that α equals the difference between the two orders of differentiation, n1 and n2 of Equation (1), i.e. the 2nd and the fractional (ηth) derivatives. Note that C is negative for solving Equation (1), i.e. the equilibrium case, whereas C would be positive for solving Equation (2), the equality case. As a result, the analytical solution of Equation (7), considering Equation (18), is
Validating Equation (20) for η = 0 and η = 1, with C = 1 and v0 = 1, yields
and
thereby mirroring the functions of Equations (13) and (15). The derivatives of Equation (20) are
where a is the acceleration, FI is the inertial force, and FA is the applied force (
Oscillations and impact were modeled in horizontal direction, to avoid any influence by the gravitational force. For evaluating the damping and impact dynamics, the solid was represented by a point mass (the solid’s centre of mass, COM) tethered to the frame by a power-law visco-elastic spring (
i.e. the equilibrium of ηth derivative of x and second derivative of x.
The term oscillation is used for simple harmonic oscillation, undamped or underdamped, whereby the centre of mass (COM) oscillates about, and therefore crosses, the zero line, i.e. x0, the COM position of the unloaded solid.
The term reciprocation is used for forward-backward motion of the horizontally moving COM without crossing the zero line; this term is used exclusively for the transition from underdamped to overdamped states (to be explained in detail below); note that this transition is not related to critical damping.
The magnitude and direction of displacement, velocity, and acceleration of the COM; and of the inertial and impact force are defined as follows:
Displacement x and velocity v are positive on compression, and negative on expansion and tension.
The fractal derivative of the displacement is initially positive. The applied or impact force FA results from the fractal derivative, if multiplied by kΓ(1 − η) and by −1. Therefore, the applied or impact force is initially negative, which corresponds to compression. Tension causes a positive applied force.
The 2nd derivative of the displacement, the acceleration a, is initially negative. The inertial FI force results from the 2nd derivative, if multiplied by m and by −1. Therefore, the inertial force is initially positive.
In this section, the displacement, velocity and forces acting on a visco-elastic solid of η = 0.5 are explained first, as the applied or impact force corresponds to the semi-derivative of the displacement. Subsequently, the viscosity is expanded to 0 ≤ η ≤ 1 to understand the effect of different viscosities on the behaviour of the solid at impact. Finally, the damping behaviour and the coefficient of restitution COR are explained as a function of η.
According to Equation (16), the applied or impact force, i.e. the semi-derivative of x, multiplied by kΓ(1 − η), is in equilibrium with the inertial force, i.e. the second derivative of x(multiplied by m). The solution of this EODE (extraordinary differential equation), shown in
mirror image of the 2nd derivative (times the mass, which is unity here), i.e. the inertial force.
Considering the two extreme cases, namely undamped oscillation if η = 0, according to Equation (13), and maximal damping (no movement, x = 0) if η = 1, according to Equation (15), it is evident that η = 0.5 yields a damped function.
After the initial positive semi-derivative spike, the signal becomes negative at t = 2.017 s (
when decaying, as the material returns to its original shape. The fractional derivative functions decay at a power of 3 − η; the fractional derivative is negative when decaying, which corresponds to a positive applied force (tensile force), a negative inertial force, a positive acceleration and a negative deceleration. This means that the COM moves away from the frame and is decelerated when decaying.
As already mentioned above, the displacement data show oscillations (underdamping) initially (at least at smaller viscosities), and terminate in a power decay (overdamping). This behaviour stands in sharp contrast to standard damping of a mass-spring-damper system (point mass in series with a Voigt model, i.e. a linear spring and a linear damper in parallel). In the latter model, damping depends on the damping ratio ζ, which defines under-, critical and over-damping, if the damping ratio is ζ < 1, ζ = 1 or ζ > 1, respectively. As ζ approaches 1, the amplitude of the underdamped oscillations asymptotes to zero and their wavelengths increase, until the oscillations vanish in an exponential
decay at ζ = 1 (critical damping, transition from under- to overdamping). Under- and over-damping of a Voigt model are exclusively disjunctive, i.e. a system is either overdamped or underdamped throughout its time history.
In a power-law visco-elastic solid, however,
1) the wavelengths of underdamped oscillations decrease as the viscosity increases;
2) under- and overdamping are not exclusively disjunctive but (co-)exist consecutively throughout the time history of the COM displacement function (underdamping at smaller times and overdamping at larger times); and
3) there is no critical damping but rather a transition phase from under- to overdamping, whereby this transition is characterised by a reciprocating COM.
The three damping phases, underdamping, transition, and overdamping are a function of time and occur consecutively after the initial displacement of the COM. At the initial deflection of an oscillating COM, or at impact, the COM approaches the frame
and reaches its maximal displacement. After this displacement peak A (
1) Underdamping: after the initial displacement peak A, the COM oscillates about the zero line (x0), until it crosses the zero line the last time and reaches a positive peak B (
2) Transition phase: after this positive peak B, the COM displacement values stay positive, and reciprocate slightly, with positive and negative gradients, until they reach a peak C (
3) Overdamping: after peak C, the COM displacement decays (negative gradient) to zero (i.e. x0), by asymptoting towards a power decay of a power of η − 1 (negative power, as the gradient of the decay is negative). The overdamping phase can already start at peak A if both underdamping and transition phased out at higher viscosities (at η = 0.57774 in
This constitutes a new definition of damping, where under- and over-damping do not only depend on the degree of viscosity but also depend on the time, thereby existing consecutively as a function of time.
The COR is the ratio of output velocity of the COM to initial (input) velocity. For impacts, these velocities are rebound and incident velocity, respectively. In a damped oscillating COM, the output velocity corresponds to the first negative velocity peak (
Comparable behaviour of damping, i.e. co-existence of under- and overdamping was described by Burov and Barkai [
1) an underdamped solution as the displacement of a particle oscillates with a non-monotonic decay and crosses the zero line;
2) a non-monotonic decay solution without zero crossing; and
3) an overdamped solution as monotonically decaying, where the decay is of the power law type.
They define a “critical exponent”, where the exponent corresponds to the fractional order of the displacement’s derivative. This exponent corresponds to the viscosity η in Equation (2) and ranges between 0 and 1. If the exponent is smaller than the “critical”
one, then “the overdamped behavior totally disappears, and the decay to equilibrium is never monotonic” [
1) underdamping, followed by the transition phase, followed by overdamping (
2) transition phase, followed by overdamping (
3) overdamping only (
with special cases of η = 0 (undamped) and η = 1 (maximally damped, rigid). In addition to that, a method was presented in the present study (
Sandev et al. [
The equilibrium of fractional derivative and 2nd derivative of a function implies that this function is a damped oscillation function. If the viscosities of a power-law visco-elastic solid are 0 or 1, then the deflection function is either undamped or maximally damped, respectively. Viscosities η between 0 and 1 result in under- and overdamping. In contrast to a mass-spring-damper system (point mass in series with a Voigt model, i.e. a linear spring and a linear damper in parallel), in which under- and overdamping are exclusively disjunctive, separated by critical damping, in a power-law visco-elastic model under- and overdamping exist consecutively as time progresses, separated by a transition phase. This analytical discovery constitutes a new definition of damping, related to non-linear visco-elastic solids, rather than to linear visco-elastic structures.
The innovation of this discovery is critical for designing non-linear visco-elastic power-law dampers and fine-tuning the ratio of under- and overdamping, considering that three phases―underdamping, transition, and overdamping―co-exist consecutively if η < 0.401; two phases―transition and overdamping―co-exist consecutively if 0.401 < η < 0.578; and one phase―overdamping―exists exclusively if η > 0.578.
The author thanks the anonymous reviewers of this paper for their insightful and valuable comments and suggestions on the manuscript, which have led to significant improvements of the paper.
Fuss, F.K. (2016) The Equilibrium of Fractional Derivative and Second Derivative: The Mechanics of a Power-Law Visco-Elastic Solid. Applied Mathematics, 7, 1903-1918. http://dx.doi.org/10.4236/am.2016.716156