Abstract
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.
Highlights
IntroductionThe equilibrium of time derivatives of the function of x is denoted as d n1 dt n1 x +
The equilibrium of time derivatives of the function of x is denoted as d n1 dt n1 x +d n2 dt n2 x = 0 i.e. d n1 dt n1 x= − d n2 dt n2 x (1)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
This paper investigates the equilibrium of fractional derivative and 2nd derivative, which occurs if the original function is damped, where the fractional derivative corresponds to a force applied to the solid, and the second derivative corresponds to acceleration of the solid’s centre of mass, and to the inertial force
Summary
The equilibrium of time derivatives of the function of x is denoted as d n1 dt n1 x +. 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 dm dt m x = dn dt n x i.e.
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