Abstract
We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system.
Highlights
Fractional calculus has undergone rapid developments in its theory, methods, and applications in recent decades due to its capability of modeling memory phenomena and hereditary properties
We focus on the effects of the fractional derivative and the distributed-order derivative on the viscosity and the inertia in a vibration system
We considered forced harmonic vibration systems with a fractional-order derivative
Summary
Fractional calculus has undergone rapid developments in its theory, methods, and applications in recent decades due to its capability of modeling memory phenomena and hereditary properties. In [34], a fractional vibration equation was considered, and it was found that if the order satisfies 1 < λ < 2, the fractional derivative contributes to both the viscosity and the inertia, i.e., the viscoinertia, corresponding to the terminology of viscoelasticity. We consider a forced harmonic vibration system with the fractional-order derivative −∞ Dtλ x (t), where 1 ≤ λ ≤ 2.
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