Technical Brief

Time-Dependent Decay Rate and Frequency for Free Vibration of Fractional Oscillator

[+] Author and Article Information
Y. M. Chen, Q. X. Liu, J. K. Liu

Department of Mechanics,
Sun Yat-Sen University,
No. 135 Xingang Road,
Guangzhou 510275, China

1Corresponding author.

Manuscript received July 29, 2018; final manuscript received October 21, 2018; published online November 14, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 86(2), 024501 (Nov 14, 2018) (6 pages) Paper No: JAM-18-1451; doi: 10.1115/1.4041824 History: Received July 29, 2018; Revised October 21, 2018

This paper presents an investigation on the free vibration of an oscillator containing a viscoelastic damping modeled by fractional derivative (FD). Based on the fact that the vibration has slowly changing decay rate and frequency, we present an approach to analytically obtain the initial decay rate and frequency. In addition, ordinary differential equations governing the decay rate and frequency are deduced, according to which accurate approximation is obtained for the free vibration. Numerical examples are presented to validate the accuracy and effectiveness of the presented approach. Based on the obtained results, we analyze the decay rate and the frequency of the free vibration. Emphasis is put on their time-dependence, indicating that the decay rate decreases but the frequency increases with time increasing.

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Grahic Jump Location
Fig. 1

Initial decay rate (|a0|) and initial frequency (ω0) for the responses of system (1) provided by the presented method, the numerical method, and the averaging method, respectively

Grahic Jump Location
Fig. 2

Time histories obtained by the presented method, the numerical method, the averaging method and the Laplace transform, respectively, for system (1) with c=0.4 and x(0)=0,x′(0)=1

Grahic Jump Location
Fig. 3

Time histories obtained by the presented method (line), the numerical method (dot), and the Laplace transform (cross), respectively, for system (1) with c=0.2 and (a) α=0.1, x(0)=1,x′(0)=−1, (b) α=0.5, x(0)=1,x′(0)=0, and (c) α=0.9, x(0)=1,x′(0)=1

Grahic Jump Location
Fig. 4

Curves provided by Pk(t)=ω0t+ϕ(t)+θ−(0.5+2k)π for determining the time (textk) when x(t) reaches extreme values. Parameters are selected as α=0.5 and c=0.2.

Grahic Jump Location
Fig. 5

Time-dependent decay rate and frequency of the free vibration for system (1) with x(0)=1,x′(0)=1 and c=0.2

Grahic Jump Location
Fig. 6

Time-dependent decay rate and frequency of the free vibration for system (1) with α=0.5 and c=0.2. The lines denote the numerical results, and the dots for the results by the presented method.



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