Abstract
In this paper, an improved short memory principle based on the Grünwald–Letnikov definition is proposed and applied in solving fractional vibration differential equations. The improved idea is to adjust the truncation of memory time in short memory principle (SMP) to the truncation of binomial coefficient terms, and the finite coefficients are repeatedly applied to the step size gradually enlarged. In this method, a very small initial step size is used to meet the accuracy requirements, and then the step size is gradually enlarged to prolong the memory time and reduce the amount of calculation. Examples of free vibration, forced vibration with a single-degree-of-freedom and a vehicle suspension two-degree-of-freedom vibration reduction model verify the method’s accuracy and effectiveness.
Highlights
The general concept of fractional derivative appears nearly at the same time as integer derivative.At this time, Leibniz expressed the derivative of integer order as dn y/dxn where n as an integer.in 1695, he and L’Hospital discussed the significance of the derivative when n = 1/2 [1].Fractional calculus did not make good progress in its early development due to the lack of a unified and widely accepted definition
An improved short memory principle (SMP) is proposed based on the fractional differential equation of the Grünwald–Letnikov definition, and the single-degree-of-freedom fractional-damped free vibration, forced vibration differential equations and vehicle suspension two-degree-of-freedom vibration reduction model are used as examples to verify its validity and reliability
If the traditional SMP is adopted, than the free vibration will not be attributed to the equilibrium position, and the improved
Summary
The general concept of fractional derivative appears nearly at the same time as integer derivative. The traditional integer-order viscoelastic differential constitutive model cannot accurately describe the mechanical behaviour of viscoelastic materials. In the past 20 years, many studies have introduced the fractional differential constitutive model into vibration analysis with viscoelastic damping. Given the history memory property of the time fractional derivative, a large amount of data is involved in the calculation to obtain more accurate results in the numerical solution process. When the fractional order tends to 0, the memory time intercepted by the SMP may be extremely large to ensure calculation accuracy. If the number of calculated data points exceeds the number of binomial coefficients, the step size will be enlarged for the limited binomial coefficients to cover a large time area This improvement aims to obtain more accurate results with as little data as possible. A single-degree-of-freedom free vibration example is used to verify the accuracy and reliability of the method
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