# Improved Short Memory Principle Method for Solving Fractional Damped Vibration Equations
## 1 Introduction  
The general concept of fractional derivative appears nearly at the same time as integer derivative. At this time, [PERSON] expressed the derivative of integer order as \(d^{n}y/dx^{n}\) where \(n\) as an integer. Then, in 1695, he and [PERSON] 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. Until the mid-19 th century, mathematicians proposed some meaningful definitions of fractional calculus, such as the Grunwald-Letnikov, Riemann-Liouville and Caputo definitions, and then the field of fractional calculus was really established. In recent decades, fractional calculus has been widely used in electrochemical processes [2; 3], dielectric polarisation [4], bioengineering [5; 6; 7], chaos [8], energy supply-demand system [9], viscoelastic mechanics [10; 11] and other emerging fields. Viscoelastic materials are widely used in vibration and noise reduction fields, such as automobiles, airplanes and buildings. For example, the elastomeric lag dampers of helicopters are a single-degree-of-freedom damping mechanism. During operation, viscoelastic materials convert mechanical energy into heat energy to achieve the purpose of vibration reduction. Many experiments have shown that the stress relaxation of many viscoelastic materials is non-exponential and have a historical memory property. The traditional integer-order viscoelastic differential constitutive model cannot accurately describe the mechanical behaviour of viscoelastic materials. However, reasonable results can be obtained by using the fractional constitutive model to describe the stress-strain relationship of viscoelastic materials [12; 13; 14; 15; 16; 17; 18; 19]. In the past 20 years, many studies have introduced the fractional differential constitutive model into vibration analysis with viscoelastic damping. On the basis of the fractional Kelvin-Voigt and Maxwell models, the free damped vibration of the oscillator was analysed in reference [20]. [21] investigated the random vibration of a frequency-dependent fractional derivative dynamic system and proposed a standard formula for predicting the random vibrationresponse. The vibration of a rectangular plate with fractional damping and the effect of viscosity on the overall vibration response were studied [22]. The dynamic response of a plane inhomogeneous anisotropic body composed of linear viscoelastic materials was investigated, and a numerical method for solving multi-term fractional differential equations was developed [23]. In reference [24], a variable fractional-order Pasternak-type viscoelastic model is proposed, and the governing equation is established. The results show that the model's fractional order has a great influence on deflection and the bending moment. The response of the Euler-Bernoulli beams with fractional order viscoelasticity under quasi-static and dynamic loads and the random vibration response of beams with fractional damping were studied in [25] and [26]. The nonstationary vibrations of linear viscoelastic single-, two- and multi-degree-of-freedom mechanical systems with a fractional derivative instead of an ordinary derivative were investigated [27]. The application of a fractional-derivative-based isolation model in seismic vibration analysis is discussed in [28]. Nonlinear fractional order systems have also been greatly developed [29; 30; 31; 32; 33]. 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. Reference [12] proposes the short memory principle (SMP) which refers to intercepting the most recent time period and ignoring the time period with less impact and farther away. SMP was used to verify that the fractional derivative of the periodic function is still a periodic function [34].