具有分数阶导数阻尼的强迫振动共振现象  

A forced vibration resonance phenomena with fractional derivative damping

作　　者：葛志新[1] 陈咸奖[2] 侯为根[3] 

机构地区：[1]安徽工业大学图书馆,安徽马鞍山243002 [2]安徽工业大学商学院,安徽马鞍山243002 [3]安徽工业大学数理科学与工程学院,安徽马鞍山243002

出　　处：《高校应用数学学报（A辑）》2015年第4期410-416,共7页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(31300125)

摘　　要：研究了一类具有分数阶导数阻尼的强迫振动共振现象.首先构造渐近解,然后利用Riemann-Liouville分数阶导数定义及性质,求出分数阶导数项的表达式.再利用多重尺度法,求出各共振的共振频率.这些共振既包括主共振也包括次共振.对于每个共振频率,引入解谐参数,消除长期项.利用数学软件画出共振振幅及初相位在不同的分数指数下的数值解的图形,发现分数指数对共振的影响,并对每个共振频率求出渐近解的一阶近似表达式.A forced vibration resonance phenomena with fractional derivative damping is studied in this paper.First, an asymptotic solution is constructed. Then, by the definition and properties of RiemannLiouville fractional derivative, the expression of the fractional derivative item is obtained. Then using the method of multiple scales, the frequency of each resonance is calculated. These resonances include the main resonance and the secondary resonances. For each resonance frequency, the detuning parameter is introduced. The secular terms are eliminated. After drawing the graph of the numerical solutions of amplitude and the relevant variables of initial phase when fractional derivatives are different by mathematical software, the influence of fractional derivatives on resonance is found. The first-order approximate expressions of the asymptotic solution for each resonance frequency is gotten.

关 键 词：多重尺度 分数阶导数 主共振 次共振 

分 类 号：O175.14[理学—数学]