【摘要】：在振動力學里,許多阻尼尤其是粘彈性阻尼,其阻尼力往往與位移的某一分數(shù)階導數(shù)成正比,這時就有必要引入分數(shù)階本構(gòu)關系來研究振動系統(tǒng)的特性。本文是在經(jīng)典振動力學的基礎上,引入分數(shù)階本構(gòu)關系,探討各種振動現(xiàn)象,對振動的基本規(guī)律進行研究。論文在第1章說明了本課題的來源、研究目的和意義,介紹了分數(shù)階理論的起源和發(fā)展,以及它需要的基礎理論。論文第2章研究了周期激勵下單自由度分數(shù)階振動系統(tǒng)的穩(wěn)態(tài)響應,得到了等效剛度系數(shù)和等效阻尼系數(shù),分析了分數(shù)階導數(shù)項對剛度和阻尼的影響。得到振幅放大因子和相位角,考慮了分數(shù)階階數(shù)和系數(shù)對振幅和相位角的影響。并且提出了用復指數(shù)的傅立葉級數(shù)形式表示周期激勵的新方法,得到周期激勵下的穩(wěn)態(tài)響應。論文第3章研究了單自由度分數(shù)階振動系統(tǒng)的瞬態(tài)響應,使用Laplace變換及復雜的反變換積分公式對穩(wěn)態(tài)響應進行了詳細分析,得到用基本解表示的瞬態(tài)響應方程�；诳挛鞫ɡ砗土魯�(shù)定理,得到基本解的方程,討論了基本解的漸近性�？紤]了對于特定的系數(shù)和分數(shù)階階數(shù)時,基本解的零點的個數(shù)和最大零點,研究了其規(guī)律。論文第4章研究了周期激勵下多自由度分數(shù)階振動系統(tǒng)的穩(wěn)態(tài)響應,得到振幅和相位角,考慮了分數(shù)階階數(shù)對振幅和相位角的影響。并且使用第2章中的新方法,得到周期激勵下的穩(wěn)態(tài)響應。論文第5章研究了多自由度分數(shù)階振動系統(tǒng)的瞬態(tài)響應,使用Laplace變換和反變換積分公式對瞬態(tài)響應進行了分析,得到用基本解表示的瞬態(tài)響應方程,利用Mittag-Leffler函數(shù)得到了基本解的方程,討論了基本解的漸近性。
[Abstract]:In vibration dynamics, many damping forces, especially viscoelastic damping, are usually proportional to a fractional derivative of displacement, so it is necessary to introduce fractional constitutive relation to study the characteristics of vibration system. In this paper, based on the classical vibration dynamics, the fractional constitutive relation is introduced to study various vibration phenomena, and the basic laws of vibration are studied. In chapter 1, the origin, research purpose and significance of this topic are explained, and the origin and development of fractional order theory are introduced, as well as the basic theory that it needs. In chapter 2, the steady-state response of fractional vibration system with periodic excitation is studied. The equivalent stiffness coefficient and equivalent damping coefficient are obtained, and the influence of fractional derivative term on stiffness and damping is analyzed. The amplitude amplification factor and phase angle are obtained, and the effects of fractional order and coefficient on amplitude and phase angle are considered. A new method of expressing periodic excitation by Fourier series of complex exponents is proposed, and the steady state response of periodic excitation is obtained. In chapter 3, the transient response of fractional vibration system with single degree of freedom is studied. The steady state response is analyzed in detail by using Laplace transform and complex inverse transform integral formula, and the transient response equation expressed by the basic solution is obtained. Based on Cauchy theorem and residue theorem, the equation of the fundamental solution is obtained, and the asymptotic behavior of the fundamental solution is discussed. The number of zeros and the maximum zeros of the fundamental solution for a given coefficient and fractional order are considered and their laws are studied. In chapter 4, the steady-state response of fractional vibration system with multiple degrees of freedom under periodic excitation is studied. The amplitude and phase angle are obtained, and the influence of fractional order on amplitude and phase angle is considered. The steady state response under periodic excitation is obtained by using the new method in Chapter 2. In chapter 5, the transient response of fractional vibration system with multiple degrees of freedom is studied. The transient response is analyzed by using Laplace transform and inverse transform integral formula, and the transient response equation expressed by the basic solution is obtained. The equation of the fundamental solution is obtained by using the Mittag-Leffler function, and the asymptotic behavior of the fundamental solution is discussed.
【學位授予單位】：上海應用技術大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O172;TB53
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本文編號：2253356