线性分数阶振动系统的分析与模拟
发布时间：2018-10-05 12:21
【摘要】：在振动力学里,许多阻尼尤其是粘弹性阻尼,其阻尼力往往与位移的某一分数阶导数成正比,这时就有必要引入分数阶本构关系来研究振动系统的特性。本文是在经典振动力学的基础上,引入分数阶本构关系,探讨各种振动现象,对振动的基本规律进行研究。论文在第1章说明了本课题的来源、研究目的和意义,介绍了分数阶理论的起源和发展,以及它需要的基础理论。论文第2章研究了周期激励下单自由度分数阶振动系统的稳态响应,得到了等效刚度系数和等效阻尼系数,分析了分数阶导数项对刚度和阻尼的影响。得到振幅放大因子和相位角,考虑了分数阶阶数和系数对振幅和相位角的影响。并且提出了用复指数的傅立叶级数形式表示周期激励的新方法,得到周期激励下的稳态响应。论文第3章研究了单自由度分数阶振动系统的瞬态响应,使用Laplace变换及复杂的反变换积分公式对稳态响应进行了详细分析,得到用基本解表示的瞬态响应方程。基于柯西定理和留数定理,得到基本解的方程,讨论了基本解的渐近性。考虑了对于特定的系数和分数阶阶数时,基本解的零点的个数和最大零点,研究了其规律。论文第4章研究了周期激励下多自由度分数阶振动系统的稳态响应,得到振幅和相位角,考虑了分数阶阶数对振幅和相位角的影响。并且使用第2章中的新方法,得到周期激励下的稳态响应。论文第5章研究了多自由度分数阶振动系统的瞬态响应,使用Laplace变换和反变换积分公式对瞬态响应进行了分析,得到用基本解表示的瞬态响应方程,利用Mittag-Leffler函数得到了基本解的方程,讨论了基本解的渐近性。
[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.
【学位授予单位】：上海应用技术大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O172;TB53
，
本文编号：2253356
[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.
【学位授予单位】：上海应用技术大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O172;TB53
，
本文编号：2253356
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