线性分数阶阻尼振动系统分析

发布时间：2019-02-19 21:47

【摘要】：一些像高聚物类的黏弹性材料本构关系存在着多种形式,在传统整数阶方程描述中,常常显得很复杂,而使用分数阶导数形式来描述则可显得既简洁而又准确。所以若将分数阶导数型的黏弹性本构关系应用到振动中,当会使得一些问题变得有意义。文中第一章叙述了分数阶导数的相关知识及其理论的发展和应用以及国内外的研究状况。第二章介绍了一些预备知识,包括分数阶微积分方面的主要定义,以及常见的三种分数阶黏弹性模型,这将是后面研究内容的知识基础。第三章研究了单自由度有阻尼受迫振动。并在给出初值的情况下对振动方程进行了 Laplace变换以及逆Laplace变换,得到了对于一般激励下的响应函数的表达式。通过数值解验证了正确性。然后,着重研究了不同分数阶导数下自由振动状态的振动特性。第四章研究了含分数阶导数项的二自由度振动,整个模型背景以车辆的悬架系统为研究对象展开,在建模后,主要研究了黏弹性悬架在简谐激励下稳态响应的一些基本特性,包括振幅和相位角。第五章考虑了连续系统的振动特性,内容包括分数阶黏弹性杆的纵向振动以及分数阶黏弹性梁的横向振动问题。对杆和梁的求解都使用了变量分离法,杆的求解过程用到了 Mittag-Leffler函数,而梁部分则是余弦激励下的稳态响应解,最后根据解的表达式给出了仿真图。最后一章对全文进行了总结,并对分数阶导数在黏弹性材料振动方面进行了展望。
[Abstract]:There are many forms of constitutive relations in some viscoelastic materials such as polymers, which are often complicated in the traditional integral order equation description, but it is simple and accurate to use fractional derivative form to describe them. Therefore, if the viscoelastic constitutive relation of fractional derivative is applied to vibration, some problems will become meaningful. The first chapter describes the related knowledge of fractional derivative and its development and application, as well as the research situation at home and abroad. The second chapter introduces some preparatory knowledge including the main definitions of fractional calculus and three kinds of fractional viscoelastic models which will be the knowledge base of the later research. In chapter 3, the damping forced vibration with single degree of freedom is studied. The Laplace transformation and inverse Laplace transformation of the vibration equation are carried out under the condition of giving the initial value, and the expression of the response function under the general excitation is obtained. The correctness is verified by numerical solution. Then, the vibration characteristics of free vibration state under different fractional derivatives are studied. In chapter 4, the vibration of two degrees of freedom with fractional derivative term is studied. The whole model background is based on the suspension system of vehicle. After modeling, some basic characteristics of the steady state response of viscoelastic suspension under harmonic excitation are studied. Including amplitude and phase angle. In chapter 5, the vibration characteristics of the continuous system are considered, including the longitudinal vibration of the fractional viscoelastic rod and the transverse vibration of the fractional viscoelastic beam. The method of separating variables is used for the solution of rod and beam. The Mittag-Leffler function is used in the process of solving the rod, while the beam part is the steady state response solution under cosine excitation. Finally, the simulation diagram is given according to the expression of the solution. In the last chapter, the whole paper is summarized, and the prospect of fractional derivative in viscoelastic material vibration 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】：2017
【分类号】：TB301

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