【摘要】：一些像高聚物類的黏彈性材料本構關系存在著多種形式,在傳統(tǒng)整數階方程描述中,常常顯得很復雜,而使用分數階導數形式來描述則可顯得既簡潔而又準確。所以若將分數階導數型的黏彈性本構關系應用到振動中,當會使得一些問題變得有意義。文中第一章敘述了分數階導數的相關知識及其理論的發(fā)展和應用以及國內外的研究狀況。第二章介紹了一些預備知識,包括分數階微積分方面的主要定義,以及常見的三種分數階黏彈性模型,這將是后面研究內容的知識基礎。第三章研究了單自由度有阻尼受迫振動。并在給出初值的情況下對振動方程進行了 Laplace變換以及逆Laplace變換,得到了對于一般激勵下的響應函數的表達式。通過數值解驗證了正確性。然后,著重研究了不同分數階導數下自由振動狀態(tài)的振動特性。第四章研究了含分數階導數項的二自由度振動,整個模型背景以車輛的懸架系統(tǒng)為研究對象展開,在建模后,主要研究了黏彈性懸架在簡諧激勵下穩(wěn)態(tài)響應的一些基本特性,包括振幅和相位角。第五章考慮了連續(xù)系統(tǒng)的振動特性,內容包括分數階黏彈性桿的縱向振動以及分數階黏彈性梁的橫向振動問題。對桿和梁的求解都使用了變量分離法,桿的求解過程用到了 Mittag-Leffler函數,而梁部分則是余弦激勵下的穩(wěn)態(tài)響應解,最后根據解的表達式給出了仿真圖。最后一章對全文進行了總結,并對分數階導數在黏彈性材料振動方面進行了展望。
[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.
【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】：2017
【分類號】：TB301

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