粘弹性轴向运动板的分数阶本构关系建模及非线性动力学分析

The proposed research is based on fractional nonlinear nonlinear dynamics theory, which to be investigate nonlinear vibration, stability ,bifurcation and chaos of axially moving plates constituted by a fractional law. By considering the power transmission belts are constitutived by rubber and polymers, this proposal uses nonlinear von Karmen thin plate theory and fractional viscoelastic constitutive relation to derive some mathematical models of the in-plane and out-plane vibration of the axially moving plates with some general boundary conditions and subjected to excitations. Semi-analytical and numerical methods will be used to solve the natural frequencies and the mode functions for the linear vibration. Then, based on the obtained linear solutions, the solution to the nonlinear oscillation would be found by using some approximate analytical methods. The main focus of the proposed research is the study of the nonlinear vibration mechanism of the plate which is constituted by fractional viscoelastic materials. Establish instability rule by approximate analytical method. Discuss the dynamical property and stability of the axially moving viscoelastic ultra-thin plates interacting with surrounding fluid. Great efforts will be made to discuss the complex dynamical behaviors, such as bifurcation and spatiotemporal chaos of the fractional dynamical system. This research will lead to the development of the fractional nonlinear dynamics and establish instability criterions, which to be served as the theoretical basis of the design and manufacture of the axially moving continuum.

发展和应用分数阶非线性动力系统理论，研究具有分数阶粘弹性本构关系的轴向运动板的建模、非线性振动特性、分岔和混沌等动力学行为。考虑到工程中的动力传送带大多数是由橡胶等聚合物材料制造，本课题以von Karmen 非线性薄板理论和分数阶粘弹性本构关系为基础，建立几类一般约束和一般载荷下轴向运动板的面内和面外振动的数学模型；利用半解析方法和数值方法求解线性自由振动的固有频率和模态函数；并以此为基础用近似解析方法研究系统的非线性振动解。用数值方法研究系统的分岔和时间-空间混沌等复杂非线性动力学行为；其中，重点研究分数阶粘弹性板在轴向运动中的非线性振动机理，用近似解析方法建立失稳判断准则；讨论超薄轴向运动板与流体相互作用下的动力学特性及其稳定性；探索含有分数导数动力系统的分岔和混沌现象。本研究的成果将会发展和完善分数阶非线性动力学理论，为工程中存在的分数阶粘弹性轴向运动连续体提供理论基础和设计准。

该项目针对轴向运动连续体进行了新的本构关系建模，即分数导数粘弹性材料。这种本构关系更加真实的反映了如橡胶传送带等材料的材料特性。我们对不同长宽比的粘弹性运动结构进行了建模和分析，分别研究了系统的线性和非线性振动行为。研究结果表明：过去基于文献中采用的Kelvin-Voigt粘弹性模型较高的估计了系统失稳的临界速度，因此以往的结果相对保守,失稳区域较小。我们发现工程实际中的传送带由橡胶材质组成，其临界速度比一般材料更加容易失去稳定性。另外，我们发现了固支边界条件的轴向运动连续体，在超临界状态下不仅仅有对称构型。我们严格的理论推导，还发现了一种“反对称”构型。对刚度较小的情况，我们对非线性参数共振进行了详细的理论分析。本课题组还针对分数阶偏微分方程提出了一种直接多尺度法来成功求解系统的各种响应。最后，我们利用非线性能量汇装置对轴向运动连续体进行了振动抑制，实现了变速情况下的宽频隔振，最高能量吸收效率达到了83%。该项目所发表论文被SCI检索文献12篇（其中10篇第一作者），三年他引共计40余次左右，1篇论文入选ESI高引用论文，两篇论文分别获辽宁省自然科学成果奖二等奖和三等奖，培养硕士生6人（毕业4人）。