基于无记忆变换的分数阶非线性振动系统分析方法研究

Based on a memory-free transformation, this project aims to investigate some analysis methods for nonlinear vibrations containing fractional derivatives. The theoretical basis will be rigorously addressed for the memory-free transformation with emphasis on the validity and effectiveness of its applications to periodic or quasi-periodic responses. The fractional derivatives of any desired order for periodic functions will be calculated analytically and efficiently. Based on these preliminary studies, the harmonic balancing, homotopy analysis and equivalent linearization methods will be extended to analyze nonlinear vibrations with fractional derivatives. By combining harmonic balancing with incremental process as well as minimum optimal techniques, a new approach will be proposed for investigating the periodic responses and bifurcations efficiently. Based on an iteration scheme, the homotopy analysis method will be extended, so that it is capable of analyzing strongly nonlinear as well as non-smooth dynamical systems. The equivalent linearization system will be deduced by introducing an equivalent transformation to fractional derivatives. Based on the equivalent system, the primary, sub-harmonic as well as super-harmonic resonances will be addressed, and special attention will be paid to possible jumping phenomena and amplitude death. A new system with fractional derivatives will be constructed to model the nonlinear flutter of an airfoil subjected to an unsteady flow. Limit cycle oscillations and bifurcations, initial conditions-dependent responses for this system will be investigated by extending the above-mentioned methods. The constructed model and the above-mentioned approaches will be validated by combining numerical simulations and wind tunnel tests. These methods could be widely applicable to the quantification of nonlinear vibrations containing fractional fractional derivatives, the dynamics modeling and analysis of various visco-elastic materials, and the aeroelastic design of aircraft and large suspended bridges, etc.

本项目拟基于无记忆变换，研究含分数阶导数的非线性振动系统的分析方法。探讨无记忆变换的理论基础，推导周期函数分数阶导数的解析表达式，分析其用于周期或准周期响应分析的合理性和有效性。在此基础上，推广谐波平衡、同伦分析及等效线性化等方法，使之能高效分析分数阶非线性振动系统。结合谐波平衡、增量过程和最小值优化，分析系统周期响应及其分岔；在同伦分析中引入迭代过程，使之能求解非光滑系统；研究分数阶导数的等效形式，推导线性化方程，研究谐波共振及其跳跃现象。建立含分数阶导数的机翼非线性颤振系统，利用所提方法研究极限环、分岔和响应的初值依赖性等现象。对所研究的非线性系统进行数值模拟，并进行风洞试验研究，以验证本项目方法的有效性、计算精度和计算效率。本项研究不仅具有重要的理论和学术意义，而且在分数阶非线性系统定量分析、黏弹性材料的动力学建模与分析、飞行器和大型桥梁的颤振设计中具有广阔的工程应用前景。

分数阶非线性振动广泛存在于黏弹性结构、软物质结构振动和控制系统中。分数阶导数的全局相关性导致，对问题直接求解需要大量的存储空间，而且计算过程的复杂程度很高。无记忆变换将分数阶导数转换为无穷积分，提供了降低计算量、提高分析效率的可行途径。为此，本项目基于无记忆变换，研究了含分数阶导数的非线性振动系统的若干分析方法。主要研究进展包括以下几个方面：（1）阐明了无记忆变换的理论基础，并分析了分数阶导数数值计算的几个问题，为相关应用和推广提供了技术保证；（2）采用无记忆变换分析了若干分数阶振动和颤振系统，揭示了分数阶振动频率的时间依赖特性、获得了颤振系统出现亚临界极限环的判别依据；（3）基于无记忆变换，推导了分数阶振动振幅死亡和跳跃等的解析判据；（4）在无记忆变换所得到的无穷积分基础上，构造了一个辅助积分，改善了被积函数的渐进行为，改进了无记忆方法用于直接数值模拟的计算精度，在没有增加计算复杂度的前提下、将方法提高到了2阶计算精度。在本项目的主要资助下，项目组共发表学术论文17篇，其中16篇被SCI收录；另外，积极参加国内外相关学术会议，发表会议论文共4篇。本项目取得的研究成果，将在分数阶非线性系统的解析、半解析和数值分析中得到更进一步的推广，并且在黏弹性材料建模分析、结构振动控制等方面将得到广泛的应用前景。