一种求非线性振动系统周期解的切比雪夫级数方法  

A method of Chebyshev series for calculating periodic solution to a nonlinear vibration system

作　　者：周薇[1] 韩景龙[1] 陈全龙[1] 

机构地区：[1]南京航空航天大学机械结构力学及控制国家重点实验室,南京210016

出　　处：《振动与冲击》2013年第24期1-5,14,共6页Journal of Vibration and Shock

基　　金：高等学校博士学科点专项科研基金(20113218110002)资助;江苏高校优势学科建设工程资助项目

摘　　要：将切比雪夫级数理论和非线性优化算法结合,提出一种求非线性振动系统周期解的方法。该方法将状态矢量中未知切比雪夫系数的求解,转化为对主周期上系统残差求最小值的无约束最优化问题,计算出了具有较高精度的切比雪夫级数周期解。所得周期解可通过积分运算直接求得系统的Floquet转移矩阵,从而分析周期解的稳定性。最后,以Duffing系统方程和直升机旋翼系统运动方程为例,验证了该方法正确、有效,也证明了将切比雪夫级数理论引入直升机气动弹性响应与稳定性研究是正确可行的。Combining Chebyshev series theory with a nonlinear optimization algorithm, a method was proposed to calculate periodic solution to a nonlinear vibration system. The method was adopted to convert solving the unknown Chebyshev coefficients of a state vector into an optimization problem solving the minimum residual value for one primary period of the system, and a higher precision Chebyshev series periodic solution was obtained. Floquet transition matrix of the system was calculated using integral operations of the periodic-solution. Then, the stability of the periodic solution was analyzed. At last, two examples of Duffing equation and a helicopter rotor motion equation were taken to demonstrate that the proposed method is correct and effective; introducing Chebyshev series theory into helicopter aeroelastic response and stability study is feasible.

关 键 词：非线性振动 切比雪夫级数 解析周期解 优化方法 稳定性 

分 类 号：O322[理学—一般力学与力学基础]