一种改进Mathieu方程动力不稳定边界的方法  

Improved method about dynamic unstable boundary of mathieu equation

作　　者：王杰方 安伟光[1] 宋向华 

机构地区：[1]哈尔滨工程大学航天工程系,哈尔滨150001

出　　处：《振动与冲击》2015年第12期182-188,共7页Journal of Vibration and Shock

基　　金：国家科技部国际合作专项(2012DFR00070)

摘　　要：基于Mathieu方程的临界频率方程式,提出了一种改进Mathieu方程不稳定边界的方法,并获得了比Bolotin近似边界更精确的前三阶收敛的不稳定边界。从改进的不稳定区域边界表达式和Bolotin近似公式的对比中发现:两种方法获得的第一、二阶不稳定区域相差不大,但相较于Bolotin的第三阶不稳定区域,改进的第三阶不稳定区域整体上移,且上移幅度随着激发系数的增大而增大。当激发系数μ取0.5时,上边界上移幅度为8.61%,下边界上移幅度为11.56%。对于受低频载荷作用的动力稳定性问题,第三阶不稳定边界公式的改进具有重要的意义。An improved method about unstable boundary of Mathieu equation was proposed according to the critical frequency equation,and the first three orders convergent unstable boundary was got,which is more accurate than the Bolotin approximate boundary.Comparing the two methods,it shows that their first two orders dynamic unstable region are almost the same,the third order unstable region of the improved method moves upward compared with the Bolotin method, and the range is amplified with the growth of excitation coefficient.When the excitation coefficient μis 0.5,the upper boundary moves upward about 8.61% and the lower boundary moves upward about 11.56%.With respect the dynamic stability problem caused by low frequency loading,the improvement on the third order unstable boundary expression has great significance.

关 键 词：动力稳定性 MATHIEU方程 临界频率 动力不稳定区域 

分 类 号：O342[理学—固体力学]