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 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。
Abstract
An improved method about unstable boundary of Mathieu equation was proposed according to the critical frequency equation, and the first three order convergent unstable boundary was got, which is more accurate than the Bolotin approximate boundary. Compare this two methods, it show that the first two order dynamic unstable region are almost the same, the third order unstable region was moved upward compared with Bolotin method, and the range was amplified with the growth of excitation coefficient. When excitation coefficient μ is 0.5, The upper boundary was moved
upward about 8.61% and the down boundary was moved upward about 11.56%. To the dynamic stability problem that was excited by low frequency load, the improvement of 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
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] 赵晶瑞，唐友刚，王文杰. 传统Spar平台参数激励Mathieu不稳定性的研究[J]. 工程力学，2010，27(3)：222-227.
ZHAO Jing-rui , TANG You-gang , WANG Wen-jie. Study on the parametrically excited mathieu instability of a classic spar platform [J]. Engineering Mechanics, 2010,27(3):222-227.
[2] 王平，王知人，白象忠.马丢方程解的稳定性在磁弹性屈曲中的应用[J]. 哈尔滨工业大学学报，2009，41(11)：222-224.
WANG Ping, WANG Zhi-ren, BAI Xiang-zhong. Applications of the stability of Mathieu equation’s solution in magnetic-elastisity buckling problem [J]. Journal of Harbin Institute of Technology, 2009,41(11):222-224.
[3] Yan Li , Shangchun Fan, Zhanshe Guo. Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors [J]. Commun Nonlinear Sci Numer Simulat, 2013,18:401-410.
[4] Bolotin, V V. The Dynamic Stability of Elstic Systems[M]. Holen–Day San Francisco.1964.
[5] 徐万海，吴应湘，钟兴福等. 海洋细长结构参数激励不稳定区的确定方法[J].振动与冲击，2011，30(9)：79-83.
XU Wan-hai, WU Ying-xiang, ZHONG Xing-fu,et al. Methods for parametric excitation instability analysis of slender flexible cylindrical structures in offshore engineering [J]. Journal of Vibration and Shock, 2011,30(9):79-83.
[6] Nayfeh A H. Introduction to Perturbation Technique［M］.New York:Wiley,1981．
[7] 冯世宁，陈浩然. 含分层损伤复合材料层合板非线性动力稳定性[J]. 复合材料学报，2006，23(1)：154-160.
FENG Shi-ning, CHEN Hao-ran. Nonlinear dynamic instability behavior of delaminated composite laminates [J]. Acta Materiae Compositae Sinica, 2006,23(1):154-160.
[8] 赵洪金，董宁娟，刘 超等．基于能量法的高温(火灾)环境下轴心受压格构柱动力稳定性分析[J]. 工 程 力 学,2011,
28(12):160－165.
ZHAO Hong-jin , DONG Ning-juan , LIU Chao,et al. Dynamic stability analysis on axial compression lattice column under high temperature condition using energy method [J]．Engineering Mechanics, 2011,28(12):160－165.
[9] 王俊荣，谢彬. 深水半潜式平台Mathieu不稳定问题研究[J].工程力学，2012，29(10)：347-353.
WANG Jun-rong, XIE Bin. Mathieu instablity study of a deepwater semi-submersible platform [J]. Engineering Mechanics, 2012,29(10):347-353.
[10] 党建军，张谋，郭芳等. 超空泡脉动对水下高速航行器壳体强度的影响研究[J].机械科学与技术，2011，30(11)：1930-1933.
Dang Jian-jun,Zhang Mou,Guo Fang,et al. A study of the influence of the super-cavit pulsation on the intensity of the shell of the high-speed underwater vehicle [J]. Mechanical Science and Technology for Aerospace Engineering, 2011,30(11):1930-
1933.
[11]王杰方，安伟光，宋向华. 超空泡运动体圆柱薄壳动力屈曲及可靠性分析[J].振动与冲击，2014,33(8):22-28.
WANG Jie-fang，AN Wei-guang，SONG Xiang-hua.Dynamic buckling and reliability analysis of a cylindrical thin shell for supercavitating vehicles[J]. Journal of Vibration and Shock, 2014,33(8):22-28.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
PDF(1452 KB)
