陀螺系统随机振动分析的辛本征展开方法  

Symplectic Eigenspace Expansion for the Random Vibration Analysis of Gyroscopic Systems

作　　者：赵岩[1,2] 李明武[1,2] 林家浩[1,2] 钟万勰[1,2] 

机构地区：[1]大连理工大学工程力学系 [2]工业装备结构分析国家重点实验室(大连理工大学),辽宁大连116024

出　　处：《应用数学和力学》2015年第5期449-459,共11页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(面上项目)(11472067);国家重点基础研究发展计划(973计划)(2014CB046803)~~

摘　　要：探讨了受随机载荷作用下陀螺阻尼系统随机动力响应问题.虚拟激励法作为随机振动分析的一种高效、精确方法已经广泛应用于结构抗震、抗风等工程领域.在以单类物理变量描述的Lagrange(拉格朗日)体系框架下,振型分解方法已被有效应用于上述随机振动问题的模型自由度缩减.然而,对于陀螺系统的随机振动问题,由于陀螺效应的存在,基于Rayleigh商本征值的振型分解方法受到很大限制.对此,首先给出了陀螺系统辛本征值问题的一般形式.然后对于受平稳随机载荷激励的陀螺系统(无阻尼或有阻尼)引入虚拟激励法,基于辛本征空间展开推导了系统随机振动响应功率谱的求解列式;对于仅考虑陀螺效应的保守系统(无阻尼),该求解列式可以表述为一个显式表达式.在数值算例中,应用该文提出的方法分析了平稳随机载荷作用下一类阻尼陀螺系统的随机振动响应问题,通过与其它方法进行对比,验证了该方法的精确性和有效性.The random dynamic responses of the damped gyroscopic system were investigated under random loads. The pseudo-excitation method, as a highly efficient and accurate method for random vibration analysis, had been widely used in the fields of structural seismic and wind engineering. In the Lagrange framework based on a single physics variable the method of modal superposition is effective to reduce the degrees of freedom for complex structures in the numer- ical random vibration analysis. However, for the random analysis of gyroscopic systems, given the existing gyroscopic effects, application of the modal superposition method based on the Rayleigh quotient eigenvalues will be quite limited. Therefore, the general description of the symplectic eigenvalue problem was introduced firstly. Furthermore, for the damped gyroscopic system subjected to stationary random loads, the pseudo-excitation method was used and the solution formulae were derived based on the symplectic eigenspace expansion. For the conser- vative gyroscopic system, the solution expression was in an explicit form. In the numerical examples, the stationary random responses of a gyroscopic system were computed with the pres- ent method, of which the accuracy and efficiency were verified through comparison of the re- suits with those out of other methods. The present method is of significance for the random vi- bration problems about mechanical engineering equipments with gyroscopic systems.

关 键 词：陀螺系统 辛本征空间 随机振动 虚拟激励法 

分 类 号：O324[理学—一般力学与力学基础]