矩形正交各向异性薄板弯曲受迫振动问题的分析解  

ANALYTICAL SOLUTION FOR THE FORCED VIBRATION OF ORTHOTROPIC RECTANGULAR THIN PLATES

作　　者：姚伟岸[1] 蔡智宇[1] 胡小飞[1] 

机构地区：[1]大连理工大学工业装备结构分析国家重点实验室

出　　处：《动力学与控制学报》2011年第1期12-17,共6页Journal of Dynamics and Control

基　　金：辽宁省高等学校科研项目计划(2009S018);国家自然科学基金(10772039);973国家重点基础研究计划(2010CB832704);国家高技术研究发展计划863(2009AA044501)资助项目~~

摘　　要：通过恰当的辛内积定义,首先将矩形正交各向异性薄板弯曲受迫振动问题导入到辛对偶体系,并应用分离变量和辛本征展开的有效数学物理方法给出其受迫振动稳态解的一个解析求解方法.然后,具体讨论了对边简支和对边固支两种典型边界条件的正交各向异性薄板弯曲受迫振动问题的辛本征问题,并给出了对应的辛本征值超越方程和辛本征向量的解析表达式.最后,应用本文的方法分析求解了两个具体算例,并将受均布谐载作用的四边简支矩形板受迫振动稳态解的分析解与传统的Navier法进行了精度和收敛性的对比,结果表明本文的方法比传统的分析求解方法具有更好的精度和收敛性,尤其是对内力.Firstly,based on the appropriate definition of symplectic inner product,the forced vibration problem of orthotropic rectangular thin plate was introduced into symplectic duality system,so an analytical approach for the steady state solution of forced vibration was presented by employing variables separation and eigenfunction expansion.Secondly,the symplectic eigenproblems for the forced vibration of the orthotropic rectangular thin plates with two typical boundary conditions,i.e.opposite sides simply supported and opposite sides clamped,were discussed.And the transcendental equations of symplectic eigenvalues and the symplectic eigenvectors were given in analytical form.Lastly,analytical solutions of two examples were presented by using this method.And the solution of a fully simply supported plate under uniformly distributed harmonic load was chosen to compare with the classical Navier＇s method,and the result shows that the new technique not only has good accuracy but also has better convergence speed than the classical methods,especially on internal forces.

关 键 词：正交各向异性 薄板 振动 辛空间 分析解 

分 类 号：O321[理学—一般力学与力学基础]