扇形板面内振动的解析解法  

Analytical solutions for in-plane vibration of circular sector plates

作　　者：袁运博 李宏亮[2] 孙丹 郭宜斌[1] 王东华 YUAN Yun-bo;LI Hong-liang;SUN Dan;GUO Yi-bin;WANG Dong-hua(College of Power and Energy Engineering,Harbin Engineering University,Harbin 150001,China;College of Aerospace and Civil Engineering,Harbin Engineering University,Harbin 150001,China;Harbin Marine Boiler&Turbine Research Institute(No.703 Research Institute of CSSC),Harbin 150001,China)

机构地区：[1]哈尔滨工程大学动力与能源工程学院,黑龙江哈尔滨150001 [2]哈尔滨工程大学航天与建筑工程学院,黑龙江哈尔滨150001 [3]哈尔滨船舶锅炉涡轮机研究所(中国船舶集团有限公司第703研究所),黑龙江哈尔滨150001

出　　处：《振动工程学报》2020年第6期1234-1244,共11页Journal of Vibration Engineering

基　　金：国家重大专项(MC-201501-D01-01);国家自然科学基金资助项目(51805106)。

摘　　要：利用亥姆霍兹分解和分离变量法推导了扇形板面内振动特性分析的解析解。利用两个直边上的边界条件,建立圆周方向上的本征值问题,同时证明仅当两个直边为简支约束时,圆周方向上才存在相容的本征值条件。借助分离变量法获得扇形板面内振动问题的位移和应力函数解析表达式,并根据圆周曲边边界条件,建立用于确定固有频率和模态振型的特征方程。通过不同边界条件和扇形角组合下的算例分析,验证了所建立的解析解法的适用性和精确性。参数影响分析表明,在所研究的各种边界条件下,扇形板的前5阶固有频率随扇形角的增大在局部范围内保持不变或单调减小。To investigate the free in-plane vibration of circular sector plates,analytical solutions are derived based on Helmholtz decomposition and the method of separation of variables.Eigenvalue problems in the circumferential direction are derived by applying the boundary conditions on both radial edges.Meanwhile,it is proved that compatible eigenvalue conditions can be obtained only when both radial edges are simply supported.The analytical solutions of displacements and stresses are derived by invoking the method of separation of variables.Subsequently,the corresponding characteristic equations for finding natural frequencies and mode shapes are derived using the boundary conditions at outer radius.Extensive results of circular sector plates with different boundary conditions and various sector angles are presented to demonstrate the proposed method.The results indicate that the proposed method has great accuracy and applicability.Parameter study shows that the first fifth natural frequencies keep constant in some region and decrease monotonically in other region as the sector angle increases.

关 键 词：面内振动 扇形板 本征值问题 分离变量法 亥姆霍兹分解 

分 类 号：O326[理学—一般力学与力学基础] O327[理学—力学]