基于三维弹性理论的功能梯度梁在任意边界条件下的自由振动分析  

Free vibration of functionally graded beams with arbitrary boundary conditions under thermal environment based on elasticity theory

作　　者：杨莎莎 沈承[2,3] YANG Shasha;SHEN Cheng(School of Construction Machinery,Nanjing Vocational University of Industry Technology,210023,Nanjing,China;College of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,210016,Nanjing,China;State Key Laboratory for Strength and Vibration of Mechanical Structures,Xi’an Jiaotong University,710049,Xi’an,China)

机构地区：[1]南京工业职业技术大学机械工程学院,南京210023 [2]南京航空航天大学航空学院,南京210016 [3]西安交通大学机械结构强度与振动国家重点实验室,西安710049

出　　处：《应用力学学报》2021年第6期2230-2235,共6页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金项目(11502110);江苏省自然科学基金项目(BK20150737);机械结构强度与振动国家重点实验室开放课题(SV2018-KF-01,SV2018-KF-22)。

摘　　要：功能梯度梁的振动问题研究具有重要的理论和工程意义,现有的研究大多局限于简支或固支等简单约束边界。基于三维弹性理论,建立了受任意边界条件约束的功能梯度梁的自由振动分析模型。具体来说,假设功能梯度梁的材料属性沿其厚度方向呈梯度变化,在梁的两端引入2组线性弹簧,通过调节弹簧的刚度常数来实现不同的边界约束。梁的每个位移分量展开为一个改进的傅里叶余弦级数,基于瑞利-里兹法计算了功能梯度梁的固有频率和固有振型。结果表明,该方法具有较快的收敛性,而且结果精度令人满意。最后讨论了材料体积分数的影响规律,结果发现材料体积分数的变化可显著改变结构的固有频率。The study of vibration of functionally graded beams has important theoretical and engineering significance, and most of the existing studies are limited to simple support or clamped and other simple constraint boundaries. In this paper, a free vibration analysis method for functionally gradient beams constrained by arbitrary boundary conditions is established based on the three-dimensional elasticity theory. Specifically, the material properties of functionally gradient beams vary with their thickness. Two linear springs are introduced at both ends of the beam to change the boundary conditions by changing the stiffness of the springs. Each displacement component of the beam is expanded into an improved Fourier cosine series, and the natural frequency of functionally gradient beams is calculated by Rayleigh-Ritz method. An example shows that the convergence speed of this method is fast and the results are satisfactory. Finally, the influence of material volume fraction is discussed, and the variation of volume fraction is found to significantly change the intrinsic frequency of the structure. The model proposed in this paper can be used as a reference for other approximate plate and shell theories and applied to the dynamics analysis of functional gradient structures.

关 键 词：三维弹性理论 功能梯度梁 瑞利-里兹法 任意边界 

分 类 号：O327[理学—一般力学与力学基础]