多孔功能梯度材料Timoshenko梁的非线性自由振动分析  

Nonlinear free vibration analysis of Timoshenko beams with porous functionally graded materials

作　　者：滕兆春[1] 马铃权 付小华[1] TENG Zhaochun;MA Lingquan;FU Xiaohua(School of Science,Lanzhou University of Technology,Lanzhou 730050,China)

机构地区：[1]兰州理工大学理学院,甘肃兰州730050

出　　处：《西北工业大学学报》2022年第5期1145-1154,共10页Journal of Northwestern Polytechnical University

基　　金：国家自然科学基金(12062010)资助。

摘　　要：基于Timoshenko梁变形理论研究多孔功能梯度材料梁的非线性自由振动问题。针对多孔功能梯度材料梁的孔隙均匀分布和孔隙线性分布2种形式,根据广义Hamilton原理推导多孔功能梯度材料Timoshenko梁的非线性自由振动的控制微分方程组并对方程组进行无量纲化。采用微分变换法(DTM)对各种边界条件下的控制微分方程组进行变换,得到等价代数特征方程。计算了多孔功能梯度材料Timoshenko梁在固支-固支(C-C)、固支-简支(C-S)、简支-简支(S-S)和固支-自由(C-F)4种边界条件下非线性横向自由振动的无量纲固有频率比值。将其退化为无孔隙功能梯度材料Timoshenko梁的非线性自由振动后,所得非线性无量纲固有频率比值与已有文献的计算结果进行对照,验证了文中方法的有效性和正确性,讨论了边界条件、孔隙率、细长比和梯度指数对多孔功能梯度材料Timoshenko梁非线性无量纲固有频率比值的影响。Based on the theory of Timoshenko beam, the nonlinear free vibration of functionally graded material beams is studied. In view of the two forms of uniform pores distribution and linear pores distribution for porous functionally graded material beams, the governing differential equations of nonlinear free vibration of porous functionally graded material Timoshenko beam are derived firstly in terms of the generalized Hamilton principle, and the dimensionless form of the governing differential equations are also obtained. Secondly, the governing differential equations under various boundary conditions are transformed by the differential transformation method to obtain the equivalent algebraic characteristic equations. Finally, the dimensionless natural frequency ratio of nonlinear transverse free vibration of porous FGM Timoshenko beam under four boundary conditions of clamped-clamped(C-C), clamped-simply supported(C-S), simply supported-simply supported(S-S) and clamped-free(C-F) is calculated. It is reduced to the nonlinear free vibration of Timoshenko beam without pores, the nonlinear dimensionless natural frequency ratio is compared with the calculation results in the literature, then the validity and correctness are verified. The effects of the boundary conditions, porosity, slenderness ratio and gradient index on the nonlinear dimensionless natural frequency ratio of porous functionally graded material′s Timoshenko beam are discussed.

关 键 词：多孔功能梯度材料 TIMOSHENKO梁 非线性自由振动 固有频率 微分变换法(DTM) 

分 类 号：O343[理学—固体力学]