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热环境中多孔功能梯度材料微圆板的自由振动分析
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Abstract:
基于经典板理论和修正偶应力理论，研究热环境中多孔功能梯度材料(FGM)微圆板的自由振动问题。首先利用Hamilton原理推导热环境中含有材料尺度参数的多孔FGM微圆板自由振动的控制微分方程并进行无量纲化。然后应用微分变换法(DTM)对多孔FGM微圆板自由振动的无量纲控制微分方程和边界条件进行变换，得到计算自由振动无量纲固有频率的代数特征方程。最后通过MATLAB编程计算并分析了梯度指数、孔隙率、不同升温方式和材料尺度参数对多孔FGM微圆板无量纲固有频率的影响。结果表明：材料尺度参数的增加，无量纲固有频率会增大；梯度指数影响频率，反映了材料从陶瓷到金属的转变特性；孔隙率会削弱刚度，进而影响固有频率。研究结果可为今后热环境中多孔FGM微圆板的设计与应用提供数据支持。
The free vibration problem of porous functional graded material (FGM) circular micro-plates in a thermal environment is investigated based on the classical plate theory and the modified coupled stress theory. Firstly, Hamilton’s principle is used to derive the governing differential equations for the free vibration of porous FGM circular micro-plates in thermal environment containing material scale parameters and to carry out the dimensionless quantization. Then the differential transformation method (DTM) is applied to transform the dimensionless governing differential equations and boundary conditions of the free vibration of the porous FGM circular micro-plate to obtain the algebraic characteristic equations for calculating the dimensionless natural frequency of the free vibration. Finally, the effects of gradient index, porosity, different heating methods and material scale parameters on the dimensionless natural frequency of porous FGM circular micro-plates are calculated and analyzed by MATLAB programming. The results show that the dimensionless natural frequencies increase with the increase of the material scale parameter; the gradient index affects the frequency, reflecting the transition characteristics of the material from ceramic to metal; and the porosity weakens the stiffness, which in turn affects the natural frequencies. The results can provide data support for the design and application of porous FGM circular micro-plates in thermal environments in the future.
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