热环境中旋转运动功能梯度圆板的强非线性固有振动

研究热环境中旋转运动功能梯度圆板的非线性固有振动问题．针对金属－陶瓷功能梯度圆板，考虑几何非线性、材料物理属性参数随温度变化以及材料组分沿厚度方向按幂律分布的情况，应用哈密顿原理推得热环境中旋转运动功能梯度圆板的非线性振动微分方程．考虑周边夹支边界条件，利用伽辽金法得到了横向非线性固有振动方程，并确定了静载荷引起的静挠度．用改进的多尺度法求解强非线性方程，得出非线性固有频率表达式．通过算例，分析了旋转运动功能梯度圆板固有频率随转速、温度等参量的变化情况．结果表明，非线性固有频率随金属含量的增加而降低；随转速和圆板厚度的增大而升高；随功能梯度圆板表面温度的升高而降低．

Strongly Nonlinear Natural Vibration of the Functionally Graded Rotating Circular Plate in the Thermal Environment

In this paper, the nonlinear natural vibration of the functionally graded rotating circular plate composed of metal and ceramic in the thermal environment was analyzed, considering the geometric nonlinearity, the effect of temperature on physical properties of the material, and the continuous variations of material properties in the thickness direction that follow a simple power-law distribution in terms of volume fractions of the constituents of the plate. The nonlinear natural vibration equations of the functionally graded rotating circular plate in a high-temperature environment were derived by using Hamilton’s principle. As for the surrounding clamped boundary conditions, the nonlinear differential equation of transverse natural vibration of the circular plate was obtained through the Galerkin method. Moreover, the static deflection induced by the rotation and functional gradient characteristics was determined. Based on the assumption of strong nonlinearity, an improved method of multiple scales was employed to solve the differential equation of natural vibration, and the nonlinear expression for natural frequency was achieved. Through numerical calculations, the changes of natural frequency of the circular plate with rotational speed, temperature, volume fraction, and plate thickness were discussed, respectively. Furthermore, the degenerated model of the system was introduced to verify the rationality of the dynamic model, where the corresponding numerical solutions obtained by the Runge-Kutta method and the periodic graph method were compared with the analytical ones, and the results were basically consistent. It is found that the nonlinear natural frequency increases with the increase of rotating speed or disk thickness, but decreases with the increase of metal content or temperature of the circular plate surface. Especially, when the surface temperatures of both the metal and the ceramic increase at the same time, the nonlinear natural frequency decreases faster. The given natural frequency and mode shape solutions are of practical importance for various engineering dynamic analyses of functionally graded circular plates.