STRONG NONLINEAR PRIMARY RESONANCE OF ROTATING FUNCTIONALLY GRADED CIRCULAR PLATES UNDER HARMONIC FORCE
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摘要: 功能梯度材料作为一种新型材料，具有良好的力学性能，近年来被广泛关注和应用。该文针对金属-陶瓷功能梯度圆板，考虑周边夹支边界约束条件，选取多项式形式的振型函数，利用伽辽金法，推得旋转运动状态和热效应作用下系统的纵横耦合非线性振动方程，求得由旋转及密度差引起的静挠度项。用改进多尺度法求解方程，得到强非线性系统的频幅响应方程和解析解。通过算例，给出功能梯度圆板的幅频曲线、幅值-激励力曲线、幅值-温度曲线，分析了不同物理量对结构共振幅值的影响规律，并且比较了解析解和数值解，两者结果较为吻合。Abstract: As a new type of material, functionally graded material (FGM) has attract wide attention and application in recent years due to its good mechanical properties. For the metal-ceramic FGM circular plate with clamped boundary condition, the mode function in polynomial form and the Galerkin method are adopted to derive the longitudinal and transverse coupling nonlinear vibration equations of the system under rotational motion and thermal effects, and the static deflection caused by rotation and density difference is also obtained. The improved multi-scale method is used to solve the equations, and then the frequency-amplitude response equation and analytical solution of the strong nonlinear system are obtained. Based on calculation examples, the amplitude-frequency curves, amplitude-excitation force curves, and amplitude-temperature curves of the FGM plate are presented, and the influence of different physical parameters on the resonance amplitude of the structure is analyzed. Reasonable consistence is evident by comparing the analytical and numerical solutions.
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Key words:
- FGM circular plate /
- rotational motion /
- strong nonlinearity /
- primary resonance /
- analytical solution
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图 7 本文解析解与文献数值解验证图
Figure 7. Verification diagram between analytical solution in this paper and numerical solution in the reference
表 1 两种材料的温度相关系数
Table 1. The coefficients of two materials related to temperature
相关系数 材料 $P_{-1} $ $P_{0} $ $P_{1} $ $P_{2} $ $P_{3} $ 弹性模量系数 $ \mathrm{Si}_{3} \mathrm{~N}_{4}$ $ 0$ $348.43 \times 10^{9} $ $-3.07 \times 10^{-4} $ $2.16 \times 10^{-7} $ $-8.95 \times 10^{-11} $ $\text { SUS304 } $ $ 0$ $ 201.04 \times 10^{9}$ $ 3.08 \times 10^{-4}$ $-6.53 \times 10^{-7} $ $ 0$ 热膨胀系数 $ \mathrm{Si}_{3} \mathrm{~N}_{4}$ $ 0$ $ 5.87 \times 10^{-6}$ $ 9.10 \times 10^{-4}$ $ 0$ $ 0$ $\text { SUS304 } $ $ 0$ $ 12.33 \times 10^{-6}$ $8.09 \times 10^{-4} $ $ 0$ $ 0$ 
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