Analysis of Free Vibration and Critical Buckling Load of Fiber Reinforced FGM Beams
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摘要: 基于经典梁理论(CBT)研究轴向力作用下纤维增强功能梯度材料(FGM)梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法(DTM)对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定(C-C)、固定-简支(C-S)和简支-简支(S-S)3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。
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关键词:
- 功能梯度材料梁 /
- 自由振动 /
- 固有频率 /
- 临界屈曲载荷 /
- 微分变换法 (DTM)
Abstract: Based on the classical beam theory (CBT), the transverse free vibration and critical buckling load of fiber reinforced functionally graded material (FGM) beams subjected to axial force are studied. Firstly, the mixing law model is considered to characterize the material properties of the fiber reinforced FGM beam. Secondly, the governing differential equations of transverse free vibration and critical buckling load of the fiber reinforced FGM beam subjected to axial force are derived by Hamilton principle, and the governing differential equations and boundary conditions are transformed by differential transformation method (DTM). The dimensional natural frequencies for transverse free vibration and dimensionless critical buckling loads of the fiber reinforced FGM beam are calculated under three boundary conditions of clamped-clamped (C-C), clamped-simply supported (C-S) and simply supported-simply supported (S-S). The problem is reduced to homogeneous material beams and FGM beams and compared with the existing literatures to verify its effectiveness. Finally, the effects of stiffness ratio, fiber volume fraction and dimensionless compressive load on the dimensionless natural frequency and all parameters on the dimensionless critical buckling load of the fiber reinforced FGM beam under different boundary conditions are discussed. -
表 1 不同边界条件各向同性材料梁的无量纲固有频率
表 2 不同边界条件下FGM梁无量纲固有频率
Pc/Pm Ω C-C C-S C-S 本文解 文献[21] 本文解 文献[21] 本文解 文献[21] Ω1 22.1080 22.108 15.2354 15.235 9.7526 9.7525 2 Ω2 60.9416 60.942 49.3725 49.373 39.0113 39.010 Ω3 119.4699 119.47 103.0117 103.01 87.7733 87.773 Ω1 21.7197 21.720 14.9678 14.968 9.5813 9.5813 3 Ω2 59.8711 59.871 48.5052 48.505 38.3251 38.325 Ω3 117.3713 117.37 101.2022 101.20 86.2315 86.231 Ω1 21.3518 21.352 14.7143 14.714 9.4190 9.4190 4 Ω2 58.8571 58.857 47.6837 47.684 37.676 37.676 Ω3 115.3834 115.38 99.4882 99.488 84.7710 84.771 Ω1 21.0199 21.020 14.4856 14.486 9.2726 9.2726 5 Ω2 57.9424 57.941 46.9427 46.942 37.0905 37.090 Ω3 113.5903 113.59 97.9421 97.942 83.4536 83.454 -
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