Nonlinear Vibration Analysis of Functionally Graded Materials Plate in Transverse Load and Thermal Environment
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摘要: 本文进行了功能梯度材料板在横向载荷和热载荷作用下的非线性振动特性研究。基于Hamilton原理, 采用Reddy的3阶剪切变形理论和Von-Karman理论建立了功能梯度材料板(物理性质随温度变化, 材料成分分布服从幂率原理)结构的非线性振动方程, 利用Galerkin方法将偏微分方程截断为常微分方程, 并利用多尺度方法求解非线性系统的主共振。分别讨论了不同参数下板的幅频响应, 观察到了多值和跳变现象。Abstract: Nonlinear vibration characteristics of functionally graded materials plates in transverse load and thermal environment are investigated. Based on the Hamilton principle, Reddy's three order shear deformation theory and Von-Karman theory are employed to establish the nonlinear vibration equation, in which the physical properties vary with the temperature and the material composition distribution obeys a power-law principle. The partial differential equations are truncated into the ordinary differential equations with the Galerkin method, and the multi-scale method is used to obtain the solution of primary resonance of the nonlinear systems. The amplitude frequency response of plates with different parameters are discussed, and the multi-values and jumping phenomena are observed.
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图 3 不同热环境下弹性模量的变化
Figure 3. Changes of elastic modulus in different thermal environments
图 4 不同热环境下热膨胀系数的变化
Figure 4. Changes of thermal expansion coefficient under different thermal environments
图 6 不同温度条件下幅值频率响应曲线
Figure 6. Amplitude frequency response curves under different temperature conditions
表 1 FGMs材料参数
Table 1. Material parameters of FGMs
属性 P-1 P0 P1 P2 P3 E/GPa Ti-6Al-4V 0 122.7 -4.605×10-4 0 0 ZrO2 0 132.2 -3.805×10-4 -6.127×10-8 0 ν Ti-6Al-4V 0 0.288 8 0 0 0 ZrO2 0 0.333 0 1.108×10-4 0 0 ρ/(kg·m-3) Ti-6Al-4V 0 4 420 0 0 0 ZrO2 0 3 657 0 0 0 α/K-1 Ti-6Al-4V 0 7.430 0×10-6 7.483×10-4 -3.621×10-7 0 ZrO2 0 13.300×10-6 -1.421×10-3 9.549×10-7 0 k/(W·(mK)-1) Ti-6Al-4V 0 6.10 0 0 0 ZrO2 0 1.78 0 0 0 
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表 2 平板的频率
Table 2. Frequency of the plate
Hz p 阶次 理论 本文 FEM 误差/% m=1, n=1 445.54 443.93 444.39 0.10 0 m=2, n=1 964.95 957.65 964.95 0.75 m=1, n=2 1 257.74 1 245.38 1 256.90 0.92 m=1, n=1 383.67 383.92 383.54 0.09 100% m=2, n=1 832.86 828.84 832.31 0.41 m=1, n=2 1 085.52 1 078.02 1 084.80 0.62
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表 3 不同体积分数的FGMs板频率
Table 3. Frequency of FGMs boards with different volume fractions
体积分数p 0.1 1 10 20 30 40 50 60 70 80 1阶频率/Hz 436.14 410.59 389.41 386.38 385.18 384.53 384.12 383.84 383.64 383.48 2阶频率/Hz 942.99 887.84 842.07 835.55 832.96 831.56 830.68 830.08 829.65 829.31
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表 4 FGMs平板频率随温度变化(p=0)
Table 4. Variation of FGMs plate frequency with temperature (p=0, ΔTL=0, ΔTU≠0)
温度/K 0 100 200 300 400 500 600 700 735 1阶频率/Hz 415.67 378.33 334.26 296.70 258.62 216.92 165.98 87.37 屈曲 2阶频率/Hz 898.69 871.18 845.75 821.56 797.77 773.60 748.30 721.18
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表 5 FGMs平板频率随温度变化(p=2, ΔTL=0, ΔTU≠0)
Table 5. Variation of FGMs plate frequency with temperature(p=2)
温度/K 0 100 200 300 400 456 1阶频率/Hz 375.74 323.72 268.64 205.74 120.18 屈曲 2阶频率/Hz 812.46 757.33 703.49 649.64 594.22
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表 6 FGMs平板频率随温度变化(p=0, ΔTL=ΔTU)
Table 6. Variation of FGMs plate frequency with temperature(p=0)
温度/K 0 100 200 204 频率/Hz 415.67 272.07 30.16 屈曲
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表 7 FGMs平板频率随温度变化(p=2, ΔTL=ΔTU)
Table 7. Variation of FGMs plate frequency with temperature(p=2)
温度/K 0 100 150 163 频率/Hz 375.74 230.76 113.31 屈曲
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表 8 屈曲温度随体积分数的变化(ΔTL=0, ΔTU≠0)
Table 8. Change of buckling temperature with volume fraction(K)
体积分数p 0.1 0.5 1 2 热屈曲温度/K 463 489 482 456
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