ANALYSIS OF THERMOELASTIC DAMPING FOR FUNCTIONALLY GRADED MATERIAL MICRO-BEAM
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摘要: 基于Euler-Bernoulli梁理论和单向耦合的热传导理论，研究了功能梯度材料（functionally graded material，FGM）微梁的热弹性阻尼（thermoelastic damping，TED）.假设矩形截面微梁的材料性质沿厚度方向按幂函数连续变化，忽略了温度梯度在轴向的变化，建立了单向耦合的变系数一维热传导方程.热力耦合的横向自由振动微分方程由经典梁理论获得.采用分层均匀化方法将变系数的热传导方程简化为一系列在各分层内定义的常系数微分方程，利用上下表面的绝热边界条件和界面处的连续性条件获得了微梁温度场的分层解析解.将温度场代入微梁的运动方程，获得了包含热弹性阻尼的复频率，进而求得了代表热弹性阻尼的逆品质因子.在给定金属-陶瓷功能梯度材料后，通过数值计算结果定量分析了材料梯度指数、频率阶数、几何尺寸以及边界条件对TED的影响.结果表明：（1）若梁长固定不变，梁厚度小于某个数值时，改变陶瓷材料体积分数可以使得TED取得最小值；（2）固有频率阶数对TED的最大值没有影响，但是频率阶数越高对应的临界厚度越小；（3）不同的边界条件对应的TED的最大值相同，但是随着支座约束刚度增大对应的临界厚度减小；（4）TED的最大值和对应的临界厚度随着金属组分的增大而增大.Abstract: Based on Euler-Bernoulli beam theory and the one-way coupled heat conduction theory, thermoelastic damping (TED) of functionally graded material (FGM) micro-beams was studied. By assuming the material properties of the rectangular cross-section micro-beams to be varied continuously along the thickness direction as power law functions and ignoring the variation of the temperature gradient in the axial direction, one dimensional and one-way coupled heat conduction equation with variable coefficients was established. By using the layer wise homogenization approach, the heat conduction with variable coefficients was simplified as a series of differential equations defined in each layer. The equation governing flexural free vibration of the FGM micro beams subjected to time dependent non-uniform heating was developed on the basis of classical beam theory. By using the boundary conditions at the top and the bottom surfaces and the continuity conditions at the interfaces, analytical solution of the temperature field in the FGM micro-beams given layer wisely was obtained. Substituting the temperature field into equation of motion of the micro-beams, the complex frequency including TED was achieved, and finally, values of the TED was extracted. Numerical results of the TED were calculated for the given values of physical and geometrical parameters of a metal-ceramic FGM beam. Effects of the material gradient, the geometry, frequency orders and the boundary conditions on TED were analyzed in detail. The results showed that:(1) if the beam length is fixed, one can arrive at the minimum of the TED by changing the volume fraction of the ceramic when the beam thickness is less than a certain value; (2) the orders of the frequency have no influence on the maximum of TED, however, the larger frequency corresponds to the smaller critical thickness (at which the TED reaches the maximum); (3) for different boundary conditions the maximums of TED are same, but the critical thickness is smaller for the stronger end constraints; (4) both the maximum of TED and the critical increase of the FGM micro beams increase along with the increment in the values of the volume fraction of the metal.
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Key words:
- functionally graded material /
- micro-beams /
- thermoelastic damping /
- energy dissipation /
- free vibration
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图 2 本文所得纯陶瓷微梁的热弹性阻尼解答与文献中的解析解的比较
Figure 2. A comparison of the present solution of TED for a pure ceramic microbeam with the analytical solution in the literature
图 3 前三阶模态下FGM微梁的热弹性阻尼随厚度的变化曲线
Figure 3. Curves of the thermoelastic damping of the FGM micro-beam versus thethickness in the first three vibrating modes
图 4 FGM简支微梁自由振动时的频移和衰减随厚度的变化的关系曲线 (一阶模态)
Figure 4. The frequency shift and the attenuation change with the thickness of anS-S FGM micro-beam (in the first mode)
图 5 给定不同材料梯度指数n时热弹性阻尼$Q^{ - 1}$与厚度h之间的关系曲线 (一阶模态)
Figure 5. Thermoelastic damping $Q^{ - 1}$ versus the thickness $h $ of S-S FGMmicro-beam for different values of $n $ (in the first mode)
图 6 给定不同厚度时FGM微梁的热弹性阻尼$Q^{ -1}$与材料梯度指数n之间的关系曲线 (一阶模态)
Figure 6. Thermoelastic damping $Q^{ - 1}$ versus the material gradient $n $ for somespecified values of the thickness (in the first mode)
图 7 不同边界条件下热弹性阻尼与厚度的关系曲线 (一阶模态)
Figure 7. Relationship curves between TED and the thickness under differentboundary conditions (in the first mode)
表 1 镍和氮化硅的物性参数 ($T_0 = 300$ K)
Table 1. The material properties of nickel and silicon nitride ($T_0 =300$ K)
表 2 一阶频率下FGM微梁不同N对应的$Q^{ - 1}$值 (S-S, $h=3 \mu $m, $l=300 \mu $m, $n=1$)
Table 2. Values of the quality factor $Q^{ - 1}$ with numbers of the divided layer of an FGM beam at thefirst order frequency (S-S, $h=3 \mu $m, $l=300 \mu $m, $n=1$)
表 3 热弹性阻尼最大值$Q_{\max }^{ - 1} (\times 10^{ - 4})$和相应的临界厚度$h_{\rm cr }(\mu $m) 值随材料梯度指数和边界条件的变化 (一阶模态，$l =300 \mu $m)
Table 3. Values of the maximum TED, $Q_{\max }^{ - 1} (\times 10^{ - 4})$ and the related critical thickness $h_{\rm cr }(\mu $m) varying with the boundary conditions and for material gradient index n (in the firstmode, $l=300 \mu $m)
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