Thermo-Elastic Vibration Analysis of Functionally Graded Material Pipes in Elastic Matrix
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摘要: 为了研究嵌入弹性基体功能梯度输流管的流固耦合振动问题，首先根据欧拉梁模型理论推导得到功能梯度输流管道的振动控制方程，然后采用微分求积法对振动控制方程进行求解，最后根据计算结果详细讨论了材料组分的体积分数、温度、长细比及弹性基体的弹性系数对系统的固有频率及临界流速的影响. 研究结果表明：（1） 内部材料组分的体积分数增大会使系统的无量纲固有频率增大，临界流速减小（指数n由0增大到10，流速为0时的固有频率增大约13%，临界流速减小约6%）；（2） 随着温度的升高，系统的固有频率和其临界流速都会降低（长径比为100时，温度升高30 K，流速为0时的固有频率减小约4%，临界流速减小约14%），减小长径比会使得系统的固有频率明显下降（长径比为100、50和20时，系统的固有频率分别为160、41.1和11.87.）；（3） 系统的固有频率随着管道外径的增大而降低，管壁越薄变化越快，管壁越厚变化越慢（外径由0.1 m增大到0.11 m时，其固有频率的下降幅度约为外径由0.19 m增大到0.2 m时的100倍）；（4） 弹性基体弹性系数k增大会提高系统的固有频率（k增大3倍，系统的固有频率提高了约74%）.Abstract: In order to investigate the fluid-structure coupled vibration problems of functionally graded material (FGM) fluid conveying pipes embedded in elastic matrix. Firstly, the vibration control equation of the FGM pipe was derived according to Euler-beam model theory. Then, the differential quadrature method was used to solve the vibration control equation. Finally, the influence of the volume fraction of the material component, temperature, slenderness ratio and elastic coefficient of elastic matrix on natural frequency and critical velocity of the system was discussed according to the calculated results.The following conclusions are obtained: (1) The increase of the volume fraction of the internal material components will lead to the dimensionless natural frequency of the system increase and the critical velocity decrease (exponential n increases from 0 to 10, the natural frequency increases by about 13% and the critical velocity decreases by about 6% when the flow velocity is 0). (2) With the increase of temperature, the natural frequency and the critical velocity of the system will decrease (when the length-diameter ratio is 100, the temperature increases by 30 K, the natural frequency decreases by about 4% wand the critical velocity decreases by about 14% hen the flow rate is 0). The natural frequency of the system decrease obviously when the slenderness ratio decreases (when the length-diameter ratio is 100, 50 and 20, the natural frequencies of the system are 160, 41.1 and 11.87, respectively). (3) The natural frequency of the system decreases with the increase of the external radius, the thinner the tube wall the faster it changes, the thicker the tube wall the slower it changes (the decrease of natural frequency caused by the increase of outer diameter from 0.1 m to 0.11 m is about 100 times that from 0.19 m to 0.2 m). (4) The increase of the elastic coefficient k will improve the natural frequency of the system (the natural frequency of the system increased by about 74% when k increases by three times).
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图 1 功能梯度输流管分析模型
Figure 1. Analysis model of functionally graded material fluid conveying pipes
图 4 长细比不同对应的无量纲临界固有频率关于流速的变化
Figure 4. Dimensionless vibration frequency corresponding to different slenderness ratio vary with the velocity
图 5 无量纲固有频率随管道外部半径变化情况
Figure 5. Dimensionless vibration frequency vary with outer radius
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