STOCHASTIC DYNAMIC RESPONSE ANALYSIS OF PIPE CONVEYING FLUID BASED ON THE PATH INTEGRAL METHOD
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摘要:随机激励下的输液管道在工程上广泛存在, 对其进行研究具有十分重要的意义. 为了预测高斯白噪声激励下输液管道系统的随机动态响应, 基于哈密顿原理建立了高斯白噪声激励下非线性输液管道的动力学模型. 采用Galerkin截断方法对输液管道的控制方程进行离散化. 采用基于Gauss-Legendre公式的路径积分法计算了输液管道随机振动响应的位移概率密度函数和速度概率密度函数. 采用Monte Carlo方法与路径积分法得到的计算结果进行对比, 验证了路径积分法在计算输液管道振动响应上具有较高的计算精度. 研究了流速、激励强度和阻尼系数对输液管道位移概率密度函数和速度概率密度函数的影响, 并确定了输液管道位移概率密度函数出现双峰时的临界流速. 结果表明, 采用路径积分法计算输液管道系统的动态响应是有效的. 流速增大会使系统可能发生的最大位移变大, 可能发生的最大速度不变; 激励强度增大会使系统可能发生的最大位移和最大速度变大; 阻尼系数增大会使系统可能发生的最大位移和最大速度变小. 此外, 研究发现流速增大是诱导输液管道发生随机分岔的因素之一.Abstract:The study of pipes conveying fluid under stochastic excitation is of great importance as they are widely used in engineering. To predict the stochastic dynamic response of pipe conveying fluid system under Gaussian white noise excitation, a dynamic model of the nonlinear pipe conveying fluid under Gaussian white noise excitation is established based on the Hamilton’s principle. The Galerkin truncation method is employed to discretize the governing equation of pipes conveying fluid. The probability density function of the displacement and the probability density function of the velocity of the pipe conveying fluid are calculated by the path integral method based on the Gauss-Legendre formula. The results of the Monte Carlo method are compared with the results obtained by the path integral method to verify the accuracy of the path integral method in the calculation of the vibration response of the pipe conveying fluid. The effects of system parameters such as fluid speed, excitation strength and damping coefficient on the probability density function of the displacement and the probability density function of the velocity of the pipe conveying fluid are investigated. The critical fluid speed when the probability density function of displacement for the pipe conveying fluid has a double peak is determined. The results show that the path integral method is effective in calculating the response of the pipe conveying fluid system. The maximum possible displacement of the system will increase and the maximum possible speed will remain unchanged with the increase of fluid speed. The maximum possible displacement and the maximum possible speed of the system will increase with the increase of excitation strength. Increasing the damping coefficient results in that the maximum possible displacement and the maximum possible speed of the system decreases. In addition, it is found that the increase of fluid speed is one of the factors inducing stochastic bifurcation of the pipe conveying fluid.
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图 1高斯白噪声激励下两端简支输液管道示意图
Figure 1.A simply-supported pipe conveying fluid under distributed Gaussian white noise excitations
图 2路径积分法和Monte Carlo法的计算结果对比
Figure 2.Comparison of results of path integral method and Monte Carlo method
图 3联合概率密度的结果对比
Figure 3.Comparison of theoretical results and simulation results of joint probability density
图 4不同流速下的位移概率密度和速度概率密度
Figure 4.Probability density of the displacement and probability density of the velocity with different fluid speed
图 6位移概率密度峰值位置随流速的变化图
Figure 6.The peak position of the probability density of the displacement varies with fluid speed
图 7不同激励强度下的位移概率密度和速度概率密度
Figure 7.Probability density of the displacement and probability density of the velocity with different excitation strength
图 8不同激励强度下的联合概率密度
Figure 8.Joint probability density with different excitation strength
图 9不同阻尼系数下的位移概率密度和速度概率密度
Figure 9.Probability density of the displacement and probability density of the velocity with different damping coefficient
图 10不同阻尼系数下的联合概率密度
Figure 10.Joint probability density with different damping coefficients
10不同阻尼系数下的联合概率密度 (续)
10.Joint probability density with different damping coefficients (continued)
Item Notation Value Young’s modulus E 2.07 × 1011Pa length of the pipe L 30 m outer diameter D 0.6 m inner diameter d 0.583 m density of the pipe ρp 7850 kg/m3 density of the fluid ρf 800 kg/m3 excitation strength Dw 0.05 fluid speed $\varGamma$ 20 m/s damping coefficient η 1 
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