覆冰输电线舞动建模及解析法分析
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摘要
覆冰输电线舞动将对整个社会造成巨大的经济损失。为了深入了解覆冰输电线舞动原理,论文提出了基于曲梁理论的三自由度舞动方程,并使用多尺度法分析了覆冰输电线随风速变化的分叉和稳定。
首先,基于zhu提出的曲梁理论和Mason提出的迁移坐标系,建立了覆冰输电线的三维拉格朗日应变表达式。并根据哈密顿原理,建立了考虑截面偏心的四自由度(轴向分量、法向分量、副法向分量和扭转分量)舞动方程。再根据覆冰输电线特性,将上述四自由度舞动模型进一步简化为三自由度(法向分量、副法向分量和扭转分量)舞动模型。该模型可以考虑覆冰截面偏心、初始扭转角和抗弯刚度对覆冰输电线舞动影响。在建立风荷载模型时,除了考虑扭转角位移、法向速度和副法向速度对攻角的影响外,还考虑了初始扭转角对攻角的影响。除此之外,根据Galerkin方法将荷载模型和舞动模型进行离散。
其次,为了验证上述提出的基于曲梁理论的三自由度舞动模型,本文还建立了基于索结构理论的单自由度(竖向分量)、双自由度(竖向分量和水平分量)和三自由度(竖向分量、水平分量和扭转分量)舞动模型。并采用经典覆冰舞动算例(D型覆冰截面),将上述四种模型计算得到和实验测得的竖向舞动幅值进行对比分析。计算结果表明基于曲梁理论的三自由度舞动模型比基于索结构理论舞动模型精度高。除此之外,还分析了上述四种模型与实验数据间存在误差原因。
最后,还利用多尺度法和基于曲梁理论的三自由度舞动模型分析1:1和2:1内共振覆冰输电线舞动性质。为了便于多尺度法计算,根据覆冰输电线特性,将三自由度舞动模型简化为两自由度(竖向分量和水平分量)舞动模型。基于以上简化模型,利用多尺度法建立1:1和2:1内共振简化幅值方程来分析覆冰输电线随风速变化的分叉情况。在此基础上,建立1:1和2:1简化幅值方程的雅可比矩阵来判断各分支的稳定性。但在分析内共振2:1单幅值舞动(分支Ⅲ)的稳定性时,2:1简化幅值方程的雅可比矩阵出现歧义现象(不能判断其稳定性)。为了解决上述问题,本文还提出了复合指数-Cartesia形式的2:1简化幅值方程。除此之外,本文还根据团队在中国空气动力研究和发展中心所做的实验数据,拟合了覆冰输电线空气阻力和升力系数关于攻角的三次曲线,并根据以上数据分别对比分析了内共振1:1和2:1考虑覆冰截面偏心和不考虑偏心的分叉和稳定情况。分析结果表明:截面偏心对舞动影响较大。最后,在2:1内共振不考虑偏心情况下,发现了多重稳定的特殊现象,并根据2:1简化幅值方程和两自由度舞动方程在时域内积分得到的随时间变化的法向和副法向位移,分析并验证了该特殊现象的存在。
The galloping of the iced transmission line will cause the great damage to the whole society. In order to better understand the theory of galloping of iced transmission line, a three-degree-of-freedom galloping model based on curved beam theory has been proposed. Also, the Multiple Scale Perturbation Method (MSM) is introduced to analyze bifurcation and stability of the iced transmission line.
Firstly,based on curved beam theory proposed by Zhu and converted curvilinear coordinates raised by Mason, three-dimension Lagrange strain tensor of iced transmission line is established. Then, according to Hamilton principle, a four-degree-of-freedom (tangential, nominal, bi-nominal and torsional) galloping model considering iced eccentricity is formulated. Moreover, due to the property of transmission line, a reduced three-degree-of-freedom (nominal, bi-nominal and torsional) galloping model is obtained in view of bending, rotation and eccentricity of cross section. Furthermore, the initial rotational angle is introduced into the model of attack angle in addition to the rotational displacement, nominal velocity and bi-nominal velocity. Finally, According to Galerkin method, three-degree-of-freedom discrete galloping model and aerodynamic discrete model are obtained.
Secondly, in order to test the accuracy of three-degree-of-freedom galloping model based on curved beam theory, three galloping models based on cable structure are established, such as one-degree-of-freedom (vertical) , two-degree-of-freedom (vertical and horizontal) and three-degree-of-freedom (vertical, horizontal and rotational) galloping models. Moreover, one classical galloping example (cross section of D shape) is chosen to analyze the forgoing four galloping models. Comparing vertical displacements of the four galloping models and the experimental data, it can be found that the accuracy of the three-degree-of-freedom galloping model based on curved beam theory is higher than the other three galloping models based on cable structure. Furthermore, the reasons for the difference between the four galloping models and experimental data are discussed.
Finally, the three-degree-of-freedom galloping model based on curved beam theory and MSM are introduced to analyze the 1:1 and 2:1 resonant cases. In order to establish the model suitable for MSM, a two-degree-of-freedom (nominal and bi-nominal) galloping model is reduced from the three-degree-of-freedom galloping model due to the property of iced transmission line. In view of the above reduced two-degree-of-freedom galloping model, 1:1 and 2:1 Reduced Amplitude Modulation Equations (RAME) are formulated with the method of MSM to analyze the bifurcations of 1:1 and 2:1 resonant cases. Moreover, the Jacobian matrix of 1:1 and 2:1 RAME are established to analyze the stability of the forgoing bifurcations of 1:1 and 2:1 resonant cases except the special branch (one-mode (nominal) galloping), in which case the Jacobian matrix of 2:1 RAME becomes singular. In order to solve this problem, a mixed polar-Cartesian form of 2:1 RAME is adopted. Furthermore, according to the experimental data recently completed in Chinese Aerodynamic Research and Development Center, the drag and lift coefficients are fitted in the form of cubic equations. In view of the above fitting data, the bifurcation and stability of the 1:1 and 2:1 resonant cases considering the eccentricity and without considering the eccentricity are discussed. The results turn out that the iced eccentricity plays an important role on bifurcation and stability. Finally, in the 2:1 resonant case without considering the eccentricity, one special phenomenon (multiple stabilities) is found. Also, from the analysis of numerical integration of RAME and reduced model in time history, it is proved that this phenomenon actually appears as the wind speed is in the wind speed range of multiple stabilities, which is obtained by analyses of bifurcation and stability of RAME.
首先,基于zhu提出的曲梁理论和Mason提出的迁移坐标系,建立了覆冰输电线的三维拉格朗日应变表达式。并根据哈密顿原理,建立了考虑截面偏心的四自由度(轴向分量、法向分量、副法向分量和扭转分量)舞动方程。再根据覆冰输电线特性,将上述四自由度舞动模型进一步简化为三自由度(法向分量、副法向分量和扭转分量)舞动模型。该模型可以考虑覆冰截面偏心、初始扭转角和抗弯刚度对覆冰输电线舞动影响。在建立风荷载模型时,除了考虑扭转角位移、法向速度和副法向速度对攻角的影响外,还考虑了初始扭转角对攻角的影响。除此之外,根据Galerkin方法将荷载模型和舞动模型进行离散。
其次,为了验证上述提出的基于曲梁理论的三自由度舞动模型,本文还建立了基于索结构理论的单自由度(竖向分量)、双自由度(竖向分量和水平分量)和三自由度(竖向分量、水平分量和扭转分量)舞动模型。并采用经典覆冰舞动算例(D型覆冰截面),将上述四种模型计算得到和实验测得的竖向舞动幅值进行对比分析。计算结果表明基于曲梁理论的三自由度舞动模型比基于索结构理论舞动模型精度高。除此之外,还分析了上述四种模型与实验数据间存在误差原因。
最后,还利用多尺度法和基于曲梁理论的三自由度舞动模型分析1:1和2:1内共振覆冰输电线舞动性质。为了便于多尺度法计算,根据覆冰输电线特性,将三自由度舞动模型简化为两自由度(竖向分量和水平分量)舞动模型。基于以上简化模型,利用多尺度法建立1:1和2:1内共振简化幅值方程来分析覆冰输电线随风速变化的分叉情况。在此基础上,建立1:1和2:1简化幅值方程的雅可比矩阵来判断各分支的稳定性。但在分析内共振2:1单幅值舞动(分支Ⅲ)的稳定性时,2:1简化幅值方程的雅可比矩阵出现歧义现象(不能判断其稳定性)。为了解决上述问题,本文还提出了复合指数-Cartesia形式的2:1简化幅值方程。除此之外,本文还根据团队在中国空气动力研究和发展中心所做的实验数据,拟合了覆冰输电线空气阻力和升力系数关于攻角的三次曲线,并根据以上数据分别对比分析了内共振1:1和2:1考虑覆冰截面偏心和不考虑偏心的分叉和稳定情况。分析结果表明:截面偏心对舞动影响较大。最后,在2:1内共振不考虑偏心情况下,发现了多重稳定的特殊现象,并根据2:1简化幅值方程和两自由度舞动方程在时域内积分得到的随时间变化的法向和副法向位移,分析并验证了该特殊现象的存在。
The galloping of the iced transmission line will cause the great damage to the whole society. In order to better understand the theory of galloping of iced transmission line, a three-degree-of-freedom galloping model based on curved beam theory has been proposed. Also, the Multiple Scale Perturbation Method (MSM) is introduced to analyze bifurcation and stability of the iced transmission line.
Firstly,based on curved beam theory proposed by Zhu and converted curvilinear coordinates raised by Mason, three-dimension Lagrange strain tensor of iced transmission line is established. Then, according to Hamilton principle, a four-degree-of-freedom (tangential, nominal, bi-nominal and torsional) galloping model considering iced eccentricity is formulated. Moreover, due to the property of transmission line, a reduced three-degree-of-freedom (nominal, bi-nominal and torsional) galloping model is obtained in view of bending, rotation and eccentricity of cross section. Furthermore, the initial rotational angle is introduced into the model of attack angle in addition to the rotational displacement, nominal velocity and bi-nominal velocity. Finally, According to Galerkin method, three-degree-of-freedom discrete galloping model and aerodynamic discrete model are obtained.
Secondly, in order to test the accuracy of three-degree-of-freedom galloping model based on curved beam theory, three galloping models based on cable structure are established, such as one-degree-of-freedom (vertical) , two-degree-of-freedom (vertical and horizontal) and three-degree-of-freedom (vertical, horizontal and rotational) galloping models. Moreover, one classical galloping example (cross section of D shape) is chosen to analyze the forgoing four galloping models. Comparing vertical displacements of the four galloping models and the experimental data, it can be found that the accuracy of the three-degree-of-freedom galloping model based on curved beam theory is higher than the other three galloping models based on cable structure. Furthermore, the reasons for the difference between the four galloping models and experimental data are discussed.
Finally, the three-degree-of-freedom galloping model based on curved beam theory and MSM are introduced to analyze the 1:1 and 2:1 resonant cases. In order to establish the model suitable for MSM, a two-degree-of-freedom (nominal and bi-nominal) galloping model is reduced from the three-degree-of-freedom galloping model due to the property of iced transmission line. In view of the above reduced two-degree-of-freedom galloping model, 1:1 and 2:1 Reduced Amplitude Modulation Equations (RAME) are formulated with the method of MSM to analyze the bifurcations of 1:1 and 2:1 resonant cases. Moreover, the Jacobian matrix of 1:1 and 2:1 RAME are established to analyze the stability of the forgoing bifurcations of 1:1 and 2:1 resonant cases except the special branch (one-mode (nominal) galloping), in which case the Jacobian matrix of 2:1 RAME becomes singular. In order to solve this problem, a mixed polar-Cartesian form of 2:1 RAME is adopted. Furthermore, according to the experimental data recently completed in Chinese Aerodynamic Research and Development Center, the drag and lift coefficients are fitted in the form of cubic equations. In view of the above fitting data, the bifurcation and stability of the 1:1 and 2:1 resonant cases considering the eccentricity and without considering the eccentricity are discussed. The results turn out that the iced eccentricity plays an important role on bifurcation and stability. Finally, in the 2:1 resonant case without considering the eccentricity, one special phenomenon (multiple stabilities) is found. Also, from the analysis of numerical integration of RAME and reduced model in time history, it is proved that this phenomenon actually appears as the wind speed is in the wind speed range of multiple stabilities, which is obtained by analyses of bifurcation and stability of RAME.
引文
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