旋转叶片刚柔耦合系统动力学研究
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摘要
本文针对旋转叶片刚柔耦合系统的动力学问题,基于广义Hamilton原理,以混合坐标法建立了由中心刚性圆盘及柔性叶片组成的旋转刚柔耦合系统动力学模型,研究了旋转叶片大范围转动对柔性叶片变形运动的影响。同时,研究了柔性叶片沿长度方向预扭角的变化、截面尺寸的变化对自身变形运动的影响。在大范围运动的柔性叶片变形场的描述中,不仅考虑了旋转平面内及垂直于旋转平面方向上的由弯曲引起的横向变形,还考虑了这两个方向上由剪切引起的横向变形,并计及了叶片大范围转动产生的离心力在横向弯曲变形引起的轴向缩短量上的离心力势能。
基于Hamilton变分原理导出了大范围运动时旋转叶片系统刚柔耦合连续动力学方程,该方程是一组非线性偏微分积分方程组,并采用有限元法对连续动力学方程进行离散,推导了带有预扭角、变截面,考虑剪切变形、截面转角及离心刚化效应的旋转叶片系统非线性、时变、耦合的有限维离散动力学方程,基于有限维离散动力学方程,研究了旋转叶片在加速过程中预扭角变化、叶片长度变化、刚性圆盘半径变化、截面尺寸变化、转速变化等物理因素对叶片动力学性态的影响。研究了旋转叶片在稳速过程中受冲击作用时,预扭角的变化对叶片动力学性态的影响。首次将恒定转速下叶片受到周期性变化的扰动力,引入到本文所推导的旋转叶片系统非线性、时变、耦合的动力学方程中,并编制了兼容性、扩展性强的通用计算程序。基于状态空间法对旋转叶片刚柔耦合系统进行了频率分析和稳定性分析,研究了中心刚体大范围运动对柔性叶片变形运动振动频率的影响,研究了叶片转速对位移响应的稳定性的影响,并给出了数值仿真结果。
研究结果表明,加速过程中叶片预扭角、截面尺寸参数的改变对叶片端部位移响应影响较小,而叶片在稳速过程中受到冲击作用时,预扭角对叶片端部在垂直于旋转平面方向上的横向位移有很大影响。并且,一个微小的低频周期性扰动力,就会使得带有预扭角的叶片在垂直于旋转平面方向上产生较大的横向位移,甚至发散。这种垂直于旋转平面方向上的横向位移会造成叶片在运行中碰擦侧壁,易发生断裂事故。同时,研究结果也表明,材料质量密度较大的叶片,其动频率随转速增加而增加,表现为动力刚化效应；材料质量密度较小的叶片,其一阶频率在某一转速范围内随转速增加而减小,出现了动力柔化现象。叶片截面尺寸沿长度方向变小,其模态频率降低,动力柔化效应提前。这种情况同样会引起叶片振动幅值增大而碰擦侧壁。
本文建立了考虑多种物理因素影响的以中心刚性圆盘及柔性叶片组成的刚柔耦合系统动力学模型及方程,该模型及方程可以用于解决非惯性系下大范围运动旋转柔性叶片刚柔耦合系统动力学问题,本文的研究方法具有一定的理论意义及工程应用价值,本文的研究结果对工程中叶片的理论分析及设计计算具有一定的参考价值。
Based on the Hamilton principle, the dynamic problem of a rotating flexible blade is studied in this paper. A dynamic model of rigid-flexible coupling system consists of a flexible blade fixed on the edge of a rigid disc is built using a Hybrid set of Coordinates. Effects of rotation of the flexible blade on its deformation displacement are discussed. And variations of pre-twisted angle and dimensions of cross-sections along the axis of blade are studied. Both trasversal deformations resulted from bending and shearing in the rotating plane and its perpendicular plane are considered in the description of deformation displacement of the flexible blade. And centrifugal force potential energy resulted from the rotation of blade is considered.
Based on Hamilton variation principle, continuous dynamic equations of the rigid-flexible coupling system of rotating blade are derived. The equations are a set of non-linear, partial differential and integral equations. The equations are discreted by finite element method. A set of discreted dynamic equations are given considering factors of pre-twisted angle, variable cross-section, shearing deformation, rotation of cross-section, and centrifugal stiffening effect. Effects of variations of pre-twisted angle, length of blade, radius of disc, dimensions of cross-section and rotating speed on dynamic behavior of blade are studied in the process of acceleration. Effects of variations of pre-twisted angle on dynamic behavior of blade subjected to impacting action are studied in the process of constant rotating speed. Periodical exciting force acting on blade at a constant rotating speed is introduced in the non-linear, time-varying and coupling dynamic equations. A general and compatible computational procedure is programmed. Based on state-space method, frequency and stability analysis of the rigid-flexible coupling system are carried out. Effects of rotation of the flexible blade on its deformation displacement frequency are studied. Effects of rotating speed on displacement response stability of blade are studied. And numerical results are given in the paper.
Results show that variation of pre-twisted angle and cross-section have little influences on the displacement response of blade tip in the process of acceleration. When the blade is rotating at a constant speed and subjected to an impacting action, the variation of pre-twisted angle has great influences on transversal displacement of blade tip on the plane perpendicular to the rotating plane. And a tiny low frequency periodical exciting force will cause great transversal displacement or divergence on the plane perpendicular to the rotating plane for pre-twisted blade. The tansversal displacement on the plane perpendicular to the rotating plane will make the blade touch inside surface and result in fracture failure in the operation. Results also show that frequencies increase with increasing rotating speed for blade with high mass density, which case is called dynamic stiffening. Meanwhile, the first order frequencies decrease with increasing rotating speed for blade with low mass density, which case is called dynamic softening. Frequencies decrease with the dimension recucing of cross-section along the axis of blade and the effect of dynamic softening occurs earlier, which case will also make vibration amplitude of the blade increase and may touch inside surface.
Considering the effects of many factors of an rotating blade, dynamic model and equations of a rigid-flexible coupling system are given in this paper. The model and equations can be used to solve dynamic problems of a rigid-flexible coupling system of a rotating blade in the noninertial system. Methods presented in the paper have certain theoretical significance and engineering application value. Results in the paper have some referrence value for theoretical analysis and design of blade in the engineering.
基于Hamilton变分原理导出了大范围运动时旋转叶片系统刚柔耦合连续动力学方程,该方程是一组非线性偏微分积分方程组,并采用有限元法对连续动力学方程进行离散,推导了带有预扭角、变截面,考虑剪切变形、截面转角及离心刚化效应的旋转叶片系统非线性、时变、耦合的有限维离散动力学方程,基于有限维离散动力学方程,研究了旋转叶片在加速过程中预扭角变化、叶片长度变化、刚性圆盘半径变化、截面尺寸变化、转速变化等物理因素对叶片动力学性态的影响。研究了旋转叶片在稳速过程中受冲击作用时,预扭角的变化对叶片动力学性态的影响。首次将恒定转速下叶片受到周期性变化的扰动力,引入到本文所推导的旋转叶片系统非线性、时变、耦合的动力学方程中,并编制了兼容性、扩展性强的通用计算程序。基于状态空间法对旋转叶片刚柔耦合系统进行了频率分析和稳定性分析,研究了中心刚体大范围运动对柔性叶片变形运动振动频率的影响,研究了叶片转速对位移响应的稳定性的影响,并给出了数值仿真结果。
研究结果表明,加速过程中叶片预扭角、截面尺寸参数的改变对叶片端部位移响应影响较小,而叶片在稳速过程中受到冲击作用时,预扭角对叶片端部在垂直于旋转平面方向上的横向位移有很大影响。并且,一个微小的低频周期性扰动力,就会使得带有预扭角的叶片在垂直于旋转平面方向上产生较大的横向位移,甚至发散。这种垂直于旋转平面方向上的横向位移会造成叶片在运行中碰擦侧壁,易发生断裂事故。同时,研究结果也表明,材料质量密度较大的叶片,其动频率随转速增加而增加,表现为动力刚化效应；材料质量密度较小的叶片,其一阶频率在某一转速范围内随转速增加而减小,出现了动力柔化现象。叶片截面尺寸沿长度方向变小,其模态频率降低,动力柔化效应提前。这种情况同样会引起叶片振动幅值增大而碰擦侧壁。
本文建立了考虑多种物理因素影响的以中心刚性圆盘及柔性叶片组成的刚柔耦合系统动力学模型及方程,该模型及方程可以用于解决非惯性系下大范围运动旋转柔性叶片刚柔耦合系统动力学问题,本文的研究方法具有一定的理论意义及工程应用价值,本文的研究结果对工程中叶片的理论分析及设计计算具有一定的参考价值。
Based on the Hamilton principle, the dynamic problem of a rotating flexible blade is studied in this paper. A dynamic model of rigid-flexible coupling system consists of a flexible blade fixed on the edge of a rigid disc is built using a Hybrid set of Coordinates. Effects of rotation of the flexible blade on its deformation displacement are discussed. And variations of pre-twisted angle and dimensions of cross-sections along the axis of blade are studied. Both trasversal deformations resulted from bending and shearing in the rotating plane and its perpendicular plane are considered in the description of deformation displacement of the flexible blade. And centrifugal force potential energy resulted from the rotation of blade is considered.
Based on Hamilton variation principle, continuous dynamic equations of the rigid-flexible coupling system of rotating blade are derived. The equations are a set of non-linear, partial differential and integral equations. The equations are discreted by finite element method. A set of discreted dynamic equations are given considering factors of pre-twisted angle, variable cross-section, shearing deformation, rotation of cross-section, and centrifugal stiffening effect. Effects of variations of pre-twisted angle, length of blade, radius of disc, dimensions of cross-section and rotating speed on dynamic behavior of blade are studied in the process of acceleration. Effects of variations of pre-twisted angle on dynamic behavior of blade subjected to impacting action are studied in the process of constant rotating speed. Periodical exciting force acting on blade at a constant rotating speed is introduced in the non-linear, time-varying and coupling dynamic equations. A general and compatible computational procedure is programmed. Based on state-space method, frequency and stability analysis of the rigid-flexible coupling system are carried out. Effects of rotation of the flexible blade on its deformation displacement frequency are studied. Effects of rotating speed on displacement response stability of blade are studied. And numerical results are given in the paper.
Results show that variation of pre-twisted angle and cross-section have little influences on the displacement response of blade tip in the process of acceleration. When the blade is rotating at a constant speed and subjected to an impacting action, the variation of pre-twisted angle has great influences on transversal displacement of blade tip on the plane perpendicular to the rotating plane. And a tiny low frequency periodical exciting force will cause great transversal displacement or divergence on the plane perpendicular to the rotating plane for pre-twisted blade. The tansversal displacement on the plane perpendicular to the rotating plane will make the blade touch inside surface and result in fracture failure in the operation. Results also show that frequencies increase with increasing rotating speed for blade with high mass density, which case is called dynamic stiffening. Meanwhile, the first order frequencies decrease with increasing rotating speed for blade with low mass density, which case is called dynamic softening. Frequencies decrease with the dimension recucing of cross-section along the axis of blade and the effect of dynamic softening occurs earlier, which case will also make vibration amplitude of the blade increase and may touch inside surface.
Considering the effects of many factors of an rotating blade, dynamic model and equations of a rigid-flexible coupling system are given in this paper. The model and equations can be used to solve dynamic problems of a rigid-flexible coupling system of a rotating blade in the noninertial system. Methods presented in the paper have certain theoretical significance and engineering application value. Results in the paper have some referrence value for theoretical analysis and design of blade in the engineering.
引文
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