旋转柔性叶片系统动力学特性研究
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摘要
本文研究的中心刚体-柔性叶片系统是一个刚柔耦合的柔性多体系统,它具有时变、高度耦合、高度非线性的特征。它在航空航天器、高速车辆、操作机械臂、机器人、复杂机构等领域有着广泛的应用。
首先,对于中心刚体-柔性叶片所组成的刚柔耦合系统,在轴向变形中考虑了由于横向变形所引起的轴向缩短,应用哈密顿原理,得到了包含动力刚化项的刚柔耦合动力学方程组。之后,利用假设模态法把柔性叶片的变形离散化,得到考虑了二阶耦合项离散的一次近似耦合动力学方程组。若不考虑二阶耦合项,则是传统的零次近似耦合模型。在一次近似耦合模型中,如果忽略纵向变形,就得到了一次近似简化模型。在数值仿真分析中,考虑了两种情况:大范围运动已知的非惯性场中的动力学特性问题和大范围运动未知的刚柔耦合问题。仿真表明,一次近似耦合模型在各种转速下都能得到稳定的结果,而零次近似耦合模型在高速大范围运动下,将出现发散。通过对一次近似模型和一次近似简化模型的仿真结果的比较,表明一次近似简化模型可以代替一次近似模型进行动力学和控制方面的研究。
其次,应用李亚普诺夫直接法,对刚柔耦合动力学运动控制方程进行稳定性分析。李亚普诺夫直接法能够研究线性和非线性方程的稳定问题,是一种普遍的稳定性分析方法。本文利用能量-动量矩法来构造李雅普诺夫函数,对传统的零次近似耦合模型和一次近似耦合模型进行了稳定性分析,并得出零次模型和一次模型之间应用的范围,结果表明,当旋转角速度大于叶片的基频时,零次近似模型将失效;而一次近似模型能在任意的角速度下保持稳定性。
最后,把一次近似简化模型进行线性化,得到中心刚体-柔性叶片模型的线性动力学方程。利用最优跟踪控制原理,把线性模型设计的控制律来控制非线性的一次近似简化模型。因为所设计的控制器是叶片的模态坐标的函数,文中给出了一种从物理测量提取模态坐标的方法。文中研究了给定运动轨迹和给定位置两种情况的跟踪仿真,结果表明大范围运动能够到达目标,并有效地对振动进行了抑制。
In this paper, a flexible blade attached to a moving central rigid body undergoing large overall rotating motion is discussed. This is a coupling system about a rigid body and a flexible body. It not only changes with time but also has the highly coupled and nonlinear characteristics. This model has wide application in many areas, for example the airplane, spacecraft, high-speed vehicles, robotic arm, the robot areas and so forth.
Firstly, the effects of the transverse deformation induced longitudinal deformation were also included in the whole longitudinal deformation. The dynamic equations including the dynamic stiffening terms was established by utilizing the Hamilton theory. The flexible blade is discretized by employing the approach of assumed modes method. This is called for the first-order approximation coupling (FOAC) model taking the second-order coupling quantity. If the second-order coupling quantity is neglect, the model is called for zero-order approximation coupling (ZOAC) model. The simplified first-order approximation coupling (SFOAC) model which neglects the effect of axial deformation of the blade is also studied. Two cases are considered in the simulations. One is the dynamics study in non-inertia system which the large motion of a system is known and the other is that the large motion of a system is unknown. The simulation indicated that the FOAC is stable in any angle velocity, but the ZOAC is emanative in high angle velocity. The SFOAC model is valid for the description of a rotating blade in dynamics and control by numerical comparisons made between the results SFOAC and FOAL.
Then nonlinear stability of hub-leaf model in large overall motions is studied by Lyapunov direct method. Lyapunov direct method can be used in linear models and nonlinear models. This is a rife method in judging the stability. Utilizing the Energy-Momentum method, nonlinear stability of the equilibrium in the first-order approximation coupling (FOAL) model and the zero-order approximation coupling (ZOAC) model is analyzed. The results show that the ZOAC will lose stability when the rotating angular velocity is greater than its fundamental frequency. However, the FOAC can keep stable in any rotating angular velocity in the coupling dynamical model.
At last, the linearization model is the linearization treatment for the SFOAC model. Active controller is designed using optimal tracking control theory. Since the controller designed is a function of modal coordinate of the blade, a modal filter used to extract tire modal coordinate from actual physical measurements is presented. Two cases are studied, the one is trajectory given, the other is angle given. By the optimal tracking controller, the desired motion target may be obtained and the vibration of flexible blade can be suppressed.
首先,对于中心刚体-柔性叶片所组成的刚柔耦合系统,在轴向变形中考虑了由于横向变形所引起的轴向缩短,应用哈密顿原理,得到了包含动力刚化项的刚柔耦合动力学方程组。之后,利用假设模态法把柔性叶片的变形离散化,得到考虑了二阶耦合项离散的一次近似耦合动力学方程组。若不考虑二阶耦合项,则是传统的零次近似耦合模型。在一次近似耦合模型中,如果忽略纵向变形,就得到了一次近似简化模型。在数值仿真分析中,考虑了两种情况:大范围运动已知的非惯性场中的动力学特性问题和大范围运动未知的刚柔耦合问题。仿真表明,一次近似耦合模型在各种转速下都能得到稳定的结果,而零次近似耦合模型在高速大范围运动下,将出现发散。通过对一次近似模型和一次近似简化模型的仿真结果的比较,表明一次近似简化模型可以代替一次近似模型进行动力学和控制方面的研究。
其次,应用李亚普诺夫直接法,对刚柔耦合动力学运动控制方程进行稳定性分析。李亚普诺夫直接法能够研究线性和非线性方程的稳定问题,是一种普遍的稳定性分析方法。本文利用能量-动量矩法来构造李雅普诺夫函数,对传统的零次近似耦合模型和一次近似耦合模型进行了稳定性分析,并得出零次模型和一次模型之间应用的范围,结果表明,当旋转角速度大于叶片的基频时,零次近似模型将失效;而一次近似模型能在任意的角速度下保持稳定性。
最后,把一次近似简化模型进行线性化,得到中心刚体-柔性叶片模型的线性动力学方程。利用最优跟踪控制原理,把线性模型设计的控制律来控制非线性的一次近似简化模型。因为所设计的控制器是叶片的模态坐标的函数,文中给出了一种从物理测量提取模态坐标的方法。文中研究了给定运动轨迹和给定位置两种情况的跟踪仿真,结果表明大范围运动能够到达目标,并有效地对振动进行了抑制。
In this paper, a flexible blade attached to a moving central rigid body undergoing large overall rotating motion is discussed. This is a coupling system about a rigid body and a flexible body. It not only changes with time but also has the highly coupled and nonlinear characteristics. This model has wide application in many areas, for example the airplane, spacecraft, high-speed vehicles, robotic arm, the robot areas and so forth.
Firstly, the effects of the transverse deformation induced longitudinal deformation were also included in the whole longitudinal deformation. The dynamic equations including the dynamic stiffening terms was established by utilizing the Hamilton theory. The flexible blade is discretized by employing the approach of assumed modes method. This is called for the first-order approximation coupling (FOAC) model taking the second-order coupling quantity. If the second-order coupling quantity is neglect, the model is called for zero-order approximation coupling (ZOAC) model. The simplified first-order approximation coupling (SFOAC) model which neglects the effect of axial deformation of the blade is also studied. Two cases are considered in the simulations. One is the dynamics study in non-inertia system which the large motion of a system is known and the other is that the large motion of a system is unknown. The simulation indicated that the FOAC is stable in any angle velocity, but the ZOAC is emanative in high angle velocity. The SFOAC model is valid for the description of a rotating blade in dynamics and control by numerical comparisons made between the results SFOAC and FOAL.
Then nonlinear stability of hub-leaf model in large overall motions is studied by Lyapunov direct method. Lyapunov direct method can be used in linear models and nonlinear models. This is a rife method in judging the stability. Utilizing the Energy-Momentum method, nonlinear stability of the equilibrium in the first-order approximation coupling (FOAL) model and the zero-order approximation coupling (ZOAC) model is analyzed. The results show that the ZOAC will lose stability when the rotating angular velocity is greater than its fundamental frequency. However, the FOAC can keep stable in any rotating angular velocity in the coupling dynamical model.
At last, the linearization model is the linearization treatment for the SFOAC model. Active controller is designed using optimal tracking control theory. Since the controller designed is a function of modal coordinate of the blade, a modal filter used to extract tire modal coordinate from actual physical measurements is presented. Two cases are studied, the one is trajectory given, the other is angle given. By the optimal tracking controller, the desired motion target may be obtained and the vibration of flexible blade can be suppressed.
引文
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2 Sunada W & Dubowsky S. The Applications of Finite Element Methods to the Dynamic Analysis of Flexible Spatial and Co-Planar Linkage Systems. ASME Journal of Mechanical Design,1971,103:643-651
3 Sunada W & Dubowsky S. On the Dynamic Analysis and Behaviour of Industrial Robotic Manipulators with Elastic Members. ASME Journal of Mechanisms,Transmissions & Automation in Design,1983,105:42-51
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6 Erdman.A.G., Sandor.G.N. Kineto-elasto dynamics-A review of the state of the art and trends, Mechanism and Machine Theory,1972,7:19-33
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