二元机翼颤振及其主动控制的研究
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摘要
颤振是动气动弹性力学中最重要的研究内容之一，本文以亚音速内二元机翼为研究对象，对机翼发生颤振时的动态特性、结构非线性对气动弹性响应的影响、颤振主动控制方法及其试验验证等方面进行了深入研究，主要研究内容如下：
（1）研究了具有扑动和俯仰两自由度二元矩形机翼的气动弹性特性。采用片条理论推导了作用于翼面上的气动升力和气动力矩，利用能量法分别建立了两自由度二元刚性机翼和四自由度二元弹性机翼的气弹运动方程；采用特征值法、图解法和二次颤振曲线法求解机翼系统的颤振频率和颤振速度，分析了气动和结构参数对机翼气动弹性特性的影响，讨论了影响颤振速度的各个因素，得到了机翼系统频率和阻尼比随风速变化的趋势图。
（2）研究了硬立方刚度结构非线性对二元刚性机翼颤振特性的影响，采用能量法建立了此类机翼系统的气弹运动方程，应用描述函数法对立方非线性进行等效线性化处理，采用传统的线性系统颤振分析方法预测了非线性系统的颤振速度和极限环幅值；进一步利用Hopf分岔理论验证了预测结果的准确性并根据Routh-Hurwitz判据研究零平衡点的稳定性。结果表明，立方非线性不会改变原线性系统的临界颤振速度；但当超过该速度时，系统出现极限环振动，不同初始条件下系统收敛于相同的极限环。
（3）研究了操纵面间隙非线性对二元刚性机翼颤振特性的影响，利用拉格朗日方程建立了三自由度二元机翼系统的气弹运动方程，应用描述函数法对间隙非线性进行拟线性化处理，通过系统的特征多项式求得极限环的幅值和频率，利用摄动法推导了极限环稳定时需满足的条件。结果表明，间隙非线性导致系统的颤振提前产生，且系统出现幅值突变现象；当风速大于系统的临界颤振速度时，不同初始值将导致系统发散或产生亚谐波周期、准周期运动等复杂动力学行为。
（4）以带操纵面的二自由度二元刚性机翼模型为研究对象，当不考虑非线性因素时，提出了一种基于实测柔度的颤振主动控制方法，建立了机翼系统的闭环控制模型，利用闭环系统的动柔度推导了实现任意极点配置时的增益。结果表明，闭环控制条件下系统颤振速度大幅度提高；当考虑立方非线性因素时，针对极点难于精确配置的情况提出了基于系统实测柔度的鲁棒控制方法，通过求解系统的极小范数最小二乘解得到了系统的控制增益，实现了极限环的准确配置和期望极点在一定范围内的配置，使系统具有很好的鲁棒性。
（5）以具有NACA0018型标准截面的二元刚性机翼模型为试验对象，设计了风洞试验装置和控制系统，对基于动柔度法的系统极点配置和颤振主动控制理论进行了验证。试验结果表明，基于测量得到的系统动柔度，结合求解控制增益的方法，可方便地实现系统的任意极点配置；相比采用提高模态阻尼的方法，采用分离系统模态频率的方法可更加有效地提高系统的颤振速度；试验证明了动柔度法在颤振主动控制中的有效性和准确性。
The flutter phenomenon is one of the most important problems in dynamicaeroelastictiy. In this dissertation, by taking two-dimensional airfoil in subsonic flow as theresearch object, the flutter characteristics, the influence of structure nonlinearities onaeroelastic response, flutter active control method and its test verification aresystematically and thoroughly studied. The main contents are included as follows:
(1) The aeroelastic characteristics of two-dimensional airfoil with flap and pitchdegrees of freedom is investigated. Applying strip theory, the expressions of unsteadyaerodynamic for lift and pitching moment acting on airfoil are derived, then aeroelasticequations of motion for two degrees of freedom rigid airfoil and four degrees of freedomflexible airfoil are established by using energy method. Eigenvalue method, graphicmethod and flutter conic method are employed to determine flutter frequencies and fluttervelocity. The effects of system parameters on aerodynamic characteristics are analyzed andsubsequently the trends of frequency and damping ratio for system are obtained.
(2) The effect of cubic hard spring stiffness in pitching direction on fluttercharacteristic of two-dimensional rigid airfoil is studied. Firstly, with the help of describingfunction method, the equivalent linear flutter motion equations of airfoil system areestablished based on energy method. Then using the traditional linear analytical method topredict the flutter speed and limit circle amplitude of original nonlinear system.Furthermore, Hopf bifurcation theory is used to verify the accuracy of the above results,and the Routh-Hurwitz criterion to determine the stability of system. Finally, it isconcluded that cubic nonlinearity has no effects on flutter speed of the corresponding linearsystem, but the nonlinear airfoil system will generate limit circle oscillation behavior whenthe air speed beyond the flutter speed, moreover the system will converge to the same limitcircle under different initial value.
(3) Considering the effects of freeplay in control surface, aeroelastic equations ofthree degrees of freedom two-dimensional airfoil system are constructed with Lagrangianequation. Similarly, by the use of describing function method to deal with the freeplaynonlinearity, then amplitude and frequency of limit circle could be deduced from thecharacteristic polynomial of system, thereafter the conditions for stabilization system arederived by perturbation technique. The numerical simulation shown that freeplaynonlinearity leads to the flutter phenomenon advanced, and the jumping behavior of limit circle amplitude for control surface deflection could be observed. In addition, the differentinitial condition of system will cause complex dynamic responses, such as period motionor divergence.
(4) A two-dimensional airfoil with a control surface is studied, in which thenonlinearities are not included. A novel flutter active control method based on measuredreceptance is put forward, the close-loop control model of airfoil system is conducted, andcontrol gains can be derived by using its receptance, with the gains the poles can beassigned arbitrarily. It is conclude that the critical flutter speed is greatly increased underthe closed-loop system by way of numerical simulation. When cubic nonlinearity inpitching direction is considered, a robust control method based on measured receptance isproposed according to the difficulty of accurately assigning desired poles, where thecontrol gains can be obtained by solving the minimal norm least square solution fornonlinear airfoil system. With the proposed method, limit circle can be assigned accuratelyand the desired poles can be assigned around a range of values, the system has very goodrobustness.
(5) Take a working section containing a NACA0018airfoil as the wind tunnel testobject, the test fig and control system are designed and built firstly. Then the verificationfor the above theoretical analysis including pole assignment and flutter active controlmethod is implemented through the wind tunnel test. The results shown that combing thesolved gains, the desired poles can be assigned conveniently based on the measuredreceptance. The flutter speed could be increased more efficiently by separating the flap andpitch frequencies method than increasing the mode damping ratio method. Experimentalresults validate the availability and accuracy of the receptance method in flutter activecontrol.
（1）研究了具有扑动和俯仰两自由度二元矩形机翼的气动弹性特性。采用片条理论推导了作用于翼面上的气动升力和气动力矩，利用能量法分别建立了两自由度二元刚性机翼和四自由度二元弹性机翼的气弹运动方程；采用特征值法、图解法和二次颤振曲线法求解机翼系统的颤振频率和颤振速度，分析了气动和结构参数对机翼气动弹性特性的影响，讨论了影响颤振速度的各个因素，得到了机翼系统频率和阻尼比随风速变化的趋势图。
（2）研究了硬立方刚度结构非线性对二元刚性机翼颤振特性的影响，采用能量法建立了此类机翼系统的气弹运动方程，应用描述函数法对立方非线性进行等效线性化处理，采用传统的线性系统颤振分析方法预测了非线性系统的颤振速度和极限环幅值；进一步利用Hopf分岔理论验证了预测结果的准确性并根据Routh-Hurwitz判据研究零平衡点的稳定性。结果表明，立方非线性不会改变原线性系统的临界颤振速度；但当超过该速度时，系统出现极限环振动，不同初始条件下系统收敛于相同的极限环。
（3）研究了操纵面间隙非线性对二元刚性机翼颤振特性的影响，利用拉格朗日方程建立了三自由度二元机翼系统的气弹运动方程，应用描述函数法对间隙非线性进行拟线性化处理，通过系统的特征多项式求得极限环的幅值和频率，利用摄动法推导了极限环稳定时需满足的条件。结果表明，间隙非线性导致系统的颤振提前产生，且系统出现幅值突变现象；当风速大于系统的临界颤振速度时，不同初始值将导致系统发散或产生亚谐波周期、准周期运动等复杂动力学行为。
（4）以带操纵面的二自由度二元刚性机翼模型为研究对象，当不考虑非线性因素时，提出了一种基于实测柔度的颤振主动控制方法，建立了机翼系统的闭环控制模型，利用闭环系统的动柔度推导了实现任意极点配置时的增益。结果表明，闭环控制条件下系统颤振速度大幅度提高；当考虑立方非线性因素时，针对极点难于精确配置的情况提出了基于系统实测柔度的鲁棒控制方法，通过求解系统的极小范数最小二乘解得到了系统的控制增益，实现了极限环的准确配置和期望极点在一定范围内的配置，使系统具有很好的鲁棒性。
（5）以具有NACA0018型标准截面的二元刚性机翼模型为试验对象，设计了风洞试验装置和控制系统，对基于动柔度法的系统极点配置和颤振主动控制理论进行了验证。试验结果表明，基于测量得到的系统动柔度，结合求解控制增益的方法，可方便地实现系统的任意极点配置；相比采用提高模态阻尼的方法，采用分离系统模态频率的方法可更加有效地提高系统的颤振速度；试验证明了动柔度法在颤振主动控制中的有效性和准确性。
The flutter phenomenon is one of the most important problems in dynamicaeroelastictiy. In this dissertation, by taking two-dimensional airfoil in subsonic flow as theresearch object, the flutter characteristics, the influence of structure nonlinearities onaeroelastic response, flutter active control method and its test verification aresystematically and thoroughly studied. The main contents are included as follows:
(1) The aeroelastic characteristics of two-dimensional airfoil with flap and pitchdegrees of freedom is investigated. Applying strip theory, the expressions of unsteadyaerodynamic for lift and pitching moment acting on airfoil are derived, then aeroelasticequations of motion for two degrees of freedom rigid airfoil and four degrees of freedomflexible airfoil are established by using energy method. Eigenvalue method, graphicmethod and flutter conic method are employed to determine flutter frequencies and fluttervelocity. The effects of system parameters on aerodynamic characteristics are analyzed andsubsequently the trends of frequency and damping ratio for system are obtained.
(2) The effect of cubic hard spring stiffness in pitching direction on fluttercharacteristic of two-dimensional rigid airfoil is studied. Firstly, with the help of describingfunction method, the equivalent linear flutter motion equations of airfoil system areestablished based on energy method. Then using the traditional linear analytical method topredict the flutter speed and limit circle amplitude of original nonlinear system.Furthermore, Hopf bifurcation theory is used to verify the accuracy of the above results,and the Routh-Hurwitz criterion to determine the stability of system. Finally, it isconcluded that cubic nonlinearity has no effects on flutter speed of the corresponding linearsystem, but the nonlinear airfoil system will generate limit circle oscillation behavior whenthe air speed beyond the flutter speed, moreover the system will converge to the same limitcircle under different initial value.
(3) Considering the effects of freeplay in control surface, aeroelastic equations ofthree degrees of freedom two-dimensional airfoil system are constructed with Lagrangianequation. Similarly, by the use of describing function method to deal with the freeplaynonlinearity, then amplitude and frequency of limit circle could be deduced from thecharacteristic polynomial of system, thereafter the conditions for stabilization system arederived by perturbation technique. The numerical simulation shown that freeplaynonlinearity leads to the flutter phenomenon advanced, and the jumping behavior of limit circle amplitude for control surface deflection could be observed. In addition, the differentinitial condition of system will cause complex dynamic responses, such as period motionor divergence.
(4) A two-dimensional airfoil with a control surface is studied, in which thenonlinearities are not included. A novel flutter active control method based on measuredreceptance is put forward, the close-loop control model of airfoil system is conducted, andcontrol gains can be derived by using its receptance, with the gains the poles can beassigned arbitrarily. It is conclude that the critical flutter speed is greatly increased underthe closed-loop system by way of numerical simulation. When cubic nonlinearity inpitching direction is considered, a robust control method based on measured receptance isproposed according to the difficulty of accurately assigning desired poles, where thecontrol gains can be obtained by solving the minimal norm least square solution fornonlinear airfoil system. With the proposed method, limit circle can be assigned accuratelyand the desired poles can be assigned around a range of values, the system has very goodrobustness.
(5) Take a working section containing a NACA0018airfoil as the wind tunnel testobject, the test fig and control system are designed and built firstly. Then the verificationfor the above theoretical analysis including pole assignment and flutter active controlmethod is implemented through the wind tunnel test. The results shown that combing thesolved gains, the desired poles can be assigned conveniently based on the measuredreceptance. The flutter speed could be increased more efficiently by separating the flap andpitch frequencies method than increasing the mode damping ratio method. Experimentalresults validate the availability and accuracy of the receptance method in flutter activecontrol.
引文
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