﻿ 间隙非线性对二元翼段颤振特性影响
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (4): 730-737  DOI: 10.11990/jheu.201709113 0

### 引用本文

LI Yufei, BAI Junqiang, LIU Nan, et al. Analysis of aeroelastic characteristics of 2-D wing with free play[J]. Journal of Harbin Engineering University, 2019, 40(4), 730-737. DOI: 10.11990/jheu.201709113.

### 文章历史

1. 西北工业大学 航空学院, 陕西 西安 710072;
2. 中国航空工业空气动力研究院, 辽宁 沈阳 110034

Analysis of aeroelastic characteristics of 2-D wing with free play
LI Yufei 1, BAI Junqiang 1, LIU Nan 2, LI Guojun 1, HE Xiaolong 1
1. School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China;
2. AVIC Aerodynamic Research Institute, Shenyang 110034, China
Abstract: By using unsteady aerodynamics method of unsteady Reynolds averaged Navier-Stokes and combining the motion equation of a structure, we established the time-domain aeroelastic analysis system to calculate the flutter speed of the configuration of a 2-D wing with three degrees of freedom and without free play.In studying the effect of free play nonlinearity on flutter characteristics, we used the description function method to solve the free play problem and determined the limit cyclic oscillation phenomenon caused by free play nonlinearity.Examining the influence of preload on the flutter characteristics showed that preload could weaken the nonlinear effect caused by free play.
Keywords: unsteady flow    aeroelasticity    flutter(aerodynamics)    transonic aerodynamics    computational fluid dynamics    time domain analysis    airfoils    free play    limit cycle oscillation    preload

1 非定常气动力求解 1.1 控制方程

 $\iiint\limits_\Omega {\left( {\frac{{\partial Q}}{{\partial t}} + \frac{{\partial \left( {F - {F_v}} \right)}}{{\partial x}} + \frac{{\partial \left( {G - {G_v}} \right)}}{{\partial y}} + \frac{{\partial \left( {H - {H_v}} \right)}}{{\partial z}}} \right){\text{d}}\mathit{\Omega }} = 0$ (1)

1.2 非定常气动力验证

 $\alpha \left( t \right) = {\alpha _0} + {\alpha _m}\sin \left( {\omega t} \right),k = \omega b/{V_\infty }$ (2)

 Download: 图 2 非定常升力系数和俯仰力矩系数随攻角变化 Fig. 2 Coefficients of lift and pitch moment change along with the angle of attack
2 结构运动方程

 $\mathit{\boldsymbol{M\ddot q}} + \mathit{\boldsymbol{D\dot q}} + \mathit{\boldsymbol{Kq}} = \mathit{\boldsymbol{f}}$ (3)

 $\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{Ax}} + \mathit{\boldsymbol{Bf}}$ (4)

 $\mathit{\boldsymbol{x}} = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{q}}\\ {\mathit{\boldsymbol{\dot q}}} \end{array}} \right];\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {\bf{0}}&\mathit{\boldsymbol{I}}\\ { - {\mathit{\boldsymbol{M}}^{ - 1}}\mathit{\boldsymbol{K}}}&{ - {\mathit{\boldsymbol{M}}^{ - 1}}\mathit{\boldsymbol{D}}} \end{array}} \right];\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {\bf{0}}\\ {{\mathit{\boldsymbol{M}}^{ - 1}}} \end{array}} \right]$

 ${{\mathit{\boldsymbol{\tilde x}}}^{n + 1}} = {\mathit{\boldsymbol{x}}^n} + \frac{1}{2}\mathit{\boldsymbol{A}}\left( {3{\mathit{\boldsymbol{x}}^n} - {\mathit{\boldsymbol{x}}^{n - 1}}} \right) + \frac{1}{2}\mathit{\boldsymbol{B}}\left( {3{\mathit{\boldsymbol{f}}^n} - {\mathit{\boldsymbol{f}}^{n - 1}}} \right)$

 ${\mathit{\boldsymbol{x}}^{n + 1}} = {\mathit{\boldsymbol{x}}^n} + \frac{1}{2}\mathit{\boldsymbol{A}}\left( {{{\mathit{\boldsymbol{\tilde x}}}^{n + 1}} + {\mathit{\boldsymbol{x}}^n}} \right) + \frac{1}{2}{\mathit{\boldsymbol{B}}}\left( {{{\mathit{\boldsymbol{\tilde f}}}^{n + 1}} + {\mathit{\boldsymbol{f}}^n}} \right)$

 $\mathit{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{c}} M&{{S_\alpha }}&{{S_\beta }}\\ {{S_\alpha }}&{{I_\alpha }}&{{I_\beta } + b\left( {{a_f} - a} \right){S_\beta }}\\ {{S_\beta }}&{{I_\beta } + b\left( {{a_f} - a} \right){S_\beta }}&{{I_\beta }} \end{array}} \right]$
 $\mathit{\boldsymbol{D}} = \left[ {\begin{array}{*{20}{c}} {{D_h}}&{}&{}\\ {}&{{D_\alpha }}&{}\\ {}&{}&{{D_\beta }} \end{array}} \right],\mathit{\boldsymbol{K}} = \left[ {\begin{array}{*{20}{c}} {{K_h}}&{}&{}\\ {}&{{K_\alpha }}&{}\\ {}&{}&{{K_\beta }} \end{array}} \right]$
 $\mathit{\boldsymbol{q}} = \left[ {\begin{array}{*{20}{c}} h\\ \alpha \\ \beta \end{array}} \right],\mathit{\boldsymbol{f}} = \left[ {\begin{array}{*{20}{c}} { - L}\\ {{M_y}}\\ {{M_\beta }} \end{array}} \right]$

 $\mathit{\boldsymbol{\tilde M}} = \left[ {\begin{array}{*{20}{c}} {M/m}&{{x_\alpha }}&{{x_\beta }}\\ {{x_\alpha }}&{r_\alpha ^2}&{r_\alpha ^2 + b\left( {{a_f} - a} \right){x_\beta }}\\ {{x_\beta }}&{r_\alpha ^2 + b\left( {{a_f} - a} \right){x_\beta }}&{r_\beta ^2} \end{array}} \right]$
 $\mathit{\boldsymbol{\tilde D}} = \left[ {\begin{array}{*{20}{c}} {{\zeta _h}}&{}&{}\\ {}&{{\zeta _\alpha }}&{}\\ {}&{}&{{\zeta _\beta }} \end{array}} \right]$
 $\mathit{\boldsymbol{\tilde K}} = \left[ {\begin{array}{*{20}{c}} {{{\left( {{\omega _h}/{\omega _\alpha }} \right)}^2}}&{}&{}\\ {}&{r_\alpha ^2}&{}\\ {}&{}&{r_\beta ^2{{\left( {{\omega _\beta }/{\omega _\alpha }} \right)}^2}} \end{array}} \right]$
 $\mathit{\boldsymbol{\tilde q}} = \left[ {\begin{array}{*{20}{c}} {h/b}\\ \alpha \\ \beta \end{array}} \right],\mathit{\boldsymbol{\tilde f}} = \frac{{V_f^2}}{{\rm{ \mathsf{ π} }}}\left[ {\begin{array}{*{20}{c}} { - {C_l}}\\ {2{C_m}}\\ {2{C_{{m_\beta }}}} \end{array}} \right]$

 Download: 图 3 带后缘操纵面二维翼型的几何和结构参数的定义 Fig. 3 Geometry and structural definition of foil with actor
3 算例与分析 3.1 间隙非线性处理

 Download: 图 4 后缘操纵面的恢复力矩和等效刚度随偏角变化趋势的示意 Fig. 4 The recovery moment and equivalent stiffness of the trailing edge control surface change along with the angle of attack

 ${\tau _\beta } = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 0,\\ {K_\beta }\left( {\beta - \delta } \right), \end{array}&\begin{array}{l} \left| \beta \right| \le \delta \\ \left| \beta \right| > \delta \end{array} \end{array}} \right.$

 $\beta = {\beta _m}\sin \left( {\omega t} \right)$ (5)

 ${\tau _{\beta ,eq}} = \frac{1}{{2{\rm{ \mathsf{ π} }}}}\int_0^{2{\rm{ \mathsf{ π} }}/\omega } {{\tau _\beta }\left( t \right)\sin \left( {\omega t} \right){\rm{d}}t}$ (6)

 ${K_{eq}} = {\tau _{\beta ,eq}}/{\beta _m}$ (7)

 ${\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}^{\rm{T}}}\mathit{\boldsymbol{D \boldsymbol{\varPhi} }} = \left[ {\begin{array}{*{20}{c}} {2{m_h}{{\omega '}_h}{\zeta _h}}&{}&{}\\ {}&{2{m_\alpha }{{\omega '}_\alpha }{\zeta _\alpha }}&{}\\ {}&{}&{2{m_\beta }{{\omega '}_\beta }{\zeta _\beta }} \end{array}} \right]$

 $\omega _i^2\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\varphi }}_i} = \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{\varphi }}_i},\;\;\;\;i = h,\alpha ,\beta$ (8)

3.2 间隙非线性带来的极限环振荡

 ${h_{{\rm{rms}}}} = \sum\limits_{n = {N_1} + 1}^{{N_2}} {\sqrt {{{\left( {h\left( {{t_n}} \right) - {h_{{\rm{avg}}}}} \right)}^2}/\left( {{N_2} - {N_1}} \right)} /\left( {b\delta } \right)}$
 ${\alpha _{{\rm{rms}}}} = \sum\limits_{n = {N_1} + 1}^{{N_2}} {\sqrt {{{\left( {\alpha \left( {{t_n}} \right) - {\alpha _{{\rm{avg}}}}} \right)}^2}/\left( {{N_2} - {N_1}} \right)} /\delta }$
 ${\beta _{{\rm{rms}}}} = \sum\limits_{n = {N_1} + 1}^{{N_2}} {\sqrt {{{\left( {\beta \left( {{t_n}} \right) - {\beta _{{\rm{avg}}}}} \right)}^2}/\left( {{N_2} - {N_1}} \right)} /\delta }$

 Download: 图 5 振荡频率及各模态位移均方根的试验结果与计算结果对比 Fig. 5 Comparison of the oscillation frequency and the mean square of the modal displacement with the experiment results

 Download: 图 6 后缘操纵面偏角的分岔 Fig. 6 Bifurcation of skew angle of trailing edge control surface
 Download: 图 7 来流速度比V/Vf=0.27时后缘操纵面偏角的相图和频谱能量分布 Fig. 7 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.27
 Download: 图 8 来流速度比V/Vf=0.40时后缘操纵面偏角的相图和频谱能量分布 Fig. 8 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.40
 Download: 图 9 来流速度比V/Vf=0.49时后缘操纵面偏角的相图和频谱能量分布 Fig. 9 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.49
 Download: 图 10 来流速度比V/Vf=0.52时后缘操纵面偏角的相图和频谱能量分布 Fig. 10 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.52
 Download: 图 11 来流速度比V/Vf=0.55时后缘操纵面偏角的相图和频谱能量分布 Fig. 11 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.55
 Download: 图 12 来流速度比V/Vf=0.60时后缘操纵面偏角的相图和频谱能量分布 Fig. 12 Phase diagram and spectral energy distribution of the skew angle of the trailing edge control surface of velocity at V/Vf=0.60
3.3 间隙非线性影响分析

 ${\tau _\beta } = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {\tau _0},\\ {\tau _0} + {K_\beta }\left( {\beta - \delta } \right), \end{array}&\begin{array}{l} \left| \beta \right| \le \delta \\ \left| \beta \right| > \delta \end{array} \end{array}} \right.$ (10)

 $\beta = {\beta _0} + {\beta _m}\cos \left( {\omega t} \right)$ (11)

βm/δ=2.0，Kβ=3.0为例，操纵面偏角和恢复力矩的变化如图 15所示，随着操纵面偏角β离开间隙非线性范围(虚线范围内)的增加，等效刚度显著增加，图 15(a)(b)(c)的等效刚度比Keq/Kβ分别为0.391，0.500和0.804。由图 16可见，在偏角振幅βm较小时，τ0可以使后缘操纵面的等效刚度显著增加，从而提高颤振速度。

 Download: 图 15 不同初始操纵面偏角β0对应的恢复力矩 Fig. 15 The restoring moments corresponding to the initial control surface deflection angle β0
 Download: 图 16 在不同初始偏角β0下等效刚度比Keq/Kβ随着操纵面偏角振幅βm的变化情况 Fig. 16 The equivalent stiffness ratio Keq/Kβ varies with the amplitude of the deflection angle βm of the control surface at different initial skew angles of β0

Ma=0.20、δ=0.39°时，预加载τ0分别为0.5Kβδ和1.0Kβδ的后缘操纵面偏角分岔图如图 17所示。结果表明:预加载τ0可以显著提高颤振速度，分别提升至无间隙非线性系统的75%和98%。

 Download: 图 17 Ma=0.20，δ=0.39°时不同预加载的操纵面偏角分岔图 Fig. 17 The bifurcation of different preloaded control surface deflection angle at Ma=0.21, δ=0.39°

Ma=0.80、δ=0.39°时，预加载τ0分别为0.2Kβδ和0.5Kβδ的后缘操纵面偏角分岔图如图 18所示，颤振速度分别提升至无间隙非线性系统的80%和98%。此外，初始扰动对预加载系统的颤振速度产生较大影响，如图 19所示，将初始扰动增大10倍和100倍后，系统的颤振速度由0.98Vf下降至0.8和0.75。因此，需要根据实际情况施加预加载来减弱间隙非线性的影响，但是预加载的不足之处在于消耗了最大承载力和响应速度，增加了飞行控制系统设计的复杂性。

 Download: 图 18 Ma=0.80，δ=0.39°时不同预加载的操纵面偏角分岔图 Fig. 18 The bifurcation of different preloaded control surface deflection angle at Ma=0.80, δ=0.39°
 Download: 图 19 Ma=0.80，δ=0.39°，τ0=0.5Kβδ时不同初始扰动的操纵面偏角分岔图 Fig. 19 The bifurcation of different initial disturbance of control surface deflection angle at Ma=0.80, δ=0.39°, τ0=0.5Kβδ
4 结论

1) 分析了低速线性和间隙非线性系统的颤振特性，并与实验结果进行对比，证明了本文颤振时域分析方法的可行性。

2) 通过对不同来流速度下气弹响应的时域计算，得到了间隙非线性影响下产生的5种典型的响应，并通过频谱分析得到了各响应的主要作用频率。

3) 探讨了间隙大小对亚音速和跨声速阶段非线性系统分岔特性的影响，亚音速情况下由于不存在激波的影响，系统主要受到间隙非线性的影响，分岔图基本一致。跨音速阶段由于激波影响，会使得分岔图有明显的变化。

4) 研究了通过预加载措施提高颤振速度可行性，通过预加载可以使得后缘操纵面等效刚度增加，提高系统颤振速度。

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