## 基于交叉簧片式铰链的变弯度机翼机构设计

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1. 哈尔滨工业大学（深圳） 机电工程与自动化学院，深圳 518000

2. 上海宇航系统工程研究所，上海 201108

## Design of wing mechanism with variable camber based on cross-spring flexural pivots

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1. School of Mechanical Engineering and Automation, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518000, China

2. Shanghai Institute of Aerospace System Engineering, Shanghai 201108, China

 基金资助: 深圳市国际合作研究资助项目（GJHZ20170313113529978）

 Fund supported: 深圳市国际合作研究资助项目（GJHZ20170313113529978）

Abstract

In order to design airfoil mechanism with continuous variable camber, a morphing wing structure and modeling analysis was proposed by using flexible trailing edge mechanism and rigid connecting rod driving mechanism. The wing mechanism was based on cross-spring flexural pivots. The theoretical mechanics model of the flexible trailing edge mechanism was established by using chained beam constraint model, and the relationship between the force and deformation of the mechanism was also obtained. Then, compared the theoretical mechanics model with finite element model. On the basis of the mechanical model, In order to improve the aerodynamic characteristics of the wing mechanism, NSGA-II multi-objective genetic algorithm was used to optimize the dimension parameters of the mechanism. After optimization, the lift-drag ratio of the morphing wing in cruise stage is increased by 1.09%, and the lift coefficient in takeoff stage is increased by 2.54%. The deformation precision and deformation range of the airfoil mechanism were tested by experiments.

Keywords： variable camber wing ; flexible mechanism ; chained beam constraint model ; multi-objective optimization ; aerodynamic performance

XU Jun-heng, YANG Xiao-jun, LI Bing. Design of wing mechanism with variable camber based on cross-spring flexural pivots. Journal of Zhejiang University(Engineering Science)[J], 2022, 56(3): 444-451, 509 doi:10.3785/j.issn.1008-973X.2022.03.003

## 1. 变弯度机翼的机构设计

### 图 1

Fig.1   Schematic diagram of wing layout

### 图 2

Fig.2   Overall schematic diagram of variable camber wing

### 图 3

Fig.3   Flexible trailing edge and sub element of wing

### 图 4

Fig.4   Driving mechanism model diagram

### 图 5

Fig.5   Parameter definition of flexible trailing edge sub element

$\theta = \frac{{15\cos \alpha (\lambda f\cos \alpha + m)}}{{(18{\lambda ^2} - 18\lambda + 15\lambda \cos {\alpha ^2} + 2)p + 120\cos \alpha (3{\lambda ^2} - 3\lambda + 1)}}.$

Wittrick[14]在研究交叉簧片式的柔性铰链时，得到结论：当交叉簧片柔性铰链的几何参数λ=87.3%或λ=12.7%时，铰链的轴漂最小，铰链右端活动刚体的运动最接近绕着交叉点的圆周运动. 本研究确定铰链的几何参数λ=12.7%，此时，将柔性铰链的轴漂忽略不计，则刚体中心点绕着交叉点做圆周运动，在XY轴上的位移Δx、Δy为转动角度θ的函数：

$\left. \begin{array}{l} \Delta y = \lambda L\cos \alpha \sin \theta ,\\ \Delta x = \lambda L\cos \alpha (1 - \cos \theta ). \end{array} \right\}$

### 图 6

Fig.6   Stress diagram of flexible trailing edge

### 图 7

Fig.7   Diagram of force and deformation of sub unit

1）后缘子单元模型：

${{\theta _i}{ = }\dfrac{{15\cos {\alpha _i}({\lambda _i}{f_i}\cos {\alpha _i} + {m_i})}}{{(18\lambda _i^2 - 18{\lambda _i} + 15{\lambda _i}\cos {\alpha _i}^2 + 2){p_i} + 120\cos {\alpha _i}(3{\lambda _i}^2 - 3{\lambda _i} + 1)}},}$

$\Delta {y_i} = {\lambda _i}{L_i}\cos {\alpha _i}\sin {\theta _i},$

$\Delta {x_i} = {\lambda _i}{L_i}\cos {\alpha _i}(1 - \cos {\theta _i}).$

LiIiλiαiE分别为第i个子单元所对应的结构几何参数和材料参数，每个子单元模型建立3个标量方程，一共有12个标量方程.

2)静力平衡方程. 对于第i个子单元，如图7所示，存在静力平衡关系：

$\left[ {\begin{array}{*{20}{c}} {{F'_{i - 1}}} \\ {{P'_{i - 1}}} \\ {{M'_{i - 1}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ {\left( {{c_i} + \Delta x_i^{}} \right)}&{ - \Delta y_i^{}}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{F_i}} \\ {{P_i}} \\ {{M_i}} \end{array}} \right] .$

$\begin{split} &\left[ {\begin{array}{*{20}{c}} {\cos {\theta _{i - 1}}}&{ - \sin {\theta _{i - 1}}}&0\\ {\sin {\theta _{i - 1}}}&{\cos {\theta _{i - 1}}}&0\\ 0&{{0_i}}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{F'_{i - 1}}}\\ {{P'_{i - 1}}}\\ {{M'_{i - 1}}} \end{array}} \right]{\rm{ + }}\\ & \qquad {\rm{ }}\left[ {\begin{array}{*{20}{c}} { - (F{u_{i - 1}} + F{l_{i - 1}})\cos {\theta _{i - 1}}}\\ {(F{u_{i - 1}} + F{l_{i - 1}})\sin {\theta _{i - 1}}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{F_{i - 1}}}\\ {{P_{i - 1}}}\\ {{M_{i - 1}}} \end{array}} \right]. \end{split}$

$\begin{split} & \left[ {\begin{array}{*{20}{c}} {\cos {\theta _{i - 1}}}&{ - \sin {\theta _{i - 1}}}&0\\ {\sin {\theta _{i - 1}}}&{\cos {\theta _{i - 1}}}&0\\ {\left( {{c_i} + \Delta {x_i}} \right)}&{ - \Delta {y_i}}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{F_i}}\\ {{P_i}}\\ {{M_i}} \end{array}} \right]{\rm{ + }}\\ & \qquad \left[ {\begin{array}{*{20}{c}} { - (F{u_{i - 1}} + F{l_{i - 1}})\cos {\theta _{i - 1}}}\\ {(F{u_{i - 1}} + F{l_{i - 1}})\sin {\theta _{i - 1}}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{F_{i - 1}}}\\ {{P_{i - 1}}}\\ {{M_{i - 1}}} \end{array}} \right]. \end{split}$

$\begin{split} &\left[ {\begin{array}{*{20}{c}} {\cos {\beta _4}}&{ - \sin {\beta _4}}&0\\ {\sin {\beta _4}}&{\cos {\beta _4}}&0\\ 0&0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{F_o}}\\ {{P_o}}\\ {{M_o}} \end{array}} \right] + \\ & \qquad \left[ {\begin{array}{*{20}{c}} { - \left( {F{u_4} + F{l_4}} \right)\cos {\theta _4}}\\ {\left( {F{u_4} + F{l_4}} \right)\sin {\theta _4}}\\ 0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{F_4}}\\ {{P_4}}\\ {{M_4}} \end{array}} \right]. \end{split}$

3）几何约束方程. 柔性后缘满足几何约束方程

$\sum\nolimits_{i = 1}^4 {\left[ {\left( {{c_i} + \Delta x_i^{}} \right)\cos {\beta _i} - \Delta y_i^{}\sin {\beta _i}} \right]} + {c_{{\text{tip}}}}\cos {\beta _{\text{o}}} = {X_{\text{o},}}$

$\sum\nolimits_{i = 1}^4 {\left[ {\left( {{c_i} + \Delta x_i^{}} \right)\sin {\beta _i} + \Delta y_i^{}\cos {\beta _i}} \right]} + {c_{{\text{tip}}}}\sin {\beta _{\text{o}}} = {Y_{\text{o}}},$

$\sum\nolimits_{i=1}^4 {{\theta _i}} = {\beta _{\text{o}}}.$

### 2.3. 柔性后缘机构模型的实例分析与验证

Tab.1  Shape and geometric parameters of flexible trailing edge sub element

 子单元编号 L/mm T/mm W/mm λ α/（°） 1 55.43 3.000 15 0.127 30 2 44.34 2.400 15 0.127 30 3 35.47 1.920 15 0.127 30 4 28.38 1.536 15 0.127 30

${\text{SE}} = \sum\nolimits_{i=1}^9 {\sqrt {{{({x_i} - x{'_i})}^2} + {{({y_i} - y{'_i})}^2}} }$

### 图 8

Fig.8   Comparison of results when flexible trailing edge subjected to bending moment

### 图 9

Fig.9   Comparison of results when flexible trailing edge subjected to longitudinal load

### 图 10

Fig.10   Flow chart of optimization algorithm

### 图 11

Fig.11   Schematic diagram of optimization results

### 图 12

Fig.12   Airfoil diagram at optimized design point

Tab.2  Optimization design variable results

 飞行阶段 θ1/(°) θ2/(°) θ3/(°) θ4/(°) Fx/N Fy/N 状态1）不考虑气动力 状态1）考虑气动力 状态2）不考虑气动力 状态2）考虑气动力 2.3 1.2 2.0 3.8 0 5.8 2.0 1.6 2.5 2.8 −198 46.0 4.6 2.5 4.0 7.5 0 12.0 4.6 2.6 4.0 9.2 −58 26.0

### 图 13

Fig.13   Driving mechanism diagram

${{\boldsymbol{l}}_{CA}} + {{\boldsymbol{l}}_{AB}} = {{\boldsymbol{l}}_{CB}}.$

${l_{CA}} + {l_{AB}}{{\text{e}}^{{\text{i}}{{{\varphi }}_{\text{1}}}}} = {l_{CB}}{{\text{e}}^{{\text{i}{(\text{π} - }}{{{\varphi }}_{\text{3}}}{\text{)}}}}.$

${l_{CB}}{\text{ = }}\sqrt {{{({l_{AB}}\sin {\varphi _1})}^2} + {{({l_{CA}} - {l_{AB}}\cos {\varphi _1})}^2}} ,$

${\varphi _3}{\text{ = arcsin}}\left(\frac{{{l_{AB}}\sin {\varphi _1}}}{{{l_{CB}}}}\right){\text{.}}$

${D_x} = {l_{CD}}\cos {\varphi _3} + {l_{AC}}\text{，}$

${D_y} = - {l_{CD}}\sin {\varphi _3}.$

### 图 14

Fig.14   Wing trailing edge trajectory

## 4. 变弯度机翼样机实验

### 图 15

Fig.15   Parameter definition of flexible trailing edge sub element

### 图 16

Fig.16   Experimental results of deformation accuracy

### 图 17

Fig.17   Deformation range experimental results

## 5. 结　论

（1）建立柔性后缘机构的力学模型，利用ANSYS对力学模型进行验证，柔性后缘标志点的位移相对误差可控制在2.15%.

（2）根据设计的机构方案和力学模型，开发基于NSGA-II遗传算法的机翼结构优化算法，经优化，机翼巡航阶段的升阻比提升1.09%，起降阶段的升力系数提高2.54%.

（3）搭建机翼实验平台，对机翼样机进行变形精度和变形范围实验。结果表明，机翼在2个设计状态处的外轮廓点坐标理论结果和实验结果的偏差之和分别为5.9、9.6 mm；变弯度机翼机构的等效偏转角为−13.5 °~13.7°，机翼具有较大的变形能力.

（4）在弦向柔性后缘变弯度机翼的研究方向，提出新颖的机构构型、系统的力学建模分析与设计方法，较好地协调了机翼变形能力与承载能力之间的矛盾，实现机翼弦向连续变弯度. 该机构可用于具有固定翼的无人机场景.

（5）本研究仅从翼型的角度出发，对提出的变弯度机翼机构进行了优化和气动仿真分析，对于分析结果，需要进一步进行风洞实验和飞行实验来验证其准确性.

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