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Shape reconstruction of self-adaptive morphing wings' fishbone based on inverse finite element method
ZHANG Ke, YUAN Shenfang, REN Yuanqiang, XU Yuesheng
Research Center of Structure Health Monitoring and Prognosis, State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: Benefiting from the ability of providing reference information for deformation-control system, shape sensing technology is considered as an important way to guarantee the safety and improve the operational performance of self-adaptive morphing structures. However, the conventional optical imaging based shape sensing technologies are unable to meet the need of real-time shape sensing of self-adaptive morphing structures. In this paper, a shape sensing technology based on the inverse Finite Element Method (iFEM) and the idea of superposing segmented displacement is proposed to reconstruct the deformation of morphing wing's major load-bearing structure of fishbone. Firstly, a four-node quadrilateral inverse-shell element is developed based on Mindlin deformation theory for the major load-bearing structure of the morphing wing. Secondly, strain sensors are used to obtain strain distribution of the structure surface as the input of the proposed method. Then the transfer function between the strain field and the displacement field can be obtained by adopting the least square variational equation. Finally, the corresponding displacement of the major load-bearing structure is reconstructed, based on which the reconstruction of wing deformation can be realized. The proposed method is verified through experiments performed on the major load-bearing structure of a morphing wing. The results show that under the deflection angles of 5°, 10°, and 15° of the morphing wing, the reconstructive displacements have a strong consistency with measured displacements, which verifies the feasibility and accuracy of the proposed method.
Keywords: structure shape sensing    shape reconstruction    inverse finite element method    self-adaptive smart structures    morphing wings

1 机翼鱼骨变形重构方法

1.1 鱼骨结构分析

 图 1 自适应变形机翼 Fig. 1 Self-adaptive morphing wing
 图 2 鱼骨结构实物图 Fig. 2 Physical structure of fishbone
 图 3 鱼骨偏转变形 Fig. 3 Deformation of fishbone
1.2 逆向有限元法的变形重构流程

 图 4 逆向有限元算法流程图 Fig. 4 Flowchart of inverse finite element method
1.3 四节点逆向壳单元

 图 5 鱼骨主梁俯视平面及传感器布局 Fig. 5 Vertical view of fishbone and layout of strain sensors

 图 6 四节点逆壳单元 Fig. 6 Four-node quadrilateral inverse-shell element
 $\left\{ {\begin{array}{*{20}{l}} {u = {u_0} + z{\theta _{{y_0}}}}\\ {v = {v_0} - z{\theta _{{x_0}}}}\\ {w = {w_0}} \end{array}} \right.$ （1）

 $\mathit{\boldsymbol{u}}_i^e = {\left[ {\begin{array}{*{20}{l}} {{u_i}}&{{v_i}}&{{w_i}}&{{\theta _{xi}}}&{{\theta _{yi}}}&{{\theta _{zi}}} \end{array}} \right]^{\rm{T}}}$ （2）

 ${\mathit{\boldsymbol{u}}^e} = {\left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{u}}_1^e}&{\mathit{\boldsymbol{u}}_2^e}&{\mathit{\boldsymbol{u}}_3^e}&{\mathit{\boldsymbol{u}}_4^e} \end{array}} \right]^{\rm{T}}}$ （3）

 ${\mathit{\boldsymbol{\varepsilon }}_{\rm{b}}} = {\left[ {\begin{array}{*{20}{l}} {{\varepsilon _{xx}}}&{{\varepsilon _{xy}}}&{{\gamma _{xy}}} \end{array}} \right]^{\rm{T}}}$ （4）
 ${\mathit{\boldsymbol{\varepsilon }}_{\rm{s}}} = {\left[ {\begin{array}{*{20}{l}} {{\gamma _{xz}}}&{{\gamma _{yz}}} \end{array}} \right]^{\rm{T}}}$ （5）

 ${\mathit{\boldsymbol{\varepsilon }}_{\rm{b}}} = \mathit{\boldsymbol{e}}({\mathit{\boldsymbol{u}}^e}) + z\mathit{\boldsymbol{k}}({\mathit{\boldsymbol{u}}^e}) = {\mathit{\boldsymbol{B}}^m}{\mathit{\boldsymbol{u}}^e} + z{\mathit{\boldsymbol{B}}^k}{\mathit{\boldsymbol{u}}^e}$ （6）

 ${\mathit{\boldsymbol{\varepsilon }}_{\rm{s}}} = \mathit{\boldsymbol{g}}({\mathit{\boldsymbol{u}}^e}) = {\mathit{\boldsymbol{B}}^s}{\mathit{\boldsymbol{u}}^e}$ （7）

1.4 基于应变的变形重构方法

 图 7 单元内应变测量结构表面离散应变分布示意图 Fig. 7 Discrete strain measurement on surface of structure inside the element
 $\mathit{\boldsymbol{e}}_j^\varepsilon = \frac{1}{2}\left( {{{\left[ {\begin{array}{*{20}{c}} {\varepsilon _{xx}^ + }\\ {\varepsilon _{yy}^ + }\\ {\gamma _{xy}^ + } \end{array}} \right]}_i} + {{\left[ {\begin{array}{*{20}{c}} {\varepsilon _{xx}^ - }\\ {\varepsilon _{yy}^ - }\\ {\gamma _{xy}^ - } \end{array}} \right]}_i}{\kern 1pt} {\kern 1pt} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,n$ （8）
 $\mathit{\boldsymbol{k}}_j^\varepsilon = \frac{1}{{2h}}\left( {{{\left[ {\begin{array}{*{20}{c}} {\varepsilon _{xx}^ + }\\ {\varepsilon _{yy}^ + }\\ {\gamma _{xy}^ + } \end{array}} \right]}_i} - {{\left[ {\begin{array}{*{20}{c}} {\varepsilon _{xx}^ - }\\ {\varepsilon _{yy}^ - }\\ {\gamma _{xy}^ - } \end{array}} \right]}_i}{\kern 1pt} {\kern 1pt} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,n$ （9）

 ${\mathit{\boldsymbol{\varepsilon }}_j} = \mathit{\boldsymbol{e}}_j^\varepsilon + z\mathit{\boldsymbol{k}}_j^\varepsilon$ （10）

 $\begin{array}{l} {\varPhi _e}({\mathit{\boldsymbol{u}}^e}) = {\left\| {\mathit{\boldsymbol{e}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{e}}^\varepsilon }} \right\|^2} + {\left\| {\mathit{\boldsymbol{k}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{k}}^\varepsilon }} \right\|^2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda {\left\| {\mathit{\boldsymbol{g}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{g}}^\varepsilon }} \right\|^2} \end{array}$ （11）

 ${\left\| {\mathit{\boldsymbol{e}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{e}}^\varepsilon }} \right\|^2} = \frac{1}{n}\iint\limits_{{A^e}} {\sum\limits_{j = 1}^n {{{(\mathit{\boldsymbol{e}}{{({\mathit{\boldsymbol{u}}^e})}_j} - \mathit{\boldsymbol{e}}_j^\varepsilon )}^2}{\rm{d}}x{\rm{d}}y} }$ （12）
 ${\left\| {\mathit{\boldsymbol{k}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{k}}^\varepsilon }} \right\|^2} = \frac{{{{\left( {2h} \right)}^2}}}{n}\iint\limits_{{A^e}} {\sum\limits_{j = 1}^n {{{(\mathit{\boldsymbol{k}}{{({\mathit{\boldsymbol{u}}^e})}_j} - \mathit{\boldsymbol{k}}_j^\varepsilon )}^2}{\rm{d}}x{\rm{d}}y} }$ （13）
 ${\left\| {\mathit{\boldsymbol{g}}({\mathit{\boldsymbol{u}}^e}) - {\mathit{\boldsymbol{g}}^\varepsilon }} \right\|^2} = \frac{1}{n}\iint\limits_{{A^e}} {\sum\limits_{j = 1}^n {{{(\mathit{\boldsymbol{g}}{{({\mathit{\boldsymbol{u}}^e})}_j} - \mathit{\boldsymbol{g}}_j^\varepsilon )}^2}{\rm{d}}x{\rm{d}}y} }$ （14）

 $\frac{\partial }{{\partial {\mathit{\boldsymbol{u}}^e}}}{\varPhi _e}({\mathit{\boldsymbol{u}}^e}) = {\mathit{\boldsymbol{k}}^e}{\mathit{\boldsymbol{u}}^e} - {\mathit{\boldsymbol{f}}^e} = {\bf{0}}$ （15）

 ${\mathit{\boldsymbol{k}}^e}{\mathit{\boldsymbol{u}}^e} = {\mathit{\boldsymbol{f}}^e}$ （16）

 $\begin{array}{l} {\mathit{\boldsymbol{k}}^e} = \iint\limits_{{A^e}} {[{{({\mathit{\boldsymbol{B}}^m})}^{\rm{T}}}{\mathit{\boldsymbol{B}}^m} + {{(2h)}^2}{{({\mathit{\boldsymbol{B}}^k})}^{\rm{T}}}{\mathit{\boldsymbol{B}}^k}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda {({\mathit{\boldsymbol{B}}^s})^{\rm{T}}}{\mathit{\boldsymbol{B}}^s}]{\rm{d}}x{\rm{d}}y \end{array}$ （17）
 $\begin{array}{l} {\mathit{\boldsymbol{f}}^e} = \frac{1}{n}\int\limits_{{A^e}} {\sum\limits_{i = 1}^n {[{{({\mathit{\boldsymbol{B}}^m})}^{\rm{T}}}\mathit{\boldsymbol{e}}_i^\varepsilon + {{(2h)}^2}{{({\mathit{\boldsymbol{B}}^k})}^{\rm{T}}}\mathit{\boldsymbol{k}}_i^\varepsilon } } + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda {({\mathit{\boldsymbol{B}}^s})^{\rm{T}}}\mathit{\boldsymbol{g}}_i^\varepsilon ]{\rm{d}}x{\rm{d}}y \end{array}$ （18）

 ${\mathit{\boldsymbol{K}} = \sum\limits_{e = 1}^{{n_{{\rm{el}}}}} {{{({\mathit{\boldsymbol{T}}^e})}^{\rm{T}}}} {\mathit{\boldsymbol{k}}^e}{\mathit{\boldsymbol{T}}^e}}$ （19）
 ${\mathit{\boldsymbol{F}} = \sum\limits_{e = 1}^{{n_{{\rm{el}}}}} {{{({\mathit{\boldsymbol{T}}^e})}^{\rm{T}}}} {\mathit{\boldsymbol{f}}^e}}$ （20）
 ${\mathit{\boldsymbol{U}} = \sum\limits_{e = 1}^{{n_{{\rm{el}}}}} {{{({\mathit{\boldsymbol{T}}^e})}^{\rm{T}}}} {\mathit{\boldsymbol{u}}^e}}$ （21）

 $\mathit{\boldsymbol{KU}} = \mathit{\boldsymbol{F}}$ （22）

1.5 鱼骨结构分段位移叠加

 ${\sigma = \frac{M}{W}}$ （23）
 ${W = \frac{{b{h^2}}}{6}}$ （24）

 图 8 分叉与主梁结合部分示意图 Fig. 8 Combined part of bifurcate and girder

 $\mathit{\boldsymbol{u}}_m^e = \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{u}}_m^{{N_1}}}&{\mathit{\boldsymbol{u}}_m^{{N_2}}}&{\mathit{\boldsymbol{u}}_m^{{N_7}}}&{\mathit{\boldsymbol{u}}_m^{{N_8}}} \end{array}} \right]$ （25）

 $\mathit{\boldsymbol{u}}_m^{{N_2}} = {\left[ {\begin{array}{*{20}{l}} {{u^{{N_2}}}}&{{v^{{N_2}}}}&{{w^{{N_2}}}}&{\theta _x^{{N_2}}}&{\theta _y^{{N_2}}}&{\theta _z^{{N_2}}} \end{array}} \right]^{\rm{T}}}$ （26）
 $\mathit{\boldsymbol{u}}_m^{{N_7}} = {\left[ {\begin{array}{*{20}{l}} {{u^{{N_7}}}}&{{v^{{N_7}}}}&{{w^{{N_7}}}}&{\theta _x^{{N_7}}}&{\theta _y^{{N_7}}}&{\theta _z^{{N_7}}} \end{array}} \right]^{\rm{T}}}$ （27）

 $\mathit{\boldsymbol{u}}_{m + 1}^e = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}}_m^{{N_2}}}&{\mathit{\boldsymbol{u}}_m^{{N_3}}}&{\mathit{\boldsymbol{u}}_m^{{N_6}}}&{\mathit{\boldsymbol{u}}_m^{{N_7}}} \end{array}} \right]$ （28）

 $\mathit{\boldsymbol{u}}_m^{{N_3}} = \left[ {\begin{array}{*{20}{c}} {{u^{{N_2}}} + {l_1} \cdot \theta _z^{{N_2}}}\\ {{v^{{N_2}}} + {l_2} \cdot \theta _z^{{N_2}}}\\ {{w^{{N_2}}} - {l_1} \cdot \theta _x^{{N_2}} - {l_2} \cdot \theta _y^{{N_2}}}\\ {\theta _x^{{N_2}}}\\ {\theta _y^{{N_2}}}\\ {\theta _z^{{N_2}}} \end{array}} \right]$ （29）
 $\mathit{\boldsymbol{u}}_m^{{N_6}} = \left[ {\begin{array}{*{20}{c}} {{u^{{N_7}}} + {l_1} \cdot \theta _z^{{N_7}}}\\ {{v^{{N_7}}} + {l_2} \cdot \theta _z^{{N_7}}}\\ {{w^{{N_7}}} - {l_1} \cdot \theta _x^{{N_7}} - {l_2} \cdot \theta _y^{{N_7}}}\\ {\theta _x^{{N_7}}}\\ {\theta _y^{{N_7}}}\\ {\theta _z^{{N_7}}} \end{array}} \right]$ （30）

2 变形重构实验

 图 9 鱼骨结构实验装配图 Fig. 9 Assembly diagram of fishbone during experiment

 图 10 鱼骨结构变形重构实验系统 Fig. 10 Experimental system for deformation reconstruction of fishbone

 图 11 鱼骨结构表面应变测量结果 Fig. 11 Results of strain measurement of fishbone surface
3 变形重构结果及分析

 $\mathit{\boldsymbol{f}}_2^e = {\left[ {\begin{array}{*{20}{c}} {2.86 \times {{10}^{ - 4}}}\\ {7.35 \times {{10}^{ - 5}}}\\ \vdots \\ { - 3.58 \times {{10}^{ - 6}}}\\ 0 \end{array}} \right]_{24 \times 1}}$ （31）
 $\begin{array}{l} \mathit{\boldsymbol{k}}_2^e = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left[ {\begin{array}{*{20}{c}} {2.25 \times {{10}^{ - 3}}}&{8.14 \times {{10}^{ - 3}}}& \cdots &{3.69 \times {{10}^{ - 5}}}\\ {5.51 \times {{10}^{ - 3}}}&{6.63 \times {{10}^{ - 3}}}& \cdots &{2.33 \times {{10}^{ - 5}}}\\ \vdots & \vdots &{}& \vdots \\ {4.42 \times {{10}^{ - 3}}}&{3.10 \times {{10}^{ - 3}}}& \cdots &{7.29 \times {{10}^{ - 6}}} \end{array}} \right]_{24 \times 24}} \end{array}$ （32）

 $\mathit{\boldsymbol{U}} = {\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ \vdots \\ {0.082}\\ 0 \end{array}} \right]_{36 \times 1}}$ （33）

 分段编号 u/mm v/mm w/mm θx/rad θy/rad 1 4.045 0 10.264 0.004 0.082 2 3.817 0 7.318 0.004 0.072 3 3.608 0 6.917 0.004 0.060 4 3.330 0 5.420 0.003 0.053

 图 12 重构位移与实测位移对比曲线 Fig. 12 Comparison of reconstructive displacement and measured displacement

 偏转角度/(°) 位移绝对误差/cm 偏转角误差/(°) 5 0.58 0.67 10 1.27 1.46 15 1.73 1.91
4 结论

1) 针对自适应智能结构变形监测的需求，本文提出了一种基于逆向有限元算法和位移分段叠加思想的变形重构方法，实现对航空自适应变形机翼鱼骨结构的变形重构。

2) 实验验证结果表明，自适应变形机翼在分别偏转5°/10°/15°的情况下，机翼末端的位移重构误差不超过1.73 cm，因此验证了该方法在变形机翼这种自适应智能结构的变形重构研究中的有效性和准确性。

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http://dx.doi.org/10.7527/S1000-6893.2019.23617

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#### 文章信息

ZHANG Ke, YUAN Shenfang, REN Yuanqiang, XU Yuesheng

Shape reconstruction of self-adaptive morphing wings' fishbone based on inverse finite element method

Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 223617.
http://dx.doi.org/10.7527/S1000-6893.2019.23617