# Multidisciplinary Design and Optimization of Variable Camber Wing with Non-Equal Chord

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of Trailing Edge Deflection Mechanism

## 3. Multidisciplinary Optimization Framework for Non-Equal Chord Morphing Wing

## 4. Comprehensive Optimization of Aerodynamics and Honeycomb Structure Parameters

#### 4.1. Aerodynamic Model of the Wing

_{1}and N

_{2}are the parameters controlling the shape of airfoil and the coordinate of control points is ${x}_{i+1}=\frac{{N}_{1}+i}{{N}_{1}+{N}_{2}+n}$, i = 0, 1, 2, …, n.

_{1}= 0.5, N

_{2}= 1 and n = 6. The thickness of the reference airfoil at the trailing edge control point was changed based on the CST method to generate trailing edge deflection airfoil 2, 3 and 4. The flight parameters and wing states under the four flight conditions are described in Table 1.

_{Df}and shape drag coefficient C

_{Dform}were calculated using the Friction program (the entire flow was solved as turbulent flow in this paper), and induced drag coefficient C

_{Di}was calculated using Cart3D. In this paper, the lift–drag ratio K is given as follows:

#### 4.2. Equivalent Parameters of Honeycomb Structure with Zero Poisson’s Ratio

#### 4.3. Comprehensive Optimization

- (1)
- Condition A

- (2)
- Condition D

_{L}.

- (3)
- Condition B

- (4)
- Condition C

_{1}.

## 5. Non-Equal Chord Wing Structure Model

#### 5.1. Wing Structure Design

#### 5.2. Definition of Structural Optimization Problems

_{ij}and LS

_{ij}; the thickness of the front and rear spar web in each zone of the inner and outer segment wing, denoted as FSW

_{ij}and BSW

_{ij}; the cross-sectional area of the front and back spar rod in each zone of the inner and outer segment wing, denoted as FSR

_{ij}and BSR

_{ij}; the thickness of the common rib web of the inner and outer segment wing, denoted as RW

_{i}(i = 1~2); the cross-sectional area of the common rib rod of the inner and outer segment wing, denoted as RR

_{i}(i = 1~2); the thickness of the strengthened rib web of the inner segment wing, denoted as SRW

_{1}; the cross-sectional area of the strengthened rib rod of the inner segment wing, denoted as SRR

_{1}; the thickness of the upper and lower flexible skin of the trailing edge of the outer segment wing, denoted as UHS

_{j}and LHS

_{j}(j = 1~4); the thickness of the upper and lower skin and the chord and span of the wedge, denoted as JPUS, JPLS, JPX1, JPX2 and JPY; the cross-section radius of the circle beam in each section of each curved beam, denoted as CB

_{mn}(m = 1~5, n = 1~4); and the thickness of each planar disk on each curved beam, denoted as P

_{mn}(m = 1~5, n = 1~3). There are a total of 84 design variables. Table 3 lists the value range of each design variable.

## 6. Results and Analysis of Multidisciplinary Optimization

- (1)
- Aerodynamic optimization

- (2)
- Optimization of honeycomb structure parameters

- (3)
- Wing structure optimization

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Improved deflection mechanism: (

**a**) top view; (

**b**) the ball is inserted into the spanwise “plate” 2.

**Figure 6.**Initial trailing edge shapes of the four airfoils: (

**a**) NACA 2412 reference airfoil; (

**b**) shape of trailing edge.

**Figure 7.**Accordion honeycomb structure parameters [24].

**Figure 17.**Stress of aluminum wing structure (“+2.5 G” in condition A): (

**a**) shell element; (

**b**) rod element; (

**c**) beam element; (

**d**) planar disk.

**Figure 20.**Z-direction displacement (condition D): (

**a**) global deformation; (

**b**) deformation of flexible skin.

**Figure 21.**Stress of aluminum wing structure (condition D): (

**a**) shell element; (

**b**) rod element; (

**c**) beam element.

Condition | Flight Altitude | Mach Number Ma Flight Speed V | Angle of Attack | Flight State | Airfoil of Outer Wing |
---|---|---|---|---|---|

A | 1 km | Ma = 0.3 | 0° | Cruise | The left and right sides are airfoil 1 |

B | 1 km | Ma = 0.2 | 0° | Cruise | The left and right sides are airfoil 2 |

C | 1 km | Ma = 0.2 | 0° | Roll | One side is airfoil 3; the other side is airfoil 4 |

D | 0 km | V = 40 m/s | 13° | Take-off and landing | The left and right sides are airfoil 4 |

Curved Beam Number | Distance/mm |
---|---|

1 | 25 |

2 | 150 |

3 | 300 |

4 | 450 |

5 | 575 |

Design Variable | Lower Limit | Upper Limit |
---|---|---|

US_{ij}, LS_{ij} | 0.5 mm | 4 mm |

FSW_{ij}, BSW_{ij} | 0.5 mm | 5 mm |

FSR_{ij}, BSR_{ij} | 10 mm^{2} | 50 mm^{2} |

RW_{1}, RW_{2} | 0.5 mm | 5 mm |

RR_{1}, RR_{2} | 10 mm^{2} | 30 mm^{2} |

SRW_{1} | 1 mm | 5 mm |

SRR_{1} | 20 mm^{2} | 50 mm^{2} |

UHS_{1}, LHS_{1} | 2 mm | 5 mm |

UHS_{2}, LHS_{2} | 2 mm | 5 mm |

UHS_{3}, LHS_{3} | 2 mm | 3 mm |

UHS_{4}, LHS_{4} | 1 mm | 2 mm |

JPUS, JPLS | 0.5 mm | 1 mm |

JPX1, JPX2, JPY | 0.5 mm | 2 mm |

CB_{m}_{1} | 0.5 mm | 4 mm |

CB_{m}_{2} | 0.5 mm | 3 mm |

CB_{m}_{3} | 0.5 mm | 2 mm |

CB_{m}_{4} | 0.5 mm | 1.5 mm |

P_{m}_{1} | 0.5 mm | 4 mm |

P_{m}_{2} | 0.5 mm | 3 mm |

P_{m}_{3} | 0.5 mm | 2 mm |

h/mm | b1/mm | b2/mm | t1/mm | t2/mm | t3/mm | |
---|---|---|---|---|---|---|

stringer 1 | 6 | 5 | 5 | 0.5 | 0.5 | 0.5 |

stringer 2 | 4 | 4 | 4 | 0.5 | 0.5 | 0.5 |

stringer 3 | 3 | 3 | 3 | 0.5 | 0.5 | 0.5 |

Flight Condition | Performance Index | Constraint Condition |
---|---|---|

condition A (0.3 Ma cruise) | displacement of wing tip in Z-direction/mm | ≤75 |

stress of wing structure using duralumin/MPa | ≤390 | |

strain of flexible skin at trailing edge of outer segment wing/με | ≤99,109 | |

first-order instability factor | ≥0.8 | |

condition D (take-off and landing) | displacement of wing tip in Z-direction/mm | ≤75 |

stress of wing structure using duralumin/MPa | ≤390 | |

strain of flexible skin at trailing edge of outer segment wing/με | ≤99,109 | |

displacement of curved beam end in Z-direction/mm | ≤9.21 | |

stress of curved beam and planar disk/MPa | ≤390 |

Variable | Initial Value | Optimization Result | Value Range |
---|---|---|---|

${u}_{1}$ | 0.0473 | 0.0473 | 0.0453~0.0493 |

${u}_{2}$ | 0.0266 | 0.0267 | 0.0246~0.0286 |

${l}_{1}$ | −0.0193 | −0.0193 | −0.0213~−0.0173 |

${l}_{2}$ | −0.0104 | −0.0104 | −0.0124~−0.0084 |

$K$ | 18.67 | 18.74 |

Variable | Initial Value | Optimization Result | Value Range |
---|---|---|---|

${u}_{1}$ | 0.0463 | 0.0462 | 0.0443~0.0483 |

${u}_{2}$ | 0.0192 | 0.0206 | 0.0172~0.0212 |

${u}_{3}$ | −0.0080 | −0.0064 | −0.0100~−0.0060 |

${l}_{1}$ | −0.0198 | −0.0194 | −0.0218~−0.0178 |

${l}_{2}$ | −0.0180 | −0.0180 | −0.0200~−0.0160 |

${Z}_{t}$ | −0.0620 | −0.0637 | −0.0640~−0.0600 |

${C}_{\mathrm{L}}$ | 1.4814 | 1.4997 |

Variable | Initial Value | Optimization Result | Value Range |
---|---|---|---|

${u}_{1}$ | 0.0453 | 0.0455 | 0.0433~0.0473 |

${u}_{2}$ | 0.0192 | 0.0203 | 0.0172~0.0212 |

${u}_{3}$ | 0.0020 | 0.0024 | 0.0000~0.0040 |

${l}_{1}$ | −0.0201 | −0.0199 | −0.0221~−0.0181 |

${l}_{2}$ | −0.0160 | −0.0164 | −0.0180~−0.0140 |

${Z}_{t}$ | −0.0200 | −0.0214 | −0.0220~−0.0180 |

$K$ | 22.79 | 22.42 |

Variable | Initial Value | Optimization Result | Value Range |
---|---|---|---|

${u}_{1}$ | 0.0493 | 0.0485 | 0.0473~0.0513 |

${u}_{2}$ | 0.0402 | 0.0387 | 0.0382~0.0422 |

${u}_{3}$ | 0.0390 | 0.0380 | 0.0370~0.0410 |

${l}_{1}$ | −0.0163 | −0.0162 | −0.0183~−0.0143 |

${l}_{2}$ | 0.0036 | 0.0023 | 0.0016~0.0056 |

${Z}_{t}$ | 0.0460 | 0.0476 | 0.0440~0.0480 |

${C}_{1}$ | 0.7520 | 0.7742 |

Variable | Initial Value | Optimization Result | Value Range |
---|---|---|---|

$h$ | 1.2 | 1.4872 | 0.5~1.5 |

$g$ | 0.5 | 0.5020 | 0.5~1.0 |

$t$ | 0.08 | 0.0799 | 0.01~0.08 |

$\eta $ | 1.0 | 1.5000 | 0.5~1.5 |

${E}_{X}$/MPa | 5.9191 | 2.8346 | |

${E}_{Z}$/MPa | 12,928 | 15,157 | |

$\left[{\epsilon}_{eq}\right]$/με | 68,902 | 99,109 |

Wing Structure | Design Variable | Initial Value | Optimal Value | Wing Structure | Design Variable | Initial Value | Optimal Value |
---|---|---|---|---|---|---|---|

upper skin | US_{11} | 2 mm | 0.993 mm | lower skin | LS_{11} | 2 mm | 0.644 mm |

US_{12} | 1.5 mm | 0.978 mm | LS_{12} | 1.5 mm | 0.5 mm | ||

US_{13} | 1.5 mm | 0.706 mm | LS_{13} | 1.5 mm | 0.5 mm | ||

US_{21} | 1 mm | 0.5 mm | LS_{21} | 1 mm | 0.5 mm | ||

US_{22} | 1 mm | 0.5 mm | LS_{22} | 1 mm | 0.5 mm | ||

front spar web | FSW_{11} | 2 mm | 4.83 mm | rear spar web | BSW_{11} | 2 mm | 1.043 mm |

FSW_{12} | 1.5 mm | 0.508 mm | BSW_{12} | 1.5 mm | 0.5 mm | ||

FSW_{13} | 1.5 mm | 0.5 mm | BSW_{13} | 1.5 mm | 0.5 mm | ||

FSW_{21} | 1 mm | 0.5 mm | BSW_{21} | 1 mm | 0.5 mm | ||

FSW_{22} | 1 mm | 0.5 mm | BSW_{22} | 1 mm | 0.5 mm | ||

front spar rod | FSR_{11} | 30 mm^{2} | 50 mm^{2} | rear spar rod | BSR_{11} | 30 mm^{2} | 50 mm^{2} |

FSR_{12} | 25 mm^{2} | 44.07 mm^{2} | BSR_{12} | 25 mm^{2} | 10 mm^{2} | ||

FSR_{13} | 25 mm^{2} | 10.3 mm^{2} | BSR_{13} | 25 mm^{2} | 10 mm^{2} | ||

FSR_{21} | 20 mm^{2} | 10 mm^{2} | BSR_{21} | 20 mm^{2} | 10 mm^{2} | ||

FSR_{22} | 20 mm^{2} | 10 mm^{2} | BSR_{22} | 20 mm^{2} | 10 mm^{2} | ||

rib web | RW_{1} | 1 mm | 0.5 mm | rib rod | RR_{1} | 20 mm^{2} | 10 mm^{2} |

RW_{2} | 1 mm | 0.5 mm | RR_{2} | 20 mm^{2} | 10 mm^{2} | ||

SRW_{1} | 2 mm | 1.016 mm | SRR_{1} | 30 mm^{2} | 20 mm^{2} | ||

wedge | JPUS | 1 mm | 0.5 mm | wedge | JPX1 JPX2 | 1 mm 1 mm | 0.997 mm 0.998 mm |

JPLS | 1 mm | 0.5 mm | |||||

JPY | 1 mm | 0.71 mm | |||||

upper flexible skin | UHS_{1} | 3.5 mm | 2 mm | lower flexible skin | LHS_{1} | 3.5 mm | 2 mm |

UHS_{2} | 3.5 mm | 2 mm | LHS_{2} | 3.5 mm | 2 mm | ||

UHS_{3} | 2.5 mm | 2 mm | LHS_{3} | 2.5 mm | 2 mm | ||

UHS_{4} | 1.5 mm | 1 mm | LHS_{4} | 1.5 mm | 1 mm | ||

curved beam 1 | CB_{11} | 2 mm | 2.129 mm | disk on curved beam 1 | P_{11}P _{12}P _{13} | 3 mm 2 mm 1 mm | 0.5 mm 0.5 mm 0.537 mm |

CB_{12} | 1.5 mm | 2.078 mm | |||||

CB_{13} | 1 mm | 1.495 mm | |||||

CB_{14} | 1 mm | 1.199 mm | |||||

curved beam 2 | CB_{21} | 2 mm | 2.413 mm | disk on curved beam 2 | P_{21}P _{22}P _{23} | 3 mm 2 mm 1 mm | 0.5 mm 0.5 mm 0.539 mm |

CB_{22} | 1.5 mm | 2.413 mm | |||||

CB_{23} | 1 mm | 1.695 mm | |||||

CB_{24} | 1 mm | 1.358 mm | |||||

curved beam 3 | CB_{31} | 2 mm | 2.336 mm | disk on curved beam 3 | P_{31}P _{32}P _{33} | 3 mm 2 mm 1 mm | 0.5 mm 0.5 mm 0.541 mm |

CB_{32} | 1.5 mm | 2.345 mm | |||||

CB_{33} | 1 mm | 1.656 mm | |||||

CB_{34} | 1 mm | 1.341 mm | |||||

curved beam 4 | CB_{41} | 2 mm | 2.256 mm | disk on curved beam 4 | P_{41}P _{42}P _{43} | 3 mm 2 mm 1 mm | 0.5 mm 0.5 mm 0.544 mm |

CB_{42} | 1.5 mm | 2.218 mm | |||||

CB_{43} | 1 mm | 1.592 mm | |||||

CB_{44} | 1 mm | 1.282 mm | |||||

curved beam 5 | CB_{51} | 2 mm | 1.823 mm | disk on curved beam 5 | P_{51}P _{52}P _{53} | 3 mm 2 mm 1 mm | 0.5 mm 0.5 mm 0.548 mm |

CB_{52} | 1.5 mm | 1.775 mm | |||||

CB_{53} | 1 mm | 1.272 mm | |||||

CB_{54} | 1 mm | 0.982 mm |

Flight Condition | Performance Index | Optimization Result |
---|---|---|

condition A (0.3 Ma cruise) “+2.5 G” | displacement of wing tip in Z-direction/mm | 72.17 |

stress of wing structure using duralumin/MPa | 387.7 | |

strain of flexible skin at trailing edge of outer segment wing/με | 18,200 | |

first-order instability factor | 0.80 | |

condition D (take-off and landing) | displacement of wing tip in Z-direction/mm | 51.6 |

stress of wing structure using duralumin/MPa | 256.4 | |

strain of flexible skin at trailing edge of outer segment wing/με | 81,770 | |

displacement of curved beam end in Z-direction/mm | 9.14 | |

stress of curved beam and planar disk/MPa | 390 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Li, X.; Wu, T.; Yin, H.
Multidisciplinary Design and Optimization of Variable Camber Wing with Non-Equal Chord. *Aerospace* **2023**, *10*, 336.
https://doi.org/10.3390/aerospace10040336

**AMA Style**

Wang Y, Li X, Wu T, Yin H.
Multidisciplinary Design and Optimization of Variable Camber Wing with Non-Equal Chord. *Aerospace*. 2023; 10(4):336.
https://doi.org/10.3390/aerospace10040336

**Chicago/Turabian Style**

Wang, Yu, Xiang Li, Tingjia Wu, and Hailian Yin.
2023. "Multidisciplinary Design and Optimization of Variable Camber Wing with Non-Equal Chord" *Aerospace* 10, no. 4: 336.
https://doi.org/10.3390/aerospace10040336