# Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Problem

## 3. Design Process and Implementation of Lattice-Filled Structures

#### 3.1. Design Process and Corresponding Theories

#### 3.1.1. Design Process

- (1)
- Finite element model modeling: model the design domain, specify the types and parameters of the filled lattice, calculate the equivalent material parameters of the specified lattice, mesh the design domain, define the material parameters as the calculated equivalent parameters, define boundary conditions, etc.;
- (2)
- Topology optimization: define the objective function, design the variables and constraints of the topology optimization formulation, carry out the topology optimization problem;
- (3)
- Geometric modeling of topology optimized results: choose several control nodes and the surface of the optimized design, build NURBS surfaces surrounding the topology optimized designs, modify the locations of control nodes, add bolt holes;
- (4)
- Lattice filling: define the filling parameters and filling domain, uniformly fill the lattices in the filling domain;
- (5)
- Structural response verification: perform structural analysis; if the performance was not satisfactory, modify the local geometric model or return to (1) to modify the filling parameters.

#### 3.1.2. Finite Element Model Modeling

#### 3.1.3. Topology Optimization

**ρ**is the relative density, C(

**ρ**) is the weighted sum of the compliances of two overload cases, N is the number of elements,

**K**is the global stiffness matrix,

**U**is the structural displacement vector,

**F**is the load vector, v

_{j}is the volume of the j-th element, V* is the value of the effective volume constraint of the structure, f

_{1}(

**ρ**) is the fundamental frequency of the structure, f* is the fundamental frequency of the initial design, which is equal to 995 Hz, ρ

_{min}and ρ

_{max}are the upper and lower limits for the design variable values, and T is the notation for the transpose. In this work, the objective function is the weighted compliance of the structure in the over-load cases.

#### 3.1.4. Geometric Modeling of the Topology Optimized Results

#### 3.1.5. Lattice Filling

#### 3.2. Lightweight Design of Lattice SkinFilling for the Support Structure

^{3}, elastic modulus E = 110 GPa, and Poisson’s ratio v = 0.30.

_{ij}is the equivalent stiffness parameter, $\overline{\rho}$ is the relative density of the lattice cells, $\varpi $ is the angle of the single rod and the horizontal plane, $\delta $ is the diameter of the rod piece, D is the outer cell envelope dimension, and E

_{0}is the elastic modulus of the base material manufactured for the additive.

#### 3.3. Machinability Verification

## 4. Numerical Simulations of the Lattice-Filled Structure

#### 4.1. Numerical Simulation Model

#### 4.2. Static Responses and Discussion

#### 4.3. Dynamic Characteristics and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The considered aircraft support, which is a thin-walled structure with several holes and is made of aluminum alloy, weighing 1.60 kg.

**Figure 2.**Structural responses of the initial design: (

**a**) Displacement distribution in the axial overload case; (

**b**) Displacement distribution in the transverse overload case; (

**c**) Vonmises stress distribution in the axial overload case; (

**d**) Vonmises stress distribution in the transverse overload case.

**Figure 5.**Analysis model in topology optimization; the yellow solid domain is the design domain, and the blue round hole domains are the non-design domains.

**Figure 9.**Lattice-filled design of the aircraft support (inner diameter of 360 mm, outer diameter of 400 mm, height of 100 mm): (

**a**) overall view; (

**b**) partial cross-section view.

**Figure 11.**Detailed FEM model with a total of about 310,000 nodes: (

**a**) entire model; (

**b**) beam elements of lattice and solid elements of bolt holes; (

**c**) internal view.

**Figure 12.**Homogenization analysis model with a total of 85,000 nodes: (

**a**) full model; (

**b**) internal model.

**Figure 13.**VonMises stress distribution: (

**a**) detailed model analysis results; (

**b**) homogeneous model analysis results.

**Table 1.**The equivalent material’s stiffness coefficients of the specified BCC lattice from different methods.

Method | Stiffness Coefficients (MPa) | ||||||||
---|---|---|---|---|---|---|---|---|---|

D_{11} | D_{22} | D_{33} | D_{12} | D_{13} | D_{23} | D_{44} | D_{55} | D_{66} | |

Analytical formulation | 435 | 435 | 435 | 435 | 435 | 435 | 435 | 435 | 435 |

NIAH | 431 | 431 | 431 | 419 | 419 | 419 | 425 | 425 | 425 |

Laser Power/W | Scan Rate/mm | Open Space/mm | Thickness/μm |
---|---|---|---|

120 | 3500 | 0.10–0.19 | 30 |

**Table 3.**Comparison frequencies of the first six modes of the detailed model and the homogenization model.

Mode | Detailed Model | Homogenization Model | Relative Error |
---|---|---|---|

1 | 1508.3 | 1506.5 | 0.1% |

2 | 1508.6 | 1506.6 | 0.1% |

3 | 1522.9 | 1517.1 | 0.3% |

4 | 1526.8 | 1527.7 | 0.0% |

5 | 1527.9 | 1527.7 | 0.0% |

6 | 1665.6 | 1658.9 | 0.4% |

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**MDPI and ACS Style**

Huang, Z.; Liu, Y.; Huang, X.; Mu, D.
Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures. *Mathematics* **2022**, *10*, 4576.
https://doi.org/10.3390/math10234576

**AMA Style**

Huang Z, Liu Y, Huang X, Mu D.
Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures. *Mathematics*. 2022; 10(23):4576.
https://doi.org/10.3390/math10234576

**Chicago/Turabian Style**

Huang, Zhou, Yong Liu, Xin Huang, and Dong Mu.
2022. "Research on the Lightweight Design of an Aircraft Support Based on Lattice-Filled Structures" *Mathematics* 10, no. 23: 4576.
https://doi.org/10.3390/math10234576