# Three-Dimensional Finite Element Modeling of Ultrasonic Vibration-Assisted Milling of the Nomex Honeycomb Structure

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## Abstract

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## 1. Introduction

## 2. Finite Element Model

#### 2.1. Material Parameters

#### 2.2. Law of Behavior Applied

^{el}and ε

^{p}represent the elastic and plastic strain tensor, respectively.

#### 2.3. Chip Separation Criterion

^{f}indicates the total strain at which the material experiences damage.

#### 2.4. Finite Element Modeling

_{x}= U

_{y}= U

_{z}= 0) along the X, Y, and Z axes as well as any rotation around these axes (U

_{Rx}= U

_{Ry}= U

_{Rz}= 0), as shown in Figure 6b. The difference between conventional milling and ultrasonic vibration-assisted axial milling is the introduction of ultrasonic vibrations along the Z axis at the end of the milling cutter. The ultrasonic rotary machining (RUM) process represents an approach in mechanical manufacturing that involves the synchronization of three types of motions. These motions include translation of the tool along the axis OY, characterized by the feed rate V

_{f}, rotation of the cutting tool around the axis OZ, expressed by the spindle speed n, and the vibration of the tool along the OZ axis, generating a sinusoidal ultrasonic wave (see Figure 6). To guarantee precise monitoring of the milling simulation process, a reference point denoted RP, was assigned along the axis of revolution of the cutting tool in accordance with the representation in Figure 5b. This point assumes an essential role in assigning cutting parameters and in assessing the cutting forces acting throughout the milling operation. In order to describe the motion of the cutting tool, the following equations are used to define its global coordinates xyz:

_{c}stands for the cutting speed; and f denotes the vibration frequency, which is fixed at 21.26 KHz.

#### 2.5. Components of the Cutting Force

_{y}and F

_{x}, are evaluated using the KISTLER-9256C2 dynamometer. The advantage of this technique lies in its ability to determine the average values in both directions using these formulas:

_{x}and F

_{y}represent the average values of the cutting force components along the X and Y axes.

## 3. Results and Discussion

#### 3.1. Mesh Size Study

_{x}and F

_{y}. However, the cutting force components recorded a weak decrease between sizes 0.7 mm and 0.8 mm, followed by a sharp increase between sizes 0.8 mm and 0.9 mm. The noticed variation can be credited to the elastic-restoring forces of the NHC core walls. Thus, as soon as the element reaches its breaking point, it is deleted, which results in a loss of contact between the wall of the structure and the milling cutter, leading to a reduction in the F

_{x}and F

_{y}components. Effectively, the force resulting from spring back due to the uncut walls resists the rotation of the tool, thereby increasing the cutting force components. Despite this, small elements have a lower capacity to resist deformation. Therefore, mesh distortion issues manifest themselves more quickly, which can lead to calculation failures. In principle, a mesh size of 0.2 mm seems to provide an ideal compromise between the accuracy of the results and the efficiency of the calculation. However, it was essential to re-evaluate this approach and adapt it to a size of 0.4 mm in order to achieve the best balance between mesh quality and the calculation time. The objective of this adaptation is to guarantee an appropriate resolution of physical phenomena while preventing the deformation of the elements.

#### 3.2. Influence of Cutting Width on Cutting Force Components

_{x}and F

_{y}components when milling the NHC core. This analysis was performed using RUM technology regardless of whether or not ultrasonic vibrations were applied. To carry out this study, four cutting width values were examined, including 4 mm, 6 mm, 8 mm, and 10 mm. Other cutting parameters remained constant throughout the analysis, including a feed rate of 2000 mm/min, a spindle speed of 3000 rpm, and an amplitude vibration is 25 µm. The obtained results are illustrated in Figure 8 [20].

_{x}and F

_{y}components, revealing a significant increase in these components as a function of the cutting width when milling the NHC structure, regardless of whether ultrasonic vibrations are used or not. Increasing the cutting width causes a distension of the contact surface between the milling cutter and the part, which results in an increase in the volume of material removed per unit of time. Similarly, a large cutting width can result in high friction resistance during the machining process, thereby causing an increase in cutting force components. During the process of cutting the Nomex honeycomb structure using rotary ultrasonic machining, it is clear that the use of ultrasonic vibrations results in a remarkable decrease in the F

_{x}and F

_{y}components, with a reduction reaching up to 42%. In this context, the cutting tool undergoes intense rotation and vibrations, which encourage the formation of cracks in the walls of the NHC structure. This eases the penetration of the tool without encountering resistance from the material constituting the NHC structure. Although increasing the cutting width can improve cutting efficiency, it is also linked to an increase in cutting force components, which can lead to other complications such as premature wear of the cutting tool and degradation of the quality of the machined surface. It is important to determine an optimal value of the cutting width in order to reconcile these two characteristics when machining the NHC core. This optimal value aims to maximize the material removal volume while keeping the cutting force components at a satisfactory level. Thus, this approach guarantees acceptable cutting efficiency and robust numerical results.

#### 3.3. Impact of Vibration Amplitude on Machined Surface

#### 3.4. Analysis of the Distribution of Stresses and Displacements in the Cutting Zone

#### 3.5. The Influence of Vibration Amplitude on the Size of the Chips Generated

## 4. Conclusions

- The study of the impact of the cutting width on the components of the cutting force is carried out, finding a significant increase in these components with increasing cutting width, both in simulations and in experiments. Our results suggest that the use of ultrasonic vibrations helps to mitigate the negative effects of the F
_{x}and F_{y}components in both directions. Furthermore, a significant agreement between the results of the numerical model and the experimental data was observed. - The amplitude of the ultrasonic vibration directly impacts the chip size, leading to a reduction in this one with increasing vibration amplitude.
- The amplitude of the vibration influences the surface quality, leading to an improvement of the latter when the amplitude of the vibrations is increased.
- Applying ultrasonic vibration to the cutting tool induces additional stress in the cutting area of the honeycomb cell wall, accelerating material deterioration while reducing cell wall deformation, facilitating a more efficient milling of the Nomex honeycomb core.
- By continuing this research, it is planned to develop the numerical model by taking into account other parameters in order to detect the burrs that form on the thin walls during the machining process.
- In the industrial context, the optimization of manufacturing processes often requires costly and time-consuming tests to evaluate different configurations. The 3D modeling presented thus offers a considerable advantage in terms of speed, efficiency, and profitability.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Dimensions of the Nomex honeycomb structure (NHC); (

**b**) Dimensions of the alveolar cell [21].

**Figure 2.**UCK cutting tool: (

**a**) UCK used in milling simulation; (

**b**) UCK used in the experiment phase.

**Figure 5.**(

**a**) Mesh of the part (NHC); (

**b**) Mesh of the cutting tool and location of reference point (RP) [21].

**Figure 6.**(

**a**) Milling planar representation of the NHC core; (

**b**) Boundary conditions adopted in the numerical modeling [21].

Mechanical Properties | |
---|---|

Density [g/cm^{3}] | 1.4 |

E [MPa] | 3400 |

Poisson’s ratio | 0.3 |

Yield strengths for simple wall thickness (MPa) | 29 |

Yield strengths for double wall thickness (MPa) | 61 |

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**MDPI and ACS Style**

Zarrouk, T.; Nouari, M.; Salhi, J.-E.; Abbadi, M.; Abbadi, A.
Three-Dimensional Finite Element Modeling of Ultrasonic Vibration-Assisted Milling of the Nomex Honeycomb Structure. *Algorithms* **2024**, *17*, 204.
https://doi.org/10.3390/a17050204

**AMA Style**

Zarrouk T, Nouari M, Salhi J-E, Abbadi M, Abbadi A.
Three-Dimensional Finite Element Modeling of Ultrasonic Vibration-Assisted Milling of the Nomex Honeycomb Structure. *Algorithms*. 2024; 17(5):204.
https://doi.org/10.3390/a17050204

**Chicago/Turabian Style**

Zarrouk, Tarik, Mohammed Nouari, Jamal-Eddine Salhi, Mohammed Abbadi, and Ahmed Abbadi.
2024. "Three-Dimensional Finite Element Modeling of Ultrasonic Vibration-Assisted Milling of the Nomex Honeycomb Structure" *Algorithms* 17, no. 5: 204.
https://doi.org/10.3390/a17050204