# Optimal Design and Experimental Verification of Ultrasonic Cutting Horn for Ceramic Composite Material

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## Abstract

**:**

## 1. Introduction

## 2. Initial Design and Finite Element Analysis of an Ultrasonic Cutting Horn

#### 2.1. Conceptual Design of a Sonotrode

#### 2.2. Finite Element Analysis

_{axial}, f

_{axial−1}, f

_{axial+1}, f

_{1st isolation}and f

_{2nd isolation}represent the axial mode frequency, one lower mode frequency, one higher mode frequency, the 1st frequency isolation and the 2nd frequency isolation. Figure 3 shows the mode shape of the axial mode and one lower and higher mode. In the axial mode, the horn vibrates along the axial direction. The one lower and higher mode of the sonotrode are the bending motion of the horn lower part and blade. The one lower mode shape is the 2nd bending mode, where both ends of the blade deform in the y-axis direction, and the one higher mode shape is the 3rd bending mode where both ends and the center of the blade deform in the y-axis direction. The axial frequency, the 1st frequency isolation and the 2nd frequency isolation were 28,520 Hz, 1588 Hz and 864 Hz, respectively.

_{low})

_{min}, (u

_{low})

_{max}and (u

_{upp})

_{max}represent the displacement uniformity, gain, minimum displacement at the horn lower face, maximum displacement at the horn lower face and maximum displacement at the horn upper face, respectively. Under the analysis conditions, the displacement of the shaft was constrained along the x- and y-axes, and the axial mode frequency was used as the excitation frequency. Figure 4 shows the calculated displacement along the horn lower face. The maximum displacement, minimum displacement, displacement uniformity and gain were 16.80 μm, 13.76 μm, 1.07 and 81.9%, respectively. These dynamic characteristics (natural frequency, frequency isolation, displacement) are changed with changes in the design variables, so it is very hard to decide the specific values of design variables using the trial and error method. Therefore, we used the optimal design method to derive the best horn model.

## 3. Optimization of the Developed Ultrasonic Cutting Horn

#### 3.1. Sensitivity Analysis

#### 3.2. Optimization

_{1}, α

_{2}and α

_{3}represent the weight factors of the output parameters, and we set each of these values as 0.333, 1 and 0.333, respectively. The weight factor of the f

_{2nd isolation}is greater than the others in order for both frequency isolations to be at the same level. (f

_{isolation})

_{lower_limit}and (a)

_{lower_limit}represent the lower limits of the frequency isolations and displacement uniformity, and these values were set as 1200 and 84. X

_{i}, (X

_{i})

_{lower_limit}and (X

_{i})

_{upper_limit}represent the design variable, upper and lower limits of the design variables. The input parameters are the design variables used in the sensitivity analysis. The optimal design problem was solved using the multi-objective genetic algorithm. In addition, finite element analysis of the optimal model was performed under the same analysis conditions as the analysis of the initial model, and the variation of each output parameter was confirmed. Table 3 shows the input and output parameters of the initial and optimal model. In the optimal model, the x-axis position and width of the slot were smaller than the initial model. Additionally, the thickness of the lower part was the same as the initial model, and the z-axis position of the slot, the thickness of middle bar, and the thickness of the side bar were larger than the initial values. Table 4 shows the output parameters of the optimal and initial models. Figure 8 shows the displacement at the lower face of each model. The optimal model has a maximum and minimum displacement greater than the initial model, by 0.75 μm and 1 μm, respectively. In addition, the difference between the maximum and minimum displacement was 0.25 μm smaller than the initial model. The 1st and 2nd frequency isolations were 1189 Hz and 1181 Hz, respectively. The 1st frequency isolation was 358 Hz less than the initial model, and the 2nd frequency isolation increased by 403 Hz. Both the 1st and 2nd frequency isolations of the optimal model were greater than 1000 Hz.

## 4. Fabrication and Experimental Verification

#### 4.1. Displacement Measurement Experiment

#### 4.2. Cutting Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Mode shape and natural frequency of the (

**a**) one lower mode (f

_{axial−1}), (

**b**) axial mode (f

_{axial}) and (

**c**) one higher mode (f

_{axial+1}).

**Figure 6.**The 1st order trend lines of the frequency isolations according to the variation of the design parameters of the ultrasonic cutting horn.

**Figure 7.**The 1st order trend lines of the displacement uniformity and gain according to the variation of the design parameters of the ultrasonic cutting horn.

**Figure 9.**(

**a**) Experimental set-up for a displacement measurement experiment and (

**b**) fabricated optimal model and measuring points.

**Figure 12.**SEM images of cutting surface (

**a**) without the application of ultrasonic vibration and (

**b**) with the application of ultrasonic vibration.

Component | Material | Young’s Modulus [GPa] | Density [kg/m^{3}] | Poisson’s Ratio |
---|---|---|---|---|

Horn | Al7075 | 71.9 | 2802 | 0.34 |

Blade | WC | 500 | 13,834 | 0.2 |

Shaft | SCM435 | 205 | 6287 | 0.29 |

Parameter Name | Symbol | Lower Boundary | Upper Boundary |
---|---|---|---|

x-axis position of slot [mm] | sp_{x} | 18.4 | 23.4 |

z-axis position of slot [mm] | sp_{z} | 42.0 | 49.0 |

Width [mm] | w | 61.5 | 64.5 |

Thickness of side part [mm] | th_{sid} | 21.5 | 25.0 |

Thickness of middle part [mm] | th_{mid} | 18.0 | 19.5 |

Thickness of lower part [mm] | th_{low} | 16.0 | 16.5 |

Parameter Name | Symbol | Initial Model | Optimal Model |
---|---|---|---|

x-axis position of slot [mm] | sp_{x} | 23.4 | 21.3 |

z-axis position of slot [mm] | sp_{z} | 42.0 | 43.4 |

Width [mm] | w | 64.5 | 63.5 |

Thickness of side part [mm] | th_{sid} | 22.0 | 22.7 |

Thickness of middle part [mm] | th_{mid} | 19.0 | 19.5 |

Thickness of lower part [mm] | th_{low} | 16.0 | 16 |

Parameter Name | Initial Model | Optimal Model | |
---|---|---|---|

Displacement [μm] | Maximum | 16.80 | 17.55 |

Minimum | 13.76 | 14.76 | |

Axial mode frequency [Hz] | 28,520 | 28,478 | |

One lower mode frequency [Hz] | 26,932 | 27,289 | |

One higher mode frequency [Hz] | 29,366 | 29,659 | |

1st frequency isolation [Hz] | 1588 | 1189 | |

2nd frequency isolation [Hz] | 846 | 1181 |

Displacement [μm] | Axial Mode Frequency [Hz] | ||
---|---|---|---|

Maximum | Minimum | ||

Simulation | 17.55 | 14.76 | 28,478 |

Experiment | 17.56 | 15.30 | 28,480 |

Difference [%] | 0.06 | 3.52 | 0.01 |

Test # | Cutting Force [N] | |
---|---|---|

Ultrasonic Cutting | Conventional Cutting | |

1 | 76.3 | 235.6 |

2 | 77.9 | 240.1 |

3 | 82.4 | 228.9 |

4 | 75.5 | 232.5 |

5 | 76.9 | 234.8 |

6 | 78.4 | 239.4 |

7 | 80.4 | 229.8 |

8 | 81.2 | 234.5 |

9 | 76.8 | 231.2 |

10 | 78.8 | 233.0 |

average | 78.5 | 234.0 |

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**MDPI and ACS Style**

Hahn, M.; Cho, Y.; Jang, G.; Kim, B.
Optimal Design and Experimental Verification of Ultrasonic Cutting Horn for Ceramic Composite Material. *Appl. Sci.* **2021**, *11*, 1954.
https://doi.org/10.3390/app11041954

**AMA Style**

Hahn M, Cho Y, Jang G, Kim B.
Optimal Design and Experimental Verification of Ultrasonic Cutting Horn for Ceramic Composite Material. *Applied Sciences*. 2021; 11(4):1954.
https://doi.org/10.3390/app11041954

**Chicago/Turabian Style**

Hahn, Mibbeum, Yeungjung Cho, Gunhee Jang, and Bumcho Kim.
2021. "Optimal Design and Experimental Verification of Ultrasonic Cutting Horn for Ceramic Composite Material" *Applied Sciences* 11, no. 4: 1954.
https://doi.org/10.3390/app11041954