## 3.1 Structural parameter design of the UVAIB device

In the process of UVAIB, the horn plays the role of transmitting sound waves and amplifying ultrasonic vibration. Therefore, the design quality of the horn has an important impact on the performance of the UVAIB device. Due to the limitation of the structural size of the brazing platform, an ultrasonic horn with a resonant frequency of 30kHz is developed, which can meet the requirements that the abrasive tool vibrate at the natural frequency when brazing. In addition, the horn with this simple shape machining tool as a load can be equivalent to a multi-step transition section stepped horn that is easy to calculate. A stepped horn is shown in Fig. 4a. The diameter and length of the input section and output section are *D*1, *D*2, and *l*1, *l*2, respectively. The frequency equation of the stepped horn is as follows:

tan(φ1 + φ2) = 0 | (1) |

φ1 = *k*1*l*1 | (2) |

\({\varphi _2}=\arctan \left[ {\frac{{{Z_{02}}}}{{{Z_{01}}}}\tan ({k_2}{l_2})} \right]\) | (3) |

where φ1, φ2 is the shape factor, *k* is the circular wave number, *k* = ω/*c*, ω is the circular frequency, ω = 2π*f*, *f* is the resonant frequency, *c* is the propagation velocity of longitudinal wave in the material, *c*=√*E*/*ρ*, *E* is the elastic modulus, *ρ* is the density of the material, and *Z* is the impedance, Z0*n* = *ρcs**n*, *s**n* is the cross-sectional area of each section.

A three-step transition stepped horn is shown in Fig. 4b. The diameter and length of the input section, transition section and output section are *D*1, *D*2, *D*3 and *l*1, *l*2, *l*3, respectively. According to the long line theory, the stepped horn with multi-step transition section can be regarded as an acoustic transmission line, and the latter section can be regarded as the load impedance of the former section. Consequently, the frequency equation of the three-step transition stepped horn can be obtained by the recursive method as follows:

tan(*k**1**l*1 + *φ*1) = 0 | (4) |

\({\varphi _1}=\arctan \left[ {\frac{{{Z_{02}}}}{{{Z_{01}}}}\tan ({k_{\text{2}}}{l_2}+{\varphi _2})} \right]\) | (5) |

\({\varphi _2}=\arctan \left[ {\frac{{{Z_{03}}}}{{{Z_{02}}}}\tan ({k_{\text{3}}}{l_3})} \right]\) | (6) |

\(\tan \left\{ {{k_{\text{1}}}{l_1}+\arctan \left[ {\frac{{{z_{02}}}}{{{z_{01}}}}\tan ({k_{\text{2}}}{l_2}+\arctan (\frac{{{z_{03}}}}{{{z_{02}}}}\tan ({k_{\text{3}}}{l_3}))} \right]} \right\}=0\) | (7) |

When each section of the horn is made of the same material, *k*1 = *k*2 = *k*3. Based on the above equation, the size of the three-step transition stepped horn can be obtained.

## 3.2 Optimizing designation parameters of the UVAIB device

All components are modeled in the three-dimensional software, as shown in Fig. 5a. Subsequently, the modal analysis of UVAIB device composed of transducer, horn and abrasive tool is carried out with finite element analysis software ANSYS, and the longitudinal vibration mode is extracted, as shown in Fig. 5b. The material properties are defined in Table 1. By adjusting the length of each section of the horn, a suitable structure for the best brazing quality can be obtained. Considering the energy loss in the process of ultrasonic propagation, it is necessary to make the displacement at the connection between transition section and abrasive tool as small as possible to reduce unnecessary friction loss. As shown in Fig. 5b, the growth rate of amplitude on both sides of the last node is different, and the closer it is to the end face of the transition section, the faster the amplitude increases. Therefore, it is hoped that the position of the node is close to the end face of the transition section. That is, the value of *x*, which is equal to *x*2 minus *x*1, should be small. Where *x*1 is the location of the last node, *x*2 is the end face position of the transition section.

In addition, in order to study the influence of ultrasonic amplitude on brazing quality, the designed ultrasonic vibration device should have a large amplification factor MP, which is defined as the ratio of the relative output displacement *ξ*3 to the relative input displacement *ξ*1, so that the amplitude can be adjusted in a large range. Furthermore, in order to obtain a higher quality brazed abrasive tool, the part of the abrasive tool covered with filler alloy should vibrate more evenly as much as possible. That is, the ratio δ of the relative vibration displacement *ξ*2 at brazing starting location to the relative vibration displacement *ξ*3 at the end face of abrasive tool should keep a large value.

Table 1

Material properties of the UVAIB device.

component | Material | Density *ρ* | Elastic modulus *E* | Poisson's ratio *v* |

Horn/Wheel Electrode sheet Front/rear cover plate Piezoelectric ceramics sheet | 316L Beryllium bronze 6061Al Piezoelectric ceramics | 7930 kg/m3 8250 kg/m3 2750 kg/m3 7600 kg/m3 | 200 Gpa 122.6 Gpa 72 Gpa 68 Gpa | 0.29 0.35 0.33 0.3 |

In practice, the size of *D*1 is determined by the transducer end face and the size of *D*3 and *l*3 is determined by the abrasive tool. Therefore, considering the influence of transition section diameter on the performance of horn, the different *D*2 are taken for calculation and the corresponding values of *l*1 and *l*2 can be obtained. According to the calculation results, the relationship between the length of *l*2 and *l*1 is shown in Fig. 6a. When the length of *l*2 increases from 20 mm to 90 mm, the length of *l*1 shows a downward trend. In addition, in the range of 20 mm to 60 mm, the larger the value of *D*2, the faster the length of *l*1 will drop. However, when the length of *l*2 is in the range of 60 mm to 90 mm, the length of *l*1 tends to be the same for different *D*2. After modeling with above values and carrying out the modal analysis, the effect of length of *l*2 on the relative distance of node is obtained, as shown in Fig. 6b. When the length of *l*2 increases from 20 mm to 40 mm, the relative distance of node keeps decreasing. When the length of *l*2 is in the range of 40 mm to 60 mm, the relative distance of the node tends to be stable, which fluctuates only in a small range. After the length of *l*2 reaches 60 mm, the relative distance of the node generally becomes an upward trend. Especially from the overall trend, when the value of *D*2 increases, the relative distance of the node will also increase. Therefore, in order to obtain a small relative distance of the node, the length of *l*2 should be in the range of 40 mm to 60 mm.

As a result, the amplification factor is studied under the above restrictions. As shown in Fig. 6c, when the length of *l*2 increases from 40 mm to 60 mm, the amplification factor for different *D*2 shows a straight upward trend. In addition, as the value of *D*2 increases, the corresponding amplification factor also increases. Therefore, for the consideration of obtaining a large amplification factor, the length of *l*2 is decided to be 60 mm. Besides, combined with Fig. 5b, in order to obtain a small relative distance of the node and a large amplification factor simultaneously, the value of *D*2 should be 19 mm.

It is known that when the length of *l*2 is 60 mm, the corresponding length of *l*1 is about 80 mm. Thus, in order to obtain a higher vibration uniformity, the influence of length of *l*1 on the vibration uniformity is studied, which is in the range of 76 mm to 84 mm. As shown in Fig. 6d, when the length of *l*1 increases from 76 mm to 84 mm, the vibration uniformity first decreases and then increases. When the length of *l*1 is 80 mm, the vibration uniformity of the UVAIB device is 92%, which is the highest value. Subsequently, the value of vibration uniformity shows a downward trend. Therefore, the length of *l*1 is determined to be 80 mm.

In summary, the size of the horn is determined as *D*2 = 19 mm, *l*1 = 80 mm and *l*2 = 60 mm. The simulation results indicate that the resonant frequency of the ultrasonic vibration device is 30171 Hz, and after setting the flange, the resonant frequency becomes 30243 Hz, as shown in Fig. 7. At this time, the ultrasonic energy loss can be reduced, the amplification factor can reach 8, and the vibration uniformity of abrasive tools can reach 92%.