1. Introduction
There is an important application value for ultrasonic technology in the arc welding of metal materials. Ultrasonic energy can be used to refine the weld microstructure [
1], reduce weld porosity defects [
2], and increase weld width and depth [
3], through which the mechanical properties of the welded joint can be enhanced [
4]. Ultrasonic energy can also improve the corrosion resistance of welded joints [
5]. Yang proposed a method of ultrasonic impact during arc welding that involves applying periodic ultrasonic energy to the weld pool, which could reduce the input of ultrasonic energy and improve the weld structure. Results have demonstrated that periodic ultrasonic energy can break grains and promote heterogeneous nucleation [
6]. Low-frequency mechanical impact is introduced in the process of ultrasonic impacts during arc welding, which can lead to a detrimental effect on welding quality, such as incomplete fusion defects in the weld, decreased weld width and penetration, coarse grains, and decreased mechanical properties [
7]. In order to mitigate the adverse effects of mechanical impacts with low frequencies, the introduction method of periodic ultrasonic vibration must be changed. Therefore, it is necessary to design a sonotrode that can roll on both sides of the weld. In this study, periodic ultrasonic power supply is applied, and the periodic ultrasonic energy is transferred from the sonotrode into the weld pool.
There are usually two methods in the design of sonotrodes: a solution for the frequency domain equation and one for numerical simulation. Yazdian and Hamed designed a booster and composite sonotrode by an analytical method, and they also performed finite element analysis [
8,
9]. Hornišová demonstrated that the transfer matrix method is more precise than the impedance matrix method and the equivalent circuit method [
10]. Yahya designed an acoustic sonotrode for assisted-rotational magnetorheological abrasive flow, which is finished through finite element simulation. The results showed that ultrasonic vibration can reduce the surface roughness of the component [
11]. The amplification factor of acoustic sonotrodes is related to the ratio of the cross-sectional areas at both ends of an acoustic sonotrode. The larger the cross-sectional ratio is, the higher the amplification factor. The stepped acoustic sonotrode has a large amplification factor, but stress concentration readily occurs at the variable section of the acoustic sonotrode, and the safety factor of the acoustic sonotrode is low [
12,
13,
14,
15]. In order to address the issue of stress concentration at the variable cross-section of the stepped horn, it is usually necessary to design a transition section at the variable cross-section. Zeng clarified the optimal single-arc transition curve for sonotrodes with various cross-section ratios through finite element analysis [
16]. Based on single-arc transition, González-Mendoza proposed that the design of a pressure relief groove can decrease the stress concentration [
17]. Tu found that the effects of double-curvature arc transition, three-curvature arc transition, elliptical transition, and streamline transition on reducing stress concentration are superior to single-arc transition [
18]. Hannemann studied the crack propagation factors on the elliptical transition segment of the stepped shoulder and found that the radius of the transition segment was one of the main influencing factors of crack propagation [
19]. Mehran designed a wide-blade ultrasonic horn based on finite element simulation, and the ultrasonic vibrations were amenable to being evenly distributed on the horn [
20]. Satpathy designed a stepped sonotrode to weld thin metal plates by using ANSYS R15.0 and COMSOL 5.2a software, and the results showed that the displacement node was the most vulnerable part [
21]. Hameed designed a sonotrode with a characteristic frequency that is greater than that of the transducer, and they found that that sonotrode had a larger amplification factor and a smaller stress distribution than the traditional sonotrode [
22].
A rotating sonotrode consists of a thin end and a roller end. The design of the roller end mainly relies on the Kirchhoff theory and the Mindlin theory. The design of the thin end mainly relies on the wave equation theory of longitudinal vibration. A rotating sonotrode can be used for ultrasonic textiles [
23,
24], ultrasonic plastic welding [
25], ultrasonic atomization [
26], and the ultrasonic cold working of metals. Kustron designed two different ultrasonic atomizing devices with different characteristic frequencies based on finite element simulation for the metal powder process, and the amplitude distribution of the two ultrasonic atomizing devices is also similar [
27]. The ultrasonic metal cold working process mainly includes ultrasonic rolling, ultrasonic cutting, ultrasonic gear honing, and ultrasonic grinding. The roller end can be classified into three categories based on their thickness-to-diameter ratio, namely the thin plate, medium plate, and thick plate. The Kirchhoff theory was applied to the design of the roller end thin plate, and the Mindlin theory was applied to the design of the roller end thin plate and medium plate. Zhang employed the Kirchhoff theory to design a rotating sonotrode for the ultrasonic rolling process. The results showed that, compared to the segmented design, it is easy to realize the resonance of an ultrasonic system when the rotating sonotrode is designed as a whole and when the roller end can amplify the amplitude [
28]. The machining of a honeycomb composite material is one of the application directions of ultrasonic cutting. The cutting blade is the roller end of a rotating sonotrode, and the characteristic frequency of the sonotrode increases with the increase of blade thickness [
29,
30,
31]. For the ultrasonic gear honing process, the roller end of the sonotrode is usually a gear. The gear is matched with the gear to be machined, and the roughness of the tooth root can be reduced through ultrasonic vibration [
32]. The size of the roller end is usually not designed because it is determined by the parameters of the gear being machined. The design of a rotating sonotrode is predominantly intended for the thin end part. In the design, the gear is simplified as a circular plate with a diameter that is identical to that of its dividing circle. Additionally, various thin-end shapes were designed and their sizes were solved. Zhang and Zhou applied the Mindlin theory to design a rotating sonotrode with a conical transition composite thin end by an analytical method [
33] and a transfer matrix method [
34], respectively. The results demonstrated that the frequency of the rotating sonotrode decreases when there is an increase in the radius of the roller end and increases with an increase in thickness of the roller end [
34]. Based on the theory of three-dimensional elasticity and the Chebyshev–Ritz method, the characteristic frequency of a rotating sonotrode composed of a cone and a gear with free vibration was obtained by Li [
35]. During the ultrasonic grinding process, the design of the rotating sonotrode was relatively complex, and the thickness and material of the grinding layer on the outer side of the roller end had to be considered. Based on the Mindlin theory, Wu created a rotating sonotrode composed of a three-annular plate-shaped roller end and a conical thin end. The roller end material included brass and gray cast iron, while the thin-end material was 45 steel [
36]. On this basis, Fu developed numerical solution software with a MATLAB/GUI interface for a rotating sonotrode, as well as designed a rotating sonotrode with an aluminum-based phenolic resin binder as the roller end material [
37].
This study aims to design a rotating sonotrode that can roll on both sides of the weld during the arc welding process of metal materials. The three-dimensional diagram is shown in
Figure 1a. In order to mitigate the impeding effect of a raised weld shape on the rolling of the rotating sonotrode, it was necessary to create an arc groove with a radius of 5 mm on the outer surface of the roller end, as shown in
Figure 1b. Both thin ends were subjected to the same pressure so that the roller end could be in close contact with the base material. The frequency of the existing sonotrode was mainly between 20 kHz and 35 kHz. In this study, in order to reduce the size of the rotating sonotrode, the frequency of the rotating sonotrode was designed to be 50 kHz.
In comparison with the existing research on rotating sonotrodes, the specific differences from this article are as follows:
(1) The application fields and assembly method for a rotating sonotrode are different. The rotating sonotrode developed in the current research was predominantly employed in the field of machining. The rotating sonotrode was assembled by the bolt connection method, as shown in
Figure 2. The application direction of the rotating sonotrode designed in this study was used to assist the arc welding process of metal materials, which is composed of double thin ends and a roller end. The rotating sonotrode in this study is a whole unit and does not involve mechanical assembly.
(2) The calculation methods for the rotating sonotrode sizes are different. In previous studies, the size of the roller end is determined, and the size of the thin end needs to be solved. In this study, a double thin-end rotating sonotrode was designed. The size of the fine end was fixed and the size of the rolling end, which is more difficult to solve, was solved.
(3) The shapes of the rotating sonotrode are different. In this study, the rotating sonotrode has a curved groove. The method for making a slotted rotating sonotrode that has a target design frequency and large amplification factor on the basis of satisfying the longitudinal–flexural resonance mode is very critical.
Based on the above analysis, in order to design a rotating sonotrode for the ultrasonic vibration-assisted arc welding process of metal materials, the design went through the following steps: Firstly, a mathematical model of the unslotted rotating sonotrode was established based on the Mindlin theory. TC4 titanium alloy was used as the material of the rotating sonotrode. The size of the rotating sonotrode was solved by MATLAB R2022a software, and the size was optimized by finite element simulation. Secondly, the effect of the transition form of the junction between the roller end and the thin end; the perforation on the roller end on the stress distribution; and the characteristic frequency and amplification factor of the sonotrode were investigated by the finite element simulation method. Finally, by optimizing the position and size of the perforation, the designed rotating sonotrode was able to reach the target design frequency and had a large amplification factor. This study provides an effective design method for the design of rotating sonotrodes with double thin-ended structures.
2. The Design Route of Double Thin-End Rotating Sonotrodes
A rotating sonotrode with a longitudinal–flexural resonance mode at a characteristic frequency of 50 kHz was designed. It was imperative to apply pressure at the node position of the thin end in order to guarantee the transmission of the ultrasonic vibration. The node positions at the two thin ends had to be symmetrical relative to the roller end. Therefore, the shape of the thin end could not be too complicated. The two thin ends of the rotating sonotrode were designed as equal cross-section rods, and the length of the thin end was fixed at half of the wavelength. Subsequently, the size of the roller end, comprising the outer diameter, inner diameter, and thickness, was resolved. In this study, the number of unknowns increased, and it was more difficult to solve the sonotrode size directly by the frequency equation. It was only possible to obtain the size of the roller end of the rotating sonotrode by solving the frequency equation numerically.
The size of the unslotted rotating sonotrode can be obtained through solving the frequency equation numerically. There could be the smallest error between the target characteristic frequency and the actual characteristic frequency for unslotted rotating sonotrodes with an optimum dimension, which had a longitudinal–flexural resonance mode. The effect of grooving on the ultrasonic vibration and propagation in the rotating sonotrode was analyzed. The influence of the transition mode at the junction between the roller end and the thin end on the stress concentration of the rotating sonotrode was studied, and then the influence of the size and position of the hole on the characteristic frequency and amplification coefficient was studied. Finally, the design of the target rotating sonotrode was completed.
Finite element simulation was employed to investigate the vibration mode, characteristic frequency, and stress distribution of the rotating sonotrode. The solid mechanics model was employed, and the control equation was as follows:
In the formula, μ is the shear elastic constant; λ is the bulk elastic constant; and s is the particle vibration displacement.
The rotating sonotrode’s material (TC4 titanium alloy) was defined as a linear elastic material. According to the different research objectives, the surface of the rotating sonotrode was given as different boundary conditions. The free triangle model was used to mesh the rotating sonotrode, and the interior of the rotating sonotrode was a tetrahedral grid with a maximum unit length of 11.3 mm and a minimum unit length of 1.41 mm.
5. Conclusions
In this paper, a rotating sonotrode for the ultrasonic vibration-assisted arc welding of metal materials was designed by combining theoretical calculation with finite element simulation. An effective design method is proposed for a rotating sonotrode with a groove and double-thin-end structure. The effects of the groove, perforation, and transition types on the characteristic frequency, stress distribution, and amplification factor of the rotating sonotrode were clarified. The main conclusions are as follows:
(1) For the calculation of the size of the rotating sonotrode without grooving, a numerical solution based on discrete intervals can obtain design results with a lower frequency error and dimensional accuracy than a numerical solution based on discrete basis values. The fifth-order mode and its characteristic frequency are obtained by finite element simulation. The longitudinal bending resonant mode is used, and the frequency error is only 0.004%. After the rolling end is grooved, the characteristic frequency of the rotating sonotrode clearly decreases.
(2) The transition mode of the connection between the roller and the thin end has a significant impact on the stress distribution and characteristic frequency of the sonotrode. The elliptic transition mode can significantly reduce the stress concentration of the sonotrode without changing the characteristic frequency of the sonotrode.
(3) The perforation on the roller end will affect the characteristic frequency and amplification factor of the rotating sonotrode. The characteristic frequency of the sonotrode decreases with increases in the size of the perforation. When the perforation size is constant, the characteristic frequency of the sonotrode also decreases with the perforation position away from the axis of roller end. The amplification factor changes irregularly. Through the coupling effect of the perforation size and position on the characteristic frequency and amplification factor of the sonotrode, a rotating sonotrode with a characteristic frequency of 49.721 kHz and an amplification factor of 3.02 is obtained by the method of a joint perforation in two positions, which meets the design requirements.