## 1 Introduction  
In modern industrial applications, cylindrical components, such as shafts, bearings, and screws, are widely used in aerospace, ship, marine, and automotive industries [1, 2]. Due to the influence of high cycle fatigue and cyclic stress, these components have to retain a long service life to reduce the replacement frequency and cost. Therefore, a surface enhancement process is required after manufacturing. One of the often-used processes is the peening process, which is a physical method [3]. In the traditional method, sand pills and iron pills are typically used [4]. They are ejected with high speed and collide with the workpiece surfaces to improve the fatigue strength and change the surface properties of metal materials by generating surface compressive residual stresses. The traditional method is a simple way to treat the surfaces of metallic rods. [PERSON] [5] found that the cylindrical workpieces of annealed medium carbon steel were peened with 140-2300% coverage by the traditional process and that the fatigue limits increased 14-25% compared to the non-peened workpieces. However, the traditional peening process tends to increase the surface roughness, causing a certain damage and limiting the improvement of surface properties. Ultrasonic needle peening is another method to peen cylindrical workpieces. An actuator is used to generate oscillation, which leads to a relative movement between the sonotrode and the workpiece surface [6]. The sonotrode end is excited with a high frequency (more than 20 kHz) and directly impacts the workpiece surface. Utilizing this method to open the cylindrical workpieces of Ti-6 Al-4V, researchers showed in their experimental results that the \(10^{8}\) cycles fatigue strength can improve by 11.7% [7]. The compressive residual stress and the surface hardness increased as well. The solid steel pin is modeled as a rigid material [8]. The simulation results showed that the surface roughness magnitude depends on the correlation of the vertical and lateral load components. In this process, a rotated equipment is required for the rotation of the cylindrical specimen. Otherwise, the treating area is relatively small. During water peening process, water cavitation jet goes through a peening nozzle with high speed [9]. When the bubbles are injected with a flow on a metal workpiece surface, a plastic deformation can be developed. Water cavitation jet peening utilizes this deformation. A Ti-6 Al-4V rod with a diameter of 5 mm was investigated using this process and it was found that the depth of the introduced compressive residual stress increased up to 230 \(\upmu\)m at a processing time per unit length of 10 s/mm [10]. The laser peening process uses high-energy pulsed laser to generate shock waves on the surfaces of the treated surfaces [11]. [PERSON] et al. [12] investigated the effects of laser peening on residual stresses in 7050-T7451 aluminum alloy rods. The results showed that the geometrical effects of the curved target surface greatly influenced residual stress fields. Compared to the traditional shot peening and ultrasonic needle peening, the mentioned shotless peening methods (water cavitation peening and laser peening) require skilled operators as well as expensive and complicated equipment. In order to achieve good surface properties with simple equipment, ultrasonic cavitation peening has been developed in recent decades [13; 14; 15]. During ultrasonic cavitation peening, a piezoelectric transducer is utilized and the sonotrode is partially immersed in a liquid. The results showed that the hardness and the residual compressive stress were significantly improved but the surface roughness was not obviously increased. Using a piezoelectric transducer has many advantages: simple equipment, compact structure, as well as a high output power [16]. Due to the vibration of the transducer sonotrode, cavitation bubbles are generated and then collapsed on the treated surfaces, which introduces the compressive residual stress and increases the surface hardness.