## 1 Introduction  
In recent years, advancements in newly developed materials have further led to increased demands for their mechanical and physical properties. Hard and brittle materials such as superalloys, ceramics, quartz glass, and sapphires are thus produced. Due to the significant differences in their physical and mechanical properties as compared with metallic materials, traditional machining methods struggle to enhance machining efficiency and cost reduction simultaneously. As a result, the development of ultrasonic-assisted machining methods has emerged. This assisted machining approach can reduce the cutting force and friction resistance between the cutting tool and workpiece, thereby improving the machining quality. Furthermore, with the evolution of machining technologies, achievinghigh rigidity and a lightweight design have become crucial research topics in machine tool structural development. When designing a machine tool structure, it is essential to consider its rigidity and avoid the resonant frequencies that the machine tool may encounter during the machining process. As a result, the possibility of a machine tool's vibrating system causing environmental resonances may be minimized. Therefore, research into machining characteristics during cutting processes to investigate the interconnected relationship between machine tool structural rigidity and process dynamics is increasingly important. This interconnection could serve as a reference for machine tool structural design and manufacturing improvements. The ultimate goal of machining performance and efficiency promotions can be achieved accordingly. Through the use of the finite element method for the design and analysis of milling tools, the principle of ultrasonic lateral vibration conversion for the transformation of vertical vibration power at the vibration node (the maximum pressure point) into lateral vibration is applied (2009). The necessary horizontal cutting force may thus be generated to optimize the removal efficiency of hard and brittle materials during abrasive grain milling. This also enables the design of more suitable cutting tools for milling processes. The milling tool shape, when expanded and scaled down during the machining stage, is fully incorporated after power conversion and can effectively enhance the cutting performance. Furthermore, under the same conditions of piezoelectric ceramics, changes in the existing vibration system can be made to achieve optimal performance. The results show that the conversion of vertical vibration power into lateral vibration effectively enhances the horizontal cutting force and the material removal is facilitated. To increase the swing of the vibration system, in addition to increasing the input pressure, a swing amplification shaft can be used to amplify the required swing. Therefore, modeling and designing the amplification shaft in finite element software for simulation analysis is necessary and the amplified value demonstrates a linear relationship with the theoretical values. The simulation of a vibration response in an ultrasonic vibration system with software could be coupled with appropriate elements such as common metal materials and quartz piezoelectric ceramics, and high accuracy could be achieved. The design time and manufacturing costs in practical manufacturing processes may thus be effectively reduced [1]. [PERSON] et al. (2023) investigated the impact of ultrasonic vibration on the machining performance of titanium alloy experimentally in the hybrid-assisted grinding of ultrasonic vibration/plasma oxidation. In this experiment, a three-axis CNC machining center with an ultrasonic spindle at a frequency of 25 kHz attached to a conductive grinding wheel was utilized to perform ultrasonic vibration. It was connected to a plasma oxidation power supply and hybrid-assisted grinding of a titanium alloy was thus achieved. These results indicate that applying ultrasonic vibration and increasing the vibration amplitude can reduce the grinding forces.