# Research on Stagger Coupling Mode of Pulse Duration and Tool Vibration in Electrochemical Machining
## 1 Introduction  
Electrochemical machining (ECM) is a non-traditional machining process that removes workpiece material via a controlled electrochemical anodic dissolution reaction. With its special characteristics (e.g., no tool wear, high material removal rate, and independence of material hardness [1; 2; 3]), ECM has been widely applied in aviation, aerospace, energy, healthcare, and other high-tech fields [4; 5]. Adjusting the coupling of pulse duration and tool vibration in ECM (PVECM) is an effective method to improve machining accuracy and surface quality [6; 7; 8; 9]. The PVECM process can achieve electrochemical machining in smaller inter-electrode gaps, and can remove electrolysis products and Joule heat from major inter-electrode gaps. Alternating the machining and scouring processes has a positive and beneficial effect on improving the electrolyte flow field.  
Numerous studies have evaluated the effect of tool vibration. For example, [PERSON] et al. presented a mathematical model that included vibration to investigate how input parameters affect the efficacy of tool vibration [10]. By using low-frequency tool vibration, [PERSON] et al. significantly improved electrolyte velocities in the inter-electrode gap to improve the removal rate of electrolytic products [11]. [PERSON] et al. investigated how vibration direction, amplitude, and tool feed rate affect the replicating accuracy and processing speed by applying ultrasonic vibration to ECM [12]. [PERSON] et al.  
proposed different feed modes with vibration to improve the stability and localization of ECM [13]. [PERSON] et al. presented an electrochemical micro-machining method with cathode vibration feed based on the mechanics of vibration and the electrochemical principle [14]. [PERSON] et al. improved electrolyte diffusion and convection by applying ultrasonic vibrations [15], and [PERSON] et al. improved the flow field of machining regions in the process with tool vibration [16].  
Previous research into PVECM focused on how pulse and vibration parameters such as vibration frequency, vibration amplitude, and pulse duration affect machining accuracy and surface quality [17; 18; 10; 19]. There are few studies on the effects of coupling modes of pulse duration and tool vibration. The traditional coupling mode of pulse duration always uses a symmetrical distribution at the moment when the cathode attains the minimum inter-electrode gap. In this coupling mode, the pulse duration consists of two different stages: one where the inter-electrode gap narrows and one where it expands. The electrolyte flow field and the electrolysis products should differ in each stage. The traditional symmetry coupling mode may not be the optimal one. Other than the traditional symmetry coupling mode, two other modes of coupling pulse duration and tool vibration are presented: the pre-position and post-position coupling modes. In the pre-position (post-position) coupling mode, the pulse occurs before (after) the cathode attains the minimum inter-electrode gap.  
For this study, simulation analysis and comparative experiments involving the three coupling modes of pulse duration and tool vibration were carried out. In this document, a multi-physical model that couples fluid flow, electricity, and heat is presented. The simulation results indicated that the electrolyte pressure and gas void fraction when the inter-electrode gap was narrowing differed clearly from those when the inter-electrode gap was expanding. Next, three coupling modes are presented: pre-position coupling, traditional symmetry coupling, and post-position coupling. Finally, the results of the experiments investigating machining localization are discussed to evaluate these three coupling modes.