# Effect of Axial Modification on the Meshing Performance of Involute Developid Gear Pair
## 1 Introduction  
The involute beveloid gear (hereinafter referred to as \"beveloid gear\") is a novel type of gear that has garnered considerable interest within the field of gear transmission [1]. It can be considered a general form of involute cylindrical gear, exhibiting a displacement coefficient that varies along the axial direction of each tooth section. The modification coefficient on each section varies linearly along the tooth width, forming a conical shape that enables the gear to realise various transmission forms, such as paralleled transmission, intersected transmission, and crossed transmission, which has generated significant interest among researchers [2]. In the context of paralleled transmission, the beveloid gear facilitates the realisation of non-clearance meshing transmission, which is characterised by high stability, high precision, high efficiency, and low wear rate. The beveloid gear pair has been widely deployed in marine gearboxes, automobile splitters, robots, and other applications. A substantial body of research has been conducted in the field of involute gear transmissions, both domestically and internationally, which has resulted in notable advancements in the field. In their research, [PERSON] investigated the parameter design andmeshing mode of the beveloid gear pair employed in the marine gearbox with a small inclination angle [3; 4]. [PERSON] proposed a method for machining the internal leveloid gear pair with straight teeth parallel to the rack cutter, where the theoretical mathematical models of both the internal and the external leveloid gears are derived [5]. [PERSON] developed a non-linear time-varying dynamic model to study torsional dynamic behaviour of a typical multi-mesh gear train [6]. [PERSON] put forth a numerical methodology for calculating the load distribution and contact pressure of the crossed leveloid gear, then established a finite element accurate meshing model of the crossed leveloid gear through the influence coefficient method [7]. [PERSON] derived a mathematical model of the leveloid tooth flank with concave and convex modifications, established a finite element and grid analytic model, and conducted a contact analysis of the tooth flank [8]. [PERSON] constructed a meshing model of a working joint cylinder and joint cone between a cylindrical gear and a crossed leveloid gear transmission. Also, they derived the tooth flank equation and investigated the variation in contact ellipse, contact trace, and principal curvature of the gear pair under varying parameters [9]. [PERSON] proposed a worm drive with adjustable side clearance, comprising an involute helical leveloid gear and a double-guide involute cylindrical worm, and presented the side clearance theory [10]. [PERSON] established a formula for calculating the error and side clearance of two-way non-load transmission and deduced a continuous side clearance curve for a gear pair [11]. [PERSON] proposed a method for calculating the magnitude and distribution of the contact force of paralleled leveloid gear pairs and investigated the influence of macro-geometric parameters on the contact force distribution [12]. [PERSON] derived the contact line equations of a paralleled leveloid gear pair and proposed an analytical algorithm for calculating the mesh stiffness and examined the influence of various parameters on the mesh stiffness [13]. [PERSON] put forth a novel analytical approach for modelling cracks and an elastic spring methodology for ascertaining the time-varying meshing stiffness of spur gears [14]. There are also many results in gear modification. [PERSON] obtained the fitting function with helical gears as the carrier based on the Kriging method and optimised the axial modification parameters with the objective of minimising the dynamic transfer error fluctuation within the modification interval [15].