Abstract This paper proposes a class of variable circular arc gears. Using the deviation function method, the complete construction process of the tooth profile is given, proving that it satisfies the fundamental law of tooth profile meshing. Basic formulas such as the tooth profile equation are derived, the characteristics of variable circular arc gears are discussed, and examples of variable circular arc tooth profiles are given. This type of gear has high load-bearing capacity and is particularly suitable for high-speed, high-precision CNC machining, and will be applied in heavy-duty transmission applications. Keywords : Gear, Tooth profile, Deviation function, Meshing principle, Gear strength Introduction With the development of science and technology, the requirements for gear transmission are constantly increasing, especially in terms of heavy load and miniaturization. Currently, three tooth profiles are commonly used in gear design: involute, cycloidal, and circular arc. Involute gears are widely used due to their simple manufacturing and insensitivity to center distance deviations; however, they have poor load-bearing capacity and are rarely used in heavy-duty transmissions. Cycloidal gears have low contact stress and a large overlap of tooth profiles, which is beneficial for improving bending strength, but they require high precision in the manufacturing and assembly of the meshing gears. Circular arc gears, which appeared in the 1950s, have low bending strength, and to achieve continuous contact, they are generally made as helical gears, which greatly limits their manufacturing and miniaturization. To improve the load-bearing capacity of gears, researchers have proposed many new tooth profiles, such as micro-segment gear tooth profiles and stepped double involute gears, which can play a certain role in improving gear strength. However, these new tooth profiles still have some shortcomings in manufacturing or assembly. This paper proposes a new type of gear—the variable circular arc gear—based on the concept of a deviation function. The paper introduces the formation principle of the variable circular arc gear, derives the tooth profile calculation formula, discusses the characteristics of the variable circular arc gear, and provides examples of the tooth profile. 1. Formation Principle of Variable Circular Arc Tooth Profile 1.1 Formation Principle of Variable Circular Arc Tooth Profile The formation principle of variable circular arc tooth profile is introduced below based on the deviation function (DF) method proposed by American scholars DCHYang et al. Let P1 be the pitch circle of a gear with radius 1, θ1 be the rotation angle, and draw a series of circles with radius e(θ1) and centers evenly distributed on the pitch circle, as shown in Figure 1 (here, e(θ1) = r0COS(2θ1); when e(θ1) satisfies certain constraints, this series of circles can Let g[sub]1[/sub](θ[sub]1[/sub]) be the smooth tooth profile enveloped by the given equation. From the diagram, we can see that e1(θ[sub]1[/sub]) = ||P(θ[sub]1[/sub]) - g(θ[sub]1[/sub])ll, and e(θ[sub]1[/sub]) is called the deviation function. We collectively refer to a class of tooth profiles that satisfy the above generation principle as variable circular arc envelope tooth profiles, or simply variable circular arc tooth profiles. Gear pairs constructed from these profiles are called variable circular arc gear pairs. When the deviation function is e(θ1) = (rcosα)θ1, the variable circular arc tooth profile is the common involute tooth profile; while when the deviation function is e(θ1) = , the variable circular arc tooth profile is the cycloidal tooth profile; and for ordinary circular arc gears, it is equivalent to the deviation function e(θ1) only taking values ​​at certain discrete points. 1.2 Calculation of Variable Circular Arc Tooth Profile As can be seen from Figure 2, the equation of the variable circular arc gear tooth profile P1 expressed in coordinate components is: where ψ is defined as the angle between the normal of the tooth profile at point g and the x-axis, which is a function of θ1. As shown in Figure 2, [align=center] [/align] In order to make the obtained tooth profile practically applicable, certain restrictions must be placed on the selection of the deviation function e(θ1). (1) To ensure that the tooth profile curves do not intersect, it is required that (2) To ensure that equation (3) has a solution, it is required that (3) To ensure that the profile curve C′ is continuous, at the intersection point Cp of the pitch circle p1 and the tooth profile g1, e(θcp) must be equal to zero. (4) In order to ensure that the tooth profile is smooth, e(θ1) must change monotonically between its minimum and maximum values. (5) In order to make the number of gear teeth N1 take an integer value and ensure a certain overlap coefficient μ, the angle range Φ1 between the two intersection points should be satisfied. 2 Meshing principle of variable circular arc gear tooth profile Consider a pair of variable circular arc gears with meshing tooth profiles, as shown in Figure 3. At a certain instant, the tooth profiles c1 and c2 of the pair of conjugate gears O1 and O2 mesh at point g. According to the principle of tooth profile meshing, the equation of the conjugate tooth profile can be written as: where g1x, g1y, g2x, and g2y are the coordinate components of the meshing point g in the gear's rectangular coordinate system O1 and O2, respectively; O1 and O2 are the rotation angles of gears θ1 and θ2; and φ1 and φ2 are the angles between the normal to the tooth profile at point g and the two x-coordinate axes. The condition for maintaining contact during the meshing of two tooth profiles is that the velocity components of point g on tooth profile c1 and point g on tooth profile c2 in the common normal direction are equal, i.e., vgn1 = vgn2. From the principle of tooth profile generation, we know that there is a common normal nn′ at any meshing point g of the two gear tooth profiles. It intersects the line connecting the centers O1 and O2 at node P. Perpendicular lines are drawn from points O1 and O2 to nn′, intersecting at points n1 and n2 respectively. As shown in Figure 3, the transmission ratio is therefore, we conclude that variable circular arc gears can achieve constant speed ratio transmission. The relative sliding velocity between the tooth profiles of a variable circular arc gear pair is generally given by the pressure angle of the variable circular arc gear, which is a variable related to the gear rotation angle. For an involute tooth profile, e(θ1) = r1cosα(θ). Substituting this into the pressure angle calculation formula, we get e = a. 4. Characteristics of the Tooth Profile of a Variable Circular Arc Gear Theoretically, variable circular arc gears include external meshing gear pairs and internal meshing gear pairs. Since external meshing involute gears have advantages such as simple manufacturing and insensitivity to errors, the variable circular arc gear discussed below is a convex-concave internal meshing variable circular arc gear used in heavy-load applications to improve gear strength. As shown in Figure 4, the variable circular arc tooth profile can be divided into two parts: convex teeth a, a′ and concave teeth b, b′, using the pitch circle as the dividing interface. a′ and b′ are mirror images of a and b. Segments a and b can be designed using different deviation functions, but they must intersect on the pitch circle and ensure at least one order of continuity at the intersection point. The main failure modes of gears are tooth breakage and pitting. The performance indicators reflecting their failure modes include tooth root bending strength and surface contact strength. The approximate formula for calculating tooth root bending stress is: where ω is the tooth width, and h and b are the distance from the point of application of force F[sub]t[/sub] to the tooth root and the tooth width, respectively. According to Hertz's contact stress theory, assuming two gears are made of the same material, the formula for calculating the contact stress at contact point g of the internal meshing tooth profile is: where ρ[sub]g1[/sub] and ρ[sub]g2[/sub] are the radii of curvature of the two tooth profiles at the contact point, respectively. Their calculation formulas are as follows: From the above formula, it can be seen that the magnitude of the tooth surface contact stress is proportional to the difference in the radii of curvature at the contact point. Both the tooth root bending stress and the tooth surface contact stress are related to the deviation function of the generated tooth profile. Therefore, by rationally designing the deviation function of the variable circular arc gear, we can obtain the required tooth root bending strength and surface contact strength. The meshing state of the variable circular arc gear at a certain instant is similar to that of the double circular arc gear. However, ordinary double circular arc gears must be made into helical gears to achieve continuous transmission. The variable circular arc gear introduced here achieves continuous transmission of the circular arc gear in the case of spur teeth by enveloping a series of circular arcs with different radii. The variable circular arc gear does not have undercutting phenomenon, and the minimum number of teeth can be two. The ability to achieve high precision and low cost manufacturing is one of the main reasons limiting the application of non-involute gears. If we take the node as the origin and select the gear rotation angle, deviation function e(θ), and pressure angle a as the three feed axes, then the normal tool radius compensation becomes very simple during CNC interpolation machining. In addition, Equation (3) has the form of the Minkowski Pythagorean speed curve, which makes it easy to achieve high-speed and high-precision CNC machining. This is a very important advantage of variable circular arc gears. Similar to other internal meshing gears, variable circular arc gears also have problems such as sensitivity to errors. 5 Examples of variable circular arc gear tooth profiles The main content of variable circular arc gear design is to design a gear tooth profile that meets the strength performance requirements under certain geometric constraints. The design of variable circular arc gears is generally divided into four steps: 1) Determine the gear pair parameters, such as: gear material properties, transmission ratio, overlap coefficient, module, center distance, and load torque. 2) Generate the deviation function. 3) Construct the tooth profile. 4) Evaluate the performance of the gear pair. If the requirements are met, end; otherwise, go to step 2. The following is an example of an internal meshing positive variable circular arc gear pair designed using the deviation function method. Let the center distance of the gear pair be l, and the pitch circle radius be l/2. The deviation function, as shown in Figure 5, depicts the continuous convex-concave contact characteristics of the gear tooth profile, with an overlap coefficient of 1.2. Due to space limitations, other aspects and methods related to the design of variable circular arc gear pairs will be further introduced in later articles. 6. Conclusion This paper proposes a novel variable circular arc gear and introduces a new method for designing such gears, providing new ideas and a broader research space for tooth profile studies. From the above derivation, demonstration, and calculation, the following conclusions can be drawn: The variable circular arc gear pair meets the basic conditions for gear transmission. It mainly adopts a convex-concave internal meshing transmission form and is used in heavy-duty transmission applications. By optimizing the design deviation function, the variable circular arc gear can obtain a high-performance gear tooth profile, especially significantly reducing the bending stress at the tooth root and the surface contact stress. It also has advantages such as compact structure and a minimum number of teeth of only two. The tooth profile design method has clear physical meaning and concise formulas, facilitating optimization design, computer-aided design, and manufacturing. In particular, it easily enables tool radius compensation and high-speed, high-precision CNC machining. Because this type of tooth profile requires point-to-point correspondence during transmission, it is sensitive to manufacturing and assembly errors, necessitating high-precision CNC machining. Other properties of this type of gear require further research. References : 1. Wu Xutang, ed. Gear Meshing Principle. Beijing: Machinery Industry Press, 1984. 2. Gear Research Laboratory, Taiyuan Institute of Technology, ed. Circular Arc Gears. Beijing: Machinery Industry Press, 1992.