Abstract
The elastic-plastic deformation in the contact deformation area during gear shaving is a crucial factor influencing the mid-concave error. Stress analysis serves as the foundation for understanding such deformation. Utilizing Hertz theory, a local coordinate system for the contact deformation zone during the shaving process is established. An elastic-plastic stress analysis model is constructed to analyze the elastic and plastic stresses in the contact deformation area. Formulas for calculating the elastic stress and the residual stress resulting from plastic deformation are proposed. The elastic yield limit stress value in the contact deformation zone during the shaving process is obtained. This value is used as a criterion to distinguish between elastic and plastic deformations in the contact deformation area of the shaved gear, providing a method for quantitatively analyzing the nature of elastic-plastic deformation.
1. Introduction
Gear shaving is a contact squeeze cutting process, theoretically involving point contact. During shaving, the shaving cutter and the workpiece gear engage with no side clearance on交错 axes, with the shaving cutter driving the workpiece gear in rotational motion. The shaving process is completed through relative sliding between the engaging tooth surfaces. However, in practice, the cutting edge of the shaving cutter must penetrate a certain depth into the tooth surface of the workpiece gear to achieve shaving. Due to the elastic and plastic deformations of the tooth surface during the actual shaving process, the engagement between the shaving cutter and the workpiece gear cannot follow the precise engagement trajectory near the pitch circle, leading to increased cutting near this area. Additionally, the number of engagement points in this region is fewer than at the tooth tip and tooth root, increasing the tooth surface pressure and causing micro-plastic deformation, which results in the mid-concave error on the tooth surface of the workpiece gear.
Many scholars at home and abroad have analyzed the contact deformation of gears using Hertz contact theory and elastic mechanics theory. However, these studies mainly focus on the deformation magnitude or the accumulation of residual stress and strain during the deformation process, which, while providing guidance for practical engineering, cannot accurately address the fundamental issue of contact elastic-plastic deformation in the mid-concave error zone during gear shaving, namely, quantitatively analyzing the conditions for plastic deformation on the tooth surface of the workpiece gear during the shaving process. In this paper, Hertz contact theory and elastic-plastic stress analysis methods are used to analyze the elastic and plastic stress changes on the tooth surface layer of the workpiece gear during the shaving process, obtain an initial theoretical criterion for distinguishing elastic and plastic deformations on the tooth surface, and analyze the impact of plastic deformation and the resulting residual stress on the mid-concave error.
2. Contact Points and Pressure Between Shaving Cutter and Workpiece Gear
The contact deformation of the workpiece gear is a significant factor affecting the mid-concave error, and the nature and magnitude of the tooth surface contact deformation depend on the load applied to the tooth surface. Therefore, it is necessary to analyze the variation laws of the number of contact points and the pressure between tooth surfaces during the shaving process.
Studies by numerous scholars indicate that the mid-concave error during shaving mainly occurs when the contact ratio is between 1 and 2. At this contact ratio, the number of contact points between the shaving cutter and the workpiece gear is between 2 and 4. The number of contact points on the tooth surface follows a cyclical pattern of approximately 3-4-3-2-3-4, with two-point contacts mostly distributed near the middle of the tooth profile, and three or four-point contacts occurring more frequently near the tooth tip and tooth root.
During the shaving process, the radial force between the shaving cutter and the workpiece gear shafts remains constant, but the number of tooth surfaces engaged simultaneously changes. The “double-tooth engagement zone” where two pairs of tooth surfaces are in contact is always located at the tooth tip or tooth root, while the “single-tooth engagement zone” where one pair of tooth surfaces is in contact is always near the pitch circle. Thus, the pressure and compressive strength between the two tooth surfaces in the single-tooth engagement zone are significantly greater than those in the double-tooth engagement zone, causing the shaving cutter to cut deeper near the pitch circle of the workpiece gear, resulting in the removal of additional metal material and the mid-concave error on the tooth profile of the workpiece gear.
The pressure between the tooth surfaces of the shaving cutter and the workpiece gear can be calculated using the instantaneous static equilibrium equations for different numbers of contact points, although this method is not sufficiently accurate. However, it can roughly indicate the distribution and range of the pressure between tooth surfaces during shaving. The calculation method and formulas are not detailed here due to their inaccuracy, but the conclusion is given: the range of pressure between contact tooth surfaces during the shaving process is approximately 1700N to 2500N. As a tooth on the workpiece gear engages from its tip to its root on one side, the pressure near the tooth tip on that side is relatively small. When engaging near the pitch circle, the tooth surface pressure is maximum. Before and after engaging at the tooth root, the tooth surface pressure gradually decreases until the tooth surface is fully disengaged.
3. Stress Calculation of the Workpiece Gear During Shaving
3.1 Elastic Contact Stress of the Workpiece Gear
During the shaving process, the elastic and plastic deformations at the contact points on the tooth surface of the workpiece gear are important factors affecting the mid-concave error. Therefore, the analysis and calculation of the contact stress on the tooth surface of the workpiece gear are fundamental. Since the axial dimension of the gear tooth is much larger than its tooth height and thickness, and the gear is not constrained in the axial direction, the contact at the engagement point during shaving can be simplified as a two-dimensional contact model in the end plane, i.e., a plane strain problem.
According to Hertz contact theory, the normal load in the contact area during the shaving process is obtained as:
p(x)=p01−a2x2
where a is the half-width of the contact area in the end plane, calculated as:
a=π4p(r11+r21)E11−ν12+E21−ν22
where r1 and r2 are the curvature radii of the two contacting bodies at the contact area, E1 and E2 are the elastic moduli of the shaving cutter and the workpiece gear, respectively, ν1 and ν2 are their Poisson’s ratios, and p is the total normal load in the contact area. The maximum unit pressure p0 is the surface contact stress. The load distribution on the contact area of the tooth surface of the workpiece gear can be calculated using Equation (1).
Throughout the engagement process, the workpiece gear is considered a flexible body, but other factors such as bending moments, lubrication, and temperature are not considered for their influence on contact stress. A coordinate system is established to analyze the stresses within the gear contact area. On the contact interface of the workpiece gear, σx=σz=−p(x), while outside the contact area, by Saint-Venant’s principle, the stress components on the surface are all zero.
The principal stresses within the workpiece gear tooth are:
σx=−p0am(1+m2+n2z2+n2)−2[m2+n2zσz]
σz=−p0am(1−m2+n2z2+n2)
τzx=p0an[m2+n2m2−z2]
where m and n are parameters related to x, z, and a, given by:
m2=21[(a2−x2+z2)+(a2−x2+z2)2−4x2z2]
n^2 = \frac{1}{12}[(a^2 – x^2 + z^2) + \sqrt{(a^2 – x^2 + z^2)^2 – 4x^2z^2}] – \frac{1}{12}[(a^2 – x^2 + z^2) – \sqrt{(a^2 – x^2 + z^2)^2 – 4x^2z^2}]
These equations define the parameters m and n, which are utilized in the stress calculations within the contact area of the shaved gear tooth surface during the shaving process. Specifically, the principal stresses and shear stress within the gear tooth can be expressed as:
σ_x = -p_0a \cdot m\left(\frac{1 + z2}{m2}\right) – 2\left[\frac{z}{m2}\right]\sigma_z
σ_z = -p_0a \cdot \frac{1 – z2}{m2}
τ_zx = p_0a \cdot \frac{n}{m2m2}
By substituting the expressions for m2 into these stress equations, we can solve for the elastic stress distribution on the surface layer of the shaved gear tooth during different meshing points in the shaving process. This provides a basis for obtaining a theoretical criterion to distinguish between elastic and plastic deformation on the shaved gear tooth surface.
It’s worth noting that these calculations assume a plane strain problem, simplifying the contact at the meshing point to a two-dimensional end-face contact model. Furthermore, the gear tooth being shaved is considered a flexible body, but other factors such as bending moments, lubrication, and temperature are not considered in the contact stress analysis.
The above content details the mathematical formulation used to calculate stresses within the contact area of the shaved gear tooth, which is crucial for understanding and predicting the elastic and plastic deformations that occur during the shaving process and their impact on the mid-concave error.