Gears are fundamental components for the transformation of motion and power, characterized by constant power transmission. Their performance significantly determines the overall capability of mechanical equipment, granting them a crucial strategic role in both national economic development and national defense. As an innovative type of gear, the geometric elements constructed tooth gear features convex-concave meshing with point contact, exhibiting high load-carrying capacity and excellent lubrication performance. It has found applications in numerous fields such as aerospace, naval vessels, mining and metallurgy, general industrial gearboxes, and small-module gear transmissions.
After years of research and development, studies on the machining methods for geometric elements constructed tooth gears have accumulated experience in areas like CNC profile machining, gear hobbing, worm wheel grinding, and form grinding. However, there has been no related research on gear shaving, which is one of the efficient and economical gear finishing processes. Therefore, this paper takes the cylindrical geometric elements constructed tooth gear as the object of study, investigating the fundamental theory of shaving cutter design and verifying the correctness of the theoretical research through experimental studies.
The primary research contents are as follows:
① Introducing the basic principle of curve-conjugate meshing for cylindrical geometric elements constructed tooth gears, providing the fundamental definition of conjugate curves, solving the relationship between relative velocity and normal vectors of conjugate curves, and deriving the meshing equation and conjugate curve equation for parallel-axis cylindrical gear pairs. The rack-generation method is employed, combined with the basic rack tooth profile in the normal plane, to present the method for solving the tooth surface equation of the corresponding gear pair.
② Taking a typical two-point meshing convex-concave cylindrical geometric elements constructed tooth gear pair as an example, analytical solutions for the complete tooth surface equations, including the working segment, root transition segment, and tip transition segment, are derived based on the basic tooth profiles of the convex and concave gears. According to the kinematic principle of gear shaving machining, a spatial coordinate system for the shaving process is established, and the tooth surface equations for the convex and concave shaving cutters corresponding to the two-point meshing gear pair are derived. Using a 6-module two-point meshing gear pair as an example, mathematical models for the gear pair and its corresponding shaving cutter are built using Matlab software. A program for calculating corresponding tooth surface data points is compiled, and the method for creating solid models of the gear pair and its shaving cutter using 3D modeling software is provided, offering a theoretical basis for the design of cylindrical geometric elements constructed tooth gear shaving cutters.
③ Based on the Python language and the Eric7 IDE development tool, a design software for cylindrical geometric elements constructed tooth gear shaving cutters is developed. It realizes functions such as basic tooth profile design, strength verification, and the design of data points and geometric parameters for the shaving cutter. The complete software development process is introduced, and a test report for the corresponding software functions is presented.
④ The developed design software is used to complete the design of the gear pair and the shaving cutter. The shaving cutter is then employed on a Y4232CNC2 shaving machine to perform gear shaving on the hobbed geometric elements constructed tooth gear. Simultaneously, the optical 3D measuring instrument Alicona InfiniteFocus G5 is used to measure the surface roughness of the gear before and after shaving. Experimental results show that the surface roughness of the geometric elements constructed tooth gear is reduced to within 0.8μm after shaving, proving the correctness and feasibility of the gear shaving method for cylindrical geometric elements constructed tooth gears.
Fundamentals of Cylindrical Geometric Elements Constructed Tooth Gear Meshing
The cylindrical geometric elements constructed tooth gear is constructed based on the fundamental principle of curve-conjugate meshing. This section elaborates on the basic principle of curve-conjugate meshing, derives the meshing equation and conjugate curve equation for cylindrical geometric elements constructed tooth gears. Subsequently, based on the conjugate curve principle, the tooth surface equations for cylindrical geometric elements constructed tooth gears generated by a rack cutter are derived using the rack-generation method, providing a theoretical foundation for subsequent gear shaving processes.
Basic Principle of Curve-Conjugate Meshing
Definition of Conjugate Curves
Conjugate curves can be described as: during motion, two smooth curves maintain tangential contact continuously along a prescribed direction according to a specified law of motion. For these two curves, if they satisfy the following four conditions:
① Curves $\Gamma_1$ and $\Gamma_2$ are both smooth and regular.
② At any instant $t$, curves $\Gamma_1$ and $\Gamma_2$ are in point contact, i.e., they are tangent at the contact point $G$.
③ Every point on curve $\Gamma_2$ enters contact at a unique instant $t$, meaning the contact point on the curve is unique.
④ Within a certain region, given the motion law, curves $\Gamma_2$ and $\Gamma_1$ are mutually conjugate.
Then curves $\Gamma_1$ and $\Gamma_2$ are a pair of conjugate curves, and the two curves are conjugate to each other.
In a gear pair, when the contact lines on gear 1 and gear 2 are curves $\Gamma_1$ and $\Gamma_2$, respectively, gear 1 and gear 2 are geometric elements constructed tooth gears. The cylindrical helix satisfies the above conditions, and its conjugate curve remains a cylindrical helix. When curves $\Gamma_1$ and $\Gamma_2$ are two conjugate cylindrical helices, gear 1 and gear 2 are cylindrical geometric elements constructed tooth gears.
Relative Velocity and Normal Vector
The coordinate systems are established as shown in the figure. $S(O-x,y,z)$ and $S_p(O_p-x_p,y_p,z_p)$ are the fixed coordinate systems representing the initial positions of pinion 1 and gear 2, respectively. Their direction vectors are $(\mathbf{i},\mathbf{j},\mathbf{k})$ and $(\mathbf{i}_p,\mathbf{j}_p,\mathbf{k}_p)$. Curves $\Gamma_1$ and $\Gamma_2$ are situated on the tooth surfaces of gear 1 and gear 2, respectively. Coordinate axes $z$ and $z_p$ coincide with the rotation axes of pinion 1 and gear 2, respectively, and the two gear axes are parallel, i.e., axes $z$ and $z_p$ are parallel and have the same direction. The $x$-axis and $x_p$-axis coincide, and the center distance is $a$, with a transmission ratio $i_{21}$. The moving coordinate system $S_1(O_1-x_1,y_1,z_1)$ is attached to pinion 1 and rotates counterclockwise about axis $z_1$ with angular velocity $\omega_1$. The moving coordinate system $S_2(O_2-x_2,y_2,z_2)$ is attached to gear 2 and rotates clockwise about axis $z_2$ with angular velocity $\omega_2$. Their direction vectors are $(\mathbf{i}_1,\mathbf{j}_1,\mathbf{k}_1)$ and $(\mathbf{i}_2,\mathbf{j}_2,\mathbf{k}_2)$, respectively.
At the initial positioning moment, the moving coordinate systems $S_1$ and $S_2$ coincide with the fixed coordinate systems $S$ and $S_p$, respectively. After a period of time, the pinion rotates counterclockwise by an angle $\phi_1$, while the gear rotates by an angle $\phi_2$. According to the definition of transmission ratio, the relationship between angular velocities and rotation angles can be expressed as:
$$ i_{21} = \frac{\omega_2}{\omega_1} = \frac{\phi_2}{\phi_1} $$
Based on relative motion relationships, the relative velocity vector for parallel-axis conjugate curves during meshing can be obtained as:
$$ \mathbf{v}^{(12)} = \mathbf{v}^{(1)} – \mathbf{v}^{(2)} = (\mathbf{v}_0^{(1)} + \boldsymbol{\omega}^{(1)} \times \mathbf{r}^{(1)}) – (\mathbf{v}_0^{(2)} + \boldsymbol{\omega}^{(2)} \times \mathbf{r}^{(2)}) $$
Here, $\mathbf{v}^{(1)}$ is the velocity vector of the meshing point as it moves with gear 1; $\mathbf{v}^{(2)}$ is the velocity vector of the meshing point as it moves with gear 2; $\mathbf{v}^{(12)}$ is the relative velocity vector at the meshing point; $\mathbf{v}_0^{(1)}$ and $\mathbf{v}_0^{(2)}$ are the initial velocity values for gears 1 and 2, respectively, with $\mathbf{v}_0^{(1)}=0$ and $\mathbf{v}_0^{(2)}=0$. $\boldsymbol{\omega}^{(1)}$ is the angular velocity vector of gear 1; $\boldsymbol{\omega}^{(2)}$ is the angular velocity vector of gear 2. The expression for the relative velocity vector in coordinate system $S_1$ is:
$$ \mathbf{v}_1^{(12)} = \mathbf{v}_1^{(1)} – \mathbf{v}_1^{(2)} = (\mathbf{v}_{10}^{(1)} + \boldsymbol{\omega}_1^{(1)} \times \mathbf{r}_1^{(1)}) – (\mathbf{v}_{10}^{(2)} + \boldsymbol{\omega}_1^{(2)} \times \mathbf{r}_1^{(2)}) $$
Let the position vector of the smooth curve $\Gamma_1$ in coordinate system $S_1$ be:
$$ \mathbf{r}_1(t) = x_1(t) \mathbf{i}_1 + y_1(t) \mathbf{j}_1 + z_1(t) \mathbf{k}_1 $$
where $t$ is the parameter of curve $\Gamma_1$. According to differential geometry, the normal vector $\mathbf{n}$ at any point $P$ on curve $\Gamma_1$ can be expressed as a linear combination of the principal normal $\mathbf{n}_\beta$ and the binormal $\mathbf{n}_\gamma$:
$$ \mathbf{n}(t) = u(t) \mathbf{n}_\beta(t) + v(t) \mathbf{n}_\gamma(t) $$
where $u$ and $v$ are coefficients for $\mathbf{n}_\beta$ and $\mathbf{n}_\gamma$, respectively, representing different contact directions in practical computation. Thus, the normal vector along any direction at the contact point in coordinate system $S_1$ can be expressed as:
$$ \mathbf{n}_1(t) = u(t) \mathbf{n}_{1\beta}(t) + v(t) \mathbf{n}_{1\gamma}(t) $$
Meshing Equation and Conjugate Curve Equation
Let smooth curves $\Gamma_1$ and $\Gamma_2$ be on the tooth surfaces of gear 1 and gear 2, respectively. When the driving gear 1 rotates counterclockwise about its axis, curves $\Gamma_1$ and $\Gamma_2$ engage in contact, meaning that at any instant, there is at least one point of tangency between the two curves.
Let point $P$ be the contact point at a certain instant. At point $P$, the relative velocity $\mathbf{v}^{(12)}$ must be along the common tangent direction, so $\mathbf{v}^{(12)}$ is perpendicular to the curve’s normal $\mathbf{n}$:
$$ \Phi = \mathbf{n} \cdot \mathbf{v}^{(12)} = 0 $$
Substituting the relative velocity and normal vector expressions into the above equation yields the meshing equation in coordinate system $S_1$:
$$ \Phi_1(t, \phi_1) = \mathbf{n}_1(t) \cdot \mathbf{v}_1^{(12)}(t, \phi_1) = 0 $$
According to the kinematic method for gears, given curve $\Gamma_1$, curve $\Gamma_2$ can be solved using the above equation. The points on the conjugate curve $\Gamma_2$ can then be obtained:
$$
\begin{cases}
\mathbf{r}_2(t) = \mathbf{M}_{21}(\phi_1, \phi_2) \cdot \mathbf{r}_1(t) \\
\Phi_1(t, \phi_1) = \mathbf{n}_1(t) \cdot \mathbf{v}_1^{(12)}(t, \phi_1) = 0
\end{cases}
$$
Here, $\mathbf{r}_1(t)$ is the position vector of the given curve $\Gamma_1$ in coordinate system $S_1$; $\mathbf{r}_2(t)$ is the position vector of the conjugate curve $\Gamma_2$ in coordinate system $S_2$; $\mathbf{M}_{21}(\phi_1, \phi_2)$ is the coordinate transformation matrix from $S_1$ to $S_2$, expressed as:
$$ \mathbf{M}_{21}(\phi_1) =
\begin{bmatrix}
\cos(\phi_1+\phi_2) & -\sin(\phi_1+\phi_2) & 0 & a\cos\phi_2 \\
\sin(\phi_1+\phi_2) & \cos(\phi_1+\phi_2) & 0 & -a\sin\phi_2 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
Constructing Cylindrical Geometric Elements Constructed Tooth Gears Using the Rack-Generation Method
Conjugate Rack Tooth Profile for Cylindrical Geometric Elements Constructed Tooth Gears
1. Rack Tooth Profile as a Circular Arc
When only one pair of conjugate curves participates in meshing on the tooth surface of a cylindrical geometric elements constructed tooth gear, a circular arc is selected as the basic rack profile. At this point, the cylindrical geometric elements constructed tooth gear is essentially a circular arc gear. For generality, let the coordinates of the arc’s center be $(e, l)$.
The left-side circular arc in coordinate system $S_{nl}$ can be expressed as:
$$ \mathbf{r}_{nl}(t) = [\rho \sin t + e] \mathbf{i}_{nl} + [-\rho \cos t + l] \mathbf{j}_{nl} $$
where $t$ is the parameter of the arc curve, determining the position of a point on the arc; $\rho$ is the radius of the arc curve. Similarly, the curve equation for the right-side profile in coordinate system $S_{nr}$ can be derived:
$$ \mathbf{r}_{nr}(t) = [-\rho \sin t + e] \mathbf{i}_{nr} + [-\rho \cos t + l] \mathbf{j}_{nr} $$
2. Basic Profile as a Parabola
When there are two pairs of conjugate curves meshing on the tooth surface of a cylindrical geometric elements constructed tooth gear (i.e., two contact points on the meshing tooth surface), let the basic rack profile for the convex tooth be a circular arc with its center at the pitch point, and the basic rack profile for the concave tooth be a parabola. The concave parabola and the convex arc are tangent to each other.
Coordinate system $S_t(O_t-x_t,y_t,z_t)$ is fixed to the rack cutter alongside $S_{nl}$ and $S_{nr}$. $O_t$ is the vertex of the parabola. The center of the left convex arc coincides with $O_{nl}$, and the linear distance from $O_t$ to $O_{nl}$ is $L$. $\rho$ is the arc radius. The two normal plane profiles are tangent simultaneously at points $A$ and $B$. The positions of $A$ and $B$ in coordinate system $S_t$ can be expressed as $A(\rho\sin\theta, L-\rho\cos\theta, 0)$ and $B(-\rho\sin\theta, L-\rho\cos\theta, 0)$, where $\theta$ is an independent parameter whose value determines the relative position of the tangent points.
The coordinate transformation matrix $\mathbf{M}_{nlt}$ from $S_t$ to $S_{nl}$ can be expressed as:
$$ \mathbf{M}_{nlt} = \begin{bmatrix}
\cos\alpha & -\sin\alpha & 0 & L\sin\alpha \\
\sin\alpha & \cos\alpha & 0 & -L\cos\alpha \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where $\alpha$ is the angle between axes $y_t$ and $y_{nl}$, and $\alpha = \alpha_1 + \theta = (\alpha_2 + \alpha_1)/2$, thus $\theta = (\alpha_2 – \alpha_1)/2$; $\alpha_1$ and $\alpha_2$ are the normal pressure angles of the gear.
The parabola equation is known as $x_t^2 = 2p y_t$. Its expression in coordinate system $S_t$ is:
$$ \mathbf{r}_t(t) = \begin{bmatrix} t \\ \frac{t^2}{2p} \\ 0 \\ 1 \end{bmatrix} $$
where $t$ is the curve parameter; $P$ is the parabola’s focal parameter. The slope at tangent point $A$ can be expressed as:
$$ k = \tan\theta = \frac{\rho \sin\theta}{P} $$
Solving this equation yields $P = \rho \cos\theta$. Substituting this into the parabola equation, at point $A$ we have:
$$ (\rho \sin\theta)^2 = 2 (\rho \cos\theta) (L – \rho \cos\theta) $$
Solving for $L$ gives: $L = \rho \cos\theta + \frac{\rho \sin^2\theta}{2\cos\theta}$.
The expression for the parabola in coordinate system $S_{nl}$ is:
$$ \mathbf{r}_{nl}(t) = \mathbf{M}_{nlt} \cdot \mathbf{r}_t(t) = \begin{bmatrix}
t\cos\alpha – \frac{t^2}{2P}\sin\alpha + L\sin\alpha \\
t\sin\alpha + \frac{t^2}{2P}\cos\alpha – L\cos\alpha \\
0 \\
1
\end{bmatrix} $$
Similarly, the expression for the right-side parabolic profile in coordinate system $S_{nr}$ can be obtained:
$$ \mathbf{r}_{nr}(t) = \begin{bmatrix}
-t\cos\alpha + \frac{t^2}{2P}\sin\alpha + L\sin\alpha \\
-t\sin\alpha – \frac{t^2}{2P}\cos\alpha – L\cos\alpha \\
0 \\
1
\end{bmatrix} $$
Tooth Surface Equation of Cylindrical Geometric Elements Constructed Tooth Gears
Taking a two-point contact cylindrical geometric elements constructed tooth gear pair as an example:
1. Convex Gear Tooth Surface Equation
Based on the generating motion relationship between the gear and its rack cutter, the tooth surface equation of the gear in coordinate system $S_{2t}$ can be expressed as follows:
$$ \mathbf{r}_{2t}(t, \eta_{2t}) = \begin{bmatrix}
(x_{nt} + \frac{x’_{nt}}{y’_{nt}} r_{2t}\eta_{2t})\cos\eta_{2t} \cos\beta_t + (r_{2t} + y_{nt} – \frac{x’_{nt}}{y’_{nt}} x_{nt})\sin\eta_{2t} \\
-(x_{nt} + \frac{x’_{nt}}{y’_{nt}} r_{2t}\eta_{2t})\sin\eta_{2t} \cos\beta_t + (r_{2t} + y_{nt} – \frac{x’_{nt}}{y’_{nt}} x_{nt})\cos\eta_{2t} \\
r_{2t}\eta_{2t} \cot\beta_t – \frac{x’_{nt}}{y’_{nt}} r_{2t} \csc\beta_t \\
1
\end{bmatrix} $$
where $x_{nt}(t)$, $y_{nt}(t)$ are the coordinate values of the basic profile in the normal plane coordinate system; $x’_{nt}(t)$, $y’_{nt}(t)$ are the first derivatives of $x_{nt}(t)$ and $y_{nt}(t)$ with respect to $t$.
2. Concave Gear Tooth Surface Equation
Similarly, the tooth surface equation for the concave gear can be derived:
$$ \mathbf{r}_{2a}(t, \eta_{2a}) = \begin{bmatrix}
(x_{na} + \frac{x’_{na}}{y’_{na}} r_{2a}\eta_{2a})\cos\eta_{2a} \cos\beta_a – (r_{2a} – y_{na} + \frac{x’_{na}}{y’_{na}} x_{na})\sin\eta_{2a} \\
-(x_{na} + \frac{x’_{na}}{y’_{na}} r_{2a}\eta_{2a})\sin\eta_{2a} \cos\beta_a – (r_{2a} – y_{na} + \frac{x’_{na}}{y’_{na}} x_{na})\cos\eta_{2a} \\
-r_{2a}\eta_{2a} \cot\beta_a \cos\beta_a – \frac{x’_{na}}{y’_{na}} r_{2a} \csc\beta_a \\
1
\end{bmatrix} $$
where $x_{na}(t)$, $y_{na}(t)$ are the coordinate values of the basic profile in the normal plane coordinate system; $x’_{na}(t)$, $y’_{na}(t)$ are their first derivatives with respect to $t$.
Since the cylindrical gear tooth surface is a helicoid, it satisfies:
$$
\begin{cases}
x_2 = x_0(u) \cos\eta_2 – y_0(u) \sin\eta_2 \\
y_2 = x_0(u) \sin\eta_2 + y_0(u) \cos\eta_2 \\
z_2 = p_2 \eta_2
\end{cases}
$$
where $p_2$ is the helical parameter of cylindrical gear 2, and $p_2 = r_2 \cot\beta$; $\mathbf{r}_0(u) = (x_0(u), y_0(u))$ is the vector equation of the cylindrical gear’s transverse section, with $u$ as the parameter.
Tooth Surface Equation of Cylindrical Geometric Elements Constructed Tooth Gear Shaving Cutter
The tooth form of a geometric elements constructed tooth gear primarily consists of a working segment and transition curves. This type of gear typically employs convex-concave contact transmission, where the pinion is made with convex teeth (convex helicoid) and the working tooth profile lies outside the pitch circle, while the gear is made with concave teeth (concave helicoid) and the working tooth profile lies inside the pitch circle. Based on requirements, geometric elements constructed tooth gears are often designed for single-point or multi-point contact meshing, meaning one or multiple contact lines exist during gear meshing. This section takes a typical two-point meshing gear pair as an example to derive the tooth surface equations for the gear and its corresponding shaving cutter.
Tooth Surface Equation for Two-Point Meshing Geometric Elements Constructed Tooth Gears
Basic Rack Profile for Two-Point Meshing Geometric Elements Constructed Tooth Gears
Taking a two-point contact (pressure angles of 15° and 35°) cylindrical geometric elements constructed tooth gear pair as an example:
1. Basic Profile of Convex Gear
The normal plane profile of the rack cutter for machining the convex gear consists of two parts: cutting edge arc I for generating the root transition segment (concave arc for the tip transition) of the convex gear, and cutting edge arc II for generating the working tooth surface of the convex gear.
The position vector of arc segment I in coordinate system $S_{nt}$ is:
$$ \mathbf{r}_{nt}^{I}(\theta_{et}) = \begin{bmatrix}
\rho_g \cos\theta_{et} + (\rho_t \sin\alpha_{tmin} + e) \\
\pm[-\rho_g \sin\theta_{et} + (\rho_t \cos\alpha_{tmin} + l_t)] \\
0 \\
1
\end{bmatrix} $$
where $\theta_{et}$ is the parameter for root transition arc I; $\rho_g$ is the radius of arc I; $\rho_t$ is the radius of arc II; $\alpha_{tmin}$ is the minimum value of parameter $\alpha_t$; $l_t$ is the distance from the arc center to the $x_{nt}$ axis.
The position vector of arc segment II in coordinate system $S_{nt}$ is:
$$ \mathbf{r}_{nt}^{II}(\alpha_t) = \begin{bmatrix}
\rho_t \sin\alpha_t \\
\rho_t \cos\alpha_t + l_t \\
0 \\
1
\end{bmatrix} $$
2. Basic Profile of Concave Gear
The basic profile of the concave gear consists of three parts: tip chamfer line segment I, parabolic working segment II, and root arc segment III.
The position vector of line segment I in coordinate system $S_{na}$ can be expressed as:
$$ \mathbf{r}_{na}^{I}(l_a) = \begin{bmatrix}
M_x – l_a \cos\delta_e \\
M_y + l_a \sin\delta_e \\
0 \\
1
\end{bmatrix} $$
where $M_x$, $M_y$ are the coordinates of point M in $S_{na}$; $\delta_e$ is the angle between the chamfer line and the symmetry axis $x_{na}$.
The position vector of parabolic segment II in $S_{na}$ is:
$$ \mathbf{r}_{na}^{II}(t) = \begin{bmatrix}
t\cos\alpha – \frac{t^2}{2p}\sin\alpha + L\sin\alpha \\
\pm(t\sin\alpha + \frac{t^2}{2p}\cos\alpha – L\cos\alpha) + l_a \\
0 \\
1
\end{bmatrix} $$
where $t \in [t_{min}, t_{max}]$; $\alpha = (\alpha_{a1}+\alpha_{a2})/2$; $p = \rho_a \cos\theta$; $L = \rho_a \frac{\sin^2\theta}{2\cos\theta} + \rho_a \cos\theta$; $\theta = (\alpha_{a1}-\alpha_{a2})/2$.
The position vector of root arc segment III in $S_{na}$ is:
$$ \mathbf{r}_{na}^{III}(\theta_{ea}) = \begin{bmatrix}
\rho_{ea}(\cos\theta_{ea} – 1) \\
\rho_{ea} \sin\theta_{ea} + h_{f2} \\
0 \\
1
\end{bmatrix} $$
where $\rho_{ea}$ is the root arc radius; $\theta_{ea}$ is the parameter.
Tooth Surface Equation for Two-Point Meshing Gears
Based on the rack-generation equations and the basic profile equations, the complete tooth surface equations for the convex and concave gears in their respective attached coordinate systems ($S_{2t}$, $S_{2a}$) can be derived. For brevity, the explicit expanded forms are omitted here, but they follow the structure of $\mathbf{r}_{2}(t, \eta_2)$ as shown in the general rack-generation equations, with the specific $\mathbf{r}_{n}(t)$ and its derivative $\mathbf{r}’_{n}(t)$ substituted from the basic profile equations above.
Calculation of Tooth Surface for Cylindrical Geometric Elements Constructed Tooth Gear Shaving Cutter
Principle of Gear Shaving
The interaction between a shaving cutter and a gear during gear shaving can be regarded as the meshing of a pair of spatial crossed helical gears with zero backlash. During the shaving process, because the direction of motion velocity at any contact point between the shaving cutter and the gear being shaved is different, the cutting edges of the shaving cutter slide along the tooth surface of the gear, thereby generating cutting action. The principle of axial shaving involves the rotation of the shaving cutter driving the gear to rotate, while the shaving cutter simultaneously performs a slow feed motion along the gear’s axis of rotation to ensure the entire face width is shaved. When the axial motion changes direction, the rotation direction of the shaving cutter also reverses, and a vertical feed is applied along the shaving cutter axis. After all stock allowance is removed, several passes without vertical feed (spark-out passes) are performed to improve the surface quality of the shaved gear.
Coordinate Systems and Transformations
Coordinate systems are established as shown in the figure. $S_w(O_w-x_w,y_w,z_w)$ and $S_f(O_f-x_f,y_f,z_f)$ are fixed coordinate systems, where $z_w$ and $z_f$ represent the rotation axes of the shaving cutter and the gear, respectively. The $x_f$ axis coincides with the $x_w$ axis, and the angle between them, $\Sigma$, is the crossing angle between the shaving cutter and the gear. The distance between origins $O_w$ and $O_f$ is $a$, the center distance. Moving coordinate systems $S_q(O_q-x_q,y_q,z_q)$ and $S_2(O_2-x_2,y_2,z_2)$ are attached to the shaving cutter and gear, respectively, initially coinciding with $S_w$ and $S_f$.
The shaving cutter (1) rotates with angular velocity $\boldsymbol{\omega}^{(1)}$ about axis $z_w$, and the gear being shaved (2) rotates with angular velocity $\boldsymbol{\omega}^{(2)}$ about axis $z_f$. Simultaneously, the shaving cutter translates along the $z_w$ axis with velocity $\mathbf{v}_0^{(1)}$. When the shaving cutter rotates by angle $\phi_1$, the gear rotates by angle $\phi_2$, and $O_wO_q = l$.
The transformation matrix from $S_2$ to $S_w$ is:
$$ \mathbf{M}_{w2} = \begin{bmatrix}
\cos\phi_2 & -\sin\phi_2 & 0 & a\cos\phi_2 \\
-\sin\phi_2 \cos\Sigma & -\cos\phi_2 \cos\Sigma & \sin\Sigma & -a\sin\phi_2 \cos\Sigma \\
\sin\phi_2 \sin\Sigma & \cos\phi_2 \sin\Sigma & \cos\Sigma & a\sin\phi_2 \sin\Sigma \\
0 & 0 & 0 & 1
\end{bmatrix} $$
The transformation matrix from $S_f$ to $S_w$ is:
$$ \mathbf{M}_{wf} = \begin{bmatrix}
\cos\phi_1 & \sin\phi_1 & 0 & 0 \\
-\sin\phi_1 & \cos\phi_1 & 0 & 0 \\
0 & 0 & 1 & -l \\
0 & 0 & 0 & 1
\end{bmatrix} $$
The transformation between $S_2$ and $S_f$ is a simple rotation about $z$: $\mathbf{M}_{f2} = \mathbf{R}_z(\phi_2)$.
Using these transformations, the gear tooth surface equation $\mathbf{r}_2(t, \eta_2)$ can be expressed in the fixed coordinate system $S_f$ as $\mathbf{r}_f(t, \eta_2, \phi_2) = \mathbf{M}_{f2} \cdot \mathbf{r}_2(t, \eta_2)$.
Relative Velocity and Normal Vector
The angular velocity and translational velocity vectors of the shaving cutter (1) can be expressed in $S_f$ as:
$$
\begin{cases}
\boldsymbol{\omega}^{(1)} = \omega_1 (-\sin\Sigma \mathbf{j}_f + \cos\Sigma \mathbf{k}_f) \\
\mathbf{v}_0^{(1)} = v_{01} (-\sin\Sigma \mathbf{j}_f + \cos\Sigma \mathbf{k}_f)
\end{cases}
$$
Similarly, for the gear (2):
$$
\begin{cases}
\boldsymbol{\omega}^{(2)} = -\omega_2 \mathbf{k}_f \\
\mathbf{v}_0^{(2)} = \mathbf{0}
\end{cases}
$$
Let the position vector of a point $M$ on the gear tooth surface in $S_f$ be $\mathbf{r}_f = (x_f, y_f, z_f)$. The velocities of $M$ as it moves with body 1 and body 2 are:
$$
\begin{cases}
\mathbf{v}_f^{(1)} = \boldsymbol{\omega}^{(1)} \times (\mathbf{r}_f – a\mathbf{i}_f) + \mathbf{v}_0^{(1)} \\
\mathbf{v}_f^{(2)} = \boldsymbol{\omega}^{(2)} \times \mathbf{r}_f
\end{cases}
$$
The relative velocity at the contact point is $\mathbf{v}_f^{(12)} = \mathbf{v}_f^{(1)} – \mathbf{v}_f^{(2)}$. Its expanded form is:
$$ \mathbf{v}_f^{(12)} = \begin{bmatrix}
\omega_1(y_f\sin\Sigma – z_f\cos\Sigma) + \omega_2 y_f \\
-\omega_1 (x_f – a)\sin\Sigma – \omega_2 x_f – v_{01}\sin\Sigma \\
\omega_1 (x_f – a)\cos\Sigma + v_{01}\cos\Sigma
\end{bmatrix} $$
where $\omega_2 = i_{21} \omega_1$ and $i_{21} = Z_1 / Z_2$; $v_{01} = \omega_1 p_1$ for the kinematics of the helicoidal motion of the shaving cutter; $p_1$ is the helical parameter of the shaving cutter.
The normal vector to the helicoidal gear tooth surface in $S_2$ is given by $\mathbf{n}_2 = \frac{\partial \mathbf{r}_2}{\partial \delta} \times \frac{\partial \mathbf{r}_2}{\partial \lambda}$, where $\delta$ and $\lambda$ are the surface parameters (e.g., $t$ and $\eta_2$). Using the property of a helicoid, the normal vector can also be related to the transverse profile $\mathbf{r}_0(u)$ and its derivative. For the convex gear working segment II, the normal vector in $S_2$ is:
$$
\begin{cases}
n_{x2}^{II} = p_2 (\rho_t \beta \cos\alpha_t \cos\eta_{2t} \pm \rho_t \cos\alpha_t \sin\eta_{2t}) \\
n_{y2}^{II} = -p_2 (\rho_t \cos\alpha_t \sin\eta_{2t} \mp \rho_t \beta \cos\alpha_t \cos\eta_{2t}) \\
n_{z2}^{II} = \rho_t \sin\alpha_t (\rho_t \sin\alpha_t + r_{2t})
\end{cases}
$$
where $\beta = \cot\beta_t$. This normal vector is then transformed to $S_f$: $\mathbf{n}_f = \mathbf{M}_{f2} \cdot \mathbf{n}_2$.
Meshing Equation
Since the shaving cutter and the gear remain in contact during shaving, at any contact point $M$, the meshing equation must be satisfied:
$$ \Phi = \mathbf{n}_f \cdot \mathbf{v}_f^{(12)} = 0 $$
Substituting the expressions for $\mathbf{n}_f$ and $\mathbf{v}_f^{(12)}$ yields a complex equation. After significant algebraic manipulation and noting that $\omega_1$ and $v_{01}$ are arbitrary constants, the condition for the meshing equation to hold at all times leads to two independent equations. For the convex gear working segment II, the meshing equation reduces to a relationship involving the parameters:
$$ \sin\alpha_t [p_2 \beta \cos\alpha_t + (r_{2t} \beta – \rho_t \sin\alpha_t)\sin(\eta_{2t}-\phi_2)] – \cos\alpha_t (\rho_t \beta \cos\alpha_t – p_2 \sin\alpha_t) \cos(\eta_{2t}-\phi_2) = c_t n_{z2}^{II} $$
where $c_t = \frac{p_{1t}(1-\cos\Sigma)}{\rho_t (p_{1t} \sin\Sigma – p_{2t})}$ is a constant, with $p_{1t}$, $p_{2t}$ being the helical parameters of the shaving cutter and convex gear, respectively.
Let:
$$ U_t = \cos\alpha_t (\rho_t \beta \cos\alpha_t – p_2 \sin\alpha_t) $$
$$ V_t = \sin\alpha_t [p_2 \beta \cos\alpha_t + (r_{2t} \beta – \rho_t \sin\alpha_t)\sin(\eta_{2t}-\phi_2)] $$
$$ W_t = c_t n_{z2}^{II} $$
Then $(\eta_{2t}-\phi_2)$ can be solved as:
$$ \eta_{2t}-\phi_2 = f_1(\alpha_t) = \arcsin\left( \frac{V_t W_t \pm U_t \sqrt{U_t^2 + V_t^2 – W_t^2}}{U_t^2 + V_t^2} \right) – \arg(U_t, V_t) $$
Similarly, for the concave gear parabolic working segment II, the meshing equation becomes:
$$ \sin(\eta_{2a}-\phi_2) \beta_a \cos[t \cos\alpha + (L – t^2/(2p))\sin\alpha] + \cos(\eta_{2a}-\phi_2) [t \sin\alpha – (L – t^2/(2p))\cos\alpha] = c_a n_{z2a}^{II} $$
where $c_a$ is a constant for the concave pair. This allows solving $(\eta_{2a}-\phi_2) = f_1(t)$.
Shaving Cutter Tooth Surface Equation
The tooth surface of the shaving cutter is the envelope of the family of gear tooth surfaces in the coordinate system attached to the shaving cutter, under the relative motion defined by the shaving process. A point on the gear surface $\mathbf{r}_f(t, \eta_2, \phi_2)$ is also a point on the shaving cutter surface at the instant of contact. Using the meshing equation $\Phi=0$, one of the parameters (e.g., $\phi_2$) can be expressed as a function of the other gear surface parameters and the axial displacement $l$ of the shaving cutter: $\phi_2 = f_2(t, \eta_2, l)$ or $\phi_2 = f_2(\alpha_t, \eta_{2t}, l)$.
For the convex gear, from its tooth surface equation, $z_{2t}^{II} = p_{2t} \eta_{2t} + \text{const}$. Combining this with the solved relation $\eta_{2t}-\phi_2 = f_1(\alpha_t)$ from the meshing equation and another derived relation from the geometry of contact, $\eta_{2t}$ and $\phi_2$ can both be expressed as functions of the basic profile parameter $\alpha_t$ and the axial position $l$: $\eta_{2t} = g_1(\alpha_t, l)$, $\phi_2 = g_2(\alpha_t, l)$.
The corresponding angle of rotation for the shaving cutter is $\phi_1 = i_{21} \phi_2 + l/p_{1t}$.
Finally, the tooth surface of the convex gear shaving cutter in its own coordinate system $S_q$ is obtained by transforming the contact point from $S_f$ to $S_q$:
$$ \mathbf{r}_q(\alpha_t, l) = \mathbf{M}_{qw} \cdot \mathbf{M}_{wf} \cdot \mathbf{r}_f(\alpha_t, \eta_{2t}(\alpha_t, l), \phi_2(\alpha_t, l)) = \mathbf{M}_{q2} \cdot \mathbf{r}_2(\alpha_t, \eta_{2t}(\alpha_t, l)) $$
where $\mathbf{M}_{q2} = \mathbf{M}_{qw} \cdot \mathbf{M}_{wf} \cdot \mathbf{M}_{f2}^{-1}$. The explicit form is lengthy but computable.
Similarly, for the concave gear shaving cutter, the tooth surface equation in $S_q$ is a function of the parabola parameter $t$ and the axial position $l$: $\mathbf{r}_q(t, l)$.
Thus, the shaving cutter tooth surface is defined by two independent parameters: one from the basic gear tooth profile ($\alpha_t$ or $t$) and the other from the axial position of contact ($l$).
Design Example
Design Parameters for the Gear Pair and Shaving Cutter
The basic profile parameters for the cylindrical geometric elements constructed tooth gear pair are summarized in Table 1.
| Parameter | Symbol | Convex Gear | Concave Gear |
|---|---|---|---|
| Normal Module | $m_n$ (mm) | 6 | 6 |
| Number of Teeth | $Z_2$ | 6 | 30 |
| Helix Angle | $\beta$ (°) | 33.8217 | 33.8217 |
| Normal Pressure Angle | $\alpha_n$ (°) | 25 | 25 |
| Whole Depth | $h_2$ (mm) | 9.00 | 9.12 |
| Addendum | $h_{a2}$ (mm) | 7.404 | 0.996 |
| Dedendum | $h_{f2}$ (mm) | 1.596 | 8.124 |
| Tooth Thickness | $s_{a2}$ (mm) | 9.24 | 9.60 |
| Face Width | $b_2$ (mm) | 80 | 80 |
| Profile Curvature Radius | $\rho_2$ (mm) | 9 | 9 |
| Profile Center Offset | $l_2$ (mm) | 3.537 | 3.578 |
| Root Fillet Radius | $\rho_g$ (mm) | 2.4 | 2.712 |
Table 1: Basic profile parameters for cylindrical geometric elements constructed tooth gear pair.
The main transmission parameters for the gear shaving process are summarized in Table 2.
| Parameter | Symbol | Convex Shaving Cutter | Concave Shaving Cutter |
|---|---|---|---|
| Shaving Cutter Teeth | $Z_1$ | 41 | 30 |
| Crossing Angle | $\Sigma$ (°) | 14 | 14 |
| Shaving Cutter Face Width | $b_1$ (mm) | 50 | 50 |
Table 2: Main parameters for gear shaving engagement.
Modeling Method and 3D Models
Using the derived mathematical models and the parameters from Tables 1 and 2, the tooth surface data points for both the gear pair and the corresponding shaving cutters were calculated via programmed algorithms in Matlab. These data points were then imported into Unigraphics NX 3D CAD software. Surfaces were constructed from the point clouds, then trimmed, stitched, and solidified to create precise solid models. The resulting 3D models of the convex gear, concave gear, convex gear shaving cutter, and concave gear shaving cutter were successfully generated, validating the theoretical design.
Software Development for Cylindrical Geometric Elements Constructed Tooth Gear Shaving Cutter Design
The design process for a cylindrical geometric elements constructed tooth gear shaving cutter is complex, involving numerous gear parameters, intricate formulas, and computationally intensive mathematical models for the cutter tooth surface. Design iterations are often required to optimize parameters. Integrating computer technology with the shaving cutter design process by developing specialized software leverages the powerful computational and visualization capabilities of computers to handle these complex tasks. This significantly improves design efficiency and accuracy, shortens product development cycles, and facilitates the broader application of geometric elements constructed tooth gears in power transmission fields.
Software Design and Development Tools
The design software was developed using the Python programming language combined with the Eric7 Integrated Development Environment (IDE). The graphical user interface (GUI) and application framework were built upon the mainstream PyQt6 library for Windows applications.
Python was chosen for its simplicity, readability, and strong support for scientific computing libraries (e.g., NumPy, SciPy, SymPy). Its object-oriented nature facilitates modular and maintainable code. PyQt6 provides a robust, cross-platform framework for creating sophisticated GUIs, with its signal/slot mechanism enabling effective interaction between the interface and the core algorithms. Eric7 is a full-featured IDE for Python and Ruby that integrates a powerful editor, debugger, and Qt designer, streamlining the development process.
Overall Software Development Workflow
The development followed a standard software engineering lifecycle: Requirements Analysis, Software Design (including Algorithm Design and UI Design), Coding, and Testing.
1. Requirements Analysis: The core requirement is to provide a tool that automates the design and analysis of cylindrical geometric elements constructed tooth gears and their shaving cutters. Key functional points were identified:
– Input interfaces for basic gear parameters with recommended default values.
Generation of tooth surface data points and geometric parameters for both standard and modified geometric elements constructed tooth gears.
– Generation of tooth surface data points and geometric parameters for the corresponding shaving cutters.
– Strength verification calculations (contact fatigue, bending fatigue).
– A configuration file for setting defaults like file save paths and data point density.
– Export of data point files (in .dat format recognizable by 3D software like UG NX) and gear parameter reports to user-specified locations.
2. Algorithm Design: The core algorithms implement the mathematical models derived in the theoretical sections. Key challenges and solutions included:
– Translating complex mathematical equations (e.g., meshing equations, conjugate curve equations) into efficient Python code.
– Implementing numerical methods like the bisection method to solve transcendental equations (e.g., for solving the operating pressure angle $\alpha’$ in the equation: $\inv\alpha’ = \inv\alpha_t + \frac{2(x_1+x_2)\tan\alpha_n}{z_1+z_2}$).
– Writing routines for the iterative calculation of the shaving cutter tooth surface coordinates based on the two-parameter ($\alpha_t$ or $t$, and $l$) solution.
– Developing logic for recommending key design parameters, such as the starting point of the contact path and the envelope surface radius $h$, based on geometric constraints (e.g., $h_{recommended} = (0.8 \sim 0.95) \rho_{min}$).
3. User Interface Design and Interaction: The GUI was designed using Qt Designer. The main window employs a `QTabWidget` to separate functionality for standard gear design and shaving cutter design. Within each tab, `QGroupBox` widgets organize input parameters logically. User input is captured via `QLineEdit` and `QComboBox` widgets. The core computation is triggered by `QPushButton` clicks (e.g., “Calculate”, “Generate Data Points”). The signal/slot mechanism connects these button clicks to the corresponding algorithmic functions. A file dialog (`QFileDialog`) allows users to choose the export path.
Software Function Realization
The developed software successfully realizes the planned functions. The main interface for standard geometric elements constructed tooth gear parameter design is shown in the accompanying figure (conceptual description). It includes sections for:
– Basic Gear Parameters: Input for module, pressure angles, number of teeth, helix angle, profile shift coefficients, addendum/dedendum coefficients, etc.
– Gear Pair Design Tools: A separate utility for preliminary gear pairing calculations based on center distance and ratio, with results transferable to the main input fields.
– Parameter Recommendation & Input: Buttons to calculate and display recommended ranges for critical geometric elements constructed tooth gear parameters (contact path start/end, envelope radius). The user can then input chosen values within these ranges.
– Action Buttons: Buttons for “Strength Check”, “Shaving Cutter Design”, and “Generate Data Points”. The “Shaving Cutter Design” section has a similar interface but includes parameters specific to the shaving process (cutter tooth number, crossing angle, etc.).
Upon clicking “Generate Data Points”, the software executes the algorithms, computes thousands of 3D coordinates defining the left and right tooth flanks, and exports them as separate .dat files. A text file containing all calculated geometric parameters (pitch diameters, tip diameters, root diameters, center distance, etc.) is also generated. These files are ready for direct import into 3D CAD software to build the solid model, as demonstrated in the design example.
Test Report
A comprehensive test was conducted on the software. The test environment consisted of standard PC hardware (AMD processor, 8GB RAM) running the Windows operating system. The testing focused on functional verification, input validation, computational accuracy, and file export functionality.
Conclusions: All primary functional modules of the “Cylindrical Geometric Elements Constructed Tooth Gear Shaving Cutter Design Software” passed the functional tests. The software fully implements the requirements specified during the analysis phase. The interface is user-friendly and intuitive. No critical (crash-inducing) errors were encountered during testing. Some minor and moderate issues (e.g., specific input validation edge cases) identified during testing were resolved. The software is functionally capable of meeting practical design needs, significantly streamlining the design process for these specialized gears and their shaving cutters.
Experimental Study on Gear Shaving of Cylindrical Geometric Elements Constructed Tooth Gears
This section describes the experimental validation of the proposed shaving cutter design theory. The developed software was used to design a specific gear pair and its corresponding shaving cutters. The shaving cutters were then manufactured, and they were employed to perform gear shaving on pre-hobbed geometric elements constructed tooth gears. The improvement in surface finish was quantitatively measured to verify the effectiveness of the shaving process.
Shaving Process Design
A design case was defined with the following requirements: input speed 2000 rpm, input power 30 kW, center distance 125 mm. Using the developed software, a cylindrical geometric elements constructed tooth gear pair and its matching convex and concave shaving cutters were designed. Key parameters for the shaving process are listed in Table 3.
| Parameter | Symbol | Convex Shaving | Concave Shaving |
|---|---|---|---|
| Spindle Speed | $s_n$ (rpm) | 2000 | |
| Feed Rate | $s_v$ (mm/min) | 50 | |
| Offset Distance | $E$ (mm) | 9 | 9 |
Table 3: Key parameters for the shaving process.
The 3D models of the gear pair and the shaving cutters were generated from the software output, and based on these models, physical shaving cutters were manufactured. The tooth profiles of the actual manufactured shaving cutters matched the design specifications.
Machining Process and Results
1. Gear Hobbing: The gear blanks (material: 40Cr, hardness 30-33 HRC after heat treatment) were first machined on a Y3180 hobbing machine to their basic shape, leaving the required stock allowance for shaving.
2. Gear Shaving: The pre-hobbed gears were then finished on a Y4232CNC2 CNC shaving machine. This machine features digital servo drives and ball screws for high positioning accuracy and stable machining precision. The physical shaving cutter was mounted on the machine spindle, and the workpiece was clamped on the worktable. After proper alignment and setting of the crossing angle ($\Sigma = 14^\circ$), the CNC program (generated based on the process parameters) was executed to perform axial shaving. The process successfully removed material from the gear tooth surfaces.
Measurement of Tooth Surface Roughness
To quantitatively assess the effect of gear shaving, the surface roughness ($R_a$) of the gear teeth was measured before and after the shaving operation using an Alicona InfiniteFocus G5 optical 3D measurement instrument. This instrument uses focus-variation technology to provide high-resolution, non-contact 3D surface topography data, including accurate roughness measurements.
Results:
– Convex Gear: The surface roughness improved from $R_a = 1.5362 \mu m$ (after hobbing) to $R_a = 0.6276 \mu m$ (after shaving).
– Concave Gear: The surface roughness improved from $R_a = 1.5127 \mu m$ (after hobbing) to $R_a = 0.6563 \mu m$ (after shaving).
The results show a clear and significant reduction in surface roughness for both gears after the shaving process, achieving a finish well within the targeted range of below $0.8 \mu m$. This experimental evidence confirms that the designed shaving cutters based on the derived theory are effective and that the gear shaving process is a viable and successful finishing method for cylindrical geometric elements constructed tooth gears.
Conclusions and Future Work
Main Conclusions and Innovations
This research focused on the cylindrical geometric elements constructed tooth gear, investigating its fundamental meshing theory, the design theory and methodology for its shaving cutter, and the development of corresponding design software. The designed shaving cutter was successfully used in experiments, validating the correctness of the design.
Main Conclusions:
1. The mathematical model for the basic meshing principle of cylindrical geometric elements constructed tooth gears was established, clarifying the concept of conjugate curves, coordinate systems, relative velocity, and leading to the meshing equation and conjugate curve equation. Based on this, tooth surface equations for gears generated from basic rack profiles (arc and parabola) were derived using the rack-generation method.
2. Based on the kinematics of the shaving process and spatial point-contact meshing theory, the coordinate systems for shaving engagement were established. The relative velocity and normal vectors were derived, leading to the meshing equation and, crucially, the mathematical model for the shaving cutter tooth surface. This provides the theoretical basis for the shaving process for these gears. A complete 3D modeling method was presented and demonstrated with a design example.
3. The design software for the cylindrical geometric elements constructed tooth gear shaving cutter was successfully developed, debugged, and tested. It automates parameter calculation, optimization, strength verification, and data point generation, greatly improving design efficiency.
4. The shaving cutters designed by the software were used to successfully shave geometric elements constructed tooth gears. Surface roughness measurements confirmed a dramatic improvement from about 1.6 $\mu m$ to below 0.8 $\mu m$, proving the practical correctness and feasibility of the gear shaving method for these gears.
Innovation: The primary innovation lies in establishing the mathematical model for the shaving cutter of a typical two-point meshing geometric elements constructed tooth gear, developing the corresponding design software, and completing experimental verification. This lays a theoretical and experimental foundation for promoting the application of geometric elements constructed tooth gears.
Future Work Outlook
While this study established the theory and initial experimental validation for shaving geometric elements constructed tooth gears, several areas warrant further investigation:
1. Integrate Tooth Contact Analysis (TCA) algorithms into the design software. This would allow designers to simulate the transmission error and contact pattern of the shaved gear pair under load, enabling performance prediction and further optimization of the shaving cutter design before manufacturing.
2. Conduct comprehensive transmission performance experiments on the shaved geometric elements constructed tooth gear pair. Comparative tests against other gear types (e.g., standard involute gears) under controlled conditions (load, speed, lubrication) would quantitatively validate the claimed advantages in load capacity, efficiency, and noise/vibration.
3. Investigate the wear characteristics and service life of the specialized shaving cutters under production conditions to optimize their geometry and material for cost-effective manufacturing.