The modified semi-generating (MSG) method, widely employed for hypoid gear cutting, offers a balance between production efficiency and quality. Traditionally, a calculation method known as the ‘GM’ method has been used to determine machine settings for gear cutting. To simplify calculations, this method selects the calculation reference point for pinion gear cutting as the mid-point of the face width on the pitch cone (point P). However, this point often does not coincide with the desired center of the tooth contact pattern (point M). This inherent theoretical discrepancy necessitates time-consuming trial-and-error adjustments on the machine tool based on contact pattern observation to achieve the correct meshing performance.
Based on established principles of gear meshing theory, this paper presents a refined computational approach. The core improvement is to designate the contact pattern center M as the fundamental reference point for all pinion gear cutting calculations. This modification directly addresses the shortcoming of the traditional method. This new methodology can be referred to as the GMM method. Notably, the traditional GM method’s results can be derived as a specific case of this new approach where the reference point offset is zero. This document uses terminology consistent with standard gear theory literature.
Machine Tool Adjustment Parameters for Cutting the Gear (Gear 2)
When the gear ratio is relatively high, the larger gear (Gear 2) is typically finished via a non-generated (formate) cutting process. The pinion is then generated to match Gear 2. The design reference point P2 corresponds to the mid-point of the face width on Gear 2’s pitch cone. The equidistant conjugate tooth surfaces of Gear 1 and Gear 2, Σ(1) and Σ(2), are tangent at point P. To achieve the specified root angle δf2 for Gear 2 during gear cutting, we introduce a pair of imaginary gears, Gear 2′ and Gear 1′. These share axes with the real gears, possess the same pair of equidistant conjugate surfaces Σ(1) and Σ(2), but Gear 2′ has a pitch angle δ2′ equal to δf2. Its reference point P2′ corresponds to point P(2) on Σ(2).
Given the basic machine setup parameters (cutter radius Rc02, blade angle αc2, etc.), the pitch cone geometry of the imaginary gears (δ1′, R1′, R2′) can be derived. The pressure angle αn2′ at point P2′ is calculated considering the asymmetric tooth form of hypoid gears:
$$
\alpha_{n2′} = \alpha_{n0} \pm \Delta \alpha_{n2′}
$$
where αn0 is the limit pressure angle for the imaginary gear pair, and the upper sign is for the convex side, the lower for the concave side. The difference between the mean blade angle and this pressure angle is:
$$
\Delta \alpha_c = \frac{\alpha_{ci} + \alpha_{ce}}{2} – \alpha_{n2′}
$$
To compensate for Δαc, on a machine with a tilting cutter head, the cutter is tilted by an angle ic2. On machines without this capability, the gear blank is rotated about the tangent to the tooth line at point P(2) by an angle Δθ. This paper focuses on the latter method, as illustrated in the geometric relationship below. From this geometry, the key adjustment parameters for finishing Gear 2 are derived.
The root cone distance at point P2 is: Rf2 = R2′ cos δf2 / cos δ2′.
The spiral angle at point P(2) after tilting the blank by Δθ is: β2 = β2′ + Δβ, where Δβ = arctan(sin β2′ tan Δθ).
The machine root angle setting for cutting Gear 2 is: δm2 = δf2 – Δβ.
The machine center setting (sliding base) adjustment ΔXB2 is: ΔXB2 = Rf2 sin Δβ / cos δm2.
The blank offset adjustment ΔE2 is: ΔE2 = Rf2 (1 – cos Δβ) / cos δm2.
The cutter radial setting Sr2 and initial angular position θc2 are calculated as:
$$
S_{r2} = \sqrt{(R_{c02} \cos \alpha_{c2})^2 + (R_{c02} \sin \alpha_{c2} \pm \Delta S)^2}
$$
$$
\theta_{c2} = \arctan\left(\frac{R_{c02} \sin \alpha_{c2} \pm \Delta S}{R_{c02} \cos \alpha_{c2}}\right)
$$
where ΔS = (R2′ sin β2′ / cos δ2′) tan Δθ. The specific calculation of the cutter position depends on the machine tool model.
Geometric Parameters of the Imaginary Gears 1′ and 2′
The equidistant surfaces Σ(1) and Σ(2), offset by a distance corresponding to the cutter edge radius, are tangent at point P’ (for the GM method) or point M (for the proposed GMM method). The offset distance l is given by:
$$
l = \frac{r_{c0}}{\sin \alpha_c} \mp \rho_a
$$
where ρa is the cutter edge radius.
The position of point M(2) on Σ(2) is defined relative to P(2) by two shifts: a profile shift Δh (positive towards the root) and a lengthwise shift Δl (positive towards the toe). These are determined from the desired contact pattern location. For a pattern centered at mid-working depth:
$$
\Delta h = \pm \left( \frac{h_w}{2} – k_{a2} m_t \right)
$$
where hw is the mean working depth and ka2 is the addendum coefficient of Gear 2.
The coordinates (rM, zM) of M(2) in the coordinate system attached to Gear 2 are:
$$
r_M = r_{P^{(2)}} – \Delta h \sin \alpha_{n2′} \mp \Delta l \cos \beta_{2′} \cos \alpha_{n2′}
$$
$$
z_M = z_{P^{(2)}} + \Delta h \cos \alpha_{n2′} \mp \Delta l \cos \beta_{2′} \sin \alpha_{n2′}
$$
where Δl’ = Δl / cos δ2′.
Using vector functions, the equation of surface Σ(2) and its unit normal can be established in the cutting machine coordinate system. By applying coordinate transformations and solving the equation of meshing between Σ(1) and Σ(2), the parameters (u, θ) defining the corresponding point M(1) on the pinion’s conjugate surface Σ(1) can be determined iteratively. This process is fundamental for establishing the precise geometric relationship needed for subsequent pinion gear cutting calculations.
Machine Tool Adjustment Parameters for Cutting the Pinion (Gear 1)
Following the principle of local contact, the pinion tooth curvature is modified. While the formulas for curvature correction in the GMM method are similar to those in the traditional GM method, the key curvature correction coefficient C is recalculated based on the new reference point M. This coefficient is vital for determining the modified machine settings that will produce the desired localized contact pattern.
The pressure angle αn1′ at the meshing point M on the imaginary pinion surface is found from the scalar product of the surface unit normal and the unit normal of the pitch plane:
$$
\alpha_{n1′} = \arcsin(\mathbf{n}^{(1)} \cdot \mathbf{k}_q)
$$
where kq is the unit vector normal to the pitch plane at M.
The main curvatures and geodesic torsion of the imaginary pinion surface Σ(1) at point M are calculated. These are then used to find the equivalent curvatures (KLs, KRs) along the tooth lengthwise and profile directions of the real pinion to be cut. The relationship involves the induced normal curvature from the gear mating and the chosen curvature correction parameters.
A crucial parameter is the angle ξ1 from the first principal direction on surface Σ(1) to the tooth lengthwise direction. This is derived from the surface’s tangent vectors and the defined tooth line orientation at M. The formula is:
$$
\xi_1 = \arctan\left( \frac{ \partial \mathbf{R}^{(1)} / \partial \theta \cdot \mathbf{e}_t }{ \partial \mathbf{R}^{(1)} / \partial u \cdot \mathbf{e}_t } \right)
$$
where et is the unit vector along the tooth line.
The machine settings for finishing the pinion—including the cutter tilt angle ic1, swivel angle jc1, machine root angle δm1, sliding base ΔXB1, blank offset ΔE1, and cradle angle increment Δφ1—are then computed using formulas that incorporate the geometric parameters of point M, the calculated curvatures KLs and KRs, and the angle ξ1. The detailed formulas for these adjustments are consistent with established modified roll methods for gear cutting.
Computational Example for Pinion Gear Cutting Adjustments
To illustrate the differences between the proposed GMM method and the traditional GM method, a standard design example is used. Calculations are performed in metric units. The following table shows the machine adjustment parameters for finishing the gear (Gear 2).
| Parameter | Symbol | Concave Side | Convex Side |
|---|---|---|---|
| Spiral Angle | β2 | 34.500° | 34.500° |
| Machine Root Angle | δm2 | 72.831° | 72.831° |
| Sliding Base Adjustment | ΔXB2 | -0.254 mm | -0.254 mm |
| Blank Offset Adjustment | ΔE2 | 0.008 mm | 0.008 mm |
| Cutter Radial Setting | Sr2 | 114.300 mm | 114.300 mm |
| Cutter Initial Angle | θc2 | 15.000° | 15.000° |
The next table presents the machine adjustment parameters for finishing the pinion (Gear 1) calculated using the GMM method for three different sets of reference point offsets (Δh, Δl). The first row (Δh=0, Δl=0) corresponds to the traditional GM method results.
| Parameter | Symbol | Concave Side (αc1 = 22°) | Convex Side (αc1 = 18°) | ||||
|---|---|---|---|---|---|---|---|
| Case 1: Δh=0, Δl=0 | Case 2: Δh=+0.5mm, Δl=0 | Case 3: Δh=0, Δl=+1.0mm | Case 1: Δh=0, Δl=0 | Case 2: Δh=-0.5mm, Δl=0 | Case 3: Δh=0, Δl=+1.0mm | ||
| Cutter Tilt Angle | ic1 | 45.112° | 45.205° | 45.108° | 42.331° | 42.255° | 42.307° |
| Swivel Angle | jc1 | 1.805° | 1.887° | 1.782° | 2.995° | 2.915° | 2.947° |
| Machine Root Angle | δm1 | 16.669° | 16.654° | 16.637° | 16.669° | 16.684° | 16.679° |
| Sliding Base | XB1 | -3.048 mm | -3.058 mm | -3.045 mm | -3.048 mm | -3.038 mm | -3.044 mm |
| Blank Offset | E1 | 19.050 mm | 19.049 mm | 19.039 mm | 19.050 mm | 19.051 mm | 19.045 mm |
| Cutter Radius (Inner/Outer) | Rci1/Rce1 | 76.200 mm | 76.197 mm | 76.197 mm | 76.200 mm | 76.203 mm | 76.203 mm |
| Cradle Angle Increment Coeff. | C | 0.4125 | 0.4131 | 0.4125 | 0.4125 | 0.4119 | 0.4123 |
Tooth contact analysis (TCA) performed by inputting these adjustment parameters into computer software confirms the correctness of the derived formulas. A critical observation from the table is the significant change in the calculated inner cutter tip radius for the pinion convex side when the reference point is shifted in the profile direction (Δh). This demonstrates that if the traditional GM method’s reference point P’ is used, but the contact center is required to be at a different point M, it is very difficult to control the size of the pinion’s convex side contact pattern during the gear cutting process. The new GMM method, by directly using the desired contact center M as the calculation reference point, allows for precise pre-determination of machine settings. This significantly reduces the time required for trial-and-error adjustments on the machine tool and ensures consistent, high-quality gear meshing performance. This approach represents a fundamental improvement in the computational preparation for the modified semi-generating gear cutting of hypoid gears.