1. Research Background and Motivation
Gear transmission is the most widely used power-transmitting mechanism in the whole field of mechanical engineering. Because of its high transmission efficiency, large torque capacity, long service life and compact structure, it has become one of the symbols of modern industrialisation, and it plays an irreplaceable role in aerospace, automobile manufacturing, construction machinery, metallurgy and mining. The transmission performance of a cylindrical gear is governed mainly by two factors: the tooth profile and the tooth trace. The involute profile is dominant because of its favourable meshing properties and its simple manufacture, whereas the tooth trace directly determines the load distribution along the face width.
Straight spur gears enter mesh and leave mesh simultaneously along the whole face width, so a small parallelism error of the two axes immediately concentrates the load at one end of the tooth. Helical gears distribute the load more gradually and therefore run more smoothly, but the inclined contact line produces an axial force that shortens bearing life. Herringbone gears cancel the axial force by symmetry, yet the undercut at the central section makes them difficult to manufacture and the left and right tooth flanks are not always exactly symmetrical. Curved tooth trace gears, and especially circular-arc tooth trace gears, combine smooth engagement with a high load-carrying capacity, but the existing manufacturing routes remain either slow or inaccurate, which prevents their industrial application.
In parallel, the upstream forming chain of gear production is also changing. Near-net-shape forged gear blanks are increasingly preferred because they reduce the allowance that has to be removed by cutting, refine the grain flow along the tooth root, and shorten the total process chain. Nevertheless, the economic benefit of forged gear blanks can only be realised if the subsequent tooth cutting operation is fast and accurate enough. A machining method that removes the whole face width in a single continuous pass is therefore highly desirable, because it matches the short cycle time of forged gear blanks and preserves the near-net-shape advantage of the forging process.
For this reason I propose a new type of curved tooth trace gear, the planar spiral toothed cylindrical gear, together with a dedicated high-efficiency machining method and the corresponding machining device. The tooth trace of this gear is a planar spiral, the profile at the mid-section of the face width is a standard involute, and the concave flank has a slightly larger radius of curvature than the convex flank. This asymmetry produces a slightly crowned tooth that keeps the contact zone in the middle 80 %–90 % of the tooth surface, suppresses edge loading, and tolerates a certain amount of manufacturing and assembly error.
2. Geometrical Definition of the Planar Spiral Tooth Trace
The planar spiral toothed cylindrical gear is defined by five principal parameters: the module, the number of teeth, the pressure angle, the face width and the radius of curvature of the spiral at the mid-section of the face width. On the pitch cylinder the concave and convex tooth traces are separated by a constant distance, so that
$$
R_{N}-R_{W}=\frac{\pi m}{2}
$$
where the subscripts refer to the concave and convex flanks respectively. The trace of the pitch cylinder, developed on a plane, is described by
$$
\begin{cases}
x_{DP}=\pm\left(R_{T}\mp\dfrac{\pi m}{4}\right)\cos\theta\\[6pt]
z_{DP}=\pm\left(R_{T}\mp\dfrac{\pi m}{4}\right)\sin\theta
\end{cases}
$$
in which the upper and lower signs correspond to the concave and convex flanks, the parameter denotes the angular position of a point on the trace with respect to the line joining the trace centre to the disc centre, and the spiral radius of the cutter is the governing quantity. A pair of gears must consist of one left-hand and one right-hand member, exactly as in the helical case. Along the mid-section of the face width the circular tooth thickness and the circular space width are equal and amount to half the pitch,
$$
s_{C}=e_{C}=\frac{\pi m}{2}
$$
At every other section the space width is larger than the tooth thickness. This is the geometric origin of the crowned shape: when the pair is loaded, the flanks touch first near the mid-section and the contact zone spreads towards the ends only under increasing load, so that the root fillets at the two ends are relieved.
| Quantity | Symbol | Relation | |
|---|---|---|---|
| Transverse module | $m$ | standard value | |
| Pressure angle | $\alpha$ | $20^{\circ}$ | |
| Number of teeth | $z$ | integer | |
| Pitch diameter | $d$ | $d=mz$ | |
| Base diameter | $d_{b}$ | $d_{b}=d\cos\alpha$ | |
| Addendum diameter | $d_{a}$ | $d_{a}=(z+2h_{a}^{*})m$ | |
| Dedendum diameter | $d_{f}$ | $d_{f}=(z-2h_{a}^{*}-2c^{*})m$ | |
| Circular pitch | $p$ | $p=\pi m$ | |
| Base pitch | $p_{b}$ | $p_{b}=\pi m\cos\alpha$ | |
| Tooth thickness at pitch circle | $s$ | $s=\pi m/2$ | |
| Space width at pitch circle | $e$ | $e=\pi m/2$ | |
| Trace radius, mid-section | $R_{T}$ | $R_{T}\geq 1.5B$ | |
| Face width | $B$ | $B<d$ | </d$ |
3. Static and Dynamic Tooth Surface Equations
The generating surface of the disc cutter is a surface of revolution. In the coordinate system attached to the cutter disc it reads
$$
\begin{cases}
x_{D}=\left(R_{T}\pm\dfrac{\pi m}{4}\mp u\sin\alpha\right)\cos\vartheta\\[6pt]
y_{D}=\left(R_{T}\pm\dfrac{\pi m}{4}\mp u\sin\alpha\right)\sin\vartheta\\[6pt]
z_{D}=u\cos\alpha
\end{cases}
$$
After the depth-of-cut pass the profile is obtained by rotating this surface about the workpiece axis through the angle
$$
\mu=\frac{\theta_{T}-\theta_{k}+2k\pi}{z}
$$
where the integer enumerates the tooth spaces. Introducing the transformation matrices between the disc frame, the workpiece frame and the stationary frame,
$$
\mathbf{M}_{CD}=
\begin{bmatrix}
1 & 0 & 0 & -R_{T}\\
0 & 0 & -1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 1
\end{bmatrix},
\qquad
\mathbf{M}_{\mu}=
\begin{bmatrix}
\cos\mu & \sin\mu & 0 & 0\\
-\sin\mu & \cos\mu & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}
$$
the tooth surface produced by the depth-of-cut pass is
$$
\begin{cases}
x=\cos\mu\left[\left(R_{T}\pm\dfrac{\pi m}{4}\mp u\sin\alpha\right)\cos\vartheta-R_{C}\right]\pm\sin\mu\left(u\cos\alpha\mp R_{T}\right)\\[6pt]
y=-\sin\mu\left[\left(R_{T}\pm\dfrac{\pi m}{4}\mp u\sin\alpha\right)\cos\vartheta-R_{C}\right]\pm\cos\mu\left(u\cos\alpha\mp R_{T}\right)\\[6pt]
z=\left(R_{T}\pm\dfrac{\pi m}{4}\mp u\sin\alpha\right)\sin\vartheta
\end{cases}
$$
During generating, the cutter is indexed to the angular position of the space and the parameter of rotation vanishes, so the surface of revolution itself becomes the meshing surface. Its unit normal is obtained from the cross product of the two first derivatives,
$$
\mathbf{e}_{D}=\cos\vartheta\,\mathbf{i}-\sin\vartheta\,\mathbf{j}\pm\sin\alpha\,\mathbf{k}
$$
and the meshing condition follows from the requirement that the relative velocity be perpendicular to the common normal,
$$
\mathbf{e}_{D}\cdot\mathbf{v}^{12}=0
$$
Writing the velocity of the cutter point and the velocity of the workpiece point in the stationary frame and substituting the normal, the parameter along the cutting edge is found to be
$$
u=\frac{\sin\alpha\left[\left(R_{T}+\dfrac{\pi m}{4}\right)\cos\vartheta-R_{C}\varphi-R_{T}\right]}{\cos\vartheta}
$$
Substitution of this expression into the surface of revolution yields the contact line, and a further rotation about the workpiece axis yields the complete tooth flank. The resulting family of surfaces is controlled by the cutter radius, the pitch radius, the angular parameter, the generating angle, the module and the number of teeth. The static surface equation is the basis of every subsequent step: the numerical plotting of the flank, the parametric solid model, the finite element mesh and the verification of the machined sample.
For a meshing pair the two members rotate in opposite directions with the ratio
$$
\frac{\xi_{1}}{\xi_{2}}=\frac{R_{2}}{R_{1}}=\frac{z_{1}}{z_{2}}
$$
and the dynamic flank equations are obtained by replacing the angular parameter with its instantaneous value in the stationary frame. The contact line of the pair is the intersection of the dynamic flank with the plane of action, whose equation is
$$
\frac{y_{L}-R_{C1}\sin 20^{\circ}}{R_{C1}\sin 20^{\circ}}=\frac{x_{L}-R_{C1}\cos 20^{\circ}}{R_{C1}\cos 20^{\circ}-a_{l}}
$$
where the centre distance is denoted by the last symbol. Solving this system simultaneously for the two members gives the instantaneous contact line, whose position moves across the face width as the pair rotates.
4. Contact Ratio of the Planar Spiral Gear
The total contact ratio is the sum of a transverse and an axial contribution,
$$
\varepsilon_{\gamma}=\varepsilon_{\alpha}+\varepsilon_{\beta}
$$
The transverse part is computed exactly as for an involute spur gear,
$$
\varepsilon_{\alpha}=\frac{1}{2\pi}\left[z_{1}\left(\tan\alpha_{a1}-\tan\alpha_{t}\right)+z_{2}\left(\tan\alpha_{a2}-\tan\alpha_{t}\right)\right]
$$
while the axial part reflects the additional length of the meshing zone created by the curved trace,
$$
\varepsilon_{\beta}=\frac{\Delta L}{p_{bt}}=
\frac{2R_{b1}\arcsin\dfrac{B}{2R_{b1}}-2R_{b2}\arcsin\dfrac{B}{2R_{b2}}}{\pi m\cos\alpha}
$$
Combining the two contributions gives
$$
\varepsilon_{\gamma}=
\frac{1}{2\pi}\left[z_{1}\left(\tan\alpha_{a1}-\tan\alpha_{t}\right)+z_{2}\left(\tan\alpha_{a2}-\tan\alpha_{t}\right)\right]
+\frac{2R_{b1}\arcsin\dfrac{B}{2R_{b1}}-2R_{b2}\arcsin\dfrac{B}{2R_{b2}}}{\pi m\cos\alpha}
$$
Two conclusions follow immediately. First, the axial contribution can be increased by widening the face width or by reducing the radius of the spiral, which is why the ratio of the spiral radius to the face width must be kept above a lower bound. Second, if the spiral radius becomes too small the circumferential tooth thickness at the two ends of the tooth becomes thinner than at the mid-section, the flank loses its crowning and the load-carrying advantage disappears. The admissible window is therefore
$$
R_{T}\geq 1.5B
$$
which I adopted as a design constraint throughout the present work.
5. Parametric Modelling System Based on SolidWorks Secondary Development
An accurate solid model is the prerequisite of any finite element, interference or meshing analysis. For straight and helical gears several commercial toolkits already exist, but for a curved tooth trace gear with a variable spiral the manual modelling effort is prohibitive. I therefore developed a dedicated parametric modelling system by coupling the SolidWorks API with a Visual Basic environment through the COM interface. The program drives the CAD kernel directly, so the user only enters the basic parameters and the system builds the complete model automatically.
The modelling sequence is as follows. A cylindrical blank is first created about the origin. A datum plane is then inserted at the mid-section of the face width and the involute space profile is generated from a set of discrete points obtained from the involute equations, after which the addendum circle, the dedendum circle and the root fillet are added to close the sketch. Twenty-five equally spaced datum planes are subsequently created along the face width, and on each of them the same space profile is placed with an angular offset computed from the planar spiral equation. A lofted cut through the twenty-five sections produces one tooth space, and a circular pattern finally reproduces it around the whole circumference. The shaft bore is cut last.
| Parameter | Symbol | Value |
|---|---|---|
| Number of teeth | $z$ | 32 |
| Module | $m$ | 4 mm |
| Pressure angle | $\alpha$ | $20^{\circ}$ |
| Face width | $B$ | 30 mm |
| Addendum coefficient | $h_{a}^{*}$ | 1.0 |
| Bore diameter | $d_{0}$ | 20 mm |
| Minimum cutter mounting radius | $R_{T}$ | 250 mm |
| Number of cutters on the disc | $n$ | 8 |
With this system a complete gear can be generated in a few seconds, and a pair of gears with different tooth numbers and opposite hands can be produced in one session. The accuracy of the generated solid was checked against the analytical surface obtained from the equations of the preceding section, and the deviation remained inside the modelling tolerance over the whole flank. The system therefore removes the bottleneck that previously prevented systematic study of curved tooth trace gears, and it provides the geometric input for every simulation reported below.
To confirm the meshing behaviour of the ideal geometry, a pair with thirty and sixty teeth, a face width of fifty millimetres and a cutter radius of one hundred and fifty millimetres was generated and assembled in the contact analysis module. The computed contact zone was confined to the middle 80 %–90 % of the flank, which agrees with the theoretical prediction and confirms that the crowned shape is reproduced correctly by the modelling system.
6. Anti-Misalignment Behaviour of the Gear Pair
Misalignment in a cylindrical gear pair originates from three sources: manufacturing error, assembly error and service deformation. Manufacturing error includes the parallelism error of the two axes and the lead error of the flanks. Assembly error arises from the fit between the bore and the shaft and is amplified when rolling bearings are interposed. Service error comes from the elastic deflection of the shaft, the bearings and the teeth themselves. Regardless of its origin, misalignment reduces the contact area, raises the local contact stress, promotes pitting and may finally cause tooth fracture.
When a parallelism error exists, a spur gear pair changes from full-width contact to edge contact at one end of the tooth, and the other end becomes completely unloaded. A helical pair behaves in a similar way but over a longer contact line, so that the peak stress at the loaded end is even higher. The planar spiral gear behaves differently: because the trace is curved, the pair possesses an additional degree of freedom in the axial direction and can accommodate a small angular error by shifting the contact zone along the spiral rather than by concentrating it at the edge.
The elastic deformation of the flank has two components. The bending component follows from the bending stress at the critical section, which is located by the thirty-degree tangent construction,
$$
\sigma_{F}=\frac{K_{F}F_{t}}{bm}Y_{Fa}Y_{Sa}Y_{\varepsilon}Y_{\alpha}
$$
and the corresponding deflection is obtained from Hooke’s law,
$$
\delta_{W}=\frac{F_{t}h_{Fc}}{BE}Y_{Fa}Y_{Sa}Y_{\varepsilon}Y_{\alpha}
$$
The contact component follows from the Hertz pressure on the curved flanks,
$$
\sigma_{H}=Z_{H}Z_{E}Z_{\varepsilon}\sqrt{\frac{F_{t}}{bR_{t}}\cdot\frac{1}{\dfrac{1-\nu_{1}^{2}}{E_{1}}+\dfrac{1-\nu_{2}^{2}}{E_{2}}}}
$$
so that the local approach of the two flanks is
$$
\delta_{H}=\frac{\sigma_{H}}{E}R_{t}
$$
The total deformation of the pair is the sum of the two contributions,
$$
\delta=\delta_{H}+\delta_{W}
$$
and it is this total deformation, combined with the crowned trace, that redistributes the load away from the tooth ends.
6.1 Behaviour under ideal meshing
Three gear pairs, one spur, one helical and one planar spiral, with identical module, tooth numbers, face width and material, were analysed in the static structural module of a general-purpose finite element package. The material was alloy steel with a Young’s modulus of $2.00\times10^{5}$ MPa, a Poisson ratio of $0.3$ and a density of $7850$ kg/m³. Hexa-dominant meshing was used and the mesh was refined to $0.2$ mm on the flanks that participate in the contact. A rotational velocity was applied to the driving gear and a torque of $100$ N·m in the opposite sense to the driven gear, the bores being supported by cylindrical supports.
| Item | Setting |
|---|---|
| Material | Alloy steel |
| Young’s modulus | $2.00\times10^{5}$ MPa |
| Poisson ratio | 0.3 |
| Density | 7850 kg/m³ |
| Mesh method | Hex dominant |
| Flank mesh size | 0.2 mm |
| Driving speed | 10 rad/s |
| Resisting torque | 100 N·m |
| Friction coefficient | 0.1 |
Under ideal meshing the maximum contact stresses were $110.1$ MPa for the spur pair, $72.35$ MPa for the helical pair and $47.19$ MPa for the planar spiral pair. The planar spiral gear therefore carries the same torque with less than half the peak stress of the spur gear. The reason is that the curved trace lengthens the contact line and spreads the load over a larger area, while the crowned shape prevents the concentration that would otherwise appear at the tooth ends.
6.2 Behaviour with a parallelism error of the axes
Two elementary misalignments were examined: a rotation of one axis by $0.1^{\circ}$ in the horizontal plane and a rotation of $0.1^{\circ}$ in the vertical plane. In reality the two occur together, but separating them makes the comparison clearer. For each case the module was varied from $3$ to $6$ mm and the tooth-number ratio from $20{:}40$ to $30{:}60$ and $40{:}80$.
| Module (mm) | Spur | Helical | Planar spiral |
|---|---|---|---|
| 3 | 220.79 | 420.77 | 147.85 |
| 4 | 168.57 | 339.90 | 112.88 |
| 5 | 126.95 | 260.97 | 98.86 |
| 6 | 105.50 | 214.63 | 71.36 |
| Module (mm) | Spur | Helical | Planar spiral |
|---|---|---|---|
| 3 | 152.85 | 306.01 | 135.27 |
| 4 | 148.14 | 294.58 | 118.43 |
| 5 | 134.35 | 272.83 | 88.45 |
| 6 | 102.26 | 200.64 | 66.60 |
| Tooth ratio | Spur | Helical | Planar spiral |
|---|---|---|---|
| 20:40 | 148.14 | 294.58 | 118.43 |
| 30:60 | 117.17 | 233.18 | 89.03 |
| 40:80 | 73.89 | 143.87 | 52.80 |
| Tooth ratio | Spur | Helical | Planar spiral |
|---|---|---|---|
| 20:40 | 168.57 | 339.90 | 112.88 |
| 30:60 | 149.39 | 306.04 | 108.81 |
| 40:80 | 125.60 | 278.46 | 90.94 |
Three trends emerge consistently from the whole data set. The peak contact stress of all three gear types decreases as the module increases, because the tooth becomes thicker and the contact ellipse grows. The peak stress also decreases as the number of teeth increases, because the contact line lengthens and the load per unit length falls. Most importantly, at every combination of module and tooth number the planar spiral gear shows the lowest stress: roughly 20 %–30 % lower than the spur gear and 40 %–50 % lower than the helical gear. The superiority is not a marginal effect but a systematic consequence of the crowned trace, which converts an edge load into a distributed load even when the axes are no longer parallel.
7. Full-Width High-Line-Speed Continuous Generating Milling
The meshing analysis demonstrates the merits of the planar spiral gear, but it also exposes the manufacturing problem. A curved tooth trace cannot be produced on a conventional hobbing or shaping machine, and the disc-type cutters used abroad for circular-arc gears cut one tooth space at a time and therefore require an indexing return stroke after every space. I therefore developed a fundamentally different process that removes the whole face width continuously.
The principle can be explained by considering the meshing of the gear with a planar spiral rack. The rack and the gear form a conjugate pair, so the rack can be regarded as the cutter and the gear as the blank. If a single cutter is mounted on a large disc and the disc rotates while the rack remains stationary, the cutter traces a circle on the rack, and the resulting tooth space is circular; the rack must then be indexed before the next space can be cut. If, instead, the rack advances along the radial direction of the disc while the disc rotates, the trajectory left on the rack is a planar spiral.
The velocity relation that produces a pitch-equal spiral is
$$
v=\frac{\pi m}{2}\omega_{D}
$$
where the rotational speed of the disc is denoted by the last symbol. When this relation holds, the cutter returns to the blank after one revolution exactly one pitch further on, so it enters the next tooth space instead of re-cutting the previous one. The trajectory on the blank is described, in the plane of the developed pitch surface, by
$$
\begin{cases}
x_{D}=R_{T}\cos\vartheta+\dfrac{m}{2\pi}\left(2k\pi-\vartheta\right)\\[6pt]
z_{D}=R_{T}\sin\vartheta
\end{cases}
$$
and the generating line of the tooth space is the spiral
$$
\begin{cases}
x_{D}=R_{T}\cos\vartheta-\dfrac{m}{2\pi}\vartheta\\[6pt]
z_{D}=R_{T}\sin\vartheta
\end{cases}
$$
For a cylindrical gear the translation of the rack is replaced by the rotation of the blank. The blank rotates about its own axis while the cutter disc rotates about the disc axis, and the two rotations are linked by the requirement that one revolution of the disc corresponds to one tooth pitch of the blank,
$$
\frac{\omega_{C}}{\omega_{D}}=\frac{1}{z}
$$
so that
$$
\omega_{C}=\frac{\omega_{D}}{z}
$$
In polar form the trajectory of the cutter relative to the blank is the superposition of a circular motion and a linear motion,
$$
r_{1}\left(\vartheta\right)=R_{T},
\qquad
r_{2}\left(\vartheta\right)=\frac{m}{2\pi}\vartheta
$$
and therefore
$$
r\left(\vartheta\right)=R_{T}\pm\frac{m}{2\pi}\vartheta
$$
which is exactly the planar spiral required by the gear geometry. Because the cutter radius is large, the cutting speed at the tooth flank is high even at a moderate spindle speed, and the flank quality improves accordingly.
8. Multi-Cutter Layout on the Rotary Disc
A single cutter removes the whole chip load and therefore wears quickly, which is unacceptable for forged gear blanks that may carry a hard as-forged skin. I therefore distribute several cutters along a planar spiral on the disc, with a pitch equal to the circular pitch of the gear,
$$
p=\pi m
$$
The polar equation of the layout spiral is
$$
\begin{cases}
x_{dk}=\left(R_{T}+\dfrac{m}{2\pi}\beta\right)\cos\beta\\[6pt]
z_{dk}=\left(R_{T}+\dfrac{m}{2\pi}\beta\right)\sin\beta
\end{cases}
$$
For a disc carrying cutters equally spaced in angle, the angular position of the cutter of index $k$ is
$$
\beta_{k}=\frac{2k\pi}{n}
$$
so that its mounting coordinates become
$$
\begin{cases}
x_{dk}=\left(R_{T}+\dfrac{mk}{n}\right)\cos\dfrac{2k\pi}{n}\\[6pt]
z_{dk}=\left(R_{T}+\dfrac{mk}{n}\right)\sin\dfrac{2k\pi}{n}
\end{cases}
$$
The cutter of smallest mounting radius is the main forming cutter and determines the final flank; the remaining cutters are mounted at progressively lower axial heights and act as roughing cutters. In each revolution the lowest cutter removes a thin layer, the next cutter removes another thin layer, and the main cutter only has to finish the surface. The cutting force per cutter is thereby reduced, the thermal load is distributed, and the surface finish improves. The axial step between consecutive cutters is kept small, of the order of one tenth of a millimetre, so that no cutter is overloaded.
Each cutter body carries an outer and an inner cutting edge. The outer edge generates the concave flank and the inner edge generates the convex flank, and their pitch radii are
$$
R_{ob}=R_{T}+\frac{\pi m}{4},
\qquad
R_{ib}=R_{T}-\frac{\pi m}{4}
$$
The difference between the two radii is precisely the quantity that produces the crowning. The mounting radius must not be reduced below the value that keeps the angular spread of the cutter within acceptable limits, otherwise the front face of the cutter sweeps a large angle, the circumferential space width at the two ends of the tooth increases, and the tooth becomes thick in the middle and thin at the ends. Such a tooth has poor bending strength and poor transmission smoothness, so the large disc diameter is not merely a convenience but a geometric requirement.
| Item | Symbol | Value |
|---|---|---|
| Main cutter mounting radius | $R_{T}$ | 200 mm |
| Number of cutters | $n$ | 8 |
| Spiral pitch of the layout | $p$ | $\pi m$ |
| Outer edge pitch radius | $R_{ob}$ | $R_{T}+\pi m/4$ |
| Inner edge pitch radius | $R_{ib}$ | $R_{T}-\pi m/4$ |
| Axial step between cutters | $\Delta h$ | 0.1 mm |
9. Depth-of-Cut and Generating Motions
The machining cycle consists of two stages. In the depth-of-cut stage the disc rotates at high speed and the blank rotates at the reduced speed given by the ratio derived above. The cutter plane is tangent to the cylindrical surface of the blank, the active cutters sweep the whole face width in one revolution, and the blank is simultaneously fed along the axis of the disc so that successive layers are removed. Since a single pass cannot remove the full tooth depth, the blank is fed in small increments, typically less than one millimetre per pass, and the number of passes is chosen so that the total depth is reached without overloading the cutters. This layer-by-layer strategy is particularly suitable for forged gear blanks, whose surface layer may be harder than the core.
In the generating stage the axial feed is stopped. The disc continues to rotate and the blank receives an additional rotation together with a translation along the radial direction of the disc, the two being coupled by the rolling condition
$$
v_{xc}=\omega_{s}r
$$
where the additional angular velocity is denoted by the first symbol and the pitch radius of the blank by the second. When the additional rotation is superimposed in one sense, the convex flank is generated; when it is superimposed in the opposite sense, the concave flank is generated. The translation reverses accordingly. Because the cutter edge describes a conical surface during this motion, the envelope of the successive positions of the edge is exactly the involute profile, and the generated flank is therefore a true involute at every section of the face width.
The whole cycle is continuous. The cutters do not leave the blank between successive tooth spaces, no indexing return stroke is needed, and the spindle never has to be stopped. This is the essential difference from the classical disc-cutting process, and it is the reason why the cycle time can be reduced so drastically.
10. Virtual Machining Verification
Before building a physical machine I verified the process by virtual machining. The complete device, including the two linear axes and the two rotary axes, was modelled in a CAD environment and exported in a neutral format. The cutter disc with its eight cutters was imported into the tool library of a machining simulation package, and the control system was configured so that the ratio between the disc rotation and the blank rotation could be imposed directly. The numerical control program was written as a subroutine that is repeated once per tooth space,
$$
\Delta A=\frac{360^{\circ}}{z},
\qquad
\Delta Z=p
$$
so that the angular increment of the blank and the axial increment of the disc correspond exactly to one tooth. The simulation ran without interference or collision, and the resulting solid showed a clear involute profile at every section of the face width, with a standard involute at the mid-section.
To check the profile quantitatively, the simulated gear was imported into the CAD environment and compared with an equation-driven standard involute curve generated on the same base circle. The two curves coincided over the whole active profile, which proves that the generating motion produces a genuine involute rather than an approximation. To check the contact behaviour, two simulated gears of opposite hand were assembled and analysed in the contact module. The theoretical contact line in the unloaded state is a point contact near the mid-section; under load the elastic deformation converts it into a line contact confined to the middle 80 %–90 % of the flank, with the two ends only lightly loaded. The same result was reproduced independently by a numerical model of the concave and convex flanks built directly from the analytical surface equations, in which the concave flank is coloured blue and the convex flank red. The agreement between the analytical model, the virtual machining result and the finite element prediction confirms both the correctness of the surface equations and the feasibility of the process.
11. Dedicated Machining Device
No machine tool on the market can execute the motion scheme described above, because the ratio between the two rotary axes must be controlled precisely and continuously while the two linear axes follow the generating law. I therefore designed and built a dedicated device on the basis of a conventional numerically controlled lathe bed. The device has four controlled axes: two linear axes for feed and generating translation, and two rotary axes for the cutter disc and the workpiece table.
The spindle of the device is driven by an AC servo motor with a rated power of 5.5 kW, a rated torque of 35 N·m and a rated speed of 1500 r/min, connected to the spindle through a belt drive with a ratio of 1:4. The belt drive is chosen because it damps the torsional vibration of the disc and because the available space inside the headstock does not allow a direct coupling. The two linear axes use precision ball screws driven by AC servo motors, which provide the stiffness and the resolution required for the depth-of-cut increments. The rotary table is driven by a servo motor with a rated power of 0.85 kW and a rated torque of 5.39 N·m, coupled to a servo drive whose positioning accuracy is one micrometre and whose rotary accuracy is five arc seconds. This accuracy is essential, because any error in the table rotation is transferred directly to the tooth spacing.
The cutter disc is made of alloy steel and carries eight radial slots. Each slot is provided with clamping bolts on both sides, which press a wedge against the cutter shank and prevent it from creeping during high-speed rotation. A radial positioning bolt sets the distance between the cutter axis and the disc centre, so that the eight cutters can be arranged exactly on the planar spiral. The cutter body is made of cemented carbide of the type W12Cr4V3Mo3Co5Si, with a hardness in the range 69–81 HRC and retained hardness at temperatures up to 1000 °C, which is necessary because the continuous cutting of forged gear blanks generates a substantial amount of heat at the cutting edge. The shanks are also made of cemented carbide, so that they resist the shear load without permanent deformation.
The control system was developed on an open numerical control platform. It provides manual mode, MDI mode and automatic mode, together with linear interpolation, circular interpolation and multi-axis interpolation. The real-time link between the computer and the servo drives is established over a field bus, so that the ratio between the two rotary axes can be maintained to within a few counts per revolution. Because the ratio determines the tooth spacing, this synchronisation is the most critical function of the control system. Before cutting, the operator enters the module, the number of teeth, the pressure angle, the face width and the spiral radius; the system computes the cutter positions on the disc, the speed ratio and the generating increments, and generates the corresponding program.
| Subsystem | Component | Specification |
|---|---|---|
| Spindle | AC servo motor | 5.5 kW, 35 N·m, 1500 r/min |
| Spindle transmission | Belt drive | ratio 1:4 |
| Linear axes | Ball screw + AC servo | resolution 1 µm |
| Rotary table | AC servo motor | 0.85 kW, 5.39 N·m |
| Rotary table drive | Servo drive | 1 kW, 5 arc s, 1 µm |
| Cutter disc | Alloy steel | 8 radial slots |
| Cutter body | Cemented carbide | 69–81 HRC |
| Control | Open NC platform | 4 axes, field bus |
12. Trial Cutting Experiment
The final validation was a trial cut on the physical device. The forged gear blanks were first turned to the required outside diameter and faced, then clamped on the rotary table with the mid-section of the face width aligned with the pre-cutting point of the main cutter. The cutter positions on the disc were adjusted according to the module and the spiral radius, and the speed ratio was set according to the number of teeth. The disc speed was 200 r/min and the main cutter mounting radius was 230 mm, which gives a cutting speed of
$$
v_{c}=\frac{2\pi R_{T}n_{D}}{1000}=\frac{2\pi\times 230\times 200}{1000}\approx 289\ \text{m/min}
$$
This is a high cutting speed for a milling operation and it is the reason why the flank finish of the samples is so good.
| Parameter | Symbol | Value |
|---|---|---|
| Module | $m$ | 5 mm |
| Number of teeth | $z$ | 63 |
| Face width | $B$ | 50 mm |
| Pressure angle | $\alpha$ | $20^{\circ}$ |
| Disc speed | $n_{D}$ | 200 r/min |
| Main cutter radius | $R_{T}$ | 230 mm |
| Cutting speed | $v_{c}$ | 289 m/min |
| Number of depth passes | — | 12 |
| Depth increment | — | ≤ 1 mm |
| Generating increment | — | $10^{\circ}$ |
| Radial stroke | — | $\pm 31.42$ mm |
The full tooth depth of 11.25 mm was removed in twelve passes, each not exceeding one millimetre, which kept the cutting force within the capacity of the disc and preserved the sharpness of the carbide edges. During the depth-of-cut stage the blank was fed continuously along the disc axis, so that the whole face width of every tooth space was machined in one revolution of the disc. In the generating stage the additional rotation of $10^{\circ}$ in one sense produced the convex flank, and the same increment in the opposite sense produced the concave flank, the disc rotation being maintained throughout.
The samples produced in this way show a clear planar spiral trace, a slightly crowned tooth with the concave flank curvature slightly larger than the convex one, and a mirror-like flank finish in which the cutter marks are almost invisible. The symmetry of the trace about the mid-section suppresses the axial force that a helical gear would generate, and the crown suppresses the edge loading that a spur gear would suffer. The measured flank roughness and the visual inspection of the tooth trace both confirm that the process is stable and repeatable.
| Process | Finishing time (min) | Relative time |
|---|---|---|
| Hobbing | 160 | 1.00 |
| Shaping | 190 | 1.19 |
| Proposed continuous generating milling | 35 | 0.22 |
The economic consequence is significant. For the same gear the proposed process requires about one fifth of the time needed by hobbing and about one sixth of the time needed by shaping. Applied to forged gear blanks, whose near-net shape already reduces the machining allowance, the saving in cycle time is compounded, and the total cost per piece falls accordingly. The process is also compatible with forged gear blanks of harder grades, because the multi-cutter arrangement distributes the chip load and the carbide edges retain their hardness at elevated temperature.
13. Conclusions
I have defined a planar spiral toothed cylindrical gear, derived its static and dynamic tooth surface equations from the generating principle, formulated its contact line and its contact ratio, and verified the equations numerically and by solid modelling. The gear has a planar spiral trace on the pitch cylinder, an involute profile at the mid-section, and a concave flank whose curvature radius slightly exceeds that of the convex flank, so that the tooth is slightly crowned. The crowning confines the loaded contact to the middle 80 %–90 % of the flank and relieves the root fillets at the two ends.
I have built a parametric modelling system by coupling the CAD kernel with a Visual Basic environment through the COM interface. The system generates a complete, accurate solid model from a handful of input parameters, and it has removed the modelling bottleneck that previously restricted the study of curved tooth trace gears.
I have quantified the anti-misalignment behaviour of the gear by finite element analysis. Under ideal meshing the peak contact stress of the planar spiral pair is less than half that of a comparable spur pair and about two thirds that of a comparable helical pair. With a parallelism error of $0.1^{\circ}$ in either the horizontal or the vertical plane, the peak stress of the planar spiral pair remains 20 %–30 % below the spur value and 40 %–50 % below the helical value over the whole range of modules and tooth numbers examined, and the peak stress of all three types decreases as the module and the number of teeth increase.
I have proposed a full-width high-line-speed continuous generating milling method. The cutter disc and the blank rotate in a fixed ratio of one tooth per disc revolution, the cutters are arranged on a planar spiral whose pitch equals the circular pitch, and the depth-of-cut stage is superimposed on the generating stage. The method removes the whole face width continuously, eliminates the indexing return stroke, and maintains a high cutting speed throughout. Virtual machining in a simulation environment confirmed that the generated profile is a true involute and that the contact zone of the resulting pair is confined to the middle of the flank.
I have designed and built a four-axis dedicated machining device and completed trial cuts on forged gear blanks. The samples show a clear planar spiral trace, a slightly crowned tooth, an almost mark-free flank and a finishing time of about one fifth of that required by hobbing. The combination of a gear geometry that tolerates misalignment with a process that removes the whole face width in a single continuous cycle provides a practical route for the industrial application of curved tooth trace cylindrical gears, and it is fully compatible with the increasing use of forged gear blanks as the starting stock.
Further work should extend the analysis from the numerical model to a dynamometer test rig, so that the misalignment behaviour of the physical samples can be measured rather than predicted, and should include an optical inspection of the machined flanks so that the deviation from the theoretical surface can be quantified. The dynamic transmission error, the wear resistance and the optimisation of the generating parameters are the natural next steps.