The planar involute profile has been used to model bevel gears without consideration toward approximation error; this profile was thusly modeled and tested using unloaded tooth contact analysis.

The planar involute profile is well known in and out of the gear community. As such, it is easy to misconstrue its relationship with conjugate action. It has been noticed that the planar involute is being increasingly used as a self-conjugate curve for bevel gears despite no mathematical basis for doing so. The amount of error from doing this is largely unquantified. In this article, a bevel gear with a planar involute profile has been simulated and analyzed to determine the validity of its use. This geometry is then compared with several standard bevel gear geometries. It is seen that a large level of transmission errors and a non-ideal contact area result from misusing the planar involute profile as a reference profile for bevel gears.

1 Introduction

The planar involute is perhaps the most common reference profile for spur gears. Its conjugate nature and ease of manufacturability have been noted for well over a century [1, 2, 3, 4]. Its ubiquity in this regard can lead to misconceptions on the nature of conjugate action, though. Consider this excerpt from Shigley’s Mechanical Engineering Design, a well-known and respected book on the engineering of mechanical components [5]:

“When the tooth profiles, or cams, are designed so as to produce a constant angular-velocity ratio during meshing, these are said to have conjugate action. In theory, at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action. One of these solutions is the involute profile, which, with few exceptions, is in universal use for gear teeth.”

While this may ring true with many readers, there is a detail that may easily mislead an engineer that is not as familiar with gear geometry. Insinuated in the phrase, “universal use for gear teeth,” is the notion that the planar involute can be used for any gear type and conjugate action is a given. This implication is not unique to this book. It’s quite common and easy to misstate, whether due to a simple mistake, oversight, or other reasons. While the delineation may seem pedantic, precision in language is as important as precision in surface.

With the necessity of computer modeling, CAD, and FEA, the need for precise geometric tooth models is ever increasing. This is further exacerbated by the growing role of additive manufacturing in the field of engineering. While a gear engineer may be aware the geometry of a tooth is no trivial matter, it is not unreasonable that a typical mechanical engineer would be entirely unaware of this issue. It is precisely at this point where the easily misconstrued relationship between the planar involute profile and conjugacy can cause issues.

You might be an engineer without a high level of technical knowledge on gear geometry and yet you are tasked with modeling a bevel gear. A quick search online for modeling this in CAD programs yields several videos showing how to make bevel gears with involute teeth [6, 7, 8]. Paired with plenty of introductory literature discussing the planar involute, it follows that a nonexpert may assume this method is correct. Engineering is also a field that is often governed by approximations. However, the crux of every approximation is an understanding of error. So, how non-conjugate is the planar involute when abused to use in a bevel gear?

2 Modeling and assessment of bevel gear tooth surfaces

Conjugacy is a kinematic constraint, dictating the presence of a constant transmission ratio between dynamic linkages [1, 2, 9]. If the geometry of a member is known and the method of meshing that member with another is also known, the conjugate geometry of the other member is given, entirely irrelevant to the reference profile of the first member. Many phenomena can be leveraged to determine conjugate geometry. Errichello and Stadtfeld presented the approach shared among several renowned gear theoreticians that considered normal vectors passing through a fixed point along the line of action [10]. Litvin and Fuentes considered an approach that took advantage of the enveloping phenomenon that occurs within kinematically constrained systems [3]. Dooner presented an approach that leverages screw theory [11]. Other approaches surely exist as well. No matter how geometry is determined, though, they all consider conjugate action at the heart of it. Real geometries are not drawn or calculated via parametric equations; they are generated and manufactured.

To assert this point, three bevel gear geometries are considered and the non-conjugacy of each measured. The first geometry is generated via a crown gear, which considers a spherical involute profile. This profile was selected because it has been extensively studied for application toward bevel gears. The second geometry is generated by simulating the Coniflex® method. This process was chosen because it simulates a common manufacturing process for straight bevel gears and results in a virtual model that is highly accurate to what can and will be manufactured. The final geometry is a planar involute profile applied to bevel gears. This is the reference profile that is commonly considered as a baseline geometry for spur gears and, as mentioned earlier, is sometimes considered to model straight bevel gears, causing a large level of transmission errors instead of the expected conjugate mesh.

Figure 1: Geometry of the gear modeled by an enveloping process.

2.1 Modeling considering the corresponding enveloping process

The first two geometries will be determined by leveraging the enveloping phenomenon within conjugate linkages. This method considers a known geometry of the generating tool and the kinematics of the process of generation by emulating the process of meshing between the generating tool and the blank geometry of the gear. The meshing process with respect to a coordinate system that is fixed to the being-generated gear is then simulated, leading to the computation of a family of curves for which their envelope defines the generated profile of the gear, as illustrated in Figure 1. In this case, the red geometry is the geometry of the tool that is considered as known and the white geometry is the generated profile of the gear, shown only as a reference. By simulating the meshing process of the tool with the being-generated gear, the envelope to the curves of the tool profile in a coordinate system fixed to the gear can be computed, and with it the geometry of the gear obtained. Points that belong to the envelope of the family of curves are given, in general, by Equation 1:

This equation considers a point on the generating tool, r0 (u,θ), that is described by two surface parameters, u and θ. The process of meshing the tool with the to-be-generated gear is simulated using the coordinate system transformation M10 (ϕ), which is a function of a generalized parameter of motion, ϕ. Applying the transformation yields a point in the envelope of the to-be-generated tooth, r1 (u,θ,ϕ), which is a function of the surface parameters and the parameter of motion. Points that are solely along the gear profile can be determined by solving the equation of mesh [3], which is given by Equation 2:

This equation may look intimidating but is in truth physically intuitive. Points on the outline are discerned by leveraging the fact that the normal vector of the gear tooth surface is orthogonal to the sliding velocity. Afterall, gear teeth do not push each other, they slide. To describe this mathematically, first consider taking the partial derivative of the point, r1 (u,θ,ϕ), with respect to each surface parameter. Doing so yields the tangent vector in the direction of the corresponding surface parameter. Crossing these two vectors provides the normal vector at the point r1 (u,θ,ϕ). The sliding velocity is obtained by taking the partial derivative of r1 (u,θ,ϕ) with respect to the generalized parameter of motion. Orthogonality is ensured by having a dot product of these two vectors equal to zero. The power in this process comes from the ability to freely manipulate the generating tool geometry and the process of meshing while maintaining a generalized solution, removing the need to solve many complex geometry problems.

The first geometry to be considered as a reference is generated via a crown rack gear with a spherical involute as given by Fuentes and Gonzalez in [12]. The spherical involute is considered an extension of the planar involute for bevel gears. The definition of the spherical involute can be seen in Figure 2. For the generation of this profile, a disk plane, bounded by the great circle of the sphere containing the to-be-generated bevel gear, rotates. As the disk rotates, a tangent cone is rotated without slippage and with a constant transmission ratio. The curve traced by this linked motion is the spherical involute. This geometry has the advantage of having insensitivity toward shaft angle misalignments [13], similar to the planar involute being insensitive toward center distance variations. It may experience a surge as a reference geometry for forged bevel gears [14] but largely remains unpopular in traditional manufacturing due to the lack of corresponding generating manufacturing processes. The crown rack geometry is given by a spherical involute. Mesh between the crown rack and the gear is simulated to generate an envelope of curves that ensures conjugacy.

Figure 2: Definition of spherical involute, taken from [12].

The second reference profile is generated by simulating the Coniflex® manufacturing method. While a great deal of literature exists on the matter, the methods given by Fuentes et al. in [15] are used. This method simulates a crown gear with disc cutters as seen in Figure 3 [16, 17]. In this case, the method is being used such that no crowning is introduced since the goal is to compare baseline geometries. However, it should be noted that through blade angle, swing angle, and complex kinematic relationships, the Coniflex® method is capable of achieving optimally localized contact and precisely adjusting the level of transmission errors.

Figure 3: Coniflex® cutters aligned to and simulating a crown gear.

2.2 Modeling of the planar involute profile for bevel gears

The process of determining the planar involute profile for spur gears is well established. The application of the planar involute profile to bevel gears, although not advised, is described here so it can be compared with the spherical involute geometry and the geometry obtained by the Coniflex® manufacturing method.

Figure 4 shows the planar involute curve generated by a point Po of a straight line that rolls counter-clockwise over a circle, known as the base-circle, with a radius of rb. Due to the condition of rolling without sliding in Equation 3:

It can be observed from Figure 4 and in Equation 4 that:

And in Equation 5:

Similarly, it can be seen that in Equation 6:

And Equation 7:

The parametric coordinates of an arbitrary point on the planar involute, Pi, are therefore given in Equation 8 as

where the ±, derived by symmetry, respectively denotes whether the right side or left side of the tooth is being considered. The range bounds of the angle θ must be chosen well. In the case that the addendum height, dedendum height, and fillet radius are known, it can be determined by finding the angle that provides the addendum radius, rh, and the form radius of the active profile, rf, as seen in Figure 4.

Figure 4: Planar involute definiton with radial bounds.
Figure 5: Bevel gear tooth blank.

Unlike spur gears, bevel gears clearly do not have a constant radius. Additionally, the teeth of a straight bevel gear taper in size from the heel to the toe, as shown in Figure 5. Fortunately, both of these characteristics are straightforward to account for, as the consistent taper allows for the use of simple proportional relationships. This taper follows in Equation 9 from

where fw is the facewidth, γ is the pitch cone angle, and rp is the pitch radius which is given in Equation 10 by

For the pitch radius, m represents the outer module, and Z is the number of teeth. For any given ri between rmin and rmax, the instantaneous module, mi, is found in Equation 11 by

With the instantaneous module known, the addendum radius of the cross-section is given in Equation 12 by

where ac is the addendum coefficient. The bottom of the active tooth height at this cross section is found in Equation 13 by

where rc and dc are respectively the fillet radii and dedendum height coefficients. The base radius at this section is finally calculated in Equation 14 as

where α is the pressure angle.

A planar involute profile can now be created for any cross-section of the bevel gear, but the profile still must be correctly positioned. In a CAD program, this can simply be done on a plane that is orthogonal to the pitch cone angle, γ, and intersects the desired cross-section. The last step is to rotate each planar involute by the angle, ε, which provides the expected tooth thickness and is obtained in Equation 15 by

where the ± respectively denotes the right and left side. This angle is visualized in Figure 6. Considering the involute I0, given by Equation 8, the goal to rotate this curve so that it aligns with I3. This is begun by first rotating the curve clockwise by the involute of the pressure angle, given by tanα — α, so that it aligns with I1. This allows the pitch-point of the involute curve to intersect with the x-axis. The curve is then rotated counter-clockwise to intersect with the y-axis and coincide with I2. The final step is to set the curve to fit its angular pitch definition. Since this curve is one half of the active portion, it should be rotated clockwise by a quarter of an angular pitch, given by π/(2〈Z), so that it aligns with I3.

Figure 6: Toward the defintion of the angle ε.

If a point cloud for the tooth is being numerically created, each point must be placed correctly via coordinate system transformation. Consider some point along the involute, Pi, corresponding to a given cross-section, with coordinates obtained by Equation 8, and expressed in Equation 16 as

These points can be transformed to the proper cross-section by M which is given in Equation 17 by

The final product of this process is a set of point clouds that correspond to a planar involute bevel gear tooth as seen in Figure 7. Duplicating these point clouds for each tooth will allow for a gear to be fully modeled. Since no fillet is being generated by any tool in this case, it can be modeled by adding a constant radius fillet between the tooth and root cone, using NURBS surfaces, Bezier curves, or following any other approach.

Figure 7: Planar involute bevel gear (5 teeth shown).

2.3 Geometric assessment

The positional deviation of the driven gear with respect to the driving gear from the theoretical position given by the transmission ratio is known as transmission error. Transmission errors can be caused by the meshing of non-conjugate mating geometries, manufacturing errors, errors of alignment, and/or deformations. Even ideally conjugated gears can suffer from some combination of errors of alignment, manufacturing errors, and deformations due to the transmitted torque, sometimes catastrophically so. As such, gears typically consider some form of crowning to make the geometry more resilient at the expense of transmission errors potentially being introduced during design. No matter the source, these errors are a cause of noise and vibration, and therefore must be carefully engineered [3, 18].

Transmission errors can be measured in a variety of ways for a computerized model. One such way is to use tooth contact analysis (TCA). The intent of this study is to assess the validity of a method to serve as a baseline geometry. So, the TCA will be performed on an unloaded, perfectly aligned pair. This is done to isolate the errors caused by non-conjugate geometry selection. In addition to the unloaded TCA, a topological comparison is performed between the planar involute profile and spherical involute profile.

3 Results and discussion

The TCA tests and topological comparison were performed with a pair of miter gears, each having 25 teeth. The outer module of the pair is 3.0; the addendum coefficient is 1.0; the dedendum coefficient is 1.25, and the fillet radius coefficient is 0.2. No crowning is being introduced into any of the geometries, since a theoretical mesh is being performed. In the case of the Coniflex® geometry, the mean cutter radius is 150 mm, and there is no blade profile angle. These design parameters were selected so that a fairly regular geometry, free of undercutting or any irregular load-sharing, is being analyzed.

The geometric comparison between the planar involute and the spherical involute geometries can be seen in Figure 8 on the left.

For this test, the spherical involute profile was taken as a reference surface, and the normal deviation between it and the planar involute profile is measured. Only one side is shown since the results are symmetric. This approximation, as can be seen, is geometrically excellent near the pitch cone definition, sub-micron in some places. Although, the planar involute rapidly separates from the spherical involute when moving in either direction along the profile. That said, the greatest difference is still below anything visually detectable for most, especially in a CAD program. Tests unavailable in CAD programs, like TCA, show the true effect of this approximation.

The geometric comparison between the spherical involute and Coniflex® geometries can be seen in Figure 8 on the right side. Although the methods of generation differ, it can be seen how the Coniflex® method excellently emulates the geometry generated via a crown rack with a spherical involute. This is because the Coniflex® method produces an octoidal geometry, which is well known to approximate a spherical involute. While the spherical involute and Coniflex® geometries are not identical, both can be self-conjugated as is shown with further testing.

Figure 8: Topological comparison between spherical involute and planar involute (left) and between spherical involute and Coniflex® geometry (right).

The first TCA is performed on the geometry that was generated via crown gear with a spherical involute. The results of which can be seen in Figure 9. The transmission errors graph shows conjugate action within the noise threshold of the algorithm employed for analysis. The unloaded tooth contact pattern shows line contact that spans the entirety of the tooth surface. This serves as a good basis for gear design.

Figure 9: TCA of miter gears generated via spherical involute crown gear.

A true pair would need some crowning to help with misalignments. While it is noted that the spherical involute has some resilience against misalignment [12], investigations have been performed toward methods of introducing crowning to improve this effect [14]. Alternatively, other crown rack geometries could be considered as the basis. The kinematic coupling condition is such that the generating tool can be freely altered. So long as conjugacy is maintained between each member and the mutual generating tool, conjugate action when the members are meshed is to be expected [3, 4]. This provides significant freedom when optimizing geometries.

The second TCA performed is on the geometry generated via Coniflex® cutters. The results of this test can be seen in Figure 10. Observably, this pair has no transmission errors and therefore exhibits conjugate action. The unloaded tooth contact pattern also shows line contact that spans the entirety of the tooth surface. Similar to the crown gear generated tooth surfaces, this would serve as a good basis for gear design. This is to be expected since the dual-interlocking cutters are designed to simulate the crown gear but have the advantage of rapid manufacturing [4].

Figure 10: TCA of miter gears generated via Coniflex® cutter.

The predominant advantage of simulating a manufacturing process to obtain gear geometry is that it is immediately manufacturable. Processes like Coniflex® have means of introducing crowning embedded into the manufacturing process. So, the designer starts with a conjugate basis. Any change from the conjugate basis results in transmission errors that are truly controlled — sometimes even entirely avoidable — all the while maintaining a surface that can be manufactured without having to resort to slow methods such as using an endmill or 3D printing.

The TCA results, performed on the gear pair with a planar involute profile, can be seen in Figure 11. The transmission errors graph shows a large parabolic function of transmission errors, with a maximum of a little over 270 arcseconds (about 1,350 microradians). Even unloaded, this gearset will be loud. Line contact is still present. However, a large portion of the tooth is never in mesh. Large loads may result in deformations that will expand this contact area. However, observing the geometric comparison of the planar involute with the spherical involute provides insight that a large deformation would be needed to expand the contact area in a significant way due to the existing large delineation from conjugate definition.

Figure 11: TCA miter gears created using planar involute.

4 Conclusions

The planar involute profile has been used to model bevel gears without consideration toward approximation error. This profile was thusly modeled and tested using unloaded tooth contact analysis. Results from testing show a significant level of transmission errors and reduced contact area despite innocuous design parameters. The planar involute profile was compared to profiles generated via a crown gear with a spherical involute and Coniflex® cutters. Both of the other methods showed no transmission errors and a full contact area, making them a good basis to introduce crowning and perform further analysis.

Conjugacy should stand as the first and foremost consideration for the baseline of a gear design. After all, conjugate action implies the presence of a transmission ratio. This linear transformation of power is the reason gears are used over other dynamic linkages. A significant body of engineering and mathematics has been invested into the initial design of a gear tooth geometry, the various methods of rapidly manufacturing it, and optimizing the tooth geometry to increase strength and misalignment insensitivity. Approximation is a necessity in engineering, but with it comes the need for mindfulness of its context of use. For hobby and non-critical usage, the level of transmission error and noise introduced by considering a planar involute to be the baseline geometry of bevel gear may very well be acceptable. For industrial application, though, an engineer must think about the impact of choosing to model teeth in this way.

Acknowledgements

The authors express their gratitude to the Gleason Works for the continued support of the Gleason Doctoral Fellowship at the Rochester Institute of Technology and members of the RIT Gear Research Consortium for their gracious financial support for gear research at RIT. 

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