##### Gear Modeling

##### Project Background

When I originally began modeling my lathe project in Creo Parametric, my goal was to create an extremely basic lathe mechanism with no bells and whistles. As I continued to model it and select materials, I realized how useful some of those accessories tended to be and slowly added new features and better materials and modified the frame to be more structurally efficient. I made some of my most important changes after watching a video series detailing the teardown, repair and cleaning of a Colchester lathe, which gave me insight into the useful functions of some components found in professionally used lathes.

For months I had been resolved to use a direct belt-drive between my motor and my spindle, and had everything modeled for that setup, but I decided finally that the benefits of using a gearbox to gear up or down the motor input was worth the effort. I don’t like copying others’ models for my projects and prefer to model everything that I need from scratch because I find it fun, I like to test my skill, and I often find new, clever ways of doing things to save time while modeling.

I discovered the involute curve and the role that it plays in constant-force-transmission gear systems while researching where to begin in modeling spur gears. At that point in time, nearly everything that I was reading or watching online pertained to gear design, manufacture, math and terminology. I learned about the work and tools involved in the two most common gear cutting techniques, hobbing and broaching, as well as gear standards and the underlying geometry of each type of gear. I was so intrigued by the many applications of gears and their complex, elegant geometries that I wanted to learn to model every kind that I was familiar with in CAD rather than spur gears alone.

Gear-modeling gave me my first introduction to relation-driven CAD models, where parameters can be modified on the fly and a 3D part can be regenerated to be completely different with its features dependent on those adjustable parameters. I used this to drive tooth patterns and gear dimensions allowing me to select a number of teeth, a gear module or diametral pitch value, pressure angle and gear width, and then within seconds, generate a gear with those features. Using these techniques, I delved into every type of gear that I knew of, starting with spur, then moving to helical spur, cross-axis helical, rack and pinion, helical worm, cylindrical worm, globoid worm and bevel gears.

I eventually used the spur gear templates that I developed to form both an eight-speed drive train gearbox that could be manufactured and used in my lathe, as well as a quick-change gearbox which would allow me to fine-tune the feed rates for a consistent surface-finish or for thread cutting. It took over two months of my free time to fully develop template models of the many kinds of gears that I might need to use in my future modeling projects, each with mathematically perfect relations driving them to truly represent their ideal real-world counterparts.

By far the most difficult gear to model was the cylindrical worm gear, requiring for me to go through many iterations of models that I thought would work before breaking down the facts about the gear’s geometry and attempting an entirely new modeling method. Gears have become a major interest of mine and I hope to acquire an indexing head in the near future with which I will be able to broach my own gears for use in my projects, particularly my lathe. Eventually, I hope to design my own hobbing machine, so I can form cylindrical and Hindley/globoid worm gears as well as helical spur gears which can’t be made using conventional broaching techniques.

I believe that the most enjoyment that I have gotten out of 3D modeling was experienced when I finally discovered how to manipulate the underlying geometry behind gear teeth. For example, I was aware that for gear pairs of a certain low number of teeth given a specific pressure angle, the teeth will interfere in their motion with teeth on the other gear if modeled the same way as gears with a larger number of teeth. The solution to this problem is to model an undercut in the root of the tooth that is slightly deeper than the path that the corners of a tooth will take as the tooth travels past the other gear. The undercut shape is a trochoid curve which can be modeled perfectly using relation-driven curves within Creo, creating the perfect undercut exactly the way that it is done in industry with gear hobbing. Discovering that I could model this and actually putting it into effect without tutorials, with only the underlying math driving these geometries, was extremely satisfying.

In the future I will be trying again to model spiral bevel gears which I was unable to get to work in my first attempt at driving them with relations in Creo largely because I jumped right into spiral bevel gears before researching any other type of gear and before developing a good understanding of Creo’s relation-driven geometries. I am curious to see if I can actually create a template for spiral bevel gears and hypoid gears but will need to wait until I have the free time to attempt such a difficult task.

##### Detailed Description

##### Spur Gears

The spur gear is the type of gear that generally comes to mind first when gears are mentioned. They are the most basic and I believe the most often used form of gear, and their fundamental geometries are adopted by nearly every other gear type. There are many terms used to define a gear with unique features. Diagrams with these terms and the relationships between them can easily be found online, but an understanding of them is necessary for sticking to the standards used today to manufacture gears.

For example, we have standardized gear sizing in two main ways: The metric, called the module of a gear, is defined as the diameter of a gear's pitch circle in millimeters (theoretical circle of contact between two meshing gears) divided by the number of teeth on that gear. The imperial solution for gear sizing is called pitch diameter and is the ratio of the number of teeth divided by the pitch diameter of the gear in inches.

Spur gears of a specific size can be modeled in full by defining the diametral pitch or module, the pressure angle, the number of teeth and the width of the gear, as all other parameters are either standardized or based on these inputs, including profile-shifted gears. The base diameter, used to generate an equation for the tooth face’s involute curve, is based on the pressure angle and pitch diameter, which is solvable through the given diametral pitch and number of teeth.

The top land of a gear’s teeth is defined by the gear’s addendum which, in most standard gears, is one divided by the diametral pitch while the bottom land, similarly, is defined by the dedendum of the gear, set to be 1.25 divided by the diametral pitch. These are the main terms needed to produce a simple spur gear, but others like clearance and flank are used in further defining a gear, both when adding fillets to sharp corners and when solving interference problems with tooth undercutting.

In a spur gear, the features created from these geometries can simply be extruded in the normal direction to the sketch plane, through the gear’s center axis, forming the entire gear. Most gearboxes tend to use spur gears and some use helical gears, with fewer still taking advantage of the benefits offered by double helical and herringbone gears. I decided to design my gearboxes based on simple spur gears because, as a hobby machinist, I will be capable of owning the tools necessary to create these gears in my home shop, rather than outsourcing them and spending large amounts of money on their production.

Spur gears are easily manufacturable either through hobbing or broaching, the latter of which can be performed with an indexing head and a rigid moving bed such as a lathe saddle or a mill table found in most home shops. I plan to acquire or design and construct my own indexing head so that I may produce gears myself for future projects, as gears open many doors in the way of driving rigid mechanisms with constant force transmission and velocity ratios. This will specifically come in handy when manufacturing the drive-train gearbox for my lathe and the quick-change gearbox that will drive the saddle for thread cutting and uniform cutting operations, both of which together require 51 gears of two differing modules and many different tooth counts.

##### Gear Hobbing

The gear-forming technique called hobbing is used in industry to produce extremely precise and accurate gears that will not interfere with each other, including but not limited to spur, helical and double helical, cylindrical worm, globoid worm, bevel and helicon gears. A hob is essentially a worm-gear-shaped cutting tool with flutes ground into it and relief cuts behind the cutting edges. When cutting spur gears with a hob, the hob’s rotating axis is skewed from perpendicular to the gear blank’s rotating axis by an angle equal to the lead angle of the hob, in order to ensure that the teeth are cut parallel to that axis as they are meant to be in spur gears. This axis skew is further increased when cutting helical spur gears by a total amount equal to the lead angle of the hob added to the helix angle of the spur gear.

The hob is formed so that its cross section at the plane made by the point of contact between the hob and gear blank and parallel to the end faces of the gear blank has the correct dimensions to form the gear teeth, its cross-section matching the cross-section through a rack of the same module, rather than the cross section through the hob’s central axis, which differs slightly because of the lead angle skew. This difference sets hobs apart from worm gears which aren’t prepared this way and have no skew in their orientation to their matching worm-wheels.

The hob and gear blank are driven past each other by the hobbing machine at the proportional speeds necessary to keep the new gear’s teeth in synchronized motion with the hob’s teeth. Essentially, the system treats the hob as a worm gear and the new spur gear as a worm wheel specifically in terms of the ratio between their angular velocities, but with different, skewed orientations.

As the hob revolves, its teeth remove material from the gear blank, and through its motion past the blank as it revolves, they shape the involute curve (generated geometrically from gear’s base diameter) into the face of each tooth. Continuously as the hob removes the material, it also moves parallel to the gear blank’s axis to cut new locations in the blank’s face and extend the cut to eventually run the entire width of the spur gear. At any given time, the cross section of the hob through the point of contact parallel to the gear’s ends resembles the same cross section found in a rack with the same module as the gear being produced, as was mentioned earlier.

The hob is designed with its addendum and dedendum values swapped relative to those on the gear blank which, when applicable, cuts the trochoidal undercut pattern into spur gears of a small number of teeth or with negative profile shifts, and otherwise cuts the clearance and proper addendum height into the gear blank. Essentially, since the hob is a cutter, it needs to be constructed with the values of addendum and dedendum swapped in order to create cuts in the gear that complement those values, effectively forming the correct dedendum and addendum on the gear blank. Because of this, the one difference between the cross section of the hob and that of a meshing rack of the same module is that the addendum and dedendum values are swapped, making the teeth pointier and shifting them away from the base.

At any given time, only hob teeth passing this cross section are cutting at the correct depth, which is why the hob is continuously moved slowly to traverse across the entire gear tooth face parallel to the spur gear’s axis, generating full teeth. This motion makes the machine capable of forming spur gears with any width in the range that the hob’s moving axis can travel.

I delved deep into the math behind trochoidal and involute curves in my attempts to model spur and helical gears using correct undercutting techniques. It is intriguing to watch an animation of a pinion gear as it meshes with a rack, specifically a rack with the cross-section of a hob, and watch the tips and surfaces of the rack’s teeth perfectly trace the involute and trochoidal curves that define the pinion. I made multiple animations within Creo Parametric to demonstrate this effect which is a result of hobbing and solves the problem of interference in gears with a small number of teeth but weakens the base of said teeth by reducing their thickness.

##### Gear Cutting & Broaching

Broaching with cutting tools that already have the trochoidal undercut and involute curves worked into their cutting pattern will also produce gears that do not interfere with each other, but broaching has its own problems. For one, broaching is a drawn-out process either performed entirely by hand with a linear cutter, or with a mill and revolving cutter, with the necessary curves built into them. The main issue that arises is that there would need to be a unique cutter for every possible number of teeth being formed on the generated gear in order to cover the whole spectrum of gears that need to be produced.

Manufacturing and distributing this range of cutters for every standard module and diametral pitch is impractical, and the solution to that impracticality used today leads to imperfections in the meshing pattern of gears produced with this method. Cutters are manufactured and sold with most standard sets having eight cutters to each cover the entire range of gear tooth count for a given module or diametral pitch.

For example, the subtle change between the involute of a 135-tooth gear and that of a rack is small enough that the eighth cutter in the set covers the entire range of teeth from 135 to infinity, where infinity is equivalent to a rack. Technically, the curve of the cutter’s tooth faces matches the involute of the lower end of the range, so if one were to cut a 135-tooth gear, the cutter would be perfect for the job. This makes the cutter slightly more inaccurate as the number of teeth diverges upwards from 135, the change becoming more subtle with each added tooth.

However, as the number of teeth gets smaller, for instance at the other end of the spectrum, the change in the involute form relative to the change in number of teeth is far less subtle. Therefore, the range that each cutter can cover decreases as the number of teeth being cut decreases. The eighth cutter in the average set, designed to cut gears with a small number of teeth, can only cover the range of 12 to 13 teeth as the form of a 14-tooth gear is too far different from that of a 12-tooth gear to mesh sufficiently. The next cutter can cut 14-16 and so-on, effectively covering the entire range of teeth possible from 12 to infinity, but the error in using an unmatched cutter to shape one’s gear teeth will make those teeth imprecise which is, in most cases, undesirable.

Hobbing is superior to broaching and shaping as it is faster, more precise, more versatile in its ability to cut the whole range of tooth count, and it can produce either straight or helical spur gears. Broaching is far more realistic for hobbyists to be able to perform correctly, though, so each has its merit. This also limits most hobbyists to only manufacturing straight spur gears, as helical and herringbone cannot be produced in simple broaching or shaping setups.

##### Helical, Double Helical & Herringbone Gears

If I could, I would design my lathe gearboxes to use some form of helical gear. The main benefit of a helical gear over a spur gear is that it spreads the contact event over time across the width of the gear so as to avoid abrupt contact which is a large problem in spur gears, causing vibration and thus noise. Both are undesirable in machinery, reducing the life of a mechanism’s components including the gears themselves and making the machinery less reliable and more dangerous to operate.

I see two main problems with helical gears. First, they are more difficult to manufacture in non-industrial settings and require hobbing or 5-axis CNC machining to generate, since the helix wraps in a circular motion around the face of the gear blank, removing the possibility of manual broaching using standard linear motion tables as an option. Hobbing is a much faster and much more geometrically perfect technique than broaching for very precisely manufacturing gear teeth which forms the involute tooth face with clever use of the motion of a moving flat-face gear cutter rather than a preformed involute broach.

In an industrial setting, helical gears are nearly always superior to straight spur, when they can be manufactured. The more practical problem that helical gears have isn’t in the ease in manufacturing them, though, but in the forces that they generate when compared with their spur counterparts. The helical angle of the path that a helical gear’s tooth profiles follow means that the tooth face of one gear contacts the tooth face of the other gear at that same angle, adding a new constant axial component of force in helical gears to the radial and tangential forces that already existed between two spur gears.

In spur gears, the pressure angle at which the faces of two teeth contact each other, which is measured from the tangent line between the two gears’ intersecting pitch circles, dictates the magnitude of the force components which are transmitted between the gears. The cosine of the pressure angle is the ratio of the force transmitted tangentially over the total force, which is the desired direction of transmission, while the remaining force, defined as the ratio of the sine of the pressure angle multiplied by the total force, goes to driving the gears away from each other radially.

The axles on which the gears ride are bolstered by design engineers to repel this force in all practical cases, for the most part eliminating that problem, but in helical gears there is a third component of that force, adding to mechanical losses. The force distributions then become three-dimensional and most force goes to energy transmission, with the remaining force divided both to driving the gears away from each other radially and driving them apart axially. This is a large issue in designing mechanisms as they then require some sort of mechanical component capable of resisting that axial force to prevent the gears from shifting out of position or the other components from failing to resist the axial force and breaking or becoming misaligned.

Double helical gears come into play here, and when the desired helix angle is large enough, or accommodations cannot be made to resist axial force generated by helical gears, but the smoothness of helical gears is desired, they are the best option. Herringbone gears are a form of double helical gear but differ slightly in manufacturing method, structure, and volumetric envelope. Both have the benefits given by helical gears, due to the helical nature of the tooth sweep and thus contact between them is spread evenly over time, however with the major difference that a double helical gear is split down the middle at which point the helix is reflected. The helical pattern is exactly the same on both sides mirrored across the center-plane, and in some versions is sometimes even staggered for further motion smoothing, forming a sort of arrow shape when looked at from above.

Double helical gears are manufactured mainly through hobbing, but due to the limitations of hobbing, there must be a clearance gap separating the halves of the gear to prevent interfering cuts. Herringbone gears, however, have no such gap in the middle, and are manufactured using broaching or shaping with complex tool paths and utilizing cutters that cut out of phase with each other to prevent them or their chips from colliding. Hobbing is a quick process, so double helical gears are easier to manufacture than herringbone gears, however the gap in the center is a clear problem in certain circumstances.

For a given size, specifically width, of a double helical gear, there is lost space in that gap that could have been used to transmit force, while the double helical gear’s herringbone counterpart is able to make use of this space. Therefore, herringbone gears with the same force-transmission capabilities are able to fit within a more compact envelope. 3D-modeling helical, double helical and herringbone gears is simple with some slight changes to the spur gear model.

Obviously because of the helical nature of these gears the tooth profile cannot be swept perfectly normal to the sketching plane and parallel to the central axis. It instead has to follow a path whose angle up from the plane normal to its axis matches the helix angle of that gear, and which sweeps at that angle linearly proportionally to the trajectory parameter from one face of the gear to the other. More simply put, a line can be swept in the axial direction attached at one end to the central axis of the gear and change its angle of rotation around that gear linearly with relation-driven modeling as a function of the trajectory parameter of the sweep, or the fraction of the sweep length at any given point through the sweep over the total sweep length. A few trigonometric calculations can be made beforehand to determine the constant multiplied by the trajectory parameter in order to obtain a curve with a constant and correct helix angle for that gear.

In the end, a gear with the tooth profile swept through the curve generated by this sweep will form a helical gear and reflecting this gear across the plane made by one of its faces will form a herringbone gear, allowing a simple revolved cut to be made through the center-plane if a double helical gear form is desired instead.

##### Bevel & Helicon Gears

Things get far more complicated when dealing with bevel and worm gears, as the profiles change size and orientation as they are swept. I attempted to model a spiral bevel gear for the 90-degree transmission that I needed to convert the motion of my lathe’s saddle-driving wheel to rotation in the lead screw driving the saddle. Though I came close when following an online set of instructions for modeling spiral bevel gears, my attempt was made before I had gained a deeper understanding of gear geometry, and its components don’t mesh properly when placed in a model mechanism. With some trial and error, I believe that I could generate such a model now, which I will eventually attempt to do to add more depth to my lathe model.

Hypoid gears have the benefit over spiral bevel gears of having a bit more tooth contact and being more versatile in how their axes are oriented. Rather than requiring that the axes of the meshing gears intersect, hypoid gears are specifically designed with non-intersecting axes to allow them to be used in situations where space is limited, and to give them more contact and higher force transmission. I have never attempted to model a hypoid gear and think that it sounds like an extremely daunting task but would love to eventually design templates for hypoid and spiral bevel gears within Creo to use in projects, and potentially 3D print.

Bevel gears are meant to transmit motion to a new axis separated from the driving-gear’s axis by a known angle and intersecting at one point with the driving gear’s axis. This is often done with perpendicular axes, as is my case, where mating two gears perpendicularly brands them as miter gears. In many cases, in addition to transmitting motion to a new axis, the gear-ratio is changed via differing numbers of teeth on the two gears. These bevel gears are what I sought to produce, as I desired to obtain one centimeter of saddle travel for every revolution of the wheel while the lead screw was designed for only half of a centimeter of travel per revolution, requiring a 2:1 ratio.

Bevel gears suffer from many of the same problems as do spur gears, which similarly can be resolved by transmitting force over time instead of all at once. Spiral bevel gears are used for this and are more difficult to model because the path that the tooth cut takes is far more complex than a straight line, and the profile size changes as a factor of the distance of the profile from the central axis of the gear.

Hypoid gears are effectively a transition gear between spiral bevel gears and cylindrical worm gears. The further the axes are from intersecting in a hypoid gear, the more similar it is to a worm gear, though the contact in a hypoid gear tends to be on one of the “end faces” of the larger gear’s cylinder rather on the side face, as it is in worm gears. Like worm gears, though, hypoid bevel gears have large amounts of sliding contact and less rolling contact, drastically reducing their lifetime and decreasing the efficiency of the system in some cases. They are very much an in-between gear in this sense because they combine rolling and sliding contact, where spiral bevel gears rely almost entirely on rolling contact and worm gears rely almost entirely on sliding contact. This allows them to transmit more force than spiral bevel gears as well, but not quite as much as worm gears, and makes them wear out more quickly than spiral bevel gears but not quite as quickly as worm gears.

##### Cylindrical & Globoid Worm Gears

Helical worm gears were where I began in my modeling of worm gear trains primarily because they are simply a helical gear matched with a worm gear and required no differences in modeling from helical gears. However, cylindrical worm gears are truly used in industry where helical worm gears rarely are because of their non-enveloping point-contact nature.

Cylindrical worm gears are called “single-enveloping” gear pairs because the face of the teeth on the worm wheel envelops the tooth face on the worm gear, where the worm wheel is the large disk and the worm gear is the usually small and long screw-like gear. This single-enveloping nature allows the contact between the two gears at any given time to be line contact, similar to two spur gears meshing. Line-contact can transmit far more force than point-contact, as the force is spread out over a larger area thus leading to a lower pressure, lowering the risk of localized plastic deformation at the interface between gears. Because of this, cylindrical worm wheels are capable of transmitting far more force for the same number of teeth in contact than helical worm wheels and can mesh with the same worm gears as their helical worm wheel counterparts.

Cylindrical worm gears are not perfect however and have the major shortcoming of being limited to essentially three or less teeth transmitting force at a given time, depending on the number of teeth on the worm wheel. The other major area in which cylindrical gears can be improved is in their method of contact. Line-contact is good but is easily surpassed by face-contact.

Worm gear geometry has been developed that does not utilize the involute-curve-based tooth profile of a cylindrical gear, thus it is not limited at any given time to point contact at a cross section or line contact at the gear interface. It is instead capable of constant line contact through the central cross section of the gear and face contact at the ends of the worm in the interface between teeth. In addition, many more teeth are in contact at a given time, since the worm’s ends are defined by the point where the tooth face is parallel to the line, generated between the axes of the gears, equal to the gear’s pressure angle, giving room for any number of teeth that all transmit force simultaneously. On both fronts, number of teeth and contact pattern, this allows for far more force to be transmitted between teeth than in a cylindrical worm set without risking permanent plastic or brittle deformation of the tooth faces of the worm or worm wheel.

This gear pair is considered to be “double-enveloping” due to the worm’s hourglass shape, wrapping around the worm wheel and allowing many teeth to be in contact throughout its motion in addition to the single-enveloping nature of the worm wheel around the worm gear, similar to the cylindrical worm wheel. Today it is called a globoid worm gear and is superior in every mechanical way to its cylindrical worm gear counterpart. My first introduction to this kind of gear had them named Hindley worm gears after a clockmaker named Henry Hindley who in the mid-1700s made the first known globoid gear with an hourglass-shaped worm and used the same tooth-cutting technique that is still used today.

I have looked into patents concerning the details of manufacture and tooth contact between the globoid worm and worm wheel which I have found very intriguing, especially with how much they surpass the capabilities of their cylindrical and straight-worm counterparts. I was forced, when modeling the Hindley worm gear in Creo Parametric, to learn new methods of feature cutting and to delve deeper than I ever had into the trigonometry within the cuts that form the worm and worm wheel teeth. It took a few days to finally get the geometry to be mathematically perfect but was well worth the effort as I ended up with template models that I can apply to future projects with both single and multi-start worms and the entire range of tooth-number possibilities.

I had a lot of fun animating the meshing between my cylindrical and globoid gear models for this site, including the failed attempts at generating a cylindrical worm wheel. I actually completed the model for the globoid worm wheel long before I ever solved the problem of the cylindrical wheel in Creo and had to go through 10 different models before fully resolving all of the issues that I encountered. I was not able to properly visualize the path that a section of the tooth on the worm gear traveled through as it passed across the cylindrical worm and was stuck making guesses for weeks before finally breaking down the motion into its base components.

I realized that effectively, the worm wheel would have to act the same as many infinitely thin spur gears rotating about points along a circle the same distance from the central axis of the worm as was the central axis of the worm wheel. By modeling a simplified version of this effect, I was able to visualize for the first time the path through which the tooth profile had to be swept and met irony in that it was to be swept along the same path as was the sweep that I used to form my globoid worm gears. Despite not using an hourglass-shaped worm, the motion of the teeth relative to each other remains the same between globoid, helical and cylindrical single enveloping gear pairs.

I discovered later that there were two main shortcomings in the methods that I used for my cylindrical and globoid worm gear models. While the worm gears were both mathematically perfect, both worm wheels didn’t truly represent the exact profiles of real-world manufactured worm wheels. First, I discovered in my cylindrical worm wheel that when the teeth mesh in my models, the line contact that should exist between teeth exists at one end of the tooth travel but not at the other end, and it fades to point contact, not truly representing the exact geometry in a real-world cylindrical worm wheel. I could not think of a way to resolve this issue with the tools given by Creo, as there is no way in Creo Parametric 3.0 to simulate the cutting path that would be given in manufacturing a worm wheel through hobbing. Cross-sectional sweeps are the only useful option given by Creo, and real-tool volume-sweeping was made available in later models of Creo that I don’t currently have.

So, I had to mathematically define my involute curve and sweep it manually, even though the real section is only defined by involute curves at one point in the gears’ motion. To boil it down, there is no easy or clean way to model the true form of a cylindrical worm’s teeth in Creo Parametric 3, and I’m only looking for elegant solutions, so I have settled with the near-win model that I achieved for demonstration and representation purposes.

In my globoid worm gears the problem is similar in that there is no graceful way to make the cuts necessary for my model to represent its real-world counterpart. The model comes so close to perfect when examining one-start worm and worm wheel combinations that it truly doesn’t matter to me much, so long as I understand its shortcomings. It even performs well in two and three-start situations but begins to clearly diverge and cause interference as the number of starts grows, as I detail in the animations that I made of the different start combinations.

The issue appears as a result of the non-constant helix angle that exists within the worm gear of a globoid gearset. The instantaneous value of the helix angle is lowest at the extremes of the worm, where its diameter is widest, because for every revolution around the central axis the circumference covered by the tooth increases as the ends are approached. The helix angle is steepest at the center of the worm’s hourglass shape, where its diameter and circumference are smallest, because the tooth coils over a very short distance relative to how it acts at the ends of the worm. One can imagine a helical spur gear with a very small diameter versus one with a large diameter, both with the same helix angle, where the tooth revolves around the small gear many times between its ends, but only some fraction of one time on the larger gear.

In this case, since the number of revolutions per change in angle across that hourglass arc needs to remain constant, being tied to the diametral pitch, the thing that changes to accommodate is the helix angle, which adjusts itself continuously through the tooth sweep. The conventional method used for modeling the necessary tooth cuts in a globoid worm takes the tooth shape at the extreme ends of the worm gear and cuts both of those sections out of the worm wheel blank centered at the same location. In a worm gear system with one start and/or a sufficient minimum worm pitch diameter, this is sufficient to prevent interference in every possible scenario between the two gears. When the helix angle only makes small changes throughout the tooth-cut, as occurs in wheels with minimum diameters that don’t vary much from their maximum diameters, no interference occurs. However, when a great difference between maximum and minimum helix-angle is present, all locations between the two ends of the worm gear that form the cuts taken from the worm wheel, if used in place of those ends, would remove extra material that the cuts used in the model do not remove.

Changing the number of starts adds a linear multiplier to the helix angle at any given point with the value of the number of starts. So, for instance, if the minimum helix angle were two degrees and the maximum were three degrees, the difference between the two would only be one degree, and interference might not be significant at all. However, in a 10-start system, the helix angles then change to a 20-degree minimum and a 30-degree maximum, the difference now growing to a significant 10 degrees, which easily causes interference between meshing teeth due to inconsistent cutting angles. This issue never appears in real hobbing, as the worm gear takes care of the cutting itself and forms the wheel exactly the way It needs to be to mesh perfectly.

An additional feature called crowning is added in real worm gears in order to allow for some play in the alignment of the axes, since it is so difficult to achieve a perfect mesh even with precision-drilled axes, especially because no system is ever perfectly rigid or perfectly aligned. Crowning hardly reduces the maximum allowed force transmission between gears, and it reduces the risk of chattering and prevents gears from effectively forming their own uneven crowning. It is quite similar to backlash in many ways, which is the small gap between gears that allows for their expansion during use, prevents imperfections from causing binding, and gives a small amount of play to the axis alignment of meshing gears.

I did not add crowning to my gears as I meant for them to be perfect geometric representations of the mechanisms utilized in globoid and cylindrical worm gear systems. As I am unable to set Creo to perform every cut from every possible position relationship between the worm gear and worm wheel, I have to settle on the next best thing, which again is limited to 2D-profile-swept material removal. I have no choice when using Creo Parametric 3 but to work with the cuts formed by the teeth at the extreme ends of the worm gear which cover the most bases and function properly for one-start gear systems. Eventually If I come to own the newer Creo 4 or Creo 5 programs, I’ll be able to move my designs over to new part files and use the new solid-tool cutting methods that they offer.

##### Lathe-Drive Gearbox

The reason that I got involved in gear design in the first place was to design the two major gearboxes in my lathe, which I have finally done, after investigating the many other types of gears besides spur gears. I ended up producing a drive-train gearbox with eight speeds meant to handle large forces for transmitting force between the motor and spindle with low energy loss, allowing for a large range of spindle speeds.

Using features in Excel, I was able to replicate a Colchester lathe’s drive-train gearbox, extrapolating from images and fundamental gear geometry the number of teeth on each gear and the degrees of freedom that each gear had, including its connections to other gears. In that gearbox, most gears were set up in pairs to add a ratio transformation in one state and remove it entirely in another state. One of the gears was set up to allow direct force transmission to another gear shaft through the same axle in one state while, in its other state, passing through two other gear pairs before reaching that same axis. The combinations of these ratio transformations allowed for an almost perfectly exponential curve of achievable output speeds.

I was able to make my own guesses regarding the module or diametral pitch of the gears, but regarding both the interactions between gears and their individual numbers of teeth, I was forced to perform analysis based on a small video and an image of an opened lathe gearbox. I found that there were three pairs of gear states that when toggled in certain combinations would yield different output speeds. In order to figure out the exact ratios and numbers of teeth, I first had to draw a sketch of the gearbox and determine which gears were allowed to interact with or were mounted directly to which other gears. Since there were three levers controlling gear positions and each lever had two states, I figured that there were 2^3 or eight speeds selectable through these levers and found the manufacturer-given speeds online. I threw them into excel and plotted them, then determined the required ratios that the gears would need to have in order to produce those speed changes.

I then closely followed that model and made matrices of the possible numbers of teeth between known gear pairs and the ratios that they produced. Using conditional formatting, I passed the matrices through filters that would color the ratios far from the expected ratios as dark red and those very close to said ratios as white. Gears mounted on one same axis meshing with gears mounted on another same axis are required by their geometry, I discovered, to have a specific number of teeth.

Two meshing gears each have their own tooth count and adding them together yields an integer number. Any other pair between then same two axes and of the same module is required to have the same total number of teeth. In the case of this gearbox, the most common pairs were 22 and 22 gears for a ratio of 1:1, adding to 44 teeth, and 31 and 13 teeth, also adding to 44 teeth and yielding a ratio of 2.38:1 which very closely matched the ratio that I was expecting to see. I used this method to find every gear’s number of teeth, then I modeled each gear in Creo and assembled them into a mechanism.

Funny enough, the fit exponential curve that my gear ratios produced was extremely close to the exponential curve given by the manufacturer-given-speed curve, however its constituent points weren’t nearly as well matched with the curve itself, despite it being the most accurate possible combination of gear ratios that could lead to that curve. The cool thing was that the points picked by the manufacturer, instead of being chosen directly from the real gear ratios, seemed to be picked from the values predicted by the exponential curve at those gear states. So, my gear ratios ended up predicting even more accurate spindle speeds than the manufacturer of the original gearbox decided to print on their lathe for whatever reason.

Upon completing that project satisfactorily, I constructed a much smaller yet more complex gearbox for cutting threads on my lathe, named a quick-change gearbox. It houses 41 gears in two separate mechanisms: One section is designed for easily halving its output speed relative to the input spindle speed, made up of 16 gears, each set of two gears revolving at half of the speed of the gear pair driving it. This way I can easily select from any one of the eight accessible output gears in this train as an input to the second system to achieve a different starting ratio from 1:1 to 1:128, which drives the input of the other section.

##### Quick-Change Threading Gearbox

There are 17 gears in the second system, increasing in tooth count at a one-tooth interval from 16 teeth to 32 teeth. Technically I only needed 16 gears to cover the entire range of gear ratios from 16:32 to 31:32 but since this is not for an industrial application, I decided to squeeze in the one extra gear at the end. I like the numbers better with that final gear, as I can achieve a range of cutting speeds from one thread per inch to 256 threads per inch, rather than ending it at the less attractive 248 threads per inch. This second section takes the output of the first section, which can be anywhere from 1:1 from the spindle to 1:128 from the spindle by factors of two, and divide it into 16 more parts. At a 1:1 input ratio, I can achieve any integer value between one and 16 tpi, by twos from 16 to 32, by fours from 32 to 64 and so on to 256. This is based on the mechanisms used already in industrial lathes, so it isn’t anything new, but since I make the rules in this project I threw in the few extra gears to give me more control.

The Colchester lathe quick-change gearbox that I used as a reference when designing mine only had its second section comprised of nine gears separated for the most part by twos, which while spatially efficient, gave a lower range of possible tpi output values, so I bumped it up to 16. It also was only capable of halving the spindle speed four times in its first section, leading to a maximum cutting speed of 4 threads per inch, and a minimum of 128 threads per inch. Again, I desired more options, so I used the same mechanism with 16 gears so that I could halve the spindle speed eight times, giving me the range from 1 tpi to 256 tpi.

Most lathes are incapable of performing both metric and imperial cutting speed ratios. Inherently, metric threading standards make it difficult to produce gearing ratios that will cut accurate threads, as many end up being (in threads per centimeter, a necessary conversion for lead screw driving) irrational or have repeating decimals. For instance, the standard of 0.35 mm/thread produces 28.571428 threads per centimeter, which is a ratio unattainable with any gear system. I attempted to cover as many standard metric threads as possible in my quick-change gearbox design, despite some being impossible with a single lead screw.

The lead screw that drives the saddle in my lathe is a ball screw with a 0.5-centimeter lead. As such, I was able to easily convert that lead to rational ratios giving me access to metric threads like 0.4, 0.5 and 1 mm per thread which convert to 25, 20 and 10 threads per centimeter respectively, easily attainable with gears. However, I desired for my lathe to be able to make imperial threads, not only for versatility but because the majority of the machining that I have witnessed in the United States seems to remain on the imperial system, as are most threaded components that I could purchase in stores here.

Rather than look up the issue of converting metric lead screw ratios to imperial, I decided to solve the problem myself and used Excel to map out potential gear ratios that would lead me to a usable ratio. I saw quickly when multiplying 25.4 mm by different integers in an array that the smallest integer value (needed for gear teeth) that was produced was 127. Dividing 127 by 50 gives the ratio 2.54, meaning that I could convert metric to imperial rotations of my lead screw through use of a 127-tooth gear meshed with a 50-tooth gear. My lead screw has a lead of 0.5 cm per revolution though, so to go directly between cm and inches I needed a ratio of 127/25 to translate the spindle output into exactly 1 tpi of saddle movement. I matched this with a pair of 76-tooth gears whose ratio is 1:1, giving me a maximum metric ratio of 0.5 threads per centimeter.

Coincidentally this is the last non-repeating thread-per-centimeter value that the realistic range of metric standard threads offers, and I can approximate the two higher standard threads of 5.5 and 6 millimeters per thread as 4.5 and 4.25 threads per inch respectively without creating large divergence error. I calculated the error of my achievable imperial approximations of metric standard threads and found that my 4.5 tpi approximation of 5.5 mm/thread accumulates one thread-width of error after 21.49 centimeters of threaded length. The 4.25 approximation performs much better only accumulating one thread of error after 152.4 centimeters of length. Depending on the play between mating threads, these are both perfectly usable approximations, and the very best that I can do with my lathe setup.

I learned only a few days after coming up with my own metric-to-imperial conversion gear system that the 127-tooth gear is in fact how things are done in industry and was excited to have found that solution myself. I have modeled both gearboxes but have yet to acquire an indexing head or table on which to broach gears, so for the time being, I won’t be focusing on the production of my gearboxes but will instead further improve their designs.

##### Internal Spur Gears

Both in my research online regarding the gearboxes that I intended to make and in my engineering classes, I was shown the many uses of the planetary gear train, used in drills and other such high-torque scenarios. They are constructed entirely from spur gears, but they make use of a type that I have not yet mentioned called an internal spur gear. The cross-section of an external spur gear is entirely carved away from a gear blank leaving behind essentially a negative gear, which is the form of an internal gear, with the teeth on the inside of a ring. External spur gears can ride within this internal gear effectively allowing a gear system to fit in a smaller envelope, and also reversing the direction of motion transmission relative to normal spur meshing, making the output revolve in the same direction as the input.

This can be further utilized by combining multiple external gears within the internal gear as is the case in planetary gearsets. A planetary gearset consists of an internal spur gear called a ring gear which resides around the entire system. Within the center of that gear, riding on the same axis but an independent axle, is the sun gear. Meshing with both the sun gear and the ring gear, placed somewhere between the two within the ring gear, is at least one gear called a planet gear. These are normally found in sets equally dispersed around the sun gear, usually with as many gears of the same size with the correct number of teeth that will fit without interfering with each other. This can be expanded, with multiple levels of planet gears exchanging motion between the sun and ring gear and reversing the direction of the sun relative to the ring each time, but the ratio between the sun and ring gear will remain the same.

However, it is often not the rotation of the sun or the ring gear that is sought, but the rotation of the carrier, which tracks the circular motion of the axes of all of the planet gears around the central axis. In a planetary gear system there is one input and are two possible outputs of motion. The sun gear, the ring gear, or the carrier may be held fixed, another of the three, driven, and the last drawn from as the output which, in systems that input and output motion in the same axis, can be very useful. This is used often in tools like drills to achieve a high-torque output in line with the motor output, or toggle from high torque to high speed by switching which gear or component is held fixed in the planetary gear system.

One of the most notable planetary gear systems that I’ve seen was in a YouTube video showing off the extremely high gear ratio of 3,616,238,492,881:1 in a small system. It utilized four in-series planetary gearbox combos each with an output ratio of 1379:1, where the input of each subsequent shaft is driven by the output of the previous, giving a total ratio of 1379 to the power of four, all within the envelope of a two-liter bottle. It cleverly takes advantage of a slight difference in rotation speed of the output ring gear by offsetting the number of teeth between its two systems by one tooth on each gear.

Two side-by-side planetary gear systems are paired together with their planet gears rigidly linked and sharing the same axes, the input to the entire system being the first gearset’s sun gear, and the output being the ring gear of the second gearset. The fact that the two carriers are effectively locked together due to the planet gears being constrained to the same axes and rotational velocities paired with the fact that the ring gear on the output gearset is one tooth larger or smaller than the ring gear on the input gearset creates an output velocity in the second ring gear characterized by the one-tooth difference, the sun gears tooth count and the output ring gear's tooth count.

Rotating the carrier one full turn relative to the fixed ring gear requires the sun gear to revolve a number of times characterized by the ratio of that ring gear's number of teeth to its sun gear's number of teeth, plus one (the one representing the one revolution made by the carrier). The input system's planet gears will revolve a number of times per revolution of the carrier characterized by the tooth count of one planet gear divided by the tooth count of the ring gear which also forces the planet gears of the output system to revolve that same number of times. Dividing the tooth count of a planet gear on the output system by previous resulting value yields the number of teeth that the output ring gear will progress relative to the carrier's motion, and if that value is different than the number of teeth on the output ring gear, the ring gear will turn.

Dividing the output ring gear's tooth count by the quantity of the difference between its tooth count and that resulting value will yield the number of carrier revolutions required to cause one rotation of the output ring gear. Multiplying that value by the ratio of the carrier's revolutions to the input sun gear's revolutions will yield the total gear ratio from input to output, which adds up extremely quickly in systems with a small tooth count on the sun gear and large count on the ring gears.

Stacking these systems in series can achieve almost any unbelievably large gear ratio in a small envelope, though the torque limitations of the gearbox are inferior to other designs if the carrier cannot hold the planet gears to their axes. I plan to model this system myself now that I understand the underlying principles but have not yet thought of a use for it. I may attempt to 3D print the mechanism for display purposes once I own a 3D printer.

##### Cycloidal Gearboxes

While researching these gearboxes and others to use in my lathe and watching the many videos online pertaining to their mechanisms, I happened across an entirely new type of gearbox that I had never heard of before, with a very unique design. I had of course learned of internal gears and their use in planetary gear systems but hadn’t ever been made aware of cycloidal gear reducers and their extremely high velocity and torque ratios, which have many similarities to planetary gear systems.

They are composed of an internal and an external spur gear, but the teeth, instead of being geometrically shaped by involute curves, are driven by a combination of matching hypocycloids and epicycloids. Say for instance one wanted a large torque output and deep reduction in the velocity of an input shaft with a 40:1 ratio and desired the output to be in the same axis as the input, similar to planetary gearsets. Certain worm gears could be used if the axis direction were supposed to change, but the output torque of worm gears is limited compared to cycloidal gearsets and cannot produce an output through the same axis as the input.

In a cycloidal gearbox, an external cycloid-based gear with 40 teeth rolls within an internal cycloid gear with 41 teeth. A cam is set up in the center of the internal gear on which the external gear rides, with its axis offset from the internal gear’s axis by the value of the difference between their pitch diameters. The cam is driven by the input and moves the central axis of the external gear in a circle around the internal gear’s axis, which in turn causes the teeth to mesh.

One major advantage that cycloidal gears have over involute gears is that a large number of teeth can mesh at one time, each exerting a different amount of force based on its angle of contact, all with no backlash whatsoever and no interference. The geometry that results from alternating complementing hypocycloid and epicycloid curves, which add to form an exact circle, is able to ride within or outside of any other cycloidal gear with the same parameters defining its curves. This allows the smooth interface between cycloidal gears. They perform much better in the sense of force transmission when used in this way, as an internal and external gear pair differing by one tooth, than in any other scenario, and are limited by this, becoming less efficient and eventually worse than involute gears in almost any other scenario.

Where in that prime situation there are many teeth in contact at any given time, effectively half (though all contributing differently to the force transmission), in other scenarios the number of teeth in contact drops rapidly, and systems with low numbers of teeth can fail to transmit any force at all in certain orientations. I designed a pump based on these cycloidal gears with a small number of teeth, but it must be driven by an involute gear on the backside in order to get the correct motion needed to pump fluid effectively. Otherwise, the teeth lose contact and do not mesh.

Cycloidal gear boxes that utilize cams to move components do have one large weakness, but it is largely taken care of with one small, clever design change. When the center of mass of a large and heavy component, such as the external cycloidal gear within the gearbox, is rapidly changing as is the case with its cam-driven rotation, it induces vibration. At high input velocities, the vibration introduced can be dangerous and cause mechanism components to fail. To largely reduce the massive swings in the inertia of the system, these gearboxes are manufactured with two external cycloidal gears riding within the internal gear, 180 degrees out of phase with each other. That way, the inertia that each introduces is essentially negated by the equal and opposite inertia introduced by the other.

If I were to design a system like this this myself, I would take it a step further and consider all system inertia in a way which I have not seen done by any of the gearboxes that I have found online. This system with two equal gears riding alongside each other works to reduce inertia changes only within their plane of motion. It does nothing to reduce the imbalance that occurs between the front and the back of the gearbox, now exaggerated by two separate gears, and therefore I would improve this design by introducing a third gear.

I would sandwich one external gear between two others, each one on the outside with half of the gear width of the one in the middle, effectively balancing the inertia in all dimensions as long as they are precisely manufactured with equal weights. The two on the outside would be out of phase with the one between them and combine to be the same mass and mass distribution as the one in the middle, and the net inertia introduced by them all simultaneously would be effectively 0. I assume that this has been done already since it isn’t a large stretch from the original solution, but I found it to be an interesting improvement to the many gearboxes of this kind built based on only two out of phase gears that can be found online.