##### 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.