All about gear design!

The module controls the size of the teeth, and thus, the size of the gear. Overall, the impact of the module on gear design can be summarized as follows:

Larger module -> Larger teeth -> Bigger gear

Gear tooth dimensions in function of the module

In the image above, the black dashed line represents the root circumference of the gear (the one where the teeth start), and the blue dashed line the pitch circle.

Here you can check a table for the DIN modules.

The pressure angle affects the load capacity, the efficiency and the transmission error of a gear system. A higher pressure angle generally results in a stronger, more efficient and more accurate transmission, but also in higher friction and noise. In practice, a pressure angle of 20° to 25° is commonly used for gears.

Pressure angle effects on tooth geometry

With an increase in pressure angle, the teeth become sharper. This, in turn, influences the minimum number of teeth required, as a higher pressure angle allows for fewer teeth in the gear.

Two dimensions come into play when manufacturing and using gears, the addendum circle and the pitch circle:

Gear radii

The respective formulas for the addendum and pitch circumferences are:

If you're planning to machine gears, the addendum circle represents the size of your material previous to the cutting. The pitch circle is just a reference for assembling gears together.

When assembling gears, the distance between centers is derived from the position they take when their pitch circles are tangent:

Distance between centers

Meaning that the distance between centers 'C' can be expressed as:

C = \( (PD_{1} + PD_{2})\over 2 \)

Where 'PD' is the pitch diameter of its respective gear.

For gear pairs, the principles of the transmission between them can be expressed as follows:

Gear pair transmission

Transmission of torque and speed between gear pairs is in function of the ratio between their amount of teeth. It can be expressed as follows:

$$ i = {z_{Driven}\over z_{Driving}} $$

Where the general expression involving torque and rotational speed is:

Where

• ω is the rotational speed (commonly in rpm or rad/s).
• T is the torque (commonly in N·m or lb·ft).
• z is the amount of teeth of the gear.

Gears work under the principle of levers, where a small force applied at one point is amplified and transferred to another point.

(1).- System in balance due to levers principle

In the image above, the system is balanced due to the levers principle, as the 10kg multiplied by its lever length (2m) is equal to the right weight (20kg) multiplied by its arm length (1m). This means that '(10kg * 2m) = (20kg * 1m)'.

Therefore, gears can be visualized as a set of levers organized in a circular configuration:

(2).- Two lever arrays in a circular configuration

The calculations for this lever array do not change, they still use the same levers principle. As an example, if 10kg of force is applied to the left array, what would be the necessary force applied to the right array to balance it out? For the sake of simplicity, consider both arrays are touching at a single tangent point, and that their center distance is the sum of half of both lever lengths (ignoring that continuous contact would be impossible for this geometry):

(2.1).- Simplified version of image (2) shown as two levers in contact

With the image above, the similarities with the system on image (1) are obvious, only this time there is no fulcrum at the center since each lever has its own rotational point (white dots) at their own center. This means that the resulting force for the right lever can be calculated as '(1m * 10kg) - (0.5m * x) = 0'. We isolate 'x' getting 'x = (1m * 10kg)/(0.5m)' meaning that 'x' is equal to 20kg.

Furthermore, that those lever arrays can be visualized as simply two tangent disks:

(3).- Two tangent disks horizontally aligned

The distance between the centers of the two disks is equal to the sum of their radii (1.5m). If there is no slipping between the disks during rotation, it can be deduced that for every one full rotation of the left disk, the right disk completes two full rotations in the opposite direction. This is due to the diameters ratio, which is determined by diving the diameter of the left disk (2m) by the diameter of the right disk (1m) equaling 2. The same would be true for the inverse, for every full rotation of the right disk, the left one would make half a rotation.

Gears can be easily understood when visualized as circles in contact at a single point, as will be demonstrated in the following sections.

The fundamentals of gear geometry are relatively straightforward, particularly when visualized as two circles in contact. When designing gears, there are four circles to consider, but mostly only one is crucial when assembling them.

(4).- Gear radii

Addendum Circle: The outer circumference of the gear, represented by a solid red circle.

Pitch Circle: The circumference where the gears make contact, represented by a blue dashed circle.

Base Circle: The circumference where the involute portion of the teeth starts, represented by a green dashed circle.

Root Circle: This is the circumference where the teeth begin, represented by black arc sections, as indicated by the arrow.

The respective formulas for the diameters of each circumference are as follows:

Where 'm' is the module, 'z' is the number of teeth for the gear (also refered as 'N'), and 'α' is the pressure angle. These concepts will be further explained in the 'Gear Geometry' sections below.

There are various types of gears, but there are certain guidelines that can be followed for the majority of them during assembly. These guidelines include:

• Mating gears must share the same module.
• They must also have the same pressure angle.
• Their pitch circles must be tangent.

By keeping these guidelines in mind, determining the distance between centers becomes simple. In the image shown, the distance between centers for two external gears, in this case spur gears, is simply the sum of their pitch radii, which can be calculated using the following equation:

(5).- Gear mesh

As seen in the image above, the distance between centers for two external (in this case spur) gears is merely the sum of both their pitch radii, which follos the expression:

Achieving perfectly tangent pitch circles can be challenging, but the involute section of the gear's teeth compensates for small errors in positioning, minimizing their impact on the gear's lifespan, even though they may shorten it.

As a gear gets larger, its teeth will resemble more and more a trapeze. Essentially, a rack can be thinked of as a section of a gear with an infinite amount of teeth:

(6).- Rack geometry

Pitch line 'PL': The imaginary line where contact occurs between a rack and a gear.

Pressure angle 'α': The pressure angle of the rack.

Pitch 'P': The distance between teeth.

Tooth thickness 'T': The thickness of the tooth at the pitch line.

Addendum 'ha': The top portion of the tooth, it is the distance between the pitch line and the tip of the tooth.

Deddendum 'hf': The bottom portion of the tooth, it's measured from the start of the tooth (the trapeze section, not the bottom of the rack) to the pitch line.

The respective formulas for the parameters above are as follows:

The geometry of a rack is crucial, as it has a significant impact on the geometry of other types of gears.

The module is a crucial factor in gear design as it determines the overall size of the gear. The module affects the size of the gear teeth, which is represented by the distance between the pitch radius and the tip of the tooth (addendum radius):

(7).- Impact of the module in the gear tooth

The image above shows the ipact that the module has on the tooth size since its total height depends on it. The total height of the tooth is known as 'h'.

(8).- Tooth total height 'h'

The total height 'h' of the tooth is given by the formula:

When selecting a module size, it's important to consider the effects it will have on the gear. A larger module results in bigger teeth and gear, as well as stronger teeth.

Table 1.- DIN Modules

The pressure angle is one of the most important parameters in gear design, as it affects the load capacity, the efficiency and the transmission error of a gear system. It is defined as the angle between the line of action, which is the line connecting the points of contact between two meshing gears, and a line perpendicular to the plane of rotation of the gears.

(9).- Pressure angle effects on tooth geometry

As depicted in the illustration, the pressure angle affects the tooth form. With an increase in pressure angle, the teeth become sharper. This, in turn, influences the minimum number of teeth required, as a higher pressure angle allows for fewer teeth in the gear.

A higher pressure angle generally results in a stronger, more efficient and more accurate transmission, but also in a higher friction and noise. In practice, a pressure angle of 20° to 25° is commonly used for gears, although this may vary depending on the specific application and the materials used.

An involute is a curve that is defined based on another shape or curve. In modern gear manufacturing, the involute of a circle is commonly used. The parametric equations for the involute of a circle are as follows:

(10).- Involute of a circle

The parametric equations for the involute of a circle are the following:

In the case of gear teeth, the involute portion starts at the base circle, and its parametric equations are as follows:

By including the σ parameter, the involute curve can be rotated 'σ' radians/degrees around the origin. This is demonstred in image (11), where σ=π/2 (or 90 degrees):

(11).- Involute of a circle rotated 90 degrees

Including the 'σ' parameter in the equations above is not necessary. An alternate approach to rotating the involute curve around the origin is to use a 2D rotation matrix:

While the length of the involute curve is typically not a critical factor in most CAD software, it can be useful to restrict the curve to a specific radius. In such cases, parameter 't' in the equations [11] can be used to directly control the extent of the involute curve to the desired radius.

It's important to note that the parameter 't' actually represents a range of the "roll angle", which is the angle at which the gear tooth's point of tangency with the pitch circle rolls along the line of action. By setting the roll angle for a specific radius, the involute curve can be effectively limited to that radius, making it a helpful tool in gear design.

(12).- Involute roll angle 'θ'

The image above is a graphical representation of the roll angle. While it looks intimidating, its equation is rather simple:

$$ \theta_{r_t} = \sqrt{\left(\frac{r_t}{r_b}\right)^2-1} $$

Here, \(r_t\) is the radius at which the involute curve coordinates are to be found. When 't' equals '\( \theta_{r_t} \)', the involute will be touching the circumference of the circle with radius '\(r_t\)' at the resulting X,Y coordinates. With this in mind, an effective range for 't' can be found for the following cases:

If \(r_b \geq r_r \) then

$$ 0 \leq t \leq \theta_{r_a}$$

If \(r_b < r_r\) then

$$ \theta_{r_r} \leq t \leq \theta_{r_a}$$

Where:

To correctly design gears using CAD software and perform FEA analysis, it is crucial to consider tooth geometry. While the properties of teeth have been discussed in the involute and rack geometry sections, it is important to note that tooth geometry varies between a rack and a real gear, especially in terms of tooth thickness. Thus, understanding the specific tooth geometry for the type of gear being designed is crucial for accurate modeling and analysis.

(13).- Tooth thickness at an arbitrary radius

The tooth thickness at an arbitrary radius, represented by the symbol \(T_t\) , is illustrated in Image (13). This value can be calculated using the following expression:

Where

$$α_t = \cos^{-1}({rb\over r_t})$$

$$inv(ψ) = tan(ψ) - ψ $$

And

Although tooth thickness measured in arc length alone may not be very useful for gear design in CAD software, it can be used to calculate the corresponding tooth thickness angle.

(14).- Tooth thickness at an arbitrary radius

$$ σ_t = {T_t\over r_t} $$

Where \(σ_t\) is the tootch thickness angle for the arbitrary radius \(r_t\).

Coming soon...

• Torque: Is a measure of the twisting or turning force applied on an object, such as a shaft or a wheel. It is typically measured in units of Newton-meters (N·m) or pound-feet (lb·ft) and is the rotational equivalent of linear force (or linear work).
• Rotational speed: Is a measure of how fast is an object is rotating. It is typically measured in units of rotations per minute (rpm).
• Driving gear: The gear that causes another gear to rotate in a mesh.
• Driven gear: The gear moves as a result of the driving gears' rotation.
• Pinion: A term used to describe the smaller gear of a pair.
• Wheel: A term used to describe the bigger gear of a pair.

Simplifying the visualization of gears as circles in contact has been a crucial step in understanding their power and speed transmission.

In a system of two tangent disks without slipping, as shown in image (3), the ratio of the diameters of both disks determines the relationship of their rotations. This relationship is called the transmission ratio, which can be expressed as:

$$i = {Driven\over Driving}$$

For example, if the left disk drives the system (causing the other to move), the transmission ratio would be 2 (i = 2m/1m). Conversely, if the right disk drives the system, the transmission ratio would be 0.5 (i = 1m/2m).

For rotational speeds, the transmission ratio can be expressed as:

$$i = {\omega_{Driving}\over \omega_{Driven}}$$

Where ω is the rotational speed (commonly in rpm or rad/s).

The torque is inversely proportional to the rotational speed and can be expressed as:

$${\omega_{Driving}\over \omega_{Driven}} = {T_{Driven}\over T_{Driving}}$$

$$i = {T_{Driven}\over T_{Driving}}$$

Where T is the torque (commonly in N·m or lb·ft).

(15).- Transmission gear mesh

To assemble gears, it's important to make their pitch circles tangent. For this to happen, they must have the same pressure angle and module. The transmission ratio can be determined using the teeth amount instead of the diameters, as the module cancels out:

$$\therefore i = {z_{Driven}\over z_{Driving}} $$

This means that the gear system's transmission ratio is a function of their teeth amount.

The general expression can be expanded to:

Where 'ω' is the rotational speed, 'T' is the torque, and 'z' is the number of teeth.

Using Image (15) as an example, if the left gear has 40 teeth and the right gear has 20 teeth, both with a 2mm module, the transmission ratio could be 2:1 or 1:2, depending on which gear is the driving gear. If the smaller gear drives, the torque will increase but the speed will decrease by the ratio. If the larger gear drives, the speed will increase but the torque will decrease by the ratio.

Gear trains are simple mechanisms where two or more gear arrangements are put to work. There are two types: simple and compund gear trains. Simple gear trains are those where all the gears are aligned alongside each other as represented in the following image:

(16).- Simple gear train

The calculations for the transmission ratio in simple gear trains are very straightforward since only the first and last gear matter. For example, in image (16), if gear 1 drives the system, what is the transmission ratio at gear 4? The gray gears have 40 teeth each, and the dark blue gears have 20 teeth each.

Using the general expression for the transmission ratio in a gear pair:

$$ i = {z_{Driven}\over z_{Driving}} $$

Substituting all the driven and driving gears:

$$ i = {z_{2} \cdot z_{3} \cdot z_{4}\over z_{1} \cdot z_{2} \cdot z_{3}} \rightarrow i = {\cancel{z_{2}} \cdot \cancel{z_{3}} \cdot z_{4}\over z_{1} \cdot \cancel{z_{2}} \cdot \cancel{z_{3}}} $$

$$ \rightarrow i = {z_{4}\over z_{1}} \rightarrow i = {20\over 40} $$

$$\therefore i = {1\over 2} $$

This proves that the transmission ratio is only affected by the first and last gear, as the middle gears serve as both driving and driven gears, they will cancel each other out in the equation.

Compound gear trains consist of gear pairs where the output gear drives the input gear of the next stage. Commonly, they are used to increase or decrease the speed or torque of the system. This section explains how they work and the calculations involved in understanding them.

(17).- Compound gear train

Image (17) shows a compound gear train, where gears 1 and 4 have 40 teeth, and gears 2 and 3 have 20 teeth and are concentric. If gear 1 rotates at a speed of 10 rpm, what is the speed of gear 4?

The transmission ratio is determined by the number of teeth on each gear, as given by the following expression:

$$ i = {z_{Driven}\over z_{Driving}} $$

In the configuration shown, gear 1 drives gear 2, and gear 3 drives gear 4, giving:

$$ i = {z_{2} \cdot z_{4} \over z_{1} \cdot z_{3}} \rightarrow i = {20 \cdot 40 \over 40 \cdot 20} $$

$$\therefore i = {1} $$

Calculating the rotational speed:

$$i = {\omega_{Driving}\over \omega_{Driven}} \rightarrow i = {\omega_{1}\over \omega_{4}}$$

$$\omega_{4} = {\omega_{1}\over i} \rightarrow \omega_{4} = {10rpm \over 1} $$

$$\therefore \omega_{4} = 10rpm $$

This means that gear 4 is rotating at the same speed as gear 1, and the same is true for torque, since \( 1^{-1} = 1 \) (in other words, 1:1 is the same as switching it around to 1:1).

Another example, in image (18), the configuration for the gear train changes. If gear 1 is driving the system, what would be the transmission ratio at gear 4 ?

(18).- Compound gear train reducer

Using the general expression for the transmission ratio in a gear pair:

$$ i = {z_{Driven}\over z_{Driving}} $$

Substituting all the driven and driving gears:

$$ i = {z_{2} \cdot z_{4} \over z_{1} \cdot z_{3}} \rightarrow i = {40 \cdot 40 \over 20 \cdot 20} $$

$$ \rightarrow i = {1600\over 400} $$

$$ \therefore i = 4 $$

This configuration is called a reducer because it reduces the speed but increases the torque. In our notation for reducers "A:B" the larger parameter will always be 'A'. Other literatures may have it switched around, but don't worry, the math will be the same, only it'll be switched around (inversed). This means for a reducer with a transmission ratio of 4, others may represent it as 0.25 or \( 1\over 4\) (\( 4^{-1} = \) \( 1\over 4\)). Or the equivalent of having our notation of 4:1 switched to 1:4.

The key feature to understand helical gears is their helix. A helix is a three dimensional curve that resembles a spiral or a coilded spring. It is a curve that lies on a cylinder or cone, and it has a constant slope or pitch along its length. The parametric equations of the helix stand as follows:

• \(X = r \cdot \cos(t) \)
• \(Y = r \cdot \sin(t) \)
• \(Z = b \cdot t \)

$$ 0 \leq t \leq 2\pi$$

Where 't' controls the circular span of the helix and 'b' is a parameter to control the vertical advance of the helix alongside 't' in the Z axis (this will be further explained down bellow).

(19).-Helix in gears

Another important parameter is the pitch, which is the distance between the start and the end of the helix at one revolution as shown in the image above. Its definition follows the expression:

$$ P_h = \pi \cdot PD \cdot {\cos(\beta)\over \sin(\beta)}$$

Where β stands for the helix angle. The helix angle controls the steepness of the helix and redirects a portion of the applied force to the axis, giving helical gears the ability to handle more load. However, because of this redirection, special bearings are required since the gears will be subjected to an axial force. The image bellow is meant to help visualize the helix angle:

(20).-Helix angle

As you can see in the image above, the teeth of the gear are represented by two gray lines, and their inclination is determined by the helix angle. The steeper the helix angle, the greater the inclination of the teeth, which for a gear holding its pitch diameter results as a reduction in the pitch. Taking all of this, the helix equations can be redefined as:

• \(X = ({PD \over 2}) \cdot \cos(t) \)
• \(Y = ({PD \over 2}) \cdot \sin(t) \)
• \(Z = ({P_h \over 2\pi}) \cdot t \)

Where

$$ 0 \leq t \leq 2\pi$$

$$ P_h = \pi \cdot PD \cdot {\cos(\beta)\over \sin(\beta)}$$

In this new set of equations 'b' was replaced for a new expression. This new definition for 'b' makes it so that the total height of the helix caps at the pitch when it completes a full revolution. It may seem strange, but this is made to ensure that at one revolution, meaning when 't' is equal to 2π, the vertical distance between the start and ending points of the helix is equal to the pitch.

In helical gear design, the manufacturing method used determines the resulting geometry of the gear. Unlike spur gears, machining for helical gears is not performed in the normal plane, but in an angled plane relative to it.

(21).-Helical gear milling setup

Image (21) depicts the setup for helical gear manufacturing by milling with a disk cutter. The angle of inclination of the cutter relative to the central axis of the gear is defined by the helix angle β. This angle affects the size of the teeth, which differs from that of spur gears. The greater the helix angle, the more protruded the teeth become, resulting in a difference in tooth size between helical and spur gears.

Depending on the plane from where it is viewed, the size of the teeth for helical gears will differ from that of spur gears. This is represented in the image bellow, where there are two modules for reference: the normal module and the transverse module. The normal module \( m_n \) is the one that the cutter has, and when viewed from the plane normal to the cut, the teeth geometry is a direct and unmodified result from the cutting tool (since the space between the teeth left by the cutter will be the same as in spur gears). However, when the gear is viewed from one of its faces, the teeth geometry is different (this is easier to visualize in gears with high helix angles) as they seem to be bigger in size.

(22).-Helical gear modules

This stems from the fact that the steeper the helix angle, the smaller the projection of the cutter will be on the plane perpendicular to the gear's axis. This leads to a smaller space between teeth, which increases their width. This can make the teeth appear to have a different module, called the transverse module, although they were made using the same cutter as their spur gear counterparts. The transverse module \( m_t \) results from the projection of the normal module onto the face plane.

It is this difference in teeth that affetcs the geometry of helical gears, as the manufacturing methods available will require to address this in different ways. Overall, there are two systems for helical gear design: the radial system and the normal system.

Helical gears designed with the radial system have the same basic dimensions as their spur gear counterparts, making them an easy replacement for spur gears in an already defined system. This means that the transverse module is equal to the normal module. However, it requires special machining tools to fabricate helical gears, with one tool needed for each helix angle.

Radial system helical gears are better suited for non-traditional manufacturing processes. For instance, 3D printing and laser-cutting with the help of a slicer are great options for producing helical gears. Although 5-axis CNC milling can also be used, it may not be ideal for cost reasons.

(23).-Helical tooth path

NOTE: Whilst the helix in the image is in the addendum circle, the helix for the calculations is defined at the pitch circle.

Helical gears belonging to this system may be viewed as spur gears twisted along their vertical axis as shown in image (23). It may seem hard to grasp the transition of a 2D image into a 3D property, so we made a gif to illustrate this:

The animation above illustrates the section view of a helical gear when viewed from above as you move up its height. You can achive the same by using a 3D slicer (like Cura) and use the preview to see the layers alongside the vertical axis. This is important because for helical gears, the involute is in the transverse plane.

Helical gears designed with the normal system can be fabricated using conventional manufacturing methods, which makes them the default choice when hobbing or making gears with a mill. However, their basic dimensions are different from that of their spur gear counterparts.

This stems from the fact that the more pronounced the helix angle, the smaller the space between the teeth. Since there is a reduction in the overall spacing of the teeth, the material size must change to compensate. This can be better understood with a machining example:

(24).-Gear cutter & milling

Image (24) shows a gear cutter used for milling, same as in image (21). Milling is probably the simplest gear making process to understand since the cutter shapes the space between teeth:

(25).-Gear cutter profile

What's interesting is what would happen if the cutter was rotated as in image (21)? How would the gear's teeth look? To understand this, the image bellow introduces a new green cutter section (which is the same as the blue cutter section) and rotates it. This rotation is how helical gears would be manufactured using milling:

(26).-Inclined gear cutter profile projection

NOTE: The blue cutter and the green cutter are exactly the same. On the right side of the image, the green cutter lines are dashed because it is a projection, not the actual geometry of the cutter.

Image (26) depicts the projection of the inclined cutter on the transverse plane. As you can see, when viewed from the top (or if you want to get technical the transverse plane) the projection of the gear cutter is reduced. The more pronounced the helix angle is, the smaller the projection of the cutter will be. This is a key concept to understand why the basic dimensions of helical gears are different; the cutter is still the same, but by inclining it there's ought to be a modification on the teeth.

(27).-Inclined gear cutter profile impact on the teeth

IMPORTANT: Same as in the normal system, the involute resides at the transverse plane.

As shown in the image above, a reduced projection of the cutter increases the width of the teeth. This is a big difference in teeth geometry when compared to spur gears. As a rule of thumb, the greater the helix angle, the more notorious this difference will be. To compensate this, modifications to the material size are performed, meaning the dimensions for helical gears are different:

[12] PD = \(m_t\) * z

BD = PD * cos(\(\alpha_t\))

[13] AD = PD + 2\(m_n\)

[14] RD = PD - 2.5\(m_n\)

Where:

[15] \(m_t\) = \(m_n \over cos(\beta)\)

[16] \(\alpha_t\) = \( \arctan ({tan(\alpha) \over cos(\beta)}) \)

\(m_n\) is the normal module, previously known simply as \(m\).

An important thing to mention is that even though the geometry of the teeth changes, the addendum and deddendum stay the same. It may be hard to grasp why, but think about it: the gear cutter is still the same as for spur gears. Yes the material size has changed, but as long as the tool stays the same the addendum and deddendum won't change.

When modeling helical gears in the normal system, knowing the tooth thickness at a certain radius may seem tricky; after all the dimensions have changed. What's important to understand is that since the module in the transverse plane is different, so will the base radius, which is used for the calculations of the tooth thickness \(T_t\). Once the tooth geometry is set on the transverse plane, you can twist it along its vertical axis, same as in the radial system explanation (although the geometry is not that of a spur gear):

Coming soon...

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