Tuesday, June 28, 2022
Home3D PrintingEasy Way of How to Design Gears

Easy Way of How to Design Gears

Hello Machine Bros!
Today are bringing you an Easy Way of How to Design Gears from scratch, to fulfill a certain function with certain requirements for a particular project.

We will also show you how to design gears in SolidWorks and how to 3D print them.

In our previous article How to Replicate and 3D Print Gears, we explained how to replicate or copy an existing gear to print it in 3D and also we explained the two most common types of gears, spur and helical, and when it is better to use one or the other.

If you missed it, we recommend that you check it out, so that you know certain concepts, and it will also help you understand this article more easily.

You will find also all the technical information so that you can 3D print this amazing crossbow composed of gears!

How to design gears
Crossbow Prototype 3D Printed
Table of Contents Hide

1st Step – Choose Between Spur or Helical Gear

The first thing you should ask yourself and define is what type of gear do I need, a spur or helical gear?

In the previous article How to Replicate and 3D Print Gears we explained why one or the other is better, but we can refresh your memory with a brief summary regarding the 3 factors that most influence the making of this decision.

  1. Axial load is the most important factor, helical gears produce axial load and spur gears do not.
Axial load on gears
Red arrows represent axial load

2. Noise. Spur gears at high speeds are much louder than helical gears.

3. And last, the way the gear teeth come into contact.

Spur gears do not come into contact “smoothly and gradually” while helical gears do come into contact “smoothly and gradually”. Allowing the helical gears to be better to transmit force and speed gradually.

Furthermore, it should be noted that, if we had a spur gear and a helical gear of the same dimensions and similar characteristics, the teeth of the helical gear would have a greater contact area due to their inclination, therefore, its teeth would be more resistant.

It is also necessary to consider that a spur gear is cheaper and easier to manufacture than a helical gear.

So, we could simplify the choice of gear to use by asking the following questions:

Do you consider that noise can be an important factor in your prototype?

If noise influences your choice a lot, because since spur gears produce more noise at high speeds, you should choose a helical gear.

Do you consider that you need a smooth and gradual power transmission in your prototype?

The teeth of the helical gears come into contact more progressively, for this and other reasons it is that race vehicles generally use spur gears and private vehicles use helical gears, as they allow for more comfort by reducing noise and allowing a smoother and gradual coupling.

Do you consider the axial load produced by helical gears a design problem in your prototype?

Helical gears produce axial loads, spur gears do not.

In other words, if we choose to use a helical gear, we will have to consider that in the design of the prototype we must use mechanisms that allow these loads to be supported (for example, axial bearings).

For this reason it is that spur gears are generally chosen in racing vehicles, having to use fewer complex mechanisms to support the axial load reduces the total weight of the vehicle, a very important factor in this type of driving.

There are two other aspects that we must mention to you:

  1. “If we had a spur gear and a helical gear of the same dimensions and similar characteristics, the teeth of the helical gear would have a greater contact area due to their inclination, therefore, its teeth would be more resistant.”

    Although this is not such a determining factor when choosing one type of gear or another, since if we wish we can simply design the spur gear a little wider, thus we would make the teeth more resistant.
  2. “It is necessary to take into account that generally, a spur gear is cheaper and easier to manufacture than a helical gear.”

    In 3D printing, this would not be so relevant, since there would not be so much difference between 3D printing a spur gear and a helical one, although at the time of design it could be said that it is indeed a bit more complex to design the helical gear.

What could really increase the cost of producing a helical gear is creating the gear using other production and manufacturing methods, such as lathing and milling.

2nd Step – Choose if Your Priority is Torque or Angular Velocity

Once the type of gear to be used has been chosen, we must define our priority in the use of the gears of our prototype. The angular velocity or the torque?

What is the Angular Velocity in a Gear?

The gears rotate at a certain speed, this is known as angular velocity (w).

There are different units to express angular velocity, but the one most used in gears is rpm (Revolutions Per Minute). This unit is easy to understand, it refers to how many turns (revolutions) the gear is capable of making in one minute.

For example, if a gear is completing 3 turns (or revolutions) in one minute, this means that the angular velocity (w) of this gear is 3 rpm (w = 3 rpm).

How do we calculate the rpm of a gear?

We simply have to divide the revolutions (or turns) that the gear has been able to do in a certain time expressed in minutes, example:

Imagine a gear that has completed 20 tourns in 10 minutes

Rpm = revolutions / time in minutes
Rpm = 20/10 = 2

This hypothetical gear has an angular speed (w) of 2 rpm (w = 2rpm), this means that this gear is rotating at a speed where it is capable of making 2 turns in a minute, which is equal to say it makes 2 turns in a minute or 2 revolutions (rev) in a minute.

What is Torque in a Gear?

Torque can be a slightly more complex concept to understand, but for practical purposes, to simplify calculations and understanding of how torque influences gears we are going to simplify the concept of it.

Let us define the torque (T) in this opportunity as the product of multiplying a force (F) applied at a certain distance (r) from the center of rotation of the gear.

Torque = (Force) x (Distance)
T = (F)x(r)

But, for this formula to be fulfilled, it is necessary that the force be considered and applied perpendicular to the distance. In gears, we will consider the tangential force to the gear, with the following image you will understand it better.

How Torque works
Torque

Let’s imagine that we have the wrench shown in the image and we are going to tighten a nut.

To tighten it, we exert a force (F) at a distance (r) from the center of rotation of the nut, which produces a torque (T).

This torque is the product of multiplying the force by the distance (T = F x r) and is the one in charge of turning and tightening the nut.

With this in mind let’s use the same logic, but now with a gear.

Torque in gears
Torque in a gear

Two things usually happen in a gear:

  1. We turn the gear by applying a force (F) on its teeth, which is what happens when one gear turns another gear. If this happens, we are producing a torque (T).
  2. The gear rotates because it is connected to a shaft with torque (T), for example, we connect the gear to a motor, a servo motor, a stepper motor, etc. If this happens, we have a certain force (F) on the gear teeth as a result of the torque (T).
Gear forces

In the previous image we have two gears coupled (gear 1 and gear 2).

Let’s imagine that the gear (1) is coupled to a motor, which allows the gear (1) to turn.

This turn, product of the torque (T1) of the motor produces a force (F1) on the gear teeth (1), this force is transmitted to the gear teeth (2), F1 being equal to F2.

Now, due to the force applied to the teeth of the gear (2), a torque (T2) is produced in the gear (2), a product of the force (F2) multiplied by the distance (r2), therefore, the gear (2) will also rotate but in the opposite direction.

All of this can be easily calculated, let’s give this example values.

Suppose that the torque in the gear (1) is 10 Kgf.m (kilograms force per meter), the distance r1 is 0.20 meters and the distance r2 is 0.10 meters, then we have the following:

T1 = 10 Kgf.m
r1 = 0.20m
r2 = 0.10m
F1 = F2 = F = Not know it, but since both forces are equal, we will call it simply (F)
T2 = Not known

We start from the torque formula

T = (F)x(r)

We clear (F)

F = (T1)/(r1)
F = (10 Kgf.m)/(0.20 m) = 50Kgf

Now we can calculate (T2)

T2 = (F)x(r2)
T2 = (50 Kgf)x(0.10 m) = 5 Kgf.m

We can see that if we apply a force (F) at a certain distance (r), the greater the distance (r), the greater the torque (T).

And inversely the following happens, if we have a torque (T), the greater the distance (r), the lower the force (F).

In other words, if we had a motor with a certain torque, to which we couple a gear, the larger the diameter of the gear, the less force we will have on its teeth.

The behavior of two coupled gears is closely related to the principle of the lever, it is said that Archimedes said in reference to the lever “Give me a place to stand on, and I will move the Earth.

How torque works on a lever
Lever
Archimedes and torque
Archimedes

Next, we will show you a video that will help you better understand torque.

In the video you will see that there is a small difference between the results obtained by calculations and the results obtained by testing, this is because the calculations do not take into account other factors such as friction force.

Even during the manufacturing of the lever there could be small errors in the distances, instead of having exactly 10cm between the ropes there could be for example 10.05cm, these small errors accumulate and deviate a bit the result, and finally the factor mentioned comes into play at the beginning “for this formula to be fulfilled, it is necessary that the force be considered and applied perpendicularly”, and we are not 100% sure that we are exerting the force perpendicular to the lever, since we are making the approximation in a visual way.

When the force is not applied perpendicular to the lever, you have to make small adjustments in the formula to calculate the torque, but this does not really matter to us for what we are going to learn in this article.

Video that helps to better understand the concept of torque
Torque examples
Results of the calculations made in the video

Now let’s clarify what happens to the direction of rotation of the gears when we couple several of them.

We will talk about two directions of rotation, the first is clockwise (CW), and the other is counterclockwise (CCW).

When we connect two or more gears, the direction of rotation is reversing or alternating, in the following image, you can understand it better.

Alternating direction of gear rotation, where CW is (Clockwise) and CCW (Counter-clockwise)
Alternating direction of gear rotation, where CW is (Clockwise) and CCW (Counter-clockwise)

Another concept that you need to know is the “idler gear“, these gears are the ones in the middle between the driving gear (Driver) and the driven gear.

If in the image above, the gear “T1” is coupled to a motor, this would be the driving gear, the gears “T2” and “T3” would be idler gears, and the gear “T4” would be the driven gear.

Once the concepts (angular velocity, torque, and direction of rotation) are understood, we can return to the second step, where we asked ourselves:

What is more important in the use of the gears of our prototype, speed or torque?

We may be looking to build a gear ratio that allows us to obtain higher torque, higher speed, lower speed, etc. But you must bear in mind that when you have two gears working together, you sacrifice one for the other.

That is, if you want to make a gear configuration with higher torque, you sacrifice speed and vice versa if you want to gain speed.

Suppose we have two gears engaged where one of them has twice as many teeth as the other. In this first example, the gear with the highest number of teeth is the gear connected to a motor, which has a torque of 1Kgf.m and an angular speed of 60rpm.

With this first configuration we will achieve that the small gear has twice the angular speed, but half the torque.

N1 = Number of teeth of the driving gear (coupled to the motor)
T1 = Torque of the driving gear (coupled to the motor)
w1 = Angular speed of the driving gear (coupled to the motor)
N2 = Number of teeth of the driven gear
T2 = The torque of the driven gear w2 = Angular velocity of the driven gear

What is a driver gear
The large gear is the driver (it is coupled to a motor) and the small gear is the driven

Now suppose that instead of the large gear being connected to the motor, the small gear is connected to the motor. With this second configuration we will have twice the torque on the large gear, but half the angular velocity.

N1 = Number of teeth of the driving gear (coupled to the motor)
T1 = Torque of the driving gear (coupled to the motor)
w1 = Angular speed of the driving gear (coupled to the motor)
N2 = Number of teeth of the driven gear
T2 = The torque of the driven gear
w2 = Angular velocity of the driven gear

What is a driven gear
The small gear is the driver (it is coupled to a motor) and the large gear is the driven

In those two examples, you can see how you sacrifice speed in order to obtain torque and vice versa.

Speed and torque relationships on gears
Results comparison

Knowing this, how speed and torque are related, you can better understand why you should have a priority.

Do you need more torque or speed for your project?

In the previous example, we did not run calculations because first, we needed you to understand how the number of teeth affects these two variables (angular velocity and torque), but once we understand this we will move on to the third step, perform calculations.

3rd Step – Calculate Gear Ratio

At this point you should have already decided what type of gear you need (first step) and if your priority is torque or angular velocity (second step).

Now we are going to perform some calculations, for this, we may need the torque formula ( T = F x r), and some new formulas that we will show you below.

How to calculate the gear ratio

Gr = Gear ratio
N = Number of teeth
w = Angular velocity
T = Torque

Everything that contains the number (1) refers to the driving gear (driver) and everything that contains the number (2) refers to the driven gear (driven)

Let’s use the previous gear combination as a reference to perform the calculations.

Gear ratio and number of teeth

The first thing we will calculate in this case is the gear ratio (Gr)

How to calculate the gear ratio with the number of teeth

Now we can calculate the rest, let us start with the angular velocity (w2), for this we will clear (w2) from the following formula and do the respective calculations.

Angular velocity and gear ratio

We proceed to do the same but this time to find the torque (T2), from the following formula we will clear (T2) and do the respective calculations.

Gear ratio and torque

Let us do another similar exercise but this time for the other gear ratio, where the small gear is the driver, since it is the one that will be coupled to the motor.

How to calculate the gear torque with the gear ratio

The first thing we will calculate in this case is the gear ratio (Gr)

Calculate the gear ratio with number of teeth

Now we can calculate the rest, let us start with the angular velocity (w2), for this we will clear (w2) from the following formula and do the respective calculations.

Calculate the gear ratio using the angular velocity

We proceed to do the same but this time to find the torque (T2), from the following formula we will clear (T2) and do the respective calculations.

How to calculate the gear torque

What is the First Thing you Should Calculate to Start Designing the Gears of Your Prototype?

In most cases, the first thing you should calculate is the Gear ratio (Gr), whether it is a function of the angular speed (w) or torque (T) that you want.

· Depending on the angular velocity:

Suppose you have a motor with an angular speed of 100rpm, but you need your prototype to have an angular speed of 20rpm. The first thing we will do is calculate that gear ratio.

Gear ratio depending on the angular velocity

Knowing the Gear ratio, we will already know how many teeth of difference the driving gear must have with respect to the driven gear.

Gear ratio of the driven gear

The driven gear must have 5 times the amount of teeth that the driving gear will have, for example, if we decide that the driving gear must have 10 teeth (N1 = 10), then the driven gear will have 50 teeth (N2 = 50).

We know this simply by entering the corresponding values in the formula above.

How big can be the driver gear

· Depending on the Torque:

Suppose you have a motor with a torque of 2Kgf.m, but you need your prototype to have a torque of 3Kgf.m.

The first thing we will do is calculate that gear ratio.

Torque calculation of gears

Knowing the transmission ratio, we will already know how many teeth of difference the driving gear must have with respect to the driven gear.

Calculate the number of teeth knowing the Gear ratio

The driven gear must have 1.5 times the amount of teeth that the driving gear will have, for example, if we decide that the driving gear must have 10 teeth (N1 = 10), then the driven gear will have 15 teeth (N2 = 15).

We know this simply by entering the corresponding values in the formula above.

How to calculate the size of a driven gear

Before proceeding to the next step, there are other things you should know.

What Happens if Two or More Gears are United in One Compound Gear or United by the Same Axis?

Two gears united into one (compound gear)
Two gears united into one (compound gear)
Gears joines by the same axis
Several gears joined by the same axis

In the examples above, it happens that, being connected in this way, the gears will turn or rotate at the same angular speed, that is, at the same rpm and will also have the same torque.

How Would you Calculate the Gear Ratio (Gr) if There are More than Two Gears Working Together?

When two or more gears work together it is known as a “Gear Train”.

When there are more than two gears, what needs to be done is to calculate by parts, that is, we start with the first driving gear and the first driven gear, then the driven gear will become the driving gear and we will have a new driven gear.

We will show you an example with which you can understand it more easily.

In the following image, the smallest gear is the driver, it is coupled to a motor, we will calculate the torque and speed of the rest of the gears that make up the system.

How to calculate the dimensions of an idle gear
In this example, gear (2) is actually an idler gear, but for calculation purposes we will consider it first as a driven gear and then as a driving gear.

We will first calculate the gear ratio between gear (1) and (2).

How to calculate the gear ratio

Second, we will calculate the angular velocity of the gear (2)

How to calculate the gear angular velocity

Third we will calculate the gear torque (2).

Calcilating the gear torque

Having then:

Calculate the gear ration among 3 gears

Now we will calculate the gear ratio between gear (2) and (3), where we will take gear (2) as the driver and gear (3) as the driven.

Gear ratio between driver gear and driven gear

We will calculate the angular velocity of the gear (3).

Math formula of gear ratio

Finally, we will calculate the gear torque (3)

How to calculate gear ratio with torque

Having then:

How to design helical and spur gears

There is another way to calculate T3 and w3 without calculating T2 or w2 (in this way we ignore the calculations of “T” and “w” of the idler gears), which we will explain to you below.

You only have to calculate the Gr between gears (1) and (2), and then multiply it by the value of the Gr between gears (2) and (3).

How to calculate the torque of a gear

Calculate the angular velocity of the gear (3)

Angular velocity calculation

Then the gear torque (3)

MAth for gear torque

As you can see, the results are the same, only in this way we will not know the values T2 and w2 of the gear (2).

How to Calculate the Gear Ratio (Gr) if There are One or More Compound Gears in the Set?

How to calculate the gear ratio (Gr) if there are one or more compound gears in the set
Gear train containing a compound gear (viewed from various angles)

Suppose we have the gear train shown in the image above. Where the smallest gear and which is not compound, is connected to a motor that rotates at 100rpm with a torque of 1 Kgf.m.

Now we want to calculate the gear ratio.

The first thing we will do is show you a drawing to analyze it.

Gear ratio calculations for a compound gear

To begin with, we will calculate the gear ratio between gear (1) and part “a” of gear (2), since it is the one coupled with gear (1)

how to calculate the gear ratio of a compound gear

Second, calculate the angular velocity of gear (2).

Angular velocity of compound gears

Third, calculate the gear torque (2).

Compound gear torque calculation

Having then:

Gear ratio math calculation for a compound gear

We must carry out the same calculations, but between part “b” of gear (2) and gear (3), since they are the parts that connect.

Take part “b” of gear (2) as the driver and gear (3) as the driven.

Start by calculating the gear ratio.

Gear ratio in function of number of teeth of a compound gear

Calculate the angular velocity of the gear (3)

Calculation of the angular velocity of a gear

We calculate the gear torque (3)

Calculation of the torque of a gear

Having then:

Formulas for compound gears

There is another way to calculate T3 and w3 without calculating T2 or w2 (in this way we ignore the calculations of “T” and “w” of the idler gears), which we will explain to you below.

You only have to calculate the Gr between gears (1) and (2a), and then multiply it by the value of Gr between gears (2b) and (3).

How to calculate the gear ratio between gears

Calculate the angular velocity of the gear (3)

Formula of angular velocity for a compound gear

Then the gear torque (3)

Calculatin of Torque of a compound gear

As you can see, the results are the same, only in this way we will not know the values T2 and w2 of the gear (2).

How to Perform these Calculations if the Gear is Helical?

Helical Gears Calculations
Helical Gears

In general, calculations are done in the same way as spur gears.

You just must take into account the following, the helical gears the more helix angle they have, the greater the axial load they produce, therefore, these gears are a little less efficient in transmitting torque than spur gears, the efficiency of the helical gears is approximately between 98% and 99.5%.

For this reason, when calculating the torque of a helical gear, it is recommended to multiply the value obtained by 0.98.

For example, let’s assume that the two gears in the following image are helical.

Helical gear formulas

Start by calculating the gear ratio.

Gear ratio calculation for helical gears

Calculate the angular velocity of the gear (2)

Calculation of angular velocity for helical gears

Then the gear torque (2)

Torque Calculations for Helical Gears

But now we will take into account the efficiency, so we will multiply the torque (T2) by 0.98.

Formulas for helical gears

Having then:

Math formulas for helical gears

4th Step – Final Decisions Regarding Gear Design

You should have already chosen what type of gear you need, know if your priority is torque or angular velocity and you should also have an idea of what gear ratio you need for your project.

Knowing all this, you will have to decide several things.

· Decide How Many Gears You Will Need and Whether You Will Use Compound Gears

This decision will usually be made based on the dimensions of your project.

If you have little space you could not use very large gears, then you would have to try to satisfy your gear ratio (Gr) by making use of several gears and even compound gears.

· Decide the Module Your Gears will Have

According to Wikipedia, we can define the module as “A characteristic of magnitude that is defined as the relationship between the measure of the primitive diameter expressed in millimeters and the number of teeth”. In other words, it is nothing more than a ratio and is given by the following formulas:

Spur Gears:

Spur Gear Module ecuation

Helical Gears:

Helical gear module ecuation

dr = Reference diameter (Millimeters)
De = External diameter (Millimeters)
N = Number of teeth
M = Module
A = Helix angle [Sexagesimal degree (DEG) (°)]

Although to choose the gear module you will not need to use these formulas, you need to know the following.

The higher the module, the greater the external diameter of your gear, but also the larger the teeth, for this reason, they will be more resistant.

You should choose a module that meets your design, that the teeth are strong enough to withstand the loads and large enough that they can be 3D printed.

3D printing in SLA you may not have so many problems printing the gear teeth, but for FDM you will have to choose a module that allows it to be printed by that technology. Even, it is obvious that you cannot exaggerate with the choice of the module to the point that the external diameter of the gear is so large as not to fit in your prototype.

If you want to know the difference between 3D printing in SLA or FDM, check out the article FDM or SLA: Which 3D printer to buy?

IMPORTANT NOTE: For two spur gears to mesh well, it is necessary that both have:

  1. The same Modulus (M)
  2. The same helix angle (A)
  3. To have opposite helix directions (one gear must have a helix direction “right” and the other “left”).

· Decide the Number of Teeth Your Gears will Have

With the gear ratio (Gr) you will already know the difference in teeth that one gear must have compared to the other, for example, you will know if the driven gear must have twice as many teeth as the driving gear, three times, etc.

But you still will not have defined the exact number of teeth, this will also depend on the dimensions of your design, you should know that the more teeth you add to a gear that already has a defined module, its external diameter will be greater.

In SolidWorks, the least number of teeth allowed is 10, the maximum 300.

· Decide How Wide your Gears will Be

This decision is also made based on the dimensions of the project, the wider the gear the teeth will also be wider, and therefore more resistant.

For this reason, what limits you to how wide your gear will be, are the dimensions of the project, how much space you have to place the gears.

Another point to keep in mind is that obviously the wider the gear, the more material you will spend.

· Decide the Pressure Angle your Gears will Have

Representation of the displacement of the normal force in a spur gear, where "Pa" is the pressure angle
Representation of the displacement of the normal force in a spur gear, where “Pa” is the pressure angle – Images taken from Wikipedia

According to Wikipedia the pressure angle is defined as ” The complement of the angle between the direction that the teeth exert force on each other, and the line joining the centers of the two gears”.

Textually it is difficult to understand, but with the image above it is easier to understand.

Normally you choose between two types of pressure angles, 20° and 14.5°, we advise you to use 20° as it is the most used.

· Decide the Helix Angle your Helical Gears will Have

The helix angles (A) most used in this type of gearing range from 15 degrees to 30 degrees, the reasonable limit being 45 degrees (integer values, that is, without decimals).

The truth is that choosing this angle is not so easy, engineers often perform tests to determine which helix angle performs best and optimally meets the requirements of the project.

There is no easy way to choose the helix angle, it depends on many factors.

If you want to create a “standard” helical gear, most gear designer catalogs offer helical gears with helix angles close to 20°.

For this reason and without major complications, if you want to create a helical gear and do not know which helix angle to choose, we recommend 20°.

There are other helical gears that work in a cross way, known as crossed helical gears. These have a helix angle of 45 °, but we could talk about these gears in another article.

 efficiency of the helix angles is studied taking into account the friction (u)
Graph taken from Wikipedia where the efficiency of the helix angles is studied taking into account the friction (u).

IMPORTANT NOTE: The helical gears the greater the helix angle they have the greater axial load they produce; therefore, these gears are a little less efficient in transmitting torque than spur gears.

The efficiency of helical gears is approximately between 98% and 99.5%. For this reason, when calculating the torque of a helical gear, it is recommended to multiply the value obtained by 0.98.

5th Step – Enter the Desired Values in SolidWorks to Generate the Model

· For Spur Gears

SolidWorks will need:

  • The module (M)
  • The number of teeth (N)
  • The width of the gear (W)
  • The diameter of the hole (Hole)
  • The pressure angle that generally we will always select 20° since it is the most used.

Remember: For two spur gears to mesh well, it is necessary that they both have the same Module (M)

· For Helical Gears

SolidWorks will need:

  • The module (M)
  • The number of teeth (N)
  • The width of the gear (W)
  • The diameter of the hole (Hole)
  • The helix angle (A)
  • The direction of the propeller (“Right” or “Left”)
  • The pressure angle that generally we will always select 20° since it is the most used.

Remember: For two helical gears to mesh well, it is necessary that they both have the same Module (M). It is also necessary that both gears have the same helix angle (A) and finally it is necessary that they have opposite helix directions (one gear must have a “right” helix direction and the other “left”).

6th Step – Calculate the Distance Between the Gears so that they Operate Correctly

Once your gears have been designed, you must calculate the distance between them so that they operate correctly. The first thing you need to do is calculate the reference diameter (dr).

Calculation of the Reference Diameter (dr)

IMPORTANT NOTE: In the formulas and calculations we will work with the following units, the units of length in millimeters (mm), the angles in sexagesimal degree (deg) (°)

· For Spur Gears

dr = (N)*(M)

dr = Reference diameter
N = Number of theet
M = Module For Helical Gears

· For Helical Gears

Calculation of the reference diameter of an Helical Gear

dr = Reference diameter
N = Number of teeth
M = Module
A = Helix angle

Calculation of Center Distance (C)

Calculation of center distance of Gears
Formula of the center distance of gears

C = Center distance
dr1 = Reference diameter of gear (1)
dr2 = Reference diameter of gear (2)

Calculation of Tolerance (Tc)

Although the separation “center distance (C)” should theoretically be sufficient for the gears to operate, the reality is that it will always be better to leave a tolerance (Tc), which will be added to the center distance (C).

Tc = (0.25) * (M)

Tc = Tolerance
M = Module

Once the calculations are made, the final distance that we will use will be:

Distance between gear centers = C + Tc

Let’s do an exercise so you can understand better, we have two spur gears, both of module 1 (M = 1), one of the gears has 10 teeth and the other 15 teeth (N = 10 and N = 15).

Calculate the distance that should be between them so that they work correctly.

Calculate the distance between gears

Let’s start by calculating the reference diameter (dr) of both gears:

dr1 = (N1) * (M) = (10) * (1) = 10mm
dr2 = (N2) * (M) = (15) * (1) = 15mm

Now let’s calculate our first center distance (C) without taking into account the tolerance (Tc).

Gears Center Distance Formula

Finally, we will calculate the tolerance (Tc) and add it to the central distance (C), which will allow us to obtain a new central distance (C).

Tc = (0.25) * (M) = (0.25) * (1) = 0.25mm

Our new central distance (C) will be:

C = 12.5mm + Tc = 12.5mm + 0.25mm = 12.75mm

That is, for our gears to operate correctly they must have a separation of 12.75mm, which would be:

Separation between gears

Let’s do the same exercise, but assuming now that they are helical gears and have a helix angle of 20 degrees (A = 20°).

Center distance between helical gears

Let’s start by calculating the reference diameter (dr) of both gears

Reference diameter calculation of a gear

Now let’s calculate our first center distance (C) without taking into account the tolerance (Tc).

How to calculate the center distance between two gears

Finally, we calculate the tolerance (Tc) and add it to the central distance (C), which allows us to obtain a new central distance (C).

Tc = (0.25)*(M) = (0.25)*(1) = 0.25mm

Our new central distance (C) will be:

C = 13.3mm + Tc = 13.3mm + 0.25mm = 13.55mm

That is, for our gears to operate correctly they must have a separation of 13.55mm, which would be:

How to calculate the separation between gears

Let’s replicate these two exercises in SolidWorks

Video of the calculation of the center distance between spur gears
Video of the calculation of the center distance between helical gears

Do You Want to Know What External Diameter your Gears will Have?

· For Spur Gears

De = (M)*(N+2)

De = External diameter
M = Module
N = Number of teeth

· For Helical Gears

External diameter for helical gears

De = External diameter
M = Module
N = Number of teeth
A = Helix angle

7th Step – 3D Print your Gears

That’s it! you just have to configure your printing in your slicer of preference, print your gears and start doing some testing to check if they work properly.

Depending on whether the result obtained is the desired one or not, it would be necessary to make modifications and adjustments to your gears or to the prototype in general.

Summary on Designing and Calculating Gears

We will provide you with a PDF in which there is a summary prepared by ourselves where we list the steps and provide the corresponding calculations so that you can download and save it on your smartphone or anywhere you want.

To download the guide, click here Short Guide to design and calculate gears

Example of Creating Gears for a Crossbow

Crossbow with gears
Crossbow with gears

1 – Choose Between Spur or Helical Gear

We will make a crossbow. To load said crossbow in an easier way, we intend to build a gear ratio that allows reducing the force necessary to perform this action. For this prototype, the best choice is the use of spur gears.

2 – Choose if your Priority is Torque or Angular Velocity

In this case, our priority will be the torque, we do not care so much about the time it takes us to load the crossbow, we want to be able to reduce the physical effort we must exert to load the crossbow.

According to the tests carried out to load our crossbow, we will need to exert an approximate force between 5Kgf and 6Kgf, which you can see in the following video.

Video of the strength test

3 – Calculate Gear Ratio

We intend to reduce the force we have to exert to load the crossbow at least three times. For this, we must reduce the torque generated three times as well.

That is, from the following formula:

Gear ratio formula

We are interested in the torque and the transmission ratio

Gear ratio in function of torque

Knowing that T1 represents the driving gear and T2 the driven gear and taking into account that we want to decrease at least 3 times the torque in the driven gear, we consider the equation as follows.

To begin with, let’s make the equation a little easier to understand by solving for T2 (our driven gear):

Calculating the gear ratio

We want T2 to have a value 3 times less than T1, which is the same as saying that T1 will have a value 3 times greater than T2, having then:

How to calculate the torque of a gear

If we return the equation to how we had it at the beginning, it would be very clear that we need a Gr value of 1/3

Torque relationship between gears

And what about the angular velocity? Although it is not our priority, we will calculate it to know what will be happening with this value, from the following formula we will keep the angular velocity and the transmission ratio:

Transmission ratio in function of the angular velocity

We already deduce that our transmission ratio will be 1/3, leaving

Transmission ratio calculation

Let’s clear the equation to analyze it more easily

Angular velocity relationship of gears

Remember that w1 refers to the driving gear and w2 to the driven gear, so the equation tells us that our driven gear will rotate 3 times faster than our driving gear, that is, when the driving gear of one turn, the driven gear will give three turns.

And what about the number of teeth? We go back to the original equation

Gear ratio in function of the number of teeth

But this time we will stick with the gear ratio and the number of teeth

How to calculate Gr having N

We know that Gr is 1/3

Number of teeth relationship of a gear

Solving to analyze the equation we have

Number of teeth of a gear

This means that our driving gear (N1) must have 3 times more teeth than the driven gear. Next, we will show you a video that demonstrates how the force diagram would look using the calculated gear ratio.

Video of the force diagram using the calculated gear ratio

4 – Final Decisions on Gear Design

· Decide How Many Gears you will Need and Whether or Not you Will Use Compound Gears

For our design we made the decision to only use two spur gears, which will not be compound.

· Decide the Module that your Gears will Have

We chose a module 2, since it is easy to print the teeth by FDM, it conforms to the dimensions of our design and is strong enough to withstand the applied load, which in exaggeration would be 6Kgf.

We review that in SolidWorks and in the following video we will show you how we do it.

Video that explains how the gear module is related to the force capable of supporting its teeth

· Decide the Number of Teeth your Gears will Have

In our case, we decided that the smallest gear (which is the driven gear) has the fewest possible teeth, which is 10 teeth (N2 = 10). And with the following formula we will calculate how many teeth our driving gear will have.

Deciding the number of teeth your gears will have

We have the gear ratio and the number of teeth

Gr = N2/N1

We know that Gr is 1/3

How to calculate the number of teeth of a gear

Clearing N1, we have

Number of teeth relationship between gears

Substituting N2 for 10, which is the number of teeth of our driven gear we have the following

How to calculate the number of teeth of a driver gear

Therefore, for the gear ratio to be fulfilled correctly, it is necessary that our driving gear has 30 teeth (N1 = 30) and our driven gear 10 teeth (N2 = 10).

· Decide How Wide your Gears will Be

Remember that this decision is made based on the dimensions available in your project. In our case, 5 millimeters thick is enough to adequately support the stresses.

we analyze this in SolidWorks in the video where we explain how to choose the gear module.

If you need to make the gears more resistant, you can perfectly achieve it by making the gear wider. We will show you in the next video how by making the gear wider it becomes more resistant.

Video that explains how the width of the gear is related to the force capable of supporting its teeth

· Decide the Pressure Angle your Gears will Have

Unless you have a special reason to choose a pressure angle other than 20°, we recommend that you choose this one because it is the most used, in this case that is precisely what we did.

· Decide the Helix Angle your Helical Gears will Have

This step does not apply to our design because we decided to use spur gears.

5th Step – Enter the Desired Values in SolidWorks to Generate the Model

We have already explained this many times during the videos, so this step does not require further explanation.

Instead we will show you how our designed gears finally turned out in the following video, which by the way we decided to print them in SLA as it is a more precise technology when building the shape of the teeth, although it should be noted that with the module that we choose (2) is large enough to be printed correctly by FDM.

It is also necessary to mention that we use a resistant resin called Siraya Tech Blu.

During the design of the gears we decided to make the pulley that loads the crossbow as small as possible. There is no need to make it the diameter of the driving gear.

By making it as small as possible, the torque generated is even less than that calculated at the beginning, this due to the fact that the distance is smaller, which in our case ends up being an advantage, we will also present these new calculations below.

Video that explains the final design that the gears will have
Video explaining the final calculations of the gears
Final results of the calculations
Final results of the calculations

6th Step – Calculate the Distance Between the Gears so that they Operate Correctly

For this step we will use the following formula

Calculate the distance between the gears so that they operate correctly

Remember that for spur gears the “dr” is calculated as follows

dr = (N)*(M)

Having then:

Distance between gears

Our gears have the following values

M = 2, N1= 10, N2 = 30

Entering those values in the formula we will have

How to find the center distance between two gears
Video about how to check that the calculated center distance is correct

Step 7 – 3D Print your Gears

Now we will show you a series of videos that explain how the prototype was finished. The first thing you should be wondering is whether the gear ratio is doing its job correctly or not. In the following video, you will see how the gears reduce the effort required to load the crossbow.

Video showing how the gears do their job

In the previous video you may have noticed that instead of being necessary to exert a force of 5Kgf to load the crossbow, the maximum force that we achieved was 1.3Kgf, and this due to the friction force, and the moments when we did not place the measuring instrument perpendicularly correctly. Otherwise the average force tends to be around values less than 1Kgf.

Now we will show you a video of the crossbow working.

Demonstration video of the complete operation of the crossbow

Finally, we will leave you two videos, one that shows how to 3D print the crossbow.

We recommend that you print in the way shown in the video, the pieces totally solid and oriented in the indicated way, otherwise we cannot guarantee that it will withstand the tension of the elastic.

The other video explains how to assemble the crossbow, it is necessary to review the holes that will carry bolts or screws with a 3mm diameter drill bit. The smallest holes are made with a 1mm drill bit.

We use Fishing Nylon that serves as a rope or strap to load the crossbow, we chose Fishing Nylon because it is strong enough to withstand the stress of carrying the crossbow.

It should be noted that some parts must be glued, other parts must be sanded a little so that they fit smoothly and finally we recommend lubricating the moving parts with a little grease. The crossbow was printed in PLA.

IMPORTANT NOTE: The Machine Bros does not recommend the use of the crossbow for children; this is not a toy and should only be used by responsible and intelligent adults. The crossbow as presented in the videos in this article can injure people, and we ask for prudence and care when operating it. NEVER point the loaded crossbow at a person. The Machine Bros is not responsible for the misuse of this.

Video explaining the best way to 3D print the crossbow
Video that explains how to assemble the crossbow

Link to Download the STL Files of the Crossbow

If you want to 3D print your own crossbow we will leave you a link so that you can download the STL files of it:

https://www.thingiverse.com/thing:4663600

Conclusions About the Easy Way of How to Design Gears

As you may have observed in this article, gears are very useful. There are an infinity of applications that we can give them, in this article we show and explain in detail how to design your own gears to fulfill a certain task, gears that fit your prototypes and requirements, but if what you want is to simply replicate a gear that already exists, you can see our article How to Replicate and 3D Print Gears.

We invite you to encourage yourself to build your own gears, to continue designing, creating, if gears were a limitation for you, they no longer have to be. If you have any questions do not think twice about asking us, we are here to help you.

Greetings.

See you soon Machine Bros!

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