Spur Gear Guide – Design, Calculations and Types

Spur gear systems are one of the most used engineering systems in the machine elements world. To design successful spur gear systems, we must adopt an engineering approach. By following the instructions in this article, you can design these systems according to your system’s parameters. 

Spur gear systems.

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Important Points to Consider Before the Design of Spur Gear Systems

To start to design a spur gear system for a specific system, you need to know about the fundamentals of these systems. Here you can find very useful and detailed information about spur gear systems. 

Torque and Speed Calculations

Also, in gear systems, power is transmitted from pinion gear which thet attach to the input shaft. And bigger gear takes the power to transmit it to the output shaft. In theory, the transmitted power is constant.

But you can adjust the moment or torques of different gear sets. There is also a strict relationship between torque and the RPM of gear. When you are using bigger output gears according to input gear, the output torque will be higher. But if you use a smaller output gear, the output torque will be lower.

But you must understand that when the diameter of output gear increases, torque increases but RPM decreases. RPM increases with the decreasing diameter of output gear also.

We stated that power is constant. So, when you divide the obtained torque with RPM, you will find out the power. But, to calculate the power and torque relations for that system, you must understand the acting forces between gear teeth.

Forces on Spur Gear Teeth

Forces on gears.
Illustration of acting forces on spur gear tooth(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).

There are three types of forces in theory between spur gear teeth pairs. These forces are; tangential force, radial force, and normal force.

Tangential Forces on Spur Gear Teeth

The tangential force between spur gear pairs is the force that is tangential to the pitch line. Power is completely transmitted with this force. You can find out tangential force if you divide the power to circumferential speed.

Tangential Force Calculator

Here we prepared a calculator to calculate the tangential force acting between spur gear pairs. You need to enter the ‘Pitch Line Velocity and ‘Power’ inside the brackets. Then hit the ‘Calculate!’ button to calculate the ‘Tangential Force’. If you want to do another calculation, click on the ‘Reset’ button, then re-enter all the parameters again.

Furthermore, you need to use proper sets of units to obtain correct results. For SI units, the pitch line velocity must be in the m/s. And power must be Joule. The obtained tangential force will be Newtons.

Radial and Normal Force in Spur Gear Teeth

Also, to calculate radial and normal force, you need to know the pressure angle of them. And if you calculated the tangential force, you can calculate both radial and normal forces.

Radial And Tangential Force Calculator

The use of the calculator above to calculate radial and normal forces is very easy just like the previous calculator. So, the unit of the angle must be in degrees.

Torque Calculation from the Data Above About Spur Gears

We stated the relation between torque and power. And also, you can easily calculate the torque of an individual gear if you know the power transmitted between gear sets and the diameter of this individual gear.

Torque Calculator For An Individual Gear

Just enter the required information above. Again, you must enter the information in the correct units.

Efficiency in Spur Gear Systems

Like all the other mechanical systems, these systems have also efficiency in power transmission. Power is not transmitted fully between shafts through gears. For each gear pair, there is a theoretical efficiency which is generally around 97-99%.

Also, for gear trains, we must multiply the efficiencies between each gear pair to find out the total power that is transmitted between input and output shafts.

For high-power transmissions, this loss of power can appear as heat. We must consider the heat seriously. If it is required, we must build some cooling systems near the gear system. Because the lubrication and force transmission can be damaged because of the excessive heat generation because of power loss.

Spur Gear Types and Tooth Form

The most general type of spur gear is that which has a solid hub. These spur gears are generally used in transmissions of machinery. The shaft is assembled inside the hole that is bored to this hub. Also, keyways are machined on this hole for the assembly of keys. You can find a small screw hole on them. These screw holes are machined because of the assembly of keys on spur gears. Solid hub spur gears can be thick compared with other types.

Solid hub type spur gear.
Solid hubbed spur gear type.

Also, there is another type of spur gear that is generally used in high diameters. Because of these high diameters, spoked design is used to save from material usage. There is a keyway that is bored on this type of spur gear and the hub section can be smaller than solid hubbed types.

Spoked design of spur gear.
Spoked design of spur gear.

Rack is another different type of spur gear which have straight geometry, unlike other spur gears. Because of this straightness, they do not have the same teeth type as other spur gears.

Rack and pinion.
Rack and pinion.

Involute Teeth Design of Spur Gears

Involute tooth form.
Source Of Image: Machine Elements In Mechanical Design, Robert L. Mott

From gear mechanisms, we do not want vibrations and noises. So, a proper design of teeth is important. For round spur gears, involute design of teeth is most appropriate to eliminate such problems like;

  • Vibrations,
  • Lack of accuracy in the movement of mechanisms,
  • Noise because of the scratching of teeth…
Involute teeth contact.
Contact of involute teeth(Source Of Image: Machine Elements In Mechanical Design, Robert L. Mott)

If you take a look at the involute design of mutual teeth that contacting to each other during rotation, there is rolling between two contacting teeth. When a straight and perpendicular line is drawn from the contact region of involute teeth, this drawn line must be tangent to each base circle of spur gears.

It is invented that this involute design of spur gear teeth compensates nearly all the problems stated above.

Pitch and Metric Module Concept

When we consider two contacting spur gear-pair, we can find out the pitch diameters of these two gears.

Pitch diameter in spur gear systems.
Pitch diameter illustration(Source Of Image: Robert L. Mott, Machine Elements For Mechanical Design).

The pitch diameter of two assembled spur gears is the tangent diameters that are shown above illustration. In a spur gear pair, one of the spur gears is driven and another one is driving gear.

How to Calculate Pitches of Spur Gears? What are the Types of Spur Gear Pitches?

In spur gear terminology, there are two types of spur gear pitch values are used. These pitch values are; Diametral Pitch and Circular Pitch.

  • Diametral Pitch Calculator: Diametral pitch values are the most used pitch value in spur gear terminology to classify the standards. It is generally used in inches. It is calculated by the division of the pitch diameter of spur gear(Dp) to the number of teeth(N).

Diametral Pitch Calculator For Spur Gears

You can use the calculator above to calculate the diametral pitch of an individual spur gear. You just need to enter pitch diameter(inches) and the number of teeth. Then click on the ‘Calculate!’ button to calculate the diametral pitch in inches. Click on the ‘Reset’ button to calculate other calculations.

In standards, diametral pitch values are given as integers. You can select a spur gear according to these standards. Diametral pitch standards for them are generally used in countries where US Customary Units are used.

  • Circular Pitch Calculator For Spur Gears: Circular pitch is another kind of standard that is not frequently used as diametral pitch. Circular pitch can also be calculated by the multiplication of diametral pitch with π, for individual spur gears.

As you must understand, two gear mates in spur gear systems must have the same pitch.

What is the Metric Module of Spur Gears?

In SI units, the counterpart of pitch standards for spur gears is the term ‘metric module’ which is stated with ‘m’. The unit of the metric module must be millimeters.

The calculation of the metric module is the same with a diametral pitch. But, you need to consider the pitch diameters in millimeters. After converting the pitch diameter value to millimeters, you can use the calculator above to calculate the module of your gear.

Gear manufacturers generally use integer values of modules when they produce their spur gears.

Diametral Pitch to Metric Module Conversion for Spur Gears

If you have the diametral pitch or metric module to convert it to another one, you just need to know that you need to make a conversion between inches to millimeters.

To obtain the metric module value of them, you need to divide 25.4 by the diametral pitch value of the spur gear. To obtain a diametral pitch, you need to do vice-versa.

Geometric Features of Spur Gears and Calculations

Designing the construction of machinery, all the geometrical features must be calculated. Contacting of pinion and gears must be calculated in detail to obtain the required construction. There are a bunch of terms related to these calculations;

  • Addendum(a) Of Spur Gears: The distance between the pitch circle and the outermost section of the gear tooth.
  • Dedendum(b) Of Spur Gears: The distance between the pitch circle and the bottom of the tooth space.
  • Clearance(c) Between Spur Gears: At assembled gears, clearance defines the space between the outermost section of the first spur gear and the bottom of the tooth space of the second one.
Three definitions.
The illustration shows the three definitions above(Source Of Image: Machine Elements In Mechanical Design- Robert L. Mott).

You can use the calculator below to calculate Addendum, Dedendum, and Clearance of a spur gear if you know diametric pitch or metric module;

The use of the calculator above to calculate addendum, dedendum, and clearance values for spur gear mates is very easy. Select the required information from the list which you know; diametral pitch or metric module. Do not forget that the unit of diametral pitch is inches and the metric module is millimeters.

Enter the required value then click on the ‘Calculate!’ button. To make another calculation, click on the ‘Reset’ button then re-enter the values.

Calculation of Outside Diameter of a Spur Gear

The outside diameter of them is a very important constructional value. It is generally referred to as ‘Da’. If you know the metric module or diametric pitch and number of teeth of your spur gear, you can use the calculator below to calculate the outside diameter.

Just like the calculator before this, you need to select the information you have about your spur gear mates; metric module, or diametral pitch…

Outside diameter calculated via diametral pitch formula is like below. Check your calculations with this formula;

Outside diameter formula

Metric module outside diameter formula;

Metric module formula for spur gears.

The diameter of spur gear is calculated as the extraction of two dedendum values from the calculated outside diameter. This value is called ‘D’.

Calculation of Other Constructional Values of Spur Gear Tooth

  • Working Depth Calculation: Working depth is calculated as multiplying addendum with 2.
  • Tooth Thickness Calculation: You can calculate the tooth thickness value of mating gears by using this equation; pi/(2*Pa). Pa is the diametral pitch of gear.
  • Root Diameter Calculation: If you extract two dedendums from the diameter of spur gear, you will find out the root diameter value.
  • Whole Depth Value: Whole depth is the summation of addendum and dedendum of a spur gear.

Center Distance Calculation of Mating Spur Gears with Calculators

Maybe center distance calculation is the most important aspect of spur gear mates. The exact calculation of center distance is very important in terms of the construction of other elements inside machinery such as shafts etc.

If you know the number of teeth of pinion and gear mates, and diametral pitch or metric module, you can easily calculate the center distance value between mating gears.

Center distance formula via diametral pitch;

Center distance formula.

Center distance formula via metric module;

Center distance for spur gears.

Contact Ratio Specification

Like bending stress, contact stress is also a very important parameter to calculate the safety of the spur gear mates. Here, you will find detailed information about the contact stress safety calculations for spur gears.

Contact Stress Calculator For Spur Gear Mates

As you can see above, there are lots of coefficients and inputs to calculate the contact stress of a spur gear mate. You just need to enter the all inputs inside brackets then click on the ‘Calculate!’ button to calculate contact stress.

If you want to do another kind of calculation, click on the ‘Reset’ button to re-calculate the contact ratio value.

Recommended unit sets for each input are typed inside brackets. You can use this Sı unit set. If you have another unit set such as US Customary, you can convert your units to another unit set by using the Unit Converter tool.

How to Define All The Parameters of Contact Stress Calculation?

Here you can find definitions of all the parameters required to calculate the contact stress of a spur gear mate.

  • Velocity Of Pinion

This is the circumferential velocity of pinion gear which is calculated at the pitch diameter. 

  • Velocity Of Gear

Gear velocity is the output gear’s pitch diameter circumferential velocity.

  • Elasticity Modulus Of Gear And Pinion Gears

Materials of gears can differ from each other. And all the engineering materials have elasticity modulus values. Enter the elasticity modulus values inside the brackets.

  • Face Width Of Contact

Face width is a very important parameter for spur gear mates. Enter the face width value inside the brackets.

  • Pinion And Gear Diameter

These are the diameter values of each pinion and gear pitch diameters.

  • Pressure Angle

In general, pressure angle values are provided from gear manufacturer data. Enter the pressure angle of the gears in the spur gear set.

  • Factors

In Mechanicalland, you can find out the required information about these all factors related to spur gear sets. Specify all these factors then enter them inside brackets.

  • Total Load

In general, three forces are acting to spur gear teeth and the resultant of these three loads is called the total load. Enter the total load value inside brackets.

Click on the ‘Calculate!’ button to see the results!

Avoiding Interference Between Spur Gear Pairs

Gear interference for spur gears.
Gear interference problem illustration(Image Source: beinginventive.typepad.com)

Interference between spur gear mates can de be described as; the interference between the tip of teeth of pinion gear which is a smaller gear in spur gear mates, and the root side of teeth of the bigger gear. In the design stage, the designer must know to alleviate this issue.

What Causes Interference between Mating Spur Gears?

The most general reason for interference between gears is the size difference between the pinion and the bigger gear. If the size difference is higher than the recommended values, interference could occur. In the design stage, the designer must consider the design recommendations for the mating of these gears.

In pinion and rack pairs, interference is the most prominent issue compared with other gear mates. Because you can think of the pinion that has an infinite pitch.

How to Alleviate Gear Interference?

In the design or gear selection stage, you need to comply with the gear mating recommendations of the manufacturer or general recommendations about standard spur gears, to prevent interference problems.

For standards of pinion gears, there are the minimum number of teeth recommendations such as;

  • If the 25 degrees, involute form, and full-depth tooth form are used on pinion gears, it is supposed that the minimum number of teeth must not be below 12.
  • If the pressure angle is 20 for the same tooth form, the recommended minimum number of teeth must be 18.
  • If the pressure angle is 14, the minimum number of teeth must be 32.

In general, 20 degrees of pressure angles are standard for them. Also according to the number of teeth of pinion gears, some recommendations are available to prevent interference between spur gears, in terms of the number of maximum teeth used for bigger gears;

  • If the number of pinion teeth is 13, which is not encountered design procedure for 20 degrees of pressure angles, the maximum recommended number of bigger gear teeth is 16. This ratio is meaningless for most machinery applications. Because of that, the number of teeth for pinions is not used as this low.
  • If the number of pinion teeth is 14, the recommended number of bigger teeth is 26.
  • For 15 teeth for pinion, the maximum number of bigger gear teeth is 45.
  • For 16, this maximum number is 101.
  • And for 17 teeth for pinion, this recommended maximum number for bigger gear is 1309.

As you understand from the values above, the use of teeth numbers above 16 is common practice. Recommended maximum teeth for bigger gear increases exponentially with the increasing number of pinion teeth number.

If There is an Interference Problem in Existing Design or Machinery?

Illustration of undercut(Image Source): khkgears.net)

You can face an interference problem with the existing design or machinery. All the systems can be installed, but there is an interference problem. You can do the operation called ‘undercut’ on bigger gear teeth.

Undercut prevents gear interference in a very efficient way. With a proper machining process, you give round geometry on the root of teeth.

But you need to be aware that this operation decreases the strength of teeth.

There is a required strength to transmit power between gear mates. Undercut processes can reduce this strength.

Velocity Ratio Calculation

You can use our calculator to see the velocity ratio of your spur gear system. 

Train Value Calculations in Spur Gear Trains

If you will use gear trains in your design it will provide transmission of power between shafts that are far away from each other. So, you need to be aware of the training value for this system. 

Idler Gear Requirement Check

Idler gears are generally used for changing the rotation direction of the output gear. You need to decide to use whether your system needs idler gears inside it. 

Stress Calculations 

In the strength design of spur gears, one kind of stress is considered for spur gear teeth. It is bending stress resulting from the tangential force acting on spur gear teeth from another mate. This stress must be calculated and compared with the gear material’s strength values to ensure the safety of the gear system.

In general, Lewis Equation is used to calculate final bending stress acting teeth. But, the place where bending stress is calculated is important. For Lewis Equation, bending stress must be calculated according to tooth root of tooth form. 

You know that there is a radius beneath the root of teeth. So, this is a very critical point to calculate gear stresses.

Bending Stress Calculator for Spur Gear Mates

Bending Stress Calculator For Spur Gear Mates

The use of the calculator above is very simple. You just need to enter the required values into brackets and click on the ‘Calculate!’ button to see ‘Bending Stress’. If you want to make another calculation, just click on the ‘Reset’ button then re-enter all values again. All the required values are explained below.

Tangential Force

Power is transmitted between spur gear teeth through tangential force. The calculation of tangential force is very simple. You can find out the tangential force calculation from here.

Diametral Pitch

You can calculate the tangential force for an individual gear by using the calculators in this link.

This is also a very important parameter for all gear types. Take a look at the article here to understand the diametral pitch and metric module.

Face Width of Tooth

Face width value for spur gears.
Illustration of face width(Image Source: engineerharry.wordpress.com).

The total thickness of teeth set on a gear. This is a very important parameter because the bending stress will be calculated along this width.

Overload Factor

The overload factor is the multiplication factor of load on gear teeth. This factor is bigger than 1 if there are vibrational or variational working conditions. Generally, the overload factor is selected from catalogs, according to the machinery and working characteristics of the gear set.

You can enter a 1 to overload factor value if you have smoothly working machinery.

Size Factor

According to the size of the gear, you can select a size factor. You can find a reference below to select the proper size factor for your system.

Size factor chart.
Size factor selection(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).

Load Distribution Factor

This is the hardest factor to specify for a gear system. This factor originated because of lots of kinds of sources. Elastic deformations of different elements in the machinery, misalignments of shafts, thermal distortions, and gear teeth inaccuracies can be the reason for the non-uniform distribution of load.

To calculate the load distribution factor, you need to specify two values before that;

If which is pinion proportion factor and Cma mesh alignment factor. You can roughly specify these two values from the graphs below.

Cfp graph.
Cpf graph(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).
Cma graph.
Cma graph(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).

But we are sharing this information for giving a reference to you. You can use another source. This information is also sourced from American Gear Manufacturers Assoc.

Load Distribution Factor Calculator

Enter the Cpf and Cma factors inside the brackets then hit the ‘Calculate!’ button to see Km.

Rim Thickness Factor

Rim thickness is also a very important factor. It is important because the stress concentration place can change into the rim of gear if the thickness of the rim is low.

Rim thickness of spur gears.
Rim thickness illustration(Image Source: engineerharry.wordpress.com).

You can calculate the rim thickness factor by using the calculator below.

Rim Thickness Factor Calculator

You just need to enter the rim thickness and depth of gear tooth values to find out the rim thickness factor.

You just need to enter the rim thickness and depth of gear tooth values to find out the rim thickness factor.

Dynamic Factor

For dynamic loadings, we include this factor also. There is a bunch of consideration to define this factor.

  • If we produce the used gears with average quality toolings, which is also the most general type, you can find the dynamic factor by using curves 5, 6, or 7.
  • If we increase the accuracies of tooth profiles with the additional grinding or other kinds of methods, we can use 8, 9, 10, and 11.
  • We do not use qualities below 5 generally in high velocities.
Kv chart.
Representative curve to select Kv(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).

Geometry Factor

You can select a proper geometry factor for your gear. But we provided data for only a pressure angle of 20.

Geometry factors chart.
Geometry factor chart(Image Source: Robert L. Mott: Machine Elements In Mechanical Design).

Contact Ratio Calculations

Contact ratio is a very important parameter for spur gear designs. When we calculate the all the parameters for mating spur gears, you need to be also sure about the contact ratio of mating spur gears is inside the acceptable range.

The minimum acceptable contact ratio recommended by spur gear manufacturers is 1.2. In general applications, contact ratios of spur gear systems are around 1.5.

You can use the calculator below to see the contact ratio of your spur gear mates.

Contact Ratio Calculator For Spur Gear Mates

As you see above, the use of a contact ratio calculator is very simple. You need to have some information about your spur gear mate system;

  • Np is the number of teeth at your pinion gear.
  • Ng is the number of teeth at your bigger gear.
  • Pd is the diametral pitch value of your gear. If you gave the metric module, the value you can convert this metric module into diametral pitch value.
  • θ is the pressure angle value of your gear tooth. In most applications, this value is 20 degrees. But you need to take a look at the catalogs of gear manufacturers.

Enter the required data above at the contact ratio calculator, then click on the ‘Calculate!’ button to see the contact ratio.

If you want to do another calculation, click on the ‘Reset’ button then re-enter the values again.


Right after the geometrical calculations and definitions of the spur gear system, we must establish stress calculations for both bending stress and contact stress to show the safety of the designed spur gear system. 

In the design process, you must apply reasoning to obtain these systems from scratch. 

Above all, Mechanicalland does not accept any responsibility for calculations made by users in calculators. A good engineer must check calculations again and again.

Also, you can find out many more calculators in Mechanicalland! Take a look at the other engineering calculators available in Mechanicalland!

Finally, do not forget to leave your comments and questions below about the spur gear design guide and share your problems. 

Your precious feedbacks are very important to us.

External Link: Design and analysis of composite spur gears using finite element method


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