Spur gears are one of the most commonly used types of gears in machinery. They consist of a cylindrical wheel with straight teeth that mesh with another gear to transmit torque and rotation. The teeth on spur gears are aligned parallel to the axis of rotation, making them ideal for applications where precise positioning is required. Spur gears are used in a wide range of machinery, including industrial machines, automotive transmissions, and power tools.
Now let’s dive into the calculations. As experts in the field, we know that accurate calculations are crucial for the successful design and operation of spur gears. In this guide, we’ll cover everything you need to know to ensure your spur gear calculations are precise and reliable.
Spur Gear Terminology
To accurately calculate the performance of spur gears, it’s essential to understand the terminology associated with them. Here are some of the key terms you should know:
| Item | Symbol | Basic Gear Item Inforamtion |
|---|---|---|
| Number of Teeth | N | To determine the number of teeth on a gear, you need to know the pitch diameter, diametral pitch, and the number of teeth on the mating gear. |
| Pitch Diameter/Pitch Circle | D | The diameter of the pitch circle. The theoretical circle around which the teeth of a gear are located. |
| Diametral Pitch | DP | The ratio of the number of teeth on a gear to its pitch diameter. |
| Pressure Angle | α | The angle between the tooth face and a tangent to the pitch circle. ( α=20° for most gears, Other less used are 14.5° and 25°.) |
| Module | m | Gear module is a fundamental parameter used to define the size of a gear. |
| Reference Diameter | d | The reference diameter is used in gear designing and calculation. It has direct connections with other important gear parameters like the module (m), center distance (c) and pressure angle (α). |
| Face Width/Tooth Height | h | The width of the gear tooth measured parallel to the axis of rotation. |
| Addendum | ha | The distance from the pitch circle to the top of the gear tooth. |
| Dedendum | hf | The distance from the pitch circle to the bottom of the gear tooth. |
| Center Distance | C | The center distance is important because it ensures that the gears mesh properly and transfer power smoothly. |
| Number of teeth on Driving Gear | N1 | The number of teeth on the driving gear is an important factor that affects the performance of the gear pair. |
| Number of Teeth on Mating Gear | N2 | The gear ratio is the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear. |
| Input Speed | RPM/S1 | Input speed in spur gear refers to the speed of the driving gear, which is the gear that transmits power to the driven gear in a pair of spur gears. |
| Desired Output Speed | S2 | Desired output speed in spur gear refers to the rotational speed that is desired or required for the output gear in a gear pair. |
| Tooth strength | S | Tooth strength in spur gear refers to the ability of the gear teeth to withstand the forces & stresses generated during operation without failing or deforming |
With these terms in mind, let’s move on to the calculations.
Equations for Standard Spur Gear
| To Find | Equation |
| Base Circle Pitch | DB = D cosφ |
| Circular Pitch | p = ( π D )/ N p = π / P |
| Center Distance | C = Np (mG + 1) / 2P C = ( Dp + DG ) / 2 C = ( NG + Np) / 2P C = (NG + Np) p / 2P C = (NG + Np) p / 6.2832 |
| Diametral Pitch | P = π / p P = N / D P = [ Np ( mG + 1) ] / 2C |
| Gear Ratio | mG = NG / Np |
| Number of Teeth | N = P D N = ( π D ) / p |
| Outside Diameter (Full Depth Teeth) | DO = ( N + 2 ) / P DO = [ ( N + 2 ) p ] / π |
| Outside Diameter (American Standard Stub Teeth) | DO = ( N + 1.6 ) / P DO = [ ( N + 1.6 ) p ] / π |
| Outside Diameter | DO = D + 2a |
| Pitch Diameter | D = N / P D = (N p ) / π |
| Root Diameter | DR = D – 2b |
| Whole Depth | a + b |
| Working Depth | aG + ap |
Standard Spur Gear Calculations
Calculating the performance of spur gears requires a thorough understanding of the gear’s dimensions, tooth profile, and operating conditions. Here are some of the questions you might have:
Formula:
Pitch Diameter/ Pitch Circle (D) = Number of teeth (N) / Diametral pitch (DP)
It is an important parameter in gear design and calculation because it affects the gear’s rotational speed, torque, and overall performance.
Formula:
Diametral Pitch (DP) = Number of teeth (N) / Pitch diameter (D)
Diametral pitch affects the size and spacing of the teeth on the gear, and also determines the gear’s strength and load-carrying capacity.
Formula:
Outside Diameter = ( Number of teeth + 2 ) / Diametral pitch
(For full depth teeth) = [ (Number of teeth + 2) Diametral pitch) / π ]
The outside diameter determines the size and shape of the gear, and also affects the speed ratio of the gear system.
Formula:
Tooth Thickness = 1.5708 / diametral pitch
1.5708 is a numerical constant, commonly known as pi/2 or half of the mathematical constant pi (π). It is approximately equal to 1.570796327 radians or 90 degrees in terms of angle measurement.
Note: This formula assumes that the gear has standard involute teeth and a pressure angle of 20 degrees. Other gear types or pressure angles may require different formulas to calculate tooth thickness.
Formula:
Addendum (ha) = Module (For standard gears, ha=m=1)
Dedendum (hf) = 1.25 x Module (For standard gears, hf=1.25m)
Once you have calculated the addendum and dedendum values, you can determine the face width of the gear by adding the two values together.
The module also has to do with the tooth height, for standard gears, the tooth height equals to h=2.25*m
Formula:
Module = (Reference Diameter (d)) / (Number of Teeth (N))
The reference diameter of a spur gear is a crucial parameter used as a foundation for determining various tooth dimensions and other significant characteristics of the gear. It is also known as the “basic” or “standard” diameter.
Formula:
Face Width/ Tooth Height (h) = Addendum + Dedendum
The tooth height/face width is an important parameter in gear design, as it affects the overall size and shape of the gear.
Formula:
Number of Teeth (N) = Pitch diameter(P) * diametral pitch(DP)
Number of teeth (N) = [ ( π * Diametral pitch( DP) ) / (Circular pitch)p ]
The number of teeth in a spur gear greatly influences its function in any machinery; the greater the force, the greater the spur gear.
Formula:
Number of Teeth on the Mating Gear (N2) = (N1 x R) / S2
To determine the number of teeth on the mating gear, you need to know the gear ratio and the desired output speed. Where N2 is the number of teeth on the mating gear, N1 is the number of teeth on the driving gear, R is the gear ratio, and S2 is the desired output speed.
Formula:
Gear ratio = Number of teeth on driving gear / Number of teeth on driven gear
The gear ratio determines the output speed and torque of a gear system.
Formula:
Input Speed (RPM/S1) = (Output Speed / Gear Ratio) x (Number of Teeth on Output Gear / Number of Teeth on Input Gear)
Input speed in a spur gear refers to the rotational speed of the input gear, typically measured in revolutions per minute (RPM).
Formula:
Output Speed (S2) = (Input speed * Gear ratio) / 60 (The factor of 60 is used to convert the output speed from seconds to minutes.)
The desired output speed determines the rotational speed of the driven gear or system.
Formula:
Center Distance (C): C = (N1 + N2)/2D
The number of teeth on the mating gear is required to determine the correct value of C, which is the center distance between the two gears. If the center distance is incorrect, the gears may not mesh properly and may result in noise, vibration, or even gear failure.
Where N1 and N2 are the numbers of teeth on the two gears. You need to know the pitch diameter(D) of both gears and the sum of their number of teeth.
Formula:
Tooth strength: S = (Y x K x Wt) / FOS
(Calculated using the Lewis equation)
Where S is the tooth strength, Y is the Lewis form factor, K is the geometry factor, Wt is the tangential force, and FOS is the factor of safety.
Calculating the spur gear pitch diameter can be done by dividing the number of teeth on the gear by its diametral pitch
Divide the number of teeth (N) by the pitch circle diameter (D, in inches)
The formula of operating gearing = Contribution / PBIT.
The pressure angle and pitch must match exactly for two gears to mesh properly.
Spur Gear Calculator
One important tool in performing spur gear calculations is the spur gear calculator. A spur gear calculator is a software program or online tool that helps engineers and designers calculate important parameters of spur gears, such as gear ratio, pitch diameter, and tooth size. The calculator can quickly and accurately provide the necessary calculations. Using a spur gear calculator can save time and reduce errors compared to performing calculations manually. There are different types of spur gear calculators available online for free including desktop software, mobile apps, and tools, Making them an accessible and convenient tool for anyone involved in gear design or selection.
Conclusion
Accurate spur gear calculations are essential for the successful design and operation of machinery. This is why understanding the spur gear calculation formula information is important. Having a good understanding of the formula allows for more accurate calculations and can help identify potential design issues early on in the process . You can ensure that your gear designs are precise, reliable, and safe.
At Diversified Specialities Inc, we specialize in providing high-quality spur gears for a wide range of applications. With over 50 years of experience and expertise in the field, we can help you with the gear design for your equipment needs.
