Formulae of gears

Question

Formulae of gears

Accepted Answer

Formulae of gears involve calculations for gear ratio, pitch diameter, and tooth profile.

Gear ratio formula: Number of teeth on driven gear / Number of teeth on driving gear
Pitch diameter formula: Number of teeth / Diametral pitch
Tooth profile formula: Involves calculations for involute curve and pressure angle

Accepted Answer

Gears are fascinating mechanical elements that transmit power and motion through interlocking teeth. To understand how gears work and design them effectively, we use a variety of formulas. Here's a breakdown of some of the most important gear formulas:
Basic Gear Terminology
Before diving into the formulas, let's define some key terms:
 * Pitch Circle: An imaginary circle that represents the point of contact between two gears.
 * Pitch Diameter (D): The diameter of the pitch circle.
 * Number of Teeth (N): The number of teeth on the gear.
 * Module (m): A measure of the tooth size, calculated as pitch diameter divided by the number of teeth (m = D/N).
 * Circular Pitch (p): The distance between corresponding points on adjacent teeth, measured along the pitch circle (p = πm).
 * Diametral Pitch (P): The number of teeth per inch of pitch diameter (P = N/D).
 * Addendum (a): The distance from the pitch circle to the top of the tooth.
 * Dedendum (d): The distance from the pitch circle to the bottom of the tooth.
 * Pressure Angle (φ): The angle between the line of action and the tangent to the pitch circle.
Key Formulas
 * Gear Ratio (i): The ratio of the number of teeth on the driven gear (N2) to the number of teeth on the driving gear (N1). It also equals the inverse ratio of their speeds (n1/n2).
   i = N2 / N1 = n1 / n2

* Pitch Diameter (D): Related to the number of teeth (N) and module (m).
   D = N * m

* Center Distance (C): The distance between the centers of two meshing gears.
   C = (D1 + D2) / 2

* Speed Ratio (n1/n2): The inverse of the gear ratio.
   n1 / n2 = N2 / N1

* Torque Ratio (T2/T1): The ratio of the torque on the driven gear (T2) to the torque on the driving gear (T1). Inversely proportional to the speed ratio.
   T2 / T1 = n1 / n2

Important Considerations
 * Units: Be consistent with the units you use in your calculations (e.g., millimeters, inches).
 * Standardization: Gears are often manufactured according to standard specifications to ensure interchangeability.
 * Pressure Angle: The pressure angle is an important parameter that affects the tooth shape and gear performance. Common pressure angles are 20 degrees and 25 degrees.
Applications
These formulas are used in various applications, including:
 * Designing gears for specific speed and torque requirements.
 * Calculating gear ratios for different applications.
 * Determining the size and dimensions of gears.
 * Analyzing the performance of gear systems.
Where to Learn More
 * Gear manufacturers' catalogs: These often include detailed information about gear terminology, formulas, and selection.
 * Engineering textbooks: Textbooks on machine design or mechanical engineering provide comprehensive coverage of gear theory and calculations.
 * Online resources: Websites and online calculators can help you with gear calculations and design.
By understanding these fundamental gear formulas, you can gain a deeper appreciation for how gears work and how to design them effectively for various applications.