Calculating Gear Displacement: Mechanical Engineering Insight

Calculating gear displacement is an important aspect of mechanical engineering, especially in applications where precise motion control and positioning are required. Gear displacement, often referred to as linear displacement or linear travel, represents the distance that an object, such as a rack or linear actuator, moves in response to the rotational motion of a gear or gear system. To calculate gear displacement, you need to consider the gear's parameters and the specific gear arrangement. Here's how to do it:

1. Understand Gear Terminology:

Before calculating gear displacement, it's important to be familiar with the following gear terminology:

• Pitch Circle: The circle through which the teeth of a gear are geometrically defined. It's also called the pitch circle diameter (PCD).

• Pitch Diameter (D): The diameter of the pitch circle. It's equal to the number of teeth (N) divided by the diametral pitch (P).

• Diametral Pitch (P): A measure of the size of the gear teeth. It represents the number of teeth per inch of the pitch diameter. The formula to calculate P is P = N / D.

• Module (M): A metric measure of gear tooth size, defined as the pitch diameter (D) divided by the number of teeth (N). M = D / N.

2. Determine Gear Ratios:

In a gear system, the displacement of one gear is transmitted to another gear through the meshing of their teeth. The gear ratio (GR) is defined as the ratio of the number of teeth on the driven gear (output gear) to the number of teeth on the driving gear (input gear). Gear displacement depends on this ratio.

3. Calculate Linear Displacement:

To calculate the linear displacement (Dlinear) produced by a rotating gear, use the following formula:

Dlinear = (Θ * D) / (2 * π * GR)

• Dlinear: Linear displacement (in units of length, e.g., millimeters or inches).
• Θ (Theta): The angular displacement of the input gear (in radians). This is the angle through which the input gear has rotated.
• D: Pitch diameter of the driven gear (in units of length).
• π (Pi): A mathematical constant approximately equal to 3.14159.
• GR: Gear ratio (output teeth/input teeth).

Make sure to use consistent units for all variables in the formula.

4. Solve for Linear Displacement:

Plug in the values for Θ (in radians), D (pitch diameter of the driven gear), and the gear ratio (GR) into the formula to calculate the linear displacement.

Example:

Let's say you have an input gear with 20 teeth (N1), a driven gear with 80 teeth (N2), and an angular displacement of 30 degrees (Θ). The pitch diameter of the driven gear (D) is 50 mm. Calculate the linear displacement.

First, calculate the gear ratio (GR):

GR = N2 / N1 = 80 / 20 = 4

Convert the angular displacement from degrees to radians:

Θ = (30 degrees) * (π radians / 180 degrees) ≈ 0.5236 radians

Now, use the formula to calculate linear displacement:

Dlinear = (0.5236 radians * 50 mm) / (2 * π * 4) ≈ 2.083 mm

So, the linear displacement produced by the gear system is approximately 2.083 millimeters.

Gear displacement = (gear width * gear height * gear pitch diameter * π) / 2

Flow rate = (gear displacement * pump speed) / 60

Gear width = 1 inch
Gear height = 2 inches
Gear pitch diameter = 3 inches

Gear displacement = (1 inch * 2 inches * 3 inches * π) / 2
= 6π cubic inches