Radius Calculation for Gear Rotation with Rack and Pinion

In summary, the gear will rotate about the angle that it has turned, from the linear displacement of the rack.
  • #1
Parsifal1
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I want to find how far a gear from a rack and pinion will rotate from the displacement of the rack and the angle that it will have turned using: s=r*angular displacement.

So what is the radius in this case?
 
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  • #2
Parsifal1 said:
I want to find how far a gear from a rack and pinion will rotate from the displacement of the rack and the angle that it will have turned using: s=r*angular displacement.

So what is the radius in this case?
Can you post a diagram to make this a bit clearer please? Also, is this for schoolwork?
 
  • #3
berkeman said:
Can you post a diagram to make this a bit clearer please? Also, is this for schoolwork?

Its a project. I don't know which type of gear I will be using yet, I just want to know what you would use for the radius in that formula in the case of a gear. Would you use the radius of gyration, using the outer and inner radius formula?

figure001-custom.jpg
 
  • #4
Gear motion calculations are based on pitch circle diameter - diameter D in your diagram
 
  • #5
Nidum said:
Gear motion calculations are based on pitch circle diameter - diameter D in your diagram
So I couldn't use the formula s=r*angular displacement to work out the rotation of a rack and pinion gear?
 
  • #8
The effective motion geometry of a rack and pinion is that of a plain cylinder of diameter equal to the gear pitch circle diameter rolling without slip against a flat surface .
 
  • #9
Nidum said:
The effective motion geometry of a rack and pinion is that of a plain cylinder of diameter equal to the gear pitch circle diameter rolling without slip against a flat surface .
So the linear displacement of the rack is equal to the linear displacement of the gear? If so I should be able to use the formula I mentioned?
 
  • #10
Linear displacement of rack = ( angle turned by pinion / 360 ) x Pi x pitch circle diameter .

or

Linear displacement of rack = ( angle turned by pinion / 360 ) x Pi x 2 x pitch circle radius .
 
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  • #11
You can also derive formulas based directly on gear module number rather than on pitch circle diameter/radius
 
  • #12
The rack's displacement is equal to the tangential displacement (arc length) of the pinion at the pitch diameter. So if the pitch diameter of the pinion is 20 mm (radius of 10 mm), and you rotate the pinion 90 degrees (π/2 radians), the rack will move about 15.71 mm (calculation here).
 
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  • #13
Mech_Engineer said:
The rack's displacement is equal to the tangential displacement (arc length) of the pinion at the pitch diameter. So if the pitch diameter of the pinion is 20 mm (radius of 10 mm), and you rotate the pinion 90 degrees (π/2 radians), the rack will move about 15.71 mm (calculation here).

I see, so that was the formula I mentioned then. Problem solved.
 
  • #14
Parsifal1 said:
So what is the radius in this case?
You do not need to measure the pitch circle radius. There is an easier way.
You only need to know the pitch of the linear gear rack and the number of teeth on the gear wheel.
That will give you the relationship between linear position and rotation angle.
 

FAQ: Radius Calculation for Gear Rotation with Rack and Pinion

1. What is the definition of the radius of a gear?

The radius of a gear is the distance from the center of the gear to the outermost point on the gear's circumference. It is typically measured in millimeters or inches.

2. How is the radius of a gear different from the diameter?

The radius is half the size of the diameter. The diameter is the distance across the widest part of the gear, while the radius is the distance from the center to the outer edge.

3. What factors determine the radius of a gear?

The radius of a gear is determined by the size and shape of the gear teeth, as well as the overall size of the gear. The number of teeth and the pitch (distance between teeth) are also important factors.

4. How do you measure the radius of a gear?

The radius of a gear can be measured using a caliper or a micrometer, which are precision tools designed for measuring small distances. The gear should be placed on a flat surface and the measuring tool should be used to determine the distance from the center of the gear to the outer edge.

5. Can the radius of a gear be changed?

Yes, the radius of a gear can be changed by altering the shape, size, or number of teeth on the gear. This can be done through various manufacturing processes such as cutting, grinding, or molding.

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