# Why is the gear ratio reversed in a planetary gear system?

• Trying2Learn
In summary, the gear ratio and torque are inverses of each other when the gears are in external mesh. This is consistent with the wikipedia article, which got the equation wrong. A thought experiment may help to visualize this.
Trying2Learn
TL;DR Summary
What is the relation between torque to teeth in an epicyclic gearing system?
Hello,

I am confused about something. I will take the time to work it out, but right now, I am using it to remember. And I am now confused.

Go here:
https://www.smlease.com/entries/mechanism/gear-train-gear-ratio-torque-and-speed-calculation/

Scroll down to the section on: Gear Ratio and Torque

In a loss-less system, I get that that ratio of output to input torque is inverse to the output to input number of gear teeth:
To/Ti = Ni/No

Now go hear for a planetary gear system:
https://en.wikipedia.org/wiki/Epicyclic_gearing

Scroll down about one third to: Torque Ratios of Standard Epicyclic Gearing

(Granted, there are three gears here and in an planetary style, not simple).

But why (say, for the first one), is this ratio reversed.

Tr/Ts = Nr/Ns

Am I missing something, or is there an error in one of these? I get the simple gear system (and I am confident of working it out), but for the planetary one, is there an inverse relation because the teeth are INSIDE on the ring gear? Can someone point me to where (assuming the wiki page is correct) it is worked out?

When gears are in normal external mesh, the angular velocity changes sign.
Running a pinion in an internal ring gear does not reverse the sign of the angular velocity.
The absolute ratio remains the same, it is not the reciprocal.

Trying2Learn said:
Can someone point me to where (assuming the wiki page is correct) it is worked out?
In the wikipedia epicyclic article, N is angular velocity, z is tooth count, and T is torque.
For the same power the product of shaft torque and shaft angular velocity should be constant.
With a fixed carrier, the planets are idlers, so Tr * Nr = Ts * Ns;
∴ Tr = Ts * Ns / Nr;
Which suggests the wikipedia article got it wrong.
My guess is that someone incorrectly thought N was the toothcount.

Baluncore!

THANK YOU VERY MUCH

May I ask you to repeat this, but just by affirming? (I am shocked that I found this error.)

Set aside your comment on the sign and reciprocity--perhaps my wording was off, and that was not my issue, anyway.

May I ask you to confirm this.

In a gearing system, where T is the torque and (in my question) N is the tooth count

T_input/T_output = N_input/N_output

Also, in the same wikipedia article, the very next line for the torque of the ring to the carrier: did they incorrectly get that one wrong, too?

Can you suggest a source that explains the input/output for the torque/teeth, simply.
(If possible, an online source?, but I will gladly accept anything)

Trying2Learn said:
In a gearing system, where T is the torque and (in my question) N is the tooth count

T_input/T_output = N_input/N_output
For tooth count N, the relative angular velocities are Vi = 1 / Ni; and Vo = 1 / No.
Power = Torque * Angular velocity.
Conservation of energy requires; Ti * Vi = To * Vo.
∴ Ti * No = To * Ni;
∴ Ti / To = Ni / No;
If there are only two shafts, one input and one output, you are correct.

Wikipedia gets things wrong. It is consistent in that article.

I will see what references I can find.

Baluncore said:
For tooth count N, the relative angular velocities are Vi = 1 / Ni; and Vo = 1 / No.
Power = Torque * Angular velocity.
Conservation of energy requires; Ti * Vi = To * Vo.
∴ Ti * No = To * Ni;
∴ Ti / To = Ni / No;
If there are only two shafts, one input and one output, you are correct.

Wikipedia gets things wrong. It is consistent in that article.

I will see what references I can find.
This is great. I am really excited I am on the right track.

Please do not forget me and please provide a reference if you can.

A thought experiment may help.
• Draw an imaginary line between the centers of the two gears
• When the Input gear turns, some number of teeth will cross that line
• Since the teeth of the gears are meshed, the same number of teeth of the other, Output, will also cross that line
• If the Output gear is bigger, then it will turn only a partial revolution while the Input gear turns turns a full revolution
• Now you can see that if you swap the Input and Output, putting the big gear on the input
• The smaller gear will turn faster than the big gear
Here is a link with lots of diagrams and explanation:
https://www.wikihow.com/Determine-Gear-Ratio

This one has formulas and a sketch:
https://www.engineeringtoolbox.com/gear-output-torque-speed-horsepower-d_1691.html

(above found with:

Just remember that if the Speed goes UP by some factor, say 3.5, then the Torque goes DOWN by the same factor.

Code:
    IN               Out
Speed Torque     Speed Torque
200   100        700    28.6     speed UP by 3.5, torque DOWN
200   100         57   350       speed DOWN by 3.5, torque UP

Cheers,
Tom

Last edited:
Baluncore said:
Hidden behind the wikipedia epicyclic page is a talk page with another set of formulas.
https://en.wikipedia.org/wiki/Talk:Epicyclic_gearing#Formulas_for_calculating_gear_ratios

Thank you Baluncore. I will look.
Tom.G said:
A thought experiment may help.
• Draw an imaginary line between the centers of the two gears
• When the Input gear turns, some number of teeth will cross that line
• Since the teeth of the gears are meshed, the same number of teeth of the other, Output, will also cross that line
• If the Output gear is bigger, then it will turn only a partial revolution while the Input gear turns turns a full revolution
• Now you can see that if you swap the Input and Output, putting the big gear on the input
• The smaller gear will turn faster than the big gear
Here is a link with lots of diagrams and explanation:
https://www.wikihow.com/Determine-Gear-Ratio

This one has formulas and a sketch:
https://www.engineeringtoolbox.com/gear-output-torque-speed-horsepower-d_1691.html

(above found with:

Just remember that if the Speed goes UP by some factor, say 3.5, then the Torque goes DOWN by the same factor.

Code:
    IN               Out
Speed Torque     Speed Torque
200   100        700    28.6     speed UP by 3.5, torque DOWN
200   100         57   350       speed DOWN by 3.5, torque UP

Cheers,
Tom

Tom.G said:
A thought experiment may help.
• Draw an imaginary line between the centers of the two gears
• When the Input gear turns, some number of teeth will cross that line
• Since the teeth of the gears are meshed, the same number of teeth of the other, Output, will also cross that line
• If the Output gear is bigger, then it will turn only a partial revolution while the Input gear turns turns a full revolution
• Now you can see that if you swap the Input and Output, putting the big gear on the input
• The smaller gear will turn faster than the big gear
Here is a link with lots of diagrams and explanation:
https://www.wikihow.com/Determine-Gear-Ratio

This one has formulas and a sketch:
https://www.engineeringtoolbox.com/gear-output-torque-speed-horsepower-d_1691.html

(above found with:

Just remember that if the Speed goes UP by some factor, say 3.5, then the Torque goes DOWN by the same factor.

Code:
    IN               Out
Speed Torque     Speed Torque
200   100        700    28.6     speed UP by 3.5, torque DOWN
200   100         57   350       speed DOWN by 3.5, torque UP`

Cheers,
Tom
Thank you, Tom. This helps a lot

## 1. How do gear ratios affect torque?

Gear ratios determine the relationship between the number of teeth on two gears, which directly affects the amount of torque produced. A higher gear ratio means that the output gear will have more teeth than the input gear, resulting in a decrease in torque but an increase in speed. Conversely, a lower gear ratio will result in an increase in torque but a decrease in speed.

## 2. What is the formula for calculating gear ratio?

The gear ratio can be calculated by dividing the number of teeth on the output gear by the number of teeth on the input gear. For example, if the output gear has 20 teeth and the input gear has 10 teeth, the gear ratio would be 20/10 or 2:1.

## 3. How does changing the gear ratio affect the performance of a machine?

Changing the gear ratio can have a significant impact on the performance of a machine. A higher gear ratio can increase the speed of the machine, making it more efficient for tasks that require high speeds. On the other hand, a lower gear ratio can increase the torque, making the machine more powerful for tasks that require more force.

## 4. What is the relationship between gear ratios and gear teeth?

The gear ratio is directly proportional to the number of teeth on each gear. This means that as the number of teeth on the output gear increases, the gear ratio will also increase. Similarly, as the number of teeth on the input gear decreases, the gear ratio will also decrease.

## 5. How do gear ratios affect the efficiency of a machine?

Gear ratios can affect the efficiency of a machine in two ways. First, a higher gear ratio can result in a more efficient use of power, as less torque is required to achieve the same speed. Second, a lower gear ratio can result in more energy loss due to friction and heat, reducing the overall efficiency of the machine.

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