Simple gear pair: efficiency and output torque

Click For Summary

Discussion Overview

The discussion revolves around the output torque and efficiency of a simple gear pair, specifically focusing on the relationship between input torque, load torque, and the effects of friction and inertia. Participants explore theoretical and practical aspects of gear mechanics, including calculations and simulations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant describes a scenario involving a pinion and gear, detailing input torque, angular velocities, and load torque, and seeks to determine output torque and efficiency.
  • Another participant questions the relationship between load torque and output torque, suggesting that if they are not equal, the system will accelerate or decelerate, affecting energy transfer.
  • Concerns are raised about the implications of having load torque significantly greater than output torque, questioning which component drives the other in such a case.
  • Participants discuss the impact of friction on output torque calculations, presenting equations for scenarios with and without friction.
  • One participant shares simulation results that contradict expected output torque values, prompting requests for clarification on the simulation setup and load conditions.
  • There is a suggestion to use hand calculations instead of simulations for clarity in understanding the problem.
  • Participants emphasize the importance of free body diagrams to analyze the system, with discussions on the role of inertial reaction torques and gear ratios.
  • There is a debate about the inclusion of inertial reaction torques in free body diagrams, with differing opinions on their relevance to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the role of inertial reaction torques and their inclusion in analysis. There is no consensus on the correctness of the output torque calculations or the simulation results, indicating ongoing debate and exploration of the topic.

Contextual Notes

The discussion includes various assumptions about friction, inertia, and the definitions of torque and efficiency. Some participants note that the problem may be over-defined in steady state but under-defined in transient cases, highlighting the complexity of the analysis.

liao
upload_2017-7-15_14-43-5.png

Hi. I have a question about output torque and efficiency of a simple gear pair as shown on the picture. So, I have a pinion and a gear. I give an input torque Tp in the clockwise direction. Therefore, the pinion will rotate with ωp angular velocity in clockwise and the gear ωg in counter-clockwise. Then, I apply a load Tl in the opposite direction of the gear motion (clockwise direction). The gear is still rotating in the counter-clockwise. I neglect the friction between the gear mesh, but I consider the bearing friction both in pinion cp*ωp and gear cg*ωg. The moments of inertia of the pinion and the gear are Ip and Ig, respectively. My question is what the output torque To is because I want to find the efficiency of this gear pair. Thank you very much.
 
Engineering news on Phys.org
liao said:
View attachment 207233
Hi. I have a question about output torque and efficiency of a simple gear pair as shown on the picture. So, I have a pinion and a gear. I give an input torque Tp in the clockwise direction. Therefore, the pinion will rotate with ωp angular velocity in clockwise and the gear ωg in counter-clockwise. Then, I apply a load Tl in the opposite direction of the gear motion (clockwise direction). The gear is still rotating in the counter-clockwise. I neglect the friction between the gear mesh, but I consider the bearing friction both in pinion cp*ωp and gear cg*ωg. The moments of inertia of the pinion and the gear are Ip and Ig, respectively. My question is what the output torque To is because I want to find the efficiency of this gear pair. Thank you very much.

@CWatters if To = Tl, I would not get a correct efficiency η.
η = Po/Pi = To ωg/(Tp ωp) = Tl ωg/(Tp ωp)

we know that rg ωg = rp ωp and ωg/ωp = rp/rg, so

η = Tl rp/(Tp rg)

In this equation, we can see that Tl/Tp has to be rg/rp in order to have η = 1. If I don't have Tl/Tp close to rg/rp, the efficiency could be much less than 1, for example 0.2, which doesn't make sense.
 
liao said:
In this equation, we can see that Tl/Tp has to be rg/rp in order to have η = 1. If I don't have Tl/Tp close to rg/rp, the efficiency could be much less than 1, for example 0.2, which doesn't make sense.

If Tl <> TO the system accelerates (decelerates) putting energy into (removing energy from) the angular momentum of the gears. Load also takes more/less power. That's where the missing energy goes (comes from) if n <> 1.

What happens if Tl >> TO ? Which end is driving which?
 
Last edited:
CWatters said:
If Tl <> TO the system accelerates (decelerates) putting energy into (removing energy from) the angular momentum of the gears. Load also takes more/less power. That's where the missing energy goes (comes from) if n <> 1.

What happens if Tl >> TO ? Which end is driving which?
Thanks again for your reply. If Tl >> To, I believe the gear will drive the pinion in the load direction. However, I'm only interested in the case where the load is small such that the pinion still drives the gear. Just to confirm, did you think that my analysis on the efficiency when To = Tl correct?
 
liao said:
did you think that my analysis on the efficiency when To = Tl correct?

Yes.
.
 
Regarding friction..

Without friction
To = Tp * rg/rp

With friction in the bearings..
To = ((Tp - cp*ωp) * rg/rp) - cg*ωg
 
CWatters said:
Regarding friction..

Without friction
To = Tp * rg/rp

With friction in the bearings..
To = ((Tp - cp*ωp) * rg/rp) - cg*ωg
Thanks for your reply. I simulated your To with friction in the bearings in Simulink, but To is incorrect. Here are the results. So, I drive the pinion for a SFUDS (simplified federal urban driving schedule). rg/rp =3. The first figure is the angular velocities of both pinion and gear. We can see that the pinion turns 3 times faster than the gear. The second figure is the torque. Let me know your thoughts. Thanks again. I really appreciate your help! :)

upload_2017-7-16_12-25-44.png

upload_2017-7-16_12-28-50.png
 
Sorry I've never used Simulink. Clearly something is wrong as you are applying an input torque and getting no (or very little) output torque. Is there a load?
 
The problem is over defined for the steady state case and under defined for the transient case .

Just for starters which case is it that are you looking at ?

In any case I strongly suggest that you use simple hand calculations for this problem rather than Simulink .
 
  • #10
CWatters said:
Sorry I've never used Simulink. Clearly something is wrong as you are applying an input torque and getting no (or very little) output torque. Is there a load?
Yes, there is. Basically I derived the equation of motion of the simple gear by Lagrange's equation, then find the time response. I have tried three other options for To. Could you check it on the attached brief write-up? Thanks a lot CWatters!
 

Attachments

  • #11
Nidum said:
The problem is over defined for the steady state case and under defined for the transient case .

Just for starters which case is it that are you looking at ?

In any case I strongly suggest that you use simple hand calculations for this problem rather than Simulink .
Hi Nidum. Thanks for your reply. Sorry, I should've said that I'm looking at the transient case. Why is it under defined?

I've done simple hand calculations. I use Simulink to validate it.
 
  • #12
I don't want to back track through what you have done so far . We'll start afresh using proper engineering principles .

So action the first : Show me two free body diagrams - one for each gear . Show all relevant information on each diagram .
 
  • #13
-- and use some good sketching software for the diagrams if you can .
 
  • #14
Nidum said:
I don't want to back track through what you have done so far . We'll start afresh using proper engineering principles .

So action the first : Show me two free body diagrams - one for each gear . Show all relevant information on each diagram .
 

Attachments

  • NewtonEuler.PNG
    NewtonEuler.PNG
    30 KB · Views: 588
  • #15
Nidum said:
-- and use some good sketching software for the diagrams if you can .
sorry, I don't have experience of sketching software for these diagrams. I will learn it though. Thanks.
 
  • #16
The inertial reaction torques are missing - the gears have moments of inertia and they are accelerating ?

Best not to mix tangential forces and torques . Use only torques when you re-draw the diagrams .
 
  • #17
Nidum said:
The inertial reaction torques are missing - the gears have moments of inertia and they are accelerating ?

Best not to mix tangential forces and torques . Use only torques when you re-draw the diagrams .
upload_2017-7-16_15-38-39.png


Yes, the gears have moments of inertia and they are accelerating. Please see the revised diagram, I hope this one is clearer. Tr is the reaction torque.
 

Attachments

  • upload_2017-7-16_15-34-39.png
    upload_2017-7-16_15-34-39.png
    20.6 KB · Views: 596
  • #18
The reaction torques are in general going to be different for the two gears ?

Also we need a way to define the gear ratio
 
  • #19
Nidum said:
The reaction torques are in general going to be different for the two gears ?

Also we need a way to define the gear ratio

Question: The reaction torques are the product of the internal constraint force F and the gear/pinion radius, right? if so, yes, they're different due to different radius.

The gear ratio is ωp/ωg = rg/rp
 
  • #20
I was referring to the inertial reaction torques as mentioned in my previous post .

Anyway we have what we need now . It's late here so I am signing off for tonight . Talk to you tomorrow .
 
  • #21
Nidum said:
I was referring to the inertial reaction torques as mentioned in my previous post .

Anyway we have what we need now . It's late here so I am signing off for tonight . Talk to you tomorrow .
Thanks a lot Nidum. I really appreciate it! Good night! :)
 
  • #22
Nidum said:
The inertial reaction torques are missing - the gears have moments of inertia and they are accelerating ?

Best not to mix tangential forces and torques . Use only torques when you re-draw the diagrams .

I'm sure we could argue about the "inertial reaction torques" until the cows come home, but I would never include such in a FBD. They are exactly analogous to m*a terms, and are not real torques (the cross product of a distance with a force) any more than the m*a terms are force. Neither will obey the third law; they have no reaction to be found anywhere in the universe.

On the second point, regarding using torques rather than forces, I disagree here as well. It is force that interact between the two gears, the contact forces between the teeth. A gear does not exert a torque on a second gear, but rather it exerts a force. I would suggest that the FBDs need to show the contact forces and the bearing forces on each gear, as well as any torques applied through connecting shafts.
 
  • #23
Dr.D said:
I'm sure we could argue about the "inertial reaction torques" until the cows come home, but I would never include such in a FBD. They are exactly analogous to m*a terms, and are not real torques (the cross product of a distance with a force) any more than the m*a terms are force. Neither will obey the third law; they have no reaction to be found anywhere in the universe.

On the second point, regarding using torques rather than forces, I disagree here as well. It is force that interact between the two gears, the contact forces between the teeth. A gear does not exert a torque on a second gear, but rather it exerts a force. I would suggest that the FBDs need to show the contact forces and the bearing forces on each gear, as well as any torques applied through connecting shafts.
Hi Dr.D. Thanks for your reply and suggestions for the FBDs. What do you think the output torque To is?
 
  • #24
@liao has now worked out the final solution to this problem .
 
Last edited:
  • #25
Congratulations to Liao. Why does someone else announce this to us all?
 

Similar threads

A novel compound epicyclic gear system I can't seem to understand
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Increasing the output torque of a DC motor with gears
  • · Replies 16 ·
Replies
16
Views
4K
Gearbox torque calculation - input/output
  • · Replies 3 ·
Replies
3
Views
35K
Can Constant Velocity Produce Zero Torque?
  • · Replies 4 ·
Replies
4
Views
4K
Planet gear efficiency. Need help
  • · Replies 1 ·
Replies
1
Views
2K
Gear Torque Basics: Questions & Answers
  • · Replies 18 ·
Replies
18
Views
46K
How Do You Calculate Gear Train Efficiency Based on Load Torque?
  • · Replies 13 ·
Replies
13
Views
8K
Torque calculations: Rotating vertical shaft
  • · Replies 5 ·
Replies
5
Views
5K
Torque created around offset wheel
  • · Replies 8 ·
Replies
8
Views
3K
Engines: Torque Lost To Friction
  • · Replies 9 ·
Replies
9
Views
12K