# Calculating gear ratios for a drivetrain

JBA
Gold Member
The above sounds in the right range but for travel over irregular terrain the average speed for most casual hikers is about 2+ mph (1 m/sec), admittedly this varies considerably with age and physical conditioning, which includes flat some sustained stretches of 3 mph and a large number of varied lower and higher speeds over varying terrain.

At some point, speed and time become totally irrelevant relative to simply conquering the impeding obstacle. Comparing travel over irregular terrain to a travel on a level road is like trying to compare a trip through a river rapids to a canoe trip across a smooth lake.

jack action
Gold Member
Does the surface which it is traveling on not have an effect on this? With it being rough terrain?
If we assume the 800 N weight stated by @sandy stone and a rolling resistance of 1% (relating to your 0.008 coefficient), that gives you a resistance of 8 N. That leaves 45 N (= 53 - 8) of "available" force. This corresponds to a 5.6% slope (= 45 / 800) or 3°, which sounds reasonable to me. Of course, the user can use more force to climb a steeper hill, he just won't be able to do it all day that's all.

If you want to consider rough terrain, well it is just a matter of considering a higher rolling resistance coefficient. With a 53 N force, that corresponds to a coefficient of 0.066 (= 53 / 800). That correspond to a stage coach (19th century) on dirt road or soft snow on road for worst case. Of course, there is no room for a slope in this case without increasing user's input effort.

If you want to consider that last case instead as your typical use, you have no choice but to reduce your designed speed. If you say the designed speed will be, for example, 0.75 m/s, then the available force will be 100 N (= 75 W / 0.75 m/s) instead of 53 N. Then you get 47 N (= 100 - 53) of "available" force which brings you back to your original leeway (but at a lower speed).

Either way, you have to find the middle ground and define your usage correctly. You won't be able to create force without reducing speed and vice versa, because you are limited by your power input. If your speed range is too wide, then you will have no choice but to use a transmission with more than one gear ratio available, like with cars or most bicycles.

JBA
JBA
Gold Member
As discussed above, now that you have a basic knowledge of the mechanical factors, you need to focus a bit more on refining the target users and region(s) for your unit and the actual terrain conditions they will encounter (ie the Serengeti plain or the Tibetan Mountains). You have already been focusing upon the key issue of the operator force, which is paramount because the most important question is "can this person get from point A to point B with this unit, regardless of the time involved?". The speed at which they can achieve this and the time required are secondary to that primary goal.

We, in our advanced societies have become obsessed with the speed at which we can accomplish something, where as in more remote regions and societies just being able to accomplish a task or being able to do that with a bit less effort is the most important factor.

To place this in the starkest context: "If you cannot walk, then any level of improved mobility is better than having to drag yourself across the ground or be carried everywhere."

I have worked in a jungle village region of Nigeria, West Africa a few decades ago; and I have seen this reality up close and personal, as some would put it.

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Sounds like a great project and you're getting some great help. After reading the thread I have some questions if you don't mind.

First I'm unsure of the lever design as described. I may have missed this but will power be delivered on both the forward and back strokes of the lever or only in one direction alternating between levers? I ask because I have no idea beyond a guess of what the better stroke is, push or pull? The pull stroke intuitively seems the more powerful stroke but the push stroke has the leverage of the chair.

"Pulley" is used in the thread and "gear" I think once. Does your design call for a pulley and belt or gear and chain? Is there any information on what's best for rough terrain?

Considering you are using a ratcheting freewheeling, I assume, hub why not use a hub with a bicycle derailleur? There are inexpensive, heavy duty mountain bike assemblies available that wouldn't add much to production complexity and would add more flexibility to the chair and possibly solve some of your gear concerns.

Finally, I've seen on racing wheelchairs that an extreme negative camber is used on the drive wheels. I assume it aids in tracking? Is there any advantage to a camber spec design for rough terrain?

If you want to consider that last case instead as your typical use, you have no choice but to reduce your designed speed. If you say the designed speed will be, for example, 0.75 m/s, then the available force will be 100 N (= 75 W / 0.75 m/s) instead of 53 N. Then you get 47 N (= 100 - 53) of "available" force which brings you back to your original leeway (but at a lower speed).

I don't understand how this relates back to the 0.066 rolling coefficient which you calculated from the design output force / weight of user and chair (53/800)?

I do want the design to be solely based on use over rough terrain (its what my project title indicates), therefore I want to consider the case of using the 0.066 rolling coefficient.

you need to focus a bit more on refining the target users and region(s) for your unit and the actual terrain conditions they will encounter (ie the Serengeti plain or the Tibetan Mountain

The terrain condition I intend is in rural Indonesia, therefore I am looking at dirt roads/paths with varying sized ruts and peaks.

We, in our advanced societies have become obsessed with the speed at which we can accomplish something, where as in more remote regions and societies just being able to accomplish a task or being able to do that with a bit less effort is the most important factor.

Speed is not my primary concern, as long as the system is able to travel around walking speed or perhaps a little higher I am happy. Its more just getting the gear ratio correct for use over rough terrain.

First I'm unsure of the lever design as described. I may have missed this but will power be delivered on both the forward and back strokes of the lever or only in one direction alternating between levers? I ask because I have no idea beyond a guess of what the better stroke is, push or pull? The pull stroke intuitively seems the more powerful stroke but the push stroke has the leverage of the chair.

"Pulley" is used in the thread and "gear" I think once. Does your design call for a pulley and belt or gear and chain? Is there any information on what's best for rough terrain?

Considering you are using a ratcheting freewheeling, I assume, hub why not use a hub with a bicycle derailleur? There are inexpensive, heavy duty mountain bike assemblies available that wouldn't add much to production complexity and would add more flexibility to the chair and possibly solve some of your gear concerns.

The wheels will only be driven when the levers are pushed forwards, then the freewheel mechanism allowing the levers to be pulled backwards so they can be pushed forward again.

The system will also use gears and chain, ideally taking straight from a standard second hand bike.

Unfortunately due to time constraints I now have on this project, it is better suited for to just stick to one gear ratio instead of changing the design to have multiple gears, thanks for the idea though.

For the type of application you describe I would just recommend that all the wheels (not just the back) have as large a diameter as possible.

For the type of application you describe I would just recommend that all the wheels (not just the back) have as large a diameter as possible.

Are you referring to the size of the gears or the actual wheels?

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It seems your gear ratio question may irrelevant. The optimum ratio has more to do with the end user's preferences and not a formula. If you have the time consider temporarily adapting your lever design to a multi-speed bicycle, grab a chair, head someplace where there are a lot of people and test. Find out what the preferred ratios are and then decide based on your knowledge of Indonesia. Then again, sometimes you just have to make a decision based on your experience. Pursuing a formula to justify or rationalize a gear ratio choice in your situation is to me what would require justification. Please explain what you expect to achieve by calculating gear ratios when it's a fairly straight ratio? With a chain it's just a matter of counting the gear teeth isn't it? It's why I asked about sprocket or pulley. Pulleys would be just slightly more involved for ratios.

I was talking about the actual wheels. Much easier to negotiate uneven terrain with large-diameter wheels.

I was talking about the actual wheels. Much easier to negotiate uneven terrain with large-diameter wheels.

The front two wheels are bicycle wheels of about 660mm diameter and then a rear castor wheel which is about half the size which allows the wheelchair to steer.

It seems your gear ratio question may irrelevant. The optimum ratio has more to do with the end user's preferences and not a formula. If you have the time consider temporarily adapting your lever design to a multi-speed bicycle, grab a chair, head someplace where there are a lot of people and test. Find out what the preferred ratios are and then decide based on your knowledge of Indonesia. Then again, sometimes you just have to make a decision based on your experience. Pursuing a formula to justify or rationalize a gear ratio choice in your situation is to me what would require justification. Please explain what you expect to achieve by calculating gear ratios when it's a fairly straight ratio? With a chain it's just a matter of counting the gear teeth isn't it? It's why I asked about sprocket or pulley. Pulleys would be just slightly more involved for ratios.

I am just trying to work out what the size of each gear should be for a use over mainly rough terrain. Is this possible? Or will the gears just be the same size with the power provided to the wheels being varied by where the user pushes the lever?

jack action
Gold Member
Or will the gears just be the same size with the power provided to the wheels being varied by where the user pushes the lever?
By moving his hand on the lever, chances are that the user will still provide the same power. But the torque and rpm output will change.

If the user applies the same force at the same velocity, then by moving his hand down the lever, he's only changing the effective length of the lever. Remember that Torque = Force X Radius and Angular Velocity = Velocity / Radius. If you change the radius, when one goes up, the other one goes down. That is why the power is always constant throughout a gearbox, no matter the ratio:

Power = Force X Velocity = Torque X Angular Velocity.

By moving his hand on the lever, chances are that the user will still provide the same power. But the torque and rpm output will change.

If the user applies the same force at the same velocity, then by moving his hand down the lever, he's only changing the effective length of the lever. Remember that Torque = Force X Radius and Angular Velocity = Velocity / Radius. If you change the radius, when one goes up, the other one goes down. That is why the power is always constant throughout a gearbox, no matter the ratio:

Power = Force X Velocity = Torque X Angular Velocity.

Is there even a point in me having two gears then? Besides from locating the lever arm away from wheel axle?

It seems that I am not getting any mechanical advantage at all from having two gears.

jack action
Gold Member
Is there even a point in me having two gears then? Besides from locating the lever arm away from wheel axle?

It seems that I am not getting any mechanical advantage at all from having two gears.
Of course you're having mechanical advantage with having 2 gears. Do you need it? That is another question. At one point, you will have to define your objectives in terms of Power In vs Torque/Velocity Out and do some calculations to get the Radius / Gear Ratio combination you need.

Although it is true that moving the arm along the lever will change the torque output, it will also affect the Force / Velocity input of the user's hand. I would be surprised that having an arm bent at 90° vs having a straight arm will produced the same capabilities regarding force and velocity. It's still a mechanical advantage thing, but on a biological point of view.

At one point, you will have to define your objectives in terms of Power In vs Torque/Velocity Out and do some calculations to get the Radius / Gear Ratio combination you need.

This is what the initial post was about, although now I'm unsure exactly where I've been helped with this due to the multiple different ideas from different users (I understand that's the point of a forum).

Are you able to clarify how I go about calculating this? Not taking into account any mechanical advantages from a biological point of view.

jack action
Gold Member
The torque at the lever (##T_L##) is ##T_L = FL##, where ##F## is the hand force (your 40 N) and ##L## is the lever length (your 580 mm or 0.58 m).

That torque is the same as the torque of the driver wheel ##T_r = T_L##.

The torque of the driven wheel ##T_n## is a ratio of the gear radius: ##T_n = T_r \frac{R_n}{R_r}##. Because it is a ratio, you could use the diameter or the number of teeth instead of the radius (for both of them).

The torque of the driven wheel is the same as the torque of the wheel ##T_w = T_n##.

So, putting it all together: ##T_w = \frac{R_n}{R_r}FL##. The force at the tire-road contact patch will be ##F_w = \frac{T_w}{R_w}## where ##R_w## is the radius of the wheel.

For the angular velocity (AV), it will be similar:

The AV at the lever (##\omega_L##) is ##\omega_L = \frac{v}{L}##, where ##v## is the hand velocity (should be equal to the power of the user divided by the lever force ##F##) and ##L## is the lever length (your 580 mm or 0.58 m).

That AV is the same as the AV of the driver wheel ##\omega_r = \omega_L##.

The AV of the driven wheel ##\omega_n## is a ratio of the gear radius: ##\omega_n = \omega_r \frac{R_r}{R_n}##. Because it is a ratio, you could use the diameter or the number of teeth instead of the radius (for both of them).

The AV of the driven wheel is the same as the AV of the wheel ##\omega_w = \omega_n##.

So, putting it all together: ##\omega_w = \frac{R_r}{R_n}\frac{v}{L}##. The velocity at the tire-road contact patch will be ##v_w = \omega_w R_w## where ##R_w## is the radius of the wheel.

You will notice that ##Fv = F_w v_w##, which means power in = power out.

You should used SI units with those equations (N, m, N.m, m/s, rad/s).

Thanks a lot for your help, I have a few questions regarding this method.

1. Am I correct in thinking I determine my own gear ratios to begin with and carry out the equations, deciding if my end result is correct and changing the ratio accordingly if required?

2. For calculating hand velocity, how do I know the power of the user to divide over the lever force (40N)?

3. How can I justify what torque and velocity I am trying to achieve with these equations in order to know when I have the ratio correct?

Besides from them 3 questions, the rest of it makes sense to me.

jack action
Gold Member
I think you need to re-read this thread and start playing with numbers in the equations, such that the info sinks in. Some of the questions you asked are already answered in posts #21 and #24.

I think you need to re-read this thread and start playing with numbers in the equations, such that the info sinks in. Some of the questions you asked are already answered in posts #21 and #24.

I've had a re-read through of the thread and answered most of my questions, thank you.

I initially did the calculations using a gear ratio of 0.2m diameter for the driven and 0.1m diameter for the driver. I worked it out as getting an output of Fv=301.7N with the method you have shown. Looking at previous posts I only need this output to be 53N, therefore I do not need as much of a difference between my gears and I should recalculate with this in mind... Am I thinking about this correctly?

jack action
Gold Member
Without more detailed info about the power input and the road conditions, that seems to be the most reasonable guess (53 N @ 1.4 m/s), at least as a first attempt for a prototype to be tested. But that is just my opinion and I have no expertise in wheel chair design.

Surely 300 N @ 0.25 m/s seems to be way to slow. It will be like trying to pedal in first gear with a 21-speed bicycle: The legs go very fast but the bicycle doesn't.

Without more detailed info about the power input and the road conditions

What kind of information would be required to take rough terrain in account?

Surely 300 N @ 0.25 m/s seems to be way to slow

Can you clarify why this would be at 0.25 m/s? I thought I was calculating the output power, and my initial calculation had found the output to be too high at 300N?

jack action
Gold Member
I'm sure 0.008 for rolling resistance is too small. How much higher should it be, it is difficult to determine for rough terrain. I doubt it would be below 0.02-0.03. Even then, this represents an average and since the power input will be very inconstant, these variations may have a greater effect. Especially if there is lots of pebbles and other irregularities on the road.

300 N times 0.25 m/s gives 75 W, i.e. the power that a human can easily produce (again, read the information already given and let it sink in). And 0.25 m/s is 0.9 km/h which is very slow. This means that, to go at a more normal pace, the user will have to push and pull the lever very fast (even if it will be extremely easy). With the little information I'm given here (Hand force = 40 N and hand power = 75 W) and the extrapolation I've done (Hand speed = 75 W / 40 N = 1.875 m/s), it is the best I can do for predictions.

Wilson123
I took the rolling resistance from the below table taken from this link http://www.engineeringtoolbox.com/rolling-friction-resistance-d_1303.html

Perhaps a higher one is more suited for rough terrain?

300 N times 0.25 m/s gives 75 W, i.e. the power that a human can easily produce (again, read the information already given and let it sink in). And 0.25 m/s is 0.9 km/h which is very slow. This means that, to go at a more normal pace, the user will have to push and pull the lever very fast (even if it will be extremely easy). With the little information I'm given here (Hand force = 40 N and hand power = 75 W) and the extrapolation I've done (Hand speed = 75 W / 40 N = 1.875 m/s), it is the best I can do for predictions.

Therefore, 75/3000=0.25

I want my answer to be 1.4 (walking speed), so I need to decrease the power out to around 50N? (75/50=1.5)