Variable Radius Pulley for Steering: looking for a nice resource or someone who likes challenges

[kahlju]

Hello,

I am working on a research project that for packaging reasons needs to steer a wheel on one side (opposite an in wheel motor). It uses "handlebars" with a gear train with a pulley output. It is actuated from the pulley input through a cable (standard cable like bike brake cable or seat back locks). While turning the steering, the output at the wheel pulls in one side while the other extends. This in turn makes the wheel turn.

Looking at the attached will help. The handlebar angle in will not exactly be the angle out at the wheel, but for solving this we could assume you want them to be the same if it helps.

The problem lies with the extension of the cables, which is why we are trying to do a variable radius pulley. If you look at the attached you will hopefully be able to see that as the wheel angle changes the extension of the cable changes differently for each side (simple trig to find the extension). If you had a constant radius pulley on the other side, it would cause a slack in the cable, which would cause a wobble. Depending on the angle range, it may also prevent it from turning. Because of this, I'm trying to map the radius based on angle change of the shaft attached to this gear. The math is giving us a headache. So if you have your equations setup for the extensions, you know a change in length that needs to be accounted for at the pulley. I tried to do this in polar coordinates to get a very continuous function, but it gets screwy when talking about the shape of the pulley's cammed profile. The shape of the cammed/variable pulley is a function involving the angle up to 360 and the radius at that angle. If you make this pulley to account for the extensions, you need to add in arc length changes, these will be your boundaries. The arc length is a function of the steering angle. Solving this ends up being an integral with bounds involving variables. If you add in arc length changes you actually need to adjust your bounds to the "new tangent point" to do the arc length bounds math. This seems to be multi-varible boundaries and it throws me for a loop.

I have not attempted this yet, but it seems very problematic. That is why before attempting to solve, I am writing on here. Does anyone have either a simpler way to do this, a resource, or the ability to solve this math? If I have someone interested in the latter I can rewrite my scribblings and upload it at that time.

Thank you for any help you can provide.

Thank you,
~Justin Kahl

 

[MikeHalloran]

Bobsledders steer with cables, but those guys are crazy.

Why do you feel obligated to do such an important job in a way that no one else does it?

Who is at risk when it fails?



Mike Halloran
Pembroke Pines, FL, USA

 

[kahlju]

It's not like a bobsled. It has to be this way for a packaging reasons. The wheel has an electric motor drive inside of it and it folds out of a compartment. Since, I am in our R&D for a project that is planned for 5-10 years out, I cannot explain much more than that about the purpose, only the mechanism.

 

[kahlju]

Also for risk, the speeds reached are only 20 mph. The risk is to the user but the crash at this speed is pretty survivable.

 

[kahlju]

Sorry, I misspoke, the brake is on the other side of the wheel, not the motor.

 

[gruntguru]

Gut feel says it can't be done without the cable going slack. Can you build the variable ratio into some other part of the mechanism - the cable runs at the top of your image or the drive to your cable drum?

je suis charlie

 

[Lnewqban]

I have not attempted this yet, but it seems very problematic. That is why before attempting to solve, I am writing on here. Does anyone have either a simpler way to do this, a resource, or the ability to solve this math? If I have someone interested in the latter I can rewrite my scribblings and upload it at that time.

Machining such a pulley is not an easy task.
Reseach Ackerman geometry for steering applications.
Cable pull-pull systems were used in the early times of aviation for remotely controlling elevators, rudder and ailerons.

Rotational to linear movement creates an asymmetry.
That is why pistons moved by a crankshaft have a max and a min translation speed, while the crankshaft keeps constant rpm's.

Using a pushrod is not problematic because the other side is free to move as it pleases.
As long as the pushrod is perpendicular to the lever from the point of rotation, the horn will deflect the same in both directions, without any differential.

Using two cables instead of one pushrod, forces us to compensate for that asymmetrical movement of one side respect to the other.
How do we do it? Introducing another asymmetry, by making the lever from the point of rotation non-perpendicular respect to the cables.

Crossed and parallel cables require different geometry, as the article of the link below shows.
The greatest the distance between pulley and arm, the less evident it is.
The practical difference in slack and movements may be small and we may live with it, but it is there.

Why the top and bottom sections of a belt moving two pulleys, or a chain moving two sprockets never develop any slack, regardless of the direction of the rotation?
Because the belt or chain is always connected to the wheels at the points at which they are perpendicular to the lever from the point of rotation.

http://www.qmfc.org/school/ackerman.htm

"Engineering is achieving function while avoiding failure." - Henry Petroski