Mathematical formula needed - caster over curb

[CPguy]

Dear all,

I am looking for a mathematical formula for the energy needed to push a caster over a curb.

My reasoning and my assumptions so far:

1.) Standard rear drive wheelchair approaching the curb in a straight line (both casters touch the curb at the same time)
2.) flat surface and a 90° curb
3.) The tire traction is sufficient to convert all motor torgue into movement
4.) Max controller limit = max torgue
5.) The curb is smaller than the radius of the caster wheel (curb hight < radius of the caster). This also means that if curb hight > radius of the caster it becomes impossible to surmount the curb. In the special case of curb hight = radius of the caster the energy requirement should infinite thus for all practical purposes it becomes impossible to surmount the curb.
6.) No friction or other losses

I started out to apply "Inelastic collision" formulas because I frequently hit curbs at speed (as a kind of run up) but to no avail.

Now I am starting to think that as at the time of impact with the curb the speed can be close to zero, it may be better to think of the movement as a straight lift. Meaning that I just have to calculate the energy needed to lift the mass of the casters including all the mass resting on the casters by the hight of the curb. Basicly asking: "How much energy is needed to lift a mass of 50kg by 5 cm?"

Am I on the right track?

[Burgerman]

I dont think so.

On a rear drive chair 2 things that are at least half of your calculation and maybe more are being ignored.

The CG position, so the actual mass on the casters changes as its higher up on a chair that you think.
So scenario 1... You slowly hit the curb, your CG position then forces the front of the chair down. How much? Well hit it at a few mph and the rear of the chair lifts off the deck. Now you have a 120kg human and a 170kg chair with all the mass on the front wheels plus the inertial mass as it is rotating forwards... So the higher the curb the greater the bump in an ever increasing exponential way.

Stop ALONGSIDE and touching the curb, and the opposite happens. As you push the stick forwards the motor torque tries to rotate the chair like a wheelie. And literally lifts the casters up the curb. Now the more REARWARD the CG is the more this happens. Since the forward acceleration is low (initially) the height of the CG isnt important.
In my chairs - rolling - most of the mass already on the drive wheels. So very little added torque is needed to lift the front. Low enough that I can wheelie and lift the front as I approach a curb and so it doesent need to "climb" it at all. The caster then serves no part in the raising the caster up over the curb. So zero push required.

Then theres the tyre size and pressure. Because my chairs are pretty rear biased and have 10 inch outer diameter tyres on the casters they are happy at a very low pressure. That has the effect of behaving as if the curb is both lower as the tyre deforms, and it removes the pointy bit (the corner) as this sinks into the tyres at 10psi... Effectively the tyre has the same affect as a ramp.



Very old BM1 chair, zero suspension, high tyre pressures, rear biased 10 inch tyres on front. 70mm curb.
https://www.wheelchairdriver.com/gopro/curb.mp4 Curbs are effortless with that. But soft tyres are way way better as below if I can find it...

OK tyres at 7 to 10 psi. Possible because they are large ross section. And rear biased chair.
Also very old BM2...
https://www.wheelchairdriver.com/doomed/soft.mp4
AT 1.40!
Curbs are effortless with that. But soft tyres are way way better.

What I am sayng is that your formula isnt going to tell us very much because these things all make a massive difference that it is ignoring.
But if the wheel diameter double your curb height the force needed is infinite in your simplistic equasion. In the wheel diameter is infinite the force needed is zero... So it should be a curve that starts very (vertical) and gradually becomes loss force needed as the curb gets smaller in a exponentially decreasing way.

[CPguy]

@burgerman: Valid points indeed. Do you think an approximation is possible? To have some practical value as a guideline.

[Burgerman]

As a purely interesting concept or an excersize in maths yes. As a practical or useful tool probably almost no help. All the other forces that it ignores will be several magnitudes greater than its predictive power.

The mid drive chairs that have the motor mounted on the front arms for e.g physically lift the arm as you apply torque. So a 6 inch wheel lifts itself up a 3 inch curb. Where a formula like you will make will say thats impossible. A rear drive chair IS that front arm... Its just that it adds many more parameters as you and the chair add mass, enertia, rotational enertia too.

[scottnj@gmail.com]

It is possible to calculate, but it can get very complex.

It's been about 25 years since I took my mechanical engineering classes, so I'm a bit rusty.

I would start with a static analysis of each stage of going over a curb, assume no motion at every point. You will need the CG in relation to the drive wheels and casters. The drive wheels will create a force in the X direction and also a torque that will help lift the front of the chair. The forces on the casters will have an X and Y component. When the drive wheels go up the curb, they will have torque and X and Y forces.

Then you can integrate the torque, or if using a computer you can iterate and sum over the full range of positions

Then if you want to get into dynamics you will need to estimate the moment of inertia and tire deformation.

[Burgerman]

And you will need to know and enter into this equasion many parameters. A list of them for every chair/curb. But yes in theory. But in reality not exactly practical.

Even tyre deformation alone depends on many different (non linear) parameters. Not just air pressure and damping, diameter etc.
And worse, these things will then feed back and change the primary calculations. Such as the rotational enertia forcing a caster downwards as the curb makes the chair decelerate. Which then causes more tyre deformation and rinse and repeat. I suspect any calculation is going to rapidly get super complicated.

[scottnj@gmail.com]

https://x-engineer.org/curb-climb-torque/

[Burgerman]

That however claims to ignore 5 parameters. And I can think of another 5. In the case of a chair which is short coupled, the length of the vehicle, the CG height and its pitch enertia, and the mass height above the curb, and a bunch of others all have a huge affect and are pretty much ignored here. So while it will give you a number, I am not sure how much use it is.

Driven (pushed) wheels

In the case of a driven wheel, there are two forces acting on the wheel: the weight G [N] and the push force F [N]. Both forces are acting in the centre of the wheel.

So it ignores all the rest!

[scottnj@gmail.com]

https://x-engineer.org/curb-climb-torque/

Whoever did this analysis made some assumptions.
Static, not dynamic (chair will have momentum and inertia), but static will give the worst case scenario for motor torque.
They only looked at the initial point of contact when the wheel is still touching the ground, this is the point with the highest loads.
They looked at each wheel individually and not the system as a whole, the force and torque from the drive wheels will change some of the forces on the caster.
Since this is a static analysis, tire deformation, suspension, momentum, inertia, etc was not looked at.

I agree, probably not worth the effort.

[Burgerman]

This reminds me of a day talking in the local model plane shop.

One spotty youth had just finished an engineering degree. And was trying to tell me what he learned. One of those things was the rate of acceleration due to gravity, the 9.8 meters per second per second thing. And his studies told him and his professor told him that whatever he dropped regardless of mass falls at the same speed. He was so sure of it that I took a £5 note from him.

I picked up twp identical small carboard film boxes. I added a heavy bolt to one.
He swore based on all his knowledge (and zero experience of the world) that when I released them they would hit the deck together. He was very smug.

His face was a picture when the one with the bolt slammed into the floor a full second faster. So he repeated it himself...

This is the problem with education... He was correct. IN A VACCUUM.
I just asked him how he though a parachute worked and took his money.

[Burgerman]

Whoever did this analysis made some assumptions.
Static, not dynamic (chair will have momentum and inertia), but static will give the worst case scenario for motor torque.

Nope! It wont.
Depending on how high the CG is, then non static (as in moving towards the curb) however slowly will cause the chair to initially decelerate as its tyre meets the resistance, That means that the chairs mass is added to the front. If the CG was 10 feet high it would make it easier to visulualise. But it will lift the rear, and push down on the front actively opposing any torque. Pushing the casters downwards. That prevents climbing, and in fact is a positive feedback loop. The faster you were moving the harder it will try and trip the chair up.

If you ever tried to push a manual chair over even tiny obstructions you find that it jams up! A small stone. Stops dead for no obvious reason. Thats why.

So yes, and they dont just ignore that... But a great bunch of things. All that intereact in horribly exponential ways.

[scottnj@gmail.com]

Good point on the height of the CG and the dynamics.
Like you said, too many variables.

For the fun of it I tried to see what AI would come up with. Basically TLDR for me, so I didn't review any of it for accuracy, but I'm including it as a pdf here if anyone else is interested.

It also generated a webpage.
https://codepen.io/scottnj/pen/ByNaKoo

[Burgerman]

All Ai appears to do is trawl the stuff it finds on the web, shuffle it all, mix it together and make a new page full of nonsense that sounds ike it knows what its talking about. Soon - next few years - the amount of content (by which I mean ever more trawled layers of official sounding "answers") will apear. Why? Because no human effort is needed. It will endlessely trawl other bots results... Soon there will be thousands of times more "information" or web pages than there is now. All of it useless but sounding great. Why? Because ads... The more automatically created content (filler) s generated, the more £££ from ads... Every marketer, and I used to be one, will download software that runs as fast as his computer can go creating more pages...

The internet will have 1000x more garbage than today as if it wasnt bad enough already. That includes video, pdf, wiki, and all the rest.

[CPguy]

Thank you to all for the insights! I never expected this to get this complicated, lol

[Burgerman]

Well it is what it is. You can ignore all the other effects. But then your results are meaningless. So theres that!

[ex-Gooserider]

AI (Artificial IDIOTs) also hallucinate... A friend recently sent me this about 'Fictional fiction" created by an AI...

I really do dislike AI, at least for replacing academia, though it is useful for other stuff. Here's a headline I love: "Fictional fiction: a newspaper's summer book list recommends nonexistent books. Blame AI."

The recommended reading list contained some works of fiction. It also contained some works that were, in fact, actually fictional.

The content distributor King Features says it has fired a writer who used artificial intelligence to produce a story on summer reading suggestions that contained books that didn't exist.

The list appeared in "Heat Index: Your Guide to the Best of Summer," a special section distributed in Sunday's Chicago Sun-Times and The Philadelphia Inquirer last week.

More than half of the books listed were fake, according to the piece's author, Marco Buscaglia, who admitted to using AI for help in his research but didn't double-check what it produced. "A really stupid error on my part," Buscaglia wrote on his Facebook page.

Among the summer reading suggestions was "The Last Algorithm" by Andy Weir, described as "a science-driven thriller following a programmer who discovers an AI system has developed consciousness" and been secretly influencing world events. "Nightshade Market," by Min Jin Lee, was said to be a "riveting tale set in Seoul's underground economy."

Both authors are real, but the books aren't. "I have not written and will not be writing a novel called 'Nightshade Market,'" Lee posted on X.

I think " "The Last Algorithm" by Andy Weir, described as "a science-driven thriller following a programmer who discovers an AI system has developed consciousness" and been secretly influencing world events.:" is especially ironically appropriate...

ex-Gooserider