I'm trying to figure out the equation of a bicycle's wheel path when the front wheel is rotated by ϴ degrees. In this problem the front wheel's rotation remains fixed with respect to the bike's frame, and the rear wheel is trailing (following) the front wheel.




Instinctively, I can tell that the front wheel will be tracing a circular path, while the rear wheel will be following a smaller circle.
How does one approach a problem like this? Of course I'm willing to do my homework and reason through the hints given by the members of this forum. It's just that it's been a very long time since I've worked on something similar. I've already done some googling and came out empty handed... maybe I didn't use the proper keywords for my search.

BTW, this is not homework, this is so I can make a few calculations for a special wheel chair I'm designing for my 16 year old daughter with cerebral palsy. Thank you all in advance.

 

My instinct tells me to draw a line normal to rear wheel at its center. Draw another similar line on the front wheel.
Where the two lines intersect is the center of rotation.

 

My instinct tells me to draw a line normal to rear wheel at its center. Draw another similar line on the front wheel.
Where the two lines intersect is the center of rotation.

Take a look at figure 2 in this link... I'm not sure I understand the author's intention but apparently the problem is more complicated than it seems.

 

Found it! BUT.... Oh, My, GOD.... this problem is far more complex than I thought at first.

Here's a math document exclusively treating the problem, and a similar one explaining the same thing in a slightly different way.

 

There's an excellent reference book on this and other biking topics called "Bicycling Science". And yes, wheel path is incredibly complicated.

You can ride a bike easily, but describing in detail HOW you ride a bike is almost beyond our abilities.

 

There's an excellent reference book on this and other biking topics called "Bicycling Science". And yes, wheel path is incredibly complicated.

You can ride a bike easily, but describing in detail HOW you ride a bike is almost beyond our abilities.

And this is considering only 2 dimensions, that is, as viewed from the top... I can't even begin to imagine what the math and physics looks like when one considers also a real bike's vertical tilt ...

 

Take a look at that book if you can, I found it quite interesting. Too much arcane information but that's the price you pay.

 

It is incredibly complicated. For instance, when you start a turn on a bicycle or motorcycle, you actually initially turn the handle bars the opposite direction -- known as "counter-steering" -- you don't even realize you do it and it is much more pronounced on a motorcycle. The counter-steering uses the conservation of angular momentum and the gyroscopic effects of the wheels to tilt the bike in the direction of the intended turn.

But depending on what you are doing, you can often ignore the complications and work with a much simpler model.

 

working a fire control solution is fun too. Roundish globe as the frame of ref. Ship is the second frame of ref. and firing platform is the third. firing platform is to remain stable while firing by subtracting all movement created by pitch, roll and yaw in the second frame of ref. from the first frame. then windage, barrel wear and temp., compass heading and barometer are compensated for. finally the earths spin is factored in and target elevation is found. Outcome is a shell from a five inch gun can find its way from a ship tossing in the ocean to a point within 20 meters of the target. Battleships did all of this by operators turning gears to input the data on an Analog computer for a point the target had not reached yet. its a wonder the shells ever hit anything if you look at all the data and calculations required.

 

The people that worked out stuff like this, given the tools they had at hand, were nothing short of geniuses. They understood applied engineering and applied math in ways that have largely been lost today -- at least on a broad scale.

 

It is incredibly complicated. For instance, when you start a turn on a bicycle or motorcycle, you actually initially turn the handle bars the opposite direction -- known as "counter-steering" -- you don't even realize you do it and it is much more pronounced on a motorcycle. The counter-steering uses the conservation of angular momentum and the gyroscopic effects of the wheels to tilt the bike in the direction of the intended turn.

But depending on what you are doing, you can often ignore the complications and work with a much simpler model.

You can probably prove me wrong on this motorcycle steering bit, but from my years of riding, it's not so much turning the bars as it is putting more weight on the side of the bar you want to turn toward. You lift the right side of the bar and put more weight on the left side of the handle bar to turn left at speed. The opposite is true to turn right.

If you were to turn a motorcycle or bicycle handle bars at the angle shown in the first post at a speed more than a walking pace you would crash.

 

Interesting mathematically, but given the wheelchair application I think MrChips' suggestion (post #2) will get you close enough for the design .

 

You can probably prove me wrong on this motorcycle steering bit, but from my years of riding, it's not so much turning the bars as it is putting more weight on the side of the bar you want to turn toward. You lift the right side of the bar and put more weight on the left side of the handle bar to turn left at speed. The opposite is true to turn right.

If you were to turn a motorcycle or bicycle handle bars at the angle shown in the first post at a speed more than a walking pace you would crash.

Next time you're on a motorcycle (or even a bicycle) do the following test. Making a conscious effort to not shift your weight or do anything else, use the palm of your hand to push one of the handlebars forward and make note of the resulting motion.

If you were to instrument the front of the bike and record the angle at which the wheel is turned, you find that everyone countersteers without even being aware of it.

 

working a fire control solution is fun too. Roundish globe as the frame of ref. Ship is the second frame of ref. and firing platform is the third. firing platform is to remain stable while firing by subtracting all movement created by pitch, roll and yaw in the second frame of ref. from the first frame. then windage, barrel wear and temp., compass heading and barometer are compensated for. finally the earths spin is factored in and target elevation is found. Outcome is a shell from a five inch gun can find its way from a ship tossing in the ocean to a point within 20 meters of the target. Battleships did all of this by operators turning gears to input the data on an Analog computer for a point the target had not reached yet. its a wonder the shells ever hit anything if you look at all the data and calculations required.

To increase the chances, the firing was done "broadside" fashion (not sure about that term applying strictly for gunnery) and eventually all towers in short but quick sequence repetition.

BTW, maybe you know that during combat condition, it was considered a must to keep every single possible light bulb on, to minimize the possibility of filaments breaking because of vibration wich was, STRONG.

A classic of all times (there is a 2nd part).

 

Particularly on the battleships (don't know about smaller ships) you had optical rangefinders that fed data into the mechanical computers and to get the most from the parallel reading you wanted those sights to be separated as much as possible which favors a broadside orientation. Lots of things favor broadside engagements for naval artillery, not the least of which is the ability to arrange ships in a line in order to bring coordinated and massed fire onto a target. This was known as the "line of battle" and the larger ships (the "capital" ships) were designed and intended to be fought this way, which is why there were referred to as "ships of the line". The term "battleship" actually derives from this ("ship of the line of battle").

 

We actually had a thread on a related topic not so long ago.

 

@cmartinez: I'm curious how such a calculation would relate to the construction of a wheelchair? Wheelchairs don't make banked turns like bicycles, and if I remember correctly, most of the ones I've seen have front wheels that can rotate 360 degrees.

 

@cmartinez: I'm curious how such a calculation would relate to the construction of a wheelchair? Wheelchairs don't make banked turns like bicycles, and if I remember correctly, most of the ones I've seen have front wheels that can rotate 360 degrees.

I'm not trying to construct a wheelchair... I'm trying to modify one. You see, my (extremely lovely) daughter was born with cerebral palsy and has some control of her left hand and arm, while her right hand is practically useless but has a little control of her right arm. She is able to maneuver the left back wheel of her wheelchair, but not the right one. So you can imagine what happens when she tries to move around a bit on her own, by handling only the left wheel... she starts going in circles, that is... but she keeps doing it anyway 'cause she finds it amusing and keeps laughing all the while (yes, she is that lovely)
Anyway... to make a long story short, my 18 year old son (who has the curious combination of being both a sports addict and a science geek) got this assignment from his physics teacher to choose a project for his final exam that would involve designing and building something useful and inexpensive demonstrating some of the concepts learned during the course.
So I suggested he found a way that would allow his sister to control her wheelchair's direction of motion, considering the natural limitations imposed by her condition... And after lots and lots of thought, the big idea finally coalesced into attaching a vertical tube to the directional axis of the small right front wheel of her wheelchair, and have that tube go through a bushing attached to one side of the armrest. Then at the tip of that tube attach a second horizontal tube ending in a small vertical grip that will work as a rotating lever. The idea is to tie her right hand to the grip using some velcro, and that way she'll be able to use what limited motion she enjoys in her right arm to control the direction of the wheel.
So... my son was asked to clarify the calculations involved in the project and... oh my God... I hadn't anticipated how hard those calculations would be... in the end what he did is he calculated the reaction forces on both the front and rear wheels (also involving friction, of course) due to the angle of rotation of the front wheel, instead of performing the far more complex calculations involved in the path that the wheelchair would follow under all possible conditions.
I hope I've answered your question.

 

I'm not trying to construct a wheelchair... I'm trying to modify one. You see, my (extremely lovely) daughter was born with cerebral palsy and has some control of her left hand and arm, while her right hand is practically useless but has a little control of her right arm. She is able to maneuver the left back wheel of her wheelchair, but not the right one. So you can imagine what happens when she tries to move around a bit on her own, by handling only the left wheel... she starts going in circles, that is... but she keeps doing it anyway 'cause she finds it amusing and keeps laughing all the while (yes, she is that lovely)
Anyway... to make a long story short, my 18 year old son (who has the curious combination of being both a sports addict and a science geek) got this assignment from his physics teacher to choose a project for his final exam that would involve designing and building something useful and inexpensive demonstrating some of the concepts learned during the course.
So I suggested he found a way that would allow his sister to control her wheelchair's direction of motion, considering the natural limitations imposed by her condition... And after lots and lots of thought, the big idea finally coalesced into attaching a vertical tube to the directional axis of the small front wheel of her wheelchair, and have that tube go through a bushing attached to one side of the armrest. Then at the tip of that tube attach a second horizontal tube ending in a small vertical grip that will work as a rotating lever. The idea is to tie her right hand to the grip using some velcro, and that way she'll be able to use what limited motion she enjoys in her right arm to control the direction of the wheel.
So... my son was asked to clarify the calculations involved in the project and... oh my God... I hadn't anticipated how hard those calculations would be... in the end what he did is he calculated the reaction forces on both the front and rear wheels (also involving friction, of course) due to the angle of rotation of the front wheel, instead of performing the far more complex calculations involved in the path that the wheelchair would follow under all possible conditions.
I hope I've answered your question.

Thanks, I understand now. Hope your project goes well and is a big help for your daughter.

 

Nice project with a lot of useful things, both educational for your son and beneficial for your daughter.

A concern that I have is that controlling the direction of one of the front wheels may not be enough to counter the turning torque of having just one of the big wheels being driven -- it may just skid the front wheel, which generally don't have decent tread on them to begin with. But worth a try!