# avoid collision of two overlapping spinning wheels

#### laurencewithau

##### New member
Hi. I once taught maths but this is beyond me. I make wind spinners from bicycle wheels and it occurred to me that it might be possible for them to interact without colliding. But I've failed every time.
To be clear: the wheels are vertical axis and in the same plane and spin on their spindles.But also they rotate around the tubes to which they are attached. I'll try to upload a video of them.
If they are next to each other on their separate tubes and if the distance between them is greater than their combined radiuses, in other words greater than their diameters, given that they are the same size, then there is no problem; but at the same time they are not interacting.
Suppose that I now run each tube through a wooden gear, the two gears being the same size with the same number of teeth. This time they are interacting and they turn at the same rate when the gears engage. This means that their position relative to each other can be controlled; for instance, I could engage the gears in such a way that the wheels are vertically parallel as the initial state, so that they are in sync and return to the same parallel position after one revolution.
Or I could start them off at right angles, one facing north or south, say, the other east or west. Given this initial angular control, I thought that there would be some way of arranging things, such that the wheel centres, in other words the tubes, would be less than one diameter apart, so that the wheels would give the illusion of passing through each other. Was I wrong or is there some way, some degree of overlap, some initial angle, some relation of gear diameter to wheel diameter, such that the wheels will always miss each other? Or can it all be answered simply with a straight " no, it's impossible"? In the video, by the way, there are no wooden gears and the wheels have wind direction arms, which would be removed. Clearly, too, the wheels are not overlapping. Thanks in advance.

#### lev888

##### Senior Member
How would the two rims pass through each other? Without rims, if you had something like 2 windmills, I think it would work. Let's say each has 4 "blades", one spinner has them initially at 0, 90, 180, 270 degrees and the other at 45, 135, ...

#### laurencewithau

##### New member
Thanks lev888. Yes, I'll get around to windmills one day. What I meant was that the illusion could be given of one passing through the other. In fact I've just done an experiment. I placed two interlocking wooden gears the same size on a bench with pins through their centres to rotate them. Then I attached "hands", on the analogy of a clock face, except that the hands were longer than the radius of the gear face. Then I spun the gears and found that the hands or arms missed each other if one was initially at midday or midnight and the other perpendicular to it. In that initial state, the hands could be as long as one diameter and still miss each other. Cheers, Laurence

#### lev888

##### Senior Member
Sorry, can't visualize it. A photo or a drawing would help.

#### Subhotosh Khan

##### Super Moderator
Staff member
Thanks lev888. Yes, I'll get around to windmills one day. What I meant was that the illusion could be given of one passing through the other. In fact I've just done an experiment. I placed two interlocking wooden gears the same size on a bench with pins through their centres to rotate them. Then I attached "hands", on the analogy of a clock face, except that the hands were longer than the radius of the gear face. Then I spun the gears and found that the hands or arms missed each other if one was initially at midday or midnight and the other perpendicular to it. In that initial state, the hands could be as long as one diameter and still miss each other. Cheers, Laurence
If the centers of the gears are at relative "rest" - then those arms will have to move at same angular speed to avoid collision.

#### laurencewithau

##### New member
Thanks lev888. Yes, I'll get around to windmills one day. What I meant was that the illusion could be given of one passing through the other. In fact I've just done an experiment. I placed two interlocking wooden gears the same size on a bench with pins through their centres to rotate them. Then I attached "hands", on the analogy of a clock face, except that the hands were longer than the radius of the gear face. Then I spun the gears and found that the hands or arms missed each other if one was initially at midday or midnight and the other perpendicular to it. In that initial state, the hands could be as long as one diameter and still miss each other. Cheers, Laurence
Sorry, can't visualize it. A photo or a drawing would help.
Hi. I haven't constructed the thing yet, but I can try to describe the problem more clearly.
Imagine two full size bicycles turned upside down and the fittings removed so that the front fork rotates freely through a circle. Take the circle and circular motion to be "ideal" and place the bikes side by side. a few inches apart,
Since the wheels are 26" diameter, say, they will collide if haphazardly turned and very little movement will be possible.
Now increase the distance between them to 27". then they will never collide.
Now use your imagination, or gears, or imagined gears, such that the wheels turn at the same rate; and at the same time decrease the distance to 20", say. Before engaging the gears, if you like, you are free to position the wheels at any relative angle you please. You could point both of them forward, each in line with the body of the bike, so that the wheels are parallel to each other.
My little experiment indicates that this is the worst choice to make.
Or you could leave one at 12 o'clock and turn the other to 9 or 3 o'clock, so that the wheels are at 90 degrees.
Then my finding is that they will always miss each other, and I suspect that this will be the case if the distance is further decreased, perhaps until the gap between the bikes is less than 13" (ideally).
My question, really, is about the mathematics and how to model the problem and solve it. My guess would be that you reduce it to a problem in two dimensions, the counterpart to straight sticks instead of wheels, and that you draw two adjacent circles, each with a line through the centre that extends beyond the circle, where this models, if you like, two clocks with elongated hands.
There must be a mathematics for this, and it will answer such questions as what if one hand is less elongated than the other, or what if the hands turn at a constant rate, as with clocks, but not at the same rate.
Hope this makes it clearer. Cheers.

#### laurencewithau

##### New member
If the centers of the gears are at relative "rest" - then those arms will have to move at same angular speed to avoid collision.
Hi. Yes, and the arms will move at the same rate if the gears are the same in size and tooth number. I just wish I could work out the mathematics. Cheers

#### Dr.Peterson

##### Elite Member
I think we really need a drawing.

Are the wheels turning around 360 degrees around a vertical axis? It would seem that they can't possibly rotate all the way around if the distance between them is less than their radius; and if it is more than the radius, they may be able to turn all the way around if they are rotating in opposite directions as your gear analogy suggests. It would be impossible to do what I think you're saying if they rotated in the same direction.

#### lev888

##### Senior Member
If they are wind spinners, shouldn't they be oriented in the same direction relative to the wind? If they do not have to be parallel, then yes, they can be less than one diameter apart. And they should rotate in opposite directions. If initially they form a right angle they can be one radius apart.
If rates are not the same sooner or later you'll get a collision.
I'm assuming it would be easy to adjust the initial positions until a full revolution goes ok. If so, why do you need exact math?

#### LCKurtz

##### Full Member
For what it's worth, here's an animation looking down from the top so 2D:

The settings for this has one lagging the other by $$\displaystyle 45^\circ$$, with radius $$\displaystyle 1$$ and distance between centers $$\displaystyle 1.2$$.

#### laurencewithau

##### New member
I think we really need a drawing.

Are the wheels turning around 360 degrees around a vertical axis? It would seem that they can't possibly rotate all the way around if the distance between them is less than their radius; and if it is more than the radius, they may be able to turn all the way around if they are rotating in opposite directions as your gear analogy suggests. It would be impossible to do what I think you're saying if they rotated in the same direction.
I think we really need a drawing.

Are the wheels turning around 360 degrees around a vertical axis? It would seem that they can't possibly rotate all the way around if the distance between them is less than their radius; and if it is more than the radius, they may be able to turn all the way around if they are rotating in opposite directions as your gear analogy suggests. It would be impossible to do what I think you're saying if they rotated in the same direction.
Hi Dr Peterson. As you say, it can't work if distance between centres less than the radius. If the wheels rotate in opposite directions, as in the LCKurtz really cool animation, it can work. But are you sure that it can't work if they are going in the same direction? I don't know how to do a drawing but I'll perform the experiment again tomorrow, this time with the gears touching but not engaging, and then I'll turn them both in the same direction, lining the teeth up as a rather jerky approximation to same constant speed, and I'll video it and upload it. If the sticks attached to the gears always collide when moving in the same direction, at least I'll know, and then I'll have to work within that limitation. Thanks for alerting me. Cheers, Laurence

#### laurencewithau

##### New member
If they are wind spinners, shouldn't they be oriented in the same direction relative to the wind? If they do not have to be parallel, then yes, they can be less than one diameter apart. And they should rotate in opposite directions. If initially they form a right angle they can be one radius apart.
If rates are not the same sooner or later you'll get a collision.
I'm assuming it would be easy to adjust the initial positions until a full revolution goes ok. If so, why do you need exact math?
Hi lev888. Thanks for your reply. As I said to Dr Peterson, I'll do the experiment again tomorrow, but you both say opposite direction only.
I need the maths in order to know in advance whether something works.
Before I did the experiment I installed the two wind spinner wheels with their connected gears, and the moment the wind caught them they crashed into each other and got damaged.
I've learnt since then, but I don't want to be limited to making only same size gears.
Much better, in fact, to have a small drive gear and a larger driven gear, because then the gears will move in less of a wind.
But I need the maths in order to work out the specs by which collisions can be avoided. It would mean, of course, that the two wheels turn at different rates, so I need the maths to tell me whether this is ever possible if the wheels overlap each other's circles.
Why bother with all this, you may ask, but it keeps me off the streets and there's no end to the kinetic wind sculpture possibilities.
To prove that point, I've attached a You Tube link to the breathtaking beauty of the Anthony Howe sculptures in America.

#### LCKurtz

##### Full Member
I don't think that if they spin in the same direction that the distance between centers can be any less then 1.5 times the radius. And then only if one lags the other by $$\displaystyle 90^\circ$$. Here's what that looks like:

And even then, you don't get any overlap to speak of.

#### laurencewithau

##### New member
For what it's worth, here's an animation looking down from the top so 2D:
View attachment 14072
The settings for this has one lagging the other by $$\displaystyle 45^\circ$$, with radius $$\displaystyle 1$$ and distance between centers $$\displaystyle 1.2$$.
Hi LCKurtz. That's impressive and thanks very much. I'm trying to imagine them going in the same direction and I still can't see why not. I'll do the experiment tomorrow and upload the footage. I'll attach the Anthony Howe video here for you to marvel at. He sells his sculptures for up to half a million dollars, which seems a lot but he makes them himself, unbelievably, and he gets them to mesh by using a software program. Thanks again.

#### laurencewithau

##### New member
I don't think that if they spin in the same direction that the distance between centers can be any less then 1.5 times the radius. And then only if one lags the other by $$\displaystyle 90^\circ$$. Here's what that looks like:
View attachment 14086
And even then, you don't get any overlap to speak of.
Hi Again. I've just noticed your second animation. They do work if it's in the same direction; and yes, my first experiment came up with 90 degrees. You say 1.5 times the radius, and it sounds like trial and error; but surely there must be the maths to work all this out and save me all the time I wasted on the first two spinners when they crashed into each other. half a radius, by the way, is quite enough to achieve the effect of one spinner passing through the other. Thanks again. Cheers, Laurence

#### LCKurtz

##### Full Member
I'll save you some time. Here's what happens if you just lag by $$\displaystyle 80^\circ$$:

#### laurencewithau

##### New member
I'll save you some time. Here's what happens if you just lag by $$\displaystyle 80^\circ$$:
View attachment 14088
Thanks again. It's very kind of you to go to do this. Looks like the spinners are colliding at the top. You've saved me the job of setting up the experiment and laboriously moving the sticks one tooth at a time if they are to move in the same direction. The gears can't be allowed to engage because then they move in opposite directions. In your animation the maths, if known, would give values connecting red stick initial angle and green stick horizontal axis distance. Thanks again. Cheers, Laurence

#### LCKurtz

##### Full Member
The smallest possible distance between centers appears to be $$\displaystyle r \sqrt 2$$, at which wheels turning the same direction with a $$\displaystyle 90^\circ$$ lag just touch at the tips. That's why 1.5 looked pretty close for $$\displaystyle r=1$$.

#### laurencewithau

##### New member
The smallest possible distance between centers appears to be $$\displaystyle r \sqrt 2$$, at which wheels turning the same direction with a $$\displaystyle 90^\circ$$ lag just touch at the tips. That's why 1.5 looked pretty close for $$\displaystyle r=1$$.
Great. Thanks again. I thought I was the only one who was up that late last night, unless you are in a different time zone. Yes, that function would fit. Thanks for all your help, in particular for the really cool animations. Cheers, Laurence