Re: The mechanics of an rolling up double cone.

Before I go into the equations, we must realize that the motion of the 
center of gravity is the only thing that can determine which direction 
the roller moves. This is because the force initiating its motion is 
gravity itself. The roller slides to the point in the track where its 
potential energy is the lowest, and that is where its center of gravity 
is lowest in height. If the track were open ended the cone could even 
fall out of it. 

Having understood that, we come to answer your last two questions: 
Rotation is not necessary for it to move on any given direction. if 
friction was low enough it would slide in the direction of lowest 
potential energy as well.

We also understand it rolls "up the slope" because the track and roller 
are designed so that motion in the direction that the vertical slope 
increases causes the center of gravity of the roller to move down. So it 
is in fact rolling *down*.

Also, notice that even though prescence of rotation does not determine 
the direction of motion, it does determine the rate of acceleration 
because not only is the object accelerating down, it's spin is also 
accelerating. This is nothing that a drop of "Magic Lube" and some 
wishfull assumptions cannot fix. So again, take away friction, but keep 
contact, and the object cannot rotate, it simply slides in the same 
direction.

Now on to the equations of motion. For this, I have taken away friction 
and rotation to simplify our work, we just want to know what direction it 
rolls anyway, and we know friction and rotation do not make any 
difference.

So, let's define our variables:

Cone dimensions (consider only one side of the roller, assume its all 
symmetrical)
Asume cone base radius R
Cone height (distance from the tip to the base) equal to H

Track dimensions (consider only one side of the tracks, assume motion is 
constrained appropriately, as if rolling on both tracks)
Slope in the vertical plane: MV
Slope in the horizontal plane (the gradual splaying of the tracks) MH


Slope in the vertical plane is determined by dividing overall upward 
travel by the overall horizontal travel.
Slope in the horizontal plane is determined by dividing overall travel in 
the direction away from the base of the cone and towards the tip, by the 
overall horizontal travel.


What we aim to find is an equation of the height (y) versus horizontal 
position (x). Let us define horizontal position as increasing in the 
direction that the tracks slope "up" and splay appart. With this facts, 
please draw yourself a diagram to follow the equations from now on. My 
initial guess is that this will be a linear equation. The derivative of 
this equation should be a constant and should tell us the direction of 
motion by its sign (once we plug in real world values for the dimensions) 
Because the roller will move in the direction that Y decreases (as the 
potential energy decreases) So if that derivative turns out positive, we 
know the roller will move in the direction that X becomes more negative, 
so that the height decreases.


The first vertical contributor is the vertical sloping of the tracks. The 
vertical motion due to vertical sloping of the tracks is: x (mv) If this 
was the only contributor, we would know right away that a positive slope 
(if mv is positive) would cause Y to increase along with X, and would 
mean the object rolls in the direction X becomes more negative... this 
would be the case for a cylinder rolling down... but there is more.


The second vertical motion contributor is the cone's shape constrained by 
the track, as it slides outward towards the cone tip.

For every unit of forward motion, the point of contact on the cone slides 
to the outside, where the radius is smaller, and in turn the cone moves 
down. Because of the conical shape, this is a linear motion. (the radius 
of a cone decreases lienarly with distance from the base)

vertical motion due to horizontal sloping of the tracks:  X (mh) (-R/H) 
where mh is the outward sloping of the tracks, and R/H is the base Radius 
of the cone divided by its height (tip to base). I plugged in the 
negative sign because I already know as x increases with a positive mh 
the cone will move down with respect to the point of contact with the 
track.

So, the overall equation of motion for its center of gravity is as 
follows: y = x[mv + mh(-R/H)]
Its derivative is:
y' = [mv + mh(-R/H)]

And right away we can see that to make this cone slide up or down we may 
choose the values for MV and MH(-R/H) so that one is larger than the 
other.


If we plug in some real world values we could have something like this:

y' = (0.1 + 0.1(-3/2)   y' is a negative number, therefore object rolls 
in the direction that tracks appear to slope up.

In this case, both slopes (the upward and the outward) are equal to a 10% 
grade, and the cone has a base that is 3 units of radius and a tip to 
base distance of 2 units.


Have fun!

-Aurelio