It _might_ be slightly easier to deal with a "discrete" version of this problem, instead of your "continuous" version. The idea is that the object has a large number of "faces", and transitions from sitting on one face to sitting on another face, but retains enough momentum to continue to move. The problem is to keep the momentum from falling too low or from "snowballing" & getting too large. One can take a cue from the little walking robots -- they still take discrete steps, but if the inclined plane is a bit steeper, they make larger excursions. One needs to incorporate some sort of automatic throttling mechanism to remove energy when the object starts to move too fast. Thus, we need some sort of _dissipation_ mechanism -- a dashpot or perhaps a fan (dissipate energy into moving air). I can imagine some sort of clockwork mechanism inside such a device with faces, such that a moving weight provide just enough of an impulse to get the object to move to the next face in the sequence. This requires a clever design of the faces to make sure that the faces are large enough so that the object can stabilize itself, but not so large that it can't make the next transition. Furthermore, the angle between the faces can't be very large, or it will take too much energy to move the object onto the next face. At 02:54 PM 4/22/2010, Allan Wechsler wrote:
Mistakenly sent this to Henry alone.
---------- Forwarded message ---------- From: Allan Wechsler <acwacw@gmail.com> Date: Thu, Apr 22, 2010 at 5:53 PM Subject: Re: [math-fun] Trying to recall funny shape To: Henry Baker <hbaker1@pipeline.com>
I'm attracted to the simpler problem: no internal moving parts, constant density (though if hollows are allowed, I think we don't lose any generality there), and a level surface. Within these constraints, I wonder what is possible. I suspect that in theory you can make it do almost anything (as long as you don't mind it rolling very slowly).
Here's a problem: Can you design a roller that goes in a straight line forever? It would have a cylindrical ring for the steady state, and a nearby hillock to provide the initial extra potential energy, but designing the transition so that the ground-trace would be straight would be a big challenge.
On Thu, Apr 22, 2010 at 5:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Very cool idea!
However, you'd have to control the speed & angle very carefully to allow the shape to fall to one side or the other at the appropriate time.
You might also have to engineer non-homogeneity into the mass to give you some more degrees of freedom.
Of course, if you also allow for internal mechanisms, where weights can move around within while "rolling" without, the possibilities seem endless.
If you also give some of the internal parts the ability to start with a spin, then gyroscopic mechanisms can also be employed.
There are already simple "walking" robots with no moving parts that can maneuver themselves down an inclined plane, so these ideas would subsume those robots.
At 12:00 PM 4/22/2010, Allan Wechsler wrote:
This business of weird rolling shapes got me musing. One could design a "ball" that, released from rest in the proper orientation on a sufficiently flat surface, would roll along a predetermined path. In principle, if one were parsimonious enough with potential energy, the path could be arbitrarily ornate, though the object would roll very slowly along it.
So, in practice, how complicated a path could be programmed into, say, a machined steel roller? Could a skilled machinist create a "signature" roller that wrote the machinist's name?
On Thu, Apr 22, 2010 at 2:38 PM, <rcs@xmission.com> wrote:
[another SBG contribution trapped by the mystery mail filter -- rich] -----------
Date: Wed, 21 Apr 2010 15:46:57 -0700 Subject: Re: [math-fun] Trying to recall funny shape From: "Stephen B. Gray" <stevebg@roadrunner.com> To: math-fun@mailman.xmission.com
On 4/21/2010 12:50 PM, Dan Asimov wrote:
I recently read of an odd solid shape, with I think an odd name,
that has recently been defined. It may be this:
Take a solid cylinder whose diameter equals its height, cut it in half along a square, rotate one half 90 degrees, and glue the halves back together again.
Does this shape have a name, or is there something similar that does? Apparently it has very interesting rolling behavior.
--Dan
It's called a gefubuwichmuttle. Next question?
-Anonymous