A puzzling issue: object with 8 degrees of freedom

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Uncle Al counts 10 external sides. Idiot.

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The pentagram was a postulated shape, I then tried a square, then a triangloziod. THe last was OK.

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__________

The formal name escapes me now.

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Better not take the other end of the bet.

A rotational degree of freedom occurs in one object capable of rotating with respect to another.

An object with several rotatable links can provide several degrees of rotary freedom for each successive link with respect to the base attachment.

An object like the Manx three-legged object, can rotate three ways at each ankle with respect to its limb.

Brian W

No: under the conventions that mathematicians and physicists use (and those *are* the relevant ones, after all, in this "math" newsgroup) you have described only three degrees of freedom, not six. These three degrees are East-West, up-down and in-out. This just says that there are three lines along which the motion can be projected, or three coordinates needed to describe velocity. For a generally-shaped so- called rigid body there can be two more degrees of freedom, associated with rotational angles and the like (i.e., the object's orientation). For non-rigid bodies there can be additional degrees of freedom, associated with vibrational modes, internal angles, etc. You need to be careful to count these correctly in order to obtain correct figures for specific heats in polyatomic gasses when doing statistical thermodynamics.

Anyway, the OP is, presumably, dealing with only orientation and internal-structure degrees of freedom. I'm still not sure about the exact answer to the OP's question.

R.G. Vickson

Funny, I am posting from the "sci.physics" group. So you do not understand that the three planes have two directions each that creates the 6 directions (degrees) of motion?

Rotation is just a curving change in the 6 degrees of freedom.

These are all still motions in the 6 degrees of freedom again. No need to add "extra" degrees of freedom at all.

Adding "degrees of freedom" and ignoring that the degrees you add are just curved or vibrational in the already 6 degrees known is a joke to science. It might be fun math but it is still just ignorant of physical motion and the scientific 6 degrees of freedom.

| > > He is thinking of objects like a deformable cubus with corners | > > whose angles are not fixed. But such a cubus has | > > - three orientational degrees of freedom | > > - three internal angles | > > which makes a total of only 6 degrees of freedom. | > > A cubus has 3fold symmetry when seen along | > > a diagonal, so that would fit; but 6 are not 8 | > > degrees of freedom. | >

| > > I brought up the idea of a tetrahedral skeleton, | > > (like a methane molecule

. | > > It has 8 degrees of freedom, | > > it has 3fold symmetry in some configurations, | > > but we do not see a way to build that in metal | > > or rubber without having more or less than 8 degrees | > > of freedom. | >

| > > On the other hand, I am not able to prove | > > that the puzzle is impossible to solve. | >

| > > Is there another solution? Where can one look for such | > > objects or related theorems? Are there books or sites | > > on these issues? | >

| > > Thanks in advance! | >

| > Hi John, | > The shape or makeup of an object does not change the freedom | > of it's motion. | > Freedom of motion has 6 directions, up- down, forward- backward,left-right. | > Those are the 6 "so called degrees" that I would call planes of motion | > instead. | | No: under the conventions that mathematicians and physicists use (and | those *are* the relevant ones, after all, in this "math" newsgroup) | you have described only three degrees of freedom, not six. These three | degrees are East-West, up-down and in-out. This just says that there | are three lines along which the motion can be projected, or three | coordinates needed to describe velocity. For a generally-shaped so- | called rigid body there can be two more degrees of freedom, associated | with rotational angles and the like (i.e., the object's orientation). | For non-rigid bodies there can be additional degrees of freedom, | associated with vibrational modes, internal angles, etc. You need to | be careful to count these correctly in order to obtain correct figures | for specific heats in polyatomic gasses when doing statistical | thermodynamics. | | Anyway, the OP is, presumably, dealing with only orientation and | internal-structure degrees of freedom. I'm still not sure about the | exact answer to the OP's question. | | R.G. Vickson

6-DOF x, y, z, pitch, roll, yaw.

6-DOF platforms:

I understand perfectly well. If you are moving to the East with speed

10 (in some units) your velocity is v = 10; but if you are going West at speed 10 your velocity is v = -10. If you 5 units to the right of the origin your coordinate is x = 5, but if you are 5 units to the left it is c = -5. That is how physicists use these concepts; mathematicians do so as well.

Nonsense.

Well, some bodies have fewer than 6 degrees of freedom, while others have more than 6. They can have as many as 8 degrees, but if counting the three "position" coordinates (which the OP did not want to do).

No, it is well-documented dynamics. For example, when studying the specific heats of dilute gasses by statistical thermodynamics, we need to worry about these thing; and if we count correctly, we get perfect agreement with experiment. That matters, because physics is an experimental science, after all.

No, it is not just fun mathematics; it is known to anybody who has taken undergraduate statistical mechanics or undergraduate classical mechanics. People who go to the library and read appropriate textbooks know about these things.

R.G. Vickson

You say you understand the above and then you prove you actually do not by stating that. Sorry.

Some bodies have less than 6 only when something is in the way. There are no extra degrees of freedom just because of a shape or position of an object.

So you can't do basic math anymore using 3 dimensions and 6 degrees of freedom?

Classical mechanics has no such bullshit at all. physical objects also have no such bullshit about extra degrees of freedom at all. You are only fooling yourself with "multiple dimension bullshit if you are finding more than 6 degrees of freedom.

Just to clarify the meaning of a degree of freedom in my thinking.

"An independent geometric variable."

Angle as a side length appear to be the issue and the basic variable where angle appears dependent was all that need be discussed.

_______ | | |

here is a right angle and it is clearly independent of side length. A constructed geometric as a whole then allows side to cause angle.

IN mechanical interpretation a machine can be described as a geometric function. MY right angle machine above has three apparent degrees of freedom. Two sides and the angle. As soon as a machine is a triangle it has three again, but the angles are side length determined and the meaning as machine itself in abstract appear to be a school idea I am trying to understand.

A function of machine applies to any design wheather a robot like device or not. A location of machine parts as function allows a complexity design such as a robot to be discussed, but exact question was apparently unclear to me.

A thoughtful man would allow all machine whether a simple constant structure to be a fuunctionally defined structure.

A robot type of machine appears the issue. A three fold symmetry mean it has three axises and so all machines with threee axis are functionally equivalent.

An inverted triangleoizoid;) was used at the National Insitute of Standards and Technology, NIST as a robot arm/platform. Cables varying the side lengths functionally could cause the tip of the volume to be motionable in an exact functional fashion. It translates left and right and up and down and spins in the center in a circle if required.

So all that was required was to allow an actual usage to be stated. I thought it was a crystallike common question and not a mechanical engineeering question. Analogy to crystal was a possible reason for the NIST discovery as a class of robot though.

If there is a question concerning the usage of crystal design in robots I would entertain them because there are few machines able to be used in crystal analogy!

Here is a small novel machine design based on a crystal geometric.

A cubic structure has like maybe ten degree of freedom. And to functionally control side length to cause the function meant the solution would be indeterminate! A rather airfcraft like control function would have to be defined for a cubic to be used as a robot and a test loop in control code would have to prevent nonsolution motion.

Making the nIST inverted triangleosoid a truely novel discovery.

Maechanical design is very interesting and the basic question here is to either talk of the method of machine analogy or not.

Thanks Doug

Hi Doug, The cubic structure still does not have more than 6 degrees of freedom, What you are doing is allowing for the degrees to have a multiple of positions or multiple motions within the normal 6 degrees. That is not "extra degrees" that is still just an extra motion in the normal

6 degrees of freedom.

Any point of the object can have no more than 6 different directions (up, down, left, right, forward, backward) of motion in the 3 planes of 3D space they reside it. Motions that combine lets say up and left, are not an extra degree of freedom. They are a combination of 2 or more of the normal degrees of freedom.

Again, The rotation at each end you speak of is still just freedom of motion in the same 6 degrees known. and each end also has a compression factor to allow

6 degrees to still exist at each end. You are merely mixing already known degrees of freedom into "extra" degrees that are not actually there. You are adding already known degrees of freedom as "extra" degrees.

A cube, has 6 degrees of freedom only, Any single point on the cube or inside the cube also only has 6 degrees of freedom. You can combine any of the degrees for different motion in such free 3D space. But it still only moves with 6 degrees of freedom.

Just because it has ends does not give it "extra" degrees by adding the same degrees. Each end can move in the same 6 degrees the other end can move in. There is no "addition of degrees occuring".

Dear freddy osbourne:

A truss member has at least three degrees of freedom:

1) change in length (+/-) 2) bending off the central axis (up/down, left/right, etc., Euler column) 3) torsion around the central axis

David A. Smith

correction, a frame member has six degree of freedom at each end which is 12 degree of freedom total. i was thinking about 2d instead of 3d solution. a cubic would have six degree of freedom at each faces making it 6 times 6 equals to 36 degree of freedom. i think yahoo answer is more fun because i can correct each answer before it is consolidated within a reasonable time...

Using only 2 degrees of the 6 known.

Using 4 degrees of the 6 known.

Again Using same 4 degrees of (2) of the 6 known. (up down left right motion of points with a variable of motion for each point)

Still only 6 degrees of actual freedom total.

Dear freddy, The 6 degrees of freedom on one side are the same 6 degrees of freedom on the other side. You have not added "actual" degrees of freedom, You are using the same 6 known degrees of freedom.

No, that is a single degree of freedom.

Only two.

No, this is three degrees of freedom.

David A. Smith

Dear Spaceman:

He has correctly described 6 additional degrees of freedom, that describes unique configurations of the total structure. You don't know physics, you don't know mechanics, you are wasting time.

David A. Smith

No, two directions of motion is 2 degrees of freedom in a single plane of motion.

No, again 4 degrees but now in two planes. I think you are confusing planes with degrees. Each plane has 2 directions of freedom. (2 degrees)

Nope. It is the same as above. It has 2 directions for for any point in one plane and 2 more directions in the other again,

4 degrees of freedom.