Degrees of freedom and rotation

[Madou]

Can anyone explain to me why does a figure which can't move from a plain has three degrees of freedom?

 

[Helios]

If you think it has two degrees of freedom, you may be forgetting rotation.

 

[Madou]

thank you

 

[HallsofIvy]

A rigid figure in the plane has three degrees of freedom. Choose any two points on the figure. If you know the position of one point, and the angle the line between the two points makes with some "reference" (say, the x-axis) then you can calculate the postion of any point on the figure. The position of one point on a two dimensional surface is given by two numbers (that's basically what "two dimensional" means) and the angle is the third number: 3 numbers to determine the position of every point= 3 "degrees of freedom".

 

[schroder]

A great deal depends upon the type of figure. If it is a sphere which is constrained by having to maintain one contact point with a plane, it will have five degres of freedom. Two of these are translational in the x and y directions, and three of these are rotational along x, y and z axis. If there are two spheres connected by a rod and it is maintaining two points of contact with a plane, then two degrees of freedom are lost, (one translational and one rotational) but four still remain. Now consider three spheres connected by three rods and maintaining three points of contact with a plane. This body will still have translational motion in two directions and rotational motion about one axis, so it still has three degrees of freedom and has lost three. In general, one degree of freedom is lost for each point of contact with the plane, excluding redundant contact points. The lesson here is to avoid redundant constraints when designing machinery as the required precision increases with the number of constraints. A three-legged chair is much easier to make even with a plane floor than a four-legged chair!

 

[Madou]

And why do the scissors on a plane have four degrees of freedom?

 

[Madou]

I guess because we can fix one part of the scissors - then the other would have 2 degrees of freedom, and if we fix the other - the first one would have 2 degrees of freedom as well. 2+2=4 degrees of freedom ;)

 

[Madou]

A rigid body in 3D-space has got 6 degrees of freedom.

 

[schroder]

And why do the scissors on a plane have four degrees of freedom?

Another way to evaluate degrees of freedom (DOF) is to count the number of valid (non-redundant) constraints. Going back to the example of the three spheres connected by three rods, on the plain; With one sphere touching the plain there is one constraint (translational in the z axis is constrained). Touch down a second sphere and rotational motion in either x or y is constrained (depending on orientation). Touch down the third sphere and now rotational in both x and y is constrained. This still leaves available translational in both x and y and rotational about the z axis. Three constraints = loss of three DOF, but three still remain. If you were to add another rod and sphere to this body and touch that down, it would not constrain the remaining three DOF so that fourth constraint is redundant. You will also find it very difficult to maintain all four spheres in perfect contact with the plain, meaning you need to precision engineer the body but get noting in return for you efforts! Now, the scissors on the plain is similar to the three connected spheres, but with a difference; Once you have touched down three points on the plane, the x and y rotations as well as z translation are constrained, same as with the spheres. To completely constrain the spheres would require three more points of contact. This could be done by having one of the spheres drop into a V-shaped slot (2 points of contact) which removes one more translational DOF. Another sphere could be dropped into triangular hole (3 points of contact) this would lock up the final translational DOF as well as the rotation about the z axis. All six DOF are constrained by using a total of six non-redundant contact points. Something similar can be done with the scissors, but because of the scissor action, the top blade would still be able to rotate around the z axis, requiring one more constraint to be added. This would mean the scissors has one extra DOF. However, a PAIR of scissors is not a single rigid body so it is not limited to the six DOF allowed to a rigid body.