Friday, November 20, 2009

Designing for Degrees of Freedom

We all think about it all the time when assembly modeling in our favorite MCAD package. We make sure to fully constrain all the degrees of freedom of the model per our design intent. If not, any change could result in unpredictable consequences and the robustness of our model compromised. Hours later, we may have our model back to the way it was prior to the change since we forgot to save it first.

But what happens when those degrees of freedom are something physical instead of virtual. We know how our model crashes, but what about our systems and machines? What happens if we over constrain our design? In the virtual world, the software either produces an error notifying us of our own ignorance, or it automatically corrects it for us and we merrily progress towards our deadlines, none the wiser.

An old concept making its way into new light is a concept called Exact Constraint being championed by James G. Skakoon of Vertex Technology, LLC.

To understand exact constraints we have to look back to our basic geometry class. There are 6 degrees of freedom in a solid body: 3 translations and 3 rotations.
We also have definitions of elements: points, lines, and planes.
  • A point is just that, a singularity within space. It has no length, area, or volume.
  • A line is defined as the distance between 2 points. It has length, but no area or volume.
  • A plane is a flat surface defined by 3 points or a line and a point (which is equivalent to 3 points since a line is defined by 2 points). Mathematically there are additional ways to define a plane, but geometrically this is the basic definition. A plane is considered infinite, but can be considered as having length and area, but no volume.
Using a cubic solid, applying...
  1. a planar constraint removes 3 degrees of freedom: 1 translation and 2 rotation.
  2. a line constraint removes 2 degrees of freedom: 1 translation and 1 rotation.
  3. a point constraint removes 1 degrees of freedom: 1 translation and 0 rotation.
If we use a planar constraint as the primary, and linear constraint as the secondary, and a point constraint as the tertiary, we fully constrain our design. Any more than that, and we over constrain the design. Overconstraining often leads to mechanism failure, obscure load paths, or other problems.

Taking a look at a shaft and pulley arrangement in the above figure*. The designer of this arrangement added a center bearing because the radial loads from the belt and pulley caused too much shaft deflection. Not knowing the design considerations, we don't know if a material change or diameter change on the shaft is possible. All we know is that the designer used 3 bearings.

As defined above, a shaft is like a line and therefore defined by 2 points. In this case, the shaft is defined by 3 points - the 3 bearing points. Inevitably, one of those bearings is not going to align and therefore cause an over constrained condition in the design and all pitfalls that come with it, including additional shaft stresses.

So how could it be designed better? If the shaft deflection truly is the forcing factor in the design, and extra bearing blocks are needed for support, then the shaft could have been sectioned and joined with a shaft coupler. This would allow for each shaft segment to be defined by only 2 points, the shafts to be joined and therefore rotating at the same velocity, and any misalignment to be compensated for by the coupling. All self-imposed stresses due to misalignment of the bearings - creating the over constrained condition - is removed and the system will be more robust.

You can find more information on Exact Constraints, including additional examples, by reading Exact Constraint: Machine Design using Kinematic Principles by James G. Skakoon or his article in ASME ME Magazine online.

*Note: The inspiriation for this post came from an article in ASME magazine written by Mr. Skakoon. One image was taken from that article because I'm too lazy to create my own.