Senior Research Scientist, MIT Center for Technology, Policy and Industrial Development
In this presentation I sketch out a model of mechanical assemblies that uses the same underlying mathematics, namely Screw Theory, to model both the nominal and varied condition of an assembly. The model represents assemblies as kinematic mechanisms which may or may not be capable of motion by intent.
Assemblies are designed with the intent of achieving one or more Key Characteristics, that is, specifications on relative position and orientation between features on possibly non-adjacent parts, and specifications on allowed variation of the Key Characteristics. Paths called Datum Flow Chains are established by the designer to carry relative position and orientation from part to part in order to establish nominal achievement of each Key Characteristic.
Parts are joined by one or more assembly features that are modelled as sets of elementary surface contacts. These features instantiate the part-to-part constraint goals established when each Datum Flow Chain was declared. Screw Theory is used to determine the state of constraint inside each feature and between features in order to characterize the state of constraint of the entire assembly. Variation analysis is conducted by assuming that one or more of the surfaces within a feature may move within its tolerance zone in ways that the tolerance specification allows. Screw Theory is then used to propagate the effect of this variation onto the assembly to see the effects on the Key Characteristics.
Only properly constrained assemblies can be correctly analyzed for the effects of variation at the feature or part level. In the case of over-constraint, a stress analysis is needed. Without taking stress and strain into account, a unique Datum Flow chain does not exist. In the case of under-constraint, there is no particular nominal condition, requiring the addition of an artificial constraint. CAD systems check for geometric compatibility but do not detect situations where locked-in stress could exist under conditions of variation. Screw Theory permits us to fill this gap.
Current design practice does not make a clear distinction between creation of a competent nominal design (that is, one that is as close as practical to properly constrained or one in which the designer deliberately inserts desired over-constraint and takes it into account) and performance of a variation analysis (too often called tolerance analysis). Similarly, CAD current systems do not support this distinction or provide adequate tools for addressing each kind of design.