Modeling Dispersions Affecting Pre-Defined Functional Requirements of Mechanical Assemblies Using Jacobian Transforms

Assuming that a method exists for exhaustively generating 3-D tolerance chains around a desired functional requirement, this paper describes a method to mathematically quantify the relationship between the effects that the functional elements dispersions implied in a chain have on the dispersions of the desired functional requirement. This is achieved by using the concept of six virtual joints associated to each functional element implied in a chain. These virtual joints then become the basis for modeling virtual kinematic chains around the desired functional requirement. By associating each virtual joint with a coordinate frame, the impact of various small displacements of the individual functional elements on the overall functional requirement can be computed using a Jacobian matrix.