Tolerance Analysis And Synthesis Using Virtual Joints

This paper presents the results of a research aiming at mathematically modeling the effects of tolerances buildups in close kinematic tolerance chains. The model consists of associating a set of six virtual joints, three for small translations and three for small rotations, to every pair of functional elements in a tolerance chain. These virtual joints can therefore simulate the effects of positional and orientational inaccuracies between two functional elements of the same part which are assumed to result from manufacturing precision limits. The mathematical model is obtained by first associating a coordinate frame to every virtual joint. Transformations matrices describing the global position and orientation of every frame with respect to a base frame are then computed. The effects of small translations and small rotations of every virtual joint on a point of interest in the chain (in particular on the functional requirement) can be computed using standard Jacobian matrices, which are easily obtained once the transformations matrices have been figured out. The model is in the form of six equations relating the new position and orientation of a point of interest in the chain (in cartesian space) to the small dispersions of the functional elements of the chain (in joint space) as simulated by possible moves about their virtual joints. A detailed example is provided that will illustrate the use of the developed model.