How geometrical tolerances affect the measurement of reciprocal alignment of two different assemblies: a case study

Often a designer has the problem to apply a suitable system of geometrical and dimensional tolerances to an assembly. The right solution is not unique, in fact it depends on the chosen parameters. If the tolerances have to be optimized, some important parameters have to be taken into account, e.g. the efficiency of each prescription, or if this last is reachable, or it can be verified and how much the realization costs. The authors opinion is that a statistical approach based on the Monte Carlo Method is very useful when the tolerances chains are complex. This paper shows an application of this method in order to verify the functional alignment between two assemblies and a critical analysis of the uncertainty in phase both of the component design and test. This study has been developed thanks to the strict requirements imposed by ESA (European Space Agency) on the components that Thales Alenia Space has to realize within the LISA Pathfinder experiment. The very critical aspect of this work is to reciprocally align two cylindrical elements of two different assemblies. The specifications require 100 μm as maximum linear displacement and 300 μrad as maximum angular displacement. Moreover this prescriptions have to be verified also when the two elements are independently moving. To be able to reach such strict accuracy level the components have been assembled in an ISO 100 class cleanroom and the work space was a 3D Coordinate-Measuring Machine (CMM). The cylindrical elements have a 10 mm diameter, so the value of the measurement uncertainty associated with the alignment check is fundamental. Starting from the different uncertainty sources, the measurability and verifiability of the alignment have been considered and evaluated. The overall uncertainty has been assessed by numerical simulations which have taken into account the dimensional, geometrical and form tolerances as well as the instrumental uncertainty of the 3D CMM. This estimation has been positively validated by a session of repeated measurements. Numerical simulations have also allowed performing a sensitivity analysis, in order to give information about which sources more contribute to the overall uncertainty.