A simple model of sheet metal assembly

This paper is meant to serve as a case study of mathematical modeling in industry. The problem, which arises in the automotive industry, is to predict the variation in the final assembly given the variation in the parts and tooling. In this paper, we present a simple model of welding and clamping of beams which demonstrates that the order in which the clamps and welds are applied influences the final shape of the assembly. The modeling of the process is done by simple mechanics. To solve the mathematical problem, we use standard ideas from constrained optimization and scientific computation. Additionally, using a statistical simulation we show that clamping and welding from the inside out leads to a smaller standard deviation in the result (as measured by the displacement of the right end of the beams) in response to normal distributions of variations in parts and welding. The findings help explain why certain welding sequences may be preferred.