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ORIGINAL ARTICLE

A Chernov bound for robust tolerance design and application Ambre Diet1,2

· Nicolas Couellan2,3 · Xavier Gendre2,4 · Julien Martin1

Received: 8 July 2020 / Accepted: 9 October 2020 / Published online: 13 November 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020

Abstract Within an industrial manufacturing process, tolerancing is a key player. The dimensions uncertainties management starts during the design phase, with an assessment on variability of parts not yet produced. For one assembly step, knowledge can be gained from the tolerance range required for the parts involved. In order to assess output uncertainty of this assembly in a reliable way, this paper presents an approach based on the deviation of the sum of uniform distributions. As traditional approaches based on Hoeffding inequalities do not give accurate results when the deviation considered is small, an improved upper bound is proposed. Then, the impact of the stack chain geometry on the bound definition is discussed. Finally, an application of the proposed approach in tolerance design of an aircraft sub-assembly is detailed. The main interest of the technique compared to existing methodologies is the management of the confidence level and the emphasis of the explicit role of the balance within the stack chain. Keywords Design · Deviation · Manufacturing · Quality · Sum of uniform distributions · Tolerance

1 Introduction The management of dimensions uncertainties is a key player in the manufacturing process of various industrial sectors such as transportation (automotive, aeronautics, . . . ) or household appliances industry.

 Ambre Diet

[email protected] Nicolas Couellan [email protected] Xavier Gendre [email protected] Julien Martin [email protected] 1

Tolerancing Department, Airbus Operations S.A.S, 316 route de Bayonne, 31060 Toulouse, France

2

Institut de Math´ematiques de Toulouse UMR 5219, Universit´e de Toulouse, 31062 Toulouse, France

3

´ ENAC, Universit´e de Toulouse, 7 avenue Edouard Belin, 31400, Toulouse, France

4

ISAE-SUPAERO, Universit´e de Toulouse, 10 Avenue ´ Edouard Belin, 31055, Toulouse, France

Dimensions may have some deviation from the designed value without significant impact on the quality and functional requirements of the final product. Tolerance intervals are defined according to engineering knowledge and scientific analysis in order to determine these acceptable variations. A deviation out of the determined tolerance bounds is considered non-compliant and implies an action such as an investigation or a modification in the process or the design. The perfect balance between functional requirements and process capability has to be found so that the specified tolerance interval is the most accurate possible. If the tolerance is too tight, the process might not have the capability to manufacture it and either there will be many rejected items or some costly improvement will be needed to produce compliant items. Otherwise, a too wide toleranc