Publication | Closed Access
Tolerances: Their Analysis and Synthesis
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1990
Year
Mathematical ProgrammingEngineeringIndustrial EngineeringProtocol ComplianceConvex PolytopeStructural OptimizationDimensioning And TolerancingOperations ResearchReliability EngineeringUncertainty QuantificationManufacturing Cost MinimizationSystems EngineeringRobust OptimizationReliabilityDesignComputer EngineeringStructural ReliabilityFault Tolerant ControlReliability ModellingOptimization ProblemTolerance SynthesisStandardization
A dimension’s tolerance is modeled as a random variable with an associated standard deviation. The study aims to synthesize tolerances by optimizing assembly stack‑up conditions to minimize manufacturing cost. Tolerance synthesis is performed by approximating the volume under a multivariate probability density function constrained by nonlinear stack‑up conditions with a convex polytope, using the reliability index to transform the probabilistic optimization into a deterministic nonlinear programming problem, and an algorithm that converges to the global optimum. The algorithm enables a trade‑off between performance and tolerance, replaces worst‑case analysis, and converges globally, as demonstrated by implementation examples.
Tolerance, representing a permissible variation of a dimension in an engineering drawing, is synthesized by considering assembly stack-up conditions based on manufacturing cost minimization. A random variable and its standard deviation are associated with a dimension and its tolerance. This probabilistic approach makes it possible to perform trade-off between performance and tolerance rather than the worst case analysis as it is commonly practiced. Tolerance (stack-up) analysis, as an inner loop in the overall algorithm for tolerance synthesis, is performed by approximating the volume under the multivariate probability density function constrained by nonlinear stack-up conditions with a convex polytope. This approximation makes use of the notion of reliability index [10] in structural safety. Consequently, the probabilistic optimization problem for tolerance synthesis is simplified into a deterministic nonlinear programming problem. An algorithm is then developed and is proven to converge to the global optimum through an investigation of the monotonic relations among tolerance, the reliability index, and cost. Examples from the implementation of the algorithm are given.