Variability is unavoidable in the realization of products. While design must specify ideal geometry, it shall also describe limits of variability (tolerances) that must be met in order to maintain proper product function. Although tolerancing is a mature field, new manufacturing processes and design methodologies are creating new avenues of research, and modelling standards must also evolve to support these processes. In addition, the study of uncertainty has produced widely-accepted methods of quantifying variability, and modern tolerancing tools should support these methods. The challenges introduced by new processes and design methodologies continue to make tolerancing research a fertile and productive area.

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Tolerancing: Managing uncertainty from conceptual design to final product

It is widely acknowledged that the allocation of part tolerances is a highly responsible task due to the complex repercussions on both product quality and cost. As a consequence, since its beginnings in the 1960s, least-cost tolerance allocation using optimization techniques, i.e. tolerance-cost optimization, was continuously in focus of numerous research activities. Nowadays, increasing cost and quality pressure, availability of real manufacturing data driven by Industry 4.0 technologies, and rising computational power result in a continuously growing interest in tolerance-cost optimization in both research and industry. However, inconsistent terminology and the lack of a classification of the various relevant aspects is an obstacle for the application of tolerance-cost optimization approaches. There is no literature comprehensively and clearly summarizing the current state of the art and illustrating the relevant key aspects. Motivated to overcome this drawback, this article provides a comprehensive as well as detailed overview of the broad research field in tolerance-cost optimization for both beginners and experts. To facilitate the first steps for readers who are less familiar with the topic, the paper initially outlines the fundamentals of tolerance-cost optimization including its basic idea, elementary terminology and mathematical formulation. These fundamentals serve as a basis for a subsequent detailed discussion of the key elements with focus on the different characteristics concerning the optimization problem, tolerance-cost model, technical system model and the tolerance analysis model. These aspects are gathered and summarized in a structured mind map, which equips the reader with a comprehensive graphical overview of all the various facets and aspects of tolerance-cost optimization. Beside this, the paper gives a retrospect of the past fifty years of research in tolerance cost-optimization, considering 290 relevant publications. Based thereon, current issues and future research needs in tolerance-cost optimization were identified.

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From tolerance allocation to tolerance-cost optimization: a comprehensive literature review

This article is devoted to the analysis of the influence of the joint clearances in a mechanism of a circuit breaker, which is a 42 degree-of-freedom mechanism made of seven links, seven revolute joints, and four unilateral contacts with friction. Spatial (3D) revolute joints are modeled with both radial and axial clearances taking into account contact with flanges. Unilateral contact, Coulomb’s friction and Newton impact laws are modeled within the framework of nonsmooth mechanics without resorting to some regularizations or compliance/damping at contact. The nonsmooth contact dynamics method based on an event-capturing time-stepping scheme with a second order cone complementarity solver is used to perform the numerical integration. Furthermore, the stabilization of the constraints at the position level is made thanks to the stabilized combined projected Moreau–Jean scheme. The nonsmooth modeling approach together with an event–capturing time-stepping scheme allows us to simulate, in an efficient and robust way, the contact and impacts phenomena that occur in joints with clearances. In particular, comparing with the event-detecting time-stepping schemes, the event-capturing scheme enables us to perform the time-integration with a large number of events (impacts, sliding/sticking transitions, changes in the direction of sliding) and possibly with finite-time accumulations with a reasonable time-step length. Comparing with compliant contact models, we avoid stiff problems related with high stiffnesses at contact which generate some issues in contact stabilization and spurious oscillations during persistent contact periods. In the studied mechanisms of the circuit breakers, the numerical method deals with more than 70 contact points without any problems. Furthermore, the number of contact parameters is small—one coefficient of restitution and one coefficient of friction. Though they are sometimes difficult to measure accurately, the sensitivity of the simulation result with respect to contact parameters is low in the mechanism of the circuit breaker. It is demonstrated that this method, thanks to its robustness and efficiency, allows us to perform a sensitivity analysis using a Monte Carlo method. The numerical results are also validated by careful comparisons with experimental data, showing a very good correlation.

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Multibody systems with 3D revolute joints with clearances: an industrial case study with an experimental validation

Tolerance specification is an important part of mechanical design. Design tolerances strongly influence the functional performance and manufacturing cost of a mechanical product. Tighter tolerances normally produce superior components, better performing mechanical systems and good assemblability with assured exchangeability at the assembly line. However, unnecessarily tight tolerances lead to excessive manufacturing costs for a given application. The balancing of performance and manufacturing cost through identification of optimal design tolerances is a major concern in modern design. Traditionally, design tolerances are specified based on the designer's experience. Computer-aided (or software-based) tolerance synthesis and alternative manufacturing process selection programs allow a designer to verify the relations between all design tolerances to produce a consistent and feasible design. In this paper, a general new methodology using intelligent algorithms viz., Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi Objective Particle Swarm Optimization (MOPSO) for simultaneous optimal selection of design and manufacturing tolerances with alternative manufacturing process selection is presented. The problem has a multi-criterion character in which 3 objective functions, 3 constraints and 5 variables are considered. The average fitness factor method and normalized weighted objective functions method are separately used to select the best optimal solution from Pareto optimal fronts. Two multi-objective performance measures namely solution spread measure and ratio of non-dominated individuals are used to evaluate the strength of Pareto optimal fronts. Two more multi-objective performance measures namely optimiser overhead and algorithm effort are used to find the computational effort of NSGA-II and MOPSO algorithms. The Pareto optimal fronts and results obtained from various techniques are compared and analysed.

Simultaneous optimal selection of design and manufacturing tolerances with alternative manufacturing process selection

Concurrent design of tolerances by considering both the manufacturing cost and quality loss of each component by alternate processes of the assemblies may ensure the manufacturability, reduce the manufacturing costs, decrease the number of fraction nonconforming (or defective rate), and shorten the production lead time. Most of the current tolerance design research does not consider the quality loss. In this paper, a novel multi-objective optimization method is proposed to enhance the operations of the non-traditional algorithms (Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO)) and systematically distribute the tolerances among various the components of mechanical assemblies. The problem has a multi-criterion character in which three objective functions, one constraint, and three variables are considered. The average fitness factor method and normalized weighted objective function method are used to select the best optimal solution from Pareto-optimal fronts. Two multi-objective performance measures namely solution spread measure and ratio of non-dominated individuals are used to evaluate the strength of Pareto-optimal fronts. Two more multi-objective performance measures namely optimizer overhead and algorithm effort are used to find the computational effort of NSGA-II and MOPSO algorithms. The Pareto-optimal fronts and results obtained from various techniques are compared and analysed. Both NSGA-II and MOPSO algorithms are best for this problem.

Concurrent multi-objective tolerance allocation of mechanical assemblies considering alternative manufacturing process selection.

Improved algorithm for tolerance allocation based on Monte Carlo simulation and discrete optimization

The purpose of functional tolerancing process is to define the geometrical specifications(tolerances) of parts ensuring functional requirements. An important distinction in tolerance process is that engineers are more commonly faced with the problem of tolerance synthesis rather than tolerance analysis. In tolerance analysis the parts tolerances are all known and the resulting geometrical requirement respect is calculated. In tolerance synthesis, on the other hand, the geometrical requirement is known from design requirements, whereas the magnitudes of the parts tolerances to meet these requirements are unknown. In this paper, we focus on the gear tolerances, and we propose an approach based statistical analysis for tolerance analysis and genetic algorithm for tolerance synthesis. Usually, statistical tolerance analysis uses a relationship between parts deviations and functional characteristics. In the case of tolerance analysis of gears, thus relationship is not available in analytic form, the determination of a functional characteristic(e.g. kinematic error) involves a numerical simulation. Therefore the Monte Carlo simulation, as the simplest and effectual method, is introduced into the frame. Moreover, to optimize the tolerance cost, genetic algorithm is improved. Indeed, this optimization problem is so complex that for traditional optimization algorithms it may be difficult or impossible to solve it because the objective function is not available in analytic form. For the evaluation of the fitness of each individual based on Monte Carlo Simulation, the number of samples is the key of precision. By a large number of samples, the precision can be improved, but the computational cost will be increased. In order to reduce the computational cost of this optimization based on Monte Carlo Simulation and Genetic Algorithms, the strategy is to adopt different precision of fitness; different numbers of samples during the optimization procedure are introduced into our algorithms.

Fangcai Wu

Papers

Improved algorithm for tolerance allocation based on Monte Carlo simulation and discrete optimization

Optimization of Gear Tolerances by Statistical Analysis and Genetic Algorithm