Tolerance Chains Research Articles

Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach

There is a widely accepted consensus that component manufacturing precision is directly correlated with improved functional performance. However, this increase in precision comes at the expense of higher manufacturing costs, resulting in a trade-off between quality and affordability. In light of this opposing behavior, low-cost products typically exhibit lower quality, whereas high-quality products tend to be more expensive. This study introduces a novel approach for optimizing the dimensional tolerances of mechanical assembly components, taking into account both their manufacturing requirements and the associated costs of non-quality. Furthermore, the method considers the functional constraints imposed by interrelated tolerance chains within the product. Instead of relying on an exact mathematical solution, the proposed solution employs a heuristic approach through a simple and flexible algorithm. This enables practical implementation, as different cost-tolerance functions can be selected based on specific requirements. To provide a comprehensive evaluation of the proposed method, a concise review of the relevant literature in the field was conducted, allowing a comparison with other state-of-the-art approaches.

In English

Boron carbide is a material of interest for personal body armor, but its low fracture toughness and amorphization limits its widespread use. Al and Si atoms in doped boron carbide reduce this problem. Passage of the substitution reaction in boron carbide powders with Al and Si vapors in vacuum was found. Certification methods: chemical analysis, full-profile XPA (Powder Cell for Windows. Version 2.4 FREE, W. Kraus & G. Nolze) and modeling in format of the 15-atomic unit cell B12(C-C-C) of trigonal syngony, spatial group R3 ̅m, Z = 3. A mixture of powders of boron carbide, aluminum or silicon is heat treated in vacuum at conventional evaporation temperatures of Al (1520 K) or Si (1640 K) for 1–5 h. The samples were purified with alkali and analyzed by arbitration chemical analysis for boron, carbon, aluminum and silicon. The formula composition of the input powders of boron carbide was determined as B12[(C-В-C)x(C-C-C)1-x], where x = 0.4–0.6. The aluminum substitution reaction takes place in both types of boron carbide chains and corresponds to the formula B12(C-Al-C) or AlB12C2. In the presence of silicon, the reaction took place exclusively at the positions of the tri-carbon chains. The composition of the obtained solid solution corresponds to - B12[(C-B-C)0.4(C-Si-C)0.6], starting powder B12[(C-B-C)0.4(C-C-C)0.6]. The absence of boron phases of silicide, such as SiB3 (SiB2.89), SiB6, SiBn (n ≈ 23) indicated greater resistance of C–B–C chains to interaction with vaporous Si. The content of Al and Si in the substituted phases is equal to 13.3 and 4.0 (% at.). Equivalent molar amounts of Al8B4C7 and SiC of gas-nano-phase origin were measured in the reaction products with vapor-like Al and Si. The area of tolerance chains of the boron carbide structure in the format of the average specific electronegativity (χN-Sh/rai) was found. It is in the range of values: 2.79 ≥ ССС ≥ СВС ≥ CSiC ≥ ВВС ≥ 2.18.

Selective assembly of planetary gear stages to improve load sharing

Selective assembly processes are already used in various applications to achieve higher accuracy or to limit the variations caused by geometrical tolerances. ZF Wind Power has implemented the Selective Assembly principle to achieve better load sharing between planets in higher-loaded multiple-planet planetary stages. Two possible approaches are considered: in case of multiple contributing parts in the tolerance chain, a high number of assembly combinations will be possible, each having a specific positional deviation (e.g., a 5-planet system with no back-up parts will have 1.7 million possible combinations). In case only one component is the main driver, geometrical variation has to be minimized by selecting the best-matching set of components out of a larger group of available parts. The first approach is called Selective Assembly, the second Planet Sorting. This article gives an overview of both the Selective Assembly and Planet Sorting processes: the theory behind the processes, a specific calculation example, the validation of the process, the implementation of the process in the different involved departments throughout the organization and the effect on gear rating assessment.

Computer-aided tolerance chain identification system for tolerance allocation

Computer-aided tolerance chain identification system for tolerance allocation

Identification of Surrogate Models for the Prediction of Degrees of Freedom within a Tolerance Chain

The computation of assembly tolerance information is necessary to fulfill robust design requirements. This assembly is computationally costly, with current calculations taking several hours. We aim to identify surrogate models for predicting degrees of freedom within a tolerance chain based on point connections between assembly components. Thus, replacing part of the current computation workflow and consequently reduce computation time. We use manufacturing tolerances set by norms and industrial standards to identifly these surrogate models, which define all relevant features and resulting output variables. We use black-box modeling methods (artificial neural networks and gradient boosted trees), as well as white-box modeling (symbolic regression by genetic programming). We see that these three models can reliably predict the degrees of freedom of a tolerance chain with high accuracy (R2 > 0.99).

An Efficient Method for Statistical and Deterministic Tolerances Synthesis Using the Jacobian Torsor Model

This research aims to create an efficient way to guide designers in selecting tolerances more cost-effectively through the Jacobian-Torsor unified model by performing an iterative statistical and deterministic analysis of geometric tolerances. This model is created based on the assembly's Functional Requirements (FR) and the starting tolerance values of the Functional Elements (FE) that comprise the tolerance chain. The model initially produces deterministic results. However, the Monte Carlo simulation allows converting it into a statistical model. The model is assessed iteratively, with each simulation iteration employing random values of the torsor components that correspond to the functional elements as inputs. The simulation ends when the number of iterations approaches the maximum value set at the star, and a collection of values representing the components of the Functional Requirement (FR) torsor is created based on repeated multiplications of the model. Finally, the percentage calculation defines the contribution of each Functional Element (FE) to the total Functional Requirement (FR). A numerical example is provided to demonstrate the use of the suggested method. A comparison of the two approaches, deterministic and statistical, reveals that the latter has a more significant economic impact.

Tolerance analysis by static analogy: numerical and experimental results

Industrial requirements to design high quality products in shorter and shorter times impose the use of numerical models to estimate the geometrical deviations of these products, which are assemblies, starting from the geometrical deviations of their components. Numerical models may support this estimation activity, thus reducing the time to market and the design costs. The free-body model is an interesting numerical method to translate the tolerance chain into a static problem solved by algebraic or graphical procedures by using the free-body diagrams of force analysis. This work presents a free-body model that is able to deal with dimensional and geometrical tolerances that involve translations of the features to which the tolerances are applied. Moreover, it validates the free-body model for tolerance analysis of rigid parts by defining and solving a case study through numerical and experimental activities. The present work uses a case study to demonstrate the effectiveness of the free-body model by underlining that the experimental results obtained are very close to those from the numerical model.

Sensitivity Analysis of Manufacturing Tolerances in Permanent Magnet Synchronous Machines With Stator Segmentation

In this article the sensitivity of manufacturing tolerances on the output of a permanent magnet synchronous machine with stator segmentation is studied. To reduce the simulation effort, a transition of the stator segment air gaps to replacement geometry was applied and the relevant parameters are discussed, whereas the uncertainty of geometric and material parameters is modeled with a normal distribution. Furthermore, tolerance chains are expressed through the convolution of independent probability distributions. For the Design of Experiments Sobol sequences are utilized. To further decrease the number of design parameters an approach is presented to reduce redundant parameters to individual design parameters. For the sensitivity analysis the torque and torque ripple serve as quality objectives. In addition, an approach is revealed to use the variance of the radial forces as quality objective without considering mechanical FE simulations. For manufacturing tolerances a linear regression is satisfying to describe the main effects on the quality objectives. Repetitive machine parameters, e.g. the remanences, have less influence on the machine's output than particular parameters such as the housing radius or static eccentricity. The influences of stator segmentation are negligibly small, whereas the housing radius has the highest sensitivity on the machine's output.

A CONCEPTUAL MODEL COMBINATION FOR THE UNIFICATION OF DESIGN AND TOLERANCING IN ROBUST DESIGN

AbstractIn design engineering, the early consideration of tolerance chains contributes to robust design. For this, a link of design and tolerancing domains is essential. This paper presents a combination of the graph-based tolerancing approach and the Contact and Channel approach to link these domains. The combined approach is applied at a coinage machine. Here it provides detailed insights into state-dependent relations of embodiment and functions, which can improve robustness evaluation of the concept. This approach shows a possibility to bridge the gap between design and tolerancing domains.

Selection of parameters in cost-tolerance functions: review and approach

The paper deals with the cost-tolerance functions used in minimum-cost tolerance allocation on mechanical assemblies. The consistency of the cost-tolerance functions associated with different dimensions of a tolerance chain is a necessary condition for a correct optimization of tolerance values. This requires a careful selection of function parameters, in order to take into account all the variables influencing the cost, and thus to compare the costs of the different dimensions in a common scale. The cost-tolerance functions proposed in the literature are reviewed, revealing that the parameters are often selected without explicit reference to empirical data or objective rules. The development of procedures for parameter selection has been addressed in several studies, which are also reviewed along with the needed cost data published in textbooks and reports. Finally, an original approach is proposed to build cost-tolerance functions for combined processes, for which fewer results or methods are available. The parameters of a reciprocal power function, chosen for the favorable balance between accuracy and ease of use, are expressed through an empirical relationship with some design variables; these include the nominal dimensions and the type and size of the part features involved in the allocation problem. An application example shows that the proposed procedure results in optimal tolerance values consistent with common design criteria.

An iterative method of statistical tolerancing based on the unified Jacobian–Torsor model and Monte Carlo simulation

Abstract This paper focuses on exploring an iterative method of statistical tolerance design to guide designers to select tolerances more economically and effectively. After having identified the assembly functional requirement (FR) and the functional elements (FEs) of corresponding tolerance chain, the expression of a unified Jacobian–Torsor model can be derived. Monte Carlo simulation is employed to generate random variables simulating the variations of small displacement torsor associated with the FE pairs with all the generated random values being within the intervals constrained by the corresponding tolerance zones. Then, the real multiplication operations are repeatedly executed to this model, a large number of real torsor component values of FR will be obtained and we can perform statistical analysis for these simulated data to get the statistical limits of the assembly FR in the desired direction. The tolerances of critical FEs may need to be adjusted to satisfy the assembly FR imposed by the designer, and the percentage contribution of each FE to the assembly FR can help determine these critical tolerances that need to be tightened or loosened. Once the calculated FR is in close agreement with the imposed FR, the iterative process can be stopped, and the statistical tolerance redesign is achieved. The effectiveness of the proposed method is illustrated with a case study. Compared with the deterministic tolerancing method, the results show that the proposed method is more economical and that can relax significantly the precision required, manufacturing and inspection costs can then be reduced considerably.

Sampling-based Tolerance-Cost Optimization of Systems with Interrelated Key Characteristics

Abstract Robustness is a key factor contributing to a high functionality of technical systems under uncertainty. In this context, existing methodologies, such as Axiomatic Design and Quality Function Deployment, can help to identify coupled functional requirements which significantly affect the robustness of a system. However, it is often not possible to decouple all in the final product design. As a consequence, multiple interrelated tolerance chains have to be considered in the subsequent tolerance design leading to multi-constrained or multi-objective optimization problems. Despite their significant influence on the optimization process and its results, interrelated tolerance chains have not been studied in detail yet, especially in the context of sampling-based tolerance-cost optimization. Moreover, a holistic framework for the tolerance-cost optimization of systems with multiple key characteristics is missing so far. In order to close that gap, this paper presents a framework to consider multiple key characteristics in both least-cost and best-quality tolerance-cost optimization using sampling techniques for tolerance analysis. Therefore, interrelated tolerance chains and their effects on the optimization process in terms of the definition and handling of multiple constraints and objectives are discussed in detail. The proposed research aims to bring all important aspects together in one common framework. Thus, it is supposed to help researchers and practitioners to properly define and solve the tolerance optimization problem. In order to show its benefits and applicability, it is applied to an illustrative case study. The novelty of this paper is to present a comprehensive method to support the tolerance engineer in creating a robust and optimal tolerance design of products with multiple interrelated key characteristics.

Tolerance management in robot-based assembly optimizes product, process and system deviations

Deviations of individual parts, assemblies, modules, processes, and equipment interact in assembly and can lead to rework or rejects of the finished product. In general, deviations cannot be avoided but must be managed. There are many analysis methods, however only few abstract methods concentrate on optimization. Thus, there is a lack of concrete, easy and feasible optimization methods for deviation reduction. This paper presents a novel method for assembly optimization by combining tolerance chains and a developed optimization tree, which gives a holistic overview of practical improvement measures. The method is validated on an automated robot process in engine production.

Monte Carlo simulation for tolerance analysis in prefabrication and offsite construction

Abstract Producing assemblies that experience little to no rework during assembly fit-up is a perennial challenge in offsite construction. While the traditional response has been to solve geometric issues onsite at the expense of large rework costs, manufacturing optimisation techniques can mitigate these issues upstream. Tolerance analysis is one such method that is widely used in manufacturing to predict if problems will occur from the accumulation of tolerances and dimensional variability. This article demonstrates Monte Carlo tolerance simulation on a prefabricated construction assembly, where variations up to 37 mm are identified compared to as-built deviations which range up to 30 mm. Process optimisation is also explored, where risk of rework related to dimensional variability is reduced by 65.6% through selection of alternate fabrication processes. To compare the Monte Carlo method to traditional analysis methods, a simplified 1-D tolerance analysis is used. Compared to an as-built deviation of roughly 11 mm, the Monte Carlo method produces a conservative value of 15.4 mm, while other traditional methods are either overly conservative (worst-case tolerance chain has a deviation of 19.8 mm), or overly ambitious (root sum square tolerance chain has a deviation of 4.6 mm). Tolerance analysis through Monte Carlo simulation is shown to be a proactive design tool with several key advantages for prefabricated and offsite construction. First, complex three-dimensional geometric interactions can be readily modelled using very basic tolerance configurations. Secondly, potential misalignments at key connection points can be identified and quantified in terms of a probability distribution of variation. Finally, design improvements can be achieved by comparing alternate construction processes to mitigate the risk of assembly rework.

Tolerance Management in a Semi-Automated and Collaborative Human-Robot Aircraft Riveting Process

<div class="section abstract"><div class="htmlview paragraph">Large aircraft sizes with high precision requirements combined with complex joining tasks are typical challenges for aircraft production. To increase competitiveness and effectiveness, the automation of such production processes seems a viable solution for companies in the aircraft sector. When implementing automation, in order to handle small batch sizes and high variation while adhering to tight tolerances, the production equipment must meet high quality standards and flexibility requirements.</div><div class="htmlview paragraph">To achieve the objectives above, tolerance management is essential: deviations are acceptable within limits, as long as they do not result in quality losses and expensive rework. For these reasons, all the interactions between the product, production process and production equipment used must be analyzed in detail. The importance of this analysis is evident in assembly where new technologies are used, such as (semi-)automation using Human-Robot-Collaboration. Despite its innovative value, this approach must be robust, within tolerances and have minimal deviations from the outset. However, current planning and optimization of deviations and tolerances lack properly developed methods and approaches.</div><div class="htmlview paragraph">This paper proposes a method for securing and achieving proper tolerance in assembly processes: characteristic trees and tolerance chains are simple methods to make tolerance management more effective and attractive. The method combination promotes understanding of interactions and communication between all those involved in the development of products and the associated processes.</div><div class="htmlview paragraph">The methods developed are validated using a semi-automated riveting process, with Human-Robot-Collaboration, to complete a joining process in the assembly of the aft section. In this scenario the pressure bulkhead is mounted to the section barrel by means of hundreds of rivets. The intention is to implement a semi-automated production process to improve ergonomics, increase process traceability and efficiency, and minimize rework while meeting tolerance requirements.</div></div>

Model reduction in geometric tolerancing by polytopes

There are several models used in mechanical design to study the behaviour of mechanical systems involving geometric variations. By simulating defects with sets of constraints it is possible to study simultaneously all the configurations of mechanisms, whether over-constrained or not. Using this method, the accumulation of defects is calculated by summing sets of constraints derived from features (toleranced surfaces and joints) in the tolerance chain. These sets are usually unbounded objects (R6-polyhedra, 3 parameters for the small rotation, 3 for the small translation), due to the unbounded displacements associated with the degrees of freedom of features. For computational and algorithmic reasons, cap facets are introduced into the operand polyhedra to obtain bounded objects (R6-polytopes) and facilitate computations. However, the consequence is an increase in the complexity of the models due to the multiplication of caps during the computations. In response to this situation, we formalized and tested a method for controlling the effects of cap facets. Based on the combinatorial properties of polytopes, we propose to trace the operand faces during the different operations. An industrial case is solved and discussed in order to show the significant gain in computational time when applying the new method. This example has been chosen to be as general as possible to illustrate the genericity of the method.

Evolving tolerance management for increased robustness of subsea installation operations

AbstractLow oil prices leads to a need for large cost reductions and improved effectiveness in the subsea oil and gas industry. At the same time, production in harsher environment and deeper water is inducing a need for more complex production systems. Increasing complexity reduces the tolerances for successful manufacturing and installation. Increasing robustness of the installation process can reduce installation costs. Robustness of the installation process is dependent on successful tolerance management. This paper investigates different methods for tolerance management. The conclusion is that successful tolerance management partly is independent on the chosen method for tolerance analysis. The research shows that successful tolerance management relies on understanding of the tolerance chain, cooperation, and early initiation. We applied system modelling as a system engineering approach to support manually calculated tolerance budgets. The research finds that this supports understanding, exploration, and communication. We recommend this as best practice. The research shows that tolerance management can contribute to increased robustness of the installation process by including manufacturing and operational factors in the tolerance analysis. System modelling and budgeting support discussion and exploration of these factors.

Managing Installation Tolerances through System Modeling and Tolerance Budgeting

AbstractContractors in the oil and gas industry are facing challenges when installing subsea production systems (SPS) in deep waters. The installation relies on engineering of the systems with high accuracy levels and narrow clearances on the interfacing surfaces to meet required installation tolerances. To ensure that all installation tolerances requirements are met, there is a need for a systematic governing process of managing, controlling, and verifying them. Engineers define installation tolerances through qualification activities of components and technologies. Extensive systems and complex installation sequences generate tolerance chains affecting the interfacing components. The verification of the installation consequently requires a significant effort.The research focus is on how system modeling and tolerance budgeting would help the process of managing installation tolerances of a subsea production system in the context of preventing late verification, potential late design changes, and errors in installation. Use of system modeling made it possible to visualize the installation of the system of interest, and systematically structure relevant information. The visualization supported the researchers to develop an understanding of the tolerance chain for the system of interest. Based on the models, we were able to put together a tolerance budget calculating the theoretical worst‐case scenario of installation on a chosen critical misalignment. Our research showed that the systems engineering (SE) effort had a positive impact on the process, considering the cost of the effort relative to the potential cost of the preventable scenarios.

Process-based tolerance assessment of connecting rod machining process

Process tolerancing based on the process capability studies is the optimistic and pragmatic approach of determining the manufacturing process tolerances. On adopting the define–measure–analyze–improve–control approach, the process potential capability index (C p) and the process performance capability index (C pk) values of identified process characteristics of connecting rod machining process are achieved to be greater than the industry benchmark of 1.33, i.e., four sigma level. The tolerance chain diagram methodology is applied to the connecting rod in order to verify the manufacturing process tolerances at various operations of the connecting rod manufacturing process. This paper bridges the gap between the existing dimensional tolerances obtained via tolerance charting and process capability studies of the connecting rod component. Finally, the process tolerancing comparison has been done by adopting a tolerance capability expert software.

Investigation of Effects (Welding Sequence, Fixturing, Welding Points) on Distortions after Spot Welding for Determining Individual and Cumulative Tolerances

Abstract: The manufacturing and GPS (Geometrical Product Specifications and Verification) have to be thought and analysed together. Welding and Metal Forming are two of manufacturing methods, which are used to create sheet metal parts assembly groups. The dimensions do not stay steady and they change during these two manufacturing methods. Because firstly, after forming (for example bending) the springback and secondly, after welding (for example spot welding) the distortions exist. In this study, the distortions after spot welding are researched and their effects on tolerance chain and assembly tolerances are analysed. These distortions are determined with analysis software after designing some sheet metal part assembly examples. After that, these distortions are achieved with real measurement (scanning) after welding of the sheet parts. At the final step, the analysis and measurement results are compared.