Tolerance Allocation Research Articles

Geometric accuracy allocation for multi-axis CNC machine tools based on sensitivity analysis and reliability theory

Machining accuracy of a machine tool is influenced by geometric errors produced by each part and component. Different errors have varying influence on the machining accuracy of a tool. The aim of this study is to optimize errors to get a desired performance for a numerical control machine tool. Applying multi-body system theory, a volumetric error model was constructed to track and compensate effects of errors during operation of the machine, and to relate the functional specifications on volumetric accuracy to the permissible errors on the joints and links of the machine. Error sensitivity analysis was used to identify the influence of different errors (especially the errors which have large influences) on volumetric error. Based on First Order and Second Moment theory, an error allocation approach was developed to optimize allocation of manufacturing and assembly tolerances along with specifying the operating conditions to determine the optimal level of these errors so that the cost of controlling them and the cost of failure to meet the specifications is minimized. The approach developed was implemented in software and an example of the geometric errors budgeting for a five-axis machine was discussed. It is identified that the different optimal standard deviations reflect the cost-weighted influences of the respective parameters in the equations of the functional requirements. This study suggests that it is possible to determine the coupling relationships between these errors and optimize the allowable error budgeting between these sources.

Optimal tolerance design for mechanical assembly considering thermal impact

In this paper, a cost–tolerance model based on neural network methods is proposed in order to provide product designers and process planners with an accurate basis for estimating the manufacturing cost. Tolerance allocation among the assembly components is carried out to ensure that the functionality and design quality are satisfied considering the effect of dimensional and geometric tolerance of various components of the assembly by developing a parametric computer aided design (CAD) model. In addition, deformations of various components of mechanical assembly due to inertia and temperature effects are determined and the same is integrated with tolerance design. The benefits of integrating the results of finite element simulation in the early stages of tolerance design are discussed. The proposed method is explained with an application example of motor assembly, where variations due to both dimensional and geometric tolerances are studied. The results show that the proposed methods are much effective, cost, and time saving than the ones considered in literature.

Design Verification through Tolerance Stack up Analysis of Mechanical Assembly and Least Cost Tolerance Allocation

Geometric dimensioning and Tolerancing (GDT) constitutes the dominant approach for design and manufacture of mechanical parts that control inevitable dimensional and geometrical deviations within appropriate limits. The stack up of tolerances and their redistribution without hampering the functionality is very important for cost optimization. This paper presents a methodology that aims towards the systematic solution of tolerance stack up problem involving geometric characteristics.Conventional tolerance stack up analysis is usually difficult as it involves numerous rules and conditions. The methodology presented i.e. generic capsule method is straightforward and easy to use for stack up of geometrical tolerances of components and their assembly using graphical approach. In the work presented in this paper, angularity tolerance has been considered for illustration of the methodology. Two approaches viz. Worst Case (WC) and Root Sum Square (RSS) have been used. An example of dovetail mounting mechanism has been taken for purpose of stack up of angularity. Based on the stacked tolerance, it can be verified with the design tolerance of the assembly. Based on the comparison, designer has to reassign the appropriate tolerances to fulfil the functionality if required. If the stacked tolerance is as per designer requirement, then reallocation of tolerances on individual components should be done. Costs versus tolerance data are available for each component. With optimization technique, the optimized cost has been calculated for the assembly.

Design and optimisation of assembly technique for auto-body components

The dimensional quality of auto-body relates to the whole external appearance, wind noise, the effect of closing the door and even driving smoothness of vehicles. The assembly dimensional quality can be improved by optimising assembly operations between parts and reasonable allocating tolerances of components. A multi-attribute directed liaison graph is proposed to describe precedence relationships and key control characteristics (KCCs) to eliminate unfeasible assembly sequences. A hybrid particle swarm optimisation and genetic algorithm is presented to optimise assembly operations for selecting the best assembly sequence. Based on the above, tolerances of the optimal KCCs are designed by using NSGA-II algorithm according to assembly tolerances and manufacturing costs. To improve optimisation effectiveness, the initial population uses the orthogonal experimental design and sensitivity coefficients of KCCs to generate chromosomes. Finally, the work applied the method by illustrating the process of assembly sequence optimisation and tolerance allocation, and the results show that this case verified the proposed method.

Effects of dimensional errors on compliant mechanisms performance by using dynamic splines

The paper deals with the description of a methodology for investigating the effects of the dimensional errors on the performance of compliant assemblies in order to perform an optimal allocation of tolerances. The proposed approach is based on the modeling of flexible elements using the dynamic spline formulation which is able to take into account the nonlinear compliance of the system using a small set of parameters. Starting from the description of the elasto-kinematic behavior of the system written in terms of Lagrange equations, the deduction of sensitivity coefficients is discussed and an algorithmic description of the tolerance allocation scheme is presented. In order to give an example of application, the methodology has been successfully applied to the optimization of a constant-force compliant mechanism.

Optimal statistical tolerance allocation for reciprocal exponential cost–tolerance function

Statistical tolerancing has been widely employed in industry as it is more practical compared with the worst-case tolerancing in achieving lower manufacturing cost while satisfying design specification. As reciprocal exponential function is one of the commonly employed cost–tolerance models in practice and current approach is difficult to allocate statistical tolerances for such a function, this article investigates a method for optimal statistical tolerance allocation with such a cost–tolerance function. The method is to minimize manufacturing cost subject to constraints on tolerance target and machining capabilities. The optimization problem is solved by applying the algorithmic approach. Particularly, a closed-form expression of the tolerance optimization problem is further derived based on the Lagrange multiplier method with integrating the Lambert W function, a multivalue complex function. In addition, for constrained minimization problems with only equality constraints, the optimal tolerance allocation can be obtained by solving simultaneous equations without the time-consuming computing on differentiating while keeping the solution accurate. An example is illustrated to demonstrate the application of this approach. Through comparisons with the regular Lagrange multiplier method applied to reciprocal exponential cost–tolerance type, the result reveals that tolerances can be allocated much faster using this proposed method.

Quality- and cost-driven assembly technique selection and geometrical tolerance allocation for mechanical structure assembly

The assembly process planning has been the subject of extensive scientific work, mainly due to the multiple aspects involved from geometrical matters to operational research concerns. However, very few issues about assembly technique selection are addressed. The aim of this paper is to propose a method to select an assembly technique for each joint of a product and to allocate geometrical tolerances accordingly. This is achieved by solving a multi-objective optimization problem to minimize the cost and the non-conformity associated with the assembly plan. The potential benefits of the method are illustrated on a case study representing the assembly of a simple mechanical structure.

Process-oriented tolerancing using the extended stream of variation model

Current works on process-oriented tolerancing for multi-station manufacturing processes (MMPs) have been mainly focused on allocating fixture tolerances to ensure part quality specifications at a minimum manufacturing cost. Some works have also included fixture maintenance policies into the tolerance allocation problem since they are related to both manufacturing cost and final part quality. However, there is a lack of incorporation of other factors that lead to increase of manufacturing cost and degrade of product quality, such as cutting-tool wear and machine-tool thermal state. The allocation of the admissible values of these process variables may be critical due to their impact on cutting-tool replacement and quality loss costs. In this paper, the process-oriented tolerancing is expanded based on the recently developed extended stream of variation (SoV) model which explicitly represents the influence of machining process variables in the variation propagation along MMPs. In addition, the probability distribution functions (pdf) for some machining process variables are analyzed, and a procedure to derive part quality constraints according to GD&T specifications is also shown. With this modeling capability extension, a complete process-oriented tolerancing can be conducted, reaching a real minimum manufacturing cost. In order to demonstrate the advantage of the proposed methodology over a conventional method, a case study is analyzed in detail.

A comparison of tolerance analysis models for assembly

Mechanical products are usually made by assembling many parts. The dimensional and geometrical variations of each part have to be limited by tolerances so that it can ensure both a standardized production and a certain level of quality defined to satisfy functional requirements. The appropriate allocation of tolerances among the different parts is the fundamental tool to ensure that assemblies work correctly at lower costs. Therefore, to ensure their functionality, assembly designers have to apply tolerance analysis. A model based on either worst case or statistical type analysis may be used. Actually, there are some different models used or proposed by the literature to make the tolerance analysis of an assembly constituted by rigid parts, but none of them is completely and univocally accepted. None of them has done an objective and complete comparison for analyzing the advantages and the weaknesses and furnishing a criterion for the choice and application. This paper briefly introduces three of the main models for tolerance analysis, the Jacobian, the vector loop, and the torsor. These models are briefly described and then compared to show their analogies and differences. Some guidelines are provided as well, with the purpose of developing a novel approach which is aimed at overcoming some of the limitations of these models.

Tolerance Analysis and Allocation for Design of a Self-Aligning Coupling Assembly Using Tolerance-Maps

A self-aligning coupling is used as a vehicle to show that the Tolerance-Map (T-Map) mathematical model for geometric tolerances can distinguish between related and unrelated actual mating envelopes as described in the ASME/ISO standards. The coupling example illustrates how T-Maps (Patent No. 6963824) may be used for tolerance assignment during design of assemblies that contain non-congruent features in contact. Both worst-case and statistical measures are obtained for the variation in alignment of the axes of the two engaged parts of the coupling in terms of the tolerances. The statistical study is limited to contributions from the geometry of toleranced features and their tolerance-zones. Although contributions from characteristics of manufacturing machinery are presumed to be uniform, the method described in the paper is robust enough to include different types of manufacturing bias in the future. An important result is that any misalignment in the coupling depends only on tolerances, not on any dimension of the coupling.

Closed-Form Optimal Tolerance for Minimum Manufacturing Cost and Quality Loss Cost

Tolerance allocation have significant influence on the manufacturing cost and quality loss cost. In order to obtain optimal tolerance, Lagrange multiplier method is used to minimize the summation of manufacturing cost and quality loss cost subject to constraints on product functional requirement. The reciprocal power cost-tolerance model with different functional constraints is considered, and closed-form optimal tolerances are obtained. Using the model proposed in this paper, the optimal tolerance can be obtained quickly and accurately. One example is used to illustrate the method proposed in this paper.

Simultaneous component and tolerance allocation through suppliers' selection considering technological capability and production capacity to minimise purchasing cost and quality loss

In this paper we investigate component and tolerance allocation through supplier selection model. The model contributes to the current literature by incorporating simultaneous allocation of component and tolerance policy which has not been considered in almost all of previous models. The objective of the model is to minimise purchasing cost and quality loss considering supplier’s technological capability and production capacity. Numerical example is provided to show the implementation of the model and sensitivity analysis is performed to explore the effect of key parameters on selected suppliers, amount of allocation and total cost. The results of sensitivity analysis indicate that the changes on technological capability and production capacity will affect the optimisation result in which the model will tend to select the suppliers which have the lower price comparing to the selected suppliers in numerical example.

Statistical Dynamic Specifications Method for Allocating Tolerances

The assembly of components coming from manufacturing process characterized by high dimensional variation could be difficult if their cumulative variation is in the order of magnitude of the allowed tolerance. The problem was approached by managing dynamically the specifications and the allocation of tolerances. By means of simulating the production of multi-component assemblies different scenarios were defined to demonstrate the advantages of the Statistical Dynamic Specifications Method (SDSM). Simulation results showed an average increment of 46% in the tolerance of the intervened component. With this, the corresponding potential capability index cp varied from 1.29 to 1.85. Thus, it was shown that SDSM is a useful tool to deal with the high variation problem without focusing directly on the variation sources.

Cost Estimation Method for Variation Management

Cost engineering key objectives is to ensure cost estimates’ accuracy and to avoid cost overruns. In such a global context, this paper focuses on the manner through which the uncertainties as well as variations, impact the cost dimensions during different phases of product lifecycle: Tolerance allocation, Process Planning, Inspection Planning.What is investigated is the adaptability of a modified Activity-Based Costing model in evaluation of cost regarding the activities in different stages of the product's life cycle. This effective model encloses the cost estimation of tolerancing, process and inspection planning, via the impact of variations and uncertainties (i.e. inspection risk). The aim is to take into account not only the cost but also the quality (quality-driven activity-based costing) of product.

The Comparison of Imperialist Competitive Algorithm Applied and Genetic Algorithm for Machining Allocation of Clutch Assembly

The allocation of design tolerances between the components of a mechanical assembly and manufacturing tolerances can significantly affect the functionality of products and related production costs. This paper introduces Imperialist Competitive Algorithm (ICA) approach to solve the machining tolerance allocation of an overrunning clutch assembly. ICA is a multi-agent algorithm with each agent being a country, which is either a colony or an imperialist. These countries form some empires in the search space. Movement of the colonies toward their related imperialist, and imperialistic competition among the empires, form the basis of the ICA. During these movements, the powerful imperialist are reinforced and the weak ones are weakened and gradually collapsed, directing the algorithm towards optimum points. The objective of present study is to obtain optimum tolerances of the individual components for the minimum cost of manufacturing using ICA.The results were finally compared with the Genetic Algorithm (GA). Based on the results, ICA has demonstrated excellent capabilities such as accuracy, faster convergence and better global optimum achievement in comparison with traditional GA.

Analysis of Pose Accuracy Reliability and Sensitivity for Six Degrees Freedom Welding Robot

With the continuous development of industrial automation, robots are applied to various fields of industry increasingly, and the robot pose accuracy become an important issues. Through the establishment of the robot parameterized virtual prototype model, the error of bar length, joint angle and joint space have been taken into account, and the Monte Carlo sampling method is used to process a number of simulation analysis. The reliable probability which is given within the error limits is calculated, and finally through sensitivity analysis, the conclusion is drawn that different error factors influent the robot end error differently, which have a certain significance for the allocation of the manufacturing tolerances of the robot and trajectory planning.

Advanced Tolerancing Based on Product Performance by Using Statistical Tolerance Index C<i><sub>pk</sub></i> and C<i><sub>c</sub></i>

In this paper, the improvement of performance design for mass production are shown by using statistical tolerance index Cpk and Cc. Recently, the demand for product performance is getting higher. Meanwhile, precision machining and measurement are also developing in manufacturing industry. To satisfy consumer in terms of product performance, we have proposed advanced tolerancing method by using statistical tolerance index (STI). In this study, it is assumed that the performance of each product can be quantified and depends on the only functional dimension. It is also assumed that the dimension distribution of each machined part follows normal distribution. Consequently, performance of mass production can be quantified statistically. To control performance, we introduce STI which is composed of process capability limitation. After all, performance design is to calculate STI so as to satisfy the demand for performance. There are two problems in performance design. One is a synthesis of STI and another is an allocation of STI. These problems are just like the conventional tolerance synthesis and allocation problems. In this paper, the former problem is formulated algebraically.

A numerical study on effect of temperature and inertia on tolerance design of mechanical assembly

PurposeDue to technological and financial limitations, nominal dimension may not be able achievable during manufacturing process. Therefore, tolerance allocation is of significant importance for assembly. Conventional tolerance analysis methods are limited by the assumption of the part rigidity. Every mechanical assembly consists of at least one or more flexible parts which undergo significant deformation due to gravity, temperature change, etc. The deformation has to be considered during tolerance design of the mechanical assembly, in order to ensure that the product can function as intended under a wide range of operating conditions for the duration of its life. The purpose of this paper is to determine the deformation of components under inertia effect and temperature effect.Design/methodology/approachIn this paper, finite element analysis of the assembly is carried out to determine the deformation of the components under inertia effect and temperature effect. Then the deformations are suitably incorporated in the assembly functions generated from vector loop models. Finally, the tolerance design problem is optimized with an evolutionary technique.FindingsWith the presented approach, the component tolerance values found are the most robust to with stand temperature variation during the product's application. Due to this, the tolerance requirements of the given assembly are relaxed to certain extent for critical components, resulting in reduced manufacturing cost and high product reliability. These benefits make it possible to create a high‐quality and cost‐effective tolerance design, commencing at the earliest stages of product development.Originality/valueWith the approach presented in the paper, the component tolerance values found were the most robust to withstand temperature variation during the product's application. Due to this, the tolerance requirements of the given assembly are relaxed to a certain extent for critical components, resulting in reduced manufacturing cost and high product reliability. These benefits make it possible to create a high‐quality and cost‐effective tolerance design, commencing at the earliest stages of product development.

Tolerance design of mechanical assembly using NSGA II and finite element analysis

The technological and financial limitations in the manufacturing process are the reason for non-achievability of nominal dimension. Therefore, tolerance allocation is of significant importance for assembly. Conventional tolerance allocation methods are limited by an assumption that all parts are rigid. Every mechanical assembly consists of at least one or more flexible parts which undergo significant deformation due to inertia effect. Finite element analysis is used to determine the deformation of components in an assembly. Therefore, integration of statistical tolerance design with finite element analysis will guarantee that the optimal tolerance values of various components of the assembly obtained as end product of the tolerance design will remain within tolerance variation. Then the product can function as intended under a wide range of operating conditions for the duration of its life. In this paper, tolerance design of a piston cylinder assembly is done to demonstrate the proposed methodology.

Modeling of Geometric Variations for Line-Profiles

The geometric variations in a tolerance-zone can be modeled with hypothetical point-spaces called Tolerance-Maps (T-Maps) for purposes of automating the assignment of tolerances during design. The objective of this paper is to extend this model to represent tolerances on line-profiles. Such tolerances limit geometric manufacturing variations to a specified two-dimensional tolerance-zone, i.e., an area, the boundaries to which are curves parallel to the true profile. The single profile tolerance may be used to control position, orientation, and form of the profile. In this paper, the Tolerance-Map (Patent No. 6963824) is a hypothetical volume of points that captures all the positions for the true profile, and those curves parallel to it, which can reside in the tolerance-zone. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. T-Maps have been generated for other classes of geometric tolerances in which the variations of the feature are represented with a plane, line or circle, and these have been incorporated into testbed software for aiding designers when assigning tolerances for assemblies. In this paper the T-Map for line-profiles is created and, for the first time in this model, features may be either symmetrical or nonsymmetrical simple planar curves, typically closed. To economize on length of the paper, and yet to introduce a method whereby T-Maps may be used to optimize the allocation of tolerances for line-profiles, the scope of the paper has been limited to square, rectangular, and triangular shapes. An example of tolerance accumulation is presented to illustrate this method.