Tolerance Chains Research Articles

Tolerance Analysis and Synthesis Using Jacobian Transforms

This paper presents a mathematical tool to help the designer in the difficult task of tolerancing mechanical assemblies. It starts with the identification of a critical tolerance chain around some functional requirement of the assembly. A mathematical relationship which quantifies the effects that possible small displacements of functional elements in the chain have on the functional requirement is obtained. This corresponds to the solution of the tolerance analysis problem. This is expressed as a set of equations in matrix form with a Jacobian matrix which provides the desired analysis relationship. The solution to the tolerance synthesis problem is obtained by simply pseudo-inverting the Jacobian matrix, where small displacements of the functional element pairs in the chain are expressed as a function of the desired small displacements of the functional requirement. The paper presents the procedure for obtaining both the analysis and synthesis equations. An example is used to illustrate the generation of tolerance equations. The paper is concluded with a discussion of how the equations could be used in a statistical tolerancing context using Monte Carlo simulations.

Tolerance Chain Detection by Geometrical Constraint Based Coupling Analysis

Assembly tolerance chains are often the root cause for low geometrical robustness and high manufacturing costs. This paper presents a framework for function means modeling in configuration design that allows for modeling and analysis of geometrical couplings, and detection of potential tolerance chains. Requirement decomposition is described, and geometrical couplings on different hierarchical assembly levels are modeled, analyzed and explained. The treatment of sub-assemblies is especially described. The presented framework includes the use of locating schemes for positioning parts in assemblies. A general positioning system for modeling datum frames and locating schemes for parts and sub-assemblies are presented and used. An automotive example is used to show how an overall geometrical product constraint is decomposed into locating schemes on individual parts and how potential tolerance chains are detected. Finally, geometrical coupling quantification and the use of fixtures as a mean for breaking tolerance chains are discussed. The proposed framework enables potential tolerance chains to be detected already in the configuration design phase. By detecting potential tolerance chains in very early configuration design, design solutions with low robustness, needing tight tolerances to fulfill their functional requirements, may be avoided.

The generic capsule approach to tolerance stack analysis

The tolerance chart is a graphical method for representing the manufacturing dimensions of a workpiece or assembly, at all stages of its manufacture. It is used to control tolerance stack up when the machining of a component involves inter-dependent tolerance chains. The build of the tolerance model is a complicated procedure. This paper aims at presenting a simple and systematic approach the generic capsule approach, to the process of tolerance stack analysis. The model is constructed, representing the given and the unknown dimensions. The proposed method uses, as the name implies, a generic capsule, which takes into account all the related aspects of the axis and surface type of tolerance stacks. Links are then established using the model, which help formulate the stack path of interest, into a linear equation. The equation is used to complete the tolerance stack analysis module. This will be explained using a variety of geometric dimensioning and tolerancing examples, as we progress. In addition, it is generic and hence simple to practice.

CAD-based integrated tolerancing system

Assembly tolerance analysis involves the determination of tolerances of critical dimensions. This paper addresses two basic areas in tolerance analysis and allocation. (1) Dimension normalization, which identifies the relevant functional dimensions. A new algorithm called the dimension block diagram (DBD) algorithm for dimension normalization is presented. (2) Dimension representation, which provides a robust framework for representing the tolerance chain. A new dimension representation framework using Assembly Dimensional Tolerance (ADT) tree is presented. The two methodologies are automated and integrated within the framework of a CAD system using an Expert System. A computer test bed software has been developed demonstrating the viability of this approach.

The influence of manufacturing tolerances on the kinematic performance of mechanisms

The well-established theory of linear dimension and tolerance chains has been extended to include non-linear systems, such as linkage mechanisms, employing a technique based on the use of kinematic constraint equations. The results are interpreted with reference to production and assembly technologies and two examples are analysed to demonstrate the application of the technique.

Optimum assembly using a component dimensioning method

Many tolerance allocation methods based primarily on cost optimisation have been developed in recent years. However, the dimensions of the assembly parts are usually determined by convenience or design constraints other than the requirements of the assembly itself. The dimensions are often determined to give sufficient clearance during assembly without much consideration given to their tolerances. This is usually followed by a tolerance analysis to check for interference. The dimensions are then adjusted to eliminate any interferences. There is no control over the size of the clearance of an assembly. This paper presents a new approach of dimensioning the components of an assembly which would allow real control over the size of clearance or interference in an assembly. A unique algorithm in conjunction with a relationship matrix is developed to formulate the dimensions in a tolerance chain into a linear equation. This linear equation is then used to develop the tolerance analysis module. The dimensioning module is accomplished using a separate algorithm in conjunction with the linear equation established. The tolerance analysis and the dimensioning modules are subsequently tested on a number of examples.

Tolerancing a solid model with a kinematic formulation

The 3D analysis of tolerances is one of the critical functional requirements of solid-modelling systems which is most lacking at the present time. 3D tolerance analysis is most difficult to carry out by hand when it involves geometrical tolerances. A computerized 3D tolerance analysis is therefore needed, and an appropriate tolerancing model must be devised. The paper first acknowledges the kinematic nature of tolerance-chain analysis. A kinematic formulation of full 3D dimensional and geometrical tolerances is then proposed. The scheme represents all tolerances within a kinematic model, which is compatible with existing standards such as ansi y14.5m. The semantics of tolerances is respected, as tolerances are automatically interpreted to yield tolerance zones and datum reference frames. A tolerance chain is modelled as a kinematic chain, thus enabling multiple tolerances to be manipulated to solve 3D tolerance-chain analysis problems. The implementation of the modelling concept on an exact solid modeller is described. An application of the tolerancing model to the visualization of tolerance zones is also shown.

Automatic Generation of Tolerance Chains from Mating Relations Represented in Assembly Models

This paper presents an algorithm for generating tolerance chains from the mating relations between components of assemblies. The algorithm is developed upon a feature-based assembly modeling strategy that represents each component in close relation to its mating features, dimensions, and tolerances. The mating relations within an assembly are described by a mating graph. Tolerance chains together with their dimensions and tolerances are generated automatically by searching through a mating graph for matching mating features. A prototype program package based on the presented algorithm has been developed, and several examples of various complexity have been tested with success.

Automated tolerance analysis for mechanical assemblies modeled with geometric features and relational data structure

The automation of tolerance analysis of assemblies is based on the development of the data structure for their representation in geometric modeling, and the computerization of procedures developed by Bjorke. The data structure contains assembly-level and component-level relationships to automate the generation of the tolerance chain for a user-selected sum dimension. A statistical analysis is performed on a sum dimension, the characteristics of which can be easily generated by the hierarchical relationships and the mating conditions between components contained within the assemblies' geometric data structure. The method uses a truncated unit beta distribution for the sum dimension and the contributions to the sum dimension by the statistical properties of 15 link types. The user supplies the tolerance range of the sum dimension and its location within the distribution. The program provides the user with a graphical display of the sum-dimension distribution, the specified range and location of the tolerance, and the results of the numerical integration of the distribution that generates the confidence limits for a given application.

Operational Dimensioning and Tolerancing in CAPP

The ideal output of operational sequencing of CAPP should include all start and end coordinates of tool movement for all operational dimensions rather than just give dimensions and toleraces unrelated to the coordinate system on the NC machine and leave additional work for the NC programmers. This paper deals with a new method of analysis of Operational Dimension and Tolerance Chain (ODTC) for CAPP systems. Based on discussion of previous works, apparent shortcomings are discussed and an extended approach provides ODTC to generate operational dimensions and tolerance for NC machines as a part of the decision methodology of CAPP systems.

Computerized tolerance techniques

The intense global competition to produce quality product at low cost has led many industrial nations to consider mechanical tolerances as a key factor to bring about cost savings as well as be competitive. In the last two decades, not enough research work has been done in the area of tolerance techniques. In this paper, a comprehensive summary of the state of the art and the projection of future trends in tolerance techniques is presented to help make a decision concerning the improvement techniques today and to aid in guiding research for the future. This paper reviews the status of theory and practice about how manufacturing and assembly processes are characterized for relating tolerances to process or production cost. The specification of tolerancing on the dimension of the manufactured part has a significant impact on the final production cost. Tight tolerances can result in excessive process cost, while loose tolerances may lead to increased waste and assembly problems. This paper systematically reviews the state of art by classifying the papers written so far into three categories. The categories are tolerance chain technique, tolerances based on analysis and synthesis, and tolerances based on cost-tolerance algorithms and design methods. Future areas of research and the unresolved issues have been presented that will serve as a guiding light for the researcher to investigate and bring about the solution in future.

Representation of assemblies for automatic tolerance chain generation

A scheme of representing assemblies and an algorithm for the tolerance chain generation are developed so that tolerance chains of assemblies can be generated automatically to accommodate tolerance analysis at the assembly level. In the hierarchical data structure representing an assembly the connectivity information is carried by the instances of components and subassemblies, and the mating relations between each pair of mating entities are described by mating links, mating paths, mating conditions, and mating features. Mating graphs are derived from the mating links before they are searched for the generation of tolerance chains. The scheme has been implemented in a prototype interactive package that allows the user to model assemblies directly without detailed object modeling. Several examples of various complexity have been tested with success.

Tolerance control in the machining of discrete components

Abstract Tolerance charting is a manual procedure for controlling tolerance stack up when the machining of a component involves interdependent tolerance chains. Such a heuristic, experience-based method of allocating tolerances to individual cuts of a process plan can be embodied in a computer based module. This paper presents a graph theoretic representation for a tolerance chart and a linear programming model for optimum allocation of tolerances among individual machining cuts.

Tolerance chart optimization

A tolerance chart is a graphical representation of a process plan and a manual procedure for controlling tolerance stackup when the machining of a component involves interdependent tolerance chains. This heuristic, experience-based method of allocating tolerances to individual cuts of a process plan can be embodied in a computer-based module. This paper introduces a graph theoretic representation for the tolerance chart. A special path tracing algorithm is used to identify tolerance chains from this graph. Optimal tolerance allocation among individual cuts is achieved using a linear goal programming model instead of existing heuristic methods. A more comprehensive mixed integer programming model is developed to incorporate linear tolerance cost functions and alternative process selection.

Specifying operational availability : Herbert Dagen. Proc. a. Reliab. Maintainab. Symp., 145 (1986)

The automation of tolerance analysis of assemblies is based on the development of the data structure for their representation in geometric modeling, and the computerization of procedures developed by Bjorke. The data structure contains assembly-level and component-level relationships to automate the generation of the tolerance chain for a user-selected sum dimension. A statistical analysis is performed on a sum dimension, the characteristics of which can be easily generated by the hierarchical relationships and the mating conditions between components contained within the assemblies' geometric data structure. The method uses a truncated unit beta distribution for the sum dimension and the contributions to the sum dimension by the statistical properties of 15 link types. The user supplies the tolerance range of the sum dimension and its location within the distribution. The program provides the user with a graphical display of the sum-dimension distribution, the specified range and location of the tolerance, and the results of the numerical integration of the distribution that generates the confidence limits for a given application.