Tolerance Analysis Method Research Articles

A solution of partial parallel connections for the unified Jacobian–Torsor model

The unified Jacobian–Torsor model is an innovative three dimensional (3D) tolerance analysis method which uses the torsor model for tolerance representation and the Jacobian matrix for tolerance propagation. It is very suitable for complex assemblies which contain a large number of joints and geometric tolerances. However, previous studies about this model only considered serial connections in assemblies. Two reasons may account for this problem. One hand, the Jacobian matrix for parallel structures is complex. On the other hand, the torsor model which represents tolerance of assembly containing parallel connections is not presented. Ignoring these partial parallel connections would result in a loss of accuracy and reliability of the unified Jacobian–Torsor model. In this paper, a solution of partial parallel connections for the unified Jacobian–Torsor model is presented. A new torsor model representing partial parallel connections caused by joints of cylindrical and planar surfaces is obtained by a composition operation of screw parameters between torsors. Meanwhile, variations of this new torsor are achieved by an intersection or composition operation between participated torsors. A calculation scheme is proposed for this solution with a deterministic way and statistical way. An example is also presented to illustrate this solution.

Tolerance Mathematical Modeling and Analysis Method Based On Control Points of Geometric Elements

ABSTRACTTolerance is one of the important factors that cannot be ignored in modern manufacturing process, it not only affects the machining accuracy and production quality of parts, but also has a vital significance in process route, testing, manufacturing costs, assembly of final products, etc. In this paper, the tolerance mathematical modeling and analysis method based on control points of geometric elements are proposed, in which the geometric tolerance is indicated according to the position variation of control points. The first step is to define the inherent direction of geometric elements. The inherent direction of point is the same with the one of the benchmark points, and the inherent direction of linear is the linear itself while the inherent direction of plane is the one of the normal directions. The second step is to establish the tolerances coordinate system according to the inherent direction of geometric elements, then defining and classifying the degrees of freedom of geometric elements, ta...

A comprehensive fuzzy feature-based method for worst case and statistical tolerance analysis

Tolerance analysis is an analytical tool for the estimation of accumulating effects of the individual part tolerances on the functional requirements of a mechanical assembly. This article presents a comprehensive feature-based method for tolerance analysis of mechanical assemblies with both dimensional and geometric tolerances. In this method, dimensional and geometric tolerance zones are described by the combination of fuzzy modelling and small degrees of freedom (SDOF) concept. In this model, the uncertainty in dimensions and geometric form of features is mathematically described using fuzzy modelling, and the kinematic effects of tolerances in assemblies are expressed by SDOF concept. In the proposed method, complicated GD&T concepts such as various material modifiers (maximum material condition, least material condition and regardless of feature size), envelope requirement, bonus tolerances and Rule No. 1 (Taylor principle) in several conditions are accurately modelled. Compatible with the proposed model, a procedure for modified worst case and statistical tolerance accumulation analysis is introduced. An algorithm is laid out that describes the steps and the procedure of tolerance analysis. The application of this method is demonstrated through an example, and the results are compared with experimental results.

Application of Monte Carlo Method in Tolerance Analysis

Monte-Carlo simulation is the most popular and simplest method for nonlinear statistical tolerance analysis. Random values for every part are got according to the part distributions, and the value of the response function is computed for each set of part values. A sample of response function values is thus got, and the moments of the sample are computed using the standard statistical formula. Atpresent, the researches on assembly tolerance analysis have been focused on dimension tolerance and hardly take geometric tolerance intoconsideration. In this paper, geometric tolerance is treated as dimension tolerance whose nominal value is zero, and the Monte Carlo SimulationMethod is applied to tolerance analysis including geometric tolerance. Finally, a case of a top column assembly verifies the validity and accuracy of the method.

Assembly Tolerance Analysis Method Based on the Real Machine Model with Three Datum Planes Location

An assembly tolerance analysis method based on the real machine model is proposed, aiming to investigate the effect of the geometric errors of datum feature on the result of tolerance chain. The real parts are simulated by the variation of the geometric features, which are represented by the control point variation model and generated by the Monte Carlo simulation. The investigated objective is the assembly success rate of pin-hole assembly at the location with the three datum planes, the assembly clearance or interference is calculated according to the assembly requirements and assembly process. The part coordinate system in assembly is constructed firstly by the datum embodiment principles, then the position of the junction planes are determined according to the assembly requirements, and finally, the position of the parts relative to machine are determined by the coordinate transformation. The assembly success rate is represented by the probability distribution of clearance or interference between pin and hole, and the influences of the datum errors on the assembly result are discussed by the example assembly.

Tolerance Analysis and Compensation Method Using Zernike Polynomial Coefficients of Omni-directional and Fisheye Varifocal Lens

There are many kinds of optical systems to widen a field of view. Fisheye lenses with view angles of 180 degrees and omni-directional systems with the view angles of 360 degrees are recognized as proper systems to widen a field of view. In this study, we proposed a new optical system to overcome drawbacks of conventional omni-directional systems such as a limited field of view in the central area and difficulties in manufacturing. Thus we can eliminate the undesirable reflection components of the omni-directional system and solve the primary drawback of the conventional system. Finally, tolerance analysis using Zernike polynomial coefficients was performed to confirm the productivity of the new optical system. Furthermore, we established a method of optical axis alignment and compensation schemes for the proposed optical system as a result of tolerance analysis. In a sensitivity calculation, we investigated performance degradation due to manufacturing error using Code V(R) macro function. Consequently, we suggested compensation schemes using a lens group decentering. This paper gives a good guidance for the optical design and tolerance analysis including the compensation method in the extremely wide angle system.

A small displacement torsor model for 3D tolerance analysis of conical structures

The small displacement torsor model is a classic three-dimensional tolerance analysis method. It uses three translational vectors and three rotational vectors to represent tolerance information in three-dimensional Euclidean space. However, the target features of this model mainly focused on planes and cylinders in previous studies. Little attention is invested to conical features and their joints which are used widely and more complex than the planar and cylindrical features. The objective of this article is to present a three-dimensional mathematical method of tolerance representation about conical surfaces and their joints based on the small displacement torsor model, and propose a mathematical model of variations and constraint relations of components of the small displacement torsor for conical surfaces caused by geometric tolerances limited by its tolerance zone. In addition, a simple example involving conical structures is used to demonstrate three-dimensional conical tolerance propagation. Both deterministic and statistical results are obtained by this model.

Impact of a behavior model linearization strategy on the tolerance analysis of over-constrained mechanisms

All manufactured products have geometrical variations which may impact their functional behavior. Tolerance analysis aims at analyzing the influence of these variations on product behavior, the goal being to evaluate the quality level of the product during its design stage. Analysis methods must verify whether specified tolerances enable the assembly and functional requirements. This paper first focuses on a literature overview of tolerance analysis methods which need to deal with a linearized model of the mechanical behavior. Secondly, the paper shows that the linearization impacts the computed quality level and thus may mislead the conclusion about the analysis. Different linearization strategies are considered, it is shown on an over-constrained mechanism in 3D that the strategy must be carefully chosen in order to not over-estimate the quality level. Finally, combining several strategies allows to define a confidence interval containing the true quality level.

The Method of Tolerance Analysis Base on the Monte Carlo

The primary goal of concurrent engineering is to minimize the time spend and to reduce quality problems encountered between a product's conception and customer use, thereby minimizing cost and maximizing product quality. Tolerance is an important index of product design and manufacture. Although they are small as compared with part dimensions, tolerances can propagate and accumulate in an assembly affecting the product assimilability. Moreover, tolerance is inevitable because manufacturing exactly equal parts is known to be impossible. This paper presents a brief review of statistical tolerance and clearance analysis for the assembly, and proposes a method of tolerance analysis by a leading commercial computer-aided tolerance (CAT) analysis system, VisVSA. This method can ensure assembly precision requirements, and design optimally tolerance distribution based on Monte Carlo simulation, take a example of brake assembly to detailed introduce the use of this method. The purpose of this article is expected to make the following contributions: (i) to help the designers to evaluate products for assimilability, (ii) to provide a new perspective to tolerance problems, and (iii) to provide a tolerance analysis tool, VisVSA, which can be incorporated into a CAD or solid modeling system.

Assembly tolerance analysis method based on the real machine model

The position errors of an objective feature of a machine is affected comprehensively by the geometric errors and assembly errors of the parts on the assembly chain. In order to investigate the effect of assembly datum errors on the transformation and propagation of the geometric errors in an assembly chain, a modeling and analysis method of assembly error accumulation based on the real machine model is proposed. In this approach, the calculation process of the real position of geometric feature is consist of four steps. Firstly, the deviation geometric feature are represented by the control point variation model , and the deviations of geometric feature from its normal position are simulated by the Monte Carlo method. Secondly, the normal position of the deviation geometric feature in the datum reference frame (DRF) is determined by theoretical dimension. The third step, the DRF establishment method is proposed, and the measure DRF of the objective feature is established with the datum geometry example. The fourth step, the assembly contact model is proposed, the relationship between the measure datum reference frame and assembly datums is calculated according to the assembly sequence and the deviation geometry example of assembly datum surface. The real position of all related parts on the assembly chain in machine coordinate system are calculated by above procedures, and the transformation and propagation of the geometric errors of assembly datum in an assembly chain are obtained. The procedure to simulate the real machine assembly is decomposed into four modular processes, which is useful in realizing the automation of assembly error analysis. A assembly error accumulation example of pin-hole assembly with the three orthogonal datum planes is investigated, and the proposed approach is verified.

Tolerance Analysis Method for Passive Alignment of Photonic Chips

This paper presents a novel tolerance analysis method for the photonic assembly domain. It systematically links the characteristics of microfabrication technology to optical coupling. A package design is described by physical tolerance elements, each one representing a specific microfabrication step. Mathematical tolerance elements are added to complete the package description. When error distributions are added to the physical tolerance elements, an optical coupling expectancy of a package design can be obtained. Moreover, it is demonstrated how this method can be used to characterize the contribution of individual design and fabrication factors to the total alignment performance. The tolerance analysis method is generic in nature since it makes use of elementary and generic fabrication steps. The main benefits of the method are its systematic approach, the insight it gives in the performance of a design as well as the major factors affecting the performance, and the analysis of design variations it allows.

Tolerance analysis of mechanical assemblies based on small displacement torsor and deviation propagation theories

Construction of the quality tolerances (Q-T) function is a key step in assembly tolerance analysis process. This paper presents a new feature-based tolerance analysis method for mechanical assemblies with GDT (2) the analysis process and acquired Q-T function is relatively concise to the other feature-based method; (3) the integrated assembly information is more intact. A case study is provided to illustrate the proposed method.

投影光刻物镜波像差的公差分析方法

A simple and reliable method of tolerance analysis is proposed to ensure the performance of lithographic projection lens and minimize the manufacturing costs of lithographic projection lens.This method defines P-V wavefront error as merit function based on RMS wavefront error,and selects the optimal compensators.Compared to present methods,this method decreases mechanical complexity and minimizes the manufacturing costs by adopting less compensators,with both RMS wavefront error and P-V wavefront error achieving the requirement.As an example the method is applied to a refractive projection lens for 90nm resolution lithography projection lens designed by us.The results show that,only with 7compensators,RMS wavefront error and P-V wavefront error at 97.7%probability are less than 0.0412λand 0.2469λ,respectively.In conclusion,this method is able to meet the performance requirement of lithographic projection lens.

A Parameterized Method for Assembly Tolerance Analysis Based on Deviation Propagation Theory

The paper proposed a new tolerance analysis method: a unified tolerance and deviation model is adopted to synthesize different kinds of tolerances; an integrated tolerance propagation model is designed to construct the functional relationship between the tolerances and quality requirements. The method is simpler and more widely applicable than the traditional linearization analysis method. A case study is provided to illustrate the proposed method.

Sample Size Determination in NT-Net Quasi-Monte Carlo Simulation

Monte Carlo (MC) technique prevails in probabilistic design simulation, such as in statistical tolerance analysis and synthesis. A quasi-Monte Carlo (e.g., number theoretic net method (NT-net)) with better computation efficiency over MC has recently attracted interests in application. In spite of a comprehensive case study (Huang et al., 2004, “Tolerance Analysis for Design of Multistage Manufacturing Processes Using Number-Theoretical Net Method (NT-net),” Int. J. Flexible Manuf. Syst., 16, pp. 65–90 and Zhou et al., 2001, “Sequential Algorithm Based on Number Theoretic Method for Tolerance Analysis and Synthesis,” ASME J. Manuf. Sci. Eng., 123(3), pp. 490–493) for comparison between NT-net and MC, a method for sample size determination of NT-net is still not available. Combinatorial theory and the solution of occupancy problem are used for estimating equivalent sample sizes of MC and NT-net, allowing the NT-net sample size determination in application. A multivariate Chebyshev polynomial with variant coefficients is used to represent generic design functions for validation. The results are verified by case studies.

Tolerance Analysis Method Using Improved Convolution Method

Convolution method is studied to analyze statistical tolerance for linear dimension chain and nonlinear dimension chain. Hybrid convolution method is proposed, which is the integration of analytical convolution and numerical convolution. In order to reduce the algorithm errors, improved convolution method is proposed. Comparing with other statistical tolerance analysis methods, this method is faster and accurate. At last, an example is used to demonstrate the method proposed in this paper.

Data-Driven Tolerance Analysis for Products Assembly

Tolerance characteristics have an important influence on products dynamic properties. To realize dynamic properties oriented tolerance analysis and design of mechanical products, a data-driven based tolerance analysis method was proposed. Firstly, based on the cloud model theory, an assemble cloud model was established, and a new method of assembly data processing and knowledge mining was proposed. Secondly, the assembly cloud model was used to analysis the measured data from practical assembly process and search the alignment laws. Finally, using the above mentioned methods, product key parts tolerance was analyzed and adjusted, and the product quality consistency could be improved dramatically.

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.

A Tolerance Analysis Method for Assembly Yield

The major objective of tolerance analysis is to check whether the resultant or assembly tolerances conform to the design specifications in individual components. Although the first four moments were used to describe assembly reject rate, there are no further applications in reliability index and assembly yield. Therefore, a novel tolerance analysis method to estimate the assembly yield based on first four moment reliability index is proposed. Firstly, some basic definitions are given. Secondly, the tolerance analysis model based on the first fourth moment reliability index (FMRI) is developed. Furthermore the assembly yield models based on FMRI, for both individual design function and multi-design functions are presented.

A novel statistical tolerance analysis method for assembled parts

Tolerance analysis of an assembly is an important issue for mechanical design. Among various tolerance analysis methods, statistical analysis is the most commonly employed method. However, the conventional statistical tolerance method is often based on the normal distribution. It fails to predict the resultant tolerance of an assembly of parts with non-normal distributions. In this paper, a novel method based on statistical moments is proposed. Tolerance distributions of parts are first transferred into statistical moments that are then used for computing tolerance stack-up. The computed moments, particularly the variance, the skewness and the kurtosis, are then mapped back to probability distributions in order to calculate the resultant tolerance of the assembly. The proposed method can be used to analyse the resultant tolerance specification for non-normal distributions with different skewness and kurtosis. Simulated results showed that tail coefficients of different distributions with the same kurtosis are close to each other for normalised probabilities between −3 and 3. That is, the tail coefficients of a statistical distribution can be predicted by the coefficients of skewness and kurtosis. Two examples are illustrated in the paper to demonstrate the proposed method. The predicted resultant tolerances of the two examples are only 0.5% and 1.5% differences compared with that by the Monte Carlo simulation for 1,000,000 samples. The proposed method is much faster in computation with higher accuracy than conventional statistical tolerance methods. The merit of the proposed method is that the computation is fast and comparatively accurate for both symmetrical and unsymmetrical distributions, particularly when the required probability is between ±2σ and ±3σ.