A Comparison Study of Mathematical Models for Tolerance Analysis

2013 ◽  
Vol 765-767 ◽  
pp. 759-762
Author(s):  
Jian Xin Yang ◽  
Zhen Tao Liu ◽  
Ben Zhao
Keyword(s):  
Tolerance Analysis ◽  
Mathematical Representation ◽  
Small Displacement ◽  
Propagation Mechanism ◽  
Comparison Study ◽  
Worst Case ◽  
Geometric Tolerance ◽  
Geometric Tolerances ◽  
Small Displacement Torsor ◽  
Comprehensive Comparison

This paper reviews two major models (Small Displacement Torsor, Deviation and Clearance Domain) for 3D functional tolerance analysis and compares them. The underlying mathematical representation of geometric tolerances can be classified as inequalities and multi-variate region. The corresponding algebraic or geometric tolerance propagation mechanism of each model is briefly introduced for worst-case and statistical tolerancing. Through a comprehensive comparison of these models, this paper gives some suggestions for choosing the appropriate method for a given tolerancing problem.

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

2014 ◽  
Vol 229 (14) ◽  
pp. 2514-2523 ◽  
Author(s):  
Sun Jin ◽  
Hua Chen ◽  
Zhimin Li ◽  
Xinmin Lai
Keyword(s):  
Three Dimensional ◽  
Tolerance Analysis ◽  
Dimensional Euclidean Space ◽  
Small Displacement ◽  
Analysis Method ◽  
Tolerance Zone ◽  
Geometric Tolerances ◽  
Small Displacement Torsor ◽  
Conical Surfaces ◽  
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.

A Comparison Study of Small Displacement Torsor and Analysis Line Methods for Functional Tolerance Analysis

2012 ◽  
Vol 605-607 ◽  
pp. 358-364
Author(s):  
Chun Li Li ◽  
Jian Xin Yang ◽  
Jun Ying Wang ◽  
Wen Xin Ma
Keyword(s):  
Computation Time ◽  
Tolerance Analysis ◽  
General Characteristic ◽  
Worst Case Analysis ◽  
Small Displacement ◽  
Worst Case ◽  
Small Displacement Torsor ◽  
Functional Tolerance ◽  
Tolerance Accumulation ◽  
Analysis Line

Tolerance analysis plays an important role in the stage of product design and has great influences on the product assembly quality and manufacturing costs. Two major methods are used for three-dimensional functional tolerance analysis, which are small displacement torsor and analysis line. A positioning mechanism with two parts is presented for tolerance accumulation calculation. Through the comparison of these two methods on computation processes and results, analysis line method can establish the explicit relationship between the functional requirement and the tolerances of the influential part, which allows finding the accumulation results in the worst-case and statistical conditions. However, it requires the determination of transfer relationship case by case. For small displacement torsor model, it permits a set of inequalities to express the tolerance zones, which yields a linear programming problem. It is applicable to different tolerance chains for its general characteristic. However it is adopted only for the worst-case analysis and requires more computation time.

A CAD model for the tolerancing of mechanical assemblies considering non-rigid joints between parts with defects

2021 ◽  
pp. 095440542110257
Author(s):  
Anis Korbi ◽  
Mehdi Tlija ◽  
Borhen Louhichi
Keyword(s):  
Tolerance Analysis ◽  
Design Stage ◽  
Small Displacement ◽  
Worst Case ◽  
Mechanical Assemblies ◽  
Proposed Model ◽  
Small Displacement Torsor ◽  
Rigid Joints ◽  
Geometrical Defects ◽  
The Ideal

During the design stage, the ideal simulation and visualization of the mechanical assemblies behavior require the modeling of parts with dimensional and geometrical defects. However, the deviations caused by parts deformations can generate an important difference between the ideal assembly and the real product. In this regard, this paper proposes a tolerance analysis method of CAD assemblies considering non-rigid joints between parts with defects. The determination of realistic rigid components with dimensional and geometrical defects is based on the worst case tolerancing approach and the Small Displacement Torsor (SDT) parameters. The Finite Element (FE) computation is executed to determine deformations of realistic non-rigid part models under external loads. Sub-algorithms to define non-rigid joints between realistic parts are developed. The tolerance analysis is established using the realistic CAD assembly. A case study is presented to evaluate the proposed model.

Semantic Representation of Flatness Based on ALC(R) and Mathematical Definition

2013 ◽  
Vol 662 ◽  
pp. 961-965
Author(s):  
Yun Feng Xie ◽  
Yu Lu Du ◽  
Yan Ru Zhong ◽  
Yu Chu Qin
Keyword(s):  
Semantic Representation ◽  
Heterogeneous Systems ◽  
Mathematical Representation ◽  
Small Displacement ◽  
3D Cad ◽  
Logical Language ◽  
Small Displacement Torsor ◽  
Tolerance Specification ◽  
Representation Language ◽  
Important Significance

The information of form tolerances in existing 3D CAD systems is just a kind of symbol and text which is lack of engineering semantic at present. Therefore, a reasonable explanation and Semantic representation of form tolerances has very important significance. In order to reduce the uncertainty and support the semantic interoperability in tolerance specification design, an approach for mathematical representation of flatness based on the Small Displacement Torsor (SDT) is proposed. Based on this, a representation of flatness using description logical language ALC(R) with concrete domain is proposed. Then flatness information is formalized using OWL, an ontology representation language devised by W3C, to be shared and interoperated between heterogeneous systems by building OWL ontology. At last, there is a practical example to verify the feasibility of this approach.

A small displacement torsor model for tolerance analysis in a workpiece-fixture assembly

2009 ◽  
Vol 223 (8) ◽  
pp. 1005-1020 ◽  
Author(s):  
J N Asante
Keyword(s):  
Effective Means ◽  
Experimental System ◽  
Tolerance Analysis ◽  
Geometric Error ◽  
Small Displacement ◽  
Small Displacement Torsor ◽  
Set Up ◽  
Workpiece Setup ◽  
Feature Constraints

Workpiece geometric error, locator geometric error, and clamping error are factors that influence workpiece setup in workpiece fixturing. These errors accumulate and propagate during fixturing. They may be the reason for a machined feature being out of tolerance after machining. This paper presents a methodology for modelling and analysing the combined effect of these errors on a machined feature. Deviation of a machined feature due to the combined errors is expressed in terms of the small displacement torsor parameters. Given a tolerance on the machined feature, constraints are specified for that feature to establish a relationship between the tolerance zone of the feature and the torsor parameters. These constraints provide boundaries within which the machined feature must lie. This is used for tolerance analysis of the machined feature. A case study example was used to illustrate the approach. An experimental system was also set up to verify the analytical model. The results show that this approach offers an effective means for fixturing tolerance analysis.

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

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Gagandeep Singh ◽  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Statistical Study ◽  
Tolerance Analysis ◽  
Worst Case ◽  
Iso Standards ◽  
Statistical Measures ◽  
Geometric Tolerances ◽  
Different Types ◽  
The Future ◽  
Tolerance Assignment

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.

Derivation of Generalized Datum Reference Frames for Geometric Tolerance Analysis

1993 ◽  
Author(s):  
Swami D. Nigam ◽  
James D. Guilford ◽  
Joshua U. Turner
Keyword(s):  
Coordinate System ◽  
Reference Frame ◽  
Reference Frames ◽  
Tolerance Analysis ◽  
Intermediate Representation ◽  
Coordinate Systems ◽  
Geometric Tolerance ◽  
Geometric Tolerances ◽  
Concise Representation ◽  
General Method

Abstract Datum reference frames define coordinate systems for use in determining part compliance with geometric tolerances. A datum reference frame is specified based on the perfect nominal geometry of the part features called out as datums. However, the actual computation of a coordinate system frame of reference from the datum callouts becomes quite challenging when the features depart from nominal location, orientation, size, and form. We present a general method for representing datum reference frames (both partial and complete), and for computing a coordinate system from a simulated varianced part and a datum reference frame specification. The method makes use of built-in construction procedures, and derived or “virtual” geometry, in conjunction with a powerful parts positioning module that simulates the placement of the varianced part in a fixture represented by the datum surfaces. The reliance on virtual geometry as an intermediate representation, permits the concise representation of not only the datum reference frame types defined in the standard, but also allows for any arbitrary datum reference frames constructed by the user.

scholarly journals Optimisation and Reliability for Geometrical Tolerance Value against Positional Characteristic in Rotational Shaft System

2020 ◽  
Vol 8 (6) ◽  
pp. 3713-3722
Keyword(s):  
Tolerance Analysis ◽  
Design Stage ◽  
Mathematical Representation ◽  
Security Level ◽  
Rotating System ◽  
Software Application ◽  
Geometric Tolerance ◽  
Size Population ◽  
System Life ◽  
Optimum Solution

In the process of manufacture and installation, geometrical dimensions and tolerances (GD&T) should be taken into consideration to improve reliability and reduce the adverse impact on critical parts of the rotating system. GD&T must be considered by manufacturers and assembly worker. This paper presents an analysis of geometrical tolerance (GT) values in rotational shaft using the genetic algorithm (GA) method. GA optimization uses a geometric mathematical model. Mathematical models were developed using the offset and algebraic methods to calculate the ideal geometric features that best fit a set of positioning points based on the standard equations criteria. The calculation using the Matlab software application will use the optimal GA parameter. There is a combination of four genetic parameters associated with size population, crossover, mutation and stop state will develop algorithm performance which will produce optimum GT value. The geometric tolerance value for the position characteristic was analyzed to determine and predict the probability and reliability shaft in rotational system. Comparative values of each GT value are compared to find out the reliability values obtained can be used and verify the GT value requirements require mathematical representation. Tolerance analysis at the design stage to evaluate and predict quality by considering the probability of failure rates. The Actual value of the radius must be small from the allowable radius (Ract <Rallow) to cope with the high failure rate throughout the operating period. Due to dynamic nature of the shaft round and the possibility of a variable size of shafts, the GT value should be analyzed to ensure that the value obtained is correct and can be optimum solution to this problem. The GT value to be considered is at the center of the shaft involved which will affect the relevant components. Impact of GT value on the destruction of system critical component such as bearings, gear and couplers as benchmark for review for the optimization of shaft geometric tolerances in rotating machines to overcome the problem and improve concentricity shaft. The contribution of this study is to examine the effect of shaft size and the value of geometric tolerance on system reliability. Estimating and predicting levels of reliability more accurately improves system life, knows the system's impact accurately, knows the security level of a rotating system and also knows the quality of the mechanism at the design level.

Geometric Tolerance Analysis for Mechanism Design

1996 ◽  
Author(s):  
Jhy-Cherng Tsai
Keyword(s):  
Mechanism Design ◽  
Tolerance Analysis ◽  
Variational Models ◽  
Factors Affecting ◽  
Geometric Tolerance ◽  
Geometric Tolerances ◽  
Statistical Tolerance ◽  
Mechanical Devices ◽  
Major Factors ◽  
Representation Scheme

Abstract Manufacturing tolerances and joint clearances are the two major factors affecting mechanism accuracy. As error analysis is one of the bottlenecks of precision machinery design, methods for geometric tolerance analysis must be investigated for mechanism design. This paper describes an approach for analyzing errors caused by geometric tolerances and clearances in mechanism design. The method consists of three parts: variational kinematic models for geometric tolerances, a systematic geometric dimensioning and tolerancing (GD&T) representation scheme, and computation methods for interval and statistical tolerances. Variational models are based on differential transformation to model kinematic errors caused by tolerances and clearances. The model is consistent with error models used in typical mechanical devices. The GD&T scheme, called the Tolerance Network (TN), employs graph theory for representing GD&T as well as fitting specifications of a design is described. Errors are propagated by traversal throughout the network and stack-up of these variational models along the dominate path in the TN. Error computation methods for both interval and statistical tolerance types are discussed. A method for computing central moments, rather than analytical distributions, of statistical tolerances is developed to reduce the computation complexity. A five-degree-of-freedom robot is used as an example at each step to illustrate this approach.

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