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.

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.

Geometric tolerance analysis through Jacobian model for rigid assemblies with translational deviations

2016 ◽  
Vol 36 (1) ◽  
pp. 72-79 ◽  
Author(s):  
Wilma Polini ◽  
Andrea Corrado
Keyword(s):  
Tolerance Analysis ◽  
Geometric Error ◽  
Functional Requirements ◽  
Content Type ◽  
Geometric Tolerances ◽  
Mechanical Assemblies ◽  
Geometric Deviations ◽  
2D And 3D ◽  
Practical Implications

Purpose – The purpose of this paper is to carry out a tolerance analysis with geometric tolerances by means of the Jacobian model. Tolerance analysis is an important task to design and to manufacture high-precision mechanical assemblies; it has received considerable attention by the literature. The Jacobian model is one of the methods proposed by the literature for tolerance analysis. The Jacobian model cannot deal with geometric tolerances for mechanical assemblies. The geometric tolerances may not be neglected for assemblies, as they significantly influence their functional requirements. Design/methodology/approach – This paper presents how it is possible to deal with geometric tolerances when a tolerance analysis is carried out by means of a Jacobian model for a 2D and 3D assemblies for which the geometric tolerances applied to the components involve only translational deviations. The three proposed approaches modify the expression of the stack-up function to overcome the shortage of Jacobian model that the geometric error cannot be processed. Findings – The proposed approach has been applied to a case study. The results of the case study show how, when a statistical approach is implemented, the Jacobian model with the three developed methods gives results very similar to those due to other models of the literature, such as vector loop and variational. Research limitations/implications – In particular, the proposed approach may be applied only when the applied geometrical tolerances involve translational variations in 3D assemblies. Practical implications – Tolerance analysis is a valid tool to foresee geometric interferences among the components of an assembly before getting the physical assembly. It involves a decrease of the manufacturing costs. Originality/value – The original contribution of the paper is due to three methods to make a Jacobian model able to consider form and geometric deviations.

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.

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

2020 ◽  
Vol 7 (2) ◽  
pp. 165-176
Author(s):  
Heping Peng ◽  
Zhuoqun Peng
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Iterative Method ◽  
Iterative Process ◽  
Simulated Data ◽  
Small Displacement ◽  
Real Multiplication ◽  
Small Displacement Torsor ◽  
Statistical Tolerance

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.

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.

Tolerance Analysis in Machining: An Approach Combining the Model of Manufactured Part and the Jacobian-Torsor Model

2008 ◽  
Author(s):  
Mojtaba Kamali Nejad ◽  
Alain Desrochers ◽  
Franc¸ois Villeneuve ◽  
Fre´de´ric Vignat
Keyword(s):  
Kinematic Chain ◽  
Tolerance Analysis ◽  
Small Displacement ◽  
Functional Requirements ◽  
Combined Approach ◽  
Functional Elements ◽  
New Approach ◽  
Nominal Model ◽  
Small Displacement Torsor ◽  
Position And Orientation

To perform tolerance analysis in machining, a combined approach which blends the benefits of the Model of Manufactured Part (the MMP model) and the Jacobian-Torsor model is proposed. The former is based on the CAD nominal model, where deviations are described relative to the nominal part using small displacement torsor. The later starts with the kinematic dimension chains and expresses the relative position and orientation of the various components of the chosen kinematic chain by Jacobian matrices. The Jacobian-Torsor model uses interval arithmetic for expressing the possible variation of the functional elements and for calculating the extreme bounds of the functional requirements. In the following sections, the two aforementioned models will first be outlined before the new combined approach for tolerance analysis in machining is presented. This new approach uses the advantages of the MMP model to simulate the machining operation, taking into account positioning and machining defects. Furthermore it takes advantages of the interval-based formulation which has been used in the Jacobian Torsor model. The combined approach is finally applied on an example.

scholarly journals Probabilistic Tsunami Loss Estimation Methodology: Stochastic Earthquake Scenario Approach

2017 ◽  
Vol 33 (4) ◽  
pp. 1301-1323 ◽  
Author(s):  
Katsuichiro Goda ◽  
Raffaele De Risi
Keyword(s):  
Effective Means ◽  
Loss Estimation ◽  
Source Modeling ◽  
Earthquake Scenario ◽  
Tsunami Disaster ◽  
Plain Area ◽  
Set Up ◽  
Tsunami Fragility ◽  
Sendai Plain

This study develops a probabilistic tsunami loss estimation methodology for enhancing community resilience against tsunami disasters. The method is based on novel stochastic earthquake source modeling and state-of-the-art tsunami fragility modeling. It facilitates the quantitative evaluation of tsunami loss for coastal community by accounting for uncertainties of earthquake occurrence and rupture characteristics. A case study is set up to illustrate an application of the developed method to the Sendai Plain area by focusing on possible tsunami events in the Tohoku region of Japan. The quantitative tsunami hazard as well as risk assessment results serve as effective means to make decisions regarding tsunami disaster risk reduction.

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