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.

Using Tolerance-Maps to Generate Frequency Distributions of Clearance for Pin-Hole Assemblies

2006 ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Frequency Distribution ◽  
Probabilistic Representation ◽  
Frequency Distributions ◽  
New Model ◽  
Iso Standards ◽  
Material Condition ◽  
Geometric Tolerances ◽  
The Future

A new mathematical model for representing the geometric variations of lines is extended to include probabilistic representations of 1-D clearance which arise from multidimensional variations of an axis, a hole and a pin-hole assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (T-Map), a hypothetical volume of points that models the 3-D variations in location and orientation for a segment of a line (the axis), which can arise from tolerances on size, position, orientation, and form. Here it is extended to model the increase in yield that occurs when maximum material condition (MMC) is specified. The frequency distribution of 1-D clearance is decomposed into manufacturing bias, i.e. toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is focused on geometric bias and manufacturing bias is presumed to be uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled pin and hole.

Using Tolerance-Maps to Generate Frequency Distributions of Clearance and Allocate Tolerances for Pin-Hole Assemblies

2007 ◽  
Vol 7 (4) ◽  
pp. 347-359 ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Statistical Method ◽  
Frequency Distribution ◽  
Probabilistic Representation ◽  
Worst Case ◽  
Frequency Distributions ◽  
One Dimensional ◽  
Material Condition ◽  
Geometric Tolerances ◽  
Case Basis

A new mathematical model for representing the geometric variations of lines is extended to include probabilistic representations of one-dimensional (1D) clearance, which arise from positional variations of the axis of a hole, the size of the hole, and a pin-hole assembly. The model is compatible with the ASME/ ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242), a hypothetical volume of points that models the 3D variations in location and orientation for a segment of a line (the axis), which can arise from tolerances on size, position, orientation, and form. Here, it is extended to model the increases in yield that occur when maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis; the statistical method includes the specification of both size and position tolerances on a feature. The frequency distribution of 1D clearance is decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is built from geometric bias, and it is presumed that manufacturing bias is uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled pin and hole. A comparison is made between the effects of choosing the optional material condition MMC and not choosing it with the tolerances that determine the allowable variations in position.

Automating Linear Tolerance Analysis Across Assemblies

1992 ◽  
Vol 114 (1) ◽  
pp. 174-179 ◽  
Author(s):  
N. P. Juster ◽  
P. M. Dew ◽  
A. de Pennington
Keyword(s):  
Mathematical Model ◽  
Manufacturing Process ◽  
Tolerance Analysis ◽  
Mathematical Framework ◽  
Worst Case ◽  
Experimental Software

One of the tests carried out by designers in an attempt to check whether an assembly of components will function correctly is tolerance analysis. Tolerance analysis, although relatively straightforward, is liable to be time consuming and error prone. It cannot be automated unless a suitable mathematical framework is developed to model the variations introduced by the manufacturing process. The designer allows for the variations by means of tolerances attached to the dimensions. This paper describes a suitable mathematical model and shows how it may be used to automate linear worst case tolerance analysis across assemblies. Experimental software has been written, based on the theory.

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.

Statistical Tolerance Allocation for Tab-Slot Assemblies Utilizing Tolerance-Maps

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Tolerance Allocation ◽  
Probabilistic Representation ◽  
Worst Case ◽  
Material Condition ◽  
Geometric Tolerances ◽  
Constant Size ◽  
Medial Plane ◽  
Statistical Tolerance ◽  
Tolerance Assignment ◽  
Case Basis

A new mathematical model for representing the geometric variations of tabs/slots is extended to include probabilistic representations of 1D clearance. The 1D clearance can be determined from multidimensional variations of the medial-plane for a slot or a tab, and from variations of both medial-planes in a tab-slot assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (Patent No. 6963824) (T-Map), a hypothetical volume of points that models the range of 3D variations in location and orientation for a segment of a plane (the medial-plane), which can arise from tolerances on size, position, orientation, and form. Here it is extended to model the increases in yield that occur when the optional maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis. The frequency distribution of 1D clearance is decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is built from geometric bias, and it is presumed that manufacturing bias is uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled tab and slot. A comparison is made between the effects of specifying the optional MMC and not specifying it with the tolerance that determines the allowable variations in position of a tab, a slot, or of both in a tab-slot assembly. Statistical tolerance assignment for the tab-slot assembly is computed based on initial worst-case tolerances and for (a) constant size of tab and slot at maximum material condition, and (b) constant virtual-condition size.

A New Mathematical Model for Geometric Tolerances as Applied to Axes

2003 ◽  
Author(s):  
S. Bhide ◽  
J. K. Davidson ◽  
J. J. Shah
Keyword(s):  
Mathematical Model ◽  
Metric Space ◽  
Convex Set ◽  
New Model ◽  
Iso Standards ◽  
Geometric Tolerances

A new mathematical model for representing the geometric variations of lines is extended to include form and accumulation (stackup) of tolerances in an assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map©, a hypothetical volume of points which corresponds to all possible locations and variations of a segment of a line (the axis) which can arise from tolerances on size, position, orientation, and form. Every Tolerance-Map is a convex set in a metric space. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Much of the detail in this paper would probably reside internally to software for designers, yet would not be included in the interface; its workings should be invisible to the user.

A New Mathematical Model for Geometric Tolerances As Applied to Rectangular Faces

2001 ◽  
Author(s):  
A. Mujezinović ◽  
J. K. Davidson ◽  
J. J. Shah
Keyword(s):  
Mathematical Model ◽  
Reference Frames ◽  
Convex Set ◽  
New Model ◽  
Iso Standards ◽  
Geometric Tolerances ◽  
Trade Offs ◽  
Level Model ◽  
Probability Methods

Abstract A new mathematical model for representing geometric tolerances is applied to rectangular faces, is extended to show its sensitivity to the precedence (ordering) of datum reference frames, and is adapted to include material modifiers. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map©1, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Methods are introduced whereby designers can identify trade-offs and optimize the allocation of tolerances. Examples are presented that illustrate important features of the new model.

A New Mathematical Model for Geometric Tolerances as Applied to Polygonal Faces

2003 ◽  
Vol 126 (3) ◽  
pp. 504-518 ◽  
Author(s):  
A. Mujezinovic´ ◽  
J. K. Davidson ◽  
J. J. Shah
Keyword(s):  
Mathematical Model ◽  
Reference Frames ◽  
Convex Set ◽  
New Model ◽  
Iso Standards ◽  
Geometric Tolerances ◽  
Trade Offs ◽  
Level Model ◽  
Probability Methods

A new mathematical model for representing geometric tolerances is applied to polygonal faces and is extended to show its sensitivity to the precedence (ordering) of datum reference frames. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map®2, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Methods are introduced whereby designers can identify trade-offs and optimize the allocation of tolerances. Examples are presented that illustrate important features of the new model.

The Effects of Different Specifications on the Tolerance-Maps for an Angled Face

2004 ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Convex Set ◽  
Angular Size ◽  
Linear Size ◽  
New Model ◽  
Iso Standards ◽  
Geometric Tolerances ◽  
Dimensioning And Tolerancing ◽  
Level Model ◽  
Size Tolerance

A new mathematical model for representing geometric tolerances is applied to a part with an angled face and is extended to show its sensitivity to different specifications for dimensioning and tolerancing the part. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map®, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. This paper develops several Tolerance-Maps for a part with an angled end face for different tolerance specifications. These specifications are linear size, angularity, angular size, “linear size & angularity” and “linear & angular size” tolerance. Comparison of Tolerance-Maps for their content for these specifications led to the following conclusions: a) only angular size tolerance is not sufficient for tolerancing an angled face; b) if the value of tolerance remains the same, the allowable variation is more in a part having only an angularity tolerance than in one having only a size tolerance.

Using Tolerance-Maps to Generate Frequency Distributions of Clearance for Tab-Slot Assemblies

2007 ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Frequency Distribution ◽  
Probabilistic Representation ◽  
Worst Case ◽  
Frequency Distributions ◽  
New Model ◽  
Material Condition ◽  
Geometric Tolerances ◽  
Medial Plane ◽  
Case Basis

A new mathematical model for representing the geometric variations of tabs/slots is extended to include probabilistic representations of 1-D clearance which can be determined from multi-dimensional variations of the medial plane for a slot or a tab, and from variations of both medial planes in a tab-slot assembly. The model is compatible with the ASME/ANSI/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map (T-Map), a hypothetical volume of points that models the range of 3-D variations in location and orientation for a segment of a plane (the medial plane), which can arise from tolerances on size, position, orientation, and form. Here it is extended to model the increases in yield that occur when the optional maximum material condition (MMC) is specified and when tolerances are assigned statistically rather than on a worst-case basis. The frequency distribution of 1-D clearance is decomposed into manufacturing bias, i.e. toward certain regions of a Tolerance-Map, and into a geometric bias that can be computed from the geometry of multidimensional T-Maps. Although the probabilistic representation in this paper is built from geometric bias, and it is presumed that manufacturing bias is uniform, the method is robust enough to include manufacturing bias in the future. Geometric bias alone shows a greater likelihood of small clearances than large clearances between an assembled tab and slot. A comparison is made between the effects of choosing the optional MMC and not choosing it with the tolerance that determines the allowable variations in position of a slot.

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