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

2002 ◽  
Vol 124 (4) ◽  
pp. 609-622 ◽  
Author(s):  
J. K. Davidson ◽  
A. Mujezinovic´ ◽  
J. J. Shah
Keyword(s):  
Mathematical Model ◽  
Convex Set ◽  
New Model ◽  
Planar Surfaces ◽  
Geometric Tolerances ◽  
Trade Offs ◽  
Level Model ◽  
Probability Methods

A new mathematical model is presented for representing the tolerances of planar surfaces. The model is compatible with the ASME Standard for geometric tolerances. Central to the new model is a Tolerance-Map®,1 a hypothetical volume of points which corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, 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. Stackup relations are developed for parts where the centers of faces are offset laterally. 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 Round Faces

2000 ◽  
Author(s):  
J. K. Davidson ◽  
A. Mujezinovic ◽  
J. J. Shah
Keyword(s):  
Mathematical Model ◽  
Convex Set ◽  
New Model ◽  
Planar Surfaces ◽  
Geometric Tolerances ◽  
Trade Offs ◽  
Level Model ◽  
Probability Methods

Abstract A new mathematical model is presented for the tolerances of planar surfaces. The model is compatible with the ASME Standard 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 plane which can arise from tolerances on size, form, and orientation. Every Tolerance-Map®2 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. Stackup relations are developed for parts where the centers of faces are offset laterally. 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 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.

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.

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.

Geometric tolerances: A new application for line geometry and screws

2002 ◽  
Vol 216 (1) ◽  
pp. 95-103 ◽  
Author(s):  
J K Davidson ◽  
J J Shah
Keyword(s):  
Mathematical Model ◽  
Cylindrical Surface ◽  
Line Geometry ◽  
Tolerance Zone ◽  
New Model ◽  
Plücker Coordinates ◽  
Geometric Tolerances ◽  
Size Tolerance

A new mathematical model is introduced for the tolerances of cylindrical surfaces. The model is compatible with the ISO/ANSI/ASME standard 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 line (the axis) that can arise from tolerances on size, location and orientation of the cylindrical surface. Each axis in a tolerance zone will be represented with the six Plücker coordinates. Cylindrical surfaces in a tolerance zone for the same hole can then be treated by attaching a size tolerance to each of the lines, thereby forming a screw. Relationships for the content of line solids for a tolerance zone are developed to correspond to the variations of locations. These are then used to obtain a measure for the increment in cost when a more refined tolerance is specified. This model is one part of a bilevel model that is under development for geometric tolerances.

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.

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.

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