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.

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.

Formation of Geometric Tolerance Zones for Polyhedral Objects

1997 ◽  
Author(s):  
Utpal Roy ◽  
Bing Li
Keyword(s):  
Variational Model ◽  
Tolerance Zone ◽  
Geometric Tolerance ◽  
Geometric Tolerances ◽  
Polyhedral Objects ◽  
Different Types ◽  
Tolerance Zones ◽  
Algebraic Constraints ◽  
Size Tolerance ◽  
Part Model

Abstract This paper presents a scheme for establishing geometric tolerance zones for polyhedral objects in solid modelers. The proposed scheme is based on a surface-based variational model. Variations are applied to a part model by varying each surface’s model variables. Those model variables are constrained by some algebraic relations derived from the specified geometric tolerances. For size tolerance, two types of tolerance zones are considered in order to reflect two different types of size tolerances. For any other geometric tolerance (form, orientation or positional), the resultant tolerance zone is defined by the combination of size tolerance and that particular geometric tolerance specifications. Appropriate algebraic constraints (on the model variables) are finally used to establish the tolerance zone boundaries in the surface-based variational model.

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.

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 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 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 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.

Effects of Size, Orientation, and Form Tolerances on the Frequency Distributions of Clearance Between Two Planar Faces

2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Gaurav Ameta ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Mathematical Model ◽  
Uniform Distribution ◽  
Frequency Distribution ◽  
Gaussian Distribution ◽  
Planar Surface ◽  
Tolerance Zone ◽  
Frequency Distributions ◽  
Geometric Tolerances ◽  
Paper Form ◽  
Truncated Gaussian

A new mathematical model for representing the geometric variations of a planar surface is extended to include probabilistic representations for a 1D dimension of interest, which can be determined from multidimensional variations of the planar surface on a part. 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. 6963824), a hypothetical volume of points that models the 3D variations in location and orientation of a feature, which can arise from tolerances on size, position, orientation, and form. The 3D variations of a planar surface are decomposed into manufacturing bias, i.e., toward certain regions of a Tolerance-Map, and into geometric bias that can be computed from the geometry of T-Maps. The geometric bias arises from the shape of the feature, the tolerance-zone, and the control used on the mating envelope. Influence of manufacturing bias on the frequency distribution of 1D dimension of interest is demonstrated with two examples: the multidimensional truncated Gaussian distribution and the uniform distribution. In this paper, form and orientation variations are incorporated as subsets in order to model the coupling between size and form variations, as permitted by the ASME Standard when the amounts of these variations differ. Two distributions for flatness, i.e., the uniform distribution and the Gaussian distribution that has been truncated symmetrically to six standard deviations, are used as examples to illustrate the influence of form on the dimension of interest. The influence of orientation (parallelism and perpendicularity) refinement on the frequency distribution for the dimension of interest is demonstrated. Although rectangular faces are utilized in this paper to illustrate the method, the same techniques may be applied to any convex plane-segment that serves as a target face.

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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