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.

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.

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.

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.

Process Capability Based Tolerance Allocation in Pneumatic Percussive Riveting

2020 ◽  
Author(s):  
Hua Wang ◽  
Jialei Zhang ◽  
Junyang Yu
Keyword(s):  
High Efficiency ◽  
Load Transfer ◽  
Process Capability ◽  
Tolerance Analysis ◽  
Tolerance Allocation ◽  
Hole Drilling ◽  
Case Method ◽  
Pneumatic Hammer ◽  
Worst Case ◽  
Mating Conditions

Abstract Pneumatic percussive riveting is an important way to join the sheet metals. In order to ensure the load transfer and the fatigue performance of riveted joint, the interference of the rivet/hole is strictly specified. The interference of the rivet/hole is highly correlated with the process capability of the pneumatic hammer and the diameter of the pre-hole. It is a critical step to choose the appropriate pneumatic hammer to ensure the interference requirements. Energy per blow of the pneumatic hammer is a proclaimed parameter from the riveting hammer manufacturer. It is difficult for the designer to choose the riveting hammer in practical riveting scheme based on energy per blow. Tolerance analysis is an efficient way to model the manufacturing variation and implement process control. This paper presents the tolerance allocation of the pneumatic percussive riveting based on the process capability of the pneumatic hammer. In order to obtain the designed interferences of the rivet/hole, a tolerance chain is built with the process capability of the pneumatic hammer, the diameter of the pre-hole and the diameter of the rivet shank. The process capability of the pneumatic hammer is represented with the interferences of the rivet/hole after riveting. It is an intuitive parameter for the designer to choose the riveting hammer in practical riveting scheme. The process capability of the pneumatic hammer is obtained from the designed riveting experiments with the pneumatic percussive riveting platform. The diameter of the pre-hole affects the interference of the rivet/hole also. The tolerance for manual hole-drilling should be determined to assure the interference requirements and high drilling operation efficiency simultaneously. The variation of the pre-hole is obtained from the manual drilling experiments and diameter measurements. Different hole-drilling results in different mating conditions between the pre-hole and the rivet. The fit conditions of different pre-holes are modeled and the final interferences after riveting are analyzed. Worst case method and statistical analysis method are two common methods for tolerance analysis. For the manual hole-drilling and the pneumatic percussive riveting, worst case method is employed to analyze the constructed tolerance chain in order to accomplish the interferences of the rivet/hole. The different analyzed dimensions, rivet-hole clearances and pre-hole drilling variation, are investigated respectively. The reported work enhances the understanding of the tolerance allocation for the pneumatic percussive riveting. The interference based process capability of the pneumatic hammer provides good reference for pneumatic hammer choosing in riveting scheme. The reported tolerance chain of the interference could be used for the tolerance determination of manual hole-drilling with good quality and high efficiency.

Application of Variational Geometry to the Analysis of Mechanical Tolerances

1989 ◽  
Author(s):  
P. M. Martino ◽  
G. A. Gabriele
Keyword(s):  
Mechanical Design ◽  
Worst Case Analysis ◽  
Prototype System ◽  
Tolerance Design ◽  
Worst Case ◽  
Geometric Tolerances ◽  
Solid Models ◽  
The Cost ◽  
Selection Of

Abstract The proper selection of tolerances is an important part of mechanical design that can have a significant impact on the cost and quality of the final product. Yet, despite their importance, current techniques for tolerance design are rather primitive and often based on experience and trial and error. Better tolerance design methods have been proposed but are seldom used because of the difficulty in formulating the necessary design equations for practical problems. In this paper we propose a technique for the automatic formulation of the design equations, or design functions, which is based on the use of solid models and variational geometry. A prototype system has been developed which can model conventional and statistical tolernaces, and a limited set of geometric tolerances. The prototype system is limited to the modeling of single parts, but can perform both a worst case analysis and a statistical analysis. Results on several simple parts with known characteristics are presented which demonstrate the accuracy of the system and the types of analysis it can perform. The paper concludes with a discussion of extensions to the prototype system to a broader range of geometry and the handling of assemblies.

Algebraic Representation of Geometric and Size Tolerances

1995 ◽  
Author(s):  
S. H. Mullins ◽  
D. C. Anderson
Keyword(s):  
Tolerance Analysis ◽  
Design Analysis ◽  
Algebraic Representation ◽  
Dimensional Metrology ◽  
Algebraic Form ◽  
Worst Case ◽  
Mechanical Component ◽  
Analysis And Synthesis ◽  
Statistical Tolerance ◽  
Statistical Tolerance Analysis

Abstract Presented is a method for mathematically modeling mechanical component tolerances. The method translates the semantics of ANSI Y14.5M tolerances into an algebraic form. This algebraic form is suitable for either worst-case or statistical tolerance analysis and seeks to satisfy the requirements of both dimensional metrology and design analysis and synthesis. The method is illustrated by application to datum systems, position tolerances, orientation tolerances, and size tolerances.

A Design-and-Build Project to Introduce Concepts Relating to Dimensional Variation

1998 ◽  
Vol 26 (4) ◽  
pp. 259-272
Author(s):  
S. M. Panton ◽  
P. R. Milner
Keyword(s):  
Classroom Environment ◽  
Tolerance Analysis ◽  
Cylinder Block ◽  
Worst Case ◽  
Good Design ◽  
Minimum Number ◽  
Dimensional Variation ◽  
Assembly Problem ◽  
Statistical Tolerance ◽  
Statistical Tolerance Analysis

A design-and-build project which has been used to introduce Year 2 students of Mechanical Engineering to the concepts of dimensional variation and the influence of dimensional variation on function and assembly. The project simulates the cylinder head cylinder block assembly problem and specifies requirements in terms of a tolerance on concentricity of the cylinders in the head and block, and the interchangeable assembly of the head and block. Materials which are easily and cheaply sourced and tools which are easily manufactured and safe to use in a classroom environment are used throughout. During the project the students are exposed to concepts such as worst-case and statistical tolerance analysis, sensitivity analysis, geometric moment effects, minimum constraint design, co-variance and gauging. The exercise also emphasizes that good design means components that function and assemble with the minimum number of tight tolerances.

scholarly journals Geometrical and dimensional tolerance analysis for the radial flux type of permanent magnet generator design

2021 ◽  
Vol 12 (2) ◽  
pp. 68-80
Author(s):  
Muhammad Fathul Hikmawan ◽  
Agung Wibowo ◽  
Muhammad Kasim
Keyword(s):  
Permanent Magnet ◽  
Manufacturing Process ◽  
Cumulative Effect ◽  
Tolerance Analysis ◽  
Air Gap ◽  
Tolerance Allocation ◽  
Design Stage ◽  
Worst Case ◽  
Radial Flux ◽  
Permanent Magnet Generator

Mechanical tolerance is something that should be carefully taken into consideration and cannot be avoided in a product for manufacturing and assembly needs, especially in the design stage, to avoid excessive dimensional and geometric deviations of the components made. This paper discusses how to determine and allocate dimensional and geometric tolerances in the design of a 10 kW, 500 rpm radial flux permanent magnet generator prototype components. The electrical and mechanical design results in the form of the detailed nominal dimensions of the generator components, and the allowable air gap range are used as input parameters for tolerance analysis. The values of tolerance allocation and re-allocation process are carried out by considering the capability of the production machine and the ease level of the manufacturing process. The tolerance stack-up analysis method based on the worst case (WC) scenario is used to determine the cumulative effect on the air gap distance due to the allocated tolerance and to ensure that the cumulative effect is acceptable so as to guarantee the generator's functionality. The calculations and simulations results show that with an air gap of 1 ± 0.2 mm, the maximum air gap value obtained is 1.1785 mm, and the minimum is 0.8 mm. The smallest tolerance value allocation is 1 µm on the shaft precisely on the FSBS/SRBS feature and the rotor on the RPMS feature. In addition, the manufacturing process required to achieve the smallest tolerance allocation value is grinding, lapping, and polishing processes.

scholarly journals Refocusing in Reduction Semantics

2004 ◽  
Vol 11 (26) ◽  
Author(s):  
Olivier Danvy ◽  
Lasse R. Nielsen
Keyword(s):  
Transitive Closure ◽  
Operational Semantics ◽  
Sufficient Conditions ◽  
Lambda Calculus ◽  
Transition Function ◽  
Evaluation Function ◽  
Abstract Machine ◽  
Worst Case ◽  
State Transition Function ◽  
Case Basis

The evaluation function of a reduction semantics (i.e., a small-step operational semantics with an explicit representation of the reduction context) is canonically defined as the transitive closure of (1) decomposing a term into a reduction context and a redex, (2) contracting this redex, and (3) plugging the contractum in the context. Directly implementing this evaluation function therefore yields an interpreter with a worst-case overhead, for each step, that is linear in the size of the input term. <br /> <br />We present sufficient conditions over the constituents of a reduction semantics to circumvent this overhead, by replacing the composition of (3) plugging and (1) decomposing by a single ``refocus'' function mapping a contractum and a context into a new context and a new redex, if any. We also show how to construct such a refocus function, we prove the correctness of this construction, and we analyze the complexity of the resulting refocus function. <br /> <br />The refocused evaluation function of a reduction semantics implements the transitive closure of the refocus function, i.e., a ``pre-abstract machine.'' Fusing the refocus function with the trampoline function computing the transitive closure gives a state-transition function, i.e., an abstract machine. This abstract machine clearly separates between the transitions implementing the congruence rules of the reduction semantics and the transitions implementing its reduction rules. The construction of a refocus function therefore shows how to mechanically obtain an abstract machine out of a reduction semantics, which was done previously on a case-by-case basis. <br /> <br />We illustrate refocusing by mechanically constructing Felleisen et al.'s CK machine from a call-by-value reduction semantics of the lambda-calculus, and by constructing a substitution-based version of Krivine's machine from a call-by-name reduction semantics of the lambda-calculus. We also mechanically construct three one-pass CPS transformers from three quadratic context-based CPS transformers for the lambda-calculus.

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