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 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.

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.

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.

Bias in Using an Age–Length Key to Estimate Age-Frequency Distributions

1978 ◽  
Vol 35 (2) ◽  
pp. 184-189 ◽  
Author(s):  
S. J. Westrheim ◽  
W. E. Ricker
Keyword(s):  
Frequency Distribution ◽  
Age Distribution ◽  
Length Distribution ◽  
Fish Population ◽  
Systematic Bias ◽  
Parental Age ◽  
Frequency Distributions ◽  
Class Strength ◽  
Key Words Age ◽  
Age Frequency Distribution

Consider two representative samples of fish taken in different years from the same fish population, this being a population in which year-class strength varies. For the "parental" sample the length and age of the fish are determined and are used to construct an "age–length key," the fractions of the fish in each (short) length interval that are of each age. For the "filial" sample only the length is measured, and the parental age–length key is used to compute the corresponding age distribution. Trials show that the age–length key will reproduce the age-frequency distribution of the filial sample without systematic bias only if there is no overlap in length between successive ages. Where there is much overlap, the age–length key will compute from the filial length-frequency distribution approximately the parental age distribution. Additional bias arises if the rate of growth if a year-class is affected by its abundance, or if the survival rate in the population changes. The length of the fish present in any given part of a population's range can vary with environmental factors such as depth of the water; nevertheless, a sample taken in any part of that range can be used to compute age from the length distribution of a sample taken at the same time in any other part of the range, without systematic bias. But this of course is not likely to be true of samples taken from different populations of the species. Key words: age–length key, bias, Pacific ocean perch, Sebastes alutus

Frequency distribution of Wuchereria bancrofti microfilariae in human populations and its relationships with age and sex

Parasitology ◽  
1990 ◽  
Vol 101 (3) ◽  
pp. 429-434 ◽  
Author(s):  
P. K. Das ◽  
A. Manoharan ◽  
A. Srividya ◽  
B. T. Grenfell ◽  
D. A. P. Bundy ◽  
...  
Keyword(s):  
Frequency Distribution ◽  
Negative Binomial ◽  
Age Distribution ◽  
Indirect Evidence ◽  
Wuchereria Bancrofti ◽  
Parameter Estimates ◽  
Human Populations ◽  
Sampling Errors ◽  
Frequency Distributions ◽  
Age And Sex

SUMMARYThis paper examines the effects of host age and sex on the frequency distribution of Wuchereria bancrofti infections in the human host. Microfilarial counts from a large data base on the epidemiology of bancroftian filariasis in Pondicherry, South India are analysed. Frequency distributions of microfilarial counts divided by age are successfully described by zero-truncated negative binomial distributions, fitted by maximum likelihood. Parameter estimates from the fits indicate a significant trend of decreasing overdispersion with age in the distributions above age 10; this pattern provides indirect evidence for the operation of density-dependent constraints on microfilarial intensity. The analysis also provides estimates of the proportion of mf-positive individuals who are identified as negative due to sampling errors (around 5% of the total negatives). This allows the construction of corrected mf age–prevalence curves, which indicate that the observed prevalence may underestimate the true figures by between 25% and 100%. The age distribution of mf-negative individuals in the population is discussed in terms of current hypotheses about the interaction between disease and infection.

Frequency distribution of space groups in soluble and membrane proteins and their complexes

2021 ◽  
Vol 77 (6) ◽  
pp. 187-191
Author(s):  
Rajneesh K. Gaur
Keyword(s):  
Membrane Proteins ◽  
Frequency Distribution ◽  
Data Bank ◽  
Frequency Distributions ◽  
Space Group ◽  
Space Groups ◽  
Group Frequency ◽  
Different Types ◽  
Analytical Assessment ◽  
Three Space

The space-group frequency distributions for two types of proteins and their complexes are explored. Based on the incremental availability of data in the Protein Data Bank, an analytical assessment shows a preferential distribution of three space groups, i.e. P212121 > P1211 > C121, in soluble and membrane proteins as well as in their complexes. In membrane proteins, the order of the three space groups is P212121 > C121 > P1211. The distribution of these space groups also shows the same pattern whether a protein crystallizes with a monomer or an oligomer in the asymmetric unit. The results also indicate that the sizes of the two entities in the structures of soluble proteins crystallized as complexes do not influence the frequency distribution of space groups. In general, it can be concluded that the space-group frequency distribution is homogenous across different types of proteins and their complexes.

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.

scholarly journals Regional parent flood frequency distributions in Europe – Part 1: Is the GEV model suitable as a pan-European parent?

2014 ◽  
Vol 18 (11) ◽  
pp. 4381-4389 ◽  
Author(s):  
J. L. Salinas ◽  
A. Castellarin ◽  
A. Viglione ◽  
S. Kohnová ◽  
T. R. Kjeldsen
Keyword(s):  
Statistical Model ◽  
Frequency Distribution ◽  
Climatic Factors ◽  
Extreme Value ◽  
Flood Frequency ◽  
Generalized Extreme Value Distribution ◽  
Statistical Hypothesis ◽  
Generalized Extreme Value ◽  
Frequency Distributions ◽  
Hypothetical Scenario

Abstract. This study addresses the question of the existence of a parent flood frequency distribution on a European scale. A new database of L-moment ratios of flood annual maximum series (AMS) from 4105 catchments was compiled by joining 13 national data sets. Simple exploration of the database presents the generalized extreme value (GEV) distribution as a potential pan-European flood frequency distribution, being the three-parameter statistical model that with the closest resemblance to the estimated average of the sample L-moment ratios. Additional Monte Carlo simulations show that the variability in terms of sample skewness and kurtosis present in the data is larger than in a hypothetical scenario where all the samples were drawn from a GEV model. Overall, the generalized extreme value distribution fails to represent the kurtosis dispersion, especially for the longer sample lengths and medium to high skewness values, and therefore may be rejected in a statistical hypothesis testing framework as a single pan-European parent distribution for annual flood maxima. The results presented in this paper suggest that one single statistical model may not be able to fit the entire variety of flood processes present at a European scale, and presents an opportunity to further investigate the catchment and climatic factors controlling European flood regimes and their effects on the underlying flood frequency distributions.

A Mathematical Model Depicting the Stress-Strain Diagram and the Hysteresis Loop

1959 ◽  
Vol 26 (1) ◽  
pp. 95-100
Author(s):  
I. R. Whiteman
Keyword(s):  
Mathematical Model ◽  
Hysteresis Loop ◽  
Yield Stress ◽  
Frequency Distribution ◽  
Elastic Region ◽  
Tangent Modulus ◽  
Strain Diagram ◽  
Cumulative Frequency ◽  
Cumulative Frequency Distribution ◽  
Stress Strain

Abstract A model is made up of elastoplastic elements, all of which have the same value of Young’s modulus E, but which have different values of yield stress. It is shown that the dimensionless tangent modulus graph Et/E represents the cumulative frequency distribution of those elements which are in the elastic region. From the frequency distribution, the equations for the stress-strain diagram and the hysteresis loop can be written.

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