ISICION DE LA CORTE INTERNACIONAL DE JUSTICIA EN EL
2. JUECES AD HOC EN EL PROCEDIMIENTO
The method of performing tolerance stackups taught in this text requires all dimensions and tolerances to be converted into equal-bilateral format.
This chapter describes the techniques for converting plus/minus dimensions and tolerances into the equal-bilateral format. This is necessary for all dimen-sions and tolerances, whether they are plus/minus or GD&T. Conversion of geometric dimensions and tolerances into equal-bilateral plus/minus toleranced dimensions will be discussed in Chapter 9. Whether dealing with U.S. inch or metric dimensions and tolerances, linear or angular units, the technique for con-version is the same.
convertIng lImIt dImensIons to equAl-bIlAterAl FormAt
The following example presents the procedure for converting limit dimensions into equal-bilaterally toleranced dimensions.
example 4.1: converting limit dimensions to equal-bilateral Format
Given a limit dimension:
•
10.00 Upper limit (metric format) 9.55 Lower limit (metric format)
Subtract the lower limit from the upper limit to obtain the total
•
tolerance.
Total tolerance = 10 – 9.55 = 0.45
32 Mechanical Tolerance Stackup and Analysis, Second Edition
Divide the total tolerance by 2 to obtain the equal-bilateral tolerance
• value.
Equal-bilateral tolerance value = 0.45/2 = 0.225
Add the equal-bilateral tolerance value to the lower limit. This is the
•
adjusted nominal value.
Adjusted nominal value = 9.55 + 0.225 = 9.775
(Note: The adjusted nominal value can also be obtained by subtracting the equal-bilateral tolerance value from the upper limit.)
Conversion complete:
Equal-bilateral equivalent = 9.775 ± 0.225
convertIng unequAl-bIlAterAl FormAt to equAl-bIlAterAl FormAt
The following example presents the procedure for converting unequal-bilaterally toleranced dimensions into equal-bilaterally toleranced dimensions.
example 4.2: converting unequal-bilateral Format dimensions to equal-bilateral Format
Given an unequal-bilaterally toleranced dimension (inch format)
•
8.50 +.25–.10 Establish upper and lower limits.
•
Add the plus tolerance to the nominal value; this is the upper limit.
Subtract the minus tolerance from the nominal value; this is the lower limit.
Upper limit = 8.50 + .25 = 8.75 Lower limit = 8.50 – .10 = 8.40
Subtract the lower limit from the upper limit to obtain the total
•
tolerance.
(Note: The total tolerance can also be obtained by adding the + and – tolerances given).
Total tolerance derived from limits = 8.75 – 8.40 = .35 or
Total tolerance derived from given tolerances= .25 + .10 = .35
Converting Plus/Minus Dimensions and Tolerances 33
Divide the total tolerance by 2 to obtain the equal-bilateral tolerance
• value.
Equal-bilateral tolerance value = .35/2 = .175
Add the equal-bilateral tolerance value to the lower limit. This is the
•
adjusted nominal value.
Establish the adjusted nominal value = 8.40 + .175 = 8.575
(Note: The adjusted nominal value can also be obtained by subtracting the equal-bilateral tolerance value from the upper limit.)
Conversion complete:
Equal-bilateral equivalent = 8.575 ± .175
convertIng unIlAterAlly posItIve FormAt to equAl-bIlAterAl FormAt
The following example presents the procedure for converting unilaterally posi-tive toleranced dimensions (plus something, minus nothing) into equal-bilaterally toleranced dimensions.
example 4.3: converting unilaterally positive Format dimensions to equal-bilateral Format
Given a unilaterally positive toleranced dimension (inch format)
•
8.50 +.25–.00 Establish upper and lower limits.
•
Add the plus tolerance to the nominal value; this is the upper limit.
The specified nominal value is the lower limit.
Upper limit = 8.50 + .25 = 8.75 Lower limit = 8.50 – .00 = 8.50
Subtract the lower limit from the upper limit to obtain the total
•
tolerance.
(Note: The total tolerance is equivalent to the plus tolerance.) Total tolerance derived from limits = 8.75 – 8.50 = .25 or
Total tolerance derived from given tolerances = .25 – .00 = .25
34 Mechanical Tolerance Stackup and Analysis, Second Edition
Divide the total tolerance by 2 to obtain the equal-bilateral tolerance
• value.
Equal-bilateral tolerance value = .25/2 = .125
Add the equal-bilateral tolerance to the lower limit. This is the adjusted
•
nominal value.
Establish the adjusted nominal value = 8.50 + .125 = 8.625
(Note: The adjusted nominal value can also be obtained by subtracting the equal-bilateral tolerance value from the upper limit.)
Conversion complete:
Equal-bilateral equivalent = 8.625 ± .125
convertIng unIlAterAlly negAtIve FormAt to equAl-bIlAterAl FormAt
The following example presents the procedure for converting unilaterally nega-tive toleranced dimensions (plus nothing, minus something) into equal-bilaterally toleranced dimensions.
example 4.4: converting unilaterally negative Format dimensions to equal-bilateral Format
Given a unilaterally negative toleranced dimension (metric format)
•
8.50 0–.25 Establish upper and lower limits.
•
The specified nominal value is the upper limit.
•
Subtract the negative tolerance from the nominal value; this is the lower
• limit.
Upper limit = 8.5 + 0 = 8.5 Lower limit = 8.5 – .25 = 8.25
Subtract the lower limit from the upper limit to obtain the total
•
tolerance.
(Note: The total tolerance is equivalent to the minus tolerance.) Total tolerance derived from limits = 8.5 – 8.25 = .25 or
Converting Plus/Minus Dimensions and Tolerances 35
Total tolerance derived from given tolerances = 0 +.25 = .25
Divide the total tolerance by 2 to obtain the equal-bilateral tolerance
• value.
Equal-bilateral tolerance value = .25/2 = .125
Add the equal-bilateral tolerance to the lower limit. This is the adjusted
•
nominal value.
Establish the adjusted nominal value = 8.25 + .125 = 8.375
(Note: The adjusted nominal value can also be obtained by subtracting the equal-bilateral tolerance value from the upper limit.)
Conversion complete:
Equal-bilateral equivalent = 8.375 ± .125
dImensIon shIFt WIthIn A converted dImensIon And tolerAnce
As presented in Chapter 3, design nominal is not always at the midpoint of the tol-erance range. Converting unequal-bilaterally and unilaterally toltol-eranced dimen-sions to equal-bilateral format changes the dimension value so it is at the midpoint of the tolerance range. The limits are not changed, only the format of the dimen-sion and tolerance(s) are changes.
In tolerancing jargon, we have effected a dimension shift. The new dimension value is different than the value specified on the drawing. The dimension value has been shifted to the midpoint of the tolerance range. Remember our earlier discussion that it makes no difference how a tolerance range is specified, that is, whether limits, equal-bilateral, unequal-bilateral or unilateral tolerances are specified, the result is the same. All that has legally been specified are upper and lower limits for a dimension. Only with equal-bilateral tolerancing is the stated dimension value on the drawing the midpoint of the range.
Where dimensions are included in the tolerance stackup, the dimension shift is little more than a curiosity, as it has no effect on the outcome of the tolerance stackup. Dimension shifts are accounted for in the tolerance stackup method, and can be ignored without consequence. Using more advanced and streamlined methods where dimensions are not included in the tolerance stackup and only the tolerances are included, dimension shifts must be accounted for. This text does not address the tolerance analysis methods where dimensions are not included in the tolerance stackup.
An easy method to determine the dimension shift for a dimension and tol-erance converted to equal-bilateral format follows. The dimension shift will be calculated for the dimensions and tolerances shown in examples 4.1 to 4.4 in the previous section.
36 Mechanical Tolerance Stackup and Analysis, Second Edition
example 4.1a: dimension shift calculation for limit dimensions converted into equal-bilateral Format
In example 4.1, limit dimensions were converted into equal-bilateral format.
Limit dimensions do not state a nominal or “mean,” so there is no dimension shift when converting limit dimensions into equal-bilateral format.
example 4.2a: dimension shift calculation for unequal-bilateral Format converted into equal-unequal-bilateral Format Given an unequal-bilaterally toleranced dimension (inch format)
8.50 +.25–.10
that has been converted into equal-bilateral format:
8.575 ± .175 The dimension shift is calculated as follows:
Converted dimension value – original dimension value = dimension shift Dimension shift = 8.575 – 8.50 = .075
The sign of the dimension shift is positive, indicating the dimension was shifted toward the high end of the tolerance range. Note: When convert-ing an unequal-bilaterally toleranced dimension to an equal-bilaterally toler-anced dimension, the dimension shift is always half the difference between the positive and negative tolerance values, and the shift is toward the larger of the two values.
example 4.3a: dimension shift calculation for unilaterally positive Format converted into equal-bilateral Format Given a unilaterally positive toleranced dimension (inch format)
8.50 +.25–.00
that has been converted into equal-bilateral format:
8.625 ±.125 The dimension shift is calculated as follows:
Converting Plus/Minus Dimensions and Tolerances 37
Converted dimension value – original dimension value = dimension shift Dimension shift = 8.625 – 8.50 = .125
The sign of the dimension shift is positive, indicating the dimension was shifted toward the high end of the tolerance range. Note: When converting a unilater-ally positive toleranced dimension to an equal-bilaterunilater-ally toleranced dimen-sion, the dimension shift is always half the positive tolerance value, and the shift is toward the high end of the tolerance range.
example 4.4a: dimension shift calculation for unilaterally negative Format converted into equal-bilateral Format Given a unilaterally negative toleranced dimension (metric format)
8.50 0–.25
that has been converted into equal-bilateral format:
8.375 ±.125 The dimension shift is calculated as follows:
Converted dimension value – original dimension value = dimension shift Dimension shift = 8.375 – 8.50 = –.125
The sign of the dimension shift is negative, indicating the dimension was shifted toward the low end of the tolerance range. Note: When converting a unilater-ally negative toleranced dimension to an equal-bilaterunilater-ally toleranced dimen-sions, the dimension shift is always half the negative tolerance value, and the shift is toward the low end of the tolerance range.
Note: In the first edition of this text the term mean shift was used instead of dimension shift. While mean shift is commonly used in industry to describe this condition, it is not the best term to use. Also, the term mean shift has been used in Six Sigma tolerance analysis for quite some time to represent cases where the mean of a distribution of a population is shifted from its starting position. This is an accurate use of the term. Chapter 21, which is new to this edition, includes an introductory discussion of mean shift as used in Six Sigma methodologies.
Therefore, this edition of Mechanical Tolerance Stackup and Analysis refers to the condition where the dimension value has been converted and shifted to the midpoint of the tolerance range as dimension shift.
To reinforce the reason that mean shift is not the correct term to use to describe a converted and shifted dimension value, consider the following. The
38 Mechanical Tolerance Stackup and Analysis, Second Edition
“mean” in this instance is really the midpoint of the range in this context. It merely lies at the middle of the range. From a mathematical point of view, the term mean shift in this context is actually a misnomer, as the “mean” requires multiple values to be determined, and for a single dimension in a tolerance stackup, there is only one specified value or range with a corresponding mid-point. According to the McGraw-Hill AccessScience Encyclopedia of Science
& Technology Online (2010):
mean [MATHEMATICS] A single number that typifies a set of numbers, such as the arithmetic mean, the geometric mean, or the expected value. Also known as mean value.
For the purposes of tolerance analysis, we are interested in the arithme-tic mean. The arithmearithme-tic mean is the average value for a group of values and is found by adding the values and dividing the sum by the number of values.
According to the McGraw-Hill AccessScience Encyclopedia of Science &
Technology Online (2010):
arithmetic mean [MATHEMATICS] The average of a collection of numbers obtained by dividing the sum of the numbers by the quantity of numbers. Also known as arithmetic average; average (av).
Therefore, strictly speaking, the mean in this case is the arithmetic mean, and the arithmetic mean requires a collection (or population) of values to be of any significance. This is the situation found in the Six Sigma methodologies intro-duced in Chapter 21, and also encountered when inspecting mass-prointro-duced parts, where the same feature on many similar parts is inspected many times. Say 100 parts are manufactured, and it is desired to know where the mean of the process lies for the diameter of a hole, with a dimension of ∅10 ±0.5 specified. The diam-eter of the hole on each of the 100 parts is measured and recorded. The values are plotted on a chart, the arithmetic mean is calculated using the method described above and found to be ∅10.2. The measured mean of the manufacturing process is ∅10.2, which shows that the result of the manufacturing process is centered 0.2 above the specified mean or nominal.
In this context, there has been a mean shift of 0.2 in the positive direction, or toward a larger value. If the mean of the manufacturing process was 9.8, there would still be a mean shift of 0.2, but it would be in the negative direction—the mean shift could be stated as –0.2. You see, this value really is a mean value—it really is the result of taking the sum of many values and dividing that sum by the number of values. In this example the number of values was 100. For these reasons, the term dimension shift is used to describe the converted and shifted dimension value in this edition instead of mean shift.
Converting Plus/Minus Dimensions and Tolerances 39
dImensIon shIFt recAp
Dimension shift in a tolerance stackup with dimension values stated is of little concern. When a dimension and its tolerance are converted into equal-bilateral format, the dimension value may be shifted up or down depending on how the dimension and tolerance were specified. As long as all the dimensions and toler-ances are treated in the same manner and included in the tolerance stackup, any dimension shift will be accounted for in the final result.
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