Tolerance Stack-up Analysis - As Explained by a Dummy Part 2/2

Welcome back dear reader. I take it that you obtained a brief background to the marvelous art and science of Tolerance Stack-up Analysis, and to our friendly prehistoric builder from part one of this article.

Special thanks to Jeff Hall and Evan Jackson, without whom I would have never been introduced to this beautiful art and science. They went out of their way to teach and train me, all while being extremely patient with me. Thank you gentleman. I would not have been the engineer I am today without your support.

Background

As a continuation to Part one, this article discusses step-by-step procedures for the two major types of tolerance stack-up analysis:

1. Worst Case Tolerance Analysis
2. Statistical Tolerance Analysis

Once again, this article is not a comprehensive guide for tolerance stack-up analysis and the material presented in the reference section are highly recommended for an in-depth understanding. What this article does provide, is a quick and easy, step-by-step procedure to perform linear tolerance stack-up analysis on a 1D stack-up.

Introduction

As a Recap of part one of this article, the prehistoric builder is faced with a conundrum of estimating the potential "gap" or "interference" between the last stone and the wall of the wooden sled. This problem is particularly tricky for him as he has a number of stones and sleds and clearly cannot evaluate each case by trial and error.

Certain assumptions are made, drawing parallels to modern day manufacturing methods.

1. The stone sizes estimated by the builder are the Nominal Size and the tolerance expected are based on his skills with the hammer and chisel. This is applicable to modern day mechanical assemblies as the CAD Nominal, which is determined by the design intent and the tolerances that are specified, partly by the requirements but mostly by the manufacturing process capabilities.
2. The builder's sleds too have a specified nominal and tolerance.
3. The stack-up is linear and 1D only.
4. The variations in the builder's manufacturing capabilities follow a Normal Distribution, quite similar to modern day manufacturing processes.

Prerequisite to Tolerance Stack-up Analysis: Tolerance Formats

Before we delve into performing tolerance stack-up analysis, it is essential to understand the popular tolerance formats that occur in design specifications.

Note: While Tolerance Stack-up Analysis can be applied to components and assemblies designed with GD&T (Geometrical Dimenstionaing and Tolerancing) specifications, this article uses the conventional dimensioning system for the sake of explanation.

Tolerance Formats

1. 1. Limit Dimension: Does not specify a Nominal value of the dimension; only specifies the maximum and minimum value.
2. 
   

2. Equal Bilaterally Toleranced Dimension: Specifies a nominal value and the amount a dimension may deviate from nominal. The tolerance values are equal in each direction.

1. 3. Unequal Bilaterally Toleranced Dimension: Specifies a nominal value and the amount a dimension may deviate from nominal. The tolerance values are not equal in each direction, and neither value is zero.

1. 4. Unilaterally Toleranced Dimension: Specifies a nominal value. The tolerance is in one direction only, either larger or smaller. The other tolerance value is zero.

Converting Tolerances to Equal Bilaterally Toleranced Dimension:

The technique that will be presented subsequently and used by our friend the builder requires all dimensions to be converted into an Equal Bilaterally Toleranced Dimension.

• Limit Dimensions:

1. Obtain Total Tolerance by (Upper Limit - Lower Limit) of the Dimension. Here it will be 22-18 = 4
2. Divide the Total Tolerance by two to obtain the Equal Bi-Lateral Tolerance. Hence 4/2 = 2
3. Subtract the Equal Bi-Lateral Tolerance from the upper limit (Maximum Value) or add the same to the lower limit (Minimum Value) to obtain the Nominal Dimension. Correspondingly in this example would be 22-2 = 20; or 18+2 = 20.
4. Finally, the Equal Bi-Lateral Tolerance Dimension would be Nominal +- the equal bilateral tolerance; which in our case is 20 +- 2

• Unequal Bilaterally Toleranced Dimension:

1. Establish the upper and lower limits for the dimension. In other words, convert the given dimension into a Limit Dimension. Upper Limit = Nominal Value + Positive Tolerance Value. Lower Limit = Nominal Value - Negative Tolerance Value. Hence in our example, Upper limit = 20+2 = 22, and Lower Limit = 20-1 = 19.
2. Obtain Total Tolerance by (Upper Limit - Lower Limit) of the Dimension. Here it is 22-19 = 3.

Note : The total tolerance can also be obtained by adding the + and – tolerances given

1. Divide the Total Tolerance by two to obtain the Equal Bi-Lateral Tolerance. Here it is 3/2 = 1.5
2. Subtract the Equal Bi-Lateral Tolerance from the upper limit (Maximum Value) or add the same to the lower limit (Minimum Value) to obtain the Nominal Dimension. Hence 22-1.5 = 20.5, or 19+1.5 = 20.5
3. Finally, the Equal Bi-Lateral Tolerance Dimension would be Nominal +- the equal bilateral tolerance which in our case is 20.5 +- 1.5

• Unilaterally Toleranced Dimension:

1. Establish the upper and lower limits for the dimension. In other words, convert the given dimension into a Limit Dimension. Upper Limit = Nominal Value + Positive Tolerance Value. Lower Limit = Nominal Value - Negative Tolerance Value. In this example: Upper limit = 20+1 = 21, and Lower limit = 20-0 = 20
2. Obtain Total Tolerance by (Upper Limit - Lower Limit) of the Dimension. Here it is 21 - 20 = 1
3. Divide the Total Tolerance by two to obtain the Equal Bi-Lateral Tolerance. Here it is 1/2 = 0.5
4. Subtract the Equal Bi-Lateral Tolerance from the upper limit (Maximum Value) or add the same to the lower limit (Minimum Value) to obtain the Nominal Dimension. Hence 21-0.5 = 20.5 ; or 20+0.5 = 20.5;
5. Finally, the Equal Bi-Lateral Tolerance Dimension would be Nominal +- the equal bilateral tolerance which in our case is 20.5+-0.5

The same procedure can be applied to unilaterally negative and positive dimensions.

Essential notes:

• When converting an unequal-bilaterally toleranced dimension to an equal-bilaterally toleranced 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.
• When converting a unilaterally positive toleranced dimension to an equal-bilaterally toleranced dimension,the dimension shift is always half the positive tolerance value, and the shift is toward the high end of the tolerance range.
• When converting a unilaterally negative toleranced dimension to an equal-bilaterally toleranced dimensions,the dimension shift is always half the negative tolerance value, and the shift is toward the low end of the tolerance range.

Step-by-Step Procedure for Tolerance Analysis

Worst Case Tolerance Analysis

1. Select the distance (gap or interference) whose variation is to be determined and label one end of the distance as A and the other end as B as shown below

2. Determine the Positive and Negative Directions for the dimensions. The positive direction is the direction from point A to point B.

3. Begin building a "dimensional loop" starting from Points A or B. In this case, we start from A. The objective of this loop is to form a path of dimensions of features/parts in direct contact with one another. Note, it is very important to form a continuous path.

Some essential rules for the loop vector:

• Loops must pass through every part and every joint in the assembly.
• A single vector loop may not pass through the same part or the same joint twice, but it may start and end in the same part.
• If a vector loop includes exactly the same dimension twice, in opposite directions, the dimension is redundant and must be omitted.

4. Next, Positive dimensions are indicated by placing a “+” sign adjacent to the dimension value and Negative dimensions are indicated by placing a "-" sign adjacent to the dimension value.

5. Convert all dimensions and tolerances to equal-bilateral format: This was discussed earlier.

6. Now all the dimensions and tolerances are entered into a chart and totaled for reporting purposes. Place each positive dimension value in the positive column on a separate line. Place each negative dimension value in the negative column on a separate line. Place the tolerance value for each dimension in the tolerance column adjacent to each dimension.

7. Add the entries in each column, entering the results at the bottom of the chart

8. Subtract the negative total from the positive total. This gives the nominal dimension or distance. In cases where all the dimensions and tolerances in the chain were not originally in equal-bilateral format this value will probably be different than the distance that is measured directly from a drawing, CAD model, or what our builder estimated.

Hence the evaluated Nominal Gap = 82-61 = 21

The evaluated total tolerance correspondingly is +- 5

Thus making the Gap dimension and tolerance as 21+-5

9. Apply the total tolerance. Adding and subtracting the tolerance from the nominal dimension gives the maximum and minimum distance values. Hence, the Gap: Worst Case Maximum = 21+5 = 26; and the Worst Case Minimum = 21-5 = 16.

WOW! That was fun! While the Worst Case Tolerance Stack-up analysis gives us extremely essential information, it is however extremely conservative and at times an overkill to design based only on it. This is due to the fact that the variations incurred in manufacturing processes are modeled as Normal distributions.

Therefore, a more "probable" maximum and minimum gap/interference size would be extremely useful.

Statistical Tolerance Stack-up Analysis: RSS Method

1. Repeat steps 1 to 8 from the earlier section and develop the table.

2.Take each tolerance value, (the absolute value) and square it. Place this value in the Statistical Tolerance or Squared Tolerance Column next to each tolerance. Sum up the Squared Tolerances column as before.

3. Take the square root of the sum of statistical tolerances (RSS). Enter this result at the bottom of the chart. This is the RSS Tolerance Value.

4. Apply the total statistical tolerance. Adding and subtracting the statistical tolerance from the nominal dimension gives the likely or probable maximum and minimum distance values. Hence in this case, Nominal = 21. RSS Tolerance Value = +- 2.739. Hence the gap = 21+-2.739. Correspondingly the probable maximum = 21+2.739 = 23.739, and the probable minimum = 21-2.739 = 18.261.

Note: If it is desired to take a slightly more conservative approach, multiply the RSS tolerance by an adjustment factor (such as 1.5 for example), substituting the larger adjusted RSS value for the RSS value. Hence in this example, 1.5 x RSS = 4.109. The corresponding gap = 21+-4.109

Well, there you have it! Now you have access to a step-by-step procedure to perform to perform Linear Tolerance analysis in 1D.

Essential additional tolerances incurred:

Our builder, while loading the sleds incurs shifts in the assembly of stones. This is universal to all mechanical assemblies, and is known as Assembly Shift. Assembly shift accounts for the freedom parts have to move from their nominal locations due to the clearance between mating internal and external features at assembly.

Assembly shift is added to the tolerance stack-up as a line item without a sign or direction, as it allows the parts to shift in both the positive and negative directions, similar to an equal-bilateral tolerance.

Hence, to incorporate assembly shift, add the shift as +- Value to the tolerance column without any dimension value. Proceed as usual after that.

Significance of the RSS Tolerance and Relation to Process Control (Standard Deviation)

The relationship between the RSS tolerance and the standard deviation in the process, is considered that level of process controls of the inputs represents the level of process controls of the output. Which would imply that if all the component tolerances are assumed to be ±1σ, ±2σ, or ±6σ, then the RSS tolerance stack-up result represents ±1σ, ±2σ, or ±6σ, respectively.

The evaluation of the standard deviation allows the engineer to evaluate the "Percentage defects" incurred for an assumed gap size. Using this, the Parts Per Million failure can be found. This article however does not delve into this evaluation and will be discussed in subsequent articles. Please refer to the references for material that explains this in great detail.

Concluding remarks:

Like all things in life, the art and science of Tolerance Stack-up analysis takes practice. I too am in the process of getting better at it every day. The process is often performed on a spread-sheet to have quick and easy calculations, yet the algorithm for the process remains the same.

In the recent future, I will share some tips and tricks that I happened to pick up during my internship that pertained to tolerance stack-up analyses.

I pray that this was as enjoyable to you to read as it was for me to write. Please comment your thoughts and share it with your friends if you liked it.

References:

1. Mechanical Tolerance Stackup and Analysis, Second Edition - Bryan R. Fischer
2. https://en.wikipedia.org/wiki/Tolerance_analysis
3. https://en.wikipedia.org/wiki/Engineering_tolerance