An Abstract Stack-up analysis during product design stages is especially important. While complex 3D stack-up analysis can be challenging, linear stack-up analysis is easier to understand and can be used in projects.

This paper focuses on linear stack-up analysis, which helps predict total variation in assemblies due to dimensional tolerances. It examines two methods: Worst Case Condition and Root Sum Square (RSS) with three sigma. The paper looks at their effectiveness, accuracy, and use in real-life situations through comparisons and case studies. It highlights the strengths and weaknesses of each method. Additionally, it provides advice on choosing the right approach based on design intent, risk tolerance, and manufacturing capability. Finally, a hybrid strategy that combines the strengths of Worst Case, the efficiency of RSS, and the precision of three Sigma can lead to optimized designs that meet both functional and economic goals.

An Overview

Stack-up analysis is an important part of the mechanical design that considers the additive effect of tolerances of multiple parts in an assembly. The figure provided shows a hierarchical classification of stack-up analysis techniques into two fundamental groups: Linear Stack-Up Analysis and 3D Stack-Up Analysis.

The linear methods are the Worst-Case Condition (WCC), Root Sum Square (RSS), and Six Sigma, which vary widely in accuracy, risk tolerance, and cost-effectiveness. The linear stack-up analysis methods are often employed in the early stages of design to ensure that the dimensional compatibility is met, meaning that the item will function properly. The 3D Stack-Up Analysis Modeling includes new methods, such as Monte Carlo Simulations and Vector Based Calculations, which consider the geometric relationships in the multi-axis variants of complex assemblies.

The 3D stack-up analysis methods are particularly helpful for complex assemblies since many of the spatial interactions are significant factors affecting an operationally designed item. This paper elaborates on the Worst-Case Scenario and Root Sum Square (RSS) methods with a 3-Sigma approach, explaining their methodologies and examining where and how they can be applied in the real-world engineering design.

Tolerance Analysis

Tolerance analysis is a detailed process that evaluates how dimensional variations impact the fit, function, and performance of mechanical assemblies. It includes two main parts: understanding individual tolerancing specifications and analyzing the total variation between multiple features, which is often called a tolerance stack-up.

Before performing a stack-up, it is important to interpret the dimensioning and tolerancing, often known as GD&T, found in an engineering drawing or 3D model. This takes skill and experience because tolerancing specifications can be complex and affect the final assembly.

The stack-up process calculates the total variation across a series of dimensions, usually to check for clearances, interferences, or alignment. Questions explored range across various aspects like will the shaft go into the hole? Will the parts interfere? Will the assembly achieve its dimensional targets in the worst-case conditions?

This type of tolerance analysis is useful for the present parts as well as parts that are being developed. It helps to discover and address problems earlier in the process, improve tolerance, and ensure assembly. Whether checking a press-fit, locating fasteners, or checking process shifts, tolerance analysis is required to do it well from an engineering design and quality control perspective.

Stack-up Analysis

A tolerance stack-up is a decision-making tool as well, that evaluates the total variance on mechanical assemblies. It supplies the numbers to engineers that help in deciding if a design will fulfil overall requirements, the usual output is minimum / maximum distances. In statistical analysis, it can also help to illustrate the different range of possibilities and provide more clarity on the variability.

After completing a stack-up, the results help in making decisions on adjusting part shape, size, tolerances, or even the assembly and manufacturing processes. For instance, if an unwanted interference is found, the designer might change tolerances, redesign parts, or add fixturing to manage the variation.

Fixtures are particularly effective at reducing assembly variation. By using tight-fitting holes and pins to position parts, fixtures ensure that the parts align consistently. However, the features for fixing must be accurately tolerated and included in the design since they become functional parts that affect the final assembly.

In a stack-up analysis, the series of dimensions is split into positive and negative directions. The process requires identifying the distance of interest, sketching the stack-up, adding dimensions in each direction, and calculating the nominal value. Then, the total variation is added and subtracted from the nominal to find the upper and lower limits.

Fixtures need to be a part of the stack-up calculation since their tolerances directly influence the results. Fixture tolerances are much tighter, around 5 to 10% of part tolerances, but they must be confirmed based on drawings or manufacturer specifications.

Dimensioning and Tolerancing Formats
There are four standard formats for expressing linear and angular dimensions and tolerances. (Refer Fig 1,2,3) These formats apply to both the U.S. inch and metric systems. According to ASME Y14.5M-1994 and ASME Y14.5-2009, the rules for angular tolerancing are consistent across both unit systems.

The Four Formats

A) Limit Dimensions

These specify only the upper and lower limits without a nominal value. In the horizontal format, the smaller value comes before the larger (e.g., 9.95, 10.05), while in vertical format, the larger value is above the smaller. This pattern is used for the internal as well as the external features.

B) Equal-Bilateral Tolerances

They have values and equal permissible values in both directions (for example,10.00 ± 0.05).

C) Unequal-Bilateral Tolerances

There are a nominal value and different deviations (MU), but none of them is zero
(e.g., 10.00 +0.10/-0.05).

D) Unilateral Tolerances

The nominal value is stated with the deviation in one direction only (i.e., 10.00 +0.10/-0.00).

Whichever approach is taken, it is supposed to ensure that the nominal is covered within the tolerances. This guarantees the attainability and practicability of the given limits in production.
Even though all four methods define the upper and lower limits, they vary in how clearly, they convey design intent. Importantly, none of these formats guarantee that manufacturing will target the nominal value; often, additional process controls are needed to achieve that.

Limit Dimension:
There are the same number of decimal places in both limits.

Equal Bilateral Tolerancing:
The number of decimal places may be different for dimensions and tolerances.

Unequal Bilateral Tolerancing:
The number of decimal places may be different for dimensions and tolerances. Bothtolerances must have the same number of decimal places.

Unilateral Tolerancing:
The number of decimal places may be different for dimensions and tolerances. Thezero tolerance has no decimal places and is not preceded by a +ve or a -ve sign.

Limit Dimension:
There are the same number of decimal places in both the limits.

Equal Bilateral Tolerancing:
The number of decimal places must be the same for dimensions and tolerance.

Unequal Bilateral Tolerancing:
The number of decimal places must be the same for dimensions and both tolerances.

Unilateral Tolerancing:
The number of decimal places must be the same for dimensions and both tolerances.

Standard deviation (σ)

The standard deviation is a measure of the amount of variation or depression of the given set of values.

Where, σ = Standard deviation

N = Number of counts to measure (Sample Size)

xi = Distance measured of the sample individually

µ = Average mean of the sample measurement

Let us look at an example to calculate the standard deviation (σ)

Example - Calculate the standard deviation of following bracket with a sample size of 10 units.

Worst Case Condition (WCC)

This method results in the maximum or minimum variation for distance or gap, regardless of the probability. This method assumes that the manufacturing will always produce all parts or features in maximum or minimum conditions.

The worst-case tolerance analysis is a technique for calculating the maximum range a dimension can lie within. The approach also assumes that all the contributing dimensions and tolerances, all reach the specified worst conditions at once (i.e., at either the maximum or minimum), even though in practice these processes occur independently of each other.

Usually, the distance being analyzed is not directly shown on the drawing as a dimension or tolerance; it might be a reference dimension or a functional requirement. The analysis tracks a chain of dimensions and tolerances, moving from one feature (point A) to another (point B), similar to links in a chain. Each dimension and its associated tolerance are either added or subtracted based on their direction in the stack-up.

The result is a total variation range, consisting of minimum and maximum values, which defines the worst-case scenario. This method is especially valuable in safety-critical applications where failure is unacceptable. However, it often results in conservative designs that require tighter tolerances, and can lead to higher manufacturing costs.

The worst-case analysis is simple and guarantees complete interchangeability, but it does not consider statistical variation or process capability. It is most effective when absolute reliability is necessary, or when statistical data is unavailable.

Mathematical Definition of WCC

Considering the above figure that contains the four blocks having each nominal size and dimensional tolerance.

1. 1. u1, u2, u3 andu4 have the nominal dimensions of each individual feature of size.
   2. t1, t2, t3 andt4 have tolerances of each individual feature of size.
   3. U±T is total nominal dimension and tolerance.
   4. A1, A2, A3 and A4 are loop considerations with a positive direction.

According to the Worst Case Condition (WCC) the mathematical formula is:

Maximum Allowable Size is 22.85mm and Least Allowable Size is 21.15mm

WCC CASE STUDY 1

This case study looks at finding the required thickness (X) of the gasket used in the wheel hub assembly. The goal is to identify the smallest and largest gaps in the assembly using the worst-case tolerance method shown in the table below. We will use these calculated gaps to find the right gasket thickness that ensures the required compression ranges from a minimum of 20% to a maximum of 50%.

To begin with, we start by defining the particular gap whose visibility is to be evaluated and set up a dimension loop around in. We then give a plus sign (+) or minus sign (-) depending on whether each dimension in the loop travels toward or away from the gap. Other than that, the nominal dimensions with their tolerances and assigned signs, are derived in a table. Taking everything together, these contributions bind the smallest and the largest possible gaps in the assembly.

These final gap values are then used to select a gasket that is thicker than the maximum gap and can be compressed down to the minimum gap.

POSITIVE TOTAL 120.0

• NEGATIVE TOTAL 118.0

-----------------------------------------------

= NOMINAL GAP 2.0 ± 1

MAXIMUM GAP = 3.0 mm

MINIMUM GAP = 1.0 mm

Results

The minimum and maximum gaps according to the WCC are specified as follows: the minimum gap is 1mm and maximum gap is 3 mm for the given tolerances. This implies that the gasket itself is thicker than 3 mm and should be compressed down to 1mm.

For example, a neoprene seal with the initial thickness of 3.5 mm can be selected and compressed up to 70% till it attains a thickness of 1 mm as required. The compression amount, should be controlled to stay in the range of 20-50%, so we can adopt this RSS 3σ method for verification to increase the dimensional accuracy.

WCC CASE STUDY 2

This case study examines whether the snaps will properly engage for PCB mounting under the worst-case tolerance conditions. In this assembly, the PCB sits inside the enclosure, held in place by a locating pin, and is then secured by snap features before being bolted. The goal is to evaluate all dimensional variations to ensure that the snaps engage in every tolerance condition.

For the procedure, please refer to the WCC Case study 1

POSITIVE TOTAL 148.9

- NEGATIVE TOTAL 146.6

-----------------------------------------------

= NOMINAL GAP 2.3 ± 0.6

MAXIMUM GAP = 2.9 MM

MINIMUM GAP = 1.7 MM

Results
Based on the Worst-Case Calculation (WCC), the minimum gap is determined to be 1.7 mm, while the maximum gap is 2.9 mm as per the given tolerances. Therefore, as per the snap design in the worst case, minimum engagement will be the Snap Thickness (4.2mm) - Maximum Gap (2.9 mm) i.e., 1.3 mm. The maximum engagement will be the Snap Thickness (4.2mm) - Minimum Gap (1.7 mm) i.e., 2.5 mm.

Considering the worst-case method, we do not have any relation to the repeatability of how many times these tolerances will be satisfied and the rejection rate of the parts. We will explore the phenomenon in the case study associated with the RSS 3σ method.

Root Sum Square (RSS) with 3σ approach

The root sum squared (or RSS) method is a statistical tolerance analysis method that allows one to simulate the expected outcome for a population of manufactured parts and their associated assemblies. Furthermore, it is even important to understand this method when specifying tolerances for production parts.

RSS stands for the root sum squared, a statistical tolerance analysis method. It is used to predict the expected outcome for a set of manufactured parts and their assemblies. Worst-case tolerance analysis is often overly conservative for high-volume production. In such cases, the manufacturing tolerances are over-specified and they are tighter than needed.

This method results in the maximum and minimum variation for distance or gap in terms of probability also, this method considers the outcome providing a more realistic analysis.

The tolerance value for each individual feature is initially a set value of 3σ.

Ti=3 σi

σi=Ti3

Where, σi - Initial Standard Deviation

Ti - Initial Set of Value

For more information we need to understand the Central Limit Theorem (CLT).

Central Limit Theorem (CLT)

A sequence of independent random variables with mean µ1, µ2, µ3 …. µn and variation σ12, σ22, σ22 … σn2 then the density function approaches the standard normal distribution as 'n' approaches infinity. (This requires at least five sets of data.)