Tolerance Stackup analysis is an design tool to analyse and optimize product assembly. It calculates the cumulative effects of part tolerances in an assembly. Therefore it helps in ensuring smooth part assembly. In this article we will discuss worst case and RSS merhods to solve tolerance analysis problems.

For good understanding of tolerance stackup analysis. We suggest you to first understand what are limit, Fits and Tolerance ?

#### Why Tolerance Analysis is Required

Production of closely mating parts, without tolerances is economically unfeasible. It results in increased rejection rate and overall product cost. Therefore tolerance are used. Now the first question that comes in mind is **how to calculate optimum tolerance?**

Answer to this question is tolerance stackup analysis. It is the best way to validate product assembly and calculate optimum part tolerances. It ensures proper part fitment during product assembly.

Optimization of part tolerances results in increasing tolerance band. It reduces part rejection rate and Cost.

Sometimes out of tolerances part gets assembled easily. But with time they degrades and their performance is get affected.

#### Types of Tolerance Stack up Analysis

Various methods to perform tolerance stack up analysis is available. But **Worst Case** and **Statistical Tolerance Analysis** methods are more popular.

##### Worst Case Analysis

It is the traditional type of tolerance stack up calculation. It is simple arithmetic (addition and subtraction). In this method all dimensions are assumed at extreme limit. For low production volumes this method is recommended.

**Advantages of worst case analysis**

- Worst-case tolerance ensures 100% of the parts will assemble and function properly.

**Disadvantage of Worst Case Analysis**

- It requires very tight individual component tolerances. Therefore manufacturing and inspection cost will increase.

**Example**

Lets consider an example of five disks stacked on one another. Tolerances for each disk is given.

To understand this in detail lets consider the example of five disks stacked on one another. We will calculate maximum and minimum thickness using worst case analysis.

Total disks thickness using worst case tolerance stack-up analysis can be calculated with simple arithmetic operations.

**Step-1 :** Calculate Total Nominal Thickness = Sum of all nominal dimensions = 67 mm

**Step-2 :** Calculate Total tolerance = Sum of all Positive and negative tolerance = ± 1.5 mm

Step-3 : Calculate Upper and lower Limit

Upper Limit = Nominal Dimension + Tolerance = 67 + 1.5 = 68.5 mm

Lower Limit = Nominal Dimension – Tolerance = 67 – 1.5 = 65.5 mm

##### Statistical Tolerance Analysis

Statistical tolerance Analysis does not focus on the extreme limits. Because there is a distribution of the variation for each dimension. Each dimension has unique distribution based on the manufacturing process.

This analysis is based on Bell curve and normal distribution. For example if we are manufacturing thousands of disks. Some of disk thickness will be low, some high and maximum will be in middle.

**Advantages Statistical Tolerance Analysis**

- It helps in increasing tolerances limits. In worst case method, tolerances become very tight that increases cost.

**Disadvantages Statistical Tolerance Analysis**

- With statistical tolerance analysis there are chances part dimensions can go out of limit.
- Statistical tolerance analysis does not give 100% guarantee for part assembly.

**Example of Root Sum of Squares (RSS) Statistical Tolerance Analysis :**

The root sum square (RSS) method works on a statistical approach. It is assumed that most of the parts fall to the mid of the tolerance zone. To understand this in detail lets consider the example of five disks stacked on one another. We will calculate maximum and minimum thickness using RSS method.

For calculation we will consider process is 3σ capable and Cpk is 1.

**Step-1:** Calculate Nominal Thickness for all Disks

Total Nominal Thickness = (15+10+15+12+15) = 67 mm

**Step-2 :** Calculate Standard deviation for each tolerances.

σ1 = 0.5/3=0.166

σ2 = 0.1/3=0.033

σ3 = 0.2/3=0.066

σ4 = 0.3/3=0.1

σ5 = 0.4/3=0.133

**Step-3 :** Calculate Standard deviation for assembly.

σ(assembly) = √ [(σ1)^2 + (σ2)^2 + (σ3)^2 + (σ4)^2 + (σ5)^2]

= √ [(0.166)^2 + (0.033)^2 + (0.066)^2 + (0.1)^2 + (0.133)^2]

**= 0.2472**

**Step-4 :** Calculate total tolerance zone.

Tolerance Zone = σ(assembly) X 3 = 0.74162

Step-5 : Calculate Upper and lower Limit

Upper Limit = Nominal Dimension + Tolerance = 67 + 0.74162 = 67.74162 mm

Lower Limit = Nominal Dimension – Tolerance = 67 – 0.74162 = 66.2584 mm

#### Tolerance Stack-up Calculator

Manual calculations is not the only way for tolerance stackup calculations. You can also use worst case and RSS method tolerance stack-up calculator for tolerance stack-up calculations.

#### Conclusion

To sum up, Tolerance stackup analysis is very important part of product design. It also helps in reducing part cost. Various tolerance stackup analysis methods are available.

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