# Tolerance Stackup Analysis : Worst Case and RSS

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Tolerance Stackup analysis is a design tool used 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 methods 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. Therefore tolerance are used. But over-defining tolerances results in increased rejection rate and overall product cost.

Optimization of part tolerances helps in increasing part tolerance band. Optimum part tolerance are calculated using tolerance stackup analysis. It is the best way to validate product assembly and deliver quality products.

#### 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. Let’s discuss these methods in detail.

##### Worst Case Analysis

Worst Case is the traditional type of tolerance stack up calculation. This method uses simple arithmetic (addition and subtraction) calculations. In this method all dimensions are assumed at extreme limit. For low production volumes worst case tolerance analysis is recommended.

• Worst Case Tolerance Analysis ensures 100% Parts Assembly on Production floor.

• Worst Case Tolerance Analysis requires very tight individual component tolerances. This increases overall manufacturing and inspection cost.

Example To understand Worst Case Tolerance Analysis, let’s consider five disks stacked on one another.

Five stacked maximum and minimum worst case thickness can be calculated using simple arithmetic operations.

Step-1 (Calculate Total Nominal Thickness)

Nominal total disk thickness is calculated by adding nominal thickness of all disks. Total Nominal Thickness = (15+10+15+12+15) = 67 mm

Step-2 (Calculate Total Tolerance)

Total upper and lower tolerance are added.

Total Upper Tolerance = (0.5+0.1+0.2+0.3+0.4) = +1.5 mm

Total Upper Tolerance = (0.5+0.1+0.2+0.3+0.4) = -1.5 mm

Step-3 (Upper and lower Limit Calculation)

Upper and lower part limits are calculated by adding nominal dimension with tolerance.

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

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

You can also use this worst case tolerance calculation sheet

##### Statistical Tolerance Analysis Statistical tolerance Analysis does not focus on the extreme limits. Because manufactured part variations considered to be distributed. Each dimension has unique distribution based on the manufacturing process.

This analysis is based on Bell curve and normal distribution. For example if one thousand disks are manufactured. Disk thickness will be manufactured on upper limit, on nominal and lower limit,

• Statistical Tolerance analysis helps in increasing component tolerance limits. In worst case, part tolerances becomes very tight that increases part cost.

• When parts are designed using statistical tolerance analysis. Manufactured part can gets out of limit and parts can get rejects. Number of parts can get rejected depending on part manufacturing capability (3σ, 4σ, 6σ
• Statistical tolerance analysis does not give 100% guarantee for part assembly.

Example for Tolerance Stackup Analysis using Root Sum of Squares (RSS) Method : 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 Statistical Tolerance Analysis lets consider five disks stacked on one another. We will calculate maximum and minimum thickness using RSS method.

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

Step-1 (Calculate Total Nominal Thickness)

Nominal total disk thickness is calculated by adding nominal thickness of all disks. Total Nominal Thickness = (15+10+15+12+15) = 67 mm

Step-2 (Calculate Standard deviation for each Tolerance Considering process is 3σ capable)

Standard Deviation = Total Tolerance / (2 X 3)

σ1 = (0.5+0.5) / (2X3) = 0.166

σ2 = (0.1+0.1) / (2X3) = 0.033

σ3 = (0.2+0.2) / (2X3) = 0.066

σ4 = (0.3+0.3) / (2X3) = 0.1

σ5 = (0.4+0.4) / (2X3) = 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]

σ (assembly) = 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

You can also Use this Tolerance stack-up calculation sheet.

#### 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. You can Download Tolerance stack-up calculation sheet here.

#### 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.

Got Questions?  We will be happy to help.

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