Tolerance Stackup Analysis
Jo?o Leite
Engineering Manager, Program Manager, Tool & Die Expert ~25 years experience in the ?? automotive industry
In the article “Process Performance (capability)” we talk about process performance indicators (aka “capability”). Where 2 types of problems were identified, they are:
- Dispersion – associated with the amplitude of the measured values;
- Location – associated with the location of the average value of the measured values, in view of the specification limits;
However, in day-to-day life, this problem is amplified when we have to link several dimensions in the same part or in an assembly of different parts.
Where a set of parts with dimensions within the tolerance range, then conforming parts. When assembled, they give rise to a set with dimensions outside the specified range.
If we think of 3 pieces, each one with a different length.
It will be mathematically correct that the length of the set formed by A+B+C is 100+50+25 = 175 mm.
In each of the pieces, the length value may suffer from the problems mentioned above (location, dispersion).
Making it very difficult to predict what we will get during the life of the product.
Solution
To minimize this problem, an analysis of tolerances should be carried out, or Tolerance Stackup Analysis.
In tolerance analysis, several activities are carried out related to the study of potential accumulated variation in mechanical parts and assemblies. This methods can be used in other types of systems subject to accumulated variation, such as mechanical and electrical systems.
This study should be carried out during the design phase, before defining the final specifications (eg geometric dimensioning and tolerances - GD&T), or before defining the manufacturing process, both for the components and for the assembly.
Methods include 2D tolerance stacks, 3D simulations, and RPS analysis.
It must be carried out, at least, for all the critical characteristics of the product.
Practical case
In this example, we are going to use the “worst case” of tolerance analysis, commonly called “tolerance stackup”.
In this method the individual variations of each component are placed within their tolerance limits. Making two sums, they are:
- Sum of the individual maximum dimensions;
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- Sum of the individual minimum dimensions;
In this case, the distribution of the individual variables is not considered, assuming that each of the variables will not exceed their respective specified limits.
This model predicts the maximum expected variation of the measurement. The tolerance requirements are defined so that 100% (conforming) parts will assemble and function correctly, regardless of actual component variation.
This method of adding dimensions and their tolerances (tolerance stackup) has two disadvantages:
However, this type of study (Worst Case) is often required by the customer for all critical dimensions.
For all non-critical dimensions statistical calculations can be applied to ensure acceptable assembly yields with increased component tolerances and reduced manufacturing costs.
Example
In the graphs below, we have histograms representing the normal distribution of each of the variables, and the respective values of Cp and Cpk.
Calculating the dimension resulting from stacking the dimensions (sum), in order to simulate the resulting set, associating the different components randomly, we would have:
From the graph of the "arithmetic sum of tolerances" we would obtain an off-center tolerance, with excellent capability indicators, if the assmebly tolerance matches the calculated sum of individual tolerances (175 +1/-0.7).
However, if for the same assembly we had a centered and slightly lower tolerance as the customer's requirement, the result in terms of capability would be significantly worse, if the assembly tolerance is cusomer defined witout taking in consideration the individual components tolerances (exampple 175 +0,4/-0,4).