Most parts look fine on their own. The drawing looks okay, and all dimensions are as per the specifications. However, just as the assembly process is complete, something arises that does not seem to fit, and no one can immediately explain the reason. This usually results from a problem with the tolerance stack. It tends to arise more frequently because tolerances are always considered individually, part by part, never across the full assembly.
Introduction: What Is a Tolerance Stack-Up?
Every manufacturing process entails some degree of dimensional tolerance. Despite having very small tolerance values, every manufactured part would never be produced to the same dimensions. For instance, a shaft might have a tolerance of 0.02mm less than nominal, while the hole that it should fit would have a tolerance of 0.02mm over. By themselves, both parts would pass; however, together, this represents a total of 0.04mm of variation. Whether this variation would be adequate or not would depend upon the functionality of the assembly.
Therefore, tolerance stack-up analysis is a technique by which all the individual variations are added to determine whether the total variation would still allow the system to function properly. This is not an issue of a specific dimension, but rather, an issue of how the variation of all dimensions at the same time affects the operation of an assembly.
The simple calculations include addition and subtraction. For each part, the nominal dimension and tolerance are taken, and the accumulated value is calculated. The key question here is whether the assembly functions when all parts are at the worst-case dimensions.
This question covers three things, and most engineers only check one:
- Will the components fit together physically? It considers issues such as clearances, interference fits, and confirms if a pin goes into a hole.
- Will the component assembly perform its intended function? A gap can be adequately large for the assembly but too large for the seal to fit or too small for a particular mechanism to function.
- Will it keep on doing both continuously in each of the units being produced? The design may be perfect in a prototype, but if it fails even in one out of twenty produced units, it clearly has a stack-up problem that didn’t show up earlier.
Why Tolerance Stack-Ups Matter in Real Production
Skipping tolerance stack-ups shows up later in your production run in very specific and costly ways, as listed below:
- Production line assembly problems: If parts don’t fit together, the production stops. Failing to do the tolerance stack-up before cutting the tools and ordering the parts means that the solution requires design changes and scrapping parts that were manufactured within tolerance, but are now incompatible.
- Field failures of the finished product: A mechanism that turns almost perfectly on the bench could be seized due to thermal expansion when in use. A seal that works during assembly will fail to hold pressure due to insufficient compression. The reason for failures like these is harder to track down, as they do not manifest themselves as a dimensional problem, but rather as a design or materials problem.
- Inconsistency of the final product: Even a marginal stack-up will mostly work, but ultimately fail 1 out of 50 or 1 out of 500 units. The problem is difficult to identify even when doing small batch tests.
- Future cost of tightening tolerances: If you discover stack-up after production has begun, you must tighten tolerances in one or more dimensions, leading to slower machining, more inspections, more scrap, and increased cost per piece. However, if you find a stack-up during the design phase, there is virtually no cost involved.
Looking for assistance with your tolerance analysis for your existing design? Get in touch with Premium Parts, and we’ll help analyze the risks of stack-up in your assembly designs.
Methods of Tolerance Analysis
There are several ways to run a tolerance stack-up, and the right method depends on how much variation the design can tolerate and how much production volume is involved.
Worst-Case Analysis
Worst-case tolerance analysis is the most straightforward technique. It assumes that each member of the tolerance chain is at worst-case dimensions all at once and verifies whether the assembly functions in these circumstances.
If the worst-case tolerance analysis passes the test, then the assembly will be functional for all units manufactured. This might seem ideal, but the compromise in this case is the high costs. This is because it requires tightening the tolerances, hence increasing machining cost and scrap rate. The more parts in the chain, the higher the possibility of requiring tolerances that cannot be practically maintained.
Worst-case tolerance analysis should be employed in cases where failure will have catastrophic effects, such as in medical devices, aviation equipment, and safety-related mechanisms.
Root Sum Squared (RSS) Analysis
Root Sum Square (RSS) analysis is a mathematical/statistical approach that considers how, in reality, it is impossible to have all of the pieces in the assembly at the same time, having the worst-case size. Some will be slightly above average, some below, and therefore the distribution will be proper throughout the production sample.
RSS starts by squaring all of the tolerances in the chain, adding up all the values, and taking the square root of the sum. Thus, we get a reduced tolerance value compared to the worst-case scenario.
The drawback of RSS is that it does not ensure that there will be no failures at all. Instead, it gives a statistical probability of failures. Assuming a normal distribution and a controlled process, RSS shows that 99.73% of assemblies will be within specifications, meaning that there should be about 2,700 failures per million units produced. For the majority of commercial items, it is an acceptable rate.
For the high-volume consumer products and highly precise items, this number is too big. RSS is used only when there are at least three dimensions in the chain and the process of production is stable and normally distributed.
Cpk and Process Capability
Calculating by any method only gives you a number, and the number only tells you what your design is able to withstand. However, for production to work, you must consider the capability of the manufacturing process to meet these tolerances.
This is where Cpk comes into consideration. It is the statistic for the capability of a manufacturing process to stay within specification relative to its variation. A Cpk value of 1.0 means that your manufacturing process barely fits within the specification. A Cpk value of 1.33 means that your manufacturing process is safe by some margin. A Cpk value of 1.67 and above means that the manufacturing process is capable of sufficient margin on both sides.
In the case of marginal results in tolerance stack-up, the Cpk analysis will help you determine whether your manufacturing process is capable of meeting the individual tolerances. Tolerance stack-up combined with Cpk analysis is what makes an assembly viable for production.
Need to specify tight tolerances in your precision assembly? Submit your blueprint to Premium Parts, and rest assured, we will check the process capability as well as the tolerances before giving you a quote.
Conclusion
Tolerance stack-ups are mandatory on any assemblies where function really matters. Tolerances that may appear adequate for each part in a separate case will lead to a nonfunctional assembly when assembled together, and learning about it after tools have been fabricated and parts have been ordered is definitely not what you want.
The choice of approach depends on the risk involved, volume of manufacturing, and available process data. However, always start from the definition of assembly requirements, identification of functional dimensions, and calculation of their tolerance stacks before any part fabrication.
At Premium Parts, tolerance stack-ups are reviewed as one of the steps during the quote generation for precision assemblies. In case your assembly requires tight functional tolerances, critical clearances, or high-risk failure modes, tolerance stack-up review should be among the first things done.
Discuss your assembly requirements with a Premium Parts engineer before going into production.
Frequently Asked Questions
When should the tolerance stack-up be performed?
Before setting tolerances on any assembly in which fit or function relies upon multiple parts. The sooner you do your tolerance stack-up, the more cost-effective it will be. Under any of the following situations, tolerance stack-up must be performed:
- When evaluating mechanical fit
- Testing/inspecting the mechanism performance
- Managing production costs
- Avoiding assembly issues
How many parts are needed for RSS to be valid?
Three or more. With only two dimensions in your tolerance chain, the statistical average used by RSS falls apart.
What’s the relationship between Cpk and tolerance stack-ups?
Tolerance stack-up defines the requirements that the manufacturing process needs to meet to manufacture your product. Cpk indicates whether the process can meet those requirements. A product with a requirement of Cpk = 1.67 from a process with a Cpk = 1.0 will fail in production, even if the tolerance stack-up math is right.
Can tolerance stack-up be performed without a CAD package?
Sure. At its simplest level, your tolerance stack-up requires only nominal values and tolerances from your print. Spreadsheets will handle your linear tolerance chain easily. 3D tolerance packages will provide additional value for complex assemblies with variations that are not strictly linear.
