Root Sum Square Tolerance Calculator | Stack Check

Root sum square is a statistical method that can be used to calculate total tolerance of an assembly. An assembly is made up of a collection of parts, and each part has a tolerance to that part. The tolerance to a part is the amount of variation that is permit in that part.

If many parts is placed into an assembly, the total variation of those individual parts create the total variation that is permit to the assembly. There are two main methods of calculating the total variation of an assembly: worst-case analysis and root sum square analysis. Worst-case analysis calculates the total variation of the assembly by assuming that each of the individual parts are at its maximum error simultaneously.

Root Sum Square for Assembly Tolerance

As such, the total variation that is calculated using this technique will be large. However, it is unlikely that each of the parts of an assembly will be at their maximum error at the same time. Instead, the errors will cancel each other out due to the independent movement of each part.

Root sum square analysis is a statistical estimation of the total variation of an assembly that takes into consideration the probability that any given part will be at a certain error. To calculate the root sum square of an assembly, each of the individual tolerances of the parts (represented as sigma) must be square, summed, and the square root must be taken of that sum. The result will be a total variation of the assembly that is 30% to 50% smaller than the total calculated through the worst-case analysis.

As such, the root sum square analysis allow for the individual tolerances of the parts to be loosened so that manufacturing of those parts is made easier. Root sum square analysis is most effective when the parts of an assembly can move independently of one another. For instance, one part may be slightly too large while another is slightly too small…

The errors cancel each other out. To utilize this method, a sigma level must be selected for each of the individual part. Sigma levels of 3σ are often used to allow for 99.7% of parts to fall within the allowed tolerance.

Additionally, sigma levels of 1σ can be used if there is limited data on the size of each part, or if sigma levels of 4σ is used to be more conservative in the allowed deviation of each part. One of the most common errors in the root sum square analysis method is failing to account for the direction of the nominal dimension of the parts. Nominal dimensions are the intended size of the parts, but the dimensions of the parts may deviate from the nominal dimension in either a positive or negative direction.

If a dimension is positive, it will contribute to the total length of the part; if the dimension is negative, it will detract from the total length of that part. If the wrong sign is assigned to the dimensions, the mean size of the parts will be incorrectly calculate, which may cause the assembly to fall outside of the specifications of the assembly. Not all parts of an assembly contribute to the total variation of the assembly equally.

Some parts may have a large impact on the total variation, while other parts may have a small impact. If one part has a very large sigma compared to the other parts of the assembly, then that part is the dominant contributor to the total variation in the assembly. As such, it is more effective to tighten the tolerance of the dominant contributors to the total variation than it is to tighten the tolerance of the parts that have a small impact on the total variation.

Worst-case analysis should be used in situations where the parts of an assembly are not independent of each other. For instance, if a component is cut from a single piece of material, all of the parts will be link to one another. Such parts are not independent of one another, and, therefore, root sum square analysis could lead to a calculated tolerance of the assembly that is too tight.

Such a tight tolerance could lead to failures in the assembly. After calculating the tolerance of an assembly, the Cpk value of that assembly should be calculated to determine how capable the process is of creating components within the required specifications. If the Cpk value is above 1.33, the process is within a comfortable margin.

However, if the Cpk value of the process is below 1.0, there will be a likelihood that the parts created by that process will fall outside of the specifications for the parts. Even with a narrow tolerance of the assembly, if the process is not center correctly within the specifications for the parts, the assembly will fail. Root sum square analysis is a method of calculating the total variation of an assembly through the consideration of probability of the individual parts.

The root sum square analysis method requires that the individual sigmas be squared, sum, and the square root taken of that sum. It is important to account for the direction of the nominal dimension for each part. In addition, it is important to determine which parts are the dominant contributors to the total variation.

Finally, it is important to remember that this method is not appropriate for applications where the parts of an assembly are not independent of one another. Through the use of these methods, engineers can create a realistic specification of the tolerances for each part of an assembly.