Tolerance studies with the Java Application ConvolutionBuilder

This is a proprietary tool for linear tolerance chain analyzes and syntheses as well as process data evaluations with the universal distribution "empirical density function" and the module for creating scatter forecasts for multidimensional and/or physical target variables, such as torques and impress forces. 

This assembly simulation is based on the statistical law that when assembling, it is very unlikely to assemble a module from individual parts that have all extreme values their dimensions. The dimensions of the assemblies fluctuate much narrower than it would be expected from a purely arithmetic perspective. 

These three program modules, see graphic on the right, can be combined as desired, e.g. to validate tolerance studies with process data from production. The graphic on the right with the three colored circles is intended to symbolize this flexibility. 

Literature

High demands are placed on complex assemblies in terms of operation and dimensional precision. As a result the component parts must meet high standards of precision. The classical arithmetic distribution of the existing clearances of the component parts would lead to unnecessarily tight manufacturing tolerances, which increases the manufacturing costs. Economically - viable solutions can only be obtained when more precise knowledge exists of the distribution of component characteristics. This knowledge may come from control charts within the framework of the SPC applications in production, which in turn could generate a statistical basis for the analysis of the dimension chain. A step towards 

DESIGN FOR PRODUCTION or DFSS ⇒ DESIGN FOR SIX SIGMA

Process data analysis with the "Density Trace" approach 

The integrated violin viewer (plot) is used to create an overall graphic of many evaluations / characteristics. The months or production days for a characteristic could be shown on the X-axis, or it could be a characteristic from several clusters, or different suppliers are placed side by side. Furthermore, this graphic can also be used for component sampling using many different characteristics. The characteristic value is on the Y-axis, with all characteristics being scaled to a 100% tolerance range so that different tolerance ranges remain comparable. Without having to read a lot of numbers in long Excel tables, the mean value of the measured values with the 50% range (thick blue line), the range of scatter and the form of distribution are immediately apparent. Overshoot and undershoot percentages are clearly visible and indicate possible problems. A possible time dependency, e.g. a trend, can be seen from the indicated individual values. The time axis runs from left to right for each violin graphic. This graphic type is extremely flexible.

Below are a few brief notes on the theory. This statistical tolerance calculation is based on the principle of convolution. The assembly is shown as an addition of several individual parts with a positive or negative sign. Each individual dimension (individual part) is subject to a certain distribution (manufacturing characteristics), which is discretized by class formation. The individual size distributions are calculated one after the other to form the overall distribution of the closed size with the help of statistical convolution (cube method). This resulting closed dimension distribution is a forecast for the scatter behavior of the assembled assemblies in the series.

For multidimensional problems, the general law of error propagation according to Gauss was implemented, which also applies to physical equations. User-defined mathematical relationships can be processed via a function compiler. This allows, for example,  tolerance forecasts for torques or press-in forces.
Furthermore, in addition to the density trace, a λ and β distribution adjustment is integrated in the module for evaluating measured values:

The Tolerance Dashboard   The Tolerance Wizard

In this Tolerance Dashboard it is possible to see the influence of each input feature to the target value by easy using the slider for each input item. Furthermore there is an multidimensional optimization tool available which works with the Evolution strategie "CORR". By using this option it is easy to close the gap between the current  tolerance and the allowed variation!

Calculation of tolerance groups inside of the tolerance to avoid to tiny, unpossible tolerances on the single parts. 

This assembly technology was created by the technical university Illmenau, Prof. Dr. Zocher.

ASM statement: Build the product quality by using a scophisticated assembly strategy.

Don't place the high quality requirements on the single part tolerances!