Sigma Magic Help Version 19

Tolerance Analysis

Overview

Tolerance analysis evaluates how variations in part dimensions (tolerances) affect an assembly's overall performance, fit, and function. In manufacturing and design, it's essential because no part is made perfectly — there will always be some variation. Tolerance analysis can determine the input components' tolerances so that the final product or assembly meets certain required specifications.

The objective of tolerance analysis:

• Ensures that parts fit together properly, even with dimensional variations.
• Helps prevent assembly issues or functional failures.
• Reduces costs by identifying where tighter (more expensive) tolerances are needed.
• Supports design decisions like part interchangeability and manufacturing methods.

There are different methods available for tolerance analysis. A brief description of the main methods is shown below.

• Worst-Case Analysis Assumes all dimensions are at their maximum tolerance limits at the same time. This method is very conservative. Even though it ensures parts will always fit, but may lead to over-tight tolerances.
• Root Sum Square Method Assumes variations follow statistical distributions. Not all the components will be at the extreme values at the same time. If the process is normally distributed, some components may be at a higher value, some may be at a lower value, and some may be close to the mean value, so the variations average. This tolerance analysis method ensures more realistic and cost-effective tolerances than the worst-case analysis method.
• Monte Carlo Analysis The worst-case Method and the RSS method are typically used when the transfer function between input and output is linear. We can use the Monte Carlo analysis method for cases where the transfer function is non-linear. This method uses random number generation to simulate complex distributions or formulas between inputs and outputs. No assumptions are required for this method, which you can use for both linear and non-linear functions.
• CAD-CAM Method For complex 3D models, there is software available that can automatically build the transfer function between inputs and outputs and perform the tolerance analysis. This approach is NOT available within Sigma Magic software - you will need the transfer function between inputs and output to use this software.

To add this tool to your workbook, click on Project and then select Tolerance Analysis.

Inputs

Click on Analysis Setup to open the menu options for this tool.

Setup

A sample screenshot of the setup menu is shown below.

1	Num Items: Specify the number of parts to be used for tolerance analysis. You can specify up to 100 parts in this study.
2	Assumed Cpk: The assumed Cpk value translates the tolerance you specify to a standard deviation value. The process is assumed to be normally distributed, and the parts produced vary between the tolerance values. For example, if the assumed Cpk value is 1, the standard deviation is derived as Tol Width / 6.
3	Model Equation: Specify the equation between the input(s) and output. Each part value is defined by the variable x1, x2, etc. Part 1 dimension is specified as x1, part 2 is specified as x2, etc. For a linear model, the equation is defined as x1 + x2 + x3, etc. For a non-linear model, you will need to define the equation used to determine the tolerance.
4	Linear Model: For a linear model, the equation is defined as x1 + x2 + x3, etc., depending on the items specified in (1) above. If there are four parts, the model equation is defined as x1 + x2 + x3 + x4. Note that the equation is automated; if you change the number of items, the model equation changes appropriately. If the suggested equation does not model your problem correctly, you must click on Custom Model and modify the equation as required.
4	Custom Model: If your model is not straightforward and linear, click this radio button to define a custom model. You can define any equation - linear or non-linear using the terms x1, x2, x3, etc. You will need to mathematically model the important gap or part and define the equation here. Note that the model equation does not change for a custom model if you change the number of items. You will need to define the right equation manually.
6	Help Button: Click on this button to open the help file on this topic.
7	View Example: Click on this button to open the example file. You can view the example to get an idea of how to fill out this tool, or you can use the example as a starting point and modify it to meet your project needs.
8	Cancel Button: Click on this button to exit without saving any changes.
9a	Create: If this is your first time using this template, click this button to format the worksheet template. You can also update the worksheet format at any time, but remember that you may Lose any data entered on this worksheet. The Create button will reformat your worksheet based on the number of items you have specified in Step 1 above.
9b	Analyze: After you have all the data defined on the worksheet, you can click the Analyze button to perform the analysis. You can also click the Compute Outputs button to perform the analysis. and generate analysis results.

Click on the Options tab to specify the options for this analysis. A sample screenshot of the worksheet is shown below:

1	Worst Case: Select this check box if you want to perform a worst-case analysis. Note that the worst-case tolerance gives you very conservative estimates for tolerances which are probably not required in the real world and drive us product costs.
2	Root Sum of Squares: Select this check box if you want to use the root sum of squares to design the tolerances. For a linear model, the root sum of squares is obtained as the square root of each square. tolerance value. For a non-linear model, Sigma Magic computes the partial derivatives of the specified transfer function and uses these to estimate the root sum of squares. However, this is just a first-order approximation and may not be valid if your model equations are significantly non-linear.
3	Safety Factor: This textbox can only be specified for the Root Sum of Squares method. The default safety factor is 1.0. If you specify a value, say 1.1, then the tolerance values obtained using the RSS method is scaled by 10% (multiplied by 1.1). You can use this safety factor to make the tolerances wider than what you would get with the pure RSS method.
4	Achieve a Specified Sigma: Select this check box if you want to determine the tolerance that gives you the given Sigma level. Note that the overall variation of the process is estimated first for either the linear model or the non-linear model. Then, this variation is used to derive the tolerance limits for the given Sigma level. This analysis assumes that the tolerances are normally distributed and the transfer function is relatively linear.
5	Sigma Level: This textbox can only be specified for Achieve a Specified Sigma option. The default Sigma Level is 6.0. The software will try to determine the tolerances in such a way as to achieve the given Sigma Level.
6	Monte Carlo Simulation: Select this check box to perform a Monte Carlo simulation of the tolerance equation. Note that this method can handle model nonlinearities and theoretically can handle any type of distribution for the input tolerances. However, in this method, all parts are assumed to be normally distributed. Suppose you want to simulate a different distribution (say uniform). In that case, you must copy the transfer function to a Monte Carlo simulation sheet and perform a more detailed analysis.
7	Num Runs: This option can only be specified if you perform a Monte Carlo simulation. The default value for the number of runs is 1000. The larger the number of runs, the more accurate the simulation results; however, it will take longer to perform the analysis.
8	Tolerance Values: This option can only be specified if you perform a Monte Carlo simulation. Specify the Tol- and Tol+ values. These values will be used to determine the capability of the process.

If you click on the Create button, the worksheet will be formatted to allow the user to enter the details of the part dimensions. A sample worksheet is shown below. On the worksheet, enter a brief description of the tolerance analysis project - for example, you can document the product number or the assembly details on this line. You may also want to document the units of this study (say mm). In the table below, enter the following details:

Item	Description
#	Row number
Part Number	Name of the part or its part number
Description	Brief description of the part if required
Dimension	The nominal dimension of the part
Tol-	The value by which a dimension can be smaller than the nominal value
Tol+	The value by which a dimension can be larger than the nominal value

Verify

If you click the Verify button, the software will perform some checks on the data you entered. A sample screenshot of the dialog box is shown in the figure below. The software checks if you have correctly specified the input options and entered the required data on the worksheet. The results of the analysis checks are listed on the right. If the checks are passed, they are shown as green checkmarks. If the verification checks fail, they are shown as a red cross. If the verification checks result in a warning, they are shown in the orange exclamation mark, and finally, any checks that are required to be performed by the user are shown as blue info icons.

1	Section: The left-hand side shows the major section categories.
2	Description: The left-hand side shows the description of the check that is performed.
3	Status: The right-hand side shows the status of the checks.
4	Overall Status: Shows the overall status of all the checks performed. If all the checked items are okay, the status image in the header will show a green checkmark. If any checks fail, the header status image will show a red cross mark.

Outputs

Click on Compute Outputs to update the outputs. You can also open the Analysis Setup dialog box and then click on the Analyze button. If all the required options have been correctly specified, the analysis is performed, and the results are shown on the worksheet. A sample set of results is shown below. The results section shows the resultant tolerance analysis numbers for each method used. It shows the method, the nominal dimension of the output variable, the tolerances derived from the analysis (Tol- and Tol+), the process capability indices like Cp and Cpk, and the defects per million opportunities (DPMO). Use the appropriate row to determine the tolerances for your study.

The graphs section shows the relative contribution to the overall variation by part. The higher the bar value, the greater the contribution of that part to the overall variation. If you perform the Monte Carlo analysis, you will also get an additional sensitivity chart. This chart also shows which part has the biggest influence on tolerancing. Say, for example, you are not happy with the tolerance and want to reduce it further, you can focus on tightening those parts that have a bigger impact on the tolerance analysis.

Examples

The following examples are in the software's Examples folder.

• An example of Tolerance analysis with three parts is shown in Tolerance 1.xlsx.

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