In this article, we will explore two examples of 2D tolerance analysis using the worst case method. This is part of a series of articles on multi-dimensional tolerance analysis, and it's highly recommended to read the Multi-Dimensional Tolerance Analysis Basics article first to get background information on how MEboost performs the analysis.

There are two use cases for tolerance analysis, and we will look at an example of each:

**Dimension Analysis** Analyze the resultant dimension on a part or an assembly of parts. This is analogous to a 1D stack height.

**Clearance Analysis** The resultant dimension represents the clearance/interference between assembled parts.

## Dimensional Analysis Example

We'll do a worst case analysis on the following plate. Given dimensions of the holes, we want to find the variation in the distance between them. The drawing shows nominal dimensions.

For worst-case, we assume that each input dimension (labeled A through E) has worst-case values of ±.010. To open the 1D/2D/3D worst case tool, click the following button on the MEboost ribbon.

The tool form will appear. Click the "Dimension Chain" tab to enter the input dimensions in the chain. Dimension A information is entered.

For 2D tolerance analysis, all dimensions are in the xy plane. For this example, we'll align the x axis with the bottom of the plate. Alpha is the angle of the vector in the xy plane from the positive x axis. Dimension A goes from right to left so its alpha angle is 180 degrees. Beta is the angle of the vector from the xy plane. Since all dimensions are in the xy plane, beta is zero.

Once all dimension information is entered, click the Add Dimension button. The dimension name will appear in the input dimension chain box. You can click on a dimension to see all of its information.

Add the remaining input dimensions.

Now click the Results tab. We can save the model for use later by selecting "New Model", entering a name, and clicking the Save button. Note that you need to also save the workbook to make the changes permanent.

For 1D analysis, the minimum and maximum resultant dimension occurs when input dimensions are at their minimums and maximums. This is straightforward to calculate.

For 2D and 3D, the extreme resultant values may occur with input dimensions at intermediate values. The problem is we don't necessarily know what intermediate values result in extreme values of the resultant.

To estimate the resultant extreme values, input dimensions are sampled as uniformly distributed random variables and the resultant dimension is calculated. This process is repeated for the number of iterations specified. The greater the iterations, the better chance of finding a close approximation of the resultant extremes.

To determine if enough iterations are used, start with a nominal value such as 50,000. Then increase iterations and see if the results are changing by an unacceptable amount. If there is little change after increasing iterations, the true extremes have been estimated.

For this example, we'll use 250,000 iterations. Click the Calculate button to start the calculation process.

The nominal, minimum, and maximum resultant dimension is shown. Delta x, y, and z are the x, y, z components of the resultant dimension.

To determine sensitivity, the resultant dimension is calculated by using the minimum and maximum of a dimension while holding all other dimensions at their nominal values. The values shown in the sensitivity box is the change in the resultant dimension from nominal when the variable is at minimum and maximum.

## Clearance Analysis Example

For the clearance example, we have a rectangular bar that must fit inside an extrusion. In this example we will also show how to model geometric tolerances. The dimensions for the bar are shown below with a tolerance of ±.005.

The extrusion drawing is shown below with a tolerance of ±.005.

The assembly is shown below with the dimensions that are to be modeled.

The geometric tolerances and their modeling values are shown in the table below. Note that we use a nominal dimension of 0.

To open the 1D/2D/3D worst case tool, click the following button on the MEboost ribbon.

The tool form will appear. Click the "Dimension Chain" tab to enter the input dimensions in the chain. Dimension A information is entered.

When all dimensions are entered, they will appear in the Input Dimension Chain box.

Click on the Results tab. For this example we will run 250,000 iterations. Since the resultant dimension is always positive, it cannot tell us for sure whether there is a clearance or interference fit. Therefore we will find extreme values of the deltas which will show this information.

Click the calculate button to run the analysis.

From the results, the minimum delta y is positive, but the minimum delta x is negative. Therefore, an interference fit is possible since minimum delta x is negative. Both minimum delta x and delta y must be positive to guarantee a clearance fit. In real life, you could adjust the nominal dimensions and tolerances to iterate until a suitable design is achieved.