preliminary tolerances.
3.Analyze the assembly tolerances.
4.Determine fitness of design, iterate as necessary.
Component dimensions are “in-specification” (in spec) if they are manufactured within their specified tolerances. The combined effect of the individual variations in an assembly may not be in spec, even if the components are. It is sometimes discovered through tolerance analysis that a concept will not achieve the assembly tolerance needed.
When a tolerance stack-up shows the assembly to be “out-of-spec,” the designer reduces and/or re-distributes component dimension tolerances using some of the following methods:
1.Redesign the assembly to reduce the number of components in the stack-up chain, or utilize features that can be produced through inherently more precise processes.
2.Change the component dimensioning scheme to better represent assembly method, apply different geometric controls, or reduce the number of dimensions in the stack-up chain.
3.Eliminate fits from the stack-up chain.
4.Utilize precision locating methods (see Chapter 4) to reduce variation introduced by fits.
5.Reassign / reduce component dimension tolerances.
THE TOLERANCE STACK-UP CHAIN
The first step in a tolerance stack-up is to determine the chain of dimensions contributing to the stack-up. Consider the assembly of components in Figure 3-17. In order for the components to be assembled, it is necessary for the tab of the left component to fit within the slot of the right component. The detail drawings illustrating relevant dimensions are shown in Figure 3-18. The designer is interested in the allowable variation on the size of the lower gap of the tab / slot features, as well as the upper gap of the tab / slot features. Two stack-up calculations are required to determine that positive, non-zero gap distances exist on both the top and bottom clearances.
For chains involving only a few dimensions, it may be tempting to take shortcuts. However, an organized approach is the best approach in determining the tolerance stack-up chain. The procedure is as follows:
1.Understand the assembly. Analyzing unfamiliar designs may require some time to get oriented.
2.Gather the detail component drawings, which may be in process.
3.Make an assembly sketch. Hand-drawn is best, as it is often helpful to illustrate gaps and clearances, show components at extreme positions and orientations, and show other dimensions out of scale.
4.Choose a sign convention (e.g., positive up, positive to the right).
Figure 3-17: Sample Assembly for Analysis
Figure 3-18: Detail Drawings of Component Parts
5.Determine which dimension needs to be solved. This requires a good understanding of the problem in order to convert a design concern into a specific dimension that must be discovered. The concern being addressed by the analysis may require that several stack-ups be evaluated. A stack-up to determine if a retaining ring will fit within its groove would seek to calculate the clearance gap dimension (groove width – ring thickness) and ensure it is positive. More complicated problems can be properly defined and understood with the aid of the sketch.
6.Draw the assembly dimension and label it ‘A’ for assembly. The assembly dimension is the unknown dimension you’re solving for. It should be the only dimension that is not obtained from the component detail drawings.
7.Identify contributing dimensions. Determine which dimensions on the component detail drawings contribute to the size, position, and / or orientation of the assembly dimension sought. These dimensions may be given with bilateral equal or unequal tolerances, unilateral tolerances, limit tolerances, title block tolerances, or geometric controls. Draw the other dimensions on the assembly sketch per the detail drawings. Label them.
8.Make each dimension into a vector by assigning an origin and destination. The choice of origin and destination is arbitrary, but it is important that this convention be maintained once it is chosen. The vector is illustrated with an arrowhead on the destination’s extension line and an origin symbol (circle) on the origin’s extension line.
9.Chain all the dimensions together head-to-tail. Ensure the chain of dimensions forms a complete loop with the destination of the final dimension connecting to the origin of the first. The term “loop” is used even if the vectors may all be one-dimensional (e.g.. all dimensions are up / down, or all dimensions are left / right). Although components are three-dimensional, and specifications made properly using GD&T fully define the component in 3D, tolerance analyses are generally 2D or even 1D problems. Depending on the assembly, the designer determines in steps 1 through 3 whether the stack-up problem is 1D, 2D, or 3D. The simplification to a 2D or 1D problem does not omit any information or oversimplify the analysis — it is done when the problem is, at its core, a 2D or a 1D problem. The vast majority of tolerance stack-up problems within machine design are 1D. In other fields, 2D and 3D analyses may be required more often. The rest of this section focuses on techniques for 1D stack-ups.
10.Write the stack-up equation. Write out the tolerance stack-up equation as the algebraic sum of dimensions, following the positive / negative sign convention. Set the sum equal to 0. Re-arrange the equation to solve for assembly dimension A.
A tolerance stack-up chain drawing clearly shows how component dimensions relate to one another and contribute to assembly dimensions. Understanding this relationship allows the design engineer to meet the functionality and cost goals. The creation of the stack-up chain drawing is the first step in analysis or assignment and is best begun when the design of the overall assembly scheme is still preliminary.
The simplified stack-up chain drawing is shown in Figure 3-19. Each dimension has a marked origin and destination, i.e., a “from” denoted by the circular origin symbol and a “to” denoted by an arrow. Each “from” connects to the next “to.” The assembly dimension connects the first “to” to the last “from.” A clearance dimension is included for each fit.
Figure 3-19: Stack-Up Chain Drawing
The stack-up equation is first written by summing the dimensions using the sign convention (“up” arrows positive, “down” arrows negative):
+G − H − A + F + E + B − C − D = 0
Re-arranging to solve for the assembly dimension A:
A = B − C − D + E + F + G − H
The assembly dimension A contains both dimensions with positive signs (positive contributors) and negative signs (negative
