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 contributors).
The stack-up equation must be created before any calculations can be done. Some tolerance stack-up problems are very simple and can be solved with simple arithmetic without the need to write the stack-up equation. These are sometimes done mentally, or written on scrap paper that is not maintained as part of the engineering records, resulting in a loss of important engineering analysis. The step-by-step process illustrated here assures that a standard methodology is followed, ensuring a commonality of approach regardless of the complexity of the problem. This method is appropriate for both simple problems and complex stack-ups involving dozens of dimensions. It also allows checking and review by other engineers seeking to verify correctness of analysis, to issue engineering changes, or to diagnose problems encountered downstream.
PRELIMINARY TOLERANCE ASSIGNMENT
Assigning small tolerances requires producing high-precision components that increase cost. The designer must be careful not to exceed the capabilities of the manufacturing process (see Section 3.1), or to exceed cost targets. When assembly variation is sufficiently controlled by limiting allowable component variation, interchangeability is achieved. Interchangeability has many benefits to the assembly process, maintenance, and field service, but it requires careful planning.
Interchangeability may be required if a firm uses globally-sourced components and its products are installed, sold, used, or serviced globally; for example if its strategy is to be able to build-to-print from anywhere any time, and ship parts to any field location. Replacement parts and serviceable parts often need to be completely interchangeable as machining-to-fit is not an option. Interchangeable parts enable lower-skilled technicians or users to service, repair, and / or replace parts without precision assembly setup procedures. Whenever it is reasonable to assume that the component in question will be exchanged for an equivalent one, designing for interchangeability should be considered.
When interchangeability is not required, or when assembly variation cannot be adequately controlled by tolerance assignment alone, several design options are available, each requiring greater skill during the assembly process:
•Setup at assembly: Adjustability is designed into the assembly, assembly instructions are provided by the design engineer, and final setup dimensions and assembly tolerances are clearly indicated. Standard shop inspection tools, portable coordinate measuring machines, or custom-designed gages may be used to achieve the assembly accuracy required.
•Assembly fixturing: Similar to setup at assembly, though fixtures are used to hold some or all components in relationship to one another during the assembly process. Many modular fixturing systems are available, or custom fixturing may be required.
•Assembly of matched sets: Components are segregated by the measurements of critical dimensions. Assemblers match small components with large ones, for example — to achieve a tighter assembly variation. The stack-up guides the assembly planner in creating assembly instructions. Ball bearings are produced as matched sets of inner / outer races and balls.
•Machining at assembly: After the components are assembled, critical dimensions are achieved by finish machining features that were only roughed in initially. Dowel pins or other permanent fitting methods are often used to lock in the critical assembly dimensions, either by preventing disassembly or by ensuring repeatable re-assembly.
Even though the assembly’s dimensional needs have been met, the task of tolerance assignment is not necessarily complete. Consider an example of seven dimensions contributing to a tolerance stack-up chain. One approach is to assign the same tolerance to each dimension in the chain, such that the total assembly tolerance is achieved. This may not be optimal because features that are simple to produce with precision could have smaller tolerances assigned, and features made using more difficult or costly operations may require larger tolerances. A second approach seeks to match each contributing dimension tolerance to the manufacturing capabilities used to produce it. Section 3.1 provides guidelines for common manufacturing processes. Good partnerships between design engineers (function, design, dimensioning) and machine shop / manufacturing personnel (process capability and cost) can prove extremely valuable. Alternative approaches to tolerance assignment incorporate manufacturing vs. tolerance cost information and sensitivity analysis to assign tolerances based on multiplicative effects of their contributions to the assembly tolerance and cost. Analysis gives just one answer, but assignment has no unique solution.
ANALYSIS AND ASSIGNMENT METHODS
Tolerance analysis and tolerance assignment are two sides of the same coin, the only difference being which dimensions are considered inputs and which are outputs. For either approach, four methods of calculation are common:
•Worst-case
•Statistical
•Combined
•Monte Carlo
Worst-Case Method
The most common method of tolerance stack-up for machine design engineers is the worst-case method. The tolerance stack-up calculation is performed twice, resulting in