Tolerance Analysis

An engineering design must perform properly in spite of dimensional variation. To achieve this, engineering design requirements must be expressed as assembly tolerance limits. The designer must assign limits to the gaps, clearances and overall dimensions of an assembly which are critical to performance.

Assembly tolerance limits are applied to the statistical distribution of the assembly variations predicted by tolerance analysis to estimate the number of assemblies which will be within the specifications. Designers need to control more than just gaps and clearances in assemblies. Orientation and position of features may also be important to performance. To be a comprehensive design tool, a tolerance analysis system must provide a set of assembly tolerance specifications which covers a wide range of common design requirements. A system of assembly tolerance specifications patterned after ANSI Y14. 5 has been proposed [Carr 93].

Those ANSI Y14. 5 feature controls which require a datum appear to be useful as assembly controls. However, there is a distinct difference between component tolerance and assembly tolerance specifications, as seen in Fig. 9. In the component tolerance specification shown, the parallelism tolerance zone is defined as parallel to datum A, a reference surface on the same part. By contrast, the assembly parallelism tolerance defines a tolerance zone on one part in the assembly which is parallel to a datum on another part. In order to distinguish an assembly tolerance specification from a component specification, new symbols have been proposed.

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The feature control block and the assembly datum have been enclosed in double boxes.

Modeling Procedures and Rules

The ability to model a system is a fundamental skill for effective engineering design or manufacturing systems analysis. Unfortunately, few engineers know how to construct variational models of assemblies beyond a 1-D stack. This is primarily because the methods have not been established. There is little treatment of assembly modeling for tolerance analysis in engineering schools or texts.

Until engineers learn how to model, tolerance analysis will never become widely used as have other CAD/CAE tools. A consistent set of modeling procedures, with some guiding rules for creating vector assembly models, allows for a systematic approach which can be applied to virtually any assembly. The steps in creating a model are: 1. Identify the assembly features critical to the assembly. Locate and orient each feature and specify the assembly tolerances. 2. Locate a datum reference frame (DRF) for each part. All model features will be located relative to the DRFs. 3.

Place kinematic joints at the points of contact between each pair of mating parts. Define the joint type and orient the joint axes. These are the assembly constraints. 4. Create vector paths from the DRF on each part to each joint on the part. The paths, called datum paths, must follow feature dimensions until arriving at the joint. Thus, each joint may be located relative to the DRF by controlled engineering dimensions. 5. Define the closed vector loops which hold the assembly together. The datum paths defined in Step 2 7 of 14 5/11/2011 4:27 PM A Comprehensive System for Computer-Aided Tolerance Analysis of 2-D... ttp://adcats. et. byu. edu/Publication/97-4/cirp_2_7_97a. html become segments of the vector loop. A vector loop must enter a part through a joint and leave through another joint, passing through the DRF along the way. Thus, the vector path across a part follows the datum path from the incoming joint to the DRF and follows another datum path from the DRF to the outgoing joint. 6. Define open vector loops to describe each assembly tolerance specification. For example, for an assembly gap, the loop would start on one side of the gap, pass through the assembly, and end at the other side of the gap. 7.

Add geometric variations at each joint. Define the width of the tolerance zone and length of contact between the mating parts as required. The nature of the variation and direction is determined by the joint type and joint axes. Other variations, such as position, may be added at other feature locations. Modeling rules are needed to ensure the creation of valid loops, a sufficient number of loops, correct datum paths, etc. For example, an important set of rules defines the path a vector loop must take to cross a joint. Each joint introduces kinematic variables into the assembly which must be included in the vector model.

Fig. 10 shows the vector path across a 2-D cylinder-slider joint. The rule states that the loop must enter and exit the joint through the local joint datums, in this case, the center of the cylinder and a reference datum on the sliding plane. This assures that the two kinematic variables introduced by this joint are included in the loop, namely, the vector U in the sliding plane and the relative angle f at the center of the cylinder, both of which locate the variable point of contact in their corresponding mating parts.

Modeling Example

The process of creating an assembly tolerance model for analysis is illustrated in the figures below for a seatbelt retraction mechanism. The device is an inertial locking mechanism for the take-up reel. One of the critical assembly features is the gap between the tip of the locking pawl and the gear, as shown in Fig. 12. The assembly is of reasonable complexity, with about 20 dimensional variations and several geometric variations as contributing sources.

The contribution by each variation source depends on the sensitivity of the gap to each component variation. Fig. 13 shows the DRFs for each part and local feature datums which define model dimensions.

In Figure 14, the kinematic joints defining the mating conditions are located and oriented. Clearance in the rotating joints was modeled by two methods.

In the first case, the shafts were modeled as revolute joints, centered in the clearance, with clearance variation added as an equivalent concentricity. In the second case, the CAD model was modified so each shaft was in contact with the edge of the hole, modeled by parallel cylinder joints, and variation was determined about this extreme position. After the joints have been located, the assembly loops can then be generated, as shown in Fig. 15. To simplify the figure, some of the vectors are not shown. Fig. 14 Kinematic joints define mating conditions. Fig. 5 Vector loops describe assembly. Models for geometric variation may then be inserted into the vector assembly model, as shown in Fig. 16. The completed CATS model, in Fig. 17, is ready for assembly tolerance analysis.

Tolerance Analysis

The analysis approach used within the CATS system is based on linearization of the assembly equations and solution for the variations by matrix algebra. A detailed description with examples may be found in [Chase 95, 96] and [Gao 97]. The linearized method provides an accurate and real-time analysis capability that is compatible with engineering design approaches and tools. Vector assembly models can be used with any analysis system. Gao used the CATS Modeler as a graphical front end for a Monte Carlo simulator [Gao 93]. An iterative solution was used to close the vector loops for each simulated assembly. Histograms for each assembly feature being analyzed were generated from the computed assembly dimensions. A comparison of the linearized approach with Monte Carlo analysis is presented in [Gao 95].

Cad Implementation

Pop-up menus present lists of joints, datums, g-tols and design specs to add to the CAD model. The model is created completely within the graphical interface of the CAD system. There are no equations to type in to define mating conditions or other assembly relationships. CATS is tightly integrated with each CAD system, so it becomes an extension of the designer's own CAD system. Current CAD implementations include: Pro/ENGINEERa (TI/TOL 3D+), CATIAa, CADDS5a, and AutoCADa; (AutoCATS).

The Analyzer has built-in statistical algorithms to predict variation in critical assembly features due to process variation. It features built-in algorithms for tolerance synthesis, which re-size selected tolerances to meet target assembly quality levels. Matrix analysis gives instant feedback for any design iteration or "what-if" study. The user interface is standard XWindows Motif, with multiple windows, scroll bars, pop-up menus, dialog boxes, option buttons, data fields and slide bars for data entry, etc. The designer is in complete control of the tolerance analysis/design process.

Graphical plots give visual feedback in the form of statistical distributions, ranked sensitivity and percent contribution plots. Engineering limits are shown on the distribution, with corresponding parts-per-million reject values displayed.

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