The Direct Linearization Method (DLM) for tolerance analysis of 3-D mechanical assemblies is presented. Vector assembly models are used, based on 3-D vector loops which represent the dimensional chains that produce tolerance stackup in an assembly.Tolerance analysis procedures are formulated for both open and closed loop assembly models. The method generalizes assembly variation models to include small kinematic adjustments between mating parts.Open vector loops describe critical assembly features. Closed vector loops describe kinematic constraints for an assembly. They result in a set of algebraic equations which are implicit functions of the resultant assembly dimensions. A general linearization procedure is outlined, by which the variation of assembly parameters may be estimated explicitly by matrix algebra.Solutions to an over-determined system or a system having more equations than unknowns are included. A detailed example is presented to demonstrate the procedures of applying the DLM to a 3-D mechanical assembly.
Geometric feature variations are the result of variations in the shape, orientation or location of part features as defined in ANSI Y14.5M-1982 tolerance standard [ANSI 1982]. When such feature variations occur on the mating surfaces between components of an assembly, they affect the variation of the completed assembly. The geometric feature variations accumulate statistically and propagate kinematically in a similar manner to the dimensional variations of the components in the assembly.The Direct Linearization Method (DLM) for assembly tolerance analysis provides a method for estimating variations and assembly rejects, caused by the dimensional variations of the components in an assembly. So far, no generalized approach has been developed to include all geometric feature variations in a computer-aided tolerance analysis system. This paper introduces a new, generalized approach for including all the geometric feature variations in the tolerance analysis of mechanical assemblies. It focuses on how to characterize geometric feature variations in vector-loop-based assembly tolerance models.The characterization will be used to help combine the effects of all variations within an assembly in order to predict assembly rejects using the DLM.
Two methods for performing statistical tolerance analysis of mechanical assemblies are compared: the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection of 2-D and 3-D vector models of assemblies were analyzed, including problems with closed loop assembly constraints. Closed vector loops describe the small kinematic adjustments that occur at assembly time. Open loops describe critical clearances or other assembly features. The DLM uses linearized assembly constraints and matrix algebra to estimate the variations of the assembly or kinematic variables, and to predict assembly rejects. A modified Monte Carlo simulation, employing an iterative technique for closed loop assemblies, was applied to the same problem set. The results of the comparison show that the DLM is accurate if the tolerances are relatively small compared to the nominal dimensions of the components, and the assembly functions are not highly nonlinear. Sample size is shown to have great influence on the accuracy of Monte Carlo simulation.
Tolerance sensitivity indicates the influence of individual component tolerances in an assembly on the variation of a critical assembly feature or dimension. Important applications include assembly tolerance analysis and tolerance allocation. This paper presents a new method for determining tolerance sensitivity using vector loop assembly tolerance models. With a vector loop model, the assembly kinematic constraints or assembly functions can be established automatically as implicit functions. This method evaluates the derivatives of the kinematic constraint equations with respect to both manufactured dimensions and assembly variables. The derivative matrices are then used to calculate the tolerance sensitivity matrix. This is a closed form method which relates the derivatives of the assembly functions to the coordinates of the joints or nodes and the orientations of the vectors and the local joint axes in an assembly. It is accurate, simple and very suitable for design iterations.
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