Comparison of Assembly Tolerance Analysis by the Direct Linearization and Modified Monte Carlo Simulation Methods

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. INTRODUCTION The linearization method and Monte Carlo simulation are the most commonly used methods for statistical tolerance analysis of mechanical assemblies, due to the versatility and speed of the linearization method and the nonlinear capability and accuracy of Monte Carlo simulation. The concern for using the linearization method is its accuracy, and so far, little information on accuracy is available to analysts. For Monte Carlo simulation, accuracy is related to the sample size, although the relationship of sample size and accuracy has not been well defined. Traditionally, both the linearization method and Monte Carlo simulation for statistical tolerance analysis of mechanical assemblies are applied to explicit assembly functions, that is, the assembly feature or dimension must be expressed in terms of the component dimensions in the assembly (Cox 1986, Shapiro & Gross 1981, DeDoncker & Spencer 1987, Doepker & Nies 1989, Early & Thompson 1989, Fuscaldo 1991). In 2-D or 3-D space, this function is usually a nonlinear implicit function of the assembly variables. It is very difficult or impossible for a designer to establish an explicit function for “real world” assemblies. The authors have developed a generalized linearization method and modified Monte Carlo simulation for 2D and 3-D assembly tolerance analysis using implicit assembly functions (Chase, Gao & Magleby 1995a, Gao 1993). The linearization method is called the Direct Linearization Method (DLM). These two methods will be described. This paper applies the DLM and Monte Carlo simulation using implicit assembly functions to a set of mechanical assemblies in both 2-D and 3-D space. The results from the two methods are compared. A Monte Carlo simulation with a very large sample size is chosen as the “exact” solution. The effect of sample size on the accuracy of the Monte Carlo simulation is also investigated. The DLM and modified Monte Carlo simulation using an implicit assembly function, were developed to take advantage of the increased use of CAD in product design. Solid modelers are used to create assembly models, and an assembly tolerance modeler is employed to include the tolerances of the components in the assembly models. Assembly tolerance analysis and allocation can then proceed on the assembly models using the DLM and Monte Carlo simulation. This section will discuss the vector-loop-based assembly tolerance modeler, and both the DLM and modified Monte Carlo simulation methods for assembly tolerance analysis. Vector-Loop-based Assembly Tolerance Modeler The DLM and Monte Carlo simulation were applied to the same vector loop based assembly tolerance models (Chase, Gao & Magleby 1995a, Chase, Gao & Magleby 1995b). Solid modelers do not generally contain the controlled dimension and tolerance data that are required for tolerance analysis. The vector loop assembly tolerance model extends the functions and data structure of the solid modeler to include tolerancing capabilities. It allows the designer to create 2-D and 3-D vector assembly tolerance models graphically and add them to the solid model as objects. The vector model is stored as part of the solid model database. The model contains the complete dimension and tolerance information required for performing tolerance analysis. The complete model may then be accessed by the tolerance analysis module which will perform statistical tolerance analysis and tolerance allocation (CATS Modeler 1994). Manufactured parts are seldom used as single parts. They are used in assemblies of parts. The dimensional variations which occur in each component part of an assembly accumulate statistically and propagate kinematically, causing the overall assembly dimensions to vary according to the number of contributing sources of variation. The resultant critical clearances and fits which affect performance are thus subject to variation due to the tolerance stackup of the component part variations. The three major sources of variation in assemblies are included in the models: 1) dimensional (lengths and angles) 2) geometric feature (ANSI Y 14.5) 3) kinematic (small internal adjustments) The model is based on a graphically generated vector chain(s) or loop(s) representing a mechanical assembly. Each vector represents a component dimension. Complex assemblies may require solving several vector loops simultaneously. Contact between mating parts is described by kinematic joints, (planar, slider, pin joints, etc.), which assist the designer to conceptually understand the adjustability within the assembly. Kinematic constraints assure that variations propagate through the assembly in a realistic way. Figure 1 shows the kinematic joints, datums and vector loop for the one-way clutch assembly discussed in a previous paper (Chase, Gao & Magleby 1995a). Hub Vector Assembly Loop