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6.0 Stacked blocks: An introduction to the multi-loop problem.

This example demonstrates how multiple vector loops are used to solve for assembly variations. Three blocks of various shapes and sizes are stacked together to form the following assembly (Figure 6.1).

Figure 6.1. Assembly of stacked blocks.

Because of the number of parts and contact points (joints) in the assembly, network graph theory dictates that we use three vector loops to solve for all of the unknown assembly variations of this problem. As explained in the Modeler Manual, we will use the three loops shown in Figure 6.2. The initial tolerances were determined by the type of manufacturing processes to be used to produce the parts. This chapter will present an introduction to tolerance editing, Six Sigma Component Drift analysis, and Six Sigma Assembly Drift analysis.

The specification for this assembly is on the dependent length A. See the Modeler manual for more details.

Figure 6.2. Loops used to solve for assembly variations of stacked blocks.

 Independent Variables B CYLINDER = 6.620 ± 0.2mm C CYLINDER = 6.620 ± 0.2mm F GROUND = 3.905 ± 0.125mm G GROUND = 4.060 ± 0.15 mm I BLOCK = 6.805 ± 0.075mm J GROUND = 28.125 ± 0.35mm K GROUND = 10.675 ± 0.125mm Dependent Variables (lengths only) A GROUND = 18.718 ± ? mm D BLOCK = 8.671 ± ? mm E GROUND = 10.048 ± ? mm H BLOCK = 2.189 ± ? mm L BLOCK = 27.297 ± ? mm

6.1 Start Up

After creating or calling up a vector-loop model of the stackblock assembly from the TI/TOL 2D Modeler, enter the analyzer environment. Do this by selecting Analyze from the TOL-2D main menu.

The main window for the TI/TOL 2D Analyzer will appear, and the user can set the desired analysis options with the Analyzer main menu commands. Three analysis option sets will be outlined for the stackblock tutorial that demonstrate the capabilities of the TI/TOL 2D Analyzer beyond those described in the previous chapter.

As was described in the previous chapter, the user should verify that the initial tolerance data was input correctly from the TI/TOL 2D Modeler. The commands in the Edit menu should be used to verify or modify the initial data. The user can then proceed to enter the desired analysis options.

6.2 Analysis Option I

The first set of analysis options will include a root sum squares (RSS) statistical analysis with geometric tolerances applied to the assembly. Select the Root Sum Squares and the Apply Geometric Tolerances commands from the Options menu to activate these options.

The target assembly sigma for the stackblock assembly will now be increased from the 3.00 default setting to 4.50. Select the Target Assembly Sigma command from the Options menu and change the sigma value in the Target Assembly Sigma pop-up window to 4.5, as shown below.

Weight factors will now be added to different dimensions of the assembly. The dimensions that need to have the loosest tolerances must receive the largest weight factors. Also, all vendor-supplied parts must have their dimensions fixed since such dimensions are not subject to change in tolerance allocation procedures. The fixed dimension and the weight factors assigned to the dimensions in the stackblock assembly are shown in the figure below.

Save the new allocation data by selecting the Save option from the File menu in the Allocation Data window.

6.3 Analysis Results I

A specification analysis will now be done on the stackblock assembly. Since there is only one specification applied to the assembly, the Select Specification command of the Analyze menu can be skipped and the Design Spec Analysis command can be selected directly. The results of the analysis are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:33:19] ============================================================================== Assembly Model File: STKBLK_TUTORIAL Geometric Tolerances: Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ i1 : 6.8050 : 0.07500 : 0.02500 : None : n/a g : 4.0600 : 0.15000 : 0.05000 : None : n/a f : 3.9050 : 0.12500 : 0.04167 : None : n/a k : 10.6750 : 0.12500 : 0.04167 : None : n/a j : 28.1250 : 0.35000 : 0.11667 : None : n/a b : 6.6200 : 0.20000 : 0.06667 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_30 : Flatness : JNT_13 : 0.08000 FC_31 : Angularity : JNT_14 : 0.05000 FC_32 : Parallelism : JNT_16 : 0.05000 FC_26 : Flatness : JNT_18 : 0.08000 FC_27 : Roundness : JNT_18 : 0.02000 FC_28 : Flatness : JNT_20 : 0.05000 FC_29 : Roundness : JNT_20 : 0.02000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 0.30000 Lower Limit: -0.30000 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- Upper : 0.30000 : 0.10036 : 2.99 : 1398.88 : 1.3989e-03 Lower : -0.30000 : : 2.99 : 1398.88 : 1.3989e-03 -------:-----------:-------------:----------:---------------:-------------<> Total : 2797.76 : 2.7978e-03

The specification results show that a root sum squares analysis of the initial tolerance data produces a reject count of 2797.76 ppm. A reject count of 2797.76 ppm corresponds to an assembly sigma of just over 3.0. The percent contributions of individual components of the assembly are shown in the plot below.

6.4 Tolerance Allocation I

A tolerance allocation procedure will now be done on the stackblock assembly. Each dimension of the assembly will be assigned a corresponding weight factor, as was entered previously. The allocated tolerance values must meet the specified target assembly sigma value of 4.50. The tolerance allocation is done by selecting the Weight Factors Allocate command from the Allocate menu. The allocation results are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:35:02] ============================================================================== Assembly Model File: STKBLK_TUTORIAL Geometric Tolerances: Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ i1 : 6.8050 : 0.07500 : 0.02500 : None : n/a g : 4.0600 : 0.15000 : 0.05000 : None : n/a f : 3.9050 : 0.12500 : 0.04167 : None : n/a k : 10.6750 : 0.12500 : 0.04167 : None : n/a j : 28.1250 : 0.35000 : 0.11667 : None : n/a b : 6.6200 : 0.20000 : 0.06667 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_30 : Flatness : JNT_13 : 0.08000 FC_31 : Angularity : JNT_14 : 0.05000 FC_32 : Parallelism : JNT_16 : 0.05000 FC_26 : Flatness : JNT_18 : 0.08000 FC_27 : Roundness : JNT_18 : 0.02000 FC_28 : Flatness : JNT_20 : 0.05000 FC_29 : Roundness : JNT_20 : 0.02000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 0.30000 Lower Limit: -0.30000 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- i1 : Y : 1.00 : 0.07500 : 1.00 : 0.07500 : 1.00 : 0.02500 : g : N : 2.00 : 0.15000 : 1.00 : 0.14493 : 1.00 : 0.04831 : f : N : 2.00 : 0.12500 : 1.00 : 0.12078 : 1.00 : 0.04026 : k : N : 3.00 : 0.12500 : 1.00 : 0.18117 : 1.00 : 0.06039 : j : N : 4.00 : 0.35000 : 1.00 : 0.67635 : 1.00 : 0.22545 : b : N : 1.00 : 0.20000 : 1.00 : 0.09662 : 1.00 : 0.03221 : ----------------:---:--------:---------:------:---------:------:---------:<> Specification Results: Specification Name: SP_25 Specification Type: Dependent Length Nominal Dimension: 18.7182 Geometric Tolerances: Applied Analysis Model: RSS Target Assy Sigma: 4.50 Spec Limit Assy Std Dev Assy Sigma Rejects (ppm) DPU -------:-----------:-------------:----------:---------------:-------------<> Upper : 0.30000 : 0.06667 : 4.50 : 3.45 : 3.4506e-06 Lower : -0.30000 : : 4.50 : 3.45 : 3.4506e-06 -------:-----------:-------------:----------:---------------:-------------<> Total : 6.90 : 6.9012e-06

Notice that the 4.50 target assembly sigma value forces the tolerances assigned to the individual dimensions to adjust until a reject count of 6.90 ppm is reached. The dimension adjustments are made in proportion to the corresponding weight factor.

6.5 Analysis Options II

It would now be instructive to change the original tolerance values to the allocated values just calculated, and to see how such a change would affect the percent contributions of the individual dimensions. This is done by returning to the Analyzer main window and selecting the Tolerances command from the Edit menu. Each tolerance value should be changed to the allocated value calculated above. The updated tolerance values are shown below.

Save the new tolerance values by selecting the Save command from the File menu in the Tolerances window as shown below.

6.6 Percent Contribution II

A percent contribution plot with the updated tolerance values can be created by selecting the Sensitivity Plots command from the Analyze menu. Select the only available specification for the stackblock assembly and the Percent Contribution option from the Plot Select pop-up window. The percent contribution plot is shown below.

Notice that the new tolerance values reduced the percent contribution of dimension variable b from 75.69 to 40.04. The percent contribution of other dimension variables in the assembly also changed significantly. This new design is more robust than the original design, because the assembly variation is not controlled by a single component variation.

6.7 Analysis Set III

The updated tolerance values produce a reject count of 6.90 ppm (equal to an assembly sigma of 4.50) when analyzed with a root sum squares method. The statistical method of analysis will now be changed to the Six Sigma Assembly Drift (SSA) method. This is done by selecting the Six Sigma Assembly Drift command from the Options menu. All other analysis options remain the same.

The SSA design spec analysis can be done by selecting the Design Spec Analysis option of the Analyze menu. The results of the analysis are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:40:50] ============================================================================== Assembly Model File: STKBLK_TUTORIAL Geometric Tolerances: Applied Analysis Model: Six Sigma Assembly Drift (SSA) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ i1 : 6.8050 : 0.07500 : 0.02500 : None : n/a g : 4.0600 : 0.14493 : 0.04831 : None : n/a f : 3.9050 : 0.12078 : 0.04026 : None : n/a k : 10.6750 : 0.18117 : 0.06039 : None : n/a j : 28.1250 : 0.67635 : 0.22545 : None : n/a b : 6.6200 : 0.09662 : 0.03221 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_30 : Flatness : JNT_13 : 0.08000 FC_31 : Angularity : JNT_14 : 0.05000 FC_32 : Parallelism : JNT_16 : 0.05000 FC_26 : Flatness : JNT_18 : 0.08000 FC_27 : Roundness : JNT_18 : 0.02000 FC_28 : Flatness : JNT_20 : 0.05000 FC_29 : Roundness : JNT_20 : 0.02000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 0.30000 Lower Limit: -0.30000 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± SSA Variation ----------------:------------:----------------- Upper : 0.30000 : 0.06667 : 4.50 : 1349.62 : 1.3496e-03 Lower : -0.30000 : : 4.50 : 1349.62 : 1.3496e-03 -------:-----------:-------------:----------:---------------:-------------<> Total : 2699.25 : 2.6992e-03

Notice that the SSA statistical method increased the number of rejects from 6.90 to 2699.25. This change is due to the assumptions of this model. That is, the component nominals are going to vary slightly during production. This results in a shift of assembly variable means, which translates into a larger overall assembly variation. The SSA model assumes this shift will reduce the calculated number of assembly standard deviations inside the specification limits by 1.5. Therefore, the assembly sigma is calculated using the RSS model. The assembly sigma value shown in the results section is 4.5, but the reject fraction calculation is based on 3.0 sigma (4.5 - 1.5), which gives ~ 2700 ppm rejects.

6.8 Analysis Set IV

The statistical method of analysis will now be changed to the Six Sigma Component Drift (SSC) method for the sake of comparison. Select the Six Sigma Component Drift command from the Options menu. Again select the Design Spec Analysis command from the Analyze menu to display the results of the analysis.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:42:04] ============================================================================== Assembly Model File: STKBLK_TUTORIAL Geometric Tolerances: Applied Analysis Model: Six Sigma Component Drift (SSC) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ i1 : 6.8050 : 0.07500 : 0.02500 : None : 0.25 g : 4.0600 : 0.14493 : 0.04831 : None : 0.25 f : 3.9050 : 0.12078 : 0.04026 : None : 0.25 k : 10.6750 : 0.18117 : 0.06039 : None : 0.25 j : 28.1250 : 0.67635 : 0.22545 : None : 0.25 b : 6.6200 : 0.09662 : 0.03221 : None : 0.25 ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_30 : Flatness : JNT_13 : 0.08000 FC_31 : Angularity : JNT_14 : 0.05000 FC_32 : Parallelism : JNT_16 : 0.05000 FC_26 : Flatness : JNT_18 : 0.08000 FC_27 : Roundness : JNT_18 : 0.02000 FC_28 : Flatness : JNT_20 : 0.05000 FC_29 : Roundness : JNT_20 : 0.02000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 0.30000 Lower Limit: -0.30000 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± SSC Variation ----------------:------------:----------------- Upper : 0.30000 : 0.08824 : 3.40 : 337.41 : 3.3741e-04 Lower : -0.30000 : : 3.40 : 337.41 : 3.3741e-04 -------:-----------:-------------:----------:---------------:-------------<> Total : 674.81 : 6.7481e-04

The number of rejects calculated with the SSC method is 674.81. The SSC model also assumes the component nominals will vary with time. This model accomodates these shifts by increasing each component standard deviation by 25%, and then running an RSS analysis. In this way, the SSC model performs an "upstream" mean-shift adjustment, versus the SSA model "downstream" approach. Which model is chosen depends largely on how much information is available about the manufacturing processes.

6.9 Ending the Session

To exit the TI/TOL 2D analyzer environment and return to the modeling environment, select the Exit command from the File menu in the analyzer main window.

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