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Figure 6.1: Schematic of the parallel blocks with dimension variables.

6.0 Problem Description

A stacked blocks assembly is traditionally considered a one-dimensional tolerance problem. In this case, though, we wish to explore the two-dimensional effects of surface variation on this assembly. The variable of interest is the gap between the vertical surface on the right and the top right corner of the top-most block. If the surfaces of each block were perfectly parallel to each other, then the gap variation would simply be the error in positioning the stack relative to the vertical surface. Since the block surfaces are not perfectly parallel, the stack may lean from side to side.

A parallelism geometric tolerance will be applied to each block. This effectively defines the degree to which each block can rotate.

Table 6.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (+/-) A 3.200 in -- * B 0.736 in 0.005 in C 1.000 in -- * D 0.800 in 0.005 in E 0.800 in 0.005 in F 0.800 in 0.005 in G 0.800 in 0.005 in H 1.000 in -- *

* These vectors are for positioning modeling elements. They are not dimensioned lengths, so they do not have tolerances associated with them. They will not appear in the sensitivity matrices.

6.1 Design Requirements

Table 6.2: Assembly Variables and Specification Limits.

 Variable Name Basic Size Upper Spec. Limit (USL) Lower Spec. Limit (LSL) Gap Width 0.7360 in 0.7460 in 0.7260 in

6.2 Modeling Considerations

• For best results, joints that have either a kinematic or a geometric rotation should be placed at the center point of the expected rotation. For this assembly, since the rotation of the blocks is important, the joints between each block should be located in the center of the mating surfaces. This requires using two redundant vectors, C and H. These two vectors are used only to position the joints in the middle of their mating surfaces, and are not dimensioned lengths. They must either be assigned a zero tolerance or equivalenced.

• This assembly does not have a closure constraint, and therefore no kinematic variables. No closed constraint loops are necessary to model it. A single open loop is all that is required.

• When using a gap specification, it is important to place the open loop endpoints at the points between which the variation is desired, even if it means inserting a dummy variable (such as A). TI TOL will calculate the variation that would occur along a vector that connects the two loop endpoints. If it is necessary to use a dummy variable, assign it a zero tolerance in order to eliminate it from the sensitivity matrices. This prevents it from biasing the calculated variations.

6.3 Design Goal

The purpose of this chapter is to illustrate the effects of the surface variations on the variation of the Gap. The parallelism geometric tolerances are used to introduce rotational variation into the assembly.

6.4 Part DRFs And Feature Datums

Figure 6.2: Diagram showing the location of the part DRFs and feature datums.

6.5 Kinematic Joints

Four rigid joints are used to model this assembly.

Figure 6.3: Kinematic joint diagram.

Table 6.3: Kinematic Joints Of The Parallel Blocks.

 Joint Number Part One Part Two Joint Type 1 Crank Block 1 Rigid 2 Block 1 Block 2 Rigid 3 Block 2 Block 3 Rigid 4 Block 3 Block 4 Rigid

Remarks>> Rigid joints are used to avoid introducing any kinematic degrees of freedom into this assembly. This allows us to solve for the Gap variation using a single open loop. Planar joints could also be used to model the contact between the blocks. The sliding plane degree of freedom is automatically turned off by the modeler because there is no vector created in the sliding plane. If a planar joint is used between block 1 and the base, there is a vector in the sliding plane and the translational degree of freedom must be turned off manually.

6.6 Network Diagram, Vector Loops, and Design Specifications

Figure 6.4: Network diagram and loop diagram for the parallel blocks assembly.

Remarks>> The direction of open loops is important when gap and position specifications are used. TI TOL assumes the first part is fixed in space and the parts "downstream" all rotate relative to it. This arises due to the non-commutative property of matrix multiplication. To generate the correct open loop direction, create the fixed endpoint first and the moving endpoint second.

Enter .01 and -.01 for the gap specification limits.

6.7 Geometric Tolerances

Figure 6.5: Geometric tolerance diagram.

The bottom of each block is parallel to the top of the block within a .004 in bandwidth.

6.8 Sensitivity Matrices

Table 6.4: C-B-1A Matrix

 B D E F G X -1.00000 0.00000 0.00000 0.00000 0.00000 Y 0.00000 1.00000 1.00000 1.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Table 6.5: G-B-1F Matrix

 1 2 3 4 X 3.2 -2.4 -1.6 -0.8 Y -1 1 1 1 -1 1 1 1

6.9 Predicted Assembly Variation

Table 6.6: Independent Variable Tolerances and Control Factors

 Dim Name Nominal +/- Tol Process Std Dev Process Description K B 0.7360 0.00500 0.00150 None 0.25 D 0.8000 0.00500 0.00150 None 0.25 E 0.8000 0.00500 0.00150 None 0.25 F 0.8000 0.00500 0.00150 None 0.25 G 0.8000 0.00500 0.00150 None 0.25

Table 6.7: Geometric Tolerances

 Name Part Name Type Joint Tolerance Band Char. Length 1 Block 1 Parallelism 1 0.00400 2.0 in 2 Block 2 Parallelism 2 0.00400 2.0 in 3 Block 3 Parallelism 3 0.00400 2.0 in 4 Block 4 Parallelism 4 2.0 in

Table 6.8: RSS Percent Contributions To Gap (Geometric Tolerances Included)

 Variable Name Contribution Statistical RSS 1 4.5511e-6 40.24 B 2.7778e-6 24.56 2 2.5600e-6 22.63 3 1.1378e-6 10.06 4 2.8444e-7 2.51

Table 6.9: Absolute Sensitivities Of Gap (Geometric Tolerances Applied)

 Variable Name Sensitivity Normalized 1 3.20000 35.56 2 2.40000 26.67 3 1.60000 17.78 B 1.00000 11.11 4 0.80000 8.89

Table 6.10: RSS Percent Rejects (Geometric Tolerances Applied)

 Gap Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.7460 0.00336 2.97 1472.85 1.4729e-3 Lower 0.7260 2.97 1472.85 1.4729e-3 Nom Dim 0.7360 Total 2945.71 2.9457e-3

Table 6.11: SSC Percent Rejects (Geometric Tolerances Applied)

 Gap Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.7460 0.003367 2.72 3219.76 3.2198e-3 Lower 0.7260 2.72 3219.76 3.2198e-3 Nom Dim 0.7360 Total 6439.52 6.4395e-3

Table 6.12: SSA Percent Rejects (Geometric Tolerances Applied)

 Gap Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.7460 0.00336 1.47 70326.75 7.0327e-2 Lower 0.7260 1.47 70326.75 7.0327e-2 Nom Dim 0.7360 Total 140653.49 1.4065e-1

6.10 Tolerance Allocation

No tolerance or nominal allocation will be performed on the parallel blocks assembly. The variations that contribute the most to the Gap variation are geometric tolerances, which can't be adjusted by the TI TOL tolerance allocation routines.

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