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Example Problems 
AutoCAD
Verification:
Parallel Stacked Blocks 

Home : Example Problems : AutoCad  Verification  Bike Crank 
PARALLEL STACKED BLOCKS
Figure 7.1: Schematic of the parallel blocks and corresponding dimension variables.
7.0 Problem Description
A stacked blocks assembly is traditionally considered a onedimensional tolerance problem. In this case, though, we wish to explore the twodimensional effects of this assembly. The variable of interest is the gap between the vertical surface on the right and the top right corner of the topmost 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.
This assembly will be modeled two ways. First, the tolerance on the thickness of each block will be considered an implied parallelism. In other words, one side of a block may be shorter than the nominal while the opposite side is longer than the nominal. The entire tolerance zone thus becomes an implied parallelism geometric tolerance. As long as both sides stay within the tolerance zone, the part is still considered okay. By using an implied parallelism, a series of rotations is introduced into the assembly. For the second case, a parallelism geometric tolerance will be applied to each block. This effectively reduces the rotation that may be introduced by each block surface.
Table 7.1: Manufactured Variables (Independent).
Variable Name 
Basic Size 
Initial Tolerance (±) 
A 
3.200 in 
.000 in * 
B 
.736 in 
.005 in 
C 
1.000 in 
.000 in * 
D 
.800 in 
.005 in 
E 
.800 in 
.005 in 
F 
.800 in 
.005 in 
G 
.800 in 
.005 in 
H 
1.000 in 
.000 in * 
* These variables 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.
7.1 Design Requirements
Table 7.2: Assembly Variables and Specification Limits.
Variable Name 
Basic Size 
Upper Spec. Limit (USL) 
Lower Spec. Limit (LSL) 
Gap Width 
.7360 in 
.7460 in 
.7260 in 
7.2 Modeling Considerations
7.3 Design Goal
The purpose of this chapter is to illustrate the effects of the parallelism constraint on the variation of the Gap. The problem will be analyzed first without the parallelism geometric tolerance (with the thickness tolerance used as an implied parallelism). The problem will then be analyzed with the parallelism geometric tolerance.
7.4 Part Names and DRFs
Figure 7.2: Diagram showing the location of the part DRFs.
7.5 Kinematic Joints
Four rigid joints are used to model this assembly.
Figure 7.3: Kinematic joint diagram.
Table 7.3: Kinematic Joints Of The Parallel Blocks.
Joint Number 
Part One 
Part Two 
Joint Type 
1 
Base 
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. If planar joints were used to model the contact between the blocks, a sliding plane degree of freedom would be introduced at each joint, which would make it impossible to solve for the Gap variation.
7.6 Network Diagram, Vector Loops, and Design Specifications
Figure 7.4: 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. CATS assumes the first part is fixed in space and the parts "downstream" all rotate relative to it. This arises due to the noncommutative property of matrix multiplication. To generate the correct open loop direction, create the final endpoint (the moving endpoint) first and the starting endpoint (fixed endpoint) second.
7.7 Geometric Tolerances
Figure 7.5: Geometric tolerance diagram.
The bottom of each block is parallel to the top of the block within a .004 in bandwidth. This value will be used for the second analysis.
7.8 Sensitivity Matrices
Constraint Sensitivities
C Matrix
B 
D 
E 
F 
G 

X 
1.0000 
.00000 
.00000 
.00000 
.00000 
Y 
.00000 
1.0000 
1.0000 
1.0000 
1.0000 
q 
.00000 
.00000 
.00000 
.00000 
.00000 
G Matrix
a1 
a2 
a3 
a4 

X 
3.2000 
2.4000 
1.6000 
.80000 
Y 
1.0000 
1.0000 
1.0000 
1.0000 
q 
1.0000 
1.0000 
1.0000 
1.0000 
Remarks>> There are no closed loops and no kinematic assembly variables, so A Matrix, B Matrix, F Matrix, and D Matrix are all zero. By applying zero tolerance to the A, C, and H vectors, they are eliminated from all the sensitivity matrices.
Tolerance Sensitivities
CB^{1}A Matrix
B 
D 
E 
F 
G 

X 
1.0000 
.00000 
.00000 
.00000 
.00000 
Y 
.00000 
1.0000 
1.0000 
1.0000 
1.0000 
q 
.00000 
.00000 
.00000 
.00000 
.00000 
GB^{1}F Matrix
a1 
a2 
a3 
a4 

X 
3.2000 
2.4000 
1.6000 
.80000 
Y 
1.0000 
1.0000 
1.0000 
1.0000 
q 
1.0000 
1.0000 
1.0000 
1.0000 
7.9 Resultant Tolerances Before Optimization
Table 7.4: Independent Variable Tolerances and Control Factors
Dim. Name 
± Tol. 
Std. Dev. 
Cp 
Dk 
Cpk 
Sk 
Wt. Factor 

Tol. 
Basic 
Fixed 

B 
.005 in 
.0015 
1 
0.25 
0.75 
0 
1 
1 
No 
D 
.005 in 
.0015 
1 
0.25 
0.75 
0 
1 
1 
No 
E 
.005 in 
.0015 
1 
0.25 
0.75 
0 
1 
1 
No 
F 
.005 in 
.0015 
1 
0.25 
0.75 
0 
1 
1 
No 
G 
.005 in 
.0015 
1 
0.25 
0.75 
0 
1 
1 
No 
Table 7.5: Geometric Tolerances (Case 1)
Feat. 
Joint 
Part Name 
Feature Type 
Tolerance Band 
Char. Length 
b1 
1 
Block 1 
Parallelism 
.010 in 
2.0 in 
b2 
2 
Block 2 
Parallelism 
.010 in 
2.0 in 
b3 
3 
Block 3 
Parallelism 
.010 in 
2.0 in 
b4 
4 
Block 4 
Parallelism 
.010 in 
2.0 in 
Remarks>> These geometric tolerances represent case 1the tolerance on the thickness of the blocks acts as an implied parallelism.
Table 7.6: RSS Percent Rejects (Case 1)
Spec. Name 
Spec. Type 
Nominal Dimension 
(±) Computed Variation 
With Geometric Tolerances 
Without Geometric Tolerances 

Gap 
Gap 
.736 
.02247 
Z 
Rej. 
Z 
Rej. 

ZASM = 3.000 
USL .746  Upper Tail  1.33  9.1e4 
6.00 
9.9e4  
(Rejects in PPM) 
LSL .726 
Lower Tail 
1.33 
9.1e4 
6.00 
9.9e4 
Remarks>> With no geometric tolerances applied, the gap variation is simply equal to the tolerance of B.
Table 7.7: RSS Percent Contributions To Gap (Geometric Tolerances Included)
Variable Name 
Variance 
Statistical RSS 

a1 
2.844e5 
50.69 

a2 
1.600e5 
28.51 

a3 
7.111e6 
12.67 

B 
2.778e6 
4.95 

a4 
1.778e6 
3.17 

Other 
.00000 
0.00 
Table 7.8: Geometric Tolerances (Case 2)
Feat. 
Joint 
Part Name 
Feature Type 
Tolerance Band 
Char. Length 
b1 
1 
Block 1 
Parallelism 
.004 in 
2.0 in 
b2 
2 
Block 2 
Parallelism 
.004 in 
2.0 in 
b3 
3 
Block 3 
Parallelism 
.004 in 
2.0 in 
b4 
4 
Block 4 
Parallelism 
.004 in 
2.0 in 
Remarks>> These geometric tolerances represent case 2the parallelism on the blocks is a specified parallelism.
Table 7.9: RSS Percent Rejects (Case 2)
Spec. Name 
Spec. Type 
Nominal Dimension 
(±) Computed Variation 
With Geometric Tolerances 
Without Geometric Tolerances 

Gap 
Gap 
.736 
.01009 
Z 
Rej. 
Z 
Rej. 

ZASM = 3.000 
USL .746  Upper Tail  2.97  1473  6.00  9.9e4  
(Rejects in PPM) 
LSL .726 
Lower Tail 
2.97 
1473 
6.00 
9.9e4 
Table 7.10: RSS Percent Contributions To Gap (Geometric Tolerances Included)
Variable Name 
Variance 
Statistical RSS 

a1 
4.551e6 
40.24 

B 
2.778e6 
24.56 

a2 
2.560e6 
22.63 

a3 
1.138e6 
10.06 

a4 
2.844e7 
2.51 

Other 
.00000 
0.00 
Remarks>> By specifying a parallelism bandwidth, the potential variation in the Gap was more than cut in half, and the rejects were reduced by nearly 180,000 PPM.
7.10 Nominals And Tolerances After Optimization
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 automatically adjusted by the CATS analyzer. If the user wishes, they can be adjusted manually through the input screens and the analysis repeated.
PROE Modeler: Clutch
 Stack Blocks
 Remote Positioner 
AutoCAD Modeler: Clutch
 Stack
Blocks  Remote
Positioner 
CATIA Modeler: Crank Slider 
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