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Example Problems 
AutoCAD
Verification:
Bicycle Crank Assembly 

Home : Example Problems : AutoCad  Verification  Bike Crank 
BICYCLE CRANK ASSEMBLY
Figure 5.1 Exploded view of the bicycle crank assembly.
Figure 5.2: Schematic of the bicycle crank with corresponding dimension variables.
5.0 Problem Description
The bicycle crank is an assembly consisting of a pedal crank bar, a shaft that turns a sprocket, and a pin that holds them together. The specified assembly variable is U2, which is the gap between the beginning of the threaded portion of the pin and the edge of the crank against which the nut tightens. The variations of the manufactured (independent) parts should be allocated to meet the assembly specification (length) on U2.
Table 5.1: Manufactured Variables (Independent).
Variable Name 
Basic Size 
Initial Tolerance (±) 
Pin Hole Dia. A 
9.520 mm 
.015 mm 
Dist betw. Centers B 
7.650 mm 
.076 mm 
Dist. betw. Edge & Cent. C 
13.550 mm 
.127 mm 
Shaft Hole Dia. D 
15.700 mm 
.025 mm 
Shaft Dia. E 
15.660 mm 
.013 mm 
Shaft Flat Depth F 
4.500 mm 
.050 mm 
Pin Narrow End Width G 
8.500 mm 
.050 mm 
Pin Bevel Angle q 
4.0 ° 
.5 ° 
5.1 Design Requirements
Table 5.2: Assembly Variables (Dependent).
Variable Name 
Basic Size 
Upper Spec Limit (USL) 
Lower Spec Limit (LSL) 
PinCrank Edge Gap U1 
8.7169 mm 
 
 
PinShaft Contact U2 
5.0852 mm 
7.1 mm 
3.1 mm 
Pressure Angle f 
4.000 ° 
 
 
Remarks>> If the pin extends too far out of the hole, the nut cannot be tightened properly and the pin will not wedge snugly between the crank and the shaft. Note that the centers of the shaft and the crank hole are not in the same location.
5.2 Modeling Considerations
5.3 Design Goal
The goal is to use the RSS model to adjust the nonfixed component tolerances until the assembly tolerance on U2 corresponds to the ±3s variation bandwidth.
5.4 Part Names and DRFs
Figure 5.3: Diagram showing the location of the part DRFs.
Remarks>> Remember to remove the rotational degree of freedom associated with the shaft center.
5.5 Kinematic Joints
Figure 5.4: Kinematic joint diagram.
Remarks>> Joints 2 and 3 are placed in the locations shown to help simplify the vector loop. Joints 2 and 3 could potentially be located anywhere along their sliding surfaces, but by placing them where thay are shown, two translational degrees of freedom are eliminated. This allows us to solve for the three remaining dependent variables (i.e. the three remaining degrees of freedom) with a single vector loop.
Table 5.3: Kinematic Joints of the Bicycle Crank.
Joint Number 
Part One 
Part Two 
Joint Type 
1 
Crank 
Shaft 
Parallel Cylinders 
2 
Shaft 
Pin 
Planar 
3 
Crank 
Pin 
Planar 
5.6 Network Diagram, Vector Loops, and Design Specifications
One loop is necessary to describe the bicycle crank assembly. A design specification has been applied to the dependent length U2.
Figure 5.5: Network and loop diagrams for the bicycle crank.
Remarks>> Note that vector C ends and vector D/2 begins at the center of the crank hole and not at the center of the shaft. Vectors E/2 pass through the center of the shaft.
A, D and E are the original dimensions, but it is A/2, D/2 and E/2 that are used in the loops. Therefore, the tolerances assigned to those two vectors are half the original dimension tolerances.
Use the +TOL and TOL options to apply nonsymmetric tolerances to the dependent length specification.
5.7 Geometric Tolerances
Three geometric tolerances have been applied to the remote positioner assembly.
Figure 5.6: Geometric tolerance diagram.
Remarks>> Only those geometric tolerances that affect U2 should be included. The flatness of the shaft notch will cause a rotation of the crank relative to the crank on the opposite side of the bicycle, but will not contribute to the variation of U2. a3 is applied to the flat surface of the pin because the rotation it causes at the shaft/pin joint will affect U1 and U2. The surface straightness of the crank/pin joint will also cause a small rotation of the assembly, but the effect on U2 is insignificant. Therefore it is not included in the model.
5.8 Sensitivity Matrices
Constraint Sensitivities
A Matrix
A/2 
B 
C 
D/2 
E 
F 
G 
Q 

X 
0 
0 
1 
.06976 
.06976 
.06976 
0 
8.5000 
Y 
1 
1 
0 
.99756 
.99756 
.99756 
1 
5.0852 
q 
0 
0 
0 
0 
0 
0 
0 
1 
B Matrix
U1 
U2 
f 

X 
.99756 
1 
12.410 
Y 
.06976 
0 
13.550 
q 
0 
0 
1 
F Matrix
a1 
a2 
a3 

X 
.06976 
.06976 
9.1081 
Y 
.99756 
.99756 
13.781 
q 
0 
0 
1 
Tolerance Sensitivities
B1A Matrix
A/2 
B 
C 
D/2 
E 
F 
G 
q 

U1 
14.336 
14.336 
0 
14.301 
14.301 
14.301 
14.336 
121.35 
U2 
14.301 
14.301 
1 
14.336 
14.336 
14.336 
14.301 
124.96 
f 
0 
0 
0 
0 
0 
0 
0 
1 
B1F Matrix
a1 
a2 
a3 

U1 
14.301 
14.301 
3.3100 
U2 
14.336 
14.336 
2.842 E14 
f 
0 
0 
1 
5.9 Resultant Tolerances Before Optimization
Table 5.4: Independent Variable Tolerances and Control Factors
Dim. Name 
± Tol. 
Std. Dev. 
Cp 
Dk 
Cpk 
Sk 
Wt. Factor 

Tol. 
Basic 
Fixed 

A/2 
.0075 mm 
.0025 
1 
0.25 
0.75 
0 
3 
0 
No 
B 
.076 mm 
.0253 
1 
0.25 
0.75 
0 
1 
0 
No 
C 
.127 mm 
.0423 
1 
0.25 
0.75 
0 
3 
0 
No 
D/2 
.0125 mm 
.0042 
1 
0.25 
0.75 
0 
3 
0 
No 
E/2 
.0065 mm 
.0022 
1 
0.25 
0.75 
0 
3 
0 
No 
F 
.050 mm 
.0167 
1 
0.25 
0.75 
0 
2 
1 
No 
G 
.050 mm 
.0167 
1 
0.25 
0.75 
0 
2 
0 
No 
q 
.5 ° 
.1667 
1 
0.25 
0.75 
0 
1 
0 
No 
Table 5.5: Kinematic Assembly Variables (No Geometric Tolerances)
Variable Name 
Degree of Freedom 
Tolerances (ZASM = 3.000) 

Worst Case 
RSS Case 
SixSigma 

U1 
Translation (mm) 
4.05246 
1.84704 
2.46271 

U2 
Translation (mm) 
4.20898 
1.86814 
2.49086 

f 
Rotation (º) 
.50000 
.50000 
.66667 
Table 5.6: Geometric Tolerances
Feat. 
Joint 
Part Name 
Feature Type 
Tolerance Band 
Char. Length 
a1 
1 
Crank 
Circularity 
.010 mm 
N/A 
a2 
1 
Shaft 
Cylindricity 
.015 mm 
N/A 
a3 
2 
Pin 
Straightness 
.010 mm 
14.173 mm 
Table 5.7: Kinematic Assembly Variables (Geometric Tolerances Included)
Variable Name 
Degree of Freedom 
Tolerances (ZASM = 3.000) 

Worst Case 
RSS Case 
SixSigma 

U1 
Translation (mm) 
4.23356 
1.85153 
2.46609 

U2 
Translation (mm) 
4.38817 
1.87261 
2.49421 

f 
Rotation (º) 
.54043 
.50163 
.66789 
Remarks>> For this model geometric tolerances do not contribute significantly to assembly variations.
Table 5.8: RSS Percent Contributions To U2 (Geometric Tolerances Included)
Variable Name 
Variance 
Statistical RSS 

q 
1.321e1 
33.91 

B 
1.312e1 
33.69 

F 
5.709e2 
14.65  
G 
5.681e2 

14.58  
Other 
1.235e2 
3.17 
Table 5.9: RSS Percent Rejects
Spec. Name 
Spec. Type 
Basic Size 
(±) Computed Variation 
With Geometric Tolerances 
Without Geometric Tolerances 

U2 
Length 
5.0852 
1.87261 
Z 
Rej. 
Z 
Rej. 
ZASM = 3.000 
USL 7.100 
Upper Tail 
3.23  623.8  3.24  607.1  
(Rejects in PPM) 
LSL 3.100 
Lower Tail 
3.18  735.5  3.19  716.5 
5.10 Nominals And Tolerances After Optimization
Proportional Scaling Tolerance Allocation
Table 5.10: RSS Proportional Scaling Tolerance Allocation (Geometric Tolerances Included).
Assembly Specs. 
Nom 
USL 
LSL 
± ZASM 

Dep. Length U2 (mm) 
5.0852 mm 
7.1000 mm 
3.1000 mm 
3.000 

Dimension 
Specified Values 
Allocated Values 

Name 
Nom. 
±Tol. 
Nom. 
±Tol. 
STDEV 
% Cont 

A/2 (mm) 
4.7600 
.00750 
4.7600 
.00801 
.00267 
0.33 

B (mm) 
7.6500 
.07600 
7.6500 
.08117 
.02706 
33.71 

C (mm) 
13.5500 
.12700 
13.5500 
.13565 
.04522 
0.46 

D/2 (mm) 
7.8500 
.01250 
7.8500 
.01335 
.00445 
0.92 

E/2 (mm) 
7.8300 
.00650 
7.8300 
.00695 
.00232 
0.99 

F (mm) 
4.5000 
.05000 
4.5000 
.05340 
.01780 
14.66 

G (mm) 
8.5000 
.05000 
8.5000 
.05340 
.01780 
14.59 

q (º) 
4.0000 
.50000 
4.0000 
.53405 
.17802 
33.93 

a1 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
0.13 * 

a2 (mm) 
0.0 
.00750 
0.0 
.00750 
.00250 
0.29 * 

a3 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
21.0E32 * 

Assem.Tot. 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

U2 (mm) 
5.0852 
1.8726 
5.0852 
1.9995 
.66650 

Min/Max 
3.2126 
6.9578 
3.0857 
7.0847 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 
3.23 
623.8 
3.02 
1251.6 

Lower Tail 
3.18 
735.5 
2.98 
1448.3 

Total Rejects 
1359.3 
Total Rejects 
2699.9 
Weight Factor Tolerance Allocation
Table 5.11: RSS Weight Factor Scaling Tolerance Allocation (Geometric Tolerances Included)
Assembly Specs. 
Nom. 
USL 
LSL 
± ZASM 

Dep. Length U2 (mm) 
5.0852 mm 
7.1000 mm 
3.1000 mm 
3.000 

Dimension 
Specified Values 
Allocated Values 

Name 
Nom. 
±Tol 
Nom 
±Tol 
STDEV 
% Cont 

A/2 (mm) 
4.7600 
.00750 
4.7600 
.01659 
.00553 
1.41 

B (mm) 
7.6500 
.07600 
7.6500 
.05605 
.01868 
16.07 

C (mm) 
13.5500 
.12700 
13.5500 
.28097 
.09366 
1.97 

D/2 (mm) 
7.8500 
.01250 
7.8500 
.02765 
.00922 
3.93 

E/2 (mm) 
7.8300 
.00650 
7.8300 
.01458 
.00480 
4.25 

F (mm) 
4.5000 
.05000 
4.5000 
.07375 
.02458 
27.95 

G (mm) 
8.5000 
.05000 
8.5000 
.07375 
.02458 
27.82 

q (º) 
4.0000 
.50000 
4.0000 
.36870 
.12291 
16.18 

a1 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
0.13 * 

a2 (mm) 
0.0 
.00750 
0.0 
.00750 
.00250 
0.29 * 

a3 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
1.0E32 * 

Assem.Tot. 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

U2 (mm) 
5.0852 
1.8726 
5.0852 
1.9995 
.66650 

Min/Max 
3.2126 
6.9578 
3.0857 
7.0847 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 
3.23 
623.8 
3.02 
1251.6 

Lower Tail 
3.18 
735.5 
2.98 
1448.3 

Total Rejects 
1359.3 
Total Rejects 
2699.9 
Nominal Allocation
U2 represents the distance between the base of the pin's threads and the surface of the crank. If that distance is too long, not enough threads will be exposed, and the nut will work free. However, U2 may safely be shorter, as long as the length does not become less than 3.1 mm. In cases such as this, upper specification limit nominal allocation may be useful.
Table 5.12: RSS Nominal Allocation (Geometric Tolerances Included).
Upper Specification Limit Justified.
Assembly Specs. 
Nom. 
USL 
LSL 
± ZASM 

Dep. Length U2 (mm) 
5.0852 mm 
7.1000 mm 
3.1000 mm 
3.000 

Dimension 
Specified Values 
Allocated Values 

Name 
Nom. 
±Tol 
Nom 
±Tol 
STDEV 
% Cont 

A/2 (mm) 
4.7600 
.00750 
4.7600 
.00750 
.00250 
0.33 

B (mm) 
7.6500 
.07600 
7.6500 
.07600 
.02533 
33.68 

C (mm) 
13.5500 
.12700 
13.5500 
.12700 
.04233 
0.46 

D/2 (mm) 
7.8500 
.01250 
7.8500 
.01250 
.00417 
0.92 

E/2 (mm) 
7.8300 
.00650 
7.8300 
.00650 
.00217 
0.99 

F (mm) 
4.5000 
.05000 
4.4901 
.05000 
.01667 
14.65 

G (mm) 
8.5000 
.05000 
8.5000 
.05000 
.01667 
14.58 

q (º) 
4.0000 
.50000 
4.0000 
.50000 
.16667 
33.93 

a1 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
0.15* 

a2 (mm) 
0.0 
.00750 
0.0 
.00750 
.00250 
0.33* 

a3 (mm) 
0.0 
.00500 
0.0 
.00500 
.00167 
1.1E32* 

Assem.Tot. 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

U2 (mm) 
5.0852 
1.8726 
5.2274 
1.8726 
.62420 

Min/Max 
3.2126 
6.9578 
3.3548 
7.1000 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 
3.23 
623.8 
3.00 
1350.0 

Lower Tail 
3.18 
735.5 
3.41 
327.0 

Total Rejects 
1359.3 
Total Rejects 
1677.0 
Remarks>> The entire assembly variable mean shift was accomplished by adjusting the nominal of f.
Using USL nominal allocation, the assembly variance was shifted so that the +3s variation corresponds to the upper specification limit of 7.1 mm. The new minimum size became 3.3548 mm, which is still larger than the lower specification limit of 3.1 mm.
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