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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) Pin-Crank Edge Gap U1 8.7169 mm -- -- Pin-Shaft 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

• Vector C must end at the shaft hole center, not the shaft center. Use the CAD system tools to find the correct point.

• A, D, and E are given as diameters with diametral tolerances but used in the model as radii. Therefore their respective tolerances must be divided in half.

• The shaft radius (E/2) is used twice. Those dimensions must be equivalenced.

• The shaft DRF is cylindrical. All cylindrical datums introduce a rotational degree of freedom into the tolerance model. In this case, though, we don't want a degree of freedom there. In the UNIX-based analyzer, it is possible to directly turn off that degree of freedom. In the AutoCATS version, the same effect can be accomplished by giving the shaft a rectangular DRF.

5.3 Design Goal

The goal is to use the RSS model to adjust the non-fixed 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 non-symmetric 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

-B-1A 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

-B-1F Matrix

 a1 a2 a3 U1 14.301 14.301 -3.3100 U2 -14.336 -14.336 -2.842 E-14 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 Six-Sigma 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 Six-Sigma 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 1.321e-1 33.91 B 1.312e-1 33.69 F 5.709e-2 14.65 G 5.681e-2 14.58 Other 1.235e-2 3.17

 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 21.0E-32 * 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 1.0E-32 * 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 1.1E-32* 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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