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Figure 5.1 Exploded view of the bicycle crank assembly.

Figure 5.2: Schematic of the bicycle crank with 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 base of the threaded portion of the pin and the edge of the crank against which the nut tightens.

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. 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 [[theta]] 4.0 [[ring]] --

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 7.1000 mm 3.1000 mm Pressure Angle [[phi]] 4.000 [[ring]] -- --

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.

5.2 Modeling Considerations

• Vector C must end at the crank hole center, not the shaft center. Use a cylindrical feature datum on the crank to link the parallel cylinders joint to the crank DRF.

• 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. TI TOL assumes all cylindrical datums introduce a rotational degree of freedom into the tolerance model. In this case we don't want a degree of freedom at the shaft center, so it must be turned off manually.

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 +/-3[[sigma]] specification limits.

5.4 Part DRFs And Feature Datums

Figure 5.3: Diagram showing the location of the part DRFs and feature datums.

Remarks>> Remember to remove the rotational degree of freedom associated with the shaft DRF.

5.5 Kinematic Joints

Figure 5.4: Kinematic joint diagram.

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.

To apply the proper specification limits, enter 2.0148 and -1.9852 at the prompts.

5.7 Geometric Tolerances

Three geometric tolerances have been applied to the bicycle crank 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. [[alpha]]3 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

Table 5.4: -B-1A Matrix

 A/2 B C D/2 E F G U1 14.33559 14.33559 0.00000 14.30067 -14.3007 14.30067 -14.3356 U2 -14.3007 -14.3007 1.00000 -14.3356 14.33559 -14.3356 14.30067 [[phi]] 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Table 5.5: -B-1F Matrix

 1 2 3 U1 14.3007 14.3007 -3.31000 U2 -14.3356 -14.3356 -2.842 E-14 [[phi]] 0 0 -1.00000

5.9 Predicted Assembly Variation

Table 5.6: Independent Variable Tolerances and Control Factors

 Dim Name Nominal +/- Tol Process Std Dev Process Description K A/2 4.7600 0.00750 0.00250 None 0.25 B 7.6500 0.07600 0.02533 None 0.25 C 13.5500 0.12700 0.04233 None 0.25 D/2 7.8500 0.01250 0.00417 None 0.25 E/2 7.8300 0.00650 0.00217 None 0.25 F 11.4300 0.05000 0.01667 None 0.25 G 8.5000 0.05000 0.01667 None 0.25

Table 5.7: Kinematic Assembly Variables (Geometric Tolerances Not Applied)

 Variable Name Nominal WC +/- Assembly Variation (ZASM = 3.00) RSS SSA SSC U1 8.7169 2.99350 1.51332 1.51332 2.01776 U2 5.0852 0.07444 1.51683 1.51683 2.02244 q 4.0000 [[ring]] 0.00000 0.00000 0.00000 0.00000

Table 5.8: RSS Percent Rejects (Geometric Tolerances Not Applied)

 Dep Len. U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.1000 0.50561 3.98 33.77 3.3767e-5 Lower 3.1000 3.93 43.14 4.3144e-5 Nom Dim 5.0852 Total 76.91 7.6910e-5

Table 5.9: Geometric Tolerances

 Name Part Name Type Joint Tolerance Band Char. Length 1 Crank Circularity 1 0.01000 N/A 2 Shaft Cylindricity 1 0.01500 N/A 3 Pin Straightness 2 0.01000 14.1732 mm

Table 5.10: Kinematic Assembly Variables (Geometric Tolerances Applied)

 Variable Name Nominal WC +/- Assembly Variation (ZASM = 3.00) RSS SSA SSC U1 8.7169 3.17459 1.51880 1.51880 2.02188 U2 5.0852 3.29767 1.52232 1.52232 2.02656 q 4.0000 [[ring]] 0.04043 0.04043 0.04043 0.04043

Remarks>> For this model geometric tolerances do not contribute significantly to the critical assembly variation.

Table 5.11: Normalized Sensitivities To U2 (Geometric Tolerances Applied)

 Variable Name Sensitivity Normalized D/2 14.3356 12.40 E/2 14.3356 12.40 F 14.3356 12.40 1 14.3356 12.40 2 14.3356 12.40 A/2 14.3007 12.37 B 14.3007 12.37 G 14.3007 12.37 C 1.0000 0.87 3 0.0000 0.00

Table 5.12: RSS Percent Contributions To U2 (Geometric Tolerances Applied)

 Variable Name Contribution Statistical RSS B 1.3125e-1 50.97 F 5.7086e-2 22.17 G 5.6808e-2 22.06 E/2 3.8590e-3 1.50 D/2 3.5679e-3 1.39 Other 8.7846e-3 1.91

Table 5.13: RSS Percent Rejects (Geometric Tolerances Applied)

 Dep Len. U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.1000 0.50744 3.97 35.87 3.5871e-5 Lower 3.1000 3.91 45.76 4.5756e-5 Nom Dim 5.0852 Total 81.63 8.1627e-5

5.10 Tolerance Allocation

Weight Factor Tolerance Allocation

Table 5.14: RSS Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A/2 N 3.00 0.00750 1.00 0.01813 1.00 0.00604 B N 1.00 0.07600 1.00 0.06124 1.00 0.02041 C N 3.00 0.12700 1.00 0.30701 1.00 0.10234 D/2 N 3.00 0.01250 1.00 0.03022 1.00 0.01007 E/2 N 3.00 0.00650 1.00 0.03143 1.00 0.01048 F N 2.00 0.05000 1.00 0.08058 1.00 0.02686 G N 2.00 0.05000 1.00 0.08058 1.00 0.02686
 Dep Len U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.1000 .66650 3.02 1251.63 1.2516e-3 Lower 3.1000 Target Sig 2.98 1448.30 1.4483e-3 Nom Dim 5.0852 3.00 Total 2699.93 2.6999e-3

Table 5.15: SSC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A/2 N 3.00 0.00750 1.00 0.01360 1.00 0.00453 B N 1.00 0.07600 1.00 0.04593 1.00 0.01531 C N 3.00 0.12700 1.00 0.23026 1.00 0.07675 D/2 N 3.00 0.01250 1.00 0.02266 1.00 0.00755 E/2 N 3.00 0.00650 1.00 0.02357 1.00 0.00786 F N 2.00 0.05000 1.00 0.06044 1.00 0.02014 G N 2.00 0.05000 1.00 0.06044 1.00 0.02014
 Dep Len U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.1000 .66650 3.02 1251.63 1.2516e-3 Lower 3.1000 Target Sig 2.98 1448.30 1.4483e-3 Nom Dim 5.0852 3.00 Total 2699.93 2.6999e-3

Table 5.16: SSA Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A/2 N 3.00 0.00750 1.00 0.01205 1.00 0.00402 B N 1.00 0.07600 1.00 0.04072 1.00 0.01357 C N 3.00 0.12700 1.00 0.20411 1.00 0.06804 D/2 N 3.00 0.01250 1.00 0.02009 1.00 0.00670 E/2 N 3.00 0.00650 1.00 0.02089 1.00 0.00696 F N 2.00 0.05000 1.00 0.05357 1.00 0.01786 G N 2.00 0.05000 1.00 0.05357 1.00 0.01786
 Dep Len U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.1000 .44428 3.04 1202.73 1.2027e-3 Lower 3.1000 Target Sig 2.97 1497.20 1.4972e-3 Nom Dim 5.0852 3.00 Total 2699.93 2.6999e-3

Table 5.17: WC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A/2 N 3.00 0.00750 0.00707 B N 1.00 0.07600 0.02387 C N 3.00 0.12700 0.11967 D/2 N 3.00 0.01250 0.01178 E/2 N 3.00 0.00650 0.01225 F N 2.00 0.05000 0.03141 G N 2.00 0.05000 0.03141
 Dep Len U2 Spec Limit WC Variation Lower 7.1000 1.98519 Satisfied Lower 3.1000 Satisfied Nom Dim 5.0852

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