bicycle crank

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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)
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
Dim. Name	± Tol.	Std. Dev.	Cp	Dk	Cpk	Sk	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)
Variable Name	Degree of Freedom	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)
Variable Name	Degree of Freedom	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

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			21.0E-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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