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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

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

 

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.

 


PRO-E

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Analyzer: Clutch | Stack Blocks | Remote Positioner
Verification: Clutch | Stack Blocks | Remote Positioner | Bike Crank | Parallel Blocks | NFOV

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Analyzer: Clutch | Stack Blocks | Remote Positioner
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CATIA

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