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Example Problems
AutoCAD Verification:
Pawl and Ratchet
Home : Example Problems : AutoCad - Verification - Pawl and Ratchet    

PAWL AND RATCHET

 

Figure 6.1: Schematic of the pawl and ratchet and corresponding dimension variables.

6.0 Problem Description

This model consists of a ratchet, pawl and a housing to which both are attached. The pawl arm has a cylindrical end that stops the clockwise rotation of the ratchet at increments equal to the spacing of the teeth. The assembly is used to rotationally position an object attached to the ratchet.

Table 6.1: Manufactured Variables (Independent).

Variable Name

Basic Size

Initial Tolerance (±)

Xpawl

.400 in.

.0015 in.

Ypawl

.300 in.

.0015 in.

Rpawl

.300 in.

.001 in.

Rtip1

.050 in.

.001 in.

Rtip2

.050 in.

.001 in.

Rroot

.313 in.

.002 in.

Thetaroot

62.0 º

.8 º

 

6.1 Design Requirements

The assembly variable of interest is the angular orientation of the ratchet, called Phitooth. The variation of this angle should be no more than ±0.4º

Table 6.2: Assembly Variables and Specification Limits.

Variable Name

Basic Size

Upper Spec. Limit (USL)

Lower Spec. Limit (LSL)

Rside1

.0832 in.

--

--

Rside2

.0832 in.

--

--

Phipawl

73.863º

--

--

Phitip1

7.069º

--

--

Phitooth

99.0679º

99.4679º

98.6679º

Phitip2

125.069º

--

--

Remarks>> It will require two closed loops to solve for these six dependent variables.

 

6.2 Modeling Considerations

6.3 Design Goal

The goal for this problem is to compare the use of geometric tolerancing vs. traditional tolerancing methods, and to see how each affects the final assembly variations. In order to explore this issue, the assembly will first be modeled using standard dimension tolerances. It will then be modeled with the appropriate dimension tolerances replaced by ANSI Y14.5 geometric tolerances, and the analysis repeated. The PPM rejects for each case will be used to compare the results.

 

6.4 Part Names and DRFs

 

Figure 6.2: Diagram showing the location of the part DRFs.

Remarks>> The cylindrical feature datum on the end of the pawl has two rotational degrees of freedom associated with it. Though CATS has built-in methods of identifying them, it may be best to create two separate cylindrical datums in that location, one for each cylindrical slider joint.

In the AutoCATS version, the user may also want to replace the cylindrical DRF of the ratchet with a rectangular DRF in order to keep from duplicating the rotational degree of freedom introduced by the revolute joint at the same point. In the UNIX versions of CATS, the user can explicitly turn off the redundant degree of freedom in the analyzer.

 

6.5 Kinematic Joints

Four joints are required to model the ratchet and pawl.

 

Figure 6.3: Kinematic joint diagram.

Remarks>> The two cylindrical slider joints are each associated with a single cylindrical feature datum, so that there are two dependent angles that occur at the cylindrical datum. It is important to place the pawl’s DRF at a location other than this shared cylindrical datum in order for the analyzer to calculate the correct sensitivities for the two dependent angles. The DRF defines a reference vector used by both angles. If the DRF for the pawl is located at the cylindrical datum, then no reference vector is defined.

Table 6.3: Kinematic Joints of the Ratchet and Pawl.

Joint Number

Part One

Part Two

Joint Type

1

Ground

Ratchet

Revolute

2

Ground

Pawl

Revolute

3

Pawl

Ratchet

Cylindrical Slider

4

Ratchet

Pawl

Cylindrical Slider

 

6.6 Network Diagram, Vector Loops, and Design Specifications

The network diagram in figure 5.4 shows that two loops are necessary to describe the pawl and ratchet assembly. A design specification has been applied to the dependent angle Phitooth.

 

 

Figure 6.4: Network diagram and loop diagram for the pawl and ratchet

Remarks>> Loop 2 contains two pairs of redundant vectors. Redundant vectors occur because each loop that passes though a part must pass through that part’s DRF. They are important in this case because they create reference vectors that define the dependent angles. If CATS does not automatically make the two redundant vectors equivalent to each other, it must be done manually or the analysis results will be invalid.

In order to generate loops like those shown, first manually create Loop_2. The autoloop generator option can then create Loop_1.

 

6.7 Geometric Tolerances

Four geometric tolerances have been applied to the pawl and ratchet assembly.

 

Figure 6.5: Geometric tolerance diagram.

Remarks>> For this comparison of geometric tolerances versus dimension tolerances, when using true position, angularity, and profile geometric tolerances , the manufactured part tolerances to which they correspond must be turned off (i.e. set to zero). In other words, it would be redundant to use these three types of geometric tolerances and their manufactured tolerance counterparts at the same time, and would effectively double the variation associated with each dimension.

In order to compare traditional and geometric tolerancing, the geometric tolerance bandwidths must be equivalent to the original dimesion tolerances. For example,

a2 = a4 = 2 * (±TolRtip1,Rtip2)

a3 = CharLen * tan(±Tolthetaroot)

For this case, the characteristic length is equal to Rside2 (remember that CATS assumes that geometric tolerances are given as a bandwidth, while dimension tolerances are assumed to be symetric). The analysis that follows will be split into two parts. For part 1, the standard length and angle tolerances will be used and the geometric tolerances will be set to zero. For part 2, the tolerances of Xpawl, Ypawl, Rtip1, Rtip2, and thetaroot will be set to zero, and the geometric tolerances will be turned on. Part 2 will be split into two cases. For case 1, the position tolerance will be equal to the tolerances of Xpawl and Ypawl. For case 2, the position tolerance will be set equal to the root sum square of the tolerances of Xpawl and Ypawl (see figure 6.6).

 

 

Figure 6.6: Two cases of position tolerance.

6.8 Sensitivity Matrices

Constraint Sensitivities

A Matrix

 

Xpawl

Ypawl

Rpawl

Rtip1

Rtip2

Rroot

thetaroot

X1

1.0000

.00000

-.96060

-.98750

.00000

-.15761

.00000

Y1

.00000

1.0000

.27794

.15761

.00000

-.98750

.00000

q1

.00000

.00000

.00000

.00000

.00000

.00000

.00000

X2

.00000

.00000

.00000

-.98750

-.32445

.00000

-.30909

Y2

.00000

.00000

.00000

.15761

.94590

.00000

.04933

q2

.00000

.00000

.00000

.00000

.00000

.00000

-1.0000

 

B Matrix

 

Rside1

Rside2

phipawl

phitip1

phitooth

phitip2

X1

-.15761

.00000

.30000

.38338

.00000

.00000

Y1

-.98750

.00000

-.40000

-.11182

.00000

.00000

q1

.00000

.00000

1.0000

1.0000

1.0000

.00000

X2

-.15761

.94590

.00000

.38338

.00000

-.38338

Y2

-.98750

.32445

.00000

-.11182

.00000

.11182

q2

.00000

.00000

.00000

1.0000

.00000

-1.0000

 

F Matrix

 

a1

a1

a2

a3

a4

X1

1.0000

.00000

-.98750

.00000

.00000

Y1

.00000

1.0000

.15761

.00000

.00000

q1

.00000

.00000

.00000

.00000

.00000

X2

.00000

.00000

-.98750

-.30909

-.32445

Y2

.00000

.00000

.15761

.04933

.94590

q2

.00000

.00000

.00000

-1.0000

.00000

 

Tolerance Sensitivities

-B-1A Matrix

 

Xpawl

Ypawl

Rpawl

Rtip1

Rtip2

Rroot

thetaroot

Rside1

.00000

.00000

.00000

.53171

1.1326

.00000

-.09425

Rside2

.00000

.00000

.00000

1.1326

.53171

.00000

-.09425

phipawl

.93334

3.2000

-.00716

-2.1758

-3.7455

-3.3071

.31168

phitip1

-3.3387

-2.5040

2.5112

4.4969

3.3965

2.9989

-.28264

phitooth

2.4054

-.69597

-2.5040

-2.3211

.34903

.30817

-.02904

phitip2

-3.3387

-2.5040

2.5112

4.4969

3.3965

2.9989

-1.2826

-B-1F Matrix

 

a1

a1

a2

a3

a4

Rside1

.00000

.00000

.53171

-.09425

1.1326

Rside2

.00000

.00000

1.1326

-.09425

.53171

phipawl

.93334

3.2000

-2.1758

.31168

-3.7455

phitip1

-3.3387

-2.5040

4.4969

-.28264

3.3965

phitooth

2.4054

-.69597

-2.3211

-.02904

.34903

phitip2

-3.3387

-2.5040

4.4969

-1.2826

3.3965

 

6.9 Resultant Tolerances Before Optimization

Table 6.4: Independent Variable Tolerances and Control Factors

Dim.

Name

± Tol.

Std.

Dev.

Cp

Dk

Cpk

Sk

Wt. Factor

Tol.

Basic

Fixed

Xpawl

.0015 in

.0005

1

0.25

0.75

0

1

1

No

Ypawl

.0015 in

.0005

1

0.25

0.75

0

1

1

No

Rpawl

.001 in

.0003

1

0.25

0.75

0

1

1

No

Rtip1

.001 in

.0003

1

0.25

0.75

0

1

1

No

Rtip2

.001 in

.0003

1

0.25

0.75

0

1

1

No

Rroot

.002 in

.0007

1

0.25

0.75

0

1

1

No

thetaroot

.800º

.2667

1

0.25

0.75

0

1

1

No

 

Table 6.5: Kinematic Assembly Variables (No Geometric Tolerances)

Variable

Degree of

± Assembly Variation (ZASM = 3.000)

Name

Freedom

Worst Case

RSS Case

6-SIG Case

Rside1

translation (in)

.00298

.00182

.00242

Rside2

translation (in)

.00298

.00182

.00242

phipawl

rotation (º)

1.3232

.59115

.78820

phitip1

rotation (º)

1.6681

.65025

.86701

phitooth

rotation (º)

.62155

.29457

.39276

phitip2

rotation (º)

2.4681

1.1936

1.5914

 

Table 6.6: RSS Percent Contributions To phitooth (No Geometric Tolerances)

Variable Name

Variance

Six-Sigma

Xpawl

1.447e-6

 

49.25

Rpawl

6.967e-7

 

23.72

Rtip1

5.986e-7

 

20.38

Ypawl

1.211e-7

 

4.12

Rroot

4.221e-8

 

1.43

other

3.181e-8

 

1.08

 

Table 6.7: RSS Percent Rejects (Geometric Tolerances Set To Zero)

Spec. Name

Spec. Type

Nominal Dimension

(±) Computed Variation

With Geometric Tolerances

Without Geometric Tolerances

phitooth

Dep. Angle

99.068

.29457

Z

Rej.

Z

Rej.

ZASM = 3.000

USL 99.468

Upper Tail

4.07 23.1 4.07 23.1

(Rejects in PPM)

LSL 98.668

Lower Tail

-4.07 23.1 -4.07 23.1

Geometric Tolerances--Case 1

Table 6.8: Geometric Tolerances--Case 1

Feat.

Joint

Part Name

Feature Type

Tolerance Band

Char. Length

a1

2

Ground

true position

.003 in

N/A

a2

3

Pawl

profile

.002 in

N/A

a3

N/A

Ratchet

angularity

.0012 in

.0832

a4

4

Pawl

profile

.002 in

N/A

Remarks>> For Case 1, the position tolerance zone is set equal to the tolerances of Xpawl and Ypawl. Remember that dimension tolerances are symetric, while geometric tolerances are given as bandwidths.

Table 6.9: Kinematic Assembly Variables (Geometric Tolerancing, Case 1)

Variable

Degree of

± Assembly Variation (ZASM = 3.000)

Name

Freedom

Worst Case

RSS Case

6-SIG Case

Rside1

translation (in)

.00302

.00185

.00185

Rside2

translation (in)

.00302

.00185

.00185

phipawl

rotation (º)

1.3314

.59464

.68213

phitip1

rotation (º)

1.6755

.65287

.73088

phitooth

rotation (º)

.62231

.29463

.32216

phitip2

rotation (º)

2.5017

1.2226

1.2660

Table 6.10: RSS Percent Contributions To phitooth (Geometric Tolerancing, Case 1)

Variable Name

Variance

Six-Sigma

a1

1.568e-6

 

53.35

Rpawl

1.239e-6

 

23.71

a2

5.986e-7

 

20.37

Rroot

4.221e-8

 

1.44

other

3.303e-8

 

1.13

Table 6.11: RSS Percent Rejects (Geometric Tolerancing, Case 1)

Spec. Name

Spec. Type

Nominal Dimension

(±) Computed Variation

With Geometric Tolerances

Without Geometric Tolerances

phitooth

Dep. Angle

99.068

.29463

Z

Rej.

Z

Rej.

ZASM = 3.000

USL 99.468

Upper Tail

4.07

  23.2

N/A

N/A

(Rejects in PPM)

LSL 98.668

Lower Tail

-4.07

  23.2

N/A

N/A

Remarks>> For Case 1, the PPM rejects is nearly identical to those that occur with traditional dimension tolerances. This is as it should be, since the two analyses are mathematically equivalent. The purpose of this analysis has been to illustrate this point.

Geometric Tolerances--Case 2

Table 6.12: Geometric Tolerances--Case 2

Feat.

Joint

Part Name

Feature Type

Tolerance Band

Char. Length

a1

2

Ground

true position

.00414 in

N/A

a2

3

Pawl

profile

.002 in

N/A

a3

D?

Ratchet

angularity

.0012 in

.0832

a4

4

Pawl

profile

.002 in

N/A

Remarks>> For Case 2, the position tolerance zone is set equal to the root sum square of the tolerances of Xpawl and Ypawl.

Table 6.13: Kinematic Assembly Variables (Geometric Tolerancing, Case 2)

Variable

Degree of

± Assembly Variation (ZASM = 3.000)

Name

Freedom

Worst Case

RSS Case

6-SIG Case

Rside1

translation (in)

.00302

.00185

.00185

Rside2

translation (in)

.00302

.00185

.00185

phipawl

rotation (º)

1.4785

.66005

.73984

phitip1

rotation (º)

1.8835

.74491

.81415

phitooth

rotation (º)

.73272

.36486

.38743

phitip2

rotation (º)

2.7097

1.2741

1.3158

Table 6.14: RSS Percent Contributions To phitooth (Geometric Tolerancing, Case 2)

Variable Name

Variance

Six-Sigma

a1

3.135e-6

 

69.58

Rpawl

1.239e-6

 

15.46

a2

5.986e-7

 

13.29

other

1.081e-7

 

1.67

Table 6.15: RSS Percent Rejects (Geometric Tolerancing, Case 2)

Spec. Name

Spec. Type

Nominal Dimension

(±) Computed Variation

With Geometric Tolerances

Without Geometric Tolerances

phitooth

Dep. Angle

99.068

.36486

Z

Rej.

Z

Rej.

ZASM = 3.000

USL 99.468

Upper Tail

3.29

  502.9

N/A

N/A

(Rejects in PPM)

LSL 98.668

Lower Tail

-3.29

  502.9

N/A

N/A

Remarks>> Increasing the size of the position tolerance zone has a large effect on the number of rejects. Geometric tolerancing has some nice features, but in this assembly, enlarging the tolerance zone greatly increases the rejects.


PRO-E

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

AutoCAD

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

CATIA

Modeler: Crank Slider

 

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