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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 builtin 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^{1}A 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^{1}F 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 
6SIG 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 
SixSigma 

Xpawl 
1.447e6 
49.25 

Rpawl 
6.967e7 
23.72 

Rtip1 
5.986e7 
20.38 

Ypawl 
1.211e7 
4.12 

Rroot 
4.221e8 
1.43 

other 
3.181e8 
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 TolerancesCase 1
Table 6.8: Geometric TolerancesCase 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 
6SIG 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 
SixSigma 

a1 
1.568e6 
53.35 

Rpawl 
1.239e6 
23.71 

a2 
5.986e7 
20.37 

Rroot 
4.221e8 
1.44 

other 
3.303e8 
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 TolerancesCase 2
Table 6.12: Geometric TolerancesCase 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 
6SIG 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 
SixSigma 

a1 
3.135e6 
69.58 

Rpawl 
1.239e6 
15.46 

a2 
5.986e7 
13.29 

other 
1.081e7 
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.
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