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Example Problems 
AutoCAD
Verification:
ONEWAY CLUTCH 

Home : Example Problems : AutoCad  Verification  Clutch 
Figure 2.1: Schematic of the oneway clutch assembly and corresponding vector model.
2.0 Problem Description
A oneway clutch assembly transmits torque in a single direction. This example consists of four component types: a hub, an outer ring, four rollers, and four springs. When the hub rotates in a counterclockwise direction, the roller jams between it and the ring, locking them together. When the hub turns in a clockwise direction, the spring is compressed by the roller, the roller slips, and the hub is allowed to rotate freely.
Table 2.1: Manufactured Variables (Independent).
Variable Name  Basic Size  Initial Tolerance (±) 
1/2 Hub Width A  27.645 mm  .050 mm 
Roller Radius C, D  11.430 mm  .010 mm 
Ring Radius E  50.800 mm  .0125 mm 
2.1 Design Requirements
Table 2.2: Assembly Variables and Specification Limits.
Variable Name  Basic Size  Upper Spec. Limit (USL)  Lower Spec. Limit (LSL) 
Pressure Angle  7.0184[[ring]]  8.00[[ring]]  6.00[[ring]] 
Contact Angle 2  172.9816[[ring]]     
Roller Contact B  4.8105 mm     
In order for the clutch to function properly the angle 1 formed by the contact point between roller and ring must be between 6.00[[ring]] and 8.00[[ring]]. Beyond these limits the clutch is likely to slip or lock improperly.
2.2 Modeling Considerations
2.3 Design Goal
Specify the optimum tolerance combination for A and E to achieve the assembly tolerance limits on 1.
2.4 Part Names and DRFs
The assembly model is created graphically by adding symbols at selected locations on an existing CAD model of the assembly. The first task is to define a local coordinate system for each part. Each part in the assembly is given a unique name and a datum reference frame (DRF).
Figure 2.2: Diagram showing the location of the part DRFs.
DRFs are the local coordinate systems in each part. They are used to locate part features (such as holes or steps) and assembly model elements (such as kinematic joints). Datum selection determines which dimensions contribute to assembly variation. Datums also influence how parts are fixtured for production and inspection.
Remarks>> The DRFs for the hub and the ring are located at the same coordinates. The modeler will place the DRF labels on top of each other. This makes it necessary to use the zoom function to ensure the correct DRF is selected for each joint.
2.5 Kinematic Joints
Kinematic joints describe the degrees of freedom within the assembly, i.e. the small adjustments that occur between mating parts to accommodate manufacturing variations. The kinematic variations CATS solves for are these degrees of freedom that exist in the different joint types. For example, if there is a revolute joint, the assembly has a rotational degree of freedom at its location, and CATS will solve for a angular variable.
Three kinematic joints are required to model a quarter section of the oneway clutch.
Figure 2.3: Kinematic joint diagram.
Table 2.3: Kinematic Joints of the OneWay Clutch.
Joint Number  Part One  Part Two  Joint Type 
1  Hub  Ring  Revolute 
2  Hub  Roller  Cylindrical Slider 
3  Ring  Roller  Parallel Cylinders 
Remarks>> Kinematic joints generally occur only at points of contact between two parts. The ring and the hub never actually touch, but they do revolve around the same point. Therefore their "contact point" is modeled as a revolute joint.
2.6 Network Diagram, Vector Loops, and Design Specifications
The network diagram describes the connectivity between assembly components. No dimensional data is required. Parts are represented by circles or ovals. They are joined by lines or arcs representing the joints. Each joint connects two mating parts.
The network diagram shows that one loop is sufficient to solve for the clutch assembly kinematic variables. The vector loop follows dimensioned lengths and passes though each part DRF and each joint. A design specification has been applied to the dependent angle [[phi]]1.
Figure 2.4: Network diagram and loop diagram for the oneway clutch assembly.
Use: A network diagram is used to determine the number of loops necessary to describe the assembly. This can be determined by inspection or by applying the formula:
Loops = Joints  Parts + 1.
Remarks>> The number of loops and the optimum paths are determined automatically by the Autoloop Option in the CATS Modeler.
Dependent lengths (such as B) and dependent angles (such as 1 and 2) are not part dimensions. They are assembly variables whose variation depends upon the variation of the manufactured component dimensions of the assembly. Assembly variations are calculated by analyzing the vector loops representing the assembly to determine the tolerance stackup. In general, up to three dependent variables may be solved for from a single closed loop.
Engineering design limits may be specified for any assembly variable. Open vector loops are used to describe assembly features like gaps, clearance, parallel surfaces, orientation, or position. The estimated variation from the loop analysis may be compared to the design limits to predict the probable number of assemblies which will fail to meet the specification limits.
For dependent angle specifications, the nominal value that CATS uses is formed by the extension of the vector into the joint and the vector out. If the specification was applied to 2, instead of the range 6.00[[ring]]8.00[[ring]], the range 172.00[[ring]]174.00[[ring]] should be specified.
The nominal angle of 1 is not centered within the specification limits, so the user is required to enter nonsymmetric tolerances. To create the 6.00[[ring]]8.00[[ring]] range, +.9816 and 1.0184 must be entered using the +TOL and TOL options.
2.7 Geometric Tolerances
ANSI Y14.5 geometric tolerances are added to account for machined surface variations. They are applied to mating surfaces. Usually, up to two surface variations may be specified at each joint.
Figure 2.5: Geometric tolerance diagram.
Use: Geometric variations in form, orientation and location can accumulate statistically and propagate kinematically in an assembly the same as size variations. Their effects can be estimated quantitatively by including them in the vector model.
Remarks>> The concentricity tolerance applies to the ring and the shaft it turns (the shaft itself is not shown).
2.8 Sensitivity Matrices
Below is the numerical analysis for obtaining the values for the sensitivity matrices. A numerical analysis is included with this first example problem only.
Loop Equations and Tolerance Sensitivities
X Equation hx(a,b,c,e,1,2) = 0
a*cos 90º+ b*cos 0º+ c*cos 90º+ c*cos (90º  1) + e*cos (270º  1) = 0
= cos 90º = 0 = cos 0º. = 1.0
= cos 90º + cos (90º  1) = 0.12219
= cos (270º  1) = 0.12219
= c*sin (90º  1) + e*sin (270º  1) = 39 075 = 0
Y Equation hy(a,b,c,e,1,2) = 0
a*sin 90º+ b*sin 0º+ c*sin 90º+ c*sin (90º  1) + e*sin (270º  1) = 0
= sin 90º = 1.0 = sin 0º = 0
= sin 90º + sin (90º  1) = 1.99251 = sin (270º  1) = 0.99251
= c*cos (90º  1)  e*cos (270º  1) = 4.8105 = 0
Rotation Equation h(a,b,c,e,1,2) = 0
90º  90º + 90º 1  180º + 2 + 90º = 0 or 1  2 = 0
= 1.0 = 1.0
The partials of the independent variables form the A constraintsensitivity matrix, and the partials of the dependent variables form the B constraintsensitivity matrix. The same process is used to build the form tolerance sensitivity matrix (F Matrix).
Constraint Sensitivities
A Matrix
A  C  E  
X  .00000  .12219  .12219 
Y  1.0000  1.9925  .99251 
.00000  .00000  .00000 
B Matrix
B  1  2  
X  1.0000  39.075  .00000 
Y  .00000  4.8105  .00000 
.00000  1.0000  1.0000 
F Matrix
1  2  3  4  5  6  
X  .00000  .00000  .12219  .12219  .00000  1.0000 
Y  1.0000  1.0000  .99251  .99251  1.0000  .00000 
.00000  .00000  .00000  .00000  .00000  .00000 
Tolerance Sensitivities
B^{1}A Matrix
A  C  E  
B  8.1228  16.307  8.1841 
1  .20788  .41420  .20632 
2  .20788  .41420  .20632 
B^{1}F Matrix
1  2  3  4  5  6  
B  8.1228  8.1228  8.1841  8.1841  8.1228  1.0000 
1  .20788  .20788  .20632  .20632  .20788  .00000 
2  .20788  .20788  .20632  .20632  .20788  .00000 
Remarks>> In A, B, and F the signs of numbers, the order of the columns, and the magnitudes of the partial derivatives with respect to angular variables will vary according to how the loops are arranged, but the magnitudes and signs of the numbers that make up each column in B^{1}A and B^{1}F should not vary from those shown above.
2.9 Predicted Assembly Variation
Table 2.4: Independent Variable Tolerances and Control Factors
Dim. Name  +/ Tol.  Std. Dev.  Cp  Dk  Cpk  Sk  Wt. Factor  
Tol.  Basic  Fixed  
A  .050 mm  .0167  1  0.25  0.75  0  1  1  No 
C  .010 mm  .0033  1  0.25  0.75  0  0  0  Yes 
E  .0125 mm  .0042  1  0.25  0.75  0  1  1  No 
Table 2.5: Kinematic Assembly Variables (No Geometric Tolerances)
Variable Name 
Degree of Freedom  Tolerances (ZASM = 3.000)  
Worst Case  RSS Case  6SIG Case  
1  Rotation ([[ring]])  .98061  .65788  .87717 
2  Rotation ([[ring]])  .98061  .65788  .87717 
B  Translation (mm)  .67151  .44945  .59927 
Remarks>> 1 and 2 occur at separate points in the assembly, but they are equivalent angles. Therefore their variations are identical.
Table 2.6: Geometric Tolerances
Feat.  Joint  Part Name  Feature Type  Tolerance Band  Char. Length 
1  2  Hub  Flatness  .025 mm  N/A 
2  2  Roller  Circularity  .003 mm  N/A 
3  3  Roller  Circularity  .003 mm  N/A 
4  3  Ring  Circularity  .010 mm  N/A 
5  1  Ring  Concentricity  .010 mm  N/A 
Table 2.7: Kinematic Assembly Variables (Geometric Tolerances Included)
Variable Name 
Degree of Freedom  +/ Assembly Variation (ZASM = 3.000)  
Worst Case  RSS Case  6SIG Case  
1  Rotation ([[ring]])  1.2837  .68018  .89402 
2  Rotation ([[ring]])  1.2837  .68018  .89402 
B  Translation (mm)  .88404  .46472  .61080 
Remarks>>Including geometric tolerances noticeably affects the dependent variations. For highprecision assemblies, surface variations should not be ignored.
Table 2.8: RSS Percent Contributions To 1 (Geometric Tolerances Included)
Variable Name  Variance  Statistical RSS 
A  1.080e4  76.66 
C  1.716e5  12.17 
1 (hub flatness)  6.752e6  4.79 
E  6.651e6  4.72 
Other  2.597e7  1.66 
Remarks>> Dimension A (hub width) is by far the largest cause of variation in 1. If the percent rejects is too high, A's tolerance should be tightened. If the percent rejects is too low, the tolerance on E (ring radius) can be loosened to reduce costs.
Table 2.9: RSS Percent Rejects
Spec. Name  Spec. Type  Nominal Dimension  (+/) Computed Variation 
With Geometric Tolerances 
Without Geometric Tolerances  
1  Dep. Angle  7.0184  .68018  Z  Rej.  Z  Rej. 
ZASM = 3.000  USL 8[[ring]] 
Upper Tail  4.33  7.5  4.48  3.8  
(Rejects in PPM)  LSL 6[[ring]] 
Lower Tail  4.49  3.5  4.64  1.7 
2.10 Tolerance And Nominal Allocation
Proportional Scaling Tolerance Allocation
Proportional Scaling adjusts dimension tolerances proportional to the original dimension tolerances.
Table 2.10: RSS Proportional Scaling Tolerance Allocation (Geometric Tolerances Included).
Assembly Specs.  Nom.  USL  LSL  +/ ZASM 
Dep. Angle 1 ([[ring]]) 
7.0184  8.0000  6.0000  3.000 
Dimension Name  Specified Values  Allocated Values  
Nom.  +/Tol.  Nom.  +/Tol.  STDEV  % Cont.  
A (mm)  27.6450  .05000  27.6450  .07777  .02592  86.06 
C_D (mm)  11.4300  .01000  11.4300  .01000  .00333  5.65 * 
E (mm)  50.8000  .01250  50.8000  .01944  .00648  5.30 
1 (mm)  0.0  .01250  0.0  .01250  .00417  2.22 * 
2 (mm)  0.0  .00150  0.0  .00150  .00050  0.03 * 
3 (mm)  0.0  .00150  0.0  .00150  .00050  0.03 * 
4 (mm)  0.0  .00500  0.0  .00500  .00167  0.35 * 
5 (mm)  0.0  .00500  0.0  .00500  .00167  0.35 * 
Assem. Total  Nom.  +/Var.  Nom.  +/Var.  STDEV  100.00 
1 ([[ring]])  7.0184  .68018  7.0184  .99848  .33283  
Min./Max.  6.3382  7.6986  6.0199  8.0169  * Fixed Nom./Tol. 
Before Optimization  After Optimization  
Rejects  Z  PPM  Z  PPM 
Upper Tail  4.33  7.5  2.95  1592.5 
Lower Tail  4.49  3.5  3.06  1107.4 
Total Rejects  11.0  Total Rejects  2699.9 
Least Cost Tolerance Allocation
The Cost Allocation feature uses an assembly cost function and a search routine to find the least expensive combination of tolerances that meet the design specification. The function used in CATS is:
Cost = A + B(Tol^{k})
where A = fixed costs, and B and k are cost parameters dependent on the dimension size and production process.
Table 2.11: Cost Data for the OneWay Clutch Components
Model Parameters  Tolerance Range  
Part  Process  Size (mm)  A  B  k  Min.  Max. 
Hub  Milled  27.645  0  .0954442  .443140  .0508  .127 
Roller  Lapped  11.43  0  .0018938  .950878  .00508  .01016 
Ring  Ground  50.8  0  .0133356  .782762  .01016  .0254 
Table 2.12: RSS Cost Allocation (Geometric Tolerances Included)
Assembly Specs.  Nom.  USL  LSL  +/ ZASM 
Dep. Angle 1 ([[ring]]) 
7.0184  8.0000  6.0000  3.000 
Dimension Name  Specified Values  Allocated Values  
Nom.  +/Tol.  Nom.  +/Tol.  STDEV  % Cont.  
A (mm)  27.6450  .05000  27.6450  .06108  .02036  53.09 
C_D (mm)  11.4300  .01000  11.4300  .01000  .00333  5.65 * 
E (mm)  50.8000  .01250  50.8000  .05225  .01742  38.27 
1 (mm)  0.0  .01250  0.0  .01250  .00417  2.22 * 
2 (mm)  0.0  .00150  0.0  .00150  .00050  0.03 * 
3 (mm)  0.0  .00150  0.0  .00150  .00050  0.03 * 
4 (mm)  0.0  .00500  0.0  .00500  .00167  0.35 * 
5 (mm)  0.0  .00500  0.0  .00500  .00167  0.35 * 
Assem. Total  Nom.  +/Var.  Nom.  +/Var.  STDEV  100.00 
1 ([[ring]])  7.0184  .68018  7.0184  .99848  .33283  
Min./Max.  6.3382  7.6986  6.0199  8.0169  * Fixed Nom./Tol. 
Before Optimization  After Optimization  
Rejects  Z  PPM  Z  PPM 
Upper Tail  4.33  7.5  2.95  1592.5 
Lower Tail  4.49  3.5  3.06  1107.4 
Total Rejects  11.0  Total Rejects  2699.9 
Remarks>> The Proportional Scaling Tolerance Allocation and Least Cost Allocation examples both optimized based on a ZASM of +/3[[sigma]], so the final number of rejects for both cases is the same. However, the portion of the variance pool allocated to the nonfixed tolerances is different for the two cases.
Nominal Allocation
The CATS nominal allocation routines provide the designer a method of estimating the changes in dimension nominals neccessary to center the design variable distribution between the specification limits. Options are also available to align the userspecified number of standard deviations (ZASM) with the upper or lower specification limits.
Table 2.13: RSS Nominal Allocation (Geometric Tolerances Included).
Assembly Specs.  Nom.  USL  LSL  +/ ZASM 
Dep. Angle 1 ([[ring]]) 
7.0184  8.0000  6.0000  3.000 
Dimension Name  Specified Values  Allocated Values  
Nom.  +/Tol.  Nom.  +/Tol.  STDEV  % Cont.  
A (mm)  27.6450  .05000  27.6458  .05000  .01667  76.66 
C_D (mm)  11.4300  .01000  11.4300  .01000  .00333  12.17 * 
E (mm)  50.8000  .01250  50.7992  .01250  .00417  4.72 
1 (mm)  0.0  .01250  0.0  .01250  .00417  4.79 * 
2 (mm)  0.0  .00150  0.0  .00150  .00050  0.07 * 
3 (mm)  0.0  .00150  0.0  .00150  .00050  0.07 * 
4 (mm)  0.0  .00500  0.0  .00500  .00167  0.76 * 
5 (mm)  0.0  .00500  0.0  .00500  .00167  0.77 * 
Assem. Total  Nom.  +/Var.  Nom.  +/Var.  STDEV  100.00 
1 ([[ring]])  7.0184  .68018  7.0000  .68018  .22673  
Min./Max.  6.3382  7.6986  6.3198  7.6802  * Fixed Nom./Tol. 
Before Optimization  After Optimization  
Rejects  Z  PPM  Z  PPM 
Upper Tail  4.33  7.5  4.41  5.2 
Lower Tail  4.49  3.5  4.41  5.2 
Total Rejects  11.0  Total Rejects  10.3 
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