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Example Problems 
PROE
Verification:
CHAPTER 2: ONEWAY CLUTCH 

Home : Example Problems : ProE 2D  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  0.050 mm 
Roller Radius C, D  11.430 mm  0.010 mm 
Ring Radius E  50.800 mm  0.0125 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 q1  172.9816[[ring]]  174.00[[ring]]  172.00[[ring]] 
Contact Angle q2  7.0184[[ring]]     
Roller Contact B  4.8105 mm     
In order for the clutch to function properly the angle q1 formed by the contact point between roller and ring must be between 172.00[[ring]] and 174.00[[ring]].
2.2 Modeling Considerations
One of the rotations must be turned off in the analyzer or modeler in order to perform the analysis.
Remember to create an assembly datum point at each location that will have a modeling element (DRF, feature datum, kinematic joint, or specification endpoint) attached to it.
2.3 Design Goal
Specify tolerance limits for A and E that will achieve a 3 reject fraction for the assembly variable q1.
2.4 Part DRFs And Feature Datums
Each part in the assembly must be given a unique name and a datum reference frame (DRF). Feature datums define the kinematic and manufactured dimensions that define the location of the joints relative to the part DRFs.
Figure 2.2: Diagram showing the location of the part DRFs and feature datums.
DRFs define 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 'query select' or 'menu select' function to ensure the correct DRF is selected when defining paths from the kinematic joints back to the part DRFs.
2.5 Kinematic Joints
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 
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 TI TOL 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 TI TOL will solve for an angular variable.
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.
Remember to create an assembly datum plane tangent to the line of contact between the Roller and the Ring. This plane is used to properly orient the parallel cylinders joint.
Remember that one vector out of the cylindrical slider joint must lie in the sliding plane and define the kinematic length variable for the joint. The other vector out of the joint must end at a cylindrical DRF or feature datum. The cylindrical DRF or datum defines the rotational kinematic variable for the 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.
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 user may find that creating a network diagram early in the modeling stage will help in determining the type and location of kinematic joints and feature datums. If the user is having difficulty defining joints and their paths back to the part DRFs, a few minutes spent creating a network diagram may help clarify the problem.
The number of loops and the optimum paths are determined automatically by the Autoloop option in TI TOL.
Dependent lengths (such as B) and dependent angles (such as q1 and q2) are determined by analyzing the closed loops of the assembly. In general, a single closed loop may be solved for up to three dependent variables.
For dependent angle specifications, the nominal value that TI TOL uses is the inside angle formed by the vector into the joint and the vector out. If the specification was applied to q2, instead of the range 172.00[[ring]]174.00[[ring]], the range 6.00[[ring]]8.00[[ring]] should be specified.
The nominal angle of q1 is 172.9816[[ring]]. It is not centered within the specification limits, so the user is required to enter nonsymmetric tolerances. To create the 172.00[[ring]]174.00[[ring]] range, +1.0184 and .9816 must be entered at the +TOL and TOL prompts.
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 geometric tolerance applies to the ring and the shaft it turns (the shaft itself is not shown).
2.8 Sensitivity Matrices
Shown below is the method used to obtain the values for the sensitivity matrices.
Loop Equations and Tolerance Sensitivities
X Equation hx(a, b, c, e, F1, F2) = 0
a*cos 90º+ b*cos 0º+ c*cos 90º+ c*cos (90º  F1) + e*cos (270º  F1) = 0
= cos 90º = 0 = cos 0º = 1.0
= cos 90º + cos (90º  F1) = 0.12219 = cos (270º  F1) = 0.12219
= c*sin (90º  F1) + e*sin (270º  F1) = 39 075 = 0
Y Equation hy(a, b, c, e, F1, F2) = 0
a*sin 90º+ b*sin 0º+ c*sin 90º+ c*sin (90º  F1) + e*sin (270º  F1) = 0
= sin 90º = 1.0 = sin 0º = 0
= sin 90º + sin (90º  F1) = 1.99251 = sin (270º  F1) = 0.99251
= c*cos (90º  F1)  e*cos (270º  F1) = 4.8105 = 0
Rotation Equation h(a, b, c, e, F1, F2) = 0
90º  90º + 90º  F1 + 180º  F2  90º = 0 or  F1  F2 + 180º = 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).
Tolerance Sensitivities
Table 2.4: B^{1}A Matrix
A  C  E  
B  8.12279  16.30691  8.18412 
q1  0.20788  0.41420  0.20632 
q2  0.20788  0.41420  0.20632 
Table 2.5: B^{1}F Matrix
1  2  3  4  5  6  
B  8.12279  8.12279  8.18412  8.18412  8.12279  1.00000 
q1  0.20788  0.20788  0.20632  0.20632  0.20788  0.00000 
q2  0.20788  0.20788  0.20632  0.20632  0.20788  0.00000 
Remarks>> The magnitudes of the numbers in the columns of B^{1}A and B^{1}F should not vary from those shown above.
2.9 Predicted Assembly Variation
Table 2.6: Independent Variable Tolerances and Control Factors
Dim Name  Nominal  ± Tol  Process Std Dev  Process Description  K 
A  27.6450  0.05000  0.01667  None  0.25 
C_D  11.4300  0.01000  0.00333  None  0.25 
E  50.8000  0.01250  0.00417  None  0.25 
Table 2.7: Kinematic Assembly Variables (Geometric Tolerances Not Applied)
Variable  ± Assembly Variation (ZASM = 3.00)  
Name  Nominal  WC  RSS  SSA  SSC 
B  4.8105  0.67151  0.44945  0.44945  0.59927 
q1  172.9816  0.98061  0.65788  0.65788  0.87717 
q2  7.0184  0.98061  0.65788  0.65788  0.87717 
Table 2.8: RSS Percent Rejects (Geometric Tolerances Not Applied)
Dep Angle q1  Spec Limit  Assy Std Dev  Assy Sigma  Rejects PPM  Rejects DPU 
Upper  174.0000  0.21929  4.64  1.71  1.7107e6 
Lower  172.0000  4.48  3.80  3.8015e6  
Nom Dim  172.9816  Total  5.51  5.5121e6 
Table 2.9: Geometric Tolerances
Name  Part Name  Type  Joint  Tolerance Band  Char. Length 
1  Hub  Flatness  2  0.02500  N/A 
2  Roller  Circularity  2  0.00300  N/A 
3  Roller  Circularity  3  0.00300  N/A 
4  Ring  Circularity  3  0.01000  N/A 
5  Ring  Concentricity  1  0.01000  N/A 
Table 2.10: Kinematic Assembly Variables (Geometric Tolerances Applied)
Variable  ± Assembly Variation (ZASM = 3.00)  
Name  Nominal  WC  RSS  SSA  SSC 
B  4.8105  0.88404  0.46472  0.46472  0.61080 
q1  172.9816  1.28374  0.68018  0.68018  0.89402 
q2  7.0184  1.28374  0.68018  0.68018  0.89402 
Remarks>>Including geometric tolerances noticeably affects the dependent variations. For highprecision assemblies, surface variations should not be ignored.
Table 2.11: Absolute Sensitivities Of q1 (Geometric Tolerances Applied)
Variable Name  Sensitivity  Normalized 
C_D  0.41420  22.21 
A  0.20788  11.15 
1  0.20788  11.15 
2  0.20788  11.15 
5  0.20788  11.15 
E  0.20632  11.06 
3  0.20632  11.06 
4  0.20632  11.06 
Table 2.12: RSS Percent Contributions To q1 (Geometric Tolerances Applied)
Variable Name  Contribution  Statistical RSS 
A  1.2004e5  76.66 
C  1.9062e6  12.17 
1  7.5022e7  4.79 
E  7.3902e7  4.72 
Other  2.5973e7  1.66 
Remarks>> Dimension A (hub width) is by far the largest cause of variation in [[phi]]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 2.13: RSS Percent Rejects (Geometric Tolerances Applied)
Dep Angle q1  Spec Limit  Assy Std Dev  Assy Sigma  Rejects PPM  Rejects DPU 
Upper  174.0000  0.22673  4.49  3.54  3.5357e6 
Lower  172.0000  4.33  7.48  7.4779e6  
Nom Dim  172.9816  Total  11.01  1.1014e5 
2.10 Tolerance Allocation
Weight Factor Tolerance Allocation
Weight Factor Tolerance Allocation adjusts dimension tolerances proportional to the userassigned weight factors. The weight factors that are large compared to the others receive a greater portion of the unused variation when there is a positive variance pool (assembly variation is smaller than the specified assembly limits) and are reduced the least when there is a negative variance pool (assembly variation exceeds the specified assembly limits). Therefore the user should assign larger weight factors to the tolerances he wants to become (or remain) as large as possible.
Table 2.14: RSS Weight Factor Tolerance Allocation (Geometric Tolerances Applied).
Dim Name  Original  Allocated  
Fix  WF  ± Tol  Cp  ± Tol  Cp  Std Dev  
A  N  1.00  0.05000  1.00  0.07777  1.00  0.02592 
C_D  Y  1.00  0.01000  1.00  0.01000  1.00  0.00333 
E  N  1.00  0.01250  1.00  0.01944  1.00  0.00333 
Dep Angleq1  Spec Limit  Assy Std Dev  Assy Sigma  Rejects PPM  Rejects DPU  
Upper  174.0000  0.33283  3.06  1107.44  1.1074e3  
Lower  172.0000  Target Sig  2.95  1592.50  1.5925e3  
Nom Dim  172.9816  3.00  Total  2699.93  2.6999e3 
Table 2.15: WC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).
Dim Name  Original  Allocated  
Fix  WF  ± Tol  Cp  ± Tol  Cp  Std Dev  
A  N  1.00  0.05000  0.02968  
C_D  Y  1.00  0.01000  0.01000  
E  N  1.00  0.01250  0.00742  
Dep Angleq1  Spec Limit  WC Variation  
Upper  174.0000  0.98161  Satisfied  
Lower  172.0000  Satisfied  
Nom Dim  172.9816 
Table 2.16: SSC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).
Dim Name  Original  Allocated  
Fix  WF  ± Tol  Cp  ± Tol  Cp  Std Dev  
A  N  1.00  0.05000  0.75  0.05691  0.75  0.01897 
C_D  Y  1.00  0.01000  0.75  0.01000  0.75  0.00333 
E  N  1.00  0.01250  0.75  0.01423  0.75  0.00474 
Dep Angleq1  Spec Limit  Assy Std Dev  Assy Sigma  Rejects PPM  Rejects DPU  
Upper  174.0000  0.33283  3.06  1107.44  1.1074e3  
Lower  172.0000  Target Sig  2.95  1592.50  1.5925e3  
Nom Dim  172.9816  3.00  Total  2699.93  2.6999e3 
Table 2.17: SSA Weight Factor Tolerance Allocation (Geometric Tolerances Applied).
Dim Name  Original  Allocated  
Fix  WF  ± Tol  Cp  ± Tol  Cp  Std Dev  
A  N  1.00  0.05000  1.00  0.10288  1.00  0.01715 
C_D  Y  1.00  0.01000  1.00  0.01000  1.00  0.00167 
E  N  1.00  0.01250  1.00  0.02572  1.00  0.00429 
Dep Angleq1  Spec Limit  Assy Std Dev  Assy Sigma  Rejects PPM  Rejects DPU  
Upper  174.0000  0.23782  3.06  1107.44  1.1074e3  
Lower  172.0000  Target Sig  2.95  1592.50  1.5925e3  
Nom Dim  172.9816  3.00  Total  2699.93  2.6999e3 
PROE Modeler: Clutch
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