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## 5.0 The One-Way Clutch: An Introduction to TI/TOL 2D Analysis

Chapter 4 of the TI/TOL 2D Modeling Manual introduces the one-way clutch problem. The clutch problem illustrates how a closed loop is used to find the resultant variation of a critical assembly variable. As explained in the Modeling Manual, this assembly is designed to allow rotation in only one direction. The hub is attached to a drive shaft as shown in Figure 5.1. When the hub rotates clockwise relative to the ring, the roller slips on the inside of the ring. If the hub rotates counter-clockwise, then the spring allows the roller to wedge between the hub and the ring, causing the two to lock and rotate together. In this analysis, the rollers are considered to be vendor-supplied, while the tolerances on the hub and ring need to be determined. Tolerances are initially selected by the designer according to the manufacturing processes he believes may be used.

Figure 5.1. One-way clutch assembly.

Figure 5.2. Vector loop for one-way clutch.

 Independent Variables: HUB/A = 27.645mm ± 0.05mm ROLLER/C = 11.430mm ± 0.010mm ROLLER/D = 11.430mm ± 0.010mm RING/E = 50.800mm ± 0.0125mm Dependent Variables: HUB/B = 4.81mm ± ? F1 = 172.982[[ring]]± ? F2 = 7.018[[ring]]± ?

The clutch must also be able to release when the hub is rotated clockwise again. This can be assured by specifying the dependent angle F1 (shown in Figure 5.2) to be between 172[[ring]] and 174[[ring]]. A vector-loop assembly model of the clutch must be created graphically with the TI/TOL 2D preprocessor (i.e. the Pro/ENGINEER TI/TOL 2D Modeler).

## 5.1 Start Up

After creating or calling up a vector-loop model of the one-way clutch with the TI/TOL 2D Modeler, execute the 2-D Analyzer. The analysis program is started by selecting Analyze from the TOL-2D main menu.

The main window for the TI/TOL 2D Analyzer will appear:

The main window menus can now be used to analyze the clutch model.

Two exercises will be outlined below for the clutch assembly that illustrate some of the important features available in the TI/TOL 2D analyzer.

## 5.2 Clutch Exercise A

The first exercise with the clutch assembly will utilize different analysis options available in TI/TOL 2D. Specific objectives of the exercise are to introduce the Root Sum Square (RSS) and Worst Case (WC) analysis options available in TI/TOL 2D, to demonstrate design spec analysis capabilities, to demonstrate the affect of geometric tolerances in tolerance analysis, and to create percent contribution plots.

### 5.2.1 Data Verification

An important first step in the exercise is to verify that all of the necessary data for the clutch assembly analysis was input correctly in the TI/TOL 2D Modeler. Select the Edit menu from the main analyzer window to display the options available for verification and modification.

The Tolerances command of the Edit menu allows the user to verify or change the tolerance values assigned to specific dimensions. The Tolerances window should show that the two roller radii have been equivalenced (variables C and D), and that only one roller radius dimension is available for modification. Changing the tolerance on the available roller radius dimension will automatically change the equivalent dimension. The initial tolerance values for the clutch problem should be as shown below. Any incorrect tolerance value can be modified by clicking on the incorrect value and keying in the correct value.

The Node DOF command of the Edit menu allows the user to verify that the correct node degrees of freedom are active and to toggle the degrees of freedom on or off. The degree of freedom associated with the ring DRF in the clutch assembly is turned off since it is redundant with the revolute joint between the hub and the ring.

The Allocation Data command of the Edit menu allows the user to fix tolerances in the assembly and to apply weight factors as needed. The clutch assembly will initially be analyzed with the default allocation data settings (i.e. no fixed dimensions and unity weight factors).

The Process Data command of the Edit menu allows the user to specify manufacturing processes for individual parts of the assembly. No manufacturing processes will be specified initially for the clutch assembly.

### 5.2.2 Analysis Options I

Once the input data has been verified or modified, then the analysis options for the assembly can be set. The Options menu allows the user to specify the analysis model to use in calculating tolerances. Options also allows the user to decide whether or not geometric tolerances will be applied in tolerance calculations. The user can also change the target assembly sigma to a value other than the default three sigma.

The first part of the exercise will use the default analysis settings. The default settings are: Root Sum Square analysis, acceptance fraction (target assembly sigma) of three sigma (~2700 ppm rejects), and no geometric tolerances included in the analysis. Verify that the Root Sum Squares option is selected and that the Apply Geometric Tolerances option is not selected from the Options menu as shown below.

Also verify that the target assembly sigma is set at 3.00. To do this select the Target Assembly Sigma option from the Options menu. The target assembly sigma can be verified or modified in the window shown below.

### 5.2.3 Design Specification Analysis I

The Analyze menu allows the user to analyze the data using the selected analysis options.

The Select Specification option of the Analyze menu allows the user to select the specification in the assembly to analyze. In the case of the clutch problem, however, there is only one specification applied to the entire assembly. The user can, therefore, go directly to the Design Spec Analysis option.

Selecting the Design Spec Analysis option will create a display showing the information inputs and the tolerance analysis outputs. With the default analysis settings (i.e. RSS analysis, no geometric tolerances applied, and 3 sigma acceptance fraction) the design spec analysis output for the clutch assembly is shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [14:56:47] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Not Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : 0.01667 : None : n/a C : 11.4300 : 0.01000 : 0.00333 : None : n/a E : 50.8000 : 0.01250 : 0.00417 : None : n/a ----------------:----------:---------:---------:----------------------:------ Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- Upper : 1.01840 : 0.21929 : 4.64 : 1.75 : 1.7460e-06 Lower : -0.98160 : : 4.48 : 3.85 : 3.8547e-06 -------:-----------:-------------:----------:---------------:-------------<> Total : 5.60 : 5.6007e-06

Notice that the originally specified tolerances produce only 5.60 rejects per million assemblies. This means the tolerances originally specified are tighter than necessary. The RSS kinematic variations are displayed along with the assembly standard deviation and the assembly sigmas for each side of the specification limit.

Save the analysis results with the Save Results As command of the File menu. Key in the desired file name for the output data in the Save Results As dialog window.

### 5.2.4 Percent Contribution Plot I

The percent contribution chart is another feature available in the TI/TOL 2D Analyzer. The chart provides a graphical representation of the percent contribution of dimensions and their tolerances to the variation of a particular kinematic assembly variable.

To create a percent contribution plot for the clutch assembly select the Sensitivity Plots option of the Analyze menu. A Plot Select window will pop up that allows the user to select the desired specification and the desired plot (i.e. sensitivity plot or percent contribution plot). Select the only available specification for the clutch assembly and the Percent Contribution Plot option from the Plot Select window as shown below.

The percent contribution plot will now be displayed as shown below.

### 5.2.5 Analysis Options II

At this point, it may be valuable to compare the closed loop analysis results obtained in the previous section with results obtained when geometric tolerances are included in the tolerance analysis. To include geometric tolerances in the analysis, select the Options menu from the Analyzer main window and click on the Apply Geometric Tolerances command. The shaded box on the left of the Apply Geometric Tolerances button will indicate that geometric tolerances will be applied to the analysis.

### 5.2.6 Design Spec Analysis II

Now return to the Analyze menu and select the Design Spec Analysis command. The tolerance analysis results with geometric tolerances applied are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:07:04] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : 0.01667 : None : n/a C : 11.4300 : 0.01000 : 0.00333 : None : n/a E : 50.8000 : 0.01250 : 0.00417 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_15 : Concentricity : JOINT1 : 0.01000 FC_16 : Flatness : JOINT2 : 0.02500 FC_17 : Roundness : JOINT2 : 0.00300 FC_18 : Roundness : JOINT3 : 0.00300 FC_19 : Roundness : JOINT3 : 0.01000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- Upper : 1.01840 : 0.22673 : 4.49 : 3.59 : 3.5856e-06 Lower : -0.98160 : : 4.33 : 7.55 : 7.5487e-06 -------:-----------:-------------:----------:---------------:-------------<> Total : 11.13 : 1.1134e-05

Notice that by applying the geometric tolerances to the analysis the number of rejects increased from 5.60 to 11.13 ppm. The change in results shows the potential impact geometric tolerancescan have on the tolerance model. In this case, the number of rejects was doubled. Other kinematic assembly variations or specification results can also be easily compared from the output.

### 5.2.7 Percent Contribution Plot II

A percent contribution plot for the second set of analysis options can be created as described previously. The percent contribution plot for the second set of analysis options (i.e. RSS analysis, 3 sigma acceptance fraction, and geometric tolerances applied) is shown below. Observe that A is by far the largest contributor to the angle variation. The other dimension, such as C and E have little influence on this assembly variable. This information will be valuable when we re-allocate tolerances.

### 5.2.8 Analysis Options III

The same procedure outlined above can be used with all of the statistical analysis methods available in TI/TOL 2D (i.e. Worst Case, Root Sum Squares, Six Sigma Assembly Drift, and Six Sigma Component Drift). To illustrate the variation in results that occurs when different analysis methods are used, a Worst Case (WC) analysis will be performed.

To perform a WC analysis, select the Worst Case command from the Options menu.

### 5.2.9 Design Spec Analysis III

Return to the Analyze menu and select the Design Spec Analysis option. Results of the WC analysis are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:08:53] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Applied Analysis Model: Worst Case (WC) ============================================================================== <<< I N P U T S >>> ------------------- Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : n/a : n/a : n/a C : 11.4300 : 0.01000 : n/a : n/a : n/a E : 50.8000 : 0.01250 : n/a : n/a : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_15 : Concentricity : JOINT1 : 0.01000 FC_16 : Flatness : JOINT2 : 0.02500 FC_17 : Roundness : JOINT2 : 0.00300 FC_18 : Roundness : JOINT3 : 0.00300 FC_19 : Roundness : JOINT3 : 0.01000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± WC Variation ----------------:------------:-----------------

Notice that a worst case analysis of the assembly tolerances violates the specification limits.

### 5.2.10 Percent Contribution Plot III

A percent contribution plot for the third set of analysis options can again be created as described previously. The percent contribution plot for the third set of analysis options (i.e. WC analysis with geometric tolerances applied) is shown below.

## 5.3 Clutch Exercise B

The objectives of this second exercise are to introduce proportional scaling allocation and weight factor allocation techniques, to demonstrate the affects of fixed assembly dimensions and variations in target assembly sigmas, and to demonstrate allocation for a worst case analysis. The clutch assembly model should be called up in the TI/TOL 2D Modeler, and the Analyzer environment should be invoked as described in section 5.1. It should be verified that the initial input data for Exercise B is the same as that used in Exercise A.

### 5.3.1 Analysis Option I

The first set of analysis options that will be employed in this exercise includes a root sum square method of analysis with geometric tolerances applied. Select the Options menu from the main analyzer window and select the Root Sum Squares and the Apply Geometric Tolerances commands as shown below.

Verify that the target assembly sigma is set to 3.00 by selecting the Target Assembly Sigma command from the Options menu. The Target Assembly Sigma pop-up window is shown below.

The dimension for the roller radius will be fixed for this exercise. Fixing the dimension of the roller is appropriate because the roller is a vendor-supplied item. Fix the dimension for the roller radius by selecting the Allocation Data option of the Edit menu. Toggle the box in the Fixed column that corresponds with the variable name of the roller radius to YES as shown below. In the example shown below the variable C corresponds with the roller radius dimension.

All of the values in the weight factor (WF) column are initially set at 1.00. This allows for proportional scaling among the assembly dimensions when allocating tolerances.

To save the changes to the allocation data select the File menu from the Allocation Data window and select the Save option as shown below.

### 5.3.2 Tolerance Allocation I

Tolerances will now be allocated to the clutch assembly dimensions by proportional scaling to achieve the target assembly sigma of 3.00 (~ 2700 ppm rejects). To allocate the assembly tolerances select the Allocate menu from the analyzer main window. Since there is only one specification applied to the clutch assembly the Select Specification option of the Allocate menu can be skipped and the Weight Factors Allocate option can be selected directly. The output is as shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:14:53] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : 0.01667 : None : n/a C : 11.4300 : 0.01000 : 0.00333 : None : n/a E : 50.8000 : 0.01250 : 0.00417 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_15 : Concentricity : JOINT1 : 0.01000 FC_16 : Flatness : JOINT2 : 0.02500 FC_17 : Roundness : JOINT2 : 0.00300 FC_18 : Roundness : JOINT3 : 0.00300 FC_19 : Roundness : JOINT3 : 0.01000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- A : N : 1.00 : 0.05000 : 1.00 : 0.07777 : 1.00 : 0.02592 : C : 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 : 0.00648 : ----------------:---:--------:---------:------:---------:------:---------:<> Specification Results: Specification Name: SP_14 Specification Type: Dependent Angle Nominal Dimension: 172.9816 Geometric Tolerances: Applied Analysis Model: RSS Target Assy Sigma: 3.00 Spec Limit Assy Std Dev Assy Sigma Rejects (ppm) DPU -------:-----------:-------------:----------:---------------:-------------<> Upper : 1.01840 : 0.33283 : 3.06 : 1107.17 : 1.1072e-03 Lower : -0.98160 : : 2.95 : 1592.46 : 1.5925e-03 -------:-----------:-------------:----------:---------------:-------------<> Total : 2699.62 : 2.6996e-03

The input data and the kinematic assembly output are displayed as in Exercise A for the clutch assembly. The allocated dimensions section of the output displays the dimension tolerances before and after allocation. Since the original tolerances produced fewer rejects than specified, the tolerances were increased by proportional scaling until the target assembly sigma was reached. The tolerance on the roller radius dimension remained constant since the roller radius dimension was fixed.

### 5.3.3 Analysis Option II

It would now be instructive to analyze the effect of applying different weight factors to assembly dimensions prior to tolerance allocation. An assembly that has a positive variance pool (i.e. a smaller assembly standard deviation than specified) can have larger tolerance values applied to the assembly dimensions until the target assembly sigma is reached. A part in an assembly that is difficult to produce with tight tolerances can be allocated a larger portion of the tolerances available in the positive variance pool. This is done by assigning that part a larger weight factor.

If the ring in the clutch assembly is difficult and/or expensive to produce with tight tolerances, then a large weight factor should be applied to the ring dimension to give it as large a tolerance as possible. Rember from Exercise A that the ring had a small influence on the assembly dimension. This means we can probably increase it more than A without affecting the quality of the assembly. To assign a weight factor of 5.00 to the ring dimension, select the Allocation Data option from the Edit menu. The Allocation Data window that pops up allows the user to change the weight factors assigned to any of the assembly dimensions. Change the value in the weight factor (WF) column that corresponds with the ring dimension to 5.00. In the example below the variable E corresponds to the ring dimension.

Save the change to the allocation data by selecting the Save option of the File menu in the Allocation Data window.

### 5.3.4 Tolerance Allocation II

Tolerances will now be allocated to the clutch assembly dimensions with different weight factors applied until the target assembly sigma of 3.00 (~ 2700 ppm rejects) is reached. To allocate the assembly tolerances select the Allocate menu from the analyzer main window. Again the Select Specification option of the Allocate menu can be skipped and the Weight Factors Allocate option can be selected directly. The output is as shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:17:23] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Applied Analysis Model: Root Sum Squares (RSS) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : 0.01667 : None : n/a C : 11.4300 : 0.01000 : 0.00333 : None : n/a E : 50.8000 : 0.01250 : 0.00417 : None : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_15 : Concentricity : JOINT1 : 0.01000 FC_16 : Flatness : JOINT2 : 0.02500 FC_17 : Roundness : JOINT2 : 0.00300 FC_18 : Roundness : JOINT3 : 0.00300 FC_19 : Roundness : JOINT3 : 0.01000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± RSS Variation ----------------:------------:----------------- A : N : 1.00 : 0.05000 : 1.00 : 0.05028 : 1.00 : 0.01676 : C : Y : 1.00 : 0.01000 : 1.00 : 0.01000 : 1.00 : 0.00333 : E : N : 5.00 : 0.01250 : 1.00 : 0.06286 : 1.00 : 0.02095 : ----------------:---:--------:---------:------:---------:------:---------:<> Specification Results: Specification Name: SP_14 Specification Type: Dependent Angle Nominal Dimension: 172.9816 Geometric Tolerances: Applied Analysis Model: RSS Target Assy Sigma: 3.00 Spec Limit Assy Std Dev Assy Sigma Rejects (ppm) DPU -------:-----------:-------------:----------:---------------:-------------<> Upper : 1.01840 : 0.33283 : 3.06 : 1107.17 : 1.1072e-03 Lower : -0.98160 : : 2.95 : 1592.46 : 1.5925e-03 -------:-----------:-------------:----------:---------------:-------------<> Total : 2699.62 : 2.6996e-03

Notice that the allocated tolerance for the ring dimension (variable E) increased from .01944 when a weight factor of 1.00 was applied to .06286 when a weight factor of 5.00 was applied. The tradeoff is that the allocated tolerance for the hub dimension (variable A) decreased from .07777 when a weight factor of 1.00 was applied to the ring to .05028 when a weight factor of 5.00 was applied to the ring. If A is easier to produce, this was a good trade-off. Both cases satisfy the 3.00 assembly sigma specification.

### 5.3.5 Analysis Option III

A tolerance allocation procedure will now be done using the worst case (WC) statistical method. Select the Worst Case command from the Options menu.

### 5.3.6 Tolerance Allocation III

Again select the Weight Factors Allocate option of the Allocate menu. The results of the WC analysis are shown below.

 TI/TOL 2D MECHANICAL ASSEMBLY TOLERANCE ANALYSIS V3.0 ANALYSIS RESULTS Tue Sep 06 1994 [15:18:10] ============================================================================== Assembly Model File: CLUTCH_TUTORIAL Geometric Tolerances: Applied Analysis Model: Worst Case (WC) ============================================================================== <<< I N P U T S >>> ------------------- Controlled Dimensions: Process Process Dimension Name Nominal ± Tol Std Dev Description K ----------------:----------:---------:---------:----------------------:------ A : 27.6450 : 0.05000 : n/a : n/a : n/a C : 11.4300 : 0.01000 : n/a : n/a : n/a E : 50.8000 : 0.01250 : n/a : n/a : n/a ----------------:----------:---------:---------:----------------------:------ Geometric Tolerances: Name Type Joint Tolerance -------------:------------------:----------:----------- FC_15 : Concentricity : JOINT1 : 0.01000 FC_16 : Flatness : JOINT2 : 0.02500 FC_17 : Roundness : JOINT2 : 0.00300 FC_18 : Roundness : JOINT3 : 0.00300 FC_19 : Roundness : JOINT3 : 0.01000 -------------:------------------:----------:----------- Specification Limits: Upper Limit: 1.01840 Lower Limit: -0.98160 <<< O U T P U T S >>> --------------------- Kinematic Assembly Variables: Variable Name Nominal ± WC Variation ----------------:------------:----------------- A : N : 1.00 : 0.05000 : n/a : 0.01653 : n/a : n/a : C : Y : 1.00 : 0.01000 : n/a : 0.01000 : n/a : n/a : E : N : 5.00 : 0.01250 : n/a : 0.02066 : n/a : n/a : ----------------:---:--------:---------:------:---------:------:---------:<> Specification Results: Specification Name: SP_14 Specification Type: Dependent Angle Nominal Dimension: 172.9816 Geometric Tolerances: Applied Analysis Model: WC Spec Limit WC Variation -------:------------:------------:------------ Upper : 1.018400 : 0.981600 : SATISFIED Lower : -0.981600 : : SATISFIED -------:------------:------------:------------

With the WC analysis the tolerance on the hub went from an original value of .05000 to an allocated value of .01653. With the larger weight factor, the tolerance on the ring went from an original value of .01250 to an allocated value of .02066. Again the tolerance on the roller radius remained constant since the roller radius dimension is fixed.

In contrast to the RSS allocation, the component tolerances were decreased rather than increased. This is because the Worst Case model assumes all components will be produced at their tolerance limits. When WC allocation is performed, the Analyzer calculates the tolerances neccessary to produce zero rejects. This is why theWC model is such an "expensive" design method.

### 5.3.7 Ending the Session

To exit the TI/TOL 2D analyzer environment and return to the modeling environment, select the Exit command from the File menu in the analyzer main window.

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