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Example Problems
AutoCAD Verification:
ONE-WAY CLUTCH
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Figure 2.1: Schematic of the one-way clutch assembly and corresponding vector model.

2.0 Problem Description

A one-way 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 counter-clockwise 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 phi2 172.9816[[ring]] -- --
Roller Contact B 4.8105 mm -- --

In order for the clutch to function properly the angle phi1 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 phi1.

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 one-way clutch.

Figure 2.3: Kinematic joint diagram.

Table 2.3: Kinematic Joints of the One-Way 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 one-way 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 phi1 and phi2) 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 phi2, 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 phi1 is not centered within the specification limits, so the user is required to enter non-symmetric 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,Phi1,Phi2) = 0

a*cos 90º+ b*cos 0º+ c*cos 90º+ c*cos (90º - Phi1) + e*cos (270º - Phi1) = 0

= cos 90º = 0 = cos 0º. = 1.0

= cos 90º + cos (90º - Phi1) = 0.12219

= cos (270º - Phi1) = -0.12219

= c*sin (90º - Phi1) + e*sin (270º - Phi1) = -39 075 = 0

Y Equation hy(a,b,c,e,Phi1,Phi2) = 0

a*sin 90º+ b*sin 0º+ c*sin 90º+ c*sin (90º - Phi1) + e*sin (270º - Phi 1) = 0

= sin 90º = 1.0 = sin 0º = 0

= sin 90º + sin (90º - Phi1) = 1.99251 = sin (270º - Phi1) = -0.99251

= -c*cos (90º - Phi1) - e*cos (270º - Phi1) = 4.8105 = 0

Rotation Equation htheta(a,b,c,e,Phi1,Phi2) = 0

90º - 90º + 90º- Phi1 - 180º + Phi2 + 90º = 0 or Phi1 - Phi2 = 0

= 1.0 = -1.0

The partials of the independent variables form the A constraint-sensitivity matrix, and the partials of the dependent variables form the B constraint-sensitivity 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
theta .00000 .00000 .00000

B Matrix

  B 1 2
X 1.0000 -39.075 .00000
Y .00000 4.8105 .00000
theta .00000 -1.0000 -1.0000

F Matrix

  alpha1 alpha2 alpha3 alpha4 alpha5 alpha6
X .00000 .00000 .12219 .12219 .00000 -1.0000
Y 1.0000 1.0000 .99251 .99251 -1.0000 .00000
theta .00000 .00000 .00000 .00000 .00000 .00000

Tolerance Sensitivities

-B-1A Matrix

  A C E
B -8.1228 -16.307 8.1841
phi1 -.20788 -.41420 .20632
phi2 .20788 .41420 -.20632

-B-1F Matrix

  alpha1 alpha2 alpha3 alpha4 alpha5 alpha6
B -8.1228 -8.1228 -8.1841 -8.1841 8.1228 1.0000
phi1 -.20788 -.20788 -.20632 -.20632 .20788 .00000
phi2 .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-1A and -B-1F 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 6-SIG Case
phi1 Rotation ([[ring]]) .98061 .65788 .87717
phi2 Rotation ([[ring]]) .98061 .65788 .87717
B Translation (mm) .67151 .44945 .59927

Remarks>> phi1 and phi2 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
alpha1 2 Hub Flatness .025 mm N/A
alpha2 2 Roller Circularity .003 mm N/A
alpha3 3 Roller Circularity .003 mm N/A
alpha4 3 Ring Circularity .010 mm N/A
alpha5 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 6-SIG Case
phi1 Rotation ([[ring]]) 1.2837 .68018 .89402
phi2 Rotation ([[ring]]) 1.2837 .68018 .89402
B Translation (mm) .88404 .46472 .61080

Remarks>>Including geometric tolerances noticeably affects the dependent variations. For high-precision assemblies, surface variations should not be ignored.

Table 2.8: RSS Percent Contributions To phi1 (Geometric Tolerances Included)

Variable Name Variance Statistical RSS
A 1.080e-4 76.66
C 1.716e-5 12.17
alpha1 (hub flatness) 6.752e-6 4.79
E 6.651e-6 4.72
Other 2.597e-7 1.66

Remarks>> Dimension A (hub width) is by far the largest cause of variation in phi1. 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
phi1 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
alpha1 (mm) 0.0 .01250 0.0 .01250 .00417 2.22 *
alpha2 (mm) 0.0 .00150 0.0 .00150 .00050 0.03 *
alpha3 (mm) 0.0 .00150 0.0 .00150 .00050 0.03 *
alpha4 (mm) 0.0 .00500 0.0 .00500 .00167 0.35 *
alpha5 (mm) 0.0 .00500 0.0 .00500 .00167 0.35 *
Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00
phi1 ([[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(Tolk)

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 One-Way 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
alpha1 (mm) 0.0 .01250 0.0 .01250 .00417 2.22 *
alpha2 (mm) 0.0 .00150 0.0 .00150 .00050 0.03 *
alpha3 (mm) 0.0 .00150 0.0 .00150 .00050 0.03 *
alpha4 (mm) 0.0 .00500 0.0 .00500 .00167 0.35 *
alpha5 (mm) 0.0 .00500 0.0 .00500 .00167 0.35 *
Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00
phi1 ([[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 non-fixed 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 user-specified 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
alpha1 (mm) 0.0 .01250 0.0 .01250 .00417 4.79 *
alpha2 (mm) 0.0 .00150 0.0 .00150 .00050 0.07 *
alpha3 (mm) 0.0 .00150 0.0 .00150 .00050 0.07 *
alpha4 (mm) 0.0 .00500 0.0 .00500 .00167 0.76 *
alpha5 (mm) 0.0 .00500 0.0 .00500 .00167 0.77 *
Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00
phi1 ([[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


PRO-E

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

CATIA

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