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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 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

• Due to the symmetry of the assembly, only one roller needs to be included in the analysis (i.e. a quarter-section is sufficient to perform the analysis).
• Roller radii C and D are not independent dimensions (i.e. if one is over-sized, the other is over-sized). They must be equivalenced for analysis.
• The roller is vendor-supplied, so the tolerance on its radius (C and D) is fixed.
• The revolute joint between the ring and the hub and the cylindrical DRF of the ring both introduce the same rotational degree of freedom into the model.

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 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

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 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 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 non-symmetric 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 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).

Tolerance Sensitivities

Table 2.4: -B-1A 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-1F Matrix

 1 2 3 4 5 6 B -8.12279 -8.12279 -8.18412 -8.18412 8.12279 1 q1 -0.20788 -0.20788 -0.20632 -0.20632 0.20788 0 q2 0.20788 0.20788 0.20632 0.20632 -0.20788 0

Remarks>> The magnitudes of the numbers in the columns of -B-1A and -B-1F 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 1.7107e-6 Lower 172.0000 4.48 3.80 3.8015e-6 Nom Dim 172.9816 Total 5.51 5.5121e-6

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 high-precision 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 1.2004e-5 76.66 C 1.9062e-6 12.17 1 7.5022e-7 4.79 E 7.3902e-7 4.72 Other 2.5973e-7 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 3.5357e-6 Lower 172.0000 4.33 7.48 7.4779e-6 Nom Dim 172.9816 Total 11.01 1.1014e-5

2.10 Tolerance Allocation

Weight Factor Tolerance Allocation

Weight Factor Tolerance Allocation adjusts dimension tolerances proportional to the user-assigned 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 1.1074e-3 Lower 172.0000 Target Sig 2.95 1592.50 1.5925e-3 Nom Dim 172.9816 3.00 Total 2699.93 2.6999e-3

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 1.1074e-3 Lower 172.0000 Target Sig 2.95 1592.50 1.5925e-3 Nom Dim 172.9816 3.00 Total 2699.93 2.6999e-3

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 1.1074e-3 Lower 172.0000 Target Sig 2.95 1592.50 1.5925e-3 Nom Dim 172.9816 3.00 Total 2699.93 2.6999e-3

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