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Figure 7.1: Schematic of the NFOV lens assembly with dimension variables.

7.0 Problem Description

The Narrow Field of Vision Lens (NFOV) assembly consists of four parts--two lenses, a housing, and a retaining ring. The assembly is symmetric about its center line. There are two assembly variables of interest. The first is the gap between the inside surfaces of Lens 2 and Lens 1. The second is the tilt of Lens 2 relative to Lens 1.

Table 7.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (+/-) Lens Thickness A 0.303 in 0.001 in Radius of Curvature B_C 10.7185 in 0.0107 in Retainer Lip Angle 84.8[[ring]] -- Retainer Flange Depth E 0.503 in 0.001 in Retainer Radius F 1.0275 in 0.0005 in Housing Length H 7.071 in 0.002 in Lens Depth I 0.459 in 0.001 in

7.1 Design Requirements

Table 7.2: Assembly Variables and Specification Limits.

 Variable Name Basic Size Upper Spec Limit (USL) Lower Spec Limit (LSL) Contact Length D 0.0563 in -- -- Lens/Retainer Gap G 0.2492 in -- -- Contact Angle [[phi]] 5.2000 [[ring]] -- -- X1 (GAP) 7.7792 7.7863 7.7737 Y1 0.0000 -- -- 1 0.0000 -- -- X2 7.7792 -- -- Y2 0.0000 -- -- 2 0.0000 .00075 (.0286[[ring]]) -.00075 (-.0286[[ring]])

7.2 Modeling Considerations

• * The assembly is symmetric about its center-line. Therefore it can be split in two, and only half of the assembly modeled. This reduces the complexity of the model without compromising the accuracy of the results.

• * The rotations of the lenses relative to each other are caused entirely by geometric surface variation. The TILT analysis will only be valid when geometric tolerances are applied. TI TOL will not allocate geometric tolerances, so no allocation will be performed based on the TILT specification.

• * In 3-D, the contact between, for example, the Housing and the Retainer, is a circle. In 2-D, the contact becomes planar (i.e. a circle viewed from the side appears to be a line). For this reason, the joints between the Retainer, Housing, and Lens 2 are all modeled as rigid joints (or as planar joints with the translational degree of freedom removed).

• * In order to solve for the variation in the gap between the inside surface of Lens 1 and the contact surface between the Retainer and the Housing, a planar joint must be placed on the center line of the assembly and used to join the Retainer and Lens 1. This introduces a translational kinematic degree of freedom into the assembly and allows TI TOL to solve for the variationbetween the inside surface of Lens 1 and the right side edge of the Retainer (i.e. the variation of G).

7.3 Design Goal

The objective of this analysis is to find the assembly variations of the GAP and the TILT. In addition, the component tolerances will be optimized to meet +/-6[[sigma]] assembly tolerances on the GAP.

7.4 Part DRFs And Feature Datums

Figure 7.2: Diagram showing the location of the part DRFs and feature datums.

Remarks>> Lens 1 has a cylindrical DRF and a cylindrical feature datum located on top of each other. The feature datum is necessary in order to introduce the correct rotational degree of freedom into the closed loop. The closed loop requires a rotation at the lens center of curvature, but that rotation should not be included in the open loops. Therefore when defining the path from the cylindrical slider joint back to the Lens 1 DRF, first take it to the cylindrical feature datum, and then to the DRF. When defining the path from the planar joint back to the Lens 1 DRF, do not include the cylindrical feature datum. Don't include the cylindrical feature datum when defining paths from the specification endpoints back to the Lens 1 DRF either. Finally, turn off the rotational degree of freedom of the Lens 1 DRF, but not that of the cylindrical feature datum. This will remove the rotational degree of freedom from the open loops but not the closed loop.

The DRFs for the Housing and Lens 2 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' functions to ensure the correct DRF is selected for each joint.

When manufacturing Lens 2, the flat contact surface is ground, and then the curvature of each lens surface is ground relative to it. Thus the DRF of Lens 2 is located on the plane formed by its contact surface.

7.5 Kinematic Joints

Four joints are required to model half of the NFOV lens assembly.

Figure 7.3: Kinematic joint diagram.

Table 7.3: Kinematic Joints of the NFOV Lens.

 Joint Number Part One Part Two Joint Type 1 Lens 1 Retainer Cylindrical Slider 2 Retainer Lens 1 Planar 3 Retainer Housing Rigid 4 Housing Lens 2 Rigid

Remarks>> The planar joint between the Retainer and Lens 1 may seem counter-intuitive. However, because we have split the lens assembly in half, one of the dependent length variables of interest is the variation in the distance between the contact surface of the retainer and housing and the left-most point of Lens 1. A planar joint introduces a degree of freedom in that direction into the model, and allows us to solve for the length variation G.

7.6 Network Diagram, Vector Loops, and Design Specifications

One loop is sufficient to describe the NFOV assembly. The vector loop follows dimensioned lengths and passes though each part DRF and each joint. A design specification has been applied to the gap between the inside surfaces of the two lenses, and another has been applied to the angular tilt between the two lenses.

Figure 7.4: Network diagram and loop diagram for the NFOV lens assembly.

Remarks> Only one open loop is shown in the diagram above because the open loops used to calculate the air gap variation and the tilt of the lenses are identical. However, TI TOL requires a separate open loop for each open loop design specification, so two identical loops must be created inside the modeler.

Within the open loops themselves, there are two sets of redundant vectors (A and B). These occur because TI TOL requires the loops to pass through the Lens 1 DRF before passing through the joint to the retainer. The analyzer does not automatically equivalence them, so the user must do it by hand in order to get valid results.

The characteristic length for the orientation specification is the diameter of Lens 2 (3.0 inches). Lens 2 is chosen because the specification is that Lens 2 be oriented relative to Lens 1 within a .0015 inch bandwidth.

For the gap specification, enter .001 and -.001 at the prompts. For the tilt (relative orientation) specification, enter .00075 and -.00075 at the prompts.

7.7 Geometric Tolerances

ANSI Y14.5 geometric tolerances are added to account for machined surface variations. They are applied to mating surfaces. Usually, one or two surface variations may be specified at each joint.

Figure 7.5: Geometric tolerance diagram.

Remarks>> The angular variations contributing to the tilt between the two lenses are caused entirely by geometric surface variations acting at the joints. The runout and perpendicularity at the rigid joints all cause rotations. In addition to those, the variation of the contact surface between the Retainer and Lens 1 may be lobed and cause an additional rotation. To model that rotation, an angularity, profile, or flatness geometric tolerance may be applied to the planar joint between the Retainer and Lens 1, and the diameter of the contact ring used as the characteristic length. This introduces an equivalent rotation into the assembly (it has not been done in this model).

7.8 Sensitivity Matrices

Table 7.4: -B-1A Matrix

 A B_C E F H I D 0.00000 -0.09101 0.00000 1.00000 0.00000 0.00000 G -1.00000 -0.00413 1.00000 0.09101 0.00000 0.00000 [[phi]] 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Table 7.5: -B-1F Matrix

 1 2 3 3 4 D -0.09101 -0.09101 0 0 0 G -1.00413 -1.00413 0 0 0 [[phi]] 0 0 0 0 0

-B-1F Matrix (continued)

 4 5 5 6 D 0 0 0 0 G 0 0 0 0 [[phi]] 0 0 0 0

Table 7.6: C-DB-1A Matrix

 A B_C E F H I X1 -1.00000 -0.00413 1.00000 0.09101 1.00000 1.00000 Y1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 X2 -1.00000 -0.00413 1.00000 0.09101 1.00000 1.00000 Y2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 2 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

Table 7.7: G-DB-1F Matrix

 1 2 3 3 4 X1 -1.00413 -1.00413 1 0 1 Y1 0 0 0 7.53 0 1 0 0 0 1 0 X2 -1.00413 -1.00413 1 0 1 Y2 0 0 0 7.53 0 2 0 0 0 1 0

G-DB-1F Matrix (continued)

 4 5 5 6 X1 0 1 0 0 Y1 7.53 0 0.459 0.459 1 1 0 1 1 X2 0 1 0 0 Y2 7.53 0 0.459 0.459 2 1 0 1 1

7.9 Predicted Assembly Variation

Table 7.8: Independent Variable Tolerances and Control Factors

 Dim Name Nominal +/- Tol Process Std Dev Process Description K A 0.3030 0.00100 0.00033 None 0.25 B_C 10.7185 0.01070 0.00357 None 0.25 E 0.5030 0.00100 0.00033 None 0.25 F 1.0275 0.00050 0.00017 None 0.25 H 7.0710 0.00200 0.00067 None 0.25 I 0.4590 0.00100 0.00033 None 0.25

Table 7.9: Kinematic Assembly Variables (Geometric Tolerances Not Applied)

 Variable Name Nominal WC +/- Assembly Variation (ZASM = 6.00) RSS SSA SSC D 0.0563 0.00148 0.00219 0.00219 0.00292 G 0.2492 0.00209 0.00283 0.00283 0.00378 5.2000 0.00000 0.00000 0.00000 0.00000

Table 7.10: RSS Percent Rejects (Geometric Tolerances Not Applied)

 Dep Len. U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.7863 0.00088 8.03 4.44e-10 0.0000 Lower 7.7737 6.25 2.05e-4 2.0478e-10 Nom Dim 7.7792 Total 2.05e-4 2.0478e-10

Table 7.11: Geometric Tolerances

 Name Part Name Type Joint Tolerance Band Char. Length 1 Lens 1 Circularity 1 0.00200 N/A 2 Retainer Runout 1 0.00040 N/A 3 Retainer Runout 3 0.00040 2.1 4 Housing Runout 3 0.00040 2.1 5 Housing Runout 4 0.00040 3.0 6 Lens 2 Perpendicularity 4 0.00040 3.0

Remarks>> The characteristic lengths are used to convert the geometric tolerance bandwidths to an equivalent rotational variation. The formula used to convert them is

+/-d[[beta]] = tan-1(bandwidth/characteristic length)

where +/-d[[beta]] is the equivalent angular variation in radians. The longer the characteristic length given by the user, the smaller the angular variation introduced into the model.

Table 7.12: Kinematic Assembly Variables (Geometric Tolerances Applied)

 Variable Name Nominal WC +/- Assembly Variation (ZASM = 6.00) RSS SSA SSC D 0.0563 0.00153 0.00219 0.00219 0.00292 G 0.2492 0.00269 0.00301 0.00301 0.00391 5.2000 0.00000 0.00000 0.00000 0.00000

Table 7.13: Normalized Sensitivities To GAP (Geometric Tolerances Applied)

 Variable Name Sensitivity Normalized 1 1.00413 11.03 2 1.00413 11.03 A 1.00000 10.98 E 1.00000 10.98 H 1.00000 10.98 I 1.00000 10.98 3 1.00000 10.98 4 1.00000 10.98 5 1.00000 10.98 other -- 1.04

Table 7.14: RSS Percent Rejects (Geometric Tolerances Applied)

 Dep Len. U2 Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.7863 0.00090 7.87 1.78e-9 0.0000 Lower 7.7737 6.12 4.56e-4 4.5641e-10 Nom Dim 7.7792 Total 4.56e-4 4.5641e-10

Table 7.15: Normalized Sensitivities To TILT (Geometric Tolerances Applied)

 Variable Name Sensitivity Normalized 3 1.00000 25.00 4 1.00000 25.00 5 1.00000 25.00 6 1.00000 25.00

Table 7.16: RSS Percent Rejects (Geometric Tolerances Applied)

 Orientation TILT Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper .00075 8.2203e-5 9.12 0.00 0.0000 Lower -.00075 9.12 0.00 0.0000 Nom Dim 0.0000 Total 0.00 0.0000

Remarks>> The TILT variation of the assembly will cause essentially no rejects. The only variations that contribute to the TILT are geometric tolerances. For this reason, no allocation can be performed on the assembly based on the TILT specification.

7.10 Tolerance Allocation

Weight Factor Tolerance Allocation

Table 7.17: RSS Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A N 1.00 0.00100 1.00 0.00098 1.00 0.00033 B_C N 1.00 0.01070 1.00 0.01049 1.00 0.00350 E N 1.00 0.00100 1.00 0.00098 1.00 0.00033 F N 1.00 0.00050 1.00 0.00049 1.00 0.00016 H N 1.00 0.00200 1.00 0.00020 1.00 0.00065 I N 1.00 0.00100 1.00 0.00098 1.00 0.00033
 Gap GAP Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.7863 0.00094 7.56 1.98e-8 1.9762e-14 Lower 7.7737 Target Sig 5.89 1.98e-3 1.9802e-9 Nom Dim 7.7792 6.00 Total 1.98e-3 1.9802e-9

Table 7.18: WC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A N 1.00 0.00100 0.00073 B_C N 1.00 0.01070 0.00780 E N 1.00 0.00100 0.00073 F N 1.00 0.00050 0.00036 H N 1.00 0.00200 0.00146 I N 1.00 0.00100 0.00073
 Gap GAP Spec Limit WC Variation Lower 7.7863 0.00551 Satisfied Lower 7.7737 Satisfied Nom Dim 7.7792

Remarks>> The Worst Case tolerance allocation option scales the component tolerances up or down until the (assembly mean) +/- (assembly variation) is equal to the specification limits if the mean is centered. If the mean is not centered, it scales the tolerances until either (assembly mean) + (assembly variation) equals the USL and the new minimum is larger than the LSL, or (assembly mean) - (assembly variation) equals the LSL and the new maximum is smaller than the USL.

Table 7.19: SSC Weight Factor Tolerance Allocation (Geometric Tolerances Applied).

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A N 1.00 0.00100 0.75 0.00074 0.75 0.00025 B_C N 1.00 0.01070 0.75 0.01049 0.75 0.00262 E N 1.00 0.00100 0.75 0.00074 0.75 0.00025 F N 1.00 0.00050 0.75 0.00037 0.75 0.00012 H N 1.00 0.00200 0.75 0.00147 0.75 0.00049 I N 1.00 0.00100 0.75 0.00074 0.75 0.00025
 Gap GAP Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 7.7863 0.00094 7.56 1.98e-8 1.9762e-14 Lower 7.7737 Target Sig 5.89 1.98e-3 1.9802e-9 Nom Dim 7.7792 6.00 Total 1.98e-3 1.9802e-9

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