AutoCAD Verification Manual: NFOV Lens

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*Example Problems*	AutoCAD Verification: NFOV Lens
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NFOV LENS

Figure 8.1: Schematic of the NFOV lens assembly with dimension variables.

8.0 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 8.1: Manufactured Variables (Independent).

Variable Name	Basic Size	Initial Tolerance (±)
Lens Thickness A	.303 in	.001 in
Radius of Curvature B_C	10.7185 in	.0107 in
Retainer Lip Angle g	84.8º	.25º
Retainer Flange Depth E	.503 in	.001 in
Retainer Radius F	1.0275 in	.0005 in
Housing Length H	7.071 in	.002 in
Lens Depth I	.459 in	.001 in

8.1 Design Requirements

Table 8.2: Assembly Variables and Specification Limits.

Variable Name	Basic Size	Upper Spec. Limit (USL)	Lower Spec. Limit (LSL)
Contact Length D	.0563 in	--	--
Lens/Retainer Gap G	.2492 in	--	--
Contact Angle f	174.8º	--	--
X1 (GAP)	7.77921	7.7863	7.7737
DY1	0.000	--	--
Dq1	0.000	--	--
DX2	7.77921	--	--
DY2	0.000	--	--
Dq2 (TILT)	0.000	.00075 (.0286º)	-.00075 (-.0286º)

8.2 Modeling Considerations

The assembly is symmetric about its center-line so it can be split in two, and only half of the assembly modeled. This cuts down on 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. CATS 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 CATS to solve for the gap variation.

8.3 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 ±6s assembly tolerances on the GAP. Finally, nominal allocation will be used to adjust the component nominals to center the GAP within its specification limits.

8.4 Part Names and DRFs

Figure 8.2: Diagram showing the location of the part DRFs.

Remarks>> 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 a zoom function (in AutoCATS) or query function (in some of the UNIX-based systems) 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.

8.5 Kinematic Joints

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

Figure 8.3: Kinematic joint diagram.

Table 8.3: 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.

8.6 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 8.4: 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, CATS 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 CATS requires the loops to pass through the Lens_1 DRF before passing through the joint to the retainer. The analyzer finds these redundant vectors and automatically equivalences them, which adds their constraint sensitivities to zero and prevents their variations from influencing the gap and orientation calculations. If for some reason the analyzer does not find and equivalence those redundant vectors, the user must do it by hand in the analyzer 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.

8.7 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 8.5: 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).

8.8 Sensitivity Matrices
Constraint Sensitivities

A Matrix

	A	B_C	g	E	F	H	I
X	-1.0000	0.00412	-1.0275	1.0000	0.00000	0.00000	0.00000
Y	0.00000	0.09063	-0.25379	0.00000	-1.0000	0.00000	0.00000
q	0.00000	0.00000	-1.0000	0.00000	0.00000	0.00000	0.00000

B Matrix

	D	G	f
X	0.09063	-1.0000	0.00000
Y	0.99588	0.00000	-10.416
q	0.00000	0.00000	1.0000

F Matrix

	a1	a2	a3	b3	a4
X	-0.99588	-0.99588	0.00000	0.00000	0.00000
Y	0.09063	0.09063	0.00000	0.00000	0.00000
q	0.00000	0.00000	0.00000	0.00000	0.00000

F Matrix (continued)

	b4	a5	b5	b6
X	0.00000	0.00000	0.00000	0.00000
Y	0.00000	0.00000	0.00000	0.00000
q	0.00000	0.00000	0.00000	0.00000

C Matrix

	A	B_C	g	E	F	H	I
X1	0.00000	0.00000	0.00000	0.00000	0.00000	1.0000	1.0000
Y1	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
q1	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
X2	0.00000	0.00000	0.00000	0.00000	0.00000	1.0000	1.0000
Y2	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
q2	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000

D Matrix

	D	G	f
X1	0.00000	1.0000	0.00000
Y1	0.00000	0.00000	0.00000
q1	0.00000	0.00000	0.00000
X2	0.00000	1.0000	0.00000
Y2	0.00000	0.00000	0.00000
q2	0.00000	0.00000	0.00000

G Matrix

	a1	a2	a3	b3	a4
X1	0.00000	0.00000	1.0000	0.00000	1.0000
Y1	0.00000	0.00000	0.00000	7.5300	0.00000
q1	0.00000	0.00000	0.00000	1.0000	0.00000
X2	0.00000	0.00000	1.0000	0.00000	1.0000
Y2	0.00000	0.00000	0.00000	7.5300	0.00000
q2	0.00000	0.00000	0.00000	1.0000	0.00000

G Matrix (continued)

	b4	a5	b5	b6
X1	0.00000	1.0000	0.00000	0.00000
Y1	7.5300	0.00000	0.45900	0.45900
q1	1.0000	0.00000	1.0000	1.0000
X2	0.00000	1.0000	0.00000	0.00000
Y2	7.5300	0.00000	0.45900	0.45900
q2	1.0000	0.00000	1.0000	1.0000

Tolerance Sensitivities

-B-1A Matrix

	A	B_C	g	E	F	H	I
D	0.00000	-0.09101	10.713	0.00000	1.0041	0.00000	0.00000
G	-1.0000	-0.00413	-0.05652	1.0000	0.09101	0.00000	0.00000
f	0.00000	0.00000	1.0000	0.00000	0.00000	0.00000	0.00000

-B-1F Matrix

	a1	a2	a3	b3	a4
D	-0.09101	-0.09101	0.00000	0.00000	0.00000
G	-1.0041	-1.0041	0.00000	0.00000	0.00000
f	0.00000	0.00000	0.00000	0.00000	0.00000

-B-1F Matrix (continued)

	b4	a5	b5	b6
D	0.00000	0.00000	0.00000	0.00000
G	0.00000	0.00000	0.00000	0.00000
f	0.00000	0.00000	0.00000	0.00000

C-DB-1A Matrix

	A	B_C	g	E	F	H	I
"X1	-1.0000	-0.00413	-0.05652	1.0000	0.09101	1.0000	1.0000
"Y1	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
"q1	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
"X2	-1.0000	-0.00413	-0.05652	1.0000	0.09101	1.0000	1.0000
"Y2	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
"q2	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000

G-DB-1F Matrix

	a1	a2	a3	b3	a4
"X1	-1.0041	-1.0041	1.0000	0.00000	1.0000
"Y1	0.00000	0.00000	0.00000	7.5300	0.00000
"q1	0.00000	0.00000	0.00000	1.0000	0.00000
"X2	-1.0041	-1.0041	1.0000	0.00000	1.0000
"Y2	0.00000	0.00000	0.00000	7.5300	0.00000
"q2	0.00000	0.00000	0.00000	1.0000	0.00000

G-DB-1F Matrix (continued)

	b4	a5	b5	b6
"X1	0.00000	1.0000	0.00000	0.00000
"Y1	7.5300	0.00000	0.45900	0.45900
"q1	1.0000	0.00000	1.0000	1.0000
"X2	0.00000	1.0000	0.00000	0.00000
"Y2	7.5300	0.00000	0.45900	0.45900
"q2	1.0000	0.00000	1.0000	1.0000

8.9 Resultant Tolerances Before Optimization

Table 8.4: Independent Variable Tolerances and Control Factors

Dim. Name	± Tol.	Std. Dev.	Cp	Dk	Cpk	Sk	Wt. Factor
Dim. Name	± Tol.	Std. Dev.	Cp	Dk	Cpk	Sk	Tol.	Basic	Fixed
A	.001 in	.00017	2.0	.25	1.5	0	1	0	No
B_C	.0107 in	.00178	2.0	.25	1.5	0	1	0	No
g	.25º	.04167	2.0	.25	1.5	0	1	0	No
E	.001 in	.00017	2.0	.25	1.5	0	1	0	No
F	.0005 in	8.3e-5	2.0	.25	1.5	0	1	0	No
H	.002 in	.00033	2.0	.25	1.5	0	1	1	No
I	.001 in	.00017	2.0	.25	1.5	0	1	0	No

Remarks>> A process capability (Cp) of 2.0 is equivalent to ±6s component tolerances (tolerance = 3Cp*s).

Table 8.5: Kinematic Assembly Variables (No Geometric Tolerances)

Variable	Degree of	Tolerances (ZASM = 6.000)
Name	Freedom	Worst Case	RSS Case	6-SIG Case
D	Translation (in)	0.04822	0.04676	0.06234
G	Translation (in)	0.00234	0.00144	0.00192
f	Rotation (º)	0.25000	0.25000	0.33333

Table 8.6: Geometric Tolerances

Feat.	Joint	Part Name	Feature Type	Tolerance Band	Char. Length
a1	1	Lens_1	Circularity	.002	N/A
a2	1	Retainer	Runout	.0004	N/A
a3	3	Retainer	Runout	.0004	2.1
a4	3	Housing	Runout	.0004	2.1
a5	4	Housing	Runout	.0004	3.0
a6	4	Lens_2	Perpendicularity	.0004	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

±db = tan-1(bandwidth/characteristic length)

where ±db 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 8.7: Kinematic Assembly Variables (Geometric Tolerances Included)

Variable	Degree of	± Assembly Variation (ZASM = 6.000)
Name	Freedom	Worst Case	RSS Case	6-SIG Case
D	Translation (in)	0.04833	0.04676	0.06234
G	Translation (in)	0.00354	0.00250	0.00280
f	Rotation (º)	0.25000	0.25000	0.33333

Table 8.8: Normalized Sensitivities To GAP (Geometric Tolerances Included)

Variable Name	Sensitivity	Normalized Sensitivity
a1	-1.0041				10.96
a2	-1.0041				10.96
A	-1.0000			10.92
E	1.0000			10.92
H	1.0000			10.92
I	1.0000			10.92
a3	1.0000			10.92
a4	1.0000			10.92
a5	1.0000			10.92
other	--		1.64

Table 8.9: Normalized Sensitivities To TILT (Geometric Tolerances Included)

Variable Name	Sensitivity	Normalized Sensitivity
a3	1.0000		25.00
a4	1.0000		25.00
a5	1.0000		25.00
a6	1.0000		25.00

Remarks>> 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.

Table 8.10: RSS Percent Rejects

Spec. Name	Spec. Type	Nominal Dimension	(±) Computed Variation	With Geometric Tolerances		Without Geometric Tolerances
GAP	Gap	7.77921	.00343	Z	Rej.	Z	Rej.
ZASM = 6.000		USL	Upper Tail	12.41	0.0	15.99	0.0
(Rejects in PPM)		LSL	Lower Tail	-9.66	0.0	-12.45	0.0

Table 8.11: RSS Percent Rejects

Spec. Name	Spec. Type	Nominal Dimension	(±) Computed Variation	With Geometric Tolerances		Without Geometric Tolerances
TILT	Orientation	0.00000	.00099	Z	Rej.	Z	Rej.
ZASM = 6.000		USL .00075	Upper Tail	4.56	2.5	N/A	N/A
(Rejects in PPM)		LSL-.00075	Lower Tail	-4.56	2.5	N/A	N/A

8.10 Nominals And Tolerances After Optimization

Weight Factor Tolerance Allocation

Table 8.12: RSS Weight Factor Tolerance Allocation
(Geometric Tolerances Included).

Assembly Specs.			Nom.		USL		LSL		± ZASM
Gap (in)			7.7792		7.7863		7.7737		6.000
Dimension	Specified Values				Allocated Values
Name	Nom.	±Tol.		Nom.		±Tol.		STDEV		% Cont.
A (in)	0.3030	.00100		0.3030		.00195		.00033		12.06
B_C (in)	10.7185	.01070		10.7185		.02089		.00348		0.02
g (Ú)	84.80	.25		84.80		.48799		.00142		0.73
E (in)	0.5030	.00100		0.5030		.00195		.00033		12.06
F (in)	1.0275	.00050		1.0275		.00098		.00016		0.02
H (in)	7.0710	.00200		7.0710		.00390		.00065		48.24
I (in)	0.4590	.00100		0.4590		.00195		.00033		12.06
a1 (in)	0.0	.00100		0.0		.00100		.00033		12.77 *
a2 (in)	0.0	.00020		0.0		.00020		.00007		0.51 *
a3 (in)	0.0	.00020		0.0		.00020		.00007		0.51 *
a4 (in)	0.0	.00020		0.0		.00020		.00007		0.51 *
a5 (in)	0.0	.00020		0.0		.00020		.00007		0.51 *