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Example Problems 
AutoCAD
Verification:
NFOV Lens 

Home : Example Problems : AutoCad  Verification  NFOV Lens 
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 partstwo 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
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 UNIXbased 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 counterintuitive. 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 leftmost 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
B1A 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 
B1F 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 
B1F 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 
CDB1A 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 
GDB1F 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 
GDB1F 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 

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 
8.3e5 
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 
6SIG 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 = tan1(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 
6SIG 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 * 

a6 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
0.00 * 

Assem. Total 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

GAP (in) 
7.7792 
.00343 
7.7792 
.00562 
.00094 

Min./Max. 
7.7758 
7.7826 
7.7736 
7.7848 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 
12.41 
0.0 
7.56 
2.0e8 

Lower Tail 
9.66 
0.0 
5.89 
2.0e3 

Total Rejects 
0.0 
Total Rejects 
2.0e3 
Table 8.13:
WC Weight Factor Tolerance Allocation
(Geometric Tolerances Included).
Assembly Specs. 
Nom. 
USL 
LSL 
± ZASM 

Gap (in) 
7.7792 
7.7863 
7.7737 

Dimension 
Specified Values 
Allocated Values 

Name 
Nom. 
±Tol. 
Nom. 
±Tol. 
STDEV 
% Cont. 

A (in) 
0.3030 
.00100 
0.3030 
.00070 
12.61 

B_C (in) 
10.7185 
.01070 
10.7185 
.00744 
0.56 

g (Ú) 
84.80 
.25 
84.80 
.17378 
3.11 

E (in) 
0.5030 
.00100 
0.5030 
.00070 
12.61 

F (in) 
1.0275 
.00050 
1.0275 
.00035 
0.57 

H (in) 
7.0710 
.00200 
7.0710 
.00139 
25.21 

I (in) 
0.4590 
.00100 
0.4590 
.00070 
12.61 

a1 (in) 
0.0 
.00100 
0.0 
.00100 
18.21 * 

a2 (in) 
0.0 
.00020 
0.0 
.00020 
3.64 * 

a3 (in) 
0.0 
.00020 
0.0 
.00020 
3.63 * 

a4 (in) 
0.0 
.00020 
0.0 
.00020 
3.63 * 

a5 (in) 
0.0 
.00020 
0.0 
.00020 
3.63 * 

a6 (in) 
0.0 
.00020 
0.0 
.00020 
0.00 * 

Assem. Total 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

GAP (in) 
7.7792 
.00714 
7.7792 
.00551 

Min./Max. 
7.7721 
7.7864 
7.7737 
7.7847 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 

Lower Tail 

Total Rejects 
Total Rejects 
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.
Nominal Allocation
Table 8.14: RSS Nominal Allocation (Geometric Tolerances
Included).
Center Justified.
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 
.00100 
.00017 
8.52 

B_C (in) 
10.7185 
.01070 
10.7185 
.01070 
.00178 
0.02 

g (Ú) 
84.80 
.25 
84.80 
.25 
.04167 
0.52 

E (in) 
0.5030 
.00100 
0.5030 
.00100 
.00017 
8.52 

F (in) 
1.0275 
.00050 
1.0275 
.00050 
.00008 
0.02 

H (in) 
7.0710 
.00200 
7.0718 
.00200 
.00033 
34.07 

I (in) 
0.4590 
.00100 
0.4590 
.00100 
.00017 
8.52 

a1 (in) 
0.0 
.00100 
0.0 
.00100 
.00033 
34.36 * 

a2 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
1.37 * 

a3 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
1.36 * 

a4 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
1.36 * 

a5 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
1.36 * 

a6 (in) 
0.0 
.00020 
0.0 
.00020 
.00007 
0.00 * 

Assem. Total 
Nom. 
±Var. 
Nom. 
±Var. 
STDEV 
100.00 

GAP (in) 
7.7792 
.00343 
7.7800 
.00343 
.00057 

Min./Max. 
7.7758 
7.7826 
7.7766 
7.7834 
* Fixed Nom./Tol. 

Before Optimization 
After Optimization 

Rejects 
Z 
PPM 
Z 
PPM 

Upper Tail 
12.41 
0.0 
11.03 
0.0 

Lower Tail 
9.66 
0.0 
11.03 
0.0 

Total Rejects 
0.0 
Total Rejects 
0.0 
Remarks>> Centering the mean was accomplished by adjusting a single component dimension (H). All other component dimensions were held constant by assigning nominal weight factors of zero (see table 8.4).
PROE Modeler: Clutch
 Stack Blocks
 Remote Positioner 
AutoCAD Modeler: Clutch
 Stack
Blocks  Remote
Positioner 
CATIA Modeler: Crank Slider 
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