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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 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

84.8║

.25║

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

174.8║

--

--

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 (.0286║)

-.00075 (-.0286║)

 

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 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

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

.25║

.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 (║)

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 (║)

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 (┌)

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

2.0e-8

Lower Tail

-9.66

0.0

-5.89

2.0e-3

 

Total Rejects

0.0

Total Rejects

2.0e-3

 

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 (┌)

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 (┌)

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).


PRO-E

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

AutoCAD

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

CATIA

Modeler: Crank Slider

 

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