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Figure 4.1: Schematic of the remote positioner with dimension variables.

4.0 Problem Description

The remote positioner is a mechanical linkage that positions point P in two-dimensional space. Point P is meant to remain in a fixed location while the angular orientation of Part 5 varies. This model demonstrates the use of two open loops with two closed loop constraints.

This model includes 10 independent variables, and solves for six kinematic variables. In addition, two open loops allow us to solve for additional assembly variations.

Table 4.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (+/-) 1 90.00[[ring]] -- A 22.000 in .005 in B 10.400 in .005 in C 22.000 in .005 in D 5.200 in .003 in E 9.0067 in .004 in 2 30.00[[ring]] -- 3 42.60[[ring]] -- F 12.900 in .005 in G 49.300 in .010 in H 12.900 in .005 in I 49.300 in .010 in 4 42.60[[ring]] -- J 22.000 in .005 in

4.1 Design Requirements

Table 4.2: Assembly Variables (Dependent).

 Variable Name Basic Size Upper Spec. Limit(USL) Lower Spec. Limit(LSL) q1 120.00[[ring]] -- -- q2 60.00[[ring]] -- -- q3 0.00[[ring]] -- -- q4 132.60[[ring]] -- -- q5 47.40[[ring]] -- -- q6 132.60[[ring]] -- -- X1 0.00 in .10 in -.10 in Y1 0.00 in .10 in -.10 in 1 0.00[[ring]] -- -- X2 0.00 in -- -- Y2 0.00 in -- -- 2 0.00[[ring]] .05 in (.26[[ring]]) -.05 in (-.26[[ring]])

Remarks>> [[Delta]]X1 and [[Delta]]Y1 are the Cartesian coordinate locations of point P relative to Ground and are used to calculate the position variation. [[Delta]][[theta]]2 is the variation in the angular orientation of Part 5 relative to Part 1 and is used to calculate the parallelism variation. [[Delta]][[theta]]1, [[Delta]]X2, and [[Delta]]Y2 can also be solved for from the open loops, but they are not necessary to estimate the parallelism and position assembly variations.

4.2 Modeling Considerations

• When kinematic angles are shared between two or more closed loops, the placement of the Datum Reference Frames (DRFs) determines which vectors are used to define the dependent angles. For example, placing the DRF for Part 2 at the joint between A and B, and the DRF for Part 3 at the joint between C and D tells the analyzer that [[phi]]2 is the angle formed by B and C. Changing the Part 3 DRF to the joint between F and G tells the analyzer that [[phi]]2 is the angle between B and F. This distinction becomes important in cases when design specifications are applied to dependent angles and when open loops are used in the model. The joint connecting Part 2 and Part 5 has the same kind of situation. Locating the DRF for Part 5 at the joint between it and Part 4 tells the analyzer that [[phi]]6 is formed by H and I.

• In order to create the most robust model, avoid placing part DRFs at joints with manufactured angle variations, or where two loops share a common dependent angle. For example, if Part 2 uses the point between B and I as its DRF in manufacturing, for modeling purposes it's better to locate its DRF at another joint and put a feature datum between B and I. The same applies to the joint between Part 3 and Part 5.

• TI TOL does not have a direct method to account for loose clearance fits in revolute joints. In order to model the variation caused by the gap between the pin and the hole of a revolute joint, a true position geometric tolerance of the same magnitude as the expected clearance is applied to the joint.

4.3 Design Goal

The first objective of this problem is to calculate the variation in the position of point P relative to Ground and the variation in the parallelism between Part 1 and Part 5. The second objective is to re-allocate the dimension tolerances to meet the parallelism specification limits.

4.4 Part DRFs And Feature Datums

Figure 4.2: Diagram showing the location of the part DRFs and feature datums.

Remarks>> Note that their are four feature datums on the assembly. They will be used to locate joints and specification endpoints relative to part DRFs.

4.5 Kinematic Joints

Seven joints are required to model the remote positioner. All of them are revolute joints in the physical device. However, with all joints free to rotate, the system is indeterminate, and TI TOL cannot solve for the variations. Therefore, one joint must be designated the input angle and its rotational degree of freedom removed. This is done by either replacing that revolute joint with a rigid joint or by "turning off" that joint's rotational degree of freedom . For this problem the input angle was at joint 1, so it was modeled as a rigid joint.

Figure 4.3: Kinematic joint diagram.

Table 4.3: Kinematic Joints of the Remote Positioner.

 Joint Number Part One Part Two Joint Type 1 Ground Part 1 rigid 2 Part 1 Part 2 revolute 3 Part 2 Part 3 revolute 4 Part 3 Ground revolute 5 Part 3 Part 4 revolute 6 Part 4 Part 5 revolute 7 Part 5 Part 2 revolute

4.6 Network Diagram, Vector Loops, and Design Specifications

Two closed loops are necessary to constrain the remote positioner assembly. A location specification relative to joint 1 has been applied to point P, as well as a parallelism specification relative to length A (Part 1). Therefore two open loops are needed, one for each design specification.

Figure 4.4: Network diagram and open and closed loops for the remote positioner.

Remarks>> Open loops are analyzed in the same manner as closed loops. They are more sensitive to modeling errors than closed loops are, so correct placement of loop endpoints and part DRFs is critical when calculating variations with open loops.

The direction of open loops is important when gap and position specifications are used. TI TOL assumes the first part is fixed in space and the parts "downstream" all rotate relative to it. This arises due to the non-commutative property of matrix multiplication. To generate the correct open loop direction, create the first endpoint (the fixed endpoint) first and the final endpoint (moving endpoint) second.

In the course of creating the loops shown, there are three sets of redundant vectors (vectors that double back on themselves) created. These vectors must be identified and equivalenced in order for the analysis to be valid. Use the `query sel' or `menu sel' options to select the correct vectors to equivalence.

The allowable position specification variation is given as a +/- radial tolerance.

4.7 Geometric Tolerances

True position geometric tolerances have been applied to the seven joints to account for clearance variations. Each position tolerance is modeled as two orthogonal, independent vectors.

Figure 4.5: Geometric tolerance diagram.

Remarks>> Applying position tolerances to the joints in this assembly is not completely accurate. In this case, the position tolerance is not related to the position of the holes (or pins). Instead, it is being used as a way to approximate the variations that occur in the assembly due to the small clearances between the pins and holes.

4.8 Sensitivity Matrices

Table 4.4: -B-1A Matrix

 A B C D E q1 0.11103 0.05551 -0.11103 -0.11103 2.52E-18 q2 -0.08479 -0.00303 0.08479 0.08479 -0.04545 q3 -0.02624 -0.05249 0.02624 0.02624 0.04545 q4 0.08479 0.00303 -0.08479 -0.08479 0.04545 q5 -0.08479 -0.00303 0.08479 0.08479 -0.04545 q6 0.08479 0.00303 -0.08479 -0.08479 0.04545

-B-1A Matrix (continued)

 F G H I J q1 0.00000 0.00000 0.00000 0.00000 0.00000 q2 0.00000 0.00000 0.00000 0.00000 0.00000 q3 0.00000 0.00000 0.00000 0.00000 0.00000 q4 0.02756 0.01865 -0.02756 -0.01865 0.00000 q5 0.04373 0.08666 0.04373 -0.08666 0.00000 q6 -0.07128 -0.10531 0.07128 0.10531 0.00000

Table 4.5: -B-1F Matrix

 1 1 2 2 3 3 4 q1 -0.11103 2.52e-18 -0.11103 2.52e-18 -0.11103 2.52e-18 -0.11103 q2 0.08479 -0.04545 0.08479 -0.04545 0.08479 -0.04545 0.08479 q3 0.02624 0.04545 0.02624 0.04545 0.02624 0.04545 0.02624 q4 -0.08479 0.04545 -0.08479 0.04545 -0.10507 0.0268 -0.08479 q5 0.08479 -0.04545 0.08479 -0.04545 0.10507 -0.13211 0.08479 q6 -0.08479 0.04545 -0.08479 0.04545 -0.08479 0.15077 -0.08479

-B-1F Matrix (continued)

 4 5 5 6 6 7 7 q1 2.52e-18 0 0 0 0 0 0 q2 -0.04545 0 0 0 0 0 0 q3 0.04545 0 0 0 0 0 0 q4 0.04545 -0.02028 -0.01865 -0.02028 -0.01865 -0.02028 -0.01865 q5 -0.04545 0.02028 -0.08666 0.02028 -0.08666 0.02028 -0.08666 q6 0.04545 1.8e-17 0.10531 1.8e-17 0.10531 1.8e-17 0.10531

Table 4.6: C-DB-1A Matrix

 A B C D E X1 5.47372 2.73686 -6.47372 -6.47372 1.47E-16 Y1 2.22E-16 -5.6E-16 -2.2E-16 -2.2E-16 -1.00000 1 0.02624 0.05249 -0.02624 -0.02624 -0.04545 X2 5.47372 2.73686 -6.47372 -6.47372 1.47E-16 Y2 2.22E-16 -5.6E-16 -2.2E-16 -2.2E-16 -1.00000 2 0.02624 0.05249 -0.02624 -0.02624 -0.04545

C-DB-1A Matrix (continued)

 F G H I J X1 0.00000 0.00000 0.00000 0.00000 1.00000 Y1 1.56822 2.31685 -1.56822 -3.31685 0.00000 1 0.07128 0.10531 -0.07128 -0.10531 0.00000 X2 0.00000 0.00000 0.00000 0.00000 1.00000 Y2 1.56822 2.31685 -1.56822 -3.31685 0.00000 2 0.07128 0.10531 -0.07128 -0.10531 0.00000

Table 4.7: G-DB-1F Matrix

 1 1 2 2 3 3 4 X1 -5.47372 1.47e-16 -5.47372 1.47e-16 -6.47372 1.47e-16 -6.47372 Y1 -2.2e-16 8.88e-16 -2.2e-16 8.88e-16 -4.4e-16 -3.31685 -2.2e-16 1 -0.02624 -0.04545 -0.02624 -0.04545 -0.02624 -0.15077 -0.02624 X2 -5.47372 1.47e-16 -5.47372 1.47e-16 -6.47372 1.47e-16 -6.47372 Y2 -2.2e-16 8.88e-16 -2.2e-16 8.88e-16 -4.4e-16 -3.31685 -2.2e-16 2 -0.02624 -0.04545 -0.02624 -0.04545 -0.02624 -0.15077 -0.02624

G-DB-1F Matrix (continued)

 4 5 5 6 6 7 7 X1 1.47e-16 0 0 0 0 -1 0 Y1 -1 -4e-16 -2.3168 -4e-16 -2.31685 -4e-16 -3.31685 1 -0.04545 -1.8e-17 -0.10531 -1.8e-17 -0.10531 -1.8e-17 -0.10531 X2 1.47e-16 0 0 0 0 -1 0 Y2 -1 -4e-16 -2.3168 -4e-16 -2.31685 -4e-16 -3.31685 2 -0.04545 -1.8e-17 -0.10531 -1.8e-17 -0.10531 -1.8e-17 -0.10531

4.9 Predicted Assembly Variation

Table 4.8: Independent Variable Tolerances and Control Factors

 Dim Name Nominal +/- Tol Process Std Dev Process Description K A 22.0000 0.00500 0.00167 None 0.25 B 10.4000 0.00500 0.00167 None 0.25 C 22.0000 0.00500 0.00167 None 0.25 D 5.2000 0.00300 0.00100 None 0.25 E 9.0067 0.00400 0.00133 None 0.25 F 12.9000 0.00500 0.00167 None 0.25 G 49.3000 0.01000 0.00333 None 0.25 H 12.9000 0.00500 0.00167 None 0.25 I 49.3000 0.01000 0.00333 None 0.25 J 22.0000 0.00500 0.00167 None 0.25

Table 4.9: Kinematic Assembly Variables (Geometric Tolerances Not Applied)

 Variable Name Nominal WC +/- Assembly Variation (ZASM = 4.50) RSS SSA SSC q1 120.0000 0.09860 0.07708 0.07708 0.10277 q2 60.0000 0.07444 0.05813 0.05813 0.07750 q3 0.0000 0.04500 0.03245 0.03245 0.04327 q4 132.6000 0.11160 0.06460 0.06460 0.08613 q5 47.4000 0.19879 0.12320 0.12320 0.16427 q6 132.6000 0.23596 0.14710 0.14710 0.19613

Table 4.10: SSC Percent Contributions To Parallelism Of P

(No Geometric Tolerances)

 Variable Name Contribution SSC G 2.1907e-7 42.42 I 2.1907e-7 42.42 F 2.5092e-8 4.86 H 2.5092e-8 4.86 B 1.3604e-8 2.64 E 6.5299e-9 1.26 other 8.0264e-9 1.54

Table 4.11: Sensitivities To Parallelism Of P (No Geometric Tolerances)

 Variable Name Sensitivity Normalized G 0.10531 19.88 I 0.10531 19.88 F 0.07128 13.45 H 0.07128 13.45 B 0.05249 9.91 E 0.04545 8.58 A 0.02624 4.95 C 0.02624 4.95 D 0.02624 4.95 J 0.00000 0.00

Table 4.12: SSC Percent Rejects (Geometric Tolerances Not Applied)

 Parallelism Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.0500 0.00791 6.32 1.27e-4 1.2735e-10 Lower -0.0500 6.32 1.27e-4 1.2735e-10 Nom Dim 0.0000 Total 2.55e-4 2.5471e-10

Table 4.13: Geometric Tolerances

 Name Part Name Type Joint Tolerance Band Char. Length 1 Part 1 True Position 1 0.00100 N/A 2 Part 1 True Position 2 0.00100 N/A 3 Part 3 True Position 3 0.00100 N/A 4 Part 3 True Position 4 0.00100 N/A 5 Part 4 True Position 5 0.00100 N/A 6 Part 4 True Position 6 0.00100 N/A 7 Part 5 True Position 7 0.00100

Table 4.14: Kinematic Assembly Variables (Geometric Tolerances Applied)

 Variable +/- Assembly Variation (ZASM = 4.50) Name Nominal WC RSS SSA SSC q1 120.0000 0.11133 0.07767 0.07767 0.10322 q2 60.0000 0.08936 0.05871 0.05871 0.07794 q3 0.0000 0.05322 0.03276 0.03276 0.04350 q4 132.6000 0.12992 0.06519 0.06519 0.08658 q5 47.4000 0.22597 0.12380 0.12380 0.16472 q6 132.6000 0.26295 0.14767 0.14767 0.19656

Remarks>> The variation in the assembly due to the small gaps in the pin joints is insignificant. For tolerance allocation, the effects of the gaps (represented by the true position geometric tolerances) will be ignored.

Table 4.15: SSC Percent Rejects (Geometric Tolerances Applied)

 Parallelism Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.0500 0.00792 6.31 1.37e-4 1.3676e-10 Lower - 0.0500 6.31 1.37e-4 1.3676e-10 Nom Dim 0.0000 Total 2.74e-4 2.7352e-10

Table 4.16: SSC Percent Rejects (Geometric Tolerances Applied)

 True Position Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.1000 0.02426 4.12 18.76 1.8732e-5 Lower -0.1000 4.12 18.76 1.8732e-5 Nom Dim 0.0000 Total 37.53 3.7526e-5

4.10 Tolerance Allocation

Weight Factor Tolerance Allocation

The Six-Sigma Component Drift (SSC) analysis model assumes that the individual component dimension nominals gradually shift due to tool wear and other manufacturing considerations. The net result of these shifts is an increase in the process standard deviations for each dimension. This increase in the standard deviations is accounted for in the analysis by reducing the process capability index (Cp). The SSC model predicts a higher reject fraction than the RSS model does.

Table 4.17: SSC Weight Factor Tolerance Allocation

(Geometric Tolerances Not Applied)

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A N 3.00 0.00500 0.75 0.00559 0.75 0.00186 B N 2.00 0.00500 0.75 0.00372 0.75 0.00124 C N 3.00 0.00500 0.75 0.00559 0.75 0.00186 D N 2.00 0.00300 0.75 0.00223 0.75 0.00074 E N 2.00 0.00400 0.75 0.00298 0.75 0.00099 F N 2.00 0.00500 0.75 0.00372 0.75 0.00124 G N 4.00 0.01000 0.75 0.01490 0.75 0.00497 H N 2.00 0.00500 0.75 0.00372 0.75 0.00124 I N 4.00 0.01000 0.75 0.01490 0.75 0.00497 J N 3.00 0.00500 0.75 0.00559 0.75 0.00186
 Parallelism Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.0500 .01111 4.50 3.40 3.4008e-6 Lower - 0.0500 Target Sig 4.50 3.40 3.4008e-6 Nom Dim 0.0000 4.50 Total 6.80 6.8020e-6

The Six-Sigma Assembly Drift model is similar to the SSC model in that it assumes that the component nominals drift from their ideal condition. Instead of increasing the component standard deviations to account for this variation, the SSA model calculates the assembly variations using the basic RSS scheme and then increases the assembly variable standard deviation by subtracting 1.5 from the specification Z-scores.

Table 4.18: SSA Weight Factor Tolerance Allocation

(Geometric Tolerances Not Applied)

 Dim Name Fix WF Original Allocated +/- Tol Cp +/- Tol Cp Std Dev A N 3.00 0.00500 1.00 0.00559 1.00 0.00186 B N 2.00 0.00500 1.00 0.00372 1.00 0.00124 C N 3.00 0.00500 1.00 0.00559 1.00 0.00186 D N 2.00 0.00300 1.00 0.00223 1.00 0.00074 E N 2.00 0.00400 1.00 0.00298 1.00 0.00099 F N 2.00 0.00500 1.00 0.00372 1.00 0.00124 G N 4.00 0.01000 1.00 0.01490 1.00 0.00497 H N 2.00 0.00500 1.00 0.00372 1.00 0.00124 I N 4.00 0.01000 1.00 0.01490 1.00 0.00497 J N 3.00 0.00500 1.00 0.00559 1.00 0.00186
 Parallelism Spec Limit Assy Std Dev Assy Sigma Rejects PPM Rejects DPU Upper 0.0500 0.00833 4.50 3.40 3.4008e-6 Lower - 0.0500 Target Sig 4.50 3.40 3.4008e-6 Nom Dim 0.0000 4.50 Total 6.80 6.8020e-6

Remarks>> For this example problem, the allocated tolerances for the SSC model and the SSA model are identical. Usually, this is not the case.

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