1. Tolerance Allocation to Optimize Process Capability

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Elmer Pitts
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It is called a scroll compressor because it operates by the action of two spiral-shaped rotors called scrolls.

1 1. Tolerance Allocation to Optimize Process Capability by Andrew M. Terry 1 A. Background The product considered in this example is part of an industrial air conditioning system compressor made by Carrier Corp. It is called a scroll compressor because it operates by the action of two spiral-shaped rotors called scrolls. The compressor operates at very high pressure and has running clearances on the order of a few microns. The part whose tolerances are analyzed is called an unloader piston. It is purchased from a supplier and fits into a bore in the compressor case. It can be in either of two states: advanced or retracted. Only the advanced position is of interest to us. The KC to be studied is the location of the end surface of the unloader piston relative to the cylindrical surface of the rotor bore when the piston is advanced. If the piston is positioned too far from the rotor bore surface, due to an error in making the unloader or its bore, there will be an escape path from a high-pressure chamber to the following helical chamber at a lower pressure. This leakage path reduces the compressor s efficiency. If it is positioned too close, it could collide with the rotor, destroying the compressor. In the past, these parts had been made under worst case tolerancing, without statistical process control. Scrap and rework were unacceptably high. This case study involved converting the supplier and Carrier to statistical tolerancing, SPC, and optimum tolerance allocation. A twodimensional analysis was performed. Figure 1-1 is a cutaway drawing of the compressor, showing two unloader pistons and the two positions they can occupy. 1 Based on Terry, Andrew M., Improving Product Manufacturability Through The Integrated Use Of Statistics, MIT Master of Science and Master of Business Administration Thesis, June, 2000.

2 Figure 1-1 Unloader Pistons in a Scroll Compressor. Left: Two unloader pistons at different locations along the rotor allows the compression chamber to be shortened to conserve energy. Right: The upper piston is in the advanced position, while the lower one is in the retracted position. Advancing both pistons provides 100% of capacity, retracting 1 piston provides 70% of capacity, and retracting both pistons provides 40% of capacity. The relationship between the end of the piston and the rotor is defined by a manageable set of five subsidiary characteristics, of which two are provided by the supplier. Because the surface being considered is three-dimensional, we must decide how to map this to a two-dimensional error model. One approach is to calculate the average perpendicular distance from the cylindrical surface of the rotor bore to the surface of the unloader piston. This approach gives a very accurate representation of the effect of different clearances on performance. However, calculating the risk of interference is not possible from an average position. Instead, it is necessary to analyze the risk point on the end surface of the piston that protrudes furthest into the bore. A star denotes the two possible locations of the risk point in Figure 1-2. Using the risk point as a proxy for the x- direction location of the surface ensures that there will never be a collision with the rotor. Figure 1-2 shows the end surface of the piston, created from the intersection of two cylinders, along with the eccentric diameter of the piston. Piston End Surface Piston Eccentric Piston Figure 1-2 The Unloader Piston. The left frame shows a male and female piston side by side. A close-up of the end surface of the piston is shown in the center frame. Also shown in the second frame are the risk points of the piston. The right frame shows a side view of a female piston. Because the male piston is shorter, the L/D ratio of the piston is smaller, making it the more restrictive constraint. All analysis and calculations were performed using the dimensions and capabilities from the male piston and extrapolated for use on the female characteristics.

3 B. Design Requirements The function of the piston is to alternately complete the seal with the rotor bore in the rotor case and to create a short circuit for the refrigerant, effectively shortening the compression chamber. In order to fulfill this requirement, the end surface of the piston must align as closely as possible with the surface of the rotor bore. As the clearance between the piston and the surface of the cylinder bore increases, performance of the compressor degrades. The functional design limits for the piston are the following: The piston must fit as closely as possible into the bore in the rotor case without interference. The end surface of the piston must align as closely as possible with the surface of the rotor bore without protruding into it. Both of these requirements are important to the compressor performance, but the second condition is the subject of this case study. C. Geometric Analysis and Relationship to Performance The surface of the piston must be as close as possible to the rotor bore to maximize performance. This requirement can be checked only by summing the individual dimensions to calculate the location of the surface. This case study focuses on assigning tolerances to individual dimensions on the rotor case and the unloader piston to ensure functionality and performance. Figure?1 3 below contains a schematic with the dimensions that contribute the most error to the position of the piston surface.

4 X 1 Unloader Piston X 2 Y Rotor Bore y X 5 X 4 X 3 z x Dimension Name Unloader Bore Depth Piston Length Unloader Piston Perpendicularity Rotor Bore Diameter Unloader Bore True Position (in y direction) KC: Unloader Piston Clearance Label Figure 1-3 Piston Layout Schematic. The area shaded area is the unloader. The clearance Y is the KC. The unloader bore true position is measured relative to the location of the rotor bore. Although true position is defined as a diameter, in this case only error in the X and Y directions affect the performance and reliability of the compressor. Piston clearance is a function of the other characteristics as shown in Equation?1 1. Equation 1-1 Y = X 1 X 2 X 3 2 X 4 2 X 5 8 Figure 1-4 shows which of these is under the control of the supplier and which are controlled by Carrier. X1 X2 X3 X4 X5 Y

5 Specifications for Materials, Dimensions, and Tolerances X2, X3 External Supplier X2, X3 X1, X4, X5 Internal Suppliers X1, X4, X5 Assembly Functional Test Figure 1-4. Supply Chain of Critical Dimensions. In the case study, Carrier controlled three of the critical dimensions, while an outside supplier controlled two. Boxes with the shadows indicate process steps where a failure in the KC can be identified. Gross failures of the KC can be identified during assembly by pistons that visibly do not align with the surface of the rotor bore. Less obvious defects are identified at functional test by physical interference with the rotor, although this event is obviously undesirable because damage could occur before the fault is detected. Experience has shown that pistons that do not fail here will function adequately for the life of the compressor. However, if the gap between the piston and the rotor is too large, the compressor will function at a lower level of efficiency. Before one can optimize performance vs. manufacturability, it is important to quantify the relationship between gap size and compressor efficiency. To this end, an experiment was designed and performed. The location of the piston was varied with a microthreaded screw and verified with a dial indicator. The compressor performance was then measured under standardized conditions on the test calorimeter. The relevant results are shown in Figure 1-5 below.

6 Performance vs. Male Piston Clearance y = -0.03x R 2 = Male Piston Clearance (mm) Figure 1-5. Experimental Results of Compressor Performance vs. Piston Position Relative to Bore. Results were collected for a variety of compressor load conditions and the relationship in every case was linear. The thick red portion of the best-fit line denotes the region of interest. This region was interpolated between the fully loaded position (0mm) and the point at 0.5mm. Compressor performance is found to vary with the male piston clearance according to Equation 1-2: Equation 1-2 y = x where y = performance and x = clearance It is possible to accommodate a lower compressor performance with additional coils in the heat exchanger of the chiller. This, of course, costs money. Previous economic analyses of the chiller system indicated that for every unit loss of performance, the added cost to the system was $68/ton of cooling capacity. This relationship is expressed with Equation 1-3: Equation 1-3 Cost = $68 Performance T where T = tonnageof compressor For a practical example, consider an increase of 30 microns in the nominal value of the clearance. From Equation 1-2, the change in performance would be P = 0.03* 0.03 = Watt /Watt. The change in system cost for an 80 ton compressor would then be $68* *80 = $4.90. In fact, this change in performance is undetectably small and does not affect the performance rating of the compressor in any way. This fact is the basis for increasing the nominal clearance, as discussed below.

7 D. Tolerance Allocation The question of tolerance allocation has to do with how much of the variation is due to a particular dimension. This step requires making assumptions about the process capability and modifying the design based on the tolerance analysis. Because our example deals with a released product, a product redesign is beyond the scope of this project. The optimization presented achieves the best balance of tolerance allocation with the existing design and the existing production processes. The expected variation in the piston clearance can be obtained by combining Equation?1 1 and the expected distributions for each of the contributors. Six months of historical data were used to estimate the mean and standard deviation of each contributor. The historical mean and standard deviation of each component is shown in Table 1-1 below. Male Unloader Piston Clearance Characteristic Name Sensitivity from Equation 1-1 Nominal in mm from Equation 1-1 Estimated Stdev USL LSL Ppk A Unloader Bore Depth B Piston Length C Unloader Piston Perp D Rotor Bore Diameter E Unloader Bore True Position σ Y = ( S i σ i ) 2 * nom /3σ Y Clearance Values * nominal + 4 ( S i M i USL i ) 2 Table 1-1. Historical values for all contributors to the male unloader piston clearance. Each feature is made by a particular machine and set of fixtures. The historical standard deviation of the process that makes each dimension is shown in the column called Estimated Stdev. It was assumed that these would not

8 change as a result of this study. Sensitivity values come directly from Equation 1-1. The clearance variance is simply the sum of the variance contribution from each of the critical dimensions. The sensitivity S i and multiplier M i determine the contribution of each dimension to the final variance. The multiplier is 1/4 for two-sided distributions governing critical dimensions A, B, D, and E, and is 1/5 for one-sided critical dimension C. Each dimension is assumed to be a normal distribution centered on the nominal dimension or a one-sided distribution with the mean 1 standard deviation from the nominal dimension as shown in Figure 1-6. Using these assumptions, and a Ppk of 1.33 as a baseline, the variance is then assumed to be (USL nominal)/4 for two-sided distributions A, B, D, and E, and (USL nominal)/5 for one-sided distribution C. LSL Target USL Target USL Case I: Two-Sided Distribution. Process Mean = Target σ=usl/4 Case II: One-Sided Distribution. Process Mean = 1σ σ=(usl-target)/5 Figure 1-6. Graphical Representation Of Assumptions For Two-Sided and One-Sided Distributions in the Optimization Spreadsheet. At Carrier, a capable process is defined as one with a Ppk of 1.33 or greater. Table 1-1 shows that many of the dimensions that controlled unloader piston clearance were not capable. The capabilities listed are those that result from the original product design using the worst-case tolerancing method. Note that the KC has Ppk of 1.25, which is much higher than most of its component dimensions. Worst case tolerancing caused the capability of the critical dimensions to be low, causing rework and scrap. The question was: could statistical tolerancing be adopted, and if so, could each process be given tolerances that would increase its Ppk to 1.33?

9 Another concern was that the largest contributor (71%) to the expected clearance variance was from a part manufactured by an external supplier. 2 This meant that no matter how much Carrier managed to reduce variation on the parts it produced, it would not be able to decrease the variation of the piston clearance by more than 29%. However, if the design could tolerate slightly more variation, the benefits averaged over the three Carrier components would have a tremendous impact on the manufacturability of the product. The performance analysis showed that the tolerances could be loosened. We saw from the previous section that recessing the piston an additional 30 microns from the rotor bore has a negligible effect on performance, a decrease of only Watts/Watt or 0.02%. In the following subsection we will see the benefits of allowing the additional clearance. E. System Optimization Tolerances for each of the contributing characteristics were reallocated using the process outlined in Chapter 6, Section E.2. The optimization process used Microsoft Excel with the built-in Solver. A separate file called tol_alloc.xls contains the Excel code. All of the variables were squared so that the variances could be added directly. This transformation enabled a linear programming optimization, which ensures that the optimal solution is achieved. One can easily check whether the optimization worked by calculating PpkX for each of the critical characteristics PpkY* for the KC. An optimized result should have a PpkY* of 1.33 and all of the PpkX should be equal. The Solver optimization was set up with the following parameters: Decision Variables Constraints Objective Function Table 1-2. Excel Solver optimization parameters. Square of the relative USL for each critical dimension Minimum squared Ppk value, MPK2 2 Y LSL Clearance variance must be less than. This constraint ensures that the clearance will not be less than zero at Ppk = For all i: MPK2 Ppki 2 0 (This constraint ensures that the final Ppk values will be equal, essentially maximizing the minimum Ppki) Maximize the minimum Ppki. 4 2 This is calculated from the spreadsheet in Table 1-3 as the sum of cells E17 and E18 divided by cell E23.

A sample optimization spreadsheet appears as Table 1-3. Table 1-3. Optimization Spreadsheet Before Applying the Solver. All of the variables are shown squared.

10 A sample optimization spreadsheet appears as Table 1-3. Table 1-3. Optimization Spreadsheet Before Applying the Solver. All of the variables are shown squared. Once the solution is obtained, the USL and Ppk values are obtained by taking the square root of the numbers returned by the optimizer. The results are shown with the original clearance value in Table 1-4 and with the additional 30 microns of clearance in Table 1-5. Male Unloader Piston Clearance Optimized Case I Estimated Characteristic Name Sensitivity Nominal Stdev USL LSL Ppk A Unloader Bore Depth B Piston Length C Unloader Piston Perpendicularity D Rotor Bore Diameter E Unloader Bore True Position

11 Clearance Values Table 1-4. New Tolerances for the Contributors to the Male Unloader Piston Clearance. The tolerance limits are labeled USL and LSL. Note that each process now has a capability of It should be understood that only the tolerances have been changed, not the machines or the nominal dimensions. The optimizer adjusts the tolerances until all process capabilities are equal and the mean clearance 4 * clearance standard deviations = Increasing the nominal unloader bore depth to puts the piston 30 microns farther from the rotor, causing, as noted above, negligible loss in efficiency. Re-running the optimization yields the results shown in Table 1-5. Here a very comfortable Ppk = 1.81 is achieved for each individual dimension and the KC. Male Unloader Piston Clearance Optimized Case II Estimated Characteristic Name Sensitivity Nominal Stdev USL LSL Ppk A Unloader Bore Depth B Piston Length C Unloader Piston Perpendicularity D Rotor Bore Diameter E Unloader Bore True Position Clearance Values Table 1-5. New Tolerances for the Contributors to the Male Unloader Piston Clearance. The optimization was run after increasing the nominal dimension of characteristic A to F. Documentation and Training The new tolerances listed in Table 1-5 were assigned to the product. The KC drawing documented the relationship between the unloader piston clearance and each of its contributors so that this information was readily available for future reference. Evaluation tools were provided to the product design, production, and quality organizations to allow for easy evaluation of the current state of assembly capability. This also provided the tools necessary to evaluate failure probability due to a single deviant piece, or a group of deviant parts. When process excursions occurred, the evaluation was straightforward and

12 objective. Decisions could be made based on failure probabilities rather than an obscure knowledge that a part with similar error had been accepted in the past. Training for the use of these tools was very straightforward. The data that go into the acceptability model were already being collected. Because the dimensions now had such a high capability, all that was required was a quarterly audit of unloader clearance capability performed by the quality assurance organization. The Solver performed the audit and required only the most recent mean and standard deviation values for each of the contributors. If the unloader clearance capability was below 1.33, corrective action was required. An internal quality control plan was developed 3 that specifies at what point corrective action is necessary. G. Results All of the steps above were also performed for the female unloader piston. With both male and female pistons recessed from the rotor bore by an additional 30 microns, the expected decrease in performance was 0.06%. For the first month after the nominal location of the unloader piston was changed, the performance audit of the compressor was watched carefully. As expected, there was no perceptible change in the performance of the compressor, yet the benefits to the manufacturing organization were plentiful. Taking advantage of statistical tolerancing brought the capability of all processes to Ppk = Increasing the piston clearance allowed a further increase in the capability of all processes to Ppk = To understand the benefit of this increase, let us compare the expected failure rate for dimension A at the original process capability of 0.74 and the improved capability of Assuming a distribution centered on nominal, the failure rate at Ppk = 0.74 is 26,400 parts per million. At a Ppk of 1.25 the expected failure rate is 177 parts per million. This is a decrease of 150 times! When considered at the final Ppk value of 1.81, the failure rate is 0.06 parts per million. This reduction was achieved with no measurable decrease in performance. During the first three months after changing the tolerances and instating SPC in the shop, no parts were scrapped due to the statistically toleranced characteristics. A severe machine excursion caused two parts to be out of tolerance. However, the analysis tools quickly verified that the probability of failure was extremely low and the parts were accepted for assembly. The statistical tolerancing method described here was eventually applied to a total of 22 characteristics. Making some assumptions about the performance of 3 VonBorstel, Steve and Terry, Andrew, Stastistical Tolerancing, Screw Compressor Operations Quality Plan, QCP 508, Carrier Corp., 1998.

13 the manufacturing processes, we can estimate some of the financial benefits. Assuming that the process variability does not increase more than 10%, the possibility of rework or scrap for these 22 characteristics will be virtually eliminated. Because the characteristics that were optimized had historically been the source of 70% of the problems, the total number of engineering disposition requests will be reduced by approximately 70%, saving $30,000 per year in opportunity cost. The scrapping of rotor cases due to problems with these dimensions will also be eliminated, saving an additional $40,000 per year.

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