Brigham Young University BYU ScholarsArchive All Theses and Dissertations 5--4 Statistical Investigation of Friction Stir Processing Parameter Relationships Jonathan H. Record Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation Record, Jonathan H., "Statistical Investigation of Friction Stir Processing Parameter Relationships" (5). All Theses and Dissertations. 8. https://scholarsarchive.byu.edu/etd/8 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.
STATISTICAL INVESTIGATION OF FRICTION STIR PROCESSING PARAMETER RELATIONSHIPS by Jonathan H. Record A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering Brigham Young University April 5
Copyright 5 Jonathan H. Record All Rights Reserved
BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Jonathan H. Record This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Tracy W. Nelson, Chair Date Carl D. Sorensen Date Timothy W. McLain
BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the thesis of Jonathan H. Record in its final form and have found that ) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; ) its illustrative materials including figures, tables, and charts are in place; and ) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Tracy W. Nelson Chair, Graduate Committee Accepted for the Department Matthew R. Jones Graduate Coordinator Accepted for the College Douglas M. Chabries Dean, Ira A. Fulton College of Engineering and Technology
ABSTRACT STATISTICAL INVESTIGATION OF FRICTION STIR PROCESSING PARAMETER RELATIONSHIPS Jonathan H. Record Department of Mechanical Engineering Master of Science Friction Stir Welding (FSW) is an emerging joining technology in which basic process understanding is still inadequate. Knowledge of FSW parameter relationships is needed to better understand the process and implement proper machine control. This study utilized a -factor, -level factorial design of experiments to investigate relationships between key process inputs and measured output parameters. All experiments utilized 775-T7 aluminum and a threaded pin tool with a 5.4 mm shoulder diameter, 4.76 mm pin length, and 7.9 mm pin diameter. Spindle speed, feed rate, and tool depth were varied throughout 54 welds while X, Y, and Z forces, X torque, three tool temperatures, and motor power were measured. Empirical models were developed to relate outputs to inputs. The relationships between inputs and outputs are nonlinear and require, at a minimum, a quadratic equation to reasonably model them. These models were further analyzed to explore possible control schemes. Tool depth was found to be
the most fundamental means of controlling weld forces and tool temperatures. This research describes the input/output relationships enumerated above for FSW as well as a discussion of possible control schemes.
ACKNOWLEDGMENTS I would like to thank my family, Friction Stir Research Laboratory (FSRL) at Brigham Young University, and my graduate committee for their support, encouragement, and advice. Financial support for this work was provided by the Defense Advanced Research Projects Agency (DARPA) contract No. MDA97--C-, and Dr. Leo Christodoulou, Program Manager.
Table of Contents Introduction... Background... Method... 9 4 Results/Discussion of Results...5 4. Statistics Background...5 4. Variable Effects... 6 4. Statistical Models...8 4.4 Residual Analysis... 4.5 Partial Derivative Analysis... 4.6 Other Observations... 5 Conclusions... 6 Recommendations for Future Work...5 7 References...7 Appendix A...4 Appendix B...47 Appendix C...5 Appendix D...57 Appendix E...6 viii
ix
List of Tables Table. DOE levels used for all experiments... Table. Coefficients for terms in statistical models...9 Table. Partial Derivative Analysis results and Control Feasibility criteria...8 Table 4. Parameters that lead to instability... x
xi
List of Figures Figure. FSW process schematic and direction definition... Figure. Relationship between process parameters (inputs) and measured outputs...7 Figure. Setup and orientation of dynamometer, cooling plate, and work piece...9 Figure 4. Locations of the thermocouple holes on the FSW tool... Figure 5. Tool holder and depth gage... Figure 6. Main Effects plot for each response...7 Figure 7. Surface depicting predicted X Force based on Spindle Speed and Feed Rate... Figure 8. X Force residual analysis to check model adequacy... Figure 9. Different outputs obtained depending on input parameters used. These plots predict Y and Z Force, respectively... Figure. Partial Derivative plot for X Force...5 Figure. Definition of Control Feasibility...6 Figure. Control Feasibility of Feed Rate in controlling Z Force, at a value of.8...7 Figure. Control Feasibility of Shoulder Depth in controlling Z Force, at a value of.6...7 Figure 4. Predictive plots for use in obtaining desired outputs... xii
xiii
Introduction Friction Stir Welding (FSW) is a relatively new joining process which has shown much promise and potential. Extensive time and effort have been expended to understand this process and its many aspects. Researchers have explored tool geometry, tool material, equipment, resulting weld microstructure, weld performance, and many other aspects of this process. As a result of these efforts, FSW has been implemented in several applications around the world. Despite the volume and breadth of research, there are gaps in understanding the fundamentals of the process which have resulted in makeshift machine control. There is a lack of understanding with regards to: ) which factors affect the weld outcome most, ) the relationships between process inputs and outputs, and ) how to best control this process. Due to these deficiencies, or the unavailability of information in literature because of proprietary issues, FSW development and application is hindered. Many fundamental areas need exploration, especially the area of understanding the basic relationships between process inputs and measured outputs. Many studies have been systematic and performed very carefully [4, 5, 7, -, 8-7]. However, these have not used a method which fully explores the basics of the process. In addition, conclusions have been qualitative, as opposed to quantitative, and have been based largely on observation. The purpose of this study is to experimentally determine the relationships between the primary inputs and measured outputs of the FSW process. These relationships are created statistically, then analyzed and discussed with a focus toward understanding machine control. Three input parameters are varied and eleven process variables are measured throughout the experiments. The results of this study are useful in many different areas. The study provides a solid foundation for other research to be built upon. It presents information which will assist in proper application of FSW. Results will also be valuable in determining ideal
control algorithms which aid in a production environment. Results will also be publicly available for comparison with other research data.
Background Friction Stir Welding (FSW) is a solid state joining process invented by The Welding Institute in England [, ]. FSW has many advantages over a traditional arc weld. Compared to arc welding, FSW produces little or no weld distortion, porosity, or fume, and has excellent mechanical properties []. It has enabled joining of materials which were previously difficult to weld. In these, weld joint efficiencies have been in excess of 95%. The FSW process utilizes frictional heating and a stirring motion to break down the interface between two workpieces yielding a solid, fully consolidated weldment. A rotating tool, which has a protruding pin, is forced into the joint to be welded until a larger concentric shoulder rests on the surface, as shown in Figure. The spinning tool is advanced along the joint line providing frictional heating which softens the material. The spinning pin and shoulder help mix and reconsolidate the material, respectively. A coordinate system showing X, Y, and Z directions is given in Figure. Z X Figure. FSW process schematic and direction definition. Y FSW has been successful in welding a wide range of materials. Plastics, metal matrix composites, aluminum, copper, steels, stainless steel, nickel alloys, and titanium
have been successfully welded []. However, current commercial applications only involve aluminum and copper. FSW has already been successfully implemented in various applications. Some of these involve primarily aluminum alloys in many industries including marine, aerospace, and railway [-6]. Other applications include semiconductor and automotive, as well as other industrial applications. FSW equipment has evolved over the last decade. The first machines were converted milling machines which required manual control. The operator was required to monitor an output, such as the amount of flash or a reading from a depth gage, and make adjustments based on experience. Spindle speed, feed rate, and tool vertical position were found to be the three most significant inputs to control [7, 8]. In an effort to remove the human factor, FSW equipment quickly moved to CNC control. Commercial machines were designed with force control in mind. Initially, force control use was preferred to achieve quality welds [9, ]. More recently, however, force control was reevaluated as the ideal control method. Other possible control schemes have been investigated and implemented; today there is no agreement on what control is ideal. In order to effectively control the FSW process, input parameters (spindle speed, feed rate, and tool depth) need to be properly adjusted to achieve desired results (outputs). The goal of this study is to identify the relationships between these input parameters and a variety of process outputs that could potentially be used for process control. Johnson [] studied the effects of parameters on different outputs. He varied aluminum alloy composition, pin geometry, plunge depth, spindle speed, and travel speed while measuring forces and torques. Johnson was able to discuss general trends such as how X Force (force in the direction of travel) increased significantly with travel speed and only slightly with increasing spindle speed. He also showed how Z Force (force in the vertical direction, normal to the workpiece surface) increases with increasing plunge depth. In a similar study, Reynolds and Tang [] confirmed that X Force increases with increasing feed rate, and added that this relationship was nonlinear in most cases. They also observed that increased Z Force, an input, did not affect the X Force significantly. These studies provided valuable information but did not include quantitative results as part of their discussion. 4
A study by Nishihara and Nagasaka [] included the measurement of tool and anvil temperatures during welding. This study was unique because few studies measure tool temperature, which may be a key element in understanding the FSW process. This study was simplified by varying only a few inputs (spindle speed, feed rate, and plunge depth) while measuring only tool and anvil temperatures. They found that higher spindle speeds resulted in higher temperatures for both the tool and anvil. As feed rate increased, tool temperature was not greatly affected. They also found that increasing plunge depth increased tool and anvil temperatures. Although very useful, this study did not analyze how interactions between inputs may affect welding temperatures. In several studies involving robotics and FSW, many relationships between inputs and outputs were discovered [4-6]. The focus of these studies was on the control needs of robotic FSW equipment. Robots generally lack the rigidity of conventional FSW machines; therefore they require more careful control of forces or positions. Mitchell et al. [4, 5] performed an initial study to determine effectiveness of an economical load cell for use in force feedback control on robots. Many trends were observed as spindle speed and feed rate were varied. Forces and temperatures in the workpiece were measured. Mitchell found that increasing spindle speed increased plate temperature. Cook et al. [6] were able to take the next step and study force feedback for robotic FSW applications. Experimental techniques were used to vary spindle speed and feed rate, with repetitions at each set of variables. Cook observed trends similar to those of other researchers [-5] and concluded that robotic FSW implementation requires force feedback. It was recommended that to minimize the axial force (Z Force), spindle speed must be high and travel speed low. This is the first study that began to address FSW control by looking at parameter relationships. However, the analysis of this study addressed only robotic FSW applications. A study by Whiting [7] was unique because of its use of statistical methods in explaining the effect of parameters on tensile strength. Feed rate and spindle speed were varied to study their effect on weld strength, heat affected zone width, and tool wear. It was discovered that feed rate had five times the effect of spindle speed on the tensile strength. It was also found that higher feed rates produced stronger welds and less tool wear. In a similar study, Record et al. [8] discovered the basic process inputs to be 5
spindle speed, feed rate, and tool vertical position. Their objective was to identify all factors that significantly affect key FSW outputs. Although results were fairly quantitative, neither study explored the process fundamentals with enough breadth and depth. There are numerous other studies that have evaluated different inputs and outputs of FSW using various experimental approaches. Inputs have included: number of flats on the tool [8], surface preparation [9], plate thickness [], welding force and tool geometry [], material configuration for lap joints and tool motion [], and alloy [, ]. Outputs have included: forces, torques, mechanical properties [9,, 4], metallography [4, 7], plate temperatures and temperature distribution [, -6], fatigue [], tool temperature [, ], and weld quality [, 6]. A detailed review of these papers is not undertaken since their objectives greatly differ from the objective of this paper. Of the FSW literature and data publicly available, much has explored aspects of various process fundamentals. Each study provides useful information about FSW. However, relatively few studies address and focus on the most essential process parameters. Many studies have made qualitative conclusions based on observed trends in data. Few of the studies mentioned above have used a statistical approach, which can create quantitative relationships between inputs and outputs. Most studies use an insufficient number of welds and parameters to really understand the process. Few studies included tool temperature which is thought to be important in understanding thermal evolution. Additionally, the wealth of information available and the advances in process equipment has not yielded an agreement on optimal control schemes. Development and application of control schemes have been based on observations rather than detailed studies. Careful experiments with an emphasis on FSW control must be performed to better understand potential control schemes. To accomplish this, an understanding of the relationships between input and outputs must be achieved. Knowledge of FSW control will make the process more reliable, repeatable, user-friendly, and applicable. The purpose of this research is to develop a quantitative relationship between the primary FSW inputs and measured outputs (responses). These relationships are 6
represented by the box labeled Model in Figure and constitute an empirical model of FSW. The measured responses are the most descriptive in understanding FSW process fundamentals and may help in understanding possible control schemes. A statistical approach is used to establish quantitative relationships and predictive equations that may be used on a similar setup. The breadth of the statistical approach is sufficient to allow the results to be truly representative of FSW. Spindle Speed Feed Rate Tool Depth Model Tool Temperatures X Force Z Force X Torque Motor Power Figure. Relationship between process parameters (inputs) and measured outputs. 7
8
Method Plates were Friction Stir Processed (bead on plate) on a retrofitted Kearney & Trecker knee mill with PLC/PC control and a data acquisition system. Both Z Depth and Z Force control were available (axis definitions from Figure ). Z Depth control was used in this study to maintain the tool at a desired distance relative to the workpiece surface. Mounted to the bed of the mill was a mm long dynamometer capable of sensing forces up to 45 kn in the X and Y directions and 9 kn in the Z direction. Fixtures for clamping the workpiece were mounted to the upper surface of the dynamometer. The work piece and dynamometer are shown in Figure. A 6 mm thick aluminum cooling plate was used between the work piece and dynamometer. A mixture of ethylene glycol and distilled water was pumped through the Figure. Setup and orientation of dynamometer, cooling plate, and work piece. 9
cooling plate at approximately C at a constant flow rate. This cooling plate helped keep the dynamometer at reasonable temperatures to facilitate more accurate measurements (Figure ). A 4.8 mm thick steel anvil was placed on top of the cooling plate for protection and to give a solid backing surface for the workpiece. The material used in this study was Al 775-T75 with a thickness of 9.5 mm. The plates were sheared to the nominal dimensions of 95 mm by mm. The oxide layer was removed with a portable disc sander and the surface was cleaned with methanol prior to processing. The thickness of the plate was predetermined so that only partial penetration welds would be run, in an effort to minimize any possible interaction that could exist between the pin of the tool and anvil. The tool used was made from heat-treated H tool steel. Key tool dimensions include a shoulder diameter of 5 mm, pin diameter of 7.9 mm, and shoulder concavity angle of 6 degrees. The pin was 4.8 mm in length and was threaded with a pitch of.9 mm/thread. A tilt angle of.5 degrees was used. The tool was modified to accommodate thermocouples for temperature measurement at three locations within the tool. An EDM drill was used to cut long, straight, flat-bottomed holes to accommodate.6 mm diameter, stainless steel sheathed, ungrounded type K thermocouples at the three locations as shown in Figure 4. The name of each location should be noted as Pin Center, Root, and Shoulder. The distance between the thermocouple and tool/workpiece interface at each location was less than. mm. Root Pin Center Shoulder Figure 4. Locations of the thermocouple holes on the FSW tool.
A liquid-cooled tool holder was used to minimize heat flow into the machine head. Access holes near the top of the tool holder allowed the tool thermocouples to be inserted through the back of the tool. A transmitting collar assembly was clamped to the rotating portion of the tool holder and housed radio frequency transmitters which broadcasted the thermocouple readings as FM signals. The signals were captured by the receiver by way of a stationary loop antenna shown in Figure 5. The data were then transferred to the data acquisition system. Figure 5. Tool holder and depth gage. Mounted to the tool holder was an electronic digital indicator for measuring tool vertical position, shown in Figure 5. Because the gage measured the displacement between the work piece and the mounting point of the gage, this measurement effectively became shoulder depth. The gage had a range of 5 mm and a resolution of. mm. An extension adapter was connected to the indicator so that shoulder depth was measured as close to the tool as possible to account for any local variation in tool depth or plate
thickness. Readout error associated with attaching such an adapter was estimated to be.5 mm or less. The indicator readings were transferred to the data acquisition system throughout the weld. Shoulder Depth measurements were also used in a control loop to ensure a target Shoulder Depth was achieved. The target Shoulder Depth was established prior to welding as part of the factorial design, and the machine adjusted the Z Position until the desired Shoulder Depth was achieved. Thus Z Force was a measured output and Shoulder Depth was an input. A statistical method, factorial design of experiments (DOE), was useful in obtaining relationships between various factors (inputs) and responses (measured outputs). To organize such an experiment, several important setup steps were performed. The first step was to identify the factors that would be varied. Next, was to determine the levels of these factors to be used in the experiment. The final steps were to perform the experiments and analyze the information. A preliminary study was performed to determine which factors are most advantageous to explore [8]. Three factors, Spindle Speed, Feed Rate, and Tool Depth, were identified as primary inputs. These three factors were varied throughout the experiments at levels based on experience. Each of the three inputs was varied at three levels as shown in Table. Analysis of data was used with the levels of each factor on a normalized scale from one to three, also shown in Table, with the high value being labeled, and the low labeled. This was done to eliminate the dependence of the analysis on units of each parameter. Two replications of every possible combination of parameters were completed for a total of 54 experiments. The measured outputs of interest, believed to be the most descriptive and fundamental, were X, Y, and Z Force, X Torque, Motor Power, and Pin, Root and Shoulder Temperatures. The force directions are defined in Figure. X Torque is the torque required by the motor to produce the required Feed Rate. Motor Power is the power used by the spindle motor. Tool Temperature locations are defined in Figure 4. Consistent with good experimental practice, the weld order was randomized to eliminate unknown variables, and all other setup and procedural aspects were held as constant as possible.
Table. DOE levels used for all experiments Factor High Medium Low (level ) (level ) (level ) Spindle Speed SS (rpm) 5 5 Feed Rate FR (mm/min) 5 78 5 Shoulder Depth D (mm).4 -.64.89 shallow deep The range of input parameters was developed by experience, and trial and error, in welds not described here. The widest possible range of parameters was established to ensure the resulting models were as encompassing and descriptive as possible. The parameters were chosen to have a large range and yield sound welds. For this study, a sound weld is defined by: ) no tool breakage, ) no holes, ) no galling, and 4) no excess flash. Plates were friction stir processed (FSP) in the following manner. A plate was secured at a predetermined location on the anvil. The plunge sequence was performed at 5 rpm and at a plunge rate of.7 mm/min to the specified Shoulder Depth. The Spindle Speed during the dwell remained at 5 rpm. A dwell time of 5 or seconds was employed to assist in adjusting the Shoulder Depth to the desired value. After the dwell, the Spindle Speed was adjusted to the value dictated by the DOE and the tool began to traverse at a rate of 5 mm/min. The Feed Rate was then increased at a constant acceleration over a distance of 76 mm until the desired Feed Rate was obtained. Weld lengths varied depending on the Feed Rate. The 5, 78, and 5 mm/min welds were 54, 559, and 88 mm in length, respectively, which was long enough to reach steadystate conditions. Desired Shoulder Depth was maintained through a feedback control loop using the digital indicator to measure real time depth. Each process variable, except Spindle Speed, Feed Rate, and Shoulder Depth, was held constant. Each plate had two welds on it, each the same distance from the edge of the plate. This was to ensure that measurements were actually a result of changes in input parameters and not some unknown source [8].
Upon completing the 54 welds, the raw data were plotted and organized using a spreadsheet. For each weld, the steady-state region was identified. For each response of interest, values were averaged over that steady-state region for use in further analysis. All extracted data were analyzed using Minitab and Excel. Minitab, statistical analysis software, was used to create and analyze multiple regression models. Excel was used to explore the models to determine their implications in FSW control. A metric was created to describe how effective an input was in controlling a response. Conclusions about FSW control were made from these metrics. It is important to note that for this statistical study, any information about the analysis only applies for the range of parameters tested and for this particular setup. This information may, or may not, apply to other FSW conditions. 4
4 Results/Discussion of Results All welding was successfully completed as planned. Weld cross-sections were polished and etched to verify there were no holes. The results of the statistical experiments and analysis are presented in this section. The results are in the form of empirical models relating outputs to inputs. Validity of these models is discussed. These models may be used to predict responses and reveal the difficulties that may occur when adjusting parameters to obtain desired outputs. Further analysis, using the statistical models as a basis is then employed to explore FSW machine control. 4. Statistics Background Description of statistical analysis in this research required the use of effects, P values, and adjusted R written as R adj. The objective of experimental designs is to produce polynomial correlation equations that relate outputs to inputs. For example, in a linear polynomial model, the effect (β i ) relates the inputs to a response, or output, as shown in Equation. [ i i ] y = y o + β X () In this equation, y represents the response, y o is the average response, X i is the value of input parameter i, and β i is the effect that input parameter i has on the response. Thus, the larger the value of an effect, the greater influence it has in determining the response. The P value is a way of describing how well a term fits into a model. A term belongs in a model if its effect, β in Equation, is nonzero. The P value is the probability that the actual effect is zero. It is desirable to obtain P values that are as small as possible. If a term s P value is.5, then there is a 95% confidence level that the 5
coefficient is nonzero, and therefore belongs in the model. This analysis included terms in models if their P values were.5 or less. The R adj term is synonymous to the fraction of variance in the data that is accounted for by a model. This measure accounts for how many terms were included to create the model. A higher R adj value is desirable and signifies a model more closely matches experimental data. All analysis was performed with the input values having a coded form, or parameter levels being, or. By transforming all input parameters into coded form, each input is varied over the same range and loses its dependence on units so effects can be directly compared. Averaged responses for each weld s steady-state region are found in Appendix A. 4. Variable Effects Minitab s Factorial Design analysis was first used to explore general trends and effects. General trends are observed by creating Main Effects plots, shown in Figure 6. The points in the plots represent the mean at each level of a factor, while all other parameters are varied. Each point is an average of 8 data points. A reference line is drawn at the grand mean of the response data. These plots were also created to describe the magnitude of change in response when a change in input is imposed. These plots reveal general trends which are consistent, for the most part, with other literature. By analyzing Figure 6, it can be seen which inputs have the dominant influence on a response, as well as which ones have little or no effect on a response. Figures 6 (a) and (g) show how Feed Rate has dominant effects on X Force and X Torque. Spindle Speed has the largest effect on Tool Temperatures as seen in Figures 6 (b), (d), and (f). Regarding Z Force, Feed Rate and Depth have equal effects while Spindle Speed has a negligible effect as shown in Figure 6 (e). Motor Power is about equally influenced by all three inputs, as seen in Figure 6 (h). 6
Main Effects Plot - Data Means for X Force kn Main Effects Plot - Data Means for Root Temp SpindleSpeed Feed Rate ShldrD SpindleSpeed Feed Rate ShldrD 8.8 468 X Force kn 7.6 6.4 Root Temp 456 444 5. 4 4. 4 (a) Main Effects Plot - Data Means for Y Force kn (b) Main Effects Plot - Data Means for ShoulderTemp SpindleSpeed Feed Rate ShldrD SpindleSpeed Feed Rate ShldrD. 468 Y Force kn -. -.4 -.6 ShoulderTemp 456 444 4 -.48 4 (c) Main Effects Plot - Data Means for Z Force kn (d) Main Effects Plot - Data Means for Pin Temp 55 SpindleSpeed Feed Rate ShldrD 5 SpindleSpeed Feed Rate ShldrD 5 485 Z Force kn 45 4 Pin Temp 47 455 44 5 (e) Main Effects Plot - Data Means for X Torque N-m (f) Main Effects Plot - Data Means for Motor Power 6. SpindleSpeed Feed Rate ShldrD 6. SpindleSpeed Feed Rate ShldrD X Torque N-m 5. 4.5.7 Motor Power 5.7 5. 4.9.9 4.5 (g) (h) Figure 6 (a)-(h). Main Effects plot for each response. 7
4. Statistical Models Statistical models for predicting each response were created by analyzing steadystate values. One method of constructing models is to use the results from Minitab s Factorial Design analysis. This approach would work well if every response had a linear relationship with its input. However this was not the case with Y Force, Root Temperature, and Shoulder Temperature. All responses were checked to determine if squared terms would create better models to account for possible nonlinearity. The resulting form of the regression equations, including squared terms, would take the form shown in Equation. response = β + βx + βx + βx + βxx + βxx + βxx + β x x x + β x + β x + x + error () β A stepwise regression was used to explore the need of squaring input terms so the models could more closely match the experimental data. Ultimately, a multiple regression was performed for each response. Terms selected for the models were any that helped maximize R adj and minimize P values. Terms in the models could have been main effects (inputs), interaction terms, and/or squared terms. No effort was made to keep certain terms in the model, to try and match results to the factorial design analysis, or to have a model imitate observations or theories. The fifty four experimental data points were modeled by equations using up to eleven terms. The resulting models for each response are shown in Table and Equations -. Table contains the coefficients for each term (from Equation ) in the model. The primary inputs (Spindle Speed, Feed Rate, and Depth) are listed, along with the interaction between these inputs labeled as SS*FR (Spindle Speed-Feed Rate interaction), SS*D (Spindle Speed-Depth interaction), and so forth. Each measured response is listed across the top. Boxes containing -- signify that that term was not included. The bottom row contains the R adj value for each model, to give an indication of how well the model describes the data points. All P values for the terms were. or less. The complete Minitab regression output is included in Appendix B. 8
Table. Coefficients for terms in statistical models. All P values were less than.. X Force (kn) Y Force (kn) Z Force (kn) X Torque (N-m) Root Temp (C) Shldr Temp (C) Pin Temp (C) Motor Power (kw) Intercept --..9 -- 7.7 4. 48.4.7 Spindle Speed -- -- -6. -- 66. 5.8 76.5 -- Feed Rate 4.6 -.7 4..8 8.4 -- --. Depth -- -.7 -- -- -- -9.9 -- -- SS*FR.4. -- -- -- 6.7 -- -- SS*D -.. -- -- -- 5.7 -- -- FR*D --. -7. -. -.8 -- -.9 -.5 SS*FR*D -- -..4 --.6 -- 4.. FR*FR -.7.4-5. -.5-6. -4. -- -. SS*SS -- -- -- -- -.5 -.9-4.8. D*D --. -- -- -- -- -- -- R adj 9.9 7. 9. 88.9 9.5 94. 95.8 95.9 X Force(kN)= 4.6FR +.4SS*FR.SS*D.7FR () Y Force(kN)=..7FR.7D +.SS*FR +.SS*D +.FR*D.SS*FR*D +.4FR +.D (4) Z Force(kN)=.9 6.SS + 4.FR 7.FR*D +.4SS*FR*D 5.FR (5) X Torque(N-m)=.8FR.FR*D.5FR (6) RootTemp(C)= 7.7 + 66.SS + 8.4FR.8FR*D +.6SS*FR*D 6.FR.5SS (7) ShldrTemp(C)= 4. + 5.8SS 9.9D + 6.7SS*FR + 5.7SS*D 4.FR.9SS (8) PinTemp(C)= 48.4 + 76.5SS.9FR*D + 4.SS*FR*D 4.8SS (9) MotorPower(kW)=.7 +.FR.5FR*D +.SS*FR*D.FR +.SS () Equations - use Spindle Speed, Feed Rate, and Depth at coded levels (,, or ) described in Table to predict outputs in units of kn, N-m, C, or kw. These are the models used for further analysis. Analysis of these models shows there is no direct : relationship between a response and any given input. Each response depends on multiple inputs. The regression models closely match the trends and main effects found in Figure 6. Implications these models have on FSW machine control will be discussed later. 9
The coefficients given in Table should not be used to identify which inputs have the greatest effect on a response. For example, the coefficient for Depth in the X Force model is zero. However, Depth does exhibit an influence on X Force through the Spindle Speed-Depth interaction term. To understand which inputs affect responses the most, Main Effects plots are utilized as in Figure 6. To aid in visualization, response surface plots were created using Excel. Figure 7 is an example, representing predicted X Force as a function of Spindle Speed and Feed Rate. These surface plots are helpful in determining what is happening for each response in the range tested. The range of X Forces predicted by the models can be easily identified as well as how X Force is affected when a single input is changed. It is also helpful to see what combination of inputs may or may not be desirable. All response surface plots are included in Appendix C. Feed Rate vs Spindle Speed for X Force (kn) 8 X Force (kn) 6 4.8.6.4. Spindle Speed.8.6.4..6. Feed Rate.8.4 Figure 7. Surface depicting predicted X Force based on Spindle Speed and Feed Rate. The statistical models are useful in describing a steady-state response given certain input parameters. Because only steady-state values were utilized in the analysis, these models do not describe the transient portions of welds. Additional testing and analysis would be needed to fully understand closed-loop control and dynamic portions of the welding process. However, these models still may be used to help understand closedloop control with certain limitations, such as low-frequency control and assuming there are no undesirable dynamic responses.
4.4 Residual Analysis When data are fit into statistical models, it is assumed they meet certain requirements. Once data are fit into models, their residuals can then be analyzed to verify they meet those requirements. Validity of these statistical models depends on how closely the underlying assumptions are met. If the assumptions are met, then the models are adequate. The inability to meet the assumptions does not necessarily imply the models are bad, but there is no guarantee they are good. Data may also be transformed so that some assumptions are more closely met. However, this study did not require any manipulation of data. The assumptions are that: ) The errors are normally distributed ) The errors have the same variance (constant variance) ) The errors are independent (or the experiment was randomized) 4) The errors have a mean of zero (i.e., the model is adequate) To check the assumptions, the residuals (difference between measured value and predicted value) are studied. Residuals for each model were analyzed and satisfactorily met each assumption. Residual analysis for each response can be seen in Appendix D. To illustrate, the residual analysis for X Force will be discussed with reference to the assumptions -4 mentioned above. Assumption is checked by using one of two different plots. First, the residuals may be plotted on a normal scale. The plotted data should resemble a straight line as shown in Figure 8 (a). Second, the residuals may be plotted versus frequency, as shown in Figure 8 (b). The shape of this plot should resemble a bell shape, representing a normal distribution. The assumption of normally distributed errors (residuals) is met as seen in these two plots. Assumption is checked by plotting residuals versus fits (model predictions) as seen in Figure 8 (c). If the assumption is poor, one will notice an increasing, decreasing, or diamond-shaped pattern. One can see two groups of predicted values in this plot, but the residuals are not correlated with the predicted value. Assumption is met in the way the experiments were setup. This assumption, independence of errors, is met by randomizing the experiments. All experiments in this study were randomized, so it is assumed all errors are independent. This assumption was
also checked by plotting data to look for any correlations that should not exist, such as a dependency on weld run order. No such correlations were discovered. Assumption 4 is checked by plotting residuals versus the observation, or run order. Violations of this assumption have occurred if there are any patterns or trends. Figure 8 (d) shows the residuals are random and have no patterns. As each model s residual analysis was performed, it was observed that some responses more closely met the assumptions than others. There could be hidden variables such as tool wear which affect these results. But each assumption was satisfactorily met for all responses. Therefore, it is concluded that the models chosen for further analysis are good descriptions of the data points recorded for this welding series.
4.5 Partial Derivative Analysis Thus far, the analysis of data has produced good statistical models representing the FSW process. These models may be used to predict what forces and temperatures may be reached given a set of input parameters under steady-state conditions. They may be used to predict what new responses may be achieved if parameters are changed. It is important to use this information in exploring possible FSW control algorithms. By looking at the different response plots, found in Figure 7 and Appendix C, it can be seen that controlling FSW may be difficult. This can be further explained using Figure 9, which is a two dimensional representation of the information found in Figure 7. Figure 9 (a) shows what Y Force may be obtained at three different levels of Feed Rate. For the same increase in Feed Rate, x, there are two possible changes in Y Force. One change in Y Force (y ) is negative and the other (y ) is positive. Both the sign and y 4 x y y y x x x (a) (b) Figure 9 (a) and (b). Different outputs obtained depending on parameters used. These plots predict Y and Z Force, respectively. magnitude of change in Y Force are different. By noting similar trends in Figure 9 (b), the same increase in Feed Rate yields two very different outcomes in Z Force. This makes FSW control more complicated to implement because responses do not maintain
the same behavior in all areas of the parameter window. Control is simplified when it will function the same in every operating condition. Further analysis of response plots was utilized to explore this issue of difficulty in controlling FSW. This was done by plotting the partial derivatives of each response model from the previous discussion. A partial derivative represents the amount of change in response for a unit change in input, related by the slope at different portions of the surface plots (see Figure 7). Partial derivatives of each response were taken with respect to Spindle Speed, Feed Rate, and Shoulder Depth. The partial derivative plots are useful in describing which input variable is ideal for controlling a given response. With respect to possible control schemes for FSW, several things would be ideal when using an input to control an output. First, it would be ideal if these partial derivative surfaces were flat, or constant. Thus, a change in a response due to a change in one parameter would be independent of the other two inputs. Second, it would be ideal if the partial derivative surfaces were far from the zero plane. If the plane is close to, or crosses zero, then a change in an input could have zero, positive, or negative effect in the output. If the two above conditions are not met, FSW control becomes more difficult. There needs to be a significant change in the output with only slight change in input for ideal FSW control. The partial derivative of X Force with respect to Spindle Speed is shown in Figure. It is seen that the vertical axis represents the amount of change in X Force per unit change in Spindle Speed. The amount of change in X Force depends on two additional inputs, Tool Depth and Feed Rate, as indicated by the other axes. Looking at point A in Figure, it can be seen that making a unit increase in Spindle Speed, X Force will increase by.667 kn (when Depth and Feed Rate = level ). If Depth = level and Feed Rate = level, then the same unit increase in Spindle Speed would cause a.46 kn decrease in X Force as seen by point B. It is important to note that in this case, as well with most others analyzed, the amount of change in a response due to a change in input depends on the levels of the other two inputs. All partial derivative plots are found in Appendix E. By analysis of response surface plots and partial derivative plots (see Figures 7,, and Appendix E), it is apparent there was no simple way to control an individual 4
d(xforce) / d(spindle Speed)...8 A.6 d(xforce) X Force (kn) d(spindlespeed).4. B. -..6 Depth..8.4.4.8..6 Feed Rate Figure. Partial derivative of X Force with respect to Spindle Speed plotted versus Depth and Feed Rate. response. Control of FSW is more complex and involved than originally thought. Most plots show that all current parameter levels significantly affect the change in a response. This makes responses more difficult to predict. This also means that previous control methods used were too simple. They may have been adequate, but should have included more complexity in light of this discussion. In an effort to quantify and rank the ability of inputs to control an output, a metric was created. This metric, Control Feasibility, utilized information from all partial derivative plots, specifically the average distance from the zero plane and flatness of the surface. To produce this metric, the partial derivative with respect to each input at all tested levels was calculated and plotted against the input levels. Measurements were then taken to produce the metric as defined in Equation and in Figure. λ ControlFea sibility = () λ max λ min 5
λ max ( output) ( input) λ λ min Input Figure. Definition of Control Feasibility. The larger the Control Feasibility value, the greater the potential the input has to control a certain response. Control Feasibility is greater when the data points cover a smaller range and when their averaged values are further from zero. Thus, Control Feasibility describes how effective an input is at controlling a given output. An example is given using Figures and, where the partial derivative of Z Force with respect to Feed Rate and Depth is shown, respectively. Figure shows that values cross the zero plane and have a range of 6 kn. Figure, which has a higher Control Feasibility value than that shown in Figure, has a range of only 5 kn and does not cross the zero plane. The higher Control Feasibility is awarded to this latter case, due to the higher averaged value and smaller range. Thus, Depth is a better input parameter than Feed Rate for controlling Z Force. Comparison of partial derivative plots of Z Force, found in Appendix E, also support this conclusion. A summary of the Control Feasibility values along with Control Feasibility criteria for all responses is found in Table. It is interesting to note that Depth has the highest Control Feasibility value for seven of the eight responses, as shown in Table. This does not mean that Depth should be used to control all responses, or that no other adjustments should be made. Depth is 6
5 5 d(z Force) / d(feed Rate) 5.5..5..5..5-5 - -5 Feed Rate Figure. Control Feasibility of Feed Rate in controlling Z Force, at a value of.8..5..5..5..5 - -4-6 -8 d(z Force) / d(depth) - - -4-6 -8 - Depth ranked highest, indicating it would be the best input to change first if only one response was in need of adjustment. Figure. Control Feasibility of Shoulder Depth in controlling Z Force, at a value of.6. When it is desirable to control up to three outputs (three responses may be controlled with three inputs), there must be a process to select which input will be used to 7
control each output. Suppose it is desired to control X Force and Z Force. By reviewing Table, it can be seen that although Depth scored highest for X Force and Z Force, it cannot be used to control both simultaneously. Depth must be chosen to control one of them, and a different input chosen to control the other. This process involves tradeoffs. But the end goal is the same as in controlling one output: change inputs to obtain desired results. Table. Partial Derivative Analysis results and Control Feasibility criteria X Force Y Force Z Force X Torque Root Temp Shldr Temp Pin Temp Motor Power Spindle Speed Range (Max-Min).9.4.5. 8.77 76.6 94.. Average.45.5 -.56. 6.5 6.4 4.56.7 Cross zero plane? Yes Yes Yes Yes Yes Yes Yes No Control Feasibility.7.5.5...4.7.65 Feed Rate Range (Max-Min).6.4 5.76. 47.87 9.97 7.86.9 Average.6. 9.9.5-5.6 -.6 -.47.75 Cross zero plane? No Yes Yes No Yes Yes No Yes Control Feasibility.7.7.8.66...8.9 Depth Range (Max-Min).7.76 4.6.4.64.4 7.86.9 Average -.7.5-8.99 -.4-9. -8.5 -.47 -.5 Cross zero plane? No Yes No No No No No No Control Feasibility..9.6..9.74.8.57 To understand how Table may be used to select a way to control FSW, consider the following example. A weld operating with an X Force at 6 kn, Z Force at 45 kn, and Shoulder Temperature at 4 C is desired. To select the inputs that should control each response, Control Feasibility values are selected so the entire solution has the highest summed total of Control Feasibility. Table shows that X Force has Control Feasibility values of.7,.7, and. for Spindle Speed, Feed Rate, and Depth, respectively. Control Feasibility values can also be found for Z Force and Shoulder Temperature. A conclusion may be to control X Force with Depth, since its Control Feasibility value is.. It would follow that Z Force would be controlled with Feed Rate (Control Feasibility of.8) and Shoulder Depth with Spindle Speed (Control Feasibility of.4). The sum total would thus be.6. However, there are other combinations which yield 8
better overall results. The best way to control X Force, Z Force and Shoulder Temperature is with Feed Rate (.7), Depth (.6), and Spindle Speed (.4), respectively. The resulting sum is.67. The next step in order to obtain the desired outputs is to select levels for each input. This can be done using the statistical models and an optimization routine. It can also be done by using the predictive surface plots seen in Figure 4. The yellow dots in Figure 4 indicate the area of interest. By looking first at Figure 4 (a), it can be seen that for an X Force of 6 kn, Feed Rate should be set at about level.6. Z Force is then considered as shown in Figure 4 (b). If Feed Rate is already set at.6, then Z Force of 45 kn will be achieved when Depth is at about level.. To obtain a Shoulder Temperature of 4 C, Spindle Speed should be set to level., as shown in Figure 4 (c). Additionally, it is advantageous to control FSW in such a manner as to avoid operating in the regions where the partial derivative plots cross the zero plane. To better understand FSW machine control, one can explore the areas near these operating conditions (FR=.6, D=., SS=.). Suppose it is desired to increase Shoulder Temperature. The solution would rest in changing Spindle Speed, as mentioned in Table it having the highest Control Feasibility. If we increase Spindle Speed from its original level of. up to, X Force will increase slightly to 6.4 kn, Z Force will decrease to 4.7 kn, and Shoulder Temperature will plateau at about 47 C. It is important to understand how changing one input affects almost all outputs. Suppose it is desired to operate at a level spindle speed and maintain the same X and Z Force (6 and 45 kn, respectively), is there a possible solution? In using the surface plots and Excel s solver function, it appears there is no solution. All three objectives cannot be met. The closest solution obtains an X Force of 6. and Z Force of 4.7 (Feed Rate =.4 and Depth = ). Or, another solution would be to achieve the X Force = 6 kn goal and forfeit the Z Force objective. In this last scenario, Z Force would be at 4.5 kn. It is possible that all three outputs may not be controlled to a desired level independent of each other. There are relationships that link all inputs to each other. Referring to the above example, it would not be possible to achieve a Shoulder Temperature of 47 C when X Force is 6 kn and Z Force is 45 kn. There may be no solution for some desired scenarios. 9