Predicting Parting Plane Separation and Tie Bar Loads in Die Casting Using Computer Modeling and Dimensional Analysis

PREDICTING PARTING PLANE SEPARATION AND TIE BAR LOADS IN DIE CASTING USING COMPUTER MODELING AND DIMENSIONAL ANALYSIS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By Karthik S Murugesan M.S.

The Ohio State University 2008 *****

Dissertation Committee: Approved by: Dr. R. Allen Miller, Adviser Dr. Jerald Brevick Adviser Dr. Khalil KabiriBamoradian Graduate Program in Industrial and Systems Engineering

ABSTRACT

Die Casting dies and machines are high performance products that are subjected to clamp load, cavity pressure loads and thermal loads during normal operation and the dies and machine deflect under the action of these loads. The ability of the dies to withstand loads and preserve the integrity of the cavity dimensions depends on the structural design of the dies. Die castings dies are expensive products with long production lead times and the structural behavior of the dies has to be predicted at the design stage. The other common problem in die casting is the tie bar load imbalance.

The machine clamp load is distributed among the four tie bars depending upon the location of the dies and the location of the cavity center of pressure on the platen. Tie bar load imbalance causes the die parting surface to close unevenly and leads to problems such as flash and premature tie bar failure. The problem is over come by adjusting the length of the tie bars between the machine platens until all the tie bars carry equal loads. Tie bar load predictions are necessary to determine the individual length adjustments needed on each tie bars.

Numerical methods such as the finite element method are the most effective way to predict the distortion of the dies and the machine at the design stage. Performing a full FEA during the initial stages of the die design is time consuming and it is not cost effective. So off the shelf design tools such as closed form expressions, design charts

and guidelines are needed to make design improvements during the initial stages of the design.

In this dissertation research work the relative contribution of the major structural design variables of the die casting die and machine to the mechanical performance of the dies and machines was investigated using computational (FEA) experiments. The maximum parting plane separation was chosen as the performance measure for the structural behavior of the dies and the machine. The computational experiments were designed using Design of Experiments approach and closed form power law models were developed to predict the maximum cover and ejector side parting plane separation. The functional form for the power law model was obtained using dimensional analysis based on Pitheorem. These power law models were then used to explain the sensitivity of maximum parting plane separation to the design variables. The power law models can also be used to compare the performance of different dies and machines and make structural design improvements of the die. In addition a methodology to characterize the stiffness of the machine platens is also developed.

In the second part of the research power law models were developed based on dimensional analysis to predict the loads on the tie bars of the die casting machine as a function of the die location, the location of the cavity center of pressure, clamp load and the magnitude of cavity pressure. The power law model to predict die bar loads can be used to determine the length adjustments needed on the tie bar to balance the

tie bar loads. The relative contributions of the die location and cavity location on tie bar load imbalance were also studied using the exponents and coefficients of the power law model. The adequacy of the model was also studied by using tie bar load measurements from a die casting machine.

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Dedicated to my parents and my brother

ACKNOWLEDGMENT

My first and foremost thanks go to Dr. R. Allen Miller for providing me this wonderful opportunity to pursue my PhD under his guidance and support. I thank him for guiding my thought process by his constant questioning and constructive criticism. I thank him for his trust, patience and enthusiasm. Working with Dr. Miller was my most valuable academic and research experience.

I am very thankful to Dr. Khalil KabiriBamoradian for guiding me in the development of finite element models and for helping me with dimensional analysis and power law model development. I also thank him for teaching me the best finite element modeling practices and research methods. I cannot overemphasis the enormous amount of time that he spent for helping me during this research work.

I would like to thank Dr. Jerald Brevick for his contribution and support as a member of my dissertation committee. I also thank him for sharing his scientific knowledge and expertise in die casting throughout the course of my graduate study.

I thank Dr. Theodore Allen for being a member of my general exam committee and for the development of experimental design used in the tie bar load prediction model.

My special thanks to my former colleague Adham Ragab, who helped me with his valuable suggestions during the initial period of this research. I am grateful to my

colleague and friend Abelardo Garza for constantly motivating and encouraging me during my toughest times.

I am extremely grateful to CedricSize and Shih KwangChen for their laboratory support. The timely completion of this project would not have been possible without

ShihKwang’s help and support. My thanks to all the IWSE staff, particularly Darline

Wine for their administrative support.

I thank the US Department of Energy for the financial support they have provided for this project. I also thank the NADCA Computer Modeling Task force for their feedback and for their encouragement for this project.

VITA

April 8, 1979……………..Born, Coimbatore, India

2000 ……………………..B.E. Mechanical Engineering, Bharathiar University, India

2003 …………………….M.S. Engineering, Purdue University, Indianapolis, Indiana

20032008 ……………….Graduate Research Associate, The Ohio State University, Columbus, Ohio PUBLICATIONS

1. K. Murugesan, A. Ragab, K. KabiriBamoradian, R. A. Miller, “Effect of Die, Cavity and Toggle Locations on Tie bar Forces, Toggle Forces and Parting Plane Separation”, NADCA Proceedings, April 2005

2. K. Murugesan, A. Ragab, K. KabiriBamoradian, R. A. Miller, “A Model to Predict Tie Bar Load Imbalance”, NADCA Proceedings, April 2006

3. K. Murugesan, A. Ragab, K. KabiriBamoradian, R. A. Miller, “An Experimental Verification of the effect of Die Location on Tie Bar Load Imbalance”, NADCA Proceedings, May 2007

4. K. Murugesan, K. KabiriBamoradian, R. A. Miller, “Effect of support pillar patterns on Mechanical Performance of Ejector Side Dies”, NADCA Proceedings, May 2008 FIELDS OF STUDY

Major Field: Industrial and Systems Engineering (Manufacturing)

Minor Fields: Applied Statistics and Mathematics

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TABLE OF CONTENTS

Page

ABSTRACT...... II

ACKNOWLEDGMENT...... V

VITA...... VII

PUBLICATIONS ...... VII

FIELDS OF STUDY ...... VII

LIST OF FIGURES ...... XII

LIST OF TABLES ...... XV

CHAPTER 1 ...... 1

1.1 DIE CASTING DIES AND MACHINES ...... 2

1.2 MECHANICAL AND THERMAL LOADS INVOLVED IN DIE CASTING ...... 6

1.3 DIE DEFLECTIONS ...... 8

1.4 IMBALANCED LOADS ON MACHINE TIE BARS ...... 11

1.5 PROBLEM STATEMENT ...... 13

1.6 RESEARCH OBJECTIVES ...... 15

1.7 RESEARCH CONTRIBUTIONS ...... 16

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1.8 DISSERTATION OUTLINE ...... 18

CHAPTER 2 ...... 20

2.1 MATHEMATICAL MODELING AND ANALYSIS OF DIE CASTING PROCESS ...... 20 2.1.1 Fluid Flow, Thermal and Solidification Analysis ...... 21 2.1.2 Thermal Stresses and Distortion Analysis in Casting and Die...... 28 2.1.3 Models accounting for Structural Loads ...... 35

2.2 PARAMETRIC DIE DESIGN STUDIES ...... 49

2.3 RELEVANT RESEARCH IN INJECTION MOLDING ...... 51

2.4 SUMMARY ...... 58

CHAPTER 3 ...... 59

3.1 INTRODUCTION ...... 59

3.2 FINITE ELEMENT MODELING ...... 60 3.2.1 Boundary Conditions and Constraints ...... 62 3.2.2 Loads and Assumptions ...... 65 3.2.3 Material Properties ...... 65 3.2.4 Finite Element Model Predictions...... 66 3.2.5 Effect of Element Types and Cover Platen Constraint on Model Predictions ...... 70

CHAPTER 4 ...... 75

4.1 INTRODUCTION ...... 75

4.2 DESIGN OF EXPERIMENTS ...... 76

4.3 DIMENSIONAL ANALYSIS AND EMPIRICAL CORRELATIONS ...... 81 4.3.1 Determination of the Model form and NonDimensional Parameters for Predicting Parting Plane Separation...... 83 4.3.2 Empirical Correlation to Predict Ejector Side Parting Surface Separation 89 4.3.3 Empirical Correlation to Predict Cover Side Parting Surface Separation.. 95

4.4 SENSITIVITY OF PARTING PLANE SEPARATION TO VARIATIONS IN STRUCTURAL

DESIGN PARAMETERS ...... 97 4.4.1 Response Surface Plots of NonDimensional Parameters...... 97 4.4.2 Response Surface Plots of Explicit Design Variables...... 103

4.5 MODEL ADEQUACY ...... 107 4.5.1 Rules to Characterize the Spans between Pillars and the Spans between Pillars and Rails...... 112

4.6 PLATEN STIFFNESS CHARACTERIZATION AND DETERMINATION OF PLATEN

THICKNESS PARAMETER TO BE USED IN POWER LAW MODELS ...... 114 4.6.1 Methodology to Determine Equivalent Cover Platen Thickness...... 117 4.6.2 Methodology to Determine Equivalent Ejector Platen Thickness ...... 120 4.6.3 Methodology to Determine Equivalent Thicknesses for Platens with Different Toggle Locations ...... 122

4.7 DETERMINATION OF EQUIVALENT STIFFNESS OF A DIE CASTING MACHINE USING

A LUMPED ELEMENT MODEL ...... 125

4.8 SUMMARY ...... 131

CHAPTER 5 ...... 133

5.1 INTRODUCTION ...... 133

5.2 DESIGN OF EXPERIMENTS ...... 134

5.3 DIMENSIONAL ANALYSIS AND EMPIRICAL CORRELATION TO PREDICT TIE BAR

LOADS ...... 138

5.4 MODEL ADEQUACY ...... 146 5.4.1 Model Adequacy Study using Experimental Measurements ...... 150 5.4.2 Comparison of Experimental Measurements and Model Predictions...... 154

5.5 RESPONSE SURFACE PLOTS FOR THE EFFECT OF DIE LOCATION AND CAVITY

LOCATION ON TIE BAR LOADS ...... 157

5.6 SUMMARY AND CONCLUSIONS ...... 161

CHAPTER 6 ...... 163

6.1 CONCLUSIONS FROM THE POWER LAWS TO PREDICT PARTING PLANE

SEPARATION ...... 163

6.2 CONCLUSIONS FROM THE MACHINE CHARACTERIZATION STUDY ...... 165

6.3 CONCLUSIONS FROM THE POWER LAWS TO PREDICT TIE BAR LOADS ...... 166

6.4 FUTURE WORK ...... 168

REFERENCES...... 170

APPENDIX A ...... 176

A.1. PROCEDURE TO SELECT SAMPLE NODES AND PREDICT PURE DISTORTION OF

THE PARTING PLANE FROM FINITE ELEMENT MODELS ...... 179

A.2. ALTERNATE METHOD TO REMOVE PSEUDO RIGID BODY MOTION USING A

LOCAL COORDINATE SYSTEM IN ABAQUS...... 181

APPENDIX B ...... 183

B.1 LINEAR MODEL FOR TOP TIE BAR ...... 183

B.2 LINEAR MODEL FOR BOTTOM TIE BAR ...... 183

LIST OF FIGURES

Figure Page Figure 1.1: Schematic of an openclose die [5]...... 3 Figure 1.2: Schematic of a Hot Chamber Die Casting Machine [66] ...... 5 Figure 1.3: Schematic of a Cold Chamber Die Casting Machine [66] ...... 5 Figure 1.4: Free Body Diagram of Cover and Ejector Dies...... 10 Figure 1.5: Free Body Diagram of Die Casting Machine and the Die...... 12 Figure 2.1: Schematic of Toggle Spring Platen Model [1] ...... 38 Figure 2.2: Schematic of the Models Considered in [30] ...... 41 Figure 2.3: Schematic of Pistons and Connecting Rods of Injection Molding Machine Clamping Mechanism [50]...... 53 Figure 2.4: Representation of the Clamping Mechanism in Multibody Dynamics Simulation [50]...... 54 Figure 2.5: Schematic of the procedure to estimate mold deflection [52]...... 56 Figure 2.6: Mold Spring Diagram used to Estimate Gap Formation [52]...... 57 Figure 3.1: Geometry of the Part used in the Study...... 61 Figure 3.2: Schematic of the Finite Element Model ...... 62 Figure 3.3: Boundary Conditions used between Cover Platen and Base...... 64 Figure 3.4: Schematic of the Finite Element Model showing the Location of Tie Bar Load Prediction ...... 67 Figure 3.5: Illustration of Ejector and Cover Side Parting Plane Separation...... 68 Figure 3.6: Pseudo Rigid Body Movement Caused by Stretching of Tie Bars...... 69 Figure 3.7: Deflection Plots of the Cover Platen ...... 73

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Figure 3.8: Deformed Plot of Cover Platen Superimposed on the Undeformed Plot.73 Figure 4.1: Side View of an Ejector Die used in the Study ...... 77 Figure 4.2: Schematic of Pillar Patterns used in the Study...... 78 Figure 4.3: Length Scales Representing the Unsupported Span behind the ...... 91 Figure 4.4: NonDimensional Cover Separation vs. NonDimensional Die Length

(П 4) and Distance between Tie Bars (П 1) ...... 98 Figure 4.5: NonDimensional Cover Separation vs. NonDimensional Die Length

(П 4) and Platen Thickness (П 2) ...... 99 Figure 4.6: NonDimensional Ejector Separation vs...... 100 Figure 4.7: Non Dimensional Ejector Separation vs. Non Dimensional Platen

Thickness (П 12) and Weighted Average of Spans (П 4a + 1.6П 4b )...... 101 Figure 4.8: NonDimensional Ejector Separation vs...... 102 Figure 4.9: Maximum Cover Separation vs. Die Thickness & Die Length...... 104 Figure 4.10: Maximum Cover Separation vs. Die Thickness & Platen Thickness...104 Figure 4.11: Maximum Ejector Separation vs. Die Thickness & Pillar Diameter....105 Figure 4.12: Maximum Ejector Separation vs. Die Thickness & Die Length ...... 106 Figure 4.13: Maximum Ejector Separation vs. Die Thickness & Die Length ...... 107 Figure 4.14: Pillar Arrangement Patterns in the three Test Cases Used to Study the Adequacy of the Power Law Models ...... 108 Figure 4.15: Illustration of Rules for Characterizing the Spans behind the Ejector Die ...... 113 Figure 4.16: Schematic of the Platen Design Chosen to Demonstrate Stiffness Characterization Methodology...... 116 Figure 4.17: Dimensions of the Platen Design Chosen to Demonstrate Stiffness Characterization Methodology (All Dimensions in Inches) ...... 117 Figure 4.18: Schematic of Finite Element Model Used to Determine the Cover Platen Stiffness...... 119 Figure 4.19: Deflection Vs Load Curves for Cover Platen...... 119

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Figure 4.20: Schematic of the Finite Element Model Used to Determine the Ejector Platen Stiffness...... 121 Figure 4.21: Deflection Vs Load Curves for Ejector Platens...... 122 Figure 4.22: Schematic of Finite Element Models used to determine the Stiffness of Four Toggle (Right) and Two Toggle (Left) Ejector Platens...... 123 Figure 4.23: Deflection Vs Load Curves for Two Toggle and Four Toggle Platens 124 Figure 4.24: Spring Stiffness Diagram for the Die and the Machine under Clamp Load...... 126 Figure 4.25: Spring Diagram with Clamp and Pressure Loads...... 129 Figure 5.1: Coordinate System and Tie bar Labels viewed from inside face of Cover Platen...... 135 Figure 5.2: Schematic of the Finite Element Model of the 1000 Ton Machine and 250 Ton Machine Used for Model Adequacy Study [1]...... 148 Figure 5.3: Schematic of the test die on the machine platens ...... 152 Figure 5.4: Schematic of the locations of tie bars, strain gauges and coordinate system, viewed from front of cover platen...... 152 Figure 5.5: Tie bar Load Measurements vs. Predictions from the Regression Model ...... 156 Figure 5.6: Effect of Cavity Location on Tie Bar Load...... 158 Figure 5.7: Effect of Cavity Location on Tie Bar Load...... 159 Figure 5.8: Effect of Die Location on Tie Bar Load...... 160 Figure 5.9: Effect of Die Location on Tie Bar Load...... 160 Figure A.1: Contact Pressure Plot...... 180 Figure A.2: Ejector Side Separation Obtained by Sampling Nodes in Contact Regions Only ...... 181 Figure A.3: Separation With Respect to a Local Coordinate System on the Parting Surface...... 182

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LIST OF TABLES

Table Page

Table 3.1: Specifications of the Machine used in the Study...... 61 Table 3.2: Material Properties for ST4140...... 66 Table 3.3: Die and Machine Parameters used in the Simulations to Test the Effect of Cover Platen Constraint Type and Element Type...... 71 Table 3.4: Effect of Element Type and Cover Platen Boundary Condition on Parting Surface Separation Prediction ...... 72 Table 3.5: Effect of Element Type and Cover Platen Boundary Condition on Tie Bar Load Prediction ...... 74 Table 4.1: Factors used in Design of Experiments...... 76 Table 4.2: Response Surface Experimental Array ...... 79 Table 4.3: Summary of Element Types and Constraints used in Computational Experiments...... 81 Table 4.4: NonDimensional Structural Design Parameters ...... 87 Table 4.5: Parameter Estimates for Ejector Side Fit ...... 90 Table 4.6: Parameter Estimates for Cover Side Fit...... 95 Table 4.7: Summary of the Parameter Values used for Test Cases ...... 109 Table 4.8: Comparison of FEA and Power Law Model Predictions for Ejector Side for the Test Cases ...... 110 Table 4.9: Comparison of FEA and Power Law Model Predictions for Ejector Side for the Test Cases ...... 111

Table 4.10: Stiffness Values to be used in Lumped Element Model for the Dies and Machine Parts...... 127 Table 5.1: Description of Variables used for Tie Bar Load Prediction Model Development ...... 135 Table 5.2: Experimental Array Used for Tie bar Load Prediction Model Development ...... 138 Table 5.3: Non Dimensional Parameters used in Tie Bar Load Prediction Model...142 Table 5.4: Parameter Estimates for Top Tie Bar Model Fit...... 144 Table 5.5: Parameter Estimates for Bottom Tie Bar Model Fit ...... 144 Table 5.6: Summary of Finite Element Models used for Model Adequacy Study...147 Table 5.7: Comparison of Model Predictions for a 3500 Ton Machine ...... 148 Table 5.8: Comparison of Model Predictions for a 1000 Ton Machine ...... 150 Table 5.9: Comparison of Model Predictions for a 250 Ton Machine (DPX=0”, DPY=3.14”, CPX=0”, CPY=0.423”, CPR=10000 PSI) ...... 150 Table 5.10: Experimental Array...... 153 Table 5.11: Comparison of Tie bar Load Measurements and Tie bar Load Predictions from the Regression Model...... 155 Table 5.12: Difference between Measurements and Model Predictions...... 157

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CHAPTER 1

INTRODUCTION

Die casting is a net shape manufacturing process in which parts with complex geometries are produced by injecting molten metal into steel molds/dies under high pressure. The molten metal is held in the die cavity until it solidifies and the final part is ejected out of the dies and the process is repeated over thousands of cycle. Die casting offers competitive advantage over other netshape manufacturing process such as forging and stamping with its ability to produce parts with complex geometric features, high surface finish and tight dimensional tolerances. A casting that is distorted and fails to meet the specified dimensional requirements is scrapped and remelted resulting in a decrease in process yield, loss of production time, labor and increase in cost and energy consumption associated with remelting and rework.

One of the major factors that contribute to the dimensional inaccuracy of the casting is the elastic deformations of the die cavity caused by the thermo mechanical loads the dies are subjected to during normal operation. Deformations caused by repeated loading and unloading of the die might also cause fatigue failure of the die. Die casting dies are expected to run several million cycles during their life time. High manufacturing costs prohibit prototyping and any serious die deformation or die

failure problems are not noticed until the first production run. A die can cost anywhere between $50000 to $1000000 and the delivery times ranges from 312 months depending upon the complexity of the dies. Therefore it is extremely important that the die distortion be predicted and controlled at the design stage.

Numerical modeling of the die casting process is the most efficient way to predict the distortion of the dies. However, the die designer should have a thorough understanding of the physical phenomenon involved in die distortion and the mathematical theory employed in the numerical models to efficiently model the die distortion phenomenon. During the initial stages of the die design, where many primitive design iterations have to be performed quickly, numerical modeling techniques become time consuming and they are not cost effective. Therefore the goal of this research work is to develop off the shelf tools and guidelines for structural design of die casting dies.

1.1 Die Casting Dies and Machines

Die casting dies are complex high performance structures with several individual components and mechanisms that are assembled together to meet its functional requirements. The dies are usually made of at least two halves, the fixed half or the cover half and the moving half or the ejector half to facilitate the removal of the solidified casting. The surface where the two halves of the dies meet is called the die parting surface. The dies can also have moving mechanisms such as slides and cores

to produce undercuts and holes in the casting. Dies with these moving mechanisms are called nonopen close dies and dies without any of these moving mechanisms are called open close dies. A schematic of an openclose die is shown in Figure 1.1. A metal feeding system composed of runners and gates are cut out on the die to evenly distribute the molten metal into the die cavity at required velocity and pressure.

Cooling lines are provided to extract the heat from the solidifying metal and the location and design of cooling lines depend on the geometry of the part.

Figure 1.1: Schematic of an openclose die [5]

Die casting machines provide the clamping mechanism to hold the die halves together when they are subjected to high pressure loads from the incoming molten metal. They also provide the injection mechanism that injects the molten metal into the die cavity at required velocity and pressure. Die casting machines are typically rated based on the clamping force it can generate and the machine rating ranges from 50 tons to 4000

tons. Regardless of their size, die casting machines are usually classified as hot chamber machines or cold chamber machines depending on the method used to inject moltenmetal into a die. A schematic of a hot chamber machine is shown in Figure 1.2.

The injection mechanism of a hot chamber machine is immersed in the molten metal bath of a metal holding furnace. The furnace is attached to the machine by a metal feed system called a gooseneck. As the injection cylinder plunger rises, a port in the injection cylinder opens, allowing molten metal to fill the cylinder. As the plunger moves downward it seals the port and forces molten metal through the gooseneck and nozzle into the die cavity. Hot chamber machines are used primarily for low melting point alloys such as zinc and copper that do not readily attack and erode the components in the machine’s injection system. In a cold chamber machine, the molten metal is poured into a cylindrical sleeve and a hydraulically operated plunger seals the cold chamber port and forces metal into the die at high pressures. A schematic of the cold chamber machine is shown in Figure 1.3. Cold chamber machines are used for alloys such as aluminum, magnesium and other alloys that chemically attack steel.

Figure 1.2: Schematic of a Hot Chamber Die Casting Machine [66]

Figure 1.3: Schematic of a Cold Chamber Die Casting Machine [66]

A typical die casting machine consists of a cover/fixed platen, a movable/ejector platen, a rear platen and four tie bars stretching between the cover and rear platens.

The tie bars are secured to the cover and rear platens and the ejector platen is free to slide on the tie bars. The cover and ejector halves of the dies are mounted on the cover and ejector platens of the die casting machine respectively. The toggle mechanism that clamps the dies together is provided between the cover and rear platens and the injection mechanism is provided behind the cover platen.

1.2 Mechanical and Thermal Loads involved in Die Casting

The major thermal and mechanical loads that act on a die casting die during normal operation are (1) The clamp load (2) Injection pressure (3) the impact load caused by the sudden deceleration of the plunger mechanism at end of fill (4) Intensification pressure and (5) the heat released by the molten metal during filling and solidification.

The clamp load applied behind the ejector platen causes the tie bars to stretch between the cover and rear platens and tensile forces are developed on the tie bars.

Thus the clamp load is transmitted to the die parting surfaces through the machine tie bars and the cover platen. The clamp load holds the two die halves together and prevents the dies from opening up during the metal injection stage. The clamp load is kept constant through out the casting cycle.

The injection pressure is applied to the molten metal by the hydraulic plunger mechanism to fill the cavity in fractions of seconds before the metal solidifies. Die

castings are characterized by thin wall sections and the fill time is usually in the order of milliseconds. If premature solidification occurs along the metal flow front, it results in an incompletely filled cavity and leads to defective parts. Therefore to avoid premature solidification the entire filling process is completed in a fraction of seconds by injecting the metal at pressures as high as 10000 PSI.

After the injection stage a pressure spike is given to the metal to force the metal into far ends of the cavity. This pressure spike also helps to reduce shrinkage defects by forcing the remaining liquid metal to solidifying areas. It is typical to have intensification pressure values twice as high as the injection pressure.

The molten metal is pushed into the die cavity by the plunger mechanism which is driven by a hydraulic piston. Initially the metal is pushed at a lower velocity to avoid gas entrapment and this phase is referred to as slow shot phase. When a certain volume of the shot sleeve is filled, the plunger velocity is increased to achieve the required metal pressure at the gates. This phase is called fast shot phase. After the fast shot phase, the plunger and the hydraulic piston are brought to a sudden stop which causes an impact loading in the die cavity.

Finally heat is released by the molten metal during the filling and solidification. The loading sequence is repeated over thousands of cycles during the entire lifetime of the die.

1.3 Die Deflections

The cyclic thermal and mechanical loads described in previous section, cause deflections of the dies during each cycle. The elastic deflections could cause dimensional variations in the die cavity and result in parts that are out of dimensional tolerances. The deflections along the edges of the cavity could cause the molten metal to escape out of the cavity and this phenomenon is called flash. Die flashing is a major problem in die casting resulting in increased cycle time and increased cost associated with the removal of flash from the final casting. Die flashing could also affect the across parting plane dimensions of the casting and it could cause uneven loading of various machine components. Flash might also get into the clearance holes between the moving mechanisms of the dies, causing process problems. As a worst case scenario, the repeated loading and unloading of the die could also lead to fatigue failure of the die.

The effect of mechanical loads on the dies is illustrated in the free body diagram shown in Figure 1.4. Ideally the parting surfaces of the die casting dies are machined flat to mate perfectly in the absence of thermal and mechanical loads. In the first stage of casting cycle (Figure 1.4a) when the clamp load is applied behind the dies, the die parting surfaces are subjected to distributed compressive forces that are equivalent to the applied clamp load. The clamp load applied between the rear and ejector platens is transmitted to the cover die through the cover platen, whereas on the ejector side the load path is through the pillars and/or rail support. The pillars and rails behind the

ejector die are shown in Figure 1.4. In the next stage of the casting cycle (Figure

1.4b), when the molten metal is injected at high pressure, the pressure load on the cavity relieves some of the compressive forces at the parting surface. The clamp force in excess of the cavity pressure load remains at the parting surface as shown in Figure

1.4c. In Figure 1.4, the clamp and pressure loads are assumed to be static.

During the metal injection and solidification stages the dies are also subjected to thermal loads from the solidifying metal. Due to the heat flux from the metal, the regions of dies that are close to the cavity attains higher temperatures than the regions far away from the cavity causing uneven thermal expansion of the die. The uneven thermal expansion could increase and redistribute the compressive stresses at the parting surface. However the uneven growth and thermal distortion of the die depends on the geometry of the casting and this issue has to be addressed by ensuring a good part design and by proper placement of cooling lines.

Figure 1.4: Free Body Diagram of Cover and Ejector Dies

If the dies were perfectly rigid as shown in Figure 1.4, the pressure load would only tend to relieve the compressive forces at the parting surface. Since the dies are not perfectly rigid, they will undergo elastic deformation due to the clamp and pressure loads. The magnitude of elastic deflections of the dies under clamp and pressure loads will depend on the structural design of the dies and the rigidity of the support provided by the machine platens. The dies are similar to flat plate structures resting on elastic supports. The degree of deflection of these elastic supports will also add up

to the deflection of the dies. On the cover side, the die is mounted directly on the cover platen and the cover platen serves as the elastic support for the die under clamp and pressure loads. A stiffer cover platen will be subjected to less bending and hence the support area available for the cover die will be more. In other words the stiffness of the cover platen and the die foot print determines the unsupported span behind the cover die. Therefore the deflection on the cover side is mostly dependent on the stiffness of the cover platen and the die foot print. But on the ejector side, due to the presence of ejection mechanisms the rails and support pillars provide the necessary support between the ejector platen and the die. Therefore the deflection on the ejector side will largely be a function of the thickness of the die and the insert and the location, size and number of support pillars. The deflections due to mechanical loads can be controlled by ensuring a good structural design of the dies and selecting a suitable machine with appropriate stiffness characteristics.

1.4 Imbalanced loads on Machine tie bars

The other problem that is often encountered during die casting operation is unbalanced loads on the tie bars of the die casting machine. The machine tie bars are constrained to the cover and rear platens and the ejector platen is free to slide on the tie bars. When the clamp load is applied the tie bars stretch between the cover and rear platen and tensile forces are developed on the tie bars. This is illustrated in

Figure 1.5 which shows the free body diagram of a die casting die and machine with the clamp and pressure loads acting on them. Figure 1.5 shows a case where the

geometric center of the dies and cavity center of pressure coincide with geometric center of the platen. If the die is centered on the platen the moments about all four tie bars will be equal and hence all four tie bars will carry equal loads.

Figure 1.5: Free Body Diagram of Die Casting Machine and the Die

However if the die is located off center on the platen it results in unequal moments and loads on the four tie bars. As mentioned before if the dies and machine are perfectly rigid the pressure load will only tend to reduce the compressive forces on the die parting surface and it will not alter the tie bar loads. Since the machine is not rigid the tie bars will be stretched further slightly after injection stage and the tie bar loads increase further. The increase in tie bar loads after the injection stage will be proportional to the stiffness of the tie bars and the location of the cavity center of pressure with respect to the platen center.

If the four tie bars do not carry equal loads, the dies close unevenly at the die parting surface and die flashing may occur. In extreme cases, poorly balanced tie bar loads could also lead to tie bar failure. The common practice to overcome the tie bar load imbalance problem is to adjust the length of the tie bars between the platens so that all the four tie bars carry equal loads. In such a case the minimum clamp load required to hold the dies together will be higher than the one that would be needed if the dies were centered on the platen thus limiting the capacity of the machine.

1.5 Problem Statement

Numerical modeling of the die casting process using techniques such as the finite element method is the most efficient way to predict die deflections and problems such as flash at the design stage and to enable suitable design modifications. Die distortion modeling using finite element method has been an active area of research at the

Center for Die Casting at Ohio State University. Numerous literatures have been published by this die casting research group on realistic approximations of die/machine geometry, loads and boundary conditions that can be used in the finite element models to predict die distortion.

However, to make efficient design improvements based on the numerical model predictions, the die designer should have a thorough understanding of how the various structural design variables contribute to the die deflection. Therefore design guidelines explaining the sensitivity of die deflection to various design parameters are essential. Moreover, some die designers lack FEA support and they are solely

dependent on tools such as closed form analytical expressions, design guidelines, charts etc. An initial set of guidelines was developed in previous research studies [1

4]. Computational experiments were conducted to study the sensitivity of the parting plane separation to the major structural die/machine design variables viz., platen thickness, die thickness, insert thickness and the location of the die with respect to the platen center. Functional relationships describing the relative contribution of these variables to the parting plane separation on cover and ejector sides were established using polynomial approximations. One of the major conclusions from those studies was that the cover platen thickness is the dominant factor on the cover side that contributes to the parting plane separation. The variables that characterize the unsupported span behind the ejector die such as the size, number and location of pillars supports were not controlled in those computational experiments. Hence the contribution of the ejector side design variables to the parting plane separation could not be confirmed from those studies and it remained an open issue.

The other issue that needs to be investigated is the relative contributions of the location of the dies on the platen and the location of cavity center of pressure to the tie bar load distribution. To determine the amount of length adjustments needed on the tie bars to balance the loads, Die Casting process engineers usually need a prior estimate of the loads on the tie bars. Herman [5] presents a methodology based on moment equilibrium calculations to estimate the loads on the tie bars. This method assumes that the cavity load is equivalent to the machine clamp load. The location of cavity center of pressure is considered for the moment calculations and location of the

die on the platen is completely ignored. This method also assumes the dies and the machine as perfectly rigid bodies. The mathematical formulation of the tie bar load prediction problem makes it a statically indeterminate system with four unknown loads and three equations viz., a force balance equation in tie bar direction and two moment balance equations in directions perpendicular to the tie bar direction. Herman

[5] approaches this problem by applying a moment balance on the operator and helper side tie bars and estimates the sum of the top and bottom tie bars on both sides.

The total load on each side of the machine are then split between the top and bottom tie bars by assuming that the sum of the loads on top and bottom tie bars on either side of the machine are equal to the cavity load. This approach clearly violates the force equilibrium constraint. Due to these drawbacks this approach generally will produce inaccurate predictions of the tie bar loads.

1.6 Research objectives

One of the main objectives of this research is to study the relative contributions of the cover and ejector side structural design variables to die deflection and develop design guidelines and tools to aid in structural design of the die casting dies. It was decided to develop closed form approximations of the relationship between the structural design parameters and the maximum distortion on the cover and ejector side parting surfaces. These expressions can also be used to make a quick estimate of the magnitude of the die deflections at the initial design stage. The closed form expressions can be used to obtain the best estimate of die design parameters that

would result in minimal parting plane separation. The other purpose of developing these closed form expressions is to obtain the sensitivity of the maximum cover and ejector parting surfaces to variations in the structural design parameters and gain an understanding of the relative contributions of these design variables to die distortion.

The second objective of the research is to develop a closed form expression to predict the loads on the tie bars of the die casting machine by taking into account the location of the dies on the platens, location of cavity center of pressure with respect to the platen center, the clamp load, the cavity pressure load and the rigidity of the die and the machine.

1.7 Research Contributions

One of the major contributions of this research work is the development of nonlinear power law models to predict the maximum deflection on the cover and ejector side parting surfaces. Though linear polynomial models have been developed in previous research studies to predict the maximum deflection on the cover and ejector parting surfaces, these polynomial models produced inaccurate predictions in certain regions of the design space. Power law models were obtained using dimensional analysis based on Buckingham Pitheorem. The use of non dimensional parameters and power law models gives a better understanding of the relationship among the various physical and geometric parameters that affect the parting plane separation and it also increases the degree of confidence in the model predictions even outside the design space considered in this study. The power law models were also used to generate

response surface plots describing the behavior of the function in response to variations in design parameters. These plots will be useful to gain a better understanding of the relative effects of the design variables on maximum parting plane separation. Since the power law models represent the stiffness characteristics of the die/machine system at the specified level of parameters, these models can be used to evaluate and compare the structural performance of different die/machine systems.

The second major contribution of this work is a methodology to characterize the stiffness of machine platens of different sizes and designs. The power law models to predict parting plane separation were developed based on an 8.9 MN (1000 ton) four toggle machine. The stiffness of the machine was represented in the power law models by the platen thickness and the distance between the tie bars. Using the platen thickness characterization method, an equivalent platen thickness parameter can be estimated for cast platens with ribs and also for two toggle platens. The equivalent platen thickness parameter can be used in the power law models to predict maximum parting plane separation and understand the performance of the die/machine. A one degree of freedom lumped stiffness model for the die/machine is also developed to determine the total stiffness of the die and machine. The lumped model also provides an understanding of how the interaction between various machine components and the dies contribute to the total stiffness of the die.

The other major contribution of this research work is the development of a closed power law model to predict the loads on the tie bars of the die casting machine. The

power law model form to predict the tie bar loads were also obtained using dimensional analysis. Unlike Herman’s approach this model takes into account the location of the dies on the platen, the location of the cavity center of pressure, the magnitude of the cavity pressure and the clamp load. The model adequacy has also been verified using tie bar load measurements form a die casting machine.

1.8 Dissertation Outline

The dissertation is organized into 6 Chapters. In the second chapter some of the recent scientific literature on die casting die design is reviewed. A review of relevant research studies in other casting processes and injection molding are also included in chapter 2. Chapter 3 presents the methodology adopted in this research. The finite element modeling procedure that was used to predict parting plane separation and tie bar loads are presented. The assumptions on loads and boundary conditions used in the finite element model are discussed.

The power law models that were developed to predict the maximum parting plane separation on the cover and ejector side are presented in Chapter 4. The experimental design, dimensional analysis, assumptions on the model form and the model development procedure are all presented in this chapter. The sensitivity of parting plane separation to variations in the design variables are described using the power law models and response surface plots. Chapter 5 describes the computational experiments that were conducted to develop the power law model to predict the loads on the tie bars of the die casting machine. Dimensional analysis and the nonlinear

model fitting procedures are also presented. Comparison between the power law predictions and FEA predictions for machines of different designs and sizes is also presented and the experimental work conducted to test the adequacy of the tie bar load prediction model is also discussed in Chapter 5. The results and conclusions of this dissertation work are summarized in chapter 6 and recommendations for structural die design are presented in this chapter.

CHAPTER 2

LITERATURE REVIEW

This chapter provides an overview of recent developments in numerical modeling of die casting and related processes and the application of these methods in die design.

Some of the relevant analytical and experimental work in die design and analysis is also included in the review. Finally some related studies on the mechanical performance of injection molding die and machine are presented.

2.1 Mathematical Modeling and Analysis of Die Casting Process

A die casting die has to meet various functional requirements: (1) Distribute the molten metal uniformly into the cavity, (2) Should allow efficient removal of heat from the molten metal and (3) Preserve the dimensional integrity of the die cavity.

The die design task is decomposed into these functional elements and design solutions that meet each of these functional requirements are sought based on the physical principles that govern these functions. There are a variety of physical phenomena that takes place during the die casting process such as metal flow, heat transfer, solidification and thermo mechanical distortion of the die, the casting and the machine. Decomposing the design tasks establishes a clear boundary between these phenomena and the analyst solves the mathematical governing equations of either one

or more of the phenomenon to obtain a design solution. Numerical techniques such as finite element method, finite difference method and boundary element method are being extensively used to model and solve these governing equations.

The focus of the die casting research community has been mostly on developing numerical/computational methods to solve heat transfer, fluid flow, solidification and thermal distortion related problems. Very few researchers have paid attention to the role of mechanical loads such as clamping force and cavity pressure in die and casting distortion. Even in the models where mechanical loads are considered, not all of them account for the stiffness of the machine parts such as the platens, tie bars and the toggle mechanism. Some of the recent improvements in the numerical modeling methods in die casting and other similar processes in heat transfer, fluid flow and structural distortion are reviewed here.

2.1.1 Fluid Flow, Thermal and Solidification Analysis

The analysis of fluid flow, heat transfer and solidification phenomenon can predict potential problems and defects in the casting such as gas porosity, shrinkage porosity, hot spots, hot tears, etc and these types of analysis helps in proper design of cooling lines, metal feeding systems and the casting geometry. Die filling analysis is usually carried out to track the advancing flow front and predict possibilities for gas entrapment in the cavity. Fluid flow analysis coupled with heat transfer helps to identify other flow related problems such as a cold shut. The temperature profile obtained at the end of filling can be used for further solidification analysis to identify

potential problems such as hot spots and uneven cooling. To model the fluid flow in casting processes the fluid is usually assumed to be newtonian with laminar or turbulent behavior. A complete flow and solidification model should solve for the

NavierStokes equation, the continuity equation and energy equation simultaneously.

The continuity or mass conservation equation, the Navier stokes equation and the energy equation are given as

∇ ⋅ u = 0 (21)

∂u T ρ + u ⋅ ∇u = −∇p + ∇ ⋅ (∇u + ()∇u )+ ρb (22) ∂t

 ∂T  ρC p  + u ⋅ ∇T  = ∇ ()k. ∇T + Qgen (23)  ∂t 

Where, ρ is the density, u is the velocity vector, t is time, P is pressure, is viscosity, b is the body force, Cp is the specific heat, T is the temperature, K is the thermal conductivity and Qgen is the heat source term.

For the solidification problem, the heat source term in the energy equation, Qgen represents the latent heat of fusion evolved during solidification and it is given by

df Q = L s (24) gen dt

Where, L is the latent heat, f s is the fraction of solid. The evolution of the fraction of solids for a particular alloy is usually determined by microscopic models such as

Lever or Scheil equations. The release of latent heat causes an increase in the enthalpy and hence the latent heat can be accounted for in the model using the following relation

H(t + t) − H (t) C = (25) p []T ()()t + t − T t

This model is often referred to in the literature as macroscopic specific heat model.

Since the specific heat capacity, C p is temperature dependent the energy equation becomes a second order nonlinear partial differential equation and analytical solutions can be obtained only by using further simplifying assumptions.

Nevertheless various numerical methods have been proposed in literature to obtain approximate solutions for solidification.

Hetu et al [6], developed a 2.5D shell finite element model to simulate both laminar and turbulent flow regimens in thin walled die castings. They also proposed 3D finite element model for thick walled castings. The 2.5D model used a shell mesh to model the thin wall sections with a thickness dimension for the shell elements and the model

does not account for the energy equation. In the 3D model both the part and the mold were included in the computational domain and a laminar flow was assumed. The

Navierstokes equations and energy equations representing the gas in the cavity and the liquid metal were solved over the entire computational domain to obtain the pressure, velocity and temperature distributions. The 3D model also accounted for the latent heat of fusion released during phase change. Once the velocity profile was obtained, the flow front was tracked using a fill factor F. A function F (x, t) denoting the distance from the interface between the liquid and the gas was defined over the entire computational domain to track the flow front. The flow front tracking equation is then given by

∂F + U.∇F = 0 (26) ∂t

Where, U is the velocity field obtained during each time step. The flow rate information based on the velocity field was used to evaluate the fill factor for each element using the mass conservation equation. When the value of F exceeded a critical value at any portion of the cavity, that portion was assumed to be filled.

Usually the fill factor ranges between 0 and 1, where zero indicates an empty element and 1 indicates a completely filled element. The iterative procedure was repeated until the filling of the die is completed.

Barone and Caulk [7], proposed a method in which the governing equations of fluid flow were integrated through the cavity thickness instead of modeling the entire three

dimensional flow region and the fluid flow was described in terms of bulk velocity and pressure resultant. The fill fraction approach shown in equation (26) was used to track the motion of the flow front. The cavity gas was modeled as an ideal gas that compresses adiabatically as the flow front advances and finite element method was used to solve the governing equations. But the energy equation was not taken into consideration in this method and isothermal fluid flow was assumed.

Jia [8], used a turbulent incompressible fluid flow model to obtain the pressure, temperature and velocity profiles during cavity filling and the governing equations were solved using a finite difference approach. Their method also employs the fill factor tracking approach to track the flow front. The gas in the cavity was modeled as ideal gas and the pressure obtained from the ideal gas equation was related to the pressure at the free surface of the flow front.

Kulasegaram et al [9], proposed a mesh free Lagrangian particle method to model the cavity fill in die casting processes. In this method the continuum is represented by a large set of particles where each particle is described by its mass, position vector and velocity vector. This method approximates the given function and its gradient in terms of the values of the function at number of neighboring particles and a kernel function. But this was a two dimensional model and the thermal effects were not accounted for in their model. The authors argue that this method is computationally less intensive than a finite element method.

Cleary et al [10] also suggested a similar particle hydrodynamics method to simulate the molten flow. This method solved the ordinary differential equations as opposed to solving the variational form described by Kulasegaram et al [9].

Bounds et al [11] described a hybrid numerical model in their paper to predict die temperature. This method utilized a boundary element formulation for the die blocks and finite element formulation for the casting. The temperature in the die blocks (Td) was represented as two additive components; a steady state component shown in equation (27) and a transient part shown in (28)

2 ∇∇∇ Td === 0 (27)

2 ∂Td αd (∇ Td )= (28) ∂t

Applying appropriate boundary conditions the first part was solved to obtain the steady state temperature. The second part was considered to be a transient perturbation about the steady state temperature. The steady state boundary conditions are taken as the time averaged boundary conditions as shown in (29) and the perturbed boundary condition is given by (210) qd = h(x,t) [Td (x,t) − T0(x,t)] (29)

pert qd = qd − qd (210)

The governing equation for heat transfer in casting is given by

 ∂∂∂Tc  ∇∇∇((()(kc∇∇∇Tc ))) === Cc   (211)  ∂∂∂t 

Cc is the effective heat capacitance that accounts for the latent heat of solidification of the casting. The first step to perform an analysis using this method is to perform a steady state analysis using time averaged boundary condition. In the next step a transient analysis over a single casting cycle is performed using perturbed boundary conditions. Using the data from the transient analysis the boundary condition for the next cycle of steady state analysis is updated. A number of transient cycles were analyzed until a convergent die and casting temperature were obtained. The model predictions showed good correlation with experimental thermocouple measurements.

However due to the updating of boundary conditions at the end of each cycle and the coupling between the FE and BE models, the computational cost and time were expensive.

Xiong et al [12] described three different methods to improve the computational efficiency and stability of the finite difference scheme used for the thermal analysis of the die casting. The temperature distribution in the casting and the die obtained from a filling analysis at the end of solidification was used as initial conditions. In their first method called a component wise splitting method, a simpler approximation in each dimension was used for the Laplacian operator in the heat conduction equation.

The method was shown to be unconditionally stable. In their second method, an

irregular mesh was used on the casting and the dies, wherein a finer mesh was used in the cavity areas with thin sections and coarser meshes were used in the areas far from the cavity. In the third method, the computational domain was divided into a transient area and steady state area. The transient area consists of the casting and the surface layer in the die beneath the cavity where the temperature variations within a cycle was faster. A smaller time step was used within the transient layer and a larger time stepping which was a multiple of the time step size in the transient area was used in the steady state area thus reducing the computational effort needed to solve the problem.

2.1.2 Thermal Stresses and Distortion Analysis in Casting and Die

The temperature predictions at the end of a thermal and solidification analysis can also be used as an input for subsequent stress analysis in the casting and the die.

Koric and Thomas [13] developed a computational algorithm to solve for the thermal stresses, strains and displacements during the solidification process in continuous casting. During each time step, the heat conduction equation including the latent heat term was solved to obtain the temperature history in the casting. Then the temperature values were used to predict stresses in the casting during the same time step. A small strain assumption was used and an elasticviscoplastic constitutive relation was used to predict the stresses in the casting. The total strain in the elastic visco plastic models is given by

ε&total = ε&elastic + ε&inelastic + ε&thermal (212)

The inelastic stress consists of the strain rate independent plasticity and the time

dependent creep and is defined by single variable, equivalent inelastic strain, ε& inelastic given by the relation,

3 σ′ij ()ε&inelastic = ε&inelastic (213) ij 2 σ

where σ′ij is the deviatoric stress component and σ is the average stress. To model the stresses in the mushy zone two different approaches were taken. In the first approach, an isotropic elasticperfectly plastic rateindependent model was chosen for elements where the temperature was greater than the solidus temperature for at least one material point. A small yield strength value of 0.03Mpa was chosen to eliminate the stresses in the mushy zone. In the second approach a viscoplastic relation was implemented as a penalty function to generate inelastic strain in the mushy zone in proportion to the difference between the average stress in the casting and a small yield stress.

Song et al [14] used a nonlinear coupled thermo mechanical finite element model to predict the distortion of the casting in rapid tooling. In their study both the ceramic matrix and the casting were treated as elasticplastic deformable bodies. The material properties of the casting were assumed to be temperature dependent and the material properties of the matrix were assumed to be temperature independent. First the heat transfer problem was solved to obtain the temperature distribution using the current boundary conditions. A thermal contact was allowed between the casting and the

matrix and the heat generated due to plastic thermal work dissipation, friction and the latent heat of solidification were accounted for in the thermal model. Then the mechanical problem was solved using the temperature distribution obtained from the heat transfer analysis. A mechanical contact boundary condition was also enabled between the casting and the die. The displacement and temperature fields were updated after the first increment to serve as a new configuration for the next increment and the procedure was repeated until convergence was achieved. A displacement checking criteria was used to check the convergence and an adaptive time stepping based on load control was used.

Dour [15], described a normalized approach to predict thermal stresses and distortion in die casting dies. The different variables involved in the thermomechanical problem was reduced to three non dimensional numbers viz., Biot Number (hL/K),

2 Normalized time (kt/L ) and Normalized stress (σ/[αET a/(1υ)]); where, h is the heat transfer coefficient, L is the characteristic length of the slab/die, K is the thermal conductivity of die material, k is the thermal diffusivity, t is the time spent between filling and air gap formation, T a is the temperature difference between the molten metal bath and the die, E is the young’s modulus, α is the coefficient of thermal expansion and υ is the Poisson ratio. These normalized parameters were used in the thermal and mechanical governing equations and analytical expressions for temperature normalized stress and radius of curvature was derived for one dimensional case assuming appropriate thermal and mechanical boundary conditions.

The stresses and radius of curvature predicted by the analytical expressions were

summarized in plots and graphs for various values of normalized time, Biot number and temperature. Since the graphs are obtained from the non dimensional form of the equations it is claimed that these charts can be used to diagnose the thermal stresses and distortion for any real casting condition.

Broucaret [16] used an inverse method based on Laplace transform of heat conduction equation to determine the heat flux exchanged between the die and the casting in gravity die casting process. One dimensional heat conduction was assumed and a time varying heat flux and a time varying temperature were chosen as the boundary conditions at the two ends of the die respectively. The heat conduction problem was solved in the frequency domain and an expression for temperature as a function of the heat flux at the front end of the slab and temperature at the back end of the slab are obtained. The expression for temperature is then transferred back to the time domain using inverse Laplace transform based on a numerical algorithm. The heat flux in the temperature expression is estimated using the temperature measurements and applying an inverse problem numerical algorithm. The temperature measurements were also used to estimate the thermal stresses in the die.

The die material was considered elastic and the thermal stresses were estimated using the following expression:

1   1 2h (3 x + h) 2h  σ(x,t) = αE − T(x,t) + T(x,t)dx + x + h T x,t dx (214)   ∫ 3 ∫()()  1− υ   2h 0 2h 0 

Where α is the coefficient of thermal expansion, E is the elastic modulus of the die material, T(x, t) is the die temperature and h is the distance from the die surface.

Based on the one dimensional stress model, the effect of initial temperature of the die and the die coating on the thermal stresses of the die was studied. It was concluded that the more conductive the coating and the lower the initial temperature of the die, higher the heat transfer and higher the stresses.

Lin [17] used finite element modeling and simulated annealing optimization method to find the optimal cooling system design parameters to minimize the thermal distortion of the dies. In this methodology a finite element software MARC/Mentant solid was used to solve the coupled thermomechanical problem and predict die distortion. A case study with fifteen different cooling system designs were analyzed and the parameters that were varied in the case study were the distance between the cooling lines and the cavity, the distance between the cooling lines and the diameter of the cooling lines. The results of these analyses were used to construct a neural network describing the relationship between the cooling system parameters and die deformation. The polynomial equations of the neural network were used as objective

functions for the constrained optimization problem to minimize the die deformation.

A simulated annealing method was then used to solve the optimization problem.

Srivastava et al [18] showed that the stress and strain predictions from finite element analysis can be utilized to predict the thermal fatigue life of a die casting dies. The model predictions were correlated with laboratory fatigue tests. In the laboratory fatigue tests, the fatigue loads were applied to test coupons by repeatedly immersing them in molten aluminum bath and then quenching them in water at room temperature. The dip times were varied and different loading conditions were studied.

The finite element model used to simulate the different test conditions and the finite element model was solved using DEFORM 3D package. The principal stresses were obtained from the finite element analysis. The directions of principal planes were determined and the laboratory fatigue tests showed that the direction of cracking was always perpendicular to the direction of the principal stress.

The stress values obtained from the finite element analysis was used to predict the fatigue life of the die using a modified coffin mason equation which is given by

1  Cε  η  f  N F =   (215)  αT − ()()1 − υ σ/ E − εT 

Where N f is the number of cycles to failure, α is the coefficient of thermal expansion,

υ is the poison ration, ε f is the fatigue strength coefficient, ε T is the net thermal strain and σ is the principal stress. The fatigue life predicted from the FEA model showed good correlation with the laboratory fatigue life tests that were conducted.

Sakhuja and Brevick [19] conducted a study to compare the thermal fatigue life of the die materials Haynes 230 and Inconel 617 that were used for copper die casting. A sequentially coupled 2D elastic thermo mechanical finite element model was used to predict the thermal stresses on the die. In this method a thermal analysis was performed first to obtain the die temperatures. The nodal die temperatures obtained from thermal analysis at appropriate sub steps were then used in the subsequent structural analysis. The method of universal slope was then used to estimate the thermal fatigue life of the dies. The expression used in universal slope method is given by

ε 0.6 z  σu  γ = D (N f ) + 3.5 (N f ) (216) 2  E 

Where D= ln (1RA) is the logarithmic ductility, RA is the percentage reduction in area and σ u is the ultimate tensile strength. Since the R A for the die materials were not available they were determined using constancy of volume relationship for plastic deformation. Park et al [20] used a elasticviscoplastic model to investigate the thermal distortion of copper mold used for thin slab steel castings. The mold wall temperatures measured from the plant were used to obtain the heat flux profiles using an inverse heat conduction model and this data was used in the die distortion analysis.

The model assumed isotropic hardening with a temperature dependent yield stress function. A good correlation was observed between the model predictions and die distortion measurements.

2.1.3 Models accounting for Structural Loads

AhuettGarza [21] and [22], conducted an elaborate study of the loads involved in die casting process and he concluded that die deflection simulations with reasonable resolution can be carried out by accounting for the clamping load, heat released during solidification, the intensification pressure, and the heat removed during lubricant spray. His study showed that the heat released during fill, the momentum during filling and the pressure surge at the end of fill can be ignored in the die deflection simulations and still results with reasonable accuracy and resolution can be achieved. By an order of magnitude analysis it was shown that the heat released during fill can be ignored when the ratio between half the thickness of the part and the fill time is greater than or equal to one seventh. This corresponds to a case where the solidification time is at least an order of magnitude greater than the fill time. The details of the scale analysis are also provided elsewhere [23]

Based on the results of his study an initial finite element modeling procedure was developed and tested [21], [24] and [25]. The preliminary model consisted of only the cover die, ejector die and the ejector support block. First a thermal analysis was carried out to obtain the nodal temperature values on the dies, which were later used in the stress analysis.

The heat flux from the part during solidification was calculated using

mt q'(t) = K.10 (217)

Where q’ (t) is the heat flux from the molten metal, t is the time between injection and ejection and K and m are empirical constants determined based on the injection and ejection temperature of the part, the surface area of the die in contact with the casting and the total heat released during solidification. The ejection temperature of the part was obtained from a solidification analysis using MAGMA software. The cavity pressure was modeled as a hydrostatic pressure with magnitude equal to that of intensification pressure. The clamp load was modeled as a pressure boundary condition behind the ejector support block. A rigid support was assumed behind the cover die and nodes on the back surface of the die were constrained in all directions.

The stiffness of the machine was not accounted for in this model. In a subsequent study Dedhia [26] compared the parting plane separation predictions of a model with rigid support behind the cover die versus the parting plane separation predictions from a model that accounted for the machine stiffness. Spring elements were used to account for the stiffness of the platens and the toggle mechanism. The clamp load was modeled by applying appropriate displacement boundary conditions to the spring elements that represented the toggle mechanism. The separation values at several locations along the edges of the cavity were used as a measure of die deflection. The maximum separation value was about 12% to 20% higher in the model with spring

elements than the values from the model with rigid support, depending upon the design features of the die.

Choudhary [27] developed a finite element model in which the three machine platens, the Cframe and the tie bars were modeled explicitly. The die was a dummy structure that consisted of two parallel plates connected by pillars on the four corners of the plates. A roller support was modeled at the bottom of cover and ejector platens. A support block was modeled at the bottom of the rear platen to prevent displacement in the vertical direction. A small sliding contact was defined at all interfaces. The nodes on the either end of the tie bars were tied to corresponding nodes on the ejector and rear platen. The clamp load was applied as a pressure boundary condition on the toggle blocks on the cover and rear platens. Thermal loads and cavity pressure were ignored in the model. The deflection of the cover platen was predicted at eight different locations behind the cover platen and the results were compared with corresponding values from the field data. The deflection pattern from the simulations was similar to the pattern observed on the field data. But the individual deflection values fell in the range of 10% to 15% of the observed field data. This model was fairly accurate given the various approximations to the boundary conditions in the model and the procedure followed to model the clamp load.

In another die distortion modeling study, Chayapathi [28] used a finite element model in which the tie bars were explicitly modeled and the toggle mechanism was represented by linear spring elements. The nodes on one end of the tie bar that are in

contact with the nodes in the cover platen were tied to the corresponding nodes on the cover platen. The nodes on the other end of the tie bar were constrained in all six degrees of freedom. The corner nodes on the bottom of the cover platen were constrained in vertical direction to prevent rigid body motion. The clamp load was applied by specifying displacements on the free end of the spring elements. The intensification pressure load was applied as a pressure boundary condition on the cavity surfaces. A schematic of the finite element model used in this study is shown in Figure 2.1

Figure 2.1: Schematic of Toggle Spring Platen Model [1]

Ragab et al [29] experimentally verified the adequacy of this finite element model shown in Figure 2.1 in predicting the contact loads between the dies and platens on the cover and ejector sides. The contact load between the platens and the dies were

measured using a total of 35 load cells, 18 load cells on the cover side and 17 load cells on the ejector side. The contact load was measured under two different loading conditions, under clamp load only and during actual casting operation. The load cells and the fixtures used in the experiments were also explicitly included in the finite element model. The summation of cover side load cell measurements decreased by

7% after intensification whereas the summation of cover side load cell predictions from simulation remained constant. On the ejector side the summation of load cell measurements and predictions remained constant. The difference between model predictions and measurements on the cover side were attributed to the fact that the model is stiffer than the die/machine actually is.

To address these observed differences between the model predictions and measurements on the cover side further modeling improvements were tested by

Arrambide [30]. Various machine components were included in the finite element models and the predictions were compared again to the experimental load cell measurements. Four different models were tested. The first model included the cover and ejector platen, dies, inserts, the load cells and fixtures all of them modeled using quadratic tetrahedral elements. The tie bars were modeled using beam elements, with the one end of the beam elements constrained to the cover platen and the other end was fixed in space. Several nodes on the bottom of the cover platen was constrained in vertical and tie bar directions. The clamp load was applied as a pressure boundary condition behind the ejector platen. A schematic of this simple model is shown in

Figure 2.2a. In the second model the rear platen was also included and the two ends

of the tie bars were constrained to the cover and rear platens. The toggle mechanism was represented by beam elements and the clamp load was applied by specifying appropriate temperature on these beam elements. This model is also shown in Figure

2.2b. In the third and fourth models the front support frame with the support frame was added to the previous two models as shown in Figure 2.2c and Figure 2.2d.

Comparison between load cell measurements and load cell predictions between simulations showed that the front support frame did not have any effect on the contact load between the dies and the platens. Including the rear platen and toggle mechanism in the model altered the load distribution on various load cell predictions by 2 34%, with an average of 11%. Also the full model showed good correlation with the experimental measurements. To test the adequacy of the full model to predict parting plane separation, a simplified model with dies, inserts and load cells only was built.

The clamp load was applied behind the load cells directly using the predictions from the full model and also the loads from the experimental measurements. The maximum separation showed a difference of 0.001” which falls within the resolution of the numerical simulation.

Figure 2.2: Schematic of the Models Considered in [30]

In all of the die distortion modeling studies discusses above, the intensification pressure was assumed to be hydrostatic and it was modeled as a pressure boundary condition on the cavity surfaces in the finite element models. In reality the liquid metal carries the hydrostatic pressure from the plunger mechanism and transfers it to the cavity surfaces. But the solid elements used in the structural finite element models cannot carry this hydrostatic pressure to the cavity.

GarzaDelgado [31] developed a two dimensional fluid structure interaction (FSI) finite element model using ADINA to predict die distortion. It was a fully coupled

thermo mechanical test model that was developed to gain an understanding of the capability of the fluid structure interaction model to predict die distortion. An FSI boundary condition was defined at the interface between the solid and liquid domain.

A solid domain was used to represent the dies and a liquid domain was used to represent the casting and the pressure load was simulated by specifying a nodal pressure boundary condition in the gate area of the fluid domain. The FSI method uses a conjugate heat transfer to calculate the heat fluxes across the interface and hence no interfacial heat transfer coefficients had to be defined between the liquid and the solid domain. Latent heat effects were included in the model by specifying temperature dependent specific heat curve. This model was developed for demonstration purposes and it is yet to be implemented in complex die distortion simulations.

Another important dynamic load that has been ignored in die distortion simulations is the dynamics impact load caused by the sudden deceleration of the plunger mechanism at the end of fill. Xue et al [32], attempted to predict the pressure distribution in the die cavity due to this impact loading using a CFD model. The goal was to use the pressure predictions from this CFD model to approximate the dynamic cavity pressure in structural die distortion simulations. FLOW3D was used to simulate the metal flow in the shot sleeve, runner and the dies. The molten aluminum was treated as slightly compressible fluid with temperature independent material properties and a Kε turbulent model was used in the simulation. Heat transfer between the metal and the dies was included in the model and the heat conduction in

the die was neglected. Pressure history at different cavity locations was investigated.

The pressure spike was found to be more than twice of the intensification pressure that is normally used in the production of this experimental cast part used in this study. It was also observed that the pressure within the cavity was almost uniform and the maximum pressure difference in the cavity was also very small (about 50 PSI) at the instant the impact occurs. It was also observed from the predictions that the pressure in the cavity was zero during the slow shot phase and it reached the peak at different locations at different instants of time during the fast shot. The maximum pressure occurred during the deceleration phase through out the cavity.

Miller at al [33], developed a finite element model to predict the deflection of the slides in nonopen close dies and the results from the model were compared with field data. The field data consisted of the slide blowback and tilt values from nominal position under different pressure loads for different slide design. The simple finite element model assumed a rigid support behind the cover die. The model predictions showed a good correlation with the experimental data except for the high pressure cases. This was due to the assumption of rigid support behind the cover die.

Vashist [34] studied the effect of different support structures for the die on the parting plane separation. The goal was to study the parting plane separation patterns on a production die that flashed severely after it was moved from a 1000 ton machine to a

2500 ton machine. The die had to be mounted lower on the 2500 ton machine due to the location of the shot hole. Therefore, to evenly spread out the clamp load, the die

foot print was increased on the cover on the machine platen by adding support structures to the dies. This study analyzed the effect of different types of support structures and different clamp loads. The results showed that the added supports stiffened the die/machine structure and increased the platen coverage area, but they did not aid in transferring the clamp load to the die faces. Flynn et al [35] measured the deflection of the same production die at various locations of the die using LVDT’s and they reported a good match between the model predictions by Vashist [34] and the experimental measurements.

GarzaDelgado [36] studied the failure of tie bolts that occurred on a hot chamber machine, using a sequentially coupled thermo mechanical model. This case study showed that nonuniform heat growth on the parting surface of the die resulted in unequal distribution of loads on the tie bolts and resulted in tie bolts failure. The machine frame, shank and bracket were modeled explicitly in the structural model.

The toggle mechanism and the tie bolts were modeled using 3D beam elements.

Milroy et al [37] used a boundary element method to predict the die deformation at the die parting surface or interface. The two die blocks were analyzed individually.

The clamp load was represented by a uniform pressure load at the die parting surface and a static pressure load was applied on the cavity surface. Few nodes on the rear surface of the die blocks were constrained to prevent rigid body motion. The steady state die temperature distribution was obtained by the boundary element method described earlier in [11].

The heat flux on the die surfaces were then approximated by

∂T K K = heff (Tcon − T ) ∂n (218)

Triangular elements were used to discretize the die surface and pipe elements were used to model the cooling lines. Though triangular elements are geometrically linear, quadratic variations of displacement and traction were used in the model and these elements with the order of geometric variation lower than that of field variables are called sub parametric elements. The temperature distribution within an element that is obtained from the thermal model is linear. Hence a linear variation of temperature coupled with quadratic variation of displacement and traction was used in the first set of analysis. Another two sets of analysis with isoparametric elements (variation of displacement and traction linear over the triangular element). Experiments were performed to evaluate the deformed profile of the die interface using transducers. The deflection of the interface from fixed datum points was measured at different locations of the die interface. The summation of the difference between the model predictions and measurements at these locations were taken as the measure of error.

The authors argue that using the deformation profiles the inverse of the deformation can be machined into the cold die so that the dies will mate perfectly under operating conditions. But this model did not account for the stiffness of the machine platens, tie bars and the toggle mechanism and the validity of the model predictions for cases with die and cavity offcenter from the platen center were not verified.

Rasgado et al [38], used the BE model described in [11], [37] to predict the thermo mechanical stresses in copper based dies. The mechanical loads and boundary conditions are same as in [37] and the thermal analysis procedure used in [11] was used to obtain the temperature history in the die. The effect of temperature on the elastic body was represented by adding a body force, b k=γT k to the corresponding nodes denoted by ‘k’ and increasing the nodal traction to γT nk, where γ=2α (1+υ) (1

2υ)1, is a Lame’s constant, α is the coefficient of thermal expansion of the die material and υ is the Poisson ratio. An experimental rig with a die block mounted on it was used to simulate the actual thermal loads in the die casting process and strain measurements were obtained at various locations on the die. The measurements were compared with model predictions. The model consistently over predicted the stresses on the die until ejection and it under predicted the stresses on the die after the casting was ejected.

Barone and Caulk [39] presented a method to predict the ultimate distortion of both the casting and the die due to thermal and mechanical loads in the die casting process.

They formulated the die distortion problem as a nonlinear thermo elastic contact problem solved by iterative boundary element method. But the casting distortion was analyzed as an unconstrained thermo elastic shrinkage using finite element method.

Their model included the dies, the ejector support, and cover and ejector platens. The tie bars and toggle mechanisms were represented by spring elements behind the ejector and cover platen respectively with appropriate stiffness values. Suitable displacement constraints were provided on selected nodes on the bottom of the cover

platen to prevent rigid body motion of the entire structure. This boundary condition is an approximation of anchoring the machine to the base. The uneven contact on the parting surface was handled using contact/gap elements whose formulation provides for load transfer between the die components only when the mutual surface traction is compressive. Friction between the contact surfaces were ignored in this model. The cavity pressure was modeled as a hydrostatic pressure with a magnitude equal to that of the intensification pressure. The modeling approach was tested on a front drive transmission case die and the results were presented. The advantage of this method as claimed by the authors is that the casting and die distortion can be analyzed simultaneously and the shrinkage allowance for the die cavity can be estimated.

Ragab et al [40] studied the effect of casting material constitutive model on the deflection and residual stress predictions on the casting. The cover and ejector platens were included in the model and the tie bars were represented using spring elements.

The clamp load was applied by specifying displacement boundary condition on the spring elements representing the toggle mechanism. The cavity pressure was modeled as a pressure boundary condition. A contact constraint was used between the die and the casting. A fully coupled thermo mechanical analysis was conducted. Three different material models were considered for the casting, viz., elastic, elasticplastic and elasticviscoplastic. The residual stress predictions were affected significantly by the material models were as the distortion predictions were less affected. The elastic model over estimated the stresses and the viscoplastic model lacked the required material property data. Therefore in a further study Ragab, [41] used the elastic

plastic model to predict the effect of key modeling factors on casting distortion predictions. The factors considered were the yield strength and strain hardening of the casting material, the heat transfer coefficient between the die and the casting and the injection temperature of the metal. The study concluded that the yield strength had a major effect on residual stresses at ejection while the injection temperature and the heat transfer coefficient had a major effect on the residual stresses at room temperature. The disadvantage of the model used in Ragab’s study was that the solid casting could not follow the distorted shape of the cavity due to the use of solid elements for the castings and hence it might affect the casting distortion prediction.

GarzaDelgado [42] addressed this issue by using a shell mesh representing the casting surface in the sequentially coupled thermo mechanical model that included the clamp loads, intensification pressure load and the thermal load. The nodal distortion values of the shell mesh were then mapped on to the surface nodes of a solid mesh for the casting. Then the solid cavity was tied to the distorted shape of the die using tied contact and the cooling stages of the casting were simulated using a fully coupled elasticplastic thermo mechanical model. Modeling the tie bars explicitly and applying the clamp loads through spring elements caused problems in establishing contact between the die and the casting. Therefore the clamp force was modeled as a pressure boundary condition behind the ejector platen in this work.

2.2 Parametric Die Design Studies

Based on the die distortion modeling methodologies reviewed in the previous section numerous parametric die design studies has been conducted at the Center for Die

Casting to understand the role of structural die design parameters on die deflection.

Some of these parametric die design studies are reviewed here.

Jayaraman [43], conducted a parametric study of the slide design variables for an inboard lock design and the work was continued by Chakravarti [44, 45] using a refined model. The variables considered were the preload, the angle of the locking surface and the pivot. Results suggested that the preload had no effect on the blowback and tilt values but it affected the fatigue life of the slides. The trends also showed that the tip separation increased with locking face angle and slides with high pivot were better supported by the ejector platen. There was negligible effect on the parting plane separation within the range of design variables.

The effect of proud inserts on the parting plane separation was first analyzed by

Dedhia [26], [46]. The study concluded that using proud inserts resulted in a lower separation during the initial cycles but at the later stages as the die warms up and grows the separation values were similar to the cases with flush inserts (insert parting surface in line with the die parting surface). A much refined model was later used by

Tewari [3], [45] to study the effect of proud inserts on the parting plane separation.

The contact pressure between the dies and the platens and the compressive stresses in the die pocket were also studied. It was observed that the parting plane separation was

reduced around the cavity by having a proud insert and the contact pressure between the dies and the platens was not affected due to the use of proud inserts. The proud insert had negligible effect on the compressive stresses in the die pocket.

Tewari [3], [45], analyzed the effect of adding a back plate behind the ejector support box. Results from his study indicate that the back plate has negligible effect on both the parting plane separation and the contact pressure between the ejector die and the platen.

A parametric study conducted by Dedhia et al [26], [46], showed that using proud pillars had no effect on the parting plane separation. The difference in parting plane separation values between the cases with proud pillars and flush pillars was less than

5%.

A series of parametric studies [13], [28] were conducted to gain understanding of the effect of important structural variables of the dies and the machine on the die distortion. The summary of the work was published in [4], [45].

The variables investigated were die size (% of platen area covered), insert thickness, thickness of die steel behind the insert and die location on the platen. Response surface models based on design of experiments were used to study the interaction between the variables and their effect of parting plane separation. The approach for the sensitivity analysis was developed by Chayapathi [1], [28] and the initial experimental design array consisted of 15 experimental runs. An additional 16 runs were further added by Kulkarni and Tewari [2], [3] to ascertain the results. The study

showed that the dominant factor that affects the die distortion on the cover side is the cover platen thickness. Thin dies performed better than thick dies. The more the steel behind the dies the lesser was the parting plane separation observed. Small or medium sized dies (covering 40% to 50% of platen area) performed better. Kulkarni [2] attempted to study the cover and ejector side performances separately. But the ejector side design variables such as the rail size, number, size and location of the support pillars were not controlled in the computational experiments. Therefore the study was inconclusive about the contribution of ejector side design to the ejector side separation.

2.3 Relevant Research in Injection Molding

Isayev[47] proposed an approach to simulate the cavity filling, packing stage and flash formation in injection molding using a finite difference method. The flow during flash formation was calculated using a power law fluid model. The mass balance equation during flash formation was given as

∂ ln ρ ∂  ∂P S  Q + −  = − ∂t ∂x  ∂x b  b∗w (219)

Where Q is the volumetric flow rate due to leakage calculated at each nodal point on the cavity edges, ρ is the density of the melt, P is the packing pressure, S is the melt fluidity, b* is the variable gap thickness, b and W are the gap thickness and width during flash formation respectively. The variable gap thickness b* was obtained by assuming the mold as a beam under uniform pressure loading with pin supports at the

location of the leader pins. Chen et al [48] developed an analytical solution to predict flash length on straight and curved parting surfaces of injection molds. The power law fluid model was assumed for the melt flow in this study.

Carpenter et al [49] studied the effect of injection molding machine stiffness on the mold deflection predictions of a finite element model. The cavity pressure distribution at the end of filling and at the instant of maximum packing pressure was obtained from coupled thermal fluid flow simulation using MOLDFLOW. The pressure predictions were used in a structural finite element models to predict the mold deflection. Three different loading scenarios were analyzed in the structural model. In the first case, only the clamp loads were modeled. In the second and third cases, in addition to the clamp load, the cavity pressure at end of filling and the maximum cavity pressure were included respectively. The three loading cases were repeated on a model that included only the mold halves and also on a model that included the machine platens and the tie bars. The strain predictions from these two types of models were compared to experimental measurements and it was shown that the predictions from a mold and machine model showed better agreement with the experimental measurements than the predictions from the mold only model.

Significant reductions in mold opening were predicted from the mold and machine model as compared to the predicted mold opening in the mold only model.

Hostert [50] demonstrated a multi body dynamics method to simulate the dynamic characteristics of a two platen HUSKY injection molding machine. ADAMS a multi

body simulation software were used to model the mechanical structure and DSH a fluid power analysis software was used to estimate the hydraulic force that drives the clamping mechanism. The four clamp pistons that engage the tie bar threads were modeled as rigid bodies and the three connecting bars were modeled as flexible bodies. A schematic of the clamp pistons and connecting rods and their representation in ADAMS along with the constraints and boundary conditions are shown in Figure

2.3 and Figure 2.4 respectively.

Figure 2.3: Schematic of Pistons and Connecting Rods of Injection Molding Machine Clamping Mechanism [50]

Figure 2.4: Representation of the Clamping Mechanism in Multibody Dynamics Simulation [50] The fluid power analysis package reads the position and velocity of the connecting rod from ADAMS, calculates the hydraulic force and returns the value back to

ADAMS. The connecting rods were originally modeled in ANSYS and their mode shapes were obtained using finite element method. The mode shape predictions were read into ADAMS using a transformation procedure that reduces the number of degrees of freedom for the multi body simulation as compared to the original number of degrees of freedom in the finite element model. The stress time histories of the connecting rod were then predicted in ADAMS. The advantage of the multi body dynamics simulation over FEA is that large non linear motions can be modeled with less computational time and cost.

Beiter et al [51] developed a decision support system that takes into account mechanical requirements, manufacturing costs and material selection for design of

injection molded parts. The mechanical requirements of the part were estimated by approximating the part geometry as a flat plate and the maximum deflection was estimated using classical plate deflection equation. The support system can also take into account deformation of the part due to creep and impact loadings. A filling analysis capability was also included to estimate the minimum number of gates and minimum wall thickness based on a defined flow length. Provisions were also provided for estimating the cycle time required for the part. Finally a cost model was included that predicts the manufacturing costs based on the material and part design selection.

Menges [52] describes methods to predict mold deflection in injection molding. The structural features are assumed as linear springs with certain stiffness. The deflection for each of the structural feature is estimated using standard design formulas. The individual deflections are then summed up to obtain the total deflection in the mold.

A schematic of the procedure is described in Figure 2.5

Figure 2.5: Schematic of the procedure to estimate mold deflection [52]

Menges [52] et al also described the effect of the relative stiffness of the mold and the machine on gap or flash formation using a force deformation diagram. In this method the mold, the parting surface, the machine clamp unit and tie bars are represented by springs with equivalent stiffness. Then the deformation of the springs representing the molds and the machine were obtained under different loads and a load deformation curve was obtained as shown in Figure 2.6. It can be observed from the figure that when the clamp load is applied the tensile forces on the press/machine are equivalent to the compressive forces on the mold faces. When the cavity load is applied, the compressive force on the mold faces are relieved causing gap formation and the machine/press is stretched further and hence the load on the machine increases. The stretching of the machine and the increase in locking force after application of cavity

load will depend on the stiffness of the machine. If the machine is perfectly rigid there will not be any increase in the locking force. This method provides a rather simple and realistic explanation of the effect of machine stiffness on mold deformation.

Figure 2.6: Mold Spring Diagram used to Estimate Gap Formation [52]

Sasikumar et al [53], investigated the premature failure of a tie bar in an injection molding machine using experimental failure investigation methods. The failed surface and samples cut from failed tie bar was subjected various visual and macro examination methods such as chemical analysis and fractography. Mechanical property measurements such as yield strength, tensile strength, elongation and impact toughness measurements were also carried out. The analysis revealed fatigue at the first thread of the tie bar as the major root cause of failure. It was concluded that the improper process conditions caused a tie bar load imbalance resulting in a torsional component of stress, misalignment of thread and consequent gouging at the threads. It was also found that material defects such as inclusions, untransformed phases and

fine cracks at the root of the thread caused the initiation of the crack and the crack propagated due to the pulsating tensile and torsional stresses.

2.4 Summary

Thus there is a vast amount of work done in mathematical modeling of the casting and similar processes. As mentioned earlier the complexity of the models and the assumptions on the physical phenomenon depends on the goals of the analyst. The goal of this research work is to study the effect of structural design variables of the die and the machine on the mechanical performance of the dies and machine and this work addresses the structural die design issues that were untouched by Chapyapathi and others [1], [2], [3] and [4]. Improved finite element modeling methods developed by Ragab [29] and Arrambide [30] were utilized in this dissertation. NonLinear power law models to predict parting plane separation are developed in this research using dimensional analysis as opposed to the linear polynomial models developed in previous research [1], [2], [3] and [4]. A nonlinear model to predict tie bar loads on the die casting machine is also developed in this research. To the author’s knowledge no other work has been done to develop power law models to predict parting plane separation and tie bar loads which make this dissertation work unique.

CHAPTER 3

RESEARCH METHODOLOGY

3.1 Introduction

Empirical correlations to predict the tie bar loads and the maximum parting plane separation were developed by conducting computational experiments and fitting regression models to the data obtained from the computational experiments. The computational experiments refer to the finite element analysis conducted at each design point in the chosen experimental design. The regression models then serve as surrogate functional approximations of the finite element model and they are also commonly referred to as response surface models or metamodels [54]. The surrogate response surface models can be used to find the optimal design parameters that would result in the desirable output of the system and these surrogate approximations has been widely used in structural design optimization [55]. The other objective in constructing these functional approximations is to predict the output of the system at an untried design point [56]. The accuracy of the predictions will not only depend on the functional form and also on the resolution and accuracy of the underlying numerical models.

In this research dimensional analysis based on Pitheorem [58], [62] was used to determine the functional relationship and nondimensional parameters for predicting the tie bar loads and parting plane separation. The functional forms obtained from the dimensional analysis for the parting plane separation prediction problem and the tie bar load prediction problem were imposed on the parting plane separation predictions and tie bar load predictions from the respective computational experiments. This chapter describes the finite element modeling methodology that was adopted in the computational experiments to predict the tie bar loads and parting plane separation.

3.2 Finite Element Modeling

A hypothetical box shaped part and a nominal 8.9 MN (1000 ton) four toggle die casting machine that have been used in previous research studies is chosen for this research. A schematic of the part geometry is shown in Figure 3.1 and the specifications of the die casting machine are shown in Table 3.1.

Figure 3.1: Geometry of the Part used in the Study

Machine Parameters Specifications

Machine Tonnage 1000 tons

Tie bar diameter 7.5”

Tie bar length 115”

Space between tie bars 44”

Cover PlatenWidth×Height 60”×70”

Ejector PlatenWidth×Height 60”×60”

Rear PlatenWidth×Height 60”×60”

Table 3.1: Specifications of the Machine used in the Study

The schematic of the finite element model used in this study is shown in Figure 3.2.

The dies, inserts, machine platens, the machine base and the tie bars are modeled explicitly using 3D solid elements. Ten node tetrahedron elements and eight or six node brick elements that are available in ABAQUS have been used for the 3D solid elements as will be explained in the next chapter. The toggle mechanism was modeled using 2D beam elements.

Figure 3.2: Schematic of the Finite Element Model

3.2.1 Boundary Conditions and Constraints

The machine base is constrained in all directions at six nodes on the bottom edges to prevent rigid body motion. A small sliding contact was defined between the ejector

platen and the base, between the ejector platen and the tie bars and also between the rear platen and the base. The nodes on the tie bars are tied to the corresponding nodes on the rear and cover platens using a tied contact formulation in ABAQUS. Similarly a tied contact constraint is defined between the nodes on the rear side of the insert and the corresponding contact nodes on the die shoes. The corner nodes of the dies were tied to the neighboring nodes on the platens using tied multi point constraint. A coulomb friction coefficient of 0.3 was assumed for all contacting surfaces.

The boundary condition that has a significant effect on the tie bar load prediction and parting plane separation prediction is the one used between the cover platen and the machine base. In actual die casting machine, the cover platen is bolted to the base and keyways are provided to prevent platen movement in the tie bar direction. The keyways are also provided with clearances. The stiffness of this constraint will affect the bending of the cover platen and hence the support available behind the cover die and the stretching of the tie bars will also be affected. Three different approaches were considered to model this constraint between the cover platen and the machine base. In the first approach the degrees of freedom of the edge nodes on cover platen and machine base were tied using a multipoint constraint formulation. A schematic of this constraint is shown in Figure 3.3a. In the second approach a tied contact formulation was used to ties the degrees of freedom of all the nodes on the contacting surfaces of the cover platen and the base. The tied contact formulation in ABAQUS ties the degrees of freedom of the slave nodes to the corresponding degrees of freedom of the neighboring master surface node [57]. A schematic of this boundary

conditions is shown in Figure 3.3b . However both of these constraint types might lead to an over constrained model. A better methodology to represent the bolted joint in FEA is to explicitly model the bolted joints. This achieved by using truss elements connecting nodes on the cover platen and the base as shown in Figure 3.3c. The head of the bolt is modeled using rigid elements connecting the reference node (node on the truss element) to the neighboring nodes on the platen or the base. The rigid elements are also shown in Figure 3.3c. A preload of 400 MegaPascals was applied on the truss elements by specifying appropriate temperature values on the reference nodes. This causes the truss elements to stretch and generate the required preload.

Figure 3.3: Boundary Conditions used between Cover Platen and Base

3.2.2 Loads and Assumptions

The clamp load was applied in the first step of the model by specifying appropriate temperature at the beam element nodes. This causes an increase in the length of the beam elements thus producing the required clamping force. The contact force between the cover die and cover platen was chosen as a measure of the clamp force.

The temperature on the beam element nodes were adjusted until the desired clamp

load was obtained. Following the same assumption proposed by Garza [21], the pressure load was assumed to be hydrostatic with a magnitude equal to that of intensification pressure. The pressure load is modeled as a pressure boundary condition on the cavity surfaces. The thermal loads are cavity geometry specific and the thermal distortion issues have to be addressed by appropriate design of cooling lines and part geometry. Therefore thermal loads are intentionally ignored in this study.

3.2.3 Material Properties

A linear elastic material model was used in the analysis. The platens, tie bars, machine base, toggle mechanism, the dies and inserts are all assumed to be made of

ST4140 steel and the material properties that are used for the ST4140 steel in the simulations are given in Table 3.2.

Material Property Value Young’s modulus 2.068×10 11 N/m 2 Poisson ratio 0.29 Coefficient of thermal Expansion 1.170 ×10 5 / οC

Table 3.2: Material Properties for ST4140

3.2.4 Finite Element Model Predictions

The predictions obtained from the finite element models were the loads on the tie bars and the maximum separation of the cover and ejector parting surfaces from their

nominal locations. To obtain the tie bar loads, the tie bars were partitioned into two different volumes at a location between the cover and ejector platens and the contacting surfaces of the two volumes were tied together using a tied contact formulation. At the end of the analysis the contact loads between these two contacting surfaces were obtained and this contact load is taken as a measure of the tie bar load.

The location were the volume partition was done and tie bar loads were predicted is shown in Figure 3.4.

Figure 3.4: Schematic of the Finite Element Model showing the Location of Tie Bar Load Prediction

The other prediction that was obtained from the finite element model was the separation of the cover and ejector parting surfaces from the nominal parting plane location. This phenomenon is illustrated in Figure 3.5. The black lines represent the undistorted nominal parting plane. The blue lines represent the distorted ejector

parting surface and the red lines show the distorted cover parting surface. The distance between the nominal location of the parting surfaces and their final location after the application of loads was obtained from ABAQUS. The nominal parting plane represented by the black lines is a plane that passes through all the nodes that on the parting surface that are in perfect contact after the application of clamp and pressure loads.

Figure 3.5: Illustration of Ejector and Cover Side Parting Plane Separation

The default nodal displacement output obtained from ABAQUS is given with respect to a global coordinate system. The dies and inserts undergo a pseudo rigid body motion due to the stretching of the tie bars as shown in Figure 3.6. The black lines in

the figure show the undeformed shape and the green lines show the deformed shape of the dies and the machine.

Figure 3.6: Pseudo Rigid Body Movement Caused by Stretching of Tie Bars

Therefore the default displacement output from ABAQUS includes this pseudo rigid body motion and the rigid body component should be removed to predict the displacement of cover and ejector parting surfaces from their nominal location which is the plane that passes through the parting surface nodes that are in contact. A methodology that was developed in previous research [2], [3] and [4], was used to remove this rigid body translation and rotation component from the displacement outputs of ABAQUS.

The equation used to estimate this transformation component is as follows:

R 0 [][][]Xs 1 = Xf 1   + E 0 (31) T 1

Where X s and X f are the starting and final coordinates of the sample nodes respectively, R is the rotation matrix, T is the translation vector and E is the distortion component. The initial coordinates X s and final coordinates X f of a few sample nodes on the parting plane were obtained from ABAQUS and a least squares method that minimizes the sum of squares 'trace(E T E)' was employed to estimate the transformation matrix in equation (31). Then the best estimate of the pure distortion component is given by

R∗ 0 E 0 = X 1 − X 1 (32) [][][]s f  ∗  T 1 Where R* and T* are the estimates of the rotation and translation components obtained from the least squares method. The matrix least squares method and the procedure to select the sample nodes are described in APPENDIX A. An alternate method to predict the displacement of the parting plane from their nominal location using a local coordinate system in ABAQUS is also presented in APPENDIX A.

3.2.5 Effect of Element Types and Cover Platen Constraint on Model Predictions

As mentioned earlier three different types of constraint between the cover platen and the machine base were considered in this research. The finite element model predictions are also sensitive to the element type used to model the platens, dies, inserts and tie bars. A nominal 8.9 MN (1000 ton) four toggle machine and a nominal

die and insert were chosen to test the effect of the cover platen constraint and the element types on the model predictions. The dimensions of the machine and the die and the magnitude of the loads used in these test simulations are shown in Table 3.3.

Table 3.4 shows the effect of the element types and the cover platen constraint on the total parting plane separation prediction of the finite element model. Total parting plane separation in the finite element model is obtained as the normal distance from the slave node to the master node of the contacting parting surfaces. It can also be viewed as the sum of the cover and ejector side parting surface separation. It can be observed from Table 3.4 that the cover platen boundary condition and the element types have a negligible effect on the parting plane separation prediction from the finite element model.

Parameter Value Platen Thickness 11" Die Length 31" Die Width 31" Die Thickness 7.5" Insert Thickness 4.125" Rail Width 3" Rail Height 6" Clamp Load 722 tons Cavity Pressure 10000 PSI

Table 3.3: Die and Machine Parameters used in the Simulations to Test the Effect of Cover Platen Constraint Type and Element Type

Constraint Element type Element type Maximum type between Maximum used for used for die Separation the cover Separation platens, base shoes and (thousands platen and (mm) and tie bars inserts of inch) base surface to 10 node 10 node surface tied 0.273 10.763 tetrahedron tetrahedron constraint 10 node 8/6 node brick Edge nodes tied 0.269 10.604 tetrahedron

surface to 10 node 8/6 node brick surface tied 0.269 10.589 tetrahedron constraint Truss elements 8/6 node brick 10 node used to model 0.269 10.604 tetrahedron bolted joint 8/6 node brick 8/6 node brick Edge nodes tied 0.262 10.310

Table 3.4: Effect of Element Type and Cover Platen Boundary Condition on Parting Surface Separation Prediction

Figure 3.7 shows the deflection plots of cover platen in the tie bar direction for cases with bolt constraint and multi point constraint. It can be seen that the deflection of the platen in the multi point constrain case is higher (0.38 mm) than the case in which the bolts were modeled explicitly (0.258 mm). The multi point constraint is stiffer than the bolt constraint and hence the bending of the cover platen is higher in this case.

The constraint is less stiff and hence the bottom of the cover platen moves slightly towards the toggle side. This trend is shown in Figure 3.8, where the deformed shapes of the platens are superimposed on the undeformed shape of the platen.

Figure 3.7: Deflection Plots of the Cover Platen

Figure 3.8: Deformed Plot of Cover Platen Superimposed on the Undeformed Plot

Table 3.5 shows the effect of cover platen boundary condition on the tie bar load predictions of the finite element model. In Table 3.5, the tie bar loads are shown as a percentage of the nominal load which is one fourth of the total clamp load.

Tie bar Load/Nominal Load*100 Constraint Top tie Top tie Bottom tie Bottom tie bar1 (T1) bar2 (T2) bar1 (B1) bar2 (B2)

Edge nodes 97.8 98.4 102.0 101.6 tied Surface to surface tie 97.4 98.0 102.4 102.0 constraint Truss 99.0 99.2 101.5 100.3 elements

Table 3.5: Effect of Element Type and Cover Platen Boundary Condition on Tie Bar Load Prediction

It can be observed from Table 3.5 that modeling the bolts explicitly results in slightly lower imbalance (approximately 2% less imbalance) between the top and bottom tie bars as compared to the other two constraints. It should also be noted that the cases shown in the Table are the ones with the dies centered between the tie bars and the cavity center of pressure centered on the geometric center of the die.

CHAPTER 4

EMPIRICAL CORRELATIONS TO PREDICT PARTING

PLANE SEPARATION

4.1 Introduction

This chapter discusses the power law models that were developed to predict maximum parting plane separation on the cover and ejector side. A design of experiments was developed based on the major structural variables of the die casting die and the machine. A static finite element analysis was conducted at each design point specified in the design array using the modeling methodology described in the previous chapter. Maximum separation of the cover and ejector side parting surfaces from their nominal location was obtained from the finite element models. Power law models were fit to the parting plane separation data and the non dimensional structural design parameters. The non dimensional parameters were obtained using dimensional analysis based on Buckingham pitheorem. The design of experiments, the non dimensional parameters and the power law models are presented in this chapter.

4.2 Design of Experiments

The design variables included in the study are the die width, die length, die thickness, thickness of the die steel behind the insert (die shoulder thickness), pillar diameter and the pattern of ejector pillar supports. The description of the factors along with their high and low values is shown in Table 4.1. The fourth factor die thickness ratio

(TR) is the ratio between the thickness of the die steel behind the insert and the total die thickness. This factor defines the thickness of the insert used. The factor is illustrated in Figure 4.1. In Figure 4.1, t denotes the die thickness and IT denotes the insert thickness. The thickness ratio, TR is given by (tIT)/t.

Factor Description High Level Low Level Pt Platen thickness 9” 13” Lx Horizontal dimension of 24” 38” Vertical dimensionthe die of the Ly 24” 38” die t Die thickness 5” 10” TR Die thickness ratio 0.4 0.5 PD Pillar diameter 1.5” 4” X Pillar Pattern (discrete 4 Levels/Patterns factor ) Table 4.1: Factors used in Design of Experiments

Figure 4.1: Side View of an Ejector Die used in the Study (Dimensions not to scale)

The sixth factor, pillar pattern, is a discrete factor. The schematic of the four different pillar arrangements behind the ejector die that were analyzed is shown in Figure 4.2 which shows the rear view of an ejector die. The back surface of the rails is hatched in the figure. The rails are 3 inches wide in all of the cases. The rails and pillars are 6 inches long in all of the cases. In pillar pattern1 there are a total of nine pillars, one directly behind the center of pressure and the outer pillars are located at a radial distance of 6.75 inches from the center pillar. In pattern1 the outer pillars are 45° apart from each other. In pattern2, there are no pillars and the ejector die has only rail support. In pattern3 and pattern4 there are five pillars, one on the center and four outer pillars each 90° apart. The difference between pattern3 and pattern4 is in the orientation of the outer pillars. A 58 run central composite response surface

experimental design was chosen based on the five continuous factors. Then the 58 runs were repeated for each level of the discrete factor. The experimental array in un coded units is shown in Table 4.2.

Figure 4.2: Schematic of Pillar Patterns used in the Study (a) Pattern1, 9 pillars, (b) Pattern2, No pillars, (c) Pattern3, Five Pillars, (d) Pattern4, 5 pillars

RUN Pt Lx" Ly" t" TR PD" 1 9 24 24 5 0.4 4 2 13 24 24 5 0.4 1.5 3 9 38 38 5 0.4 1.5 4 13 38 38 5 0.4 4 5 9 24 24 10 0.4 1.5 6 13 24 24 10 0.4 4 7 9 38 38 10 0.4 4 8 13 38 38 10 0.4 1.5 9 9 24 24 5 0.5 1.5 10 13 24 24 5 0.5 4 11 9 38 38 5 0.5 4 12 13 38 38 5 0.5 1.5 13 9 24 24 10 0.5 4 14 13 24 24 10 0.5 1.5 15 9 38 38 10 0.5 1.5 16 13 38 38 10 0.5 4 17 9 31 31 7.5 0.45 2.75 18 13 31 31 7.5 0.45 2.75 19 11 24 24 7.5 0.45 2.75 20 11 38 38 7.5 0.45 2.75 21 11 31 31 5 0.45 2.75 22 11 31 31 10 0.45 2.75 23 11 31 31 7.5 0.4 2.75 24 11 31 31 7.5 0.5 2.75 25 11 31 31 7.5 0.45 1.5 26 11 31 31 7.5 0.45 4 27 11 31 31 7.5 0.45 2.75 28 11 31 31 7.5 0.45 2.75 29 11 31 31 7.5 0.45 2.75 30 11 31 31 7.5 0.45 2.75

Continued

Table 4.2: Response Surface Experimental Array

Table 4.2 continued

31 11 31 31 7.5 0.45 2.75 32 11 31 31 7.5 0.45 2.75 33 13 24 38 5 0.4 1.5 34 13 38 24 5 0.4 1.5 35 9 24 38 10 0.4 1.5 36 9 38 24 10 0.4 1.5 37 9 24 38 5 0.5 1.5 38 9 38 24 5 0.5 1.5 39 13 24 38 10 0.5 1.5 40 13 38 24 10 0.5 1.5 41 9 24 38 5 0.4 4 42 9 38 24 5 0.4 4 43 13 24 38 10 0.4 4 44 13 38 24 10 0.4 4 45 13 24 38 5 0.5 4 46 13 38 24 5 0.5 4 47 9 24 38 10 0.5 4 48 9 38 24 10 0.5 4 49 11 24 31 7.5 0.45 2.75 50 11 38 31 7.5 0.45 2.75 51 11 31 24 7.5 0.45 2.75 52 11 31 38 7.5 0.45 2.75

The maximum cover and ejector side parting plane separation were predicted for each case in the experimental array using the finite element modeling methodology described in previous chapter. As it can be seen from, the last five runs are repeated runs with mid level setting of each factor. The runs 27 to 32 in the experimental array were utilized to account for these uncertainties in the mesh size, element type and the boundary condition between the cover platen and the base. A summary of the element

types and cover platen constraints used among the 52 cases in the experimental array is shown in Table 4.3.

Element type used Constraint type Element type used for Runs for die shoes and between the cover platens, base and tie bars inserts platen and base 127, 10 node tetrahedron 10 node tetrahedron Edge nodes tied 3258 surface to surface 28 10 node tetrahedron 10 node tetrahedron tied constraint 29 8 or 6 node brick 10 node tetrahedron Edge nodes tied surface to surface 30 8 or 6 node brick 10 node tetrahedron tied constraint Truss elements 31 8 or 6 node brick 10 node tetrahedron used to model bolted joint 32 8 or 6 node brick 8 or 6 node brick Edge nodes tied

Table 4.3: Summary of Element Types and Constraints used in Computational Experiments

4.3 Dimensional Analysis and Empirical Correlations

As mentioned in the first chapter, the objective of the computational experiments is to develop empirical correlations to predict the maximum parting plane separation as a function of the design variables involved. Dimensional analysis was used to determine the nondimensional parameters for the empirical correlations.

Dimensional analysis is based on the principle that any valid physical relationship should be dimensionally homogeneous. Based on this principle Buckingham [58] showed in his famous Pitheorem that any equation describing the physical

relationship between the variables can be reformulated as a function of dimensionless products of variables.

For example if the original relationship of n dimensional variables is written as

f (x1, x2,....xn ) = 0 (41)

The Pitheorem states that we can express this as a new function of a set of dimensionless parameters that are conventionally represented by П’s

f (Π1,Π2 ,....Πn−m ) = 0 (42)

Equation (42) shows that there are only nm variables in the relationship as compared to the original n variables in equation (41). Here m represents the number of fundamental or independent dimensions in the relationship such as mass, length, time, temperature etc. While the number of П’s is fixed there can be any number of sets of these products. Based on the knowledge of the governing physical principles these products can be manipulated by multiplying or dividing them by one another.

Thus by the nature of dimensional analysis the number of variables required for experimentation is reduced. The dimensionless parameters also capture the fundamental nonlinear relationships between the original variables. Omitting a variable in the dimensional analysis does not invalidate the nondimensional parameters or the functional relationship obtained from it, but it only restricts the generality of the solution obtained.

In addition to the measured variables, dimensional constants such as viscosity, density, young’s modulus, Poisson ratio etc may enter the nondimensional parameters. There might be also some appropriate nondimensional values such as ratios or scale factors. Again the selection of these constants and ratios will depend on the background knowledge in the physical phenomenon. Vignaux et al., [59] demonstrated how dimensional analysis can be used to simplify regression models and how the regression model obtained based on dimensionless parameters remains dimensionally homogeneous regardless of the measurement units and transformations to the original data.

4.3.1 Determination of the Model form and NonDimensional Parameters for

Predicting Parting Plane Separation

The choice of variables to be used in the dimensional analysis and the grouping of variables that form the nondimensional groups can be determined using the knowledge of the physical phenomenon and engineering judgment. The dies and inserts are assumed as prestressed flat plates that are resting on elastic supports and subjected to uniform loading. The semi analytical empirical equation for predicting maximum deflection of a flat plat under uniform loading is given by [60]

αWL4 Maximum deflection = (43) Et 3

W is the uniformly distributed load, l is the length of the plate, E is the young’s modulus, t is the plate thickness and α is a constant that depends on the aspect ratio

and boundary conditions. The unsupported span behind the cover die is characterized by the length and width of the die and a model for cover side separation will take a form similar to equation (43). However on the ejector side the unsupported span is characterized by the distance between the pillars and rail supports. Jofreit [61] developed a semi empirical equation to predict the maximum deflection of concrete flat plates supported on straight beams around the periphery and a grid of pillars in between the beam supports. The equation to predict the maximum deflection within any inner grid of pillars is given by

4  l + 3l  αW s n   4  Maximum deflection = (44) Et3

Where ls and ln are the long span and the short span between the pillar supports. It can be observed that the span variable L in equation (44) is replaced by the weighted average of the shorter and longer span between the pillar supports in equation (44).

Therefore a model for the ejector side separation will take a form similar to the one shown in equation (44). Equations (43) and (44) suggest that the empirical correlations to predict the maximum parting plane separation on the cover and ejector side will take a power law model form. Based on the structure of the equations (43) and (44), the physical relation between the maximum parting plane separation and the design variables can be written as follows:

f(((δmax ,Pt,Ltb t,, TR,L, l n ,PD, RW, Distributed Load,E))) === 0 (45)

Where, δmax is the maximum parting plane separation, Ltb is the distance between the tie bar centers, L is the length scale representing the length and width of the die or the span between the pillars, ln is the length scale for span between pillars, PD is the pillar diameter, RW is the width of the die and E is the young's modulus of the die material. The term Distributed Load denotes the distributed contact load at the parting surface and it is approximates as follows:

Total Clamp Load Cavity Pressure × Projected Area of Cavity (46) Distrubted Contact Load = − L x × Ly Lx × Ly

Both on the cover side and ejector side, the platens behave as an elastic foundation for the dies. Therefore the deflection of the dies will also depend on the stiffness of the platens which in turn depends on the thickness of the platen and the distance between the tie bar centers. Therefore the distance between the tie bar centers is also included in the functional relationship shown in equation (46). The scaling/repeating variables to be used in the dimensional analysis were determined based on linear elastic stress strain relation. By linear elastic theory, the relation between load and deflection is given by,

F = εE A (47)

Where, F is the total load, A is the area over which the load acts, ε is the strain and E is the young's modulus.

For the problem of predicting the maximum separation, δ max , equation (47) can be written as,

F δmax = E (48) A t

δmaxE = Constant (49) t × Distributed Load

The left hand side of Equation (49) is a non dimensional number which suggests the choice of scaling variables to be used in the dimensional analysis. The variables, E, distributed load and t were chosen as the scaling variables for the dimensional analysis. Then the remaining variables were written as a product of the repeating variables raised to an exponent. For example, the nondimensional parameter involving the distance between the tie bars, L tb can be written as

a b c E × DistLoad × t × Ltb = ∏ (410)

Writing equation (410) in terms of the basic dimensions of Mass (M) length (L) and

Time (T) we have

1 2 a 1 2 b c 0 0 0 [ML T ] × [ML T ] × [][]L × L = M L T (411) Equating the exponents of Mass (M) length (L) and Time (T) on both sides of the equation (411), we obtain a=0, b=0 and c=1. Therefore the non dimensional parameter involving the distance between the tie bar centers is given by

Ltb ∏ = (412) t The other nondimensional groups can also be obtained in a similar manner and equation (45) reduces to

 δ E Pt L Lx Ly Lcx Lcy l PD RW  f  max , , tb ,TR, , , , n , ,  = 0 (413)  t × Distributed Load t t t t t t t t t 

The Lx and Ly in equation (413) is the width and length of the die respectively and

Lcx and Lcy are the characteristic dimensions of the cavity respectively. A linear combination of the parameters, L/t, ln/t, PD/t, and RW/t represents the unsupported span behind the dies and these parameters can be represented by a single non dimensional number, say, ‘L/t’ for the purposes of dimensional analysis. Therefore the nondimensional parameters in (413) can be summarized as shown in Table 4.4.

П1 (L tb /t) or (Distance between tie bar centers/Die thickness)

П2 (Pt/t ) or (Platen thickness/Die thickness)

П3 (TR) or (Die shoulder thickness/Die thickness)

П4 (L/t ) or (Span/Die thickness)

П5 (Max Separation/t)× (Young’s Modulus/Distributed contact load)

П6 (Lx/Ly) or (Aspect ratio of the die)

П7 (Lcx/Lcy) or (Aspect ratio of the cavity)

Table 4.4: NonDimensional Structural Design Parameters

All non dimensional parameters have some physical significance. The parameter П1, indicates that increasing the distance between the tie bar center has the same effect as decreasing the die thickness. If the distance between the tie bars increases for a given

platen thickness, the platen bends more and the support available for the dies is reduced and the dies deflect more. Decreasing the die thickness will also result in an increase in the deflection of the die. Though the variable Ltb was not explicitly varied in the computational experiments, the parameter П 1 varies among the experimental cases due to the variation in die thickness among the experimental cases and the effect of Ltb can be captured from the computational experiments.

The second parameter П2 can be combined with the first parameter П1, to obtain a new non dimensional parameter, say, П12, which is given by

∏1 Pt ∏1−2 = = (414) ∏2 Ltb

The parameter П12, in equation (414) shows that there is always a combination of values of platen thickness and distance between tie bar centers for which the platen stiffness remains a constant. The most important non dimensional parameter is П4 which represents the ratio between the unsupported span and the die thickness. This parameter suggests that larger the span behind the die, thicker should be the die. A right combination of the span and die thickness can always be chosen to obtain a desired magnitude of parting plane separation. On the ejector side, the parameter П 4 represents the ratio of the span between the pillar supports to the die thickness. In our experiments the pillar locations with respect to the center of pressure were fixed. But the thickness of the die was varied and hence the parameter П 4 varies among the different experimental cases. This enables the power law model fit to capture the

inherent variability in die deflection caused by changes in unsupported span and die thickness.

4.3.2 Empirical Correlation to Predict Ejector Side Parting Surface Separation

As mentioned in the previous section, the nondimensional parameter П4=L/t is a linear combination of the other span variables, ln/t , PD/t , and RW/t . Therefore seven common length scales ( L1x , L 1y , L 2, L 3, L4, RWX and RWY) that characterize the span between pillars in each pillar pattern were identified as shown in Figure 4.3 and a general closed form expression that can be extended to any arbitrary pillar pattern was developed. The model form used in the regression for the ejector side data is shown in equation (415). The term L iavg in equation (415) represents the average of the internal spans ( L2, L 3 and L 4) between the pillar supports in each of the four pillar support cases. For pillar pattern1, Liavg was taken to be the average of L2, L3 and L4.

For pillar pattern2, there are no pillar supports and hence Liavg was taken to be zero.

For pattern3, L iavg is the average of L2 and L4 and for pillar pattern4 it is the average of L3 and L4. The schematic of these length scales for the four pillar patterns are illustrated in Figure 4.3. The variable np was introduced as a correction for the number of pillar supports. The power law model was obtained by non linear regression using sequential quadratic programming optimization algorithm in SPSS

[67]. The parameter estimates, the standard error values and the confidence intervals for the parameter estimates are shown in Table 4.5. The values in Table 4.5 were

obtained from SPSS. The standard error values and the confidence intervals were estimated using the linearized form of equation (415).

7c  1c +++  6c Max Ejector Sep E   PT  1+++np  L tb  ××× === 8c +++ 0c 1 +++    ×××   ××× t Dist load  L tb   t    c4 (415) c3+++  L1x +++ L1y RWX +++ RWY  L iavg  1+++np     ×××  +++  +++ 2c   ×××  2t 2t   t  6c   Lx Lcy  ××× 1 +++  ×××    Ly Lcx 

Parameter Estimates 95% Confidence Std. Parameter Estimate Interval Error Lower Upper Bound Bound c0 26.586 6.25 14.261 38.91 c1 0.954 0.1 1.15 0.758 c2 1.62 0.078 1.467 1.773 c3 3.785 0.113 3.563 4.008 c4 1.484 0.084 1.319 1.649 c5 0.407 0.108 0.62 0.193 c6 1.488 0.119 1.722 1.253 c7 3.053 0.354 2.356 3.751 c8 20.205 3.672 12.964 27.446 Adjusted R squared = 0.982.

Table 4.5: Parameter Estimates for Ejector Side Fit

It can be observed from Table 4.5 that the estimates are at least an order of magnitude higher than the standard error. The lower and upper bounds for the 95% confidence interval does not include zero for any of the estimates and hence it can be concluded that all the estimates are significantly different from zero. An adjusted rsquared

value of 0.982 was obtained for the ejector side fit which indicates that the model explains 98.2% of the variability in the data.

The ejector side power law model in the form of non dimensional parameters is shown in equation (416) and the same model is shown in the form of explicit variables in equation (417).

Figure 4.3: Length Scales Representing the Unsupported Span behind the Ejector Die

3.1  −−−0.95+++  −−−1.5 ∏∏∏ 5 === 2 2.0 +++ 26.61+++ ((()(∏∏∏ 21 ))) 1+++np  ×××[[[][∏∏∏ 1 ]]] ×××   1.5 8.3 +++ ×××[[[][((()(∏∏∏ 4a )))(+++ 1.6((()∏∏∏ 4b )))]]] 1+++np ××× (416) −−− .0 41 ×××[[[][1+++ ((()(∏∏∏ 6 ××× ∏∏∏ 7 )))]]]

1.3  −−−0.95+++  −−− 5.1 Max Ejector Sep E   Pt  1+++np  L tb  ××× === 20 2. +++ 26 6. 1 +++    ×××   ××× t Dist load  L tb   t    1.5 (417) 8.3 +++  L1x +++ L1y RWX +++ RWY  Liavg  1+++np     ×××  +++  +++ 1.6  ×××  2t 2t   t  −−− .0 41   Lx Lcy  ××× 1+++  ×××    Ly Lcx 

It can be observed from equations (416) and (417) that the term П4a is the average of the span between the pillar supports and the rail support. This span is represented by the explicit variables L1x and L 1y in equation (417). Similarly the parameter П 4b in equation (416) is the average of the spans between the pillars. The average of the span between the pillars is represented by the explicit variables Liavg in equation

(417). The relative contributions of the design variables on the parting plane separation can be inferred from the magnitude and signs of the exponents of those variables in equation (417). It can be observed from equation (417) that the largest contribution to the ejector side separation comes from the unsupported span behind the die which is characterized by the number, size and location of the support pillars and rails. The term ‘np’ in the exponent for the average span variable suggests that as

the number of pillars increases, the value of the exponent decreases and the separation becomes less sensitive to the span between the pillars as opposed to case which has fewer pillar supports. When ‘np’ decreases the value of the exponent increases and the model prediction becomes more sensitive to the span between the pillars.

The next major variable affecting the separation on the ejector side is the thickness of the die shoe. It can be shown from equation (417) that for a case with no pillar supports, the effect of die thickness on the maximum ejector side separation is in the order, Ө (t 2.8 ) while all other factors are held constant. Increasing the thickness of the die results in an increase in the stiffness of the die and hence thicker the die, less the separation.

The dimensionless parameter representing the platen thickness, (Pt/Ltb), by itself had no effect on the maximum ejector separation and hence a first order correction term

(1+ (Pt/Ltb) c) was included to represent the effect of platen stiffness. Thus the platen thickness has only a first order effect on the ejector side separation. The parameters

П6 and П 7 represent the aspect ratio of the die and the cavity respectively.

The variables Lcx and Lcy represent the characteristic horizontal and vertical dimensions of the cavity respectively. The largest of the horizontal and vertical dimension of the bounding box of the cavity can be chosen as fixed length scale and the other length scale can be obtained from the projected area of the cavity as shown below:

Projected Area of the Cavity Lcy = (418) Lcx

It can also be noted that the parameter die thickness ratio (П3=TR) does appear in the ejector side power law model. Though the term was included in the initial curve fitting procedure, the error involved in the estimate for its exponent was higher than the estimate itself. In other words the amount of variability in the maximum separation data caused by the thickness ratio alone was negligible. This might be due to the tied constraint used between the rear surface of the insert and the die shoe surface in the finite element model. Due to the tied constraint the insert and the die shoe behave as a single block and hence the insert thickness/die shoulder thickness has a negligible effect on the maximum separation. Therefore this parameter was dropped in the final power law model fit.

4.3.3 Empirical Correlation to Predict Cover Side Parting Surface Separation

The model form used for the cover side regression is shown in (419) and the parameter estimates, the standard error values and the confidence intervals for the parameter estimates of the cover side are shown in Table 4.6. The values in Table 4.6 are also obtained from SPSS.

c4 1c c2 c3 Max Cover Sep E L tb   Pt  Lx   Lx Lcy  (419) ××× === 0c   ×××   ×××   ×××  ×××  t Dist load  t   t   t  Ly Lcx 

Parameter Estimates 95% Confidence Interval Std. Lower Upper Parameter Estimate Error Bound Bound c0 0.359 0.016 0.327 0.391 c1 1.908 0.023 1.954 1.862 c2 3.449 0.028 3.395 3.504 c3 1.822 0.021 1.863 1.782 c4 1.763 0.033 1.828 1.698 Adjusted R squared = 0.995

Table 4.6: Parameter Estimates for Cover Side Fit

The adjusted rsquared value for the cover side fit is 0.995 which shows that the model explains 99.5% of the variability in the data obtained from the computational experiments. Similarly none of the confidence intervals include zero which indicate that the parameter estimates are significantly different from zero or in other words the magnitude of the parameter estimates are larger than the standard error involved in their estimates.

The power law model obtained for the cover side is presented in the form of non dimensional parameters and in the form of explicit variables in equations (420) and

(421) respectively.

−−− 8.1 −−− 9.1 5.3 −−− 8.1 ∏∏∏5 === 0.4 [[[∏∏∏1]]] ××× [[[∏∏∏2 ]]] ××× [[[∏∏∏4 ]]] ××× [[[∏∏∏6 ×××∏∏∏7 ]]] (420)

−−− .1 81 −−− 9.1 5.3 −−− 8.1 Max Cover Sep E L tb  Pt  Lx  Lx Lcy  (421) ××× === 0.4   ×××   ×××   ×××  ×××  t Dist load  t   t   t  Ly Lcx 

The last term in the equations (420) and (421) is the correction factor for the aspect ratio of the die and the cavity geometry. It can be inferred from equations (420) and

(421) that the die length is the most important factor that contributes the separation on the cover side. The model suggests that the larger the die with respect to the area between the tie bar centers, the higher is the maximum separation on the cover side.

The next important factor that contributes to the separation on the cover side is the platen thickness. It can be seen from these equations that a thicker platen will result in lower parting plane separation on the cover side. While all other factors are held constant the effect of platen thickness and distance between tie bars on the cover separation are in the order of Pt~Ө (1.9) and L tb ~Ө (1.8) respectively. Similarly the effect of die thickness is in the order, t~Ө (1.2). In summary, a large thick platen performs better on a small thin die. The die thickness ratio parameter (П3=TR) showed no effect on the cover side separation data also. Therefore the term was dropped in the final edited cover side power law model.

4.4 Sensitivity of Parting Plane Separation to Variations in

Structural Design Parameters

In the previous section, the sensitivity of the parting plane separation to various structural design variables was explained using the magnitude and direction of the exponents of the variables. The sensitivity of the maximum separation to the design variables can also be better understood by studying at their surface/contour plots which are also referred to as response surface plots. These plots are presented in this section, first in terms of the nondimensional parameters and then in terms of the explicit design variables. The power law models obtained for predicting the maximum cover and ejector side parting surfaces were used to generate response surface plots of maximum cover and ejector side parting plane separation.

4.4.1 Response Surface Plots of NonDimensional Parameters

The response surface plots are presented in the form of nondimensional parameters in this section. The plots of nondimensional parameters not only gives a better understanding of the interaction between the individual design variables, but the effect of individual variables can also be inferred from fewer plots.

3.5

4 3 3.5

3 2.5 2.5

2 2

1.5

1.5 1

0.5 (Max Cov Sep * E)/(t * Dist Load) Dist(Max * Cov Sep E)/(t * 8 1 0 7 9 8 6 7 0.5 6 5 5 4 4 Ltb/t Lx/t

Figure 4.4: NonDimensional Cover Separation vs. NonDimensional Die Length (П4) and Distance between Tie Bars (П1) (П2=4.5, П6=1, П7=0.58)

Figure 4.4 shows the effect of the non dimensional parameter П1=L tb /t and П4=Lx/t on the non dimensional cover separation П5 while other parameters are held constant.

The figure shows that small dies result in less separation on the cover side. It can also be observed that the maximum cover separation is less sensitive to the distance between the tie bars for a small die and it is more sensitive to the distance between the tie bars for a large die. Figure 4.5 shows the effect of the non dimensional parameters П2=Pt/t and П4=Lx/t on the maximum cover separation. The plot shows that the maximum cover separation decreases as the platen thickness increases and this effect is more pronounced for a large die than for a small die. It can be concluded

from Figure 4.4 and Figure 4.5 that small thin dies on larger thick platens offer the best mechanical performance on the cover side.

8 5.5

5 6

4.5

4 4

2 3.5

3 0 (Max Cov Sep / t) * (E / Dist Load) Dist (Max/ (E Cov * Sept) / 2.3 2.5 2.2 2 2.1 7.5 7 2 6.5 1.5 6 1.9 5.5 5 1 Pt/t 1.8 4.5 Lx/t

Figure 4.5: NonDimensional Cover Separation vs. NonDimensional Die Length (П4) and Platen Thickness (П2) (П1=8.8, П6=1, П7=0.58)

Figure 4.6, Figure 4.7 and Figure 4.8 show the effect of non dimensional parameters on the maximum non dimensional ejector separation. Figure 4.6 is a plot of the ratio between the weighted average of the spans vs. the non dimensional ejector separation.

The plot was obtained for the pillar pattern1 shown in Figure 4.2. A higher value of the weighted average indicates a higher value of span between the center of pressure and the pillars and it also indicates a lower value of span between the outer most pillars and the rail support. In other words as the inner spans increase, the pillars 99

move closer to the rail support and the outer spans decrease. It can be inferred from the plot that an increase in span for a given die thickness results in an increase in the ejector side separation. The plot also shows that for any given span between the supports, a decrease in thickness results in an increase in the maximum ejector separation. Since the die thickness is usually determined during the cooling line design, an appropriate arrangement of the pillar and rail supports should be chosen to minimize the separation on the ejector side.

250

200

150

100 (Max Eje Sep / t)* (E / Dist Load) Dist / (E t)* Eje Sep / (Max

50 3.5 4 4.5 5 5.5 Weighted Average of Span / t

Figure 4.6: NonDimensional Ejector Separation vs. NonDimensional Weighted Average of Span (П4a + 1.6 П4b ) (П12=5.9, П1=8.8, П6=1, П7=0.58)

Figure 4.7 shows the effect of the stiffness of the platen and the stiffness of the die on the maximum ejector separation in the form of the non dimensional parameters. The

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parameter Pt/L tb is the ratio between the platen thickness and the distance between the tie bar centers which represents the stiffness of the platen. The ratio between the weighted average of the span and the die thickness governs the stiffness of the die.

The plot shows that the stiffness of the platen has a negligible effect on the ejector separation even for thin dies with larger unsupported spans.

650

600

550 700

500 600

450 500

400 400

300 350

200 300

(Max Eje Sep / t) * (E / DistLoad) / (E (Max* EjeSep t) / 100 250 5.5 1.35 5 200 1.3 4.5 4 1.25 150 3.5 1.2 Weighted Average of Spans/T 3 1.15 Pt/Ltb

Figure 4.7: Non Dimensional Ejector Separation vs. Non Dimensional Platen Thickness (П12) and Weighted Average of Spans (П4a + 1.6П4b ) (П1 =8.8, П6=1, П7=0.58)

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3200

3000 3500

2800 3000

2600 2500 2400 2000 2200 1500 2000

(Max Eje Sep / t) * (E / DistLoad) / (E (Max* EjeSep t) / 1000 1.32 1800 1.3 1600 1.28 1.26 4 5 1400 1.24 6 1.22 7 8 1200 1.2 9 Pt/Ltb Ltb/t

Figure 4.8: NonDimensional Ejector Separation vs. NonDimensional Distance between Tie Bars (П12) and Platen Thickness (П1) (П4a + П4b =6.4, П6=1, П7=0.58)

The interaction between the platen thickness and the distance between the tie bars and its effect on ejector separation is shown in Figure 4.8. It can be observed from this figure that the ejector separation decreases as the distance between the tie bar centers increases. In other words as the distance between the tie bar centers increases, the die foot print becomes smaller relative to the available platen area. Therefore this observation confirms that small foot print dies result in less separation. It can be concluded that a large thick platen provides the best support for the dies on the ejector side.

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4.4.2 Response Surface Plots of Explicit Design Variables

The sensitivity of parting plane separation to the explicit design variables is also shown in Figure 4.9, Figure 4.10, Figure 4.11, Figure 4.12, Figure 4.13 and Figure

4.14. They convey the same information as the plots of nondimensional parameters.

Figure 4.9 shows the variation of maximum cover side separation in response to changes in die length and die thickness for various platen thicknesses while all other variables are kept constant. The figure shows that small thin dies on thicker platens perform better on the cover side. It can also be observed that the slopes of the surface representing the 9” platen are much steeper than the slopes of the surfaces representing the other two thicker platens. This shows the significance of the platen stiffness in reducing the separation on the cover side.

Figure 4.10 shows the response surface plot of maximum cover separation as a function of the platen thickness and die thickness for different die lengths. It can be observed from Figure 4.10 that the slope in the platen thickness direction is steeper than the slope in the die thickness direction which again indicates the significance of the cover platen stiffness in reducing the cover side parting plane separation.

It can be inferred from Figure 4.9 and Figure 4.10 that small thin dies get squeezed on to the platen surface and receives better support from the platens. Therefore thin dies result in smaller separation on the cover side.

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Figure 4.9: Maximum Cover Separation vs. Die Thickness & Die Length (L tb = 44 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)

Figure 4.10: Maximum Cover Separation vs. Die Thickness & Platen Thickness (L tb = 44 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)

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Figure 4.11 shows the effect of die thickness and the distance between the tie bar centers on the maximum cover side separation for three different die sizes. It can be observed from Figure 4.11 that as the distance between the tie bar centers decreases the maximum cover side separation increases and the effect is more pronounced for a thick die. In other words as the distance between the tie bars decrease, the dies become larger relative to the available platen area and results in a larger cover side separation.

Figure 4.11: Maximum Cover Separation vs. Die Thickness & Distance between Tie Bar Centers (Pt= 11 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)

Figure 4.12 shows the response surface plot of maximum ejector separation with respect to pillar diameter and die thickness for the four different pillar patterns

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considered in this study. It can be observed that for pattern2 that has no pillar supports the parting plane separation varies nonlinearly along the die thickness direction only. Since the maximum vertical and horizontal span between pillars in pillar pattern1 and pattern4 are less than the maximum horizontal and vertical span in pattern3, the maximum separation is always less for pattern1 and patter4 as compared to pattern3. Moreover, pattern1 and pattern4 behaves identically with respect to the maximum parting plane separation.

Figure 4.12: Maximum Ejector Separation vs. Die Thickness & Pillar Diameter

Figure 4.13 and Figure 4.14 shows the variation in maximum ejector separation with respect to changes in die length and die thickness for pillar pattern1 and three different platen thicknesses. In Figure 4.13, the pillar diameter was held at 4 inches

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and in Figure 4.14 the pillar diameter was held at 1.5 inches. It can be observed from these figures that the slope in die thickness direction is much higher than the slope in the die length direction. Comparing the highest points (large thin die) in Figure 4.13 and Figure 4.14, it can also be seen that the platen thickness has a slightly higher effect on the maximum separation for cases with 1.5 inch diameter pillars as compared to cases with 4 inch diameter pillars. This indicates that the effect of platen thickness increases as the pillar diameter decreases.

Figure 4.13: Maximum Ejector Separation vs. Die Thickness & Die Length (L tb = 44 inches, PD=4”, Lx/Ly = 1, Lcx/Lcy = 0.58)

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Figure 4.14: Maximum Ejector Separation vs. Die Thickness & Die Length (L tb = 44 inches, PD=1.5”, Lx/Ly = 1, Lcx/Lcy = 0.58)

4.5 Model Adequacy

The power law models for predicting maximum parting plane separation were developed using parting plane separation data from cases with pillar arrangement patterns shown in Figure 4.2. Dimensional analysis was then used to characterize the length scales representing the span between the pillar and rail supports. To study the adequacy of this model for pillar arrangement patterns other than the ones shown in

Figure 4.2, three test cases with pillar arrangement patterns as shown in Figure 4.15 were chosen. The summary of the platen thickness, die thickness, die size, pillar diameter, clamp load and cavity pressure used for the three test cases are shown in

Table 4.7

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Figure 4.15: Pillar Arrangement Patterns in the three Test Cases Used to Study the Adequacy of the Power Law Models

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Variable Value used in Test Cases Platen Thickness 11" Die Thickness 7.5" Horizontal Die Dimension 31" Vertical Die Dimension 31" Pillar Diameter 2.75" Clamp Load 800 tons Cavity Pressure 10000 PSI

Table 4.7: Summary of the Parameter Values used for Test Cases

The power law models were used to predict the maximum ejector and cover parting plane separations respectively for the three test cases. The length scales used in the ejector side power law model to describe the span variables for the three test cases is also shown in Figure 4.15. The span between the outer most pillars and the rail support is denoted by L 1x and L 1y for all three test cases as shown in Figure 4.15. The spans between the center pillar and the outer pillars are represented by L 2 and L 3.

Since there is only a center pillar in test case1, the length scales representing the span between the pillars is ignored in the power law model.

In test case2, L 2 is the only length scale representing the span between the pillars. To be consistent with the length scales that were defined during model fitting procedure, the span between the outer pillars are always defined between the edge of the center pillar and the edge of the outer pillar when a center pillar is present. In the absence of a center pillar, the span is defined between the center of pressure and the edge of the

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outer pillar. These two scenarios are demonstrated in test case2 and test case3 respectively. Test case3 has two length scales L 2 and L 3 representing the span between the pillars. The maximum ejector side parting plane separation was also predicted for the three test cases using finite element analysis. The comparison between the FEA predictions and power law model predictions are shown in Table

4.8.

FEA Prediction Power Law Prediction Test Case (mm/inches) (mm/inches)

1 0.183/0.0072 0.130/0.0050 2 0.182/0.0072 0.172/0.0068 3 0.159/0.0063 0.124/0.0049

Table 4.8: Comparison of FEA and Power Law Model Predictions for Ejector Side for the Test Cases

It can be observed from Table 4.8 that the difference between the FEA and power law predictions ranges from 0.01 mm to 0.05mm (0.0004 in to 0.0022 in). The maximum difference of 0.05 mm (0.0022 in) was observed in test case1, which has a center pillar only. The average of the absolute value of residuals from the power law model fitting procedure was also in the order of 0.05 mm (0.002 in). Given the three test cases lie outside the model domain, the magnitudes of these observed differences between the FEA and power law predictions are very reasonable. The average of the absolute value of the residuals of the power law model fit was also found to be 0.056 mm (0.002 inches).

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The comparison between the finite element model predictions and power law model predictions on the cover side for these three cases are summarized in Table 4.9.

FEA Prediction Power Law Prediction Test Case (mm/inches) (mm/inches)

1 0.162/0.0064 0.152/0.0059 2 0.162/0.0064 0.152/0.0059 3 0.163/0.0064 0.152/0.0059

Table 4.9: Comparison of FEA and Power Law Model Predictions for Ejector Side for the Test Cases

It can be observed from Table 4.9 that the difference between the FEA and power law predictions on the cover side is 0.01 mm (0.0005 in). Since the parameters on the cover side are the same for all three test cases, the power law model predicts the same value of cover side separation for all three test cases. The average of the residuals from the model fitting procedure was also found to be in the same order of magnitude as the observed difference of 0.01 mm (0.0005 in).

Therefore the predictions from the ejector side power law models can be expected to differ from a corresponding finite element model in the range of ±0.05 mm (±0.002 inches). Similarly the predictions from the cover side power law model can be expected to differ from a corresponding finite element model in the range of ±0.01 mm (±0.0005 inches).

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4.5.1 Rules to Characterize the Spans between Pillars and the Spans between

Pillars and Rails

It was observed during the model adequacy study that the ejector side parting plane separation is very sensitive to the span between the pillars and the definition of spans should be consistent with span definitions used in the power law model development.

The following rules serve as guidelines to characterize the spans between the pillars and between the pillars and the rails.

1. The span between the pillars and the rails should be defined from the edge of

the outermost pillar and the inner edge of the rails. This is illustrated in Figure

4.16a, where the Lo’s denote the spans between the pillars and rails.

2. If a pillar is present directly behind the center of pressure, the inner spans are

defined as the horizontal and vertical distances from the edge of this center

pillar to the edge of the outer pillars as shown in Figure 4.16a where the inner

spans are represented by Li’s.

3. If a pillar is not present behind the center of pressure, the inner spans Li

should are defined as the horizontal and vertical distances from the center of

pressure to the edge of the outer pillars as shown in Figure 4.16c.

4. If only one pillar is present then the outer spans Lo are defined between the

outer edge of the pillar to the inner edge of the rails and the inner spans Li are

defined from the center of pressure to the outer edge of the pillar. This is

illustrated in Figure 4.16d

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Figure 4.16: Illustration of Rules for Characterizing the Spans behind the Ejector Die

The rules summarized above can be used as a guideline to characterize the spans between the pillars and the rail supports to be used in the power law model to predict the ejector side separation.

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4.6 Platen Stiffness Characterization and Determination of Platen

Thickness Parameter to be Used in Power Law Models

The power law models were developed based on finite element modeling of an 8.9

MN (1000 ton) four toggle die casting machine. These models also represent the stiffness characteristics of the die casting die and the machine. The stiffness of the machine is represented by the platen thickness parameter. Certain machine designs consist of cast platens with rib like structures for enhancing the stiffness of the platen and reducing the material simultaneously. For the power law model to be applicable to machines with such platen designs, an equivalent platen thickness parameter has to be developed. The equivalent platen thickness parameter will represent the constant thickness of an 8.9 MN (1000 ton) machine platen that has the same stiffness characteristic as that of the machine platen under consideration.

The location of the toggle mechanism behind the ejector platen could also vary among different machine designs. The location of the toggle mechanism acts as a constraint on the ejector platen and the stiffness of the ejector platen to resist the clamp and pressure loads will depend on the toggle location. Two platens with same thickness but different toggle locations could have different stiffness characteristics due to the differences in the mechanical constraints imposed by the toggle mechanism. For the power law models to be applicable for machines with toggle locations other than the four toggle mechanism, an equivalent constant thickness of an 8.9 MN (1000 ton) four toggle machine platen should be determined. In this

115

section the methodology to characterize the stiffness of the machine platens and determine an equivalent platen thickness parameter using finite element modeling is presented.

The support available for the dies from the platen is affected by the magnitude of the deflection of the platens in the tie bar direction. The deflection of the platens in the two directions perpendicular to the tie bars is negligible. Therefore the equivalent thickness parameter to be used in the power law models are determined by comparing the stiffness of the platen in the tie bar direction alone. The deflection of the platens will depend on the size and geometry of the platen, constraints acting on it and the location and magnitude of the loads. The relation between the load, deflection and stiffness is given by

F = Kx (422)

Where F is the load, K is the stiffness and ∆x is the deflection. The clamp load is transmitted to the machine platens through the die shoes. The stiffness of the platen can be estimated by obtaining the deflection of the platen at a chosen location under loads of different magnitudes. The stiffness of the platen is given by the slope of the load deflection curve.

In the proposed methodology a static finite element analysis was used to obtain the load deflection data for the platens. The deflection of the platen under different magnitudes of load is obtained from a finite element models that has the exact geometric representation of the platen under consideration and the platen stiffness is

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determined from the load deflection curve. Then the thickness of a solid 8.9 MN

(1000 ton) machine platen that possess approximately the same stiffness value is also determined using finite element modeling and this value of the platen thickness is chosen as the equivalent thickness parameter. The location of the loading area in the finite element model is the same as the location of the dies on the platen. The constraints used in the models are different for the cover and ejector platens.

An arbitrary four toggle ribbed platen design shown in Figure 4.17 and Figure 4.18 was chosen to demonstrate the stiffness characterization methodology. The dimensions of this platen are shown in Figure 4.18. The nominal thickness of this platen is 6 inches and the solid rib on the center is 15 inches thick. The procedure to determine the equivalent thickness parameter for cover and ejector side platens is described in section 4.6.1 and section 4.6.2 respectively.

Figure 4.17: Schematic of the Platen Design Chosen to Demonstrate Stiffness Characterization Methodology

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Figure 4.18: Dimensions of the Platen Design Chosen to Demonstrate Stiffness Characterization Methodology (All Dimensions in Inches)

4.6.1 Methodology to Determine Equivalent Cover Platen Thickness

The cover platen is constrained to the machine tie bars and the bottom of the cover platen is constrained to the machine base. The other ends of the tie bars are secured to the rear platen. In the proposed method the tie bars are modeled explicitly and one end of the tie bars is constrained to the platen using a tied contact. The nodes on the

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other end of the tie bar are constrained in space in all degrees of freedom. Twelve nodes on the bottom of the platen are constrained in all directions. The constraints are shown in Figure 4.19. Uniformly distributed load is applied on the platen center over an area of 31” by 31” which is the size of a nominal die on a 1000 ton machine.

However this loading area can be changed depending on the size of the die in the case for which the equivalent stiffness is being determined. The magnitudes of loads considered are 250 tons, 500 tons, 750 tons and 1000 tons. The maximum deflection of the ribbed platen is obtained for all of the four loading cases. The deflection is obtained with respect to a coordinate system that was described using the three corner nodes on the inside face of the platen. The coordinate system is also shown in Figure

4.18. Then the deflections of solid platens with thicknesses of 0.2795 meters (9 inches), 0.254 meters (10 inches) and 0.2286 meters (11 inches) are also obtained from finite element models. The deflection values for the solid platens are obtained at a location which is same as the location of maximum deflection on the ribbed platen finite element model.

The load deflection curves for the ribbed platen, and solid platens with thickness of

0.2795 meters (9 inches), 0.254 meters (10 inches) and 0.2286 meters (11 inches) are shown in Figure 4.19 along with the equations for best fit lines.

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Figure 4.19: Schematic of Finite Element Model Used to Determine the Cover Platen Stiffness

y_rib = 4470805440.54x - 17.88 Deflection Vs Load y_9" = 4551731632.71x - 6.83 y_10" = 5318714310.27x - 5.32 y_11" = 6047360126.79x + 9.07 10000000 9000000 8000000 7000000 6000000 Ribbed 5000000 Solid-9" 4000000 Solid-10"

Load(Newton) 3000000 Solid-11" 2000000 1000000 0 0 0.0005 0.001 0.0015 0.002 0.0025 Deflection (meters)

Figure 4.20: Deflection Vs Load Curves for Cover Platen

It can be observed from Figure 4.20 that a solid 9 inch platen exhibits the same stiffness characteristic as the ribbed platen shown in Figure 4.17 and Figure 4.18. It

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can also be seen from the slopes of the equations in Figure 4.20 that the stiffness of the 9 inch platen is approximately equal to the stiffness of the ribbed platen.

Therefore the equivalent thickness of the cover side ribbed platen can be chosen as 9 inches and this thickness parameter can be used in the power law model.

4.6.2 Methodology to Determine Equivalent Ejector Platen Thickness

The procedure used to determine the stiffness of the ejector platen is similar to the procedure used for the cover platen. However the constraints in the finite element model used to determine the ejector platen stiffness are different from the ones in the finite element models used to determine the cover platen stiffness. The ejector platen is free to slide on the tie bars and the toggle mechanism acts as a constraint behind the ejector platen. The toggles were modeled using 3D beam elements in the finite element models and one end of the beam elements is constrained in space in all degrees of freedom. The schematic of the finite element model and the constraints are shown in Figure 4.21.

The maximum deflection of the ribbed platen under loads of 250 tons, 500 tons, 750 tons and 1000 tons were obtained from the finite element model. The loads were applied as uniformly distributed load over an area of 31” by 31” on the center of the platen. The deflection is obtained with respect to the coordinate system that was defined using the three corner nodes on the inside face of the platen. The coordinate system is also shown in Figure 4.21. The deflections of solid platens at the same

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location as that of the location of maximum deflection of ribbed platen are also obtained under same loading conditions.

Figure 4.21: Schematic of the Finite Element Model Used to Determine the Ejector Platen Stiffness

The load deflection curves for the ribbed platen, and solid platens with thickness of

0.2795 meters (9 inches) and 0.254 meters (10 inches) are shown in Figure 4.22. It can be observed from Figure 4.22 that a solid 9 inch platen exhibits the same stiffness characteristic as the ribbed platen shown in Figure 4.17 and Figure 4.18. It can also be seen from the slopes of the equations in Figure 4.22 that the stiffness of the 9 inch platen is approximately equal to the stiffness of the ribbed platen. Therefore the equivalent thickness of the ejector side ribbed platen can be chosen as 9 inches.

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Deflection Vs Load y_rib = 3578847885.62418x y_9" = 2852503674.45180x + 2.8 y_10" = 3793433298.8133x 10000000

9000000

8000000

7000000

6000000 Ribbed 5000000 Solid-9" Solid-10" 4000000 Load (Newton) 3000000

2000000

1000000

0 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Deflection (meters)

Figure 4.22: Deflection Vs Load Curves for Ejector Platens

4.6.3 Methodology to Determine Equivalent Thicknesses for Platens with

Different Toggle Locations

As mentioned in the previous section the location of the constraints on the platen affects the stiffness of the platen and hence a two toggle platen will have a different stiffness characteristic as compared to a four toggle platen of same thickness. While using the power law models to predict parting plane separation for a die on a two toggle machine an equivalent thickness has to be determined for a corresponding four

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toggle machine. The schematic of the finite element models used to determine the stiffness of four toggle and two toggle ejector platens are shown in Figure 4.23.

Figure 4.23: Schematic of Finite Element Models used to determine the Stiffness of Four Toggle (Right) and Two Toggle (Left) Ejector Platens

The toggles are modeled using 3D beam elements and the node on the free ends of the beam elements are constrained in space in all three directions. The maximum deflection of the two toggle platen is obtained under loads of 250 tons, 500 tons, 750 tons and 1000 tons. The loads are modeled as uniformly distributed load on an area representing the foot print of the dies (31” by 31” inches in this case used for demonstration of the method). The deflections of the four toggle platen at the same location are also determined under same loading conditions.

A 0.254 meters (10 inch) two toggle platen is chosen as an example. The load deflection curve for this two toggle platen is shown in Figure 4.24. The load

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deflection curves for four toggle platens with 0.2286 meters (9 inches) and 0.2032 meters (8 inches) thicknesses are also shown in Figure 4.24.

y_10"_2TG = 11874368631.68x - 89.05 Deflection vs Load y_9"_4TG = 15348952508.01x - 222.54 y_8"_4TG = 10950266433.03x - 164.25 10000000 9000000 8000000 7000000 6000000 5000000 4000000 2-Toggle-10"

Load(Newton) 3000000 4-Toggle-9" 2000000 4-Toggle-8" 1000000 0 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 Deflection (meters)

Figure 4.24: Deflection Vs Load Curves for Two Toggle and Four Toggle Platens

It can be observed from Figure 4.24 that the stiffness of the 0.2032 meters (8 inch) thick four toggle platen is approximately same as the stiffness of the 0.254 meters (10 inches) thick two toggle platen. Therefore the equivalent four toggle platen thickness for a 0.254 meters (10 inches) thick two toggle platen is 0.2032 meters (8 inches). It can also be observed that a two toggle platen is less stiff than a corresponding four toggle platen with the same thickness.

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4.7 Determination of Equivalent Stiffness of a Die Casting Machine

Using a Lumped Element Model

The method described in the previous section can be used to obtain the stiffness values for the platens. Using the same procedure the stiffness values can also be obtained for the dies, inserts, toggles and tie bars. The equivalent stiffness of the die/machine system can then be determined using lumped element modeling. In the lumped element modeling of static mechanical systems the various components of the system are represented with springs of appropriate stiffness values. The springs are connected in series and/or parallel depending upon the behavior of the corresponding components under the loads acting on the system.

When the clamp load is applied by the die casting machine, the toggles are compressed between the ejector and rear platens and the tie bars are stretched between the cover and rear platens. The dies and inserts are compressed between the cover and ejector platens. Therefore the die/machine system under the action of clamp load can be represented by the lumped spring stiffness diagram as shown in

Figure 4.25.

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Figure 4.25: Spring Stiffness Diagram for the Die and the Machine under Clamp Load

The top portion of Figure 4.25 has three springs connected in series, K RP , K TB and

KCP, representing the stiffness of the rear platen, tie bars and cover platen respectively. One end of the top circuit is grounded and the clamp load acts on the other end of the series circuit. The bottom part of the diagram has five springs connected in series, K CD , K PS , K ED , K EP and K TG , representing the stiffness of the cover die, parting surface, ejector die, ejector platen and toggle respectively. The clamp load stretches the tie bars, cover platen and the rear platen and it compresses the cover die, parting surface, ejector die, ejector platen and the toggle. The pressure load applied in the next stage will stretch the parting surface spring, K PS between the cover die and the ejector die and the displacement due to the pressure load on the top circuit is negligible.

The behavior of the system shown in Figure 4.25 is illustrated using an example. This example also serves as a validation for the model behavior. The stiffness of the platens, tie bars, toggles were approximated using average displacement values predicted from the finite element model described in Chapter 3. Since the stiffness

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values of the dies and the machine parts are lumped in the linear springs with only one degree of freedom, average displacements were chosen for the lumped model as opposed to maximum displacement used in the development of power law models.

The stiffness was calculated using the relation

FClamp K = (423) X

Where ∆X is the average displacement obtained from the finite element model under clamp load only. The stiffness values thus obtained for the dies and the machine parts are summarized in Table 4.10. The value of clamp load used in the simulations is

6.432 MN.

Part Stiffness (N/mm) Cover Platen 3×10 7 Rear Platen 2.3×10 7 Tie Bars 0.82×10 7 Cover Die 17.1×10 7 Ejector Die 25×10 7 Ejector Platen 240×10 7 Toggle 2.6×10 7

Table 4.10: Stiffness Values to be used in Lumped Element Model for the Dies and Machine Parts

Since the tie bars, cover platen and rear platen are all in series, the equivalent stiffness of the top circuit in Figure 4.25 is given by

1 1 1 1 = + +

K A K CP K RP K TB (424)

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Using the stiffness values in Table 4.10, we obtain

6 K A = 5.03×10 N/mm (425)

Therefore the stretch due to the clamp load on the top portion of the circuit in Figure

4.25 is given by

6 FClamp .6 432 ×10 δClamp = = = 3.1 mm (426) A 6 KA .5 03×10 Similarly, the stiffness of the springs to the left of the parting surface in Figure 4.25 are combined to obtain an equivalent stiffness given by

1 1 1 1 = + + K 2B K ED KEP K TG (427)

Using the equivalent stiffness values K A and K2B , the circuit shown in Figure 4.25 can be reduced to one shown in Figure 4.26. In Figure 4.26, an additional load,

FClamp Fpressure , acts on either side of the parting surface, which is equivalent to the clamp load on the parting surface in excess of the pressure load applied during the metal injection stage. A pressure load of 6.8948 Mpa was used in this example. Based on the projected area of the cavity used in this research (0.0775 sq. meter), the magnitude of the pressure load is calculated as follows:

FClamp F Pressure = FClamp (Pressure × Projected Area of Cavity) (428) = 1.331× 106 Newton

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Figure 4.26: Spring Diagram with Clamp and Pressure Loads

The displacement of the cover die spring, K CD , between nodes 3 and 4, due to the net load on the parting surface is given by

(429) C P FClamp −FPressure δCD = = .0 008 mm KCD

Similarly, the displacement of the spring K 2B , between nodes 1 and 2 due to the net load on the parting surface is given by

C P FClamp − FPressure δ2B = = − .0 06 mm (430) K 2B Therefore the total displacement in the bottom part of the circuit, between the nodes 1 and 4 is given by

C P C P C P δB = δCD δ2B = .0 068 mm (431)

This total displacement of the bottom circuit due to the pressure load adds up to the displacement on the top of the circuit, shown in equation (426).

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Therefore the total displacement on the top of the circuit after clamp and pressure is given by

C P Clamp C P δA = δA + δB = 3.1 + .0 068 mm = 1.37 mm (432) Therefore the total load on the top of the circuit after clamp and pressure is given by

CP CP 6 FA = K AδA = 6.891×10 N (433)

The initial applied clamp load was 6.432 MN and the clamp load in the tie bars, cover platen and rear platen after the pressure stage is 6.891 MN as shown in equation

(433). Therefore there is 7% increase in the clamp load after the application of the cavity pressure in the lumped model where as the finite element model shows a 3% increase in the clamp load after the cavity pressure stage. Given that the stiffness of the springs are based on the average displacements obtained from the finite element model, a 4% difference between the finite element and a one degree of freedom lumped model is reasonable. This also shows that the lumped element model behaves as intended.

The equivalent stiffness of the die/machine system can be obtained from the spring diagram shown in Figure 4.25. The equivalent stiffness of the bottom circuit is given by

1 1 1 = + K B K 2B K CD (434)

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The top and bottom circuits are connected in parallel. Therefore the equivalent stiffness of the die/machine system is given by

K = K + K eq A B (435)

The methodology described in previous section to determine the stiffness of the machine platens can be used to obtain the stiffness of the dies and the other machine parts as well. Thus the lumped element model provides a better understanding of the interaction between the dies and machine parts. It also provides a means to compare the stiffness characteristics of different machines.

4.8 Summary

Power law models to predict the ejector and cover side parting surface separation were developed using dimensional analysis and data from finite element analysis experiments. These power law models can be used to predict maximum cover and ejector parting plane separation and compare the performance of different die designs and machines. The sensitivity of parting plane separation to the structural design parameters were explained from the magnitude of the exponents and coefficients of the power law model. Response surface plots are also provided as a visual aid in understanding the relative contributions of the structural design variables on parting plane separation. The accuracy of the ejector side power law model is within ±0.0022 inches and the accuracy of the cover side model is within ±0.0005 inches. Rules and guidelines to characterize the unsupported spans on the ejector side are also provided.

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A methodology to characterize the stiffness of platens of various designs and sizes was described in this chapter. This method can be used to determine the equivalent platen thickness parameter to be substituted in the power law models to predict parting plane separation. A one degree of freedom lumped element model is also provided to estimate the total stiffness of die casting machines and this model also provides an understanding of the interaction between various machine parts and the dies.

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CHAPTER 5

EMPIRICAL CORRELATIONS TO PREDICT TIE BAR

LOADS

5.1 Introduction

As mentioned in the first chapter, the current approach used in industry to predict the tie bar loads ignores the location of the die with respect to the platen center and it assumes that the machine and the dies are perfectly rigid. The current approach also violates the force equilibrium constraint thus leading to inaccurate predictions. To address these issues, a nonlinear power law model was developed to predict the tie bar loads of the die casting machine based on the location of the die and cavity center of pressure with respect to the tie bars. The model was obtained by curve fit to tie bar load prediction data from computational experiments. The computational experiments were conducted using the finite element modeling method described in chapter 3. An experimental design was developed based on the horizontal and vertical dimension of the die, the locations of the die and cavity center of pressure with respect to the platen center and the magnitude of cavity center of pressure. Dimensional analysis was used to incorporate other important scale factors and obtain the nondimensional

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parameters. The nonlinear model was then fit to the nondimensional form of the location, scale and load variables. Experimental tie bar load measurements were then compared to the power law model predictions to check the adequacy of the power law models. The design of experiments, the dimensional analysis, the nonlinear regression model and the experimental verification are described in this chapter.

5.2 Design of Experiments

The factors that were considered are the die length, die width, location of the die with respect to the platen center and the location of the cavity center of pressure with respect to the platen center and the magnitude of the cavity pressure. The description of the variables and their levels are shown in Table 5.1. The location variables in

Table 5.1 are defined with respect to a coordinate system with origin on the center of the platen area between the tie bars. The schematic of the coordinate system and the tie bar labels are shown in Figure 5.1.

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Level Level Level Level Level Factor Factor Description 1 2 3 4 5

LX Die Width (inches) 26.49 30.4 32.6 34.6 36.5

LY Die Height (inches) 26.49 30.4 32.6 34.6 36.5

Die location in X DPX 4 2 0 2 4 direction (inches)

Die location in Y DPY 4 2 0 2 4 direction (inches) Location of center of CPX pressure in X 4 2 0 2 4 direction (inches) Location of center of CPY pressure in Y 4 2 0 2 4 direction (inches)

CPR Cavity Pressure (KSI) 2 4 6 8 1

Table 5.1: Description of Variables used for Tie Bar Load Prediction Model Development

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Figure 5.1: Coordinate System and Tie bar Labels viewed from inside face of Cover Platen The initial goal of the study was to develop a linear polynomial model to predict the tie bar load imbalance as compared to the nominal load (one fourth of total clamp).

Allen et al [63, 64], proposed an experimental design methodology for developing polynomial response surface model that minimizes both the model bias and random error. This method assumes that the true model is a third order polynomial and the fit model is a second order polynomial. The proposed criterion is called Expected

Integrated Mean Square Error (EIMSE) criteria and it is given by

 2  EIMSE(){}ξ = E [Yˆ ε,x,η(x),ξ − η(x)]  (51) η,x,ε 

Where Yˆ {ε,x,η(x),ξ} and η(x) are the predicted value and true model value at the prediction point x respectively. The experimental design is ξ and ε is a vector of

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random errors. The symbol ‘E’ denotes the statistical expected value taken over the variables and the EIMSE optimal method attempts to minimize this expected value using optimization method. The optimization method searches over various candidate experimental designs and finds the one that yields the minimum EIMSE with minimum number of runs.

For our study a full factorial design based on seven factors with five levels was chosen as the initial candidate set and this full factorial design comprised of 78125 runs. Some of these runs were eliminated due to the constraints imposed on the die and cavity location by the platen size and the die size respectively. The reduced candidate set was given as the input for EIMSE optimal design selection and the best subset of points were chosen based on the criteria described in equation (51). Finally an experimental design with fifty runs was obtained and a linear polynomial model to predict the tie bar loads was developed [65]. The linear polynomial model is presented in the APPENDIX B.

One of the limitations of the linear polynomial model is that the accuracy of the model predictions is dependent on the range of the design variables within which the experimental data are collected. The model cannot be extrapolated beyond the domain it is intended to be used. The range of cavity loads used in the experimental design fell within 13%83% of the applied clamp load. The linear polynomial models that were initially developed to predict the tie bar loads showed poor predictions when the static cavity load approached 100% of the clamp load. Therefore it was decided to

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develop a nonlinear power law model using dimensional analysis to capture the inherent nonlinearity in the tie bar load data.

A few additional cases were also added to the experimental array to include cases with a cavity load equal to 100% of clamp load. The initial experimental design obtained from EIMSE criteria was an unbalanced design and hence the selection of additional cases with high cavity load values was left as an arbitrary choice.

Therefore the best we could do without seriously violating the properties of the experimental matrix was to repeat the cases in the experimental array with 2 KSI cavity pressure values at 12 KSI (corresponds to 100% of clamp load). This added an additional 10 cases to the experimental array. It was also decided to add a few cases with no cavity pressure load. Therefore the tie bar load predictions were obtained under clamp load only for cases in the experimental array with 10 KSI and these ten cases represent the zero cavity pressure cases. Thus an experimental design with a total of 70 cases was finally arrived at and the experimental array is shown in Table

5.2.

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CPR Run LX " LY " DPX " DPY " CPX " CPY " (KSI) 1 26.49 34.6 4 0 4 0 12 2 26.49 30.4 2 2 2 0 12 3 34.6 34.6 0 0 4 4 6 4 26.49 26.49 0 4 0 4 12 5 32.6 32.6 2 0 2 2 10 6 32.6 34.6 0 0 2 4 6 7 34.6 34.6 0 0 4 4 12 8 32.6 30.4 0 2 0 0 8 9 32.6 32.6 2 2 0 2 4 10 30.4 26.49 4 4 4 4 8 11 26.49 32.6 4 0 4 0 6 12 36.5 32.6 0 2 4 2 12 13 36.5 32.6 0 0 4 2 10 14 34.6 26.49 0 4 4 4 10 15 36.5 26.49 0 0 4 0 6 16 26.49 30.4 2 2 2 4 8 17 26.49 26.49 0 2 0 2 10 18 32.6 30.4 2 2 2 2 8 19 30.4 30.4 4 2 4 4 6 20 36.5 32.6 0 2 0 0 4 21 34.6 32.6 0 2 2 4 10 22 36.5 32.6 0 2 2 0 12 23 34.6 26.49 0 0 4 0 4 24 32.6 26.49 0 4 2 4 8 25 30.4 30.4 4 2 4 4 4 26 36.5 34.6 0 0 4 4 6 27 26.49 26.49 4 4 4 4 8 28 30.4 36.5 2 0 2 4 4

Continued

Table 5.2: Experimental Array used for Tie Bar Load Prediction Model Development.

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Table 5.2 continued

29 30.4 34.6 4 0 4 4 6 30 26.49 30.4 2 2 2 2 10 31 34.6 36.5 0 0 2 2 6 32 30.4 36.5 4 0 4 2 10 33 26.49 30.4 2 2 2 0 4 34 32.6 34.6 2 0 0 4 8 35 30.4 36.5 4 0 4 4 10 36 26.49 36.5 0 0 0 4 4 37 26.49 36.5 0 0 0 0 10 38 32.6 34.6 2 0 0 2 12 39 36.5 26.49 0 0 0 0 4 40 36.5 26.49 0 2 2 2 10 41 36.5 30.4 0 2 4 0 4 42 36.5 36.5 0 0 2 2 6 43 30.4 32.6 4 2 4 4 6 44 26.49 32.6 4 2 4 2 8 45 36.5 36.5 0 0 4 4 6 46 36.5 26.49 0 4 2 4 12 47 32.6 32.6 2 0 4 2 12 48 34.6 34.6 0 0 4 4 10 49 32.6 26.49 2 2 4 2 12 50 30.4 36.5 4 0 4 4 4 51 26.49 34.6 4 0 4 0 0 52 26.49 30.4 2 2 2 0 0 53 30.4 26.49 4 4 4 4 0 54 26.49 32.6 4 0 4 0 0 55 30.4 30.4 4 2 4 4 0 56 34.6 32.6 0 2 2 4 0 57 32.6 26.49 0 4 2 4 0 58 30.4 30.4 4 2 4 4 0 59 26.49 26.49 4 4 4 4 0 60 26.49 30.4 2 2 2 2 0 61 26.49 34.6 4 0 4 0 2 62 26.49 30.4 2 2 2 0 2 63 26.49 26.49 0 4 0 4 2

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Table 5.2 continued

64 34.6 34.6 0 0 4 4 2 65 36.5 32.6 0 2 4 2 2 66 36.5 32.6 0 2 2 0 2 67 32.6 34.6 2 0 0 2 2 68 36.5 26.49 0 4 2 4 2 69 32.6 32.6 2 0 4 2 2 70 32.6 26.49 2 2 4 2 2

5.3 Dimensional Analysis and Empirical Correlation to Predict Tie

Bar Loads

The functional relationship between the tie bar loads and the variables involved in the physical phenomenon is given by

f (T,Lx, Ly,DPX,DPY,CPX, CPY, L tb ,Clamp,CPR,A) = 0 (52)

Where, T is the load on the tie bar, Lx and Ly are the horizontal and vertical dimensions of the die, DPX and DPY are the horizontal and vertical location of the dies with respect to the platen center, CPX and CPY are the horizontal and vertical location of the cavity center of pressure, L tb is the distance between the tie bar centers, CPR is the pressure load and A is the projected area of the cavity. This functional relation can be reduced to one involving fewer nondimensional parameters using dimensional analysis.

The load on the tie bars is essentially a percentage of the total clamp load and the percentage of clamp load on individual tie bars depend on the location of the die and

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the cavity center of pressure. Ideally, if the dies and cavity center of pressure were centered on the platen, the load on each tie bar should be equal to one fourth of the clamp load. Therefore the tie bar load T, in equation (52) is scaled by one fourth of the clamp load. During the metal injection stage, the cavity pressure load relieves a fraction of the initial clamp load depending on its magnitude. The pressure load also redistributes the contact force on the die parting surface depending on the location of the cavity center of pressure. Hence the moments on each tie bars are also redistributed during the metal injection stage. Therefore the pressure load was approximated as the product of the cavity pressure and the projected area of the cavity and the pressure load was scaled by the total clamp load. The remaining length scale variables were scaled by the square root of the projected area of the cavity. Therefore equation (52) can be rewritten as

 T Lx Ly DPX DPY CPX CPY Ltb CPR × A  f  , , , , , , , ,  = 0 (53)  0.25× Clamp A A A A A A A Clamp 

The moments caused by the die locations DPX, DPY and the cavity location CPX and

CPY about the tie bars will also depend on the distance between the tie bar centers.

Therefore the nondimensional parameters involving the locations variables DPX,

DPY, CPX, CPY and the nondimensional parameter involving the distance between the tie bars, L tb can be combined together and the functional relationship in equation

(53) can be reduced further as follows:

 T Lx Ly DPX DPY CPX CPY CPR × A  (54) f  , , , , , , ,  = 0  0.25× Clamp A A Ltb Ltb Ltb Ltb Clamp 

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The nondimensional numbers in the equation (54) are summarized in Table 5.3

П1 Lx/√A

П2 Ly/√A

П3 DPX/L tb

П4 DPY/L tb

П5 CPX/L tb

П6 CPY/L tb

П7 (CPR × A) / Clamp

П8 T/ (0.25 × Clamp)

Table 5.3: Non Dimensional Parameters used in Tie Bar Load Prediction Model

Though the cavity geometry and the machine stiffness were not varied in the experiments, the use of nondimensional parameters shown in Table 5.3 ensures that the model could be extended to machines of different sizes and tonnages.

The loads on the four tie bars were obtained for all of the 70 cases in the experimental array and a nonlinear regression model was fit to the nondimensional parameters using the tie bar load data from the computational experiments. Nonlinear model was fit to the load data for each individual tie bars. The same model form was obtained for all the four tie bars and the magnitudes of the coefficients and exponents for the two top tie bars were approximately equal with different signs. Similarly the magnitudes of the exponents and coefficients for the two bottom tie bars were almost the same.

Therefore the tie bar load data for the top tie bars1 & 2 were pooled together and the data for the bottom tie bar1 & 2 were pooled together and the same model form was

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fit to the top tie bar data and bottom tie bar data. The models forms used for the top and bottom tie bars in terms of the nondimensional parameters are given by equations (55) and (56) respectively. It can be noted that the signs for the non dimensional parameters П4 and П6 are reversed between the equations (55) and (56).

These parameters represent the vertical location of the die and center of pressure respectively and hence their signs are positive for the top tie bars and negative for the bottom tie bars. The sign convention is explained in detail in the following section.

c1 c2 Top Tie bar load   DPX    DPY  === 0c 1 ±±±   1 +++   ××× Nominal Load  0.5L   0.5L  (55)   tb    tb   CPR ××× A   CPX   CPY  CPR ××× A  ××× 1 +++ 3c exp1 ±±±  +++ 4c 1 +++   ×××  CLAMP   0.5L   0.5L  CLAMP     tb   tb     CPR ××× A  CPR ××× A  ××× 1 −−− 5c  ××× exp    CLAMP  CLAMP 

1c 2c Bottom Tie bar load   DPX    DPY  === 0c 1 ±±±   1 −−−   ××× Nominal Load  0.5L   0.5L    tb    tb  (56)  CPR ××× A   CPX   CPY  CPR ××× A  ××× 1 +++ 3c exp1 ±±±  +++ 4c 1 −−−   ×××  CLAMP   0.5L   0.5L  CLAMP     tb   tb     CPR ××× A  CPR ××× A  ××× 1 −−− 5c  ××× exp    CLAMP  CLAMP 

SPSS was used to perform the nonlinear regression and the sequential quadratic programming algorithm in SPSS was used to solve the nonlinear regression problem.

The parameter estimates, the standard error in the estimates and the confidence intervals for the estimates for the top and bottom tie bars are provided in Table 5.4 and Table 5.5 respectively.

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Parameter Estimates 95% Confidence Interval Std. Lower Upper Parameter Estimate Error Bound Bound c0 1.005 0.002 1.001 1.009 c1 0.354 0.011 0.333 0.374 c2 0.303 0.012 0.279 0.327 c3 0.063 0.005 0.052 0.074 c4 0.886 0.045 0.796 0.976 c5 0.098 0.006 0.109 0.086 Adjusted Rsquare = 0.96

Table 5.4: Parameter Estimates for Top Tie Bar Model Fit

Parameter Estimates 95% Confidence Interval Std. Lower Upper Parameter Estimate Error Bound Bound c0 1.04 0.002 1.036 1.044 c1 0.294 0.012 0.271 0.318 c2 0.256 0.014 0.228 0.284 c3 0.062 0.005 0.051 0.073 c4 1.025 0.052 0.922 1.128 c5 0.106 0.005 0.117 0.095 Adjusted RSquare = 0.95

Table 5.5: Parameter Estimates for Bottom Tie Bar Model Fit

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The power law models to predict top and bottom tie bar loads are given by equation

(57) and (58) respectively.

0.354 0.303 Top Tie bar load   DPX    DPY  = 1.0051 ±   1 +   × Nominal Load  0.5L   0.5L    tb    tb  (57)  CPR × A   CPX   CPY  CPR × A  × 1 + 0.063 exp1 ±  + 0.8861 +   ×  CLAMP   0.5L   0.5L  CLAMP     tb   tb     CPR × A  CPR × A  × 1 − 0.098 × exp    CLAMP  CLAMP 

0.294 0.256       Bottom Tie bar load  DPX   DPY  = 1.041±   1−   × Nominal Load   0.5L t b    0.5L t b  (58)  CPR × A   CPX   CPY  CPR × A  × 1 + 0.062 exp1±  +1.031 −   ×  CLAMP   0.5L   0.5L  CLAMP     t b   t b     CPR × A  CPR × A  × 1 − 0.106 ×exp    CLAMP  CLAMP 

The term nominal load on the left hand sides of the equations (57) and (58) is defined as one fourth of the total clamp load. The ± sign before the terms involving

DPX and CPX indicates that the signs of these variables depend on the location of the die/cavity and also on the tie bar for which the load is calculated. If the die and/or cavity is positioned towards the tie bar for which the load is to be predicted, a positive sign should be chosen for the corresponding variables and if the die and/or cavity is positioned away from the tie bar for which the load is to be predicted a negative sign should be chosen. For example, if the die is horizontally offcenter towards the helper side, a positive sign should be used before the variable DPX to predict the loads on the helper side tie bars and a negative sign should be used before DPX to predict the

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loads on the operator side tie bars and vice versa. The same sign convention applies to the terms involving the horizontal cavity location CPX.

The third term in the equation represents the moments caused by the cavity pressure load. The moment terms and the load term appear as exponential terms in the model.

This indicates that the tie bar loads increase or decrease in an exponential fashion as the cavity center of pressure is moved towards or away from the respective tie bars.

The parameters involving the die dimensions, Lx/√A and Ly/√A were found to have a negligible effect on the model predictions and hence they were ignored in the model.

These nondimensional parameters were included in the initial model fitting. But the error involved in their estimates for the corresponding exponents were much higher than the value of the exponents itself. Therefore this model behaves as a lumped model where the clamp and pressure loads are approximated by point loads acting on the die center and cavity center of pressure respectively.

5.4 Model Adequacy

The power law models shown in equations (57) and (58) were obtained by curve fitting to tie bar load data from an 8.9 MN (1000 ton) four toggle machine. It can be seen from equations (57) and (58) that the location variables are all scaled by the distance between the tie bar centers. The distance between tie bar centers is determined by the machine manufacturer based on the size and clamping capacity of the die casting machine. To study the adequacy of the model to predict the tie bar loads on machines of other designs and clamping capacity, the power law model

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predictions were compared against the finite element model predictions of tie bar load on machines of other designs and tonnages. Three different machine finite element models were considered, viz, a 3500 ton four toggle machine, 1000 ton four toggle machine and a 250 ton two toggle machines. The die location, cavity location, the clamp load and magnitude of cavity pressure for these three cases are summarized in

Table 5.6

Cavity Clamp Machine DPX DPY CPX CPY CPR L tb Load Load Design (in) (in) (in) (in) (PSI) (in) (tons) (tons) 3500 ton4 4 0 4 0 0 84.25 0 3500 toggle 1000ton4 0 1.25 0 3.63 10000 44 602 722 toggle 250ton2 0 3.14 0 0.423 10000 21.75 135 250 toggle

Table 5.6: Summary of Finite Element Models used for Model Adequacy Study

The boundary conditions used in the finite element model of 3500 ton machine is same as the boundary conditions in the computational experiments that yielded the data for power law model development. However the finite element models of the

1000 ton machine and 250 ton machine are different from the finite element models in the computational experiments. A schematic of the finite element model used for the 1000 ton and 250 ton machines are shown in Figure 5.2. In this model the corner nodes on the bottom of the cover platen are constrained in vertical direction only. One end of the tie bars is constrained to the cover platen and the other end is constrained in space. The rear platen is not included in the model and the toggle mechanism was

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modeled using spring elements of appropriate stiffness. The clamp load was applied by specifying displacements on the nodes of the free ends of the spring elements.

Since the tie bars are constrained in space and the compliance of the rear platen is not included in the model shown in Figure 5.2, the model is stiffer than the finite element models used for power law model development. Therefore comparison of the power law model predictions and predictions from the finite element model shown in Figure

5.2 will provide insight into the sensitivity of the tie bar load predictions to the boundary conditions used in the finite element models.

Figure 5.2: Schematic of the Finite Element Model of the 1000 Ton Machine and 250 Ton Machine Used for Model Adequacy Study [1]

The comparison between the FEA predictions and power law predictions for the 3500 ton machine, 1000 ton machine and 250 ton machine are shown in Table 5.7, Table

5.8 and Table 5.9 respectively.

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Tie Bar Load/Nominal Load Prediction FEA Power Law Top Tie Bar1 3.8% 3.0% Top Tie Bar2 2.8% 3.8% Bottom Tie Bar1 2.9% 2.6% Bottom Tie Bar2 4.0% 3.1% Table 5.7: Comparison of Model Predictions for a 3500 Ton Machine (DPX=4”, DPY=0”, CPX=4”, CPY=0”, CPR=0 PSI)

For the 3500 ton machine shown in Table 5.7, the die is offcenter to the right, towards the top tie bar2 and bottom tie bar2 and hence these two tie bars carry higher loads than the nominal. Top tie bar1 and bottom tie bar2 carry less than the nominal load. It can be observed from Table 5.7 that the differences between the FEA predictions for the top tie bar1 and bottom tie bar1 of the 3500 ton machine is 1.1% and the differences between the FEA predictions for top tie bar2 and bottom tie bar2 is 1.2%. Since the same types of constraints were used in both the FEA and power law models, the difference between the predictions is only about 1% which is negligible. This shows that the non dimensional power law model is adequate to predict the tie bar loads well independent of the size of the machine. Though the die is not offcenter vertically there are differences in the FEA predictions between the top and bottom tie bars. This is due to the constraint imposed on the bottom of the cover platen by tying the nodes on the platen to the machine base.

Table 5.8 shows the FEA and power law predictions for a 1000 ton four toggle machine. Since the die and cavity are both offcenter towards the top tie bars, the top tie bars carry loads higher than the nominal in this case and the bottom tie bar loads are lower than the nominal. In the FEA, two nodes on the bottom of the cover platen

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were constrained in vertical direction alone. Therefore the FEA predictions for the top and bottom tie bars are symmetric about the nominal load. However the power law model shows that the top tie bar loads are 6.7% higher than the nominal while the bottom tie bar loads are only 2.1% lower than the nominal load. The constraint used between the nodes of the cover platen and the base in the computational experiments causes this asymmetry in the tie bar load predictions between the top and bottom tie bars. The same trend is observed on the tie bar load predictions for the 250 ton two toggle machine shown in Table 5.9. The die and cavity are offcenter vertically towards the bottom tie bars in the 250ton machine and hence the top tie bars loads are lower than the nominal and the bottom tie bar loads are higher than the nominal.

Both the FEA and power law model are able to capture this trend. However the power law predictions for the bottom tie bar are higher than the FEA predictions. The boundary conditions used in the FEA for the 250 ton machine are same as those shown in Figure 5.2 and hence the observed symmetry in FEA prediction between the top and bottom tie bars. The higher predictions of the power law model on the bottom tie bars are caused due to the constraint on the bottom of the cover platen.

Tie Bar Load/Nominal Load Prediction FEA Power Law Top Tie Bar1 4.1% 6.7% Top Tie Bar2 4.1% 6.7% Bottom Tie Bar1 4.1% 2.1% Bottom Tie Bar2 4.1% 2.1% Table 5.8: Comparison of Model Predictions for a 1000 Ton Machine (DPX=0”, DPY=1.25”, CPX=0”, CPY=3.63”, CPR=10000 PSI)

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Tie Bar Load/Nominal Load Prediction FEA Power Law Top Tie Bar1 8.2% 8.3% Top Tie Bar2 8.2% 8.3% Bottom Tie Bar1 8.2% 14% Bottom Tie Bar2 8.2% 14% Table 5.9: Comparison of Model Predictions for a 250 Ton Machine (DPX=0”, DPY=3.14”, CPX=0”, CPY=0.423”, CPR=10000 PSI)

5.4.1 Model Adequacy Study using Experimental Measurements

The predictions from the power law model are not expected to match exactly the tie bar load measurements from a die casting machine due to various approximating boundary conditions in the finite element model. Comparison with measurements on a die casting machine provide further insight into the adequacy of this model and also its limitations.

Therefore experiments were conducted on a 250 ton twotoggle machine by varying the die location and obtaining the tie bar loads under clamp load only. The schematic of the test die on the machine platens is shown in Figure 5.3. The test die measures

13.38 inches by 18 inches and the distance between the tie bar centers is 21.75 inches.

Four uniaxial strain gauges were attached to each tie bar to measure the longitudinal strains. The strain gauges were attached to the tie bars at a distance of 267 mm (10.5 in) from the inside face of the stationary platen so that the strain gauges are halfway between the stationary and movable platens when the test die is closed. The schematic of the strain gauge locations is shown in Figure 5.4. The four strain gauges on each tie bar are 90 ο apart. Thirteen different die setups were studied, one with a die centered

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on the platen, four cases with vertically off center dies, four cases with horizontally off center dies and four cases with diagonally off center dies. The thirteen cases are summarized in Table 5.10. Not all combinations of diagonally off center dies could be studied due to the limitation of the space available between the tie bars.

Figure 5.3: Schematic of the test die on the machine platens

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Figure 5.4: Schematic of the locations of tie bars, strain gauges and coordinate system, viewed from front of cover platen

DPX Run DPY (inches) (inches) 1 0 0 2 0 2 3 0 4 4 2 0 5 4 0 6 0 2 7 0 4 8 2 0 9 4 0 10 2 1.875 11 4 1.875 12 2 1.875 13 4 1.875 Table 5.10: Experimental Array

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Each case was repeated three times and a constant clamp load of 2500 KN was applied in all cases. Though the machine was programmed to apply a clamping load of 2500 KN, the actual clamp load applied by the machine varies slightly from the nominal. The metal injection stage was ignored in these experiments due to the practical difficulty of moving the shot sleeve for each test. The strains on each tie bar were obtained under clamp load only. The average of the strains measured by the four strain gauges on each tie bar was calculated to obtain the strain on each tie bar. The tie bar loads were then calculated from the strain values. The nominal clamp load per tie bar was assumed to be the average of the four tie bar loads and the load on each tie bar was estimated as the ratio between the tie bar load and the nominal load.

5.4.2 Comparison of Experimental Measurements and Model Predictions

The tie bar loads were also calculated for all the thirteen cases using the power law model. The experimental measurements and power law predictions of tie bar loads for the thirteen cases are shown in Table 5.11. Ideally the loads on all four tie bars should be equal for case1 where the dies are centered on the platen. However the measurements show that the loads on bottom tie bars are lower than on the top tie bars. This could be due to inaccuracy in positioning the dies on the platen causing the measurements to be biased towards the top tie bars. Further, the squareness of the machine was not checked before the experiments and lack of squareness and perfect flatness of the machine platens could also contribute to this observed difference.

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Experimental Polynomial model Case DPX DPY Measurements Predictions T1 T2 B1 B2 T1 T2 B1 B2 1 0 0 1.03 1.02 0.99 0.97 1.00 1.01 1.04 1.04 2 0 2 1.10 1.09 0.92 0.90 1.05 1.06 0.99 0.99 3 0 4 1.16 1.15 0.85 0.83 1.10 1.11 0.93 0.93 4 2 0 0.99 1.09 0.90 1.02 0.94 1.07 0.98 1.09 5 4 0 0.94 1.14 0.85 1.07 0.86 1.12 0.91 1.13 6 0 2 0.96 0.95 1.05 1.04 0.94 0.95 1.08 1.08 7 0 4 0.92 0.90 1.10 1.08 0.87 0.88 1.12 1.12 8 2 0 1.09 0.98 1.03 0.91 1.06 0.94 1.09 0.98 9 4 0 1.16 0.92 1.09 0.83 1.11 0.86 1.13 0.92 10 2 1.875 1.06 1.15 0.84 0.95 0.98 1.12 0.93 1.04 11 4 1.875 1.03 1.22 0.76 1.00 0.90 1.17 0.87 1.08 12 2 1.875 1.02 0.90 1.11 0.97 1.00 0.89 1.13 1.02 13 4 1.875 1.08 0.84 1.17 0.91 1.05 0.82 1.18 0.95

Table 5.11: Comparison of Tie bar Load Measurements and Tie bar Load Predictions from the Regression Model

The comparison between model predictions and experimental measurements for the

13 cases and for all four tie bars is shown in Figure 5.5. It can be observed from these charts that the model predictions are consistently lower than the experimental measurements for the top tie bars and they are consistently higher than the measurements for the bottom tie bars in all of the 13 cases. This can be attributed to the constraint type used between the cover platen and the machine base. The edge nodes of the cover platen and the base were tied using a multi point constraint in the computational (FEA) experiments. Though the 250ton machine used for the experiments has a welded joint the multipoint constraint used in the FEA might be

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stiffer than the actual welded joint on the machine. The lack of flatness and squareness of the dies could also have contributed to some of these differences.

Figure 5.5: Tie bar Load Measurements vs. Predictions from the Regression Model

The differences between the measurements and model predictions are shown in Table

5.12. The differences between the model predictions and the load measurements vary from 0.1% to 12% depending on the die location. As expected, the worst cases are the diagonally off center cases, case10 and case11, where the measurements show that the load on the top tie bar1 (T1) is higher than the nominal and the model predictions show that the loads on top tie bar1 (T1) is lower than the nominal. Similarly the load measurements on bottom tie bar1 (B2) are lower than the nominal in case10 and

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case11, and the model predictions show that they are higher than the nominal.

Overall, the pattern of results is very good.

Difference between Measurements and Model Case DPX DPY Predictions (%) T1 T2 B1 B2 1 0 0 2.54% 1.26% 4.83% 7.40% 2 0 2 4.50% 2.84% 6.77% 9.17% 3 0 4 6.52% 4.59% 7.20% 9.76% 4 2 0 5.40% 2.03% 7.72% 6.95% 5 4 0 7.63% 2.24% 6.75% 6.03% 6 0 2 1.98% 0.10% 2.75% 4.90% 7 0 4 4.28% 2.13% 1.80% 4.32% 8 2 0 2.80% 3.37% 5.10% 7.81% 9 4 0 5.19% 5.90% 3.89% 9.10% 10 2 1.875 8.15% 3.25% 9.65% 9.23% 11 4 1.875 12.33% 4.66% 11.54% 8.60% 12 2 1.875 2.49% 1.02% 2.25% 5.41% 13 4 1.875 3.39% 2.29% 0.91% 4.14% Table 5.12: Difference between Measurements and Model Predictions

5.5 Response Surface Plots for the Effect of Die Location and Cavity

Location on Tie Bar Loads

The relative effects of the die location and cavity location on the tie bar load imbalance can be explained using the power law models shown in equations (57) and

(58). These equations can be used to generate the response surface plots showing the tie bar load imbalance as a function of two variables. Figure 5.6 shows the effect of the location of cavity center of pressure on the load on top tie bar2, when the pressure load is 20% of the clamp load. Based on the sign conventions used in the power law models, a positive value for CPX and CPY means the center of pressure is

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oriented towards the top tie bar2 and a negative value indicates a movement away from top tie bar2.

The dies are centered on the platen for the case shown in Figure 5.6. Since the cavity load is very negligible the tie bar load is almost equal to the nominal load irrespective of the location of the center of pressure.

Figure 5.7 shows the same case but with a cavity load that is 80% of the clamp load.

It can be seen from this figure that there is a maximum of 9% imbalance in the positive and negative directions, when the center of pressure is 4 inches offcenter from the platen center both horizontally and vertically.

1.1

1.08

1.06

1.04

1.02

0.98 Top Tie Bar Load / Nominal Load Nominal / Load Bar Tie Top

0.96 4

2 4 2 0 0 -2 -2 -4 -4 CPY (inches) CPX (inches)

Figure 5.6: Effect of Cavity Location on Tie Bar Load (Cavity Load = 20% Clamp Load)

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1.07

1.1 1.06

1.05

1.05 1.04

1.03

1.02 1

1.01

1 Loadon Top NominalTieBar-2 / Load 4

2 4 0.99 0 2 0 0.98 -2 -2 -4 -4 CPY (inches) CPX (inches)

Figure 5.7: Effect of Cavity Location on Tie Bar Load (Cavity Load = 80% Clamp Load)

Figure 5.8 and Figure 5.9 show the effect of die location on the tie bar load imbalance when the cavity load was 20% of clamp load and 80% of clamp load respectively. In both of these cases the center of pressure is located on the geometric center of the platen. It can be seen from these figures that a die that is offcenter by 4 inches from the platen center causes an imbalance of approximately 10% irrespective of the magnitude of the cavity load.

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1.1

1.2

1.05

1.1

1 1

0.9 0.95

Loadon Top Tie Nominal Bar-2 / Load 4 2 4 0 2 0.9 0 -2 -2 -4 -4 DPY (inches) DPX (inches)

Figure 5.8: Effect of Die Location on Tie Bar Load (Cavity Load = 20% Clamp Load)

1.1 1.2

1.1 1.05

0.9

Loadon Top Tie NominalBar-2 / Load 4 0.95 2 4 0 2 0 -2 -2 0.9 -4 -4 DPY (inches) DPX (inches)

Figure 5.9: Effect of Die Location on Tie Bar Load (Cavity Load = 80% Clamp Load)

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It can be concluded from these plots that the effect of the location of the dies on the platen has the primary effect on the tie bar load imbalance. When the clamp load is applied, it is distributed among the four tie bars depending upon the location of the dies on the platen. In the next stage when the pressure load is applied, it relieves and redistributes the contact load on the parting surface. The redistribution of the contact load on the parting surface will depend upon the location of the cavity center of pressure and the magnitude of the cavity pressure. Due to the redistribution of the contact load, the moments on the tie bars are altered and the tie bar load is also redistributed. Therefore ignoring the location of the die could lead to poor predictions of tie bar load.

5.6 Summary and Conclusions

Power law model to predict tie bar loads as a fraction of the nominal load has been developed using dimensional analysis and finite element modeling experiments. The functional form obtained from the dimensional analysis ensures that the model predictions would be independent of the machine size and tonnage. The adequacy of the model has also been verified against experimental tie bar load measurements from a 250ton two toggle machine. The power law models were constructed using the tie bar load data obtained by finite element modeling of a 1000 ton fourtoggle machine.

Considering this fact and the magnitude of the loads involved, 0.1%12% difference between the model predictions and the load measurements from a 250 ton twotoggle

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machine is very reasonable. The model seems to predict well independent of the machine size.

The model predictions were also compared against predictions from finite element model with boundary conditions and constraints that are different from the ones used in the computational experiments. The comparison showed that the power law model predictions for the bottom tie bar are about 2% higher than the nominal load due to the tied constraint between the nodes on the bottom of the cover platen and the machine base. The power law also shows that ignoring the die location, as done in the current approach in industry, will lead to inaccurate tie bar load predictions. The significance of the effect of the die location on the tie bar load imbalance is also illustrated by the response surface plots.

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CHAPTER 6

CONCLUSIONS AND FUTURE WORK

Empirical correlations to predict parting plane separation and tie bar loads were developed in this research. In addition, a methodology to characterize the platen stiffness and determine the equivalent stiffness of dies and machine was also developed. These findings can be used together as tools for structural die design and machine selection. The closed form models also provide an understanding of the contribution of the structural design variables of the die and the machine on their mechanical performance. The major conclusions from this study are summarized in this chapter.

6.1 Conclusions from the Power Laws to Predict Parting Plane

Separation

The model forms for the power laws to predict parting plane separation were obtained using dimensional analysis and semi empirical plate deflection equations. These models can be used during the initial die design stages to compare the mechanical performance of dies and machines and they can be used as a decision tool for structural die design and machine selection.

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The following conclusions can be made about the relative contribution of the structural design variable to the parting plane separation:

• The ejector side parting plane separation has the highest sensitivity to the ratio

of the weighted average of unsupported span between the pillar supports and

the die thickness.

• The platen thickness has only a first order effect on the ejector side separation.

The effect of platen thickness decreases as the number of support pillars is

increased.

• A die that is smaller relative to the platen area between the tie bars performs

better than a large die

• The largest contributor to the cover side separation is the length and width of

the die. A smaller die results in less separation. As the cover platen wraps

around the cover die, a smaller die will have less unsupported span behind it as

compared to a large die and hence it will result in lower separation

• The second major factors affecting the separation on the cover side are the

cover platen thickness and the distance between the tie bar centers. The

sensitivity of the cover side separation to these two factors is of the same order

of magnitude. A thicker platen provides better support to the dies and results in

less separation. The distance between the tie bar centers has an interaction with

die footprint. A larger platen area relative to the die foot print provides better

support and results in less cover side separation.

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• The die thickness is the next significant variable affecting the cover side

separation. Thin compliant dies are squeezed on to the platen surface and they

follow the deflection path of the platen surface. Therefore thin dies receive

better support from the platens on the cover side.

• Small thin dies on a large thick platen performs better on the cover side,

whereas on the ejector side, the span between the pillars and the die thickness

alone play the major role on the parting surface separation.

• The die thickness ratio was found to have no effect on the ejector and cover

side parting surface separation. This could be due to the tied contact

formulation used between the rear of the insert and the front face of the die

shoe pocket. Due to the tied contact constraint the dies and inserts behave as a

single monolithic block and hence the effect of die thickness ratio or the effect

of the insert thickness could not be confirmed from these computational

experiments.

The model adequacy study showed that the predictions of the ejector side model have a standard deviation of ±0.0022 inches and the cover side models have a standard deviation of ±0.0005 inches even for the test cases that are outside the model domain.

6.2 Conclusions from the Machine characterization Study

A methodology to characterize the stiffness of the machines was developed in this study. This method can be used to determine the stiffness of the platens with complex geometric features and also to obtain an equivalent thickness parameter of a solid

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platen with same stiffness characteristic. The equivalent stiffness parameter obtained by this method can also be used in the power law models to predict the performance of the dies on the chosen machine. This study has also shown that the location of the toggle mechanism has a significant effect on the stiffness of the ejector platen. A four toggle platen has a higher stiffness than a two toggle platen of same thickness.

However this does not imply that two toggle platens are less stiff than a four toggle platen of same tonnage on the actual machines. In the actual machines two toggle platens are usually made thicker or ribs are added to the platens to achieve the required stiffness.

A static lumped element model that uses springs with appropriate stiffness values to represent the dies and machine was also presented and this model can be used to determine the equivalent stiffness of the die/machine system.

6.3 Conclusions from the Power Laws to Predict Tie Bar Loads

Power law models to predict the loads on the tie bars as a fraction of the nominal load were presented in the previous chapter. These models show that the die location on the platen is the primary factor affecting the distribution of the loads on the tie bars.

The cavity load redistributes the contact load on the parting surfaces and changes the moments on the tie bars. The redistribution of clamp load after the pressure stage will depend on the location of the cavity and the magnitude of cavity pressure.

The die length and die width were found to have no effect on the tie bar load predictions. The power law model interprets the clamp load from the dies and the

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cavity load as point loads acting on the geometric center of the die and the center of pressure of the cavity respectively.

The predictions from the power law models were compared against tie bar load measurements from a 2.2 MN (250 ton) two toggle die casting machine by varying the die locations. The comparisons showed a 0.1% to 12% difference between the predictions and the measurements depending upon the location of the dies on the platen. The power law models were obtained by curve fitting to tie bar load data from finite element models of an 8.9 MN (1000 ton) four toggle machine. The 12% difference was observed for one of the extreme offcenter cases. A part of these differences could have been caused due to the lack of perfect flatness on the machine platen and the inaccuracies in positioning the die on the platen during experiments.

Given the differences in the machine design, clamping capacity and the magnitude of loads involved, 0.1%12% difference between the predictions and measurements are reasonable and it can be said that the model predicts well independent of the machine size.

The power law predictions were also compared against FEA predictions for machines of different sizes. Though the boundary conditions were different in two of these test

FEA models from the ones used in the computational experiments, the power law predictions and FEA predictions showed a good correlation. This comparison also provided an insight that the power law predictions for the bottom tie bars might be

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slightly higher than the nominal value (approximately 2% higher than nominal for the a case with die and center of pressure centered on machine platen)

6.4 Future Work

The power law model to predict parting plane separation are specific to single cavity open close dies. It is possible to extend this model to multiple cavity dies by characterizing the span between the supports with reference to the centers of pressure of the cavities. However the adequacy of the model for multiple cavity dies has to be verified.

The models to predict tie bar load are also specific to openclose dies. The slide lock mechanism in nonopen close dies could cause asymmetric loads on the parting surface depending on the locations of the slide carriers. Therefore the power law models can be reconstructed by including tie bar load data from nonopen close dies with different locations of the slide carrier.

Currently the pressure load in the die cavity is modeled as a pressure boundary condition on the cavity surfaces with a magnitude equal to that of the intensification pressure and the dynamic impact load caused by the deceleration of the plunger mechanism is ignored. There are instances where the pressure hike in the cavity due to the impact load might exceed the intensification load. Therefore including the pressure distribution in the cavity caused by the impact load might give better predictions of parting plane separation from the finite element models.

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Finally the constraint between the cover platen and the machine base is an issue that needs further investigation. Though the explicit modeling of bolted joint showed tie bar load predictions that are closer to the nominal, the constraint is still stiffer than the joint in the actual machine. Including the clearances of the bolted joint and the keyways might improve the tie bar load predictions of the finite element model.

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APPENDIX A

DESCRIPTION OF METHODS TO REMOVE PSEUDO

RIGID BODY MOVEMENT

The pseudo rigid body translation and rotation of the dies and inserts caused by the stretching of the tie bars was described in Chapter 3. The equation used to define this affine transformation is given by

R 0 [][][]Xs 1 = Xf 1   + E 0 (A1) T 1

Where X s and X f are the starting and final coordinates of the sample nodes respectively, R is the rotation matrix, T is the translation vector and E is the distortion component. In this method X s and X f are obtained for a few sample nodes on the parting surface of the dies. The rotation matrix R and translation vector T are given by

 cosΦ.cosγ cosΦ.sinγ sinΦ  R =  sinθ.sinΦ.cosγ + cos θ.sin γ sinθ.sinΦ.sinγ + cos θ.cos γ sinθ.cosΦ (A2)    cosθ.sinΦ.cosγ + sin θ.sin γ cosθ.sinΦ.sinγ + sin θ.cos γ cos θ.cos Φ

′ T = [x y z] (A3)

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The variables θ, Φ and γ in equation (A2) are the angles of rotation about the X, Y and Z axis respectively. The variables x, y and z in equation (A5) are the translations along the X, Y and Z axis respectively.Least squares method that minimizes the sum of squares 'trace(E T E)' was employed to estimate the transformation matrix in equation (A1). By differentiating the 'trace(E T E)' and equating it to zero and doing a few matrix manipulations it can be shown that, the best estimate for the translation T is the difference between the average starting and finishing position after rotation and it is given by

* T = X s − X f * R (A4)

Where, Xs and Xf are vectors with the average starting and finishing coordinates as their components. It follows from (A1) and (A4) that

E = [Xs −1* X f ]− [( Xf − 1 * Xf )* R] (A5)

The ‘1’ in equation (A5) indicates the homogenous coordinates used to represent the affine transformation which is a translation followed by a rotation. Equation (A5) is used to minimize the sum of squares 'trace(E T E)'.

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Therefore the necessary conditions for minimum error based on the angles of rotation are given by

∂f (ϕ T, )  ∂ET ∂E   T  (A6) = tr E + E  = 0 ∂θ  ∂θ ∂θ  ∂f (ϕ T, )  ∂ET ∂E   T  (A7) = tr E + E  = 0 ∂Φ  ∂Φ ∂Φ  ∂f (ϕ T, )  ∂ET ∂E   T  (A8) = tr E + E  = 0 γ∂  γ∂ γ∂ 

Using (A5) in (A6), (A7) and (A8)we obtain,

 T  ∂f (ϕ T, )  T T ∂R x (θ)  (A9) = trR z ()()γ R y Φ [cov()()()()()Xf ,Xs − cov Xf ,Xf R x θ R y Φ R z γ ] = 0 ∂θ  ∂θ 

 T  ∂f (ϕ, T)  T ∂R y ()Φ T  (A10) = trR ()γ R ()()()()()()θ [cov X , X − cov X , X R θ R Φ R γ ] = 0 ∂Φ z ∂Φ x f s f f x y z  

 T  ∂f (ϕ T, ) ∂R z (γ) T T  (A11) = tr R y ()()()()()()()Φ R x θ [cov Xf ,Xs − cov Xf ,Xf R x θ R y Φ R z γ ] = 0 γ∂  γ∂  Equations (A9), (A10) and (A11) are only a function of the coordinate data of the sample nodes and the rotation component R. The term ‘cov’ in these equations denotes the covariance between the respective variables. These equations are solved for the angles of rotation and the rotation matrix is formed. Then equation (A4) can be used to estimate the translation component. The estimates for the affine transformation will be sensitive to the sample nodes that are used. As a result the location of the nominal parting plane will also be sensitive to the sample nodes.

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A.1. Procedure to Select Sample Nodes and Predict Pure Distortion of the Parting Plane from Finite Element Models

The nominal location of the parting plane is defined as the plane that passes through all of the contact nodes. Figure A.1 shows the contact pressure plot of an ejector parting surface superimposed on the undistorted nominal parting plane. The blue region in the plot is the region that is not in contact. The sample nodes from this region should be avoided and the sample should be selected from the corner regions that are in contact. These sample nodes are then input to a Mathcad program that estimates the transformation matrix. Once the transformation matrix is obtained from the Mathcad program, the pure distortion component is obtained using the relation

R∗ 0 E 0 = X 1 − X 1 (A12) [][][]s f  ∗  T 1

Where, X s and X f are the initial and final coordinates of the entire node set of the dies and inserts and R* and T* are the best estimates for the rotation and translation components respectively. The pure distortion values were then input as nodal boundary conditions to a static finite element model with dies and inserts only. A dummy step with no other loads was run in this finite element model and the displacement plots were obtained from this finite element model. This pure distortion of the parting surface is equivalent to the separation of the respective parting surfaces from the nominal location. The ejector side separation plot obtained by this method is

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shown in the Figure A.2. It can be observed from Figure A.2 that the nominal plane after removing the transformation passes through all of the contact nodes.

Figure A.1: Contact Pressure Plot

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Figure A.2: Ejector Side Separation Obtained by Sampling Nodes in Contact Regions Only

A.2. Alternate method to Remove Pseudo Rigid Body Motion Using a

Local Coordinate System in ABAQUS

To test and compare the affine transformation procedure, local coordinate systems were created internally in ABAQUS using three nodes for each coordinate system definition. The three nodes should be in contact and they were chosen on three different corners of the parting surface, consistent with a right handed coordinate system. Then the Zdisplacements of the parting surface nodes with respect to each of these local coordinate systems were directly obtained from ABAQUS and the displacements were averaged. The average Zdisplacement is now the measure of the

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ejector side separation and the contour plot for the displacement is shown in

Figure A.3. Comparing Figure A.2 and Figure A.3, it can be seen that the affine transformation method and the local coordinate system methods give similar predictions of parting plane separation. The difference in maximum separation predictions among the two methods varied from 0.1% to 5% for the cases in which the dies and the cavity center of pressure were centred on the platen. For cases with offcenter dies and/or center of pressure, the parting surface might be completely out of contact on one or more corners. In such cases the affine transformation procedure will yield better predictions of the nominal location of the parting plane.

Figure A.3: Separation With Respect to a Local Coordinate System on the Parting Surface

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APPENDIX B

LINEAR MODELS FOR TIE BAR LOAD PREDICTION

B.1 Linear Model for Top Tie Bar

         DPX   DPY   CPX   CPY  T = 99 1. + 70±  + 60 4.   −11 9. ±  − 7.8    Ltb   Ltb   Ltb   Ltb       ± CPX×CPR   CPY×CPR  + .0 23(CPR) + 7.3   + 8.2    Ltb   Ltb  (B1)

B.2 Linear Model for Bottom Tie Bar

         DPX   DPY   CPX   CPY  B =101 6. + 59 4. ±  + 58 1.   −12 2. ±  −17 5.    Ltb   Ltb   Ltb   Ltb       ± CPX×CPR   CPY×CPR  + (4.0 CPR) + 1.4   + 4.4    Ltb   Ltb  (B2)

185