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PAGE 1 1 OPTIMIZATION OF A CITRUS HARVESTING SYSTEM BASED ON MECHANISTIC TREE DAMAGE AND FRUIT DETACHMENT MODELS By SUSHEEL KUMAR GUPTA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Susheel Kumar Gupta PAGE 3 3 To my parents for supporting me in every possible way to achieve my goals and fulfill my dreams PAGE 4 4 ACKNOWLEDGMENTS Firstly, I would like to thank my parents for their encouragements blessings and never ending supports which has helped me achieve one of the most cherished dreams of my life to study and participate in the cutting edge res earch in USA. I would like to thank Dr. Nam providing his unstinted guidance, inputs and supervision during the course of my thesis. I am grateful to Dr. Reza Ehsani for giving me chance to work on this cha llenging, demanding yet exciting research project and supervising and motivating throughout the research. Without their support and guidance, I would have not able to grown and matured as a student and a researcher. I would like to thank Dr. Peter Ifju, an d Dr. Arthur Teixeira for their guidance and assistance in the experiments. Very special thanks to Ji nsang Chung and Garret Waycaster who helped me in understanding the concepts and principles required for my research. I am grateful for the suggestions and feedbacks from my lab mates in the mdo group especially Shu Sang, Yong Min Chung, Anirban Choudhuri, Diane C. Villanueva and Taiki Matsumara. I am would also like to thank my labmates from CREC especially, Amanda L Valentine, Joe Mari Maja, Sherrie Buch anon, Sweeb Roy, Ali Mirzakhani Ja a far Abdulridha Luv K hot, and Ying S he for guiding me and assisting me in experiments, and data collection. A special thanks to my friends Mayank Amin, Rajesh Metta, Chirag Mistry, Bhavin Parikh, Bhavesh Patel, Mitesh Pa tel, Thirumalesh Rao, Akshay Panchal, Varun for their continued support and motivation. Lastly, I would like to thank my brothers : Sudhir and Satish because without their support and encouragement; I would have not been able to achieve my endeavors. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENT S ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ ........ 10 LIST OF ABBREVIATIONS ................................ ................................ ........................... 14 ABSTRACT ................................ ................................ ................................ ................... 16 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 18 Mot ivation ................................ ................................ ................................ ............... 18 Design Problem ................................ ................................ ................................ ...... 20 Design Idea ................................ ................................ ................................ ............. 21 Objective of this Study ................................ ................................ ............................ 22 Methodology Adopted ................................ ................................ ............................. 23 Report Organization ................................ ................................ ................................ 24 2 HISTORY AND LITERATURE REVIEW ................................ ................................ 25 History ................................ ................................ ................................ ..................... 25 Mechanical Harvesting ................................ ................................ ..................... 25 Mechanical Harvesting of Citrus ................................ ................................ ....... 26 Current Harvesting System for Citrus ................................ ............................... 28 Related Literatur e ................................ ................................ ................................ ... 30 3 DETERMINATION OF PROPERTIES OF CITRUS WOOD ................................ ... 34 Materials and Methods ................................ ................................ ............................ 34 Mechanical Properties ................................ ................................ ...................... 34 Modulus of elasticity ................................ ................................ ................... 35 Modulus of rupture ................................ ................................ ..................... 36 Stress at proportional limit ................................ ................................ .......... 36 Work to maximum load in bending ................................ ............................. 36 Physical Properties ................................ ................................ ........................... 37 Volume and density ................................ ................................ ................... 37 Moisture content ................................ ................................ ........................ 39 Damping coefficient ................................ ................................ ................... 40 Results and Discussion ................................ ................................ ........................... 42 Mechanical Properties ................................ ................................ ...................... 42 Physical Properties ................................ ................................ ........................... 44 PAGE 6 6 4 MODELING OF TREE LIMBS ................................ ................................ ................ 46 Material and Methods ................................ ................................ ............................. 46 Interpolating Technique: Continuo us Piecewise Cubic Hermite Interpolation .. 46 Surrogate Technique: Polynomial Response Surface (PRS) ........................... 47 Residual Analysis: Cross Validation Error ................................ ........................ 47 Experimental Setup: Acquiring Data for the Statistical Model ........................... 48 Procedure ................................ ................................ ................................ .. 50 Assumptions ................................ ................................ .............................. 50 Classification of Primary Tree Limbs ................................ ................................ 51 Prediction of Spatial Coordinates of Limb Prototypes ................................ ...... 52 Sectional Properties of Limb Prototypes ................................ ........................... 53 Distribution of Secondary Branches ................................ ................................ 54 Distribution of Fruits on the Primary Limbs ................................ ....................... 54 Fruits on secondary branches ................................ ................................ .... 55 Fruits in fruit bearing region ................................ ................................ ....... 55 Results and Discussions ................................ ................................ ......................... 55 Primary Tree Limbs Classified in Three Zones ................................ ................. 55 Spatial Coordinates of Limb Prototypes ................................ ........................... 56 Error Analysis and PRS model for limbs in the top zone ............................ 59 Error Analysis and PRS model for limbs in the middle zone ...................... 60 Error Analysis and PRS model for limbs in the bottom Zone ..................... 61 Distribution of Secondary Branches and Fruits on the Primary Tree Limbs ..... 62 Distribution of secondary branches ................................ ............................ 62 Distribution of fruits on primary limbs ................................ ......................... 64 5 FORMULATION O F FINITE ELEMENT ANALYSIS ................................ ............... 66 Finite Element Model ................................ ................................ .............................. 67 Product ................................ ................................ ................................ ............. 67 Geometric Modeling ................................ ................................ ......................... 67 Material Model ................................ ................................ ................................ .. 68 Damping Model ................................ ................................ ................................ 68 Con tact Formulation ................................ ................................ ................................ 70 Tube to Tube Elements ................................ ................................ .................... 70 Master Slave Assignment ................................ ................................ ................. 70 Contact Property Assignment ................................ ................................ ........... 71 Pressure overclosure relationship ................................ .............................. 71 Friction model ................................ ................................ ............................ 72 Contact Constraint Enforcement Method ................................ .......................... 72 Contact Interface ................................ ................................ .............................. 72 Dynamic Analysis ................................ ................................ ................................ ... 72 Analysis Method ................................ ................................ ............................... 72 Loading and Boundary Conditions ................................ ................................ ... 74 Finite Element Model Verification and Validation ................................ .................... 75 Material and Methods ................................ ................................ ....................... 75 Tine ................................ ................................ ................................ ............ 75 PAGE 7 7 Branch ................................ ................................ ................................ ....... 76 Acceleration acquisition ................................ ................................ ............. 77 Strain acquisition ................................ ................................ ........................ 7 8 S imulation ................................ ................................ ................................ .. 79 Results and Discussion ................................ ................................ .................... 79 6 MECHANISTIC MODELS ................................ ................................ ....................... 81 Mechanistic Tree Damage Model ................................ ................................ ........... 81 Mechanistic Fruit Detac hment Model ................................ ................................ ...... 84 7 MULTI OBJECTIVE OPTIMIZATION ................................ ................................ ...... 87 Problem Formulation ................................ ................................ ............................... 88 Design of Experiments (or Design Domain) ................................ ............................ 90 Design Variables: Stiffness (s) ................................ ................................ ......... 90 Design Variables: Length of Insert ( x ) ................................ .............................. 92 Design Variables: Shaking Frequency ( v ) ................................ ........................ 92 Design Variab les: Shaking Amplitude ( a ) ................................ ......................... 92 Estimation of Cost of Analysis for Optimization ................................ ...................... 93 Shaker Optimization: Phase 1 ................................ ................................ ............. 94 Materials and Methods ................................ ................................ ..................... 95 Results and Discussion ................................ ................................ .................... 96 Middle zone ................................ ................................ ................................ 97 Top and bottom zone ................................ ................................ ................. 99 Shaker Optimization: Phase 2 ................................ ................................ ............ 102 Materials and Methods ................................ ................................ ................... 102 Results and Discussion ................................ ................................ .................. 103 Middle zone ................................ ................................ .............................. 103 Top zone ................................ ................................ ................................ .. 104 Bottom zone ................................ ................................ ............................. 105 8 CONCLUSION ................................ ................................ ................................ ...... 107 Summary of Conclusions ................................ ................................ ...................... 107 Recom mendation for Future Work ................................ ................................ ........ 110 APPENDIX A LABVIEW PROGRAM ................................ ................................ .......................... 112 B PHASE 1 OPTIMIZATION ................................ ................................ .................... 116 C PHASE 2 OPTIMIZATION ................................ ................................ .................... 118 LIST OF REFERENCES ................................ ................................ ............................. 132 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 139 PAGE 8 8 LIST OF TABLES Table page 3 1 Summary of the mechanical properties of the green citrus wood. ...................... 43 3 2 Summary of the physical properties of the green citrus wood. ........................... 44 4 1 Meta models to predict the diameter of limbs of the top zone. ........................... 60 4 2 Meta models to predict the diameter of limbs of the middle zone. ...................... 61 4 3 Meta models to pr edict the diameter of limbs of the bottom zone. ..................... 61 4 4 Average mass of the secondary branches in three zones. ................................ 63 4 5 Average number of fruits on the secondary branches. ................................ ....... 64 4 6 Configuration of fruit bearing region of a citrus tree canopy in three zones. ....... 65 5 1 Pressure Over closure relationship. ................................ ................................ ... 71 7 1 Geometry and material configuration of the designs of the insert. ...................... 91 7 2 Design of experiments for numerical analysis and optimization. ........................ 93 7 3 Allowable Fruit Detachment Index. ................................ ................................ ..... 96 7 4 Optimum configuration of tine for middle section of the canopy shaker. ............. 97 7 5 Opti mum configuration of tine for top and bottom section of the canopy shaker. ................................ ................................ ................................ .............. 100 7 6 Optimum operating parameters for the middle section of a canopy shaker. ..... 104 7 7 Optimum operating parameters for the top sec tion of a canopy shaker. .......... 105 7 8 Optimum operating parameters for the middle section of a canopy shaker. ..... 105 C 1 Error norms of RBNN meta models used to predict objective functions for the middle zone tine designs ................................ ................................ .................. 118 C 2 Optimum operating parameters of the tines in the middle section of canopy shaker ................................ ................................ ................................ ............... 118 C 3 Error norms of RBNN meta models used to predict objective functions for the top zone tine designs. ................................ ................................ ....................... 123 PAGE 9 9 C 4 Optimum operating parameters of the tines in the top section of canopy shaker. ................................ ................................ ................................ .............. 123 C 5 Error norms of RBNN meta models used to predict objective functions for the bottom zone tine designs ................................ ................................ .................. 127 C 6 Optimum operating parameters of the tines in the bottom section of canopy shaker ................................ ................................ ................................ ............... 127 PAGE 10 10 LIST OF FIGURE S Figure page 1 1 Mechanical harvested acreage of Citrus in Florida ................................ ............ 20 1 2 Design idea based on the distribution of fruits and thick branches in a tr ee canopy. ................................ ................................ ................................ ............... 21 1 3 Classification of the shaker tines and a citrus tree canopy based on the limbs configurations and the fruits distribution. ................................ ............................ 22 1 4 Schematic of process involved in the realization of adaptive shaking of tree canopy using numerical simulation. ................................ ................................ .... 23 2 1 Continuous canopy shake and catch (CCSC) harvester. ................................ ... 29 2 2 Tractor drawn canopy shake (TDCS) harvester. ................................ ................ 29 2 3 CAD model of a canopy shaker showing hub and tines. ................................ .... 30 3 1 A three point bending test of a citrus wood specimen. ................................ ....... 35 3 2 High precision electronic weighing scale. ................................ ........................... 38 3 3 Ultra Pycnometer to measure the actual volume of the wood samples. ............. 38 3 4 The wood samples of the Valencia orange used in the determination of moisture content. ................................ ................................ ................................ 40 3 5 Under damped response of a system having singe degree of freedom. ............ 41 3 6 An accelerometer mounted on a primary limb of a citrus tree. ............................ 42 3 7 A citrus wood specimen under 3 point flexural bending. ................................ ... 43 3 8 The best fit and load deflection curves of green citrus wood specimens. ........... 43 3 9 Re sponse of a citrus limb measured using an accelerometer. ........................... 45 4 1 Interaction of a canopy harvester with a row of citrus trees. ............................... 49 4 2 Interaction of a canopy shaker with a citrus canopy on YZ plane. ...................... 51 4 3 Showing a tree limb having diameter at any branching node as a function of the length and the angle of that node from the limb origin. ................................ 53 4 4 Three dimensional view of the tree limbs of citrus and their classification in three distinctive zones. ................................ ................................ ....................... 56 PAGE 11 11 4 5 Tree limbs classified in three sets and plotted on the plane Y Z. ........................ 56 4 6 Spatial modeling of the limb prototypes for the top zone. ................................ ... 57 4 7 Spatial modeling of the limb prototypes for the middle and bottom zone. ........... 58 4 8 Spatial distribution of the limb prototypes of a hypothetical tree. ........................ 58 4 9 Interaction of a canopy shaker with the limb prototypes of a hypothetical tree. .. 59 4 10 Contour plot to predict the sectional diameter of a limb in the top zone. ............ 60 4 11 Contour plot to predict the sectional diameter of a limb in the middle and bottom zone. ................................ ................................ ................................ ....... 62 4 12 Distribution of mass of a secondary branch for all three zones. ......................... 63 4 13 Distribution of the fruits attached to the secondary branches. ............................ 65 5 1 Flow chart of process involved in finite element analysis based optimization of canopy shaker. ................................ ................................ ............................... 66 5 2 Finite element model of a tree limb and a tine modeled as beam element ........ 67 5 3 Finite element model of a tree limb prototype with secondary branches (brown marker) and fruits (yellow marker) modeled as the lumped mass. ......... 68 5 4 Modeling interaction between a tine and a tree limb in Abaqus. ......................... 70 5 5 Pressure overclossure relationship used in the contact formulation. .................. 71 5 6 Prescribed displacement to the tines of a canopy shaker. ................................ .. 74 5 7 Schematic of a laboratory test equipment used to validate the FE model parameters. ................................ ................................ ................................ ........ 76 5 8 Branch specimen fixed to the solid frame with a bracket and a clamp. .............. 76 5 9 MMA7260Q Acceler ometer for sensing acceleration ................................ .......... 77 5 10 A strain gauge installed on the top and bottom surface of the branch specimen. ................................ ................................ ................................ ........... 78 5 11 Comparison of RMS of maximum strain of branch specimen obtained from FEA and experiments. ................................ ................................ ........................ 79 5 12 Comparison of RMS acceleration of the branch specimen obtained from FEA and experiments. ................................ ................................ ................................ 80 PAGE 12 12 6 1 A finite element model of a tree limb showing damage region. .......................... 82 6 2 Section points of a finite beam element of the tree limb. ................................ .... 82 6 3 A finite element model of a tree limb showing fruit bearing region. ..................... 85 7 1 Proposed two pie ce design of at tine for adaptive shaking of tree canopy ......... 89 7 2 Variation of stiffness of multiple designs of insert nor malized wrt stiffness of current design of a tine. ................................ ................................ ...................... 91 7 3 Distribution of citrus fruits in the three zones of a citrus tree canopy. ................. 96 7 4 Pareto frontier to predict the optimum tine configuration of the middle section of a canopy shaker. ................................ ................................ ............................ 97 7 5 Phase 1 optimization: Contour plots for the Damage Index and Fruit detachment Index to predict optimum tine config uration for the middle section of a canopy shaker. ................................ ................................ ............................ 98 7 6 Pareto frontier for the top zone (left) and the bottom zone (r ight) to predict optimum tine configuration for the top and bottom section of a canopy shaker. ................................ ................................ ................................ .............. 100 7 7 Comparison of a cost and tensile strength of the polyamide variants. .............. 101 A 1 LabVIEW instrument program to acquire data from an accelerom eter. ............ 113 A 2 LabVIEW instrument program to acquire strain data from a strain gauge left half portion. ................................ ................................ ................................ ....... 114 A 3 LabVIEW instrument program to acquire strain data from strain gauge right half portion. ................................ ................................ ................................ ....... 115 B 1 Phase 1 optimization: Contour plots for the Damage Index and Fruit detachment Index for the top zone tine design. ................................ ................ 116 B 2 Phase 1 optimization: Contour plots for the Damage Index and Fruit detachment Index for bottom zone tine design. ................................ ................ 117 C 1 Response surfaces used to predict the damage index and fruit detachment index of best tine design D1 of middle zone. ................................ .................... 11 9 C 2 Legend information used in the Pareto fronts and contour plots. ..................... 119 C 3 Pareto front and contour plots of the tine design D1 of middle zone. ............... 120 C 4 Pareto front and contour plots of tine design D2 of middle zone. ..................... 121 PAGE 13 13 C 5 Pareto front and contour plots of the tine design D3 of the middle zone. ......... 122 C 6 Response surfaces used to predict the damage index and fruit detachment index of best tine design D1 of top section of a canopy shaker. ....................... 124 C 7 Pareto front and contour plots of the best tine design D1 to predict optimum configuration of top section of a canopy shaker. ................................ ............... 125 C 8 Pareto front and contour plots of the best tine design D2 to predict optimum configuration of top section of a canopy shaker. ................................ ............... 126 C 9 Response surfaces used to predict the damage index and fruit detachment index of tine design D1 of bottom zone. ................................ ........................... 128 C 10 Pareto front and contour plots of the optimum tine design D1 of the bottom section of a canopy shaker. ................................ ................................ .............. 129 C 11 Pareto front and contour plots of the optimum tine design D2 of the bottom section of a canopy shaker. ................................ ................................ .............. 130 C 12 Pareto front and contour plots of the optimum tine design D3 of the bottom section of a canopy shaker ................................ ................................ ............... 131 PAGE 14 14 LIST OF ABBREVIATIONS A L Aluminum CAD Computer aided design C ANOPY Aboveground portion of a plant or crop, formed by plant crowns CCSC Continuous canopy shake and catch harvester CREC Citrus research and e ducation c enter C V Coefficient of v ariation DOM Drawn o ver m andrel DI Damage index FEM Finite element model FDI Fruit detachment Index Hz Hertz L IMB Main branches of a tree L IMB P ROTOTYPE Non physical or virtual model of a tree limb M ETA M ODEL Analytical models to predict the function when true function is unknown PA Polyamide PAN Polyacrylonitrile PCHIP Piecewise cubic Hermite interpolating polynomial PF Pareto front PO Pareto o ptimal PRESS RMS Root mean square of p rediction residual sum of s quares P RS Polynomial response surface RBNN Radial b asis n eural n etwork R ESPONSE S URFACE m eta m odel RMS Root mean s quare PAGE 15 15 RSM Response surface methods S CAFFOLD BRANCHES Primary limbs radiating form the trunk of a tree TDCS Tractor drawn canopy shake harvester T INE Long tube or rod like structures of canopy shaker which perform the shaking of a tree canopy during harvesting UTM Universal testing machine PAGE 16 16 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science OPTIMIZATION OF A CITRUS HARVESTING SYSTEM BASED ON MECHANISTIC TREE DAMAGE AND FRUIT DETACHMENT MODELS By Susheel Kumar G upta August 2013 Chair: Nam Ho Kim Major: Mechanical Engineering Mechanization of fruit and nut harvesting is becoming increasing ly important because of a significant rise in the cost of manual harvesting. T imely adaptation of technological advancement and innovation is utmost important in the fruit and nut crop industries to ensure the ir continued profit and global competitive ness However, the slow pace of research in the design and development of these harvester s is barely able to satisfy the demands of the se industries. The classical method of designing a harvester based on the field experiments is tedious, time consuming and expensive. The go al of this study is to propose a framework for the design and optimization of a harvester using numerical methods: the goal is pursued by optimizing a continuous canopy shaker for citrus crop s specifically o range ( Citrus sine n sis ) A progressiv e design app roach is presented consisting of determining the properties of wood, statistical modeling of tree limb prototypes developing mec hanistic models and, integrating numerical simulation with optimization tools Response Surface Methods (RSM) and Pareto optimal solution search techniques were applied i n order to obtain the optimal designs. PAGE 17 17 Three sets of machine parameters which consist of three different configurations of t ine s or shak ing member s and operating parameters of a shaker were propo sed to minimize the tree damage and maximize the fruit removal based on the configuration s of the tree limbs and distribution of the fruit s in a medium size citrus tree. As reference to the current co nfiguration of the tines in a canopy shaker, it was foun d that by changing the configuration of tine s in the middle and the bottom section of a can opy shaker to the tine made of polyamide rods having 50% long glass fiber and hardened steel tube in the ratio of 3:1 by length and vibrating at high frequency of 7 8 Hz and lo w amplitude of 1.5 2.5 inches has resulted in 25 30% reduction in the damage of limbs which are long, thick and hang down sharply due to the weights of fruits. Changing the top section of a canopy shaker with the tine s made of polyamide rods h aving 60% long glass fiber and vibrating at a frequency of 6.5 7.5 Hz and amplitude of 3 3.5 in has result ed in a 40 45 % reduction in the damage of limbs which are thick, long, and grow straight up to a height of 100 130 inches before curving down sli ghtly due to the weight of fruits. PAGE 18 18 CHAPTER 1 INTRODUCTION Mechanical harvesting equipment is becoming increasingly important due to current manual harvesting costs and labor shortages. Continued profit and competitiveness in the global market deman ds an efficient harvesting system. M any mechanical harvesters for harvesting fruits and nuts have been developed in the past five decades, but only few of them have been successfully adopted and commercialized (Futch and Whitney, 2004). Most current resear ch focuses on increasing the am ount of mechanical harvesting either by redesigning existing systems or by developing an entirely new harvesting mechanism Many commercialized harvesters for fruits and nuts are designed to induce either free vibration or fo rced vibration to the trunk or tree limbs/canopy (Fridley et al. 1971 ; Peterson et al ., 1972; Markwardt et al. 1964; Halderson 1966). These vibrations are then transmitted throughout the tree, creating the necessary force to cause the detachment of fruits. The harvesters essentially comprise of two parts: a mechanical system which generates vibratory motions, and a mechanical int erface which transmits vibrational energy to the tree crops. Examples of mechanical interface are: a movable tongs which grip the trunk or limbs and force them to vibrate ; or a vibrating rod which impacts the tree limbs t o cause them to vibrate freely. Mot ivation Most accomplishments in a design of vibratory harvester have been achieved using heuristic methods (Fridley, 1966). This involves designing the harvesting system using rules of thumb, educated guess es or assumptions validated by extensive field tri als. However, field trials are usually very demanding and time consuming. Moreover, the specific problem areas are very difficult to evaluate by experimental studies because PAGE 19 19 of high variability and r andomness associated with the tree s and their interaction with a harvesting machine. However, e xperience from these field trials ha d provided a general understanding of the dynamic response of tree crops but the parameters that can be used to design a shaker has not yet been formulated The explanation for good results under some conditions and excessive tr ee damage in other situations has found to be related to the tree structure and their interaction with a machine ( Fridley and Adrian, 1966). Extensive field trials are necessary to formulate these interactions in order to find the optimum harvesting parameters but would be prohibitively expensive and time consuming. An alternative approach for designing an efficient harvesting system is to use numerical methods (Archer 1965; Hurt y and Rubinstein 1964). Experimentally verified numerical model s allow designer to economically iterate various designs to select the optimum one F inite elements methods are a widely used numerical technique in structural design. However the application of these methods in the design of a harvesting system is limited b ecause of the complicated and non uniform natur e of a biological structure such as tree limbs The randomness in the s patial configuration of the tree limbs and their interaction with a har vesting system makes this physical phenomenon very difficult to simulate using numerical methods. Some assumptions and generalizations should be made to efficiently approximate the physical models and their interactions Th ese method s can prove to be an economical way to optimize a harvesting system because of their capability to iterate a large number of designs in a significantly less time as compared to the experimental methods. PAGE 20 20 Design Problem Mechanical harvesting of citrus specifically oranges (Citr us sene n sis) in Florida was started in the late 1990s. Figure 1 1 shows that the harvest ing of citrus crop rose gradually thereafter because of implementation of various mechanical means in the harvesting of citrus crops as summarized by Savary (2009). How ever, with the advent of a canopy shaker (Peterson, 1998) the mechanical harvesting of citrus has reached to its peak in the years spanning from 2005 2009. The continuous canopy shakers are the most recent and widely used type of mechanical harvesting sys tem in Florida for the harvesting of citrus This system has gained popularity and acceptance over any other harvesting equipment because it does not s top at each tree during harvest and provi des a high harvesting yield of approximately 95 96 % (Whitney, 1999). Figure 1 1. Mechanical harvested acreage of Citrus in Florida (Data provided courtesy of Florida Departm ent of Citrus) However, m echanical harvesting of citrus in Florida has been declining sharply after the seasons of 2008 2009 because of probl ems associated with extensive use of PAGE 21 21 canopy shakers The operation of these machines causes excessive tree damage or injury The tree damage or injury makes trees vulnerable to disease which girdle s and kill s not only scaffold branches or a whole tree but sometimes a whole orc hard if unnoticed. Therefore, Florida growers are reluctant to use canopy shaker for harvesting of citrus crops as they severely damage the scaffold branches and reported to reduce subsequent years fruit yield (Spann and Dany luk 201 0 ) In order to increase the mechanical harvesting, the growers concerns should be addressed. Design Idea The reluctance of the growers to employ these machines for mechanical harvesting can be solved by either modifying the existing machines or developing a new machine with can minimize the tree damage without significant ly reducing h arvesting efficiency. The idea of adaptive shaking of the tree canopy has been proposed to improve c oncerns. The adaptive shaking of tree was realized by providing a variable shaking force to the tree limbs depending on the distribution of fruits and limb s (spatial configuration and stiffness) in a tree canopy as shown in Figure 1 2. Figure 1 2. Design idea based on the distribution of fruits and thick branches in a tree canopy. PAGE 22 22 The variable shaking force to different part s of the canopy was provided by setting dif ferent configuration of a canopy shaker The limbs with similar dynamic response owing to their spatial configuration and property distributions were classified in in three zones of a tree canopy as shown in Figure 1 3. Three set s of tine (or shaking member) configuration s and combination s of fre quency and amplitude of the canopy shaker w ere proposed based on dynamic response of the three sets of tree limbs Figure 1 3. Classification of the shaker tines and a citrus tree canopy based on the limbs configuration s and the fruits distribution. Ob jective of this Study The purpose of this research is to develop a methodology for the optimization of a canopy shaker employing computer aided numerical and optimization techniques In a canopy shaker : t his study reports an economical way of defining the machine tree interactions and modeling the tree limbs based on parameters derived from the statistical prototypes rather than the random individual trees. A progressive design approach is adopted PAGE 23 23 which involves the determination of the properties of the wood, accumulating and organizing statistical information for modeling tree limbs, performing dynamic analysis, and building mechanistic models for the optimization a canopy shaker The s chematic of the design process employed to predict an optimal set of machine structural and operating parameters is shown in Figure 1 4. Figure 1 4 Schematic of process involved in the realization of adaptive shaking of tree canopy using numerical simulation. Methodology Adopted The proposed design idea is implemented using computer aided techniques rather than experimental means as the latter is more expen sive and time consuming. C omputer simulation provides an efficient tool to determine the response of the whole tree to practically any vibratory force by dividing it into a large number of small element sections, with the mass and stiffness properties of each section known from the measurements (Philips 1970; Fridley and Yung 1975; Savary and Ehsani 2010). However, it is unlike that a harvester designed based on th e response of few sets of trees will perform satisfactorily for all other trees. Thus, instead of analyzing a whole Wood Properties Modeling Tree Limbs Mechanistic Models Numerical Analysis Optimization PAGE 24 24 tree, the statistical prototypes of limbs derived from random individual trees were used This process involves the prediction of the parame ters of the primary limbs based on the statistical information accumulated from a large number of trees. The response of the tree limbs with known properties of mass, stiffness and damping is obtained using numerical techniques. The response of limbs is th en used to optimize the structural and operating variables of the harvester This methodology provides an economical way to optimize a canopy shaker for harvesting citrus crops and can be successfully extended to harvest other fruits such as citrus, peache s, apple, almonds, blueberrie s, raspberries, olives, grapes, and coffee Report Organization Chapter 2 summaries the previous studies done for the design of mechanical harvester s and various techniques used in this research. Chapter 3 is about determination of physical and mechanical properties of a green citrus wood. Chapter 4 explains the methodology to predict the statistical prototype s of a tree limb to be used in the finite element analysis. Chapter 5 describes the finite element m odeling of machine, tree limbs and their interaction. Chapter 6 identifies and quantifies the objective functions for the optimization of a shaker using mechanistic models. Chapter 7 is about the multi objective optimization of the canopy shaker. The conc lusions of this study are summarized in Chapter 8 A ppendix A provides the lab view program to acquire strain ga uge and acceleration data. A ppendices B and C provide the tables and the f igure s associated with phase 1 and phase 2 optimization of canopy shak er PAGE 25 25 CHAPTER 2 HISTORY AND LITERATURE REVIEW History Mechanical H arvesting Mechanical harvesting has been successfully a dopted for many crops including some fruit and nuts crops. Harvesting method varies with crop s. T herefor e, a specialized harvesting is n eeded depending on the type of a crop. Wide varieties of harvesters are available for grain crops, vegetables, forage crops and some fruit and nut crops. For grain crops, harvesting is a process of cutting and threshing the crop to separate the grain from stalk ( Kutzbach and Quick, 1999). In case of forage crops, baling for grasses and cutting for forage cereals (Cavalchini 1999) is required For vegetables, the harvesting is very complex process and depends on the type of vegetable being harvested. For r oot crops harvesting is accomplished by digging as in potatoes or pulling as in the case of leeks (Manfred i and Peters 1999 ). For surface crop combing (peas green beans ), stripping (cucumbers ), shaking (tomatoes), de stemming (onions garlic ), threshing (peas from their pods) and cutting (cabbage, cauliflower etc. ), operation are employed for harvesting Mechanical harvesting of fruits is mostly done for process industry. Small fruits and wine grapes are harvested by a combination of contact and non con tact methods. etc. are harvested using combination of shaking and soft combing. This is achieved by shaking the crops with a finger like structure mounted radially on oscillating drums. Grapes are harvested using straddle type harvester which shake the vines to remove the grape cluster s from them. The grape harvester uses horizontal rods to shake the vines with frequencies ranging from 10 20 Hz. PAGE 26 26 For fruit and nut crops, the most common method of harvesting is to shake the When a fruit vibrate s it experiences traction twisting, bending, and shear force These forces result s in creating necessary stress at the point of contact, either at the branch stem junctions or at the stem calyx which cause the detachment of the fruit. Various fruits such as c itrus, prunes, apples, olives, almonds, blueberries, raspberries etc. are currently harvested using the similar methods Mechanical Harve sting o f Citrus Mechanical harvesting of citrus was started in 1950s in an attempt to re duce the manual labor (Whitney, 1995). Efforts were made to mechanize citrus handling as summarized by Hedden and Churchill (1984). First ever mechanization of citrus handlin g was achieved by employing two wheel trailer for handling fruits to both process industry and fresh fruit market. In the late 1950s, loader boom mounted on high lift truck was used for the handling of the processing fruits. In later years, fruit collectio n system based on vacuum was developed for fruit handling. In such a system, fruits are transferred to the closed cylindrical hopper by the picker, which are dumped directly into a roadside truck by vacuum system. While the handling of citrus was being mechanized, studies were being conducted to improve the efficiency of the machines which pick s the fruits from a tree. Jutras and Coppock (1958) conducted time and motion studies on hand harvesting. They investigated different scenarios to design harvestin g aids to cut down on non productive time. Different type of harvesting aids were studied to make manual harvesting efficient by part ially mechanizing the process of harvesting ( Coppock and Jutras, 1960). Various harvesting aid such as v ertical lift worker positioner multi boom vertical position, and multi worker positioner was developed in the course of time. PAGE 27 27 However, they were not economically very advantageous over conventional methods of picking In the 1960s, efforts were made to duplicate manual harv esting by developing harvesting machines. Lenker (1970) developed an auger based harvester. This system used a series of parallel augers having flexible flights to gently twist the fruit from citrus branch and convey it away from the tree to a collector. L ater Chen et al (1982) developed a contact harvester which had flexible fingers to harvest the mature fruits selectively. Two other detachment devices have been developed for the removal of individual citrus fruit (Schertz and Brown 1968, Brown et al. 1969 ). A vacuum twist device was developed which pull the fruits into a large rotating rubber sock like tube and deta ch the fruit s by twisting. The amount of twisting torque applied depends on the friction between the rubber sock and the fruit. To harvest a ci trus fruit a rotating cut off device which uses pair of concentric cylinders was developed This machine was composed of outer rotating cylinder with a hood that engaged the stems and draws them against a knife supported on the stationary inner cylinder. Another concept for citrus was the roller head harvester concept. This concept consisted of a series of rubber covered rollers equally spaced in a horizontal bank. All rollers rotate in the same direction and were position ed to comb vertically through the tree canopy. However, due to low harvesting yield, the focus was shifted from contact harvesting to mass harvester like trunk shaker and limb shaker Beginning in early 1960s, investigation into development of mass fruit harvester started. Research and dev elopment to harvest citrus has resulted into trunk shaker, limb shaker, foliage, and canopy shaker. One of initial inertial shaker to harvest citrus was PAGE 28 28 designed based on the work of Adrian and Fridley (1965) Some of the early mass harvesters were primarily trunk and limb shakers. Various studies were conducted to investigate the efficiency of these harvesting systems which was found to be significantly low. Various studies were performed analytically and experimentally to impro ve the performance of these systems. In the 1970s, a vertical foliage shaker w as designed which applies force to the canopy of a tree in an attempt to increase the fruit harvesting. Comparative trails performed by Hedden and Coppock (1971) concluded that t he foliage shaker had performed better than any other harvester tired in their study The air shaker designed by Whitney 1968, and Whitney and Patterson, 1972 did not come in contact with fruits, therefore does not cause any fruit bruising. H owever, these systems were efficient only with abscission chemicals. In late 1990s, a prototype similar to the current canopy shake was designed and developed by Peterson (1998 ). Harvesting trial using canopy shaker had indicated the fruit removal efficiency ranging for 80 90%. Current Harvesting System for C itrus The current mechanical harvesting system to harvest citrus fruits in Florida is a continuous canopy shaker which is enhanced version of the design proposed by Peterson (1998) There are two versions of this sys tem : (1) C ontinuous canopy shake and catch harvester as shown in Figure 2 1 which has a catch frame for collecting harvested fruits; (2) T ractor drawn canopy shaker as shown in Figure 2 2 which is only equipped to harvest the fruits and the harvested fruits are hen picked up from the ground manually or by pick up machines. The principle mechanism to detach the fruits is same in both the harvester s and is accomplished by impacting the tree limbs periodically by sinusoidally vibrating tines Th e tines ar e 78 in ches long and arranged PAGE 29 29 radially on 12 wh eels mounted on a cylindrical frame as shown in Figure 2 3 The sinusoidal motion of tines is provided by slider crank mechanism which is powered by a hydraulic motor. Figure 2 1 Cont inuous canopy shake an d catch ( CCSC) harvester. (Photo courtesy of Ehsani and Sajith.) Figure 2 2 T ractor drawn canopy shake (TDCS) harvester (Source: http://citrusmh.ifas.ufl.edu/images/history/current/025.jpg Last accessed July, 2013.) PAGE 30 30 Figure 2 3. CAD model of a canopy shaker showing hub and tine s ( Source: Courtesy of Oxbo International Corp.) Related Literature In the past, analytical models have been developed for modeling tree crops and their interaction s with various types of harvesting machine s such as trunk shakers, limb shakers, canopy or foliage shakers and over the row harvesters. T he dynamic characteristics of the tree limb and fruit syst em were predicted using various types of numerical techniques Yung and Fridley (1975) used finite element methods to model a whole tree, including the fruit stem system. They have developed three special finite elements to mathematically describe a tree s ystem and have evaluate d the natural frequencies, mode shapes, and dynamic internal stress of the complete tree structure for steady state forced vibration. Rumsey (1967) modeled the fruit stem system as an elastic beam with a concentrated load at one end and studied the effect of inertial forces due to mass of the frui ts on the bending and shear of a limb model as a beam. Pestel and Leckie (1963) studied the vibration response of non uniform beams using matrix methods. Later the method of transfer matrice s was applied by Rumsey (1967) to solve the forced response of non uniform beams using finite element methods. Fridley and Lorenzen (1965) simulated the tree shaking by modeling the limb as a four cell PAGE 31 31 cantilever beam. They used classical Euler Bernoulli b eam theory to formulate partial differential equations. The equations were solved using finite element methods to predict the beam respo nse when vibrated with constant force and varying frequency. Schuler and Bruhn (1973) applied Timoshenko beam theory wit h structural damping to formulate the differential equations for the dynamic response of the limb s They concluded that rotary inertia ha s very little effect on the beam response while the presence of deflection due to shear had a significant effect. Phili ps et al (1970) used Euler Bernoulli theory and the Rayleigh Ritz method to formulate the forced vibration of tree limbs having variable cross section s Also, in their study, various methods to model secondary branches were investigated and a computer algo rithm was program m ed to determine the vibrational characteristic of limbs with secondary branches. Hussain et al. (1975), Ruff et al. (1980), Upadhay a and Cooke (1980), and Upadhyaya et al. (1980b) studied the fruit equation. U padhyaya et al. (1980b) used Galer kin approach to solve the partial differential equations formulated based on the Bernoulli Euler beam theory. Transient response of the direct integration method. Adrian and Fridley (1965) modeled the tree limbs as a single degree of freedom cantilever beam with viscous damping. Ebner and Billington (1968) investigated the response of an internally damped non uniform beam under force vibr ations. Hoag and Hutchinson (1970) studied the effect of proportional damping (internal damping proportional to stiffness or mass distributi on), non proportional damping ( external damping p roportional to the leaf distribution), and non linear external PAGE 32 32 damp ing (or viscous damping ) which is proportional to the n th power of the velocity of the system on the dynamic response of the tree limbs. The objective quantification in the damage analysis using mechanistic model s is widely employed in the earthquake scien ce. The design based on mechanistic models was first reported by Park and Ang (1985) where they evaluated structural damage in reinforced concrete structures under earthquake ground motions. Veletsos and Newmark (1960) applied these concepts to formulate d amage index in terms of ductility ratio defined as the ratio of the maximum deformation to the yield deformation. Lybas and Sozen (1977) has proposed a similar model to estimate the damage potential in structures using the ratio of the pre yield stiffness to the secant stiffness corresponding to the maximum deformation. Roufaiel and Meyer (1987) defined a damage index based on the flexural flexibility, which is the ratio of the rotation to moment before and after earthquake and the ultimate flexibility. Pow ell and Allahabadi (1988) presented two concepts for damage assessment: one was based on demand versus capacity, and the other was on the degradation of structural properties. The demand vs. capacity assessment includes strength, displacement, deformation, and energy dissipation whereas a degradation concept uses degradation in stiffness, strength, energy dissipation capacity. The numerical techniques for the optimization were developed and successfully employed in the structural design (Arora 1995 ; Baier 1977; Leitmann 1977; Stadler 1988 and 1992; Koski, 1979 and1980; Carmichael, 1980; Choi and Kim, 2005). The application of numerical techniques for the optimization of a structure has increased and gained popularity with the development of the finite el ement method (Kristensen and PAGE 33 33 Madsen 1976; Pedersen and Laursen 1983; Santos & Choi 1989; Bathe 1996; Kim 2009). M any engineering problems in the structural design consist of more than one objective, thus require a special techniques Marler and Arora (2004) described the main characteristics, advantages, and drawbacks of various numerical and random methods used to solve the multi objective (MO) problems. Messac et al. (2003) provided a review and comparison of several MO algorithm based on numerical optimization. Das and Dennis (1997), Cheng and Li (1999), Das and Dennis (1998), and Messac and Ismail Yahaya (2003) have respectively developed weighted sum algorithm, compromise programing, normal boundary intersection method and normalized normal const raint method to find optimum designs in multi objective optimization. PAGE 34 34 CHAPTER 3 DETERMINATION OF PROPERTIES OF CITRUS WOOD The material properties of the wood were determined from the fresh samples cut randomly from the primary limbs of Valencia or ange trees ( Botanical name: Citrus Sinensis ). The samples were machined to the sizes recommended by ASTM D143 09 The dimensions were measured at the two edges and at the center of specimen using electronic digital calipers of 0.000 5 inch resolution. The m ean values were used in the calculations of the wood properties An electronic weighing instrument was employed for accurately weighing the samples. The average values of mechanical and physical properties of citrus wood were used in the numerical model. M aterials and Methods Mechanical P roperties Six samples of wood were cut from the citrus tree s growing at the University of Florida CREC research orchard. The samp les were machined to a size of 1.5 in. x 1 in. x 18 in. A 3 p oint bending testing of the wood samples were performed using a Universal testing machine (UTM) (S eries 313, Tabletop 11,250 lb. Test Resources Inc, Shakopee, MN, USA). The samples were simply supported between the knife edges of the 3 poi nt bending support as shown in Figure 3 1 The crosshead of the instrument was adjusted to come in proper contact to the top surface at the mid plane of samples. The load was applied continuously throughout the test at a rate of motion of movable crosshead of 0.1 in ches per min ute The load cell of the instrument was set to continuously record the reaction force until the specimen fracture d or fail ed to support a load of 50 lbf. The fracture due to static bending failure was identified as the appearance of brash or f iber delamina tion of the wood spe cimens. The load deflection PAGE 35 35 data was acquired from each specimen using Test Resource R series software ( Test Resources Inc, Shakopee, MN, USA) The wood was assumed to be isotropic and homogenous and the mechanical properties were calculated from the load deflection curve as discussed below. Figure 3 1. A t hree p oint bending test of a citrus wood specimen (Photo courtesy of S K. Gupta.) Modulus of e lasticity Elas ticity implies that deformation produced by low stress is completely recoverable after the load is removed. The modulus of elasticity ( E ) was calculated from load deflection curv e of the specimens expressed as: (3 1) where M is the bending moment, I is the moment of inertia, L is the length of the specimen, and is the deflection at the mid span of the specimen. For the 3 point bending test, the bending moment ( M ) is calculated as given in Equation 3 2. ( 3 2) PAGE 36 36 Modulus of rupture Modulus of rupture ( R ) is equivalent to the fracture stresses and reflects the maximum load bearing capacity of a member in bending. It is an accepted criterion of strength for wood and is computed as: ( 3 3) wh ere b and h are the width and the height of wood sample respectively. Stress at proportional limit The stress at the proportional limit ( ) is a stress proportional to the load at which the load deflection curve is a straight line. The point at which curve is no longer linear is called the elastic limit and is expressed as: ( 3 4) where is the load at the proportionality limit. Work to m aximum load in bending Work to max imum l oad in bending ( ) is the ability of wood to absorb shock with some permanent deformation. Work to maximum load is a measure of the combined strength and toughness of wood under bending stress. It is calculated as the total area under the load d eflection curve to the maximum bending load per unit volume o f specimen as given in Equation 3 5 and is expressed in units of kJ per cubic meter ( 3 5) w here A is the area under curve to maximum load. PAGE 37 37 Physical Properties Volume and d ensity Three samples of citrus wood were cut from the bending specimen after the 3 point bending test The weight of the samples was measured using an electronic weighing machine as shown in Figure 3 2 with an accuracy of 0.2%. The change in the pressure values obtained using a n Ultra Pycnometer 1000 (Quantachrome Corporation Boynton Beach, FL, USA) as shown in Figure 3 3 were used to calculate the volume of the specimens and subsequently the density of the citrus wood. The instrument was calibrated before the experiment s with a sphere of a known volume. The sample was placed in the cell and stream of helium is passed through the cell, sample and an additional chamber named of the instrument to purge air from the system. All the valves of the instrument then closed and the system was allow ed to reach equilibrium. After reaching equilibrium the valve allowing helium gas to enter the sample cell was opened. The sample cell was pressurized to a pressure approximatel y equal to that in the regulator of the UHP hel ium tank. Then the valve to the chamber was opened resulting in the increase of the total volume of system and thus, decrease in pressure since no further gas was admitted to the system and the te mperature was maintained constant. Eliminating the term from the ideal gas equation s corresp onding to two states results in an equation for which volume displaced by the sample can be calculated : ( 3 6) ( 3 7) is obtained by solving E quation s 3 6 and 3 7. PAGE 38 38 ( 3 8) Each sample was tested five times a nd the average value of density calculated using E quation 3 9 was used in the numerical model. ( 3 9) Figure 3 2 High precision electronic weighing scale (Photo courtesy of S.K. Gupta.) Figure 3 3 Ultra Pycnometer to measure the actual volume of the wood samples (Photo courtesy of S.K. Gupta .) PAGE 39 39 Moisture c ontent The moisture content of the samples was not required in the formulat ion of a numerical model but essential in validating the properties of the green citrus wood measured in laboratory experiments As per Wilson ( 1932 ), most of the mechanical properties of wood such as modulus of rupture, the fiber stress at the elastic limit in bending, the flexural modu lus, and the fiber stress at 3 % deformation; all vary with moisture content below the fiber saturation point and become invariant to the moisture above this point where all the free water in the cell cavity has been removed, but cells are still fully saturated. For most tree species, the fiber saturat ion point lies between 25 30%. Thus, it is essential that the moisture content of samples used in the bending test should be more than the fiber saturation point in order to accurately simulate the physical phenomena using numerical model Nine samples were cut from the bending test specimen as shown in Figure 3 4 The weights of the samples were measured with an electronic weighing machine having an accuracy of 0.01 grams. The samples were dried to a temperature of 105 C until any significant de crease in the mass was noticed : the mass of wood corresponding to th is point is called oven dry mass of wood The moisture content of the specimen was determined from the loss in mass, expressed in percent of the oven dry mass: ( 3 10) where and is mass of a green sample and oven dry sample of wood respectively. PAGE 40 40 Figure 3 4. The w ood samples of the Valencia orange used in the determination of moisture content. (Photo courtesy of S.K. Gupta .) Damping c oefficient The dynamic responses of tree limbs unlike other mechanical structures are greatly affected by the amount and type of damping present in a system (Hoag and Fridley 1971 ; Hoag, 1970). The amount of damping determines the amount of vibrational energy dissipated and the amount of energ y transferred from the harvesting machine to the fruit bearing branches of a tree Physically, the damping can be internal, i.e. inhe rent in the wood and bark, and/ or external due to air resis tance. The viscous damping due to air drag on the secondary branches, twigs, leaves and fruits contribute significantly to the damping of the tree crops. The viscous damping is also mathematically convenient to model because it requires the formulation of linear secon d order differential equation only Thus, equivalent viscous damping, which models the overall damped behavior of structural system as being viscous, was adopted ( Tompson, 1993). The damping is expressed in terms of damping ratio ( which is a percentage of the fraction of critical damping ( as given in Equation 3 11. ( 3 11) PAGE 41 41 A system is said be under damped if < 1 critically damped if =1 and overdamped if > 1 In all the above cases the response of a system set into motion will eventually decay to zero with time except when = 1. For a single degree of freedom (SDOF) under damped system subjected to an impulse of t=0 exhibit s a response x(t) as shown in Figure 3 5 and decay s expone ntially as given by Equation 3 12. ( 3 12) Figure 3 5 Under damped response of a system having singe degree of freedom The l ogarithmic decrement method measures the damping of a system as the rate of decay of response for consecutive cycles of the vibration referred as the log decrement ( ): ( 3 13) The damping ratio for th e system is computed from the log decrement using Equation 3 14. ( 3 14) An experiment was setup in CREC research orchard to measure the damping of the tree limbs of citrus crop The response of the tree limbs under dynamic test ing was measured using an accelerometer (Horvarth & Sitkei, 200 5 ; Chopra, 1995). Two limbs PAGE 42 42 each from the top, middle and bottom section of the citrus trees were selected and accelerometers (X250 2 Data logger, Gulf Coast Data Concepts LLC, Waveland, MS, USA ) were installed on the tree limbs as shown in Figure 3 6 The tip o f the limbs were displaced laterall y to approximately 30 in. using an in house mechanism and then released quickly to allow the limbs to vibrate naturally. The responses of the limbs were acquired using a accelerometer data logger and the experiments were repeated thrice for each limb. The damping ratio was estimated using logarithmic decrement method. Figure 3 6 An a ccelerometer mounted on a primary limb of a citrus tree. (Photo courtesy of S.K. Gupta.) Results and Discussion Mechanical Properties A summary of the mechanical properties calculated from the 3 point bending test as shown in Figure 3 7 is listed in Table 3 1. The mean ( ) and coefficient of variation ( C V ) of mechanical properties were calculated from the load deflection curves using Equ ations 3 1 to 3 4 The e lastic modulus and modulus of rupture was only used in the formulation of a numerical model The other two properties provide the knowledge of elastic behavior and shock absorbing capacity of the citrus wood. Figure 3 8 shows the best fit curve and the load deflection curve of 3 point bending test of the citrus wood PAGE 43 43 samples The values of the mechanical and physical properties of the citrus wood computed in this research is significant and can also be used in other or fu ture research i nvolving citrus wood. Figure 3 7. A c itrus wood specimen under 3 point flexural bending. (Photo courtesy of S.K. Gupta.) Table 3 1. Summary of the mechanical p roperties of the green citrus wood Mechanical Property Mean () Coefficient of variation (CV) Flexural m odulus (GPa) 8.5 11% Modulus of r upture (MPa) 67.3 4% Stress at p roportional limit (MPa) 45.3 13% Work to maximum load in b ending (kJm 3 ) 193.2 28% Figure 3 8 The best fit and l oad deflection curve s of green citrus wood specimens. PAGE 44 44 Physical Properties A summary of the physical properti es measured are reported in Table 3 2. The mean and coefficient of variance are reported for each physical quantity. It was found that the moisture content of the samples of the bending test was more than the fiber saturation point, thus validating the measured mechanical properties of the citrus wood are invariant to moisture content and thus, can be used in numerical formulation Figure 3 9 shows the under damped response of a c itrus limb when displaced and allowed to vibrate naturally. The average value of damping ratio o f citr us limbs was found to be approximately 11. The magnitude of the dampin g ratio measured by Hoag et al ( 1970 ), Hoag et al ( 1969 ), and Horvath and Sitkei ( 2 004 ) for limbs of other species was found to be close to that of citrus limb Thus, this suggests that the damping in tree limbs is highly dependent on the air drag on fr uits, leaves and twigs (viscous damping) rather than the material property of the wood (structural damping). The friction coefficient between wood and steel was not measured but taken from the research work of McKenzie (1968). Table 3 2. Summary of the p hysical properties of the green citrus wood Physical Property Mean () C oefficient of v ariation (C V) Density (g/cc) 1.451 2.8% Moisture c ontent (%) 42.4 14.15% Damping ratio (%) 10.78 24.70% Friction 0.36 20.97% PAGE 45 45 Figure 3 9 Response of a citrus limb measured using an accelerometer. PAGE 46 46 CHAPTER 4 MODELING OF TREE LIMBS The statistical information based on the data collected from the citrus trees was used to predict the configu ration of a citrus limb prototype A meta modeling technique (or response surface methodology) was adopted to predict the sectio nal properties of a tree limb prototype The tree limbs used to design the limb prototypes were first classified in the multiple sets or zones based on the vibration transmissibilit y. The vibration transmissibility of a tree limb is defined as the ability of the limb to tran smit energy from the input point of excitation to the fruit bearing regions and is a function of mass, stiffness and damping which in turn depends on the configuration of the tree limb ( spatial coordinates and location of fruit bearing region) and material properties of wood ( ). The tree limbs having different vibration transmissibility requires diff erent shaking inputs to effectively detach fruits from them Therefore, a vibratory harvester equipped to provide multiple values of s haking force effectively minimize the tree damage and maximiz e the fruit removal. However only three distinctive sets of machine configuration providing three different shaking forces were proposed to minimize the computational and manufacturing cost The tree limbs c lassified in the three zones are analyzed separately to obtain the three set of optimum machine configurations. Material and Methods Interpolating Technique : Continuous Piecew ise Cubic Hermite Interpolation P iecewise cubic Hermite interpolat ing polynomial ( PCHIP ) is an interpolation technique which is based on piecewise polynomials. These are shape preserving piecewise cubic polynomials whose first order derivatives are continuous and are used PAGE 47 47 to generate smooth curves passing through a series of data point s. The polynomial function P(x) on the interval expressed in terms of local variables and is: ( 4 1 ) ( 4 2 ) The inbuilt MATLAB pchip was used to construct hermit interpolation passing through data points using the polynomial given in E quation 4 1 and satisfying the conditions given by E quation 4 2. Surrogate T echnique : Polynomial Res ponse Surface (PRS) Th e PRS approximation is a meta modeling technique which uses polynomial functions to predict the best fit for the data when the true response is unknown (Viana 2009). The parameters of an approximate polynomial function are obtained by the least squares me thod A second degree polynomial model is expressed as given below : (4 3) ( 4 4 ) w here is a vector of consta nt coefficients obtained by minimizing the residual error ( ) between the prediction ( ) and true response ( ) Residual Analysis : Cross Validation Error C ross validation error is the error at a data point when the surrogate is fitted to a subset of the data points not includi ng that data point (Allen, 1971; Khuri, 1996). A vector of error ( ) is obtained by fitting the surrogate to all the n 1 points and evaluating PAGE 48 48 the error at that point. This vector of error is called the cross validation error or PRESS Residual The measure used to calculate the predictive capability of a response surface is calle d PRESS RMS which is r oot mean square of PRESS error and expressed as: ( 4 5 ) ( 4 6 ) ( 4 7 ) W here Eii is the diagonal matrix of the idempotent matrix E and X is vector of data points. Experimental Setup: Acquiring Data for the Statistical Model Numerical technique to find the dynamics response of a tree limb requires the p rediction of the limb configuration The limb configuration would then be used to model a tree limb as a truncated conical elastic beam. The true response of a limb having infinite degree of freedom is approximated by modeling the limb as a finite set of elements using numerical methods. The information about the distribution of secondary branches and fruits must be included in the numerical model as they considerably modify the response of the tree limbs by changing the overall mass and dissipative properties of the primary limbs. The model and setup presented here are designed in view of its use with FEM; however, it can be extended to any other numerical technique. A field experiment was set up to find the following informatio n for modeling and analyzing a tree limb using numerical method. PAGE 49 49 Spatial co ordinates of the primary tree limbs Sectional property distribution along the length of the tree limbs Distribution of secondary branches on the primary tree limbs Distribution of the fruits on the primary tree limbs The tree information were collected from 54 tree limbs of 10 rand omly chosen medium sized tre es of the Valencia orange variety ( C itrus sinensis ) from the research orchard located at University of Florida Citrus Researc h and Education Center from 28 Feb ruary to 08 Mar 2013. A referenced system bas ed on the C artesian coordinates with the origin at the trunk base was employed to measure the limb data as shown in Figure 4 1 It was observed that m edium size citrus trees generally have 6 to 8 scaffold branches or primary limbs and measuring each and every branching node in detail would be tedious and time consuming. Therefore, a procedure was laid out which uses the following information to construct the tree limbs models: the l ength of the limb segments between two branching nodes, the angle from the Z axis (called vertical angle), and the angle from the Y Z plane (called the horizontal angle) Figure 4 1 I nteraction of a canopy harvester with a row of citrus trees PAGE 50 50 Pro cedure The following procedure was adopted during the measurement of the tree limb properties : The primary tree limbs were discretized into the node s (junction where a branch split into two or more branches) which are either the poin t of origin of secondary branch or other primary limb. The length, vertical angle and horizontal angle of each segment of a primary limb were measured. The sectional perimeter of the limb segment was measured near the branching nodes a nd at the center of the segment. Th e overall length, vertical angle, and horizontal ang le of all the secondary branches were measured. The sectional perimeter at the origin center point, and at the tip of the secondary branches was measured. The fruit s on the secondary branches and on the last segment of the primary limb s w ere counted. Assumption s The following assumptions have been made to predict the configuration of the statistical limb prototypes. The section of the branch was assumed to be circular. A branch originating from a primary limb having base diameter more than 1 in. and less than 2.5 in. w as considered as a secondary branch The twigs, stems, and leaves were not considered explicitly in the numerical model but their effect on the response of the limbs was modeled using viscous damping T he tines of a canopy shaker interact with t he tree limbs in the plane Y Z as shown in Figure 4 2 Therefore, to simplify the limb model and economically optimize the machine paramete rs using finite element methods the measured three dimensional data of tree limbs were project ed on the Y Z plane. PAGE 51 51 Figure 4 2 Interaction of a canopy shaker with a citrus canopy on YZ plane. Classification of Primary Tree Li mbs The coordinates of the branching nodes o f the tree limb were generated from the measured data using a program designed in MATLAB 2011 (Math Works, Natick, MA, USA). The tree limbs were constructed and classified into three distinctive sets based on the spatial configuration of their fruit bearing region ( which is generally the last segment of a primary limb having maximum number of the fruits ). Mathematically, E quation 4 8 was used in the classification of the limbs in the three zones, referred to as top, middle and bottom zone : ( 4 8 ) where is the height of the fruit bearing region of a tree limb measured from the ground level. PAGE 52 52 Prediction of Spatial Coordinates of Limb Prototypes Since a canopy shaker interacts more or less with the tree limb s in the plane of the tine s (Y Z) as shown in Figure 4 2 it is therefore numerically economical to model a tree limb as a 2 dimensional finite element T h us, the three dimensional data were projected on the Y Z plane and projected data were used to co nstruct the individual limbs. H owever, it is also feasible to model the tree limb s as three dimensional finite element by adding few more parameters but improvement in results will barely able to justify the computational expenses in modeling and analyzing the limbs The interpolating polynomial, PCHIP was used to create the smooth and continuously differential curve passing through measured data points to model the tree limbs The interpolated limbs were then used to predict the spatial co ordinates of th e tree limb prototypes or representatives The foll owing procedure was adopted in the modeling of a limb prototype which is a non physical model of the tree limb An axis ( Y or Z ) corresponding to the maximum value of coordinate of any tree limb s in a zone was chosen as the primary axis for that zone a nd the other axis is referred as the secondary axis. Equally spaced points were generated between minimum and maxi mum value of primary axis coordinate for each zone In this research, 30 predefined values of coordinate were chosen along the Y axis for the top zone and along the Z axis for the middle and bottom zone. The parameters of PCHIP inter polating function of a tree limb were determined from the coordinates of th e branching nodes of that tree limb. The value of secondary axis coordinate of a limb corresponding to a predefined value of primary axis coordinate was predicted using the PCHIP parameters calculated for that limb. Assuming normal distribution of the seco ndary axis coordinates of all limbs in a zone the 5 th 25 th 50 th 75 th and 95 th percentile s values were predicted for that zone PAGE 53 53 The five limb prototypes were constructed for each zone by plotting the values of predefined primary axis coordinate and the secondary axis coordinate values taken each from the 5 th 25 th 50 th 75 th and 95 th percentile s of the limb data. Sectional Properties of Limb Prototypes Perimeter values measured along the tree limbs were used to calculat e the diameter of the limb s assuming that the limb s to have a circular cross section. It was observed that the diameter of a limb at any section decreases as the distance of that section from the limb origin increases Also, the sectional diameter along the length of a tree limb varies depending on the spatial orientation in a tree structure Thus, it is concl uded that diameter of a limb at any cross section d epends on the actual distance and vertical angle of that cross section from the limb origin as labeled in Figure 4 3 The actual distance of a branching node and vertical angle of the position vector joining that branching node from the limb origin were measured for all limbs and used to predict the sectional diameter of limbs expressed mathemat ically as: ( 4 9 ) w here is the actual length of nth branch ing node from the limb origin and is the vertical angle of the position vector joining the node with the limb origin. Figure 4 3 Showing a tree limb having diameter at a ny branching node as a function of the length and the angle of that node from the l imb origin PAGE 54 54 A m eta m odel technique ( or response surface method ) w as used to construct regression models to predict the diameter at any section of a tree limb A p olynomial response surface was employed to fit the measured data and predict the diameter of a limb prototype. The s urrogat e t oolbox ( Viana, 2010 ) was used to create a response surfaces with full and s tepwise regression. The response surfaces were selected based on the i ndicator of predictive performance and indicator of quality of fit : cross validation error and adjusted correlation coefficient ( ) respectively The response surface wit h full regressio n involves all coefficient terms of polynomial response surface whereas stepwise only involves coefficient terms having Distribution of Secondary Branches The s econdary branc hes modify the dynamic response by changing the overall mass and dissipative properties of a primary tree limb. The overall length, c ircumference and coordinates of origin of secondary branch were measured. Based on field observation and data analysis it was found that around 4 5 second ary branches radiate from the primary limb at the intervals of approximately 20 25 i nch This information was used to add secondary branch effect in the n umerical analysis based on some assumptions and idealizations Distribution of Fruits on the Primary L imbs Including the effect of fruits is pivotal in determining the dynamics response of a fruit bearing limb under v ibratory excitations. The fruits ac t as an inertia damper and attenuate the dynamic responses of the limbs. The distribution of fruits on a p rimary limb can be cl assified in two distinctive group s : Fruits attached t o the secondary branches of a primary limb Fruits attached to the last segment called fr uit bearing region of a primary limb PAGE 55 55 Fruits on secondary branches The number of fruits on the secondary branches was counted. T he average mass of a citrus fru it 0.186 kg was used to compute the total mass of fruits on the secondary branches of the primary limb. Fruits in fruit bearing region A fruit bearing region of a primary limb is the region ne ar the tip of the limb where the limb branch e s into large nu mber of small twigs and bear s large number of fruits. The information regarding the configuration and the number of fruits in the f ruit bearing region is significant as numerical models are evaluated in these region s of the tree limbs to optimize the parameters of a canopy shaker Results and Discussions The tree limbs were classified into three zones based on their spatial distribution The models were formulated to predict the spatial confi guration and properties of the limb prototypes. The statistical information from the tree limbs data were analyzed and used to predict the distribution of secondary branches and fruits along the primary limbs. Primary Tree L imbs Classified in Three Zones T he tree limbs classified in th ree distinctive sets have similar dynamic responses owing to its property, distribution of s econdary branches, and configuration of fruit bearing region Figure 4 4 shows a three dimensional representation of the citrus l imbs measured to predict the limb prototypes. PAGE 56 56 Figure 4 4 Three dimensional view of the tree limbs of citrus and their classification in three distinctive zones. Spatial Coordinates of Limb P rototypes Figure 4 5 shows the tree limbs s egregated into the three disti nctive sets corresponding to the three zones in a citrus tree canopy Modeling of the tree limbs in the numerical analysis is time consuming an d expensive: therefore sets of representative or prototypes of tree limbs were defined for each zone. These representatives of limbs were modeled and analyzed in finite element analysis to optimize a canopy shaker Figure 4 5 Tree limbs classified in three sets and plotted on the plane Y Z. PAGE 57 57 Figure s 4 6 and 4 7 show the spatial distribution of the limb prototypes capturing the 5 th 25 th 50 th 75 th and 95 th percentile s of the tree limb s in the top, middle, bottom zone. T he dynamic response of f ive limb prototypes in each zone was used to appropriately balance the co st of optimization and variability in the tree limbs. The dashed lines show PHCIP interpolated limbs passing through measured data points represented by circular markers. It is noticed that the limbs in the top zone radiate at an angle of 0 20 and grow straight up to a height of 120 130 inches and then curve down slightly becau se of the weight of the fruits. However, for the middle zone, the limbs are thick, long and hang down gradually after growing approximately 4 5 feet. The limbs in the bot tom zone originate and grow near the ground and are comparatively thin ner, shorter and hang down steeply at the tip Figure 4 6 Spatial modeling of the limb prototypes for the top zone PAGE 58 58 Figure 4 7. Spatial modeling of the l imb prototypes for the middle and bottom zone Figure 4 8 compares the spatial distribution of the m ean and 90% distribution of the tree limbs for the three zones. The tree limbs from number of trees can be statistically interpreted to form a h ypothetical tree structure consisti ng of the tree limb prototypes. The shakers parameters will be optimized based on th e interaction of the machine with the limbs of this hypothetical tree rathe r than any actual tree. Figure 4 8 Spatial distribution of the limb prototypes of a hypothetical tree Bottom Zone 90% Bounds Middle Zone 90% Bounds Top Zone 90% Bounds PAGE 59 59 Figure 4 9 shows the interaction of the shaker and hypothetical tree limbs. Thus, the tines which interact with the top zone of limbs will have a different configuration based on the amount of shaking force required by those limbs to minimum damage and maximum fruit removal as compared to the tines in the middle and bottom zone s Figure 4 9 Interaction of a canopy shaker wit h the limb prototypes of a hypothetical tree. Error Analysis and PRS model for limbs in the t op z one Various metal models were formulated for predicting the sectional diameter of the limbs. Table 4 1 shows the error norms used in determining the best fit m eta model The third degree polynomial response surface with stepwise regression was chosen as the best fitting model for the limbs of top zone because it has the lowest value of cross validation error and highest value of adjusted R square. The parameters of the model are given in E quation 4 10 Figure 4 10 graphically compares the measured diameter for the limbs of top zone shown as marker with a prediction from the meta model shown as contour plot. Y Z PAGE 60 60 Table 4 1 Meta models to pred ict the diameter of limbs of the top zone 9.4 9.1 86.5 87.3 (4 10) where the units of the diameter (D), the length (L) and the degrees. Figure 4 10 Contour plot to predict the sectional diameter of a limb in the top zone. Error Analysis and PRS model for l imbs in the middle z one Wi th the lowest va lue of PRESS RMS and highest value of adjusted coefficient of determination, the three degree polynomial wit h a full regression was used to predict the diameter of the limbs of middle zon e. The error norms in the models for predicting the sectional diameter of the limbs of the m iddle zone are tabulated in Table 4 2 The parameters of this model are given in E quation 4 11 PAGE 61 61 Table 4 2 Meta models to predict the diameter of limbs of the middle z one 10.6 10.0 ( 4 11 ) Error Analysis and PRS model for l imbs in the b ottom Zone Similarly, for the limbs of the bottom zone, the three degree polynomial response surface with a full regression is the best predictive model among all models considered The error norms are compared in Table 4 3 The paramet ers of best model are given in E quation 4 12 Table 4 3 Meta models to pre dict the diameter of limbs of the b ottom z one 9.6 8.9 ( 4 12 ) Figure 4 1 1 graphically compares the measured data shown as the maker with predicted values from the best fitting model as a contour plot for middle and bottom zone s The spatial distribution of the 25 th 50 th and 75 th limb prototypes are shown in L Th e information about the spatial co ordinates and the sectional PAGE 62 62 diameters of the limb prototypes obtained from these models was used to model the limbs in finite element analysis. Figure 4 1 1 Contour plot to predict the s ectional diameter of a limb in the middle and bottom zone Distribution of Secondary Branches and Fruits on the Primary Tree Limbs Modeling secondary branches and performing dynamic simulation is expensive, thus numerical model was simplified by aggregating mass of the secondary branches and fruit s on the primary limbs. Distribution of secondary branches Figure 4 11 shows the distribution of the mass of secondary branches on the primary limbs of the top middle, and bottom zone s It is concluded from Figure 4 1 2 and f rom the field observations that the secondary branches are thick and long near the limb origin and become thinner near the tip of the limb. It was observed that secondary branching generally star ts at a distance of 20 40 in. and cease s at a distance of 100 120 in. from the limb origin for the top and middle zone s and 80 100 in for limbs in the bottom zone. On an average, the limbs of the top and the middle zone radiate five PAGE 63 63 secondary branches whereas limbs in the bottom zone owing to shorter length bifurcate to only fo ur secondary branches. The average mass of secondary branches was taken over every 20 in. segment of a primary limb and model as a lumped mass on the primary limb in FEM analysis Table 4 4 provides the informa tion about the average mass of secondary branches in each zone. This indicates that the primary limbs of the top and the middle zone not only have more sec ondary branching but also weigh 12% and 18 % respectively, more than the mass of the secondary branches in the bottom zone. Figure 4 1 2 Distribution of mass of a secondary branch for all three zones. Table 4 4. A verage mass of the secondary branch es in three zones Zone Mean (k g) Top 0.4036 Middle 0.4236 Bottom 0.3602 PAGE 64 64 Distribution of f ruits on primary limbs The mass of fruits attached to the secondary branches was modeled by aggregating them at the location of secondary branches on the primary limbs. Figure 4 1 3 shows the distribution of fruits attached to the secondary branches of the primary limbs for three zones. Table 4 5 provides the average number of fruits attached to the secondary branches. There is no subtle distinction in the distribution of fruits on the secondary branches radiating from the primary limbs for the three zones. However, the average nu mber of fruit s attached to the secondary branches in the top and the middle zone s are respectively 30% and 21 % more than that of bottom zone. Table 4 5. Average number of fruits on the secondary branches. Zone Mean Top 12.6 Middle 11.7 Bottom 9.7 Table 4 6 provides the information to con figure the fruit bearing region of a primary limbs for each zone The information regarding this region is significant because the numerical model would be evaluated at these regions to optimize the shaker. It was n oticed that the length of the fruit bearing region is approximately the same for almost all limbs irrespective of zone ; however, the distance of fruiting zone from the limb origin is maximum for top an d minimum for the bottom zone. The maximum fruiting occ urs in the middle zone with an average fruit count of 41% and 13% more than that of top and bottom zone respectively. PAGE 65 65 Figure 4 1 3 Distribution of the fruit s attached to the secondary branches. Table 4 6. Configuration of fruit bearing region of a c itrus tree canopy in three zones Zone Distance from limb origin Length of fruit bearing region Fruit c ount s Top 132 30.6 14.4 Middle 124 32.8 20.3 Bottom 112 32.6 17.9 The information regarding the modeling of l imb prototype describes in Chapter 4 is vital in the formulation of numerical model. Various models and idealization formulated here will be used for the optimization of the mechanical harvester in Chapter 7 PAGE 66 66 CHAPTER 5 FORMULATION OF FINITE ELEMENT ANALYSIS The non linear dynamic beha vior of the tree limbs was analyzed based on the continuum model. The finite element methods were employed to find the response of limbs under the impact load of the tine s of a shaker. The responses extracted from numerical analysis of the tree limb protot ypes were used in the formulation of mechanistic models for the optimization of a harvester. Figure 6 1 shows the schematic of the process involved in the optimization The finite element modeling of the intera ction between the tree limb and the tine of a shaker will be discussed in Chapter 5. Figure 5 1. Flow chart of process involved in finite element analysis based o ptimization of canopy shaker Limb prototypes Contact Formulation Dynamic analysis Modeled as Beam Element Shaking member Idealizations Mechanistic Models Statistical Model Optimization PAGE 67 67 Finite Element Model Product The non linear dynamic response of a tree limb under the excitation force was solved using ABAQUS/Standard. Abaqus input code files were written and integrated with a program in MATLAB to simulate limb prototypes. The Abaqus input code is composed of a set of commands to perform analysis using finite element methods and output analysis result s The software Abaqus was chosen because of the convenience rather than limitation. Other numerical method or analysis software can be used. Geometric Modeling The li mb prototypes were modeled as truncated conical three dimensional cantilever beam s The configuration s of the limb prototypes were defined for FE Modeling as discussed in Chapter 4. The limb f inite element model was meshed with 160 two node linear beam elements (B31). The sectional pro perties of the limb finite element models are defined from the meta models develo ped in Chapter 4. The t ine s of a canopy shaker were modeled as three dimensional beam s composed of 100 two node linear beam elements (B31). The sectional properties of ti ne s were defined as discussed in Chapter 7. Figure 5 2 shows the typical configuration of a limb and a tine in Abaqus. Figure 5 2. Finite element model of a tree limb and a tine modeled as beam element ( shown with section and without section ) PAGE 68 68 As discussed in Chapter 4, the secondary branches and fruits were idealized and model as lumped mass on the primary limb to minimize the cost of computation. The mass of secondary branches and fruits were computed and distributed on the limb prototypes as discussed in Chapter 4. Figure 5 3 shows 50th percentile limb prototype of the m iddle zone lumped with the mass of secondary branch es and fruits. Figure 5 3. Finite element model of a tree limb prototype with secondary branch es ( brown marker ) an d fruits ( yellow marker ) modeled as the lumped mass Material Model The isotropic linear elastic material model was considered for both the limb and the tine model s The branch properties wer e set based on the experiments performed on the green citrus wood as discussed in Chapter 2. The value of Poisson ratio was taken from the studies done by Savary and Ehsani (2010 ) on citrus wood. The m aterial property of the tine of a shaker was considered as a design variable and will be discussed in Chapter 7. Damping Model hesis was used to introduce damping in the model. Rayleigh damping is composed of the two parameters as given below: ( 5 1 ) PAGE 69 69 w here M, K, and C are the mass, stiffness and damping matrices of system respectively. The parameters and is mass and stiffness proportional damping of the system respectively Mass proportional damping defines the damping contribution proportional to the mass matrix of an element The damping forces are caused by the absolute velocities of the nodes of a system The resulting effect of this damping would be like a model moving through a viscous fluid and motion of any point in the model triggers damping forces S tiffness proportional damping defines damping proportional d ) p roportional to the total strain rate is introduc ed, using the following formula: ( 5 2 ) w here is strain rate and D is initial (Hyperealstic and Hyperfoam Materials) or current elastic stiffness (for all other materials). For linear analysis stiffness proportional damping is exactly the same as defining a damping matrix equal to times the stiffne ss matrix. In this research, mass proportional damping was used to account for the overall damping in the tree limbs. I t is appropriate to use this hypothesis because the branch damping ratio is largely correlated to the mass of the main branch es, seconda ry branches and fruits (Moore, 2002). The in the damping expression was neglected because the response s of the limbs are predominately in the first mode of vibration (Mayers, 1987). The stiffness dependent part of damping is effective only fo r high frequency modes. The parameters of the damping were calculated so that the PAGE 70 70 mean of the damping ratio measured in the experiments were equal to the damping ratios computed from the test simulations. A value of 87.25 was set for based o n the paramet ric studies of the 50 th percentile limb prototypes of each zone. However, was set to a low value of 0.0001 rather than null value to prevent the numerical noise. Contact Formulation The surface interaction between the limb and the tine which decides the dynamics response of a system when they are in contact was defined using Abaqus tube to tube element (ITT3). Tube to Tube Elements The tube to tube are slide line contact elements used to model the finite sliding interaction between two tube s or rod s like structures which contact each other along their either inner or outer surfaces. Master Slave Assignment Master surface was defined by the slide line constructed using the nodes of a tine model, whereas the slave surface was defined by the co ntact elements attached to the nodes of a branch model at the impact region. Figure 5 4 shows FE model of a branch limb system with the master and slave surfaces. In this setup the nodes of the branch are constrained not to penetrate the slide line defined on the tine. Figure 5 4 Modeling i nteraction between a tine and a tree limb in Abaqus PAGE 71 71 Contact Property Assignment Pressure over closure relationship Soft contact was defined which allows small amount of penetration at constraint location Soften ed contact was used to resolve the numerical difficulties in applying contact conditions by creating a soft thin layer on one or both surfaces. The p ressure over closure relationship was prescribed by using a tabular piecewise linear law. The softened cont act was specified in terms of over closure (or clearance) versus contact force as shown in Figure 5 5 The values given in Table 5 1 were chosen based on the observation during the bending test of the citrus wood samples and numerical testing so as to not significantly compromi se the accuracy of analysis. Table 5 1 Pressure Over closure relationship kP a M m 0 0E+00 3 .0E 11 1E 08 0.02902 1E 02 0.1275 5E 02 0.22 1E 01 1.5 5E 01 13 1E+00 Figure 5 5 P ressure overclossure relationship used in the contact formulation 0 2 4 6 8 10 12 14 0E+00 2E01 4E01 6E01 8E01 1E+00 1E+00 Contact Pressure ( kN m 2) Overclosure ( mm) Pressure Overclosure PressureOverclosure PAGE 72 72 Friction m odel Classical isotropic coulomb friction model with a coefficient of friction of 0.36 was used to model the frictional behavior of the limb tine system Contact Constraint Enforcement Method Direct methods based on L agrange multiplier was used to enforce pressure oveclosure constraint. However the frictional constraint was enforced using Abaqus penalty method. The choice of method was made to achieve maximum accuracy and minimum convergence time. Contact Interface The radi al clearance between the tine and the limb prototype at point of impact was defined using Equation 5 3 ( 5 3 ) Dynamic Analysis Analysis Method Non linear dynamic response of a branch was analyzed using an implicit direct integration. The implicit direct integration uses implicit operators to integrate the equations of motion. The integration operato r matrix is inverted at each increment to solve the second order non linear differential equation s of dynamic motion. The dynamic motion of a system is expressed as: ( 5 4 ) w here M, C, K and F is mass matrix, damping matrix, stiffness matrix, and vector of nodal forces of system respectively. is a set of generalized coordinate s used to represent the configuration of the system. PAGE 73 73 Hilber Hughes Taylor operator is widely used implicit operator defined based on the parameters and solved using following equations: ( 5 5 ) ( 5 6 ) w here h is an integration step size. The equations 5 5 and 5 6 are used to describe the time t n+1 using E quation 5 4 The equation of motion at time t n+1 is given as: ( 5 7 ) Equation s 5 5 5 6 , and are function of the acceleration which is solved implicitly to find the solution of Equation 5 7 However to counter the high frequency oscillation and to achieve A stability, Hilber et al (1977) induces the numerical damping into the system of equations by defining one more parameter ( ) defined as: ( 5 8 ) As studied by the H ughes (1987) HHT is stable and achieve second order accuracy when ( 5 9 ) 0.05, 0.275625, and 0.55 respectively in dynamic simulation using Abaqus PAGE 74 74 Loading and Boundary Condition s Th e branch was fixed at its base whereas the tine was subjected to sinusoidal vibration defined by frequency ( v ) and amplitude ( a ) as shown in Figure 5 6 The operating parameters such a frequency and amplitude were chosen as the design variable s for the optimization of a shaker. The values of these parameters are discussed in Chapter 7. The forward motion of the canopy shaker and rotation of h ub were also considered to accurately model the physical phenomena. Ba sed on the information from citrus growers and studies done by Roka et al. ( 2008) it was found that the canopy shaker travels at the speed of approximately 0. 5 miles per hour (or ~ 200 trees per hour) down a row of medium size trees and harvest upto 95% of mature fruits Field observation and analyzing the video of the canopy shakers it was found that the tine were allowed to interact with a citrus canopy for approximately one second and during that process, hub rotates at a speed of approximately one cycle per minute. Thus, the tine was subjected to a forward speed of 0.223 m/s and an angular speed of 0.105 rad/sec in the numerical model The limb and tine were allowed to interact in numerical model for one second with a maximum step time increments set to 0.0005 sec ond The results were extracted and us ed to compute me chanistic index required for the optimization of a shaker. Figure 5 6. Prescribed displacement to the tine s of a canopy shaker PAGE 75 75 Finite Element Model Verification and Validation The tree limb prototypes used to optimize the shaker are non physical tree limb s which are derived from the statistical data. Therefore, it is impossible to verify the FE model experimentally using limb prototypes because of the uncertainties associated wi th a structure of bio logical origin. An alternative must be devised to verify the parameter s of finite element model so as to accurately simulate the physical phenomena. Thus, a small scale setup was developed which employed the same vibratory mechanism as used in a canopy shaker to provide the excitation force to the tree limbs. The dynamic response of the tree limbs was measured in terms of longitudinal normal strain and acceleration and compared with the FE simulation s Material and M ethods The labora tory test equipment utilizing a slider crank mechanism was built in house as shown in figure 5 7. An electric motor powers the piston of the test equipment to oscillate the tine up and down with a stroke of 1 in The supply voltage of the test equipment wa s adjusted using a voltage regulator to provide four different frequencies: 2.4cps, 3.8cps, 5.1cps, and 6.5 cps. The dynamic response of a branch was measured using strain gauges and accelerometers mounted at the selected test locations on the branch speci men. The configuration of the tine and branch used in the experiments are as follows : Tine Galvanized coated elec tric metallic tube (EMT), 11/16 in. OD, and wall thickness 0.0625 in., 28 in. length and weighing ~0.7 lbs. PAGE 76 7 6 Branch A clear branch specimen was cut from the tree of Valencia variety of orange growing at University of Florida CREC research orchard The total mass of the laboratory test specimen was ~ 0.85 lbs. The branch specimen was 70 in. long and ha s maximum diameter of 21/32 in. which reduced to 17/16 inches at the tip of branch. During the experiment the branch specimen was fixed horizontally with maximum diameter end clamped to solid frame using adjustable clamp as shown in Figure 5 8. Figure 5 7. Schematic of a laboratory test equipment used to validate the FE model parameters. (Photo courtesy of S. K. Gupta ) Figure 5 8. Branch specimen fixed to the solid frame with a bracket and a clamp (Photo courtesy of S. K. Gupta ) PAGE 77 77 Acceleration a cquisition A set of accelerome ters (Tri axis, Model MMA7260Q, Freescale Semiconductor Incorporated, Austin, TX, USA) shown in Figure 5 9 were fixed at test locations of 30 inches and 60 inch es from the fixed end of the branch specimen. The data were acquired using compact data acquisition system (Model NI 96211, National Instruments Inc., Texas, and USA ). A Lab V IEW program was designed as given in Figure A 1 (Appendix A) to acquire acceleration data from the accelerometers. The sampling frequency for acquiri ng the acceleration responses of the bra nch specimen was set to 100 samples per second. The acceleration was computed using formulas as given below : ( 5 10 ) The output voltage range of the accelerometer is 0 3.3 V. The 0 g value is reading of accelerometer at zero acceleration. Theoretically, the 0g value is 1.65V; however, this value might have change from sensor to senor, so the accelerometer reading at rest was used as 0 g values. The mean of the first 50 reading of the accelerometer was used as 0g value of that sensor. The sensitivity of 0.2V /g based on 6g setting of the accelerometer was t aken for computing acceleration of the branch. Figure 5 9 MMA7260Q Accelerometer for sensing acceleration. (Source: http://www.robotshop.ca/Images/big/en/sfe mma726 0q triple axis accelerometer.jpg. Last accessed July, 2013) PAGE 78 78 Strain a cquisition Measurements, Wendell, NC, USA ) as shown in Figure 5 10 were installed near the fixed end of branch specimen to record the strain developed due to dynamic bending of the branch specimen. The strain gauges were installed on both the top and bottom surface of the branch to continuously record tensile and compressiv e longitudinal strains. The LabVIEW Virtual Instrument (National Instruments Inc., Texas, USA) was designed to communicate between strain gauge and DAQ (Data Acquisition system) The LabVIEW program used to acquire strain data is provided in Figure A 2 and A 3 The data was acquired at the sampling rate of 1000 Hz using quarter bridge strain gage m odule (Model NI 9236, National Instruments Inc., Texas, USA). Figure 5 10 A strain gauge installed on the top and bottom surface of the branch specimen. (A photo courtesy of S.K. Gupta) The branch and tine were installed to form the cross shape a ssembly as shown in figure 5 7. The tine was allowed to impact the branch at the two sets of locations: 7 The e x periments were conducted at both the impact locations with varying vibrating frequencies of 2.4, 3.8, 5.1, and 6.5 cps. PAGE 79 79 Simulation The branch specimen used in the laboratory experiment was modeled with a finite element model. The branch and tine were model ed as an elastic beam element and their mechanical and physical properties were defined in the FE model. The interaction of the branch and the tine was modeled using the tube to tube contact element. The FE parameters similar to the parameters described fo r the FE analysis of tree limb prototypes were set in the simulation of laboratory experiments. The simulation results were computed using Abaqus/Standard and compared with the experimental results. Results and Discussion The values of acceleration and strain used to define objective functions w ere computed and compared with experiments to validate the FE model Figure 5 11 shows the comparison of strain computed from the FE model with the experiment al results The regression line for root mean squa re of longitudinal strains in the branch specimen computed for all loading conditions h a s a slope of 0.7394, an intercept of 0.0002 and R square of 93.93 % Figure 5 11 Comparison of RMS of maximum strain of branch specimen obtained from FEA and experiments y = 0.7394x + 0.0002 R = 0.9393 0.00E+00 5.00E04 1.00E03 1.50E03 2.00E03 2.50E03 3.00E03 0.00E+00 1.00E03 2.00E03 3.00E03 RMS of Longitudinal Strain from Exp RMS of Longitudinal Strain from FEA FE Vs Experimental Strain Linear (FE Vs Experimental Strain) PAGE 80 80 The ro ot mean square of acceleration of the branch at test location s was calculated and compared using FE simulation s and experiments. The r egression line between FE and experiments ha s a slope of 0.9392 and intercept of 5.19 and R square of 85.05% as shown in Figure 5 12. Figure 5 12 Comparison of RMS acceleration of the branch specimen obtained from FEA and e xperiments The longitudinal strain and acceleration computed at the test points for all loading conditions from FE and experiments are highly correlated and respective ly have a Pearson coefficient of 0. 97 and 0.92. Thus, the proposed FE model is able to predict the mechanical behavior of the branch to acceptable accuracy u nder the dynamic impact loading conditions and therefore, similar FE parameters were used to analyze the tree limb prototypes for the optimization of a canopy shaker. y = 0.9392x + 5.1932 R = 0.8505 0.0 20.0 40.0 60.0 80.0 100.0 0.0 20.0 40.0 60.0 80.0 100.0 RMS Acceleraton from Exp. RMS Acceleration from FEA FEA Vs Experimental Acceleration Linear (FEA Vs Experimental Acceleration) PAGE 81 81 CHAPTER 6 MECHANISTIC MODELS The objectives for the optimization of shaker were formula ted mathematically in terms of mechanistic index. The mechanistic index is a function of one or more parameters of the objectives, which are structural response quantities that can be computed and correlated with the objectives. The indices are defined bas ed on the demand vs. capacity of a system. The results from the finite element analysis were used in the formulation of the mechanistic models. Mechanistic Tree Damage Model Consistent with the dynamic behavior of the tree limbs due to a vibratory excitation, the dynamic structural damage is estimated using damage index. The damage index is expressed as the ratio of the maximum response of the limbs to the maximum allowable deformation or strength of the tree limbs as described in E quation 6 1. Bran ches are ( 6 1 ) The response of limbs computed from FEA is used to formulate the damage index. The structural damage response of each limb is expres sed in terms of normal stress ( S11) obtained by solving partial differential equation of dy namic motion of a system using FEM The root mean square of normal stress was calculated and used to formulate damage index in order to account for the variati on s in the stress over a period of cyclic loading. The goal is to minimize the damage of tree limbs or scaffold branches having diameter more than certain critical diameter. For the medium size citrus tree, this value is taken as 2.5 inches because breakag e of limbs with more than this diameter PAGE 82 82 was observed to affect next year fruit yield. Figure 6 1 shows a branch limb with critical region defined by series of dots. Figure 6 1 A f inite element model of a t ree limb showing damage region The normal stress S11 is evaluated at each section points of the beam element as shown in Figure 6 2. The stress S11 at section points designated as 1 2 3, and 4 is given as: Figure 6 2. Section points of a finite beam element of the tree limb Symmetr ic bending about y axis ( 6 2 ) Symmetr ic bending about z axis ( 6 3 ) PAGE 83 83 where My, Mz, Iyy, and Izz are the bending moments and the second area of moment about the Y and Z direction s, respectively; y and z are the distance of outer fib er from the neutral axis along the Y and Z direction respectively; and N x and A is the axial force and the are a of cross section respectively. The stress is a vec tor of maximum sectional stress ( S11 ) computed at all the critical points in the damage region for which and is given by: ( 6 4 ) w here i equals to 1, 2, 3, and 4 corresponding to the section points of beam element. The response of the individual limb prototypes is computed as the maximum of a section stress in the damage region of the tree limb prototype and is calculated as: ( 6 5 ) Damage index of an individual limb prototype is determined by normalizing t he maximum response by the capacity of the limb The capacity is either taken as the strength of wood defined in terms of m odulus of rupture ( ) or the value of maximum response of limb prototypes when analyze with current machine configuration ( ) The current machine configuration use s the 78 inches long, DOM 4130 steel t ube hardened to 45 Rc +/ 2 and vibrates at the frequency of 4 Hz with 4 in amplitude. Minimum of these two capacities was used to formulate the damage index of the limb prototype (D) as given by Equation 6 6 ( 6 6 ) PAGE 84 84 The damage index (D) of all limb prototypes in a tree zone was averaged to provide the damage index of that zone (DI) and is expressed as: ( 6 7 ) w here k=1, 2, corresponds to the top, middle, and bottom zone respectively. Mechanistic Fruit Detachment Model Fruit detachment model estimates the amount of fruit detached from the limbs when shaken. The mechanism of fruit detachment has been investigated analytically and exper imentally by Fridley and Adrian, 1966; Wang and Shellenberger 1967; Cooke and R and, 1969; Diener et al ., 1969 ; Liang et al. 1971; Parchomchuk and Cooke 1972; Miller and Morrow 1976; Berlage and Willmorth 1974; Savary and Ehsani, 2010. They fo und that the amount of fruit removal is highly correlated with acceleration of the fruits. The past studies have suggested that the ratio (the tensile force required to detach the fruit divided by the fruit weight) is a good indicator of fruit detachment by shaking. The typical value of this ratio for fruits like citrus and prunes varies from 1 to 50 depending on the variety and the size of the fruit Savary (2009 ) measured the force required to detach H amlin and Valen cia varieties of orange He concluded that the average static force of 96.1 N and shaking force of 17.1N are required to remove the se varieties of fruits Therefore, a system which provide s an acceleration of approximately 9g would more likely to detach citrus fruits based on the ave rage weight of 0.186 Kg. However, using this value to define the maximum capacity of fruit detachment model would not be accurate pertaining to high variability in the fruit weight Thus, maximum capacity is defined in terms of the acceleration obtained by analyzing the limb PAGE 85 85 prototypes at the current configuration of the canopy harvester which in the field harvesting provides 96 99% of fruit removal (Roka, 2008). The fruit detachment finite element model compute s the fruit detachment index of the limb proto types as shown in Figure 6 3. Figure 6 3 A f inite element model of a tree limb showing fruit bearing region The fruit detachment response ( ) in the fruit bearing region of a limb prototypes was computed as the root mean squar e of resultant acceleration (a) as given below: ( 6 8 ) The resultant acceleration is computed at every node in the fruit bearing region using E quation 6 9. ( 6 9 ) j = 1 j = n PAGE 86 86 The fruit detachment response of an individual limb prototype ( ) was calculated as the mean o f responses computed at all nodes of the fruit bearing region of the limb prototype and is given as: ( 6 10 ) The fruit detachment index of a limb prototype ( ) was obtain ed by normalizing the fruit detachment response ( ) by the response obtained using the current machine configuration ( ) and is e xpressed below: ( 6 11 ) The fruit detachment response of a zone ( FDI ) was obtained by taking the average of response s of all the limb prototypes in that zone and is computed as: ( 6 12 ) w the top, middle, and bottom zone, respectively. The tree damage index and fruit detachment index were computed from the dynamic analysis of the limb prototypes using Abaqus and MATLAB program to obtain the optimal set of machine parameters PAGE 87 87 CHAPTER 7 MULTI OBJECTIVE OPTIMIZATION An efficient harvester is that which can provide high fruit removal with either no damage or minimal damage to the scaffold branches or primary limbs. I ncreasing the shaking force to the tree limbs having large number of fruits results in high fruit removal ; however, it also cause high amount of tree damage. Therefore, a shaker should be designed to provide the optimum shaking force to the tree limbs based on the distribut ion of the tree limb and fruits in a citrus tree canopy. This type of design problem generally involves finding the best trade off between the two conflicting objectives and is hence classified as multi objective problems. T he most widely accepted procedure to solve multi objective problems is to employ Pareto optimal solution search technique (Pareto 1906). Pareto optimal solutio n guarantees that, if moving from it, no improvement can be achieved on any objective functio n without worsening others (Deb, 2001). Pareto technique requires significantly high number of functions evaluations to solve multi objective optimization problems Thus, a computationally efficient strategy should be developed for the optimization of structures. One of the most effective approaches to minimize the cost of optimization in the recent years is founded on response surface met hodology (Myers and Montogo mery, 2002). Response surface based optimization methods are based on approximation of a given objective function to be optimized through a set of points belonging to domain of variation of the independent variables the function itself depends on. PAGE 88 88 The fo llowing steps have to be developed in order to implement a multi objective approach: Identification and formulation of different objectives Variable definition Defining proper design of e xperiments (DOE) Defining numerical model to be evaluated at DOE Col lection of numerical data to compute objective functions Meta modeling step to analytically develop response surface describing each objective function as a function of design variables. Multi objective optimization formulation to determine optimal Pareto solutions. Problem Formulation The fruit detachment and the tree damage depend on the amount of shaking force applied to the tree limbs by a harvester machine. The shaking force in return, is a function of structural and operating parameters of a harvester The operating parameters: frequency and amplitude of vibration of a system are only considered in the optimization of a mechanical harvester. In a canopy shaker, the frequency of the vibration can be easily controlled by changing the speed of motor conne cted to shaft of vib ratory mechanism. However, in order to change the amplitude of vibration, the crank design of the slider crank mechanism should be changed. The structural parameter of a harvester is the configuration (geometry and material) of the interface which interacts with tree canopy. For the canopy shaker, the tine acts as an interface and transmits the vibrating motion from machine to tree. PAGE 89 89 A two piece design of a tine as shown in figure 12 was proposed to provide the adaptive shaking of a tree canopy considering the following guidelines: Easy to adopt and implement in the purchased machines Can be tested and verified with the available laboratory equipment Overall lost cost of designs Figure 7 1 Proposed t wo piece design of at tine for adaptive shaking of tree canopy The proposed tine design consists of (1) a current design (steel pipe ) forming base and attached to the hub of a canopy shaker; (2) an insert in the form of rod or tube. The configuration of insert was designed to provide variable shaking force to the various parts of a citrus tree canopy. The stiffness (s) of insert which is the property of geometry and the material, and the percentage length of insert (x) were defined as the design variables in the opt imization of shaker besides the operating variables : frequency ( v ) and amplitude ( a ) The formulation used in the optimization of a canopy shaker is expre ss ed in Equation 7 1. x s PAGE 90 90 Subjected to: Structural variables: Operating variables: (7 1) Design of Experiments (or Design Domain) The design of experiments to predict optimum configurations of machine were designed based on past research work and field experiments. D esign V ariables : Sti ffness (s) The flexural stiffness of an insert is function of the geometry and material of the insert. The different designs of the insert were chosen based on the market availability and conclusion drawn from the various tine configurations experimented b y Oxbo International Corporation (a leading manufacturer of canopy shaker and other farm equipment s) and Florida growers. The different variants of polyamide (PA, also called N ylon) having same cross section denoted as P1 P5 and different cross section ge ometries of Aluminum denoted as A6 A9 was chosen as listed in table 7 1. Figure 7 2 PAGE 91 91 shows the variation of stiffness of the various designs of insert considered with respect to stiffness of the current design. Figure 7 2 Variation of stiffness of multiple designs of insert normalized wrt stiffness of current design of a tine. Table 7 1. Geometry and material configuration of the designs of the insert Designs Geometry Dimension Mat. Prop. Material Type r o (mm) t (mm) E (GPa) P 1 Rod 18.98 0 3 PA Cast, Molding and Extrusion and 1 5% Glass P 2 18.98 0 8 PA 30% Long Glass, 40% Glass and Mineral P 3 18.98 0 14 PA 50% Long Glass P 4 18.98 0 20 PA 60% Long Glass, 30% PAN Carbon P 5 18.98 0 28 PA 50% PAN Carbon A 6 Pipe 20.64 1.5 80 Cast Wrought Al. Alloy A 7 20.64 2 80 Cast, Wrought Al. Alloy A 8 20.64 2.5 80 Cast, Wrought Al. Alloy A 9 20.64 3 80 Cast, Wrought Al. Alloy Current Design 20.64 1.65 210 DOM 4130 STEEL PAGE 92 92 Design Variables: Length of Insert ( x ) The length of insert is the percentage length of an insert in a proposed two piece design of the tine The values of the design variable were chosen as listed in Table 7 2 to balance the computational cost and degree of exploration in the design domain for optimum design T he design value of 0 and 100 corresponds to 0 % and 100 % of insert respectively. The 0% of insert means that the tine is made of only current design and 100 % of insert means tine is made of only new design. Design V ariables: Shaking F requency ( v ) Shaking frequency corresponds to the number of time shaker tine knocks the tree limbs per second. Design Variables: Shaking Amplitude ( a ) Shaking amplitude is a maximum displacement of a shaker and determines the amount of flexural d eformation imparted to the tree limbs. Past experiments on many fruit and nut crops have indicated that high frequencies of 20 40 Hz and short strokes of 20 25 mm is effective for the tree s having relatively rigid structure. How ever, for willowy trees which have long and slender branches that curve down sharply due to the weight of fruits, the long stroke (100 125 mm) and low frequencies (1.5 6 Hz ) are effective for the fruit removal B ased on the experiments conducted on citrus using various harvester s the good results was achieved using a stroke of 100 1 25 mm at a frequency of 1.6 5.9 Hz B rien and Fridley, 1983) The conclusions from the past studies were used to sample the frequency and amplitude of a canopy shaker. The value s of the design variab les are l isted in Table 7 2. PAGE 93 93 Table 7 2. Design of e xperiments for numerical analysis and o ptimization Design Variables Lower bound Upper bound Design Values Length of Insert in % (x) 0 100 0:5:100 2 8 2:1:8 Shaking Amplitude (a) 1 6 1:1:6 Estimation of Cost of Analysis for Optimization The total optimization time consist of the finite element (FE) analysis calculation time multiplied by the number of optimization iterations. The dynamic FE analysis of a tree limb using single design of a tine takes about 5 min. therefore, the total time required for the optimization of single zone would be the total time required for whole FE simulation s of design domain multiplied by number of limb prototype s as shown in E quation 7 2 (7 2) The total computation time of about 138 days for one zone hardly justified the practical value derived from su ch a long and tedious analysis. Besides high computational effort required by dynamic simulation s the multidimensional optimization requires special techniques for finding the optimum design such as gradient based optimization techniques, and genetic or e volutionary al gorithms. T hese techniques further require special programs as well additional computational resources. Thus, a strategy was hereby put forward which aim ed to minimize the computation time and to employ a classical graphical optimization tech nique instead of special optimization algorithm. The strategy consists of solving the optimization problem in two phases: first PAGE 94 94 phase of optimization involves only structural variables ( s, x ); and in the second phase optimization, the best design s of phase 1 were used to further improve the objective s defined in terms of machine operating variables ( v, a ). The proposed strategy has an additional advantage of being able to provide designs based on change in structure of the machine and change in the machine operating parameters. Structural parameters of the machine are easier to test and implement as it only involves the procurement of new design s whereas machine operating parameters involve changes to be made in the machine vibratory mechanism. Thus, operating designs are costlier and difficult to implement in the exciting machine but may result in significant improvement in the objective functions Shaker Optimization: Phase 1 The phase one of optimizatio n involves the optimization of a shaker base d on only structural variables: the stiffness and the percentage length of an insert. The operating variables were fixed to the current machine setting with a frequency and amplitude of 4 Hz and 4 in. respectively The design of a shaker for the phase 1 o f optimization is fo rmulated in Equation 7 3. PAGE 95 95 Subjected to: (7 3) Materials and Methods The following procedure was adopted to find the optimum tine configuration for each zone. The re sp onse of the limb prototypes was compute d using finite element analysis. The fruit detachment index and damage index were calculated for each zone as described in Chapter 6 The P areto frontiers were constructed for each zone to choose optimum design based on the tra de off between DI and FDI. The P areto frontier was constructed based on dominance principle where a set of non dominated design points are chosen such that no objective functions can be improved further without impairing the other objective The MATLAB function by Freitas (2012 ) was used to create Pareto fronts. To minimize the computation cost t he b i objective formulation was converted into a mono objective formulation by converting the fruit detachment formulation into a constraint as give n by E quation 7 4 The best P areto optimal design was selected to obtain at least of 15 % reduction in the tree damage. ( 7 4 ) The value of FDI allowable was chosen based on the fruit distribution in a citrus tree. The fruit harvesting efficiency can be increased by providing large shaking force to the parts of canopy that have comparatively high fruiting. Theoretically, it can be achieved by setting high value of FDI allowable to those parts of tree canopy. Whitney and Wheaton (1984) have studied the fruit distribution pattern of the citrus trees and they have concluded that most of the fruiting occurs in the middle and outer part of the canopy. To PAGE 96 96 corrobor ate their finding for a case of medium sized citrus tree, an experiment was set up and fruit distribution was analyzed for 361 trees. Figure 7 3 shows the distribution of fruits in the three zone s of a tree. It is found that the average fruit density in th e middle section of a tree canopy is two times as in the top and bottom section s Thus, overall harvesting efficiency can be improved by setting the comparatively high value of FDI allowable for the middle zone as compared to the bottom and top zone s The v alues of FDI allowable selected for the phase 1 of optimization for each zone of t ree canopy are listed in Table 7 3 Figure 7 3 Distribution of citrus fruit s in the three zones of a citrus tree canopy Table 7 3. Allowable Fruit Detachment Index Top Zone 0.8 Middle Zone 0.9 Bottom Zone 0.8 Results and Discussion O ptimization results based on structural variables of the harvesting equipment for each zone are presented below The different optimal design configurations of the tine were proposed. The d esign of a tine is selected among the proposed designs of the tine PAGE 97 97 based on the degree of machine improvement desired and amount of expenditure, one is willing to invest. Middle zone Figure 7 4 shows the Pareto fron t between objective functions, damage index (DI) an d fruit detachment index (FDI), of the middle zone. The contour plots for the FDI and DI for the middle zone of the limbs are shown in Figure 7 5. Figure 7 4. Pareto frontier to predict the optimum tine configuration of the middle section of a canopy shaker Table 7 4. Optimum configuration of tine for middle section of the canopy shaker ZONE Best design Design Configuration Reduction in Damage (%) Fruit Detachment (%) MIDDLE D1 Design P 5 70% 28 GPa PA 15.2 100 D2 Design P 4 70% 20 GPa PA 20.1 96.4 D3 Design P 3 65% 14 GPa PA 24 91.5 PAGE 98 98 Figure 7 5. Phase 1 optimization: Contour plots for the Damage Index and Fruit detachment Index to predict optimum tine configuration for the middle section of a canopy shaker. T he upper bound on DI and the lower bound on FDI were set to select only those designs which result in at least of 15 % reduction of tree damage and minimum of 90% fruit detachment. Three optimum designs of tine have configurations D1, D2 and D3 shown as six pointed star in Figure 7 4 and listed in table 7 4 were proposed. The tine configurat ion of 70% 28 G P a p olyamide means that the tine composed of 70% of new design of insert and 30 % of current design by length. As in table 7 4, the tine design D1 results in 15.2 % reduction of tre e damage and fruit removal of approximately 1 00 %. Physically it means that employing the new design will result in approximately 15% reduction in maximum stress in the critical region of limbs and an average acceleration PAGE 99 99 of about 100% of the average acceleration in the fruiting zone of the tree limbs in the middle zone of the tree canopy when computed using current design of the tines In the field experiments, the current tine configuration of a canopy shaker results in the fruit harvesting efficiency of 95 96 % (Roka, 2008). A signi ficant amount of reduction in tree damage can be accomplished by employing the designs D2 and D3; however, this will also result in the considerable reduction in the fruit detachment. Top and b ottom zone Figure 7 6 show s the P areto frontier constructed fo r the limbs of the top and bottom zone of a citrus tree canopy The contour plot s for the FDI and DI are provided in the appendix B ( Figure B 1) The designs of the top and bottom zone were selected to achieve high reduction in tree damage (more than 20%) as compared to that of the middle zone. Two optimum designs of tine D1 and D2 and three designs of the tines D1, D2, and D3 as listed in table 5 were selected for the limbs of top and bottom zone respectively to mi nimize the tree damage and maximize the fruit removal It should be note d that the design D1 of top zone results in the fruit detachment of 102 % which simply means t hat the average acceleration in the fruiting bearing region of the limbs of the top zone i s 2% more than that co uld be obtained by the current design of the tines Physically, more than 100 % of fruit de tachment index means that there is higher probability of achieving fruit harvesting efficiency of 100% during real time harvesting It is notice d that the substantial reduction in tree damage (~3 0 35 ) can also be achieved using tine design D2 in the top zone and design D3 in the bottom zone. However, employing the tine design D2 of the top zone and D3 of the bottom zone would result in 15 20% reduction in the fruit removal. This reduction in the fruit removal PAGE 100 100 can be compromise d because of significantly less number of fruits in the top and bottom zone as compared to middle zone. Figure 7 6 Pareto frontier for the top zone (left ) and the bottom zone (right ) to predict optimum tine configuratio n for the top and bottom section of a canopy shaker. Table 7 5. Optimum configuration of tine for top and bottom section of the canopy shaker ZONE Best design Tine Configuration Reduction in Damag e (%) Fruit Detachment (%) TOP D1 Design P5 100% 28 GPa PA 22.5 102 D2 Design P4 100% 20 GPa PA 35 85 BOTTOM D1 Design P5 80% 28 GPa PA 20.6 91.5 D2 Design P4 80% 20 GPa PA 23.8 85.2 D3 Design P3 70% 14 GPa PA 28.2 81.7 Considering all the proposed designs of all three sections for a canopy shaker, the choice of employing any design is made based on trade off between the total cost of design and the degree of improvement required in the machine. Figure 7 7 compares PAGE 101 101 the cost of various po lyamide invariants used in the optimization with their respective flexural strength The flexural strength is the measure of capacity of a structure to bear flexural deformation; and choosing a tine design with high value of flexural strength is desirable. The information regarding the cost and the tensile strength of various variants of polyamide were obtained from the web resources and information available in CES Education Package (version : 2 005 ) The information is listed from the data published in 2005 ; therefore, absolute value may have changed now but the relative cost would be approximately same. More market exploration would be helpful in making comprehensive decision regarding the choice o f desi gn for the three sections of a citrus canopy. Figure 7 7 Comparison of a cost and tensile strength of the polyamide variants. It is noticed that both the cost and the tensile strength of polyamide variants increases from the design P1 to P 5 However, design P3 and P 4 are about 8 times cheaper than design P5 and have only marginal difference in the tensile strength The decision to choose a tine configura tion for each section of a canopy shaker depends on 0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 0 2 4 6 8 10 12 14 16 18 20 0 4 8 12 16 20 24 28 Flexural Strength (MPa) Cost USD/lb Elastic Modulus of Designs (GPa) PAGE 102 102 the amount of financial resources one is willing to put in and degree of improvement one desired to get in the canopy shaker Shaker Optimization : Phase 2 In this phase, the canopy shaker is further improved by providing the optimum combination of operating parameters: frequency and amplitude. The best design s from the phase 1 were used in the optimization to find t he optimum operating parameters The problem formulation of the phase 2 of the shaker optimization is expressed in Equation 7 5 Subjected to: (7 5) Materials and Methods The following procedure was adopted to find the optimum operating parameters of the machine for the best design s of the phase 1. The FDI and DI were computed for each combination of frequency amplitude in the design of experiments. Performing dynamic analysis is generally computationally expensive, thus meta modeling technique was implemented to minimize the cost of optimization. PAGE 103 103 Radial Basis Neural Network ( RBNN) (Park and Sandberg, 1991) was used to construct the analytical response surface of fruit detachment index and damage index as a function of the design variables: shaking frequency and amplitude. The Matlab program for surrogates designed by Viana ( 2010) was used to construct RBNN response surface. The parameters of RBNN were chosen based on the minimum PRESS RMS The Pareto frontiers for the optimum tine designs of each zone were constructed from the FDI and DI response surfaces. The best operating combination of frequency and am plitude was determined using Equation 7 6. ( 7 6 ) Result s and Discussion The o ptimum operating parameters of a canopy shaker for each zone of tree canopy are presented below. The response surfaces were constructed based on results of numerical simulation s evaluated for few combinations of frequency and amplitude. These response surfaces were then used to predict objective functions for any combination of operating variables (frequency and amplitude) in the design domain. The Pareto frontiers were constructed to predict the best combination of operating variables for all designs of tines proposed in phase 1 M iddle zone The parameter s of the response surfaces were set to maximize the predictive performances of the response surface obtained by minimizing the PRESS RMS The value of error norm s of RBNN response surface for both the objective funct ion s are listed in Table C 1 (appendix C) The r esponse surface of the objective functions for the tine design D1 for the middle zone of a canopy shaker is s hown in Figure C 1. Figure s C 3 C 4 and C 5 show Pareto frontier for the tine design D1 D2 and D3 computed in PAGE 104 104 phase 1 of the shaker optimization. The optimum operating parameters for each design of the tine fo r the middle section of a canopy shaker are listed in Table C 2 and summarized in Table 7 6 Table 7 6. Optimum operating parameters for the middle section of a canopy shaker Designs Frequency Amplitude Reduction in Damage (%) Fruit Detachment (%) (cps) (inches) D1 and D3 7.8 1.5 2 20 3 0 90 100 D2 3.6 4.6 20 102 6.4 2 26 91 It is found out that the tine design D1 and D3 if operated at high frequency (~7.8 Hz) and low amplitude of vibration (1.5 2) can result in 20 3 0 % reduction in tree damage and a fruit removal of 90 100 %. For design D2; however, two different opt imum combination of frequency was found. The tree shaking with a high frequency of around 6.5 Hz and low amplitude of 2 in. has resulted in 26 % reduction in the tree damage w ith approximately 91 % of the fruit removal; however, decreasing the shaking frequency to about 3.6 Hz and increasing amplitude to about 4.6 in. has resulted in more than 100 % fruit removal with 20% reduction in tree damage. Top zone The value of error norm s of the response surface s used in predicting the objective functions for the design s of the top zone are listed Table C 3. Figure C 6 show s the response surface constructed for the tine design D1 of the top zone as a function of operating variables ( v a ). The optimum operating parameter for both the design s of the top section of a canopy shaker are listed in Table C 3 and summarized in PAGE 105 105 Table 7 7. Figure s C 7 and C 8 show Pareto frontier and contour plot s for the optimum tine designs D1 and D2 of phase 1 respectively. Table 7 7 Optimum operating parameters for the top section of a canopy shaker Designs Frequency (Hz) Amplitude (inch) Reduction in Damage (%) Fruit Detachment (%) D1 and D2 6.5 7.5 3 3.5 40 55 > 100 It is can be concluded that for both the optimal designs, high frequency in the range of 6.5 7.5 and mid range value of amplitude of 3 3.5 inches of vibration has result in the 40 55 % reduction in the tree damage. The designs of tine ( D1 and D2 ) at these combination s of frequency and amplitude have resulted in more than 100% of fruit detachment index. B ottom zone The value of error norm s of the response surfaces constructed to predict the objective functions of designs of the tine of bottom zone are listed in the Table C 5 Figure C 9 shows the response surface constructed for the ti ne design D1 of the bottom zone The optimum operating parameter s and improvement in the objective functions for the designs of the tines proposed for the bottom section of a ca nopy shaker is given in Table C 6 and summarizes in Table 7 8 Figure s C 10, C 11, and C 12 show Pareto frontier constructed for the tine designs D1, D2, and D3 computed in phase 1. Table 7 8 Optimum operating parameters for the middle section of a canopy shaker. Designs Frequency (Hz) Amplitude (inch) Reduction in Damage (%) Fruit Detachment (%) D1, D3 and D3 3 3.5 5.5 6 35 40 ~ 80 D2 and D3 7.5 2.5 20 25 ~ 100 PAGE 106 106 Two sets of opt imum combinations of frequency were observed for tine corresponding to design D2 and D3. The t ine s configuration s (D2 and D3 ) when operated at the mid range frequency of 3 3.5 Hz and high amplitude of 5.5 6 in. has result ed in 35 40% reduction in tree damage and 80% of fruit detachment in the bottom section of tree canopy However, around 100% of fruit detachment was computed at the high frequency of 7.5 Hz and low amp litude of vibration (~2.5 in. ) with a 20 25% reduction in damage. The design D1 of the tine has only one optimum combina tion of frequency and amplitude (3 3.5 Hz and 5.5 6 in. ) and results in approximately 35% reduction in the tree damage with a fruit removal of only 80% when operated at this combination PAGE 107 107 CHAPTER 8 CONCLUSION In this study, the application of finite elemen t analysis and computer aided optimization tech niques for the design of a harvester was presented. The main goal of this study was to provide an overview of analytical possibilities available to improve the performance of a vibratory harvester: this goal w as pursued by employing numerical methods and optimization techniques in the design of a canopy shaker. A progressive design approach consisting of determining the properties of the wood, modeling statistical prototypes of tree limbs, developing mechanisti c models and, integrating numerical simulation with optimization to ols was presented. Statistical m odeling, objective q uantification using mechanistic m odeling response surface m ethod ology and Pareto optimal solution search techniques were applied in order to obtain optimum machine parameters The proposed design methodology consists of solving the optimization problem in two phases in order to reduce computational effort. Although t he pro posed framework has been developed in regards to the optimization of the citrus canopy shaker it can be easily and effectively applied to the design and op timization of other fruit crop harvesters. Summary of Conclusions The design idea of providing adap tive shaking of a tree based on the distribution of fruits and spatial configuration of the tree limbs w as realized in this study. The a daptive shaking of the tree was accomplished by designing three set of tines vibrating at different combination s of freq uency and amplitude corresponding to three sets of tree limbs in a tree canopy. The optimal design of the machine was proposed employing PAGE 108 108 numerical method instead of extensive experiments to reduce the high cost associated with setting up field trials. Lab oratory e xperiments were set up to find the properties of the green citrus wood to be used in the numerical analysis. The elastic modulus (8.5 GPa), the modulus of rupture (67.3 MPa), specific gravity (1.4508 g/cc ) and damping ratio (10.78%) were calculate d. The limbs to be modeled in the numerical analysis were defined based on the statistical distribution of the tree limbs rather than random individual trees. The spatial configuration of the limbs were predicted and modeled in t he finite element analysis as a three dimensional beam element The sectional properties of the tree limbs were predicted using response surface methodology. The effect of secondary branches and fruits on the dynamic response of the primary limbs was consi dered by modeling them as a lumped mass on the limbs. The distribution of secondary branches and fruits were obtained by analyzing the me asured data from the field experimentation The fruit bearing region, where the objective function was evaluated, was c on f igure d based on statistical data of the tree limbs. The finite element model was developed and the parameters were derived from the experimental results and research literature. The FEA model was verified by setting up small scale laboratory experiment ation. Pearson correlation coefficient between the simulations and experiments of 0.97 and 0.92 were observed for the physical quantities: strain and acceleration, respectively. The research objectives (tree damage and fruit removal) for the optimization o f shaker harvester were identified and quantified in terms of stress and acceleration of the tree limbs using mechanistic models. The computational cost involved in numerical simulations and optimization was minimized PAGE 109 109 using response surface methodology and two phase optimization. Pareto fronts for phase 1 optimization were constructed based on structural variables: stiffness and percentage length of the insert; and Pareto fronts of the phase 2 optimization were designed based on machine operating variables: shaking frequency and shaking amplitude. Tines made of low stiffness material (polyamides) and high stiffness material (steel) in the approximate ratio of 3:1 by length work best for limbs which are long, thick and hang down due to the weights fruits, typ ically seen in the middle and bottom secti ons of a citrus tree canopy. Considering least expensive of the proposed de signs, the tine made of 70% of p olyamide (PA) rods having 50% long g lass fiber (Elastic modulus, E=14 GPa) and 30% of steel tube (DOM 4130) by length when installed in the middle section of canopy harvester, has shown approximately 24 % reduction in tree damage and about 92 % of fruit detachment as compared to current tine configuration. However, using the same configuration of a tine in the bottom section of the can opy shaker, around 28 % reduction in tree damage and approximately 82% fruit removal was computed. Around 100% of fruit removal can be achieved in the middle section of canopy using a tine made of polyamide rods having 50% PAN c arbo n fiber (E=28GPa) but they are expensive and result in comparat ively less reduction in the tree damage. The tree limbs which are thick, long and grow straight up to a height of 100 130 in. and then slightly curve down due to the weig ht of the fruits; approximately 35% reduction in the tree damage and about 85% fruit detachment can be obtained by installing ti nes of polyamide with 60% long glass fiber or 30% PAN Carbon (E=20GPa) in the top section of a c anopy shaker. PAGE 110 110 The phase 2 of opt imization proposed different combinations of frequency and amplitude c orresponding to each zone of a citrus tree canopy to further minimize the tree damage and maximize the fruit removal. From the manufacturing point of view, these combinations of frequenc y and amplitude are obtained by employing three different vibratory mechanisms in a harvesting system; however, the cost of installing these modifications could be high but can be considered owing to the improvement achieved in the machines Considering th e least expensive of the tine configura tions : the tines made of rod of PA having 50% long glass fiber and tube of steel (DOM 4130) in the ratio of 3:1 when vibrates at a high frequency of 7 8 Hz and low amplitude of 1.5 2.5 inches has resulted in 20 30% of reduction in tree damage and 90 100% of fruit removal of the limbs for the middle zone and 20 25% reduction in the tree damage and ~100% fruit removal for the limbs of b ottom zone. However, the tines made of rods of only polyamides having 6 0% long glass f iber or polyamides having 30% PAN c arbon fiber and vibrating at a frequency of 6.5 7.5 Hz and amplitude of 3 3.5 inches has resulted in the 40 55% reduction in tree damage and approximately 100% fruit removal for the limbs of t he top section of a citrus tr ee canopy. Thus, the proposed study offers design alternatives based on adaptive shaking of a tree canopy to improve the current continuous canopy shaker for the citrus crop. We expect this methodology will open up a novel way to optimize the other vibratory shakers based on modeling and analyzing the tree limb prototypes using numerical methods. Recommendation for Future W ork In future work, the proposed designs will be evaluated by field trials to determine their efficacy. After evaluation, j udgment will be made to determine which optimum PAGE 111 111 machine configurations should be chosen for the commercial harvesting of citrus. The possibility of redesigning a harvester may be explored further based on the experimental results and the proposed theoretic al model s The presented theoretical model, verified in the controlled environment of laboratory experiments, may be refined further depending on its correlation with the field trials. The following recommendations are listed to further improv e a canopy shaker, and validate and refine the proposed analytical models to optimize other vibratory harvesters. A small scale fruit removal system which works on the principles similar to that of a continuous citrus canopy harvester should be built to validate the simulations. The field experiments should be setup to evaluate the proposed design modifications. The proposed analytical models can be refined further based on experimental results. The techniques of experimental based design optimization (EDO) may be us ed to further improve the performance of a harvester. Effect of vertical shaking of a tree canopy can also be explored using numerical simulation. PAGE 112 112 APPENDIX A LABVIEW PROGRAM PAGE 113 113 Figure A 1 LabVIEW instrument program to acquire data from an accelerometer PAGE 114 114 Figure A 2 LabVIEW instrument program to acquire strain data from a strain gauge left half portion PAGE 115 115 Figure A 3. LabVIEW instrument program to acquire strain data from strain gauge right half portion PAGE 116 116 APPENDIX B PHASE 1 OPTIMIZATION Figure B 1 Phase 1 optimization: C ontour plots for the Damage Index and Fruit detachment Index for the top zone tine design PAGE 117 117 Figure B 2. Phase 1 optimization: Contour plots for the Damage Index and Fruit detachment Index for bottom zone tine design PAGE 118 118 APPENDIX C PHASE 2 OPTIMIZATION Table C 1. E rror norms of RBNN meta models used to predict objective functions for the middle zon e tine designs MODEL Fruit Removal Index Damage Index Error Norms PRESS RMS RMS PRESS RMS RMS D1 Design P 5 0.0483 0.0423 0.0874 0.0429 D2 Design P 4 0.107 0.0519 0.0748 0.0521 D3 Design P 3 0.0641 0.0575 0.0784 0.055 Table C 2 O ptimum operating parameters of the tines in the middle section of canopy shaker ZONE Phase 1 Best Designs Best Operating Range Frequency Amplitude Reduction in Damage (%) Fruit Removal *(%) MIDDLE D1 OP1 7.8 1.6 19.4 99.3 OP2 7.2 1.6 27 89.3 D2 OP1 3.6 4.6 20.1 102.4 OP2 6.4 2 26.5 90.6 D3 OP1 7.6 2 21.5 103.4 OP2 7.6 1.8 32.1 89.9 PAGE 119 119 Figure C 1 Response surfaces used to predict the damage index and fruit detachment index of best tine design D1 of middle zone Figure C 2 Legend information used in the Pareto fronts and contour plots PAGE 120 120 Figure C 3 Pareto front and contour plots of the tine design D1 of middle zone PAGE 121 121 Figure C 4 Pareto front and contour plots of tine design D2 of middle zone PAGE 122 122 Figure C 5 Pareto front and contour plots of the tine design D3 of the middle zone PAGE 123 123 Table C 3 E rror norms of RBNN meta models used to predict objective functions for the top zone tine designs MODEL Fruit Removal Index Damage Index Error Norms PRESS RMS RMS PRESS RMS RMS D1 Design P5 0.037 0.022 0.29 0.17 D2 Design P4 0.022 0.01 0.021 0.14 Table C 4 Optimum operating parameters of the tines in the top section of canopy shaker ZONE Phase 1 Best Designs Best Operating Range Frequenc y (Hz) Amplitude (inch) Reduction in Damage (%) Fruit Removal *(%) TOP D1 OP1 6.6 3 40 131.1 OP2 6.8 3.6 50.6 102.3 D2 OP1 7.25 3.75 40.1 142.8 OP2 7.5 5.5 55 100.7 PAGE 124 124 Figure C 6 Response surfaces used to predict the damage index and fruit detachment index of best tine design D1 of top section of a canopy shaker PAGE 125 125 Figure C 7 Pareto front and contour plots of the best tine design D1 to predict optimum configuration of top section of a canopy shaker PAGE 126 126 Figure C 8 Pa reto front and contour plots of the best tine design D2 to predict optimum configuration of top section of a canopy shaker PAGE 127 127 Table C 5 E rror norms of RBNN meta models used to predict objective functions for the bottom zone tine designs MODEL Fruit Removal Index Damage Index Error Norms PRESS RMS RMS PRESS RMS RMS D1 Design P5 0.040 0.032 0.036 0.013 D2 Design P4 0.063 0.037 0.064 0.037 D3 Design P3 0.057 0.049 0.054 0.031 Table C 6 O ptimum operating parameters of the tines in the bottom section of canopy shaker ZONE Phase 1 Best Designs Best Operating Range Frequency (Hz) Amplitude (inch) Reduction in Damage (%) Fruit Removal *(%) BOTTOM D1 OP1 3.6 5.4 21.3 99.35 OP2 3.2 5.4 33.7 81.5 D2 OP1 7.6 2.6 20.6 102.3 OP2 3.2 5.8 38.9 82 D3 OP1 7.6 2.6 25.2 100.4 OP2 3.2 6 39.2 81.8 PAGE 128 128 Figure C 9 Response surfaces used to predict the damage index and fruit detachment index of tine design D1 of bottom zone. PAGE 129 129 Figure C 10 Pareto front and contour plots of the optimum tine design D 1 of the bottom section of a canopy shaker PAGE 130 130 Figure C 11 Pareto fro nt and contour plots of the optimum tine design D 2 of the bottom section of a canopy shaker PAGE 131 131 Figure C 12 Pareto front and contour plots of the optimum tine design D3 of the bottom section of a canopy shaker PAGE 132 132 LIST OF REFERENCES Abaqus 2010. Finite Element Software Ver. 6.10. Providence, RI, USA. Dassault Systmes Simulia Corp. Adrian, P. A., and R. B. Fridley. 1965. Dynamics and design criteria of inertia type tree shakers. Trans. ASAE 8(1):12 14. Allen, D. M. 1971. Mean square error of prediction as a criterion for selecting v ariables. Technometrics 13, 469 475 Archer, J. S. 1965. Consistent matrix formulation of structural analysis using finite element techniques. AIAA J 3(10): 1910 1918. Arora, J. S., O. A. Elwakeil, A. I. Chahande, and C. C. Hsieh. 1995. Global optimization metho ds for engineering applicati ons : A review. Struct. Optim. 9: 137 159 ASTM. D143 03.: Standard Test Methods for Small Clear Specimens of Timber. Baier, H. 1977. Uber Algorithmen zur Ermittlung und Charak terisierung Pareto optimaler L¨osungen bei Entwurfsaufgaben Elastischer Tragwe rke. Z. Angew. Math. Mech. 57, 318 320. Bathe, K. J. 1996. Finite Element Procedures in Engineering Analysis Englewood Cliffs, NJ: Prentice hall. Berlarge, A. G and F. M. Willmorth.1974. Fruit removal potential of high frequency vibrations. Trans. ASAE 17(2). 233,234. Brown, G. E., P. F. Burkner, and C. E. Schertz 1969. Mass harvest problems in California citrus. Proc. Int. Citrus Sym. 2 : 667 674. Carmichael, D. G. 1980. Computation of Pareto optima in structural design. Int. J. Numer. Methods Eng 15 : 925 952. Cavalchini, A. G. 1999. Harvesters and threshers: Forage crops. In CIGR handbook of agricultural engineering Vol.III Plant production engineering, 348 380. International commission of agricultural engineering ed. St. Joseph, Mich. : ASAE. Chen, P., J. J. Mehlschau, and J. Oritz Canavate. 1982. Harvesting Valencia oranges with flexible curved fingers. Trans. ASAE 25(3): 534 537. Cheng, F. Y. and D. Li 1998. Genetic algorithm development for multi objective optimization of structure s. AIAA J 36: 1105 1112. Choi, K. K., and N. H. Kim. 2005. Structural Sensitivity Analysis and Optimization I: Linear Systems New York, N.Y.:Springer. Choi, K.K., and N.H. Kim. 2005. Structural Sensitivity Analysis and Optimization II: Nonlinear Systems and Applications New York, N.Y.:Springer. PAGE 133 133 Choi, K.K., and N.H. Kim. 2005. Structural Sensitivity Analysis and Optimization II: Nonlinear Systems and Applications New York, N.Y.:Springer. Chopra, A. K. 1995. Dynamics of Structures:Theory and Applications to Earthquake Engineering Prentice Hall. Englewood Cliffs. NJ. Cook, J.R. and R. H. Rand.1969. Vibratory fruit harvesting: A linear theory of fruit stem dynamics. J. Agric Eng. Res 14(3), 195 209. Coppock, G.E., and P. J. Jutras. 1960. An investigation of the mobile picker's platform approach to partial mechanization of citrus fruit picking. Proc. Fla. State Hort. Soc 73:258 263. Das, I., and J. E. Dennis. 1997. A closer look and drawbacks of minimizing weighted sums of objectives for Pareto set genera tion in multi criteria optimization problems. Struct Optim 14:63 69. Das, I., and J. E. Dennis. 1998. Normal boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim. 8: 631 657. D eb, K. 2001. Multiobjective Optimization using Evolutionary A lgorithms Wiley. New York. Diener, R. G., J. H. Levin, and R. T. Whittenberger. 1969. Relation of frequency and length of shaker stroke to the mechanical harvesting of apples. ARS 42 148. US Dep t. Agric. Ebner, A. M., and D. P. Billington. 1968. Steady state vibrations of damped Timoshenko beams. Jour nal of the Structural Division ( ASCE ) Vol. 3: 737 Freescale Semiconductor Inc. 2008. 1.5g 6g three axis low g micromachined accelerometer (Docu ment number: MMA7260QT). Austin, TX: Freescale Semiconductor Inc. Available at: http://www.freescale.com/files/sensors /doc/data_sheet/MMA7260QT.pdf Accessed 01 July 2013. Freitas, A. D. 2012. Pareto fronts according to dominance relation http://www. mathworks.com/matlabcentral/fileexchange/37080 pareto f ronts according to dominance relation Fridley, R. B., and C. Lorenzen. 1965. Computer analysis of tree shaking. Trans. ASAE 8(1): 8 11, 14. Fridley, R. B. and C. Yung. 1975. Computer analysis of fruit detachment during tree shaking. Trans. ASAE 18(3) : 40 9 415. Fridley, R. B., H. T. Hartmann, J. J. Mehlschau, P. Chen, and J. Whisler. 1971. Olive harvest mechanization in California. Calif. Agric. Exp. Sta. Bull. 855. PAGE 134 134 Fri dley, R. B., and P. A. Adrian 1966. Mechanical harvesting equipment for deciduous tree fruits. Calif. Agric. Sta. Bull. 825. Futch, S. H., J. D. Whitney, J. K. Burns, and F. M. Roka. 2005. Harvesting: From manual to m echanical. Institute of Food and Agricultural Sciences. University of Florida Pub.: HS 1017 Halderson, J. L. 1966. Fundamental factors in mechanical cherry harvesting. Trans. ASAE 9 (5): 681 684. Hedden, S. L., D. B. Churchill, and J. D. Whitney. 1984. Orange removal with trunk shakers. Proc. Fla. State Hort. Soc 97: 47 50. Hedden, S. L., and G. E. Coppock. 1971. Comp arative harvest trials of foliage and limb shakers in Valencia oranges. Proc. Fla. State Hort. Soc 84:88 92. Hilber, H. M., T. J. R. Hughes, and R. L. Taylor. 1977. Improved numerical dissipation for time integration algorithms in structural dynamics. Ear thquake Eng. and Struct. Dynamics 5: 283 292. Hoag, D. L., J. R. Hutchinson, and R. B. Fridley. 1969. Effect of proportional, non proportional and non linear damping on the dynamic response of tree limbs. Trans of the ASAE 13(6): 879 884. Hoag D. L., R. B Fridley, and J. R. Hutchinson. 1971. Experimental measurement of internal and external damping properties of tree l imbs. Trans. ASAE 14(1): 20 28. Horvath, E., and G. Sitkei. 2005. Damping properties of plum trees shaken at their trunks. Trans. ASAE 48(1 ): 19 25. Hughes T. J. R. 1987. Finite Element Method Linear Static and Dynamic Finite Element Analysis. Englew ood Cliffs. New Jersey: Prentice Hall. Hurty, W.C., and M.F. Rubinstein. 1964. Dynamics of Structures Englewood C liffs. New Jersey: Prentice Hall. Hussain, A. A. M., G. E. Rehkugler, and W. W. Gunkel. 1975. Tree limb response to a periodic discontinuous sinusoidal displacement. Trans. ASAE 18(4): 614 617. Jutras, P. J., and G. E. Coppock. 1958. Mechanization of citrus fruit picking. Proc. Fla. State Hort. Soc 71:201 204. Khuri, A. I. and J. A. Cornell, 1996. Response Surfaces: Designs and analyses 2nd edition, New York: Marcel Dekker. Kim N. H. and B. V. Shankar. 2009. Introduction to Finite Element Analysis and Design New York: Wiley PAGE 135 135 Kos ki, J. 1979. Truss Optimization with Vector Criterion. Report Number 6, Tampere University of Technology, Tampere, Finland. Koski, J. 1980. Truss optimization with vector criterion, e xamples. Report Number 7, Tampere University of Technology, T ampere, Finl and. Kristensen E.S., and N.F. Madsen.1976. On the optimum shape of fillets in plates subjected to multiple in plane loading cases. Int. J. Numer. Methods Eng .10:1007 1019. Kutzbach, H. D., and G. R. Quick. 1999. Harvesters and threshers: Grain. In CIGR h andbook of agricultural engineering Vol.III Plant production engineering, 311 347. International commission of agricultural e ngineering ed. St. Joseph, Mich : ASAE. Leitmann, G. 1977. Some problems of scalar and vectorvalued optimization in linear viscoela sticity. J. Optim. Theory. Appl. 23 : 93 99. Lenker, D.H. 1970. Development of an auger picking head for selectively harvesting fresh market oranges. Trans. ASAE 13(4):500 504, 507. Liang, T., D.K. Lewis, J.K. Wang, and G.E. Monroe. 1971. Random function mo deling of macadamia nut removal by multiple frequency vibration. Trans. ASAE 14(6) : 1175 1179. Lybas, J., an d M. A. Sozen. 1977. Effect of beam strength ratio on dynamic behavior of reinforced concrete coupled w alls R eport SRS No. 444, University of Illinois, Urbana Champaign. Manfredi, E., and R. Peters. 1999. Harvesters and threshers: Root crops. In CIGR handbook of agricultural engineering Vol.III Plant production engineering, 381 408. International commission of a gricultural engineering ed. St. Joseph, Mich.: ASAE. Markwardt, E.D., R.W. Guest, J.C. Cain, and R.L. Labelle. 1964. Mechanical cherry harvesting. Trans. ASAE 7(1): 70 74, 82. Marler, R.T., and J.S. Arora. 2004. Survey of multi objective optimization meth ods for engineering. Struct. Multidisc. Optim. 26 : 369 395. Mayer, H. 1987. Wind induce d tree sways. Trees 1: 195 206. McK enzie W. M.,and H. Karpovich. 1968. The frictional behavior of w ood. Wood Science and Technology Vol. 2: 139 152. Messac, A., A. Ism ail Yahaya, C.A. Mattson. 2003. The normalized normal constraint method for generating the Pareto frontier. Struct. Multidisc. Optim. 25: 86 98. PAGE 136 136 Miller, W. M., and C.T. Morrow.1976. Vibrational characterization of the apple stem system with respect to stem separation. Trans ASAE 1 9(3): 409 411,416. Moore, J. R. 2002. Mechanical behavior of coniferous trees subjected to wind loading. Ph.D. dissertation, Oregon, USA: Oregon State University. Myers, R.H., D.C. Montgomery. 2002. Response surface methodology pr ocess and product optimization using designed experiments 2nd edn., New Work: Wiley. Brien M., B. F. Cargill, and R. B. Fridley. 1983. Principles and Practices for Harvesting and Handling Fruits and Nuts Westpo r t, Connecticut: The Avi Publishing Company, Inc. Par chomchuk, P., and J.R. Cooke.1972. Vibratory harvesting: an experimental analysis of fruit stem dynamics. Trans. ASAE 15(4) : 598 603. Pareto, V. 1906. Manuale di Economica Politica, Societa Editrice Libraria. Milan; translated into English by A.S. Schwier as Manual of Political Economy, edited by A. S. Schwier and A. N. Page, 1971. New York: A.M. Kelley. Park, J. and I. W. Sandberg. 1991. Universal approximation using radial basis function networks. MIT, Neural Computation 3 : 2 46 257. Park, Y.J., and A. H. S. Ang.1985. M echanistic seismic damage m odel for reinforced c oncrete. Journal of Structural Engineering, ASCE Vol. 111, No. 4, pp.722 739. Pedersen, P., and C.L. Laursen. 1983. Design for minimum stress concentration by fini te elements and linear programming. Journal of structural mechanics .Vol. 10, No.4 pp. 376 391. Pestel, E. C. and F. A. Leckie. 1963. Matrix methods in elastomechanics New Yor k, NY: McGrawHill Book Company. Peterson, D. L. 1998. Mechanical harvester for processed oranges. Applied Eng. In Agric 1 4(5): 455 458. Peterson, D. L., T. Liang, and A. L. Myers. 1972. Feasibility study of shake harvesting macadamia nuts. Int. Conf.Trop.Subtrop.Agric., ASAE spec. Publ. SP 01 72 Phillips A. L., J.R., Hutchinson, an d R. B. Fridley. 1970. Formulation of forced vibration of tree limbs with secondary branches. Trans. ASAE 13(1) 138 142. Powell, G.H., an d R. Allahabadi. 1988. Seismic damage prediction by deterministic methods: concepts and p rocedures. Earthquake Engineer ing and Structural Dynamics .16( 5 ): 719 734. PAGE 137 137 Roka, F., J. Burns, J. Syvertsen, and R. Ebel. 2008. Benefits of an abscission a gent in mechanical h arvesting of c itrus. Doc. n o. FE752. Food and Resource Economics Departme nt.IFAS. University of Florida. Roufaiel, M.S.L., and C Meyer.1987. Analytical m ode ling of hysteretic b ehavior of R/C f rames. Journal of Structural Engineering 113( 3 ): 429 444. Ruff, J. H., R. P. Rohrbach, and R. G. Holmes. 1980. Analysis of the air suspension stem vibration strawberry harvesting concept. Trans. ASAE 23(2): 288 297. Rumsey, J. W. 1967. Response of the citrus fruiting system to fruit removing actions. Rumsey, T. R. 1967. Simulation of forced vibration of a tree limb by finite element Santos, J. L. T., K. K. Choi. 1989. Integrated computation consideration for larger scale structural design Se ns analysis and optimization. Discretization methods and structural optimization procedure and applications, H.A. Eschenauer and G Thierauf), pp. 22 9 307, springer Verlag, Berlin. Savary. S. K. J. U. 2009. Study of the force distribution in the citrus canopy during harvest using cont inuous canopy shaker. MS thesis. Gainesville, Florida: University of Florida, Department of Agricultural and Biological Engineering. Savary, S. K. J. U., R. Ehsani M. Salyani, M. A. Hebel ,and G. C. Bora. (2011). Study of force distribution in the citrus t ree canopy during harvest using a continuous canopy shaker. Computers and Electronics in Agriculture 76(1): 51 58. Savary, S. K. J. U., R. Ehsani J. K. Schu eller and B. P. Rajaraman (2010). Simulation study of citrus tree canopy motion during harvesting using a canopy shaker. Trans. ASABE : 53(5): 1373 1381. Schertz, C.E., and G.K. Brown. 1968. Basic considerations in mechanizing citrus harvest. Trans. ASAE 11(3): 343 346. Schuler, R.T., and H. D. Bruhn. 1973. Structurally damped Timoshenko beam theory applied to vibrating tree limbs. Trans. ASAE 15(5): 886 889. Spann T. M., and M. D. Danyluk 2010. Mechanical h arvesting increases leaf and stem debris in loads of mechanically harvested citrus fruit. Horts cience 45(8): 1297 1300. Stadler, W. 1988. Fundamentals of multicriteria optimization In: Stadler, W. (ed.) Multicriteria Optimization in Engineering and in the Sciences, pp. 1 25. New York: Plenum Press. PAGE 138 138 Tompson, W.T. 1993. Theory of Vibration with app lication 3rd Ed. Englewood. Cliff. New Jersey: Prentice Hall. Upadhyaya, S. K., and J. R. Cooke. 1980. Limb impact harvesting: Part III. Model studies. Trans. ASAE 24(4):868 871, 878. Upadhyaya, S. K., J. R. Cooke, R. A. Pellerin, and J. A. Throop. 1980a. Limb impact harvesting: Part II. Experimental approach. Trans. ASAE 24(4):864 867. Upadhyaya, S. K., J. R. Cooke, and R. H. Rand. 1980b. Limb impact harvesting: Part I. Finite element analysis. Trans. ASAE 24(4):856 863. Veletsos, A. S., and N. M. Newmark .1960. Effect of inelastic b ehavior on the response of s imple systems to Earthquake m otions. Proceedings of the 2nd World Conference on Earthquake Engineering vol. 2, pp. 895 912 Viana F. A. C., R. T. Haftka, V. Steffen Jr. 2009. Multiple surrogates: ho w cross validation errors can help us to obtain the best predictor. Struct. Multidisc. Optim 39:439 457. Viana, F. A. C. 2010. uide : http://sites.google.com/site/ fchegury/surrogatestoolbox. Wangaard, F. F. 1950. The mechanical properties of wood New York: Wiley. Whitney, J. D. 1968. Citrus fruit removal with an air harvester concept. Proc. Fla. State Hort. Soc 81:43 47. Whitney, J. D. 1995. A review of citrus harvesting in Florida. Trans. of the Citrus Engineering Conference, Florida Section, ASME 41:33 59. Whitney, J. D. 1999. Field test results with mechanical harvesting equipment in Florida oranges. Applied Eng. in Agric 15(3):205 210. Whitney, J. D., and J.M. Patterson. 1972. Development of a citrus removal device using oscillating forced air. Trans. ASAE 15(5):849 855, 860. Whitney J. D., and T. A. Wheaton. 1984. Tree spacing affects citrus fruit distribution and y ield Proc. Fla. State Hort. Soc 97:44 47. Wilson, T. R C. 1932. Strength moisture relations for wood. U.S. Department of Agriculture Technical Bulletin, Number 282. Yung, C. and R.B. Fridley. 1975. Simulation of vibrations of whole tree system using finite elements. Trans. ASAE 18(3): 475 481. PAGE 139 139 BIOGRAPHICAL SKETCH Susheel Kumar Gupta was born in 1985, to Daya Shankar Gupta and Geeta Gupta in Uttar Pradesh, India. He received his Bachelor in Engineering from Madhav Institute of Technology and Science (M.I.T.S), Gwalior, India in 2007. He worked as a design eng ineer at Larsen and Toubro Limited from 2007 2011. He joined University of Florida in fall 2011 to pursue Master of Science degree in mechanical and aerospace engineering He received his MS degree in August 2013. 