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The cutting behavior of bandsaws Lehmann, Bruce 1993

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THE CUTTING BEHAVIOR OF BANDSAWSByBRUCE FREDRIK LEHMANNB.A.Sc, The University of British Columbia, 1983M.A.Sc, The University of British Columbia, 1985A THESIS SUBMITTED AS PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Mechanical EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© Bruce Fredrik Lehmann, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the Universityof British Columbia, I agree that the Library shall make it freely available for reference and study. Ifurther agree that permission for extensive copying of this thesis for scholarly purposes may be granted bythe head of my department or by his or her representatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without my written permission.Department of Mechanical EngineeringThe University of British ColumbiaVancouver, CanadaOctober 7, 1993The Cutting Behavior of BandsawsBruce F. LehmannAbstractA model of a bandsaw, subjected to lateral cutting forces on the teeth and restrained by the sawnsurfaces of the wood, has been developed. The blade model includes the effects of bladedimensions, bandmill strain, in-plane stresses, tooth (gullet) depth, tooth bending stiffness, bladespeed, strain system parameters, and the span between the guides. The lateral cutting forces alongthe length of the cut were found to have dominant low frequency components and are modelledby functions having spectra that are inversely proportional to the frequency.The results of the model simulation show that when there is little or no contact between the bodyof the blade and the sawn surfaces, the sawing accuracy is governed by the tooth-tip stiffness andthe magnitude of the lateral cutting forces. When the clearance gap is small compared to the bladedeflection, the contact forces dominate and poor cutting accuracy results because the blade cannotrecover quickly from disturbances. The clearance gap between the blade and a sawn surface isshown to be less than the side clearance of the teeth because of sawdust spillage and surfaceroughness. A formula is developed that defines how blade stiffness, the clearance gap, and thecutting forces affect sawing deviation. It is found that the tooth-tip stiffness is the blade parameterthat most significantly affects cutting accuracy. An example of determining the optimal sideclearance and some practical implications of the results are presented.iiiTABLE OF CONTENTSABSTRACT ..^ .TABLE OF CONTENTS^ iiiLIST OF FIGURES ..LIST OF TABLES ..^ viiNOMENCLATURE .. viiiACKNOWLEDGMENTS^ xii1. INTRODUCTION 1ObjectivesMethod and ScopeBackground and Previous Research2. INTERACTIONS OCCURRING DURING CUTTINGObservations from Cutting ExperimentsEquilibrium Deflection of a Bandsaw BladeSawmill ConditionsOther Practical ConsiderationsConcepts and Assumptions..^173. MODEL GENERATION AND ANALYSIS PROCEDURES ^.. 29The Model of the Bandsaw BladeThe Contact between the Blade and the Sawn SurfacesRepresentation of the Lateral Cutting ForcesSimulation of the Sawing System4. NUMERICAL RESULTS ..^ .. 49The Effect of Blade Parameters on Tooth-tip StiffnessCutting through a Simulated KnotResponse to Fractal Lateral Cutting ForcesThe Contact GraphThe Effect of the Depth of Cut5. DISCUSSION AND APPLICATIONS^ .. 90Minimization of Fibre LossThe Practice of SawingComplete Data Set6. CONCLUSIONS ..^ .. 97ivLIST OF REFERENCES .. 102APPENDICESA Inertial and Gyroscopic Effects on Blade Behavior^ .. 105B Verification of Blade Stiffness and Natural Frequency Calculations^.. 107C Contact Algorithm^ .. 111D Derivation of the Statistical Representation of the Cutting Forces^.. 120LIST OF FIGURESV1-1 General Arrangement of a Bandsaw 51-2 The Geometry of the Tooth and the Tooth Marks 132-1 A Schematic of the Sawing System^.. 182-2 Experimental Arrangement for Cutting Tests .. 212-3 Blade Deflection during Cuttinga) Displacement at the Front and Back Probesb) Interpolated Position of the Blade222-4 Spectrum of the Blade Deflection 243-1 Idealized Model of the Blade .. 303-2 In-plane Stresses in a Bandsaw Blade .. 333-3 Free Body Diagram of the Blade and a Tooth 363-4 Blade Deflection During Cutting: Example 1 ..a) Displacement at the Front Probeb) Spectrum of the Displacement at the Front Probe413-5 Blade Deflection During Cutting: Example 2 ..a) Displacement at the Front Probeb) Spectrum of the Displacement at the Front Probe..^423-6 Locations of the Contact Nodes on the Blade 463-7 Locations of Nodes on the Sawn Surfaces ^.. 474-1 Effect of Blade Parameters on Mid-span Tooth-tip Stiffness 514-2 Deflected Shape of the Blade for a Load on the Front Edge of the Blade 544-3 Simulated Cutting Forces near a Knot.. 574-4 Simulated Cut Path around a 'Knot'^.. 584-5 Blade Deflections and Contact Locations for a Simulated Cut 594-6 The Deflected Shape of the Blade during a Simulated Cut^.. 614-7 Effect of Blade Thickness on the Cut Path^.. 624-8 Effect of Bandmill Strain on the Cut Path^.. 63clis►unnig vu iuc l.ul raw ..4-10 Effect of Clearance Gap on the Cut Path ..^66vi4-11 Effect of Blade Parameters on Cutting Deviation ^.. ..^674-12 Blade Deflections and Contact Locations for a Simulated Cuta) Example 1^.. 69b) Example 2^.. 70c) Example 3^.. 714-13 Effect of Clearance Gap on Cutting Deviation 724-14 Probability Curve for Contact .. 764-15 Contact Graph .. 784-16 Normalized Cross-Section of Blade Deflectiona) Effect of Tensioningb) Effect of Thicknessc) Effect of Bandmill Straind) Effect of Blade Width804-17 The Deflected Shape of the Blade during a Simulated Cut 824-18 Progression of Contact 834-19 Effect of Load Variation Through the Depth of Cut .. 854-20 Effect of Depth of Cut on the Equivalent Stiffnesses 874-21 Effect of Depth of Cut on the Contact Graph .. 895-1 Effect of Side Clearance on Total Fibre Loss .. 92B-1 Analytical and Experimental Blade Stiffness .. 109B-2 Analytical and Experimental Normalized Blade Deflection 110C-1 Geometry of the Constraining Surfaces 115C-2 Geometry of the Interpolation for the Contact Algorithm 117viiLIST OF TABLES1-I^Typical Cutting Forces^ 112-I^Conditions for Experimental Cutting Tests^ 204-1^Parameters for the Blade Model^ 52A-I^Inertial, Gyroscopic and Stiffness Terms^ .. 106NOMENCLATUREa^height of the tooth marksA^flexibility (inverse of stiffness)[A]^flexibility matrixb^blade width; bite per toothc^blade speedd^tooth depth; difference between s and gD^plate bending stiffness = Eh 3/12(1-v2)Dc^depth of the cute^component of the cutting force about the mean force; width of tensile stress zoneE^Young's modulusEmn(y)^shape functions of blade deflectionf^frequency in cycles per second; force acting on a toothf^mean lateral cutting forcefc^filter cutoff frequencyF^lateral force on the bladeFc^net forces acting on fixed degrees of freedomFe^force on a free degree of freedomFi^lateral cutting force at the i-th location along the length of the cutFL^lateral cutting forceFn^external forces acting on the n-th degree of freedomg^clearance gapGFI^gullet feed index[G]^gyroscopic matrixh^plate thicknessviiik^stiffnessix[K]^stiffness matrixKeq^equivalent tooth-tip stiffnessKo, Q0^effective tooth-tip stiffnessesKt^bending stiffness of a toothKtt^tooth-tip stiffness of a single toothL^span between the guidesLc^length of the cutm^average cutting force at an instant in time[M]^mass matrixNc^number of increments along the cutNt^number of teeth in the cutNY^number of contact nodes across the width of the bladep^planer allowanceP^tooth pitch{P}^Load vector for Gal erkin formulationq(x,y,t)^lateral loading per unit arear^distance from the center of a knots^side clearance{S}^Generalized coordinates for Galerkin formulationSit^cut path at the i-th increment into the cut for the t-th toothSo^standard deviation of blade deflection when no contact occursSe^standard deviation of eSf^standard deviation of the cutting forcesSm^standard deviation of mST^standard deviation of blade deflection when contact occursT^bandmill strain (axial preload) Un,Vn^constraint locationsw(x,y,t)^lateral blade deflection(x,y,z)^coordinates on the blademean tooth deflection for a cutxc^deflection of a free degree of freedomxf^deflection of a fixed degree of freedomXT^tooth-tip deflectionz^a Gaussian random number; probability variableZo^nondimensional clearance gapa^factor relating ST to oToutward shift of the constraintsimbedment of a nodeov,ou^imbedmentsside clearance anglea small lever armstrain system parameterwave length; eigenvaluemass per unit areav^Poisson's ratiop^density013^standard deviation of the between-board thickness; stress due to in-plane bendingac^stress in center of the blade due to tensioningaG^gullet stressaR^roll tensioning stress at the edges of the bladeow^standard deviation of the within-board thickness°T^standard deviation of the total board thickness; stress due to bandmill strainax^axial stresses a stationary bladeT, AT^distance along the cut, and its incrementhardness functioncontact force at a nodefrequency in radians per secondGalerkin shape functionxixiiACKNOWLEDGMENTSI would like to acknowledge the support provided by the Science Council of British Columbiathrough its G.R.E.A.T Award program, in which MacMillan Bloedel Ltd and CAE Machinery Ltdacted as industrial cooperators and sponsors.I would also like to thank the following people for their support, advice and friendship: Dr StanleyHutton, my supervisor, Dr Gary Schajer, John Taylor, Leonard Valadez of California Saw andKnife, and Mr Pat Crammond of CAE Machinery....the most lasting contribution to the growth of scientific knowledge that atheory can make is the new problems which it raises, so that we are led backto the view of science and the growth of knowledge as always starting from,and always ending with, problems - problems of ever increasing depth, andever increasing fertility in suggesting new problems.Sir Karl PopperThe Growth of Scientific KnowledgeCHAPTER 1INTRODUCTIONBandsaws have been used to cut wood since the middle of the nineteenth century. Bandsaws aremore useful than circular saws for deeper cuts because thinner blades can be used, and hence, lesswood is lost to sawdust. Until 1970 there was little interest in North America in improving sawingaccuracy or in increasing the amount of solid wood recovered from a log because timber wasrelatively cheap. At present, in 1993, that situation does not exist; the cost of raw material,delivered to the sawmill, now accounts for fifty to eighty percent of the total production costs.Low lumber recovery factors are no longer acceptable. In addition, many purchasers aredemanding that sawn lumber (i.e., unplaned) has a smooth finish and be manufactured to specifieddimensional tolerances. Lastly, as ecological concerns becomes more important to society, thecosts of timber and harvesting are likely to increase.There are two categories of fibre loss. The first is the kerf loss, which is the volume of woodremoved by the teeth. The second is called the planer loss, which is the extra thickness allowanceadded to the sawn board thickness so that the board, when planed, will be completely surfaced onall slues. I he total loss is the sum of the kerf and planer losses.2One method for reducing the fibre loss is to use thin saws to reduce the kerf loss. However,thickness reduction can be carried only so far before the blade is weakened to the point whereaccurate sawing is no longer possible. The resultant inaccurate sawing causes larger planer losses.Clearly, there is an optimum thickness where the total loss is minimized. However, the effect ofblade thickness on sawing deviation needs to be known for such an analysis to be conducted.Blade stiffness is expected to be one indicator of the ability of the saw to cut in a straight line, butthere are other factors such as side clearance (the amount by which the teeth protrude from thesides of the blade) which are known to affect sawing accuracy. Sawing is a very complex processwith many interacting phenomena. The trial and error development of bandsaw technology thatcontinues to this day is in part due to the lack of understanding of the mechanics involved.A primary issue that confounds sawing research is the variability of the wood. Since no two piecesof wood are the same it is impossible to repeat a cutting experiment. Statistical methods such asan analysis of variance of board thickness are available for evaluating experimental results. Thesestatistical measures do quantify known trends in saw behavior and can be used as indicators ofcertain sawing problems, but they do not help to explain the mechanics of sawing or lead topredictions of how changes in blade design will affect cutting accuracy.The problems in sawing research at this time are:1) to determine the essential variables that encapsulate the characteristics of the wood, theblade, and the performance of the cutting system as a whole.2) to learn how the wood and the blade interact during sawing and to express theseinteractions with the above variables.3) to determine how cutting performance is affected by the characteristics of the woodand the blade.1 o solve these problems, the mechanics of sawing need to be described, understood and, where3possible, quantified.OBJECTIVEThe objective of this thesis is to investigate the mechanics of sawing wood with a bandsaw with aview to predicting how blade and operating parameters affect cutting accuracy. Specifically, theeffects of blade stiffness, the character of the cutting forces, and the side clearance on bladebehavior and cutting accuracy will be investigated.METHOD AND SCOPEBecause of the problems inherent in doing repeatable cutting experiments, this study is basedupon a simulation of the sawing system. Simulation makes possible a systematic exploration of themechanics of sawing. The danger of using simulation, especially on a system that has not beenthoroughly studied before, is that important effects that cannot yet be expressed mathematicallyare ignored. To mitigate this shortcoming, a broad approach is taken in the literature review, inthe initial investigations, and in the interpretation of the results.The approach taken in this thesis consists of the following steps:a) to identify, via a literature review and empirical observation, the interactions that occurbetween the components of the sawing system,b) to construct a numerical model of the sawing process that includes the dominantinteractions,c) to investigate the mechanics of blade behavior by simulating cutting through prescribed4disturbances,d) to discuss the mechanisms, and their relative importance, that explain why changes inblade or operating parameters affect cutting accuracy, ande) to compare trends predicted by the model to known or experimental results and tosuggest future work that could improve the predictive ability of the model.The investigation and discussion concentrate on the effects of blade stiffness and side clearancebecause they are found to dominate sawing behavior.BACKGROUND AND PREVIOUS RESEARCH Few aspects of the process and practice of sawing are precisely quantified. The information in theliterature on bandsaws is not yet comprehensive enough to be applied in practice. Much anecdotalinformation on mill practice is available, but one must keep in mind that the effect of certaincommonly recommended procedures is to compensate for, but not to correct an underlyingproblem. The result is that cause and effect are confused. This is probably one reason why achange in blade or operating parameters improves sawing behavior on one bandmill, but has noeffect or is detrimental on another. All of these factors must be considered when reviewing sawingliterature.For the purpose of presenting the literature on bandsaws and bandsawing, the information ispartitioned into six areas of interest.1. BANDSAW OPERATION AND PREPARATIONA bandsaw blade is a strip of steel with teeth along one or both edges that is welded to form aWHEEL CROWNOVERHANGSTRAINCYLINDER6continuous loop. This band runs on two wheels, one of which can translate to apply an axialpreload to the blade. See Figure 1-1. The force pushing the wheels apart is termed the "bandmillstrain". Adjustments are also available for changing the relative tilt angles and axial alignment ofthe wheels. The wheels are generally crowned to ensure the tracking stability of the band. The tiltadjustment is used to set the amount the teeth overhang the edge of the wheels. Guides arelocated just above and below the cutting region to provide support for the blade in the cuttingarea.To improve the blade stiffness and tracking stability, residual stresses are put in the blade byplastically deforming the blade between two narrow pressure rollers. This process is called"tensioning" and a blade with residual stresses is said to be "tensioned". The result is that beforestraining, the edges of the blade have a tensile residual stress while the center is in compression.Additionally, the residual stresses are purposely made to be asymmetric so that the back of theblade is longer that the front edge. The blade is then said to have "backcrown"; the purposes ofwhich are to increase the stiffness of the front edge and to compensate for thermal expansion ofthe front edge resulting from the heat generated during cutting.2. EVALUATION OF SAWING PERFORMANCEThe quality control method developed to monitor the sawing process is important to sawingresearch for two reasons. Firstly, the method forms a basis, but not a complete method, forevaluating experimental data. Secondly, the method separately evaluates, to some extent, howwell the saw is performing and the condition of the feed system.The evaluation is an analysis of thickness variation of typically fifty  boards [Brown, ed., 1982].The thickness of the boards is measured at six places on each board. These data are used to findthe mean thickness, the within-board variation, owl, and the between-board variation, og2 (thevariation of the mean board thicknesses). The total variation, o', is the sum of the within and7between-board variations. Generally, the within-board variation is considered to be a measure ofthe alignment and condition of the blade, the condition of the wood (knots, pitch, frozen, etc.) orthe looseness of the feed system. The between-board variation is considered to be a measure ofthe positioning accuracy and repeatability of the setworks (the mechanism that positions the saw).3. STATICS, DYNAMICS AND BUCKLING OF BANDSAW BLADESPrevious researchers have based their analyses of the handsaw blade on a beam or a plate modelthat is simply supported at both guides and free on the other two edges [Mote, 1965, 1968;Foschi and Porter, 1970; Ulsoy and Mote, 1980, 1982; Taylor, 1985; Wang and Mote, 1985].These models include the effects of bandmill strain (axial preload), the speed of the blade, and rolltensioning. The blade parameters are the blade width, the plate thickness and the span lengthbetween the guides. A further factor that affects the bandsaw is the mechanical method used tosupport the top wheel and apply the strain. Mote [1965] developed a non-dimensional stiffnessparameter, K, which varies between 0 and 1. For a deadweight strain system there is no stiffness inthe top wheel, so it is free to move. In this case lc = 0. For a fixed top wheel, the stiffness of thestrain system is infinite, and K = 1.The governing equation of an axially moving band is [Mote, 1965]a 2 W^a 2W^a 2wDV 4 w + 2µc^+ [A, , + K[IC 2 - hox )^— q(x,y,t)axat^a`t^ rxwhereEh 3 12( 1 — v2 )11^= mass per unit areac^= blade axial velocityh^= blade thicknessy^- dXldl Jt1CSS UIM11UU11011 At LC1U VGlUGit y8= strain system parameterq^= lateral loadingw(x,y,t) = lateral deflectionThe terms on the left-hand side represent, respectively: the flexural stiffness, the gyroscopiccoupling of the axial and lateral velocities, the inertial force, the centripetal forces and the effectof the in-plane axial stresses. The Ritz [Ulsoy, 1979], Galerkin [Mote, 1965; Ulsoy, 1979], andthe finite element [Ulsoy, 1979; Taylor and Hutton, 1991] methods have been used to solve thisequation. The types of analysis performed are the determination of natural frequencies and modeshapes, the lateral stiffness and the critical buckling loads caused by the cutting forces.The natural vibration of bandsaws has been extensively investigated, mostly because resonancewas thought to be a major cause of sawing inaccuracy [Ulsoy et al, 1978]. This is true for circularsaws, but no experimental data have been presented to justify this assumption for bandsaws. Thisis not to say that bands do not resonate, especially during idling, but no one has shown that boardthickness variation corresponds to the resonant vibration of the bandsaw.The buckling of bandsaws caused by the feed and tangential cutting forces has been studied byPahlitzsch and Puttkammer [1974, 1976], Foschi and Porter [1970], and by Ulsoy [1979]. Ulsoyet al [1978] concluded that for wide bandsaws used in sawmills, the cutting forces are too smallto cause buckling.The lateral stiffness of bandsaws has not been extensively studied. Pahlitzsch and Puttkammer[1974, 1976] and Porter[1971] used a beam model with axial loading for the bandsaw, for whichthe cross-section of the blade remains rectangular after the blade has deflected. For this model tobe valid the span must be much greater than the width of the blade. This model is appropriate forthe narrow resaw bands used by Pahlitzsch and Puttkammer for their verification tests, but may9not be appropriate for the wide bandsaws used for primary log breakdown. Taylor and Hutton[1991] used finite element analysis of a plate model of a wide blade to calculate the stiffness at thetooth-tip, the bottom of the gullet and the back edge of the blade. There was good agreementbetween the numerical and experimental results. The parameters that affected tooth-tip stiffness,in order of decreasing importance, were thickness, span length, strain and blade width.4. FATIGUE ENDURANCE OF BANDSAWSFatigue is mentioned here because the parameters that govern fatigue life, blade thickness, strainand roll tensioning, also affect the stiffness, natural frequencies and critical buckling loads. Achange in any parameter that increases stiffness, will, in general, result in a reduction in bladefatigue life.Bandsaw blades frequently develop cracks in the gullet region that sometimes cause catastrophicfailure of the blade. These cracks are known to result from metal fatigue [Junes, 1965, 1968]. Thestresses that produce cracking are of two types: the constant stresses produced by bandmill strainand roll tensioning and the oscillating stress resulting from the blade being bent over the wheels.[Allen, 1973 a, b; Hutton and Taylor, 1991]5. ACTION AT THE TEETH AND CUTTING FORCESThe cutting force acting upon the teeth may be considered as the sum of three orthogonalcomponents: the feed, tangential and lateral forces. The feed and tangential forces act in the planeof the blade and have been studied extensively because they determine power requirements[Franz, 1957; McKenzie, 1964]. The lateral force has not received as much attention.1 he primary problem encountered when investigating the cutting of wood is the variation in the10properties of wood. Szymani [1985] listed specific gravity, moisture content, and temperature asthe determinants of the cutting forces. These properties have large variations, both between andwithin species, that are apparent in growth related characteristics such as spiral and interlockedgrain, knots (irregular density and grain direction), and reaction wood. Also, sawing can releaseresidual growth stresses that could cause the wood to pinch the blade.Pahlitzsch and Puttkammer [1976] and Fujii et al [1984, 1986] measured blade deflection and thethree components of the net force acting on short blocks as they were cut by a bandsaw. Theshapes of the sawn surfaces were also measured. Both groups of researchers produced similarexperimental results. The graphs of the cutting forces and the blade deflection show that the feedand tangential forces are fairly constant during the cut and then return to zero when the cut isfinished. The lateral force, and the blade deflection graphs have similar shapes. From Pahlitzschand Puttkammer's data Ulsoy et al [1978] produced typical and range values of the cutting forces,which are reproduced in Table 1-I. Fujii et al [1984] concluded that the blade deflection increasedwith increasing lateral force and assumed that the deflection was a result of the lateral force.Pahlitzsch and Puttkammer, however, noted that at the end of the cut the lateral force and theblade deflection did not always return to zero when the teeth left the wood. The deflection andlateral force became zero only after the back edge of the blade had cleared the end of the wood.They concluded that, in this instance, the wood was restraining the blade from returning to itsequilibrium position. In other words, the lateral force measured by the dynamometer supportingthe wood block was a result of the blade deflection and that the lateral forces do not always actsolely on the teeth.St. Laurent [1970, 1971] examined the effect of sawtooth edge defects and knots on the cuttingforces. In his experiments, a single tooth was mounted on a three component force dynamometer.The three forces were recorded while a block of wood was pushed onto the tooth. The resultsshowed that small defects on the corner of a tooth can cause the lateral force to be as large as11Table 1-I. Typical Cutting Forces [Ulsoy, et al, 1978]Direction^Range^Typical tangential^100-1000 N (23-225 Lbs)^500 N (112 Lbs)feed^0-600 N (0-135 Lbs)^250 N (56 Lbs)lateral^0-25 N (0-5.6 Lbs)^15 N (3.4 Lbs)27% of the tangential force (i.e., FL = 12 lbs.). For a perfect tooth the range of values of thelateral force was only 1 or 2 lbs. In the study on the effect of knots St. Laurent measured thetangential cutting force as well as the average and maximum lateral force near or in a knot. Forsoftwoods the average lateral force was about 20-30% of the tangential force in the surroundingclear wood (i.e., 4 lbs. < FL < 10 lbs.). However, the peak lateral force was between 30-100% ofthe tangential force in the clear wood (i.e., 40 lbs. < FL < 60 lbs.).Preliminary data showing the effect of density variations and fibre direction on the cutting forceshas been presented by Axelsson et al [1991]. They used computer tomography to provide animage of the wood density and compared these images to the three components of the cuttingforce when the wood was later machined with a single tooth cutter. The in-plane forces werenoticeably affected by wood density, especially around knots, but the lateral force only showedthe effect of density when one of the tooth corners was purposely damaged. The effect of toothdamage was similar to that reported by St. Laurent [1970].The action of the teeth produces the two sawn surfaces and the sawdust. Most of the sawdust iscarried away in the gullet of the tooth but some escapes and falls into the space between the sawnsurface and the side of the blade. Sawdust spillage could result in blade heating, and, hence, inpoor cutting accuracy because the thermal stresses could reduce the stiffness of the blade.12The flow and spillage of sawdust has been studied by Chardin [1957], who used high speedphotography to photograph the tooth and gullet as the tooth emerged from the bottom of a cut.Reineke [1956] described in detail how sawdust compacted in the gullet, the conditions for theonset of spillage and the characteristics of the sawdust chips. See Figure 1-2. The followingobservations and conclusions were made:a) The density of compacted sawdust is about 70 percent that of solid wood. Therefore,the volume of solid wood to be removed by one tooth can be only 70 percent of thevolume of the tooth gullet, otherwise sawdust will be forced out of the gullet. TheGullet Feed Index (GFI) is the ratio of the area of the wood removed by a tooth (equalto the advance or bite per tooth multiplied by the depth of cut) to the area of the gullet.Hence, if the GFI is greater that 0.7, spillage should be expected.b) There is a wide range of particle sizes for any given bite per tooth. Some of these fineparticles could escape the gullet. Also, some particles could lodge in irregularities of thesawn surfaces.c) The spilled sawdust shows a large percentage of fine particles. Of the larger spilledparticles, there were thin flakes produced by the flanks of the teeth. These flakes couldeasily fall between the blade and the sawn surfaces.The flanks of the teeth sever the chips from the kerf walls. Because of the clearance angle of theteeth, each tooth leaves a mark on the sawn surfaces. See Figure 1-2. The height of this mark hastwo components: Firstly, a geometric effect due to tooth shape equal to a=bTan(y), where b is thebite per tooth and y the side clearance angle. Secondly, an effect due to the deformation of thewood fibres during the cut and their subsequent "springback". Johnston and St. Laurent [1975]measured the springback with a single tooth apparatus and found, for example, that when cutting13Figure 1-2. Geometry of the Tooth and the Tooth Marks.14wet spruce at a bite per tooth of 0.050 inches and a side clearance angle of 3 degrees, thespringback was 0.0053 inches. The geometric effect, 'a', was 0.0026 inches for this case, so theclearance gap between the blade and the sawn surfaces is reduced by a total of 0.0079 inches.Breznjak and Moen [1972] calculated the kerf width by measuring the width of the wood blockbefore cutting and then subtracting the total width of the remainder of the block and the board.They also calculated the kerf width based upon the difference in weight between the original blockand the sum of the remaining block and board. Their result was that the width of the tooth wasgreater than the kerf width based on weight difference, which was in turn greater than that basedon thickness difference. The kerf based on weight difference was about 0.007 inches smaller thanthe tooth width, and for the thickness difference method, 0.014 inches (at a bite per tooth of0.047 inches). The reduction in kerf width decreased as the bite was decreased. The authorsconcluded that surface roughness, tooth marks and fiber springback caused the reduction in kerfwidth.6. RESULTS FROM CUTTING TESTSThere have been few systematic full scale cutting tests carried out. Statistical analysis requires alarge number of cuts to be made (20-100) [Brown, ed., 1982, Allen, 1973b] under controlledconditions for each condition being tested. It is a time consuming and expensive task to conductextensive experiments. On the other hand, many boards can be cut in the sawmill, but the condi-tions are not under the control of the researcher.As a part of a larger study on bandsaw vibration during cutting, Breznjak and Moen [1972]showed that there is a relationship between log quality, expressed by the number of knots per unitsurface area of the log, and the maximum blade deflection. Taylor and Hutton [1991] used thesum of the knot diameters on the sawn surface to account for variation in experimental results.15Basically, if more knots are present then the sawing accuracy is worse.Pahlitzsch and Puttkammer [1976] measured the deflection of a bandsaw blade during the cuttingof short blocks of wood. In the traces of blade motion that were presented, the frequency of theoscillation was not greater than 2 Hz. They made the following general observation concerningrecorded blade motion:"The blade, in general, oscillated about the ideal cutting line with a relatively large periodin the feed direction and a more or less large amplitude in the direction perpendicular tothe cutting plane."The wave lengths of the oscillations were 200 mm and larger, while the lateral deflections were 0-2 mm (0-0.050 inches). Feed speeds up to 0.4 m/s (79 in/s) were used. The blade width was 100mm (3.4 inches).Allen [1973 a, b] presented data showing the effect of blade thickness, strain, and feed speed onthe within-board sawing deviation. The sawing deviation was approximately inversely pro-portional to the square root of strain. As the feed speed was increased, such that the Gullet FeedIndex increased from 0.65 to 0.90, the sawing deviation increased. In a set of the cuttingperformance tests, a 0.058 inch thick by 8 inch wide blade run at 15000 lbs. strain had the samesawing accuracy as a 0.042 inch by 7.5 inches blade running at 24000 lbs. strain, but the thinnersaw saved 0.030 in kerf.Kirbach and Stacey [1986] showed that there is a minimum critical side clearance, below which thesawing deviation increases rapidly. They ascribed the reduced sawing accuracy to the increasedlikelihood of contact between the blade and the sawn surfaces that would cause heat, and hence,thermal stresses in the blade that would reduce the blade stiffness. The example given was of a160.058 inch thick by 8 inches wide blade at 18000 lbs strain, cutting a 7.3 inches thick block ofwestern red cedar: the critical side clearance was 0.028 inches. They also noted that there is amaximum allowable side clearance. If the side clearance is larger than the bite per tooth thenexcessive amounts of sawdust will spill in to the space between the blade and the sawn surfaces, so,they postulated, that blade heating and increased sawing deviation will occur.GrOnland and Karlsson [1980] conducted extensive cutting tests with a small resaw at theSwedish National Sawmilling School, where six boards were cut from each of 850 cants. Theseresults clearly showed the undesirable effects of having a Gullet Feed Index greater than 0.7. Theeffect of small spring-setting on the teeth was found to be undesirable, as Kirbach and Staceyshowed for swaged teeth. Kirbach [1985] presented experimental results that showed the within-board deviation beginning to increase when the GFI is between 0.5 and 0.6, and increasing veryrapidly when the GFI reaches 0.7.A set of tests conducted in the University of British Columbia Sawing Laboratory confirmed thetrend that reduced side clearance increases sawing deviation. For the smallest side clearancetested, the blade cut fairly straight for the first 18 inches then suddenly deflected more than 1 inchover a distance of 6 inches, after which the blade left the wood. It seems unlikely that thetemperature of a blade that is cooled by the guide lubricant water and by the convection coolinggenerated by a blade moving at 9400 fpm could increase so much in the half second the blade wasin the cut. Also, the power required to heat the blade to temperatures that would cause asignificant loss in blade stiffness is greater than the power of the saw motor. This observationdoes not disprove that heat is a factor in causing the wild cutting, but it does indicate thepossibility of other mechanisms.17Research is to see what everybody else has seen, and to think what nobodyelse has thought.Albert Szent-GyorgyiNobel LaureateImagination is a good servant, and a bad master.Agatha ChristieThe Mysterious Affair at StylesImagination is more important than knowledge.Albert EinsteinCHAPTER 2INTERACTIONS AND PROCESSES OCCURRING DURING CUTTINGThe purpose of this chapter is to generate the concepts and assumptions that will form the basis ofthe mathematical model of the sawing process developed in this thesis. Also included is adiscussion of the conditions under which a bandsaw must operate and the consequences of theassumptions used. This is accomplished by presenting a systematic view of sawing based oninformation in the literature, and, where necessary, by conducting experiments to fill any gaps inthe information.The sawing system is composed of a number of entities that interact with one another. See Figure2-1. The entities are:.4•40510 IWO/ AKA,^ ....... AI/CONTACT \^ SOMEFORCES SAWDUSTSPILLEDFROMGULLETSLATERALCUTTINGFORCESSAWN SURFACESWOOD^  PROPERTIESCREATINGTHE SAWNSURFACESFEED SYSTEMSAW BLADEFORCES ON TEETH DEFLECTS THE BODY BODY <^ >TEETHFORCES ON BODY DEFLECTS THE TEETH18Figure 2-1. A Schematic of the Sawing System.191) the wood, which includes the sawn surfaces,2) the feed system that carries the wood through the saw,3) the saw blade, which includes the body of the saw as well as the teeth, and4) the sawdust.The interactions and processes in which the entities participate are:1) the lateral cutting forces that act between the wood and the tooth,2) the creation of the sawn surfaces by the teeth,3) the spillage of the sawdust from the gullets,4) the contact forces caused by the body of the blade touching the sawn surfaces5) the contact forces caused by the sawdust coming between the blade and the sawnsurfaces,6) the support of the wood provided by the feed system, and7) the structural connection between the body of the blade and the teethOf the entities, processes and interactions listed, only the saw blade has been previously modelledaccurately. The definition of the sawn surfaces can be understood as the instantaneous location ofthe teeth, although some allowances must be made for surface roughness and springback. Theapproximate range of the magnitude of the lateral cutting force is known, but not the details ofhow it varies along the length and through the depth of the cut.Although some information is available on sawdust spillage, there are so many variables involvedthat few general statements can be made about the amount and eventual location of the spilledchips. Spillage is known to increase when the Gullet Feed Index becomes greater than 0.6. Morespilled sawdust ends up near the bottom of the cut because this is where the gullet becomes fulland because the saw drags chips downward. There is no literature that quantitatively describes the20rigidity or the accuracy of travel for the various feed systems used in sawmills.The effect of the contact between the sawn surfaces and the blade on blade equilibrium is onlybriefly commented on by Ulsoy et al [1978]. All other articles and books consider this interactiononly as a source of blade heating.OBSERVATIONS FROM CUTTING EXPERIMENTSOne of the gaps in the knowledge of sawing is the effect of the contact between the sawn surfacesand the blade. To investigate this and other effects, the motion of the blade during cutting wasexamined. A bandsaw was instrumented with non-contacting displacement probes, thearrangement of which is shown in Figure 2-2. Figure 2-3(a) is a typical set of traces of the motionof the blade measured simultaneously near the front and back edges of the blade. The total time ofthe cut was 1.20 seconds. Note that the measurements were made above the cut: the deflection ofthe blade in the wood could be as much as 50% greater than that measured at the probes. Theblade dimensions and the operating conditions are given in Table 3-I.Table 2-I. Conditions for Experimental Cutting TestsStrain^15000 lbsBlade thickness 0.065 inchesBlade width^9.375 inchesTooth width^0.140 inchesFeed speed^200 fpmBite per tooth^0.037 inches21Figure 2-2. Experimental Arrangement for Cutting Tests.z0R-0 0.020La_aLa 0.010.220^10^20^30^40^50^60^70^80DISTANCE ALONG THE CUT (INCHES)Figure 2-3. Blade Deflection During Cutting a) Displacement at the Front and Back Probesb) Interpolated Position of the Blade23The spectrum of the trace of the front probe is given in Figure 2-4. The lowest natural frequenciesof the blade correspond to the peaks at 60 Hz. Although the blade is vibrating before and after thecut (see Figure 2-3), the natural vibration is damped out during cutting. The peaks at 10 and 20Hz. correspond to the wheel rotation frequency and its first harmonic. The dominant bladedeflection occurs at frequencies less than 5 Hz.The first observation is that the low frequency content dominates the character of the signal. Thecutting deviation is occurring at frequencies less than 5 Hz, as compared to the lowest naturalfrequency of 60 Hz. One can, therefore, conclude that blade stiffness, not blade dynamics, is themost significant factor in determining cutting deviation. A comparison of the relative effects of theinertial, gyroscopic and stiffness forces is given in Appendix A. It is shown that for excitationfrequencies below 30 Hz. the stiffness effect dominates. Resonant behavior of the blade does notaffect the cutting deviation.A second observation is that just after the front of the blade deflects the back of the blade will alsodeflect. In fact, the signal from the back of the blade is a delayed version of the signal from thefront. The delay approximately equals the distance between the probes. To aid visualization ofhow the blade deflected in the cut, the blade position at several points during the cut wasextrapolated from the deflection traces. This is shown in Figure 2-3(b). (It will be seen in Chapter4 that the cross-section of the blade does not stay straight when the blade deflects, so the straight-line extrapolation is not exact.)In two regions of the cut, the deflection of the back of the blade is greater than that of the front. Itis not possible that the cutting forces acting upon the teeth could, by themselves, produce thisdeflection. There must be lateral forces acting on the back half of the blade. The only entity thatcould provide such a force is the sawn surface, which acts as a geometric constraint on the motionUI IC UTAUC.1240.010to0.008LirzZ 0.006tl0.0040.0020^10^20^30^40^50^60^70^80^90^100FREQUENCY (HZ)Figure 2-4. Spectrum of the Blade Deflection.25The body of the blade must follow in the path made by the teeth. If the teeth deflect by an amountgreater than the side clearance, it is certain that the blade will contact the sawn surfaces at somepoint during a cut.EQUILIBRIUM DEFLECTION OF A BANDSAW BLADESince contact is occurring between the blade and the sawn surfaces, the deflection of the blade isnot solely the result of cutting forces acting upon the teeth. It has also been shown that the bladedynamics are not involved in the sawing accuracy of a bandsaw, although blade vibration couldaffect surface roughness. These observations lead to the conclusion that at every instant in time,or at every point along the cut, it may be assumed that the cutting force, the contact forces andthe resistance of the blade to deflection are in quasi-static equilibrium. This assumption applieseven though the lateral cutting forces will vary as the blade advances into the cut. The resultingdeflection of the tooth-tips defines the newest addition to the cut path.SAWMILL CONDITIONS The operating conditions imposed on a bandsaw in a sawmill are much more adverse than thoseseen in a laboratory setting, primarily because the condition and alignment of the machinerycannot be maintained so precisely. As mentioned above, the rigidity and alignment of the feedsystem are important. There are four basic types of feed systems, each having their owncharacteristic problems.1) The carriage system is the most rigid because the log is held at several points, but it isnot economical for small diameter logs.262) For the sharp chain system, the log is carried on a chain running in a guide way and heldon the chain by press rolls. The alignment of this system is difficult because the chain isgenerally loose in its guide way. In addition, the press rolls do not stop the log fromrolling laterally as it is fed into the saw, which results in the log leaning against theblade.3) In the line-bar system a sawn edge is pressed against a flat plate and/or a bank ofrollers. Problems arise if the infeed and outfeed are not aligned to each other, if therollers are misaligned, worn or loose, or if the press rolls are applied after the cut hasstarted or released before the wood has cleared the saw. In any event the wood willjump and not have a straight motion through the saw. In practice, the saw oftenmitigates these jumps.4) In the end-dogging system the log is held at its ends, and then travels on a carriagearrangement. The middle of the log is unsupported and will flex as residual stresses inthe log are released by the sawing. This problem also applies to the sharp chain feeds.Whichever system is used, there is the likelihood that the wood will move independently relativeto the blade and even press against the blade.The blade can be damaged by rocks and pieces of metal that are occasionally found in a log.Damage can also occur if the bandmill is still being set when the cut starts. The blade must berugged enough to survive these incidents. Allowances must also be made in designing a blade forvariations in the skill of the sawfilers and of the sawyers.The preceding discussion on the effects of sawmill conditions give rise to two conclusions. Firstly,saw blades may have to be oversize to limit damage or to make the blade insensitive to variations in27their preparation and operating conditions. Secondly, the predictive ability of a model of the sawingprocess may depend upon how well these non-ideal conditions are represented. This is speciallytrue concerning the rigidity and alignment of the feed system.FURTHER PRACTICAL CONSIDERATIONS1) The blades are never perfectly flat. Peak to peak lateral runout is typically between 0.002 to0.005 inches.2) Because of the bending moments generated when the blade bends over the guides and thewheels, the span between the guides does not form a plane; instead, the blade bows outward.Also, because of the backcrown and wheel tilt, the front edge of the blade bows out less thanthe back edge. Hence, at mid-span, the blade is not parallel to the carriage tracks. The cross-section of the blade is curved, presumably because of the combined effects of tensioning, anti-elastic bending and the stretching of the band over the wheel crown. These distortions from theplane are small: typically 0.002 inches maximum.3) The teeth are frequently not aligned in the lateral direction. Either the teeth have more clearanceon one side on the blade than the other, or the clearance is not uniform from tooth to tooth.The above non-ideal aspects of the blade may have measurable effects on the contact between theblade and the sawn surfaces, either because the clearance gap is reduced or is less on one side ofthe blade than the other.4) The teeth not only overhang the wheels, they also overhang the guides, so that the blade is not e gu e me. oo muc over ang (more than 0.5 inches) has a•^I^•^1,11^'I 1128noticeable effect on tooth-tip stiffness and cutting accuracy.CONCEPTS AND ASSUMPTIONS The representations of the lateral cutting forces and the contact between the blade and the woodwill be developed in the next chapter. With these representations it is possible to construct anumerical model of all the important entities, processes and interactions present in sawing.The following assumptions were made when constructing the model of the sawing system:1) the feed system and the wood are perfectly rigid,2) the blade is simply supported at the guides and it lies in a plane between the guides,3) the process of cutting is quasi-static, so that inertial and gyroscopic effects on the bladecan be ignored,4) the effects of heat generated when the blade rubs against the sawn surfaces or from thecutting process will not be included,5) the clearance gap in the model will not be equal to the side clearance of the teeth becauseof tooth marks, springback and spilled sawdust, but it will be the same on both sides ofthe blade and constant through the depth of the cut, and,6) the behavior of the blade between the guides is isolated from other parts of the bandmilland the rest of the blade.29The sciences do not try to explain, they hardly even try to interpret, theymainly make models. By a model is meant a mathematical construct which,with the addition of certain verbal interpretations, describes observedphenomena. The justification of such a mathematical construct is solely andprecisely that it is expected to work.John Von NeumannCHAPTER 3MODEL GENERATION AND ANALYSIS PROCEDURESIn this chapter the numerical models representing the bandsaw blade, the contact between theblade and the sawn surfaces and the lateral cutting forces are developed. These representationsare then combined to form the model of the cutting process.THE MODEL OF THE BANDSAW BLADEThe blade is assumed to be a flat plate, simply supported at the guides and free on the other twoedges. The governing equation (repeated from Chapter 1) isa 2wDV 4w + 2[1c a2w^d2w + (cttc 2 — hox ) ^ — q(x,y,t)axat^32 t^a`x (3.1)The coordinate system, which is fixed space, and the blade geometry are shown in Figure 3-1. Theboundary conditions are no moments or shears forces on the free edges; expressed as30Figure 3-1. Idealized Model of the Blade.a2wa2y ^)M = 0 = Da3w^a3wV = 0 = D ^ +(2 v) 2a xayay( a2w + v a2xand no deflection or bending moment at the guides31y = b/2^ (3.2)W = 0a2wMx = = ^exx = oa2w and+v^a2y x = L(3.3)The Galerkin method was used to solve the governing equation. The displacement of the blade wasassumed to be the summation of shape functions,(x, y) = Emn (y)Sin(nmA)^ (3.4)The total deflection was expressed asM Nw(x,y,t)— 2 2s,,(01.(x,y)m=1 n=1M N= 2 Ismn wEnin (y)sin(nnyL)m=1 n=1(3.5)Each shape function, Winn, satisfies, implicitly, the boundary conditions at x=0 and x=L. Thefunction Emn(y) must satisfy the boundary conditions at y=± b/2. Because of the symmetry of theboundary conditions about the centerline of the blade, there are only two independent boundaryequations. A polynomial series was used for E mn(y), but it was necessary to separate E mn(y) intoeven and odd functions due to the character of the boundary equations. These functions were:Emn (Y)(1) +G(2)(kYlm + (-1(3)(Ylm+2mn^mil/b^—nin /b ;m even32(3.6)(m21)1(yx)m +G(2)n(yx)m+2 +G()(y)m+4mn b ;m oddThe constants G mn are choosen to satisfy the boudary condistions at y=±b/2. An arbitratry thirdcondition E (b/2) = 1 makes the solution for the constants G mn deterministic.The axial stress in a stationary blade, ox(y), has three components: the stress from bandmill strain;the residual stress from roll tensioning; and the in-plane bending stress resulting from theinteraction of the effects of backcrown, wheel tilt, wheel crown and blade overhang. These stressesare shown schematically in Figure 3-2. The change in axial stress that occurs because the blade isforced to accelerate around the wheels is accounted for in the term ictici nu2^2 in thea xgoverning equation. Note that these are the stresses when the band is running on the wheels, andare assumed to be the net result of the above stresses. The following relationships exist:01, = T/2bh^ (3.7)2eoR = (b-2e)oc^(3.8)whereT^= bandmill strain0T^= stress due to bandmill straine^= width of the tensile stress regionoR^= tensile stress on the edges of the blade (gullet stress)oc^= compressive stress in the centre of the bladeSTRESS FROM BANDMILL STRAINa^a f*^a QT0 b ^Am33li^.i11MIIIIIIIIIIIIII 0acROLL TENSIONING STRESSIN- PLANE BENDING STRESSTOTAL STRESS DISTRIBUTIONFigure 3-2. In-plane Stresses in a Bandsaw Blade.34The lateral loads acting upon the blade are assumed to be point loads of magnitude Ff acting atthe coordinates (x0f,y0f)• This is expressed mathematically asq(x,y)=^Fo(x-x of )8(y —y of )f =1(3.9)where SO is the Dirac delta function.The Galerkin procedure is carried out by multiplying both sides of the governing equation by eachshape function,= E id (y)Sin( k = 1, 2, M1 = 1,2,...,N (3.10)and integrating over the area of the plate. This generates MN number of equations which can beexpressed in matrix notation byEmbl+EG61 +EKli sl={ 13 }where {S} =The deflection at any point on the blade caused by static loads is found by first solving[1(]{S} = {P}and then substituting the S mn into Equation 3.5.(3.11)(3.12)The natural response is found by first defining{X}==iXieISso that(3.13)35{[M0 mG]k [ M0 -0K]} {} . 0^(3.14)The eigenvalues, k, and eigenvectors {X} of this equation are complex. The natural frequency w nis the imaginary part of the eigenvalue. Since the model has no damping (the G matrix containsonly gyroscopic terms), the real part of the eigenvalue is generally zero. However, if the K matrixlooses its positive definiteness, as when the axial stresses are arranged so that the blade buckles,the real parts will be non zero.The mode shape is taken to be the real part ofV(t)= [9i(S)+0(S)][Cos(w ri t)+ iSin(w n t)]^ (3.15)which is equal to9i(V(1))= 91(S)Cos(u) n t)- ;:s(S)Sin(cu n t)^ (3.16)where^} is the eigenvector in the coordinates of {S}.This model of the bandsaw as a moving rectangular plate has been well developed by previousresearchers. A comparison of the results of the computer program written for this work to theresults in the literature is given in Appendix B. Also included in Appendix B is a comparison ofexperimental and calculated blade stiffnesses.Modelling Tooth Deflection The Galerkin method is not convenient when irregularly shaped boundaries, such as the toothededge of a bandsaw, must be represented. For this reason, the following approximation is used forcalculating the tooth-tip stiffness. A free body diagram showing the cross-section of the blade andthe tooth is shown in Figure 3-3. The tooth is cantilevered from the edge of the blade and has alateral force, F, acting on the tooth-tip. The equilibrium equations arc:^1 ^a^W2^WIEDGE OF THE BLADE36e(b) R2 Figure 3-3. Free Body Diagram of the Blade and a Tooth37M = Fd^ (3.17)R = F (3.18)where d is the depth of the tooth. The moment, M, and force, F, acting on the edge of the bladeand the rotation of the edge, 0, can be approximated by a wrench made up of two loads, R1 andR2, separated by a small distance, E.whereR = R1-R2M = (R1-R2)E/20 = (w1 -w2)/ER1 = F(1/2 + d/E)R2 = F(1/2 - d/E)(3.19)(3.20)(3.21)(3.22)(3.23)The deflections w1 and w2 can be calculated by applying the loads R1 and R2 to the plate model.The tooth itself has a cantilevered bending stiffness Kt which is proportional to the cube of theplate thickness. The deflection of the tooth-tip iswt = w1 + 0*d + F/Kt^(3.24)= w1 + (w1-w,)d/E + F/KtThe tooth-tip stiffness is equal to F/wt .Flexibility MatrixIn the development that follows a flexibility matrix [A] for the part of the blade that is in possiblecontact with the sawn surfaces is needed. The degrees of freedom in this case are the lateraldeflections rather than the generalized Galcrkin coordinates. The flcxilrility inanix is generated byfirst specifying the locations of the degrees of freedom, say a grid pattern. A column of [A] is the38vector of the deflections for a unit load at one node. These deflections are calculated with theGalerkin development given above. The flexibility matrix includes the degrees of freedom of theteeth.THE CONTACT BETWEEN THE BLADE AND THE SAWN SURFACESThe problem of determining the amount of contact between a bandsaw blade and the saw surfaceswill be described in terms of constraining an elastic body between two arbitrary parallel surfaces.(For the analysis developed here the surfaces are assumed to be perfectly rigid and frictionless.)The difficulty in solving contact problems is that the zone of contact is not known. Since thecontact area changes, the relationship between force and deflection is nonlinear.A number of methods, including variational methods with penalty functions and load incrementtechniques, have been developed for solving contact problems. The method chosen for this thesisfollows, with some modifications, the method developed by Okamoto and Nakazawa [1979] andexpanded upon by Tseng [1980], for solving contact problems in conjunction with a finite elementprocedure (or any method that produces a stiffness matrix for a discretized body). In this studythe method is slightly altered in that the flexibility matrix instead of the stiffness matrix is used.This flexibility matrix for the bandsaw is generated from the Galerkin solution of the continuousplate model by the method descibed in the previous section.The details of the contact algorithm are given in Appendix C.39REPRESENTATION OF THE LATERAL CUTTING FORCEThere are, as yet, no experimental results showing how the lateral cutting forces vary along thelength of the cut. The only available indicator of the lateral cutting force is the motion of the frontedge of the bandsaw during cutting, as shown in Figure 2-3. If there were no contact between theblade and the sawn surfaces then the deflection of the blade would be directly proportional to thelateral cutting force. However, contact does occur, so the indicator is imperfect, but is the bestavailable.In the absence of any direct information on how the lateral cutting forces vary along the length ofthe cut, two artificial methods of generating a disturbing force are used.Simulation of the Cutting Force Around a Knot The first disturbing force to be considered is one similar to that existing when cutting around aknot. This disturbance furnction will be used in much the same way that a unit step input is usedto evalulate the response of control systems To approximate the variations in grain direction andwood density around a knot that would cause a saw to deviate, a potential or hardness function isused and defined asC//r. ; r <q) = ; rl srsr 0 (3.25)0 ; r > rowherer = 2 + z 2is the radial distance from the center of the 'knot'. The axis of the knot is parallel to the x-axis ofthe blade. Outside the radius ro the knot has no effect and within the inside radius, ri, the knot hasa high uniform hardness. To convert hardness to a lateral cutting force the gradient of thehardness function in the lateral, z, direction is taken:40;ri s r re,;otherwise(3.26)The basis for this approach is that the change in density results in different forces acting on thetwo sides of the tooth. The net lateral force is then proportional to the gradient of the density.For values of zoo the function F(y,z) first increases from zero to a maximum when y=0 thendecreases to zero as y is increased. This corresponds to the expected lateral forces around a knotwhere in the clear wood on either side of the knot the lateral forces are near zero but reach amaximum near the center of the knot.General Cutting ForcesThe 'knot' function presented above provides a reasonable representation of a general feature seenin much of the blade deflection data. However, the experimental data shows that blades aresubjected to multiple disturbances of varying lengths and amplitudes. To investigate the characterof the actual cutting forces the specta of the blade deflection data were taken. Typical cuttingdeflection data and their spectra are shown in Figures 3-4 and 3-5. The spectra show a 1/fbehavior, where f is the frequency (measured in units of cycles/sec. or cycles/unit length). Thisresult is typical and applies for straight as well as cuts with high deviations.The spatially dependent characteristics of wood that affect the cutting forces, such as density andthe direction dependent strength properties appear to be well suited to being represented byslowly varying functions. An examination of the surface of a piece of wood shows the generaldirection of growth, but there are variations and irregularities. un a smaller scale these variationshave smaller variations superimposed on them.1 10FREQUENCY (HZ)100(/3••■■••• 0.0001b)0.001a)0.0200 04,kiyaki,-0.010 --0.020^START OF CUT END OF CUT1TIME (SECONDS)20Figure 3-4. Blade Deflertinitniiring Cutting: Examplc 1.a) Displacement of the Front Probeb) Spectrum of the Displacement at the Front Probe41a)0^ 1^ 2TIME (SECONDS)1.7)Li.,i 0.010UZ.......z01.:UW—JLI- - 0.010C)b)ri)LU2C.)Z....-LIJCI 0.0001DI--:3a.2<0.0 0 0 0 I0.0 01 _ 0.004 ,----.e ''fyV44210^ 100FREQUENCY (HZ)Figure 3-5. Blade Deflertion Diming rutting: Example 2.a) Displacement at Front and Back Probesb) Spectrum of the Displacement at the Front Probe43It is proposed that the lateral cutting force along the length of the cut be simulated with byfunction having a 1/f characteristic. If the deflected paths of the blade produced by the simulationare comparable to measured blade deflections, then it is possible to estimate the effect of changesin blade design on sawing variation. These concepts are discussed in the next chapter.The algorithm used to generate a function with a 1/f frequency characteristic follows the one givenby Peitgen and Saupe [eds., 1988]. First a series of Gaussian random numbers, zi, is generated thathas a mean of zero and a variance of 02. If yiti are random numbers uniformly distributed on theinterval [0,A], then zi will be approximately Gaussian if it is calculated as[ 1 1 12 n= —A n— Eyil — Nrjid (3.27)The values A = 1 and n = 12 were used.To produce the series representing the force along the length of the cut, Fi, the series of zi is thenpassed through a low pass filter with a 1/f roll-off and a very low cutoff frequency (measured incycles per unit distance), fc. A recursive filter was used:F1 = z1F = ^27tf At   Zi^_1 ;i = 2,3,4,...,N c(1+ 2nfc6a)whereAt = the incremental distance along the cutNc = the total numbere of increments along the cut.(3.28)The magnitude of the fractal force is controlled by the standard deviation, 0, and differentfunctions can he generated by changing the ei.c1 value supplied to the random number generator.The lateral force on each tooth is assumed to be equal, which is a reasonable assumption for44shallow depths of cut, but is probably not a good assumption for deep cuts. An assessment of thisassumption is given in Chapter 4.SIMULATION OF THE COMPLETE SAWING SYSTEMMathematical descriptions of the elements of the sawing process have been developed in thischapter, but they have not yet been assembled into a system that simulates the whole process. Theconstruction of the simulation model is presented by describing the main steps of the computerprogram.In practice, the sawn surfaces are parallel and separated by an amount equal to the width of thetooth less springback and the tooth marks. However, in the geometry of the contact problem, it isonly the distance that the blade can move laterally before it contacts the sawn surface that isessential. This distance is, by definition, the clearance gap, g. From the point of view of thegeometry of the contact problem the thickness of the blade is not important. In this model,therefore, the sawn surfaces are assumed to be separated by an amount equal to twice theclearance gap.The sawn surfaces are created by the teeth. If the variable S represents the deflection of the teethfrom the ideal cutting plane at all points through the cut, then the sawn surfaces are defined asV=S+g^ (3.29)andU=S-g.^ (3.30)Because the solution algorithm works with discrete points along the length of the cut and withdiscrete locations on the blade and the sawn surfaces, one task of the computer program is to45generate grid systems that define these points.Step 1) The user sets all the blade parameters, the clearance gap, the variables that define thecutting forces, and the dimensions of the "wood".Step 2) The user sets the forward increment that the blade moves between each time step, At. Thetotal number of increments along the cut is equal to N c.Step 3) Determine the maximum number of teeth, N T=Int(Dc/P)+1, that could be in the cut. (D cis the depth of cut and P is the tooth pitch.) The teeth are positioned symmetricallythrough the depth of the cut at the instant that the blade deflection is calculated.Step 4) A grid of nodes on the blade that could come into contact with the sawn surfaces isformed consisting of Ny nodes across the width of the blade and NT nodes through thedepth of cut. See Figure 3-6. Also, a grid of nodes on the sawn surface is formed of NTnodes located symmetrically through the depth of cut and N e nodes along the length ofthe cut. The nodes are the locations of the teeth at each increment along the cut. SeeFigure 3-7. The deflection of the teeth at these nodes defines the sawn surface. Thesedeflections are given the variable Sit, where i = 1,2,3,...,Nc and t = 1,2,3,...,NT.Step 5) Generate the flexibility matrix, [A],of the blade for the nodes, including the teeth and anyarbitrarily located "probe" positions.46Figure 3-6. Locations of the Contact Nodes on the Blade.aalrir.4 - Dc - (NT - OP 248Step 6) Generate the lateral force, Fi, for each tooth for the whole length of the cut (i =1,2,3,...,1•10). Either the 'knot' function or a fractal function is used. The same lateral forceis applied to each tooth for each increment.Step 7) Start to incrementally advance the "wood" onto the blade. For the first increment only theteeth are in contact with the "wood": the body of the blade is unconstrained.Step 8) Determine the constraints for each node on the blade. These constraints are U nt = Sit - gand Vnt = Sit + g, where j is the number of the position along the sawn surface physicallyclosest to the node n on the blade.Step 9) Calculate the deflection at each node of the blade with the contact algorithm. SeeAppendix C for details.Step 10) The deflections of the teeth, xi-, are stored as the newest addition to the cut path, Sit.Step 11) Advance the "wood" one increment, AT.Step 12) Determine the position of the sawn surfaces that constrain the nodes. The teeth are neverconstrained as they have the assumed ability to cut laterally.Step 13) Repeat steps 8 though 12 until the whole length of the "wood" has advanced past theblade.The output of the program is the cut path Sit. The time histories of the blade deflections atarbitrary "probe" locations are also available. The mean and standard deviation of the surface Sitand the effective stiffnesses Ko, Q0, and Keg (as discussed in Chapter 4) are also calculated.49Science is built with facts, as a house is built with stones. But a collectionof facts is no more a science than a heap of stones is a house.Jules Henri PoincareCHAPTER 4NUMERICAL RESULTS AND DISCUSSIONThis chapter has five objectives:1) To show and discuss how bandsaw blade design parameters affect the tooth-tip stiffnessand the cross-stiffness between the tooth-tip and points on the body of the blade.2) To use the cutting simulation model to investigate the mechanics of blade deflectionand recovery caused by the knot-like cutting force.3) To explore the use of a fractal cutting force representation to estimate the effect ofchanges in blade design on cutting accuracy.4) To present the results in nondimensional form.5) To examine how the depth of cut affects sawing behavior.50EFFECT OF BLADE PARAMETERS ON TOOTH-TIP STIFFNESS Of the many blade parameters involved in bandsawing, most are controlled by the saw filer andothers are dictated by the process requirements. The filer chooses the thickness of the blade, thebandmill strain, the tooth design, and the amount of roll tensioning. The selection of blade width issomewhat arbitrary because band steel is produced in standard widths and because the bladebecomes narrower at each sharpening. The decision then is to specify the initial blade width andhow narrow the blade becomes before it is discarded. The filer also controls the in-plane bendingstresses resulting from the interacting effects of backcrown, wheel tilt, wheel crown and bladeoverhang. These effects are not fully understood, but the result is that the front edge of the bladeis stiffer than the back, and stiffer than a blade without these in-plane bending stresses.The filer does not control the span length, which must be greater than the depth of cut. There are,however, devices that automatically adjust the position of the top guide as the depth of cutchanges. The blade speed is generally set by the bandmill manufacturer and is usually between8000 and 10000 fpm.For the conditions specified in Table 4-I (based on a common five foot bandmill) the blade modeldeveloped in Chapter 3 predicts a tooth-tip stiffness of 182.4 lbs/inch. This stiffness wascalculated for a tooth located at mid-span. The predicted effect of a change in any of the aboveparameters is shown in Figure 4-1. These results are similar to those obtained by Taylor andHutton [1991]. The parameters can be grouped by their relative significance, starting with themost significant, as:1) Plate thickness, span and strain parameter2) Strain, in-plane bending and tensioning3) Blade width and tooth depth4) Tooth bending stiffness and speedZ-s- 225cnm-J(t)Cr)zLLI 200L._L.L.I::coa.I= I75I-0^(7R0I-1500.0501^10.060 0.070h - THICKNESS (INCHES)1250.08010 40002000c -SPEED (IN./SEC)51ia^9^io^II^12b- WIDTH (INCHES)20^2I5^30^35L- SPAN (INCHES)0^0:5^1.0k- STRAIN SYSTEM10000 15000T - BANDMILL STRAIN (LBS)i0^2500^5000^ 7500crR - TENSIONING STRESS (PSI)-5000^-2500^6^2500oh - IN-PLANE BENDING (PSI)20000I10000500090000.751000^3000^5000^7000KT - TOOTH BENDING STIFFNESS (LBS/IN)0.5^ 0.625d - TOOTH DEPTH (INCHES)Figure 4-1. Effect of Blade Parameters on Mid-span Tooth-tip Stiffness.Table 4-I. Parameters for the Blade Model 52StrainBlade widthBlade thicknessSpanTensioningBending stressTooth bending stiffnessTooth depthStrain parameterSpeedT = 15,000 lbsb = 10 inchesh = 0.065 inchesL = 30 inchese = 1 inchescIT = 5,000 psiGB = 0 psiKt = 5,000 lbs/inchd = 0.625 inchesK = 0.0c = 2,000 inches/secThere are three components to the tooth-tip stiffness of a stationary (c=0) bandsaw blade. Theseare, 1) the string or tie-rod stiffness due to the tensile axial load on the blade; 2) the torsional ortwisting stiffness of a rectangular plate; and 3) the local bending stiffness of the plate or the teeth(i.e., the cross section of the blade does not stay rectangular). The span and the axial stress in theband determine the size of the string stiffness effect. The roll tensioning stresses, other axialstresses and the blade dimensions determine the size of the torsional stiffness effect. The bladethickness determines the size of the local bending effect.When the blade is running there are some additional effects that affect the tooth-tip stiffness andthe lateral stiffness of the blade in general. Firstly, there are extra tensile axial stresses in the bladedue to the centrifugal action of the blade running around the wheels. Secondly, there is an inertialeffect that affects the blade stiffness [Mote, 19651 This inertial term is proportional toix2 (a 2 w / a2x). Lastly, the straining mechanism may not be able to keep the pressure betweenthe wheels and the band constant if the speed is increased.In the following paragraphs the effects of changes in a blade parameter on tooth-tip stiffness aredescribed in more detail. The large loss of stiffness due to reduced blade thickness accounts forthe difficulty in using thin blades. The reduction in stiffness has three causes; all related to the factthat the bending stiffness of a plate is proportional to the cube of the thickness:531) The deflection of the edge of the blade that supports the tooth increases as the bladebecomes thinner. For thinner blades, the deflection of the blade becomes more localized inthe region near the point of application of the load. In other words, a thick blade does notbend much when viewed in cross-section, whereas a thin blade shows much bending. SeeFigure 4-2.2) The bending stiffness of the tooth itself, Kt, decreases as the blade thickness decreases. As aresult, the tooth-tip deflects relative to the edge of the blade. For the blade parametersexamined, it is acceptable to model the tooth as a rigid cantilevered beam that rotates aboutthe edge of the blade as long as the tooth bending stiffness is greater than 2000 lbs/inch .3) The moment stiffness of the edge of the blade that supports the tooth also decreases as theblade thickness decreases. This allows the whole tooth to rotate, which increases tooth-tipdeflection. The moment on the edge of the blade increases as the depth of the tooth, d,increases, so increasing the depth of the tooth increases tooth-tip deflection.Since increasing tooth bending stiffness does not have much effect unless the tooth is very weak,it would be better to reduce the tooth depth, d, as this reduces the moment on the edge of theblade, and, hence, the rotation of the tooth and the deflection of the tooth-tip. Using shallowerteeth would also increase the bending stiffness of the teeth.The effect of the in-plane stresses, whether caused by tensioning, strain or in-plane bending, issignificant. If the axial stress at the front edge of the blade is increased then the tooth-tip stiffnessalso increases.54Figure 4-2. Deflected Shape of a Blade for a Load on the Front Edge of the Blade.55The effect of blade width, which shows that narrower blades are stiffer than wider blades, issomewhat non-intuitive. Firstly, the strain was not changed, so the axial stress is increasing, whichcauses the lateral and torsional stiffness of the blade to increase. Secondly, the deflection of wideblades caused by a load on the edge is more localized. In other words, the load is not transmittedto the center or back of the blade.The effect of blade speed, c, on tooth-tip stiffness is dependent on the strain system parameter, K.When K=O, which is the case for a well maintained bandmill, the top wheel of the bandmill cantranslate to keep the pressure on the band constant. As the speed increases the component of bandaxial stress due to centrifugal effects increases, which increases the overall stiffness of thebandsaw. At the same time the inertial component of the blade stiffness due to the axial translationof the blade is decreasing. The effect of the additional axial stress due to the centrifugal effectsexactly compensates for the loss in inertial component of the stiffness. Hence, when K=O, theblade stiffness is independent of speed. If K is not zero then, as the speed is increased, the topwheel does not move enough to maintain a constant pressure between the wheels and the band.Consequently, the sum of the axial stresses due to wheel pressure and the centrifugal effects doesnot increase enough to counter the loss in inertial component of the stiffness. The net result is areduction in lateral blade stiffness as the band speed is increased.CUTTING THROUGH A SIMULATED KNOTIn this section, the response of the blade to a simple disturbance that could represent the forcesencountered when cutting near a knot is investigated. With such a disturbance it is possible toobserve the mechanisms that govern blade deflection. In the results that follow, the disturbanceforce function will not be changed. The parameters of the 'knot' function (Equation 3.25) are C =40 lb-inl, ri = 1 inch, re, = 10 inches. The centre of the knot is located at a point 20 inches along56the cut. Figure 4-3 shows how the lateral force varies along the cut for paths that are 2, 3, and 4inches away from the center of the knot. Only the results from the path 2 inches from the center ofthe knot will be used because for this case the maximum force is 10 lbs., which is in the range ofthe peak lateral force near a knot found by St. Laurent [1971]. The maximum force is reached at apoint 20 inches into the cut.A plot of the resulting cut path for the base bandsaw specified in Table 4-I is shown in Figure 4-4.The force acts on a tooth at mid-span and the contact nodes are on a line across the mid-span ofthe blade, (i.e., the geometry of cutting a 1 inch thick board at mid-span). As would be expected,the blade begins to deflect as the 'knot' is approached and reaches a maximum just as the forcereaches a maximum. In the deflection stage the shape of the cut path is the same as that of thelateral force function: the proportionality constant being the tooth-tip stiffness.The recovery stage is much different from the deflection stage. The blade deflection does notdecrease as quickly as the force decreases. Also, the blade overshoots the ideal cutting line in thefinal part of the recovery stage. To understand what is happening, more information is neededthan is provided by the cut path. Since contact between the blade and the sawn surfaces isoccurring, it is necessary to know where the contact is occurring and how the blade hasdeflected.This information is provided in Figure 4-5. The lower section shows the deflections of the tooth,the front and back edges of the blade, and the middle of the blade at each point during the cut.The upper section shows where the contact occurred relative to the front and back edges of theblade. The direction of the arrow indicates the direction of the contact force acting on the blade.For example, at about 33 inches into the cut, contact is occurring only at the back edge of theblade and is restraining the blade from deflecting toward the positive direction.>-57Figure 4-3. Simulated Cutting Forces near a Knot.-0.040-0.060-0.100 '^'^ I^ I0^10 20^30^400.020Z-f3w1 -(..) -0.020z- 0.080 CL "KNOT si58DISTANCE ALONG THE CUT ( INCHES)Figure 4-4. Simulated Cut Path Around a 'Knot'.• FTOOTH /FRONTMIDDLEBACK;14• St_ B••5901—ZOLL0.080w0wti0- 0.080 ^0 4010^20^30DISTANCE ALONG THE CUT (INCHES)Figure 4-5. Blade Deflections and Contact for a Simulated Cut. Arrows show the Location andDirection of the Contact Forces. Lines in Lower Section show the Deflections of theBlade at Instants Along the Cut.60Figure 4-5 is a very compact representation of what happened in the cut, but some visualimagination is needed to place the blade between the two sawn surfaces because the figurecontains no visual information about the clearance gap or the width of the blade. The position ofthe blade at various instants during the cut is shown in Figure 4-6 to aid the reader in interpretingFigure 4-5. The points labeled T, F, M, and B are for the instant when the tooth is 30 inches intothe cut. Note how the blade is dragged over the bump in the sawn surface at 20 inches along thecut, and over the smaller bump at 22 inches. This progression of the contact from the front to theback of the blade is what produces the straight line groups of contact indicators. It is clearlyshown that when the cut path is not proportional to the lateral force, then contacting is occurring.Contact is guaranteed to occur at some point during the cut once the deflection is greater than theclearance gap. However, contact on the front 1/3 of the blade has the greatest effect on toothdeflection. For instance, as the tooth moves from 20 to 25 inches along the cut, the deflectedshapes of the blade show that the contact at the back of the blade has a large effect on thedeflection of the back of the blade, but has almost no effect on tooth deflection.Contact does not affect the deflection stage significantly unless the gap is very small because thereis very little contact, but contact does amplify and prolong the recovery stage. Contact does notaffect the deflection stage as much because the blade must deflect an amount greater than the gapbefore contact can occur. On the other hand, the blade is almost certainly in contact when therecovery stage begins.The effect of changes in bandsaw parameters on the behavior of the blade will now be investigated.Figure 4-7 shows how blade thickness affects the cut path. As would be expected, the thinnestblade deflects the farthest because its stiffness is the lowest. The effects of bandmill strain andtensioning are also as expected: more strain or tensioning results in more stiffness and smallerdeflections. See Figures 4-8 and 4-9.rf)U0LIJ-0.050C.."KNOT"0^ 10^20^30^40DISTANCE ALONG THE CUT (INCHES)0.020F.)20- 0.020-0.040I-tLLLI - 0.060- 0.080- 0.1000 10^20^30DISTANCE ALONG THE CUT (INCHES)4062BLADE THICKNESS0.072 IN.0.0650.0580.049Figure 4-7. Effect of Blade Thickness on the Cut Path. All other Parameters are as in Table 4-I.BANDMILL STAIN20,000 LBS17,50015,00012,50010,0000.020L.L.1- 0.020O -0.040_Jw - 0.060-0.080- 0.100 ^630 10^20^30^40DISTANCE ALONG THE CUT (INCHES)Figure 4-8. Effect of Bandmill Strain on the Cut Path. All other Paramters as in Table 4-I0.020w-0.0200 -0.04017-Lu_JLL10 -0.060-0.080TENSIONING STRESS10,000 PSI7,5005,0002,500064I^I^I^I^I^I^i 10 20 30 40DISTANCE ALONG THE CUT (INCHES)-0.1000Figure 4 - 9. Effect of Tensioning on the Cut Path. All other Parameters are as in Table 4-165Note that the maximum deflection often occurs after the center of the 'knot' has been passed andthat the blade overshoots the ideal cutting line when returning. The contact between the blade andthe sawn surfaces causes both of these phenomena.The effect of the clearance gap is shown in Figure 4-10. The differences in blade behavior areentirely a result of the contact between the blade and the sawn surfaces. The smaller the gap themore the blade is inhibited from changing direction in the cut. This is shown most clearly for thegap of 0.005 inches, where the constraining effect of the sawn surfaces did not allow the blade tochange direction even after the disturbing force had dropped away. The only reason the bladeturns back is that the restoring forces of the blade pulled the front of the blade around. The sameovershooting occurs when the blade returns to the ideal cutting line.RESPONSE TO FRACTAL LATERAL CUTTING FORCESIn the previous section the response of the blade to a simple disturbance was investigated. Such afunction is useful for investigating the mechanics underlying blade behavior. However, thissimulation does not provide information that quantifies the cutting accuracy of the blade becausethe character of the lateral cutting forces is generally more complex than that of the 'knot'function. The purposes of this section are to present the simulated blade behavior with fractalforce excitation and to ascertain if the results correspond to observed saw behavior.The effects of changes in blade design on the cutting deviation, ST (the standard deviation of thetooth deflection), are shown in Figure 4-11. These results were obtained from the simulatedcutting of twenty sixteen foot long blocks of "wood". Each block had a different seed andstandard deviation for the fractal force generator. The advance per time step was At = 0.1 inches;CLEARANCE GAP0.050 IN.0.0300.0200.0150.0100.005a660.020U,wz-0.020-0.040—JoLt. -0.060-0.080-0.100400^10^20^30DISTANCE ALONG THE CUT (INCHES)Figure 4-10. Effect of Clearance Gap on the Cut Path. All other Parameters are as in Table 4-I. 670.035-630zz 0.0300w 0.025CDzI—I—n0 0.020KTKT0.0150.050 0.060^0.070h - THICKNESS (INCHES)0.0807^8^9^10 12b - WIDTH (INCHES)20^25^30^35L- SPAN (INCHES)0 2000 4000c - SPEED (IN./SEC)0^0.5^1.0K- STRAIN SYSTEM10000^ 15000^ 20000T - BANDMILL STRAIN (LBS)0^2500^5000^7500^10000o-R - TENSIONING STRESS (PSI)-5000^-2500^0^2500crb - IN-PLANE BENDING (PSI)50001000^3000^5000^7000^9000KT - TOOTH BENDING STIFFNESS (LBS/IN)0.5^ 0.625^ 0.75d - TOOTH DEPTH (INCHES)Figure 411. Effect of Blade Parameters on Cutting Deviation68the cutoff frequency was fc = 1/192 cycles per inch; and the number of nodes across the blade wasNy=50. Only one tooth located at mid-span is assumed to be cutting: the effect of depth of cut isdiscussed later. The standard deviation of the cutting force is SF3.36 lbs. The clearance gap was0.015 inches. The results are as expected: any change in blade design that improved the tooth-tipstiffness also decreased the sawing deviation.Examples of some cut paths are shown in Figure 4-12. The similarity of these paths to theexperimental traces (Figure 2-3) is quite good in that the deflections have a slow, meanderingcharacter and the back of the blade deflected some time after the front of the blade deflected.The effect of clearance gap on sawing deviation is shown in Figure 4-13 for four different bladethicknesses. There are two significant behavior patterns shown in these results. Firstly, for largegaps a change in the gap has no effect on the cutting deviation. This occurs because for large gapsthere is not much contact between the blade and the sawn surfaces. In this situation the effect ofthe contact forces is negligible compared to that of the cutting forces, so changing the gap will nothave a measurable effect.The second effect occurs for small gaps where the contact forces begin to have a significant effecton saw deflection. This is the region investigated by Kirbach and Stacey [1986] in their criticalside clearance study. In this region (e.g., 0.010 inches < g < 0.015 inches for the 0.065 inchesthick blade), there is a transition from contact having almost no effect (g > 0.015 inches) to thecondition where the contact forces dominate (g < 0.010 inches). The results of the simulationagree with Kirbach and Stacey's experimental results in that once the side clearance (or clearancegap) becomes smaller than some critical value, the sawing deviation increases rapidly for smallchanges in the clearance.Although there is agreement between the behavior pridic't&d by the model and the results obtainedPiet'irtfe.4t1^411.10^20^30DISTANCE ALONG THE CUT (INCHES)40- 0.0500U)0Co^00_JLL0TOOTHFRONTMIDDLE#, BACKF90_JV4I-0cr0.0500004fv`l' ie////14/ leC.)4CD0z00^t—z0crLL0.050w000w0BACKMIDDLEFRONTTOOTH-0.050 ^0 10^20^30 40DISTANCE ALONG THE CUT (INCHES)40- 0.0500 10^20^30DISTANCE ALONG THE CUT (INCHES)OI-2O0.050MIDDLEBACKLi]0U-JLL'gum 4-13. Effect of Clearance bap on cutting Deviation. Horizontal Lines ShowAsymptotic Limits for g --> cx ^.0.010^0.020CLEARANCE GAP (INCHES)0 0.0300.0800.070w 0.060—z 0.0500I-w 0.0400.030(r) 0.0200.01007273by Kirbach and Stacey, the explanation of the behavior is different. Kirbach and Stacey state thatless side clearance causes more contact, which is true, but they assumed that the sole cause of thepoor cutting behavior was due to the heating of the blade. The present model does not include theeffects of thermal stresses in the blade. Nevertheless, the model predicts the critical side clearancephenomenon. This would indicate that the longer recovery times that occur when the gap is smallare the primary cause of the increase in sawing deviation and that the heating produced by thecontact is a secondary effect that may further aggravate the deflection. There is also the possibilitythat spilled sawdust may pack very tightly, thus causing heating and, hence, increased sawingdeviation. However, it seems more likely that this scenario would cause a relatively uniformchange in the temperature and stress distributions across the blade, which would not affect theblade stiffness if the strain system is well maintained (i.e., K-0).THE CONTACT GRAPHIn the previous sections the simulated behavior of specific bandsaws was investigated. In thissection two dimensionless variables are presented that determine how a bandsaw behaves. Theprincipal benefit of these variables is that they have a physical interpretation, a feature thatsignificantly contributes to the understanding of blade behavior. It is important to know theconcepts that led to their development.Nondimensional Sawing Deviation When the clearance gaps are so large that no contact occurs then the tooth deflection isproportional to the cutting force. It is then possible to say that the standard deviation of the toothdeflection, when no contact occurs, isS tt = Lim STg--400(4.1) where74Sf^= standard deviation of the lateral cutting forceKtt^= tooth-tip stiffnessST^= standard deviation of the tooth deflection.So = ST when no contact occursSince ST converges to So (see Figure 4-13), it is convenient to nondimensionalize the sawingdeviation asSo^SfST^STKtt^ (4.2)Nondimensional Clearance Gap Contact is guaranteed to occur at points during the cut whenever the deflection of the tooth, xT,equals or exceeds the clearance gap, g. There is an infinite number of combinations of the lateralcutting force and the contact forces that could cause the tooth deflection to equal the clearancegap. The complicating factor is the variety of contact conditions that could occur. However, theeffect of contact is generally to amplify and prolong the blade deflection. Contact is stillguaranteed to occur as long as the lateral cutting force is, by itself, large enough to produce atooth deflection equal to the clearance gap. This force is fg = gKtt•The fraction of time that the lateral cutting force equals or exceeds fg is therefore a crudeindicator of how much contact occurs during a cut. Mathematically, this concept is expressed asthe probability that the lateral cutting force, fL, exceeds the range ±fg.In the following development it is assumed that no contact occurs. Hence, the lateral cutting forceis proportional to the tooth-tip deflection and the probability distribution curves for the lateralcutting force and the tooth deflection are identical. Hence, the variable75z= fL AT^=KttSf^Sf^Socan be used for normalizing the lateral force and the blade deflection. See Figure 4-14. Anormalized contact variable can therefore be defined asZ = fg = g^g° Sf Sf S oThis variable is the nondimensional form of the clearance gap. The physical interpretation of Z o isthat it is an inverse indicator of the fraction of time that the lateral cutting force exceeds fg, or thefraction of time that the tooth deflection exceeds the clearance gap.As expected, the clearance gap, g, governs the amount of contact that occurs. However, gap, as ameasure or indicator of the amount of contact, must be taken relative to how much the bladedeflects. For instance, a stiffer blade will not deflect as much and hence will not allow as manydeflections that would result in contact. Also, if the magnitude of the cutting force were todecrease, the blade will not deflect as much and, hence, there will be fewer deflections that wouldresult in contact.The area outside z = ±Zo, A, in Figure 4-14 would be an elegant measure of the amount ofcontact. This area, A, could be called the probability of contact, but there are too manyassumptions separating A from the actual amount of contact for this variable to be consideredreliable. These assumptions are 1) the probability distribution is for the no-contact case; and 2) thelimits of the area at z = .-L-Zo are somewhat arbitrary. The variable Z o is considered a more reliableindicator of the amount of contact due to its directness and simplicity.In this thesis the clearance gaps on each side of the blade are assumed to be equal and the averagelateral cutting force is near-zero. A non-zero value of the mean clearance gap, -g , means that thexT (4.3)(4.4)blade deflection ounng cutting could be biased towards the side of the blade having the largergap. A non zero value of the mean cutting force, f , means that the wood properties, a grinding0I"71.....170Cco4,.—..,I":V0C7'II:CT.—.....t....-.nCCDrp...,0-,nCc7,--.1CT77problem and/or a misalignment of the blade has caused a bias in the cutting force. Since theseproblems are generally correctable (although very common), their effects will not be examined inthis work.Generality of the Contact GraphThe dimensionless form, sixttisf versus gKtt/Sf, of the data in Figures 4-11 and 4-13 is shownin Figure 4-15. Essentially, all of the data lie on the best fit curveSTKtySf = 1+ 0.089( gKtt 3f(The regression coefficient, r2, was 0.983.) Although there is some scatter about this curve,Equation 4.5 predicts ST within five percent.It is important to realize that this curve applies for as wide a range of blade parameters as is usedin practice, at least for a 5 foot bandmill. This means that the blade design, in the structural sense,is totally represented by the tooth-tip stiffness, Ktt. It does not matter how the value of K tt wasgenerated, whether it was from any combination of blade thickmess, strain, tensioning, etc.: onlythe resulting value of K tt matters.Insensitivity to Blade DesignThe dominant effect of Kt t infers that the effect of the contact forces on ST is insensitive tochanges in blade design. To see why this is so, consider writing the tooth deflection asFT Fn^1_^+ Fn Antx T =^+KttFTKnt )^Ktt^n.1{Attn -1 (4.6)where—2.755(4.5)FT = the lateral force on the tooth'71....(10=..,n1"....-vi(")0=anC)"c")=32I-i-Y1-(/)U)0 2 3540S---.10079Fn = the contact force on node nKnt = 1/Ant = the cross stiffness between node n and the tooth tip.At t = 1/Ktt = tooth-tip flexibilityAlthough the contact force, Fn, is affected by present and past values of the cutting force actingon the tooth, FT, the next most important factor of how contact affects XT is the ratio Ant/Att.Since this ratio is a function of the node location across the width of the blade, the influence of thecontact forces will vary depending on the location of the contact force. By Maxwell's reciprocaltheorem, Ant/At t is equivalent to the normalized shape of the blade caused by a load on the tooth-tip. This shape is fairly insensitive to changes in blade design (see Figure 4-16) because the effectsof lateral and torsional ('rigid body') bending effects remain fairly constant relative to each other.The local bending is more affected by changes in blade design, but the local bending has a smallereffect on the shape of the cross section of the blade.The variation of the cross-stiffness across the width of the blade has many features that helpexplain the effect of contact on tooth deflection. Firstly, contact forces acting on the back half ofthe blade have little effect and may even reduce the tooth deflection because At < 0. At a pointabout two thirds of the blade width from the front edge, the cross stiffness is zero and a contactforce acting here will have no effect on tooth deflection. It is only when the contact forces act onthe front third of the blade that they have much effect. Contact forces acting on the gullet line willhave the greatest effect.Progression of Contact One can conclude that the closer the contact gets to the front of the blade the more effect it willhave on tooth deflection. There is also the concept, discussed earlier, of the likelihood of contact.The effect of contact, therefore, has two components: 1) the probability that a certain contact modewill occur, and 2) the severity of the effect of the contact mode. By contact mode is meant anything01.0b)01-7-o 0w1.0w< 0.5cowNJ2• 1.0tx0c)T = 10,000 lbsT = 15,000 lbsT = 20,000 lbs0• d)0.5^11.0 a)0R = 10,000 psiQR = 5,000 psicfR = 0 psi0.5h = 0.072"h= 0.065"h = 0.058"h = 0.049"0.57"b 10"b.: 12"0Yla0.5- 0.5c^iuss-s Limn of ade veriection.a) Effect of Tensioning b) Effect of Thicknessc) Effect of Bandmill Strain d) Effect of Blade Width.80Fis81from no contact to binding. These modes were identified from examining the blade deflectionsproduced by the simulation, an example of which is shown in Figures 4-6, 4-7 and 4-17. There is aprogression of contact modes and these are shown in Figure 4-18. Generally, the severity of thecontact increases as the deflection increases and/or the curvature of the cut path increases. In bothcases the contact moves closer to the front of the blade.In summary, reducing the value of gKtt/Sf increases the likelihood that contact will occur and thatmore severe modes of contact will occur. The effect of the contact forces relative to that of thecutting forces is quantified by the value of ST/S o . The relationship between ST/So and gKtt/Sf isa quantification of the effect of contact. Hence, a graph of ST/S o versus gKtt/Sf has been termedthe contact graph.From a practical point of view, this development, for the first time, provides an equation(Equation (4.5)) that relates the saw design to its cutting performance. The implications of thisrelationship will be investigated in the next chapter.THE EFFECT OF THE DEPTH OF CUTIn the analysis and simulation presented it was assumed that only one tooth was cutting. Arealistic simulation of the multi-tooth case can only be achieved if information on how the lateralcutting force varies through the depth of cut is available. The approach adopted here, instead, isto investigate the consequences of assuming that the forces on each tooth are equal.A question that arises is how to define tooth-tip stiffness when each of the tooth-tips in the cuthas a different stiffness, depending on its distance from the guides. This is an important questionct,ause t ie tooth-up stiffness has men identified as the parameter that encapsulates the effects ofaa t aa a I 110^20^30DISTANCE ALONG THE CUT (INCHES)40ViLuI0z...._,„ZO_0ILI_JLi-LU0DECREASING RADIUS OF CURVATUREIfINCREASING DEFLECTION-at^CONTACT MOVING CLOSER TOTHE FRONT EDGE OF THE BLADE84blade design.Effect of Non-uniform Load PatternsA simple method for investigating the effect of variations in the cutting force through the depth ofcut is to calculate how different load patterns affect the blade deflection. The assumed loadpatterns, which have triangular wave forms, and the resulting tooth deflections are shown inFigure 4-19. The wavelengths of the applied force distribution,k , are expressed as multiples ofthe tooth pitch, P, which was 1.75 inches. The peak-to-peak amplitude of the load patterns was 2lbs. and the mean force of each load distribution was adjusted to be zero. For the case k=00, auniform load of 1 lb. on each tooth was used. The blade parameters are specified in Table 4-I.Clearly, the effect of the mean force (k=00) will dominate the response of the blade. Variations inthe load shorter than =6P will have to have huge magnitudes to produce deflections comparableto that of the mean force.General Case of Unequal Forces on the Teeth Since the deflection caused by the mean cutting force dominates the deflection response of theteeth, it is convenient to separate the force distribution into two components: a mean force andthe variant from the mean.fit = mi + eit ; t = 1,..,NT^ (4.7); i =where fit is the force on the t-th tooth at the i-th increment along the cut; mi is the mean of theforces on the teeth during the i-th increment; and ei t is the variant on the t-th tooth during the i-thincrement. It follows that the standard deviation of the cutting force isSc2 = Sm2 + Se2^(4.8)A = 2P A = 4 P4 tN\JtA = 6P47\ = 0085 4 4t^A = 8P^A = I2Pgurc 4-19. Effcct of Load Vanation Through the Depth of Cut. The Arrows show theMagnitude and Direction of Forces on the Tooth-tips.86where Sm2 = Var(m) and Se2 = Var(e) for all teeth and for all increments. A derivation given inAppendix D shows that the following two equations apply when no contact occurs:2^22 S m SSo =^+K(2)_(4.9)fx — ^Keg (4.10)where Ko, Q0 and Keq are equivalent blade stiffnesses, f is the mean cutting force, and x is themean deflection.The effect of depth of cut, expressed as the number of teeth in the cut, NT, on K o, Q0 and Keq isshown in Figure 4-20. The cutting zone is symmetrical about mid-span. For NT = 1,Ko = 00 = Keq = Kttso the formulae collapse to the single tooth cutting case presented earlier. For NT > 7 (a 12 inchdepth of cut), Q0 levels off and Ko is about one third of 0 0. Assuming Se = Sm, and 3K0 = Q0, Secontributes only five percent to the value of S o. This result reinforces the earlier conclusion thateffect of the mean cutting force, mi, dominates the cutting deviation.Consequences of Assuming Equal Forces on the Teeth The model assumes that the forces on the teeth through the depth of cut are equal (i.e., S e = 0).Although the analysis will not be given here, it can be shown that when S e is not known So can bequite reasonably approximated by S m/Ko, even if Se = 2Sm. This approximation underestimatesSo . The assumption of Se = 0 does not, therefore, significantly affect the results of the simulation.In any case, So remains the parameter for nondimensionalizing ST and g on the contact graph.0 K0• Q oX K egCUTTING ZONE SYMMETRIC ABOUT MID-SPANTOOTH PITCH = 1.75"SPAN = 30"287^ 14^ 21DEPTH OF CUT (INCHES)0100IMOMt^',....1 --...^ 41— —)11,Iol.—— 10--_111Pc— —10— - -II— --X---01---)11— --OE— --14-- 4.-4-6I^I^I^I^a^I^I^1^I^I^I^I^I^I^I^a^I87Figure 4-20. Effect of Depth of Cut on the Equivalent Stiffnesses.88The effect of depth of cut on the contact graph is small, as can be seen in Figure 4-21. For valuesof gl(0/Sf greater than 0.7 the curves for NT > 1 conincide with the curve developed from thesingle tooth simulation. For values less than 0.7, there is some deviation from the curve, but thesimulation is not expected to be that accurate in this region anyway because heating effects arenot accounted for.There are three important conclusions from this analysis of the effect of depth of cut.1) The variations in the lateral cutting force through the depth of cut, ei t, will have to be largecompared to the mean, mi, before their effect will be comparable to that of the mean force. If eit(or Se) is small in reality, then the shape of the blade between the guides will be bow-shaped, as inFigure 4-19 (A.=00), and will not bend into an S-shape. In experiments [Taylor, 1985; Alexandru,1967], the bow-shape is always seen, which indicates that the variant force, eit, is small.2) If the effect of eit is small, as it appears to be, then it is not important to measure it accurately.Variations in the cutting force that have wavelengths less than ?. = 6P could be ignored.3) Since the blade always deflects into a bow-shape, the deflections measured by a probe locatedabove the cutting zone will give a good indication of how the blade is deflecting in the wood.Probe data would be useless if the blade bent into an S-shape. 892o(yI) ..-I-- (I)^I00 0.5 1.0 1.5gKoSfFigure 4-21. Effect of Depth of Cut on the Contact Graph.90An idea, like a ghost, according to the common notion of ghosts, must bespoken to a little before it will explain itself.Charles DickensTo know a thing well, know its limits. Only when pushed beyond itstolerance will true nature be seen.Frank HerbertCHAPTER 5DISCUSSION AND APPLICATIONSA set of variables for quantifying the basic sawing system and a mathematical relationshipbetween these variables has been found. Although the model is an incomplete representation ofthe sawing process, it provides enough information to explore the optimization of blade designand to better understand the practical aspects of sawing.MINIMIZATION OF FIBRE LOSSThe fibre loss (the sum of the kerf width and the amount removed by the planer), allowing for fivepercent planer skip [Brown, ed., 1982], can be written asF = a(1.64)ST + h + 2(g+d) + p^ (5.1)where91a = a factor to account for the difference between the board thickness deviation, cy T, andtooth deflection deviation, ST.h = blade thicknessg = clearance gapd = difference between side clearance and the clearance gapp = planer allowanceThe cutting deviation, ST, can be expressed by Equation 4.5 so that Equation (5.1) can be writtenin as,l' 641aS f (1+ 0.089(gKV 12.755F=  K )Sf^+2g+2d+h+p0 (5.2)For a given blade thickness, increasing the tooth stiffness, Ko, always reduces fibre loss. The fibreloss is also sensitive to the ratio S o=Sf/Ko, which means that the factors that affect Sf, such as thetooth sharpness and the characteristics of the wood, need to be monitored as well.Figure 5-1 shows how the fibre loss varies with side clearance and a. The following values wereused: h = 0.065 inches, d = 0.015 inches, p = 0.060 inches, and So=Sf/K0 = 0.020 inches. For agiven blade thickness there is an optimum side clearance that minimizes the fiber loss. It isimportant to see that fibre loss is insensitive to changes in the side clearance as long as the value isnot smaller that 90 percent and not larger that 120 percent of the optimum side clearance. This isfortunate because this allows for variations in the clearance gap caused by tooth grinding, sawdustspillage and surface roughness.The optimum side clearance is somewhat sensitive to 'a', so it is important to reduce the betweencut (and between-board) variation, which is affected by the repeatability of the setworks, and to0.20.0600 0.020^ 0.040SIDE CLEARANCE (INCHES)0.50.4(7)wIo 0.3z0.I093know the relationship between board thickness variation and blade deflection deviation.The predicted optimum side clearance is close to the clearances used in practice. Typically, theside clearance ranges between 0.025 inches and 0.035 inches. For 'a'=1, the optimum clearance is0.028 inches. To keep the fibre loss no larger than 2% of the optimal minimum (i.e., within a 1%change in lumber recovery), the clearance would have to be within 0.025 to 0.033 inches. Tointroduce production costs into the optimization, the effect of bite on Sf and g needs to be known.THE PRACTICE OF SAWING The purpose of this section is to discuss, in terms of the principles presented in this thesis, howthe sawing process is monitored and controlled in contemporary sawmills.As described in Chapter 2, blade design is mostly a trial and error process. Given the commonemphasis on production volume rather than lumber recovery, the general objectives are typically to:1) improve blade reliability to minimize production stoppages;2) increase the feed speed to increase production volume;3) improve recovery if and only if production volume is not compromised; and4) improve surface finish.These goals limit the usefulness of the optimization developed in the previous section.Whether or not the drive for production volume dominates, the managers and mill personnelaffect the design of the saws. Firstly, the type and quality of the logs available or chosen by themanagers and/or log buyers affects not only the grade of the lumber, but also the ease or difficultyin processing the logs. Wood properties affect the cutting forces and the straightness of the logsaffects flow well the feed system can hold the logs. For example, a low grade log will have more94knots so either thicker (i.e., stiffer) blades or slower feed speeds are necessary. Also, since theblade must recover from more disturbances, more side clearance is needed.The sawfiler controls the stiffness of the blade, and is the only person to do so. The filerinfluences a) the cutting forces by selecting the tooth geometry, the method of sharpening theteeth and, in consultation with management, the saw replacement schedule; b) the clearance gap,by setting the side clearance and the gullet area; c) the maximum feed speed, by setting the toothpitch and gullet area; d) the reliability of the blade by preparing consistent quality blades. Incooperation with the millwrights, the filer is also responsible for the condition and alignment ofthe bandmill and the feed system.The sawyer, in most cases, controls the feed speed, which is adjusted to account for the depth ofcut, the condition of the wood and the quality of the saw. The sawyer watches how the sawresponds, directly controlling the sawing deviation. In this way, the sawyer compensates forvariations in blade preparation, and in the wood properties. By setting the feed speed the sawyercontrols the cutting forces and the amount of sawdust spillage by changing the bite per tooth.For systems where the feed speed is determined from measurements of the log diameter, thecompensating feedback of the sawyer is removed. Consequently, the saw must be oversized toallow for the worst log (worst cutting conditions) and for variations in blade preparations.COMPLETE DATA SETThe complexity of the sawing system is partly caused by the large number of variables involved. Ithas been impossible to apply the theories developed in this thesis to published cutting data, otherthan the comparison of general trends, because important information was not included in the95recorded data. The data were omitted because they were not considered important, or becausethere is no reliable method of measurement.To focus the need for measuring and recording the many variables involved in sawing, and fordeveloping measurement procedures, the following is suggested as a complete set of data forcutting tests. The necessary accuracy of the measurements has yet to be determined. Knowing thisinformation not only ensures that experimental results can be repeated, but will also aid indiagnosing problems in the laboratory or sawmill. Experimental repeatability corresponds directlyto consistent sawing in the mills.1) Blade Properties. One should record the blade dimensions, speed, strain, strain systemparameter, K, span, tensioning, backcrown, wheel tilt and crown, tooth dimensions and toothstiffness, temperature distribution, location of the cutting zone. The equivalent stiffnesses K o, Qoand Keg are the minimum parameters that should be measured. Ideally, the flexibility or stiffnessmatrix of the blade, including the tooth-tips and the probe locations should be measured.2) Cutting Forces. Record the wood species, and the mean and variation of the density andmoisture content. If possible, measure Sf Sm, and Se. A good estimate is Sf = SoKo. Othervariables are the bite, the tooth geometry and the sharpness.3) Clearance Gap. Record the side clearance, gullet area, gullet feed index, the surface roughness,depth of tooth marks, the quantity and location of spilled sawdust, blade flatness and toothrunout, the alignment and the looseness of the feed system. Keep in mind that the wood maydeform during cutting and either lean against saw or pinch the blade.4) Gutting Accuracy. Board thickness data is only an indicator of blade behavior. The morecorrect measurement of blade behavior is the shape of the sawn surface, from which a standarddeviation of deflection, ST, can be calculated.9697We are not interested in establishing scientific theories as secure, or certain, orprobable. Conscious of our fallibility we are interested only in criticizing themand testing them, in the hope of finding out where we are mistaken; oflearning from our mistakes; and if we are lucky, of proceeding to bettertheories.Sir Karl PopperTruth and Approximation to TruthExperience never errs; what alone may err is our judgment, which predictseffects that cannot be produced by our experimentsLeonardo da VinciCHAPTER 6CONCLUSIONSA model of a bandsaw, subjected to lateral cutting forces on the teeth and restrained by the sawnsurfaces of the wood, has been developed. The blade model includes the effects of bladedimensions, bandmill strain, in-plane stresses, tooth (gullet) depth, tooth bending stiffness, bladespeed, strain system parameters, and the span between the guides.The spectrum of the blade deflection during cutting was experimentally found to be inverselyproportional to the frequency. From this observation it was argued that lateral cutting forces alsohave this character. Since the blade motion is dominated by motions at frequencies much lowerthat the natural frequencies of the blade, it is possible to ignore the effects of the inertia of theblade when calculating the blade deflection during cutting.The cutting simulation model was used to determine how blade parameters, the cutting forcesand the clearance gap affect blade deflection. These simulations showed that any change in theblade that increased the tooth-tip stiffness reduced the deflection amplitude during a cut. Asensitivity study showed that plate thickness, bandmill strain and the guide span had the greatest98effect on tooth-tip stiffness.Three equivalent tooth-tip stiffnesses, Ko , 00, and Keq, define how the cutting deviation and themean deflection of the blade are affected by blade design, depth of cut, and the location of thecutting zone. The combination of blade parameters used to get these stiffnesses does not have asignificant effect on cutting deviation.The simulations conducted show that reducing the clearance gap hinders the ability of the blade tochange direction in the cut. This effect is caused completely by the increased amount of contactbetween the blade and the sawn surfaces as the clearance gap is reduced. The closer the contactoccurs to the front of the blade, the more the cut path is affected. Consequently, reducing theclearance gap amplifies the amplitude of blade deflection and prolongs the length of the recoverystage. Contact, therefore, can have a significantly negative effect on blade behavior during cutting.Saw behavior can be separated into two broad cases. The first occurs when there is little or nocontact between the blade and the sawn surfaces. In this case the blade deflection is determinedsolely by the magnitude of the cutting forces and the tooth-tip stiffness. The second case occurswhen the effects of the contact forces have a significant effect. It was found that the degree ofcontact is governed by a dimensionless variable, g/So.The cutting deviation can be nondimensionalized as ST/So. A relationship was found betweeng/So and ST/So. This relationship defines how the design and operating parameters and thecutting forces affect the sawing deviation. The relationship between g/S o and ST/So predicts theexistance of a critical side clearance below which cutting deviation increases rapidly.99FUTURE WORK1) Quantify, in statistical form, the factors that affect the lateral cutting forces. The definitions ofSf, Sm, Se and f provide a rationale for measuring the lateral cutting force. There are five issuesto be resolved:a) the variation of the lateral force along the length of the cut;b) the variation of the lateral force through the depth of the cut;c) the effects of wood properties such as density, moisture content, number of knots, andgrain direction ( and how they can be monitored);d) the effect of bite per tooth; ande) the effect of tooth geometry and tooth sharpness.The use of functions having a 1/f characteristic to model how the lateral force varies along thelength of the cut appears to be very promising. A possible experiment is use with a blade with avery large side clearance so that contact does not occur. The shape of the sawn surface can bemeasured, from which So can be calculated. The variation of the cutting force, Sf, is proportionalto So. The facts that Q0 is large compared to Ko and that the low frequency componentdominates the character of the lateral cutting force means that the spatial resolution for measuringthe lateral cutting forces can be quite coarse.2) In the model, the lateral cutting force does not change if the tooth deflects, which is not thecase in practice. The change in the lateral cutting force, F, with deflected position, xT, can beaccounted for in the current model by defining a process stiffness k p = aF/axT. Springs of thisstiffness can be attached to the tooth-tips in the model. The flexibility matrix can be appropriatelymodified to account for these springs. The contact algorithm and other aspects of the simulationmodel do not need to be changed.1003) Determine how the clearance gap is affected by the side clearance, bite, sawdust spillage (gulletshape, depth of cut), springback, and tooth marks. It is expected that the clearance gap will varyalong the length and through the depth of the cut.4) Develop a device for measuring the effective stiffnesses of the blade, or some other measure ofblade stiffness that quantifies how well the blade is prepared. A consequence of Q 0 being largerthan Ko means that Ko has the most influence on blade behavior. Hence, Ko should be used as aquality control parameter for blade preparation.5) Determine how much the feed system allows or causes the wood to move during sawing. Thiswould include the effect of pinching caused by the release of residual stresses in the wood.6) The model does not consider blade heating, which is commonly assumed to be the cause ofpoor sawing accuracy, yet it does predict the critical side clearance phenomenon. Furthermore,the common assumption of the Gullet Feed Index (GFI) theory is that heat is generated when thesawdust spills from the gullets. The results of this work indicate that the loss in clearance gap thatoccurs when sawdust is spilled could also be the cause of the poor cutting accuracy. There issome doubt, therefore, whether heat causes sawing problems or heat is generated because of thecontact that occurs when blade deflections are large. It is important, therefore, to determine howmuch blade heating occurs and how much it affects blade stiffness. The rigidity of the feed systemand the amount of sawdust packing are expected to have the major influences.7) Determine how blade stiffness can be increased without causing cracking or reducing thereliability of the blade.8) Determine the relationship between board thickness variation, cIT, and the deviation of bladedeflection, ST. _Board thickness variation is more relevant to the process and easy to measure,whereas saw deviation better describes saw behavior.101102LIST OF REFERENCESAxelsson, B., Grundberg, SA., and Gronlund, J.A., 1991 "The Use of Gray Scale Images whenEvaluating Disturbances in Cutting Force Due to Changes in Wood Structure and ToolShape," Holz als Roh- and Werkstoff, Vol. 49, No.12, pp. 491-494.Alexandru, S., 1967. "Automatic Regulation of the Feed Speed in Log Bandsaws in Relation toBlade Deviations in the Cutting Plane", Industria Lemnului, Vol. 18, No. 2, pp. 41-48.Allen, F.E.,"Bandsaw Tooth and Gullet Design", unpublished engineering notes.Allen, RE., 1973a. "High-strainfThin Kerf," Proceedings of the First North American sawmill Clinic, Portland, Oregon.Allen, F.E., 1973b. "Quality Control in the Timber Industry", Australian Forest Industries Journal,(September).Alspaugh, D.W., 1967. "Torsional Vibration of a Moving Band," Journal of the Franklin Institute,Vol. 283, No. 4, pp. 328-338.Breznjsk, M. and Moen, K., 1972. On the Lateral Movement of the Bandsaw Blade UnderVarious Cutting Conditions, Norsk Treteknisk Insitutt, No. 46.Brown, T.D. (Ed.), 1982. Quality Control in Lumber Manufacturing, Miller FreemanPublications, San Francisco.Chardin, A., 1957. "L'Etude du Sciage par Photographie Ultra-rapide," Bois Forets desTropiques, No. 51, pp. 40-51.Foschi, R.O., and Porter, A.W., 1970. Lateral and Edge Stability of High-strain Band Saws,Canadian Western Forest Products Laboratory, Report VP-X-68, 17 pages.Foschi, R.O., 1975. "The Light-Gap Technique as a Tool for Measuring Residual Stresses inBandsaw Blades," Wood Science and Technology, Vol. 9, pp. 243-255.Franz, N.C., 1957. An Analysis of the Wood Cutting Process, University of Michigan, AnnArbor, Mich.Fujii, Y., Hattori, N. and Noguchi, M. and Okumura, S., 1984. "The Force Acting on Band Sawand the Sawing Accuracy," Bulletin of the Kyoto University Forests, No. 56, (November),pp. 252-260.Fujii, Y., Katayama, S. and Noguchi, M., 1986. "Measurement of Bandsaw Deformation withMoire Topography, "Journal of the Japanese Wood Research Society, Vol. 32, No. 7, pp.498-504.Gro ndlund, A. and !Carlsson, L., 1980. "Praktiska rad vid Bandsagning", STFI-Meddelande serieA 666. 41 pages.Hutton. S G and Taylor, J., 1991. "Operating Strcsacs in Dambaw DIddeS and their Effect onFatigue Life," Forest Products Journal, Vol. 41, No. 7/8, pp. 12-20.103Johnston, J.S. and St. Laurent, A., 1975. "Tooth Side Clearance Requirements for High PrecisionSaws", Forest Products Journal, Vol. 25 No. 11, pp. 44-49.Jones, D.S., 1965. "Gullet Cracking in Saws," Australian Timber Journal, (August), pp. 22-25.Jones, D.S., 1968. "A Psychiatric Examination of a Bandsaw...(or Why a Bandsaw Cracks,"Australian Timber Journal, (June), pp. 63-69.Kirbach, E., 1985. A Procedure to Accurately Determine the Minimum Side Clearance for Wood Cutting Saws, FORINTEK Canada Corp., 11 pp.Kirbach, E. and Stacey, M, 1986.  Problems and Solutions in Maintenance and Operation of Band Saws, FORINTEK Canada Corp. 25 pages.Koch, P. ,1964. Wood Machining Processes, Ronald Press, New York.McKenzie, W.M., 1961u. Fundamental Analysis of the Wood-Cutting Process , Department ofWood Technology, School of Natural Resources, University of Michigan, Ann Arbor, 151pp.Mote, C.D., 1965. "Some Dynamic Characteristics of Band Saws", Forest Products Journal, Vol.15, No. 1, pp. 37-41.Mote, C.D., 1968. "Divergence Buckling of an Edge-loaded Axially Moving Band", J. Applied Mechanics, Vol. 33, pg. 463.Mote, C.D. and Holoyen, S, 1975. "Confirmation of the Critical Speed Theory in SymmetricCircular Saws" ASME, J. of Engineering for Industry, Vol. 97(B), No. 3, pp. 1112-1118.Okamoto, N. and Nakazawa, M., 1979. "Finite Element Incremental Contact Analysis withVarious Frictional Conditions", International Journal for Numerical Methods Engineering,Vol. 14, pp. 337-357.Pahlitzsch, G. and Putkammer, K., 1976. "Ermittlung der Steifhiet von Bandsageblattern", Holz alsRoh- und Werkstoff, Vol. 31 , pp. 161-167.Pahlitzsch, G. and Putkammer, K., 1974. "Beurteilungskriterien fur die Auslenkungen vonBandsageblattern", Holz als Roh- und Werkstoff, Part 1: Vol. 32 , pp. 52-57; Part 2: Vol.32, pp. 295-302; Part 3: Vol. 34, pp. 413-426.Pahlitzsch, G. and Putkammer,K., 1976. "Schittversuche beim Bandsagen", Holz als Roh- und Werkstoff, Part I: Vol. 34 , pp. 17-21, Part II: Vol. 34, pp. 17-21.Peirgen, H-0, and Saupe, D. (Eds.), 1988. The Science of Fractal Images, Springer-Verlag, NewYork.Porter, A.W., 1971. "Some Engineering Considerations of High-Strain Band Saws," Forest Products Journal, Vol. 21, No. 4, pp. 24-32.Reineke, L.H., 1956. "Sawing Rates, Sawdust Chambering and Spillage", Forest Products Journal, Vol. 6 No. 9, pp. 348-354.Soler, A.I., 1968. "Vibrations and Stabilit of a Moving Band " J:I, 10.II 1_,^1^1104St. Laurent, A., 1970. "Effects of Sawtooth Edge Defects on Cutting Forces and SawingAccuracy", Forest Products Journal, Vol. 20 No. 5, pp. 33-40.St. Laurent, A., 1971. "Influence des noeuds sur les forces de coupe dans le sciage du bois",Canadian Journal of Forest Research, Vol. 1, pp. 43-56.Szymani, R., 1985. "Wood Properties and Characteristics Which Are of Concern in Machining",Wood Machining News, March/April, pg. 2.Taylor, J., 1985. "The Dynamics and Stresses of Bandsaw Blades," Master's thesis, Department ofMechanical Engineering, University of British Columbia, Vancouver, British Columbia,Canada.Taylor, J. and Hutton, S.G., 1991. "A Numerical Examination of Bandsaw Blade ToothStiffness," FORINTEK Canada Report, 17 pages.Tseng, J., 1980. "The Application of the Mixed Finite Element Method to the Elastic ContactProblem." Master's thesis, Department of Civil Engineering, University of BritishColumbia, Vancouver, British Columbia, Canada.Ulsoy, A.G., 1979. Optimizing Saw Design and Operation - Vibration and Stability of Band Saw Blades: A Theoretical and Experimental Study, Technical Report 35.01.330, University ofCalifornia Forest Products Laboratory, Richmond California, 117 pages.Ulsoy, A.G., Mote, C.D., 1980. "Analysis of Bandsaw Vibration", Wood Science, Vol. 13, No. 1.Ulsoy, A.G., Mote, C.D. and Szymani, R, 1978. "Principle Developments in Band Saw Vibrationand Stability Research", Holz als Roh- and Werkstoff, Vol. 36, pp. 273-280.Wang, K.D. and Mote, C.D., 1985. "Vibration Coupling Analysis of Bandsaw Systems",Proceedings of the 8th Wood Machining Seminar, Richmond, California, October 7-9, pg.416.105APPENDIX AInertial and Gyroscopic Effects on Blade BehaviorThe purpose of this appendix is to show that the inertial and gyroscopic effects are smallcompared to the effect of the blade stiffness for the range of excitation frequencies encounteredduring sawing.During cutting the blade generally deflects to a bowed shape, similar to the lowest vibrationmode. Assuming that the Galerkin shape function for the lowest lateral vibration mode is111(x,y) = Sin(nx/L)the terms in the matrices of the Galerkin formulation for the lowest vibration (n=1) mode are1113L^8p.cbM — G — ^2 3K=D2 I-((704(ii) —(Kpbc2LFor typical values of the blade parametersb = 10 inches^K = 0h = 0.065 inches^p = 7.34(104) Lb-sec2/in4L = 30 inches^E = 30.0(106) psiT = 15000 lbs.^v = 0.3c = 2000 in/sec106the sizes of the inertial force, Mw2, the gyroscopic force Go), and the stiffness, K, for variousexcitation frequencies are given in Table A-1.Since the cutting motion is almost entirely below 5 Hz., the stiffness effect dominates thegyroscopic effect by more than an order of magnitude and the inertial effect by more than twoorders of magnitude. This result is consistent with those of Taylor [19851, who measured thetransfer function for a bandsaw at various speeds. His results, for a blade similar to that consideredhere, show that the transfer function is constant over the range of 0 to 30 Hz.Table A-1. Inertial, gyroscopic and stiffness termsExcitationFrequency(Hz)Inertial(Lbs-sec2)Gyroscopic(Lbs-sec)Stiffness(Lbs)in in in0 0.0 0.0 12341 0.3 16.0 12345 6.7 73.5 123410 27.0 146.9 123420 107.9 294.0 123430 243.0 441.0 123440 431.0 588.0 123450 674.0 735.0 123460 971.0 882.0 123466.1 1234.0 971.0 1234107APPENDIX BVERIFICATION OF CALCULATION OFBLADE STIFFNESS AND NATURAL FREQUENCIESCOMPARISON TO MOTE'S UPPER AND LOWER BOUNDS FOR NATURALFREOUENCYMote [1965] developed equations for the upper and lower bounds of the first lateral naturalfrequency of a bandsaw. The lower bound assumes the blade has no bending stiffness, and theupper bound is a solution of limited series Galerkin approximation.For typical values of the blade parametersb = 10 inches^x=0h = 0.065 inches^p = 7.34(104) Lb.-sec2/in4L = 30 inches^E = 30.0(106) psiT = 15000 lbs.^v = 0.3c = 2000 in/secMote's lower and upper bound frequencies are 59.0 Hz and 60.1 Hz., respectively. The computerprogram developed for this thesis calculated the natural frequency to be 59.7 Hz., which isbetween the bounds estimated by Mote. In addition, by setting Young's modulus, E, equal to zeroin the program, which is Mote's condition for the lower bound, a frequency of 59.0 Hz wascalculated.108BLADE STIFFNESS AND DEFLECTED SHAPEIn this section the model is tested for its ability to predict how a blade deflects in response tolateral loads. A system of weights, cords, pulleys and bars applied a lateral load to a precise pointin the blade. The resulting deflections were measured with a dial gauge.The dimensions and parameters for the blade were:Blade thickness^0.049 inches^Wheel width^9 inches (crowned)Blade width^9.47 inches^Tooth pitch^1.75 inchesSpan^30 inches^Tooth depth^0.65 inchesBandmill strain^13,840 lbs.^Tensioning^NoneBlade overhang on the wheels: front^1/8 inchesback 5/16 inchesFigure B-1 shows how the stiffness varied across the width of the blade at mid-span. Theanalytical prediction compares well with the actual stiffness. The most notable difference is theasymmetry of the experimental results. It is likely that this was caused by the crown, which had itshighest point about one third of the wheel width from its front edge, thus causing the frontportion of the blade to be slightly tighter than the back.Figure B-2 shows how the blade deflects for a lateral load applied to the tooth-tip. The deflectionis normalized to the tooth-tip deflection as is done in Figure 4-16. Again, the correlation betweenthe analytical and experimental results is good, except that the front of the blade is resistingdeflection more than the back.8 90^I^2^3^4^5^6^7DISTANCE FROM GULLET BOTTOM (INCHES)v 500Z■--cnm--I 400u)(I)wzu_ 300u.I—U)I.0••0.5p< ••- MODEL• EXPERIMENT•0•0^I^2^3^4^5^6^7^8^9DISTANCE FROM GULLET BOTTOM (INCHES)1■4C111APPENDIX CTHE CONTACT ALGORITHMOVERVIEWThe purpose of this appendix is to describe the theory and structure of the algorithm developed tocalculate the constrained deflection of the blade. During sawing, the loads on the blade are appliedby both the external loads (i.e., the lateral cutting forces) and the constraints. The problem is todetermine which portions of the blade come into contact with the sawn surfaces and the magnitudeof the contact forces.A common method for solving nonlinear problems, such as the contact problem, is to incrementallyincrease the load, starting from zero, until the full load is attained. However, it is likely that thelateral cutting force could be zero when the body of the blade is in contact with the sawn surfaces.Hence, the load increment method cannot be used. An alternative method is to apply the fullexternal loads to the blade, initially assuming that the constraints are withdrawn, and thenincrementally move the constraints towards the deflected blade until they are in their final position.In the case of sawing, the constraints are two parallel sawn surfaces which pinch the blade betweenthem.The principle of the contact solution algorithm is to determine the order in which the nodes comeinto contact or lose contact as the constraining surfaces are moved inward. A characteristic ofnonlinear problems is that the final solution is dependent upon the order that the loads (or in thiscase, constraints) are applied to the structure. If the correct order of contact is not found then thealgorithm could give an incorrect solution or cyclically fix and free a set of nodes without everbreaking out of the loop. A correct solution is guaranteed only if the order of contact is correct.The algorithm determines the amount that the constraints should be incremented inward so that112only one node will just come into contact or just lose contact. Between these points where thestatus of a node (fixed or free) just changes, the system is piece-wise linear because the bladedeformation is linear. Hence, between points where the status of a node does not change thechanges in the deflections of the structure and in the magnitude of the contact forces will varylinearly with the movement of the constraints. The basis of the algorithm is to use linearinterpolation to determine the size of the increment that causes a node to change status.THE ALGORITHMThe first step is to determine which degrees of freedom of the blade, {x}, could come into contactwith the surface and the location of the constraints, Un and Vn, for each node. These steps aredescribed in Chapter 3. The algorithm presented here is general in that an external force can beapplied to any node on the blade whether or not that node is constrained. In this application,however, the only external forces are those applied to the tooth tips, which are assumed to haveno constraints.Equilibrium Conditions for the Blade The discrete equilibrium equation is[A]{F} = {x} (C.1)where [A] is the flexibility matrix of the blade. In the references [Okamoto and Nakazawa, 1979;Tseng, 1980], the contact problem is formulated as a mixed problem, meaning that bothdisplacements and forces can be independent variables. Assume for now that the nodes that are incontact are known. The equilibrium equation can be rewritten as(C.2)113whereXf are the free (unconstrained) degrees of freedomXc are the fixed (constrained) degrees of freedomFe are the external loads acting on free degrees of freedom.Fc are the net loads required to enforce the constraint that act on the fixed degrees offreedom.The external loads {Fe} are known. The displacements of the constrained degrees of freedom,{Xc} are also known because these nodes are known to be in contact with the constraint. Thevectors {Fc} and {Xf} are unknown, but can be solved by evaluating the following equations:[A22] {Fc} = {Xc} [A211{Fe}^ (C.3){Xf} =^11 {Fe} + IAl21{Fc} (C.4)Since Equation C.3 is solved with a simultaneous equation routine, this formulation results infaster computer solutions if the number of degrees of freedom in contact is less than half of thetotal number that could come into contact. However, if most of the degrees of freedom come incontact, then the formulation^K 11 K12 Xf^Fe^[1(21 K22 X c^Fc (C.5) should be used because of the large number of times these equations must be solved. For thecontact between the blade and the sawn surfaces, few nodes are expected to come into contact sothe flexibility formulation was used.deflection equal to the position of the constraint. The contact force is equal to F c only if there is114no external load Fn acting on that node. If the contact force, cp, is taken to be positive incompression, then{Fn - Fc contact on the positive x side of bodycp = Fc - Fn contact on the negative x side of body0^no contact(C.6)Conditions Consistent with Contact The conditions that define whether the deflections of the elastic body are consistent with theconstraint are:1) none of the constraints are exceeded2) where contact occurs, the contact force is compressive (i.e., cp > 0)These conditions must apply for each increment of the algorithm. When these conditions andEquation C.2 are satisfied for the imposed constraints, then the contact problem is solved.Relationships for Piece-wise Linear Contact In the following derivation the index n is the node number and i is the increment number. Theamount that the constraints have been shifted outward is 13. For the first increment (i=1), pi is setso that only one node is in contact. This situation is shown in Figure C-1.The algorithm takes advantage of the property of piece-wise linear systems that changes indisplacements are proportional to changes in forces. Therefore, when nodes that are fixed to aconstraint are moved inward an amount efi, then the changes in the contact forces at the fixednodes, 4, and the changes in the deflections of the free nodes, Ax, are proportional to AO.The proportionality constant between Ax or Acp and A must be determined for each node. This> Ca115Figure C-1. Geometry of the Constraining Surfaces.116can be done by calculating {X f} and {cp} for two values of 13. Note that status of the nodes mustbe the same for both calculations. For this algorithm the two values of 0 are p=pi and 13=0. Todifferentiate the two solutions the variables for f3=0 have a '—' above them (i.e., ^cp,- etc.),while the variables for f3=i3i have no mark (i.e., X, cp, etc.).The amount by which a free node exceeds its constraint is termed the imbedment, which is definedas:xv"ni = x ni — Vn'n1 =Unxni—Pi—Pi(C.7)A- v^5cni^(C.8)uni = U n 5c niIn the following development öni is the general variable for O uni and Ovni; 8 ni is the generalvariable for 8 uni and b vni.Because the contact conditions at a node are determined by the imbedment at that node, it isconvenient to formulate the algorithm in terms of the imbedments rather than the nodaldisplacements. This can be done because the imbedments are also proportional to A.The values of 8 and cp as a function of t can found by interpolating between their values for f3=0and 13•=i. See Figure C-2(a) and (b). The relationships are:bni -^Pi (Oni - 8)^(8,31 :810(C.9) • -P (C.10)((p ni -cp)^(Tin - (Trin i)1176ni — 6^6 ni^6 ni Pi - R^Pi6ni0^PicPni — (1)^Tni^'ni 0i -^Pibni •0O000iSHIFT IN THE CONSTRAINTSFigure C-2. Geometry of the Interpolation for the Contact Algorithm.118Determining the Size of the IncrementThe size of the incremental change in 13 should be the smallest that causes the status of only onenode to change. A change in status occurs when either S or cp to becomes zero.1) Free Nodes: When 3 is reduced so that an imbedment ô ni goes from negative to positive, thenthat node is a candidate for being fixed. From Equation C.9 the reduction in 13i to have animbedment, S, of zero isAl3 =^oni Pi ^(C.11)(ô ni^)2) Fixed Nodes: Wheni3i is reduced, so that the contact force cpnl of a fixed node goes frompositive to negative, then that node is a candidate for being freed. From Equation C.10 thereduction in I to have a zero contact force, cp, iscP niPi 613 ni —(9)ni ci)ni(C.12)The node corresponding to the smallest value of 6,13ni (= Af3in) is then found. This node, andonly this node, is fixed or freed as required.The value of p for the next increment will be1341 = Pi^ (C.11)The values of 60 +1 and^can be found by solving equations (C.3) and (C.4), but it is fasterto use the interpolations of Equations C.9 and C.10. For the free nodesAPmin 8 n,i +1^uni^pi (uni - um ) (C.12)and for the fixed nodes119L3min (cPni - cT) ni )99 n, i +1 = (Pill^pi (C.13)The incrementing procedure given above is carried out until 13 = 0 and the contact conditions areconsistent.120APPENDIX DDERIVATION OF THE STATISTICAL REPRESENTATIONOF THE CUTTING FORCESSince the deflection caused by the mean cutting force dominates the deflection response of theteeth, it is convenient to separate the force distribution into two components: and mean force andthe variant from the mean.fit = m. + e.^; t = 1,2,..,N T(D.1); i = 1,2,...,N,where fit is the force on the t-th tooth at the i-th increment along the cut; mi is the mean force;and eit is the variant on the i-th tooth.The deflection of the teethl can be calculated, assuming no contact forces are present, by{X} = [Al{f}or^ (D.2)NTxit = jt fitwhere A is the flexibility matrix for the teeth. Substituting Equation D.1 into Equation D.2 results inNT^NTxit = mi ajt + Eait eitj-1^j-1(D.3)NT= mict + Iajt ettj-11 In the following development, xit is equivalent to the cut path Si t used in Chapter 3.121The values of ct are the deflections of the teeth when there is a unit load acting on each tooth.The cutting force fit has a mean f and a standard deviation, Sf, defined asN,N,f = 2 2 jNTNC (D.4)_ 2Ni N2 Sf =^2 ^ (D.5)j..1.i...1(NTNc —1)Both of these values depend on tooth geometry, sharpness, wood species and characteristics.There are also variations in m t and eit, defined ass2^1( 111 —1)2(Nc —1)2Se= NY eitj..1i.4(NT1sic —1)It can be shown that(D.6)(D.7)Si =^ (D.8)The mean deflection of the teeth is thenf NT— 2 ct —NT t=1^Keqand the standard deviation of the tooth deflection isf(D.9)^2 S v4 2^Nx; Nx: 2So =^2,Ct 2, zajtNT t..1(D.10)

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