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The effect of bandsaw stresses of blade stiffness and cutting accuracy Taylor, John 1993

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THE EFFECT OF BANDSAW STRESSESON BLADE STIFFNESS ANDCUTTING ACCURACYbyJohn TaylorB.A.Sc. The University of British Columbia, 1980.M.A.Sc. The University of British Columbia, 1986.THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS OF THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Mechanical Engineering.We accept this thesis as conformingto the required standard.THE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© John Taylor, 1993.In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of^Mechanical EngineeringThe University of British ColumbiaVancouver, CanadaDate^4th. October 1993.DE-6 (2/88)ABSTRACTThis thesis consists of an experimentally based study of theeffect of bandsaw stresses on blade stiffness and cuttingaccuracy for the bandmills used for primary breakdown in NorthAmerican sawmills. The effect of bandsaw roll-tensioning andbandmill axial forces on the stresses in the blade, on theinteraction of the blade with the bandmill and on the stiffnessand cutting accuracy of the sawblade are examined. The stressesdue to roll-tensioning are measured and empirical relationshipsdeveloped that relate the rolling load to the stresses induced.Investigations into effect of roll-tensioning on the torsionalfrequency and lateral stiffness of the sawblade show that rollingin the centre 60% of the sawblade will increase blade stiffnesswhile rolling outside this region will decrease it. The optimumrolling location was shown to be the sawblade centre-line. Thestresses in the blade due to the interaction of the blade andwheel were analyzed and a model was developed that successfullypredicted the effects of roll-tensioning and overhang on thetracking stability of the sawblade. Experiments were conducted todetermine how the stresses, frequency and stiffness of thesawblade relate to its cutting accuracy. The experiments indicatethat cutting accuracy is strongly related to the lateral edgestiffness of the sawblade and that improved cutting.accuracy canbe obtained by confining the roll-tensioning to be close to theblade centre-line.iiTABLE OF CONTENTSABSTRACT ^TABLE OF CONTENTS ^  iiiLIST OF TABLES LIST OF FIGURES ^  viNOMENCLATURE ACKNOWLEDGEMENTS ^  xiv1. INTRODUCTION  ^11.1 Background1.2 Objective1.3 Approach1.4 Previous Work2. EXPERIMENTAL APPARATUS  ^ 142.1 Laboratory Bandmill2.2 Roll-Tensioning Machine.2.3 Data Acquisition.2.4 Cutting Accuracy.3. STRESSES INDUCED BY ROLL TENSIONING^  193.1 Experimental Work3.2 Analysis and Discussion3.3 Conclusions4. SAWBLADE INTERACTION WITH THE BANDMILL^ 464.1 Experimental Work4.2 Analysis and Discussion4.3 Conclusions5. THE EFFECT OF SAWBLADE STRESSES ON FREQUENCY ANDSTIFFNESS ^  895.1 The Effect of Rolling Position on Lateral Stiffnessand Torsional Frequency.5.2 The Effect of Bandmill Strain on Lateral Stiffness.5.3 Analysis and Discussion.5.4 Conclusions6. CUTTING EXPERIMENTS ^  1066.1 The Effect of Roll-Tensioning on Cutting Accuracy ^6.2 The Effect of Gullet Stress on Tooth Stiffness andCutting Accuracyiii7. CONCLUSIONS ^  131LIST OF REFERENCES  135APPENDIX I ^  138Finite Element Analysis Program and Models UsedAPPENDIX II  142Analysis of the Effect of Roll-Tensioning on TorsionalFrequencyAPPENDIX III ^  148Analysis of the Effect of Blade OverhangivLIST OF TABLESTable 3.1 Principal Strains Close to Rolled Zone^(AO.^.^. 27Table 3.2 Rockwell Hardness of Plates used in Rolling LoadExperiments^ 29Table 3.3 Rolling Loads for Professionally Tensioned Blade. 34Table 4.1 Strains in Blade due to Wheel Tilt Angles^ 57Table 4.2 Measured Values of Wheel Tilt, Backcrown andOverhang. ^ 58Table 5.1 Comparison of Final Frequency and Stiffness Valuesfor Partitioned Blade^ 93Table 6.1 Sawblade Data and Cutting Test Results. ^ 117Table 6.2 Lateral Tooth and Blade Stiffnesses BeforeTensioning. ^ 125Table 6.3 Increase in Edge Stress due to Roll-Tensioning.^. 127Table 6.4 Equivalent Bandmill Strains for Equal GulletStresses.  ^127Table 6.5 Lateral Tooth and Blade Stiffnesses AfterTensioning. ^ 127vLIST OF FIGURESFigure 1.1 Diagram of Bandmill^ 2Figure 1.2 Details of the Roll-Tensioning Process. ^ 3Figure 1.3 Measuring Roll-Tensioning in a Band^ 4Figure 1.4 Motion of a Sawblade During Cutting 8Figure 1.5 Frequency Spectrum of a Sawblade During Cutting.^. 9Figure 1.6 Damping of Sawblade Vibration During Cutting.^.^. 12Figure 2.1 Schematic of Roll-Tensioning Machine. ^ 15Figure 2.2 Instrumentation and Data Acquisition Equipment.^. 17Figure 3.1 Plate Rolling Geometry and Strain Gauge Locations. 21Figure 3.2 Measured Strains due to a Centre and Off-centre Ro1122Figure 3.3 Measured Strain due to Multiple Rolls^ 23Figure 3.4 Predicted Strain Distribution for a Tensioned Blade.24Figure 3.5 Strain Distribution for a Tensioned, StrainedBandsaw^ 25Figure 3.6 Magnitude and Direction of Measured Strains Close toRoll Path and at Plate Edges. ^ 26Figure 3.7 Measured Strain as a Function of Plate Width.^. 28Figure 3.8 Plate Rolling Geometry and Strain Gauge Locations. 29Figure 3.9 Measured Strain as a Function of Plate Thickness. 30Figure 3.10 Measured Strain as a Function of Rolling Load.^. 31Figure 3.11 Narrow Strip Rolling Geometry and Strain GaugeLocations. ^ 31Figure 3.12 Induced Strains vs Rolling Load. ^ 32Figure 3.13 Strain Gauge Locations and Rolling Geometry for aProfessionally Tensioned Blade 34Figure 3.14 Roll-Tensioning Stress Distribution Model.^. 36viFigure 3.15 Experimental and Theoretical Stress Distributions forTwo Rolling Positions. ^  38Figure 3.16 Longitudinal Plastic Strain Integral vs TransverseStrain Integral. ^  41Figure 4.1 Measurement Geometry of Backcrown Investigation. ^ 50Figure 4.2 Change in Backcrown and Wheel Tilt vs Bandmill StraifilFigure 4.3 Contact Zones and Relative Angles of the Blade andWheel^  52Figure 4.4 Strain Gauge Positions for Tilt Angle Experiment.^55Figure 4.5 Tilt Angle Strains. ^  56Figure 4.6 Measured Moment in Blade due to Overhang and Wheel^Tilt.  59Figure 4.7 Cylindrical Model of Blade on Wheel and Coordinates ofCylindrical Shell Element^  61Figure 4.8 Stress due to Interaction of Tensioned Blade andWheel^  66Figure 4.9 Forces and Moments in Strained Blade with Backcrown.67Figure 4.10 Sawblade Stresses due to Backcrown and Backcrown plusBandmill Strain. ^  69Figure 4.11 Cylindrical Model of Blade with Backcrown.^70Figure 4.12 Geometry of Crowned Wheel. ^  74Figure 4.13 Wheel and Blade Tilt Angles.  78Figure 4.14 Moment in Blade due to Overhang for Four BandTensions^  82Figure 4.15 The Effect of Blade Width and Blade Thickness onOverhang Moment. ^  84Figure 4.16 Comparison of Experimental and Analytical OverhangMoments. ^  85Figure 5.1 Rolling Geometry for Partitioned Blade.^. 89Figure 5.2 Blade Stiffness Measurement Geometry.  ^90Figure 5.3 Variations in Torsional Frequency and LateralStiffness vs Rolling Positions.  ^91viiFigure 5.4 Measured Changes in Light Gap and Torsional Frequencyvs Rolling Pattern. ^  93Figure 5.5 Comparative Stiffness of a Tensioned vs Un-TensionedBlade^  96Figure 5.6 Stiffness of a Tensioned and Un-Tensioned Blade vsBandmill Strain^  97Figure 5.7 Frequency and Lateral Edge Stiffness vs RollingPosition.  99Figure 5.8 Comparison of Analytical and Experimental Frequenciesvs Rolling Position^  100Figure 5.9 Comparison of Analytical and Experimental EdgeStiffness vs Rolling Position^  101Figure 6.1 Details of Centre Region Tensioning used in CuttingTests^  106Figure 6.2 Sample of Cutting Data. ^  108.Figure 6.3 Comparative Saw Performance: 14000 lb. Strain. . ^ 109Figure 6.4 Comparative Saw Performance: 19000 lb. Strain. . ^ 110Figure 6.5 Comparative Saw Performance: 25000 lb. Strain. . ^ 111Figure 6.6 Comparative Saw Performance Averaged Over the ThreeBandmill Strains. ^  112Figure 6.7 Cutting Accuracy vs Gullet Stress^ 113Figure 6.8 Cutting Accuracy vs Tooth Stiffness  116Figure 6.9 Mean Blade Displacement vs Tooth Stiffness.^. ^ 117Figure 6.10 Cutting Accuracy vs Gullet Stiffness^ 118Figure 6.11 Cutting Accuracy vs Gullet Stress.  119Figure 6.12 Diagram of Stresses for Conventional and CentreTensioned Saws^  121Figure 6.13 Strain Gauge Locations and Rolling Profile for CentreTensioned Blades  124Figure 6.14 Cutting Accuracy of Centre Tensioned vsConventionally Tensioned Saws. ^ 127viiiFigure 6.15 Cutting Accuracy vs Gullet Stiffness^ 128Figure 1(1) Models of Blade Used in Analysis  137Figure 11(1) Stresses and Displacements in Torsional FrequencyModel^  141Figure III(1) Cylindrical Model of Blade on Wheel. ^ 147Figure 111(2) Pressure, Forces and Moments Associated withOverhang^  147Figure 111(3) Coordinates, Forces and Moments used in OverhangAnalysis  150Figure III(4) Stresses, Displacements and Associated Pressures ofBlade on Wheel^  152^Figure 111(5) Approximation of Anticlastic Displacement.   155ixNOMENCLATUREChapters 1, 2 and 3.A = indentation area across roll patha = wheel radiusb^plate widthC i constants associated with roller dimensions and materialyield characteristicsd^distance of roll path from blade centre-lineE = Youngs modulusF r rolling loadh = plate thicknessI z transverse strain integral (2A/h)I o"= total axial plastic strainL,L,^span lengthm, = exponentn number of displacement samples per cut.N number of cuts.Rt = radius of transverse curvaturet = roller path widthVm the variance of the meansV = the mean of the variancesvj variance of sawblade displacement for cut 'j'.x i^a sample of sawblade displacement in cut 'j'.xJ = mean of sawblade displacement for cut 'j'.3i. = the average of the meansE = strainE,, = elastic strain in the z direction etc.E l„, = plastic strain in the z direction etc.E z = E p-z + E,, = total strain in the z = compressive stress in the roll patham = stress due to bending momenta, = tensile stress due to roll-tensioningax = stress in the x direction etc.Chapter 4 (additions)bw = width of blade on the wheelc = distance from wheel centre-line to wheel crownD = Eh3/12 (1-v 2 )d = distance from blade centrelined0 = small angleE = Youngs modulusLc = Chord length of backcrown gaugeLT = Total blade lengthM = bending momentMbc = moment due to backcrownM 1 ,2 = moments applied to satisfy equilibriumMt = moment due to wheel tiltNo = circumferential in-plane forces per unit width.RbC = radius of backcrowns = overhangxiw = radial displacement.wbc = deflection due to backcrownwe = height of wheel crownwo = minimum deflection to achieve full contactZ = radial pressureA = displacement of blade due to backcrown in centre of chordE 0 = circumferential residual strain.0 = angle of blade at tangent pointv = Poissons ratio0 = wheel tilt anglea s = stress due to bandmill strainChapter 5 (additions).x = meana = standard deviationAppendices I and II (additions).e l = non-dimensional roll path widthg = gullet depthG = shear modulusL,1 = span lengthP12 = sawtooth pitch on modelr/ = non-dimensional distance to roll pathT = kinetic energyUQ = strain energyW,W(x ,z) = torsional displacement of bladexi iE, strain in axial direction0 = angle of twist of blade= mass per unit volumea(x)^stress as a function of xQZ = stress in axial directionw = frequency (rad/s)Appendix III (additions).F = line load due to overhang moment M olFw = force on wheel excluding overhang effectsM0 edge moment due to overhangP^edge load to support overhangP = pressure on wheel excluding overhang effectss = blade overhangT^force on wheel excluding blade overhang effectsTo = band tensionwoo = displacement of bladewi (x)^initial shape of bladew0 = displacement of wheelw = distance of blade neutral axis from wheel= 3 (1 _ v 2 ) / a.2h26(x-0)^Dirac delta functionACKNOWLEDGEMENTSI feel privileged to have been associated with the people thatform the Wood Machining Group at U.B.C., I shall miss theirsupport and companionship. I would particularly like to thankProfessor Stan Hutton, the architect of the group, for hissupport and guidance throughout my studies. To fellow studentBruce Lehmann, thankyou, for all the discussions that we had, forreviewing my thesis and for your valued friendship. To my wifeand son, I could not have done it without you, your continuingsupport and understanding has meant a great deal to me.Finally, I would like to acknowledge with thanks, the financialsupport provided by the Science Council of British Columbiathrough the GREAT scholarship program and the generous suppportin time and money provided by my employer, Forintek Canada Corp.xiv1. INTRODUCTION1.1 BackgroundFrom information provided by the Council of Forest Industries ofBritish Columbia (COFI 1988), it is estimated that bandsawsproduce two to three billion board feet of lumber annually'.Because such a large volume of lumber is involved, losses inrevenue due to poor cutting accuracy can be of a significantproportion. As an example, using 1992 dollars, a moderate 2%improvement in recovery of lumber from logs would provide anincrease in annual revenue of about $250,000 for an average size(200,000 bf/day) mill.Bandsaws are widely used for the primary breakdown of logs,either as single headrigs, breaking large logs into manageablepieces (cants) for the other sawing centres in the mill, or intwo's or four's (twin or quad), breaking smaller logs into amixture of cants and lumber.A bandmill, schematically shown in Figure 1.1 , consists of acontinuous band which travels around two wheels. A loadingmechanism forces the wheels apart and applies tension to theband. The total force applied between the wheels is termed'This is based on bandsaws producing 20% to 25% of the totallumber production.1Figure 1.1 Diagram of Bandmill.bandmill "strain". The wheelsare generally crowned a smallamount for blade trackingstability and one wheel tiltsso the blade can be positionedwith the teeth overhanging theedge of the wheel.One of the principal factorsgoverning bandsaw performanceis the magnitude anddistribution of stresses inthe blade. These stresses aredue to the bandmill strain,the bending of the blade overthe wheel, the blade preparation techniques (roll-tensioning),the blade velocity, the tilting of the wheel, the crown on thewheel, the amount that the blade overhangs the edge of the wheeland the cutting forces. The largest are those due to bending,bandmill strain and roll-tensioning, as can be confirmed by thework of Eschler (1982), Pahlitzsch and Puttkammer (1972) andThunell (1972).The tensioning process used in the preparation of bandsaws hasbeen in existence for about 100 years. The process is still verymuch 'an art, with the skills being handed on by experienced2Figure 1.2 Details of the Roll-Tensioning Process.filers to apprentices working in the sawmill filing room. Theprocess of tensioning involves the cold rolling of the blade inthe longitudinal direction between two narrow rollers(Figure 1.2). The pressure between the rollers causes plasticdeformation of the blade material resulting in permanent residualstresses in the blade. The art of roll-tensioning involves theintroduction of these stresses in a manner that is conducive toimproving the performance of the bandmill. The roll-tensioningprocess increases the stiffness of the blade in a favourable way.During tensioning, the blade is rolled symmetrically with respectto its centre-line with the rolling proceeding from the centre-line outwards; also, an in-plane curvature is introduced intosingle cut blades (blades with teeth on only one edge) by rolling3TransverseCurvatureCircleTemplateBlade^4101111111111111111Rt^^ Light Gapy CircleA" Template:de raisednear to the back edge of the blade. This in-plane curvature iscalled "backcrown" and its effect is to elongate the back of theblade more than the front, this creates a higher stress andstiffness in the region of the teeth when strained by thebandmill. Backcrown is introduced to compensate for thetemperature gradient across the blade due to the dissipation ofthe cutting energy and is normally measured by determining the'height' of the curvature in the centre of a given chord lengthalong the back edge of the blade, e.g. 0.005 inches in 36 inches.In this thesis the radius of the backcrown curvature is alsoused.The sawfiler does not consider the tensioning process in terms ofinduced stresses. Instead, the degree of tensioning is assessedby measuring the deformations of the blade when the blade is bentinto a curve. Specifically, the curvature of the blade across itsFigure 1.3 Measuring Roll-Tensioning in a Band.4width is measured by shining a light through the gap between theblade and a specified radius template (Figure 1.3). The sawfilerthus assesses the amount of tensioning (cold rolling) required byensuring that some specified radius or "light gap" is obtained.Although the importance of bandmill strain and roll-tensioningare generally recognised, there is little consensus on what theoptimum levels of stress should be. The choice of bandmill straincan vary by a factor of two from one mill to another. The amountof stress in the sawblade due to roll-tensioning is assessed fromthe transverse curvature of the blade. Due to the subjectivenature of the process, and the lack of uniformity in training,this curvature varies from one filer to another. Additionally,Foschi (1975) has shown that small differences in transversecurvatures can be obtained with widely differing distributions ofstress, thus it is apparent that there are significant variationsin sawblade stresses from roll-tensioning.The process of tensioning is generally considered to be of utmostimportance to the satisfactory operation of the bandsaw. Theprocedure redistributes the stresses such that the edges of thesaw have increased stiffness, leading to improved cuttingaccuracy. Variations in the tensioning procedure will vary thedistribution and magnitude of stress, consequently:-^there will be variations in the stiffness and cuttingaccuracy of the sawblade,5- the cutting accuracy cannot be optimised with respect tothe distribution of stress in the blade,- the fatigue life of the blade cannot be controlled andfatigue cracks may occur.To overcome these problems, it is necessary to have a knowledgeof the stresses in the blade and understand how these stressesaffect the cutting accuracy of the saw and the fatigue life ofthe blade.1.2 ObjectiveThe objective of this thesis is to determine the relationshipbetween the stresses in a bandsaw, induced during bladepreparation by roll-tensioning and in operation by bandmillstrain, and the cutting accuracy of the saw. This requires thatwe determine:1. How roll tensioning affects the stress distribution in thesaw.2. How the stresses and displacements from roll-tensioningaffect the interaction of the sawblade with the bandmill.3. How the resulting stresses affect the natural frequenciesand lateral stiffness of the saw.4. How the stresses, natural frequencies and lateral stiffnessof the saw relate to its cutting accuracy.1.3 Approach6The approach used in this thesis has been to conduct experimentsto empirically determine the relationships described by the foursteps in the objectives. The results are then discussed andcompared to previous analytical and experimental investigations.Where necessary, analytical methods have been developed forcomparison with the experimental results. These include a finiteelement model of the cutting span of the sawblade for theanalysis of the natural frequency and lateral stiffness, and ashell model of the blade over the wheel for the analysis of theblade-wheel interaction.In this investigation the static stiffness of the blade isconsidered to be one of the most important factors associatedwith cutting accuracy. The motion of the sawblade under typicalcutting conditions is shown in Figure 1.4 and is composed of asmall high frequency element superimposed on a large lowfrequency oscillation. The low frequency motion is related to thestatic stiffness of the saw and accounts for the majority of theblade displacements in the cut. Figure 1.5 shows the frequencyspectrum of the sawblade for a typical cutting condition and themajority of the blade motion can be seen to occur below 10 Hz.,well below the fundamental lateral and torsional frequencies.Thus this investigation focusses upon the stresses andstiffnesses of a stationary sawblade.7Figure 1.4 Motion of a Sawblade During Cutting.One of the problems identified in the introduction was the effectof the stresses in the blade on the fatigue life of the saw. Thework in this thesis investigates the effect of the roll-tensioning process on the stresses in the saw and how thesestresses affect the stiffness and cutting accuracy for a range ofstress levels. The effect of the magnitude of stress on thefatigue life of the blade has been investigated by Allen (1985),Hutton and Taylor (1990) and Pahlitzsch and Puttkammer (1972) andis not included here.835 30...^ ..A.4111116......s..41..... -0^10^20^30^40^50^60^70^80^90^100Frequency (Hz.)As previously discussed, the largest stresses in the cuttingregion of the saw are due to roll-tensioning and bandmill strain.Once the relationship between these stresses and the cuttingaccuracy of the saw have been established, it is natural toconsider the optimization of these stresses. This has beenconsidered in this investigation and sawblades with optimaltensioning and bandmill strains were shown to reduce the standarddeviation of the sawblade deflections, measured during cutting,by 30 to 40%.1.4 Previous WorkThe relationship between the tensioning process and the resultingstress distribution in the bandsaw has been discussed by Anderson(1975), Aoyama (1971a), Bajkowski (1967) and Schajer (1981).However, the effect of multiple rolls (>3), incrementallyincreasing rolling loads and conventional tensioning by sawmillpersonnel has not been examined.The relationship between the in-plane stress distribution and thetransverse curvature of the sawblade was analyzed by Foschi(1975). From this work, Allen (1985) developed methods ofestimating the amount of roll-tensioning stress in a saw bladefor an assumed parabolic distribution. However, Foschi (1975)concluded that the transverse curvature is not a good indicatorof the distribution of stress in the saw, consequently, reverse10analysis to obtain the stress distribution from the transversecurvature is not reliable.The stresses in the blade due to the blade overhanging thebandmill wheel have been examined by Sugihara (1977). However,the rigid nature of the wheel was not included in the analysis.This results in a large peak in the contact pressure at the frontedge of the wheel, an effect that has been observed inexperimental results by Fujii, Hattori and Noguchi (1988) andSugihara, Hattori and Fujii (1981). This effect is included inthe model presented here. The stresses in the blade due to theinteraction of the blade and wheel, in the presence of roll-tensioning and backcrown, has been the subject of experimentalexaminations by Fujii et al. (1988). An analytical examination ofthe pressures and stresses has not been conducted.The natural frequencies of the bandsaw blade have been thesubject of many investigations, Alspaugh (1967), Anderson(1974), Kirbach and Bonac (1978), Mote (1965a,b), Soler (1968),Tanaka and Shiota (1981) and Ulsoy and Mote (1980). These havebeen directed at determining the effect of blade speed and in-plane loads on blade resonance. On occasion, reductions in kerfand improved cutting accuracy are cited as justification for theinvestigation. However, the relationship between vibrationalstability and improved performance has not been demonstrated. Infact, once the saw enters the wood, the natural damping11associated with cutting will significantly reduce the magnitudeof blade vibration (Figure 1.6) and the displacement of the bladewill be primarily due to the cutting forces. The comparison ofblade frequency with lateral stiffness and cutting accuracy hasnot been examined.Foschi and Porter (1970), Garlicki and Mirza (1972) and Saito andMori (1970) have examined in-plane buckling as a guide to theeffect of the various operating parameters on saw performance.The investigation by Saito and Mori (1970) also included theeffect of teeth in the experimental buckling measurements.However, the relationship between buckling loads and feed forces,and the behaviour of the sawblade compared to blade bucklingbehaviour, has not been established.Foschi and Porter (1970) also examined the lateral stiffness ofthe sawblade centre-line as a guide to the effect of the bladeparameters on lateral stability. The lateral stiffness of thecutting edge of the blade as a guide to cutting accuracy has notbeen examined.132. EXPERIMENTAL APPARATUSThe experiments conducted during the course of this investigationrequired the use of a number of different machines, items ofequipment and instruments. In this section each of these aredescribed and the relevant information given. The methods used todetermine cutting accuracy are also presented.2.1 Laboratory BandmillThe bandmill was a standard, single column, 5 ft. re-saw withseven-stage setworks. The welded steel wheels were 5 ft. indiameter by 9 in. wide with a crown of approximately 0.005 in.The distance between the wheel centres was 94 in. Tension wasapplied to the band via a hydraulic straining system that couldprovide up to 30,000 lb. of force (or "strain"). The bottom wheelof the bandmill was driven by a hydraulic motor fed from a swashplate type hydraulic pump and 100 horsepower electric motor. Theblade speed could be varied from zero to 11,000 feet per minute.The blade was guided in the cutting region by a pair of pressureguides set 0.25 in. to 0.375 in. out from the wheel tangentpoints. The span between the guides was normally set at 30 in.The wood to be fed into the bandsaw was mounted on a rigidlyconstructed carriage running on precision aligned rails. Thecarriage was driven by a separate hydraulic system via a cable14RollingWheelsPlateLoad appliedby handand drum. The carriage feed speed could be varied from zero to480 feet per minute.2.2 Roll -Tensioning Machine.In the roll-tensioning process the bandsaw blade is rolled in theaxial direction between a pair of small hardened steel rollers(Figure 2.1). The load is applied to the rollers via a lever andFigure 2.1 Schematic of Roll-TensioningMachine.screw mechanism and the pressure between the rollers is such thatthe sawblade material is locally deformed. This deformationinduces residual stresses into the sawblade. The magnitude of the15load was determined from bonded resistance strain gauges, locatedat a stress concentration point on the support frame, and acalibrated Wheatstone Bridge circuit. The stresses in the bladewere also measured with strain gauges attached to both sides ofplate. The rollers were 4.0 inches in diameter by 0.5 inchesthick with a 10 inch radius crown. The rolling speed was 9.3inches per second.2.3 Data Acquisition.The instrumentation and data acquisition equipment are shownschematically in Figure 2.2. Non-contacting displacement probeswere used to monitor blade displacement, a loadcell monitoredbandmill strain and a tachometer monitored the carriage speed.The signals were initially collected on a multi-channel,instrument quality, tape recorder. The signals were subsequentlyfed, via a Neff high speed data acquisition system, into a VAX11/750 computer for processing and graphical interpretation.A second, PC controlled, data acquisition system was developedduring the investigation reported here and used for some of thecutting experiments. This system monitored the blade deviationjust above the lumber surface and recorded the mean and standarddeviation of the blade motion for each cut.16CarriageCarriage-TachometerAmplifierMulti-channel A/D Converter^Data I/O toTape Recorder Multiplexer^ComputerisplacementTra sducersComputerTerminalPA-p".,41/MEM VAX 750 ComputerDigitalPlotterFigure 2.2 Instrumentation and Data Acquisition Equipment.2.4 Cutting AccuracyCutting accuracy was determined from the lateral displacement ofthe front edge of the sawblade during the cutting process. Thedisplacement was measured with a non-contact displacementtransducer, located just behind the gullet line of the saw andimmediately above the surface of the lumber being cut. For eachcut the mean and variance of the sawblade displacement weredetermined:17Ex,^E (xi -111 ) 2.5E• =^17.7 n-1 (2.1)Where:^i^1,2,3...,n; j^1,2,3...Nn number of displacement samples per cut.N number of cuts.x i = a sample of sawblade displacement in cut 'j'.mean of sawblade displacement for cut 'j'.vv variance of sawblade displacement for cut 'j'.Thus, for a set of cuts, several measures of blade deviation areavailable:r '7=^;^- E (573.--)71 2N(2.2)_LV =^; a = V+Vm) 2Where:^x = the average of the meansVm the variance of the meansV = the mean of the variancesa = the total standard deviation of the set of cutsIn this thesis both a and^are used to define the cuttingaccuracy of the sawblade.183. STRESSES INDUCED BY ROLL TENSIONING.This Section of the work determines the magnitude anddistribution of stresses in the sawblade due to roll-tensioning.The relationship addresses the first step in the sequence ofevents outlined in Section 1. In this section experiments areconducted to determine the effect of rolling position, number ofrolls, plate width, plate thickness and rolling load on themagnitude and distribution of stress in sawblade material. Theroller paths are confined to previously unrolled sections of thesawblade. The results are compared to other analytical andexperimental investigations.3.1 Experimental WorkThe aim of the experimental work was to measure the stressesinduced in sawblade material for a range of rolling conditions.The axial and bending stresses induced by rolling were determinedfrom the surface strains measured with bonded resistance straingauges attached to both sides of the plate. The rolling load wasdetermined from strain gauges attached to the roll-tensioningmachine and a commercial signal conditioner based on a WheatstoneBridge circuit. The rolling load was displayed in lbs.193.1.1 Single and Multiple Rolls.In this experiment, a section of saw plate is rolled seven timesin the axial direction and the strains induced in the plate byeach roll are recorded. The section of saw plate was 10.25 in.wide by 0.058 in. thick by 48 in. in length. The location of therolls and the order of rolling is shown in Figure 3.1a, alongwith the location of the strain gauges used to measure thestrains.The strains induced by the centre roll and one of the off-centrerolls are presented in Figure 3.2. The cumulative effect of theseven rolls is shown in Figure 3.3. It is apparent from theseresults that the distribution of tensile stress outside therolled region is approximately linear. The distribution ofcompressive strain in the rolled region is unknown, the strainsshown in the experimental results were obtained from equilibriumconsiderations, assuming a distribution of compressive strain inthe form of a truncated cone, and may not be representative ofthe actual values. The dimensions for the truncated conical shapewere 0.25 inches in width at the zero stress point and 0.125inches at the maximum compressive stress point.Figure 3.4 shows the calculated residual stress distribution in abandsaw that has been rolled 19 times in a symmetric fashion.Such a rolling pattern corresponds to industrial practice for theblade considered. In this case it has been assumed that the20Figure 3.1 Plate Rolling Geometry and Strain Gauge Locations.21Figure 3.2 Measured Strains due to a Centre and Off-centreRoll.22• 03C-500-W-1000--12000200 -100 -01 1 1 1 1 1 1 1 1 11 2 3 4 5 6 7 8 9 10Blade Width (in.)6^4^2^1^3^5^7111111111111AFigure 3.3 Measured Strain due to MultipleRolls.rolling proceeded from the centre-line such that after every oddnumbered roll the pattern was symmetric.Figure 3.5 shows the stress distribution in the same sawbladeafter mounting on the bandmill and strained such that a uniformtensile stress of 12000 psi. is superimposed upon the stressesshown in Figure 3.4. Thus the result of the rolling process is tomodify the stress distribution of an untensioned blade such thatregions of uniform tensile stress (higher than provided bybandmill strain) exist at both edges of the blade.23500 -375 ----...,To'c111C4%)-500 --700 - L19 Rolls @ 3900 lb.-1 000 IIIIIII^III0 1 2 3 4 5 6 7 8 9 10Blade Width (in.)Figure 3.4 Predicted Strain Distributionfor a Tensioned Blade.When measuring the strains due to rolling some variation wasnoted in the strain distributions close to the rolled region. Toinvestigate the behaviour of the induced stress in this region,an experiment was conducted to measure the principal strainsclose to the roll path and to compare them to the axial strainsin the edges of the blade. A section of 16 ga. bandsaw steel, 36in. long by 10.25 in. wide was used for the experiment.Figure 3.6 shows the location of the six pairs of strain gauges(numbered 1,4,5,8,9 & 10) and four pairs of strain rosettes(numbered 2,3,6 & 7) used to measure the strains in'the plate.The distance from the centre of the roll path to the centre ofthe rosettes was 0.58 in., this left 3/16 in. between the edge of241000 -775 "---47--,■ " a eN, IS, I., •,.. I, I, 4, " .., r, I,C-E0Toc17500 - 0M----to^400 - cU)c FEla. E65 Cmm30019 Rolls @ 3900 lb.-500 IIIIIIIIII1 2 3 4 5 6 7 8 9 10Blade Width (in.)Figure 3.5 Strain Distribution for aTensioned, Strained Bandsaw.the roller and the edge of the rosette. The roller was 1/2 in.wide, the roll path was approximately 1/4 in. wide.The plate was rolled along the centre-line at a load of 4000 lb.and then 5000 lb. The measured strains and their co-ordinatedirections are presented in Table 3.1 and are also shown onFigure 3.6.From the data it is apparent that the principal stresses, whetherclose to the roll path or in the body of the plate, are in theaxial direction. The average strains close to the rolling lineare 950 of the strains at the edge of the plate. Consequently,250x 35^90°--1' 0 0 --^0 0 ---■37iso-^L 00 . 31 - axial microstrain -10 0c? 1.90 t-^ 37 @ -3°^at-* 34 @ 1°CD*-► 32 @ 0°eg0® 4-* 30 @ 0°bco@^amicdrgtrainndirectionO036 c) 035 -,..Figure 3.6 Magnitude and Direction of Measured Strains Close toRoll Path and at Plate Edges.95% of the transition from compressive stress in the rolledregion to tensile stress in the plate occurs in a narrow region(0.58 in.) close to the roller path. The existence of thetransverse compressive strain is due to the Poisson's ratioeffect, though the measured values are only about 16% of theaxial strain. This is possibly due to the error in measurementcoupled with the very small values of strain.3.1.2 Plate Width, Plate Thickness and Rolling Load.To measure the effect of plate width on induced stress fourdifferent widths of 17 gauge plate were used. Each width wasrolled three times at 3,900 pound load (Figure 3.1b), one roll onthe centre-line and one roll 1-1/4 inches each side of the26Table 3.1 Principal Strains Close toRolled Zone (AE).centre. The results arepresented in Figure 3.7.The curve running throughthe data points is thepredicted strain based onthe results of the multipleroll experiment in theprevious section. It isapparent that the axialstrains in the rolled zoneare essentially a functionof rolling load and thetensile stresses in theremainder of the blade(determined fromequilibrium considerations)are inversely proportional to the blade width.To determine the effect of plate thickness and rolling load oninduced stress, five plate thicknesses were rolled with loadsranging from 3,000 pounds to 6,000 pounds (Figure 3.8). The platethicknesses were 17 ga. (0.057 in.), 16 ga. (0.065 in.), 15 ga.(0.072 in.), 14 ga. (0.083 in.) and 13 ga. (0.095 in.). The loadswere distributed over two plates to reduce the cumulative stressand avoid in-plane buckling.GaugeNo.Strain(10Dir'n01 31 0°2, 34 1022 -5 91°3 1 30 0°32 -5 90°4 35 0°5 35 0°61 37 -3°62 -8 87°7 1 32 0°72 -4 90°8 36 0°9 37 0°2717 ga. Plate3 Rolls at 3900 lb.. .0S.O Data^.— — Predicted Strain2^1^I^1^1^i4 6^8^10 12Plate Width (in.)20-0220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -11 0xXFigure 3.7 Measured Strain as a Function ofPlate Width.The load vs strain data are presented in Figure 3.9. It wasexpected that the induced stress for a given rolling load woulddecrease with increasing thickness. This is supported by the 17,16 and 15 gauge results but not by the 14 and 13 gauge results.Subsequent hardness measurements of the plates used in the testare presented in Table 3.2 and show that the hardness of the 14and 13 gauge plates are respectively two points and four pointslower than the 17, 16 and 15 gauge plates. This would indicate alower yield point and thus greater induced stress from rolling,as witnessed in the experimental results.28Rolling Pattern for Each Gauge.Plate Rollinq Line Load (lb)1 1 and 2 30001 3 and 4 60002 1 and 2 40002 3 and 4 5000< 1" //^i n/ / / in^1"^1" 1 '4 Rolling linesI^10 0 0®a2 Pairs of straingauges5 Plate thicknesses13 ga. to 17 ga.Plate Thickness and VariableLoad Experiment.Figure 3.8 Plate Rolling Geometry and Strain Gauge Locations.Figure 3.10 shows the induced stress as a function of rollingload for the 17, 16 and 15 gauge plates. For the range of rollingloads considered, linear approximations closely fit the datapoints for all three gauges with small variations in slope andlarge variations in intercepts. The intercept being the linearprediction of the rolling load at which yielding commences.29 5045 -40 -VJ35 -30 -25 -20 -15-10-5 1^1^ 1^1^1^I^10.055^0.065^0.075^0.085^0.095Plate Thickness (in.)0 3000 lbs. 0 4000 lbs. 0 5000 lbs. A 6000 lbs.Figure 3.9 Measured Strain as a Function ofPlate Thickness.Table 3.2 Rockwell Hardness of Plates used in Rolling LoadExperiments.Go 17ga 16ga 15ga 14ga 13ga1a44.330.5344.700.6644.550.5142.300.5740.850.75As the three thinner gauges had similar hardnesses one wouldexpect the yield points also to be similar. To examine thematerial behaviour in this region a narrow strip of bandsaw steel(Figure 3.11) was rolled in 250 lb. increments from . 0 lb. to 4000lb. The results (Figure 3.12) indicate the non-linear behaviourin the early stages of yielding changing to an almost linear3 050 -40 -0CC30 --S 20 —Esin io -Gauge^Slope(ge/1000 lb.)Interceptowo 17 ga. 10.11 1491A 16 ga. 9.91 1750o 15 ga. 9.62 2051CulTRolled Zone•^  36"i^2 Strain GaugesA0 1^3000 1^0-I^I4000 5000Rolling Load (Ib.)6000Figure 3.10 Measured Strain as a Function of Rolling Load.Figure 3.11 Narrow Strip Rolling Geometryand Strain Gauge Locations.relationship as the load increases.The last two experiments have looked at the effect of twoseparate rolls at specific loads and the effect of incrementally31Figure 3.12 Induced Strains vs Rolling Load.increasing the rolling load on a single line. If the roller loadis held constant, and repeated rolling is conducted on the sameline, Schajer (1981) has shown that the plastic deformation inthe rolled region decreases with increasing number of rolls,consequently, the induced stress will also decrease with eachroll. For rolling loads of 15 kN (3375 lb.) the indentation areadecreases in approximate proportion to the square root of thenumber of rolls. This becomes important when considering theeffect of rolling previously rolled sawblades. Should the rollerpath coincide with that of a previously rolled portion of thesawblade the plastic deformation will be reduced and less stresswill be induced into the blade than that predicted by the resultsof this section.32To this point, the experiments have been designed to examine theeffect of the basic variables of the roll-tensioning process onthe stresses induced in the sawblade. To help relate the resultsto industry practice, the stresses induced in a professionallytensioned section of blade were also measured. A 6 ft. long by10.25 in. wide section of 17 gauge bandsaw steel was equippedwith four pairs of strain gauges as shown in Figure 3.13. Aprofessional benchman was asked to tension the strip in the sameway that he would normally prepare a sawblade. The approximaterolling locations and the rolling loads were recorded as well asthe total stress induced in the edges of the sawplate.The location and magnitude of the individual rolls are shown inFigure 3.13 and Table 3.3. The stress induced in the edges of theblade by this rolling strategy was 12,940 psi. This comparesTable 3.3 Rolling Loads for Professionally Tensioned Blade.ROLL 1 2 3 4 5 6 7• S 9LOAD(lb.)3485 4130 4385 4455 3775 4270 4280 3523 3978ROLL 10 11 12 13 14 15 16 17LOAD(lb.)2385 2383 3640 3816 3835 3570 3800 3600reasonably well with the predicted value of 11,514 psi. from thelinear relationship in Figure 3.10. The maximum transversedisplacement (light gap) was in the centre of the blade and331"4_ .44 pairs of4'^strain gauges14^in0T..I(a) Location of Strain Gauges0 05)^0--..1,—..--C6)^0000o a+001 1(b) Approximate Rolling Positons.Figure 3.13 Strain Gauge Locations and Rolling Geometry for aProfessionally Tensioned Blade.34measured 0.080 in. Thus, for the 10.25 in. blade, the transversecurvature conformed to a 27 ft. circle.The professionally prepared strip was considered to be heavilytensioned, consequently, the magnitude of stress can beconsidered to be close to the upper level of tensioning stressfor this dimension of blade. The heavy level of tensioning wasindicated by: the transverse curvature corresponding to a 27 rather than the 35 ft. to 40 ft. normally quoted byindustry; severe in-plane buckling due to the magnitude of thestresses; and the edge rolls (10 and 11), put there to reduce thelevel of stress in the gullet and back edge of the blade to stopfatigue cracking.35RollerPathEqu'libriumTensileStressPrescribedCompressiveStressCombined Stress- • DistributionaTX 4—ac4EquilibriumMomentStress3.2 Analysis and DiscussionIt is apparent from the plate rolling results in Figure 3.2 thatthe distribution of stress follows a linear relationship withrolling position. A linear approximation of the stress is shownFigure 3.14 Roll-Tensioning Stress Distribution Figure 3.14 and is analogous to the distribution of stressfrom a narrow heated axial strip (Timoshenko and Goodier 1970).Force and moment equilibrium for the stresses shown inFigure 3.14 leads to the following relationship:36+ a —t + 12 a —tdy tcb^(3.1)ax(y) = -(7 c (y)^c b^c b 3Where: t = width of rolled zoneb = plate widthd = distance of rolling zone from plate centre-lineac = compressive stress magnitude in rolled zoneax = axial tensile stressThe first term on the right hand side is the compressive stressin the rolled region. The second term is the average tensilestress to balance the compressive rolling stress and the thirdterm is the moment stress due to the non-symmetry of the rollingposition and is related to the backcrown curvature. An averagevalue for the compressive stress can be determined empiricallyfrom the multiple roll results, where the mean tensile straininduced in a 10.25 in. wide by 17 gauge plate, for a 3900 lb.roll, was 20 microstrain. Approximating the stress in the rolledregion as a rectangular strip 0.25 in. wide, equilibrium dictatesan average compressive stress of 23,652 psi.Substituting this value of compressive stress in Equation (3.1),the distribution of tensile stress due to non-symmetric rollingwas calculated for rolling positions 6 and 7 (Figure 3.1). Theresults of these calculations are shown on Figure 3.15 along withthe experimental data. The results compare well with the measuredstress distributions indicating that the linear model is valid.37Figure 3.15 Experimental and Theoretical Stress Distributions forTwo Rolling Positions.From the plate thickness and rolling load experiments,Figure 3.10 shows that for the range of rolling loads considered(3000 lb. to 6000 lb.) the induced stresses are linear with load.This linear relationship has been demonstrated by Bajkowski(1967) and, with some manipulation, is evident in the work bySchajer (1981).Bajkowski, in his work with frame saws, experimentally determinedthe roller path widths for a range of rolling loads. A stripslightly narrower than these widths was then removed from theroller paths and the axial extension measured, this axialextension was found to follow a linear relationship with rollingload.38From Schajer (1981), the relationship between rolling load andinduced stress was not given directly and has been obtained bycombining two results. The effect of rolling load on the rollerpath indentation area and the effect of indentation area oninduced axial plastic strain in the rolled region.From Schajer, the indentation area was shown to follow a powerlaw relationship with rolling load:A = CI F,mi^ (3.2)Where:^A = indentation areaC 1 = constantFr = rolling loadm l = 2.2 (determined empirically)The relationship between the indentation area and the inducedstress is defined by the relationship between the transverse andaxial strains in the roller path. The total transverse strain andtotal axial plastic strain were defined by Schajer as:39-T. =t/2f c zdy = 2A-t/2^h(3.3)t/24' = f €p_xci_y-t/2Where: 6, = transverse strainE p.., = axial plastic straint = rolled zone widthI z = total transverse strainI on = axial plastic strain integrated across roll pathThe relationship between them was determined empirically and ispresented in Figure 3.16.To assist in obtaining a direct relationship between rolling loadand induced stress, the empirical curve of Schajer's results inFigure 3.16 was fitted to a power law relationship:IV' = C2I:2^(3.4)Where:^C2 = constantm2 = 0.46 (scaled from Figure 3.16)40Figure 3.16 Longitudinal Plastic Strain Integral vs TransverseStrain Integral.41The stress induced in the plate can be determined from the totalplastic strain and equilibrium considerations:are'at - ^b(3.5)Where:^Q, = axial tensile stress in the plate.b = plate widthI o" = axial plastic strain integrated across path widthThe relationship between rolling load and induced stress isobtained by combining Equations (3.2),(3.3),(3.4) and (3.5) andnoting that the exponents m 1 and m2 are very close to beingreciprocals of one another:C3 EFr6 t -bh in2(3.6)Where:^C3 = constantF r = rolling loadm2 = 0.46 (see above)Thus, with some manipulation, it is shown that the work ofSchajer also predicts a linear relationship between rolling loadand induced stress.42From the present experimental results this linear relationship isclearly demonstrated in Figure 3.10. Examination of this figureshows that the slopes of the data are almost independent of bladethickness, with the principal variation being in the intercepts(the predicted yield loads). For a strictly linear behaviour thiswould imply that the yield load varies significantly withthickness. However, Schajer has shown that the indentation areain the roller path due to plastic deformation is independent ofthickness (for the range from 0.060 in. to 0.100 in.), thus, itwould seem reasonable to assume that the yield loads are similarfor all three thicknesses and the effect of thickness is to alterthe behaviour in the early stages of yielding. The results of theincremental rolling loads (Figure 3.12) show non-linearities inthe early stages of yielding that would support this assumption.Consequently, the yield loads in Figure 3.10 will be higher thanthe actual values and the predicted value of induced stress willbe invalid close to these yield loads.Estimates of the load at which the onset of yielding occurs canbe determined from the analysis of the contact stresses betweentwo ellipsoids (Lipson and Juvinall 1961). For the geometry ofthe rollers in our machine, and assuming a yield stress of180,000 psi., yielding was predicted to occur at 1100 lb. Thiscompares well with the experimental results in Figure 3.12.43Anderson (1975) analyzed the effect of roll-tensioning using afinite element approach. This work is of interest because itrelates rolling load directly to induced stress and the initialrolling load and the plate and roller dimensions are very closeto one of the rolling load experiments in this work. For arolling load of 6300 lb. Anderson predicted an induced tensilestress of 1640 psi., compared to 1460 psi. from the results ofthis section (Figure 3.10). The difference in the results may bedue to the flow rule condition used by Anderson, where thelateral stress cry was set at 10% of ax .In the experiment to examine the stresses close to the rolledzone, the stresses induced in the edges of the plate were 23.4micro-strain for the 4000 lb. roll and 34.9 micro-strain for boththe 4000 lb. and 5000 lb. rolls. These results compare well withthe predicted values of 22.3 micro-strain for a 4000 lb. roll and32.2 micro-strain for a 5000 lb. roll, determined from the linearrelationship in Figure 3.10. The implication from these resultsis that the final induced stress is essentially a function of thefinal rolling load and is unaffected by the two stage approach.The predicted value of induced stress for the professionallytensioned blade compared well with the measured values. Thissupports the earlier conclusion that the effect of the rolls islinearly independent and also that the results of this sectionare applicable to the industrial application of roll-tensioning44and can be used to draw conclusions about the magnitude anddistribution of stresses in bandsaw blades.3.3 ConclusionsIt may be concluded that the process of roll-tensioning isessentially that of a stress redistribution that increases thelateral stiffness of the blade to loads applied at the front orback edges. It does so by plastically deforming the blade suchthat the centre region of the blade is longer than the edges withthe result that after tensioning the front and back edges aretighter than the centre region.The tensile stress distribution induced by roll-tensioning obeysthe linear relationship described by Equation (3.1). When therolling pattern is symmetric about the blade centre-line theinduced stress is uniform, except in the narrow rolled regionswhere it is highly compressive, and the in-plane curvatureremains unchanged. When the rolling pattern is not symmetric, thestress is linearly distributed with a maximum at the edge closestto the roll and an in-plane curvature or backcrown is induced.The majority of the transition from compressive stress in therolled region to tensile stress in the blade occurs very close tothe rolling line, with 95% of the maximum tensile stress havingbeen achieved within 0.5 in. from the edge of the rolled zone.The accumulation of stress due to multiple rolls closely followsa linear superposition assumption.45The magnitude of the tensile stress due to roll-tensioning is afunction of the rolling load, the roller and plate geometry andthe yield strength of the material. Where the stresses due torolling are just above the yield point of the material theinduced stress is non-linear with respect to rolling load, thischanges to a linear relationship as the rolling load isincreased. This linear relationship has been observed previously(Bajkowski 1967) and, for the range of rolling loads considered(3000 lb. to 6000 lb.), is supported by the experimental resultspresented here. Thus, both the magnitude and distribution oftensile stress induced by roll-tensioning can be described bysimple linear relationships. For previously rolled sawblades,where the location of the rolls is unknown, the roller path maycoincide with a previously rolled portion of the blade, in thiscase the magnitude of induced stress may be less than thatpredicted here.464. SAWBLADE INTERACTION WITH THE BANDMILLThis part of the work was designed to determine how the stressesand displacements introduced by roll-tensioning affect theresulting stress distribution as the saw is strained on thebandmill. This relationship addresses the second step in thesequence of events outlined in Section 1. In this sectionexperiments are conducted to measure the geometry of the bladeand wheel in the presence of bandmill strain, wheel tilt, roll-tensioning, backcrown and overhang, and to measure the stressesin the blade due to wheel tilt. The results are then analyzed insome detail.The conventions used in this section are as follows: the toothededge of the blade is also referred to as the front of the blade;a forward curvature in the blade is associated with a convextoothed edge; a forward wheel tilt causes the toothed edge of theblade to move forward and introduces a backward curvature intothe blade.4.1 Experimental WorkFor a long narrow rectangular plate stretched by the action offorces at the ends: the stress away from the ends of the platewill be uniform, providing the forces are symmetricallydistributed about the axis of the plate. For the case of a47bandsaw blade stretched between the bandmill wheels: the stressin the cutting region will be symmetric, providing any roll-tensioning stresses in the sawblade are symmetric, the blade andwheel crown are centred on the bandmill wheels and the wheels arevertical. The total stress in the cutting region will then be asuperposition of the roll-tensioning stresses and the uniformstress due to bandmill strain.Factors that affect this symmetry are the non-symmetric roll-tensioning stresses that produce backcrown, an off-centre wheelcrown, wheel tilt and blade overhang. The aim of the experimentalwork is to provide information on how these factors affect theinteraction of the blade and bandmill such that the behaviour canbe analyzed.4.1.1 The Effect of Bandmill Strain on a Blade with Backcrown.When a blade with a non-symmetric distribution of roll-tensioningstresses, as in the case of a blade with backcrown, is mounted ona bandmill, the distribution of stress from the blade-wheelinteraction will not be symmetric across the blade. The back edgeof the blade is longer than the front, therefore, the pressurebetween the blade and wheel will depend upon the magnitude of thebackcrown curvature and the tilt angle of the wheel. The tiltangle of the wheel will be governed by the kinematics of thesystem.48In order to investigate the geometric relationship between theblade and the wheel, an experiment was conducted to measure thein-plane curvature of the blade, the tilt angle of the wheels andthe contact region between the blade and wheels, for a wide rangeof bandmill strains.The blade used was 9.5 in. wide from gullet to back by 0.057 in.thick and had a backcrown radius of 1350 ft. The bandmill strainwas varied from 2,000 lb. to 20,000 lb. in increments ofapproximately 2000 lb. A one inch overhang was maintainedthroughout the experiment. This ensured that the back edge of theblade did not overhang the back edge of the wheel and affect thecontact region measurements.The blade was run on the bandmill and positioned with a 1 in.overhang from the gullet to the edge of the wheel. This procedurewas repeated for each change in strain. Once a stable situationhad been reached, the bandmill was stopped, and the blade andwheel positions were measured with respect to a plumb-line asshown in Figure 4.1. Additionally, the backcrown was measured infive places along the back span and the blade wheel contact zoneswere determined by tapping the blade over the wheel area.When a blade with backcrown is mounted on a bandmill thebackcrown curvature affects the interaction of the blade andwheel as well as the shape of the blade between the wheels.49a)4^ CContact zonesdetermined bymnr, tapping• orACaCU Cm2a•^•^Wheel D ^•^ont11).WheelBladeBecause the back of theblade is longer than thefront, the pressurebetween the wheel andblade is higher at thefront of the blade and theresulting moment tends toreduce the backcrowncurvature. The change inbackcrown with strain isshown in Figure 4.2a andit is apparent that themajority of the change incurvature has occurred bythe 8000 lb. strain level.A simple curved tie-rodanalysis of the bladebetween the wheelspredicts a 3% reduction inbackcrown for an 8000 lb.bandmill strain. Hence,Figure 4.1 Measurement Geometry ofBackcrown Investigation.the tie-rod effects can be largely ignored and the reduction inbackcrown can be attributed to moments generated by the non-symmetric contact between the blade and wheel.50I^I^I^I^I^I^I^I^I^12^4^6^8^10 12 14 16 18 20Strain (1000 lbs.)I111111110^2^4^6^8^10 12 14^16 18 20Strain (1000 lbs.)....77.= 1500 —u)m 2500—CCas 35003c 4500 —2 5500 -Uaso 6500 ^COas32 —28 —CC 24 —0 20 —16-12 —8-4-0--4—Figure 4.2 Change in Backcrown and Wheel Tilt vs BandmillStrainThe tilt angle of the top wheel relative to the bottom wheel isshown in Figure 4.2b. The individual wheel tilts and the bladecontact regions are shown schematically in Figure 4.3a. At thelow strain levels both wheels were tilted backwards, the bladetended to ride on the front corner of the top wheel and theposition of the blade was insensitive to changes in the tiltangle. This was considered an extreme condition but is includedbecause of the large change in backcrown that occurred at the low51 1IL".111;;;;;;;;L■a) Relative Wheel Tilts & Blade Contact Zones0^2^4^6^8^10^12^14^16^18^20Strain (1000 lbs.)b) Wheel Tilt vs. Angle of Blade from Backcrown5.0 —-cs•• 4.0 —cC 3.0 —p 2.0 —r 1.0 —a)0) 0.0 —c< _ 1.0 —Angle of Bladets.Combined Wheel Tilt Angle0^2^4^6^8^10^12^14^16^18^20Strain (1000 lbs.)Figure 4.3 Contact Zones and Relative Angles of the Blade andWheel.strain levels. As the strain increased above 4000 lb., the angleof the blade and wheels and the location of the contact zonesbecame similar for the top and bottom wheels. In the 6000 lb. to10,000 lb. strain levels there are three contact regions betweenthe blade and wheel. The contact at the front edge is required tosupport the blade overhang, the contact one third of the way back52from the front edge is most likely due to the crown of the wheelat this location and the contact at the back edge is minimal asmight be expected for a blade with backcrown. Above the 10,000lb. strain level the membrane stresses are high enough such thatthe blade is progressively pulled down onto the wheel surfaceexcept at the back edge, where, even at the highest strainlevels, a small gap existed. This was attributed to theanticlastic effects.When a blade with backcrown is mounted on the bandmill thesection of blade between the wheels is curved in-plane. Forstable operation it is necessary that, at the wheel tangentpoints, the angle formed by this in-plane curvature is matched bythe angle of the wheels, i.e. the axis of the blade and the planeof the wheel are co-linear at the tangent points. Figure 4.3bcompares the relative tilt angle of the wheels to the angleformed by the curvature of the blade and, once above 6000 lb.strain, these can be seen to be quite close. This co-linearity isa kinematic requirement for stability, providing no slippageoccurs.It is apparent from the experimental results in Figure 4.2 thatthe majority of the reduction in blade curvature (backcrown)occurs at the low strain levels and is due to the initialalignment of the blade and wheel for stable operation. At the midstrain levels (6000 lb. to 12000 lb.) further reductions in53backcrown occur that require a reduction in tilt angle tomaintain the blade position. Above the 12,000 lb. strain levelboth the backcrown and wheel tilt increase, this was thought dueto the blade overhang and is discussed later in the chapter.4.1.2 The Effect of Wheel Tilt on Blade Stresses.Wheel tilt is used to position the blade in the desired locationwith respect to the wheels. This is usually with the gullet lineof the blade overhanging the edge of the wheel by a small amount.From the previous results it is apparent that the tilt angle willimpart moments to the blade to compensate for such factors asbackcrown and overhang and to ensure precise tracking. Thefollowing experiment was conducted to measure the relationshipbetween the stresses in the blade and the tilt angle of thewheel.A plain band 10.25 in. wide by 0.057 in. thick was used for theexperiment. Strain gauges were used to measure the stressesassociated with the wheel tilt at three locations across theblade (Figure 4.4). The strain-tilt relationship was determinedfor bandmill strain levels of 8000 lb. and 15000 lb.Figure 4.5 presents the results of the experiment. The slopes ofthe individual traces are presented in Table 4.1. As the range ofwheel tilt used in operation is much less than that for theexperimental results, the slopes of the traces were taken from544.625" 0.5"4--10.25°t.i4 4.625'4- 1^+30.5'--O.0.057° FL3 pairs ofstrain gauges ------....,.....,„.....,....,...........................Figure 4.4 Strain Gauge Positionsfor Tilt AngleExperiment.the five readings centredabout the crossover point. Thechange in strain due to tiltfollows a simple linearrelationship. The small slopeof the centre trace is due tothe change in bandmill strainas the tilt varies.4.1.3 The Effect of Overhang.It has been observed that whenbandsaws are locatedsymmetrically on the wheelsthey tend to be very sensitiveto changes in tilt angle andto the application of cuttingforces, especially when the blades are untensioned. Possiblybecause of this, bandsaws are operated with the toothed edge ofthe blade overhanging the edge of the wheel. Certainly the bladeis much less sensitive to wheel tilt when overhang is present,indicating a stabilizing effect.The overhang, combined with the wheel tilt to maintain it,imparts moments to the blade that will effect the stresses in thecutting region of the saw. The following experiment was conductedto determine the relationship between wheel tilt, overhang and5515000 lb Strain•■••141 III^IIIIIIIII II I I I—Strain Gauge Pair01&2 03&4 05&65003450^ 8500 lb. Strain1••400S7 350c 3005 2502001505•Strain Gauge Pair1&2 o3&4 05&610050700650600550500c450171. CT) 4°03503002502000^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1 0Tilt Angle (Degrees)0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0Tilt Angle (Degrees)Figure 4.5 Tilt Angle Strains.56Table 4.1 Strains in Blade due to Wheel Tilt Angles.Bandmill Strain (1b) Strain Gauges Slope (tie/degree) Average Slope(le/degree)8000 1 & 2 +452If 3 & 4 +^1.5 444II 5 & 6 -43515000 1 & 2 -445/I 3 & 4 - 13.6 425,, 5 & 6 +406.5the in-plane curvature of the blade. The stresses in the bladecan be determined from the changes in blade curvature.Two blades were used for the experiment. One was a plainuntensioned band, 8.065 in. wide by 0.057 in. thick, withouttensioning or backcrown. The other was a toothed blade,7.2 in. gullet to back by 0.049 in. thick, with a backcrownradius of 13500 ft. and a light gap of 0.027 in. The bandmillstrain was 12000 lb.The overhang and in-plane curvature were measured for a range ofwheel tilt values and the results are presented in Table 4.2. Thebackcrown measurement is the 'height' of the in-plane curvaturein the centre of a 3 ft. chord, a positive value being associatedwith a convex curvature of the back edge of the blade. As one57Table 4.2 Measured Values of Wheel Tilt, Backcrown and Overhang.Untensioned Blade(8.065 in. by 0.057 in.)Tensioned Blade(7.2 in. by 0.049 in.)Overhang(in.)Wheel Tilt(Deg.)Backcrown(in.x 10 3 )Overhang(in.)Wheel Tilt(Deg.)Backcrown(in x 10-3)0.21 0.025° -0.5 0.22 0.028° -1.00.72 0.049° 0.0 0.51 0.077° 0.00.90 0.070° 0.5 0.78 0.140° 2.01.11 0.105° 2.0 0.96 0.189° 3.51.28 0.140° 3.0 1.21 0.280° 6.251.44 0.175° 4.5 1.35 0.336° 8.01.61 0.210° 5.5 1.51 0.392° 10.01.97 0.280° 8.0 1.64 0.448° 10.752.3 0.350° 9.5 - - -would expect, the backward curvature of the blade increases withincreasing overhang. Bending moments were determined from the in-plane curvature and wheel tilt 2 values, at each level ofoverhang, and these are presented in Figure 4.6 as the totalmoment and the tilt moment. The moment due to overhang is takenas the difference between the two sets of values. Typical valuesof overhang are in the 0.5 inch to 1.0 inch range, thus, for thetwo blades used in the experiment, total moments of 500 lb-in.are not unrealistic. This would reduce the gullet stress by2The effect of wheel tilt is discussed later in Sections 4.2.4and 4.2.5. The results are shown here to demonstrate the relativesignificance of wheel tilt and blade overhang.58a) Untensioned Blade 8.065 in. by 0.057 in.+ Total Moment0 Overhang Moment0 Tilt Moment0.2 0.4 0.6 0.8^1^1.2^1.4^1.6^1.8^2^2.2Overhang (in.)b) Tensioned Blade 7.2 in. by 0.049 in.+ Total MomentO Overhang MomentO Tilt Moment0.2^0.4^0.6^0.8^1^1.2^1.4^1.6Overhang (in.)Figure 4.6 Measured Moment in Blade due to Overhang andWheel Tilt.59approximately 4%-6%,. It is apparent from these results that theoverhang should not be allowed to become too large as this wouldsignificantly reduce the stresses in the front edge of the bladeand affect the blade stiffness.The experimental results also reflect the change in stability ofthe blade as the overhang increases. At the lower overhang valuesthe moment changes quite slowly with respect to the overhang,this means that for a given change in wheel tilt quite largechanges in overhang occur before the blade stabilises. Indescriptive terms, as the tilt is increased the blade movesforward on the wheels and the backcrown curvature increases. Theblade will continue to run forward until the curvature of theblade is such that the axis of the blade is co-linear with theangle of the wheels and stability is achieved. Thus, the steeperthe moment vs overhang curve the more stable will be the blade.In this regard the tensioned blade can be seen to be more stablethan the untensioned blade at the lower values of overhang(Figure 4.6).4.2 Analysis and DiscussionIn this section a method of analyzing the displacements, stressesand moments in the blade, due to the interaction of the blade andbandmill wheel, is presented. The experimental and analyticalresults are compared and discussed.60To analyze the interaction of the blade and wheel it is initiallyassumed that the blade, free of any external forces, has formed ahoop and, at zero axial strain, just fits onto a cylinder(Figure 4.7a). The effect of bandmill strain is then modelled byFigure 4.7 Cylindrical Model of Blade on Wheel and Coordinatesof Cylindrical Shell Element.the radial expansion of the cylinder. The governing differentialequation is that determined from the equilibrium of a cylindricalshell element with symmetrical deformation with respect to theangular coordinate 0 (Figure 4.7b) (Timoshenko and Woinowsky-Krieger 1959):612 ^dx 2^a Z^(X)= -D d2k7^Ele4, + )dx 2^=^aSubstituting, we obtain:Eh ED d4w + Eh w = Z -dx4^a 2^a(4.1)(4.2)with boundary conditions:d e w _ v^d3w— ; - 0dx-2^a^cbc3 (4.3)Where:No= circumferential in-plane forces per unit widthD = Eh 3/12 (1-v 2 )^flexural rigidity of plateE = Young's modulush = plate thicknessw radial displacementv = Poisson's ratioa cylinder radiusZ = pressure normal to the surfacec= circumferential residual strainFor the condition that Z = E 0^0 the solution to Equation (4.2)that satisfies the boundary conditions is given by Foschi (1975),62where the bending moment boundary condition accounts for theanticlastic effects:w = —v (y icosh13xcosf3x+y 2 sinhi3xsin(3x)awhere: y i , y2 -sinhf3:11 cosf3-1 T c o sh 13 2° s in 11 -2 (4.4)13 2 (sinhi3b+sin(3b) To simplify the analysis the anticlastic effects are onlyincluded in the section on blade overhang, where the displacementat the edge of the blade is an important factor.4.2.1 Effects of Roll - Tensioning.At the beginning of the experimental section it was stated thatthe stresses due to the interaction of the blade and wheel wouldbe symmetric with respect to the blade axis, providing thetensioning stresses were also symmetric. This situation isanalyzed using Equation (4.2).Let the residual strain in the blade be approximated by theexpression:E . ( x ) = '21 ,s-. [ 1 2 ( ic r — i I^(4.5)63Where:x = coordinate associated with the width of the bladeb = blade widthE.(x) = axial straina c = maximum value of compressive stress in the centre ofthe sawbladeSubstituting Equation (4.5) into Equation (4.2) and examining thecases for zero pressure and zero displacement (where zerodisplacement corresponds to the blade being flat on the wheel):a (^A7 2^)woo = -a —e 12 --- -1E^b2 (Z=0)Z(x) = ac a12^b(12 2f2 -1)^(w=0)(4.6)The above expression for pressure contains both positive andnegative components. As a negative pressure is not possiblebetween the blade and wheel the wheel displacement must be largeenough such that Z z 0 for the analysis to be correct:2(Z (X) = Eh w +a — 12 A7— -1) 0a 2 ° Cha^b2 (4.7)This occurs when:64a cW a--0^E (4.8)Where wo is the radial expansion of the wheel.Consequently, the circumferential in-plane stress associated withfull contact is:1,2cr (x) = [a c(12 — -1 + -nE = 120 X2b2^a^c b 2(4.9)This is the stress induced in a tensioned blade by the action ofthe bandmill strain, after the blade has been placed over thewheels. The stress distribution predicted by Equation (4.9)compares reasonably well with the experimental results of Fujii,Hattori and Noguchi (1988) (Figure 4.8). The value of a c wasselected such that the shape of blade approximated that used inthe experiment. The experiment measured the stresses induced in atensioned blade as it was pulled onto the wheel by the bandmillstrain. The stresses shown were measured at the apex of thebandmill wheel.The bandmill strain required to obtain full contact can bedetermined from the integration of (4.9):Bandmill Strain = bha,^ (4.10)651 1.4^1.8^2.2^2.6Blade Width (in.)0.2 0.6 3 3.420191817..-.. 16—-ena0 1500 142-I-u) 12011109876N..."C/3' 13a Experiment— Analysis.CICIFigure 4.8 Stress due to Interaction of Tensioned Blade andWheel.For the case when a symmetrically tensioned blade is in fullcontact with a flat wheel, it is apparent that the stresses are acombination of the residual roll tensioning stresses and auniform stress due to the bandmill strain.To determine the stresses in the span between the wheels, thecylinder and blade are separated into two halves and externalforces and moments are applied to maintain equilibrium66Figure 4.9 Forces and Moments in Strained Blade withBackcrown.(Figure 4.9), where, in this case, F = 0.5bhu, and M2 = 0. Takingtwo pieces of blade, applying the residual and axial stressesdescribed by Equation (4.9), and fitting these two pieces intothe spaces between the two halves of the cylinder, the in-planeequilibrium conditions are satisfied. The moment M 1 , shown in theedge view in Figure 4.9a, must also be balanced by a moment inthe span, this will cause a slight outward curvature of the bladebetween the wheels. Typically this effect is small and can beneglected (Garlicki and Mirza 1978). Thus, for a symmetricallytensioned blade, the stresses in the cutting region are a67combination of the residual roll-tensioning stresses and auniform stress due to bandmill strain. This confirms theassumption at the beginning of the experimental section.4.2.2 The Effect of Backcrown.When non-symmetric roll-tensioning stresses exist in the blade,as in the case when backcrown is present, the shape of the bladewith respect to the wheel will change. The resulting interactionbetween the blade and wheel will generate in-plane moments in theblade that will modify the stresses in the cutting region. Thefollowing analysis examines the stress distribution in the bladedue to the interaction of a blade containing backcrown and thebandmill wheel.When backcrown is introduced into a bandsaw the stresses canaffect the geometry of the blade in two ways: if the blade isheld flat it will curve in-plane (Figure 4.10a); if the blade isallowed to form a hoop, free of any external forces, it will takethe shape of a truncated cone (Figure 4.11). Providing thetransverse curvature (light gap) is symmetric about the bladecentre-line this is a reasonable approximation of the change inshape of the blade due to backcrown.68Figure 4.10 Sawblade Stresses due to Backcrown andBackcrown plus Bandmill Strain.The magnitude of the displacement wb, in Figure 4.11 can beexpressed in terms of the backcrown radius R ix in Figure 4.10a:27c (wbc+a) = Rbc11►;^27ta = (Rbc-b)*^(4.11)a  b eliminating ily; wbc " Rbc-b(4.12)69Figure 4.11 Cylindrical Model ofBlade with Backcrown.where:wk = blade displacement due to backcrowna = radius of wheelRk = backcrown radiuslk = swept angle of backcrown radius to equal length of bladeb = blade widthThe shape of the blade is then:w(x) = a b  (1 xlRbc-b\ 2 b) (4.13)Expanding the cylinder until full contact is achieved results inan expression for displacement that, when substituted into70E b ahMbc Rbc-b 12 (4.15)E hb 3hrbc Rbc-b/2 12 (4.16)Equation (4.2), determines the pressure. The resultingexpressions are:w (x) = ab11 + x\Rbc -b\ 2 bj(4.14)7^. Eh  b  (1 .2c\a Rbc-b\2 bland the in plane moment due to this pressure distribution is:Comparing this moment to the moment in the blade due to backcrown(Figure 4.10):It is apparent that the backcrown put into the blade duringpreparation is removed by the blade wheel interaction and theblade is straight between the wheels, the slight difference inthe two equations is due to the displacement wbc being definedwith respect to the front edge of the blade. The final stressdistribution will be as shown in Figure 4.10b. In the cuttingregion of the blade, any increase in stress from backcrown willbe due to the magnitude of the uniform tensile stress componenta, that results from the rolling procedure used to introduce thebackcrown.71The magnitude of this uniform stress is dependent upon therolling geometry that was used to introduce the backcrowncurvature. The equations that relate the in-plane stress with therolling geometry can be determined from Figure 4.10a and forceand moment equilibrium considerations:6dtb2t)(b 3 - t 3 ) -12 tbd2=a  6 db 2 ,."((b 3 -t 3 )(4.17)Where:um bending stress due to the in-plane compressive stress in the rolled region.a, tensile stress induced by rolling.d^distance of rolled region from blade centre-line.t^width of rolled region.b^blade width.The tensile stress (vi) is related to the backcrown curvature andthe distance of the rolled zone from the blade centre-line. Inpractice, the backcrown curvature is fixed and the rollingposition and rolling load are the variables. Taking Equation(4.17) above, writing the bending stress um in terms of a thebackcrown curvature and letting t 3^b3 we obtain:72Eb 2 a t - 12RbcdSubstituting for:a c t 2Lcat ^b-t' Rbc = 8A •( 4 .18)(4.19)3L 2 crA - ^c ctd21, 3EWhere: Lc = chord length along the back of the blade.A = displacement of blade in centre of chord.This is the same as the expression obtained by Aoyama (1971b) forthe magnitude of the backcrown due to rolling position androlling load.Returning to Equation (4.18), for the range of values t/2 s d <b/2 the tensile stress falls within the region:Eb/6RbC < a t 5 Eb2/6tab,Hence the stress due to the application of a fixed backcrown willincrease as the rolled zone moves towards the centre of theblade. In practice, the value of a t will be towards the lower endof the range. This is because very large compressive stresses areneeded when rolling near the centre of the blade and bucklingwill most likely occur.734.2.3 Wheel Crown EffectCrown is added to a bandmill wheel to improve the stability ofthe blade and to enable the blade to be precisely positioned onthe wheel. The centre of the crown is normally placed 1/3 of theway across the wheel from the front edge, rather than on thewheel centre-line. This provides stability for blades narrowerthan the wheel width and compensates for reductions in width dueto the sharpening process. In this section the effect of wheelcrown on the moment in the blade will be examined.The shape of the crown can be modelled as a bi-lineardistribution (Figure 4.12) where:t  c+b/2 Ix+b/2\^bw(x) = wc^--2 sxsc\lx-b/21ww = wckc-b/2)^csxs—2b(4.20)Where we is the height of the wheel crown.Figure 4.12 Geometry of CrownedWheel.74EhZ (x) = a 2 (x)(4.21)Following the previous method the pressure distribution for thisdisplacement is:It should be noted that the pressure distributions obtained fromthe initial shapes of the blade, due to roll-tensioning,backcrown, wheel crown etc., will be independent of one another.As the wheel crown is non-symmetric, a moment will be introducedinto the blade that will increase the stress at the front edge.The magnitude of this moment can be determined by integrating thepressure distribution over the wheel area:Eh cbw wMoment - 6^ a (4.22)For a typical blade geometry of: b,^b^8.25 in., h^0.057in., We^0.005 in., a^30.0 in. and c^b/6. The moment in theblade is 539 lb/in.Thus for a full width blade the wheel crown introduces a momentinto the blade that increases the stress at the front edge by asmall amount. The backward tilt required to stabilize the bladedue to this moment will add to this value. Reductions in thewidth of the blade will reduce the distance between the centre of75the blade and the centre of the crown and the moment willdiminish accordingly.4.2.4 Wheel Tilt EffectsFrom the previous work in this section it has been shown that theeffects of wheel crown and blade overhang create in-plane momentsthat cause the blade to be curved between the wheels. As thewheels rotate this curvature will cause the blade to track,either forward or backward depending upon the direction of thecurvature, and the top wheel must be tilted to stabilize theblade at the required location. In this section, the effect ofwheel tilt on the stresses in the blade are investigated. Thestability of the blade with respect to wheel tilt is discussed inthe next section.Recognising that only the top wheel tilts and assumingfrictionless contact and constant bandmill strain, the followingexpression for the strain in the blade due to wheel tilt isobtained from membrane considerations:€ _ 2stdLTwhere: (I) =wheel tilt angled =distance from centre of contact widthLT = total blade length(4.23)766 LTMt _ 4thE(b-s) 3 (4.24)Substituting blade and wheel dimensions from the wheel tiltexperiment presented earlier in this Chapter (d = 4.625 in., LT =377 in.) results in a value of 428 microstrain/degree. Thiscompares well to the experimental results of 444 and 425microstrain/degree.Later in this chapter the effect of blade overhang on the stressin the blade is analyzed. To assist in the comparison of theanalytical results with experimental values, the effect of wheeltilt on the moment in the blade is required. The moment due totilt is determined from the change in pressure between the bladeand wheel: from Equation (4.23), with an allowance for thediminishing width of blade on the wheel, the moment due to tiltis:Where: s = overhang.4.2.5 Blade Stability MechanismThe effect of tilting the wheel is used to illustrate thestability mechanism of the blade on the wheel. The relationshipbetween the rotation of the wheel due to a change in tilt and therotation of the blade at the tangent point is analyzed and bladestability is discussed.77Figure 4.13 Wheel and Blade TiltAngles.The section of blade between the wheels can be modelled as acantilever beam. Assume the beam is built in at one end (say thebottom wheel tangent point) and has a moment applied to the freeend due to the wheel tilt (Figure 4.13). Recalling Equation(4.23), the relationship between the angle of the blade at thetangent point and the wheel tilt is:2ILTE^MLM - ^ ; 8^sEI (4.25)= 24 L5LT78Where:b = width of blade.Ls = length of the span.LT = length of blade.e = strain in blade due to wheel tilt0 = rotation of blade due to wheel tilt0 = wheel tilt rotation angle.In many bandmills L s . LT/4 and the relationship between therotation of the blade and the tilt of the wheel is 0 . 0/2.The blade will be in a steady state on the wheel when thevelocity of the blade and the velocity of the wheel are co-linearat the point of no-slip contact. If instead the blade is led ontothe wheel at some small angle, measured with respect to theperpendicular to the axis of rotation, the blade will trackacross the face of the wheel (Chardin 1979; Swift 1932). When thewheel is tilted the rotation angle of the wheel is approximatelytwice that of the blade at the tangent point (see above analysis)and a small angle is introduced between the axis of the blade andthe perpendicular to the axis of the wheel, the blade will thentrack in the direction of tilt. The motion of the blade acrossthe wheel will cause the blade to overhang the edge of the wheeland this will introduced an additional moment into the blade,increasing the in-plane curvature of the blade (see Section4.1.3). The forward motion will continue until the curvature of79the blade is such that the angle between the blade and wheel isremoved and the blade is stable on the wheel.4.2.6 The Effect of Overhang.Another aspect of the interaction between the blade and wheel,that will influence the stability of the blade and the stressdistribution in the cutting region, will be the amount of theblade overhanging the front edge of the wheel. The overhang willchange the pressure distribution between the blade and wheel(Sugihara 1977; Sugihara, Hattori and Fujii 1981), introducing anin-plane moment into the blade, and reducing the stress in thefront edge. The effect of overhang is also analyzed using thedifferential equation for a cylindrical shell (Equation (4.2)).However, in this case, the displacements are unknown and must bedetermined from the applied forces and moments. Details of theanalysis are presented in Appendix III.A program was written to solve the equations developed inAppendix III. The analysis determines the in-plane moment in theblade, taking into account the geometry of the blade and wheel inthe presence of overhang, the anticlastic curvature of the bladeand the residual roll-tensioning stresses. The effect of wheelcrown and wheel tilt angle are not included at this time.The analysis is used to examine the effect of bandmill strain,blade width, blade thickness and roll-tensioning on the moment in80the blade due to overhang. A positive moment being associatedwith a convex curvature of the back edge of the blade. Theresults are based upon a 8 in. wide by 0.057 in. thickuntensioned band.Figure 4.14a shows the relationship between overhang and bendingmoment for four different levels of bandmill strain. The analysispredicts the unstable behaviour of untensioned blades for smallvalues of overhang. As the blade starts to overhang the wheel aforward curvature (negative moment) is introduced into the blade,this will causes the blade to run forward in an unstable manneruntil the magnitude of overhang is large enough such that themoment in the blade becomes positive and a stable condition isreached (see Section 4.2.5). The stable overhang of 0.7 inches to1.0 inches compares well with experimental values.The overhang causes an in-plane moment in the blade that willsignificantly reduce the stress in the front half of the blade ifallowed to become too large. The moment would appear to be almostlinearly related to bandmill strain, this may explain theexperimental results in Figure 4.2a, above the 12000 lb. strainlevel, where the in-plane curvature increased with increasingbandmill strain.The effect of roll-tensioning is shown in Figure 4.14b. Aparabolic distribution of stress was used (Equation (4.5)) with a81a) Untensioned Blade8 in. by 0.057 in.10,000 lb8000 lb6000 lb4000 lb 0.1^0.3^0.5 ' 0.7^0.9^1.1^1.3^1.5^1.7^1.9,c_68 30......•-I-C 20EO28Overhang (in.)7 b) Tensioned Blade10,003 lb.8 in. by 0.057 in.8000 lb.65 6000 lb.4 4000 lb.32100.1 0.3 0.5 0.7^0.9^1.1^1.3 1.5 1.7 1.9Overhang (in.)Figure 4.14 Moment in Blade due to Overhang for Four BandTensions.82compressive stress of a = 5000 psi. For roll-tensioned bladesthe analysis predicts a stable condition for all values ofoverhang. A result that is supported by the observed behaviour oftensioned blades.Figure 4.15 demonstrates the small effect of blade width andblade thickness on the moment due to overhang. The larger momentoccurring for the narrowest or thinnest blade.To validate the analysis, the experimental results from Section4.1.3 (Figure 4.6) are compared to analytically predicted values.The initial negative moments (concave back edge) in theexperimental results are likely due to the crown on the wheelwhich introduces a moment into the blade in this direction. Tocompensate for this effect, the experimental results were shiftedsuch that the initial value was zero. The results of thecomparison are presented in Figure 4.16. Reasonable correlationwas obtained for both sets of results, the small offset betweenthe analytical and experimental values is possibly due todifferences between the estimated effects of wheel crown andresidual blade stresses and the actual values.4.3 ConclusionsWhen a blade is bent over a bandmill wheel the distribution ofresidual stresses in the blade will affect its transversedeflected shape (the curvature of the blade across its width). As830.8 1.8 20 0.2 0.4 1^1.2 1.4 1.60.6Overhang (in.)_-Untensioned Bladeb) Blade Thickness^ 2_1. 0.049 in.2. 0.057 in.- 3. 0.065 in.0.2 0.4 0.6 0.8^1^1.2 1.4 1.6 1.8^2Overhang (in.)---1Untensioned Blade^23a) Blade Widths1.8 inch.2. 10 inch.3. 12 Inch.43.532.521.510.50-0.53.530-0.5Figure 4.15 The Effect of Blade Width and Blade Thickness onOverhang Moment.84b) Tensioned Blade7.2 in. by 0.049 in.43.531, 2.5E iO2 0.50-0.5__aa) Untensioned Blade:8.06 in. by 0.057aaain.^13Oa0^Experiment— Analysis-1-..0 2.50^0.2 0.4 0.6 0.8^1^1.2 1.4 1.6 1.8^2Overhang (in.)a00 Experiment— Analysis0.50^0.2 0.4 0.6 0.8^1^1.2 1.4 1.6 1.8^2Overhang (in.)043.5,--:,^3catiFigure 4.16 Comparison of Experimental and Analytical OverhangMoments.85bandmill strain is applied, the interaction of this deflectedshape with the bandmill wheel will generate an axial stress and,in most cases, a bending moment. The axial stress depends uponthe magnitude of the bandmill strain, the bending moment dependsupon any non-symmetric conditions that exist such as backcrown,wheel crown, wheel tilt and overhang.When the deflected shape of the blade and the shape of the wheelare symmetric the stresses due to bandmill strain aresuperimposed upon the stresses due to roll-tensioning. These twostress components account for the major portion of tensile stressin the edges of the saw. When the deflected shape of the blade isnot symmetric, as in the case when backcrown is present, thereare two moments to be considered, the moment in the blade due tothe off-centre roll-tensioning that caused backcrown and themoment due to the interaction of the deflected shape of the bladeand the wheel. Providing the bandmill strain is large enough toensure full contact between the blade and wheel, these twomoments cancel one another out and the backcrown curvature isentirely removed. The resulting stress distribution is acombination of a uniform tensile stress and a narrow region ofcompressive stress, located in the rear half of the blade, wherethe rolling occurred (Figure 4.10b). For a fixed backcrown radiusthe magnitude of this uniform tensile stress is primarilydependent upon the distance of the rolling position from theblade centre-line.86The moment in the blade due to wheel tilt can be readilydetermined from membrane considerations. For the dimensions ofthe bandmill used in the example, the ratio between the change inwheel tilt angle and the rotation of the blade at the tangentpoint is 2 to 1. The angle of the blade and wheel were observedto be co-linear at the tangent point for stable operation.When the shape of the wheel is non-symmetric, as is the case whenthe wheel crown is located 1/3 of the way from the front edge,the moment generated by the blade-wheel interaction introduces aforward curvature into the blade. This forward curvature wasobserved in both experimental blades. To stabilize the blade abackward tilt of the wheel is required which increases thismoment slightly.The model successfully predicted the effects of overhang on bladestability and on the stresses in the cutting region of the blade.Untensioned blades were shown to be unstable for overhang valuesless than 0.7 to 0.9 inches, depending on the level of bandmillstrain. The addition of tensioning improved blade stability. Foroverhang values greater than 0.7 inches the moment in the bladecan increase rapidly and large reductions in stress occur in thefront edge of the blade.875. THE EFFECT OF SAWBLADE STRESSES ON FREQUENCY AND STIFFNESSIn this Section the effect of the stresses in the cutting regionof the sawblade on the natural frequencies and lateral stiffnessof the blade are examined. Experiments are conducted to determinethe effect of roll-tensioning and bandmill strain on the lateralstiffness and torsional frequency of the sawblade. The resultsare analyzed using finite element methods and a closed formfrequency equation is developed.5.1 The Effect of Rolling Position on Lateral Stiffness andTorsional Frequency.To help understand the effect of roll-tensioning on bladestiffness an experiment was performed in which the effect ofrolling distance from the blade centre-line was examined. A plainband was partitioned into five 6-foot sections, each section wastensioned differently and the resulting lateral edge stiffnessand torsional frequencies measured. A total of 25 combinations(16 unique) were examined. Figure 5.1 shows the location of therolling lines that were used, and, in each case, the rolling wassymmetrical with respect to the centre-line.The blade used was 10.25 in. wide by 0.057 in thick by 31 ft. 5in. long. A rolling load of 3400 lb. was used throughout. Thelateral stiffness was measured by applying point loads 1/4 inch88Section0 Unrolled111 & 2 - 1 roll each side3,4 & 5 - 2 rolls each sideIRolling line No.^I8 7 6 5 4 3 2 1 11 1 1 I 1 1 1 1 :5^4^31^1^I^I 1^24^5^3I^I^2^1^I3^4^51^2^1^1^If3^5^411^II^II...................IIIIIII1^2 3 4 5 6Blade Width (in.)2111Band divided into5 sections plus 18"around weld.Ii . 4— Rolling sequence1111 114-- Rolling patterns1^symmetric aboutcentre-line.111^1^17^8^9^10Figure 5.1 Rolling Geometry for Partitioned Blade.89IN 11ro^di0.25'^I 0.25'♦14- I^7•^- 1- -F.^ + ^ +A Ili3^C^Y ^Ib 0 1^0^I33 1^1.25'1I HI ,I-114^10.25'4-4-4-H1.25'Figure 5.2^Blade^StiffnessMeasurement Geometry.from the edge of the bladeand measuring the resultingdisplacements. Figure 5.2shows the arrangement of theguides and points at whichthe stiffness was measured.The strain applied to thebandmill was 16,000 pounds.In Figure 5.3 are plottedthe changes in torsionalfrequency and lateralstiffness as a function ofthe rolling pattern. As maybe seen from this figure,rolling at locations 1, 2, 3or 4 increases stiffness,locations 5 and 6 have negligible effect, whereas rolling atlocations 7 and 8 decrease the lateral stiffness. In general thechanges in torsional frequency match the changes in lateralstiffness, therefore, the torsional frequency can be used as anindicator of the effect of rolling on blade stiffness. The majorinstance where the change in frequency and stiffness differ is atrolling location 7 & 8 in region 4. From the results in the otherthree regions ( and results later in this chapter) the change instiffness at this location is thought to be an anomaly.905^6^7 &8 3 &4Polling Line40>.2 (1)70 c01 & 2 6Blade Region 4.. Stiffness40a) 30cn 20a)0) 100-c0 o3^4^1&2 7 & 8Rolling Line8^5 &6 1&2Rolling Line8 Ki6 0a)4 2C2 a)C0_c5&6 0 2010c6)-: 0a)0) -100.c(-) 207Blade Region 5.4>-2 0ca)0 CDLLc-2 a)0-43 & 4 02^3 &4 5 &6Rolling Line10 2 e: 20^>. ^a)8 0N E 10'..-0'6 W^ut^c.--^0a)c cs)c^4 a)^0 -10a) r0a2 -c07 & 8PiCDrt <w1-hHI 11H-p)rrCD H.0) 0U)Pzi0,3Ej Flco(Q H.0Pulo wrrcnH- '11"(DUIIiC)wwrrH1-1At the completion of the experiment the four rolled regions ofthe band (regions 2-5) had been tensioned in the same way exceptfor the sequence of the rolls. The final values of the frequencyand stiffness for these regions were combined and the means andstandard deviations are shown in Table 5.1. The untensionedregion (region 1) is included because the frequency and stiffnesshad been measured at each of the five different steps in theTable 5.1 Comparison of Final Frequency and Stiffness Values forPartitioned Blade.Region Variables X a1 Frequency 77.95 0.54Stiffness 269 32-5 Frequency 81.81 0.5Stiffness 313 5.7tensioning process and the results provide some indication of thevariation in the experimental measurements. From the smallstandard deviations of the final values it is concluded that theorder of rolling is not a significant factor in the distributionof stress.The tensioning process introduces stresses into the blade thatresult in a transverse curvature when the blade is formed into aradius in the axial direction. This transverse curvature is used920^1-^I^I^I0^1^2 3&4 5&6Rolling Line783IBlade Region 4Ught Gap‘N1114rAddlill7.6^3.4Rolling LineFrequency2 0C.)C0 70 7( T2LL5&63I^T^i 04^1&2 7818Rolling Line5&6^1&2^3&4Rolling Line..c- • 400^- 6 i^v - 20a0-4^›. ^CD0ca)^-CD co=^-2 CY^+20a)ut.I^I,•P0 • 60-C:..7.,a - 40-8....c- • 20-as=0r-NNI>,•0Ca) the industry as a method of assessing the amount of roll-tensioning in a sawblade, a straight edge or specified radiustemplate is placed across the blade and the amount of tensioningis assessed by shining a light through the gap between the bladeand template. Hence the term "light gap". During the experimentalinvestigation the maximum amplitude of the curvature was recordedat each stage of the tensioning process and the results have beenplotted, along with the change in torsional frequency of thesawblade, as a function of rolling position (Figure 5.4). For thesymmetric roll-tensioning condition it is apparent that the lightgap is an excellent indicator of the effect of rolling on thefrequency of the blade, which in turn is an indicator of theblade stiffness, and helps explain why this method is used by theindustry.5.2 The Effect of Bandmill Strain on Lateral Stiffness.The lateral stiffness of the teeth is expected to be an importantfactor in the cutting accuracy of the bandsaw. Increasingbandmill strain increases the stiffness of the teeth and has beenshown to improve cutting accuracy (Allen 1973). In thisexperiment the effects of bandmill strain on the lateralstiffness of an untensioned and a tensioned blade weredetermined.The blades used for the experiment were 9.6 in. wide from gulletto back by 0.057 in. thick. The tooth pitch was 1.75 in. and the94gullet depth was 0.6 in. The span length between the guides was30 in. One saw had been levelled only and had a zero light gapand a backcrown radius of 4500 ft. The other saw had beentensioned and had a light gap of 0.047 in. and a backcrown radiusof 964 ft. (less than half the industry standard of 2400 ft.).The lateral stiffness of the tooth, gullet and back edge of eachblade was measured at the mid-span line for a range of bandmillstrains.Figure 5.5 compares the lateral stiffnesses of the tensionedblade with that of the untensioned blade. For the tooth andgullet regions the almost constant increase in stiffness due toroll-tensioning can be seen to be superimposed upon the stiffnessdue to bandmill strain. In Figure 5.6 the stiffnesses of eachblade are shown separately and the effect of backcrown is quiteapparent: for the untensioned blade the stiffness of the gulletand back edge are almost the same, for the tensioned blade thestiffness of the gullet is much greater than the back edge due tothe large amount of backcrown in this blade. There is also alarger drop in stiffness between the gullet and tooth tip for thetensioned blade than for the untensioned blade, indicating thatthe ability of the tensioned blade to withstand the momentapplied to the front edge by the tooth has been affected by thetensioning process.9518 20 22 24 26 288 10 12 14 16500Bandmill Strain (1000 lb.)5002'-■-c) 4002a)„4= 300C13+ i200CD1008 10 12 14 16 18 20 22 24 26 28Bandmill Strain (1000 lb.)500,--:c.7.7._a 400z...7.,2a)c 300:4=cn.P. 2000_1100 8 10 12 14 16 18 20 22 24 26 28Bandmill Strain (1000 lb.)Figure 5.5 Comparative Stiffness of a Tensioned vs Un-TensionedBlade.96100 ^8 10 12 14 16 18 20 22 24 26 28Bandmill Strain (1000 lb.)Figure 5.6 Stiffness of a Tensioned and Un-Tensioned Blade vsBandmill Strain.5.3 Analysis and Discussion.At the beginning of the Chapter an experiment was conducted todetermine the effect of rolling position on the torsionalfrequency and lateral stiffness of the sawblade. The lateralstiffness is important because it is expected to be related tothe cutting accuracy of the blade. The sawblade frequencies are97important because they are related to blade stiffness, are easyto measure and will give insight into the state of roll-tensioning. To determine the relationship between the edgestiffness of the blade and the fundamental torsional frequency,an analysis was conducted to compare the effect of differentrolling positions on both these characteristics.Details of the finite element model used to conduct the analysisare in Appendix I (Figure I(1)a). The two rolling lines weresymmetrically located with respect to the blade centre-line. Theanalysis examined the effect of varying the distance between therolling lines on the lateral stiffness and torsional frequency ofthe blade. The predicted frequencies and stiffnesses were andfound to be in good agreement with Lehmann (1991).The results of the analysis are presented in Figure 5.7 and, forsymmetric tensioning stresses, show a reasonable correlationbetween the square of the torsional frequency and edge stiffness.It is apparent that the effect of rolling decreases as therolling position moves away from the blade centre-line. Rollingon the blade centre-line is optimal for increasing bladestiffness while rolling in the centre 60!1 of the blade willincrease the edge stiffness and rolling outside this centralregion will reduce the edge stiffness. It is expected that a dropin stiffness will also reduce the cutting accuracy of the saw: asit is common practice in the industry to roll to within one inch98120>. 20o-n2 -2A Frequency Squared0 Stiffness-100.5^1.0^1.5^2.0^2.5^3.0^3.5^40Rolling Distance From Centre-Line (in.)Figure 5.7 Frequency and Lateral Edge Stiffness vs RollingPosition.of the edge of the blade, it would seem that some improvement inperformance may be available by restricting the rolling to thecentre 60% of the sawblade.Using the data from the first experiment, it is possible toobtain measured values for the effect of rolling position on thefrequency and stiffness of the blade. The experimental resultscan then be compared to analytically predicted values. Forexample, taking the two pairs of rolls closest to the bladecentre-line in Figure 5.1 (rolling lines 1 & 2), these are rollsequences 1 and 2 in blade section 2, roll sequence 3 in bladesection 3, roll sequence 5 in blade section 4 and roll sequence 499 •Experiment— Analysis1 & 2^3 & 4^5 & 6^7 & 8Rolling blade section 5. The change in frequency due to these fourrolls, averaged over the four sections, is then determined.Repeating the example for rolling lines 3 & 4, 5 & 6 and 7 & 8,four changes in frequency as a function of rolling position areobtained and can be compared to the analytically predictedvalues. The rolling position was assumed to be midway between therolls.The results of the frequency comparison are presented inFigure 5.8 and excellent correlation between the analytical andexperimental results is obtained. Figure 5.9 presents the resultsFigure 5.8 Comparison of Analyticaland Experimental Frequenciesvs Rolling Position.of the comparison between the analytical and experimentalstiffness of the band: although not as precise as the frequencyresults, the correlation can still be considered good. This isbecause the stiffness is much more sensitive to the distribution100_ ••• Experiment— Analysis.•1 & 2^3&4^5&6^7 &8Rolling Position._ 12C4=c -4a)a -8cCrlFigure 5.9 Comparison of Analyticaland^Experimental^EdgeStiffness^vs^RollingPosition.of stress, the support conditions and the displaced shape of the .band and, as such, is harder to both predict and measureaccurately. From these comparisons it is considered that theexperimental results support the conclusions given aboveregarding the rolling position.It is apparent, where the rolling lines are symmetrically locatedwith respect to the blade centre-line, that insight into theeffects of roll-tensioning on blade stiffness can be determinedfrom the torsional frequency of the sawblade. To avoid having toresort to numerical techniques, a relatively simple closed formequation for the torsional frequencies can be determined using aRayleigh Quotient approach. This has been AppendixII and includes the effect of a pair of rolling lines, singlerolling lines and non-symmetric stress distributions.101From the analysis in Appendix II, the effect of a pair of rollssymmetrically located with respect to the blade centre-line, onthe fundamental torsional frequency, was examined. The analysissupports the previous analytical and experimental results.Rolling within the centre 58% of the blade increases thetorsional natural frequency and hence its torsional stiffness,whereas rolling outside of this region decreases the frequency.The optimum rolling location coincides with the blade axis, wherethe negative effects of the compressive zones are minimised byplacing them closest to the axis of rotation. Exactly the sameresult is obtained for the single off-centre roll except that thechange in frequency will be half that of the two roll case.For the case where the roll-tensioning is symmetrically locatedwith respect to the blade centre-line, it has been demonstratedthat insight into the effect of roll-tensioning on bladestiffness can be obtained from the torsional frequency. With thenon-symmetric case, examination of the frequency alone will notprovide the same information. For example, when the interactionof the blade and wheel produces a pure bending moment in theblade, as is the case with wheel crown, wheel tilt or overhang,the change in frequency will be small compared to the changes inthe lateral stiffness of the edge of the blade. Another exampleis when the stress in the blade is modified by rolling on theneutral line in the back half of the blade, this will providelittle or no change in frequency but the front edge stiffness102will increase. In this case, if backcrown is known to have beenput into a blade, the change in frequency can be used to indicatethe effectiveness i.e. the further from the blade centre-line thebackcrown rolling has occurred the less tensile stress is inducedand the smaller will be the change in frequency. In general, itis necessary to use additional geometric information, or know therolling history, prior to using the torsional frequency as aguide to lateral stiffness.5.4 ConclusionsThe changes in torsional frequency and lateral stiffness as afunction of rolling position were investigated experimentally andanalytically. The magnitude of the changes were found to bedependent upon the rolling load and the distance of the roll pathfrom the sawblade centre-line. The change in frequency andstiffness is a maximum when rolling on the centre-line, zero whenthe rolling distance was approximately 60% - of the half-width3 anda minimum when rolling at the edge.The finite element analysis presented here accurately predictedthe stiffness and natural frequency characteristics of bandsawblades under the influence of bandmill strain and roll-tensioningstresses. A closed form frequency equation was developed that3The zero point is 58 9; for the frequency and approximately 64%,for the stiffness.103predicted the effect of roll-tensioning on the fundamentaltorsional frequency of the sawblade.Qualitative predictions of the effect of symmetric roll-tensioning4 on sawblade stiffness can be made from a knowledge ofthe change in torsional frequency, an increase in frequencyalways being accompanied by an increase in stiffness. When non-symmetric roll-tensioning stresses are involved, it is necessaryto use geometric information or know the rolling history beforeusing the torsional frequency as a guide to the lateralstiffness.4The symmetry of the tensioning stresses can be determined fromthe in-plane curvature of the blade.1046. CUTTING EXPERIMENTSThis part of the work was designed to determine how the stressesin the sawblade, due to roll-tensioning and the interaction ofthe blade with the bandmill, affect cutting accuracy. In thissection, experiments are conducted to determine the influence ofroll-tensioning, gullet stress, lateral tooth stiffness,torsional frequency and the optimised distribution of stress, onthe cutting accuracy of the saw.6.1 The Effect of Roll-Tensioning on Cutting Accuracy.The objective of this portion of the work is to determine therelationship between the stresses in the sawblade, due to roll-tensioning and bandmill strain, and cutting accuracy. Thestresses in the saw were varied using three different roll-tensioning conditions and three bandmill strains.The method used compared the cutting accuracy of an untensionedblade to a blade progressively tensioned in three stages. Thethree stages of tensioning were:1) Tensioned as shown in Figure 6.12) Tensioned as shown in Figure 6.1 plus a 2400 ft.backcrown radius3) Professionally tensioned.The three bandmill strain were 14000 lb., 19000 lb. and 25000 lb.105Figure 6.1 Details of Centre Region Tensioning used inCutting Tests.For the commercially tensioned blade the transverse curvaturefitted a 20 ft. radius and the backcrown curvature fitted a 900ft. radius. The radius of transverse curvature can be considereda typical value, however, the radius of backcrown curvature isconsiderably less than the industry standard of 2400 ft. and thestresses will be correspondingly higher. An idea of the stressdistribution in a commercially tensioned blade can be obtainedfrom Chapters 3 and 4 (Figure 3.4 and Figure 4.10).Three sets of cutting tests were completed. Each set consisted ofthree cuts, with each blade, at each strain level (18 cuts perset). The untensioned blade and the tensioned blade cut each cantsequentially. The feed speed, bandmill strain and bladedeflections were recorded for each cut.106The two saws used for the above cutting tests were 9.63 in.gullet to back by 0.058 in. thick. The sawtooth pitch was 1.75in. with a gullet depth of 0.69 in. and a kerf width of 0.128 in.The cutting span length was 30 in. The cants were built up fromthree 4-inch deep by 48 inches long cants of Douglas fir to givea 12-inch depth of cut. Feed speed was 200 fpm. and blade speedwas 9425 fpm.The gullet stresses, due to the tensioning profile in Figure 6.1and the standard backcrown radius of 2400 ft., were calculatedusing the results of Chapters 3 and 4 (Figure 3.10 and Equation(4.17)) The rolling distance 'd' for the introduction ofbackcrown was 3.25 in. The calculated stresses were 5846 psi. forthe centre tensioning and 8320 psi. for the centre tensioningplus backcrown. The stresses in the commercially tensioned bladewere estimated to be 14000 psi. This was determined from thebackcrown radius and stress measurements from other commerciallytensioned blades.A sample of the cutting data are shown in Figure 6.2. Traces one,two and three correspond to the displacements of the gullet,centre and back edge of the saw measured just above the topsurface of the lumber. The reduced displacements of the centreand back of the blade are due to the blade moving to the 'inside'of the cut path, the cut path being wider than the blade. Theblade displacements were recorded from (approximately) 0.210704N.42PI•atNtoDi0PO1■I00HI2FtFtW0U1tvftSWseconds before the cut to 0.5 seconds after the cut and have beenconditioned with a low pass (below 10Hz.) filter. The cuttingaccuracy was determined from the displacement of the gullet lineof the saw while in the cut and is the total standard deviation(a) of the blade displacements (see Chapter 2). This includes thestandard deviation of the blade motion plus the standarddeviation of the means.Figure 6.3 Comparative Saw Performance: 14000 lb.Strain.The results of the cutting tests at the 14000 lb. strain levelare presented in Figure 6.3. The bar graph compares the cuttingaccuracy of the untensioned blade (Z0) to the tensioned blade (Y 1to Y3 ) at each stage of tensioning. Y 1 , Y2 and Y3 correspond to the1094^VI^4^‘12^4^`13Blade and Stage of TensioningFigure 6.4 Comparative Saw Performance: 19000 lb.Strain.three stages of tensioning described at the beginning of thissection. Figure 6.4 and Figure 6.5 present the cutting testresults for the 19000 lb. and 25000 lb. strain levels. The changein cutting accuracy of blade Z o from stage to stage is mostlikely due to changes in density of the different cants. Forcomparison purposes one cant was used for each stage and bothblades cut the same cant.At the 14000 lb. bandmill strain level the commercially tensionedsawblade, with the highest level of gullet stress, showed themost improvement in performance (530) and cut most accurately. Atthe 19000 lb. bandmill strain level all the tensioned saws cut110Z0^V1^4^‘12^4^‘13Blade and Stage of TensioningFigure 6.5 Comparative Saw Performance: 25000 lb.Strain.well, however, the saw with the lowest level of gullet stressshowed the most improvement and the saw with the mid-level ofgullet stress cut most accurately. At the 25000 lb. bandmillstrain level the saw with the mid-level of gullet stress showedthe most improvement and cut most accurately. If the results areaveraged over the three strain levels the saw with the mid-levelof gullet stress shows the most improvement and cuts mostaccurately. Thus, it would seem from these results, that at thelow strain levels a sawblade with high levels of gullet stresswill cut most accurately, while, at the mid and high strainlevels, the best accuracy is obtained by having a moderate amountof gullet stress due to roll-tensioning and backcrown.111Figure 6.6 Comparative Saw Performance AveragedOver the Three Bandmill Strains.It should be noted that the width of the tensile stress region inthe gullet area is different for the two types of tensioning. Thecentre tensioning results in a 2.7 in. wide strip of tensilestress in the gullet area compared to a 1 in. wide strip in thecommercially tensioned saw. It is possible that the wider regionof tensile stress provides a better support base for the toothand, at the higher bandmill strains, the saw is able to cut moreaccurately than the commercially tensioned saw, even though thetensile stress is lower.The relationship between the stress in the blade and the cuttingaccuracy of the saw is of prime importance in this investigation.Figure 6.7 presents the total standard deviation of each of the11216^20^24^28^32Gullet Stress (1000 psi.)Figure 6.7 Cutting Accuracy vs Gullet Stress.eighteen sets of cuts from this experiment plotted against thestress in the gullet, a power curve is fitted to the data. Thereis some scatter in the data, possibly due to the inhomogeneity ofthe wood and variations in the preparation of the saws, however,it is apparent that there is a reasonable correlation between themagnitude of stress in the gullet and the cutting accuracy of thesaw.6.2 The Effect of Gullet Stress on Tooth Stiffness and CuttingAccuracyMuch of the work completed to this point has demonstrated thatthe lateral edge stiffness of the blade is an important factor indetermining the cutting accuracy of the bandsaw. For example:displacements of the blade during cutting are primarily113associated with the static stiffness of the blade; and increasesin the stress in the front portion of the blade, whether it be byroll-tensioning or bandmill strain, are known to improve cuttingaccuracy. In this section the cutting accuracy of the blade isinvestigated with respect to the stresses, the lateral stiffnessand the torsional frequency of the blade.The same sawblades were used as for the previous experiment, oneuntensioned and one commercially tensioned. Prior to use for thisexperiment, the sawblades were resharpened and checked for leveland tension. The dimensions of the blades were 9.6 in. by 0.058in. For the untensioned blade the backcrown radius was 4500 ft.and there was no transverse curvature, for the tensioned bladethe backcrown radius was 964 ft. and the transverse curvaturefitted a 20 ft radius. The gullet stresses in the blades due tothe levelling and tensioning were estimated to be 1124 psi. and13245 psi. for the untensioned and tensioned blades respectively.The cutting tests were conducted at bandmill strain levels ofapproximately 16000 lb., 22000 lb. and 29000 lb. The strainlevels were selected such that the torsional frequency of theuntensioned blade, at the mid and high strain level, matched thatof the tensioned blade at the low and mid strain level.For this experiment nine cuts were made by each blade at eachstrain level and the bandmill strains, blade displacements,114carriage speeds, torsional frequencies and lateral stiffness wererecorded. The cants, cutting speed and blade speed were asdescribed for the previous experiment.The cutting test data and results are shown in Table 6.1.Examining first the relationship between the torsional frequencyand blade stiffness it is apparent that the torsional frequencyTable 6.1 Sawblade Data and Cutting Test Results.Blade BandmillStrain019GulletStress(isi)ToothStiffness(lbtin)TorsionalFrequency' (Hz)Mean(iin.10-3)StandardDeviation(in.10-3)Z 15879 15588 188 73 29.9 9.8Z 22038 21198 230 85 21.9 8.5Z 28974 27516 263 97 12.6 8.2Y 15724 27568 274 86 8.8 2.0Y 21928 33219 322 96 6.5 6.6Y 29239 39879 360 106 1.4 2.8is not a good indicator of stiffness. The large amount ofbackcrown in the tensioned blade will increase the gullet stressand the tooth stiffness, however, the change in torsionalfrequency will depend upon the location of the compressive stressfrom the rolling process and will not necessarily change. Becauseof this the torsional frequency cannot be used as a measure ofthe lateral stiffness of the blade without some knowledge of the115120c 104-65 8`?° 6xE0n 4U)o^2a Experimental Data— Regression AnalysisaOa0180 200 220 240 260 280 300 320 340 360Tooth Stiffness (lb/in)stresses in the blade. It could, however, be used to compare theconsistency of blades prepared in the same manner.One of the objectives of this part of the work was to determinethe relationship between the lateral tooth stiffness and cuttingaccuracy. Figure 6.8 presents the cutting accuracy as a functionFigure 6.8 Cutting Accuracy vs ToothStiffness.of lateral tooth stiffness with unremarkable results. Furtherexamination of the results in Table 6.1 shows that the totaldeviation of the sawblade motion is not well related to the toothstiffness, while the mean blade deviation would appear to beclosely related. Figure 6.9 presents the mean of the sawblademotion as a function of the tooth stiffness and a linearapproximation of the relationship can be seen to closely fit thedata. In this case, it would seem that increasing the tooth116200 220 240 260 280 300Tooth Stiffness (lb/in)Figure 6.9 Mean Blade Displacement vs ToothStiffness.stiffness has primarily affected the mean blade displacement.A persistent non-zero mean value in the sawblade displacementwhile cutting is most likely due to inaccuracies in the sawbladelevelling or tooth preparation, or errors in the alignment of theblade and feed system. Although any of these factors could beresponsible, it should be noted that the feed system used wasextremely rigid and the sawblade had been carefully aligned priorto the tests, consequently, the sawblade preparation isconsidered the most likely cause of the problem. Because thecants were built up from three 4 inch slabs the grain directionand knots were distributed more evenly and the cants were thoughtunlikely to produce a persistent offset in the sawblade.117275^325^375^425^45Gullet Line Stiffness (lb./in.)When the cutting direction of the blade is biased toward oneside, the blade will move away from the ideal path until thecutting forces are balanced by the restoring force due to theblade stiffness. The blade will then tend to oscillate about thismean value in response to the changing characteristics of thewood. If the sawblade is well prepared and accurately aligned,such that the mean deflection is close to zero, it is expectedthat increasing the stiffness of the front edge will reduce thedeviation of the saw more effectively than when the blade ismisaligned and a significant mean value is present.Figure 6.10 Cutting Accuracy vs Gullet Stiffness.Figure 6.10 is the combination of Figure 6.8 and Figure 6.9 andpresents the relationship between the stiffness of the teeth andthe magnitude of the mean plus the standard deviation of thesawblade. Thus, the two portions of the blade displacementaffected by changes in the tooth stiffness are combined and the118Figure 6.11 Cutting Accuracy vs Gullet Stress.correlation between the fitted power curve and the data is muchimproved. The cutting tests from Section 6.1 were compared in thesame way and the results are shown in Figure 6.11. The fittedcurve can be seen to represent the data more accurately than thatshown in Figure The Effect of Optimising the Roll- Tensioning Stresses onCutting AccuracyIn the previous section it was demonstrated that the lateralstiffness of the blade plays an important part in minimisingsawblade deviations. Additionally, analytical and experimentalinvestigations have shown that rolling in the centre 60% of thesawblade will increase the edge stiffness while rolling outside119this region will decrease the edge stiffness. Commerciallytensioned blades are often rolled outside this region. Thisprocedure decreases the edge stiffness while increasing thelevels of tensile stress. This is possibly the reason that asawblade with the roll-tensioning confined to the centre regionof the saw performed as well as a professionally tensioned saw,even though the stress levels were lower (Section 6.1). Tofurther investigate this phenomenon a simplified blade stressdistribution designed to maximize the lateral tooth stiffness wastested.The stress distribution in a traditionally tensioned blade wasshown in Chapter 3 (Figure 3.4). This type of rolling pattern hasthe effect of elongating the centre of the saw and producesregions of high tensile stress at the edges. These regions ofstress are responsible for the increased stiffness of the edgesof the blade. The combined effects of the stress due to roll-tensioning and bandmill strain are shown, idealized, inFigure 6.12a, with some typical stress levels. The higher thestress level in the edges of the saw the stiffer will be theteeth, however, the magnitude of this stress will limit thefatigue life of the blade and there will be a maximum level thatcan be tolerated by a particular system.The work conducted in this study has indicated that the furtherfrom the blade centre-line the roll-tensioning is carried out the120111117111iRolled Zone1401.iiiiiiii•••• •••••Xi=JAllt o--------jS. (c7) ..--------4.r-71 ------D)12700 psi12000 lb.Bandmill Strain8500 psiRoil-Tensioning^021200 psiCombined Stress2370 psi0Centre Tension18830 psiIncreased BandmillStrain (17700 lb.)21200 psi —rrrrz .ra-rr-ri-Combined Stress1Figure 6.12 Diagram of Stresses for Conventional and CentreTensioned Saws.lower will be the tooth stiffness even though the same gulletstresses are induced. To determine the increase in stiffness thatcan be achieved by revising the traditional rolling patterns,the effect of restricting the rolling to the centre 20% of theblade is investigated (Figure 6.12b). Allowing for theintroduction of four rolling lines in this region an estimated2370 psi. of tensile stress would be induced into the unrolledportion of the saw. This is some 6130 psi. lower than the fullyrolled example shown in Figure 6.12a. Increasing the bandmill121strain until the gullet stress matches that of the conventionallytensioned example results in the revised stress distributionshown. The torsional frequencies and gullet line stiffnesses forthe two stress profiles were determined analytically, the narrowcentre tensioning, coupled with the higher bandmill strain,provides a 12% increase in torsional frequency and a 16% increasein gullet stiffness over the idealized conventional tensioning.To determine if the predicted increases in stiffness for themodified roll-tensioning profile actually occurred, andtranslated into improved cutting accuracy, four sawblades wereprepared and cutting tests conducted.The four saws used for the cutting tests were 8.34 in. gullet toback by 0.057 in. thick. The sawtooth pitch was 1.75 in. with agullet depth of 0.66 in. and a kerf width of 0.125 in. Thecutting span length was 20 in. Amabalis fir cants 10 in. by 10in. by 8 ft. long were used for the cutting tests. The feed speedwas 244 fpm. and the blade speed was 9425 fpm., this provided asolid wood removed to gullet area ratio (gullet feed index) of0.7. In this experiment the initial set of cutting tests wererepeated at a higher strain level, each saw making twenty cuts ateach strain level.The method used compared the cutting accuracy of twoconventionally tensioned saws with that of two saws tensioned in122the centre 1.625 in. only. The tooth stiffness and blade stresseswere monitored throughout the experiment. Two different basebandmill strains were used for the conventionally tensioned saws,12000 lb. and 17000 lb. When testing the centre tensioned sawsthese strains were adjusted such that the gullet stresses werethe same as the conventionally tensioned saws.The saws were prepared in a similar manner, levelled, swaged,side-ground and sharpened, but without tension or backcrown. Thesaws were checked and found to have virtually identicalTable 6.2 Lateral Tooth and Blade Stiffnesses Before Tensioning.Blade Tooth(lb/in.)Gullet(lb/in.)Centre(lb/in.)Back(lb/in.)RT- 1 146 262 678 279RT-2 147 260 667 271RT-3 146 266 670 268RT-4 147 269 678 274dimensions, transverse curvatures and backcrown profiles. Eachsaw was then run on the bandmill, positioned such that the gulletoverhung the wheel by 0.25 in., and the tooth and bladestiffnesses measured (Table 6.2). At this stage it was apparentthat all saws exhibited very similar stiffness characteristics.An important consideration when analyzing or testing the effectof the two different types of roll-tensioning was to ensure that123Matching gauges ..1on reverse side^ ► -0]-H--►1/2^ 1/2'Two paird of straingauges in threelocations. around theblade (1 gauges)Roll-Tensioning Profile■NMOMMEMMIMMO;5 rolls at3000 lb(approx.)iiii;;;MMOMEM4- 2 rolls @2000 lb(approx.)at0 11____-------Ar----------the final gullet stresseswere the same in all saws.For the blades used in thecutting tests it wasnecessary to keep track ofthe stress induced by thetwo types of roll-tensioning and the bandmillstrain. To do this eachblade was instrumented withstrain gauges as shown inFigure 6.13a and the changein stress in the blade ateach stage was thenrecorded.The conventionally preparedblades were tensioned to alevel where the transversecurvature fitted a 45 ft.Figure 6.13 Strain Gauge Locationsand Rolling Profile forCentre Tensioned gauge. The centre tensioned blades were prepared as shownin Figure 6.13b. Care was taken to keep the backcrown the samefor all the saws, the end result was a backcrown radius R a 6750ft. for all four blades.124The changes in edge stress due to the roll-tensioning are shownin Table 6.3. The amount of gullet stress induced by theconventional roll-tensioning is in the order of 7000 psi. andthat for the centre tensioning around 2000 psi. Table 6.4 showsthe bandmill strains calculated to give equal gullet stresses inall four blades. The measured tooth and blade stiffnesses foreach blade at these strains are shown in Table 6.5 and anincrease in gullet stiffness of 13% was recorded for the centretensioned saws. This is slightly less than the predicted value of16% for the idealized case and is due to the experimental stresslevels being slightly lower than those used in the analysis.To determine if the increase in lateral tooth stiffness led to anincrease in cutting accuracy a series of cutting tests wereperformed using 10 cants (Test A). Each saw cut each cant twice(20 cuts per saw) for a total of 80 cuts, giving 40 cuts for eachtype of tensioning. To determine if the results were sensitive tothe level of bandmill strain, this set of cutting tests wererepeated at an increase in strain level of 5000 lb. for all fourblades (Test B). The results of both sets of tests are shown inFigure 6.14. Combining the results to give one performance figurefor each type of tensioning, the centre tensioned saws appearedto perform some 40% better than the conventionally tensionedsaws.125Table 6.3 Increase in Edge Stress due to Roll-Tensioning.Blade Tension GulletStress (psi)Back EdgeStress (psi)RT-1 Standard 6675 8865RT-2 Standard 7590 7770RT-3 Centre 2115 2745RT-4 Centre 1665 2025Table 6.4 Equivalent Bandmill Strains for Equal Gullet Stresses.Gullet Stress (psi) BandmillBlade StrainBandmill Strain Roll Total (lb)TensioningRT-1 11865 6675 18540 12000RT-2 10950 7590 18540 11074RT-3 16425 2115 18540 16612RT-4 16875 1665 18540 17067Table 6.5 Lateral Tooth and Blade Stiffnesses After Tensioning.Blade Tooth(lb/in.)Gullet(lb/in.)Centre(1b/in.)Back(lb/in.)RT-1 164 304 611 324RT-2 163 293 591 310RT-3 170 326 746 339RT-4 175 340 779 360The results are also presented as a function of the gullet linestiffness in Figure 6.15. As with the previous set of cuttingexperiments it is apparent that the cutting accuracy is stronglyinfluenced by the gullet line stiffness.126Tension 1 = Conventional tensioningTension 2 = Centre tensioningTension 1^Tension 1^Tension 2^Tension 2(Saw 1)^(Saw Z (Saw 3) (Saw 4)Tensioning Method and Saw NumberFigure 6.14 Cutting Accuracy of Centre Tensioned vsConventionally Tensioned Saws.6.4 DiscussionAt low levels of bandmill strain sawblades with high levels ofgullet stress cut most accurately. At the mid and high strainlevels the best accuracy was obtained by having a moderate amountof gullet stress, due to roll-tensioning in the centre region ofthe sawblade, coupled with standard backcrown.The results of the cutting experiments indicate that the cuttingaccuracy is strongly related to the lateral edge stiffness of thesawblade. In the first and second set of cutting experiments thecorrelation between cutting accuracy and gullet stress or lateral127300^310^320^330Gullet Stiffness (lb/in)o Experimental Data.— Regression AnalysisFigure 6.15 Cutting Accuracy vs GulletStiffness.stiffness was improved when the sum of the mean and standarddeviation of the blade deflection was used to define cuttingaccuracy.A simplified tensioning profile confined to the centre 17% of thesawblade, coupled with an increase in bandmill strain, was foundto improve edge stiffness by 13% over that for conventionallytensioned blades with similar gullet stress levels. Subsequentcutting experiments demonstrated a 30% to 40% reduction in totalstandard deviation of the sawblade for the centre tensionedblades.1287. CONCLUSIONSThe experimentally based study presented here quantifies thestresses induced in bandsaws by roll-tensioning and identifiesthe stabilizing effect of roll-tensioning, and blade overhang, onthe interaction of the blade and wheel. The relationship betweenrolling position, blade stiffness and cutting accuracy isdemonstrated and an optimum roll-tensioning configuration isshown to improve cutting accuracy.For roller paths located on previously unrolled material theresults of the rolling experiments enable the stresses introduced.into the blade by roll-tensioning to be quantified. The magnitudeand distribution of the tensile stresses due to individual roll-tensioning lines are essentially linear. The magnitude isdescribed by empirically determined co-efficients that are validexcept close to the yield loads, where some non-linearity exists,the distribution is described by a simple formula.The analysis of the stresses in the blade due to the interactionof the blade and bandmill has shown that overhang can be animportant factor in determining blade stress and stability.Normal levels of overhang (say up to 0.7 inches) have a minimaleffect on the stresses in the blade, however, if the overhangshould be allowed to become too large, this will reduce thestress, and hence the stiffness, of the cutting edge of the129blade. Untensioned blades are shown to be unstable whensymmetrically located on the bandmill wheels and large values ofoverhang are required to achieve stability. Roll-tensioningimproves blade stability and enables small values of overhang tobe used.The investigation into the effect of roll-tensioning on thefrequency and stiffness of the sawblade demonstrated that thechange in frequency and stiffness is a maximum when rolling onthe centre-line, zero when the rolling position is approximately30% of the blade width from the centre-line and a minimum whenrolling at the edge. Thus, to ensure the increase in stress isaccompanied by an increase in stiffness, rolling should beconfined to the centre 60% of the sawblade.When the stresses are symmetric with respect to the sawbladecentre-line, the torsional frequency is a good indicator ofsawblade stiffness. When the stresses are not symmetric, as inthe case when backcrown is present, additional information on therolling geometry and in-plane curvature are required before thetorsional frequency can be used as a guide to the stiffness ofthe saw.Cutting accuracy is strongly related to the lateral edgestiffness of the sawblade. Correlation between sawblade deviationand lateral blade stiffness may be improved by using the mean130plus the standard deviation of the blade deflections during thecutting process.By combining some of the results of this thesis it is possible toimprove the lateral stiffness of the sawblade, and hence thecutting accuracy, without increasing the level of stress in thegullet region of the saw. This is important because the gulletregion is most susceptible to fatigue cracks. For example, theresults of the investigation show that the blade overhang shouldbe kept below 0.7 in. otherwise the reduction in stiffness of thecutting edge can be excessive. To maintain stable operation withsmall levels of overhang some roll-tensioning is required, byconfining this roll-tensioning to a narrow centre region of theblade the stiffness of the blade is maximised with respect to thelevel of stress while providing stable operation. The remainderof the stress, up to the fatigue limits of the sawblade, shouldthen be provided with bandmill strain. The increase in bandmillstrain is proportional to the difference in gullet stress betweenthe conventionally tensioned blades and the centre tensionedblades.An experiment was conducted based upon this approach. Asimplified tensioning profile, confined to the centre 170 of thesawblade, coupled with an increase in bandmill strain, was foundto improve edge stiffness by 13% over that for a conventionallytensioned blade. Subsequent cutting experiments demonstrated a13130% to 40% reduction in sawblade deviation without any increasein gullet stress.132LIST OF REFERENCESAllen, F.E. 1973. "High-strain/thin kerf". Proc. First NorthAmerican Sawmill Clinic, Portland, Oregon. Feb. 1973.Allen, F.E. 1985. "High Strain Theory and Application".Proceedings of the 8th Wood Machining Seminar. Univ.Calif., Richmond, CA. p.384-404.Alspaugh, D.W. 1967. "Torsional Vibrations of a Moving Band".J. Franklin Inst., V283(4).Anderson, D.L. 1974. "Natural Frequency of Lateral Vibrationsof a Multiple Span Moving Band Saw". Res. Rep., WesternFor. Prod. Lab. (now Forintek Canada Corp.), Vancouver,Canada.Anderson, D.L. 1975. "Tensioning Stresses in Bandsaw BladesInduced by Roll Stretching". Res.Rep, Civil Eng. Dept.,U.B.C., Vancouver.Aoyama, T. 1971a. "Tensioning of Bandsaw Blades by Rolls, III:Effect of Saw Blade Thickness". J. Jap. Wood Res. Soc.17(5), p188-195.Aoyama, T. 1971b. "Tensioning of Bandsaw Blades by Rolls, IV:Effect of Saw Blade Width". J. Jap. Wood Res. Soc. 17(5),p196-202.Bajkowski, J. 1967. "Distribution of Stresses in Frame-sawBlades Prestressed by Rolling". [Translated fromHolztechnologie, (Wood Technology) 8(4):258-262.]Environment Canada Library, Ottawa, ON (Translation OOENVTR 867, 1975, 12p).Chardin, A. 1979. "Displacement of Bandsaw Blades on Wheels:An Experimental Approach". Sixth Wood Machining Seminar,U.Cal., Richmond, Cal. p209-221.COFI, 1988. "British Columbia Forest Industries Fact Book".Council of Forest Industries (COFI), Vancouver.Eschler, A. 1982. "Stresses and Vibrations in Bandsaw Blades".M.A.Sc. Thesis, Univ. British Columbia, Vancouver, B.C.,Canada.Foschi, R.O. and A.W. Porter. 1970. "Lateral and EdgeStability of High-Strain Band Saws". Dept. of theEnvironment, Western Forest Products Lab. (now ForintekCanada Corp.), Vancouver, BC. 17p.133Foschi, R.O. 1975. "The Light Gap Technique as a Tool forMeasuring Residual Stresses in Bandsaw Blades". WoodSci. Technol. 9:243-255.Fujii, Y., Hattori, N. and M. Noguchi. 1988. "The StressDistribution in the Band Saw in Tension". Res. Rep. (J.lang.), Bull. Kyoto Univ. No.60.Garlicki, A.M. and S. Mirza. 1972. "The Mechanics of BandsawBlades". Res. Rep., Dept. of the Environment, EasternForest Products Lab., Ottowa, Canada.Garlicki, A.M. and S. Mirza. 1978. "Lateral Stability of WideBand Saws". Res.Rep., Dept. of the Environment, EasternForest Products Lab. (now Forintek Canada Corp.),Ottawa, ON.Hutton, S.G. and J. Taylor. 1990. "Operating Stresses inBandsaw Blades and their Effect Upon Fatigue Life".Submitted to For. Prod. J. June 1990.Kirbach, E. and T. Bonac, 1978. "The Effect of Tensioning andWheel Tilting on the Torsional and Lateral FundamentalFrequencies of Bandsaw Blades". Wood Fibre 9:245-251.Lehmann, B.F. 1991. "Bandsaw Blade Frequency and StiffnessProgram - SERRA". Mech. Eng. Dept., U.B.C., Vancouver.Lipson, C. and R.C. Juvinall. 1961. "Application of StressAnalysis to Design and Metallurgy". Eng. SummerConference, Univ. of Michigan, Ann Arbor, Michigan.Martec Ltd. 1989 "Vibration and Strength Analysis Program".,Version 5, Martec Ltd., Halifax, Nova Scotia.Mote, C.D. 1965a. "Some Dynamic Characteristics of Bandsaws".For. Prod. J., V15-1.Mote, C.D. 1965b. "A Study of Bandsaw Vibrations". J. FranklinInst., V279.Pahlitsch, G. and K. Puttkammer. 1972. "The Loading of BandsawBlades: Stresses and Strength Factors". Holz als Roh-Und Werkstoff. Vol. 30. p.165-174.Saito, Y. and M. Mori, 1970. "On the Buckling of BandsawBlade. Part 1 and 2". [Translated from Wood Industry8(7):20-23 (1953) and 8(8):21-27 (1953), Tokyo. Transl.No. (DOFF TR 125 ], Dept. of Fisheries and Forestry,Ottowa, On.134Schajer, G.S. 1981. "Analysis of Roller-Induced ResidualStresses in Circular Discs and their effect on DiscVibration". Ph.D. thesis, U. Cal., Berkeley.Soler, D.I. 1968. "Vibrations and Stability of a Moving Band".J. Franklin Inst., Vol.286(4) p295-307.Sugihara, H. 1977. "Theory of the Running Stability of BandsawBlades". Fifth Wood Machining Seminar, For. Prod. Lab.,U. Cal. Berkeley, CA.Sugihara, H., Hattori N. and Y. Fujii. 1981. "Contact PressureBetween Band Saw and Wheel". Res. Rep.(Jap. lang.)Faculty of Agriculture, U. of Kyoto, Japan.Swift, H.W. 1932. "Cambers for Belt Pulleys". Institute ofMechanical Engineers, Proceedings, Vol.122., p627-659.Tanaka, C. and A. Shiota, 1981. "Experimental Studies in BandSaw Blade Vibration". Wood Sci.Techn. 15:145-159.Thunell, B. 1972. "The stresses in a Band Saw Blade". PaperiJa Puu-Papper och Tra, Vol.54(11), p759-764.Timoshenko, S. and S. Woinowski-Krieger. 1959. "Theory ofPlates and Shells". Second Ed., McGraw-Hill.Timoshenko, S.P. and J.N. Goodier. 1970. "Theory ofElasticity"., Third Ed., McGraw-Hill, N.Y.Ulsoy, A.G. and C.D. Mote. 1980. "Analysis of BandsawVibration". Wood Sci. 13(1):1-10.Young, W.C. 1989. "Roark's Formulas for Stress and Strain."6th. Ed., McGraw-Hill, Table 26.135APPENDIX IFinite Element Analysis Program and Models UsedWe are interested in determining the lateral edge stiffness andnatural frequencies of the cutting span of a bandsaw blade. Ingeneral this is modelled as a rectangular plate either simplysupported or built in at two opposite ends with the other twoedges free. The stiffness can be determined by measuring thedisplacements associated with the application of a point loadnormal to the plane of the plate. There is no closed formsolution to this problem and the behaviour of the plate has beenanalyzed using a finite element program (Martec Ltd. 1989). Theanalysis examines the effect of roll-tensioning stresses andbandmill strains on the lateral stiffness and natural frequenciesof plane and toothed edge plates. Figure I(1)(a, b & c) shows theblade models used in the analysis. An 8 node rectangular shellelement was used throughout as it provided a good rate ofconvergence without an excessive number of elements.The stresses from bandmill strain and roll-tensioning wereincluded as preload stresses. The finite element programincorporates the effect of these stresses in the elementstiffness formulation.136451JoLf^(a)L 4--elPrescribed compressivestresses due to rolling incentre two elements.AD^•• •All three models use 8 node rectangular plateelements with 5 degrees of freedom per node.P2 P2 P, P2 P2--.14-.14-.14-1.14-1.14-1.1.--J_I_(c)L = 30 in.b= 9.469 in.P, = 1.75 in.P2 = 1.766 in.g = 0.688 in.(b) LAd All All 411 All AfFigure I(1) Models of Blade Used in Analysis.137Validation of the analysis was accomplished by modelling a plateproblem with a known solution and comparing the results. Theproblem chosen was that of a uniformly loaded plate with one longedge built in and the other three edges simply supported. For themodel shown in Figure I(1)c the toothed side was built in. Thepredicted displacement was 3% greater than the published solution(Young 1989).For the model shown in Figure I(1)a the lateral and torsionalnatural frequencies were calculated and found to be in goodagreement (0.2 Hz.larger) with those from the closed formsolutions (Alspaugh 1967; Mote 1965a). For a range of rollingpositions, the lateral stiffnesses were within 1.5% of thosedetermined by Lehmann (1991).When analyzing the effect of a pair of roll-tensioning lines onedge stiffness and natural frequency it is necessary to revisethe element configuration and residual stresses for every changein rolling position. To expedite this part of the analysis, aseparate interactive mesh generation program was written thatconstructed the model to suit the dimensions and stressesprescribed.In general, the finite element models were based on a standard 9inch or 10 inch wide re-saw for a 5 ft. bandmill. The dimensionswere chosen to match the bandsaw used for the experimental data.138The span and width were adjusted from these base dimensions byeither adding elements or relocating the boundary conditions.Tooth stiffness was determined from the application of a pointload on the back of the tooth. Gullet line, centre line and backedge stiffnesses were determined from the application of a pointload at the midspan line.139APPENDIX IIAnalysis of the Effect of Roll-Tensioning on Torsional FrequencyOne of the objectives of the thesis is to determine therelationship between the stresses in the sawblade and thefrequency and stiffness of the blade. In this Appendix a closedform frequency equation is developed that predicts the effect ofroll-tensioning stresses on the torsional natural frequency ofthe sawblade.It will be assumed that the roll tensioning is conducted byrolling 2 lines at a time each symmetrically located with respectto the centre line. Figure II(1)a shows the resulting idealizedstress distribution in one half of the blade for a roll width tat a distance d + t/2 from the centre line. Figure II(1)b showsthe displaced configuration of the band between the guides.Consider that the band has an axial tensile stress distributiona(x) existing when w a 0. Then, if az and e z are the stressesand strains resulting from the displacement, the additionalstrain energy stored in the band is given by:L b/2U = 2 L fo (_2a zez+a z (x) ez)hcbcdz140Figure 11(1) Stresses and Displacements in TorsionalFrequency Model.where:az = E E z and ez = 1 ( z122 11(2)The strains due to bending have been shown to be small (Mote,1965a) and are neglected.Assuming that the displacement of the blade varies linearly withx i.e. w(x,z) = x0(z) where 8(z) is the angle of twist. Then:1412^I1 rIl  G 33b + 0 hx 2 az (x) dx1fh42w - \ L b/2fo phx2dx11(8)U f 0Lfob/2hX 2 0, (X) (d8)2 + Eh( —(2 k dz 1dO 12)2=^ dxdzx2b/2^dey^L=^hx2a z (x)(--di dxdz+ 10 E6hb405( ddz0)4 dzfoL fo"(3)11(4)For small values of dO/dz the second term may be neglected withrespect to the first term. Then adding the strain energy due tothe torsionally induced shearing stresses gives:U = f L{f b/2 hx 2 a z (x) dx+ Gh3b1( d0)2 dz0 6 ,fiz II (5)The total kinetic energy of the band is given by:T = ('L /' b/2 phx 214 2 dxdza 0 d 0^k dt/ 11(6)Assuming that in the fundamental mode of vibration:0 = Oosinir —z sinwt^ 11(7)Then, Rayleigh's Quotient is:142For the stress distribution given in Figure II(1)a, it may beshown that:b/2^ut*b3f 0 x2 az (x) cbc - 24 11(9)where:*at = at [1 - 4 (3 I 12 + 3 rieii-e/2 )]/^t .^1^dEe = - '^i- . _bHence the torsional frequency of a band, subjected to a uniformaxial stress a plus the stress distribution given inFigure II(1), is given by:w2 = I-2121 14 GI-1-2 \ 2 + a s + at* 1\ ii pL^\ JD'From Equation 11(12) it can be seen that if 4 is negative theeffect of the rolls will be to decrease the frequency and hencedecrease the stiffness of the band. Such a situation occurs when:4 ( 3r /2 +3 r / e / 4. en) > 1For the general case when e 1 << r 1 an increase in frequency willoccur when r 1 < 0.29.143where: an, = at( ^6(ri+0 .5 ei)1 - 12(e /r /2 + r le n +0 .25 e i3 ) )For the case where the blade is rolled non-symmetrically, e.g.when backcrown is introduced into the blade, the frequency changewas also investigated using the same approach as for thesymmetric case. The tensile stress due to rolling was comprisedof a uniform stress, to balance the compressive stress in therolled zone, and a linearly distributed moment stress, due to theroll being off-centre. Completing the integration the followingequation is obtained:et = a t( 1-4(3r /2 +3r /e i +e i2) -6 --1--n°^ 4(4r /3e6r /2 e /2 +4r ie /3 + e")ar Er (14)The first term is identical to the symmetric case. The secondterm is the correction for the small area of tensile momentstress missing due to the rolling line and can be droppedproviding e' « r'. The result is the same as for the symmetriccase; providing the rolling takes place within the region r' <0.29 the frequency will increase, otherwise it will decrease.144APPENDIX IIIAnalysis of the Effect of Blade OverhangIn this Appendix the effect of blade overhang is analyzed usingthe differential equation for a cylindrical shell loadedsymmetrically with respect to its axis. The equation wasintroduced in Chapter 4 and is repeated here along with a diagramof the co-ordinates (Figure III(1)):D d'w + —Eh^E.12w = Z -^E0d,^dx 4^a 2 a ""Where:w radial displacementD Eh3/12(1-p 2 )^flexural rigidity of plateE Young's modulush plate thickness^p^Poisson's ratioa cylinder radiusZ = pressure normal to the surfaceE 0= circumferential residual strainThe analysis examines the pressure distribution on a bandmillwheel, in the presence of bandmill strain, overhang, roll-tensioning stresses and anticlastic curvature, and calculates thestresses in the cutting region of the band. The approach usedfollows that of Sugihara (1977) with the addition of a rigid145wheel assumption, roll-tensioning and anticlasticcurvature. The effects of wheelcrown and wheel tilt are notincluded.When the sawblade overhangs oneedge of the bandmill wheel(Figure III(2)a) the centre ofpressure is biased towards one Figure III(1) Cylindrical Modelof Blade on Wheel.Figure III(2) Pressure, Forces and Moments Associated withOverhang.146side of the blade causing a change in the in-plane stressdistribution in the blade. To analyze the effect of overhang, theoverhanging portion is separated from the section on the wheeland the wheel is given an initial displacement wo(Figure III(2)b), leading to a pressure distribution dependentupon the initial shape and hence the displacement of the blade'.Forces and moments are then applied to the overhanging portionsuch that continuity of displacement and slope are obtainedbetween the two parts of the blade (Figure III(2)c). Thecorresponding force and moment to support the overhang are nowapplied to the edge of the blade on the wheel (Figure III(2)d)and the pressure distribution and moment determined.Assumptions made in the analysis are:a) The blade can be modelled as a cylindrical shell.b) The wheel is assumed to be a rigid surface inside the shell.c) The forces and displacements applied to the blade by thewheel can be modelled by the application of radial forces,displacements and pressures to the shell.d) The lateral in-plane forces Nx are zero.e)^The thickness of the shell is constant.51t is assumed that at zero bandmill strain the blade justtouches the bandmill wheel. The displacement w o is then equal to orgreater than the maximum displaced shape of the blade and fullcontact is assumed.147From III(1), using the coordinates in Figure III(3)a, thedisplacement of the overhung portion of the blade due to a forceP and a moment M. is:w -  M° rA coshroccosf3x+ sinhf3xsinPx2 DP 2 L 2- A3 (coshi3xsini3x + sinh.f3xcos(3x)]P^ {2AicoshPxcosf3x + (1-A2 ) coshPxsinPx-- (A2 +1) sinhf3xcos 13x]where:A1 , A3 _ (  coshPssinhPsTcosPssinf3s)inh2 Ps - sin2 PsA2 - (  sinh2 Ps+ sin2 ps)sinh2 f3s - sin2 PsR4 - 3 (1-v 2 ) a 2h2For displacement compatibility:Pfc, Pwx:=0 = wo - 2 Dp. 2^^2D P 3A2 - ^ Al-Assuming that full contact occurs between the blade and wheel:1 P^Mowx.--0 = 0 - ^ A2 -^A2 Dp 2^DP 3The overhung portion is now 'attached' to the other section ofblade by applying the force P and moment M o to the edge of the4 D13 3148Figure 111(3) Coordinates, Forces and Moments usedin Overhang Analysis.blade on the wheel. The displacement due to the force P will beresisted by the rigid nature of the wheel, this will create anarrow region of increased pressure at the front corner of thewheel. The moment M o has no effect on the pressure due to thezero slope boundary condition. In reality this is a contactproblem, the application of a moment to the front edge of theblade would result in a change in pressure distribution betweenthe blade and wheel with the blade possibly lifting off the wheelclose to the front edge. This is a non-linear problem and cannotbe investigated with the methods used here, therefore, theanalysis is confined to the zero slope boundary condition. In149support of the method used, examination of the effect of the zeroslope and natural slope boundary conditions on P and M e has shownthat the zero slope boundary condition increases the value of P,over that of a free slope boundary condition, and this tends tocompensate for the loss of effect from M0 .Modelling the pressure due to the force P as a line load at thefront edge of the wheel, the resulting pressure distribution forthe case of a blade with overhang is:P(x) = Pw(x) + PS (x-0)Where 6^the Dirac delta function.The first component of the right hand side of this equationrepresents the pressure distribution across the wheel due to theradial displacement w0 . The distribution of this pressure willdepend upon the shape of the blade due to such factors as roll-tensioning and anticlastic curvature and will be addressed next.When roll-tensioning stresses are introduced into the sawblade,and the blade is bent over the bandmill wheel, the transverseshape of the blade will depend upon the magnitude anddistribution of these stresses. When bandmill strain is appliedthe pressure between the wheel and blade will depend upon thisinitial shape and will affect the magnitude of the moment due toblade overhang.150The displaced shape of blade, as it is bent over the wheel, isrelatively insensitive to the large stress gradients associatedwith roll-tensioning (Foschi, 1975) Because of this, reasonableFigure III(4) Stresses, Displacements and Associated Pressuresof Blade on Wheel.approximations of the transverse shape of the blade can beobtained by modelling the stress distribution as a smooth curve.In the following analysis the in-plane stress distribution in afully tensioned blade is modelled as a parabola (Figure III(4)a),providing the high tensile stress regions near the edges of theblade characteristic of roll-tensioning. The displaced shape ofthe blade due to this stress when bent over the wheel is given byFoschi (1975):151wi(x)^= ,a {1-12 (i)2 124 aca i(yicoshf3xcos Px+v[ ay 2 sinh(3xsinPx)Eb 2bsinhP^cos (3 -- - coshP -2- sing b 111(6)Y1 - 13 2(sinhPb+ sin(3b)sinhP P- cos p 1-31 + coshP^sinP2^2 2-Y2 f32(sinhPb+ sinPb)and is the combination of the homogeneous solution (anticlasticcurvature) and the particular solution due to the residualstresses.The deflected shape of the blade with respect to the bandmillwheel, prior to the application of bandmill strain, is shown inFigure III(4)b. To achieve full contact the displacement of theblade is:w(x)^wo - (lg.+ w i (x))where:w(x)^the displacement of the bladew0^= the displacement of the wheel= the distance from the surface of the wheel to theneutral axis of the bladewi (x)^the initial displaced shape of the blade152Substitution of the blade displacement w(x) into Equation III(1)will only provide the wheel pressure distribution for theparticular solution part of Equation III(6) (see Figure III(4)c),the homogeneous part will provide zero pressure by definition.The homogeneous solution (anticlastic curvature) is due to thePoisson's ratio effect, associated with the bending stresses inthe blade as it is bent over the wheel, and is determined frombending moments applied to the edges of the blade. To obtain thecorrect blade displacement, that includes the initial anticlasticshape and satisfies Equation III(1), a uniform pressure and twoedge moments are required as shown in Figure III(4)d. Where:P. = Ehwo/a2 and M = vD/aAs there is no mechanism for applying these edge moments, and asthere is no pressure distribution that will provide a homogeneousdisplacement, the pressure distribution must approximate theeffect of these moments. If a pressure distribution is determinedsuch that the displacement of the blade closely approximates thatdue to the two edge moments then the pressure will approximatethe actual pressure and the effect of anticlastic curvature canbe included in the analysis.A 4th. order polynomial was chosen to model the anticlasticcurvature in the blade. The shape of the blade, along with thepolynomial approximation, is shown in Figure III(5). For the case1530.20.18_2 0.1E 0.088 0.060 0.040.020-0.02-0.040.125^0.25^0.375^0.5Blade Width (x/b)— Anticlastic Displacement• ApproximationFigure 111(5) Approximation of AnticlasticDisplacement.shown the maximum error in displacement is 5%, this is only 2% ofthe total blade displacement indicating that the shape of theblade is reasonably well approximated by the expression:w“M = a --11 -12( -7- 1 +A + Bx 2 +Cx 4Where:cr, maximum value of compressive stress in the parabolicapproximation of the stress distribution.A, B and C are determined from a least squares approximationof the homogeneous solution in Equation 111(6).The pressure on the wheel due to a displacement w. is:154^d 4^EhP , , =^\x,^YVwkx,^dx4 a 2Differentiating and combining terms:TAVI-17.K 2 -173x1Pp., (.20h2a2 )^ll^(i_v2)Where: V1lV2 =(B-1 2 - aab 2E V3 =CThe force on the wheel due to a displacement yo is:EhFw^I P , Nth.b^w‘x/a -- +s2Ehivi(b-s) 3^+s)31-1-M-)5^+s)511a^ R 2 /The equations required to solve the problem are: Equations 111(3)and 111(4), from continuity of displacement and slope between theoverhang and the portion on the wheel; Equation 111(10), from theforce on the wheel due to a displacement wo ; and Equation111(11), from the equilibrium of the system.Fw+ aP = To^III (11)The five unknowns are the force and moment due to the overhang (Pand M0), the force on the wheel and the displacement of the wheel155(F, and w0) and the tension in the band (Td. As one of theunknowns is a prescribed variable, usually the bandmill strain(2T0) , the system of equations can be solved.The final step is to determine the moment in the blade due to theoverhang so that the stresses in the cutting region can beestablished. The moment is determined from the integration of thepressure on the wheel (Equation II1(9)) about the blade centre-line:= Eh fa^pw,x) x cbcJ -22= Eh [ V1 (b-s) - -V2 ri .,4, 1^12 + s141_ V3 10:2 k 6^\ 6 1a 2^4 R 21^2 )^6 R21^+ s2^/The results of the analysis are presented in Chapter 4.156


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