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The effectiveness of the light-gap method for monitoring saw tensioning Lister, Peter F. 1993

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THE EFFECTIVENESSOF THELIGHT-GAP METHODFORMONITORING SAW TENSIONINGbyPETER F. LISTERBASc, University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Mechanical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1993© Peter F. Lister, 1993In 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 Engineering The University of British ColumbiaVancouver, CanadaDate  March 31, 1993 DE-6 (2/88)1 1ABSTRACTThis thesis investigates the effectiveness of the light-gap method for evaluating the sawtensioning process. In particular, it examines the ability of the light-gap profile to indicatechanges in saw natural frequency and stiffness caused by the tensioning stresses. The light-gap method for bandsaws is studied in detail. Many of the results are also expected to beapplicable to circular saws. Although the light-gap method is a poor indicator of the details ofthe tensioning stress distribution, it does provide a good measure of changes in naturalfrequencies and stiffness caused by the tensioning process. In particular, when a bandsaw istensioned to the traditional light-gap profile, the frequency and stiffness of the saw can becharacterized by the curvature of the light-gap profile. Additionally, the light-gap methodoffers important practical advantages over other tensioning evaluation techniques. The light-gap method indicates local non-uniformities in the tensioning state, guides roll path locationand indicates levelling defects. For these reasons, the light-gap method is confirmed to be ahighly effective and rational way of monitoring the saw tensioning process.111TABLE OF CONTENTS ABSTRACT^ iiLIST OF TABLES vLIST OF FIGURES^ viNOMENCLATURE viiiACKNOWLEDGEMENT^ x1.0 INTRODUCTION1.1 Background and motivation ^  11.2 The light-gap method 51.3 Objectives and scope 62.0 GENERAL BANDSAW RELATIONSHIPS2.1 Chapter overview^ 92.2 Stresses induced by toll tensioning bandsaw blades^ 92.3 Relationship between lateral displacement and stress 122.4 Relationship between natural frequency and stress  192.5 Relationship between stiffness and stress^ 242.6 Simplified relationships^ 283.0 INFLUENCE OF ROLL TENSIONING ON LATERAL DISPLACEMENT,FREQUENCY AND STIFFNESS3.1 Chapter overview^ 323.2 Influence of roll path position^ 324.0 THE LIGHT-GAP METHOD FOR BANDSAWS4.1 Chapter overview^ 374.2 The light-gap method as a guide for the saw tensioning process^ 374.3 Rules-of-thumb for obtaining the desired light-gap profile 39iv5.0 ADDITIONAL FACTORS INFLUENCING LIGHT-GAP MEASUREMENTS5.1 Chapter overview^ 445.2 Influence of applied curvature on the light-gap profile^ 445.3 Significance of tire size 506.0 THE EFFECTIVENESS OF THE LIGHT-GAP METHOD FOR BANDSAWS6.1 Chapter overview^ 546.2 Relationships between light-gap profile, frequency and stiffness^ 546.3 The light-gap profile as an indicator of frequency and stiffness 607.0 PRACTICAL ADVANTAGES AND LIMITATIONS OF THE LIGHT-GAP METHOD7.1 Chapter overview^ 637.2 Advantages and limitations^ 638.0 CONCLUSIONS^ 66REFERENCES 69VLIST OF TABLESTable 2.1^Rectangular plate dimensions and parameters used in the examples.^ 13Table 6.1^Plate frequencies and stress series coefficients for the tensioning cases inFigure 6.1^ 57Table 6.2^Plate frequencies and stress series coefficients for the tensioning cases inFigure 6.2^ 57viLIST OF FIGURESFigure 1.1^Wavy saw cut caused by critical speed instability of a circular saw.^ 2Figure 1.2^Typical roll tensioning machine for bandsaw blades^ 2Figure 1.3^Schematic relationship between the saw tensioning process, cuttingaccuracy and light-gap profile.^ 4Figure 1.4^The light-gap method for a bandsaw. 7Figure 1.5^The light-gap method for a circular saw.^ 7Figure 2.1^The bandsaw roll tensioning process and resulting residual stressdistribution.^  11Figure 2.2^Rectangular plate model used for the light-gap calculations^ 13Figure 2.3^Lateral displacement components for a bandsaw.^ 15a) Component of lateral displacement associated with the stress-free plate.b) Components of lateral displacement associated with each tensioningstress term shown in c).Figure 2.4^Curvatures of the lateral displacements shown in Figure 2.3^ 18a) Curvature of the stress-free component, w o(y).b) Curvatures of the stress-related components, wn(y).c) Tensioning stresses repeated from Figure 2.3.Figure 2.5^Rectangular plate model for the natural frequency calculations^ 21Figure 2.6^Influence of the individual tensioning stress terms on torsional frequencyratio when the tensioning stresses are as shown in Figure 2.3c. ^ 21Figure 2.7^Rectangular plate model for the stiffness calculations.^ 25Figure 2.8^Influence of the individual tensioning stress terms on stiffness ratio whenthe tensioning stresses are as shown in Figure 2.3c. ^ 25Figure 3.1^Stress distribution for five single roll path pairs. 33Figure 3.2^Influence of roll path position on a) lateral displacement profile andb) lateral displacement curvatures. ^ 33viiFigure 3.3^Influence of roll path position on frequency and stiffness^ 36Figure 4.1^Exaggerated light-gap profiles for a bandsaw^ 38a) Traditional light-gap profile.b) Tight-spot.c) Loose-spot.Figure 4.2^Tensioning state at three stages of the tensioning process. ^ 41Figure 5.1^Influence of applied curvature on the light-gap size 46Figure 5.2^Influence of the applied curvature on the light-gap profile.^ 49a) Lateral displacement profiles.b) Curvature of the displacement profiles in a).Figure 5.3^Influence of tire size on frequency, stiffness and stresses.^ 51Figure 6.1^Lateral displacement profiles for three different stress distributionsproducing the same specified values of frequency and stiffness^ 55Figure 6.2^Frequency and stiffness values for three different stress distributionsproducing the same specified light-gap profile^ 59Figure 6.3^Relationship between tension gauge diameter and frequency andstiffness ratio.^ 61NOMENCLATUREa = length of the rectangular plate in the x-directionA,B = parameters for w o(y)Ad ,Cd = constants for Wsynt and WskewAf,Cf = constants for wiat and wtorA*,B* = parameter for the 5/h calculation^An^magnitude of the nth term of the tensioning stress functionb = width of the rectangular plate in the y-directionbn^magnitude of the nth term of wn(y)applied curvature parameter for the 5/h calculationD = Eh3/12(1- v2), plate stiffness parameterE = Young's modulus for the plate materialF(x) = line load used in stiffness calculationFt^magnitude of F(x), in force per unit lengththickness of the rectangular plateK = "cutting stiffness" of the platen = term number for the tensioning stress functionR = applied radius of curvature used in the light-gap calculation^W(y)^approximate displacement function used in the stiffnesscalculationWskew = approximate displacement function for the skew-symmetricpart of W(y)vii'ix^sym^approximate displacement function for the symmetric part ofW(Y)w(y) = lateral displacement profile of the rectangular plate modelwo(y) = component of w(y) associated with the stress-free plate^Aw(y)^component of w(y) associated with the tensioning stressesw„ (y) nth component of Aw(y)Wlat = approximate mode shape for the first lateral frequency^Wtor^approximate mode shape for the first torsional frequency^X, y, z^coordinates for the rectangular plate calculationsan^stress parameter for the 5/h calculation^04^3(1-v2)/(R2h2)6^size of the light-gap profile6 = parameter for the 5/h calculationy = constant for the 5/h calculation= curvature of w(y)2`-i^frequency parameter for the first lateral frequency^X12^frequency parameter for the first torsional frequencyv = Poisson's ratio for the plate material^CD tat^natural frequency for the first lateral mode(Dtor = natural frequency for the first torsional modep = density of the saw plate materialax(Y)^residual tensioning stress distributiona/b, length to width ratio of the rectangular platexACKNOWLEDGEMENTThis thesis is dedicated to my parents, Jim and Christine, and my sister, Debby. Yourlove and encouragement have made this possible.Also, I would like to thank the many people who have had an influence on this work. Inparticular, I would like to thank Dr. Gary Schajer, Dr. George Shipley, Dr. RicardoFoschi, Dr. Eb Kirbach, John Taylor, Bruce Lehmann and Jan Aune. Thanks to all.The work reported here was supported by the B.C. Science Council and MacMillianBloedel Ltd. through the GREAT program and by the Natural Sciences and EngineeringResearch Council of Canada (NSERC).11.0 INTRODUCTION1.1 Background and motivationSawmills use band and circular saws to cut raw logs into saleable lumber. Each time a sawcut is made, a certain amount of solid wood is lost as sawdust. The amount of sawdust wasteproduced annually is considerable. In British Columbia alone about fifty million cubic metersof raw logs are processed each year [1]. Of this amount, approximately 10% of the wood iswasted to sawdust during the cutting process. This large amount of wasted wood presents aconsiderable economic and environmental cost to the forest industry.To reduce the amount of wood lost to sawdust, "thin-kerf' saws are being adopted byindustry. These thinner saws produce less sawdust because the width of the saw cut isreduced. Even small reductions in saw thickness can result in a considerable reduction insawdust and an even larger increase in lumber recovery [2]. For example, studies by the U.S.Forest Service have shown that a 0.012 in. reduction in the saw kerf of a typical 4/4 (1 inchboard) mill can increase lumber recovery by 1 percent [3]. Because raw logs account for 75-80 percent of the total production cost for many sawmills, even small increases in lumberrecovery can represent a considerable increase in revenue [4].Reductions in saw kerf are limited by the reduced cutting accuracy and stability associatedwith thinner saw blades. Thinner saws are less stiff than their thicker predecessors and areless able to resist the lateral forces generated during cutting. Large lumber dimensionvariations can result when the lateral forces cause the saw blade to deviate from the desiredpath. Poor dynamic behavior caused by the critical speed phenomena can also occur as sawthickness is reduced [5]. A saw operating at or near its critical speed will often snake slowlyFigure 1.1 Wavy saw cut caused by critical speed instability of a circular saw.2Figure 1.2 Typical roll tensioning machine for bandsaw blades.3from side-to-side and cut grossly inaccurate lumber. Figure 1.1 shows a typical example of awavy saw cut produced by a circular saw suffering from critical speed instability."Tensioning" is a process used to enhance the stiffness and dynamic stability of thin-kerf saws[6,7]. The process involves inducing residual stresses into a saw plate, typically by mechanicalmeans such as hammering or rolling [8,9]. Roll tensioning is the preferred method because itis faster and gives more uniform results than hammering. Other tensioning methods, such asshot peening, thermal tensioning and laser tensioning, have also been investigated but are notgenerally used by industry [10,11,12].Figure 1.2 shows a typical roll tensioning machine used for tensioning bandsaws. Similarmachines are also used for circular saws. Roll tensioning machines have two narrow crownedrollers which squeeze a small portion of the saw plate. One or both of the rollers is driven sothat the saw slowly moves and a narrow squeezed path is created in the longitudinal directionfor bandsaws or in the circumferential direction for circular saws. The deformations causedby rolling induce residual stresses into the saw plate. When these stresses are induced in afavorable manner, the lateral stiffness of the saw increases. This stiffness increase improves theaccuracy of the saw cut [13]. Favorable tensioning stresses also improve the dynamic stabilityof a saw because they raise the natural frequencies which control critical speed [14]. Figure1.3 schematically illustrates the relationship between the saw tensioning process, residualstresses and cutting accuracy.Achieving effective results from the tensioning process requires a method of evaluating theeffect of the residual stresses. A number of different non-destructive methods are available formeasuring the residual stresses directly [15]. In general, these methods have been developedfor experimental stress analysis and are either too costly, too time consuming or too awkwardfor day-to-day use.ControllerIndicatorMIIIM■111111111101M - - 1.0 NINAResidual StressesNatural FrequenciesandCritical Speedv- i^Light-gap Profile^)4Cutting Accuracy4Figure 1.3 Schematic relationship between the saw tensioning process, cutting accuracyand light-gap profile.5The most common industrial technique for monitoring the saw tensioning process is the "light-gap" method [16]. Very rarely, natural frequency measurements are also used [17]. Figure1.3 schematically shows how both the light-gap and the natural frequencies are controlled bythe residual stresses induced by the tensioning process.Previous studies of saw tensioning have focused on the relationships between the tensioningstresses and frequencies, and the tensioning stresses and light-gap profile. The frequencyrelationships have been studied in detail for circular saws [6,14,18,19], but have been lesswidely studied for bandsaws [20]. The light-gap relationships have not been well studied.Foschi [21] studied the relationship between residual stresses and the light-gap profile forbandsaws. Similar studies for circular saws have not been reported in the literature.The relationships developed by Foschi show that the light-gap method is a rather poorindicator of the details of the tensioning stress distribution. Because of this feature, the light-gap method has been criticized as being a poor indicator of the tensioning state of a saw.However in practice, the light-gap method is used to indicate changes in saw stiffness andnatural frequencies caused by the tensioning stresses. This is indicated in Figure 1.3 by thedashed lines. The effectiveness of the light-gap method for indicating these changes in sawstiffness and frequencies has not been previously studied.1.2 The light-gap methodThe light-gap method is used for both band and circular saws. The procedure is essentiallythe same for both types of saws, with a few minor differences.6Figure 1.4 illustrates the light-gap method for a bandsaw. The procedure involves inducing acurvature along the length of the saw blade which in turn induces a corresponding curvatureacross the saw width. The shape and size of the lateral displacement across the saw widthdepends on the local tensioning stresses and can be used to monitor the tensioning process. Inpractice, the lateral displacement is evaluated by placing a curved tension gauge across thewidth of the plate as shown in the figure. The tension gauge has a shallow circular profilewith an empirically chosen curvature. The gauge contacts the saw at two or more "highpoints". The resulting clearance between the saw and the gauge is called the "light-gap" and isused to monitor the tensioning process.Figure 1.5 illustrates the light-gap method for a circular saw. The procedure is basically thesame as for a bandsaw except the induced curvature is applied by supporting the saw at twopoints across a diameter and letting the saw sag under its own weight. The lateraldisplacement is then evaluated by placing a tension gauge across a perpendicular radius.Both the size and shape of the lateral displacement profile are known to be importantindicators of the tensioning state for both band and circular saws. The size of the lateraldisplacement is important because it indicates the overall amount of tensioning. The shape isimportant because it is used to guide the distribution of the tensioning stresses.1.3 Objectives and scopeThe objective of this study is to investigate the effectiveness of the light-gap method forevaluating the tensioning state of a saw. The light-gap method for a bandsaw is studied indetail because the mathematics are simpler than for a circular saw and an exact solution forthe lateral displacement is available. Thus, the characteristics and significance of the solution7Figure 1.4 The light-gap method for a bandsaw.Figure 1.5 The light-gap method for a circular saw.8can be interpreted more clearly and simply. Although the results obtained apply specifically tobandsaws, the general features are also expected to apply to circular saws.A discussion of the light-gap method for bandsaws is presented in Chapters 2 to 7.In Chapter 2, a rectangular plate model representing a bandsaw is presented. The stressesinduced by roll tensioning a bandsaw are briefly discussed and relationships between thetensioning stresses and lateral displacement, natural frequency and stiffness are given. Theserelationships are used in Chapter 3 to show how roll path position influences bandsaw lateraldisplacement, torsional frequency and stiffness.Chapter 4 describes the practical use of the light-gap method as a guide for the saw tensioningprocess. Empirically developed rules-of-thumb used by sawfilers to obtain the desired light-gap profile are discussed and are illustrated with an example case of bandsaw tensioning.Some additional factors concerning the practical use of the light-gap method are discussed inChapter 5.In Chapter 6, the effectiveness of the light-gap method is investigated. In particular, thesignificance of the size and shape of the light-gap profile in terms of bandsaw frequency andstiffness is studied.Chapter 7 discusses some of the more important practical advantages and limitations of thelight-gap method. Some simple ways of improving the results of the light-gap method arepresented.92.0 GENERAL BANDSAW RELATIONSHIPS2.1 Chapter overviewIn this chapter, a rectangular plate model representing a bandsaw blade is introduced and isused to illustrate the influence of tensioning stresses on the lateral displacement, frequencyand stiffness of a saw.The residual tensioning stresses resulting from roll tensioning a bandsaw are briefly presentedand the relationships between lateral displacement and stress, frequency and stress, andstiffness and stress are given.Finally, approximate simplified relationships are developed and are used to compare theinfluence of tensioning stresses on saw lateral displacement, frequency and stiffness.2.2 Stresses induced by roll tensioning bandsaw bladesBandsaws are commonly tensioned using a roll tensioning machine of the type illustrated inFigure 1.2. A part of the saw is squeezed between two crowned rollers creating a narrowrolled path along the length of the saw surface as the rollers turn. The part of the saw bladewithin the roll path becomes slightly thinner and the displaced material spreads longitudinallyand laterally. The lateral spreading has a negligible influence on the tensioning stressesbecause the saw plate is free to expand laterally. The longitudinal spreading, however, isresisted by the surrounding plate material. As a result, compressive stresses are inducedwithin the roller path and tensile stresses are induced in the remainder of the plate.10Figure 2.1 shows a typical stress distribution caused by rolling a rectangular plate of width balong its centreline, y=0. Large compressive stresses occur within the roller path which arebalanced by tensile stresses in the adjacent regions. These stresses are self-equilibratingbecause no external in-plane loads are applied. The magnitude of the compressive stressdepends on the force applied to the rollers, and has a maximum of about 40% of the yieldstress of the material [18]. This limit occurs when the compressive stresses become largeenough to inhibit any additional longitudinal deformations in the roll path which would furtherincrease the compressive stresses. Once the limiting compressive stress is reached, anyadditional rolling along the same roller path simply makes the saw plate wider and provideslittle additional tensioning effect.Because the maximum stress level within a single roll path is limited, the desired level oftensioning can most effectively be achieved by using a number of roll tensioning paths. Theresulting stress distribution is the superposition of the stress distributions resulting from eachseparate roll path.Typically the back edge of a bandsaw blade is rolled slightly more than the front in order toinduce a very small curvature along the back of the saw. This curvature, called "backcrown",helps to stiffen the front edge of the saw when it is mounted on a bandmill. The component ofthe tensioning stresses associated with backcrown is non-symmetric with respect to the sawcentreline. However, these non-symmetric stresses are usually quite small and can beneglected for the types of calculations considered here. The remaining tensioning stresses arereasonably symmetric and can be approximated by a Fourier cosine series,6 x (Y)^/A n cos 2nicY ^n = 1,2,3,...^(2.1)Rolling Process Roller LoadTopRollerSaw BladeBottomRoller11 Roller LoadTensioning Stresses^ax(Y)111M1111111111110MIMMINIM^‘1111111111111111101MIMI1011111-b/2^b/211Figure 2.1 The bandsaw roll tensioning process and resulting residual stress distribution.12where b is the width of the plate in the y-direction and y is measured from the plate centreline.The magnitude of each term in Equation (2.1) is found from4 y^2nnyAn = —b f° 2 ax(y)cosb dywhere lax(y) is the actual stress distribution.2.3 Relationship between lateral displacement and stressFigure 2.2 shows a rectangular plate model that can be used to study the relationship betweenthe tensioning stresses and the lateral displacement of a bandsaw. The plate shown has widthb in the transverse ("y") direction and is considered to be long in the longitudinal ("x")direction. It contains in-plane tensioning stresses and is initially flat. When the plate is bent toa radius R along its length, a lateral displacement is induced along the transverse direction.The shape and magnitude of the lateral displacement profile depends on the local tensioningstresses and can be used to monitor the tensioning state of the saw. Both the size and theshape of the lateral displacement profile are important indicators of the tensioning state of thesaw.An exact solution for the lateral displacement, w(y), of the bent plate in Figure 2.2 isdescribed by Foschi [21]. The governing equation is 3(1 - v2 )04 _ ^R2h2(2.3)(2.2)z, w(y)Lateral Displacement ProfileLight-GapTension Gauge13Figure 2.2 Rectangular plate model used for the light-gap calculations.Table 2.1^Rectangular plate dimensions and parameters used in the examples.Plate width, b = 0.254 mPlate span, a = 0.508 mThickness, h = 0.002 mYoung's Modulus, E = 210 GPaPoisson's Ratio, v = 0.3Density, p = 7850 kg/m3and the boundary conditions at y = + b/2 are14. 0dY3dew^vdyed3w (2.4)A detailed discussion of the governing equation and boundary conditions is given by Conwayand Nickola [22] for the case when no tensioning stresses are present.When tensioning stresses are present the lateral displacement, w(y), satisfying the governingequation and boundary conditions is the sum of two partsw(Y) = wo (Y) Aw(Y)^(2.5)The first part, wo(y), is the displacement of the stress-free plate and the second part, Aw(y), isthe additional displacement due to the tensioning stresses.The displacement of the stress-free plate is given by^wo (y) = A cosh (3y cos (3y + B sinhf3y sin fly^(2.6)whereb.^f3b^f3b^.nsinh^ cos^ + cosh siA B =^2 2 2^2,RO2^sinhr3b + sin 13b(2.7)Figure 2.3a shows the displaced shape, w o(y), described by Equations (2.6) and (2.7) for arectangular plate having dimensions given in Table 2.1. Only half of the plate is shown1.81.5 —^ n=11.2 —0.9 -4- n= 20.6 -P-0.3 —-0.3 —-0.6 -^ n=3-0.9 -- n=4-1.2a)0.90.6 -0.3 -E0 ^o -0.3 --0.6 --0.915b)EECc)^120 ^80 -20-^40^0^Nco-40 -th. -80 -\ /^ ‘,./N //\\N ,*---.n =3n=4-1200^0.5Distance from Plate Centreline, y/bFigure 2.3 Lateral displacement components for a bandsaw.a) Component of lateral displacement associated with the stress-free plate.b) Components of lateral displacement associated with each tensioningstress term shown in c).16because the displacement is symmetric with respect to the plate centreline, y/b=0. Thedisplaced shape describes an anticlastic surface because the lateral displacement is convexwhen the applied curvature in the x-direction is concave.The change in displacement caused by the tensioning stresses, Aw(y), is the sum of thedisplacement curves, wn(y), associated with each term of the stress series in Equation (2.1).The change in displacement is given byAw(Y) =^wn (y)^n = 1,2,3,...^(2.8)whereY ) = b —R[cos2 b^( 1)n7CY^n R (2n712n E^w 0 0 1)v bAn41+4 mu13bbn =(2.9)(2.10)and An is the magnitude of the nth stress term in Equation (2.1).Figure 2.3b shows the individual displacement curves, wn(y), associated with each of the firstfour terms of the stress series in Equation (2.1). Again, the plate has dimensions given inTable 2.1 and only half of the plate is shown because of symmetry. The sign and magnitude ofeach stress term has been chosen so that the stresses are all the same size and are tensile at theplate edge as shown in Figure 2.3c. The tensile edge stresses result in w n(y) curves whichhave a generally concave shape when the applied curvature in the longitudinal direction is alsoconcave.17Figure 2.3b shows that the wn(y) curve associated with the n=1 stress term has the largest sizeand therefore has the largest influence on the lateral displacement. As the value of n increases,the size of the wn(y) curves decrease rapidly. For large values of n, the wn(y) curves becomevery small and have a negligible influence on the lateral displacement of the saw, even thoughthe associated stresses may be quite large.The shape of the lateral displacement profile of a bandsaw can be more easily seen by lookingat the curvature across the width of the saw. The curvature, K, is given by the secondderivative of Equation (2.5),K = W"(y) = VVVy) Aw''(y)^ (2.11)wherew'c',(y) = 2B0 2 cosh f3b cosf3b – 2A$32 sinhr3b sinf3b^(2.12)Aw"(Y) =^wn(y)^n = 1,2,3,...^(2.13)andw:;(Y) = – bnR 2nb712 [ 2nb^, ,n R " , ,1cos^+ ( –1) —v w 0 (.31/ (2.14)Figure 2.4a shows the curvature of the stress-free component of displacement shown in Figure2.3a. The curvature is described by Equation (2.12) and is shown in the figure for half of theplate width. The curvature of the stress-free plate is positive and increases towards the plateedges because the shape of wo(y) becomes increasingly convex near the outside of the plate.Figure 2.4b shows the curvatures of the four w n(y) displacement components shown in Figure2.3b. The curvatures are described by Equation (2.14) for each of the stress terms reproduced0-0.6 n=2n=1n=4n=318a)^0.2co-0.2b)^0.2C)^120 ^-,--.,80 - \ s, -. `N73^-^N,,2o_ 40 -n=n=2NCn=3 n=40 0.5Distance from Plate Centreline, y/bFigure 2.4 Curvatures of the lateral displacements shown in Figure 2.3.a) Curvature of the stress-free component, w o(y).b) Curvatures of the stress-related components, w n(y).c) Tensioning stresses repeated from Figure 2.3.19in Part (c) of the figure. The curvatures of the wn(y) components are different from oneanother indicating that each wn(y) curve has a unique shape. Each curve has negative valuesof curvature corresponding to the compressive regions in each stress term. The negative areasof curvature indicate concave shaped displacements and occur because the compressivestresses tend to push the plate out of plane when it is bent along its length. The areas of theplate which contain tensile stresses have a less concave shape because tensile stresses tend topull the plate back into plane.Because each of the wn(y) curves have a unique shape, the shape of the lateral displacementprofile, w(y), of a tensioned bandsaw depends on the relative amount of each of the first fewwn(y) components. The first few stress terms in Equation (2.1) associated with thesedisplacement components describe a slowly varying "average" of the actual tensioning stressdistribution. The higher stress terms describe the details of the tensioning stresses, but hardlyinfluence the lateral displacement profile. In practice this means that stress distributions whichhave the same "average" produce similar light-gap profiles, even though the details of thestress distributions may be quite different.2.4 Relationship between natural frequency and stressNatural frequency measurements are sometimes used to monitor the saw tensioning processbecause the stresses induced by tensioning affect the frequencies associated with certainvibration modes. For circular saws, the vibration mode which controls the lowest criticalspeed of a saw is typically monitored. Although the frequency measurement method isgenerally used only on circular saws, it could also be used for bandsaws if the saw blade weresupported in a suitable manner.20The effectiveness of the frequency measurement method depends on which natural frequenciesare monitored. Theoretically, a rectangular plate has an infinite number of natural frequencies.However in practice, only the lower frequencies are of interest because these have the largestinfluence on sawing performance. The two lowest frequencies are for the first lateral and firsttorsional modes illustrated in Figure 2.5 [23]. The influence of tensioning stresses on thefrequencies of these two modes is investigated in this section.The rectangular plate shown in Figure 2.5 provides a simple model for studying therelationships between the tensioning stresses and the natural frequencies of an unstrainedbandsaw blade. The plate has width b, length a, and thickness h. It is simply-supported alongthe x=0 and x=a edges and free on the other two edges. The tensioning stresses are assumedto be symmetric and described by Equation (2.1).In general, a simple exact solution for the frequencies of the rectangular plate model is notpossible when tensioning stresses are present. However, an upper-bound estimate of thelower frequencies can be obtained using Rayleigh's method [24] and appropriate mode shapefunctions. The accuracy of this method depends on the similarity of the assumed shapefunction and the actual mode shape. Because the general nature of the solution is of primaryinterest here, the simplest admissible shape functions are used in this analysis. For the firstlateral and first torsional modes, the chosen approximate shape functions areandltxw !at( x , Y ) = A f sin —a(2.15)7CXW tor (X , Y) = Cf y sin —a(2.16)a1st Torsional Mode1st Lateral ModeFigure 2.5 Rectangular plate model for the natural frequency calculations.21 .80 1.6ccc1.41.21c 0.8g 0.6u_ 0.40.20n = 1 n = 2^n = 3^n = 4Tensioning Stress Termstress-freeFigure 2.6 Influence of the individual tensioning stress terms on torsionalfrequency ratio when the tensioning stresses are as shown inFigure 2.3c.2122where Af and Cf are constants.Following the procedure for Rayleigh's method, (2.15) and (2.16) can be used to calculate themaximum potential and kinetic energies of the vibrating system. Equating the potential andkinetic energies gives the following estimates for the frequencies of the first lateral and firsttorsional modesand2^D^4^2n2a2h 6/ ]w /at = a4ph It + ^ fD ° 26x (Y)dYD  [ 4 + 247C^N 2 + 247C 2a2h  j ry 2 axz 1 _ri. .CO t2o r = 4^20 vA^2(Yitva ph 7C^Db3 ° y(2.17) (2.18)where the integral terms are associated with the tensioning stresses.The integral term in Equation (2.17) is identically zero because the tensioning stresses givenby Equation (2.1) are self-equilibrating. Therefore, the first lateral frequency reduces tox11 11 DCI) lat =— a 2^ph (2.19)whereXII = 7E 2 '''. 9 . 87 (2.20)Equations (2.19) and (2.20) show that the tensioning stresses have no effect on the first lateralfrequency. This result occurs because the mode shape has a constant displacement along the23width of the plate. Consequently, the first lateral frequency is not an effective indicator of thetensioning stresses.When the tensioning stresses are given by Equation (2.1), the integral term in Equation (2.17)is easily evaluated and the first torsional frequency iswherec° tor =^—Aa2l2 Iph (2.21)11x12 = 714 + 247E2 6a2h (—On(1 - vA 2 + ^E^An^n = 1,2,3,...^(2.22)D n n 2The summation term in Equation (2.22) is the change in the first torsional frequency caused bythe tensioning stresses. This term is sufficiently large to enable the first torsional frequency tobe used successfully to infer the tensioning state.Figure 2.6 shows the effect of tensioning stresses on the first torsional frequency for arectangular plate with dimensions given in Table 2.1. The frequency of the stress-free plate,53.2 Hz, is used to normalize the frequencies of the plate containing the example tensioningstress distributions. Each of the example stress distributions have the same peak value and arethe same as the stresses shown in Figure 2.3c. The stresses are tensile along the plate edges,y=+b/2, which cause the frequency of the plate to increase. However, the amount of the in-crease diminishes as the value of n in Equation (2.1) increases.Figure 2.6 shows that, in a manner similar to the lateral displacement, the torsional frequencyis most influenced by the lowest terms in Equation (2.1). These terms describe a slowly24varying "average" of the actual stress distribution. Additionally, the summation term inEquation (2.22) shows that the torsional frequency depends only on the cumulative effect ofthe stress terms and not on the size of each term individually.2.5 Relationships between stiffness and stressFigure 2.7a shows a rectangular plate with the same dimensions, support conditions andtensioning stresses as that shown in Figure 2.5. A line load, F(x), is distributed along one freeedge. The loaded plate approximately represents an unstrained bandsaw blade subjected to alateral cutting force acting on the saw teeth. It provides a model for investigating theinfluence of tensioning stresses on the "cutting stiffness" of the plate.The cutting stiffness of the plate is chosen in a manner which provides a measure of the abilityof the saw plate to resist lateral cutting forces. Therefore, an increase in a plate's stiffnesswould be expected to result in improved cutting accuracy. For mathematical simplicity, thelateral cutting force, F(x), acting on the plate edge, y=b/2, is approximated asTCXF(x) = Ft sin —a(2.23)The cutting stiffness, K, is defined as r,F(x)K = ^w (x „ a/2, .1, , b/2) (2.24)where W(a/2, b/2) is the maximum displacement of the plate.25a) li td ievi i,...--44-4■^”Ikk,Ok tk1/4.Symmetric PartSkew-Symmetric PartFigure 2.7^Rectangular plate model for the stiffness calculation.1 .41 .21gocc0.8a)c 0.6.4■in 0.40.20n = 1 n = 2^n = 3^n = 4Tensioning Stress Termstress -freeFigure 2.8 Influence of the individual tensioning stress terms on stiffness ratiowhen the tensioning stresses are as shown in Figure 2.3c.26The deflection of the plate in Figure 2.7a may be found by superposing the displacements ofthe symmetric and skew-symmetric parts shown in Figure 2.7b. In general, a simple exactsolution for the displacements of each part is possible only when no tensioning stresses arepresent. When tensioning stresses are present, the displacements can be found using anapproximate method such as Rayleigh-Ritz [25]. The accuracy of this method depends on theability of the assumed shape function to model the exact displacement and, in general, themagnitude of the deflection is slightly underestimated. Here, the exact displacement is not ofprimary interest, but rather the nature of the solution. Therefore, the simplest admissibledisplacement functionsWsym (x, y) = A d sin 7—r—X^(2.25)aand7CXWskew (X) y) = Cd y sin—a(2.26)are used for the symmetric and skew-symmetric cases respectively.Using the Rayleigh-Ritz method, the constants Ad and Cd in Equations (2.25) and (2.26) arefound. When the tensioning stresses are given by Equation (2.1), the approximate symmetricand skew-symmetric displacements become respectivelyand3—1 aFt^tWSym (x ) y) = 2 ^sin—nxn 3D it^a(2.27)Fta3^ —1Wskew (X, Y) = 2^ [^ + 2(1— v) + b2h^(-1)n^A n y sin—^(2.28)nx7C 3D 1g 2^27C3D n n an = 1,2,3,...K = 7C2Da227Equation (2.27) shows that the tensioning stresses have no effect on the symmetricdisplacements. This result occurs because the assumed displacement shape has a constantdisplacement along the width of the plate. In reality, the actual displacement shape does varyslightly across the width of the plate. Therefore, the tensioning stresses do have a smallinfluence on the displacement. However, this influence is very small and is neglected here. Incontrast, Equation (2.28) shows that the tensioning stresses significantly affect the skew-symmetric displacements. The influence of the stresses on the skew-symmetric displacementsis given by the summation term in Equation (2.28).The displacement of the plate in Figure 2.7a can be found by superposing Equations (2.27)and (2.28). The maximum displacement is then given bya b^Fia.3^+ 1W(—, —2 ) = 7C 3D it^7t 8(1— v)t^2a 2 h (-1)n A± + , E  2^n3^7C^n-IX n n_n = 1,2,3,...(2.29)Substituting Equations (2.23) and (2.29) into Equation (2.24), the stiffness becomes- —1± ^127t^2n16(1— vA^4a2h (-17+ ^ +  , E  2 A n3t^it^irDt n nn = 1,2, 3 ,... (2.30)Figure 2.8 illustrates the influence of tensioning stresses on the plate stiffness described byEquation (2.30). The dimensions of the plate are given in Table 2.1 and the values of stiffnesshave been normalized with respect to the value for the stress-free plate, 13300 N/m. The28change in stiffness caused by each of the first four terms of Equation (2.1) is shown for thecase when the stresses are the same as in Figure 2.3c.Figure 2.8 shows that the tensile edge stresses increase the stiffness of the plate. However,the increase becomes less as the value of n increases. In a manner similar to the lateraldisplacement and torsional frequency, the stiffness is most influenced by the lowest terms inEquation (2.1) which describe the "average" of the actual stress distribution. Additionally, thesummation term in Equation (2.30) shows that the value for stiffness depends on thecumulative effect of each of the lower stress terms and not on the size of each termindividually.2.6 Simplified relationshipsFigure 1.3 shows that in practice, the light-gap profile is used to indicate changes in thestiffness and frequency of a bandsaw caused by tensioning. In the previous sections,relationships are given which describe how the tensioning stresses influence lateraldisplacement, frequency and stiffness. In this section, these relationships are simplified inorder to directly establish how the stress parameters in Equation 2.1 influence bandsaw lateraldisplacement, torsional frequency and stiffness. By studying the effect of the stressparameters, an indication of the effectiveness of the light-gap method can be obtained.The expression for lateral displacement given in Equations (2.5) to (2.10) can be simplified byusing a binomial expansion for the magnitude term, b n . By expanding bn as(2.31)29and retaining only the first term, Equation (2.5) becomes approximately134 2w(y)bR A,:i wo (y) – —R E [1344n 47E4 n4  cos 2n7tY + (E^b^on 7t2 V n;' wo (y)]n = 1,2,3,...(2.32)where the first component, wo(y), describes the displacement of the stress-free plate and thesecond component is the approximate change in displacement caused by the tensioningstresses in Equation (2.1).Equation (2.32) shows that the first term of the change in displacement component isproportional to Ann-4 where An is the size of the nth term in Equation (2.1). The secondterm in the change of displacement component is proportional to A nn-2 . Therefore, as nincreases, the first term decrease much faster than the second. For values of n larger thanabout two, the first term becomes small compared to the second and the change indisplacement becomes essentially proportional to A nn-2 . This relation shows why the size ofthe wn(y) curves in Figure 2.3b decreases with increasing values of n.Equation (2.21) can be simplified to show the influence of the stress parameter, n, and size,An, on the torsional frequency of a bandsaw. By expanding the frequency parameter inEquation (2.22) using a binomial expansion and retaining the first two terms, the frequency isapproximately given by1CI) for '' a2_2^_36(1 – v2 )NV7c4 + 247[ 2 (1– v)e + ^h ^E  (-1)n  A nV7c 4 + 241E 2 (1 — N/) 2 n 11 2^E 11:h^(2.33)n = 1, 2,3, ...30The first term in Equation (2.33) represents the torsional frequency of the stress-free plate andthe second term is the approximate change in frequency caused by the tensioning stresses. Thesecond term shows that the change in frequency is approximately proportional to A nn-2 whichis the same approximate relationship obtained for the lateral displacement.To establish a relationship between the stress parameter, n, and the cutting stiffness, Equation(2.30) can be rewritten as K 7r2Da 2n9702hE ( 12) A,713 + 24n(1 - v)e ^n \ 22 TC2 + 12(1 - v)2^4134(71-2 + 6(1- v)e) (2.34)n = 1,2,3,...The details of this calculation are given in reference [261The first term in Equation (2.34) is the stiffness of the stress-free plate. The second term isthe approximate change in stiffness caused by tensioning stresses given by Equation (2.1), andis proportional to Ann-2 . This is the same relationship obtained for the lateral displacementand first torsional frequency.The approximate relationships given above show that the lateral displacement, frequency andstiffness of a bandsaw all consist two components. The first component describes thebehaviour of the stress-free plate and the second component describes the changes caused bythe tensioning stresses. In each case, the changes caused by the stresses are approximatelyproportional to Ann-2 . These relationships indicate that the same stress components with the31lowest n values most significantly influence the lateral displacement profile, the first torsionalfrequency and the stiffness. Since all three features respond to tensioning stresses in similarays, a change in the lateral displacement profile (measured by the light-gap) will be a goodindicator of changes in the frequency and stiffness. This theme is explored more fully insubsequent chapters.323.0 INFLUENCE OF ROLL TENSIONING ON LATERAL DISPLACEMENT, FREQUENCY AND STIFFNESS 3.1 Chapter overviewThis chapter describes how roll tensioning influences the lateral displacement, frequency andstiffness of a bandsaw. Although saws are also commonly tensioned by hammering, rolltensioning is the preferred method because it provides faster, more uniform results. Inpractice, the location of each roller path is known to affect the tensioning state of the saw.The influence of roll path position on lateral displacement, torsional frequency and stiffness isexamined in this chapter.3.2 Influence of roll path positionFigure 3.1 schematically shows a stress distribution resulting from roll tensioning a bandsawblade. The stresses resulting from several single pairs of roll paths symmetrically placed withrespect to the plate centreline are shown for half of the plate. The location of each roll path isindicated by the compressive stress regions which, for simplicity, are modeled as stepfunctions. Each roll path pair is located at a different distance from the plate centreline buthas the same magnitude of stress. The roll paths shown in the figure can be used to study theinfluence of roll path position on the lateral displacement, torsional frequency and stiffness ofa bandsaw having dimensions given in Table 2.1. For convenience, each roll path pair isidentified by a number from 1 to 5.Figure 3.2 shows the influence of roll path position on the lateral displacement of a bandsaw33500-43• -50-100-150-200-250y/b 0.5Figure 3.1^Stress distribution for five single roll paths pairs.a)b)E0.15010.050-0.05-0.1-0.15Figure 3.2 Influence of roll path position on a) lateral displacement and b)lateral displacement curvatures.34having dimensions given in Table 2.1. Part (a) of the figure shows the change in displacement,Aw(y), associated with each individual roll path pair shown in Figure 3.1. Each displacementcurve is labelled with the number of the roll path which is associated with that curve. Again,only half of the displacement profiles need to be shown because of symmetry. The change indisplacement curves are calculated from Equation (2.6) after the stress series coefficients, A n,in Equation (2.1) have been calculated for the associated roll paths using Equation (2.2). TheAw(y) curves labelled "1" and "2" are associated with the two innermost roll path pairs andare concave shaped while the curves labelled "4" and "5" are associated with the twooutermost roll paths and are convex shaped. The magnitude of the displacements is largestfor roll paths at the centre and edges of the plate and decreases as the roll path locationapproaches a "neutral zone" located at approximately y/b = 0.3. In the area near the neutralzone, the Aw(y) curves change from the concave shapes of curves 1 and 2 to the convexshape of curves 4 and 5. The curve labelled "3" is associated with the roll path in Figure 3.1located within the neutral zone. This curve is very small in size and has a convex shape in thecentral area of the saw and a concave shape near the plate edges.Figure 3 .2b shows the curvatures of the displacement curves in Part (a) of the figure.Negative values of curvature indicate a concave shaped Aw(y) curve while positive valuesindicate a convex shape. Each Aw(y) curve has its most concave curvature at the location ofthe roll path associated with that curve. This occurs because the large compressive stresses inthe roll path tend to push the saw blade out of plane when the plate is bent along its length.The curvatures in the remainder of the plate are smaller because the tensile stresses in theseareas are relatively small and tend to pull the plate back into plane. The increased concavecurvature at the roll path position is important in practice. It allows the shape of a bandsaw'slight-gap profile can be adjusted by placing roll paths at various locations along the width ofthe saw blade. This effect is explored further in subsequent chapters.35Roll path position also influences the frequency and stiffness of a bandsaw. Figure 3.3 showsfrequency and stiffness ratios for a tensioned bandsaw having dimensions given in Table 2.1.The saw is tensioned by single pairs of roll paths symmetrically placed on each side of theplate centreline as shown in Figure 3.1. The values of frequency and stiffness are calculatedfrom Equations (2.21), (2.22) and (2.30) for different locations of the individual roll paths.These values are then normalized with respect to the frequency and stiffness of the stress-freeplate. Figure 3.3 shows that both the frequency and stiffness increase when the roll paths arein the central region of the plate and decrease when the roll paths are near the plate edges.The change in frequency and stiffness is greatest for roll paths at the centre or edges of theplate and decrease as the roll path position approaches a neutral zone, again located atapproximately y/b = 0.3.The analogous behaviors illustrated in Figures 3.2a and 3.3 clearly show the similar ways inwhich tensioning stresses influence bandsaw lateral displacement, frequency and stiffness.Rolling tensioning in the central region of the saw causes concave shaped displacements andincreases frequency and stiffness. Rolling outside of the neutral zone causes convex shapeddisplacements and decreases frequency and stiffness. Thus, the size and shape of the lateraldisplacement profile indicates the amount of change in frequency and stiffness. This featuresupports the hypothesis illustrated in Figure 1.3 that the light-gap profile is a useful indicatorof saw blade frequency and cutting edge stiffness.frequency0.2"Neutral Zone"1.41.2 —^07.E1cov)NC=+7,0.8 —8c.,cz0 0.6 —0-it0.4 —stiffness0.0^0.536Roll path position, y/bFigure 3.3 Influence of roll path position on frequency and stiffness.374.0 THE LIGHT-GAP METHOD FOR BANDSAWS 4.1 Chapter overviewThis chapter describes how the light-gap method can be used to guide the saw tensioningprocess. The general procedure for the light-gap method for bandsaws is described in somedetail. The traditional light-gap profile is illustrated and significant examples of deviationsfrom this profile are described.Rules-of-thumb for achieving the desired light-gap profile are given. These rules areexplained and are illustrated using an example case of bandsaw tensioning.4.2 The light-gap method as a guide for the saw tensioning processLight-gap measurements provide an important guide for the saw tensioning process. Thegeneral procedure for the light-gap method involves adjusting the tensioning stresses in a sawso that a specific light-gap profile is achieved. In practice, bandsaws are usually tensioned sothat the saw's lateral displacement profile matches the circular shape of a tension gauge in thecentral region of the saw. The edges of the plate are allowed to fall away from the tensiongauge profile and the point where the saw profile leaves the tension gauge curve is called the"tire line". Figure 4.1a shows a tension gauge resting on the surface of a displaced crosssection of a saw and illustrates a traditional light-gap profile for a bandsaw.Localized areas of the saw which do not match the desired light-gap profile are called "tightspots" if they rise toward the tension gauge or "loose spots" if they fall away from the gauge.Tire Line Tire Line38a)Tension GaugeliSaw Blade Tight-spotFigure 4.1 Exaggerated light-gap profiles for a bandsaw.a) Traditional light-gap profile.b) Tight-spot.c) Loose spot.b)39These features are illustrated in Figures 4.1b and 4.1c respectively. In practice, tight andloose spots are considered to indicate local defects in the tensioning state of the saw. Thesedefects are corrected by additional tensioning. By adjusting the position of the additionaltensioning rolls, the shape of the light-gap profile can be controlled.4.3 Rules-of-thumb for obtaining the desired light-gap profileIn order to achieve the traditional bandsaw light-gap profile shown in Figure 4.1a, rules-of-thumb have been empirically developed by sawfilers to enhance the utility of the light-gapmethod. These rules-of-thumb are used to interpret the information contained in the light-gapprofile and to guide the location of roll paths induced during the tensioning process.The general procedure for tensioning an initially stress-free bandsaw involves placing anumber of roll paths in the central region of the saw blade until the light-gap profile assumesthe approximate concave shape of the tension gauge. The details of the light-gap are thenadjusted by additional rolling to remove any tight or loose spots which may be indicated bythe light-gap profile.The exact locations of the initial roll paths are not important because these rolls are inducedonly to obtain the basic concave shape of the desired light-gap profile. However, Figure 3.2indicates that these initial rolls must be placed in the central region of the plate, between theneutral zones at each edge of the plate. Rolls placed in this region have the desired effect ofencouraging a concave shaped light-gap. A number of roll paths are usually required toachieve the desired effect.Once the basic concave light-gap profile is achieved, any tight or loose spots indicated by the40light-gap profile are removed by additional rolling. Tight spots, for example, are localizedareas where the curvature of the saw's lateral displacement profile is less than the curvature ofthe tension gauge. Such tight spots are corrected by rolling the saw plate at the location ofthe tight spot. Figure 3.2b shows that rolling on the tight spot is effective because themaximum curvature increase is produced at the roll path location. This localized increase incurvature lowers the tight spot relative to the adjacent areas of the saw and enables the lateraldisplacement profile to match the tension gauge shape.Loose spots, on the other hand, are areas where the curvature of the lateral displacementprofile is greater than the curvature of the tension gauge. By rolling beside the loose spot, thecurvature at the loose spot is decreased relative to the curvatures in the rest of the saw. If theloose area has adjacent tight spots, removing the tight spots will often remove or reduce theloose spot. If the loose spot extends across the width of the plate the existing tensioning isexcessive. Rolling near the plate edges will decrease the curvature in the central region of thesaw and will restore the desired level of tensioning.Figure 4.2 illustrates an example of the tensioning process described above. The tensioningstate for a bandsaw having dimensions given in Table 2.1 is shown at three stages of thetensioning process. At each stage, the stresses, lateral displacement, frequency and stiffnessare shown. The compressive regions of the stress distributions indicate the location of eachroll path. The roll paths are placed symmetrically on each side of the plate centreline becauseno backcrown is desired in this case. Only half of the plate are shown in the figure because thelateral displacement profile and the rolling are symmetric about the centreline.Figure 4.2a shows the state of the saw blade before roll tensioning. The stress-free saw has aconvex shaped lateral displacement profile and a first torsional frequency of 53.2 Hz and astiffness of 13350 N/m.c) Initial tensioning100 -m 50 -a• 0ta -50 -a -100 -can -150--200 -0• -0.5-1Saw Blade ProfileTension Gauge ProfileTight spotFrequency = 80.7 HzStiffness^= 15840 N/my/b^0.5y/b^0.5-200 -Saw Blade ProfileTension Gauge Profiley/bFrequency = 83.0 HzStiffness^= 15960 N/m0.5a) Before tensioningstress-free saw1-E.^0.5y/b^0.50-0.5-1 Frequency = 53.2 HzStiffness = 13350 N/mc) Final tensioning2"27.u)a)v)100500-50-100 --150---^ y/b 0.541Figure 4.2 Tensioning state at three stages of the tensioning process.42Figure 4.2b shows the tensioning state after the initial roll paths have been placed in thecentral region of the saw. The first roll paths are placed near the plate centre because rollingat this position results in the greatest change in the lateral displacement. Each additional rollis placed slightly outside the last so that no two roll paths overlap. The four pairs of roll pathsshown in the figure result in a lateral displacement profile which is concave in shape androughly matches the profile of the tension gauge. The tension gauge touches the saw surfacein the area around y/b = 0.25. This area is identified as a tight spot.To obtain the desired light-gap profile, the tight spot in the light-gap profile in Figure 4.2bmust be removed. Following the rules-of-thumb for correcting tight spots, two additional rollpath pairs are placed at about y/b = 0.25. The resulting tensioning state is shown in Figure4.2c. The tight spot is relieved and the desired light-gap profile is achieved.Figure 4.2 shows that tensioning a bandsaw to the traditional light-gap profile increases thefrequency and stiffness of the saw. The initial tensioning results in the largest increase infrequency and stiffness while the final adjustments to the light-gap profile typically result inonly a small change in these values. For example, the initial tensioning shown in Figure 4.2bresults in an increase in frequency and stiffness of 52% and 18% respectively over the valuesfor the stress-free plate in Part (a). However, the final adjustments to the light-gap profileonly increase the frequency by an additional 4% and stiffness by an additional 1%.Although the tensioning example in Figure 4.2 is fairly simple, practical corrections of tightand loose spots is usually an iterative process requiring several cycles of observing the light-gap profile and corrective rolling. The difficulty of obtaining the desired light-gap is furthercomplicated by non-uniformities in the flatness of the saw plate. These "levelling" defectsmust also be corrected during the tensioning process.43The rules-of-thumb used with the light-gap method are effective in practice because theyguide the placement of the roll paths so that the traditional light-gap profile may be readilyachieved. This is a significant advantage over the frequency method of saw tensioningevaluation, which does not provide any guidance to corrective actions. Furthermore, bytensioning a bandsaw to the traditional light-gap profile, the torsional frequency and stiffnessof the saw are increased and cutting performance is enhanced.445.0 ADDITIONAL FACTORS INFLUENCING LIGHT-GAP MEASUREMENTS 5.1 Chapter overviewThis chapter discusses some of the additional factors which influence light-gap measurements.The effect of the curvature applied along the length of a bandsaw blade on the light-gapprofile is discussed, as well as the significance of the size of the "tire-lines" in the traditionallight-gap profile.5.1 Influence of applied curvature on the light-gap profile. The light-gap profile for a bandsaw blade is produced by bending an initially flat saw bladealong its length. The applied curvature, R, shown in Figure 2.2 causes a corresponding lateraldeflection across the width of the saw. Equations (2.6) to (2.10) show that the value of Rinfluences the size and shape of the light-gap profile of a bandsaw. Foschi investigated thiseffect for the special case of a rectangular plate containing parabolically distributed tensioningstresses [21]. However, in general, tensioning stresses are more easily approximated using theFourier cosine series in Equation (2.1). This section investigates the influence of the appliedcurvature, R, on the size and shape of the light-gap profile when the tensioning stresses aregiven by Equation (2.1).The size of the light-gap profile, S can be defined as5 = w(0) — w(b 2)^ (5.1)45where w(0) and w(b/2) are the values of lateral deflection given by Equation (2.5) at thecentre and edge of the plate respectively. When the tensioning stresses are given by Equation(2.1), Equation (5.1) can be rewritten in non-dimensional form as6 = Cvc – Ena n 4 [(1^( —1 )n ) 4112 7C 2 (- 1)n E] n = 1,2, 3,... (5.2)C + C[ mcy AFC'whereb2= — (5.3)RhAn r b i2ad(5.4)n^Ey = 413(1— v2 ) (5.5)E = A y ra^B*^Ara— cosh^cos y^sinh 7^sin 7 (5.6)2^2 2 2andIC^7F.Ve^Are.sinh 7^cos^cosh ^sin2 22(5.7)y 2 C (sinhy^+ siny,106/h in Equation (5.2) is the size of the light-gap profile relative to the plate thickness, h, whilethe parameter, C, provides a non-dimensionalized measure of the applied curvature, R.Figure 5.1 shows the relationship between the size of the light-gap profile, 6/h, and the applied0.8 –^b^ n=146total0.7 –n=2^=40.6 –^ c0.5 –E 0.4 –w0.3 –0.2 –0.1 –0 —^i^I^I^I^I^IC-0 1 – 10 20 30. -0.2 –^ N stress-free } I an I = 0Figure 5.1 Influence of applied curvature on light-gap size.47curvature parameter, C. Several curves are shown in the figure. The first curve, labelled"stress-free", shows the size of the light-gap profile for an untensioned saw. This curve isdescribed by just the first term in Equation (5.2) because the stress parameters, a m are allidentically zero. The light-gap size for the stress-free plate is negative because the displacedshape of the saw is anticlastic.The curves labelled "n=1", "n=2" and "n=3" in Figure 5.1 show the influence of the first threecomponents of tensioning stress on the size of the light-gap profile. These curves aredescribed by the second term in Equation (5.2) for the case when the stress parameter, a m hasa magnitude of +4. The curve labelled "n=1" shows the effect of just the first component ofstress, while the curves labelled "n=2" and "n=3" show the individual effects of the second andthird components of stress. Each of these curves are positive because the sign of the stressparameter is chosen so that the resulting displacement component is concave in shape.The curve in Figure 5.1 labelled "total" is the sum of the other curves in the figure. This curveshows the influence of the applied curvature on the size of the light-gap profile for the specialcase of a rectangular plate containing the tensioning stress components indicated in the figure.Figure 5.1 shows that when the bandsaw blade is initially flat, that is C=0, the light-gap size isalso zero. As the applied curvature is increased, the size of 8/h rapidly increases to amaximum value before decreasing at higher values of C. The value of C which results in themaximum light-gap, C * , depends somewhat on the tensioning stresses in the saw. This isbecause each curve describing the effect of the nth component of stress has a maximum at aslightly different value of C. For example, the n=1 curve in Figure 5.1 has a maximum atabout C=7 while the n=3 curve has a maximum at about C=8.5. Therefore, the appliedcurvature which results in the maximum light-gap size depends on the size of the first fewstress terms in Equation (2.1) which describe the tensioning stresses. In the example given in48Figure 5.1, the maximum light-gap indicated by the "total" curve occurs at about C* =7.Figure 5.2 shows the influence of the applied curvature, R, on the shape of the light-gapprofile. Part (a) shows half of the lateral displacement profile for a bandsaw havingdimensions given in Table 2.1 and containing the same tensioning stresses as the example inFigure 5.1. The curves labelled "a", "b" and "c" show the lateral displacement profile atcorresponding points on the O/h curve in Figure 5.1. Curve b in Figure 5.2 shows the lateraldisplacement profile when the value of the curvature parameter, C *, corresponds to thelargest light-gap size. Curves a and c show the lateral displacement profile when thecurvature parameter is, respectively, less than and greater than C * .Figure 5.2 shows that the applied curvature influences the shape as well as the size of thelight-gap profile. This effect can be more easily seen from the curvatures of the lateraldisplacement profiles which are shown in Part (b) of the figure. The curvatures of curves aand b have different values because they have different light-gap sizes. However, the shape ofthe curvature profiles is almost the same indicating that the two lateral displacement profiles inFigure 5.2a have essentially the same shape. This implies that for values of C less than C *, theshape of the light-gap profile is relatively constant. For values of C greater than C * , the shapeof the light-gap profile changes. The curvature of curve c is much different than for curves aor b. Curve c is less concave in the central region of the saw and is more convex near theplate edges. This indicates that the central region of the saw becomes flatter and the plateedges become more curved when the curvature parameter increases beyond C*.Because the applied curvature influences both the size and shape of the light-gap profile, thesize of the applied curvature must be controlled to achieve repeatable results. The empiricalchoice of a tension gauge curvature implicitly takes into account the value of C used. Thus,once the tension gauge curvature and the value of the applied curvature have been established,10.80.6E 0.4— 0.2S.-0.2 - --0.4 --0.6 —a)49b)0.150.10.05°2 0co• -0.05z• -0.1-0.15-0.2-0.25Figure 5.2 Influence of the applied curvature on the light-gap profile.a) Lateral displacement profiles.b) Curvatures of the displacement profiles in a).50consistency becomes the issue.In practice, the size of the applied curvature can be controlled in several ways. One methodinvolves adjusting the curvature so that the maximum light-gap size is obtained. Anothermethod involves supporting the saw blade between two fixed points and allowing the saw tosag under its own weight. Either of these methods can produce good results. As long as thesame method is consistently used, the exact value of the applied curvature is not of practicalconcern.5.3 Significance of tire sizeIn the traditional light-gap profile shown in Figure 4.1 a, the edges of the tensioned bandsawfall away from the tension gauge profile. The point where the plate falls away from thetension gauge is called the "tire line" and the region outside this point is called the "tire". Inpractice, there is often considerable variation in the size of the tire used by different sawfilers.The significance of these variations in tire size is not well understood. This sectioninvestigates the influence of tire size on bandsaw frequency, stiffness and tensioning stress.Figure 5.3 shows two different tensioning states for a bandsaw having dimensions given inTable 2.1. The tensioning stresses, light-gap profile and frequency and stiffness are given foreach case. Each of the tensioning stresses have been adjusted so that the resulting lateraldisplacement profiles match the same sized tension gauge, but have a different sized tire.The stress distribution shown in Figure 5.3 a results from roll paths placed in the central twothirds of the saw plate, between the neutral zones near each edge of the plate. Roll pathsplaced in this region increase both the frequency and stiffness of the saw. The resulting light-y/b^0 . 5a) Typical tire size200El 1002 0^man -100tii -200-300"E" 0.5E0^-0.5^Logi^y/b^0.53.-1 tire --ON-1^ Saw Blade ProfileTension Gauge Profile 1^1^ Saw Blade Profile0.5^Tension Gauge Profile0EEtire>-. y/b^0.5b) Small tire size-200Tob. 1002u)toNvi0-100-200-300y/b0.5Frequency = 83.0 HzStiffness^= 15960 N/mFrequency = 88.3 HzStiffness^= 16220 N/m51Figure 5.3 Influence of tire size on frequency, stiffness and stresses.52gap profile matches the tension gauge shape in the central region of the saw and has a tire areaextending from y/b = 0.33. The frequency and stiffness of the saw plate are 83.0 Hz and15960 N/m respectively.The stresses shown in Figure 5.3b have been adjusted so that the resulting lateral displacementprofile matches the same size tension gauge shown in Part (a) of the figure but has a smallertire. In this case the tire extends from y/b = 0.44, the frequency is 88.3 Hz and the stiffness is16220 N/m. These values of frequency and stiffness are about 6 percent and 1 percent,higher, respectively, than the values for the plate in Part (a).The stress distribution in Figure 5.3b associated with the small tire size has roll paths placedoutside of the neutral zones of the saw. Because rolling outside the neutral zone reducestensioning, excessively large roll paths are required in the central region of the saw tocounteract the loss of tensioning. The large compressive stresses induced by heavy rollingresult in correspondingly large tensile stresses in the remainder of the plate. For example, theplate in Figure 5.3b has tensile stresses of 176 MPa which is more than three times higher thanthe plate in Part (a) which has tensile stresses of 55.3 MPa.In general, achieving a small tire size by roll tensioning outside of the neutral zone results in aslight increase in the frequency and stiffness of a bandsaw but also results in large tensiletensioning stresses. In practice, large tensile stresses at the plate edges are undesirable inbandsaws. Bandsaws are subject to fatigue related cracking at the saw tooth gullets caused bya combination of stress concentrations from the tooth grinding, the tooth profiles and largetensile stresses at the plate edges. The tensile stresses during saw operation are caused by acombination of cyclical bending stresses which occur as the saw blade passes over thebandmill wheels and uniform tensile stress caused by bandmill strain. Large tensioningstresses further increase the tensile stresses at the plate edges and can significantly reduce the53fatigue life of a bandsaw. Therefore, in order to control the size of the tensile stresses in abandsaw, tire size should be wide enough so that roll tensioning in the area outside the neutralzones can be avoided. This can be done by specifying that the tire lines should beapproximately at the neutral zones. An approximate rule-of-thumb is, therefore, that the sawshould fit the tension gauge profile over the middle two thirds of the blade.546.0 THE EFFECTIVENESS OF THE LIGHT-GAP METHOD FOR BANDSAWS 6.1 Chapter overviewThe light-gap profile can be used to indicate changes in saw plate frequency and stiffnesscaused by the tensioning stresses. Section 2.6 showed that the components of stress that mostinfluence the frequency and stiffness of a bandsaw similarly influence the size of the light-gap.However, the shape of the light-gap profile is also an important indicator of the tensioningstate. In this chapter, the relationships between the shape of the light-gap profile and thefrequency and stiffness of a bandsaw are investigated. The effectiveness of the light-gapmethod for indicating changes in the frequency and stiffness of a tensioned saw is alsodiscussed.6.2 Relationships between light-gap profile, frequency and stiffnessIn this section, the relationships between the light-gap profile, frequency and stiffness areinvestigated for two cases. The first case involves a bandsaw tensioned to a specificfrequency and stiffness while the second case involves the same saw tensioned to a specificlight-gap profile.Figure 6.1 shows three possible tensioning states for a rectangular plate having the dimensionsgiven in Table 2.1. In each case, the tensioning stress and resulting lateral displacementprofile, w(y), are shown. Each of the stress distributions have been chosen so that thefrequency and stiffness of the plate are the same for each case. Each stress distributionprovides an example of how a bandsaw blade could be roll tensioned to a specified frequency,Tensioning Specification:Frequency^=Stiffness^=83.0 Hz15960 N/mTensioning150To'^10013-^52^0A---y/b 0 . 5y-100c -150 -c+i) -200 --250 -1 -E^0 .5 •0:5".^-0.5 y/b0.53^-1Tensioning B15013^100CI-^502^00^-50u)^-100 y/b0.5)^-150can -200-250E,^1E^0 . 52...^-^00.5 0.5y/b3^-1Tensioning C150TO^100g3-^502^0,,,^ 50u)^-100y/b2^-150en' -200-2501T^0.5E0  .^-0.5 y/b 0.53^-1Figure 6.1^Lateral displacement profiles for three different stress distributionsproducing the same specified values of frequency and stiffness.5556rather than the more usual practice of tensioning to a specified light-gap profile.Tensioning A stresses represent a typical case where the bandsaw has been rolled alongseveral paths in the central region of the plate. Tensioning A is the same as the exampleshown in Figure 4.2c and provides an example of typical tensioning stresses for a new saw.Tensioning B stresses result from placing adjacent roll paths halfway between the centre andedge of the plate. Tensioning C stresses result from excessive rolling at the plate centrefollowed by rolling at the plate edge to reduce the frequency and stiffness values to the desiredlevel.Figure 6.1 shows the lateral displacement profiles for each of the three different tensioningstates. The shape of each lateral displacement profile is different, even though the frequencyand stiffness is the same in each case. Therefore, tensioning a saw to a specific frequency orstiffness produces light-gap profiles that are not unique.The non-uniqueness in the lateral displacement profiles in Figure 6.1 is caused by differencesin the associated stress distributions. Table 6.1 lists the first five values of A n for each of thestress distributions in the figure. The columns labelled "frequency" list the natural frequencyof the plate subject to the cumulative influence of the stresses corresponding to thecoefficients, An. The first listed frequency, 53.2 Hz, is for an untensioned saw. The nextfrequency adds the effect of A1, the next includes both Al and A2, and so on. Although thethree stress fields illustrated in Figure 6.1 have different individual coefficients, Table 6.1shows that the frequencies of the plates quickly converge to approximately the same value.However, as discussed in Section 2.3, the size and shape of the lateral displacement profiledepends on the size of each of the lower An coefficients individually. The three tensioningstates have different lower values of A n, and therefore have different "average" stressdistributions. These differences in the "average" stress distribution cause the differences in the57Table 6.1^Plate frequencies and stress series coefficients for the tensioning cases inFigure 6.1.nTensioning A Tensioning B Tensioning CAn[MPa]Freq[Hz]An[MPa]Freq[Hz]An[MPa]Freq[Hz]none - 53.2 - 53.2 - 53.21 -49.7 81.0 -17.4 64.4 -88.3 97.32 25.6 83.9 166.7 85.3 -125.1 84.43 12.4 83.3 22.7 84.1 -34.9 86.14 -10.7 83.0 -25.1 83.4 -57.6 84.55 9.7 82.9 10.0 83.3 27.0 84.0all - 83.0 - 83.0 - 83.0Table 6.2^Plate frequencies and stress series coefficients for the tensioning cases inFigure 6.2.nTensioning A Tensioning D TensioningAn[MPa]EFreq[Hz]An[MPa]Freq[Hz]A.[MPa]Freq[Hz]none - 53.2 - 53.2 - 53.21 -49.7 81.0 -48.0 80.2 -51.0 81.62 25.6 83.9 16.9 82.2 15.8 83.43 12.4 83.3 -14.2 82.9 10.5 82.94 -10.7 83.0 -0.9 82.9 -12.8 82.55 9.7 82.9 -38.2 83.6 0.00 82.5all - 83.0 - 83.2 - 82.658lateral displacement profiles. The stiffness of a saw is closely related to its natural frequencyand behaves in a similar manner.Figure 6.2 shows three possible tensioning cases which result when the rectangular plate inTable 2.1 is roll tensioned to the specific light-gap profile shown in the figure. For each case,the size and location of each roll path is chosen so that the lateral displacement of the platematches the curvature of the tension gauge in the central region. Again, Tensioning A is thesame as in Figure 4.2c, while Tensioning D and E are two examples of the many differentstress distributions which could also be used to obtain the same specified light-gap profile.The frequency and stiffness values resulting from the three stress distributions are given in thefigure and are virtually the same for each case.In spite of the differences in the details of the stress distributions in Figure 6.2, Table 6.2shows that the size of the first two stress coefficients are almost the same in each case.Because the shape of the displacement profile depends most strongly on the first twocoefficients, the light-gap profiles are also essentially the same. Similarly, the frequency andstiffness also depend most strongly on the lowest terms in the stress series. Table 6.2 showsthat the first two stress terms result in a frequency which has already almost converged to thefinal value. In a similar manner, the stiffness of each plate also rapidly converges.Figures 6.1 and 6.2 show that although the lateral displacement profile for a given frequencyand stiffness is non-unique, the opposite relationship does exist. That is, when a saw istensioned to a specific light-gap profile, the frequency and stiffness are effectively determined.This behaviour occurs because the frequency and stiffness depend on a weighted sum of theAn terms. The individual An values can be varied as long as the weighted sum remainsconstant. In contrast, the lateral displacement profile depends on the individual values of A n .A particular light-gap shape automatically specifies a specific weighted sum and therefore aTensioning1E 0.5Specification:Saw Blade ProfileTension Gauge Profile0-0.5-1y/b^0.5Tensioning150To.^1000-^502^0r,^-5-10„ ,..,=^-150ti -200-250Ay/b 0 . 5FrequencyStiffness==83.0 Hz15960 N/mTensioning150'a^1000-^502^0co-100e -150CZ -200-250Dy/b 0.5FrequencyStiffness==83.2 Hz15970 N/mTensioning15071-3^100502^0(0)^-100E.'^-150ti5 -200-250Ey -50 y/b 0.5FrequencyStiffness==82.6 Hz15940 N/mFigure 6.2^Frequency and stiffness values for three different stress distributionsproducing the same specified light-gap profile.5960specific frequency and stiffness. Clearly, the reverse is not true. A particular weighted sum(frequency and stiffness) does not specify the individual A n values (light-gap).6.3 The light-gap profile as an indicator of frequency and stiffnessThe results of the previous section show that when a saw is tensioned to a specific lateraldisplacement profile, the frequency and stiffness are effectively determined. This relationshipis crucial in practice. It means that the light-gap method does provide an effective way ofevaluating the influence of the tensioning stresses on saw performance.Although a number of different lateral displacement profiles could be used with the light-gapmethod, the traditional circular profile has a number of practical advantages. Tensioning asaw to this profile requires roll tensioning in the central region of the saw blade which resultsin a stress distribution with a large n=1 stress term. Figures 2.6 and 2.8 shows that this stressterm has the largest influence on frequency and stiffness for a given level of stress. Therefore,tensioning a bandsaw to the traditional light-gap profile results in a large increase in sawfrequency and stiffness. Additionally, a circular shaped tension gauge is convenient because itcan be used with different widths of bandsaws. Other tension gauge shapes would requirespecial gauges for each bandsaw width.Most importantly, however, by tensioning a bandsaw to the traditional circular shaped light-gap profile, the curvature of the tension gauge can be used to indicate the frequency andstiffness of the saw. Figure 6.3 illustrates the relationship between tension gauge diameter andsaw frequency and stiffness for the specific case of a plate having dimensions given in Table2.1. The values of frequency and stiffness are normalized with respect to the values for thestress-free plate. Each point on the curve corresponds to a plate tensioned to the traditional^ Tensioning A^o Tensioning D^A Tensioning E1 ^I-^I^1^ i^1^0^5^10^15^20^25Tension Gauge Diameter [m] 30^35•FrequencyStiffness••• • ■■ • • • • •2.261Figure 6.3 Relationship between tension gauge diameter and frequency and stiffness ratio.62light-gap profile and having one of the three stress distributions shown in Figure 6.2. Themagnitudes of the stresses have been adjusted so that the resulting lateral displacementmatches the various tension gauge diameters.Figure 6.3 indicates that by specifying the shape of a bandsaw's lateral displacement profile,the size of the profile indicates the frequency and stiffness of the saw. For a saw tensioned tothe traditional light-gap profile, as is the case here, the frequency and stiffness of the plate areeffectively indicated by the tension gauge diameter. As the tension gauge diameter decreases,the frequency and stiffness of the saw increase. At very small gauge diameters, however, thetensioning stresses may become large enough to buckle the unstrained saw blade, in whichcase the results in the figure would not be applicable. As the tension gauge diameter becomesvery large, the tension gauge shape approaches a straight-edge and the associated frequencyand stiffness ratios approach values somewhat greater than unity. This occurs because astress-free plate has an anticlastic shape and therefore, a small amount of tensioning isrequired to achieve a flat lateral displacement profile.The results from Figure 6.2 and 6.3 confirm the hypothesis in Figure 1.3 that the light-gapprofile is an effective indicator of saw blade frequency and stiffness. When a saw is tensionedto a specific light-gap profile, the frequency and stiffness are effectively determined. Inparticular, when a bandsaw is tensioned to the traditional light-gap profile, the frequency andstiffness are indicated by the curvature of the associated tension gauge. Because of thesefeatures, the light-gap method provides a rational and effective way of monitoring the sawtensioning process.637.0 PRACTICAL ADVANTAGES AND LIMITATIONS OF THE LIGHT-GAP METHOD7.1 Chapter overviewThe light-gap method is widely used in industry is because it has a number of importantpractical advantages over other tensioning evaluation techniques, such as natural frequencymeasurements. However, the light-gap method also has some limitations. Several of theseadvantages and limitations are discussed in this chapter.7.2 Advantages and limitationsOne of the most important advantages of the light-gap method is its ability to indicate thetensioning state within a localized part of the saw. The light-gap profile indicated by a tensiongauge is influenced by the tensioning stresses in the area near the gauge. Therefore,undesirable variations in the tensioning stresses can easily be identified by moving the tensiongauge along the length of the bandsaw. Frequency measurements, on the other hand, provideonly a global average of the tensioning state of a saw. Non-uniformities in the tensioningstresses are not indicated.Another advantage of the light-gap method is its ability to indicate deviations from plateflatness. These "levelling defects" are readily identified by moving a straight-edge over thesurface of a saw when it is resting on a flat surface. Because correcting levelling defectsinfluences the tensioning of a saw, levelling and tensioning are usually performedsimultaneously. The light-gap method provides a convenient way to monitor the progress ofboth of these operations. Frequency measurements provide almost no indication of levelling64defects.One of the limitations of the light-gap method is that the evaluation of the shape of the light-gap profile is somewhat subjective in nature and relies on the skill and judgment of thesawfiler. This can lead to significant variations among saws tensioned by different sawfilers.However, these variations can be reduced by following good industrial practices. Forexample, saws should be sufficiently flat so that their lateral displacement can be exactlymatched to the shape of the tension gauge. Sawfilers often leave the central region of the sawslightly loose so that levelling defects do not interfere with the light-gap profile. This practicelimits the accuracy of the light-gap method.Unlike the light-gap profile, frequency measurements have the advantage of providing aquantitative measure of the tensioning state of a saw and are not affected by small levellingdefects. Acceptable ranges for saw frequencies can be specified and used as controls for thesaw tensioning state.In general, the light-gap method is preferred over other tensioning evaluation techniquesbecause it is simple and fast to use, and it provides localized tensioning and levellinginformation simultaneously. However, the frequency measurement method, by providing aquantitative measure, has the potential to reduce the variability in tensioning among saws.Ideally, a combination of the light-gap and frequency measurement methods would beextremely effective for controlling the tensioning process. However, because of the difficultyin supporting a bandsaw effectively, frequency measurements are only practical for circularsaws. For a circular saw, the light-gap method would be used to tension and level the sawand to ensure that the tensioning state is reasonably uniform. Frequency measurements wouldthen by made as a final check to ensure that the tensioned saw had frequency values falling65within a small specified range. If frequency measurements indicated adjustments to the sawtensioning were required, these adjustment could be easily made by additional roll tensioning.Because the required frequency adjustments would typically be fairly small, the additional rolltensioning would not significantly affect the light-gap profile.668.0 CONCLUSIONSThe effectiveness of the light-gap method for monitoring the saw tensioning process has beeninvestigated in this thesis. The relationships between tensioning stresses, lateral displacement,first torsional frequency and cutting edge stiffness were studied for the specific case of abandsaw containing symmetric in-plane tensioning stresses. Although the results presentedapply specifically to bandsaws, the general features of the light-gap method are also expectedto apply to circular saws.The light-gap method has been criticized as being a poor indicator of saw tensioning becauseof its subjective nature and because it is insensitive to the details of the tensioning stressdistribution. However the light-gap method is widely accepted in industry, and is used withsubstantial success to monitor the saw tensioning process. The relationships given in thisthesis show that the light-gap method provides an effective measure of bandsaw frequencyand stiffness. This occurs because the same components of stress which most influence thetorsional frequency and cutting edge stiffness have a similar influence on the light-gap profile.Both the size and shape of the light-gap profile are important indicators of the tensioning stateof a saw. The light-gap profile for a bandsaw depends on the individual size of the lowest fewterms in the Fourier series which describes the tensioning stresses. Specifying the size andshape of the light-gap profile controls the size of each of the first few stress terms. Thisautomatically specifies the weighted sum of these terms, on which the plate frequency andstiffness depend. Therefore, by tensioning a bandsaw to a specific light-gap profile, thefrequency and stiffness of the saw are indicated.To achieve effective results with the light-gap method, the shape of the light-gap profile must67be specified. The traditional circular shaped light-gap profile provides a convenient shape. Itencourages the tensioning stress components which result in the largest increase in frequencyand stiffness for a given stress level. However, the size of the tire areas in the traditionalprofile must be controlled in order to avoid excessive levels of tensioning stress which canlead to saw cracking problems. Most importantly, however, by using the traditional circularshaped light-gap profile, changes in saw frequency and stiffness are indicated by the size of thecurvature of the associated tension gauge. In this manner, the light-gap method provides areliable and highly effective means of evaluating the affect of the tensioning process on sawperformance.Empirical rules-of-thumb used with the light-gap method guide roll path location in order toachieve the traditional light-gap profile. These rules-of-thumb are effective because roll pathposition influences the size and shape of the light-gap. Rolling in the central region of the sawplate encourages a traditional concave shaped light-gap profile and increases the torsionalfrequency and stiffness of a bandsaw. Rolling near the plate edges has the opposite effect.The largest effects are obtained by rolling at the plate centre or edges and decrease as roll pathposition approaches a "neutral zone". Rolling within the neutral zone has a negligible effecton the light-gap, frequency or stiffness.In addition to its ability to indicate saw frequency and stiffness, the light-gap method also hasa number of practical advantages over other tensioning evaluation techniques such asfrequency measurement. The light-gap profile indicates local non-uniformities in thetensioning state, can be used to indicate levelling defects, and is simple and easy to use. Incontrast, natural frequency measurements provide only a global average of the tensioning stateand so do not indicate local variations. Additionally, frequency measurements do not indicatelevelling defects and require sophisticated equipment.68Although the light-gap method has the limitation of providing a somewhat quantitativemeasure of saw tensioning, this difficulty can be reduced by following good industrialpractices. Levelling defects should be reduced as much as possible and the saw should betensioned to exactly fit the tension gauge shape. The size of the curvature applied along thelength of a bandsaw must also be controlled to achieve consistent results with the light-gapmethod.Ideally, a combination of light-gap and frequency measurements could improve the quality ofsaw tensioning. A saw would first be tensioned using the light-gap method and would then bechecked using the more quantitative frequency measurements. This procedure has thepotential to reduce tensioning variation between saws. However, it would be only practical forcircular saws because of the difficulties in making bandsaw frequency measurements.In summary, the light-gap method provides a reliable and highly effective means of monitoringthe saw tensioning process. It can indicate changes in saw frequencies and stiffness caused bythe tensioning process and has a number of important practical advantages over othermethods.69REFERENCES 1. Statistics Canada. 1989. Wood Industries. Catalog 35-250, Statistics Canada,Ottawa, Ont.2. Hallock, H. 1962. A mathematical analysis of the effects of kerf width on lumberyield for small logs. Report No. 2254, USDA Forest Service, Forest ProductsLaboratory, Madison, WI.3. Clapp, V. W. 1982. Lumber recovery: how does your mill's performance rate?Forest Industries 109(3): 26-27.4. Higgs, M. 1989. Economic advantages of saw management. Forest Industries166(5): T17-T19.5. Mote, C. D.; Nieh, L. T. 1973. On the foundation of circular saw stability theory.Wood and Fiber 5(2): 160-169.6. Dugdale, D. S. 1966. Theory of circular saw tensioning. International Journal ofProduction Research 4: 237-248.7. Schajer, G. S. 1984. Understanding saw tensioning. Holz als Roh-und Werkstoff 42:425-430.8. Quelch, P. S. 1970. Armstrong Sawfiler's Handbook, 2nd Edition. ArmstrongManufacturing Co., Portland, OR. pp. 1-6 and pp. 75-85.9. Stephenson, E.; Plank, D. 1972. Circular Saws. Thomas Robinson and Sons Ltd.,Rochdale, England. pp. 68-94.10. Umetsu, J.; Noguchi, M.; Matsumoto, I. 1987. A new technique of tensioningcircular saws using shot peening. Mokuzai Gakkaishi 33(10): 818-820.11. Mote, C. D.; Schajer, G. S.; Holoyen, S. 1981. Circular saw vibration control byinduction of thermal membrane stresses. Journal of Engineering for Industry 103(2):81-89.12. Huber, H. 1991. Straightening and tensioning of circular saws by means of laser.Proceedings of the Tenth International Wood Machining Seminar. Forest ProductsLaboratory, Richmond, CA.13.^Szymani, R.; Mote, C .D. 1977. Circular saw stiffness as a measure of tension.Forest Products Journal 27(3): 28-32.7014. Mote, C. D.; Szymani, R. 1977. Principal developments in thin circular saw vibrationand control research. Part I: Vibration of circular saws. Holz als Roh-und Werkstoff35: 189-196.15. Szymani, R.; Mote, C .D. 1974. A review of residual stress and tensioning in circularsaws. Wood Science and Technology 8: 148-161.16. Schajer, G. S. 1992. North American techniques for circular saw tensioning andlevelling: Practical measurement methods. Holz als Roh-und Werkstoff 50: 111-116.17. Szymani, R.; Trinchera, L.; Turner, J. 1987. Use and maintenance of thin circularsaws at California Cedar Products Company. Holz als Roh-und Werkstoff 45: 319-322.18. Schajer, G. S. 1981. Analysis of roller induced residual stresses in circular discs andtheir effect on disc vibration. PhD Thesis, University of California, Berkeley, CA.19. Schajer, G. S. 1985. CSAW guided circular saw vibration and stability program,version 2.4. Science and Process Control Unit, Weyerhaeuser Technology Center,Tacoma, WA.20. Taylor, J.; Hutton, S. G. 1990. The automation of bandsaw roll tensioning. Phase I:Roll tensioning stresses and performance. Project No. 12-71-K-404, Forintek CanadaCorp., Vancouver, B.C.21. Foschi, R. 0. 1975. The light-gap technique as a tool for measuring residual stressesin bandsaw blades. Wood Science and Technology 9: 243-255.22. Conway, H. D.; Nickola, W. E. 1965. Anticlastic action of flat sheets in bending.Experimental Mechanics 5(4): 115-119.23. Leissa, A. W. 1969. Vibration of Plates. Technical Information Division, NationalAeronautics and Space Administration, Washington DC. pp. 53-58.24. Tse, F. S.; Morse, I. E.; Hinkle, R. T. 1978. Mechanical Vibrations, Theory andApplications, Second Edition. Allyn and Bacon Inc., Boston, MA. pp. 288-289.25. Iyengar, N. G. R. 1988. Structural Stability of Columns and Plates. Ellis HorwoodLtd., Chichester, England. pp. 163-178.^26.^Lister, P. F.; Schajer, G. S. 1991. The effectiveness of the light-gap and frequencymeasurement methods for evaluating saw tensioning. Proceedings of the TenthInternational Wood Machining Seminar. Forest Products Laboratory, Richmond, CA.

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