cutting force model including effects of

International Journal of Machine Tools &Manufacture 41(2001)227–236Bandsawing.Part I:cutting force model including effects ofpositional errors,tool dynamics and wearC.Andersson,M.T.Andersson,J.-E.Sta ˚hl*Lund University,Dept of Productions and Materials Engineering,Box 118,S-22100Lund,SwedenReceived 8July 1999;accepted 10July 2000AbstractThis article presents a mechanical cutting force model for bandsawing.The model describes the variation in cutting force between individual teeth and relates it to initial positional errors,tool dynamics and edge wear.Bandsawing is a multi-tooth cutting process,and the terminology of the cutting action is discussed and compared with other cutting processes.It will also be shown that the setting pattern and the preset feed govern the cutting data.©2000Elsevier Science Ltd.All rights reserved.Keywords:Bandsawing;Cutting force model;Cutting force variation;Positional error1.IntroductionIn the field of machining there are a number of processes where more than one cutting edge is in contact with the workpiece at the same time.These are called multi-toothed cutting processes,and include milling,sawing,drilling and broaching.They differ in “average”cutting parameters and in the type of main motion.They also differ regarding the cutting edge engagement with the workpiece,being continuous or interrupted.This group of processes has been studied earlier,milling having received the most attention.Sawing,and especially bandsawing,has not been studied in the same extent.Sarwar and Thomp-son [6–9]in the 1970s studied the hacksaw process in order to analyze the discrepances they noticed in what they call the cutting constant,M .They found that the tooth pitch and the width of the workpiece cause these discrepancies.They developed a cutting force model based on mean undeformed chip thickness and the cutting constant.They also looked into the cutting action when machining with blunt tools.Ahmad et al.[3]studied effects of machining parameters for a group of teeth on the cutting constant,defined by Sarwar and Thompson [6].Milberg and Rehbein [5]*Corresponding author.Tel.:+46-46-222-4592;fax:+46-46-222-4529.0890-6955/01/$-see front matter ©2000Elsevier Science Ltd.All rights reserved.PII:S 0890-6955(00)00064-X228 C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236Nomenclatureb10approximate undeformed width of chip for neutral teeth,mmb1ffictive width of chip,same as width of saw kerf,mmb1i undeformed width of chip for an arbitrary tooth i,mmb1n nominal width of chip,mmb1s approximate undeformed width of chip for right-and left set teeth,mmC ri Cutting resistance,N/mm2d e distance between cutting edge and center of rotation,mmd VBi change in cutting edge distance due toflank wear,mme t tooth pitch,mmF Hi main cutting force,NF Mi feed force,NF Pi passive force,Nh1n nominal undeformed chip thickness,mmh1i undeformed chip thickness cut by an arbitrary tooth i,mmh2deformed chip thickness,mmn na number of teeth in a row not in contact with the workpiecen z number of teeth in a row between two equally set teethr i distance from tool center point to cutting edge,mms h setting height,mmv cutting speed,m/minv f feed speed,m/minVBflank wear land,mmy pi positioning error due to tool dynamics,mma clearance angle,degreesb wedge angle,degreesd distance the saw kerf is widened due to the setting,mmg rake angle,degrees␬S setting angle,degreesj MH force ratio F M/F Hj PH force ratio F P/F Halso presented a cutting force model for bandsawing.Their model describes the sawing process as a square function with a Fourier series containing three overtones.The study does not,however, include any cutting data.Vibrations have been studied by To¨nshoff and Westphal[11].Ellis and Mote[4]have been looking into vibration of circular saws.Ulsoy et al.[12]carried out a literature survey of vibrations in bandsaw blades and concluded that increased prestressing of the bandsaw blade and minimization of the distance between the guide jaws reduce the tool vibrations.No studies have been found dealing with the cutting force of each individual tooth working in a group of teeth in bandsawing.This paper presents a cutting force model which is valid for multi-toothed cutting processes in general,and which is based on individual cutting forces.The229C.Andersson et al./International Journal of Machine Tools &Manufacture 41(2001)227–236model deals with variations in individual cutting forces due to errors in cutting edge position,tool dynamics and edge wear.2.Terminology for the sawing processOur cutting force model is a mechanical load model in which the magnitude of the main cutting force depends on the size of the undeformed chip section cut and the cutting resistance of the workpiece material.A general presentation of the model has been given earlier [1,2].The current article will concentrate on bandsaw operations.The force model only deals with individually set teeth,which is the most common method of setting teeth for bandsaw blades.The three orthogonal cutting force components F Hi ,F Mi ,F Pi acting on an arbitrary cutting edge i are given by:F Hi ϭC ri ·h 1i ·b 1i(1)F Mi ϭj MH ·F Hi(2)F Pi ϭj PH ·F Hi (3)where the symbols are defined in the Nomenclature.The cutting resistance of the workpiece material should be regarded as a process parameter since its magnitude is influenced not only by undeformed chip thickness,h 1i ,but also by rake angle,inclination angle,cutting speed,nature of the cutting fluid and tool wear.The index i indicates that the equations describe cutting forces for each individial tooth.The nominal undeformed chip thickness is:h 1n ϭ(n z +1)·v f ·e t v (4)where n z is the number of teeth between two equally set subsequent teeth.This parameter varies for different tool geometries.A common setting for teeth with equal pitch is neutral,right,left,neutral,right,left,…which gives n z =2for all three differently set teeth.Bandsaw blades with differential pitch usually have a more complicated setting pattern,adjusted to the number of teeth included in a setting group.Fig.1shows two different tool geometries,one with an equal pitch inch (P2)and one with a differential pitch (34TPI).For the differentially pitched tool the setting group consists of seven teeth,whose pitch and setting are repeated along the total band length.Since the teeth are set to prevent jamming of the tool,the saw kerf is wider than the bandsaw blade.This is due to the set teeth machining at different positions in the kerf.This also causes each tooth to cut a chip that is thicker than the preset value.For the P2-geometry the nominal chip thickness is three times the preset value,shown in Fig.2.The figure also illustrates that the total cutting edge length of each tooth is not continuously active during the machining,instead the cutting action is concentrated to the corners of each tooth.230 C.Andersson et al./International Journal of Machine Tools &Manufacture 41(2001)227–236Fig.1.Two different bandsaw blade goemetries including setting pattern.The cutting action of neutral teeth is symmetric,while the right-and left-set teeth are subjected to a larger load on the outer corners,both at the main and at the secondary cutting edge.The wear pattern for each differently set tooth shows that the cutting action is distributed unevenly over the width of the tooth.Fig.3shows the flank wear land when the tooth is monitored from the flank face,for three differently set teeth.The active cutting edge length can be calculated if the setting angle ␬s setting height s h and nominal cutting depth b 1n ,shown in Fig.4,are known.Some authors,for example [5–8],use the term “cutting depth”to refer to the undeformed chip thickness.This terminology is confusing when comparing sawing with other cutting processes.The cutting depth should be related to the cutting edge length,as it is for all other cutting processes.In this article cutting depth will be refered to as chip width aiming to not confuse the readers.Fig.4makes it clear that both the nominal undeformed chip width,b 1n ,and the nominal unde-formed chip thickness,h 1n in the feed direction vary during sawing.Fig.2shows that the active cutting edge length,or chip width varies for differently set teeth.The chip widths for the right-set and left-set teeth,b 1s ,are approximately the same size,while the chip width for the neutral teeth,b 10,is slightly different.The following equations are valid for the tool geometry with equal pitch,P2.b 1s ϭb 1n ͩ1Ϫ12cos ␬s ͪ(5)b 10ϭb 1n +2d cos ␬s Ϫb 1n (6)231C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236Fig.2.Undeformed chip cross-sections for two different tool geometries.Fig.3.Flank wear land for three differently set teeth for an equal P2-geometry[10].The above equations are limited to small and moderate chip thicknesses for two reasons.Firstly, the exact value of the active cutting edge length varies for all three differently set teeth;and secondly.the chip width varies with the nominal feed per tooth.as illustrated in Fig.5.Thefigure shows that there is a big difference in active cutting edge length between,for example,h1n=10232 C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236Fig.4.(a)Definition of cutting data.(b)Definition of setting parameters and tool geometry.Fig.5.Variation in chip cross-sections for different values of feed per tooth.C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236233µm and h1n=50µm.The smaller value is commonly used in practice while the larger value is still likely to occur due to variations in undeformed chip thickness,which will be discussed in the next section.Our geometrical study of the cutting action when sawing shows that the undeformed chip thickness involves a factor times the nominal preset value.The factor n z,is the number of differ-ently set teeth passing in a row between two equally set teeth.The undeformed chip width is divided mainly into two parts located at the corners of each tooth,and is therefore always less than both the total width of the tooth b1n,and the width of the kerf b1f.The main cutting force acting on each an arbitrary tooth is calculated from:F HϭC ri·(n zϩ1)h1n·b1i(7) were b1i is calculated from Eq.(5)or Eq.(6)depending on weather the tooth is set or neutral.3.Mechanical cutting force modelUnder ideal conditions,all teeth are intact and are positioned at the same distance from a reference point on the tool and the tool movement coincides with the main movement,i.e.no tool vibrations exist.These conditions result in equal cutting forces for all teeth.However,this is never the case when machining in a workshop.The undeformed chip cross-section together with the cutting resistance determine the magnitude of the individual cutting forces,and so a variation in these parameters during machining also causes a variation in individual cutting force values.The cutting force model presented here includes three different factors which influence the size of the undeformed chip thickness.The variation in chip width is very small compared to h1n and is therefore neglected.The factors which influence the undeformed chip thickness are the follow-ing,Fig.6.Fig.6.Factors influencing individual cutting force values.234 C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236țErrors in cutting edge positionțTool vibration(movement)in the feed directionțContinuous tool wear and deteriorationThe deviation in cutting edge position between two subsequent equally set teeth influences the size of undeformed chip thickness according to:h1iϭh1nϩr iϪr i−1(8)It is likely that some cutting edges never gain contact with the workpiece at all,these are called non-active teeth.The uncut chip thickness due to the existence of non-active teeth is treated mathematically as it arose from thefirst active tooth.If there are n na non-active teeth in a row, the influence of errors in cutting edge position can be included in Eq.(8)to give: h1iϭ(n naϩ1)·h1nϩr iϪr i−n na−1(9)The movement of the saw blade in the feed direction is modelled as a value,y pi,for the position of each individual tooth.The effect of tool vibration on the variation in thickness of the unde-formed chip is then:h1iϭh1nϩr iϪr i−1ϩy piϪy pi−1(10) or,including the effects of non-active teeth:h1iϭ(n naϩ1)·h1nϩr iϪr i−nna−1ϩy piϪy pi−nna−1(11)All types of deterioration of the cutting edge alter the geometry along both the major and the minor cutting edge.The loss of tool material decreases the cutting ability,and subsequent undam-aged cutting edges must cut the excessive workpiece material left behind by damaged teeth. All types of wear that affect the major cutting edge are included in the distance d VB,which is calculated from:d VBϭVBtan(g+b)−tan g(12)where g=rake angle,b=wedge angle and VB=width offlank wear land.The equation for h1i(Eq.(8))including the effects of cutting edge deterioration becomes:h1iϭh1nϩr iϪr i−1ϩy piϪy pi−1Ϫd VBiϩd VBi−1(13) or,including the effects of non-active teeth:h1iϭ(n naϩ1)·h1nϩr iϪr i−nna−1ϩy piϪy pi−nna−1Ϫd VBiϩd VBi−nna−1(14)The individual main cutting force is determined by workpiece material and chip cross-section area given by Eq.(1).The chip width b1i is calculated from Eq.(6)for neutral teeth and from235C.Andersson et al./International Journal of Machine Tools &Manufacture 41(2001)227–236Eq.(5)for right-set and left-set teeth.The individual undeformed chip thickness including errors in cutting edge positions,tool vibration in the feed direction and cutting edge deterioration,is for n na =0given by:h 1i ϭz ·v f ·e t v ϩr i Ϫr i −1ϩy pi Ϫy pi −1ϪͩVB i −VB i −1tan(g +b )−tan gͪ(15)or,including the effects of non-active teeth:h 1i ϭ(n na ϩ1)z ·v f ·e t v ϩr i Ϫr i −n na −1ϩy pi Ϫy pi −n na −1ϪͩVB i −VB i −n na −1tan(g +b )−tan gͪ(16)All parameters are defined in the Nomenclature.4.Conclusions Studying the saw tool geometry,comparing it,with other cutting tools,the width of the tooth should be called the nominal depth of cut or chip width b 1n .The distance a tooth moves in the feed direction over the distance of one tooth pitch in the cutting direction is then called the nominal undeformed chip thickness h 1n .The nominal or preset cutting data are not the values existing in practice,because the teeth on a saw tool are set.The real depth of cut,b 10or b 1s ,is smaller than b 1n and has different values for differently set teeth.The real undeformed chip thickness is always larger than the nominal chip thickness,and may very well be fifteen times larger than h 1n depending on the setting pattern.Manufacturing tolerances and instabilty of the equipment ensure that there exist,a variation in cutting data for each individual tooth,which causes variations in individual cutting forces.Eqs.(15)and (16)allow us to calculate the individual main cutting force under the influence of errors in cutting edge position,tool vibrations in the feed direction and continous wear and deterioration.These equations will be verified in a subsequent article about bandsawing,also describing a method for measuring individual cutting forces.References[1]C.Adolfsson,Multi-tooth cutting processes —models and methods for cutting force analysis.Doctoral thesis atDept.of Prod.and Mat.Eng.,Lund University,May 1996(in Swedish).[2]C.Adolfsson,J.-E.Sta˚hl,Cutting force model for multi-toothed cutting processes and force measurement equip-ment for face milling,Int.J.Mach.Tools Manufact.35(1995)1715.[3]M.M.Ahmad,B.Hogan,E.Goode,Effect of machining parameters and workpiece shape on a bandsaw process,Int.J.Mach.Tools Manufact.29(1989)173.[4]R.W.Ellis,C.D.Mote,A feedback vibration controller for circular saws,Trans.of ASME 101(1979)44.[5]berg,H.P.Rehbein,Periodizita ¨ten gemessener Schnittkraftverla ¨ufe beim Horizontalmetallbandsa ¨gen,Indus-tri-Anzeiger 67(1988)26.[6]M.Sarwar,P.J.Thompson,Simulation of the action of a single hacksaw blade tooth,The Production Engineer(1974)195.236 C.Andersson et al./International Journal of Machine Tools&Manufacture41(2001)227–236[7]M.Sarwar,P.J.Thompson,Cutting action of blunt tools,Int.J.of MTDR,Conference Manchester,1982,p.295.[8]P.J.Thompson,A teoretical study of the cutting action of power hacksaw blades,Int.J.of Mach.Tools Des.andRes.14(1974)199.[9]P.J.Thompson,Factors influencing the saw rate of hard ductile metals during power hacksaw and bandsaw oper-ations,Metals Technology(1974)437.[10]H.Thoors,H.Chandrasekaran,Patch welding and SKIM—two methods for wear testing of band saws,Report1M–2931at Swedish Inst.for Metals Research,August1992.[11]H.K.Tiinshoff,R.Westphal,Noise reduction in circular sawing of metals,Annals of the CIRP27(1978)339.[12]A.G.Ulsoy,C.D.Mote,R.Szymani,Principal development in bandsaw vibration and stability research,Holz alsRoh-und Werkstoff36(1978)273.。