@prefix vivo: .
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@prefix ns0: .
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@prefix dc: .
@prefix skos: .
vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mechanical Engineering, Department of"@en ;
edm:dataProvider "DSpace"@en ;
ns0:degreeCampus "UBCV"@en ;
dcterms:creator "Wong, Darrell C."@en ;
dcterms:issued "2009-03-16T20:03:45Z"@en, "1996"@en ;
vivo:relatedDegree "Master of Applied Science - MASc"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description """This theoretical and experimental study examines the tracking behavior and stability of
bandsaw blades. Tracking describes the in-plane "front-to-back" motion of a bandsaw as it
runs on the bandmill wheels. Bandsaw tracking stability returns the sawblade to its initial
position after any in-plane side-to-side displacement caused by a cutting force. The tracking
behavior and stability of the sawblade are determined by the geometry of the saw and bandmill
wheels. Fourteen factors affect this geometry. They are the cutting force, wheel profile, tilt,
cross line and coefficient of friction, and the saw backcrown, overhang, strain, tensioning,
thickness, width, guides, rotational speed and temperature distribution. In practice, many of
these factors are present at the same time.
A theoretical model is presented here that accurately and reliably predicts the behavior and
stability of the band. This model incorporates and quantifies the tracking stability of cutting
force, wheel profile and tilt, and band overhang, strain, thickness and width. Both the
experimental and theoretical results show that wheel crown and overhang are the only true
stability factors and that the other factors such as width and thickness only modify the stability
of crown and overhang. It was also found that the detail of the wheel profile has a substantial
effect on tracking stability and when wheel crown is combined with overhang, the tracking
stability of overhang is improved. The model can now be applied to investigate different
wheel crown profiles. It provides an important tool for the task of improving bandsaw cutting
performance while at the same time reducing the required saw and bandmill maintenance."""@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/6068?expand=metadata"@en ;
dcterms:extent "4173159 bytes"@en ;
dc:format "application/pdf"@en ;
skos:note "FACTORS AFFECTING BANDSAW TRACKING BEHAVIOR AND STABILITY by DARRELL C. WONG B.A.Sc, The University of British Columbia, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR THE D E G R E E OF MASTER OF APPLIED SCIENCE in T H E F A C U L T Y OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1996 © Darrell Wong, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholariy purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of i The University of British Columbia Vancouver, Canada Date (3cro.fteA~ IS 1% DE-6 (2/88) Abstract This theoretical and experimental study examines the tracking behavior and stability of bandsaw blades. Tracking describes the in-plane \"front-to-back\" motion of a bandsaw as it runs on the bandmill wheels. Bandsaw tracking stability returns the sawblade to its initial position after any in-plane side-to-side displacement caused by a cutting force. The tracking behavior and stability of the sawblade are determined by the geometry of the saw and bandmill wheels. Fourteen factors affect this geometry. They are the cutting force, wheel profile, tilt, cross line and coefficient of friction, and the saw backcrown, overhang, strain, tensioning, thickness, width, guides, rotational speed and temperature distribution. In practice, many of these factors are present at the same time. A theoretical model is presented here that accurately and reliably predicts the behavior and stability of the band. This model incorporates and quantifies the tracking stability of cutting force, wheel profile and tilt, and band overhang, strain, thickness and width. Both the experimental and theoretical results show that wheel crown and overhang are the only true stability factors and that the other factors such as width and thickness only modify the stability of crown and overhang. It was also found that the detail of the wheel profile has a substantial effect on tracking stability and when wheel crown is combined with overhang, the tracking stability of overhang is improved. The model can now be applied to investigate different wheel crown profiles. It provides an important tool for the task of improving bandsaw cutting performance while at the same time reducing the required saw and bandmill maintenance. ii TABLE OF CONTENTS Abstract u Table of Contents i» List of Figures v Acknowledgment Nomenclature ix 1.0 Introduction 1 1.1 Background 1 1.2 Previous Work 4 1.3 Objectives and Scope 8 2.0 Bandsaw Tracking Mechanism 10 2.1 Tracking Movement 10 2.2 Transient and Steady State Tracking 17 3.0 Bandsaw Tracking Factors and Stability 23 3.1 Factors Affecting Tracking 23 3.2 Bandsaw Tracking Stability 32 4.0 Theoretical Model 36 4.1 Assumptions 37 4.2 Wheel Taper and Band Strain, Thickness and Width 38 4.3 Bandmill Wheel Tilt 48 4.4 Bandmill Wheel Profile 50 4.5 Bandsaw Overhang 51 4.6 Cutting Force 57 5.0 Verifying the Theoretical Model 60 5.1 Equipment 60 5.2 Equipment Set-up and Accuracy 62 5.3 Wheel Taper and Saw Strain, Thickness and Width 65 5.4 Bandmill Wheel Tilt 71 5.5 Bandmill Wheel Crown and In-feed \"Cutting\" Force 73 5.6 Saw Overhang 85 iii 6.0 Conclusions 91 7.0 Work for Future Projects 95 References 97 iv List of Figures Figure 1.1 Bandsaw Machine and Sawblade 2 Figure 2.1 Flat Wheel Tracking 11 Figure 2.2 Tapered Wheel Tracking 12 Figure 2.3 Tapered Wheel Band Shape 13 Figure 2.4 Figure-eight Pattern 14 Figure 2.5 Four Sections of Band 15 Figure 2.6 Transient Tracking Around Entire Band 16 Figure 2.7 Straight Band on Two Flat Wheels 17 Figure 2.8 Cutting Force on Straight Band 18 Figure 2.9 Geometry After Rotation by Ax 19 Figure 2.10 Geometry After Second Rotation by Ax 19 Figure 2.11 Flow Chart of Transient Tracking Process at the Entry Point 20 Figure 2.12 Unwrapped Tapered Wheel 22 Figure 3.1 Crowned Wheel 25 Figure 3.2 Double Tapered Wheel 25 Figure 3.3 Tracking Effect of Wheel Tilt 26 Figure 3.4 Tracking Effect of Overhang 27 Figure 3.5 Band Overhanging Both Sides of the Wheel 28 Figure 3.6 Tracking Effect of Strain 29 Figure 3.7 Curvature Due to Tensioning 31 Figure 3.8 Effect of Tensioning on Overhang 32 Figure 3.9 Stability of Wheel Crown and Overhang 34 Figure 3.10 Wheel Crown & Overhang Balancing Cutting Force 34 Figure 3.11 Effect of Wheel Tilt on Overhang 35 Figure 4.1 Basic Band Model 39 Figure 4.2 Straight Tapered Wheel 42 Figure 4.3 Boundary Conditions Due to Wheel Tilt 49 Figure 4.4 Cylindrical Shell Model of Overhang 52 Figure 4.5 Uniform Pressure on Band 52 Figure 4.6 Overhang of a Flat Cylindrical Wheel 55 Figure 4.7 Cutting Force Model 57 Figure 5.1 Table Top Model 61 Figure 5.2 Wheel and Band Dimensions 62 Figure 5.3 Tapered Wheel Experimental Set-up 66 Figure 5.4 Tapered Wheel Tracking 67 Figure 5.5 Effect of Band Width on Tapered Wheel Entry Angle 69 Figure 5.6 Effect of Band Strain on Tapered Wheel Entry Angle 70 Figure 5.7 Effect of Wheel Tilt on Tapered Wheel Entry Angle 72 Figure 5.8 Crown Profiles f2, f3 and f4 73 Figure 5.9 Wheel Crown & In-feed \"Cutting\" Force Equipment Set-up 75 Figure 5.10 Crowned Wheel Transient Tracking 76 Figure 5.11 Effect of Feed Force on Band Distance from Symmetric Crown 78 Center vi Figure 5.12 Effect of Width on Band Distance from Crowned Wheel Center 79 Figure 5.13 Effect of Band Width on the Stable Zero Force Tracking Position 80 Figure 5.14 Stable Tracking Position on an Asymmetric Crown 82 Figure 5.15 Effect of Feed Force on Band Distance from Asymmetric Crown 83 Center Figure 5.16 Band Buckling from the Feed Force 84 Figure 5.17 Overhang Experimental Set-up 85 Figure 5.18 Effect of Wheel Tilt on Overhang 87 Figure 5.19 Combined Tracking Stability of Overhang and Wheel Crown 89 against Feed Force Figure 5.20 Tracking Stability of Wheel Crown against Feed Force from 89 fig. 5.11 vii Acknowledgment I would like to express my gratitude to those individuals and organizations who have supported this work. A special thank you to everyone in my family who enthusiastically supported my return to university. My wife Shelley for her love and patience, my mother Jane for her incredible dedication to raising my brothers and I, and my mother and father-in-law Elizabeth and Ted for their love and caring. My work on this project was made possible by the financial support of the BC Science Council, Natural Science and Engineering Research Council of Canada (NSERC), Forintek Canada Corporation and MacMillan Bloedel Ltd. In particular, I would like to thank John Taylor and Jan Aune for supporting this work from the beginning. Finally, I would like to thank my faculty sponsor, Gary Schajer, who was the key to my \"tracking\" back to university. Thank you for your encouragement, support and most of all, for your confidence in me. viii Nomenclature a = crown height adjustment factor Bi , B 2 , B 3 , B 4 = beam equation integration constants b = band width Ci, C 2 , C 3 , C 4 = cylindrical shell equation integration constants D, Di , D 2 = wheel diameters d = overhang distance dw / dy = overhang slope dy / dx = band slope d2y / dx2 = band curvature E = band material elastic modulus F = lateral cutting force front-to-back = in-plane band motion in the workpiece feed direction f(y), f2(y), f3(y), f4(y) = wheel profile function H, J, k, r\\ = equation constants h = band thickness I = band second moment of area u = offset from crown center L = band length between wheels L T = total circumferential band length L 0 = original band section length before it's wrapped around wheel M , M(z) = bending moment M A = bending moment at wheel entry point A P = uniform pressure between band and wheel q = overhang coordinate R = curvature radius of unwrapped wheel R A = band reaction force at wheel entry point A r = wheel radial coordinate with the datum at wheel center r 0 = mean radius of profiled wheel s = wheel width T = band axial tensile force / span (strain force) V(d) = shear force at wheel edge due to overhang V L = band longitudinal velocity V T = band transverse velocity across wheel w> w (y) = curling of overhang x = coordinate along length of band from wheel entry point A Ax = incremental longitudinal movement or rotation of the band y = coordinate perpendicular to band y(x) = band position as a function of x y G = band position at wheel exit point G z = wheel profile radial coordinate with datum at band centefline Y, Y i , Y2 = wheel taper angle P = wheel tilt angle x , <|)A = band entry angle, top wheel band entry angle <|>G = bottom wheel band exit angle (j)S = Swift's steady state band entry angle v = band material Poisson's ratio do = base uniform stress in band from wheel profile a x = longitudinal stress in band xi 1.0 Introduction 1.1 Background Global competition demands that sawmill managers and maintenance personnel continue to pursue ways to improve lumber production quality and reduce operating costs. Each step in the overall log breakdown process is being examined to contribute to this goal [1,2]. Bandsaws are major machines in many mills, and they process large volumes of wood. Therefore, they are good targets for production quality improvements and operating cost reductions. A bandsaw machine or bandmill consists of a continuous band traveling around two wheels as shown in figure 1.1. The preparation of the bandsaw can be a time consuming process which includes tensioning, leveling and sharpening of the sawblade [3]. The maintenance and set-up of the bandmill includes machining the surfaces of the wheels to a flat or crowned profile, setting the \"strain\" force to pull the sawblade firmly onto the wheels, and adjusting the position of the sawblade by setting the \"tilt\" angle between the axes of the top and bottom wheels. The maintenance and set-up of the bandmill and bandsaw are performed with the goal of achieving good cutting performance at a high production rate while consuming a minimum amount of maintenance time. The dilemma is that typically an improvement in one operational area can only be achieved at the cost of another [4]. For example, good cutting performance 1 can be achieved through careful saw tensioning and lower production rates. Less maintenance time spent on tensioning and also higher production rates typically reduce cutting performance. F r o n t - t o - b a c k M o v e m e n t Figure 1.1: Bandsaw Machine and Sawblade A n understanding of bandsaw tracking may contribute to a solution to this dilemma. The term \"tracking\" describes the in-plane \"front-to-back\" movement of the saw as it runs between the wheels o f a bandmill [5,6]. During sawing, the cutting forces create an in-plane bending moment and curvature along the length of the sawblade. As the sawblade rotates around the wheels the curvature creates an angle between the saw and bandmill wheel at the point where 2 the saw first contacts and moves onto the wheel. This angle is called the \"entry angle\" (J). The entry angle is a key quantity that describes tracking. It defines the ratio of the in-plane sideways speed of the saw relative to its longitudinal speed. When the entry angle is non-zero the sawblade is not perpendicular to the wheel axis. As a result, the sawblade moves sideways in a screw-like motion, with the entry angle corresponding to the thread angle of the screw. This sideways motion is tracking. When the entry angle is zero the saw is perpendicular to the wheel axis and as a result, it runs in a stable tracking position without any sideways motion. The ability of a moving bandsaw to maintain its tracking position against the cutting forces that tend to push the saw off the wheels is referred to as tracking stability [6,7,8]. The sawblade relies on subtle details such as the bandmill wheel profile and saw overhang to generate a bending moment to counteract the moment created by the cutting forces. The balance between these moments determines the tracking displacement of the saw and it keeps the saw on the wheels without any direct mechanical constraint. A sawblade with a high tracking stability moves a small distance in response to a cutting force while one with a low stability moves a larger distance. Many other factors also contribute to the balancing moment including the saw tensioning, thickness and width and the bandmill strain and the wheel tilt angle. It is generally believed that front-to-back movement of the sawblade impairs cutting accuracy and increases the wear on the bandmill wheels and sawblade. Therefore, a saw with a higher tracking stability that responds with minimum movement to the cutting force will improve 3 cutting accuracy without a penalty in production rate. An understanding of bandsaw tracking may also provide a means of reducing sawblade maintenance time by reducing the amount of tensioning required in sawblades. Previous tracking studies have focused mainly on the stable tracking position of bandsaws. This is \"steady state\" tracking. In practice, this condition occurs when a bandsaw is idling between saw cuts. However, when a bandsaw is cutting, its tracking position is continually changing. This is \"transient\" tracking. Therefore, to maximize cutting accuracy and minimize equipment maintenance time, bandsaw transient tracking must also be examined. This study will extend the previous steady-state tracking analysis to include the general transient case. 1.2 Previous Work Swift's [8] work in the 1930's on the steady state tracking of belts around pulleys was among the first systematic studies to be published on tracking. His theory modeled the portion of the belt between the wheels as a beam in bending with an applied axial tensile load. The bending moment was calculated by assuming the stress distribution across the sawblade matched the profile of the pulley. The steady state entry angle was determined from the end slope of the beam. Swift's theory includes the effects of the wheel taper and circular crown and the belt width and strain. Wheel tilt and cross line were also included in Swift's theory as adjustments to the beam boundary conditions. 4 Schajer [9] also did work on tracking when he developed a geometrical model of how a guided circular saws hunts on a rotating shaft. The mechanism of saw hunting is the same as that for belt tracking. Schajer described in detail the movement of the saw as it enters onto and moves around the shaft. He explained that at the instant when the saw enters onto the shaft, it appears that the saw and shaft will follow different paths but since they are in contact, the saw must follow the rotation of the shaft. The effect of wheel crown profde on the tracking stability of belts was experimentally examined by Renner [10]. He showed that a pulley with a flat center section and steeply curved edges has a higher tracking stability than a pulley with a center peak that gradually slopes towards the edges. In addition, he also showed that a band mounted on the pulley with the flat center section has a more uniform stress distribution across its width. Sugihara [6] published some of the first work on the tracking of bandsaw blades. He theorized that the bending moment in the sawblade determined its tracking behavior. His work focused on the bending moments created by the cutting force, saw strain, saw width, wheel tilt and overhang. Like Swift, Sugihara modeled the band between the wheels as a beam under an axial tensile load. The cutting force was included as a concentrated force at the center of the beam. Saw overhang of a flat wheel was modeled as a cylindrical shell with a uniform line force and moment on one edge and the other edge free. The line force and moment were determined from the assumption that the bandmill wheel behaves as a beam on 5 an elastic foundation with the strain applied as a uniform pressure across the wheel. They are calculated by balancing the force and moment acting on the wheel edge. Taylor [11] extended the work done by Sugihara. He developed theoretical models for the bending moments created by backcrown, tensioning and wheel crown and extended Sugihara's work on the bending moments generated by wheel tilt and overhang. Taylor's cylindrical shell theory of saw overhang modeled the flat wheel as a rigid body and included the effects of saw tensioning and anticlastic curvature. Chardin [12] began the first of many experimental studies of bandsaw tracking that included coarse measurements of transient tracking but focused mainly on six factors that affect an industrial saw's overhang. It was found in this study that flat wheels were more stable than crowned wheels. This comparison will also be performed here in section 5. It was also found that tensioning and the coefficient of friction between the sawblade and wheel had little effect on the overhang. Wheel tilt caused the saw to track from the higher to the lower portion of the wheel axis. When an out of plane (lateral) load was applied to one edge of the saw or the edge was heated, the saw tracked in-plane from the loaded or heated edge towards the opposite edge. An in-plane load applied to the edge of the saw in the feed direction caused the saw to track in the same direction. Chardin also noted that the transient tracking caused by an in-plane load or feed force followed an inverse exponential relationship with the longitudinal movement of the saw. 6 Chardin [7] followed up the industrial studies with laboratory experiments on a metal band or a saw with no teeth. The previous experiments on the industrial bandmill were repeated reaffirming most of the initial findings except it was found that highly tensioned saws were more stable than untensioned saws. The effects of strain, thickness, width and wheel size on the saw's overhang were also examined. It was found that strain increased the stability of the band while the thickness and width had little effect. The tracking stability of a double cut bandmill with 8' wheels was found to be higher than a single cut resaw with 5' wheels. Chardin and Sales [13] then performed a laboratory study confirming the results from Chardin's earlier laboratory experiments as well as examining the combined effects of wheel size, tensioning and strain on band tracking stability. It was found that typically a minimum amount of tensioning was required for maximum tracking stability but that tensioning beyond this amount provided no additional benefit. Sales, Guitard, Fournier and Garin [14] also performed experiments on a industrial bandmill and a sawblade with teeth. The effect of edge heating, backcrown and the combined effects of strain, tensioning and flat / crowned wheels on tracking stability were examined. As found earlier by Chardin, heating one edge of the saw caused it to track in the direction of the opposite edge and strain increased the stability of the saw. It was also found that typically tensioned saws were more stable on flat wheels except for highly tensioned and untensioned saws which were more stable on crowned wheels. When the tracking movement was large, 7 crowned wheels were found to be better than flat wheels at preventing the sawblade from coming off the wheels. Rivat, Sales, and Martin [15] also did laboratory experiments using more accurate equipment to examine the effect of wheel tilt, strain and longitudinal velocity on the band's overhang. Confirming Chardin's findings, it was found that wheel tilt caused the band to track from the higher down towards the lower portion of the wheel axis and that strain reduced the overhang. It was also found that the longitudinal blade velocity decreased the overhang. 1.3 Objectives and Scope The objective of this research study is to develop a comprehensive theoretical model of bandsaw tracking including steady state and the more general transient tracking. The four steps required to achieve this objective are: 1. Apply the tracking mechanism identified by Swift [8] and Schajer [9] to bandsaw blades. 2. Isolate the bandsaw tracking factors identified by Chardin, Fournier, Garin, Guitard, Rivat, Sales, Sugihara and Swift [6,7,8,12,13,14,15]. Determine the tracking effect of these factors and place them into a logical framework. 3. Develop a theoretical tracking model that incorporates both the steady state and the transient tracking behavior of the sawblade. Swift's steady state tracking model of wheel taper, band thickness, width and strain will be developed into a basic model for transient tracking. Next, Swift's wheel tilt theory will be incorporated. Swift's and Taylor's [8,11] 8 wheel crown bending theory will be incorporated after it is extended to include more general shapes. Sugihara's and Taylor's cylindrical shell model of the sawblade overhang will also be developed for transient tracking. Finally, the initial beam theory will be extended to include Sugihara's model of the cutting force. 4. Test and experimentally verify the accuracy of the theoretical model on a table top bandmill model and metal band. Detailed measurements will be made of the band's in-plane side-to-side tracking position relative to its longitudinal movement. Beyond these objectives, the theoretical model will be utilized to determine which tracking factors have the most significant effect on bandsaw tracking stability. Machine and sawblade set-up factors that determine cutting accuracy and equipment maintenance requirements will be given special attention. A particular area of interest is the possibility of improving the bandmill wheel profile to increase tracking stability and reduce the amount of saw tensioning needed. This would allow the saw strain and saw stiffness to be increased and may also reduce the frequency of fatigue cracks in the sawblade. This change has the attractive potential of improving cutting accuracy at the same time as reducing bandmill wheel wear and sawblade maintenance requirements. 9 2.0 Bandsaw Tracking Mechanism The tracking or in-plane \"front-to-back\" movement of a bandsaw is determined by the geometries of the wheels and sawblade at the point where the saw first contacts (\"enters\") the wheel [16,17]. This tracking has two phases, transient and steady state. Transient tracking is the more general case during which both the side-to-side position and tracking speed of the band are continuously changing. After a few band rotations, the sideways tracking speed typically converges to a steady state value. This is called steady state tracking. 2.1 Tracking Movement Bandsaw tracking movement occurs continuously and simultaneously around the entire saw as it rotates on the wheels. The direction and rate of movement of the saw is determined by saw's angle at the point where it first enters onto the wheel. Mechanism of Tracking Movement To illustrate the mechanism of bandsaw tracking movement, consider the straight band running on the flat (cylindrical) wheel shown in figure 2.1. In this simple case, the longitudinal force on the band, commonly called \"strain\", is evenly distributed across the width of the band. As a result, there is no bending or curvature, and the band remains straight and perpendicular to the wheel axis. Consider the instant when a point on the band first contacts the wheel at A. After contact, there is assumed to be no slip between the band and wheel. 10 Therefore, the band is constrained to follow the rotational path of the wheel. Here, the path of the band and the path of the wheel coincide because they are both destined for point B. Therefore, after a quarter or more turns of the wheel, the side-to-side position of the band on the wheel remains unchanged. Band Posit ion & Rotational Path Band Wheel Axis S t r a i n Figure 2.1: Flat Wheel Tracking Now consider the initially straight band tracking on a tapered (conical) wheel shown in figure 2.2. The tapered wheel applies the strain unevenly to the band, resulting in a bending moment and a slight curve in the band [8]. The band axis is no longer perpendicular to the wheel axis. Consider the instant when a point on the band first contacts the wheel at A. At this instant, the path of the band appears destined for point B. However, point A on the wheel follows a path perpendicular to the rotational axis and is destined for point C. Since there is no slip between the band and the wheel, the band moves up the taper following the rotation of the 11 wheel [9]. Therefore, after a quarter turn the band has shifted sideways towards the wider side of the taper, as shown in figure 2.2. [nitial Position 0 Ro ta t iona l Pa th Position After 1 / 4 Turn Band Axis S t r a i n S t r a i n Figure 2.2: Tapered Wheel Tracking Four basic concepts are important when considering band tracking movement. They are: 1. A bending moment creates a curvature along the length of the band. 2. Tracking occurs when the band rotational path is not perpendicular to the wheel axis at the point where the band first contacts (\"enters\") the wheel. The resulting perpendicular angle between the band axis and the wheel axis is called the \"entry angle\" <)). Figures 2.2 and 2.3 illustrate this angle. 12 3. The band tracking speed V T depends proportionally on the entry angle § and the longitudinal speed of the band V L V T = <|>VL (2.1) For a fixed longitudinal speed, the tracking speed is proportional to the entry angle. 4. There is assumed to be no slip between the wheel and the band. Therefore, the side-to-side position and slope of the band at the wheel entry are maintained or stored by the wheel as it rotates. These appear at the wheel exit after a half turn. Thus, the band position and slope around the wheel contain the history of the band movement during the previous half turn. As the wheel rotates, points on the band move to the wheel exit in succession as new points are added at the wheel entry. Figure 2.2 shows the position, shape and tracking of only a portion of a band. The position and shape of the remainder of the band can be developed by continuing the rotation of the \\ \\ \\ Figure 2.3: Tapered Wheel Band Shape 13 band shown in figure 2.2. The result is the band shown in figure 2.3. Point A is at the entry and point D is at the exit (at the far side of the wheel). The horizontal distance between points A and D corresponds to the distance tracked by the band during the previous half turn. The slope of the band at D (the exit angle) is the same as the slope at A (the entry angle). They appear to have opposite signs because opposite sides of the band are shown at points A and D. Next consider a band running on two tapered wheels. The curved shape shown in figure 2.3 is repeated on both wheels. Since the band is continuous, the result is the figure-eight pattern shown in figure 2.4. The tracking of this band on two wheels follows the same four basic concepts presented earlier. Figure 2.4: Figure-eight Pattern 14 Simultaneous Tracking Movement Around Two Wheels To illustrate the simultaneous tracking movement of the band around its entire circumference, again consider the band shown in figure 2.4. The band can be divided into four sections. Two spans which are on the wheels and two free spans which are between the wheels as shown in figure 2.5. As described previously, each wheel stores the positions and slopes of the band section in contact with it, and progressively moves them to the exit point as the wheel turns. In contrast, the free spans of the band between the wheels only transmit the position and slope of the band from the exit point of one wheel to the entry point of the opposite wheel. This is done by the in-plane bending stiffness of the two free spans of the band. Wheels Figure 2.5: Four Sections of Band 15 Consider the band mounted on two tapered wheels shown in figure 2.6. The arrows numbered 1 to 10 mark the positions and slopes of the band that are in contact with the wheels. As the band and wheels rotate, the position and slope at the wheel entry points, A and E , evolve as shown in figure 2.2. Simultaneously, the stored band positions and slopes on the wheels move towards the exit points as shown in figure 2.6. The values at 1 move to 2, the values at 2 move to 3 and so on. Also at the same time, points 4 and 9 move to the wheel exit points , D and G, and their effects are transmitted by the band between the wheels to the entry points on the opposite wheels. These then affect the entry position and slope of the band during the next rotation. Figure 2.6: Transient Tracking Around Entire Band 16 2 . 2 Transient and Steady State Tracking Transient and steady state tracking describe different phases of the tracking process. During transient tracking the entry angle and tracking speed of the band vary with time. However, after a number of band rotations the entry angle and tracking speed converge asymptotically to steady state values. This is steady state tracking. Later it will be shown that on a crowned wheel the steady state entry angle and tracking speed are zero. As a result, the band maintains a stable tracking position on the wheel. On a tapered wheel, the steady state entry angle and tracking speed are non-zero. As a result, the band tracks across the wheels at a constant non-zero speed. To illustrate the mechanism of transient tracking, consider the straight band running on two flat wheels shown in figure 2.7. It was shown in figure 2.1 that in this simple case the band Figure 2.7: Straight Band on Two Flat Wheels 17 maintains a stable tracking position on the wheels as it rotates. At the instant when the band begins cutting, a feed force pushes on the center of the band between the wheels as shown in figure 2.8. Since there is no slip, the bending moment and curvature in the band created by the feed force is limited to the portion of the band between the wheels. There is no immediate effect beyond points A and G in figure 2.8. As the band rotates a small increment Ax, the band curvature changes the entry angle as shown in figure 2.9. Again due to the no slip assumption, the band slope changes only occur over the length Ax at the entry of the top wheel. At the bottom wheel exit point G, the position and slope of the band previously Ax upstream move to the exit point as explained in section 2.1. During the next rotation Ax, the non-zero entry angle creates tracking in the direction of the feed force and again the band curvature changes the entry angle as shown in figure 2.10. Again, the side-to-side position Figure 2.8: Cutting Force on Straight Band 18 Figure 2.10: Geometry After Second Rotation by Ax 19 and slope of the band at the exit of the bottom wheel become those values previously Ax upstream. As a result, during transient tracking the evolution of the entry angle due to the band curvature causes the tracking speed and position to change continually. The sequence of band tracking shown in figures 2.6, 2.7, 2.8, 2.9 and 2.10 shows that there is a delay of half a turn of the top wheel and half a turn of the bottom wheel before the feed force effect travels all the way around the band and further influences the entry angle. Figure 2.11 shows a summary of the transient tracking process. Exit Angle Exit Position after a Delay after a Delay Bending l Moment changes ^ creates Curvature changes Entry Angle I changes Entry Position Figure 2.11: Flow Chart of Transient Tracking Process at the Entry Point 20 After a number of complete band rotations the entry angle and tracking speed asymptotically converge to steady state values. This is steady state tracking. It appears from figure 2.11 that steady state tracking requires the bending moment and curvature in the band to be zero. This is true when a band is tracking on flat wheels. However, consider the band which is steady state tracking on tapered wheels shown in Figure 2.12. An examination of the band at the entry point A shows that both the curvature and entry angle are non-zero. However, unwrapping the surface of the tapered wheel, as shown in figure 2.12, shows that the curvature of the wheel matches the curvature of the band at the entry point A. Therefore, steady state tracking occurs when the bending moment in the band creates a curvature which matches that of the wheel. As a result, the entry angle and tracking speed do not vary with time. Now reconsider the band tracking on flat wheels shown in figure 2.7. The flat wheel has zero curvature when unwrapped. As a result, steady state tracking occurs on flat wheels when the band curvature at the entry point is zero. 21 Figure 2.12: Unwrapped Tapered Wheel 22 3.0 Bandsaw Tracking Factors and Stability Bandsaw tracking is the mechanism by which a saw can move sideways on the bandmill wheels during machine operation. Stable tracking occurs when the band has the tendency to return to an equilibrium position after some disturbance to its sideways motion. In practice, the most common disturbance is an in-feed (in-plane) cutting force. In section 2.3, it was shown that a in-feed cutting force creates tracking in the direction of the force. The ability of the sawblade to resist this tracking movement is a measure of its tracking stability [12,17]. A saw's tracking stability is determined by its geometry and the geometry of the bandmill wheel. In this chapter the factors that affect tracking will be examined and placed into a logical framework. The effect of these factors on the tracking stability of the bandsaw will then be examined. 3.1 Factors Affecting Tracking The characteristics of the bandsaw blade and bandmill wheel that affect the bending moment, curvature and entry angle of the sawblade directly determine its tracking behavior. Fourteen characteristics or factors which affect tracking have been identified [6,7,8,10,12,13,14,15]. They are: 1. Wheel Profile 2. Wheel Tilt 3. Band Overhang 4. Band Strain 23 5. Band Width 6. Band Thickness 7. Band Tension 8. Cutting Force 9. Band Backcrown 10. Band Guides 11. Wheel Cross Line 12. Wheel and Band Coefficient of Friction 13. Band Rotational Speed 14. Band Temperature Distribution Factors one to seven are part of a bandmill's and handsaw's regular maintenance and set-up. Therefore, these factors will be the focus of the discussion and theoretical analysis in the next sections. The eighth factor, cutting force, will also be included in the analysis because it provides a measure of a saw's tracking stability. For simplification, only the cutting force component parallel to workpiece feed direction will be examined. Although the ninth and tenth factors, band backcrown and saw guides, are also maintenance and set-up factors, for simplification they will be excluded here. Simplifying assumptions will also be made for the final four tracking factors in the development of the theoretical model in the next section. Wheel Profile The tracking effect of wheel taper was illustrated earlier in figure 2.2. The wheel taper generates uneven stresses in the band, which create a bending moment, curvature and entry angle in the band. As a result, the band tracks from the small diameter up the taper to the large diameter [8]. 24 Figure 3.1: Crowned Wheel Figure 3.2: Double Tapered Wheel In practice, bandmill wheels are rarely tapered. Typically the wheel profde consists of a smooth curved (crowned) shape similar to the one shown in figure 3.1. The tracking behavior of a band on a crowned wheel can be understood by considering the composite wheel shown in figure 3.2, consisting of two opposite tapers joined at the center. These two tapers produce tracking in opposite directions. Consider the band tracking off-center on the double tapered wheel, as shown in figure 3.2. At any time, the band can track in only one direction. Since the larger proportion of the band is running on the left taper, this side has more influence on the bending moment, curvature and entry angle. The band therefore tracks towards the wider side of the left taper, that is, towards the center of the wheel. As the band approaches the center of the wheel, the influences of the left and right tapers begin to equalize. When the band is centered on the wheel the tracking effect of the two tapers cancel and the band reaches an equilibrium position. Similarly, a band initially running more on the right taper moves 25 towards the wider side of the right taper, that is, towards the center of the wheel. Therefore, a band running on a crowned wheel has a natural tendency to remain at the top of the crown. This is referred to as \"self centering\". The bending moments and curvatures which create the self-centering effect increase with the distance that the band is displaced from the center of the crown. As a result, the self centering effect also increases with the distance between the band and the crown center. Wheel Tilt Wheel tilt creates an uneven stress distribution, bending moment and curvature across the band as shown in figure 3.3 [6]. The slope of the band at the entry to the top wheel creates the impression that the band will track towards the left. However, a closer examination of the Figure 3.3: Tracking Effect of Wheel Tilt 26 band at the wheel entry shows that the angle of the band axis is less than the angle of wheel tilt [11]. Since the entry angle is measured relative to the wheel axis, its orientation is similar to that shown for the tapered wheel in figure 2.2. As a result, the band will track towards the right or from the higher portion down towards the lower portion of the wheel axis Band Overhang Figure 3.4 shows a band overhanging a flat wheel. The portion of the band overhanging the wheel curls down towards the wheel as shown in figure 3.5 [6,11,18]. The curling action simulates the effect of a wheel taper, creating an uneven stress distribution, a bending moment and a curvature. As a result, the band tracks back onto the wheel. The curling action and the curvature of the band increases with overhang. As a result, the tracking effect also increases with overhang. [6,7,8,12,13]. Saw Tooth Wheel Imag ina ry Tapered Wheel Figure 3.4: Tracking Effect of Overhang 27 Figure 3.5 shows a band that is overhanging both sides of the wheel. The overhang on the left side of the wheel generates a tracking effect in the opposite direction from the overhang on the right. This behavior is the same as that of the double tapered wheel shown in figure 3.1. When the overhang on the left side of the wheel is larger than that on the right, the left overhang has a larger influence on the band's bending moment, curvature and entry angle. The band, therefore, tracks to the right, that is, towards the center of the wheel. Similarly, when the overhang on the right is larger than that on the left, the band tracks towards the left, that is, towards the center of the wheel. When the band is centered on the wheel and the left and right overhangs are equal, the tracking effects cancel and the band reaches an equilibrium position. Therefore, like wheel crown, overhang has a self-centering effect that increases with the distance between the band and the wheel center. L J Figure 3.5: Band Overhanging Both Sides of the Wheel 28 Cutting Forces The tracking effect of the in-feed cutting force was shown earlier in figures 2.8, 2.9 and 2.10. The cutting force creates a bending moment and curvature in the band. As a result, a entry angle develops between the axis of the band and the axis of the wheel and the band tracks in the direction of the cutting force. Band Strain Band strain by itself does not cause tracking. However, it does modify the tracking behavior caused by other factors. In the case of taper and cutting force, strain acts to straighten the band. This reduces the entry angle and correspondingly reduces the tracking speed. However, in the case of tilt, the straightening increases the entry angle and increases tracking speed. This is shown in figure 3.6. When tracking is caused by overhang, strain increases 0 0 Low Stra in High Strain Low Stra in High S tra in Figure 3.6: Tracking Effect of Strain 29 the downward curling of the overhang and as a result, also increases the bending moment that it generates. This increases the tracking effect of overhang [15]. Band Thickness and Band Width Band thickness and width are like band strain in that they also do not individually cause tracking. However, they do affect the tracking induced by other causes. Thicker or wider bands have a higher second moment of area and therefore a higher bending stiffness. As a result, when the tracking effect is due to wheel taper, the entry angle and tracking rate are increased. However, when the tracking effect is due to a cutting force, overhang or wheel tilt, the entry angle and tracking rate are reduced. Band Tensioning Tensioning is a routine process used to stretch the center of a bandsaw blade by hammering or rolling its surface [3]. This process changes the shape of the saw across its width when it is wrapped around the wheel. Band tension by itself also does not cause tracking. However, it does affect the way in which the band contacts the surface of a crowned wheel as well as the shape of the overhanging portion of the sawblade [11]. Figure 3.7 shows that tensioning can give a band a lateral profile that matches the shape of the wheel crown. This profile allows the edges of the band to contact the wheel, thereby maximizing the tracking behavior of the band. Kirbach [19] 30 found that flat wheels required the lowest saw tensioning and that the optimum amount of tensioning for crowned wheels increased with increasing crown height. Figure 3.7: Curvature Due to Tensioning When the sawblade overhangs the wheel, tensioning increases the downward curling of the overhanging edge. This is shown in figure 3.8. The steeper edge has the same effect as a steeper taper. Therefore, tensioning increases the tracking effect of overhang which pushes the sawblade back onto the wheel. 31 Saw Saw T o o t h T o o t h W h e e l W h e e l N o T e n s i o n i n g T e n s i o n i n g Figure 3.8: Effect of Tensioning on Overhang 3.2 Bandsaw Tracking Stability The tracking stability of a bandsaw can be observed during sawing. Typically, when a saw is idling between cuts, it maintains a stable side-to-side position on the wheels. When the saw begins cutting, the cutting force creates a bending moment that causes the saw to track in the direction of the cutting force, as described in section 2.2. As the saw tracks from its initial position, the tracking factors generate an opposing bending moment. When the moments balance, the saw maintains a new stable position on the wheels. When the cutting force is removed the saw returns to its original position. This process of moving and generating a bending moment in response to a cutting force and then returning to the original position after the force is removed is a measure of tracking stability. A saw with a high stability is characterized as one which moves very little in response to a cutting force. 32 Out of the fourteen tracking factors, only wheel profile, specifically wheel crown, and overhang are true stability factors. The self centering effects of wheel crown and overhang give them the ability to: 1. Maintain the band at a stable tracking position on the wheels between cuts. 2. Generate a tracking effect in the opposite direction of the cutting force that increases as the band moves from the wheel center. When the tracking effects equalize, the band maintains a new equilibrium position. 3. Return the band to its original position after the cutting force is removed. To illustrate the tracking stability of wheel crown and overhang consider the two straight bands shown in figure 3.9. Band A is running on crowned wheels and band B is running on flat wheels. The bands track at the center of the wheels because of the self centering effects of wheel crown on band A and overhang on band B. When the bands begin cutting, the feed forces create bending moments and curvatures in the bands. Through the process of transient tracking, bands A and B both track across the wheel in the direction of the cutting force. As bands A and B move off the centers of the wheels, the bending moments and curvatures created by the wheel crown and overhang begin to balance those created by the feed forces. Thus, bands A and B both slow down and stop at the location where the moments and curvatures balance as shown in figure 3.10. When the bands finish cutting and the feed forces are removed, the self centering effects of the wheel crown and overhang cause bands A and B to return to their original stable positions. 33 Crowned Wheel A (A) Crowned Wheel Flat Wheel Flat Wheel A 1 'I l l (B) I I I I I I I I Figure 3.9: Stability of Wheel Crown and Overhang Figure 3.10: Wheel Crown & Overhang Balancing Cutting Force The other factors such as wheel tilt, strain, width, thickness and tensioning only modify the tracking stability of wheel crown and overhang. Wheel tilt generates a constant non-zero bending moment, entry angle and tracking speed. Since this moment does not vary with tracking displacement, tilt by itself is unable to maintain the saw at a stable position on the wheels. However, when combined with wheel crown or overhang, the bending moment created by tilt does change the stable saw position on the wheels. For example, consider the band overhanging both sides of the wheel shown in figure 3.5. When the top wheel is tilted, the band tracks to the right down towards the lower portion of the axis, increasing the right overhang. When the tracking effect of the right overhang balances that of the wheel tilt, the band maintains a new stable position, as shown in figure 3.11. This becomes the new stable \"center\" position for the band. The remaining tracking factors, strain, width, thickness and tensioning only indirectly influence tracking stability because they are unable to generate bending moments in the band by themselves. These factors can not resist the effect of variable cutting forces if present alone. Tilt Tracking — ^ Effect r*=-= Overhang Tracking Effect Figure 3.11: Effect of Wheel Tilt on Overhang 35 4.0 Theoretical Model A theoretical model of bandsaw tracking is developed in this chapter. This model will help quantify how the tracking factors affect tracking stability. Since bandsaw tracking is affected by fourteen factors, it is possible to obtain stable tracking in many different ways. It should be possible to choose a combination of factors that produce superior tracking stability but at the same time requires less time to implement and maintain. For example, reducing the needed tensioning is a desirable goal because it is a time consuming maintenance process. Reduced tensioning has the added advantage that it makes leveling easier. The objective of the theoretical model developed in this chapter is to understand the mechanism of band tracking and to improve bandsaw tracking stability. This should allow improvements in cutting accuracy and a reduction in wheel wear and saw maintenance time. Development of the theoretical model requires five steps. They are: 1. Make simplifying assumptions for the set-up, geometry and mechanics of the sawblade. 2. Determined the initial position and slope of the saw. 3. Develop a model of the tracking effects of the basic tracking factors. The basic factors are the wheel taper and band strain, width and thickness. The model for these factors as well as the other factors will be based on their bending moment, curvature and slope effects on the bandsaw. 36 4. Model the tracking of the saw as it rotates on the wheels. This will be done by using the initial position, slope and curvature of the saw and a numerical integration method to update these values as the band rotates. 5. Extend the basic model to include the effects of wheel tilt, wheel crown, cutting force, and band overhang. 4.1 Assumptions The theoretical model will be developed with the following simplifying assumptions: • There is no cross line between the axes of the bandmill wheels. That is, the axes lie in the same plane. This is the typical set-up for a bandmill [2]. • There are no saw guides. • The saw has no backcrown. • The bandsaw teeth do not significantly affect tracking and may be omitted. As a result, the saw may be considered as a uniform band. • The coefficient of friction between the wheel and band is large enough to prevent slipping. • The band rotational speed is slow enough that the dynamic effects are negligible. • The temperature distribution across the band is uniform. • The tensioning in the band allows it to contact the wheel across its entire width. • The portion of the band between the wheels behaves as an Euler beam under axial tension [6,8], 37 • The portion of the band overhanging the wheels behaves like a cylindrical shell [6,11]. • The resultant of the cutting force acts in the workpiece feed direction. • The initial position and slope of the band around the wheels can be determined reliably. 4.2 Wheel Taper and Band Strain, Thickness and Width The simplest model of band tracking involves just four of the fourteen factors. They are wheel taper, band strain, thickness and width. This tracking model describes the side-to-side position, slope and curvature of the band at any instant in time. The model requires the initial position and slope of the band, a method of calculating the effect of the tracking factors and a method for updating the band position, slope and curvature as the band rotates and the operating conditions change. Initial Position The initial position and slope of the band can be determined by measuring the shape of the band on the wheels and developing an analytical equation for the position and slope of the band between the wheels. The portion of the band between the wheels can be modeled as a beam in bending under an axial tensile load [6,8] as shown in figure 4.1. The boundary conditions that determine the shape of this beam are the measured position and slope of the band at the wheel entry and exit points, A and G. 38 \"Exi t Point' Entry Point y @ x = 0 x = L Figure 4.1: Basic Band Model The bending moment at any point x along the length of the beam shown is figure 4.1 is d 2 y E I - 4 = Ty + R A x - M A dx (4.1) where T is the axial tensile \"strain\" force, RA is reaction force and M A is the moment at point A. The general solution for equation (4.1) is [20] y(x) =Bj sinhkx + B 2 coshkx + B 3 - B 4 x (4.2) where y is the side-to-side position of the band at a distance x from the entry point A and where 39 B, (4.3) B, (4.4) E I (4.5) The integration constants Bi, B 2 , B 3 and B 4 are calculated by applying the position and slope boundary conditions at the wheel entry and exit points. The results are: B, = -1 -k L sinh(k L)) cosh(kL) - 1 - *G + y ( k sinh(kL) cosh(kL) - 1 1 - cosh(kL) + sinh(kL) ( sinh(kL) - k L ) cosh(kL) - 1 (4.6) B , cosh (k L) ^ — — [Bj ( sinh (kL) - k L ) + <|>AL - Y o ] B 2 - - B , B 4 = (Bj k - (j>A ) (4.7) (4.8) (4.9) where $ is the slope and subscripts A and G refer to the entry and exit points of the band. Therefore, by measuring the position and slope of the band around the wheels and applying the wheel entry and exit boundary conditions to equation (4.2), the initial position of the band can be determined. 40 Modeling the Tracking Effects The tracking effects of band strain, thickness and width are already modeled in the beam equation (4.2) and as a result, only the tracking effect of wheel taper requires additional consideration. An analytical equation for the effect of wheel taper can be developed by examining the bending moment created in the band. The profile of a straight tapered wheel is defined by f(y) = y tan(y) (4.10) where y is the taper angle shown in figure 4.2 and r and y are the radial and axial coordinates of the wheel surface profile. The zero datum for y is at a convenient point on the surface of the wheel. This point can be at the center of the wheel as shown in figure 4.2 or the top of a wheel crown. At this point, r 0 is the radius of the wheel and f(0) = 0 (4.11) As a result, the radius, r, of the tapered wheel at any point, y, is given by r = r0 + f(y) (4.12) A band that is wrapped around a tapered wheel and is in complete contact with it develops a curve along its length with the same radius \"R\" as that of the unwrapped wheel. 41 Figure 4.2: Straight Tapered Wheel The length of the band in contact with the wheel is [8] contact length = % (r0 + f(y)) (4.13) If the original length of that section of the band, assuming no backcrown or tensioning was Lo, then the strain in the band as a result of being wrapped around the tapered wheel is 7c (r 0 +f (y ) ) -L 0 s(y) = — 1 — - — (4.14) Lo and the stress in the band is given by 42 a(y) - Ee(y) = E — ^ + E ^ '- (4.15) Since the strain is small, the original length can be approximated as L 0 = 7 t r 0 (4.16) Substituting equation (4.16) into the first term of equation (4.15), the stress profile becomes ( \\ ~ -c TCf(y) _,_ r 0 ~ L o) (A 1 T V a(y) = E — + E - (4.17) rcr0 L 0 Re-writing equation (4.17) a(y) = - f ( y ) + a 0 (4.18) where O o . E(^1±A (4.19) The size of the base stress o 0 depends mainly on the applied axial tensile \"strain\" force. Equation (4.18) shows that the stress profile across the width of the band matches the profile of the wheel when the band is in complete contact with the wheel across its entire width. The 43 bending moment generated by the wheel profile is calculated by integrating the stress profile given by equation (4.18) across the width \"b\" of the band about its centerline [8,11] 2 M = la(y) h y dy (4.20) where h is the thickness and the datum, y = 0, is at the centerline of the band. After substituting equation (4.18), equation (4.20) becomes M = — |f(y)ydy (4.21) The second constant term in equation (4.18) has no contribution to the moment M because of symmetry. 2 a 0 dy = 0 (4.22) Therefore, the tracking effect of wheel taper is determined by substituting the wheel taper equation (4.10) into equation (4.21). Later, the bending moment given by equation (4.21) will be used in the numerical integration model of tracking to update the entry angle as the band rotates. 44 Modeling the Simultaneous Tracking Movement The simultaneous tracking movement around the entire band is modeled as shown in figures 2.6 and 2.11. The measured position and slope of the band around the wheel and the calculated shape of the band between the wheels provides the initial position of the entire band. As shown in figure 2.6, when the band rotates, the position and slope the band on the wheels moves incrementally towards the wheel exit. At the same time, the values appearing at the wheel exit are transmitted to the entry point of the opposite wheel by the portion of the band between the wheels. This is modeled using equation (4.1). Therefore, the remaining step is to model the evolution of the band position and slope at the wheel entry point. This can be modeled and calculated with sufficient accuracy using the Euler numerical integration method. The basic steps of the Euler method are shown in figure 2.11. After the band rotates a distance Ax, the band's new position at the wheel entry is calculated from the current (\"old\") dy position and slope, y0ia and — ow . Since the slope is approximately dx dy ^ Ay _ y n e w - y o l d dx Ax Ax (4.23) the new position is calculated by dy ynew = y 0 i d + — o w Ax (4.24) dx 45 The new slope or entry angle of the band is calculated from the initial slope and the curvature of the band and wheel. In section 2.2, it was explained that the difference between the curvature of the band and the curvature wheel of the wheel, —, at the entry point acts to R change the entry angle as the band rotates. Therefore, since the difference between the curvature of the band and the wheel is approximately dx 2 band dy dy dxJ _ dx dy dx old Ax Ax (4.25) the new entry angle is calculated by dy dx' dy dx old + V y dx2 • band Ry Ax (4.26) The curvature of the band at the wheel entry point is calculated by differentiating equation (4.2) twice and substituting in the location of the entry point dx 2 band (0) = B 2 k2 (4.27) The bending moment generated in the band by the tapered wheel is calculated from equation (4.21). Finally, the beam curvature produced by this bending moment is 46 R 1 dx2 Wheel — E I M f(y)y dy (4.28) b 2 Applying this numerical integration method to the band at the entry points of both wheels, updating the position and slope of the band around the wheel and calculating the shape of the band between the wheels using equation (4.2), allows the tracking of the entire band to be calculated. In summary, the tracking process is calculated here using the following iteration scheme: 1. Determine the current position and slope of the band from the initial shape. 2. Calculate the curvature of the band at the wheel entry from equations (4.2) and (4.27). 3. The band is moved by a small amount Ax. The position and slope of the band on the wheels are moved towards the exit points. The position and slope of the band at the exits point take on the values previously existing at a point Ax upstream. 4. Calculate the new position and slope at the wheel entry from equations (4.24) and 5. The new band position and slope in equations (4.24) and (4.26) become the old values. 6. Calculate the band shape between the wheels from equation (4.2) using the old boundary conditions at the wheel entry and exit points. (4.26). 47 Repeated application of steps 2 to 6 models the simultaneous and continuous tracking of the band around its entire length. After a number of band rotations, the entry angle converges to a steady state value cj)s. This value coincides with the value determined by Swift [8] s = 2 L ( sinh(kL) 2 ) (2y{ 2y2 (4.29) L T k (cosh(kL)-l kLj [ DY D 2 J where L T is the total length of the band and subscripts 1 and 2 refer to the top and bottom wheels respectively. 4.3 Bandmill Wheel Tilt Wheel tilt can be added to the basic tracking model by examining its effect on the portion of the band between the wheels. Figure 2.4 shows the initial shape of a band tracking on two tapered wheels. Figure 4.3 shows the shape after the top wheel is tilted by an angle p. As explained in section 3.1, at the instant when the wheel is tilted, only the portion of the band between the wheels is affected. That is, the bending moment and curvature changes are isolated to the portion of the band between the wheels. Consequently, wheel tilt can be modeled as a change in the slope boundary condition for the portion of the band between the wheels [8]. The wheel tilt slope boundary conditions at the entry and exit points of the wheel are dy dx dy dx + P (4.30) 48 The change in the shape and curvature of the band between the wheels is calculated by applying the new boundary conditions to equations (4.2) and (4.27). As in the basic model, after a number of band rotations, the entry angle converges to a steady state value 4>s. This value again coincides with the value determined by Swift [8] **s 2 L sinh(kL) __2j [ 2 Y l 2 y 2 ] LfT 1 k ,cosh(kL)-1 D 2 J L T v. 2 ( sinh(kL) - kL)) k L ( cosh(kL) - 1 )J .(4.31) P- + /3 ,y @x = 0 , y @ x=L Figure 4.3: Boundary Conditions due to Wheel Tilt 49 4.4 Bandmill Wheel Profile The tracking model developed in the first three sections describes the tracking of a band on tapered wheels. On tapered wheels the bending moment and curvature generated in the band by the wheel profile are constant, as shown by equations (4.21) and (4.28). On crowned wheels the bending moment and curvature generated in the band changes with the distance between the band centerline and the crown center. To extended the tracking model to include more general wheel profiles such as wheel crown, consider the crowned wheel profile described by where \"a\" is a constant that adjusts the crown height. Substituting equation (4.32) into equation (4.18) gives the stress profile across the width of the band when it is tracking on the center of the crown. When the band is tracking at a distance, u, from the center of the crown, the bending moment and curvature in the band are calculated by f(y)= a y 2 (4-32) b u + -2 (4.33) b u - — l — (u) Wheel = M(u) E I (4-34) R dx2 50 Replacing equations (4.21) and (4.28) with equations (4.33) and (4.34) extends the tracking model to include wheel profiles where the bending moment and curvature generated by the wheel profile change as the band moves across the wheel. 4.5 Bandsaw Overhang In section 3.1 it was shown that overhang behaves like a tapered extension to a wheel. A close examination of the overhang in figure 3.4 shows that it is a smooth curve similar to a wheel crown. Therefore, overhang is modeled here as a wheel crown extension to the wheel surface. The equivalent crown shape is modeled by evaluating the profile of the overhanging part of the band. For simplicity, the slope of the wheel surface is assumed to be zero at the edge where the overhang occurs. Sugihara [6] and Taylor [11] modeled the overhang as a cylindrical shell. One edge of the shell is free while the other edge has an applied shear force and bending moment. The shear force and bending moment were derived from the displacement of the bandmill wheel. Using a different approach, the overhang is modeled here as a cylindrical shell rigidly clamped at one edge and free at the other, as shown in figure 4.4. A uniform external pressure P is applied to the shell. This pressure corresponds to the radial force per unit area acting on the band required to maintain equilibrium with the \"strain\" force T around the curved surface, as shown in figure 4.5. (4.35) 51 where b is the band width and r 0 is the mean wheel radius. Over the portion of the band in contact with the wheel, this radial force per unit area is balanced by the contact pressure between the band and wheel. Area i n Contac t w i th Rig id Wheel Wheel Edge (modeled as r i g id c lamp) Area Overhanging Wheel Figure 4.4: Cylindrical Shell Model of Overhang Band Figure 4.5: Uniform Pressure on Band 52 The equation for the radial deformation, w, of a circular cylindrical shell symmetrically loaded by a uniform pressure P is [22] dq 4 ' D where q and v are the axial radial coordinates and d w + 4 n 4 w = iL (4.36) V = ^ (4-37) r 0 ' h^ E h 3 D = / ,x (4-38) 12 (l - v 2 ) The general solution for equation (4.36) is w(q) = e 1 1 «[C, cos(Tiq) + C 2 sin(nq)] + e\"11 q [ C 3 cos(nq) + C 4 sin(riq)] - (4.39) where Ci , C2, C3 and C 4 are integration constants that are determined from the boundary conditions. The deformation boundary condition for the clamped end of the shell is w(0) = 0 (4.40) and assuming that the edge of the wheel has zero slope |^(0) = 0 (4.41) 53 At the free end of the shell the boundary conditions are moment d 2w dq2 M(d) = E I ^ - f (d) = 0 (4.42) shear V(d) = E I ^ ( d ) = 0 dq (4.43) where d is the overhanging length. If constants H and J are defined as e^sinfjid) - e _ 1 l d [ 2cos(rid) - sin(rid) ] (e^ +e _ T , d) cos(rid) (4.44) -P 4r{ — e 11 d [ sin(r)d) - cos(rid) ] ) cos(r|d) (4.45) then the results of applying the boundary conditions are C, = 2 - ^ - ^ e _ , l d cos (T id) - jje^cosfjid) - sin(rid)) - e -11 d (cos(r)d) + sin(r|d))] 1 0 T , d e^f (H-l)cos(rjd) - (H + l)sin(rid) ] - e~nd[ (H + 3)cos(r]d) + (H +1)sin(rid) ] (4.46) C 2 = H Cj + J (4.47) U V D C I J (4.48) 54 C < = ^ - 2 C ' - C ' ( 4 4 9 ) Figure 4.6 shows overhang on a flat wheel. Again, r and y are radial and axial coordinates at Figure 4.6: Overhang of a Flat Cylindrical Wheel a convenient location on the surface of the wheel, \"s\" is the distance from y = 0 to the overhang wheel edge. The length, d, of the overhanging section is (4.50) where \"u\" is the distance between the band center and wheel center and the singularity function \" (T) \" is defined as 55 and (x) = 0 when x < 0 (T) = x when x > 0 Therefore, the overhang coordinate q and the overhang deformation can be written as functions of y q = (y - s) (4.51) w(q) = w((y - s)) (4.52) Since overhang behaves as an extension of the wheel profile f(y), the combined profile is given by F(y) = f ( y ) ( s - y ) ° + w ( y - s ) f l (4.53) equivalent on wheel overhang wheel profile Since the stress profile in the band matches the wheel profile, the bending moment is calculated by M(u) E h s J' + — 2 f f(y) (y-u)dy + w(y-s) (y-u)dy (4.54) 56 when the overhang d > 0 or u + - >s . Otherwise, equation (4.33) is used. The tracking effect of overhang can be incorporated into the tracking model by calculating the curvature resulting from the bending moment using equation (4.34) and including it as the wheel curvature in equation (4.26). 4.6 Cutting Force The cutting force can be included in the theoretical model as a concentrated force acting on the edge of the band between the wheels as shown in figure 4.7. Equation (4.2) for a beam in bending and under an axial tensile load can be extended to included a concentrated edge force F [22]. For simplicity, here the force is modeled as being at the center of the band length. Figure 4.7: Cutting Force Model 57 The general beam equation (4.2) becomes [23] Y(X) = y A + ^ s in(kx) + ^ r - ( l - cos(kx)) + ^ ( k x - sin(kx)) - ^ .(4.55) where the moment at the entry point A is M k s in(kL) 9 B - 9 A c o s ( K L ) - ^A_(1 - cos(kL)) + ^(1 - c o s ) ( y ) .(4.56) and the reaction force at the same point is R A = y A +e, (l-cos(kL)) 2 (kL-sin(kL)) Pksin(kL) Pk sin(kL) cos(kL) - cosz (kL) ksin(kL) (l-cos(kL)' ^ ksin(kL) J + (1 - cos(kL))[^l ' f k L ^ 1 - cos K2J Pksin(kL) . kL T sin — L 2 2T kT (4-57) 58 In summary, equations (4.30), (4.32), (4.33), (4.39), (4.54) and (4.55) model the tracking effects of cutting force, wheel profile and tilt and band overhang, strain, thickness and width. Equations (4.24), (4.26) and (4.34) enable updating of the band position and slope at the wheel entry point and complete the model of band tracking. 59 5.0 Verifying the Theoretical Model Several series of experiments were performed in order to explore the tracking behavior of a moving band and to validate the theoretical model of tracking. These experiments include an examination of the basic tracking model of wheel taper and band strain, thickness and width followed by an examination of the wheel tilt, wheel crown, cutting force and overhang. Band tensioning will not be examined because it is difficult to produce reliably using the available equipment. A table-top bandmill model was designed and built to allow detailed measurements of band tracking behavior. For these exploratory measurements, a table-top model is preferred over a full-size bandmill because the various machine adjustments and tracking measurements can be made more accurately and conveniently. 5.1 Equipment Figure 5.1 shows the table-top bandmill model. The model consists of a mounting frame, a flat wheel, a profiled wheel, a thin straight band made from brass shim stock, and a feed force applicator to simulate the cutting force. Figure 5.2 summarizes the main dimensions. The flat wheel is mounted in fixed bearings. The profiled wheel is mounted parallel to the flat wheel in two take-up bearings. The take-up bearings can be adjusted so as to move the wheels apart and apply strain to the band. The size of the strain is determined by measuring the extension 60 of calibrated coil springs attached to the take-up bearing housing. Finally, the cutting force applicator consists of a pulley wheel pushing against the side of the band at the midpoint of its free span between the wheels. # ITEM 1 Flat Wheel 2 Profiled Wheel 3 Metal Band 4 Strain Applicator (take-up unit and spring) 5 Cutting Force Applicator 6 Linear Optical Encoder 7 Rotary Optical Encoder 8 Computer Figure 5.1: Table Top Model 61 Wheel Diameter, in Profiled Flat 11.5 11.6 Band# Width, in Thickness, in Length, in 1 0.50 0.002 99 2 0.64 0.002 99 3 0.75 0.002 99 4 0.93 0.002 99 5 1.13 0.002 99 Figure 5.2: Wheel and Band Dimensions Four linear optical encoders measure the tracking position and movement of the band at the entry and exit points of the wheels. A rotary optical encoder attached to the flat wheel measures the longitudinal motion. A computerized data acquisition system monitors these signals and records the tracking behavior of the band. The entry angle § is determined from the ratio of the sideways and longitudinal movements sideways movement longitudinal movement 5.2 Equipment Set-up and Accuracy Accurate geometric set-up of the experimental equipment is critical because a band's tracking behavior depends on the subtle details of the geometry of the wheels and band. For most of the factors such as the wheel profile, band strain, width and overhang, this was a straightforward process. However, the band's tracking behavior was found to be especially sensitive to band backcrown, wheel tilt and wheel cross line. As a result, these factors required extra attention. 62 The wheel tilt set-up and measurement can be done reliably by controlling the average tilt angle. An imperfect fit or tolerance between the wheel and shaft, the shaft and bearing or a tolerance in the bearing itself can create a misalignment between the axis of the shaft and wheel and the axis of rotation. As a result, the axis of rotation may not coincide with the axis of the shaft and wheel. As the shaft and wheel rotate, their angle changes symmetrically about the axis of rotation. To ensure that the tilt angle coincides with the axis of rotation, the tilt angle was measured by taking the average of the shaft tilt angle over one revolution. A band can be checked for backcrown by utilizing the geometry of backcrown. When a straight band is turned inside-out the geometry of the band does not change. When a band with backcrown is turned inside-out, its backcrown and curvature are reversed. As a result, backcrown can be detected in a band by comparing the steady state entry angle and tracking position of band in its normal configuration and when it is turned inside-out. A band with backcrown generates different values in each orientation. Bands with significant backcrown were not used in the experiments. The cross line between the wheel axes can be set to zero by utilizing the tracking effect of cross line. When the cross line is not zero, the geometry of the band and bending moment change when the direction of band rotation is changed. As a result, the steady state entry angle and tracking position of the band also change. Therefore, the wheel axes can be set with zero cross line by adjusting the cross line until the steady state entry angle and tracking position are the same in both directions of band rotation. 63 Repeated alignment and set-up of the equipment showed the accuracy of the entry angle measurement on a tapered wheel to be within 0.01° or 8%. On a crowned wheel, the stable tracking position was repeatable within 0.005\". In summary, the initial set-up procedure for each experiment was as follows: 1. Securely mount the wheel on the shaft and the shaft in the bearing. 2. Adjust the wheels axes to zero cross line using a machinists level. 3. Mount the desired band on the wheels. 4. Set the band strain. 5. Set the wheel tilt from the average value of four angular shaft positions. 6. Rotate the band until the entry angle reaches steady state or the band tracking position is stable on the wheel. Measure the entry angle or position. 7. Reverse the band rotation until the entry angle reaches steady state or the band tracking position is stable on the wheel. Measure the entry angle or position. 8. Adjust the wheel axes cross line to eliminate the difference in the measured entry angles and positions. Repeat steps 6 and 7 until there is no difference. 9. Release the band and turn it inside-out. 10. Repeat steps 3 to 6. 11. A difference in the measured entry angles or band positions between steps 6 and 10 indicates that the band has backcrown. 12. Rotate the band until the entry angle reaches steady state. 13. Zero the tracking position and longitudinal motion measurements. 64 5.3 Wheel Taper and Saw Strain, Thickness and Width A series of three experiments was carried out to examine and validate the basic tracking model of wheel taper, band strain, thickness and width described by equations (4.1) to (4.28). The first experiment examines the complete tracking process of the band including both the transient and stead state behaviors. The second and third experiments examine the effect of band strain and width on the steady state entry angle. For these experiments, band tracking movement was generated by the profiled wheel with a taper given by fi(y) = ytan(o.991°) (5.2) The basic equipment set-up and initial position are shown in figure 5.3. After the initial set-up of the equipment as described in section 5.2, the procedure for each experiment was the same. 1. Begin recording the band tracking positions at the entry and exit points of the wheels and the distance traveled by the band. 2. Rotate the band until the entry angle reaches steady state. 3. Stop and note the position and entry angle of the band. These will be used as the initial position and slope for the theoretical model. 4. Continue rotating the steady state tracking band. 5. Stop the band, and change the direction of rotation. 65 6. Rotate the band as it goes through transient tracking and continue rotating until the entry angle converges to a steady state value. 7. Stop and re-measure the band strain. Figure 5.3: Tapered Wheel Experimental Set-up Tapered Wheel Tracking Figure 5.4 gives a sample of the results from a typical tapered wheel tracking experiment. The measured band tracking positions on the wheel shown by the solid points closely follow the theoretical data shown by the dashed curves. The left side of the graph, where band travel equals 0 inches, corresponds to step 3 in the above procedure. At 10 inches of travel the band rotation was reversed. This corresponds to step 5. Notice that at 10 inches of travel, the 66 tracking positions at the exit points lead those at the entry because of the direction of band rotation up to this point. When the direction of rotation is reversed, the process o f transient tracking reverses the situation. After 20 inches of travel in the new direction, the entry positions on both wheels now lead those at the exit and the slopes have changed from negative to positive. Band Travel, inches Figure 5.4: Tapered Wheel Tracking (Strain = 55 lbs, Tilt = 0°, Width = 0.93\") 67 The slope of the curves in figure 5.4 equal the band angle at the associated entry or exit points. The band position at each wheel exit point lags behind the position at the corresponding entry point. This lag reflects the longitudinal motion required for a point on the band to move around the wheel from the entry to the exit points. A horizontal line drawn between the entry and exit curves has approximately the same length as half the wheel circumference. The lag in position of the band on the flat wheel behind the band on the tapered wheel is the result of the bending moment generated by the tapered wheel. Since the flat wheel does not generate a bending moment, the bending moment generated by the tapered wheel alone drives the tracking motion of the band. The position of the band on the flat wheel follows behind the band position on the tapered wheel. All the curves in figure 5.4 indicate that the entry angle and tracking speed converge to steady state values after about 60 inches of band travel. The measured entry angle converged to a value of 0.20° compared to the theoretical value of 0.21°. The effects of band width and strain on the steady state entry angle will be examined in the next two experiments. 68 Effect of Band Width on Wheel Entry Angle Figure 5.5 shows the results of the second experiment on the effect of band width on the steady state entry angle. Again, the measured and theoretical results compare very well. The graphs show that the error between the measured and calculated results is within a 8% range. The effect of band width on entry angle is fairly weak. A 124% increase in band width produces only a 79% increase in entry angle. The effect of band thickness on entry angle is similar in kind, but very much weaker. This latter factor was omitted from the experimental program. 0.0 J 1 h 1 -1 0.4 0.6 0.8 1.0 1.2 Band Width, inches Figure 5.5: Effect of Width on Tapered Wheel Entry Angle (Strain = 28.3 ksi, Taper = 0.991°, Tilt = 0°) 69 Effect of Band Strain on Entry Angle Figure 5.6 shows the results of the third experiment on the effect of band strain on the steady state entry angle. Again the measured and theoretical results match very closely. The graphs show that the error between the results is within a 5% range. The trend of both the measured and theoretical results show that strain has very little effect on entry angle. A 90% increase in strain causes only a 17% reduction in entry angle. The results in figure 5.4, 5.5 and 5.6 show that the basic tracking model of wheel taper, band strain, thickness and width is realistic. The errors between the measured and theoretical results are within the accuracy expected from the equipment. 0.3 + (0 •2 Ui 0) \"O 0.2 o c < £ 0.1 + c UJ 0.0 • Measured —•— Calculated i i i 30 40 50 Strain, lbs 60 Figure 5.6: Effect of Strain on Tapered Wheel Entry Angle (Taper = 0.991°, Tilt = 0°, Width = 0.93\") 70 5.4 Bandmill Wheel Tilt An experiment was performed to examine and validate the extension of the basic tracking model to include the effect of wheel tilt. The model was evaluated by comparing the experimental and theoretical effects of wheel tilt on the steady state entry angle. The basic equipment set-up and the initial band position are shown in figure 5.3. The tapered wheel was tilted by various specified amounts. After the initial set-up of the equipment as described in section 5.2, the procedure for each experiment was a follows: 1. Begin recording the band tracking positions at the entry and exit points of the wheels and the distance traveled by the band. 2. Rotate the band until the entry angle reaches steady state. 3. Stop the band and note the position and entry angle of the band. These are defined as the \"initial\" position and slope for the theoretical model. 4. Set the wheel tilt angle between the axes of the top and bottom wheels. 5. Rotate the band until the entry angle reaches steady state. 6. Stop and re-measure the band strain. Figure 5.7 shows the effect of wheel tilt on the band steady state entry angle. The measured data shows the same trend as the theoretical data. At the extreme values the theoretical model underestimates the effect of wheel tilt on the entry angle by approximately 0.02° or 7%. The close match between the measured and theoretical results indicates that the tracking model is realistic. 71 The trend of the curves in figure 5.7 shows that negative wheel tilt increases the entry angle. Conversely, positive wheel tilt decreases the entry angle. At a tilt angle of 1.1° the tracking effect of tilt balances that of taper, causing the entry angle to become zero. A comparison of figures 5.5, 5.6 and 5.7 shows that wheel tilt has a significantly larger effect on the entry angle than that of band strain or width. • Measured 0.3 - —•— Calculated (/) Q) O D) 0.2 -0) TD U) 0.1 -C < 0.0 -c LU \\ • -0.1 - 1 — 1 1 -0.5 0.0 0.5 1.0 1.5 Wheel Tilt, degrees Figure 5.7: Effect of Wheel Tilt on Tapered Wheel Entry Angle (Strain = 48 lbs, Taper = 0.991°, Width = 0.93\") 72 5.5 Bandmill Wheel Crown and In-feed \"Cutting\" Force A series of five experiments was carried out to verify and examine the extension of the tracking model to include the effects of wheel crown and in-feed \"cutting\" force. In practice, the most common crowned profile used on bandmill wheels is an asymmetric crown [2,3]. Typically, the asymmetric crown has a peak that is not in the center of the wheel, with different profiles on the left and right sides of the peak. This type of crown is shown in figure 5.8. The examination of the tracking behavior of wheel crown begins with a series of three experiments on the general characteristics of symmetric crowned wheels. Two symmetric crowned profiles f2 and f3 are shown in figure 5.8. With the general tracking behavior established, a final series of two experiments was carried out to examine the tracking behavior of asymmetric crowned wheels. Asymmetric crown profile ft is also shown in figure 5.8. fs(y) f*(y) Crown Peak 0.006\"--0.003\"--0.006\" -0.003\" -0.006\" -0.003\" -0 1.5\" y 0 - y 0 1.5' symmetr ic symmetric asymmetr ic Figure 5.8: Crown Profiles f2, fj and f4 73 The five experiments done to investigate the effect of wheel crown are: 1. Examine the complete tracking and self-centering effect of a symmetric wheel crown, including both the transient and steady state tracking behaviors. Since the in-feed \"cutting\" force can be adjusted independently, it can be used to generate band tracking movement and to test the tracking stability of the bands. 2. Examine the effect of the feed force on the tracking displacement of the band from the center of the wheel crown. 3. Examine the effect of band width on its tracking stability, as measured by the band's displacement from the center of the wheel crown caused by a feed force. 4. Examine the effect of band width on the zero load tracking position on an asymmetric crowned wheel. 5. Examine the tracking stability of the asymmetric wheel crown and how the stability depends on the direction of band displacement on the wheel. For these experiments the profiled wheel was prepared with crowns f2, f3 and f4 f 2(y) = 0.024 y 2 (5.3) f3(y) = 0.013 y 2 (5.4) f 4(y) = 0.0034 [(y-0.6484)5 - 1.7 (y - 0.6484)] (5.5) These crowns are shown in figure 5.8. The basic equipment set-up and initial position of the band are shown in figure 5.9. After the initial set-up of the equipment as described in section 5.2, the procedure for each experiment was as follows: 74 1. Begin recording the band tracking position at the exit point of the profiled wheel and the distance traveled by the band. 2. With no in-feed \"cutting\" force applied, rotate the band until a stable tracking position on the crown is reached. 3. Stop rotating the band. Use the tracking position of the band at this stopping point as the datum for subsequent tracking measurements as well as the initial position for the theoretical model. The procedure for the fourth experiment ends here. 4. Apply a feed force at the center of the free span of the band between the wheels. 5. Rotate the band until a new stable tracking position on the wheel is reached. 6. Stop the band and remove the feed force. 7. Rotate the band until it returns to the center of the wheel crown. 8. Stop recording the tracking position and longitudinal motion of the band. Crowned Wheel A Feed Force Flat Wheel G Figure 5.9: Wheel Crown & In-feed \"Cutting\" Force Equipment Set-up 75 Crowned Wheel Self Centering Figure 5.10 gives a sample of the results from a typical crowned wheel transient tracking experiment. The profiled wheel has a crown f2 given by equation (5.3). The solid jagged line indicates the measured tracking movement of the band. The origin of figure 5.10 corresponds to step 3 of the measurement procedure when the band is at the zero load stable crown center position. The rising curve on the left shows the tracking displacement caused by the application of a feed force (steps 4 & 5). This displacement reached a stable value after about 500 inches of band travel, or five complete rotations of the band. After 600 inches of band travel, the feed force was removed, and the band tracked back to its original stable position at the top of the wheel crown (steps 6-8). The jaggedness in the measured data is caused by unavoidable irregularities along the edges of the 0.002\" thick band used in the experiment. 0.20 CD Measured Calculated c/> b 0) u c (0 O 0 400 800 Band Travel, inches 1200 Figure 5.10: Crowned Wheel Transient Tracking (Feed force = .96 lbs, Crown = f2> Strain = 55 lbs, Tilt = 0°, Width = 0.93\") 76 The dotted line in Figure 5.10 shows the calculated tracking response of the band. The theoretical line corresponds very closely with the experimental measurements. The theoretical model successfully predicts the approach to the new stable position from the crown center after the feed force is applied. At this new stable position, the bending moment created by the feed force is balanced by the moment created by the wheel crown. The measured stable tracking position was 0.161\" while the predicted position was 0.148\". The theoretical model also successfully predicts the overshoots at 40 inches and at 640 inches of band travel shown in figure 5.10. These overshoots occur because the sudden addition (or removal) of the feed force greatly bends (or straightens) the band at the entry point, and temporarily creates an excessively large entry angle. The band at the wheel entry is able to track off the crown center for half a rotation of the top and then half a rotation of the bottom wheel before the effects of the feed force progresses around to the exit of the flat wheel. At this instant the boundary conditions at the exit of the flat wheel begin to change and the bending moment in the band begins to decrease and change sign. After a few inches of rotation, the band stops tracking away from and begins to head back towards the crown center. Again, after a delay of half a rotation of the top and bottom wheels, the boundary conditions from the new movement appear at the flat wheel exit, the bending moment reverses, the band stops and again begins to track away from the crown center. 77 Effect of In-feed \"Cutting\" Force Figure 5.11 shows the results of the second experiment on the effect of in-feed \"cutting\" force on the stable tracking displacement from the center of the wheel crown. The wheel crown had profile f3 given by equation (5.4). Figure 5.11 indicates that the theoretical and experimental results follow the same trends. The theory under-estimates the stable displacement by 12 -20%. Larger feed forces create larger bending moments which require larger offsets from the wheel crown center to balance. The relationship between the magnitude of the feed force and the distance tracked off the crown center is linear for the parabolic crown profile f3. This linear relationship can be developed theoretically by substituting equation (5.4) into equation (4.33). o c 0) G> O j§ £ 0.00 *