{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mechanical Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Zhan, James Jianlong","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-09-27T23:44:43Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1990","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Applied Science - MASc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"This study investigates some of the basic dynamic problems in the use of handsaws in industry. As the main purpose in bandsaw study is to understand and improve the cutting performance of the sawblade, this study emphasizes those dynamic problems that most affect the sawblade performance. These include the bandsaw structural dynamics, the top wheel dynamics, the global behavior of sawblades, and the washboarding problem in cutting.\r\nThe study of bandsaw structural dynamics investigates the possible structural modes affecting the performance of the sawblade. Three structural modes within a frequency range of interest from 0 to 60 Hz were found and analyzed. The corresponding critical structural elements are indentified.\r\nThe role of top wheel straining mechanism in bandsawing is discussed in the study of top wheel dynamics. Top wheel displacement during cutting and its effect on the performance of the sawblade are also investigated. The causes of top wheel displacement are the speed change of handsaws and temperature rise of the sawblades.\r\nIn the global behavior of sawblades, the sawblade between the guides is modelled as a narrow, axially moving, thin beam with end loading. The Euler's equation of motion is developed by applying Hamilton's Principle to the model. A simplified finite element method by using a modified cubic beam element is used to solve the eigenproblem. These results are verified by tests with an idling sawblade. A finite element approach capable of calculating the natural frequencies and mode shapes is obtained.\r\nThe washboarding phenomenon is investigated. Extensive experimental results are obtained and an analysis of the resulting washboarding is presented. Two types of wash-boarding have been observed and identified with the tooth impacting problem and the chatter vibration problem.\r\nThe results of this study will be of interest to those who wish to improve their understanding\r\nof the dynamic behaviors of bandsaws and also to improve the performances of handsaws.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/28734?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"D Y N A M I C S O F B A N D S A W S By James Jianlong Zhan B.Sc. Zhejiang University, 1984 M.Sc. Zhejiang University, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1990 (c) James Jianlong Zhan, 1990 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 scholarly 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 The University of British Columbia Vancouver, Canada DE-6 (2\/88) Abstract This study investigates some of the basic dynamic problems in the use of handsaws in industry. As the main purpose in bandsaw study is to understand and improve the cutting performance of the sawblade, this study emphasizes those dynamic problems that most affect the sawblade performance. These include the bandsaw structural dynamics, the top wheel dynamics, the global behavior of sawblades, and the washboarding problem in cutting. The study of bandsaw structural dynamics investigates the possible structural modes affecting the performance of the sawblade. Three structural modes within a frequency range of interest from 0 to 60 Hz were found and analyzed. The corresponding critical structural elements are indentified. The role of top wheel straining mechanism in bandsawing is discussed in the study of top wheel dynamics. Top wheel displacement during cutting and its effect on the performance of the sawblade are also investigated. The causes of top wheel displacement are the speed change of handsaws and temperature rise of the sawblades. In the global behavior of sawblades, the sawblade between the guides is modelled as a narrow, axially moving, thin beam with end loading. The Euler's equation of motion is developed by applying Hamilton's Principle to the model. A simplified finite element method by using a modified cubic beam element is used to solve the eigenproblem. These results are verified by tests with an idling sawblade. A finite element approach capable of calculating the natural frequencies and mode shapes is obtained. The washboarding phenomenon is investigated. Extensive experimental results are ii obtained and an analysis of the resulting washboarding is presented. Two types of wash-boarding have been observed and identified with the tooth impacting problem and the chatter vibration problem. The results of this study will be of interest to those who wish to improve their under-standing of the dynamic behaviors of bandsaws and also to improve the performances of handsaws. iii Table of Contents Abstract \" List of Tables viii List of Figures ix Nomenclature xii Acknowledgement xvi 1 Introduction 1 1.1 Background 1 1.1.1 Purpose of Study 4 1.2 Previous Research 5 1.2.1 Bandsaw Structural Dynamics 5 1.2.2 Top Wheel Dynamics 5 1.2.3 Global Behavior of Sawblades 5 1.2.4 Washboarding 6 2 Equipment And Instrumentation 8 2.1 Equipment \u2022 \u2022 8 2.2 Instrumentation 12 2.2.1 Bandsaw Structural Dynamics 12 2.2.2 Top Wheel Dynamics 12 2.2.3 Idling Tests 15 2.2.4 Washboarding Tests 15 3 Bandsaw Structural Dynamics 21 iv 3.1 Brief Introduction 21 3.2 Structural Model And Testing Method 25 3.3 Measuring Methods 27 3.3.1 Impacting Technique 27 3.3.2 Modal Shape Measurement 32 3.4 Disscussion 40 4 Top Wheel Dynamics 42 4.1 Top Wheel Straining Mechanism 42 4.2 Theoretical Considerations 44 4.2.1 Top Wheel Force System 44 4.2.2 Straining Mechanism Stiffness 44 4.2.3 Top Wheel Displacement Due to Centrifugal Force 47 4.2.4 Effect of Heating on Top Wheel Displacement 49 4.3 Experimental Investigation 53 4.3.1 Test Procedures and Operational Conditions 53 4.3.2 Results 55 4.3.3 Discussion 64 4.4 Conclusions 67 5 Global Behavior of Sawblades 68 5.1 Sawblade Behavior 68 5.2 Simple Finite Element Model 69 5.2.1 Equation of Motion For Free Vibration 69 5.2.2 Finite Element Method Analysis 72 5.2.3 Eigenproblem 77 5.2.4 Effect of Top Wheel Support Stiffness 77 v 5.2.5 Effect of Non-linear Stress Distribution on Torsional Natural Fre-quencies 78 5.2.6 Effect of tensile force on the torsional rigidity 79 5.3 Experimental Verifications 80 5.4 Calculation Example: Natural Frequencies of 18GA Sawblades 81 6 Washboarding 95 6.1 Washboarding Phenomenon 95 6.2 Coordinates and Terms Used to Describe Washboarding 97 6.3 Experimental Results 99 6.3.1 Experiment Procedure 99 6.3.2 Geometric Results 101 6.3.3 Washboarding Occurrence At Various Cutting Conditions 107 6.4 Explanation and Analysis of Washboarding 112 6.4.1 Geometric Analysis 112 6.5 Spectrum Analysis of Washboarding 123 6.5.1 The First Type of Washboarding 123 6.5.2 The Second Type of Washboarding 126 6.6 Conclusions 126 7 Summary of Conclusions 129 Appendices 132 A Review of Beating Vibration 132 B Chatter Theory Review: Chatter in Metal Cutting 1 3 3 vi C Washboarding Cut Pictures 138 Bibliography 144 vii List of Tables 3.1 Measured Bandsaw Resonance At 12Hz 35 3.2 Measured Bandsaw Resonance At 30Hz 37 6.3 Washboarding Test Matrix 24,000 lbs strain 101 6.4 Washboarding Test Matrix 101 6.5 Values of {pxtPy) for Different Speeds and Bites 102 6.6 Values of (px,Py) for Different Strain Levels 106 6.7 Effect of Blade Speed and Bite 106 6.8 Effect of Strain on Maximum Wave Depth 106 6.9 Effect of Blade Speed and Bite 107 6.10 Effect of Strain on Maximum Wave Depth 107 6.11 Effect of the Blade Speed and Bite on the Second Type of Washboarding 111 6.12 Effect of Strain on the Second Type of Washboarding Ill viii List of Figures 1.1 Schematic of a Bandsaw System 7 2.2 The Five Foot Bandsaw 9 2.3 Bandsaw Straining System and Tiltangle Motor 10 2.4 Dimensions of Sawblades 11 2.5 The Bansaw Cutting System 13 2.6 Experimental Set Up For Impacting Test 14 2.7 Experimental Set Up for Top Wheel Motion Test 16 2.8 Probes Calibration Curves 17 2.9 Experimental Set Up for Idling Tests 18 2.10 Experimental Set Up for Washboarding Tests 20 3.11 Bandsaw Geometrical Model 22 3.12 Transfer Function of Strained Sawblade 24 3.13 Prehminary Structural Model of Bandsaw 26 3.14 Typical Measuring Points for Impacting Test 29 3.15 Trasfer Function of Vertical Column 30 3.16 Trasfer Function of the Bottom Part of the Structure 31 3.17 Sawblade Transfer Function 33 3.18 Bandsaw with an Unbalance Weight 34 3.19 Spectrum Excited by Unbalanced Weight 36 3.20 Bending Mode of Vertical Column (12Hz) 38 3.21 Top Wheel Support Mode (42Hz) 39 ix 4.22 Simplified Bandsaw System 45 4.23 Bandsaw Strain System 46 4.24 Bandsaw Heating System 50 4.25 Fraser Mills: Top Wheel Displacement Under Single Cut 56 4.26 Fraser Mills: Top Wheel Displacement Under Successive Cuts 57 4.27 Fraser Mills: Different Top Wheel Displacement Shapes 58 4.28 Top Wheel Displacement Due To Centrifugal Force 59 4.29 Loadcell Strain Change Due To Centrifugal Force 60 4.30 UBC Set 2: Cut Deviation of Sawblade 62 4.31 UBC Set 2: Top Wheel Displacement in Cutting 63 4.32 UBC Set 2: Loadcell Strain Change in Cutting 65 5.33 Mathematical Model of Sawblade Between Guides 70 5.34 Coordinate System Used in Analysis 71 5.35 The Modified Cubic Beam Element 73 5.36 Typical Sawblade Transfer Function At Zero RPM (14,250 lbs) 82 5.37 Typical Coherence at Zero RPM (14,250 lbs) 83 5.38 Sawblade Transfer Function At 100 RPM (18,000 lbs) 84 5.39 Coherence At 100 RPM (18,000 lbs) 85 5.40 Sawblade Transfer Function At 200 RPM (18,000 lbs) 86 5.41 Coherence At 200 RPM (18,000 lbs) 87 5.42 Sawblade Transfer Function At 300 RPM (18,000 lbs) 88 5.43 Coherence At 300 RPM (18,000 lbs) 89 5.44 Natural Frequencies At Different Strain Levels (Zero RPM) 90 5.45 Natural Frequencies At Different Rotation Speed (18,000 lbs) 91 5.46 Natural Frequencies of 18GA Sawblade At Different Speeds 93 x 5.47 Effect of compliance coefficient on the first bending mode 94 6.48 Cutting Process: Formation of Cut 96 6.49 Coordinates and Pitches of The Surface 98 6.50 Terms for Cutting Process 100 6.51 Two Types of Washboarding 103 6.52 A Cut Surface With Both Types of Washboarding 104 6.53 Coupled Washboarding 105 6.54 Effect of Bandsaw Speed On Occurrence of the First Type of Washboarding 108 6.55 Effect of Bite Per Tooth on Occurrence of the First Type of Washboarding 109 6.56 Effect of Strain Level on Occurrence of the First Type of Washboarding . 110 6.57 Washboarding Versus Beating 113 6.58 Both Cut Surfaces of Washboarding 114 6.59 Cut Surface Profile . . . 117 6.60 Details Of Washboarding Cut Surfaces 118 6.61 Relative Tooth Positioning 119 6.62 Probe Positioning 124 6.63 Sawblade Vibration Spectrums for the First Type of Washboarding . . . 125 6.64 Sawblade Vibration Spectrums for the Second Type of Washboarding . . 127 A. 65 Beating 132 B. 66 Basic Diagram of Chatter 134 B.67 Vibration x[t) Produces Changes of Chip Thickness 134 B.68 Chip-thickness Variation Effect for a Lathe Tool in Orthogonal Cut . . . 136 B.69 Chatter Model for Washboarding 136 xi Nomenclature Chapter 4: Top Wheel Dynamics P: cylinder pressure V: accumulator gas volume AP: pressure change AV: volume change AF: force change of cylinder Acyi: area of cylinder piston AX: displacement of piston K: straining mechanism stiffness Pprecharge' precharge pressure of cylinder Vmax' maximum gas volume of accumulator A: blade cross section area p: mass density E: modulus of elasticity c: blade velocity \/: length of blade Al: blade elongation Fc: centrifugal force R: tensile force of blade x: top wheel displacement u: bandsaw rotation speed M: mass of blade xii Q: heat generation per unit time Cp: heat capacity of steel Ac: convection surface area Ca: convection coefficient of air To: room temperature Tt'. blade temperature at time t AT = Tt \u2014 To', temperature change AT,: steady state temperature change ATo: initial temperature of cooling curve r: time constant of temperature change L: lenght of primary span Re'. Reynolds number 7: viscosity of air Pr: Prandtl number k: heat conduction coefficient of air a: thermal expansion coefficient of steel Chapter 5: Global Behavior of Sawblades T: kinetic energy V: potential energy w: lateral deflection $: torsional deflection x,y,z: coordinates m: mass c: blade velocity xiii Cr'. critical velocity of blade p: mass density J: polar moment of inertia Ip: torsional rigidity \/: moment of inertia E: modulus of elasticity G: bulk modulus R: dynamic tension in sawblade RQ\\ static tension in sawblade B: blade width H: blade thickness \/: span legnth between guides le: element length \u00a3: nondimensional length of element [Nw], [Ne]: shape functions [M]: mass matrix [C]: gyroscopic matrix [ii\"]: stiffness matrix [U]: element coordinate vector A: eigenvalue [Q]: eigenvector n: comphance coefficient K\\ straining mechanism stiffness t: strain a: stress cr0: axial stress ap: parabolic stress Fbi first lateral frequency Ft\\ first torsional frequency Acknowledgement To everyone who helped with this project, thank you. I would particularly like to ac-knowledge my advisor, Dr. S. G. Hutton, for his continued enthusiasm and encourage-ment; Bruce Lehmann, for his valued assistance with experiments and suggestions; John Taylor, for his valued assistance with experiments; Victor Lee, for his assistance with experiments; Longxiang Yang, for his help with experiments. I would also like to acknowledge the Science Council of British Columbia for its funding for the washboarding problem; Crown Forest Industries for the assistance with sawmill testing on the top wheel motion problem. I thank my parents, for giving me the opportunity and support that were invaluable to my pursuit of an education. xvi Chapter 1 Introduction 1.1 Background The bandmiU is one of the most widely used type of saws in the wood manufacturing industry with duties ranging from primary log breakdown in sawmilling to small dimen-sion work in furniture manufacture. The main advantages of the bandmill are its ability to handle most log sizes, its high cutting speed and its relatively thin kerf. The size of a bandsaw is described by the diameter of the wheels that support the blade and, for sawmilling, range from five feet to nine feet in diameter. The blade is guided in the cutting region with pressure guides which displace the blade laterally. The crowned top wheel, supported hydraulically or pneumatically, supplies tension to the blade and can be tilted to control the blade position. The larger size handsaws are used as headrigs, the first saws in the sawmill production line, which break the logs down into large rectangular cants. The smaller size handsaws are used as resaws. These break the large cants down into multiples of the required thickness for further reduction to dimensioned lumber, usually by use of circular saws. The operating details of a bandsaw depend on many factors: the type of wood being cut, the required quality and accuracy of cut, the volume throughput required, the gauge and the tension of blade, the type of tooth, and the type of straining mechanism. Some average opiating details are included at this stage.as background information. A nine foot headrig would have a 250 to 300 Hp motor and cut logs of up to six feet in diameter 1 Chapter 1. Introduction 2 at speed up to 400fpm. Some headrigs use double cut blades with teeth on both edges, cutting the log as it travels in either direction. The smaller size five foot or six foot diameter resaws, driven with 100 to 150 Hp motors, will cut cants up to two feet thick at speeds from 100 to 300fpm. Resaws are usually run with single cut blades but are often grouped in pairs or quads to improve lumber throughput. The blades are fabricated from high quality steel. The blades range in size from 16 inches wide by 0.085 to 0.109 inches thick, for the nine foot bandsaw, to 10 inches wide by 0.049 to 0.065 inches thick, for the five foot handsaws. The tooth shape, pitch and gullet capacity are usually one of several standard patterns chosen to suit the duty of the saw. A side clearance is allowed to stop the saw binding in the cut and is created by swaging each tooth the require amount at the tip. This clearance is very important as it directly affects the amount of wood lost with each cut. Side clearances are typically slightly greater than the blade thickness, giving a total cut width of more than twice the blade thickness. Guides are used to align the sawblade. Lubrication fluid is added to the guides to decrease the guide wear and blade heating between the guides and moving sawblade. The stress or tensile force is provided and maintained by a straining mechanism lo-cated between the saw frame and top wheel. The fundamental requirement of a straining mechanism is the capacity of applying a tensile force to the sawblade in such a manner that the stress in the blade is constant, regardless of operating conditions. This means that the critical edge buckling load of the saw would also be constant, and performance always at an optimum level. This would indicate that the strain should respond under all operating conditions with a zero response time and zero damping time without in-creasing friction. Different types of straining mechamsr;- ;?ceding mechanical type and hydraulic type have been developed and used in industry with different advantages and disadvantages. Chapter 1. Introduction 3 The top wheel rests on the straining mechanism and the bottom wheel is connected to the saw frame through bearings.The motor is connected to the bottom wheel and turns the wheel at a certain speed. Due to their large inertia the wheels acts as flywheels to stabilize the sawblade motion. The sawblade is held by both the top wheel and the bottom wheel. The shape of the wheel surface is very important to properly hold the blade. Crowned wheel surface is a typical surface type. The motion of the top wheel during cutting can either optimize the blade stress or worsen the blade stress depending on the amount and the direction of the motion. Bandsaw performance is generally estimated from the measured mean and standard deviations of the thickness of the lumber produced. As a rough guide, a standard devi-ation of 0.010 inches to 0.012 inches would be considered very good, while, for a large headrig, standard deviations of up to 0.025 inches are acceptable. Much of the expertise in setting up and operating handsaws is based on experience and empirical relationships. For a bandsaw to operate successfully, it relies on the main-tenance of a balance between the following factors: the reduction of the blade thickness without increasing the deviation; the increase of the axial strain without inducing fatigue failure; and the correct roll-tensioning of the blade for the prevailing conditions. Some of the problems incurred with handsaws are: poor surface finish of the lumber such as washboarding and wood tearing; snaking of the sawblade at high feed speeds; and the formation of gullet cracks and failure. The subject of this thesis involves in the study of dynamic problems incurred in bandsawing. As the main purpose in bandsaw study is to understand and improve the cutting performance of the sawblade, this study emphasizes those dynamic problems that most affect the sawblade performance. These dynamic problems include the band-saw structural dynamics which deals with the vibration of bandsaw structure, the top wheel dynamics which deals with the behavior of top wheel during cutting, the global Chapter 1. Introduction 4 behavior of sawblades which deals with the dynamic characteristics of 6awblade with-out considering the teeth, and the washboarding which deals with the local behavior of sawteeth in cutting. 1.1.1 Purpose of Study The purposes of this study on the dynamics of handsaws are fourfold: (a) . The objectives of the structural dynamics investigations are: \u2022 to measure the structural modes that affect the sawblade performance \u2022 to provide some information for a better structural design (b) . The objectives of the top wheel dynamics investigations are: \u2022 to study the role of straining mechanism on the performance of sawblade \u2022 to measure the top wheel displacement of handsaws during cutting and the effects of rotation speed, cutting force, and heating \u2022 to provide an understanding of a properly functioning straining mechanism (c) . The objective of the investigation of the global behavior of sawblades is: \u2022 to build a model which is capable of calculating the sawblade natural frequencies (d) . The objectives of washboarding investigations are: \u2022 to study the washboarding phenomenon \u2022 to measure the washboarding occurrence at various rotation speeds, bites per tooth, and strain levels Chapter 1. Introduction 5 1.2 Previous Research 1.2.1 Bandsaw Structural Dynamics Although there axe many publications on structural dynamics of some other machines such as lathe, milling machine etc. there is no information on the analysis of bandsaw structures. 1.2.2 Top Wheel Dynamics The studies on bandmill top wheel support system have been concentrated on the design of the straining system. Few publications are available on the dynamics of the top wheel system which is composed of the top wheel, top wheel shaft, and staining mechanisms (top wheel support). Claassen [4] discussed the theoretical static stiffness of the CanCar straining system. Allen [11] describes the fundamental requirements for proper performing straining mechanisms. Hutton and Allen [12] conducted testing of a 5ft CanCar bandmill straining system at UBC and obtained the static stiffness. The hysteresis effect of hydraulic straining system was also experimentally observed by Hutton and Allen [12]. 1.2.3 Global Behavior of Sawblades Many researchers have considered the dynamics of sawblades. Of primary importances, Mote and his students [1] did many theoretical analyses on this problem. Figure [1.1] shows a schematic of a bandsaw system. Early bandsaw models consisted of an axially moving string, beam or plate. The boundary conditions, located at wheels or.'she-guides are assumed to be either clamped or simply supported. This model assumes that the cutting span is straight and is mechanically isolated from the remainder of the bandsaw. The model has been validated [2,3] by Chapter 1. Introduction 6 comparison of theoretically predicted and experimentally observed natural frequencies of handsaws moving at low speed under high tension. Later, in 1982, Ulsoy and Mote [5] again published a paper on the vibration and stability of wide bandsaw blades using an axially moving plate model which includes the effect of in-plane stresses on stiffness. The equation of motion are developed from Hamilton's principle and approximate solutions are obtained using both the classical Ritz and finite element Ritz methods. The results of the approximate analyses were verified by experimental results with good agreement. Alspaugh [5] studied the problem of the torsional vibrations of a thin strip moving at a constant speed in a longitudinal direction. The effect of point load is included. A simple formula to predict the torsional frequencies stretched by a uniformly distributed initial axial tension was developed. In the case of existing pre-stressed roll tensioning blade this formula predicts torsional frequencies which are much lower than the experimental results. Taylor [7] developed a modified formula including the effect of parabolic roll tensioning. Taylor's formula used a stress variable which was determined by equalizing the calculated natural frequency to the measured natural frequency at zero rpm. The formula was then used to predict the frequencies of the moving blade. Eschler [8] conducted vibration measurements of stationary and running sawblade including the effects of tilting of the top wheel, the roll tensioning. The experimental results were compared with results from theoretical solution of Mote's [9] and Alspaugh's [6] solutions. 1.2.4 Washboarding A saw is said to washboard when the resulting sawn lumber has a surface finish that resembles a washboard. The washboarding condition is commonly encountered in the use of both circular saws and bands .v^s and is undesirable in that axcessive planing is required in order to produce an acceptable finish board. A computer literature search shows that there is no information on this particular problem at all. Chapter 1. Introduction 7 Figure 1.1: Schematic of a Bandsaw System Chapter 2 Equipment And Instrumentation 2.1 Equipment A five foot production bandsaw manufactured by CanCar was used for the experiments (Figure 2.2). The saw is driven hydraulically via a swash plate type hydraulic pump and 100 Hp electric motor. The saw could be run from zero up to 700 rpm. The normal operating speed was 600 rpm. The saw is equipped with a hydraulic straining system as shown in Figure 2.3 which can provide an axial prestressing force from 10,000 lbs up to 32,000 lbs. The sawblade is guided by two pressure guides above and below the cutting region. The guides are lubricated with water. The top wheel could be tilted by an electric motor to align the running sawblade, and the whole bandmill could be moved by hydraulic setworks to adjust the width of the lumber. The top wheel is supported by a straining mechanism. When the saw is running the contact force between the wheels and sawblade is decreased as a result of the centrifugal force of the moving sawblade. Therefore, the straining mechanism will force the top wheel to move up to maintain the constant stress in blade. Two sawblades were used for the tests, a 16 Gauge roll-tensioned blade and an 18 Gauge untensioned blade. Dimensions for both blades are shown in Figure 2.4. Standard fixed guides were used. 8 Chapter 2. Equipment And Instrumentation Figure 2.2: The Five Foot Bandsaw Chapter 2. Equipment And Instrumentation 10 Figure 2.4: Dimensions of Sawblades Chapter 2. Equipment And Instrumentation 12 For the cutting tests, the wood was fed into the saw via a specially designed log carriage on precision aligned rails. The carriage was driven hydraulically and the feed speeds could be selected as desired from zero to 480 fpm. The whole cutting system is shown in Figures 2.5. 2.2 Instrumentation 2.2.1 Bandsaw Structural Dynamics To measure dynamic characteristics of the bandsaw structure the impacting technique was used. Figure 2.6 shows the experimental set up: a Kistler impacting hammer with a force transducer attached to its head was used to impact the structure at a desired position. A piezoelectric accelerometer was used to pick up the vibration. Both the force signal and vibration signal were fed into a Nicollet 660 dual channel frequency analyzer via two charger amplifiers. Also, in order to excite the individual modal shapes an eccentric mass was attached to the bottom wheel to provide excitation to the bandsaw while it was idling. By idling the wheel at certain speeds certain modal shapes were excited and measured using an accelerometer. More details are provided in Chapter 3. 2.2.2 Top Wheel Dynamics The top wheel displacement was measured by a eddy-current displacement probe under-neath the top wheel shaft. The displacement probe was located in the middle of top wheel shaft and mounted on the top of the vertical column of the bandsaw. To measure the stressing force provided by the hydraulic cylinder, a 'loadcell 'was mounted between the hydraulic cylinder and the top wheel shaft. The loadcell was a four arm strain gauge bridge. The output of loadcell strain gauge and displacement probe were recorded by a top guide bottom q u i d * bladt c u t t i n g f t o d l n g ,'e\u00abnt ^ 3 t r a c k Figure 2J>: The Bansaw Cutting System Figure 2.6: Experimental Set Up For Impacting Test Chapter 2. Equipment And Instrumentation 15 TEAC FM tape recorder. Later, the recorded data was analyzed using a Nicollet 660A FFT frequency analyzer as shown in Figure 2.7. The calibration curves for the probes (two displacement probes needed for the washboarding tests) are shown in Figures 2.8. 2.2.3 Idling Tests Figure 2.9 shows the experimental set up for the idling tests. The untensioned sawblade was used. Excitation of the blade was provided by an electromagnet. The magnet was driven by a Bruel & Kjaer No:1024 frequency generator. The generator signal was amplified with a 100 Watt power amplifier. The excitation force of the electromagnet was measured with a Bruel & Kjaer No:8200 piezoelectric force transducer and the signal was amplified with a B & K No:2635 charge amplifier. The displacements of the sawblade were measured with two non- contacting displacement probes with their matched proximitors. The data from the force transducer and displacement probes were recorded by a TEAC FM tape recorder and were analyzed later with a Nicollet 660A dual channel FFT frequency analyzer. The analyzer sampled and stored the information received on each channel and, once calibrated, would calculate and display the transfer function and coherence. Results could be plotted on a Tektronix 4662 plotter. 2.2.4 Washboarding Tests Washboarding tests were conducted on the five foot CanCar Bandsaw (Figure 2.2) with the 18 Gauge untensioned sawblade. Figure 2.10 show the experimental set up of the tests. 16 inch deep cants (two pieces of 8 inch deep cants) were mounted on the carriage . The feeding system on the carriage feeds the wood into the running sawblade. The cut surfaces were saved and labeled for measuring the washboarding profile and the motions of the sawblade were detected by two eddy-current probes positioned underneath the Chapter 2. Equipment And Instrumentation Figure 2.8: Probes Calibration Curves OO O 9 Figure 2.9: Experimental Sel Up for Idling oo Chapter 2. Equipment And Instrumentation 19 wood and monitoring the motions of the tooth tips and sawblade at a position right behind the gullet line. The signals from the probes and the signal from the trigger switch monitoring the time for the wood to enter and leave the cut were fed directly into a TEAC FM tape recorder. The cut surfaces and taped data were used for later processing. A depth micrometer and a ruler were used to measure the cut surfaces. A Nicollet 660A frequency analyzer was used to process the taped data. top guide Probes T j ' i i I I ' \\ I bottoa golde 5fl b l e d * Frequency Analyser P l o t t e r c u t t i n g Probe* feed!no .'cent c a r r i a g e | 1 T r i g g e r t t . c k Proxlm Iters t m TEAC (TO Figure 2.10: Experimental Set Up for Washboarding Tests Chapter 3 Bandsaw Structural Dynamics 3.1 Brief Introduction The main structure of a bandsaw can be divided into vertical column, top wheel and its support mechanism ( straining mechanism for the 5 ft CanCar bandsaw ), bottom wheel and its shaft, hydraulic power motor connected to the bottom shaft, sawblade, and foundation. Figure 3.11 shows a geometrical model of the 5 ft CanCar bandsaw. In order to move the saw closer to or farther away from the cant the whole machine rests on two guide ways . . The vertical column which supports the top part of the machine rests on two simply supported beams. The bottom part which basically is the bottom wheel and driving system is connected to the middle of these two beams. The top part of the machine consists of a steel plate with a pivot on its one side which is attached to the top of the vertical column . The top wheel shaft on the other side is supported by the hydraulic cylinder of the straining mechanism. The top wheel is mounted on the shaft. By changing the pressure inside the cylinder, the shaft with the top wheel can change its position to change the strain level of the sawblade. The machine also has a tilt motor mounted on the top part of the machine to change the tilt angle of the top wheel shaft as shown in Figures 2.2 and 2.3. This change of the shaft tilt angle will change the position of the sawblade on both the top wheel and the bottom wheel. Figure 3.12 shows the transfer function of the strained sawblade (15,0001bs strain 21 Chapter 3. Bandsaw Structural Dynamics 22 Figure 3.11: Bandsaw Geometrical Model Chapter 3. Bandsaw Structural Dynamics 23 level) between the middle of the cutting span and the top wheel shaft. The lowest mode of sawblade occurs at 58Hz. There are three major structural modes before the lowest mode of the sawblade. While the saw is in operation one or some of these modes can be possibly excited and will eventually affect the performance of the sawblade. For the 5' bandsaw in the laboratory, the vibration of the top part of the bandsaw can be clearly observed at the normal operating speed of 600rpm. This chapter is to investigate and identify these structural modes of handsaws. Theoretically, a bandsaw has infinite degrees of freedom and then has infinite struc-tural modes. Since low frequency vibration usually associates with large vibration am-plitude the investigation is restricted to certain range of low frequency referred to the frequency range of interest and to those vibration modes whose vibration may affect the sawblade vibration. Figure 3.12 shows that these structural modes are well separated. This makes it possible to consider these modes separately. In this chapter, each structural mode is considered as an single degree of freedom vibration system ( a mass-spring-damper system). Various excitations exist in a running bandsaw. The major excitation sources are the cutting excitation from the sawblade, the unbalances of the rotating top and bottom wheels, the excitation from the motors, and the excitation transfered from the neighbouring machines through the foundation. Depending on the structural rigidities and relative locations to the excitation sources, some elements con-tribute to the structural modes and some elements do not. The elements associated with structural modes are called critical elements. Modal testing technique is able to identify those critical elements. Chapter 3. Bandsaw Structural Dynamics 25 3.2 Structural Model And Testing Method In structural analysis, even the most complicated machine can be considered as consisting of many basic machine elements such as beams, plates, springs, and dashpots. In order to understand the composition of the machine and to find possible critical elements a preliminary structural model of the bandsaw was drawn as shown in Figure 3.13. Before selecting possible critical elements it should be noted that only elements whose vibrations are thought to affect the sawblade vibration are considered. Based on this criterion 4 beams or rods, 4 springs, two masses, and one dashpot are chosen to construct the model. Element HDBE represents the vertical column resting on two simple-simple supported beams GF and AC. Element MI represents the top part of the structure which consists of the top wheel shaft and top plate. MI is connected to HDBE through spring-dashpot IH. .Masses N and 0 represent top wheel and bottom wheel together with part of the blade. Springs MN and OP represent the bearing stiffnesses of both top wheel and bottom wheel. Spring NO represents the stiffness of blade. Totally, 7 displacement degrees of freedom and 1 rotation degree of freedom are included. After selecting the possible critical elements and determining the degree of freedom of the motion modal testing was used to measure the displacements and rotation corre-sponding to these degrees of freedom. Based on these measurements the critical elements were indentified and a better structural model of the bandsaw was determined. In order to confirm this information two methods were used. Firstly the impacting technique was used to find the number of structural modes and their frequencies within a frequency range of interest. By moving the accelerometer around the key positions of the structure the frequencies and modal shapes were roughly determined. After this preliminary investigation these modes were individualy excited by running the bandsaw at corresponding rotation speeds. Because of the existence of unknown inbalances of the Chapter 3. Bandsaw Structural Dynamics Figure 3.13: Preliminary Structural Model of Bandsaw Chapter 3. Bandsaw Structural Dynamics 27 wheels the running wheels were able to excite the machine at a frequency equivalent to the wheel rotation speed. A measurement of the vibration displacements all over the machine surfaces revealed the modal shape at that frequency. 3.3 Measuring Methods 3.3.1 Impacting Technique The impacting technique is widely used in determining the structural modal parameters because it is convenient to use and its force spectrum range can be adjusted by using hammer heads made of different materials. By applying an impacting force to a structure the structure will respond to the impacting with certain accelerations. Based on the measurements of both the force and the acceleration the transfer function is calculated. By examining the transfer function in the frequency domain modal parameters of the structure such as natural frequency, modal shape, and the damping ratio can be determined. The impacting force is measured through a built in force tranceducer and the structure acceleration response is measured by a piezo-electric accelerometer attached to the structure at a desired point. Both the force and response signals are fed into a frequency analyser for transfer function processing. Thirteen measuring points as shown in Figure 3.14 were selected to get an approximate idea about the natural frequencies and corresponding modes in a frequency range of zero to 60Hz. Point 13 is located in the middle of the span between guides. Point 7 (in dashed line letter) is located in the middle of the bottom plate of the vertical column. Point 11 (also in dashed line letter) is located on the back surface of the machine. Because only the low frequency mode? are responsible for the large scale displacement of the entire structure the frequency range 0\u201460Hz was sufficient to cover the lowest modes of the sawblade. For most of the measurements the impacting point was point 5 Chapter 3. Bandsaw Structural Dynamics 28 and the measuring point was moved until it covered all the relative points. Figure 3.12 shows the transfer function of the strained sawblade (15,0001bs strain level) with hitting point 13 and measuring point 12. Four basic peaks appear with frequencies at 12\/13Hz, 33Hz, 45Hz, and 58Hz. Figure 3.15 shows the transfer function with hitting point 5 and measuring point 11. Both frequencies at 12Hz and 13Hz have phase angle \u201490\u00b0. Keeping the hitting point at 5 and moving the measuring point through 5,6, 9,7 and 10 the peaks at 12Hz and 13Hz always have same phase angle signs with both phase angles are \u201490\u00b0 at points 5, 9 and 90\u00b0 at points 6,10. The transfer function at point 7 did not show clear peaks at both 12\/13Hz which indicates a nodal line. From this information we would say the 12\/13Hz mode is the bending mode of beam HDBE with rod BE rested on both simple-simple supported beams AC and FG. The mode at 33Hz is the next to be investigated. This time the hitting point and measuring point were same at point 8. The sawblade strain was not applied. The only two significant peaks (Figure 3.16) are 12\/13Hz and 28Hz. It was believed that the 28Hz one was the bottom wheel support mode at low strain level. When the sawblade was strained up to 15,0001bs this frequency went up to 33Hz because both spans were stiffened. The 44Hz mode was revealed as the top wheel support mode by the same method and reasons for the 33Hz mode. The remaining mode was the one at 58Hz. A measurement was conducted by hitting at point 13 and measuring at point 13 on the sawblade between guides. Figure 3.17 shows the obtained transfer function. Two siginificant peaks were found at 58Hz and 75Hz. Si.v ft \"ihe sawblade is relatively flexible in he lateral direction the transfer function is dominated by the sawblade modes. From this picture combined with information available [7] we found out that the 58Hz one was the first bending mode of the sawblade Figure 3.14: Typical Measuring Points for Impacting Test Chapter 3. Bandsaw Structural Dynamics 30 1 1 1 . 3 DG VLG 1 2 . 0 0 0 0 0 HZ 2 2 6 . - 0 3 T Chapter 3. Bandsaw Structural Dynamics 32 and 75Hz was the first torsional mode of the sawblade. 3.3.2 Modal Shape Measurement After approximately measuring the structural modes, the modes at 12Hz and 44Hz were confirmed by exciting the machine at the desired frequency. Because the bandsaw is very big it is very difficult to excite the machine by a conventional shaker or exciter. In this case the machine was excited by attaching an unbalanced mass to the bottom wheel. Figure 3.18 shows the bandsaw system with a unbalance mass attached to bottom wheel at a position e from the centre. The bandsaw ran at circular speed fl. The unbalance force produced is F = meO2 sin fit The system vibration equation is (assuming an one d-o-f system) Mi + Kx = mefi2 sin Ctt where M is the equivalent mass of the machine, K is the equivalent stiffness of the machine, and x is the dispacement. If the bandsaw has a critical speed at flo resonance will occur while running at that speed. Peaks may also appear at the multiples of the excitation frequency in the RMS spectrum plot. Large vibration will be measured when vary fi through Clo in the vicinity of Oo-In order to provide enough excitation force to the structure an 11.16 lbs (5.06 Kg) unbalance mass was attached to the bottom wheel at a position 21 inches from the centre. This force provided enough energy to excite the whole structure. Figure 3.19 shows an example of the spectrum measured on the top wheel shaft. The top curve was obtained when the bandsaw ran at 510 rpm and the bottom one was at 420 rpm. The top wheel \u20225 Figure 3.17 Sawblade Transfer Function CO co Chapter 3. Bandsaw Structural Dynamics 34 vobal&nee weight Figure 3.18: Bandsaw with an Unbalance Weight Chapter 3. Bandsaw Structural Dynamics 35 Table 3.1: Measured Bandsaw Resonance At 12Hz Speed Speed 2nd multiple Displacement (rpm) (Hz) (Hz) at the 2nd multiple (mm) 300 5 10 0.00175 345 5.75 11.25 0.0398 360 6 12 0.0787 375 6.25 12.5 0.0607 390 6.5 13 0.0179 420 7.0 14 0.00206 450 7.5 15 0.00200 465 7.75 15.25 0.00206 shaft is located far from the excitation force located on the bottom wheel. So this mass was enough to excite the whole machine. In order to measure the modal shape fully, 84 measuring points were selected all over the entire machine surfaces. Focus were placed on the top plate and the vertical column. Because of the geometric restraint, few points on the bottom part were selected. During the testing the machine was run at a selected speed and one accelerometer was moved around until it covered all these 84 measuring points. Within the 60Hz frequency range of interest three natural frequencies \u2014 12Hz, 30Hz, 44Hz \u2014 were expected. The 12Hz (720rpm) resonance was expected to occur as the twice multiple if the bottom wheelwas rotated at or through 6Hz (360rpm). 720 rpm was beyond the limit of the bandsaw rotation speed. The 30Hz (1800rpm) one was expected to occur as sixth, fifth, fourth, and third multiples etc. if the bandsaw was rotated at 300rpm, 350rpm,450rpm,and 600rpm etc.. The 44Hz (2640rpm) one was expected to occur as the sixth, fifth, and fourth multiples etc. if the bandsaw was rotated at 440rpm, 528rpm, and 660rpm etc.. Table 3.1 shows the measured displacement as the bandsaw was running through 340rpm to excite the 12Hz mode. The measuring point was on the top wheel shaft. Table 3.2 is the measured acceleration as the bandsaw was running 16 #AVG 1 . 9 7 - 0 3 V V L G 5 1 0 . 0 0 0 0 CPM 1 4 . 4 - 0 6 V C A M H 1 1 1 1 1 1 1 - f ;u 6 i :s M 6 . a J\\ J\\J\\ * A A A A A A A | A A\u2014[A A \/^ i i J\\A |t 4 ^ A A| AA 2 . 0 A 5 . 0 A - B \/ 4 HZ 2 0 0 Figure 3.19: Spectrum Excited by Unbalanced Weight Chapter 3. Bandsaw Structural Dynamics 37 Speed (rpm) 5th multiple (Hz) Displacement at 5th multiple (mm) 300 25.00 0.00350 309 25.75 0.00310 324 27.00 0.00458 342 28.50 0.00580 357 29.75 0.00821 372 31.00 0.00415 378 31.5 0.00394 Hz through 360rpm whose fifth multiple provided the excitation to the system at 30Hz. The measuring point was on the shaft of the bottom wheel. The peak was not as clear as the case at 12Hz because the bottom part is stiffer than the top part. Figure 3.20 shows the measured modal shape at 12Hz. At this frequency, the whole machine vibrates as a cantilever beam with the fixed end located on the foundation surface. The top part of the machine vibrates as the free end of the cantilever beam. The vertical vibration at this mode was measured to be only 17% of the horizontal one. The mode at 30Hz was found to be the up-down mode of the bottom wheel support. This frequency is lower than the 33Hz measured by the impacting test (section 3.3.1) because the sawblade strain at this time is lower than in impacting test. Figure 3.21 shows the modal shape at 42Hz. This mode is only a local mode of the top wheel and its shaft.- Detailed measurement by putting an accelerometer on both ends of the hydraulic cylinder showed that this vibration mode was from the hydraulic straining mechanism. The same reason as before applied here for the difference of the frequency 42Hz measured here and the frequency 44Hz measured in section 3.3.1. Chapter 3. Bandsaw Structural Dynamics 38 Figure 3.20: Bending Mode of Vertical Column (12Hz) Chapter 3. Bandsaw Structural Dynamics 39 Figure 3.21: Top Wheel Support Mode (42Hz) Chapter 3. Bandsaw Structural Dynamics 40 3.4 Disscussion The previous sessions revealed that from 0 up to 60Hz there were three major band-saw structural modes with frequencies at 12Hz,30Hz, and 42Hz. Because they are well separated each mode can be considered as a single degree of freedom vibration system. The mode at 12Hz is bending mode of the vertical column. The vertical column as the main connection between two wheels has three basic vibration modes: the bending mode, the torsional mode, and the logitudinal mode. Since the stiffness in longitudinal direction is normal much larger than the bending stiffness and the torsional stiffness, the natural frequency of the longitudinal mode is too high to be considered within 60Hz frequency range of interest, which is the case here. The torsional mode was not shown in the 60Hz range of interest because it is quite stiff in the torsional direction too. For the vertical column, the only weak directionis the bending direction. Because the heavy top wheel overhang on the free end of the vertical column the low bending stiffness implies that the vertical column needs to be stiffened accordingly. The mode at 30Hz is the bottom wheel support mode. This mode is believed to be the bottom wheel support mode or as it is usually called the bottom wheel and bearing assembly mode. As the bottom wheel is supported rigidly through the bearings. The only souce of this mode is the bearing itself. Normally the bearing has very rigid radial stiffness. However, the large mass could possibly bring the natural frequency of this bottom wheel and bearing assembly mode down a lot. The possibly misassembl ed bearing could also contribute to the low natural frequency of this mode. The mode at 42Hz is the mode of the top wheel and its shaft. At this mode, the whole top part together with top wheel rests on the hydraulic cylinder of the top wheel Bupport to construct a single degree of freedom vibration system. The stiffness of the hydraulic cylinder acts as the spring of the vibration system. The mass of the whole top Chapter 3. Bandsaw Structural Dynamics 41 part, top wheel, and portion of the hydraulic cylinder acts as the mass of the vibration system. The damping of the hydraulic system and the friction of the hydraulic cylinder act as the damper of the vibration system. Combining above information with the structural model in Figure 3.13, it was found that rod HDBE, spring and dashpot HI, and the bearing spring OP were the three critical elements. If one wants to avoid these structural mode for a certain range of interest these critical elements are suggested to be modified. During cutting the sawblade is loaded with complicated cutting forces and various impacts. If any structural mode is excited large lateral displacement of 6awblade may occur. This large displacement will result in the large amount of cutting waste and poor surface finish. By avoiding these structural modes bandsaw design ^d cutting performance could be improved. Chapter 4 Top Wheel Dynamics 4.1 Top Wheel Straining Mechanism The sawblade is stretched by the top wheel and bottom wheel. The stretching force or tension of the 6awblade is supplied by the top wheel straining mechanism. The top wheel with its shaft is supported by the hydraulic cylinder of the straining mechanism. Figure 4.23 shows the strain pivot assembly. The strain pivot assembly can be described as consisting of only a mass supported by a spring with damping, which results from friction within the hydraulic cylinder. The hydraulic cylinder is rigidly connected to the top wheel shaft and supports the top wheel. During tensioning, the pressure inside the hydraulic circuit moves the hydraulic cylinder up and down and then moves the top wheel up and down. Because the bottom wheel is simply supported the distance between two wheels is changed by the change of top wheel position and then changes the stretching force i.e. the tension of the sawblade. Due to the wide use of bandmill with hydraulic straining mechanism in sawmills it is very important to understand the role of straining mechanisms and the top wheel displacement during cutting. The problem that happened to the Crown Forest Elk Falls sawmill is an example. Over a period of two years the Crown Forest Elk Falls (now Fletcher Challenge Canuua) sawmill near Campbell River B.C. had a persistent gullet cracking problem on its CanCar Chip-N-Saw quadbandmills. The problem appeared to occur on all four 42 Chapter 4. Top Wheel Dynamics 43 bandmills. One suggested cause for the cracking was that the top wheel strain mechanism may not be acting properly. Technicians from Crown Forest's research section monitored the motion of the top wheel during cutting, but found very little motion compared to motion measured at Crown Forest Fraser Mills sawmill. At this point in time the Crown Forest technicians came to UBC and asked for opinion on the cause of the gullet cracking and the proper top wheel motion of a functioning straining system. The tension in the sawblade can be very greatly influenced by the stiffness of the strain support. If in one extreme the strain support were a rigid mounting (accumulator diaphragm frozen in place) the initial strain pressure would gradually reduce as the saw speed is increased until at certain speed the pressure would be zero and the entire tension in the sawblade is due to dynamic tension caused by the centrifugal force of the blade running over wheels. At this point the blade would run complete clear of the wheels and the fundamental resonant frequency of top wheel system would not be related to the sawblade at all. If at the other extreme the support had zero stiffness the top wheel would always keep good contact with the running sawblade and then keep the supporting force of top wheel constant. After conducting tests on the hydraulic straining mechanism of a CanCar bandmill, Claassen [4] concluded that the stiffness and damping of the hydraulic straining mechanism attenuate the sawblade vibrations by acting as a vibration absorber. Some researchers agree with Claassen and some disagree. During cutting, a certain amount of cutting load inevitably acts on the sawblade. Therefore, it is necessary to know how the top wheel and the straining mechanism behaves under such cutting load. This chapter studies the top wheel displacement, the causes of such displacement, and ftr-ally the effect of this displacement on the blade performance and therefore, provides some answers to above questions. Chapter 4. Top Wheel Dynamics 44 4.2 Theoretical Considerations 4.2.1 Top Wheel Force System Figure 4.22 (a) shows a simplified bandsaw system. The bottom wheel is simply sup-ported and the top wheel is supported by a cylinder which is the straining system of the bandsaw. The sawblade is travelling at speed c. Its corresponding force system is shown in Figure 4.22(b). 4.2.2 Straining Mechanism Stiffness The theoretical static stiffness of the strain mechanism can be calculated as follows [4]. During dynamic behavior an isentropic relationship applies to the gas volume PV 1A = constant (4.1) Therefore, the gas volume change occurs as the result of a pressure change (A-ft) = 1.4^(V2-V,) or AP = 1.4^ AV where subscripts 1 and 2 stand for the two isentropic stages of the system and AP and AV stand for the pressure change and volume change. The force change of cylinder AF will be AF = APA^ = lA^AcytAV Since AV = A^AX Chapter 4. Top Wheel Dynamics 45 Figure 4.22: Simplified Bandsaw System (a). Bandsaw System (b). Force System Chapter 4. Top Wheel Dynamics Figure 4.23: Bandsaw Strain System Chapter 4. Top Wheel Dynamics 47 and The stiffness of the cylinder K is AF . .P K = XX = (4-2) Since we have Then we have V = V Pprcchorge max -p VmazPprtcharge For the five foot CanCar Bandsaw: Acyi = 6.257T in2 V\u2122, = 126 in3 = 800 psi P = 1000 psi So K = 5354\/65\/incft (4.3) The measured static stiffness of the top wheel straining mechanism is 6000 lbs\/in [12], which is quite consistent with the calculated value. 4.2.3 Top Wheel Displacement Due to Centrifugal Force For the force system shown in Figure 4.22(b), we have equlibrium conditions as followings: 2R = FC- Kx (4.4) Chapter 4. Top Wheel Dynamics 48 AR R = \u2014j-Al (4.5) and Fc = 2pAc2 (4.6) Also, we have geometric relationship (4.7) Hence, the top wheel displacement due to centrifugal effect is 2pAc For the five foot bandsaw, we have * = T ^ F ^ (4-8) \/ = 386 inches A = 0.6175 in2 p = 490 lbs\/ft3 c = 0.261u> f\/s \u00a3 = 30(106) psi where u> is the bandsaw rotation speed in rpm. _ 8.9873(10-3)0,* 1.92(10*) + 1 ; Figure 4.28 in section 4.3.2 shows the comparison between the theoretical top wheel displacement calculated by equation 4.9 and the experimental results. The theoretical value is quite close to the experimental one. Chapter 4. Top Wheel Dynamics 49 4.2.4 Effect of Heating on Top Wheel Displacement It was found from the Fraser Mill tests (see section 4.3 for details) that the top wheel displacement versus time curve is exponential. This could probably be explained due to heating effect since the rise curve and decay curve of temperature in a heating system are exponential. Figure 4.24 shows the simple bandsaw heating system. A sawblade stretched by two wheels and driven by one of the wheels is running downwards through the cant being cut. The heat is generated through the rubbing between the sawblade and wood. The corresponding heat equation is as follows[15]: C PM^ = Q- CaMTt - To) (4.10) The solution of the above equation is as follows (assuming initial temperature of blade is equal to room temperature): AT4(1 - e~r) heating AT0e\"r cooling AT = < Then, we have AT. = Q CaAc CPM CaAc ATQ = -^-{l-e-*) where to is the time at which the bandsaw start to be cooled down. Calculation of Convection Coefficient (Air Only) The following equation is selected to calculate the convection coefficient Ca [15] (page 202). % ^ = Pr'(0.037i2\u00b0-8 - 850) (4.11) Chapter 4. Top Wheel Dynamics 50 5 f t CanCar Bandmill Figure 4.24: Bandsaw Heating System Chapter 4. Top Wheel Dynamics 51 Reynolds number 7 For the six foot bandsaw used in Fraser Mill, we have c = 47.9 m\/s L = 98m = 2.5m 7 = 15(10\"6) at 20\u00b0C So, * = i S ^ T 1 \" 7 - 9 8 ( 1 \u00b0 6 ) ( 4- 1 2 ) Assume Pr = 0.7 (at about 20\u00b0C ) and k = 0.025uj\/m2.\u00b0C (at about 20\u00b0C ). Then, we have Ca = 101.6iy\/m2.\u00b0C The heating system shown in Figure 4.24 is a very simple one which only takes into account the air convection. Actually, heating problem is much more complicated in the bandsaw because of the existence of radiation losses, mass transfer of water, water evaporating. The real convection coefficient can be calculated based on the measured decay curve as in the following. Steady State Temperature Rise convection coefficient C0 = 101.6w\/m2.\u00b0C mass of blade M = 33.6Kg convection area Ac = 5394in2 = 3.48m2 heat capacity of steel Cp = 470J\/Kg.\u00b0C Chapter 4. Top Wheel Dynamics 52 Then, the time constant is t = ^\u2014r- = 44.7 seconds 0aAc and the half time of the decay curve will be Thaif = rln2 = Zlstconds. This half time is not consistent with the measured half time of 5 seconds because the effects of radiation losses, mass transfer of water , and water evaporation etc. on the convection coefficient is not included. The convection coefficient Ca can be determined from the measured half time of 5 seconds, i.e. time constant of 7.2 seconds, as follows. Ca = \u00a3^ = 6Z0w\/m2.\u00b0C Total power used to run the bandsaw is 150 HP. Assuming 10 % of the total power which is 15 HP is used to heat the bandsaw i.e. Q = 15HP = 15,000 Kg.m\/s. Then, the steady state temperature rise is AT, = -Q\u2014 = 6.84\u00b0C C\/nJT.c Top Wheel Displacement Per \u00b0C Blade Temperature Rise The blade elongation per \u00b0C blade temperature rise can be calculated as follows: Al = al (4.13) This blade elongation results in a upward displacement of top wheel x. x = 0.5A\/ = 0.5\/a (4.14) For the six foot bandsaw, we have \/ = 422inches a = 6.0(10\"6) 1\/\u00b0C Chapter 4. Top Wheel Dynamics 53 So, top wheel displacement per \u00b0C blade temperature rise is x = 1.266(10-3)in\/\u00b0C (4.15) Top Wheel Displacement The top wheel displacement caused by the steady state temperature rise AT, (which is 6.84 \u00b0C) is x = 1.266(10-3)AT. = 0.00866mc\/ies 4.3 Experimental Investigation The tests in Fraser Mills sawmill were done by Dr. S.G. Hutton, Mr. Bruce Lehmann, and me. The tests in Elk Falls sawmill were done by Dr. S. G. Hutton and Mr. Bruce Lehmann. The data from both sawmills were analyzed by myself. Due to the size of the thesis these tests are not presented in detail. Only some typical results which are useful for the laboratory investigations of the problem are presented. A Kockums 6 ft bandmill was cutting 24 feet long by 20-30 inches deep cant at 600rpm in the Fraser Mills sawmill. The bandmills were running well and the cut surface finish was acceptable. In comparison with the bandmills in Fraser Mills, the bandsaw system at Elk Falls consisted of four bandmills in a Chip-N-Saw quad. The Cutting surface finish was also acceptable. But the gullet cracking problem occurred persistently. 4.3.1 Test Procedures and Operational Conditions Several tests, including both idling and cutting were conducted at UBC. The bandsaw used has the following specifications: Bandsaw type: 5 foot CanCar Wheel size: 5 feet Chapter 4. Top Wheel Dynamics 54 Blade gauge: 16 GA Straining Mechanism: hydraulic with piston accumulator Strain: 14,000 lbs The running condition varied with tests. In the idling tests, no cooling except guide lubrication was used and the rotation speed was adjusted from zero to 600 rpm. In the cutting tests, the running conditions are listed as below: SET1: Depth of cut: Length of cant: Rotation Speed: Cooling condition: 8 inches 4 feet 200 rpm water spray on guides SET2: Depth of cut: Length of cut: Rotation speed: Feed speed: 17 inches 4 feet 600 rpm 54 fpm, 200 fpm SET3: cutting across grain, through large knot Depth of cut: 8 inches Length of cut: 8 inches Rotation speed: 600 rpm Chapter 4. Top Wheel Dynamics 55 4.3.2 Results Significant top wheel displacement was measured at Fraser Mills as shown in Figures 4.25 to 4.27. Figure 4.25 shows the top wheel displacement under a single cut. The average upward displacement of the top wheel during cutting is about 0.010 inches. The maximum amount of displacement measured is 0.019 inches. The average time for the top wheel to move up during each cut is about 9.5 seconds. The response time and amount of displacement depends on the length of the cant. The difference between a single cut and successive cuts can be seen in Figures 4.25 and 4.26. Figure 4.25 also show that the time for the top wheel to move up (rise curve) is 9.6 seconds and that the time required for the top wheel to return to its original position (decay curve) is 23.9 seconds. A study of the top wheel displacement for other cuts show that the decay time of the top wheel displacement is always longer than the rise time. If successive cuts are so close together that the top wheel could not completely come down between cuts, the top wheel displacement would increment up with each cut. This can be seen in Figure 4.26 too. Figure 4.27 shows representative plots of different curve shapes of top wheel displace-ment measured at Fraser Mills. It is found that the decay curve is always exponential but the rise curve was variable. Very little top wheel displacement was measured in Elk Falls sawmill, compared to that of Fraser Mills. The average displacement measured was 0.0035 inches and the maximum displacement of the top wheel moved during cutting was 0.0093 inches. The results of tests conducted in the laboratory are presented below. (a). Effect of Rotation Speed on Top Wheel Motion Figure 4.28 shows the hysteresis curve of top wheel displacement as a function of 40 30 20 10 ^^^^^^^ \u202210 J \u202220 -30 -40 \"I 1 1 1 1 1 1 10 20 30 40 50 60 70 60 Time (seconds) Figure 4.26: Fraser Mills: Top Wheel Displacement Under Successive Cuts Chapter 4. Top Wheel Dynamics Figure 4.27: Fraser Mills: Different Top Wheel Displacement Shapes 17.5 15 . 12.5 . 10 7.5 2.5 . -2.5 Experimental Curve (Loading) Q---G ExperImental Curve (Unloading) Theoretical Curve (K-6,000 lb\/In) 0 l^[~:.;:is.zJ3-**%*% T T 100 200 300 400 Rotation Speed (rpm) 500 600 Figure 4.28: Top Wheel Displacement Due To Centrifugal Force 5ft CanCar Bandsaw at UBC 15,000lbs Strain 0 100 200 300 400 500 600 Rotation Speed (rpm) Figure 4.29: Loadcell Strain Change Due To Centrifugal Force 5ft CanCar Bandsaw at UBC 15,000 lbs Strain Chapter 4. Top Wieei Dynamics 61 rotation speed. The test was done by using non-contacting displacement probe, which was located under top wheel shaft, to measure top wheel displacement and using a speed controller to change the rotation speed from zero to 600 rpm. The strain level was set at 15,000 lbs. The test was repeated three times. These three results are consistent to each other and close to the theoretical curve. The hysteresis curve shows that there existed energy loss due to the friction and sticking of the straining mechanism. Figure 4.29 shows the change in strain measured by the loadcell caused by centrifugal force of blade as the rotation speed is changed. The test was repeated twice and the results were repeatable. A clear hysteresis effect is presented, most likely caused by friction and sticking in the straining mechanism. (b). Cutting Tests Three sets of cutting tests were conducted. In the first set, both a single cut and eight successive cuts were done with one or two seconds interval between successive cuts for setting the bandsaw. The bandsaw ran at 200 rpm to expand the cutting time. In successive cuts the bandsaw ran for about five minutes. It was cooled by water and the temperature change during running was not measurable (less than 1 0 C). The top wheel did not move at all in either case. In the second set, one 4 feet long and 17 inches deep cant was used for normal speed cutting (600 rpm). Three tests were conducted. In the first test, a feed speed of 54 fpm was used. After cutting the sawn board fell down and was left against the running blade for a few seconds. The top wheel moved 0.0005 inches to 0.0006 inches immediately after cutting. No noticeable strain change occurred during cutting. The top wheel moved down as the bandsaw speed was decreasing. In the second test, a piece of wood was pushed against the running blade in order to generate heat by friction. No displacement was observed to take place. In the third test, a feed speed of 200 fpm was used. The bandsaw Chapter 4. Top Wheel Dynamics 62 \u2022 1 \u2022 t t 0 n ( 0 6 o i \\ n c h e s ) Hor i z o n t a l P o s i t i o n (Inches) Figure 4.30: UBC Set 2: Cut Deviation of Sawblade (a). Cut Deviation in Horizontal Direction (b). Cut Deviation in Vertical Direction -100 J Begin o Cut EnVof Cut 40 T -30 T -60 70 \"T\" 10 ~T~ 20 30 t i n e (seconds) 80 Figure 4.31: UBC Set 2: Top Wheel Displacement in Cutting Chapter 4. Top Wheel Dynamics 64 rotation speed went down to 570 rpm during cutting and went up to 690 rpm after cutting. Sawdust packed on the cut surfaces, washboarding appeared, and snaking occurred near the bottom of the cut surfaces, which indicated an over-fed operating condition. The cut surface was measured to see the cut deviations across the surface with the results shown in Figure 4.30. The cut deviation was showed both vertically and horizontally. The vertical deviation of (b) was measured at 12 inches from the cut beginning end. It is found that the sawblade did not deviate too much (0.0025 inches) vertically. The horizontal deviation of (a) was measured at two locations \u2014 5 inches and 13 inches from the bottom edge. It is found that the sawblade deviated about 0.006 inches horizontally, compared the 0.0025 inches vertically. The top wheel displacement and loadcell strain change are shown in Figure 4.31 and 4.32. About 0.002 inches downward displacement occurred during cutting and 0.004 inches upward displacement after cutting were measured. This top wheel displacement could be explained as the results of rotation speed change during cutting. The loadcell measured a strain change of about 800 lbs (maximum value) or a maximum cutting force of 400 lbs during cutting (Figure 4.32). In order to provide the saw teeth with a large cutting force, a cut across the grain with a large knot was done in the third set of the tests. Seven cuts were done with about 8 seconds interval between cuts. No noticeable top wheel displacement occurred. About 100 to 200 lbs strain change measured by loadcell occurred for each cut. 4.3.3 Discussion (a). Effect of Rotation Speed on Top Wheel Displacement A top wheel displacement of 0.012 inches occurs when the speed is changed from zero to 600 rpm for the 5ft CanCar bandsaw in the laboratory. The top wheel displacement is proportional to the square of rotation speed. Any increase or decrease of speed by 100 rpm from 600 rpm during cutting could cause about 0.003 inches top wheel displacement. Chapter 4. Top Wheel Dynamics 66 In the laboratory test (set 2), the bandsaw slowed down from 600 rpm to 570 rpm during cutting. This variation of rotation speed resulted in about 0.002 inches downward displacement of top wheel. (b) . Effect of Cutting Force on Top Wheel Displacement The value of cutting force was estimated to be 500 lbs as the maximum for the 16 inches hemlock cant. This force is not enough to cause significant top wheel displacement because of the existence of friction and sticking in the straining mechanism. The cutting force slows down the bandsaw and this change in speed will affect top wheel displacement as discussed previously. (c) . Effect of Heating on Top Wheel Displacement Much heat is generated in the bandsaw especially in poor cooling and heavy cutting conditions. The heat generated by the friction between the wood and the blade could be a major cause of top wheel displacement. The existence of sawdust between blade and wood aggravates this problem. The production of sawdust depends on the Gullet Feed Index (GFI). Also, a large amount of heat could be generated if the sawn board falls down and jams against running blade after cutting, which sometimes happen in sawmills. The results in laboratory tests set 2 does not contradict this heating effect. Because the bandsaw was well cooled and not much heat was generated, tests did not show measurable top wheel displacement. (d) . The cases in Fraser Mills and Elk Falls sawmills A top wheel displacement of maximum of 0.019 inches (average 0.010 inches) measured at Fraser Mills sawmill and 0.009 inches (average 0.0035 inches) at Elk Falls sawmill. As discussed, the top wheel displacement is mainly caused by the change of rotation speed ant. Seating. The cutting forces with a maximum value of 400 lbs had no effect on the top wheel displacement and blade tension. The analysis on top wheel displacement at Fraser Mills shows that 0.00866 inches top wheel displacement would be caused by the Chapter 4. Top Wheel Dynamics 67 heating effect. This top wheel displacement of 0.00866 inches is in good agreement with the measured 0.010 inches average top wheel displacement. The heating effect did not happen to the Elk Falls because of the proper cooling arrangement there. Since they both produced acceptable surface finish their straining mechanisms functioned properly. 4.4 Conclusions Top wheel displacement is mainly caused by the speed changes of the bandsaw and the temperature rise in the blade. The amount of top wheel displacement depends on the cutting duty and cooling condition. For the 5ft CanCar bandsaw, a speed change of 100 rpm from the normal operation speed of 600 rpm can cause about 0.003 inches top wheel displacement. A blade temperature change of 1\u00b0C can cause 0.0013 inches top wheel displacement for a bandsaw with total blade of about 35 feet. The cutting force with an observable value of about 400 lbs does change the blade tension but is not enough to cause significant top wheel displacement because of the friction of the hydraulic cylinder of the straining mechanism. In the case there is no friction of the hydraulic cylinder, this force will only cause about 0.002 inches top wheel displacement because of the large stiffness of the sawblade. The change of the blade tension caused by the cutting force is very small with a value of only 6% of the strain for the test conducted. Small top wheel displacement does not necessarily indicate an improperly functioning straining mechanism. For a bandmill having enough power to cut wood, a large top wheel displacement is mainly caused by the heating due to the improper cooling system. For a bandmill with limited power and carrying heavy cutting, slowing down or speeding up of the bandmill will also contribute to the top wheel displacement. As long as the straining mechanism is functioning the top wheel displacement does not have too much effect on the sawblade cutting performance. Chapter 5 Global Behavior of Sawblades 5.1 Sawblade Behavior The performance of the sawblades directly affects the cutting accuracy and surface finish of the board produced. The bandsaw structural modes and top wheel displacement can affect the cutting performance of handsaws only by affecting the dynamic characteristics of the sawblade. After studying the structural dynamics and top wheel dynamics of handsaws the emphasis is shifted to the study of the sawblade itself. Sawblade Behavior can be discussed in two aspects: the global behavior of sawblades and the local behavior of sawteeth. The sawblade is modelled as either a travelling plate or a travelling band or a travelling string. The dynamic characteristics associated with plate, band, or string characteristics such as natural frequencies etc. are investigated as the global behavior of sawblades. Sawblade behavior associated with the sawteeth vibration is called the local behavior of sawteeth. They are discussed in Chapter 5 and Chapter 6 respectively. Although the sawteeth that actually cut the wood, it is still necessary to investigate the global behavior before considering the effect of the sawteeh. The purpose of this chapter is to develop a narrow band model for a sawblade, in-cluding the effects of rolling stress and top wheel stiffness, which is capable of calculating the natural frequencies and modal shapes of sawblades. The finite element approach is used here and verified by the experimental studies of the two lowest modes. 68 Chapter 5. Global Behavior of Sawblades 69 5.2 Simple Finite Element Model 5.2.1 Equation of Motion For Free Vibration A sawblade between the guides is modelled as a narrow, axially moving, thin beam with end loading as shown in Figure 5.33. The coordinate system to be used in the analysis is defined in Figure 5.34. Only the lateral vibration in the direction of the Z-axis w, and the twist angle w.r.t. the X-axis 9 are considered as variables. The equation of motion for free vibration can be developed by using Hamilton's Principle as follows: Applying Hamilton's Principle to a system with kinetic energy T and potential energy V yields f\\ST - 6V)dt = 0 (5.16) For the thin beam model in Figure 5.33, we have T = \\ l\\mc2 + m(w,t + cw,x)2]dx + \u00b1 \/' pj(9,t + c9,x)2dx (5.17) z Jo I Jo (\"8) Substituting eqn 5.17 and 5.18 into eqn 5.16 yields \\ fh ['{2EIw,xxSw,xx + 2GIpe,x69,x + 2Rw,x6w,x + \\rB28,x66x6wiX -2pJ6it69tt - 2pJc9XX + 2pJ0,tt + 4pJcO,xt + 2PJc29rXX)89}dxdt -{-boundary terms = 0 The boundary terms disappear by the virtue of Hamilton's Principle. Hence, the Euler's equation of motion is EIw^xxx -Rw^x \u2014 mwitt + 2mcwtXt + mc2tu \u201e = 0 -GIp6,xx -\u00a7R8,XX + pJ6,tt + 2pJc$,xt + pJc?8,xx = 0 5.2.2 Finite Element Method Analysis (5.20) (5.21) To solve the Euler's equation of motion eqn 5.21 the finite element method with a modified cubic beam element is used. The modified cubic element is composed of a cubic beam element for lateral deflection analysis and a linear truss element for torsional motion analysis as shown in Figure 5.35. Assume an approximate solution is in the form \u00ab> = [\u00ab){\u00ab(<)} [0 = {'(*)} (5.22) where, 1 - 3\u00a32 + 2f Kt - 2\u00a32+e) 3\u00a32 - 2\u00a33 i - \u20ac t (5.23) (5.24) Chapter 5. Global Behavior of Sawblades 73 Figure 5.35: The Modified Cubic Beam Element Chapter 5. Global Behavior of Sawblades 74 M O ) = w w wa(*) ,x = ft = 3 ^ = >\u2022\u00ab _ 1, (5.25) (5.26) (5.27) 0.* = Substituting eqn 5.22 with eqns 5.23,5.24,5.25, 5.26, and 5.27 into eqn 5.20 and writing it into matrix form yields [h {6wTU){[MJ^j)]{w(i)} + [CW>]{u>W> + +MT\u00ab([Jlf,]{*\u00ab} + + [iY^K^})}* = 0 (5.28) where, [Jlf(j>] = m\u00a3[Nw)[Nn]Tdx [} + [C\u00ab] {u,\u00ab)} + {\u201e,\u00ab} = {0} [Mf] + [CP] {0(0} + {tf(0} = {0} (5.30) [Mw] [0] [0] [Me] W , % (0 + + [0] \" (\u00bb\u2022), \\ ( W J . [0] [C]m 1 w J '[if.] [0] ' I M VI > = < ' {0} . [o] [Ke] . 1 W t . {0} Rearranging the order of the variables yields [M]e {iiy + [C]e {uy + [K]e {uy = {0} where {U}eT = {w1,6wi,6l,w2,8W2,(>2} Element matrices are defined as below (a), consistent mass matrix 156m\/ 420 22\/2m 420 0 54m\/ 420 -13m\/2 420 0 22m\/2 420 4m\/s 420 0 13m\/2 420 -3m\/\u00bb 420 0 0 0 pjl 3 0 0 pjl 6 54m\/ 420 13m\/2 420 0 156m\/ 420 -22m\/2 420 0 -13m\/ J 420 -3m\/ 3 420 0 -22m\/2 420 4ml' 420 0 0 0 til 6 0 0 pjl 3 (\u00bb\u2022) (5.31) (5.32) (5.33) (b). lumped mass matrix Chapter 5. Global Behavior of Sawblades [MY (c). gyroscopic matrix [cr = c -ml 5 0 \u2014m ml 5 ml 2 0 0 0 0 0 0 0 0 0 0 0 0 0 pjl 2 0 0 0 0 0 0 ml 2 0 0 0 0 0 0 0 0 0 0 0 0 0 pjl 2 0 f 0 0 5 mil 30 0 -which is skew-symmetric, (d). stiffness matrix 0 0 0 0 0 PJ m ml 5 0 0 5 -ml 5 -mP 30 ml S 0 0 0 0 PJ 0 0 0 12EI i 6a l\u00bb \"r 5\/ 6E\/ P + a 10 0 \\2El \/\u00bb 6a 51 6EI , P a 10 4EI I + 2ol 15 0 6EI P a 10 2EI I al 30 12E7 I 6a \/\u00bb 51 6EI P a 10 4EJ I 2a\/ \/ 15 where, a = R \u2014 mc2 Chapter 5. Global Behavior of Sawblades 77 1 = GIp + \u00b1RB2-pJc2 5.2.3 Eigenproblem Considering following global equation [M}{U} + [C]{U} + [K}{U} = 0 (5.37) let {q}T = {Q,Q} then above equation becomes (5.38) where, [M*) = [M] [C] [0] [I] [K') = [0] -[K] W [0] Assume the solution has form {5} = {Q}eXt- Substituting into equation 5.38 yields following eigenproblem \\{M*]{Q} = [K']{Q) (5.39) The free vibration mode shapes and the corresponding natural frequencies are ob-tained by solving above eigenvalue problem. The computer package called MATRIV is used to obtain the eigenvalues and corresponding eigenvectors. 5.2.4 Effect of Top Wheel Support Stiffness The effect of top wheel support which is the straining mechanism of the five foot bandsaw can be included through the axial tension in blade. According to Mote's investigation Chapter 5. Global Behavior of Sawblades 78 [9], the axial tension force R generally depends on the top wheel support stiffness and axial velocity of the sawblade. R = Ro + rime2 (5.40) where RQ is the initial axial tension force when the blade is stationary and r? is the compliance coefficient of the system. 77 can be expressed in the following form: V = TTZET (5-41) 1 T\" 2BHE Usually, 0 < n < 1. 5.2.5 Effect of Non-linear Stress Distribution on Torsional Natural Frequen-cies Assume a non-linear stress distribution a(x, y) acting on the beam cross section. Strain energy: S = Hfj^adedx 1 re = - I Torque(x)d6 2 Jo \u2014 - I Torquelx)\u2014dx 2 Jo v ' dx Then, Torque(x) = HJ *& y2o{\u2014)dy Torsional rigidity is expressed as follows: _ . Torque(x) Torsional Rigidity = ^ 1- GJP Chapter 5. Global Behavior of Sawblades 79 That is, Torsional Rigidity = H ft y2ady + \\BH3G (5.42) J-f 3 (a) , parabolic stress distribution (roll-tensioning)[7] 1 4 2 Substituting above equation into eqn 5.42 yields 1 1 4 Torsional Rigidity = -BH3G + \u2014HB3(a0 + \u2014ap) (5.43) 3 12 15 where, <7o \u2014 mean stress in blade <7P \u2014 stress due to parabolic roll-tensioning (b) . uniform stress contribution a{y) = vo Torsional Rigidity = \\BH3G + ^-HB3a0 (5.44) 5.2.6 Effect of tensile force on the torsional rigidity Tensile stress (assuming uniform distribution across the cross section) caused by tensile force R in blade is <7o = so, the resulting torsional rigidity is Torsional Rigidity = \\BH3G + ~~RB2 (5.45) Chapter 5. Global Behavior of Sawblades 80 5.3 Experimental Verifications Stationary and idling tests were conducted in order to verify the effectiveness of the Finite Element Model. Two sets of data were obtained in comparison with the corresponding theoretical results from the simple finite element model. The tests were conducted with an 18GA untensioned sawblade. In idling testing, the sawblade was cooled by cold water sprayed over both the top and bottom guides from a water jet in order to keep temperature of the sawblade constant. Several factors such as strain level, blade travelling speed, tensioning status, tilt motor angle, and heating condition would have effects on the measurement. Among them, only the strain level and travelling speed were considered as variables. The blade was originally untensioned. The heating condition was controlled by the proper water-spray cooling system. The remaining factor was the tilt angle. The tilt angle of the top wheel shaft was not be measured. In tests, the tilt angle was set in such a way that the tooth tips of the sawblade were about one half inch away from the edge of top wheel. The tilt angle at this setting may not be zero. This will affect the natural frequencies measurement. But since this effect was very small it was simply ignored [8]. Only the two lowest modes \u2014 first bending and first torsional modes \u2014 were mea-sured. The electromagnet was set close to the back side and |\/ away from the bottom guide. Two probes were set at the front and the back of the blade and was about |\/ away from the top guide. \/ is the cutting span length. Since the sawblade was untensioned the bending mode and torsional mode were so close together that it was very difficult to separate them. The narrow bandwidth noise signal was firstly used to get these two close peaks. Then a sine wave signal was used to drive the sawblade by tuning the frequency by hand to get the local resonance. In the Chapter 5. Global Behavior of Sawblades 81 stationary test, finger touch was used to feel the mode shapes of the sawblade to distin-guish one from the other. With this information in hand, the tests with the travelling sawblade were done by keeping track of these frequencies. Figure 5.36 is the typical transfer function of the blade. Figure 5.37 is the corre-sponding coherence function. Although the driving point and measuring point were not the same quite good coherence was still seen here at either the first bending mode or the first torsional mode. While the blade was running,various aerodynamic forces and some impulses caused by running the weld butt over both guides and eccentric effect of the wheels acted on the sawblade. These forces will have significant effect on the measurements. Figures 5.38,5.40, and 5.42 show the transfer function at 100 RPM, 200RPM,and 300 RPM. Figures 5.39, 5.41, and 5.43 are the corresponding coherence data. As the speed increases the transfer function becomes messier anH the coherence at resonance decreases. Figures 5.44 and 5.45 show the experimental results and theoretical results. The theoretical results were obtained by using 30 elements. These results are in good agreement and that means that the simple finite element model is valid to calculate the dynamic characteristics of the sawblade. 5.4 Calculation Example: Natural Frequencies of 18GA Sawblades The natural frequencies of 18GA untensioned sawblade are calculated at various strain levels and rotation speeds. Figures 5.46 and 5.47 show some of the results. Figure 5.46 shows only the bending mode with 18,000 lbs strain and 0.0 for r\\. The torsional modes are very close to corresponding bending modes. Figure 5.47 shows the effect of r] with strain at 18,000 lbs on the first bending mode. It is found that the natural frequencies decrease as the blade speed is increasing. The n increases the natural frequencies of the Chapter 5. Global Behavior of Sawblades Chapters. Global Behavior of Sawblades Chapter 5. Global Behavior of Sawblades 84 Figure 5.39: Coherence At 100 RPM (18,000 lbs) Figure 5.40: Sawblade Transfer Function At 200 RPM (18,000 lbs) 00 CD Chapter 5. Global Behavior of Sawblades 44 #AVG 180.0 DG VLG 31.6+00 T Figure 5.42: Sawblade Transfer Function At 300 RPM (18,000 lbs) ft so I i\" \u2022 .1 i r \u2022 i , r i 13000 17900 20000 22500 29000 27900 30000 Strain (lbs) Figure 5.44: Natural Frequencies At Different Strain Levels (Zero RPM) o Figure 5.45: Natural Frequencies At Different Rotation Speed (18,000 lbs) Chapter 5. Global Behavior of Sawblades 92 sawblade. The effect of n on natural frequency increases as the blade speed is increasing. H Q O tJO S-H o C\/l Nort_\u00abJIitwn\u00ablonal Sp^d (C\/Cr> Figure 5.46: Natural Frequencies of 18GA Sawblade At Different Speeds CD co 0 1 I I 1 I I I 1 I I 0 0.2 0.4 0.6 0.8 1 Non-dfmanatcnol SpMd (C\/Cr) Figure 5.47: Effect of compliance coefficient on the first bending mode Chapter 6 Washboarding 6.1 Washboarding Phenomenon Washboarding is a term used to describe a cutting process which will produce wavy cut as the surface of wash boards. For handsaws, washboarding occurs when they are operated at certain running conditions. The basic controlling running conditions for washboarding are believed to be rotation speed, bite per tooth, and strain level of handsaws. Washboarding is widely seen in industry. It is believed to result from the dynamic behavior of sawteeth. The investigation and explanation of washboarding in this chapter is to help researchers to understand this particular phenomenon. The washboarding analysis here is prehminary and experimentally oriented since the objective of this thesis is to consider various dynamic problems in bandsawing. Figure 6.48 shows the cutting process of bandsawing (please refer to Figure 2.5 for a schematic diagram of the bandsaw cutting system). The sawblade with teeth on its front side is travelling down at a speed c. Meanwhile, the wood is moving to the right with the carriage. The kerf is the width of wood removed by the sawteeth. For an ideal cut without the lateral vibration of the tooth, the cut surfaces are perfect flat surfaces. But in reality the sawteeth not only carry the required downward movement but also some lateral vibration. At certain cutting condition the teeth will produce sine wave as shown on cut surfaces . i i i d washboarding cut surfaces are produced. Sometimes only part of the cut surface was affected. Sometimes the whole cut surface 95 Chapter 6. Washboarding 96 (b) Figure 6.48: Cutting Process: Formation of Cut (a). Ideal cut (b). Washboarding cut Chapter 6. Washboarding 97 was affected. Within the same cut surface the contour slope usually changes a little. But the variation of the slopes depends on the operational and cutting conditions. Because of this variation and the uncertainty of the location where washboarding occurs the best way to measure the washboarding board is to take a picture of each washboarding board and measure the maximum depth of washboarding wave. 6.2 Coordinates and Terms Used to Describe Washboarding Figure 6.49 describes the coordinate system of the cut surface. Three directions \u2014 cutting direction (y ), feeding direction (x ), and lateral direction (w ) \u2014 are defined in order to describe the washboarding. The cutting direction is defined as the direction which is parallel to the tooth travelling path. The feeding direction is denned as the direction which is perpendicular to the cutting direction. The lateral direction is perpendicular to the plane of cutting and feeding directions xoy i.e. the cut surface. It is necessary to note that the cutting direction is not the vertical line of the cut surface but intersects with the vertical line with a angle \/?. The angle (3 can be calculated from a speed triangle as shown in Figure 6.49. However,the \/? angle is usual very small ( 1.2\u00b0 for cutting at 600 rpm rotation speed and 200 fpm feed speed ) and therefore is ignored. In order to clearly describe the washboarding surface, two pitches \u2014 Pv and Px are defined too. As shown in Figure 6.49, Pu is the surface wave length in cutting direction and Px is the one in feeding direction. The other terminology used axe the locations on cut surface, the contour lines, and the slopes of the contour lines as shown in Figure 6.49. For a cut surface shown in Figure 6.49, there are three locations on the cut surface need to be denned. These three locations are the beginning of cut, the middle of cut, and the end of cut. The beginning of the cut is the end of the cut surface or the cant that is firstly cut after the saw enters Figure 6.49: Coordinates and Pitches of The Surface i 00 Chapter 6. Washboarding 99 the wood. The middle of the cut locates the middle of the cut surface. The end of the cut is the end of the surface that is the last part of the cant to be cut before the saw leaves the wood. The contour lines are the lines connect the highest spots of the wavy surface without crossing the low spots of the surface. The slopes of the contour lines are the slopes of the projections of these contour lines on the plane of xoy. The terms regarding teeth and cutting process are defined as shown in Figure 6.50. The tooth contains three cutters: one front cutter and two side cutters. Side contact lengths li, I2 axe the actual contact lengths between wood and the side cutters of tooth for each tooth's biting wood. Normally, the side contact lengths are same and equal to the bite per tooth. In the case of the existence of lateral vibration, they are not necessary to be same. 6.3 Experimental Results The whole tests were conducted in according with the test matrices in the following section. The pictures of every cut were taken and presented in the Appendix of this thesis. The results of pitches, maximum surface wave depths, and washboarding occurrences are presented and discussed here. The experiment data for spectrum analysis is presented and analyzed in the \"Explanation and Analysis of Washboarding\" session. 6.3.1 Experiment Procedure Washboarding tests were conducted as described in session 2.2.4. Tables 6.3 and 6.4 summarize the running conditions for the washboarding tests con-ducted. Notation 'X' in the tables represents the test conducted. By choosing three parameters \u2014 rotation speed, bite per tooth, and strain level \u2014 as variables, washboard-ing occurrence against these variables was fully tested. Chapter 6. Washboarding Figure 6.50: Terms for Cutting Process Chapter 6. Washboarding 101 Table 6.3: Washboarding Test Matrix 24,000 lbs strain RPM Bite per tooth 0.032in 0.028in 0.019in 0.009in 600 X X X X 500 X X X X 400 X X X X 300 X X X X Table 6.4: Washboarding Test Matrix RPM Bite Strain (lbs) 30,000 28,000 24,000 20,000 15,000 600 0.028in X X X X X 400 0.019in X X X X X 6.3.2 Geometric Results Types of Washboarding It was found that washboarding with different characteristics occurred, in particular two distinct patterns could be identified. One of them has an average about 1.5 inches pitch in cutting direction and an average about 2.5 inches pitch in feeding direction. The other one has an average about 1.80 inches pitch in cutting direction and less than 0.25 inches pitch in feeding direction. For the convenience of analysis, they are simply called the first type of washboarding and the second type of washboarding. Figure 6.51 shows typical pictures of two types of washboarding. Under some conditions washboarding of both types appeared during the same cut at different locations of the board or cut surface and sometimes both were coupled together. Figure 6.55 shows a pattern with the second type of washboarding only in the middle and coupled washboarding in the end of the cut surface which is the beginning of that particular cut. Figure 6.53 shows the details of Chapter 6. Washboarding 102 Table 6.5: Values of (px,pv) for Different Speeds and Bites RPM Bite per tooth 0.032in 0.028in 0.019in 0.009in 600 (3.7in,1.5in,m) (3.4in,1.2in,m) (1.9in,1.5in,b) 500 (3.5in,1.3in,m) (4.2in,1.5in,m) (1.7in,1.5in,b) 400 (3.8in,1.4in,m) (4.0in,1.8in,e) (1.7in,l-5in,b) 300 (4.3in,1.5in,m) (3.9in,1.5in,m) (4.7in,1.5in,e) the coupled washboarding. Pitches and Wave Depths Table 6.5 and 6.6 summarizes the values of px and py as measured for the first type of washboarding. In these tables, the first number is Px,the second number is py, and the third one indicates the measuring location, i.e.'b' indicates measuring in the beginning of the cut, 'm' the middle of the cut, and 'e' the end of the cut. The contour lines of the cut surface can be classified as two typical types. One of them has straight contour lines with (P*> Pv) independent of the location. The other one has curved contour lines with almost same pv and increasing px from the beginning to the end. In practice, the values of px and pv vary from one part of the surface to another and for some cuts washboarding only occurs in part of the surface as one probably can see from the pictures in the Appendix and thus the values presented here are average values for that cut. Surface measurement shows that pv is essentially independent of blade speed c. The pitches for the second type of washboarding vary too much between cuts. The only evidence can be summarized is that pv ranges from 1.0 inches to 2.6 inches with average of 1.8 inches and p.. ranges from 0.25 inches to zero. Tables 6.7 and 6.8 summarizes the maximum wave depth of the first type of wash-boarding measured from the cut surfaces. Some cut surfaces have steady-state Chapter 6. Washboarding (a). The first type of washboarding (p*,P\u00bb)=T2^in, 1.5in) (b). The second type of Washboarding Figure 6.51: Two Types of Washboarding Figure 6.52: A Cut Surface With Both Types of Washboarding o Chapter 6. Washboarding 106 Table 6.6: Values of (px,pv) for Different Strain Levels RPM Bite Strain (lbs] \u2022 30,000 28,000 24,000 20,000 15,000 600 0.028in (3.4in, (3.4in, (3.3in, (3.6in, (3.2, 1.3in,m) 1.4in,m) 1.4in,m) 1.2in,m) 1.4in,m) 400 0.019in (2.4in, (2.7in, (1.8in, (1.8in, (2.5in, 1.7in,b) 1.6in, e) 1.7in,b) 1.6in,b) 1.5in,m) Table 6.7: Effect of Blade Speed and Bite RPM Bite per tooth 0.032in 0.028in 0.019in 0.009in 600 0.021in 0.030in 0.030in 0 500 0.025in 0.018in 0.020in 0 400 0.020in 0.020in 0.035in 0 300 0.025in 0.035in 0.017in 0 washboarding pattern and their wave depths are constant for the same cut surfaces. Some cut surfaces only involves in small percentage of washboarding and the washboard-ing wave depths vary from the maximum amount to zero depending on the locations. Hence, only the maximum depths of washboarding wave (peak-peak value of washboard-ing wave) were measured. Table 6.8: Effect of Strain on Maximum Wave Depth for the First Type of Washboarding RPM Bite Strain (lbs) 30,000 28,000 24,000 20,000 15,000 600 0.028in 0.030in 0.028in O.OSOin 0.030in 0.026in 400 0.019in 0.015in 0.020in 0.020in 0.030in 0.025in Chapter 6. Washboarding 107 Table 6.9: Effect of Blade Speed and Bite on Maximum Wave Depth for the Second Type of Washboarding RPM Bite per tooth 0.032in 0.028in 0.019in 0.009in 600 0 0 0 0.012in 500 0 0 0.020in 0.007in 400 0 0 0.030in 0.005in 300 0 0 0 0.005in Table 6.10: Effect of Strain on Maximum Wave Depth for the Second Type of Washboarding Strain (lbs) RPM Bite 30,000 28,000 24,000 20,000 15,000 600 0.028in 0 0 0 0 0 400 0.019in 0.005in 0.005in O.OlOin 0.005in 0.005in Tables 6.9 and 6.10 summarize the same results for the second type of washboarding. 6.3.3 Washboarding Occurrence At Various Cutting Conditions The First Type of washboarding The effect of bandsaw speed on the occurrence of the first type of washboarding is shown in Figure 6.54. These results, obtained for a strain level of 24,000 lbs and a bite of 0.028 inches, show that the bigger the rotation speed is, the more the washboarding occurs and washboarding occurrence is almost linearly related to the blade speed. More intensive washboarding was seen at higher speed \u2014 about 700 rpm \u2014 cut. A cut at 200 rpm failed because of the lack of power of the sawmill. The effect of tooth bite at a strain level of 24,000 lbs and a speed of 600rpm is shown in Figure 6.55. It shows that washboarding occurrence increases with the increase of * A S H B 0 A R D I N G t X ) 100 8 f ti-er o fe b on Figure 6.54: Effect of Bandsaw Speed On Occurrence of the First Type of Washboarding o CO 100 2 K fcr cr o B cm BITE PER TOOTH (0.001 INCHES) Figttre 6.55: Effect of Bite Pet Tooth on Occurrence of the First Type of Washboarding o to STRAIN (l.OOOLBS) Figure 6.56: Effect of Strain Level on Occurrence of the First Type of Washboarding Chapter 6. Washboarding 111 Table 6.11: Effect o: the Blade Speed and Bite on the Second Type of Washboarding Bite per tooth RPM 0.032in 0.028in 0.019in 0.009in 600 0% 0% 0% 20% 500 0% 0% 25% 15% 400 0% 0% 60% 90% 300 0% 0% 0% 20% Table 6.12: Effect of Strain on the Second Type of Washboarding Strain (lbs) RPM Bite 30,000 28,000 24,000 20,000 15,000 600 0.028in 0% 0% 0% 0% 0% 400 0.019in 7% 15% 60% 15% 3% bite per tooth. The tests of washboarding occurrence at various strain levels were only conducted at strain level between 15,000lbs and 32,000lbs because of the limitation of the hydraulic straining system. Figure 6.56 shows washboarding occurrence versus the strain levels. Overall, the percentage value of washboarding occurrence is quite high for all these five strain levels. The Second Type of Washboarding Tables 6.11 and 6.12 summarize the results for the second type of washboarding at 24,000 lbs strain level. Since they usually happen at small area the overall occurrence is small and the trend is not so clear. Chapter 6. Washboarding 112 6.4 Explanation and Analysis of Washboarding 6.4.1 Geometric Analysis Two possible mechanisms - beating and chatter - were suspe cted to be applied to the causes of washboarding. The theories of beating vibration and chatter vibration are reviewed in the appendix of this thesis. The geometric analysis is to analyze the evidence of washboarding on cut surfaces and relate the washboarding phenomenon to above two mechanisms. Beating and Washboarding Figure 6.57 shows the kerf constructions and their associated cut surface patterns in both washboarding and beating. In the case of washboarding, the vibration curve of teeth is the same as the cut surfaces curves and the highest spot on cut surface 1 corresponds to the lowest spot on sux\"face 2. In the case of beating, the teeth vibrate in a frequency higher than that of cut surfaces to construct a typical beating phenomenon. The cut surface patterns are the envelopes of the beating vibration and the highest spot on surface 1 corresponds to the highest spot on surface 2. Figure 6.58 shows both surfaces of washboarding cut. The top part of the picture is one face of the cut and the bottom part is the other. It is found that the highest spot of one face corresponds to the lowest spot of the other. The matching patterns of both surfaces of cuts reveal that both types of washboarding are not related to the beating phenomenon in vibration at all. Chatter and Washboarding Sawblade chatters when regenerative force exists. In wood cutting, this regenerative force results from the imbalanced side contact forces of a tooth. Since the regenerative force Chapter 6. Washboarding | tooth front cutter tooth tip center lateral vibration Figure 6.57: Washboarding Versus Beating Chapter 6. Washboarding 114 Figure 6.58: Both Cut Surfaces of Washboarding (a). The first type of washboarding (b). The second type of Washboarding Chapter 6. Washboaxding 115 mainly acts in lateral direction and only the lateral vibration of teeth are considered. This regenerative force are simply called lateral force. Chatter only occurs when regenerative lateral force exists. Different surface profiles are expected to be seen on the cut surfaces between the case of chatter and the case of no chatter. In the case of chatter, regenerative lateral force caused from the lateral vibration regenerates the lateral vibration. In the case of no chatter, no such lateral force exists. Figure 6.59(a) shows the relative position of the sawblade and wood being cut. During cutting, the sawblade with teeth on its front side travels down and the wood is fed against the sawteeth. Tooth marks are left when each tooth move by and remove certain amount of wood. If we look downward from a position above the cut we can see the kerf and sawteeth as shown in Figure 6.59(b) and(c). Actually during cutting the sawblade with teeth on its front edge not only carry the desired downward movement at speed c but also some lateral vibrations. Figure 6.59(b) shows three successive teeth one by one. When no lateral vibration or a little lateral vibration exists each tooth leaves a clear mark on the cut surfaces and side contact lengths li and \/j for each tooth remain same and equal to the bite per tooth b of the bandsaw. The lateral cutting force introduced by the side contact forces are perfectly balanced. Equal spaced and continuous tooth marks are shown on the cut surfaces as the case of Figure 6.59(b). On the other hand, if the lateral vibration of the tooth is large enough that the side contact lengths \/i and \/2 are no longer same side contact forces are not balanced. Due to the geometry, one of the side contact lengths has to equal to the bite per tooth b and the other one to be larger than b. The previous tooth mark on the longer side contact side is chipped off. The tooth marks are not equal spaced and are discontinuous as shown in the c,-\\se of Figure 6.59(c). Regenerative lateral force exists because of the imbalanced side contact forces. Figure 6.60 shows the details of the typical cut surfaces of two types of washboarding. Figure Chapter 6. Washboarding 116 6.60(a) is correspondent to the case of equal \/j and l2 and Figure 6.60(b) is correspondent to the case of unequal l\\ and l2. The cut surface of the second type of washboarding shows that this is a chatter phe-nomenon. The chatter frequencies can be measured by the measurements of the pitches on the cut surfaces. The Conditions for Chatter to Occur Figure 6.61 shows the relative tooth positions during cutting, where the x coordinate is in the feeding direction and w is in the lateral direction. In order to determine the involvement of lateral force caused by various side contact lengths in the washboarding process, consider the situation shown in Figure 6.61. Case (a) shows the situation of successive teeth each taking a bite b when tooth lateral vibration does not exceed certain limit (roughly straight) to cause unequal side contact forces. The shaded area represents the side contact between the wood and the tooth and in this case be seen to be balanced. It would thus be expected, if the tooth were moving down in the plane of the undeformed blade, that the lateral forces on the tooth would be balanced. Case (b) shows the situation in which a tooth deviates from a straight line, but may still be assumed to be moving vertically down in a plane parallel to that of the undeformed blade, is shown. In this case the lateral contact between the tooth and the wood is no longer balanced and some resulting lateral force is applied to the tooth. From the geometry of the situation it is seen that such a situation occurs if the lateral displacement exceeds 6 tan a during a bite size of by i.e. if the lateral displacement of the teeth has a slope larger than tan a, lateral force from this effect will be induced, but if the slope is lower than tana they will not. Chapter 6. Washboarding equal side contact lengths -\u2014 wood teeth EVEN BITES KERF h (b) A \u2014 A continuous tooth marks unequal side contact lengths wood UNEVEN BITES k r.lteeth A A discontinuous tooth marks KERF *\u2014A (c) Figure 6.59: Cut Surface Profile (a), sawblade and board (b). equal side contact lengths (c). unequal side contact lengths Figure 6.60: Details Of Washboarding Cut Surfaces (a) The First Type Of Washboarding (b) The Second Type Of Washboarding Chapter 6. Washboarding 119 Figure 6.61: Relative Tooth Positioning Chapter 6. Washboarding 120 The slope of the washboarding surface wave can be obtained based on measuring the surface pattern. Above statement can be well formulated in the following mathematical form. If we assume that the centers of tooth tips washboard in a wave with a pitch of px and an amplitude of d we have washboarding pattern expressed as follows w \u2014