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Parametric vibrations of traveling plates and the mechanics of washboarding in bandsaws Luo, Zhusan 2001

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PARAMETRIC VIBRATIONS OF TRAVELING PLATES AND THE MECHANICS OF WASHBOARDING IN BANDSAWS by ZHUSAN LUO B. A. Sc. Nanjing University of Aeronautics and Astronautics, China 1982 M. A. Sc. Nanjing University of Aeronautics and Astronautics, China 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 July 2001 © ZhusanLuo, 2001 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 of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, Canada V6T 1Z4 Date: Abstract This thesis presents a study on the high frequency dynamic behavior of traveling plates subjected to in-plane stresses. The effects of damping, parametric and modulated excitations on the vibration characteristics of the plates are considered. The application of this work to the explanation of the mechanisms responsible for washboarding in bandsaws is presented. A high frequency mode of a traveling plate is defined using the envelope of its mode shapes. Modal analysis of a traveling plate is conducted theoretically and experimentally. The effects of the traveling speed, the in-plane stresses and the plate geometry on the natural frequencies are examined. The self-excited vibration of a smooth band subjected to lateral regenerative forces is studied and the instability regions are determined. In order to model the tooth profiles and the moving lateral cutting forces applied at the teeth of a bandsaw blade, a finite element model is built and three moving plate elements are developed. The stability of the blade is investigated based on this model. The analytical results based on the smooth band model and the finite element model are consistent with the experimental results from modal testing. The vibration responses and stability of a damped spring-mass system and a damped smooth band subjected to both parametric and modulated excitations are investigated. The maximum magnification factor corresponds to the excitation at the lowest exciting frequency due to the effect of regenerative forces. The instability regions of this system are reduced by increasing damping or decreasing regenerative forces. A kinematic model of washboarding is built based on the loci of teeth so that a washboarding pattern can be simulated and the washboarding mode can be determined by decoding the pattern. Two types of washboarding patterns are observed in the cutting tests. Type I washboarding is explained as the result caused by the modulated and forced vibration due to the displacement excitations from the guides. Type II washboarding is caused by the self-excited vibration due to the regenerative cutting forces. ii Table of Contents Abstract 1 1 Table of Contents H i List of Tables v i List of Figures v » Nomenclature x » Acknowledgments xv"> 1 Introduction 1 1.1 Background 1 1.2 Previous Research 7 1.3 Objectives and Scope 9 2 Vibration Characteristics of Traveling Bands 11 2.1 Model of the Traveling Band 12 2.2 Mode Coupling and Mode Orders 15 2.3 Effects of the Guide Constraints 20 2.4 Effects of the Blade Speed 25 2.5 Effects of Internal Stresses 27 2.6 Effects of the Tooth Gullet 39 2.7 Summary 46 3 Stability of Traveling Bands Subjected to Parametric Excitations 48 3.1 Model of Regenerative Cutting Forces 49 iii 3.2 Stability Analysis of a Smooth Band 52 3.3 Instability Regions of a Smooth Band 57 3.4 summary 62 4 Modal and Stability Analyses of Moving Blades Using FEA 64 4.1 Modeling Moving Cutting Forces Applied at Tooth Tips 64 4.2 4-Node Moving Bilinear Plate Element 67 4.3 3-Node Triangular Plate Element 71 4.4 Variable-Domain Rectangular Plate Element 80 4.5 Stability Analysis of the Toothed Blade 86 4.6 Analytical Results 91 4.7 Experimental Results from Modal Testing 100 4.8 Summary 113 5 Responses of Damped Systems to Parametric and Modulated Excitations 114 5.1 Modulated Cutting Forces 114 5.2 Stability and Response of a Damped Spring-Mass System 117 5.3 Stability and Response of a Traveling Band 124 5.4 Summary 132 6 Kinematics of Washboarding 134 6.1 Washboarding Patterns Caused by a Single Traveling Wave 135 6.2 Washboarding Patterns Caused by Multi-Traveling Waves 140 6.3 Summary 144 7 Mechanics of Washboarding 146 7.1 Washboarding Caused by Modulated Vibration 147 7.2 Washboarding Caused by Self-Excited Vibration 162 7.3 Comparison of Two Types of Washboarding Mechanisms 168 iv 8 Conclusions Bibliography Appendix A Structure and Parameters of the Bandsaw Appendix B The System Matrices and the Generalized Force Vector in the Equations of Motion of a Moving Plate Appendix C Shape Functions and Explicit Expressions for MDKT Element Matrices List of Tables 2.1 Natural Frequencies of Smooth Bands and Toothed Blades 41 3.1 Effects of the Order of Fourier Series 57 4.1 Natural Frequencies of a Stationary Plate 91 4.2 Natural Frequencies of a Stationary Plate with Internal Forces 92 4.3 Natural Frequencies of a Moving Plate with Internal Forces . . . 93 4.4 Natural Frequencies of a Moving Plate with Internal Forces 94 4.5 High Natural Frequencies of a Moving Plate with Internal Forces 94 6.1 Washboarding Patterns from Metal Cutting 139 7.1 A Comparison of Washboard Pitches (Type I) 157 7.2 Vertical Pitches of Different Modes 164 7.3 A Comparison of Washboard Pitches (Type II) 166 7.4 Properties of Two Types of Washboarding Patterns 168 A . l Dimensions and Main Parameters of the Bandmill 179 A.2 Geometry of the Bandsaw Blades 179 vi List of Figures 1.1 An industrial bandsaw mill 3 1.2 Schematic of the bandmill 4 1.3 A wide bandsaw blade 5 1.4 Washboarding patterns in bandsaws: (a). Type I; (b). Type II 6 2.1 Bandsaw blade over the cutting span 12 2.2 Mode shapes: (a). Mode (1, 1); (b). Mode (2, 2) 16 2.3 Mode shapes and their envelopes: (a). Mode (1, 1); (b). Mode (2, 2) 17 2.4 Mode shapes and their envelopes: (a). Mode (9, 4); (b). Mode (2, 6) 19 2.5 Typical frequency response functions: (a). Between the upper span and the cutting span; (b). Between the lower span and the cutting span 21 2.6 Effect of the span length on the first lateral and torsional modes 23 2.7 Effect of the span length on high frequency modes 24 2.8 Effects of blade speed on lateral and torsional modes 26 2.9 Effects of blade speed on high modes 27 2.10 Lateral (LI) and torsional (Tl) frequencies increase with strain 28 2.11 Effect of the blade thickness on natural frequencies: (a). First bending mode; (b). First torsional mode 29 2.12 Effect of the blade thickness on high modes 30 2.13 Typical residual stresses induced by roll-tensioning 31 2.14 Effects of tensioning stresses on the first bending and torsional modes . . . 31 2.15 Two lowest modes of the stationary band: 33 (a). First torsional mode (1, 2); (b). First lateral mode (1,1) 2.16 Mode shapes, tensioning stresses and strain energy density at the first 33 (a) torsional mode and (b) lateral mode 2.17 Effects of tensioning stresses on high modes 35 2.18 Mode shapes of the blade installed on a tilted wheel (0C=0.4°): vii (a) . Lateral mode; (b). Torsional mode 36 2.19 Effect of the tilt angle on bending and torsional modes 37 2.20 Effect of the tilt angle on high modes 37 2.21 Wheel crown induced stress distribution 38 2.22 Two types of gullets 40 2.23 Meshes of a toothed blade (725 nodes and 914 elements) 40 2.24 Smooth Blade (42A): (a). Mode shape of the smooth blade; (b) . Mode shape at the cross-section at x=L/2 42 2.25 Toothed Blade (42A): (a). Mode shape of the smooth blade; (b). Mode shape at the cross-section at x=L/2 43 2.26 Mode shape of smooth blade (50A) at x=L/2 44 2.27 Mode shape of toothed blade (50A) at x=L/2 44 2.28 Frequency response functions of two toothed blades: (a) . Gullet Type 42A; (b). Gullet Type 50A 4 5 3.1 Regenerative cutting force: (a). Two balanced side cutting forces; (b) . Two unbalanced side cutting forces 50 3.2 Model of the cutting forces 51 3.3 (a) Natural frequency and (b) real part of the first mode 58 3.4 Stable vibration of a smooth band (F, = (2F„/3, F„)) 59 3.5 Unstable vibration of a smooth band (F, - (Fnl2, 2F„/3)) 59 3.6 (a) Natural frequencies and (b) real parts of eigenvalues of the blade subjected to regenerative cutting forces (Ro=2Q kN, K /=10 kN/m) 61 3.7 Effects of the cutting force coefficient on (a) natural frequency and (b) real part of Mode (9, 2) 62 4.1 Finite element model of the blade 66 4.2 Isoparametric bilinear element: (a). 4-node plate element; (b). Natural coordinates 68 4.3 (a) Triangular plate element and (b) area coordinates ' 75 4.4 Variable-domain rectangular plate element 81 viii 4.5 Eigenvalues of the first ten modes: 95 (a). Real parts; (b). Natural frequencies 4.6 Real parts of the first two modes: (a). Mode (1, 1); (b). Mode (2, 1) 96 4.7 Eigenvalue of Mode (1, 6): (a). Real part; (b). Natural frequency 97 4.8 Eigenvalue of Mode (10, 3): (a). Real part; (b). Natural frequency 98 4.9 Comparison of the real parts of Modes (8, 3) and (2, 6) based on the finite element model and the smooth band model 99 4.10 Effects of the depth of gullet of the blade on its stability 100 4.11 Locations of probes over a blade 101 4.12 Frequency response function of the first eight modes 102 4.13 Natural frequencies of the first eight modes 102 4.14 Mode shapes of (a) Mode (1, 1) and (b) Mode (2,2) 103 4.15 Natural Frequencies of High-order modes 104 4.16 Frequency response function of high-order modes 104 4.17 Mode shapes of (a) Mode (1,6) and (b) Mode (2, 6). . 105 4.18 Natural frequencies of the first four modes varying with the blade speed: (a) . Mode(l, 1) and (1, 2); (b). Modes (2, 1) and (2, 2) 1 0 7 4.19 Envelopes of low frequency modes: (a). Mode (1, 1); (b) . Mode (1, 2); (c). Mode (2, 1); (d) Mode (2, 2) 1 0 8 4.20 Natural frequencies of high-order modes varying with the blade speed: (a). 110 Modes (1, 5) and (2, 5); (b). Modes (1, 6) and (2, 6) 4.21 Envelopes of high frequency modes: (a). Mode (1, 5); (b). Mode (2, 5); (c) . Mode (1,6); (d) Mode (2,6) I l l 5.1 Lateral cutting forces due to (a) guide motion and (b) weldment or dished areas 5.2 A single-degree spring-mass-dashpot system subjected to a 117 regenerative force and a dynamic force 5.3 (a) Natural frequency and (b) real part of the system with different damping ratios (k] = 10 kN/m) 120 •^4 Real part of eigenvalues of (a) the undamped and (b) the damped systems 121 with different regenerative force coefficients ix 5.5 Magnification factors vs. excitation frequency 122 5.6 Variation of magnification factor with frequency 123 5.7 Frequency spectrum of the response 124 5.8 Effect of the damping ratio on the instability region: (a) . Mode (1, 2); (b). Mode (10, 2) 1 2 6 5.9 Effect of the regenerative force coefficient on the unstable region 126 5.10 Magnification factors of the responses to three excitations: (a). £ = 0.002 at (0.25, B); (b). C = 0.006 at (0.25, B); (c). C = 0.010 1 2 9 5.11 Envelopes of a mixed Mode from Modes (9, 3) and (2, 6) 130 5.12 Dominant modes: (a). (xst, xena) = (0.114, 0.495); (b) . (xst, xend) = (0.343, 0.495). 131 6.1 Washboarding pattern 134 6.2 Motion of two points on a moving string 135 6.3 Loci of two teeth 138 6.4 Washboarding pattern caused by Mode (10, 1): (a). Washboarding pattern; (b). Loci of three teeth 142 6.5 Washboarding pattern caused by Mode (1, 6): (a). Washboarding pattern; (b). Loci of three teeth 143 6.6 Washboarding pattern caused by Mode (2, 6): (a). Washboarding pattern; (b). Loci of three teeth 144 7.1 Set-ups for the cutting tests (Blades T l and T2) 147 7.2 A typical washboarding pattern (Type I) 148 7.3 Power spectra of the vibration signal (Washboarding occurs) 149 7.4 Band-pass filtered signal (Washboarding occurs) 152 7.5 Envelope of the filtered signal (Washboarding occurs) 152 7.6 Spectrum of modulation functions of this signal (Washboarding occurs).. 153 77 Spectrum of the original signal (Washboarding occurs) 153 7.8 (a) Power spectra and (b) band-pass filtered signal (without washboarding) 154 7.9 Magnification factors of the responses to three excitations 156 X 7.10 Simulated washboarding pattern 157 7 1 1 Loci, dynamic profiles and envelopes of two teeth 153 7.12 Variation of average depth of washboards with the blade speed 160 7.13 Extent of washboarding vs. bite per tooth 161 7.14 A typical washboarding pattern (Type II) 162 7 - 1 5 The band-pass filtered signal (F, = 1176 Hz) 1 6 3 7.16 Spectrum of the captured signal 163 7.17 Primary unstable region of Mode (7, 1) 165 7.18 Mode shape of Mode (7, 1) 165 7.19 Simulated washboarding pattern (Type II) 166 7.20 Loci, dynamic profiles and envelopes of two teeth 167 A . l Structure of the Bandmill: (a). Front view without the band and two wheels; (b). Side view without the band and the bottom wheel 180 XI Nomenclature a amplitude of modulation function a, b Fourier series coefficient vectors of transverse displacements c blade speed/damping coefficient (Section 5.2) Cb, Cf backward/forward wave velocity c complex Fourier Series coefficient vector of transverse displacements dc depth of cut dg depth of gullet d complex Fourier series coefficient matrix of regenerative force matrix / element displacement vector (Section 4.3) eg distance between the tooth tip and the mid-point of tooth root f distributed force fb, ff backward/forward wave frequency f, fe element nodal force vector g, g e q element gyroscopic matrix h blade thickness k stiffness k] regenerative force coefficient (Section 5.2) k, ke q element stiffness matrix l(t) length of variable-domain element l0 average length of variable-domain element m number of mode functions along axis x/mass (Section 5.2) mw mass of wheel m element mass matrix m0 average element mass matrix n number of mode functions along axis y nc number of velocity increments in Region I or III p ratio of the wheel rotating frequency to the tooth passing frequency xii p(x,y,t) distributed transverse force q(x,y,t) distributed transverse load q modal coordinates r ratio of response frequency to natural frequency rc radius of wheel crown rw radius of wheel s coordinate of tooth starting from xst t time to time at which a tooth starts its bite u displacement along axis x u element displacement vector v displacement along axis y w transverse displacement of plate x coordinate of plate/blade xc centroidal coordinate of an element along axis x xsl, xend coordinates of the top and bottom lumber surfaces x,j coordinate of 7-th tooth along axis x at time t y coordinate of plate/blade y,j coordinate of 7-th along axis y at time t z coordinate of plate/blade A area of triangular element Ao vibration amplitude Ak Fourier series coefficient vector of regenerative forces (Section 4.5)/ real part of response vector (Section 5.3) B blade width Bc distance between the wheel crow center and the wheel rim Bg guide width Bt distance between the wheel center and the tilting center Bw wheel width Bx Fourier series coefficient vector of regenerative forces (Section 4.5)/ xiii imaginary part of response vector (Section 5.3) c damping matrix D time-delay operator DE bending rigidity E Young's modulus F vibration frequency Fn natural frequency Fr wheel rotating frequency Ft tooth passing frequency FwOi Fwj modulated excitation amplitude F lateral cutting forces regenerative cutting force applied to j-th tooth F system force vector G shear modulus G system gyroscopic matrix G e condensed system gyroscopic matrix G(t) time-dependant system gyroscopic matrix G average condensed system gyroscopic matrix Hg height of guide H coefficient matrix in characteristic equation It indicator matrix for cutting force locations Ite-condensed indicator matrix J Jacobian Matrix Ki regenerative cutting force coefficient Ks top wheel support stiffness regenerative force coefficient matrix K system stiffness matrix K e condensed system stiffness matrix K average condensed system stiffness matrix K(t) time-dependant system stiffness matrix L length of cutting span L], L2, L3 area coordinates xiv total blade length Lw center distance between two wheels Myj, MyXj moments about axis y and x M system mass matrix M e condensed system mass matrix M 0 average condensed system mass matrix M(t) time-dependant system mass matrix M average condensed system mass matrix Ni shape function of triangular element Nx in-plane force in the direction x effective axial force Nxy in-plane shear force Ny in-plane force in the direction y N coordinate transform matrix/shape function of variable-domain element P tooth pitch PX,Py horizontal/vertical pitches Q generalized force vector R0 axial strain T tooth passing period/kinetic energy u, strain energy due to axial forces U global displacement vector displacement vector u 2 rotation vector V Strain energy/feed speed (Chapter 6) vh bending strain energy wnc virtual work w0 response amplitude X(x) mode function along the longitude X M magnification factor Y(y) mode function across the width z Coefficient matrix (Section 5.3) XV a, P constants for Rayleigh damping matrix 8 Dirac-delta function e strain £/, bending strain <l> vibration response phase Y shear strain coefficient of the centrifugal force/natural coordinate (Chapter 4) backward/forward wave phase angles K shear correction factor X complex eigenvalue Xb, Xf backward/forward wavelength V Poison's ratio 0 mo phase angle rotations about axes x and y 9/a tilt angle P mass density of the blade material oo stress due to axial strain ob stress due to blade bending over the wheel Oc stress due to centrifugal force Ocrown stress due to the wheel crown Ocut stress due to cutting force °P tensioning stress amplitude Otemp stress due to temperature change Oten tensioning stress Otilt titling stress Ota stress amplitude due to tilting U) circular frequency co„ natural frequency modulating/wheel rotating frequency xvi co, tooth passing frequency natural coordinate shape function of bilinear element c damping ratio A eigenvalue matrix O product of mode functions X(x) and Y(y) z summation modified mode function V 4 bi-harmonic operator of plate Superscripts T Transpose of vector or matrix — complex conjugate Subscripts 5 partial differentiation w.r.t coordinate / to n indices X V l l Acknowledgement I feel extremely fortunate to have the opportunity to work with the people at the Wood Machining Laboratory at U. B. C. I would particularly like to thank my supervisor, Professor Stanley Hutton, for his support, guidance and encouragement throughout my studies. I would also like to thank Professor Gary Schajer for his patient guidance during my supervisor's sabbatical year, Professor Mohamed Gadala for many useful discussions on the development of finite element model, and Professor Bruce Dunwoody for his kind suggestions on this thesis topic. I am very grateful to Dr. Guoping Chen and Dr. Jifang Tian for many informative discussions and their valued friendship. Dr. John Taylor and Mr. John White deserve special thanks for their valuable advice and assistance during the cutting tests. I am a indebted to my fellow student, Suresha Udupi, for his useful help in my experimental work. I would like to acknowledge the financial support by the Natural Sciences and Engineering Council of Canada. Finally, my special thanks go to my wife, my parents and my daughters. I could not have done it without their continuing understanding, encouragement and support. xviii Chapter 1 Introduction 1.1 Background A five-foot industrial bandsaw mill used for resawing in sawmills is shown in Figure 1.1. This bandmill cuts wood using the relative motion of wood held on a carriage to that of a bandsaw blade driven by a band-wheel system. The band-wheel system, as shown in Figure 1.2, primarily consists of a top wheel, a bottom wheel that drives the whole system, two guides and a bandsaw blade that runs on the wheels. The two wheels and two guides are supported on a vertical column of the mill frame. The whole frame with the band-wheel system can move along two guide rails on the mill base by hydraulic setworks so that the whole mill can be in the correct position for a certain width of lumber during cutting. The bottom wheel is driven by a hydraulic motor through a belt drive. This wheel can run at speeds up to 650 rpm. The normal operating speed is 600 rpm, and the corresponding saw blade speed is approximately 50 m/sec. The top wheel can be raised by a hydraulic cylinder so that the blade can be stressed and thus stiffened. This process, called straining, is applied before the mill runs. An axial tensile force from 10 kip up to 32 kip can be applied to the blade by the hydraulic straining system. The top wheel can also be tilted by an electric motor to position the running saw blade properly. Tilting the blade, measured by a tilt angle, not only increases the tracking stability of a running blade, but makes the tooth side of the blade tighter. Two wheels are properly crowned to prevent the saw blade from pulling off the wheels. The profile of wheel is a circular arc (Eschler, 1982). Two pressure guides made of reinforced rubber are supported on two guide arms connected with the vertical column of the mill frame. The guides are aligned to be 3/8" proud of the tangent line of the two wheels to make the blade more stable in the cutting region. During cutting, wood held on a log carriage passes through this span of blade. Water is used l Chapter 1 Introduction 2 to lubricate the guide surfaces and the blade and to remove some of the heat generated due to friction and cutting. A wide bandsaw blade shown in Figure 1.3 is manufactured from a steel strip. A typical material of the steel strip is nickel alloy (Simmonds 1980). This material is capable of withstanding the shock loading of the teeth striking timber and the flexing as it passes the wheels. It is also tough and ductile enough for further doctoring. The teeth are formed by cutting out the gullets and swaging the tooth tips. Then two ends of the bandsaw ribbon are brazed together to produce an endless band. The weldment must be properly ground. This blade is described mainly by the geometric parameters shown in Figure 1.3. The tooth pitch is the distance between two teeth. A gullet is the area where the material has been removed from the strip. It is reserved for sawdust during cutting. The gullet size is measured by the depth of gullet. A hook angle, a sharpness angle and a clearance angle are used to define the shape of a tooth tip. A kerf is the front edge of a tooth face, which is wider than the blade thickness. The difference between them is twice the side clearance. Other parameters will be explained where necessary. Before a bandsaw blade is installed on to the wheels, it must be well prepared by leveling, sharpening and tensioning. Tensioning is the process in which residual stresses are introduced by rolling the central region of the blade between pinch rollers that cause local plastic deformation of the blade. As a result, compressive stresses are applied to the central region and tensile stresses to the edges of the blade. The purpose of tensioning is to make both the tooth side and the backside stiffer, and to stabilize the running blade on the wheels. An axial strain is applied to the blade by raising the top wheel. The top wheel must have the correct tilt angle for a given strain level. During cutting, the saw is subjected to the cutting forces shown in Figure 1.2. The structure and dimensions of the bandmill with three bandsaw blades used in this study are described in Appendix A. Figure 1.1 An industrial bandsaw mill Figure 1.2 Schematic of the bandmill Chapter 1 Introduction 5 Figure 1.3 A wide bandsaw blade A bandsaw is one of the extensively used saws in the wood cutting industry. It can be found in nearly every phase of wood machining, from primary log break down to various resawing operations in sawmills, and to furniture manufacture in carpentry shops. The wide application of bandsaw results from its capacity to handle a wide range of primary log sizes, operate at a high cutting speed, and produce relatively thin kerf. Although the bandsaw has some advantages over other saws, it may experience some problems with poor cutting performance. The cutting performance is generally evaluated by means of the deviation of the board thickness from the target size. Poor cutting performance means a large mean or standard deviation of the board thickness. Figure 1.4 shows two pieces of wood with regular sinusoidal-like patterns on the sawn surfaces. Each surface looks like a washboard. This phenomenon, that of a sinusoid-like pattern produced on the sawn surface during woodcutting, is called washboarding. The effect is similar to chatter in metal cutting. Chapter 1 Introduction 6 Washboarding has three negative effects on cutting performance. First, to produce flat lumber of a given size, lumber will need to be removed by planing and more wood will be removed. This will consume material and energy and decrease the wood recovery rate. Secondly, thin kerf saws have been developed to effectively reduce the amount of kerf. But when reducing the blade thickness washboarding is frequently encountered. So the use of thin saw blades is limited. Finally, the feed speed is determined based on the area of the gullet. To increase production, a deep gullet is preferred. However, washboarding is more serious in a blade with deeper gullets. (b) Type II Figure 1.4 Washboarding patterns in bandsaws Washboarding is a serious problem in sawmills. It can occur in both bandsaws and circular saws. Once it is encountered, it is difficult to get rid of it. In some cases, changes of operation conditions or modification of saw blades can help stop the washboarding. But the same measures do not work in all cases. It is known that washboarding occurs when the saw blade experiences unstable high-frequency vibrations. But the kinematics and mechanics of washboarding in bandsaws are not fully understood. Chapter 1 Introduction 1 1.2 Previous Research It has been common to assume that the vibration of the blade in the region between the guide pads was uncoupled from the remainder of the system (Mote 1972; Ulsoy, 1978a, b; D'Angelo III, 1985). Naguleswaran (1968) modeled a bandsaw blade as an axially moving string and studied the self-excited vibration and stability of the blade due to periodic variations in the band tension. Mote (1965a, b) first analyzed the dependence of the band tension and the lowest natural frequency upon the axial velocity and the straining system using a beam model. After that, Mote (1966) and Ulsoy (1982) modeled a wide bandsaw blade as an axially moving plate. Both ends of the beam or plate were simply supported. They investigated the contributions of axial velocity, the stiffness of wheel support system, blade damping, and in-plane stresses to band vibration and stability. Ling (1989a, b, c) investigated forced vibration of a band excited by the eccentricity of wheels, unbalancing of bearings and butt weld of the saw blade. Hutton (1989) and Zhan (1990) first studied the causes for washboarding and self-excited vibration during sawing and observed two types of washboarding patterns (as shown in Figure 1.1). They suggested that one may be caused by forced vibration due to tooth impacts, the other by self-induced vibration due to regenerative effects of side cutting forces. Yokochi (1990) and Okai (1996a, b; 1997) found from their cutting tests that when the tooth passing frequency is slightly higher than the natural frequency, the lateral cutting force is in phase with the instantaneous velocity of the blade vibration and energy will be given to the saw blade. Then the saw blade vibration is excited and washboarding occurs. Lengoc (1999) reviewed the experimental results (Okai, 1996) by modifying their analytical model. He concluded that all the washboards were produced by torsional modes rather than both bending and torsional modes. It was suggested that squeeze film forces existing between the blade and the sawn surfaces might be large enough to damp out all the bending modes. Okai (1996a, b; 1997) also explained how a washboarding pattern is kinematically produced and built the relationship between the patterns and tooth passing frequency and Chapter 1 Introduction 8 natural frequency based on standing wave motion. Lehmann (1997) proposed another kinematic model of washboarding by considering lateral tooth displacement and indicated that a washboarding pattern is associated with traveling wave motion in the saw blade. It is conventionally believed by Hutton (1990), Okai (1996a) and Tian (1998) that washboarding occurs when the system is unstable due to parametric excitations. Based on an axially moving beam model, Wu (1986) studied the parametric excitation caused by periodic in-plane edge loading normal to the longitudinal axis of the band and the instability at low frequencies excited by higher frequency edge forces. Lin (1997) investigated the stability of a moving plate and predicted the traveling speed at the onset of instability based on linear plate theory. Lengoc (1995) studied stability of a moving plate subjected to distributed cutting forces. Since their studies focus on the dynamic behavior of the first few modes, the results can not be applied to the explanation of washboarding in bandsaws. Tian (1998) modeled a circular saw as a rotating disk and investigated both washboarding and instability in circular saws subjected to multiple regenerative and follower cutting forces. He concluded that washboarding was caused mainly by regenerative forces. In his model the cutting forces were applied on the rim of a disk. It can be seen by reviewing the research literature: 1) . A bandsaw blade has been modeled as an axially traveling media over the cutting span. An industrial wide bandsaw blade was modeled as a traveling plate. Its transverse vibration was studied with the emphasis on low frequency regions. High frequency vibration characteristics of the blade that are closely related to washboarding have not been studied. 2) . The stability of a bandsaw blade subjected to multiple regenerative forces has not been studied. 3) . Many analytical methods have been developed to predict the dynamic performance of smooth bands under ideal boundary conditions and loading. To model a real sawblade, however, analytical methods are not as efficient as a finite element based approach which provides the maximum flexibility in modeling tooth gullets and pressure guides. Chapter 1 Introduction 9 4) . There is no literature pertaining to the study of vibration response of a bandsaw blade subjected to both parametric excitations due to the effects of regenerative forces and time-dependent excitations due to tooth impacts and unbalanced centrifugal forces from wheels. 5) . Washboarding patterns were predicted based on either standing waves or lateral blade motion and the assumption that the tooth passing frequency is close to the natural frequency. Traveling waves in a blade were not included in the reviewed kinematic models. Thus these models not only limited the application to the case where the tooth passing frequency is far from the natural frequency, but also possibly misunderstood the information on the blade vibrations contained in washboarding patterns. 1.3 Objectives and Scope The objectives of this study are: 1) to explain the mechanisms of washboarding in bandsaws; 2) to understand the effects of a variety of primary factors, and 3) to propose practical solutions to stop washboarding. In order to achieve these goals, an industrial bandsaw blade will be modeled as a traveling plate over the cutting span. Several models will be built to deal with different issues. High frequency vibration of the saw blade over the cutting span will be theoretically and experimentally investigated. The study will determine the effect of primary factors, such as axial strain, pre-tensioning, tilting, and blade speed, on the vibration behavior of the blade, and examine the effectiveness of the developed model in the prediction of the dynamic behavior of the blade. A dynamic model for stability analysis of a blade subjected to multiple cutting forces will be developed. Based on the model, the unstable regions will be determined and the effects of some factors on the stability will be examined. In order to efficiently model the moving forces applied at tooth tips, the geometry of tooth profiles and actual boundary conditions, a finite element model of a toothed blade will be developed. This model will require the development of several types of axially traveling plate elements. A 4-node bilinear plate Chapter 1 Introduction 10 element, a 3-node triangular plate element and a variable domain rectangular plate element will be developed. In order to examine the effects of tooth impacts and dynamic disturbances from unbalanced wheels on the high-frequency vibration behavior of the blade, the vibration responses to both parametric excitations and the dynamic disturbances will be studied. It is noted that washboarding occurs when the blade experiences severe vibrations that may be caused by both regenerative cutting forces and other time-dependent dynamic disturbances. The kinematics of washboarding in bandsaws will be studied and a kinematic model will be built based on teeth loci and traveling waves in the blades so that washboarding patterns can be predicted and decoded. Based on these models, the mechanics of washboarding in bandsaws will be studied and the conditions under which washboarding occurs will be determined. Modal and cutting tests will be conducted to verify the kinematic model of washboarding and the dynamic models of the blade used in this study. Chapter 2 Vibration Characteristics of Traveling Bands The bandsaw blades considered in this study will be modeled as a traveling plate. The vibration behavior of the blade in the cutting region governs the cutting performance of the bandmill. Under normal cutting conditions, the standard deviation of board thickness, for example, is mainly dependent on the tooth-tip stiffness of the blade (Taylor, 1993). The first lateral and torsional modes of the blade are associated with the static stiffness. Ulsoy (1982) modeled the blade as a smooth band model, i.e. a bandsaw blade without teeth, and studied the dynamic behavior of the first few modes of the band. It is known that the washboarding in a narrow bandsaw (Okai, 1996) or a circular saw (Tian, 1998) occurs when the tooth passing frequency (900 ~ 1,200 Hz) is slightly higher than some natural frequency. Thus in this case very high frequency modes are of specific interest. The aim of the work presented in this chapter is to systematically investigate the high frequency vibration characteristics of a smooth band. Since a mode shape of a traveling plate is a linear combination of many modes of a stationary plate and varies with time, it is difficult to describe this mode, especially modes of high orders, in a simple way. In order to overcome this difficulty, a new definition of mode orders will be proposed to correctly define these high-order modes. This definition makes it possible to examine the effects of various factors, including the blade geometry, the blade speed, the internal stresses and the guide constraints, on the dynamic behavior of the blade in both the lowest lateral and torsional modes as well as in the high modes. The work done by the internal stresses in different modes will be introduced to explain the effects of different stress distributions on the natural frequencies and mode shapes of the blade. The transmissibility of vibrations from the adjacent spans to the cutting span will also be discussed. This study will provide an opportunity to look into these important factors which will be incorporated into the dynamic model of the blade. 11 Chapter 2 Vibration Characteristics of Traveling Bands 12 2.1 Smooth Band Model A bandsaw blade over the cutting span, as shown in Figure 2.1 can be modeled as be an axially moving rectangular plate subjected in-plane stresses of the basis on the following assumptions: 1) . All Kirchoff assumptions are valid in this model. This band is a thin plate and the lateral displacement is much smaller than the plate thickness. The shear strain and rotatory inertia are negligible. 2) . The plate thickness h is uniform and all teeth are ignored. 3) . The blade speed c is constant. Al l internal stresses are time-invariant. The structural damping is negligible. y Figure 2.1 Bandsaw blade over the cutting span The vibration of the blade is governed by the equation (Ulsoy, 1982) DEV4w + 2phcwxl + phwtt = f(x, y,t) + (Nx - phc2)wxx + 2NxyWiXy + Nyw,yy (2.1) Chapter 2 Vibration Characteristics of Traveling Bands 13 where w represents the transverse displacement, DE is the flexural rigidity, p is the mass per unit volume,/is the distributed lateral load per unit area, Nx, and Ny are the internal in-plane forces per unit length in the x direction and the y direction, Nxy is the in-plane shear force per unit length in the x-y plane. The boundary conditions are assumed to be two opposite simply supported edges at the top guide (x=0) and the bottom guide (x= L), i . e. w(x,y,t) = 0 (2.2a) y, 0,« + v w(x, y, t)yy = 0 (2.2b) and two free edges at the back side (y=0) and the tooth side (y=B), i . e. w(x,y,t)xx + v w(x,y,t) yy = 0 (2.3a) y, t)yy + (2 - v )w(x, y, 0^], , = 0 (2.3b) The first term of the left side of (2.1) represents the restoring force due to the flexural stiffness of the plate. The second term is the gyroscopic force resulting from the lateral acceleration due to the change in the slope of the deflected shape of the moving plate. The third term stands for the force due to the lateral acceleration of the plate. The in-plane forces Ny and Nxy result mainly from the cutting forces. These forces and their effects on the vibration behavior of the blade will be discussed in the later chapters. The axially distributed force Nx can be expressed by Nx(x,y)=ax(x,y)h (2.4) where ox(x, y) is the stress uniformly distributed through the thickness of plate. This stress is given by ox(x,y)=o0+o c +o lm +o ,m +o crown +o cu, +o ,emp (2.5) where a o represents the initial axial strain induced stress, a c is the stress due to the centrifugal force resulting from the moving blade over the wheels, a ten denotes the tensioning stress, a the stress due to tilting, a cr0Wn the wheel crown induced stress, a cut and a temp are the stresses introduced by the cutting forces and the differential temperature changes that occur during cutting. Chapter 2 Vibration Characteristics of Traveling Bands 14 The bending stress o b (z) in the blade due to the bending of the blade over the cylindrical wheels can be expressed by (Eschler, 1982) where rw is the radius of the wheel. Although this bending stress affects the internal stress distribution in the band, it is not included in (2.5). It will be seen in Section 2.6.2 that this stress dose not affect the natural frequencies of the blade. It is assumed that the transverse displacement can be expressed in the form of m n ; = 1 k = 1 (2.7) Ol(x,y) = Xj(x)Yk(y), i = 1, 2 , m X n (2.8) where m and n are the highest orders of eigenfunctions in two directions to be incorporated in to the solution. X/x) are the eigenfunctions of a simply supported beam Xj(x) = sin(JKx), j = l,2,...,m; x = [0, 1] (2.9) Yk(y) are the eigenfunctions of a free beam Y1(y) = l, (2.10a) F2(y) = Vl2(0.5-y), (2.10b) y t ( y ) = c o s ( p \ y) + cosh(p, y) + rk[sm($k v) + sinh(P, y ) ] (2,10c) where sin B j. + sinh B t P* Vk-,k = 3, 4,..., n; y = [0,l] (2.11) cos B k - cosh B k Using the Rayleigh-Ritz procedure leads to the discretized equations of motion, M q(t) + G q(t) + K q(t) = Q(t) (2.12) where q(t) are the generalized coordinates. M , G and K are the mass matrix, gyroscopic matrix and stiffness matrix, respectively. Q(t) is the force vector. Their expressions are given in Appendix B (Ulsoy, 1982). Chapter 2 Vibration Characteristics of Traveling Bands 15 2.2 Mode Coupling and Mode Orders In the case of a moving plate subjected to a uniform tensile stress, an individual mode shape can not be simply expressed by one of the mode functions that are used to describe the modes of a stationary plate. Since the plate is moving, a mode shape contains many components of the mode functions of a stationary plate. This phenomenon is called mode coupling in this thesis. In the later discussion, Mode (j, k) is used to describe a mode of a stationary (or moving) plate consisting of the y'-th and the k-th mode functions (or envelopes) along the length and across the width of the plate. Figure 2.2 (a) shows the mode-coupling phenomenon in the first bending mode at a specific time. When the blade speed c = 50 m/s, the first bending mode, Mode (1, 1), is a backward traveling wave and contains both first-order and second-order mode functions of a simply supported beam. The primary component is the first bending mode. The ratio of the secondary component to the primary mode is 0.25 and the second mode has a phase lag of 90° to the first mode. The ratios of other components are less than 4% in this case. Similarly, the second torsional mode (backward traveling wave), Mode (2, 2) shown in Figure 2.2 (b), contains the first three modes of a simply supported beam. The primary component is the second mode. The ratios of the first and the third mode components to that of the second one are 0.59 and 0.57 and both have the phase lag of 90°. It is found that the high-order modes of the plate traveling at a high speed have complex shapes. The main cause for this phenomenon is that travelling waves in a moving plate are distorted due to the supports on the wave-travelling path. Many mode functions of a stationary plate contribute to an individual mode of a moving plate. Therefore, it is difficult to define high-order modes of a moving rectangular plate by using stationary mode functions in two orthogonal directions. This phenomenon can not be seen in a rotating disk that has no constraints except that at the disk center, such as an unguided circular saw blade. But it does occur in a constrained rotating disk, e.g. a guided circular saw blade. Schajer (1992) studied the similar phenomenon in circular saws and found that the envelope of a mode shape of a guided Chapter 2 Vibration Characteristics of Traveling Bands 16 circular saw maintains a certain shape, although the mode shape varies with time. This interesting result is also found in the modes of traveling rectangular plates. Figure 2.3 shows the envelopes and shapes of Modes (1,1) and (2, 2) along the length (y = B) and across the width (x = LI2 for Mode (1, 1) or LI A for Mode (2, 2)) at the given times. It is found that although the mode shapes are time-dependent their envelopes are similar to those of mode shapes of a stationary plate. Mode(1,1) ,F n = 56.2 Hz 1 ..i : ; ; t = 2.96 ms x(m) (a) Mode(l, 1) x(m) (b). Mode (2, 2) Figure 2.2 Mode shapes (Ro = 26.7 kN, c = 50 m/sec) er2 Vibration Characteristics of Traveling Bands Mode (1,1), F= 56.3 Hz 0.3 0.4 x (m) 0.25 0.2 •g- 0.15 ^ 0.1 0.05 0 1 1 i 1 1 I M o d e Shape" 1 | Envelope i i i i i 2 1 ™ o X -1 -2 0.25 0.2 0.15 ' 0.1 0.05 0 -1 0 1 w(L72, y, t) (a). Mode (1, 1) Mode (2,2), F= 124.0 Hz t= 1.34 ms 0 0.1 0.2 0.3 0.4 x (m) 0.5 0.6 0.7 \\ / / Mode Shape . Envelope / • I i i - 2 - 1 0 1 2 3 4 5 w(L/2, y, t) (b). Mode (2, 2) Figure 2.3 Mode shapes and their envelopes (RQ = 26.7 kN, c = 50 Chapter 2 Vibration Characteristics of Traveling Bands 18 As shown in the bottom diagram in Figure 2.3 (b), the mode shapes across the width of plate have stationary nodes because of no traveling wave motion in this direction. So their envelopes have the same number of nodes. On the other hand, the mode shapes, shown in the top diagram in Figure 2.3 (b), along the plate length may have more nodes than the mode shapes of a stationary plate do. Some nodes appear and move with time due to the wave motion in this plate. The others are stationary and at the same locations of the nodes of mode shapes in the stationary plate. Therefore, their envelopes have the same number of the stationary nodes. For high-order modes, however, their nodes along the plate length disappear and are replaced with the valleys shown in Figure 2.4. The displacement at the node of Mode (2, 6) (backward traveling wave) shown in Figure 2.4 (a) is non-zero and a deep valley is formed. The nodes A and B of Mode (9, 4) (backward traveling wave) also become the shallower valleys shown in Figure 2.4 (b). In fact, a high-order mode of this traveling plate contains many mode components of a stationary plate. Among them, the primary component dominates the mode shape and others modify this mode so that the nodes are transited to the valleys. If the plate travels at a higher speed, the secondary components play more important role in this mode and the valleys will become shallower. The mode-coupling phenomenon causes many difficulties in the study of the high frequency vibration behavior of the blade. In order to compare modal parameters from different sources, the same modes should be first identified. To exactly describe a mode, the eigen-vector of the mode must be specified. Apparently, this is an inconvenient way to define the mode. Fortunately, it is noted from the above discussion that although the mode shapes are time-variant their envelopes keep either the same shapes as the mode envelopes of the stationary plate if the nodes still exist, or similar shapes if the nodes are reduced to valleys. Therefore, a mode of a traveling plate can be defined by two mode numbers based on the shape of its envelopes along its longitude and across the width. Some examples are demonstrated in the previous figures. 'er 2 Vibration Characteristics of Traveling Bands (b). Mode (9, 4) Figure 2.4 Mode shapes and their envelopes (RQ = 26 J kN, c = 50 m/sec) Chapter 2 Vibration Characteristics of Traveling Bands 20 2.3 Effects of the Guide Constraints The edges of the blade constrained by the guides or the wheels are conventionally idealized to be simple supports for low frequency modes (Mote, 1965a, b, 1985; Ulsoy, 1982, 1986; Eschler, 1982; Taylor, 1986). The inside-to-inside length between the guides is taken as the length of the span. However, the slope of a deflected blade along its axis is a continuous function of blade coordinates. The slope on a guide surface must be zero if the contact is always kept. If the blade at this guide is modeled as a simple support, the slope would jump from zero to some value. This would violate the continuity of the blade. In addition, this blade at the guide can definitely bear moments. Therefore, the assumption that this band is simply supported by two inside edges of the guides should be questioned. Since the blade overhangs on the pressure guides and it is not fully clamped by them, the vibration of a stationary blade can be transmitted from one span to others. Impact testing was conducted to measure the responses at the point Qc=307 mm, y=15 mm) in the cutting span to the exciting forces applied at two points. One point is rightly above the top guide and the other rightly below the bottom guide. It is seen from the frequency response function shown in Figure 2.5 (a) that two torsional modes are at 70 Hz and 140 Hz. This implies that the blade overhangs can effectively transmit the energy through the torsional modes. The peak amplitudes of these two modes shown in Figure 2.5 (b) are much lower than those shown in Figure 2.5 (a). This indicates that the bottom guide constrains the blade more tightly than the top guide does. The two guides are subjected to different pressures, which affect the vibration transmissibility. If the blade travels at a high speed, it will be able to carry more energy from one span to others. Chapter 2 Vibration Characteristics of Traveling Bands 21 10.000 ~ 1.000 E E t ? 0.100 w <o u. or 0.010 0.001 , 1 A,A / 1 f V ! V 100 200 300 400 500 600 700 800 Frequency (Hz) (a). Between the upper span and the cutting span 10.000 1.000 in to DC u. 0.010 0.001 0 100 200 300 400 500 600 700 800 Frequency (Hz) (b). Between the lower span and the cutting span Figure 2.5 Typical frequency response functions between two spans Chapter 2 Vibration Characteristics of Traveling Bands 22 The vibration transmissibility demonstrates that the edges at the blade are not fully clamped. Since the clamped edges eliminate all possible motion of the blade relative to the guides, the vibration should not be transmitted from one span into others. According to previous research work (Mote, 1965a, b, 1985; Ulsoy, 1982, 1986; Eschler, 1982; Taylor, 1986), the model with two simply supported edges and the inside-to-inside span length was used for the study of the first few modes. It can give reasonable results compared with experimental results. If two opposite edges of the blade were fully clamped, however, its natural frequencies would be higher than those of the blade with the simply supported edges. It can be deduced that there are some other factors relaxing the system. One is related to the guide wear that leads to an extension of the length of span. The other may be the blade overhang, which is about 10 ~ 15 mm outside the wheel rims or the guide surfaces for this mill. The overhung blade runs more stably than a blade that is narrower than the wheels because the blade rim is subjected to lower tensile stress and its circumference becomes smaller than that in the central region of the blade. Thus the guides or wheels do not fully constrain the edges of blade. To understand the high frequency behavior of this blade, it is important to properly model its boundary conditions at the guides. An alternative model of a pressure guide is an elastic support. Due to the complexity of this model, it will not be used in this study. The effect of the boundary conditions, both simply supported and fully clamped, can be examined using a finite element approach. In industrial bandsaws, the bottom guide is fixed and the top guide can move up and down to suit logs of different size. The length of the cutting span is thus changeable. Worn guides also reduce the span length. The wearing occurs firstly at the edges of guide surfaces. The length of the cutting span would be continuously reduced until the guides are replaced. Figure 2.6 shows the sensitivity of the lowest lateral and torsional frequencies of the blade to the length of span. The frequencies of the blade with a short span are more sensitive to the span length. The span length affects both the lateral and torsional frequencies of the blade Chapter 2 Vibration Characteristics of Traveling Bands 23 without tensioning stresses in the same way. For a given axial strain, the frequencies of the stationary blade are more sensitive (8 Hz/100 mm for c=0 and Ro=25 kN) to the span length than the moving blade (7 Hz/100 mm for c=50 m/s and Ro=25 kN). Due to the speed effect (see Section 2.5), the moving blade becomes more flexible than the stationary blade so that the frequencies are less sensitive to the span length of the moving blade. On the other hand, increasing the axial strain increases the stiffness of the system. In this case, the variation of span length has greater influence on the natural frequencies of the blade. For example, the sensitivity is 7 Hz/100 mm for a moving blade (c=50 m/s) with Ro=25 kN and 10 Hz/100 mm with R0=50 kN. Figure 2.6 Effect of the span length on the first lateral and torsional modes Chapter 2 Vibration Characteristics of Traveling Bands 24 The effect of the span length on high frequency modes is illustrated in Figure 2.7. The modes of high orders across the width, such as Mode (1, 6), are insensitive to the variation of span length. This is because the frequencies of these modes are mainly dependent on the plate bending rigidity across the width. On the other hand, the modes of high orders along the length, such as Modes (9, 1) and (10, 1), are very sensitive to the span length because these modes depend on both the plate rigidity and the span length. In addition, when the mill runs, the guides also vibrate due to unbalanced masses on the wheels and other factors. The blade can then be excited by the displacement excitations from the guides. This factor affects the behavior of the blade and the components of cutting forces at the wheel rotating frequency. B fc 13 <3 1200 1100 1000 900 800 700 600 : \ : I -t *V - - - - \ i - - - i-: \ X . (io,i) (9, 1) \ 1 1 1 500 600 700 800 Length of Span (mm) 900 1000 Figure 2.7 Effect of the span length on high frequency modes (R0 = 25 kN, c = 50 m/sec) \ Chapter 2 Vibration Characteristics of Traveling Bands 25 2.4 Effects of the Blade Speed The axial force produced by the initial static strain Ro, the blade speed c and the top wheel support stiffness Ks (Figure 2.1) is expressed by Nx=^+y\9hc2 (2.13) £> where the nondimensional coefficient r| reflects the influence of the centrifugal force on the axial force in the band. It was originally defined by Mote (1965) without considering the elongation of the part of blade over the wheels. Here this coefficient is modified by considering this effect and expressed in the form of ^ii+(L"Zrfr (2.i4) 2BhE If the straining system is a dead weight one with a lever mechanism, the wheel support stiffness Ks = 0 and T| = 1. The top wheel can freely displace until the additional in-plane force is induced in the band to balance the centrifugal force. If the two wheel centers are fixed, i.e. Ks—>°s then t| = 0 or the total length of the band does not change with the band velocity. The contact pressure between the band and the wheels decreases with the velocity until the band slips over the wheels. For the hydraulic straining system, 0 < t| <1. According to the equation of motion, the blade speed affects its natural frequencies in two ways. One is that the effective axial force in the band defined by Nxeff = N x - 9 h c 2 = ^ - ( \ - T \ ) 9 he2 (2.15) decreases with the speed, except when the top wheel support stiffness Ks = 0. Thus the natural frequencies of the blade also decrease. In addition, the gyroscopic effect is enhanced with increasing the speed. The gyroscopic effect depends on the inertial force due to Coriolis acceleration that is proportional to the product of the blade speed and the angular acceleration of the deflected blade shape. This force acting on a point on the blade is always in phase with the lateral velocity at the same point. Thus this force, like a driving force, does positive work so that the total kinetic energy in this blade becomes higher than that without the inertial force. In other words, this force Chapter 2 Vibration Characteristics of Traveling Bands 26 makes the blade more flexible. These natural frequencies of the blade will therefore decrease with the speed. Figure 2.8 shows the lowest lateral and torsional frequencies of the blade decreasing with the blade speed. When the axial static strain is lower, greater changes of the frequencies result from the fact that the gyroscopic force does more work in a more flexible system. 100 } Ro=50kN } Ro=25kN 40 60 80 100 Blade Speed (m/sec) Figure 2.8 Effects of blade speed on lateral and torsional modes As shown in Figure 2.9, the natural frequencies of some modes do not vary greatly with the speed. In this case, the bending rigidity of the band dominates its vibration behavior and the effects of the axial stress induced by the centrifugal force and the gyroscopic motion become very small at operating speeds. However, the natural frequencies of other modes increase with the blade speed. It is known that the mode shape of a high-order mode is a combination of many stationary modes. With the increase of the blade speed, this mode may have greater weights of higher-order mode components. The similar mode shape may be observed, but the mode components may have been changed with different blade speeds. Chapter 2 Vibration Characteristics of Traveling Bands 27 1400 N >> O a 3 CT1 3 1200 1000 800 600 R 0 = 2 0 k N h=1.45 m m (7, 5) •-""(3, 6) (3, 5) (1.5) 10 20 30 40 Blade Speed (m/sec) 50 Figure 2.9 Effects of blade speed on high modes 2.5 Effects of Internal Stresses 2.5.1 Axial Static Stresses Sufficient axial static strain is necessary for properly running the bandsaw. This strain increases the tracking stability, the stiffness and the low natural frequencies of the blade. Figure 2.10 shows the effect of the axial strain on the lowest natural frequencies of the blade in the cutting span. The uniform tensile stress increases both lateral and torsional frequencies because this stress increases the strain energy in the blade at each mode. At a low strain level, the natural frequencies are governed by the rigidity of the blade, the tensile in-plane stress, and the blade speed during running. At high strain level, however, these frequencies are dominated by the tensile stress. The frequency sensitivity to strain is about 1.0 Hz/kN for the stationary blade and 1.2 Hz/kN for the moving one at c =50 m/s in the strain range of 20 ~ 40 kN. It appears that the moving blade has a somewhat higher sensitivity because the effect of the centrifugal force in this region reduces with the strain. When the strain is very high, the frequencies of a stationary or moving blade increase with the strain level. Chapter 2 Vibration Characteristics of Traveling Bands 28 Figure 2.10 Lateral (LI) and torsional (Tl) frequencies increase with strain For a given strain level within a certain range, a thinner blade may have higher frequencies than a thicker blade as shown in Figure 2.11. It is known that the thicker blade without internal stresses has higher frequencies because of its higher bending rigidity. When the strain exceeds a certain level, the stiffness of the blade is dominated by the strain, instead of the bending rigidity. Since the thicker blade has higher mass density per unit area, the natural frequencies will be lower than those of the thinner blade subjected to the same strain level. If two blades are thick enough, their bending rigidity will govern the dynamic behavior, and the natural frequencies of the thicker blade will be higher than those of the thinner blade. For high frequency modes, the bending rigidity always governs the behavior of the blade. The strain energy produced by the axial force contributes a small portion of total strain energy in the blade. Consequently, the natural frequencies of high order modes of a thicker blade (Figure 2.12) are always higher than those of a thinner blade. In addition, these frequencies do not increase greatly with the strain level. Chapter 2 Vibration Characteristics of Traveling Bands 29 0 1 1 ' 1 1 1 1 0 10 20 30 40 50 60 Strain (kN) (a). First bending mode (b). First torsional mode Figure 2.11 Effect of the blade thickness on natural frequencies Chapter 2 Vibration Characteristics of Traveling Bands 30 N X O c 3 cr S-l 3 1400 1200 1000 800 600 400 h=1.45 m m h=1.65 m m _(3,_6_) (1,6) (1,5) (1,6) (3,6) (3,5) (1,5) 10 20 30 40 50 60 70 Strain (kN) Figure 2.12 Effect of the blade thickness on high modes 2.5.2 Roll-Tensioning Stresses Roll tensioning can greatly improve the cutting performance of a bandmill by introducing residual stresses into the blade to increase the stiffness of the teeth. The residual stresses change the natural frequency spectrum of the band (Ulsoy, 1982; Taylor, 1986, 1993). Properly applied roll-tensioning will increase the low torsional frequencies and decrease the low transverse frequencies (Wang, 1994). A physical explanation, based on the strain energy variation introduced by the tensioning stresses, will be given below. It is assumed that the distribution of the residual stresses shown in Figure 2.13 is parabolic across the width of the band (Taylor, 1986) and can be expressed by ° <en(y) = o „ ( 4 y 2 - 4 y + f) (2.16) where o p is the amplitude of the stresses, y (= [0, 1]) is a nondimensional variable corresponding to the coordinate [0, B] across the width of the band. As a result, tensile stresses are applied to the edges of band, and compressive stresses to the central region. Chapter 2 Vibration Characteristics of Traveling Bands 31 GO c '3 c H \2op/3 + \ B/2 / + - 0 ^ B -Op/3 Width of Band Figure 2.13 Typical residual stresses induced by roll-tensioning Figure 2.14 shows how the tensioning stresses affect the natural frequencies of the blade. The lowest lateral frequency decreases slightly with the residual stress amplitude, while the lowest torsional frequency increases almost linearly. It is noted, but not shown in Figure 2.14, that the frequency of the second bending mode decreases more rapidly and is possibly lower than the first torsional if both stress amplitude and the blade speed are sufficiently high. 100 50 100 150 200 250 Tensioning Stress Amplitude (MPa) 300 Figure 2.14 Effects of tensioning stresses on the first bending and torsional modes Chapter 2 Vibration Characteristics of Traveling Bands 32 The effects of the residual stresses on the natural frequencies are dependent on the changes in the strain energy in the system introduced by the stresses. In other words, the extent to which the stresses affect the vibration behavior of the band depends on the proportion of the corresponding strain energy to the total strain energy of the system. The strain energy introduced by the axial stresses is given by U,=$\lNxw*dxdy (2.17) D The strain energy introduced by the tensioning stresses is distributed over all possible modes. The first two modes of the stationary blade are shown in Figure 2.15. It is seen that the first torsional mode is almost the same as that of the blade without the tensioning stresses. When the blade vibrates at this mode, the strain energy variation can be expressed by 6n2Bon fi „ n UTI= j-^)Q cos2nxdx\o (4y2-4y + ±)(05-y)2dy (2.18) The function of the second integral gives the strain energy density along the width of the blade (Figure 2.16 (a)). Obviously, the integration of this part of strain energy is positive. This means that at this mode, the tensioning stresses introduce additional strain energy into the system and make the band stiffer. This is why the torsional frequencies increase with the residual stress amplitude. The sensitivity of the frequency to the stress amplitude depends on the ratio of the integral to the total strain energy of the band. Figure 2.15 (b) shows that the first lateral mode is mainly comprised of [Xi(x) Yi(y)] and [Xj(x) Ys(y)]. Since the compressive stresses lie in the central region of the band, the deformation in this region will be greater than that at two edges. If the effective tensile stresses, or the sum of static strain induced stress and the tensioning stresses, in the band is sufficiently high and smoothly distributed, the bending stiffness of the blade is nearly proportional to the stresses. Generally, the deflected shape along the width approximates the profile of the tensile stresses except that some sharp stress variations exist. Therefore the deformation along the width can be approximated by Fjfy) that is the best approximation of Chapter 2 Vibration Characteristics of Traveling Bands 33 Mode Shape of the Plate (F= 76.1 Hz) Mode Shape of the Plate (F= 51.2 Hz) (a). First torsional mode (1, 2) (b). First lateral mode (1, 1) Figure 2.15 Two lowest modes of the stationary band (a) Torsional mode (b) Lateral mode Figure 2.16 Mode shapes, tensioning stresses and strain energy density at the first torsional mode and lateral mode Chapter 2 Vibration Characteristics of Traveling Bands 34 the tensioning stress distribution (Figure 2.17). The strain energy variation at this mode can be expressed by Un = j-^-jQ cos2n xdx)Q(4y2-4y + i)(Cx + C2YJy)fdy (2.19) where Cy and C2 are constants and C/C2 > 0. This integral gives a negative value at this mode. This implies that the system vibrating at this mode has less strain energy and becomes more flexible. Therefore, the lateral frequency of the blade decreases with the tensioning stress amplitude. The sensitivity is also related to the ratio of the integral to the total strain energy in the blade. Generally, the strain energy variation at higher modes is given by UT=L 2- Cjk p-\ cos2nxdx) (4y2-4y + i)Y2(y)dy (2.20) j=\ k=\ L It is seen that the strain energy increases at the rate of j 2 (j = 2, 3 , n ) . It seems that the higher modes are more sensitive to the tensioning stresses. This is true for the first several modes. At very high modes, however, the natural frequencies shown in Figure 2.17 do not vary with the stress amplitude greatly. In fact, the bending stiffness of the band dominates its vibration behavior. Furthermore, the second integral in (2.20) becomes smaller and smaller with increasing order if the tensioning stress distribution is symmetric to the central line of the band. Taking advantage of the strain energy analysis, one can qualitatively examine what effects are caused by a given tensioning stress distribution. An optimal stress distribution could be found with this method to improve the cutting performance of the blade. Chapter 2 Vibration Characteristics of Traveling Bands 35 1200 N X >, o c <u 3 cr 3 1000 800 600 (3.6) ~ ~ ~ ~ ~ (7,5) CI76) (6,5) (3,5) 0,5) 50 100 150 Tensioning Stress Amplitude (MPa) 200 Figure 2.17 Effects of tensioning stresses on high modes 2.5.3 Wheel Tilting Stresses The top wheel tilting shown in Figure A . l changes both the magnitude and the distribution of the static tensile stress in the blade. It is assumed that the titling induced stress is linearly distributed along the width and the static in-plane force is equal to the product of the stress at the center of the band and the cross-section area of the band. In this sense, the tilting introduces a linear stress distribution with zero average, i.e. otilt(y) = ola(2y-\) (2.21) where the stress amplitude o ta is function of the tilt angle 0 ,a and can be expressed by 0 >°=oa l « r & 7 ^ — 1 ) + B»e»] ( 2 - 2 2 ) 2(LW+7C rj 2 cos0 u When the tilt angle is small, the stress amplitude is nearly proportional to the angle (Figure 2.19). Chapter 2 Vibration Characteristics of Traveling Bands 36 Figure 2.18 shows the mode shapes of the first two modes. As shown in Figure 2.18 (a), the tighter edge or tooth side, experiences smaller displacement at the lateral mode, while this edge has larger displacement at the torsional mode (Figure 2.18 (b)). The first lateral frequency of the blade decreases with the tilt angle, while the first torsional frequency increases with the angle. This phenomenon can also be explained by analyzing the strain energy variation. It will be found that the work done by this stress is negative in the lateral mode and positive in the torsional mode. This also implies that if the static strain remains the same, but the tilt direction is opposite, the natural frequencies of the blade do not change, but the mode shapes are different. This implies that the natural frequencies of the blade indicate its global behavior rather than the local changes in the internal stresses. (a) Lateral mode (b) Torsional mode Figure 2.18 Mode shapes of the blade installed on a tilted wheel (0C=O.4°) Chapter 2 Vibration Characteristics of Traveling Bands 37 Figure 2.19 Effect of the tilt angle on bending and torsional modes At very high modes, however, the natural frequencies are almost constant in the range of tilt angle shown in Figure 2.20. This means that the wheel tilting can not change the vibration behavior at high modes, i.e. at the washboarding frequencies. The smaller influence on high modes results from the domination of the bending rigidity and the small variation of the strain energy introduced by the tilting stresses. X o c <D 3 cr l a -u 3 -4—* a Z 1200 1000 800 600 - (3,6) ( 7 ,5 ) - ( 6 ,5 ) (1,6) -( 3 ,5 ) _ 1 1 , 5 ) 0.2 0.4 0.6 Tilt Angle (deg) 0.8 Figure 2.20 Effect of the tilt angle on high modes Chapter 2 Vibration Characteristics of Traveling Bands 38 ^y=^7—rA-^-) (2-23) 2.5.4 Wheel Crown Stresses The wheel crown is assumed to be a circular arc of radius rc. The bending stress along the width is given by (Eschler, 1982) Eh 1 v 2 ( 1 - V 2 ) V C rw This stress, like the wheel curvature induced bending stress, does not affect the vibration behavior of the blade because the strain energy variation at each mode produced by these stresses is zero based on (2.18). The longitudinal stress shown in Figure 2.21 can be approximated by a croWn= T " f <tf-Bw2(y-h2]-(rc-rw)-y0} (2.24) This stress, which is superimposed on the static tensile stress, affects the tensile stress distribution, instead of the stress amplitude. Based on this stress distribution, the first lateral and torsional frequencies of the blade decrease from 54.3 Hz and 59.1 Hz to 52.6 Hz and 58.8 Hz, respectively. Width of Wheel (mm) Figure 2.21 Wheel crown induced stress distribution Chapter 2 Vibration Characteristics of Traveling Bands 39 2.6 Effects of Gullet Depth The phenomenon of washboarding in woodcutting with a wide bandsaw blade can be induced by the instability of self-excited vibration at high frequencies (Hutton, 1989; Okai, 1996a; Tian, 1998). The instability of the blade is attributed to one or two modes whose frequencies are slightly below the tooth passing frequency (900 ~ 1,100 Hz). Thus the high frequency vibration characteristics of the blade should be directly related to the washboarding phenomenon. Generally, the vibration characteristics of the blade at high frequencies can be very different from those at low frequencies. Some factors, such as the geometry of teeth and the guide stiffness, can be neglected without loss of accuracy when the fundamental modes are in consideration. These factors, however, can greatly affect the vibration behavior of the blade at high modes. In sawmills, for example, it is known that the depth of gullet is an important factor in washboarding. A blade with deeper gullets would more likely produce a washboarding pattern on sawn surfaces. Thus the effect of these factors on the high frequency modes should be considered in the prediction of vibration instability of saw blades. : In almost all theoretical research work on the vibration and stability of bandsaw blades, only smooth blades were studied. The contribution of teeth to the high frequency vibration characteristics of a blade has not been understood. The purpose of the study of stationary toothed blades is to examine the effects of the geometry of teeth on its high frequency vibration characteristics. Meanwhile, in order to reasonably model a toothed blade using the finite element procedure, mesh parameters should be carefully chosen. The finite element program NASTRAN (MSC, 1997) was used in this study to analyze stationary toothed blades without internal stresses. The toothed blade was meshed using 3-node triangular plate elements and 4-node bilinear elements. Chapter 2 Vibration Characteristics of Traveling Bands 40 2.6.1 Model of Stationary Toothed Blades Two toothed blades with gullet depths of 15.9 mm (Armstrong gullet No. 42A) and 20.6 mm (No. 50A) are shown schematically in Figure 2.22. A typical mesh of a blade is illustrated in Figure 2.23. In order to examine the effects of teeth on the natural frequencies, two smooth blades, the width of which is defined as the width of the blade without teeth, were also studied. 280 270 260 250 240 230 220 210 200 0 10 20 30 40 50 60 70 80 Figure 2.22 Two types of gullets vi L2 CI y i. / / A 1 / S / /• \ \ \ \ \ / \ / \ S / \ \ / s 4 \ V-Gullet No. 50A (20.6 mm) Figure 2.23 Meshes of a toothed blade (725 nodes and 914 elements) Chapter 2 Vibration Characteristics of Traveling Bands 41 2.6.2 Effects of Tooth Gullets on Stationary Blades Since the mass of an individual tooth, whether 42A or 50A, is very small, it does not greatly affect the natural frequencies of the fundamental modes. As shown in Table 2.1, the frequency differences of the first two modes between a smooth blade and a toothed blade are less than 1%. This is also true for the modes of the first order along the width of blade, such as modes (3, 1) and (5, 1). At low frequencies, therefore, the teeth have very little effect on the natural frequencies and, in practice, can be neglected. Table 2.1 Natural Frequencies (Hz) of Smooth Bands and Toothed Blades Mode Gullet 42A Gullet 50A Smooth Toothed Diff. (%) Smooth Toothed Diff. (%) (1,1) 5.94 5.92 0.3 5.94 5.90 0.7 (2, 1) 23.99 23.87 0.5 23.98 23.79 0.8 (1,2) 26.12 25.28 3.2 26.61 25.33 1.1 (3, 1) 54.39 54.09 0.6 54.37 53.90 0.9 (2, 2) 56.22 54.57 2.9 57.13 54.58 4.5 (1,3) 148.5 141.3 4.8 152.4 142.5 6.5 (5,1) 152.4 151.4 0.7 154.0 150.8 2.1 (1,6) 1,193 1,142 4.3 1,241 1,147 7.8 Regarding high frequency modes, such as (1, 6), however, the effects of teeth are apparent and strongly dependent on the tooth profile or gullet of the blade. From Table 2.1, it is seen that the toothed blades have lower frequencies. For the blade with Gullet 42A, the frequency is 51 Hz lower than that of the smooth blade and for the other blade with a deeper gullet, this frequency decreases by 94 Hz. Figures 2.24 (a) shows the mode shape of Mode (1, 6) of the smooth blade. The mode shape at the cross section at x-L/2 is shown in Figure 2.24 (b). Compared with the mode shapes of the toothed blade (42A) shown in Figure 2.25, the magnitude of the mode shape of the smooth blade on the tooth side is very different from that of the toothed blade. This can also be seen from the mode shapes, shown in Figures 2.26 and 2.27, of the smooth and toothed blades (50A). Furthermore, the deeper the gullet, the larger the relative magnitude of Chapter 2 Vibration Characteristics of Traveling Bands peformed(0.0142) : T3 Translation (a) Mode shape of smooth blade (b) Mode shape at the cross-section at x=L/2 Figure 2.24 Smooth blade (42A) Chapter 2 Vibration Characteristics of Traveling Bands (a) Mode shape of the toothed blade VI C l |Deformed(0.0238) : T3 Translat ion (b) Mode shape at the cross-section at x=L/2 Figure 2.25 Toothed blade (42A) Chapter 2 Vibration Characteristics of Traveling Bands VI ci z1 Output Set: Mode 10 1240.513 Hz Deformed (0.014): Total Translation Figure 2.26 Mode shape of smooth blade (50A) at x=L/2 vi C l O u t p u t Set: M o d e 16 1146.678 Hz D e f o r m e d (0.024): Total Translat ion Figure 2.27 Mode shape of toothed Blade (50A) at x=L/2 Chapter 2 Vibration Characteristics of Traveling Bands 45 the mode shape on the tooth side. Therefore, the teeth effects can not be neglected if the high frequency vibration characteristics are considered. From the mode shapes shown in Figures 2.25 and 2.27, it appears that the blade with the deeper gullet will experience a higher vibration level. To qualitatively examine the gullet effects, the Frequency Response Functions (FRFs) of the blades were computed. The excitation is applied to the tip of the middle tooth. The response is evaluated at the same point. A damping ratio of 0.01 was chosen for the evaluation of the FRFs. Figure 2:28 shows the FRFs at the middle tooth tips of the blades (42A and 50A) . Mode (1, 6) of 1,142 Hz has the magnitude of 10.4 um/N. For the blade (50A), the magnitude of the same mode of 1,147 Hz shown in Figure 2.28 (b) is 14.3 NJD/N. There are also two modes at about 1190 Hz. But the magnitudes of their FRFs are the same for both gullets. It appears that the natural frequencies are insensitive to the gullet depth, but that the frequency response functions may be affected by the change of gullet depth. The effect of the gullet depth on the vibration behavior of a moving blade subjected to in-plane stresses will be discussed in Chapter 4. 1080 1100 1120 1140 1160 1180 1200 1220 1100 1120 1140 1160 1180 1200 1220 Frequency (Hz) Frequency (Hz) (a). Gullet Type 4 2 A (b). Gullet Type 5 0 A Figure 2.28 Frequency response functions of two toothed blades Chapter 2 Vibration Characteristics of Traveling Bands 46 2.7 Summary A bandsaw blade has been modeled as a traveling plate subjected to in-plane stresses. The vibration behavior of the blade has been discussed over a wide frequency range. Since a mode of a traveling plate is a linear combination of many mode functions of a stationary plate, it is difficult to define a complex mode in the traditional way. In order to overcome this difficulty, the envelopes of the mode shapes along the length and across the width of the plate are used to define a mode of this moving plate. The orders of the envelope are the mode numbers in two directions. The mode envelope across the plate is the same as that of a stationary plate because the plate does not move in this direction. The envelope along the length is similar to that of a stationary plate. For low frequency modes of the plate moving at a low speed, their envelopes are the same as the mode envelopes of a stationary plate. For high frequency modes of a plate moving at a high speed, however, the nodes on their envelopes will disappear and be reduced to valleys. These envelopes are still useful in the definition of these modes. The main factors affecting the vibration behavior of the blade are: the guide constraints, the blade speed, the internal stresses and the tooth profile. These factors have different effects on low frequency and high frequency modes. The variation of the span length affects the low frequency modes with a sensitivity of 7 ~ 10 Hz/100 mm in the span range of 700 ~ 900 mm. The modes of high orders across the width, such as Mode (1, 6), are insensitive to the variation of span length because the frequencies of these modes are mainly dependent on the plate bending rigidity across the width. On the other hand, the modes of high orders along the length, such as Modes (9, 1) and (10, 1), are very sensitive to the span length because these modes dependent on both the plate rigidity and the span length. The blade at the pressure guides is usually modeled with simply supported edges. In practice, the vibration of a stationary blade from the upper or lower span can be transmitted into the cutting span through the guides. The blade overhang on the guides and the low contact pressure between the guide surfaces and the band result in Chapter 2 Vibration Characteristics of Traveling Bands 47 this transmittance. Therefore, the blade at the guides is neither simply supported, nor fully clamped. In general, the natural frequencies of the moving blade decrease with the blade speed due to the reduction of the effective axial tensile stress and the gyroscopic effect. But some high frequency modes are not greatly affected by the blade speed. The natural frequencies of some modes increase with the speed due to mode coupling effects. The internal stresses, including static tensile stress, tensioning stresses, and stresses induced by the wheel tilt and crown, can either increase or decrease the natural frequencies. The influences of these stresses on the lateral and torsional modes depend on these stress distributions. Generally, the lowest natural frequencies of fundamental modes increase with the axial static stress. For a given stress level, however, a thinner blade has higher natural frequencies in a certain stress range than a thicker one. Tensioning stresses can decrease lateral frequencies and increase torsional frequencies. The effects on the fundamental frequencies of the stresses induced by the wheel tilting are the same as those due to the tensioning stresses. But the natural frequencies of the smooth band are independent of the tilting direction. The tighter side experiences larger displacement when the blade vibrates at the torsional mode. For very high frequency modes, the bending rigidity of the band dominates the stiffness of the system. The strain energy variation introduced by various stresses becomes smaller with increasing frequency. The natural frequencies of these modes do not greatly vary with the axial strain, the tensioning stress, and the stress due to the top wheel tilting. The masses and stiffness of teeth have minor effects on the low frequency modes. For high frequency modes, the additional masses of the teeth can decrease the natural frequencies by 5 ~ 8%. The magnitudes of frequency response functions of some modes are sensitive to the depth of gullet. If high frequency vibration characteristics of a bandsaw blade are of interest, the effects of teeth should be considered. Chapter 3 Stability of Traveling Bands Subjected to Parametric Excitations A bandsaw blade during cutting is subjected to fluctuating lateral cutting forces and internal forces produced by imperfections in the band-wheel system. A serious vibration and instability of the blade could cause washboarding during cutting (Tian, 1998). In order to understand the mechanisms of washboarding caused by these cutting forces, the instability of a smooth band will be investigated. Naguleswaran (1968) studied the transverse vibration of a moving beam subjected to a periodic tension produced by the eccentricity of the wheels and joints and flaws in the band. Wu (1986) considered the instability problem in a moving beam caused by a normal periodic edge load. Lengoc (1995b, c) investigated the stability of a moving plate subjected to uniformly distributed and periodic edge loads and non-conservative cutting forces by using the Galerkin method. These studies were limited to the first few modes that do not contribute to the washboarding. The stability in a rotating disk fixed at the center was studied by Tian (1998). In his model the multi-regenerative cutting forces and follower tangential and radial edge forces in the cutting zone were included. He found that the regenerative forces played a primary role in washboarding. The study on the instability of a moving plate subjected to parametric excitations due to multi-regenerative cutting forces is presented in this Chapter. The regenerative cutting forces acting on a toothed blade are modeled as the lateral forces applied on the front edge (tooth-side) of a smooth band. The system equations of motion are established by using the Rayleigh-Ritz method. The characteristic equations of the time-variant system are derived from the equations of motion by using a Fourier series method. Finally, the eigenvalues are solved using Muller's algorithm and thus the instability regions of the smooth band can be determined. 48 49 Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 3.1 Model of Regenerative Cutting Forces It is known that the lateral cutting forces acting on the blade are quite complicated. In general, there are several lateral cutting forces applied at a tooth tip. A regenerative cutting force is the lateral resultant force produced by the unbalanced side-cutting forces. A follower cutting force and a follower feed force result from the lateral components of the tangential force and feed force. Other lateral forces"due to the guide motion and the tooth impact will be discussed in Chapter 5. Compared with the follower forces, the regenerative cutting force plays a more important part in washboarding (Tian, 1998). Thus the follower forces are taken to be negligible here. In addition, the typical magnitudes of the tangential cutting force and feed force are about 500 N and 250 N (Ulsoy, 1978). Their variations can affect the in-plane stresses and act as parametric excitations. But these cutting forces are much smaller in magnitude than the static strain in the blade and will not be considered. To model these lateral cutting forces, the following assumptions are made: 1) . All teeth have the same geometry and the tooth pitch is a constant. 2) . All teeth are subjected to the same cutting force if they experience the same vibration. 3) . The blade speed and the feed speed are constant. Figure 3.1 shows the physics of a regenerative cutting force produced by tooth flank cutting. If the blade does not oscillate laterally, all the teeth go straight and two flanks of each tooth cut wood with the same length (Figure 3.1 (a)). Two identical lateral cutting forces acting on the two flanks in two opposite directions will be canceled. No regenerative force acts on the saw blade. If the blade oscillates laterally, as shown in Figure 3.1(b), two flanks of each tooth cut wood with different lengths. Consequently, two side cutting forces are unbalanced and the resultant, called the regenerative cutting force, will be produced. It is assumed that this force is directly proportional to the cross-sectional area of a chip. Thus it is a function of the relative displacement of the present tooth to the prior tooth (Tian 1998), i.e. Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations SO (a). Two balanced side cutting forces (b) Two unbalanced side cutting forces Figure 3.1 Regenerative cutting force Fzj=-Kl[w(x,y,t)-w(x,y,t0)}8 (x-xJ8 (y-y,,) (3.1) = -K, [w(xtj, y,., t) - w(xt], y,., t0)] where Fxj is the regenerative cutting force applied at the j-th tooth tip. K~i is the regenerative cutting force coefficient. w(x, y, t) is the lateral displacement of the blade at the point (x, y) and time t. (x,j, y,j) are the coordinates of the y'-th tooth tip. 8 is Dirac delta function. t0 is the time when the last tooth passes the same point (xtj, y,j). Clearly, to = t-T where T is the tooth passing period. The displacement at a tooth tip was defined as the displacement at a point on the front edge of a smooth band (Lengoc 1995) or on the rim of a rotating disk (Tian 1998). This simplification may be reasonable for low frequency modes. For high frequency modes, however, the depth of gullet dg affects the frequencies and mode shapes of the blade. The 51 Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations tooth stiffness is mainly dependent on the stiffness of the band. Then this tooth can be considered a rigid beam attached to the edge of the smooth blade. Thus, as shown in Figure 3.2, the displacement of the j'-th tooth tip can be approximated by the displacement at the edge point J, the mid-point of the tooth root, and the slopes at the same point. It has the form k z Figure 3.2 Model of the cutting forces w(xtj. y,j»f) = w(xj > yj ^) + eg w,x U;»yj , 0 + ^ wy (*;, y., t) (3.2) where dg is the depth of gullet, eg the longitudinal distance between point J and the j-th tooth tip. (XJ, yj) is the coordinates of point J. w ^ and w ?y are the derivatives of w(x, y, t) with respect to x and y, respectively. Therefore, this regenerative cutting force acting at the y'-th tooth tip can be approximately expressed by Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 52 FZJ(xj,y],t) = -Kl{leg dg}\ wiXjJj^-wiXj^jjJ wAxj>yj^-w,x(xJ>yjJo) w,y(xj'yjj)-w,y(Xj,yjj0) (3.3) In order to take advantage of the convenience of a smooth blade, the lateral force Fy can be equivalently converted to a force F y and moments Myi and Myxj acting at Point J shown in Figure 3.2. These moments are Myj(Xj, y., 0 = Fzj(t) dg, Myxj(t) = Fz,(*,, y , , 0 eg (3.4) It is noted that the coordinate of the mid-point of each tooth xj in (3.3) and (3.4) is a function of time, i.e. Xj(t) = x0+ct + (j-l)P (3.5) where xo is the position of the mid-point of the first tooth in the cutting region at t=to, c is the blade speed and P the tooth pitch. The force F y is nonzero only if Xj(t)=[xst, xemj\, where xs, and xend are the start and end of the cutting zone, respectively. Actually, this force also is a function of the bite per tooth and the blade speed, the geometry of tooth edges and the wood properties. If a force function is prescribed, their effects on the instability of the blade can be examined using the approach described in next section. 3.2 Stability Analysis of a Smooth Band 3.2.1 Equations of Motion — Rayleigh-Ritz Method The virtual work in an undamped smooth band performed by a distributed force j\x,y,i) and moments mx(x, y, t) and my(x, y, t) can be expressed in the form § W n c = \ A U(x,y,t)5w + mx(x,y,t)8wx+my(x,y,t)dwy]dA (3.6) The equation of motion (2. 1) of the band can be modified into the form DEVAw + 2phcwxt +phw„ =f + mxx+myy + (Nx-phc2)w_xx + 2Nxywxy +Nyw,yy Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 53 where mXiX and myy are the derivatives of moments mx and my with respect to x and y, respectively. The solution can be assumed to be mXn w(x,y,0 = X < £ , W k ( 0 (3-8) i = i and 0,.(x,y) = Xj(x)Yk(y), j = l,2,...,m; k = l,2,...,n (3.9) Using the Rayleigh-Ritz method leads to the equations of motion of the undamped system expressed by the modal coordinates q(0 Mq(f)+Gq(0 + Kq(0 = Q(0 (3.10) where M, G and K are the mass matrix, the gyroscopic matrix and stiffness matrix of the band, respectively. They are given in Appendix B. The generalized force vector Q(t) depends on the types of external forces acting on the blade. If the blade is subjected to the regenerative forces Flk, k=l, 2,N, (N, is the number of teeth in the domain of interest), the generalized forces Q(r) are given by QW = t f U(x,y,t)Oj(x,y) + mxOjx(x,y) + myOjy(x,y)]dxdy = X ^(xk,ykJ)^j(xk,yk) + MxkOjx(xk,yk) + Myk0.y(xk,yk)] (3.11) k=l j-1, 2,mx n Substituting (3.3), (3.4) and (3.9) into (3.11), the generalized force vector Q(r) is expressed by Q(0 = K,(f0)q(f0)-K,(Oq(0 (3.12) where Kqij(t) = Klfj ¥ , ( W , , 0 ^ ( W * , 0 (3-13) k=l Kqij(t0) = Klj? W,.(*,,y,,f 0)¥.(x,,y,,? 0) (3.14) k=\ where % (xk ,yk t) = ®, (** ,yk J) + e8®t,x (** ,yk ,0 + dg®iiy (xk ,yk ,0 (3.15) Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 54 It can be proven that Kq(t) is a periodic function, i.e. K,(f0) = K,(f-r) = K , ( 0 (3.16) where T = 2rc/u), u) is the tooth passing frequency. Therefore, Q(t) = -Kq(t)[q(t)-q(t-T)] (3.17) The equations of motion of the system can be rewritten in the form M q(0+ G q(0 + K q(r) + (1 - e~TD) K (t)q(t) = 0 (3.18) 9 TD TD where e~ is a time-delay operator, e.g. e~ q(t)=q(t-T). These equations will be used for the stability analysis of the system. 3.2.2 Characteristic Equation It is assumed that the displacements q(t) in (3.18) can be expressed in terms of a Fourier series, i.e. q(r) = e x ' ]T ckeik("= JT cke(X+ika)' (3.19) k——°o lc=—o° and oo q(t0) = q(t-T)= Y ckeH-T)+ikw (3.20) The periodic function Kq(t) can also be expanded into the Fourier series K _ ( 0 = £ &,eih" (3-21) / = - o o where d^^K^e^'dt (3.22) Substituting (3.19) to (3.22) into (3.18) leads to CO CO ^ iku)2M+(X+ ika)G+K + (1 - e~TX)£ d , /" ' Jc^ to'= 0 (3.23) If the et have a non-zero solution, the determinant of their coefficient matrix must be zero. In the specific case, where k=l=0, the equation is det{X2M + >.G + K + (l-e"^)d0>=0 (3.24) When k= 0, ±1 and /=0, ±1, ±2, (3.24) becomes . Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 55 (V H 0 0 0 « 0 d, d_, 0 H , 0 + (\-e~TX) d-i d0 d_2 0 0 Hj .d, d 2 d0 where H 0 = k2M + \G + K Hj = (k+ /co)2M + (X+ *to)G + K H_, = (K- ito)2M + (X- «o)G + K = Hj It is noted that is a complex vector and d* a complex matrix. Let c * = a * + . ^=0, ±1 d, = A i + I B K , £=0,±1,±2 H , =nkR+mk , *=o,±i the (3.31) can be rewritten in the form ^ 0 0 0 Hf H[ 0 Hf Hf + (l-e-Tk) 2 A l B, ^ 0 + ^ 2 = 0 (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) The characteristic equation is given by the determinant of the coefficient matrix of (3.32) that is equal to zero. In general, when k= 0, ±1,±n and 1=0, ±1,+2n, (3.32) becomes where ( H , + ( I - ^ ) D J = 0 H I = % ( H C , H , , H | ) . . „ H_„,H„) H±k = (X± ika)2M + (k± ika)G + K (3.33) (3.34) (3.35) Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 56 d 0 d, d„ d„ d0 d_2 • d„_, d - ( » + i ) d, d 2 d 0 •• d„+, d - ( » - i ) d_„ d - ( » - i ) d - ( „+ i ) • d 0 d_2n d„ d n + 1 d„_, d 2 n d0 (3.36) The characteristic equation of the system can be derived from (3.33). It is | H A + ( l - e - ™ ) D j = 0 (3.37) 3.2.3 Solution to Eigen-Problems The instability regions of a bandsaw blade subjected to the regenerative cutting forces can be determined by solving for the eigenvalues from (3.24) for zero-order Fourier series solutions and from (3.32) for the first-order Fourier series solutions. The characteristic equations contain an exponential function of the complex eigen-value A.. Most conventional methods for modal analysis of a linear system cannot be applied to these cases. Muller's algorithm (Muller, 1956), a complex root solver by optimization, is chosen for solving the non-linear eigen-problem. An instability region of the system for a given mode is defined as a tooth passing frequency band in which the real part of this eigenvalue is positive. If an eigenvalue has been found, the corresponding eigenvector can be determined by solving the homogenous linear system of rank N, such as (3.32). A new linear system of rank (7V-1) can be built. The new coefficient matrix is formed by removing the last row and column in the original coefficient matrix. The right-hand vector is the last column vector except the last element if a unit magnitude at the last degree-of-freedom is assumed. 3.2.4 Approximation of the Regenerative Cutting Forces The regenerative cutting forces can be approximated in terms of Fourier series with zero-order or higher orders. Tian (1998) determined that the eigen-values of a rotating disk are Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 57 dominated by the zero-order approximation of the regenerative forces. The effect of the approximation for the smooth band are examined by considering the zero-order (LL = 0) and the first order (LL =1) series. Table 3.1 lists the mode frequencies and real parts of the eigenvalues of the selected modes for the regenerative cutting force coefficient Ki=10 kN/m and the tooth passing frequency F,= 100 Hz, and 1000 Hz. They hardly vary with the different orders because this coefficient, compared with the blade stiffness, is small. This implies that the regenerative cutting forces can be well approximated by the zero-order Fourier series. Table 3.1 Effects of the Order of Fourier Series on System Eigenvalues (Ki-10 kN/m, Ft = 100 Hz for the first three modes, 1,000 Hz for the rest) Mode Zero-Order First-Order Real Freq. (Hz) Real Freq. (Hz) (1,1) 0.021 67.19 0.020 67.19 (2, 2) -0.025 135.62 -0.026 135.62 (2 ,3) -6.67 200.64 -6.68 200.25 (7,4) 86.12 963.42 88.85 962.44 (8 ,3) 3.98 976.90 3.96 976.88 (2, 6) -59.80 1071.0 -55.19 1070.4 (10,1) -6.90 1086.2 -6.89 1086.2 (8,4) -30.92 1122.8 -29.93 1122.0 3.3 Instability Regions of a Smooth Band 3.3.1 Unstable Regions Figure 3.3 shows the natural frequency Fn and the real part of the first mode eigenvalue for the given static strain Ro and the regenerative cutting force coefficient Ki. The natural frequency of mode (1, 1) is almost constant. The instability regions are where the real parts of the eigen-values are positive, for example, Primary instability region: F, = (Fn , 2 F„). Other instability regions: Ft = (F„ Ik, Fn f(k-0.5) ),k=2,3, ... (3.38) Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 58 1^60 I o § 40 cr a 20 to t: 1 CO t o to rr -1 F = 55.4 Hz n 0 20 40 60 80 100 120 140 160 (a) I I I I I I A 2|F /3 / V / Prirrjary Unstable Regibn \ I \ > / ^ » 2 F n / 5 ^ V \ X f n 2 F n F n / 2 \ | I I I i i i 20 40 60 80 100 120 140 160 Tooth Passing Frequency (Hz) (b) Figure 3.3 Natural frequency (a) and real part (b) of the first mode (Ro = 20kN,Kt = 10 kN/m) Figure 3.4 shows that the direction of the cutting forces at a given instant is opposite to the vibration velocity of the blade in the stable frequency region F, = (3F„/3, Fn). While the cutting edge is going up, the cutting forces point down to resist the deflection of the blade and thus absorb the lateral vibration energy. On the other hand, in the unstable region F, = (Fn/2, 2Fn 13) shown in Figure 3.5, the cutting forces and the vibration velocity are in the same direction or in phase. The energy provided by the travelling blade will switch into the transverse vibration energy through the cutting process. N Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations Mode (1,2). Fn=97.6 Hz, Ft= 85 HzJ=..3,41 ITS ^ Cutting Forces ^ j ^ ve loc i ty 0 0.1 0.2 0.3 0.4 0.5 0 6 0.7 X Mode (1,2), Fn=97.6.Hz, Ft=,85 Hz,.t ?. 4.27 ms Figure 3.4 Stable vibration of a smooth band (F, = (2Fn/3, Fn)) Mode (1,2), Fn=94 :1 Hz Ft= 53 H/, t- 16 67 1715. Ur Cutt ing Forces 1 : 1 X ^ ^ ^ V j i + Ve loc i ty 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 X Mode (1,2), Fn=94.1. Hz, Ft=.5CHz. V- 18.33 m s . . . Cutting Forces Ve loc ity -1 0.2 0.1 0 0 0.1 ~0.2 '0.3 0.4 0.5 0.6 0.7 X ~ ~ Figure 3.5 Unstable vibration of a smooth band (F, = (FJ2, 2Fn/3)) 60 Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations Under the operating speed of the blade, the tooth passing frequency ranges from 800 to 1,100 Hz. The unstable regions in this frequency range are of special interest. Figure 3.6 (a) illustrates the frequencies and real parts of the modes in this frequency range. The frequencies of Modes (11, 1) and (1, 6) do not vary greatly with the tooth passing frequency or the blade speed. The axial motion of the blade does not greatly affect the vibration frequencies of these modes. The reason is that the bending rigidity of the blade will dominate the behavior of these modes. The frequencies of Modes (11, 2) and (7, 5) increase with the tooth passing frequency. With the increase of the blade speed, this mode contains more high-order mode components of a stationary plate though the superimposed envelopes are the same as the defined mode. The influence on the system is equivalent to changes in the system stiffness and the frequency of this mode increases. Since the eigen-values are searched from the non-linear characteristic equation by the Muller's algorithm, it is difficult to trace one specific mode with an increase of the blade speed, especially in the frequency regions where many modes are closely spaced. To overcome this difficulty, one can use a small frequency increment or solve for as many roots as possible in these regions and choose interesting ones. After the eigen-values have been found, the corresponding modes will be identified based on the eigen-vectors. Figure 3.6 (b) shows the real parts of the eigen-values of these modes. The primary unstable regions of these modes start at Points A, B, C and D, respectively. Since tens of modes in this system are closely spaced in the frequency range of interest the primary unstable regions could cover the whole frequency range. Theoretically, the instability of the blade could occur at any tooth passing frequency in this frequency range. Having reviewed the equation of motion and the characteristic equation of the system, a stability analysis is derived to determine the unstable regions where flutter instability could occur. But this is only an essential, not a sufficient condition for the instability. Other conditions, such as initial conditions, damping and forced vibrations, also play an important role in the stimulation of the instability in the system. Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 61 1200 900 700 750 800 850 900 950 1000 1050 1100 1150 1200 (a) DC -5 \ (1,6) -10 700 750 800 850 900 950 1000 1050 1100 1150 1200 Tooth Passing Frequency (Hz) (b) Figure 3.6 Natural frequencies (a) and real parts (b) of eigenvalues of the blade subjected to regenerative cutting forces (i?0=20 kN, Ki=10 kN/m) 3.3.2 Effect of Regenerative Force The regenerative force coefficient has a great effect on the magnitude of the real parts of a mode in circular saws (Tian, 1998). This is also true for this smooth band. The variation of the natural frequencies of the band is very small with the regenerative force coefficient (Figure 3.7 (a)) and thus the unstable regions do not change with respect to the regenerative force if damping is not considered in this band. The maximum amplitude of the real part shown in Figure 3.7(b) increases with the coefficient and the crest of the curve shifts to the natural frequency. If damping is included, it will be found in Chapter 5 that the primary instability region of this band is extended with the increase of the regenerative forces. Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 62 iq-800 I 700 800 900 1000 1100 1200 (a) > «L K,=40 k lWi \ 10 kN/m / T T T T 700 800 900 1000 1100 1200 Tooth Passing Frequency (Hz) (b) Figure 3.7 Effects of the Cutting Force Coefficient on (a) natural frequency and (b) real part of Mode (9, 2) (R0 = 20 kN) 3.4 Summary The following conclusions can be drawn from the above discussion: (1) . Washboarding in bandsaws can be caused by the regenerative cutting forces applied at the tooth tips. These forces can be modeled as the lateral cutting forces applied on the front edge of a moving smooth band. (2) . A dynamic model of the smooth band subjected to regenerative forces has been built by using the Rayleigh-Ritz method. The unstable regions of the system can be determined from the real parts of eigenvalues solved from the non-linear characteristic equation using Muller's algorithm. The primary unstable region is where the tooth passing frequency is higher than a natural frequency and below twice the frequency. In this region, the regenerative cutting forces and the vibration velocities at the points exerted by the forces are Chapter3 Stability of Traveling Plates Subjected to Parametric Excitations 63 in the same direction. The energy from the interaction between the blade and the workpiece will be switched into the system through the forces. (3) . Since many modes are spaced in the frequency range of interest, the unstable regions cover the whole frequency range. In sawmills, washboarding occurs under certain operating conditions and disappears under others. Therefore, the stability analysis does not disclose the whole mechanisms of washboarding. It is possible that other factors also play a significant role in exciting washboarding modes. (4) . The cutting force coefficient affects both the magnitude and the location of the crest of the real parts of the eigen-values. This might be one of the reasons why washboarding occurs when the tooth passing frequency is slightly higher than the natural frequency. Because this coefficient is closely related to the bite per tooth and the tooth geometry, these factors have great effects on the instability and washboarding. Chapter 4 Modal and Stability Analyses of Moving Blades Using FEA It is found in sawmills that washboarding is closely related to the depth of tooth gullet and the deeper the gullet, the more likely washboarding is to occur during cutting. The effects of the depth of gullet on the dynamic behavior of a moving blade have never been considered in the previous studies because of the difficulty in modeling moving forces applied to an irregular moving plate. In addition, a bandsaw blade was always modeled as a smooth band with ideal boundary conditions. In an industrial bandsaw, however, guides may not support the saw blade with ideal constraints. In order to properly describe this dynamic system, the actual boundary conditions must be considered. A toothed blade subjected to moving cutting forces and its actual boundary conditions can be efficiently modeled by finite element approaches. Three types of moving plate elements, four-node bilinear element, three-node triangular element and variable-domain rectangular element, will be developed to discretize the toothed blade. Based on this finite element model, the modal parameters and instability regions of the moving blade subjected to parametric excitations will be determined and compared with those obtained from the smooth band model. Modal tests of both stationary and moving bands are conducted to verify the smooth band model and the finite element model. 4.1 Model of Moving Cutting Forces Applied at Tooth Tips The configuration of a traveling bandsaw blade over the cutting span is periodic. One tooth comes into this span at the top guide while the other goes out this span at the bottom guide. The regenerative cutting forces applied at the tooth tips are also periodic if the depth of cut is constant. Thus a dynamic model describing this blade must be time-variant. In order to model this system, the following assumptions are made 64 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 65 (1) . This is a linear system. (2) . The material is homogeneous and isotropic. (3) . Al l Kirchoff assumptions are valid in this model. (4) . The blade speed is constant. The feed speed, or the bite per tooth, is constant. (5) . All internal stresses are time-invariant. The structural damping is negligible. Based on these assumptions, this model is governed by a linear variable-coefficient partial differential equation. In order to discretize this periodic structure and model the moving forces applied at the tooth tips, a finite element approach will be used due to the versatility of this method. There are three ways to investigate the vibration and stability of this system when subjected to moving forces. (1) . Use a time-invariant model with the cutting forces applied at the tooth tips. (2) . Use a time-invariant model and decompose a moving cutting force into two forces applied at two adjacent tooth tips. (3) . Use a time-variant model. One drawback of the first approach is that the cutting forces are always applied at certain tooth tips. They no longer move with the blade and loose their periodicity. Regarding the second method, the structure is assumed to be frozen at a moment in space and all the teeth are fixed in space. A moving cutting force will be decomposed into two forces applied at two adjacent teeth based on the equality of the work done by these forces. The computation of the displacement at the point of the force action requires an interpolation of the displacements of two adjacent teeth. This interpolation may introduce a large error into this model, especially for high frequency modes. Figure 4.1 shows a finite element model of this blade built by using the third approach. The configuration of the blade and the positions of the nodes are periodic at the tooth passing frequency l/T. The blade is simply supported at two guides. The lateral cutting forces are applied at certain nodes and move with the nodes. The blade is divided into three regions: Figure 4.1 Finite element model of the blade 67 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA Region I is the left part of the blade with smooth edges. The elements in this region elongate with time. Region II is the middle part of the blade with teeth. The elements in this region move with time at a constant speed, but do not change shape. Region III is the right part of the blade with two smooth edges. The elements in this region contract with time. Two teeth close to each guide in Regions I and III will be removed to keep the total number of degrees-of-freedom of the system constant. Since these teeth are close to the fixed boundaries, they will have minor effects on the global dynamic behavior of the system. When a new cycle starts, the same procedure will be repeated. The advantage of this model over others is that moving forces can be accurately and easily modeled. The application of this method, however, leads to a difficulty that the size of some elements varies with time. This type of elements is called a variable-domain plate element in this thesis. The system discretized with these elements is time-variant and handling this system is a significant challenge. 4.2 4-Node Moving Bilinear Plate Element A moving smooth band can be discretized by using rectangular plate elements that were first formulated by Ulsoy (1982). The formulation of the elements was based on Mindlin plate theory and the simplicity of this C°-element was employed though one does not have to apply this theory to the thin band. If the teeth on a blade are discretized using triangular elements and the blade body using rectangular elements, as shown in Figure 4.1, the transition elements must be bilinear elements. Consequently, the bilinear element will be formulated in this section. The procedure for the formulation of the rectangular element (Ulsoy, 1982) will be extended to the formulation of a 4-node moving bilinear plate element. The 4-node bilinear plate element shown in Figure 4.2 (a) moves at a constant speed c in the direction of the axis x and is subjected to distributed transverse load and internal forces. Each node has three degrees of freedom, the transverse displacement w(x, y, t) and two 68 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA rotations Qx(x, y, t) and Qy(x, y, t) about axes x and y. The natural coordinates (£,, r\) of the isoparametric element are shown in Figure 4.2 (b). (a) 4-node plate element (b) Natural coordinates Figure 4.2 Isoparametric bilinear element The total kinetic energy of this element is D 1^ The strain energy of the element is given by v =4/1 { D * [ e - + 2 v e * A , + 0 ^ 0 * , + e ^ ) 2 ] + /V^w2 - 2 / w - 2m,9 2myQ y }dxdy . and D = E 12(l-v 2 ) where w, 6 x, 6 y are the transverse displacement and rotations about the axes y and x p is the density of the plate material per unit area, v Poisson's ratio, E and G are the Young's modulus and the shear modulus, K (= TC2/ 12) is the shear correction factor, h the plate thickness, c the blade speed. Nx, Ny, and Nxy are the internal forces. /, mx, and my are the distributed transverse force and moments. D is the integral domain, DE is the plate rigidity. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 69 For a thin plate, all Kirchoffs assumptions are satisfied, i.e., rotatory inertia and transverse shear deformation are negligible. Then, 9 y , 0 = 0, Q y!(x,y,t) = 0 and 9 x (x, y,t) = w x (x, y , 0, 9 y(x, y,t) = w y (x, y, t) The displacements and rotations can be expressed in terms of the shape functions, i.e., w [ N / 1 • = • N ; • u N ; 0 0 \|f2 0 0 \|r 3 0 0 0 0 \ | / 2 0 0 \|f3 0 \ | / 4 0 0 0 0 \|f4 0 u 0 0 \|/, 0 0 \ | / 2 0 0 \|r3 0 0 \|/\ where N W , and Ny are the row vectors, u is the displacement vector u = { w 1 e , 1 e y l w2ex2ey2 w3dx3Qy3 w4Qx4dy4yT-The shape functions are a - i ) ( i - T i ) 1 • . ( l + S X l - T l ) > 4 (1 + ?)(1+T1) .(1-1X1+T1 ). (4.2.3) (4.2.4) For the isoparametric bilinear element, the coordinates at any point can be expressed in the form (4.2.5) 1=1 /=i and (4.2.6a, b) x = 2. y = 2-/=] f8 ] ' a ' fa j [a 1 d x ' a = J 1 a or • a = j a x a ' W ari ^ 1 J ar) (4.2.7) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 70 where J is the Jacobian Matrix, i.e. J = d x d y d x d y (4.2.8) For a rectangular element, J = 2 Ax 0 0 Ay where Ax and Ay are the coordinate differences of the element along the axes x and y. The application of Hamilton's principle 8 I"2 (T-V)dt = 0 J ' i to this element leads to its equations of motion mii + gii + ku = f where the element mass matrix is m = p f c J J [NwNj+^-(NxNxT+NyNyT)]dxdy D ^ The element gyroscopic matrix is g = 2pAcjJ NwNw/dxdy D The element stiffness matrix is k = J J {DE[NXtXNj +N, f,N,/+vNX i J tN, i, 7' + v N w N / ' D + ^ ( N ^ + N ^ ( N , , y + N ^ ) r ] + K G A [ ( N W I X - N , ) ( N W T X - N X ) T + (N W I , - N , ) ( N W I , - N , ) R ] + (Nx - phc2) N WtXN J + Ny (N N J) + Ny (N W i , N J) + (N ^  N „ / + N ^ N J ) } dxdy The nodal forces can be obtained by the integral f = \\ ( / N w + mxN x + myN y)dx dy (4.2.9) (4.2.10) (4.2.11) (4.2.12) (4.2.13) (4.2.14) (4.2.15) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 71 In the above equations, where A= w, x, and y. All the integrals in these equations are evaluated using the Gauss integral scheme. The element stiffness matrix consists of bending, shear, and internal stress terms. For the bending term, 2x2 points are used. For the shear term, to avoid shear locking, the reduced and selective quadrature (lxl) is used. It is assumed that the internal stress in an element is constant. If the stress distribution in an element must be considered, the number of points is determined based on the order of the stress distributions. 4.3 3-Node Moving Triangular Plate Elements In order to model the teeth of a bandsaw blade, triangular plate elements will be needed so that a tooth profile can be efficiently discretized. A great number of triangular plate elements have been developed for both thin and thick plates (Cook, 1989; Yang, 1986; Zienkiewicz, 1989). According to plate theory, these plate elements can be classified into two categories. One is based on Kirchhoff theory in which C1 continuity is satisfied for inter-elements and within elements. This type of elements is valid only for bending of thin plates with small deformation. The other is derived from Mindlin theory (Mindlin, 1956), in which the rotation and the shear deformation of the plate are considered. These elements can be used to model both thick plates and thin plates. Mindlin-type triangular elements have both advantages and disadvantages compared with Kichhoff-type elements. In these elements, independent displacements and rotations are introduced. Since only C° continuity is needed over inter-elements, lower order shape functions can be used which enhance the efficiency and simplicity of this type of element. These simple and efficient elements are extremely useful in the analysis of transient and nonlinear problems, in which computation time and memory are often crucial factors. 72 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA However, the use of low-order shape functions results in the shear locking effect if exact integrals of stiffness matrix are calculated. For instance, the displacement and rotations of an isoparametric linear triangular element can be expressed in the form of linear functions of the coordinates x and y. It is desirable that the shear strains of the element vanish when the thickness of plate h —> 0. That is to say, each term in the expressions for shear strains needs to approach zero. This requirement will introduce six additional constraints into the element (Tessler, 1985) and the normal rotations are either zero or have constant values. This will cause the bending energy to vanish. In practice, the stiffness contributed by the shear strain energy will be very high and the element will become excessively stiff. Consequently, the solution is "locked" due to spurious shear constraining and useless though all of the fundamental convergence requirements (i.e. rigid body motion, constant strain and compatibility criteria) are fulfilled. To overcome this difficulty, the stiffness caused by shear strain energy must be modified. One approach is to reduce the number of integration points of the stiffness caused by the shear strain energy by using one-point quadrature (Hughes, 1977). Further research shows that it is insufficient to use reduced shear integration in order to develop a successful C° element (Batoz, 1980). Another approach is to decompose the deformation into bending and shear modes (Belytschko, 1984) and fit the bending mode to an equivalent Kirchhoff mode. In addition, Tessler (1985) used a penalty parameter, called a shear correction factor, to modify the shear term of the stiffness. In general, these approaches can give good solutions for thin plates. But the results obtained based on these methods are strongly dependent on the orientations of the mesh if the number of elements is small. Since the displacements and rotations of these elements do not conform to each other, even at the nodes and edges, excessive errors would be introduced into the stiffness caused by internal forces and gyroscopic terms. 73 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA Since a bandsaw blade is a typical thin plate, transverse shear strains are negligible. Thus the Kirchhoff theory is good enough for this plate. If either a completely or partly conforming element is used, this element may be used to model the effects of internal forces and gyroscopic forces. Furthermore, the solutions based on this type of elements are not as sensitive to mesh patterns as those using a Mindlin-type element. Some displacement-based Kirchhoff theory elements are reviewed in the references (Yang, 1986; Zienkiewicz, 1989). One of the first compatible triangular elements is the well-known HCT element (Clough, 1965). This element is formulated by the assembly of three sub-triangles of the complete element, and the displacement is expressed by an incomplete cubic polynomial (9 terms). However, the formulation involves very cumbersome algebraic manipulations and the element is rather stiff (Batoz, 1980). Another two elements are a non-conforming element and a conforming element presented by Bazeley (1965). The former does not converge to the exact solution for some mesh patterns due to its incompatibility. The latter is a compatible element formulated by appropriate superposition of polynomials and rational shape functions in area-coordinates. A very high-order (16-point) integration scheme is needed due to the presence of these rational functions. Discrete Kirchhoff theory elements (DKT element) have demonstrated their good behavior in several analyses (Batoz 1982). The formulation of these elements is based on the discrete Kirchhoff theory for thin plate bending and is obtained by first considering a theory including transverse shear deformations. In this element, the independent quantities for each node are the displacement w, and rotations 0* and 0^  and only C° continuity is satisfied. Then the Kirchhoff hypothesis is introduced in a discrete way along the edges of the element to relate the rotations and the transverse displacements. The transverse shear strain energy is totally neglected. This approach has been used to formulate 9-dof triangular plate elements. It is proved that the displacements and the free vibration eigenvalues converge quadratically to the C1 Kirchhoff solution of thin plates (Batoz, 1980). Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 74 The work presented in this section aims at modifying the DKT element in such a way that the internal forces applied to the element and the axially moving speed of the element can be included in this element. The formulation of this axially moving plate element with internal stresses will be presented in detail. 4.3.1 Assumptions A nine-degree-of-freedom moving triangular plate element and its area coordinates are shown in Figure 4.3 (a). There are three vertex nodes (1,2, and 3) and three mid-edge nodes (4, 5, and 6). These mid-edge nodes are used only for the formulation. Each node has three-degrees-of-freedom, w, w?x and w>y. The element moves in the direction of the x axis at the constant speed c. The natural coordinates Lj, L2 and L3 (area coordinates) are shown in Figure 4.3 (b). The formulation of this moving triangular plate element with internal forces is based on the following assumptions: (1) . The element is subjected to the internal forces Nx (along the x axis), A^ (along the y axis) and Nxy (shear force) in the neutral plane. These forces are uniformly distributed over the element. (2) . The element moves in the positive direction of the axis x at the constant velocity c. (3) . Rotations 0 x and 0 y of a mid-surface normal line are interpolated from nodal rotations 0X(- and Qyi (/ = 1,2, 6) using a complete quadratic polynomial: e , = t Npxi (4.3.1a) 1=1 6 = X Nfi yi (4.3.1b) 1=1 where the shape functions A7,- are given in the reference (Batoz, 1980) (also see Appendix C) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 75 X, u (a) (b) Figure 4.3 Triangular plate element (a) and area coordinates (b) (4). The Kirchhoff hypothesis is imposed at the corner nodes and mid-nodes. The transverse shear strains yyz and yxz vanish at corners 1, 2 and 3, i . e. 9 W,yi and 0 .= -w, The transverse shear strain ysz vanishes at mid-nodes 4, 5 and 6, i . e. 0 .= w . SI ,51 where s is an edge-tangent coordinate. (4.3.2) (4.3.3) (5). Normal slopes vary linearly along each side. Thus 1_ 2 0 = —(w • + w ) nk „ V ,m ,nj / The variation of the displacement w along the sides is cubic, i.e. ,sk — (w ; -w.)--(w s , .+w j ; . ) (4.3.4) (4.3.5) where /y is the length of the side ij and k = 4, 5 and 6 and j7 = 21, 32 and 13. (6). Since the displacement w is not involved in the formulation of a DKT element (Stricklin, 1969; Batoz, 1980 and 1982 and Cook, 1989), the displacement function proposed by Stricklin (1969) is conforming to the slopes only at the six nodes. Thus this displacement function can be modified if the updated displacement field satisfies the same requirements. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 76 It is noted that Assumptions (1) and (2) characterize the new moving DKT element (called MDKT element later). Assumptions (3) to (5) are the basis of derivation of a classic DKT element (Batoz, 1980). Assumption (6) implies that a displacement function should be chosen in such a way that the mass matrix and gyroscopic matrix based on the function would be as compatible as possible to the stiffness matrix based on the DKT element and the internal forces. 4.3.2 Shape Functions According to the above assumptions, the independent displacement w and rotations 0 x and 0 y can be expressed in terms of the nodal variables and shape functions (Yang, 1986; Stricklin, 1969; Batoz, 1980 and 1982). Hence, w(x, y) = Nd where w(x, y) = w(x, y) w,x(x,y) w,y(x,y)^ <wi w,XI W,XI WJ W xj W,xj ™k ™,xk W,xk and N ; N = - > = ^xl ^x2 ^x3 ^xl ^xS ^x9 w ^y3 • •• ^yl ^yS *y9 (4.3.6) (4.3.7) The displacement shape functions (Yang, 1986) and the rotation shape functions (Batoz, 1980) are given by (C.l) to (C.9) in Appendix C 4.3.3 Element Matrices of D K T Element I. Stiffness Matrix The bending strains are Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA e h=ze=z< w, w +w ,xy ,yx = zBd where B is the strain-displacement transformation matrix: 1 B = 2A C2^X,L2 + c 3 N ^ +b2Ny^ + b,Ny^ where the derivatives w.r.t. the natural coordinates L2 and L? are calculated. The bending strain energy is given by Vb =^jA eTDb e dxdy where 1 v 0 1 0 1-v (sym) and DE is the bending rigidity, i . e. D, EH> 12(1-v 2) Hence, the stiffness matrix of the DKT element can be written in the form r1 r1-^ KDKT=2A[\~ B r D A B dL2dL3 77 (4.3.8) (4.3.9) (4.3.10) (4.3.11) (4.3.12) (4.3.13) In order to obtain the explicit form of the stiffness matrix, the matrix B can be decomposed into the product of two matrices L and B c (Batoz, 1982), i.e. B = — L B 2A c with L = L c 0 0 0 L c 0 0 0 L where L c — •{! L*2 L<2 y (4.3.14) (4.3.15) (4.3.16) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 7 8 Hence, the stiffness matrix can be given by 1 and B c is the coefficient matrix that is independent of L2 and L3. c can be g: B r L n B I L ° = i ' f where and K DKT 2A"C ^D"c L DbLdL2dL3 D E 24 R v R 0 R 0 1-v (sym) R R = 2 1 1 1 2 1 1 1 2 (4.3.17) (4.3.18) (4.3.19) II. Additional Stiffness due to Internal Forces It is assumed that the element is subjected to the uniform internal forces N x , N y and N x y . The bending strain energy in the element will be modified by adding the following terms VN =jA (Nxw^x2 +2Nxyw^y +Nyw*)dA (4.3.20) Thus the stiffness matrix of the MDKT element is the sum of the stiffness matrix KDKT of the DKT element and the stiffness matrix K N , i.e. k = K D „ . + K „ (4.3.21) In order to obtain the explicit expressions for K N , the rotation shape functions N x and N y are each decomposed into the products of the coefficient matrix N x c or N y c , independent of the area coordinates L2 and Lj, and a vector L a . They are N , = N J L f l and N y = N y e L f l (4.3.22) and La = {1 L 2 L 3 L 2 2 L 2 L 3 L 3 2 }r (4.3.23) Letting L „ = £ L f lL f l rrfA (4.3.24) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 79 then the additional stiffness due to the internal forces is given in the form where N X C , N Y C and L N are given by (C.27) to (C.29) in Appendix C K„ = NXNXCTIJNNxc + 2NNXCTLNN + N N T~LNN (4.3.25) III. Mass Matrix The mass matrix of the MDKT element can be expressed by m = ph\A [ N I V N J + ^ ( N , N ; + N , N ; ) ] J A (4.3.26) Similarly, the mass matrix can be written in the form m = p M N K L M N : + ^ ( N J C L X + N , L X J (4-3.27) where L M is obtained from L*=J A L . L / d A (4.3.28) and 1^ w — "(1 -^3 ^2 ^2^3 ^3 ^2 ^2 ^3 ^2^3 ^3 ^ (4,3.29) N W C and L M are given by (C.25) and (C.30) in Appendix C . IV. Gyroscopic Matrix The gyroscopic matrix is given by g = 2phc\A NWNWJ dA (4.3.30) where N ^ = ^ = NM C i J [L B (4.3.31) ox Hence, G can be expressed in the form g = N l i e L c N ) ( „ (4.3.32) where L G = \ A ^ w L a r dA (4.3.33) NWCiX and L G are given by (C.26) and (C.31) in Appendix C . 80 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 4.4 Variable-Domain Rectangular Plate Element Dynamic analysis of spacecraft antennas and robotic manipulators gives rise to variable-domain beams. The earliest work on variable-domain beams is due to Tabarrok (1974). He modeled an axially moving beam with variable length in the domain of interest. Wang (1987) studied the vibration response at the tip of a moving flexible robotic arm. Yuh (1991) simulated the motion of a similar moving beam and experimentally verified his results. These studies were conducted by using either an analytical method, a Galerkin method or an assumed-mode method. The variable-domain beam was also studied by the use of the finite element approach (Stylianou, 1994). He used variable-length beam elements to discretize an axially moving beam and investigated the vibration response at the beam tip and instability regions due to the changes in the beam length and axial velocity. However, research on variable-domain plate elements has not been initiated. In order to model a moving toothed blade, a variable-domain plate element will be developed in this section. A variable-domain rectangular plate element is shown in Figure 4.4. The global coordinate system OXY is stationary in space. The local coordinates oxy attached to the plate moves at the constant plate velocity c. The left-hand and right-hand edges of the i-th element move at the constant velocities c,-.y and <:,• respectively. The dimension along axis y remains constant. The elongation rate is /(0 = (ci-c,._1)r, 0<t<T (4.4.1) where 0•-l)c ic . . . c,._, = = nclc, ct = — = nc2 c (4.4.2) nc nc nc is the number of velocity increments from zero to the blade velocity, i.e., the number of columns of the meshes in either Region I or III. T is the period of the tooth motion. The ratio of the length to the average length IQ and its reciprocal are defined as (4.4.3) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 81 Figure 4.4 Variable-domain rectangular plate element K 0 = 77- (4.4.4) The kinetic energy of a moving rectangular element can be expressed in the form 2 DO) 1 2 where c is the material velocity, D(t) is a variable domain. The strain energy of the element v = ^j\ {De®1 + 2 v e X i X e , , , + e , 2 i , + i - ^ - ( B X i , + e „ ) 2 ] 2 .2 + K G A [ ( w , - e j z + ( w i y - e y r ] + iV,w i; + 2iV JVw i J twy (4.4.6) + Ny w2 - 2/vv - 2mx0 x - 2my0 y }dxdy The displacement at any point in the element can be expressed in terms of the shape functions, i.e. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 82 w V • = • • u where \|/, 0 0 v 2 0 0 ¥ 3 0 0 \ j / 4 0 0 0 ^ , 0 0 \|/2 0 0 \ | f 3 0 0 v 4 0 0 0 0 0 \ | f 2 0 0 \ | f 3 0 0 \ | f 4 u u •{Wl Qxi Qyi w2 Qx2 Qy2 w3 Qx3 Qy3 w4 9 x 4 Qy4}T (4.4.7) N„ = V i " '(1-^)(1-T])' _ 1 (i+ixi-Ti) ¥ 3 "~ 4 ^ 4 . (1-^Xl+Tl), (4.4.8) Since the width of the element varies with time, the natural coordinate is expressed by (*.0 =—(*-*c(0) 1 \ t) 1 / (0 = Xt - , Xc (0 = - (*. + ) (4.4.9) (4.4.10) It is noted that the shape functions are functions of both the space coordinate x and time t. Therefore, _ ( " c i - " 0 2 ) ( 4 . 4 . 1 1 ) a;(0/0 _ (wel ~nc2)c^ (4.4.12) at(t)l0 Taking the derivative of the shape functions with respect to the natural coordinate leads to N a , x V," - ( l - T l ) ^ ¥ 2 _ 1 ( 1 - T 1 ) ¥ 3 ~ 4 (1+Tl) ¥ 4 . >i - ( 1 + T 1 ) 2 /(0 (4.4.13a) (4.4.13b) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 83 N = N (ncl -nc2)c a,(t)l0 ( l+S)N f l -N =N . | = 2 ^ ~ n c 2 ) 2 c 2 ' a^<" (a{(t)l0f { 2( "c i - n c i ) c 2 N f l > t (a,(0/0). Substituting (4.4.7) into (4.4.5) and (4.4.6), the Lagrangian is written in the form L = T-V 1 (4.4.14) (4.4.15) (4.4.16) where • T • I T I l r - l ' T T u min— u ku + —u g u+—u g u 2 2 2 . 2 m = p/zj] N / N w ^ y + £ - ^ j ] (N J ISX + IS yT IS y)dxdy 0(0 1 2 0(0 k — k | ~\~ k 2 ~^"^3 ^0 K = P h J] [IS J N „ + cN / N ^  +cN J N w + c 2 N W / N „ Jdxdy 0(0 k, = p h3 12 p fr3 12 J J N x T N x ^ y 0(0 JJ N / N ydxdy 0(0 + ^ ( N „ + N r , ; c ) r ( N A . J + N ^ ) ] + K G / z [ ( N l v , - N J r ( N ^ . - N ; c ) + ( N W J - N y ) r ( N ^ - N y ) ] + ( N M r N , , , + N w / N J ><&dy O(z) (4.4.17) (4.4.18) (4.4.19) (4.4.20) (4.4.21) (4.4.22) (4.4.23) (4.4.24) O(0 84 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA If a distributed transverse force p(x, y, t) is applied to the element, the work done by the force is W K = J J p(x, y,t)Nwudx dy (4.4.25) 0(0 Using Lagrange's equation leads to the equations of motion of the rectangular element mu + g e qu + k e q u = f (4.4.26) where g e q =rh + g T - g (4.4.27) k e q = k 0 - ( k 1 + k 2 + k 3 ) + g T (4.4.28) dm m = dt h2 = p h\\ [(N/N„ + Nj N J + — ( N / Nx + N / N x + N / Ny + N , r N y D(0 1 Z + p x2(t)[KjNw + ^ ( N / N x + N , r N , ) ] ^ , ^ - p *J* Xl(tWjNw + ~ ( N / N x + N, r N , ) ]^ ( 0 dy 12 dg dt = Phjj [(cNwJ N w + c N w > / N, v + Nj N„ + Nj NJ 0(0 (4.4.29) h1 + — ( N / N , + N / N y + N / Nx +NyTNy)]dxdy (4.4.30) + p h£ x2(t)[cNHJ N w + N / N w + ^ - ( N / N x + N , r N , ) ] ^ , , ^ - p h£ i l W [ c N ^ N H . + N / N l v + ^ ( N / N x + N / N y ) U i ( 0 J y f= JJ p(x,y,t)Nwdxdy (4.4.31) 0(0 For this isoparametric bilinear element, the coordinates at any point can be given by x = , y = ,. y ; (4.4.32a, b) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 85 and where ra i ra ] ' a ' ra i x < a =J- ] a ; or • a = j x a ' ar| dxdy = J d^,dr\ J = ax ay a x a y ar| ar| For this rectangular element, J « ) = -&x(t) 0 0 Ay (4.4.33a, b) (4.4.34) (4.4.35) (4.4.36) It is seen from the equations of motion of this element that m is a conventional mass matrix. It is symmetric and positive definite. Since dm/dt is symmetric, g e q no longer is a skew-symmetric matrix. Because dg/dt is a general matrix, k e q is not symmetric. When considering the Gauss quadrature for each term, it can be seen from the above equations that some terms in these expressions are independent of lit), some are directly proportional to lit), others are inversely proportional to lit). The mass matrix is proportional to l(t), i.e., m = m 0 l(t) (4.4.37) Where mo is the average mass matrix corresponding to the element with an average size. The gyroscopic matrix is independent of the variable length l(t). The stiffness matrix contains three mentioned cases. During computation, therefore, these properties will be utilized so that the stiffness matrix is kept symmetric and the equivalent gyroscopic can be easily computed. 86 Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 4.5 Stability Analysis of the Toothed Blade 4.5.1 System Equations of Motion with Variable-Domain Elements The system equations of motion of the plate shown in Figure 4.4 can be expressed in the form M U + G U + K U = F (4.5.1) Each system matrix can be divided into a constant matrix and a matrix that is a function of time, i.e. (M 0+ AM(O)t) + (G0+ AG(f))U + (K 0 + AK(f))U = F (4.5.2) The constant matrices Mo, Go and Ko are obtained by assembling the corresponding matrices of the three regions in which all the elements have their average sizes (i.e. l(t)=lo). The matrices AM, AG and AK result from the varying domains in Regions I and III and are time dependant. Having examined these matrices, it is found that the incremental mass matrix is directly proportional to the element length 1(f). Since the effect of the variable element length on the gyroscopic matrix is cancelled for this C° rectangular element, the incremental gyroscopic matrix is zero. As for the incremental stiffness matrix, some terms are proportional to the element length, some are inversely proportional to the length and others do not vary with the element size. When these matrices are evaluated, one can find that some terms are proportional to the blade velocity c, some to c , and others have nothing to do with the velocity. The terms in the matrices related to either the element length or the velocity are evaluated respectively so that the analytical expressions for the effects of the size and the velocity can be obtained. Since the cutting forces are always applied at the tooth tips that move at the blade velocity, these forces will be modeled as concentrated forces applied at the nodes at these tooth tips in Region II. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 87 A regenerative cutting force applied to one tooth tip is directly proportional to the relative displacement of this tooth tip to that of the prior tooth tip at the time one period before. Since the transverse displacements at these nodes, measured in the moving coordinate system oxy (Figure 4.4), are the same as those in the global coordinate system OXY, these regenerative cutting forces can be expressed in the form F = {o-o/ n l o - o / „ 2 o-o/;, 0 - 0 } 7 -= - ^ ; i t ( 0 [ U ( 0 - U ( r - r ) ] (4.5.3) where and l,(t) = diag{0-0I/n0-0In20-0Inl 0 - 0 } at nx n2 nt (4.5.4) Ik(t) = < . , k = nl, n2,..., n, (4.5.5) [0 otherwise and n , is the number of teeth in the cutting region. 4.5.2 Static and Dynamic Condensations The stability problem of the blade subjected to regenerative cutting forces is associated with solving eigenvalues from a set of nonlinear and complex algebraic equations. Large number of degrees-of-freedom will introduce difficulty into the solution, or even lead to failure of the solution. Therefore it is extremely important to reduce the number of degrees-of-freedom and assure acceptable accuracy of the solution. Here both static and dynamic condensations will be employed to achieve the goal. I. Static Condensation The purpose of static condensation is to formulate a condensed system with a much smaller number of degrees-of freedom so that a modal analysis can be conducted more quickly. Since all the moments of inertia of a thin plate is negligible, all degrees-of-freedom Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 88 in rotations can be removed through static condensation without introducing unacceptable errors. Let U = J U l U J l , l u ^ P U , (4.5.6) Then (4.4.39) can be rewritten as MjU, +G,U, +K,U, =F, (4.5.7) where Mj = P T M 0 P (4.5.8) G, = P T G 0 P (4.5.9) K , = P T K 0 P (4.5.10) F 1 = P T F (4.5.11) II. Dynamic Condensation Dynamic condensation is used to make use of the modal matrix corresponding to a small number of eigenvalues and convert the large number of physical coordinates to a reasonable number of modal coordinates. This modal matrix <X> will be employed to further condense (4.4.44). Now the eigen-problem of the conservative system M , U , +K,U, =0 (4.5.12) is in consideration. If a number of modes around the frequency of interest are selected, the modal matrix can be constructed. Then the system equations of motion can be condensed in the form M e q + G e q + [K e + K, (1 - e~TD ) I t J q = 0 (4.5.13) where q = O r U , (4.5.14) Me M , 0 (4.5.15) G = 0 T G , 0 (4.5.16) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 89 Ke - <&T K , <B (4.5.17) I t e = 0 T I t O (4.5.18) It is noted that this coordinate transformation will greatly reduce the number of degrees-of-freedom of the original system to an acceptable number of that described by the modal coordinates. This will benefit the solution of the eigenvalues in the stability analysis. Actually this transformation will be conducted after the integration of the incremental matrices with respect to time over one period so that the characteristic equation for the stability analysis can be derived. 4.5.2 Stability Analysis The condensed system of equations of motion of the toothed blade is given by [M + M a (?)] q(0 + [G + Ga (0] q(0 + [K + K a (0 + K, (1 - e~TD) ITE (r)] q(0 = 0 where M=(Mt(t)dt, M a(0 = M e ( 0 - M JO G=[ Ge(t)dt, Ga(0 = G e ( 0 - G K = [ Ke(t)dt, K a (f) = K e ( f ) - K These are a linear, variable-coefficient and ordinary differential equation system. It is assumed that the solution is: q = ex'^ [ak cos(£(u 0 + b k sin(^ O) t)] (4.5.23) Substituting (4.5.23) into (4.5.19), multiplying 1, cos(km), sin(kti)t), k=l, 2, ... and integrating these equations over one period with respect to time result in a linear algebraic system in terms of the Fourier coefficients ao, a* and b ,^ k=l, 2, .... The nonlinear characteristic equation of the system can be obtained under the conditions of the existence of non-trivial solutions. (4.5.19) (4.5.20) (4.5.21) (4.5.22) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 90 If a zero-order solution q = e"a0 is assumed, for instance, we have X 2M + X G + (K + ( l -e~™)I t )a 0 =0 where (4.5.24) (4.5.25) (4.5.26) If the first order of the Fourier series of the solution is assumed, q = eu (a0 + a, costo t + b, sin to t) one can have where ' 13 ' 2 2 Sn S12 S13 <s <s <s 1 3 21 ° 2 2 c ' 2 3 ^ 3 1 ^ 3 2 ^ 3 3 a n = 0 S 2 i -S u = 7TA, 2 M + A. G + (K + (l-e~™)I t)] S 1 2 =£" {Me(f)[(A. 2 -co 2)costof-2to A.sintof] + Ke(Ocoscof}c# f {Me(t)[(k 2 -co 2)sinto r + 2to A.costo >] + Ke(r)sinto f } ^ £" [A, 2 M e ( 0 + Ke(0]coscor<ir JT {Me(0[(A.2 -co 2)costo?-2to A,sintof] + Ke(f)coscof}cosuH<# f {M e(0[(A, 2-co 2)sinco t + 2co A,cosco f] + K e(0sinco r}cosco r A JO £" [k 2Me(t) + Ke(t)]smu>tdt f {Me(t)[(k 2 -co 2)coscof-2co A,sincof] + Ke(Ocosco?}sincor<ir f {Me(t)[(k 2 -co 2)sinco ? + 2toA.costo r] + K e (r) sin to f}sin to f dt JO S,„ — S->, — Sv> — (4.5.27) (4.5.28) (4.5.29) (4.5.30) (4.5.31) (4.5.32) (4.5.33) (4.5.34) (4.5.35) (4.5.36) (4.5.37) The integrals in the above equations can be analytically evaluated before the static and dynamic condensations. In fact, these transforms to the system equation are linear. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 91 4.6 Analytical Results 4.6.1 Modes of Smooth Bands Four examples are given in this section to verify the formulations and programs of the moving bilinear plate element, triangular plate element and variable-domain rectangular plate element. Case I: A Stationary Plate without Internal Forces The natural frequencies of a smooth band (Appendix A) are listed in Tables 4.1. Ten frequencies listed in the second column based on the Ritz method are taken as the reference results because the program based on this method was verified previously. The maximum difference listed on the bottom row is the maximum value of ten relative frequency differences between the second column and one of the columns on which the frequencies are calculated using different elements or meshes. The results from Nastran (mesh 76x26, plate elements) are in good agreement with those from the Ritz method. Regarding the bilinear and triangular plate elements, the coarser mesh (16x8) introduces the maximum difference of about 5% into the last two frequencies. The finer mesh (50x20) gives very good results. Table 4.1 Natural Frequencies of a Stationary Plate (L = 762 mm, 5=261 mm, h=\A5 mm, R0=0, c=0) Mode Ritz* (8x6) NASTRAN (76x26) Bilinear Element (16x8) (50x20) Triangular Element (16x8) (50x20) (1,1) 5.91 5.86 5.87 5.86 5.87 5.87 (2,1) 23.89 23.70 23.65 23.70 23.67 23.72 (1,2) 24.06 24.00 23.84 23.96 23.75 24.04 (2, 2) 52.52 52.33 51.51 52.24 51.63 52.46 (3,1) 54.13 53.77 54.53 53.78 53.94 53.86 (3,2) 88.73 88.32 87.04 88.15 87.15 88.62 (4,1) 96.69 96.17 98.68 96.20 96.55 96.43 (1,3) 128.28 128.38 122.54 127.59 122.04 128.39 (4, 2) 134.74 134.06 132.93 133.78 132.29 134.69 (5,1) 151.52 150.96 147.38 151.04 148.45 151.57 Max. Diff. (%) 0 -0.67 -2.73 -0.74 -4.86 -0.50 * Reference results Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 92 Case II: A Stationary Plate Subjected to Internal Forces Tables 4.2 lists the natural frequencies of a stationary plate subjected to an internal tension. If the finer mesh is used, the results obtained from the finite element analysis are satisfactory. Table 4.2 Natural Frequencies of a Stationary Plate with Internal Forces (L = 762 mm, B=26\ mm, ^ =1.45 mm, #0=20 kN, c=0) Mode Ritz* Bilinear Element Triangular Element (8x6) (16x8) (50x20) (16x8) (50x20) (1,1) 54.33 54.24 54.34 53.90 54.26 (1,2) 59.31 58.13 59.12 57.85 59.04 (2,1) 110.63 109.94 110.67 109.51 110.53 (2, 2) 120.11 117.57 120.13 117.31 119.99 (1,3) 139.19 132.60 139.55 132.27 139.26 (3,1) 170.84 168.72 171.00 168.60 170.79 (3,2) 184.73 178.59 184.87 179.61 184.67 (2, 3) 189.96 179.75 190.25 180.34 190.10 (4,1) 236.39 232.43 237.22 232.85 236.90 (4, 2) 254.61 237.88 255.09 241.45 254.63 Max. Diff. (%) 0 -6.57 0.35 -5.17 0.22 Reference results Case III: A Moving Plate Subjected to Internal Forces The natural frequencies of a moving plate subjected to an internal tension are listed in Table 4.3. In analyzing these cases, consistent mass matrices were used. If the finer mesh is used, the results obtained from the finite element analysis are consistent with those based on the Rayleigh-Ritz method. It can be seen from the three-case studies: (1) . The results obtained in one method are in good agreement with those in other methods, especially for lower frequencies. (2) . The solutions approach the values obtained from the Ritz method when finer meshes are used. This implies that the convergence properties of these elements are good. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 93 Table 4.3 Natural Frequencies of a Moving Plate with Internal Forces (L = 762 mm, 5=261 mm, h=lA5 mm, R0=20 kN, c=50 m/s) Mode Ritz* Bilinear Element Triangular Element (8x6) (16x8) (50x20) (16x8) (50x20) (1,1) 46.54 46.70 46.52 46.34 46.45 (1,2) 51.69 51.23 51.67 51.51 51.61 (2,1) 97.15 97.89 97.18 98.24 97.06 (2, 2) 107.06 106.38 107.07 108.28 106.97 (1,3) 127.35 122.63 127.67 128.58 127.49 (3,1) 155.04 157.42 155.23 160.19 155.04 (3,2) 169.01 167.26 169.13 173.53 169.01 (2, 3) 175.15 168.79 175.45 179.66 175.45 (4,1) 222.10 226.97 222.54 234.84 222.18 (3,3) 238.72 227.97 239.16 244.96 238.88 Max. Diff. (%) 0 -4.50 0.18 5.74 0.17 Reference results Case IV: Modal Analysis of a Moving Plate Using Variable-Domain Element The modal anlysis of this moving plate is conducted using both an Eulerian formulation (the mesh is not attached to the plate and it is fixed in the space) and a variable-domain FEM. The natural frequencies of the first ten modes are listed in Table 4.4. The results obtained from the time-variant FE model are consistent with those based on the Eulerian formulation, in which the coordinate frame is fixed in the space. The number nc of columns of moving meshes in Region I or III has a minor effect on these modes. The natural frequencies of high order modes of the plate are listed in Table 4.5. The same models are used for the high-frequency modes. The frequencies based on the Eulerian formulation are listed in the second column. The third and fourth columns show the results based on the variable-domain FEM. In this model, either 60 or 100 modes are used for the dynamic condensation. The eigenproblem of the condensed system (without cutting forces) is solved using Muller's algorithm. It is seen from this table that the natural frequencies based on the variable-domain FEM are in agreement with those based on the Eulerian formulation. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 94 It is also implied that 80 modes might be sufficient for the dynamic condensation without introducing unacceptable errors. Table 4.4 Natural Frequencies of a Moving Plate with Internal Forces (L = 762 mm, 5=261 mm, h=lA5 mm, R0=20 kN, c=50 m/s) No. Mode Eulerian FEM (nc=2) FEM (nc=4) 1 (1,1) 46.51 46.87 46.70 2 (1,2) 51.40 51.74 51.57 3 (2,1) 97.14 97.88 97.64 4 (2, 2) 106.43 107.15 106.93 5 (1,3) 125.22 125.44 125.34 6 (3,1) 155.11 156.28 156.00 7 (3,2) 168.00 169.14 168.86 8 (2, 3) 171.51 172.08 171.89 9 (4,1) 222.28 223.96 223.45 10 (3,3) 233.24 234.14 233.91 Max Diff. (%) 0 -0.08 -0.06 The mesh (34x12) is used and 40 modes are used for the dynamic condensation. Table 4.5 High Natural Frequencies of a Moving Plate with Internal Forces (L = 762 mm, 5=261 mm, h=\ .45 mm, fl0=20 kN, c=50 m/s) Mode Eulerian FEM (N=60) FEM (N=100) (7, 5) 977.3 980.3 978.9 (11,1) 992.2 1016.8 996.4 (11,2) 1014.1 1037.8 1018.2 (1,6) 1030.7 1030.8 1030.8 (2, 6) 1050.8 1051.1 1050.9 (8,5) 1079.3 1093.4 1081.5 (3,6) 1083.8 1084.8 1084.1 (11,3) 1093.9 1115.1 1097.6 (10, 4) 1096.1 1114.4 1099.0 (4, 6) 1129.6 1131.0 1130.0 Max Diff. (%) 0 2.48 0.04 The mesh (34x12) is used and N modes are used for the dynamic condensation. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 95 4.6.2 Instability Regions of the Smooth Band In order, to examine the reliability of the FE model for a stability analysis, the blade used for the modal analysis is now subjected to regenerative cutting forces. Its unstable regions are determined on the basis of the smooth band model and the FE model. Figure 4.5 shows the real parts of the eigenvalues and the natural frequencies of the first ten modes of the blade. Their real parts have small differences while the frequencies are almost identical. 0 -5 -10 -15 -20 t* -25 -30 -35 -40 -45 oj 4 5 6 Mode (a) Real parts ORitz • FEM 10 300 ^ 250 g 200 3 CT 03 150 100 50 0 4>-5 6 Mode ORitz • FEM (b) Natural frequencies Figure 4.5 Eigenvalues of the first ten modes (listed in Table 4.4) (R0=20 kN, #,=10 kN/m, c = 31.1 m/sec, F, = 700 Hz) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 96 Two low frequency modes, (1,1) and (2, 1), are taken as examples to demonstrate the instability regions of this blade. The real parts of the eigenvalues obtained from the Ritz method and the FE model are compared in Figure 4.6. The real parts of Mode (1,1) shown in Figure 4.6 (a) almost overlap each other. Although a small difference between the real parts of Mode (2, 1) exists, as shown in Figure 4.6 (b), they indicate that the two methods give the same natural frequencies and the same unstable regions for these two modes. . 2 I , , , L _ 1 1 0 50 100 150 200 250 300 Tooth Passing Frequency (Hz) (a) Mode(l, 1) ••jj 4 Ritz FEM _ _ . _ i 1 1 0 50 100 150 200 250 300 Tooth Passing Frequency (Hz) (b) Mode (2, 1) Figure 4.6 Real parts of the first two modes Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 97 The real parts and frequencies of Mode (1, 6), a high-frequency mode, are illustrated in Figure 4.7. The frequency range chosen, 700 ~ 1200 Hz, corresponds to the tooth passing frequency range for the bandsaw of interest in this study. The unstable regions are slightly different due to the frequency difference. Figure 4.8 shows the eigenvalue of Mode (10, 3). It is seen that the eigenvalues based on the two models are in good agreement. 1200 1 100 1000 900 Ritz FEM 700 800 900 1000 1100 1200 Tooth Passing Frequency (Hz) (b) Natural frequency Figure 4.7 Eigenvalue of Mode (1,6) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 98 Pi 2 0 -2 -4 700 800 900 1000 1100 Tooth Passing Frequency (Hz) (a) Real part 1200 fa 1200 1100 1000 900 700 800 900 1000 1100 Tooth Passing Frequency (Hz) (b) Natural frequency Figure 4.8 Eigenvalue of Mode (10, 3) 1200 4.6.3 Instability Regions of the Toothed Blade Figure 4.9 shows the differences of the real parts of Modes (8, 3) and (2, 6) based on the finite element model of a blade with teeth and the smooth band model of a blade. Compared with the smooth band, the toothed blade has lower frequencies and generally larger real parts. The frequency difference varies with different modes and can typically reach 5~8%. Because Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 99 the toothed blade has a larger real part it more likely suffers from dynamic instability. If damping is considered in this model, the toothed blade would have wider instability regions than the smooth band (see Figure 5.8 on page 126). Consequently, if a good prediction of the natural frequencies and the instability regions of a tooth blade is required, the tooth blade model should be used so that the effect of the tooth profile can be effectively evaluated. -80' 1 1 1 1 1 1 1 1 1 700 750 800 850 900 950 1000 1050 1100 1150 1200 Tooth Passing Frequency (Hz) Figure 4.9 Comparison of the real parts of undamped Modes (8, 3) and (2, 6) based on the finite element model and the smooth band model Figure 4.10 illustrates that the same modes of two blades with different depths of gullet have different real parts. In general, the deep-gullet blade (dg =19.1 mm) has larger real parts than the shallow-gullet blade (dg = 16.2 mm). This analytical prediction matches the actual phenomenon that the former blade more frequently experiences washboarding and usually makes the washboarding worse. In addition, it is known in sawmills that the length of the tooth face (see Appendix A) is an important parameter to the washboarding in bandsaws. Apparently, the longer is the face, the more flexible is the cut edge of the tooth so that the tooth tip experiences a larger displacement during cutting. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 100 80 700 750 800 850 900 950 1000 1050 1100 1150 1200 Tooth Passing Frequency (Hz) Figure 4.10 Effects of the depth of gullet of the blade on its stability 4.7 Experimental Results from Modal Testing The purpose of the modal testing of the smooth band (Blade SI) is to verify the analytical and finite element models. The set-up of the testing shown in Figure 4.11 is for both static testing and idling testing. Non-contact displacement probes were installed at the locations indicated and other locations if needed. Several grids of measuring points were used for the identification of different modes. The internal stresses in the band were calculated based on the following parameters: (1). Strain R = 26.7 kN on each side of the band; (2). Tensioning stress amplitude op = 0; (3). The wheel crown is included; (4). The wheel tilting and the back crown of the band are equivalent to the tilting angle 0 = -0.08°, which is determined in such a way that the first five modes are in agreement with the measured modes. The experimental results from modal testing of both the stationary and traveling bands and the predicted modes using Ritz and finite element methods are summarized in this section. Each mode shape is normalized by dividing it by the maximum magnitude of the shape. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 101 Figure 4.11 Locations of probes over a blade 4.7.1 Modes of the Stationary Band Figure 4.12 shows a frequency response function whose frequency band covers the first eight modes. It can be seen in Figure 4.13 that the measured frequencies of these modes are in good agreement with the predicted by using both Ritz and finite element methods. The mode shapes of Modes (1,1) and (2, 2) are shown in Figure 4.14. The mode shapes across the width are not symmetric because the band is subjected to non-uniformly distributed internal stresses due to the wheel crown, the wheel tilting and the band back-crown. Figure 4.15 shows the good agreement in the natural frequencies of high-order modes that are measured and predicted using both analytical and finite element approaches. The frequency response function containing Modes (1, 6) and (2, 6) is shown in Figure 4.16. The Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 102 measured and calculated mode shapes are shown in Figure 4.17. It can be concluded from these results that the mode frequencies, orders and shapes of high-order modes measured in the modal testing are also in reasonable agreement with the predicted on the basis of the analytical and finite element models. These two theoretical models can therefore provide reliable natural frequencies of a stationary smooth band. 0 20 40 60 80 100 120 140 160 180 200 220 Frequency (Hz) Figure 4.12 Frequency response function of the first eight modes 250 (1,1) (1,2) (2,1) (2,2) (1,3) (3,1) (2,3)" (3,2) Mode Figure 4.13 Natural frequencies of the first eight modes Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 103 Mode ( 1, 1) at X = 172 and Y = 0.94B -1 o 1 W(L/2, Y) (a) Mode (1, 1) Mode ( 2, 2) at X = 0.25L and Y = 0.94B 1 1 • -1 1 1 1 ) 1 1 0 0.1 0.2 0.3 0.4 X 0.5 0.6 1 0.7 « 0.5 as o 0 X " -1 133.6 Hz (Ritz) — 131.9 Hz (FEM) • 136.1 Hz (Exp.) 0 1 2 3 W(0.25L, Y) (d) Mode (2, 2) Figure 4.14 Mode shapes of Modes (1,1) and (2, 2) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 104 (1,5) (2,5) (3, 5) (4, 5) Mode (1,6) (2,6) Figure 4.15 Natural Frequencies of High-order modes i 1 1 i r 920 960 1000 1040 1080 1120 Frequency (Hz) Figure 4.16 Frequency response function of high-order modes Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 105 Mode ( 1, 6) at X = 1 7 2 and Y = 0 . 9 4 B -a • • • • -0 0.1 1 0.2 0.3 i 0.4 X 0.5 0.6 0.7 °" o X % -0-5 -1 0.25 0.2 0.15 0.1 0.05 0 — 1047.3 Hz (Ritz) - 1040.1 Hz (FEM) • 1074.0 Hz (Exp.) - 2 - 1 0 1 2 W ( L / 2 , Y ) (a) Mode (1,6) Mode ( 2 , 6) at X = 0 . 2 5 L and Y = 0 . 9 4 B 1075.2 Hz (Ritz) —- 1064.1 Hz (FEM) • 1098.0 Hz (Exp.) 1 2 3 W ( 0 . 2 5 L , Y ) (b). Mode (2, 6) Figure 4.17 Mode shapes of Modes (1,6) and (2, 6) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 106 4.7.2 Modes of the Traveling Band Figure 4.18 shows the first four natural frequencies of the traveling plate. The frequencies decrease with the blade speed due to the gyroscopic effect. The measured frequencies match those predicted using both the Ritz and the finite element approaches within a difference of less than 7%. A mode of the moving blade is expressed in terms of complex functions. Although the mode shape varies with time due to travelling waves in the blade, the envelope of the mode keeps a certain shape. Figure 4.19 shows the envelopes of the first four modes along the length and across the width. The points on the envelopes can be measured by the magnitude of the frequency response functions. The envelopes match the mode shapes of the stationary plate. The experimental results demonstrate that the saddles, called generalized nodes in this thesis, of the envelopes exist along the blade length. Since the moving blade has no motion across the width, the mode shapes across the width are the same as those of a stationary plate. The number of generalized nodes of the envelope along the length and the number of the nodes across the width are used to define the order of an individual mode of the moving band. Figures 4.20 and 4.21 show that the measured high-frequency modes have the same behavior as that of the first few modes. The generalized nodes exist on Modes (2, 5) and (2, 6), respectively. It must be noted that some high-order modes may have different behavior. Their frequencies increase with the band speed based on the modal analysis. These modes, unfortunately, were not excited by the shaker installed close to the bottom guide in this testing. It seems that a concentrated force can excite low-order modes (1 or 2) along the length of the band rather than high-order modes (9 or 10) along the length. Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 107 80 Figure 4.18 Natural frequencies of the first four modes varying with the blade speed Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 108 Mode ( 1, 1) at X = L/2 and Y = 0.94B i [——•— — i 1 1 r W(L/2, Y) (a) Mode(l, 1) Mode ( 1, 2) at X = 172 and Y = 0.94B ON ° o X 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 X W(L/2, Y) (b) Mode (1,2) Figure 4.19 (1) Envelopes of low frequency modes (c=39.8 m/sec) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 109 Mode ( 2, 1) at X = 0.25L and Y = 0.94B ™ 0.5 ° 0 £ -0.5 -1 119.1 Hz (Ritz) — 117.4 Hz (FEM) • 114.6 Hz (Exp.) 1 2 3 4 W(0.25L, Y) (c) Mode (2, 1) Mode ( 2, 2) at X = 0.25L and Y = 0.94B -• • / 1 0 0.1 0.2 0.3 0.4 X 1 0.5 0.6 i 0.7 125.3 Hz (Ritz) — 123.3 Hz (FEM) • 131.2 Hz (Exp.) 1 2 3 W(0.25L, Y) (d) Mode (2, 2) Figure 4.19 (2) Envelopes of low frequency modes (c=39.8 m/sec) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 110 750 700 h N £ 650 cr 600 550 • • o Mode (1,5) Mode (2, 5) •Ritz Ritz -FEM -FEM Exp. Exp. 10 20 30 Speed (m/sec) (a) Modes (1,5) and (2, 5) 40 50 1150 (b) Modes (1, 6) and (2, 6) Figure 4.20 Natural frequencies of four high-order modes varying with the blade speed Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 111 Mode ( 1, 5) at X = 172 and Y = 0.94B 624.0 Hz (Ritz) — 618.9 Hz (FEM) -• 638.7 Hz (Exp.) -0.25 0.2 0.15 0.1 0.05 0 -1 0 1 W(L/2, Y) (a) Mode (1,5) Mode ( 2, 5) at X = 0.25L and Y = 0.94B ra £ 0.5 ON X -D -u— -- -0 0.1 0.2 0.3 0.4 X 0.5 0.6 0.7 656.1 Hz (Ritz) — 648.5 Hz (FEM) • 667.8 Hz (Exp.) 1 2 3 4 5 6 W(0.25L, Y) (b) Mode (2, 5) Figure 4.21 (1) Envelopes of high frequency modes (c=39.8 m/sec) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 112 Mode ( 1, 6) at X = 172 and Y = 0.94B O O.l 0.2 0.3 0.4 0.5 0.6 . 0.7 X W(I72, Y) (c) Mode (1, 6) Mode ( 2, 6) at X = 0.25L and Y = 0.94B 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 X 1057.6 Hz (Ritz) — 1048.3 Hz (FEM) J • 1079.0 Hz (Exp.) j -2 -1 0 1 2 3 4 5 6 W(0.25L, Y) (d) Mode (2, 6) Figure 4.21 (2) Envelopes of high frequency modes (c=39.8 m/sec) Chapter 4 Modal and Stability Analysis of Moving Blades Using FEA 113 4.8 Summary The moving regenerative cutting forces applied at the tooth tips of a traveling blade are modeled by a finite element approach. In order to build this model, three types of traveling plate elements, i.e., bilinear plate element, triangular plate element and variable- domain rectangular plate element have been developed. The modal and stability analyses of a moving smooth blade demonstrate the reliability and capacity of these types of plate elements and the finite element model. One of the advantages of this finite element model over the smooth band model using the Rayleigh-Ritz method is that it is more convenient to accurately model moving cutting forces applied at the tooth tips. With this model the effects of the depth of gullet of a blade can be examined. In addition, this approach can be used to model the boundary conditions at,the pressure guides and the overhangs of the blade. The toothed blade has lower natural frequencies (about 5~8%) than the smooth band. The frequency difference varies with different modes and the configuration of the smooth band. A blade with different depths of gullet does not have a significant difference in its natural frequencies. But the real parts of its eigenvalues may be very different. The blade with a deep gullet, for instance, has larger real parts and wider instability regions if damping is included in this model (See Chapter 5). In order to verify both continuous and finite element models, modal testing of a stationary smooth band and a traveling band was conducted. The measured natural frequencies are in good agreement with the analytical predictions based on the two models. It is found that the moving blade has both traditionally defined nodes for some modes at a low blade speed or with a short wavelength and genelized nodes for others at a high blade speed and with a long wavelength. Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations The purpose of this section is to study the vibration of a blade subjected to both parametric and modulated excitations. A model of modulated cutting forces will be developed first. In order to understand the dynamic behavior of these systems, the stability and the response of a single degree-of-freedom spring-mass system will be studied. Then the dynamic behavior of a bandsaw blade subjected to regenerative cutting forces and modulated cutting forces will be investigated to determine its relationship to the mechanism of washboarding. 5.1 Modulated Cutting Forces When a bandmill runs at a high speed, the mill frame and two guides experience vibration due to the eccentricity of the wheels. As a result, the two vibrating guides can transversely excite the saw blade, as shown in Figure 5.1 (a). In addition, a weldment and a small dished area on the blade surface can also stimulate the lateral motion of the blade when they pass over the guides (Figure 5.1 (b) ). These excitations, consequently, force the saw blade to oscillate at the wheel rotating frequency or the band wrapping frequency. If the workpiece is fixed in the lateral direction, the blade teeth are subjected to extra lateral cutting forces. Clearly, the moving forces applied at the tooth tips depend on the displacement excitations exerted on the blade. In order to model these forces, in addition to the assumptions made in Section 3.1, two more assumptions are made here: (i) . The guides are in harmonic motion. (ii) . Each tooth is subjected to an identical lateral force. The time-dependent force applied at each tooth tip as a result of these effects can be expressed by 114 115 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations Fwj=Fw0+Fwicosurt, ./ = 1,2, N, (5.1) where Fwo and Fwj are the constant component and the amplitude of the lateral force, respectively, to r is the wheel rotation speed and N, is the number of teeth within the depth of cut of a cant. -«*—*» Guide Motion (a) Weldment or Dished Areas (b) Figure 5.1 Lateral cutting forces due to (a) guide motion and (b) weldment or dished areas Taking into account these lateral forces, regenerative forces and damping, this system is governed by the equation of motion Mq(f) + (C + G)q(0 + Kq(0 + K Q (0[q(0 - q(f - T)] = Q(t) (5.2) where M , G and K are the mass, gyroscopic and stiffness matrices, respectively and they are defined in Appendix B. C is the damping matrix. Kg,(t) is a regenerative force coefficient matrix. Q(t) is the exciting force vector Q(t) = (F^+Fwicosart)J£®(xj,y0) (5.3) where O is a vector of mode functions of the traveling plate, (XJ, yo) are the coordinates at the j-th tooth. For a smooth band, Xj=xs,+(j-l)P + ct, j = \,2,...,Nt y0=B 116 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations where xs, is the distance between the top guide and the top surface of the lumber. B and P are the band width and the tooth pitch, respectively, c is the blade speed. Since these forces move with the blade, the summation in (5.3) is a periodic function of time t and can be expanded into Fourier series. If the first-order terms are reserved and other higher-order components are negligible, the exciting forces can be approximated in the form Q(f) = (Fw0 +Fwl coscor0(a0 +a, cosco,r + b, sin(0,0 (5.5) where ao, a± and bi are the vectors of the Fourier coefficients and C O , is the tooth passing frequency. Since the low-frequency components do not have any contributions to the instability of the blade, they can be ignored and the forces (5.5) can be further simplified to Q(r) = (Fw 0 + Fwl cosco r t) (aj cosco, t + bl sinco, t) = Fw0 (a, cosco ,t + b1 sinco, t) F + -^-{ [(a, cos(co, -to r)f + bj sin(co, -co r)f] (5.6) + [(a,cos(co, -t-cojr + b js in^ , +cor)f] } 3 = S ( a . v * cosco, r + b ^ sinco, 0 k=l where p a w i = ^ o a i » a W 2 = a W 3 = ^ L a i ( 5 - 7 a ) b.,=^ob,, bw2=bw3=^bl (5.7b) co,=co,, co 2=co,-co r, co3 =co,+cor (5.7c) F , = F , , F23=F,+Fr = Fl(l + p) (5.7d) where F, and Fr are the tooth passing frequency and the wheel rotating frequency, p is the frequency ratio, p = FJFt = 0.00928 for this mill. It is seen from (5.6) that the forces at the tooth passing frequency C0t are modulated by the rotating frequency component (5.1). These forces are called modulated cutting forces in this thesis. Apparently, the system is subjected to three time-dependent excitations with three exciting frequencies given by (5.7c). 117 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 5.2 Stability and Response of a Damped Spring-Mass System 5.2.1 Modeling the System A single-degree of freedom spring-mass-dashpot system shown in Figure 5.2 has mass ra, damping coefficient c, spring constant k and is subjected to a regenerative force where kj is the regenerative force coefficient, x(t) is the displacement and T is the period of a disturbing force. It is assumed that the primary exciting force has the frequency CO, (=2JI/T) and is modulated by a low frequency co r («co t). Thus, the exciting force in this model has the form where Ao is the amplitude of the force function, a the amplitude of modulation function, to r the modulating frequency and CO, the excitation frequency (corresponding to a tooth passing frequency, for example), p is the ratio of the modulating frequency to the excitation frequency, i.e., /?=tor/co, . F. reg = -kl[x{f)-x(t-T)\ (5.8) f(f) = Aj(l + flcosto ,. Ocosto , t = AjCosto , t + A^costo ,(1 - p)t+ cos(x> ,(1 + p)t] (5.9) Figure 5.2 A single-degree spring-mass-dashpot system subjected to a regenerative force and a dynamic force (m = 1 kg, k = 500 kN/m, k, = 10 kN/m, £ = 0.002) 118 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations This system is governed by the equation of motion mx(t) + cx(t) + kx(t) + [x(t) - x(t - T)] = f(t) (5.10a) or Jc(0 + 2Cco nx(t)+u ]x(t)+(x) ][x(t)-x(t-T)} = f(t) (5.10b) where m m 2m(x> n CO, m The homogeneous equation is * ( 0 + 2£ co „x(0 + co 2x(r)+co 2 [ x ( 0 - 4 ? - r ) ] = 0 (5.12) and the characteristic equation is then given by X 2+2C co NX +co 2+co 2 ( l - e ^ r ) = 0 (5.13) The complex eigenvalues A, are related to the damping ratio, the regenerative force constant and the period or tooth passing frequency. The tooth passing frequencies at which the real parts of its eigenvalues are positive can be used to define the unstable regions of this system. The exciting force fit) in (5.9) can be decomposed into three components at frequencies (x),(l-p), co , , and CO t(l+p), respectively. The homogeneous solution of this equation will be attenuated due to the damping as time advances. The steady-state solution of this linear system is the sum of the solutions to the three exciting forces. Each of the terms can be expressed in the general form, fit) = F0 cosco t = F0 COS2TC Ft (5.14) where co=co,, co,(l-p) or co,(l + p) F=Fn Ft(l-p) or F,(l + p) Fo = Ai or \ a Thus the steady state response to this general excitation can be expressed in the form x(t) = Acosco ? + 5sincor (5.15a) 119 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations and for a causal system, x(t-T) = Acosco (t-T) + Bsmb)(t-T) = (A cos co T - B sin to T) cos tor (5.15b) + (Asintor + /icostor)sinto t where A and B are two constants given in the form [to 2-to -hto 2(l-costx)]F 0/ra A = —z 5 ~ — (5.16a) [to „-to -t-to ^ (1-cosa)] + (2L, to „ to + to 2sina) (2^ to „ to H-to 2sina )F0/m [to 2-to 2+to 2 ( l -cosa)] 2 + (2£ to „to+to 2sina )2 B=— v ^ ™ . ™ • > » o ^ , 1 0 > w ••--2 . . . 2 . . . . 2 / i \ l 2 . /ny , . - . . . . . . . 2 • „ \2 v ' where a = 0, -2K p or +2K p (5.17) The magnitude of response to the exciting force is given by \x(t)\ = jA2+B2 =X(to )F0/k (5.18) where the magnification factor X(to) = r (5.19) {[1 - r2 + r2(l - cosa )]2 + (2C r + r2 sina ) 2} 5 w to c r = , rc= w n &>„ The response to the excitation (5.9) can be expressed in the form x(t) = ^-{X(pt)cos(fti,t + ty0) +aX(to ,(1 - jp))cos[(to ,(1 - p)t + ck,] (5.20) +aX((ji ,(1 + p))cos[(to ,(1 + p)t + § 2]> where (j) ,• , i - 0, 1, 2 are the phases. It is seen that the vibration amplitude is directly proportional to the force amplitude Ao, the modulation magnitude a and the magnification factors X(to). 120 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 5.2.2 Instability Regions Figure 5.3 (a) shows that the natural frequencies of the system with different structural damping ratios (C, = 0, 0.002, 0.006 and 0.012) are almost identical. The unstable regions of the system are defined as the frequency range in which the real parts of the eigenvalues are positive. It is seen from Figure 5.3 (b) that the undamped system (t, = 0) has the widest primary instability region, i.e., 1IT = (Fn , 2Fn). Other primary unstable regions of the damped system shrink with an increase of damping. Furthermore, there is a critical damping ratio corresponding to the lightest damping required to make the system stable. £ 1 40 ai 120 CT <D ^ 100 CO | 80 10 •c 0 to Q . "5 o ec -10 50 100 150 200 (a) -20 _£_=0 i o c B ^ * " " ^ - ^ O.OOS~ ^ ^ • f A * 5 §! - • y 0.012* - ^ ' n ' \ \r. • \.r n N 4 • * -50 100 150 200 1/T (Hz) (b) Figure 5.3 Natural frequency (a) and real part (b) of the system with different damping ratios {k\ = 10 kN/m) The instability of the system is also affected by the regenerative force. A greater regenerative force increases the real part of eigenvalue of an undamped system (Figure 5.4 (a)). It does not change the unstable region (Tian, 1999). In a damped system, however, this 121 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations force can not only increase the real part of its eigenvalue, but also enlarge the unstable regions (Figure 5.4 (b)). In addition, it is noted that there is the critical regenerative force coefficient that is the smallest value of the coefficient for which the system is stable for the given level of damping. The effects of the damping and the regenerative force on the instability of this system are coupled. 10 kN/m k - — ( V 1 i 50 100 150 200 250 1/T (Hz) (a)C = 0 = 10 kN/m 5~ " ^ , A -• 50 100 150 200 250 1/T (Hz) (b) £ = 0.005 Figure 5.4 Real parts of eigenvalues of the undamped (a) and damped (b) systems with different regenerative force coefficients 5.2.4 Magnification Factors Figure 5.5 shows the magnification factors for the response around the natural frequency (Fn = 112.5 Hz) of the system to three excitation components at the frequencies F, (l-p), F, 122 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations and Ft (l+p). From equations (5.19) it can be seen that the peak response of the system is affected by the characteristics of the regenerative force model. These peaks are at the frequencies Ft (l-p), F,, and F, (l+p), respectively. They correspond to the excitations at the frequencies F, (l+p), F,, and F, (l-p). Without the amplitude modulation, the peak magnitude occurs at the damped resonance frequency Fj. Due to the behavior of the regenerative force, the peak magnification factor at the frequency Ft = Fa 1(1- p) is greater than those at the resonance frequency Fd and at the frequency F, = Fd 1(1+p). In fact, it can be seen from (5.19) that if F, = Fd/(l+p), i.e., a > 0, then the regenerative force within a certain range is equivalent to a positive damping in this system. On the other hand, if a < 0, the regenerative force within a certain range is equivalent to a negative damping in this system. In addition, if the excitation frequency F, is higher than the natural frequency (as shown in Figure 5.3), the real part of eigenvalue is positive and therefore the system experiences instability. 123 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations Figure 5.6 (a) shows how the magnification factors vary with the excitation frequency for the different frequency ratios. Ifp < 0.01, the peak magnification is also small. If p> 0.06, the peak response is also small. When p = 0.03, the peak magnification factor achieves a maximum value. The frequency ratio at which this peak occurs is related to the damping ratio and the amplitude of regenerative force coefficient. 1000 800 600 o CO LL c o 8 400 8> 200 0 100 • r • F,(I+P) ;F=F • -j • y v - -- - A 7 F,(1-P) * I ' \ —' ^ s • ^ 1 105 110 115 F, (Hz) 120 125 0.06 005 (S 0.04-g CD 0.03.5 0.02 a o' 0 . 0 1 1 3 0 130 Figure 5.6 Variation of magnification factor with frequency ratio (C = 0.002, iky — 10 kN/m) Figure 5.7 shows the case where the excitation frequency F, is higher than the natural frequency. The peak system responses to the three excitations for a given frequency difference (F, -F„) are plotted. The response with the maximum magnitude is always at the lower frequency F, (l-p). Their values depend on the behavior of the transfer function at this frequency and the excitation amplitude. 124 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations Ft(1-P 1 F n ' F,(1+P) 1 L — | -! \h i A 80 85 90 95 100 105 110 115 120 125 130 Frequency (Hz) Figure 5.7 Frequency spectrum of the response (Fn = 112.5 Hz, p = 0.025, £ = 0.004, kj = 10 kN/m) 5.3 Stability and Response of a Traveling Band 5.3.1 Instability of the Damped and Traveling Band The instability of a single degree-of-freedom damped spring-mass system is reduced with an increase of damping. In the previous stability analysis of a traveling band, however, the damping is excluded in the smooth band model. The effect of damping on the unstable regions of a moving band will be examined in this section. During cutting a saw blade may experience high damping provided by the workpiece so that washboarding does not always occur. In general, the damping in this system is associated with the properties of the workpiece and the blade as well as the contact of the blade to the 125 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations cant. But the damping ratio is not available for this study. The following discussion will be based on the application of the Rayleigh damping to this system. In the homogeneous equation of motion of the band, Mq(0 + (C + G)q(0 + Kq(f) + K Q (t)[q(t) - q(t -T)] = 0 (5.21) The Rayleigh damping matrix C is given by C = a M + (3K (5.22) where a and (3 are the constants that can be determined from two given damping ratios that correspond to two unequal natural frequencies (Bathe, 1996). Figure 5.8 (a) shows the real part of Mode (1,2) varying with the tooth passing frequency for different damping ratios. Increasing the damping ratio reduces the value of real part and the unstable region. This is true for high frequency modes, such as Mode (10, 2) shown in Figure 5.8 (b). If the damping is sufficiently larger, the real part of this mode is always negative and the unstable region disappears. The influence of the regenerative force coefficient on the stability of this band is illustrated in Figure 5.9. If £ = 0, all the eigenvalue curves of real parts versus the tooth passing frequency pass through point A. At this point the real parts are zero no matter how large the force coefficients are. The on-set frequency of instability is the natural frequency. Thus this force coefficient does not change the unstable region and it affects the real parts of the eigenvalues only. In fact, the damping always exists in this system. Damping reduces both the real parts by shifting them toward the negative side and the unstable regions. The on-set frequency of instability increases. A smaller regenerative force coefficient produces a smaller unstable region. For a given damping ratio, if the force coefficient is sufficiently small, the real part is always negative in the original unstable region and the system is always stable. Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 126 10 •c o CO Q_ "cC 8. -io -20 ; s. /C = 0.0P5 C=o.oi \v /•/ W 60 80 100 120 140 160 Tooth Passing Frequency (Hz) (a). Mode (1, 2) (Ki =30 kN/m) 180 200 20 t 0 co "co . o n -40 -60** ( I f f C;=QOO2 5 — ° — . c;=Q0D4 900 1000 1100 1200 1300 Tooth Passing Frequency (Hz) 1400 (b). Mode (10, 2) (K, =60 kN/m) Figure 5.8 Effect of the damping ratio on the instability region (RQ=36.1 kN) 20 CO ± -20 co CD EC -40 -60 'K, = 100 k N / m ^ — A . ^ - ^ 'K , = 60kN/m T t f 900 1000 1100 1200 1300 Tooth Passing Frequency (Hz) 1400 Figure 5.9 Effect of the regenerative force coefficient on the unstable region of Mode (10, 2) (R0 = 36.7 kN, £ = 0.002) 127 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 5.3.2 Response Magnitude of the Traveling Band The traveling band subjected to both regenerative and modulated excitations is governed by the equation of motion Mqft) + (C + G)q(0 + Kq(0 + K Q (t)[q(t)-q(t - T)) ^ (5-21) = 2- (awtcoscoAr + b w tsina) tO k=\ where =co(, co2 = co, -co r , and co3 =co, +cor. Since this is a linear system, the steady-state system response is the sum of the responses to three excitations at the frequencies toco, -co r, and co t +co r . For each excitation, therefore, the response can be expressed in the generalized form, q, (t) = A , cosco k t + Bk sinco, t (5.22) and qk(t-T) = Ak cosco, (t-T) + B K sinco, (t-T) ~(Ak cosco, T-Bk sinco, 7/)cosco, t (5.23) + (A, sinco, T + Bk cosco, r)sinco, t where Ak and B k are two constants to be determined. T is the tooth passing period. Substituting (5.22) and (5.23) into (5.21) leads to - Z 1 2 Z,, where Z n = (K - Mco ) + K O 0 (1 - cosco T) Q (5.25) Z 1 2 =(C + G)co+KG Osincor and KQO is the zero-order approximation of the regenerative force coefficient matrix KQ. From (5.24), the solution is A , = ( Z n +Z 1 2 Z- ' Z l 2 ) - ' (a* - Z 1 2 Z- 1 bwk) (5.26) B ^ Z ^ b ^ + Z . A J (5.27) Therefore the displacement at (x, y) is given by 128 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 3 w(x, y,t) = <5>T O, y)q(0 = cpT (x, y) £ q, (0 = ®T (x,y)^ (A, cosco, t + Bk sinco, 0 (5.28) k=l = W0k sin(co, ?+()>,) The response magnitude and phase to each excitation are W0k=^TAk)2 + ^TBk)2 (5.29) (5.30) where k= 1,2 and 3. I. Effects of Damping Figure 5.10 (a) and (b) show the variation of the magnification factors at point (0.25, B) with the tooth passing frequency. The damping attenuates the peak magnification factors so that the difference between three peak factors at the three frequencies is reduced. The damping also modifies the mode shapes. It can be seen from Figure 5.10 (a) that many modes of the lightly damped system (C, = 0.002) are closely scattered in this frequency region. For each mode, the response magnification to the modulated excitation at frequency Ft(l-p) is larger than those at the other two excitations at frequencies Ft and Ft(l+p). With the increase of the damping, however, some modes are greatly attenuated and even disappear, such as Modes (1, 6) and (2, 6) shown in Figure 5.10 (b). Others are mixed with their adjacent modes and modified as shown in Figure 5.10 (c). To evaluate the magnification factors at the given points on the tooth side, the RMS value of these magnification factors at these points is used to measure the magnification at a specific tooth passing frequency. A typical mixed mode, Mode (9, 3), at Ft = 1119 Hz is shown in Figure 5.11. The left half that is in the cutting zone is a part of Mode (9, 3) and the right a part of Mode (2, 6). The deflected shape across the width is simultaneously modified. The deformation of the plate within the depth of cut can be different from that of the plate outside the lumber. 129 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 0.04 850 900 950 1000 1050 1100 1150 1200 1250 1300 Tooth Passing Frequency (Hz) (a) C = 0.002 at (0.25, B) 0.01 850 900 950 1000 1050 1100 1150 1200 1250 1300 Tooth Passing Frequency (Hz) (b). £ = 0.006 at (0.25, B) 850 900 950 1000 1050 1100 1150 1200 1250 1300 Tooth Passing Frequency (Hz) (c). c = 0.010 Figure 5.10 Magnification factors of the response to three excitations (R0 = 36.7 kN, Ki = 60 kN/m, p=0.0093) 130 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations II. Effect of the Locations of Regenerative Forces Among the modes of the traveling plate (762x261mm) with light damping, which modes have comparatively large magnification factors depend on the locations at which the regenerative forces are applied. Figure 5.12 (a) shows the magnification factors of the plate whose front edge (y=B) is subjected to regenerative forces in the region of xsl = 114 mm to Xend =495 mm (or 16"-depth of cut in sawing). The factors of Mode (2, 6) are larger than those of Modes (1, 6) and (3, 6). If these forces are applied in the middle of the cutting span from xst = 343 mm to xemi =495 mm (or 6"-depth of cut in sawing), Mode (2, 6) disappears and Mode (1, 6) becomes a dominant mode. The peak magnification factors of Modes (1, 6) and (3, 6) greatly increase. Under the condition of high damping, such as ^ = 0.01, the 131 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations dominant modes would be mixed with its adjacent modes to form high-order modes, as shown in Figure 5.10 (c). 0 . 0 1 2 2 0 .01 E ~ 0 . 0 0 8 to E ~ 0 . 0 0 6 i o « 0 . 0 0 4 | 0 . 0 0 2 0 8 5 0 9 0 0 9 5 0 1 0 0 0 1 0 5 0 1 1 0 0 1 1 5 0 1 2 0 0 1 2 5 0 1 3 0 0 Tooth Passing Frequency (Hz) (b).(xst,xend) = (0.343, 0.495) Figure 5.12 Dominant modes (L=0.762 m, R0 = 36.7 kN, Kt = 60 kN/m, £ = 0.004) 132 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations 5.4 Summary The forced vibration response and the stability of two damped systems, a single degree-of-freedom spring-mass-dashpot and a traveling plate, subjected to regenerative and modulated forces have been studied. The relationships between the response, the damping ratio, the regenerative force, and the modulated excitation are summarized below. Increasing the damping in these systems greatly reduces the unstable regions and the magnitude of the real parts of eigenvalues. The largest unstable regions appear when no damping exists in this system. In handsaws, the structural damping and the damping from the interactions between the teeth and the lumber reduce the washboarding regions and increase the onset tooth passing frequency at which the washboarding occurs. In addition, there is a critical damping ratio above which the unstable regions of the system, or the washboarding in bandsaws disappear. The unstable regions are extended and the real parts of the eigenvalues increase with an increase of the regenerative force coefficient. If this coefficient is sufficiently small, the damped system is always stable. This indicates that the occurrence of instability is also dependent on the amount of the energy switched into the system. In the case of a bandsaw, a shallow cut can not produce enough energy to excite a washboarding mode. This cut is always stable. A deep cut, however, can switch more energy into the blade. It has a better chance to experience washboarding. Among three vibration response components corresponding respectively to three modulated excitations, the maximum component appears at the frequency that is the sum of the natural frequency and the modulating frequency. This is because the regenerative forces are equivalent to a negative damping in the systems if they are subjected to the modulated excitation at the frequency a bit lower than the original exciting frequency. For the purpose of a bandsaw, the forced excitation at the tooth passing frequency can be modulated by the excitation at the wheel rotating frequency. As a result, three excitations are generated at three 133 Chapter 5 Responses of Damped Systems to Parametric and Modulated Excitations different frequencies: the difference between the tooth passing frequency and the wheel rotating frequency, the tooth passing frequency and the sum of the two frequencies. Among the three response components, the largest magnification factor corresponds to the modulated excitation at the frequency that is the difference between the tooth passing frequency and the wheel rotating frequency. It will be found in Chapter 6 that this behavior of the plate subjected to modulated excitations is in agreement with the kinematics of washboarding. Increasing the damping in the traveling plate modifies the shape of the deflected plate. Under higher damping, the vibration response contains several modes as if some modes were mixed with others. This response looks like an irregular higher-order mode transited from a mode mixing with its adjacent modes. It is also found that the locations of regenerative forces determine which modes are dominant in the responses. The deformation shape of the plate in the loaded area could be different from those in other unloaded areas. Chapter 6 Kinematics of Washboarding The transverse motion of a bandsaw blade during cutting can produce a regular sinusoidal-like pattern on the sawn surfaces (Figure 6.1). The pattern looks like a washboard. This phenomenon that occurs in woodcutting is called washboarding and is similar to chatter in metal milling. The typical depth of a washboard varies from 0.13 mm to 0.5 mm (Lehmann, 1997) but can be greater than 1.2 mm. A washboarding pattern is formed by the relative motion of the blade to the workpiece and contains much information about tooth vibrations. Hence, the purpose of this section is to decode washboarding patterns by developing the kinematics of washboarding so that its mechanisms can be better understood. Feed Direction Cutting Direction Figure 6.1 Washboarding pattern Okai (1996a, b) and Lehmann (1997) studied the kinematics of washboarding based on observed washboarding patterns. They built different kinematic models and established the relationship between the tooth pitch, vertical washboarding pitch and wave length of the tooth deflection on the basis of the assumption that the tooth passing frequency is close to the wave frequency. Their models are valid only for some specific washboarding patterns. Actually, the cutting forces in both wood cutting and metal milling could excite some modes whose natural frequencies are far from the tooth passing frequency. In addition, a wave in a 134 Chapter 6 Kinematics of Washboarding 135 blade may not be an ideal traveling wave and the wave velocity varies with time and the location at which the wave arrives. Therefore, a more general kinematic model of washboarding in both bandsaws and circular saws will be developed in this Chapter. 6.1 Washboarding Patterns Caused by a Single Traveling Wave The relative motion of a tooth tip to a moving workpiece produces the geometry of the sawn surfaces. The motion of a tooth tip is analogous to the motion of a specific point on the moving string as shown in Figure 6.2. It is assumed that there is only one single backward or forward traveling wave in the blade. The backward wave motion of a moving string can be expressed in the form 2 1.5 „ 1 E E, — 0.5 c 0) E 8 0 J2 |-0.5 -1.5 -2 i i i i f,=1077 H z , f b =1066 H z Z k ( t ) 2 k + 1 t=1.102 m s (t) \ *• k ' \ S \\ ' \ i 1 N . \j/ i / \ z k ( t + T ) / i ! * f i 1 i \ i i ',' i / i z k + i (t+T) / ' 1 i ! ' ! V • C 0 100 200 300 400 500 600 700 800 Pos i t i on (mm) Figure 6.2 Motion of two points on a moving string 2n z(x,t) = sin—(x + cbt+yb) K (6.1) where X b is the backward wave length, c b is the wave speed and c b -fb X where ft, is the wave frequency, x is the position of the point and cp t, is the phase angle. Chapter 6 Kinematics of Washboarding 136 The locus of the k-th point on the string can be described by zk(t) = z(xk,t) = sm—(xk + cbt+<pb) (6.2) K where xk is the coordinate of the k-th tooth and expressed by xk ~ xo +(k-\)P + ct (6.3) where xo is the coordinate of the first point (tooth tip), P the pitch of the blade and c the travelling velocity or blade speed and c = /, P, where / , is the tooth passing frequency. The frequency of the locus of any point is f.=^(f,P + fbK) (6-4) The distance that the locus moves within a period Tw is equal to the vertical pitch of washboarding pattern, i.e. f P A. P*=cTw=-?r= , r \ (6.5) "b " "W fw fP + fb^t or ^ = ^ + ^ ( 6 . 6 ) P, K f , p (6.6) is the relationship between the tooth pitch and passing frequency, wave length and frequency, and the vertical washboarding pitch. If the wave frequency is close or equal to the tooth passing frequency, i.e.fb=ft, (6.6) becomes -L_-L + -L (6.7) This is identical to Okai's (1996) or Lehmann's (1997) equations. This equation indicates that the vertical washboarding pitch is always smaller than the tooth pitch if the washboarding is caused by a backward wave.. lff,>fb, e.g., r=f,lfb -2, then P y < 2P. In this case, the vertical washboarding pitch would be larger than the tooth pitch. lff,<fb, e.g., r= f, I f b =0.5, then P y < PI2. In this case, the vertical washboarding pitch would be much smaller than the tooth pitch. Chapter 6 Kinematics of Washboarding 137 The horizontal washboarding pitch Px is obtained by examining the wave motion of two adjacent teeth (Figure 6.2) from (6.2) and (6.3) 2K zk (t) = z(xk,i) = sin—[x 0 + (k- l)P + (cb + c)t+<pb] (6.8) b 2K zk+l ( 0 = z(xk+1 ,t) = sm—[x0 + kP + (cb+c)t+q>b] (6 .9) After time T (T=l/fi), the motion of the k-th tooth is given by zk (t + T) = z(xk, t + T) = sin ^  [x0 + (k - 1) P + (cb + c)(t + T) + cp b ] (6.10) The phase shift between (6.10) and (6.9) is 2K 2nfb Acp = —l(cb +c)T-P] = ^ (6.11) This phase difference indicates that the crest produced by the k-m tooth at time (t+T) has a phase-lag to the crest produced by the (&+i)-th tooth at time t. The two crests do not override each other on the sawn surfaces because the workpiece has moved a bite during the period T. The distance between two crests in the cutting (vertical) direction is Ap>=ltp> <6J2) Having considered the phase lead or phase lag of (6.11), the minimum distance between two crests produced by the adjacent teeth can be generally expressed in the form APy=A-round(^)]Py (6.13) where round is a function that rounds a real number to its closest integer. Thus this distance results in the crest of the washboarding pattern being off the line normal to the blade edge when the workpiece moves. If AP y < 0, a phase lag in space exists. This indicates that the crest produced by the ft-th tooth at time t+T is delayed to the crest produced by the (k+l)-th tooth at time t. Figure 6.3 shows the phase lag of the trace of the (k-2)-th tooth to that of the (k+l)-th tooth. Chapter 6 Kinematics of Washboarding 138 2r 1.5 f,=1077 Hz, fb=1066 Hz t=3.424 ms] Zk+i(t+3Tj zk.2(t+3T) 100 200 300 Position (mm) 400 Figure 6.3 Loci of two teeth When time has elapsed by nT, the (k-n+l)-th tooth will produce a crest overriding the crest produced by the (k+l)th tooth at time t. When the workpiece moves during the same time period (nT), the whole horizontal pitch will be Px =VnT (6.14) where n is the number of increments of AP y within a vertical pitch P y. It is P n = —y— (6.15) APy If/, > fh, and A / = / , -fb «f„ then Px=VnT = —^— (6.16) If the washboarding pattern is caused by a forward wave, the wave motion is governed by 271 zk(t) = z(xk,t) = sm—(xk -cft + <pf) (6.17) k f Similarly, the relationship between the vertical washboarding pitch, the tooth pitch and frequency, and wave length and frequency is given in the form Chapter 6 Kinematics of Washboarding 139 J _ ff If f , p \ (6.18) In (6.17) and (6.18), subscript/represents the parameters of a forward wave. The horizontal pitch can also be given by (6.14). Two examples are given below to demonstrate the validity of this kinematic model. I. Bandsaw Washboarding Pattern (Lehmann, 1996) P= 44.45 mm, c=47.88 m/s, V=1.016m/s, /= 1065.7 Hz, cb =284.2 m/s, \ b =0.2667 m, fb =1077.1 Hz The vertical and horizontal pitches can be found from (6.7) and (6.16). P>=38.45mm (Measured 38.1 mm), ^=89.12 mm (Measured 88.9 mm) These results are in agreement with Lehmann's (1996). II. Metal Cutting Circular saws are also used for metal cutting. The washboarding pattern is different from those encountered in wood cutting. From the view of kinematics of washboarding, the patterns varying with different cutting speeds can be predicted by the proposed formula. Since the rim speed is low, the traveling wave effects are neglected. The main parameters of the cutter, predicted and measured pitches are listed in Table 6.1. Table 6.1 Washboarding Patterns from Metal Cutting (P=27.49 mm, A. b =550 mm (2 nodal diameters)) Rim Speed c (m/s) Tooth Frequency/, (Hz) Wave Frequency/, (Hz) Calculated Pitch Py (mm) Measured Pitch Py (mm) 1.32 45 251 5.31 5.3 1.66 59 237 6.94 7 2.00 71 222 8.95 9 2.20 80 220 10.2 10 Chapter 6 Kinematics of Washboarding 140 It is seen from Table 6.1 that the tooth frequencies are much lower than the wave frequencies. This implies that low frequency forces can also excite high frequency modes and cause instability of the cutter. This phenomenon has never been recorded in wood cutting. The predictions are consistent with the measured patterns. Therefore, the proposed kinematic model can be used for general cases. 6.2 Washboarding Patterns Caused by Multi-Traveling Waves A washboarding pattern caused by a single wave motion in the blade is a regular sinusoidal wave. However, many cutting tests show that washboarding patterns are not symmetric and the pitches are not consistent. For instance, one side of a wave peak could be flatter and the other steeper. One of the reasons for this type of pattern is that the wave speed of a specific mode in the cutting zone changes with the location of the wave. The wave in this traveling blade is the combination of many sinusoidal waves, instead of a pure sinusoidal wave. Consequently, a general approach is needed for the simulation of washboarding patterns caused by severe vibrations of the teeth. The vibration of the moving blade can be expressed in the form M N Hx,y,t) = Y X Xm(x)Yn(y)[qmne^'+qmne^'] (6.19) m=\ n=\ where Xm(x) and Y n(y) are mode functions and qmn is an element of the complex eigenvector of the given eigenvalue and can be expressed in the form qmn=^Kn^- (6.20) The mode shape function belonging to this eigenvalue can be expressed by M N Hx,y, 0 = £ £ Xm(x)Yn(y)Amne°'cos(at + $mn) (6.21) m=l n=l This equation governs the wave motion in the blade or the mode shape varying with time. The motion at a specific point (xk, yo) is described by M N w 4(* 4 ,y 0 ,0 = S X Xm(xk)Yn(y0)Amne°'cos(at + Qmn), £ = 1,2,... (6.22) m=\ n-\ Chapter 6 Kinematics of Washboarding 141 Since the factor ec' affects primarily the amplitude, instead of the frequency, this term can be neglected when the vibration characteristics of the blade in the frequency domain are considered. Further simplifying the equation leads to M mn x sin m=l where (xk,y0,f) = Yd sin——L(Bm cosco t + Cm sinco 0 (6.23) Bm=Y Am„F n(v o)cos0m„ (6.24) n=l N C = - V A Y (y„) sinG (6.25) m / • mn n w O ' mn v ' It is noted that Bm and Cm are constant and related to both the magnitudes and phases of the complex modal vector. The displacement wk at the point (xk, yo) is a linear combination of M pairs of standing waves with different phases. The wave motion depends on the coefficients Bm and Cm , or the elements, Amn and 6m„, of the eigenvector. This wave might behave like either a forward traveling wave, or a backward traveling wave, or even a standing wave. Washboarding patterns produced by an individual mode result from the relative motion of the tooth tips to the moving work-piece. The tooth profiles mark two sawn surfaces. The envelope of all tooth marks constructs a wave-like washboarding pattern. The trace of a tooth shown in Figure 6.2 can be determined from (6.23) in which xk is replaced with xk(t) = xk(0) + ct (6.26) The vertical pitch Py of the pattern can be estimated from the tooth trace. The horizontal pitch Px can be determined from the phase shift of two adjacent teeth in the space domain. Suppose this phase shift is expressed in terms of the vertical pattern shift APy, the horizontal pitch is given by PX=-^APX (6.27) Chapter 6 Kinematics of Washboarding 142 Figures 6.4 through 6.6 show three washboarding patterns produced by three different modes (backward traveling waves) and the same tooth passing frequency. Since the tooth passing frequency (1077 Hz) is much higher than the natural frequency (870 Hz) of Mode (10, 1), the vertical pattern shifts significantly (Figure 6.4 (b)). The washboarding pattern shown in Figure 6.4 (a) is hard to see due to a very small horizontal pitch and rugged pattern. Washboarding Pattern (Fn=870 Hz, Ft=1077 Hz, Bite=0.710 mm) P x = 3.7 mm r P y = 56.2 mm Feed Direction (a) Washboarding pattern Traces of Three Teeth (Fn=870 Hz) Position of Tooth (m) (b) Loci of three bites Figure 6.4 Washboarding pattern caused by Mode (10, 1) Chapter 6 Kinematics of Washboarding 143 The pattern illustrated in Figure 6.5 (a) is characterized with a large inclination angle. In this case, the tooth passing frequency is higher than the natural frequency. The phase shift or vertical pattern shift is quite big so that the horizontal pitch is small. This pattern is called a Type II washboarding pattern. Figures 6.6 (a) shows a pattern with a smaller inclination angle, which is a Type I washboarding pattern. The tooth passing frequency is slightly higher than the natural frequency. The phase shift is very small and the horizontal pitch is quite large. Washboarding Pattern (F n=1022 Hz, F,=1077 Hz, Bite=0.710 mm) P x = 13.9 mm P y = 44.6 mm (a) Washboarding pattern Traces of Three Teeth (Fn=1022 Hz) 0.01 1st Bite r y s 2nd Bite / A Y 3rd Bite / w/'<' 0.4 0.42 0.44 0.46 0.48 Position of Tooth (m) 0.5 0.52 0.54 (b) Loci of three teeth Figure 6.5 Washboarding pattern caused by Mode (1, 6) Chapter 6 Kinematics of Washboarding 144 Washboarding Pattern (Fn=1050 Hz, Ft=1077 Hz, Bite=0.710 mm) P x = 28.4 mm P y = 41.8 mm Feed Direction (a) Washboarding pattern Traces of Three Teeth (Fn=1050 Hz) 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 Position of Tooth (m) (b) Tooth marks and side shapes Figure 6.6 Washboarding pattern caused by Mode (2, 6) 6.3 Summary A washboarding pattern is produced from the continuous interactions between the tooth profiles that are in wave motion and the work-piece that moves at a constant feed speed. This pattern can be predicted by tracing the loci of teeth in the cutting zone. A washboarding pattern can be described by a vertical pitch and horizontal pitch and its depth. The vertical Chapter 6 Kinematics of Washboarding 145 pitch is dependent on the frequency and shape of the mode that is excited. The horizontal pitch is determined by the feed speed and the phase shift between the loci of two adjacent teeth during two bites. These pitches can be estimated from the proposed kinematic model of washboarding caused by a single traveling wave. This model can be applied to the patterns caused by a single traveling wave and the more general cases where the tooth passing frequency may be far from one of the natural frequencies of the blade. Wave motion in a bandsaw blade is different from that in a center-clamped disk where the center constraint does not affect the traveling wave along the rim. A mode in the bandsaw is a combination of several wave components, instead of a pure sinusoidal wave. The wave speed of a mode varies with the time and locations where the wave front arrives. Thus, the loci of teeth are needed to predict washboarding patterns. Furthermore, if the displacement responses of the teeth are given, the washboarding pattern can also be simulated using the same approach. Although a washboarding pattern is mainly produced by a specific mode or a vibration response, other modes or responses with certain frequency differences might produce a very similar pattern. Hence, it is difficult to find out which mode dominates washboarding only from a measured pattern. This mode can be identified by using the information about the measured pattern and from the stability analysis of the blade. Chapter 7 Mechanics of Washboarding Washboarding in sawing is the result of the interaction between the seriously vibrating teeth and the moving lumber. Washboarding occurs if the following two conditions are simultaneously satisfied during cutting: (1) . excessive vibrations of the teeth have been excited during sawing, and (2) . a specific combination of the blade speed, the vibration frequency and the feed speed exists so that the envelopes of teeth generate the pattern. The first condition is essential for washboarding during cutting. It is associated with the vibration response of the blade and thus is called the dynamic condition. Under the latter condition, a specific washboarding pattern is dependent on the blade speed, the vibration frequency and the feed speed. This condition indicates the kinematics of washboarding and it is called the kinematic condition. Washboarding in narrow bandsaws and circular saws has been attributed to self-excited vibrations of those systems (Okai, 1996; Tian, 1998). If some mode is excited and washboarding occurs during cutting, the tooth passing frequency is slightly higher than the natural frequency because the blade is running in an unstable region under the given operating conditions. However, this explanation does not explain the fact that the washboarding occurs only in a narrow frequency band above the natural frequency. Washboarding in wide industrial bandsaws is more complicated. In order to disclose the mechanisms, a series of cutting tests with kinematic and dynamic analyses were conducted. It was found that two types of vibrations, self-excited and forced vibrations, could cause the washboarding in bandsaws. Their dynamic and kinematic mechanisms will be described in this chapter. 146 Chapter 7 Mechanics of Washboarding 147 The cutting tests were conducted with two blades, T l and T2 (Appendix A), running with a 5' Cancar bandmill and a 5' Cetec bandmill, respectively. An eddy-current probe was used to pick up vibration displacement signals during the cutting tests. The set-ups of the tests and the location of probe are illustrated in Figure 7.1. Dimension Blade Blade (mm) T l T2 L 762 502 114 191 Dc 267 235 12 15 Figure 7.1 Set-ups for the cutting tests (Blades T l and T2) 7.1 Washboarding Caused by Modulated Vibration The cutting tests conducted with Blade T l and the Cancar bandmill show that the washboarding in this saw could be caused by the combined effects of the modulated and forced vibration due to the transverse displacement excitations from two guides and the self-excited vibration due to the regenerative cutting forces. This postulation will be presented in this section. Chapter 7 Mechanics of Washboarding 148 7.1.1 Washboarding Pattern and Vibration Signals A typical Type I washboarding pattern cut with Blade T l is shown in Figure 7.2. The washboarding covers most of the sawn surface except at the beginning of the cut and in the area of rotten wood. The horizontal and vertical pitches vary with different cuts or different areas on a sawn surface. Under the given operating conditions, the average horizontal pitch is Px = 68 mm and the vertical pitch is Py = 31 mm. The horizontal pitch depends on the feed speed (or the bite per tooth) and the phase shift between the loci of two adjacent teeth (or the difference between the tooth passing frequency and the vibration frequency). Since a vertical pitch is independent of the feed speed or bite and closely related to the excited mode and tooth passing frequency, it is a good indicator of the wave motion in a blade. Figure 7.2 A typical washboarding pattern (Type I) (Blade Tl , /? 0 = 18klbs,F ;= 1166 Hz, F=l 155 Hz, Bite = 0.635 mm, Depth of Cut =381 mm, Px = 68 mm, Py = 31 mm) The power spectrum of the vibration signal corresponding to this washboarding pattern is shown in Figure 7.3 (a) and does not show any natural frequencies close to the tooth passing frequency. The mill slowed during cutting and the tooth passing frequency decreased with time. The frequencies of the spikes above 1,000 Hz shown in Figure 7.3 (b) also decrease. 149 Chapter 7 Mechanics of Washboarding PSD of (c86525a.600) at Probe 2 (n=650 rpm, Bite=0.025 in) 0.4 H x | 0.2-Q CL • I ; ; 1 ft*.*- A LA J I, f rr^ —_ A , 1 _ A J M _~^r /W*WV\ A - A I 1.6 1.2 =l 3 0.8 Z CO CD O -0.4 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 Frequency (Hz) (a) Power spectrum (50 ~ 1,200 Hz) PSD of (c86525a.600) at Probe 2 (n=650 rpm, Bite=0.025 in) 1000 1050 1100 1150 Frequency (Hz) 1200 1250 (b). Power Spectrum (1,000 ~ 1,250 Hz) Figure 7.3 Power spectra of the vibration signal (Washboarding occurs) 150 Chapter 7 Mechanics of Washboarding Since a natural frequency of this blade does not greatly vary with the tooth passing frequency, these spikes are the vibration responses to the periodic cutting forces at the frequency F that decreases with time. The tooth passing frequency F, shown in Figure 7.3 (b) was measured by an additional probe above the top guide. The vibration frequency F is slightly lower than the tooth passing frequency F,. The average frequency difference (Fr F) is about 11 Hz, which corresponds to the wheel rotating frequency Fr. Some components below 200 Hz are shown in Figure 7.3 (a). Apparently, these components can not produce washboarding patterns. Since a low frequency component produces a pattern of very long vertical pitch, an almost straight tooth mark can be observed on the sawn surfaces. A visual examination of tooth marks on the sawn surface shows that the components at about 400 Hz produce the tooth marks that cross over more than one washboarding pattern. These components do not have contributions to washboarding, either. Consequently, this washboarding must have been caused by the high frequency component, i.e. a forced vibration response. It can be seen from the band-pass-filtered signal shown in Figure 7.4 that this cut starts at x=0, washboarding starts at JC=4" and covers the remaining board. The corresponding vibration signal has large amplitude. The deep tooth marks produced by bad teeth match the areas where the signal has small amplitude. When the washboarding occurs, this high frequency component is strongly modulated by a low frequency signal. The amplitude modulation function shown in Figure 7.5 can be obtained by demodulating this signal (Luo, 1996). This signal is equivalent to the envelope of the filtered signal. From the frequency spectrum of the modulation function shown in Figure 7.6, it is found that there is a primary frequency component at 11 Hz, which is the wheel rotation frequency, and a component at 5.5 Hz, which is the band rotatation frequency. This indicates that the vibrations at the wheel rotation frequency due to the unbalanced wheels, the weldment of the band joint, and the dents on the band can modulate the high frequency component if their Chapter 7 Mechanics of Washboarding 151 amplitude are sufficiently large. Due to this amplitude modulation, some side-bands exist beside the high peak at the tooth passing frequency in the spectra. If the signal is seriously modulated, the magnitudes of these side bands can be higher than the original peak. The frequency difference between these spikes is equal to the wheel rotation frequency. Due to the effect of the regenerative cutting forces applied to the teeth of this blade, the primary forced vibration response component is at the frequency below the tooth passing frequency rather than at other frequencies (see Chapter 5). In this way, the saw blade vibrates primarily at the frequency slightly lower than the tooth passing frequency. This shift makes the kinematic condition of washboarding satisfied and the washboarding pattern produced. Consequenctly, the washboarding in this testing is possibly caused by the forced vibration of the blade at the frequency slighly lower than the tooth passing frequency. The frequency shift is due to the amplitude modulation of the blade vibration made by the vibrations of the wheels and band at the rotating frequency. The power spectrum density of the original displacement signal during cutting is shown in Figure 7.7. The displacement components at the frequencies Fr and 2 Fr indicate that the bandsaw blade has been excited at the wheel rotation frequency Fr by the guides or other factors This frequency component modulates the blade vibration at the tooth passing frequency. Figures 7.8 shows the spectrum of the signal and the filtered signal from a different cut. It is seen that the high frequency vibration level of this blade is low and the modulation effect is weak. In fact, there was no washboarding observed during this cut. It must be noted that a threshold of the magnitude of the spectrum is difficult to be established because the magnitude is not the direct measurement of the displacement at the tooth tips and may be affected by the properties of the material. In addition, it is possible that a washboarding pattern produced by the first tooth might be removed by the next tooth. For instance, although one peak appeared in this spectrum sometimes, no washboarding was observed. Chapter 7 Mechanics of Washboarding 152 M o d u l a t i o n F u n c t i o n of C 8 6 5 2 5 a . 6 0 0 at P o i n t 2 - 0 . 0 5 | 0 . 0 5 [ 10 12 14 18 20 1 °m4HW a - 0 . 0 5 ' 0 . 0 5 - 0 . 0 5 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 D i s t a n c e f rom the R igh t E n d (in) Figure 7.4 Band-pass filtered signal (Washboarding occurs) (FT = 1166 Hz, Band Width = 200 Hz) Modulation Function of C86525a.600 from Probe 2 -0.05 -0.05 8 10 12 14 16 18 20 0.05 22 24 26 28 30 32 34 -0.05 36 38 40 42 44 46 Distance from the Right End (in) 48 50 Figure 7.5 Envelope of the filtered signal (Washboarding occurs) Chapter 7 Mechanics of Washboarding x 1 o'3Spectrum of Modulation Function of C86525a.600 from Probe 2 0 • J 0 20 40 60 80 100 Frequency (Hz) Figure 7.6 Spectrum of modulation functions of this signal (Washboarding occurs) 10 20 30 40 50 60 70 80 Frequency (Hz) Figure 7.7 Spectrum of the original signal (Washboarding occurs) Chapter 7 Mechanics of Washboarding N 0 - 3 X 0.2 E. O 0.1 CO ° - 0 P S D of (C86225C.600) at Probe 2 (n=620 rpm, Bite =0.025 in) X.k.Jl..... J^I t LLJ& . i . . . At*... t Jt A.. .<Jlu. -1 1 1 r -200 400 600 800 Frequency(Hz) JL 1000 -1.8 -1.6 -1.4 1.2 H -1 3 CD - 0.8 oT CD O 0.6 *"* 0.4 0.2 0 1200 (a) M o d u l a t i o n F u n c t i o n o f C 8 6 2 2 5 C . 6 0 0 a t P o i n t 2 -0.05 | 0.I CD O a. co Q -0.I 0.05 0 -0.05 8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 D i s t a n c e f r o m the R i g h t E n d (in) 46 (b) Figure 7.8 Power spectra (a) and band-pass filtered signal (b) (without washboarding) Chapter 7 Mechanics of Washboarding 155 7.1.2 Mechanism of the Washboarding The calculated magnification factors (RMS) of Blade 1 subjected to regenerative cutting forces and three modulated excitations at frequencies F, and Ft (l+p) (p=FrlFt) are shown in Figure 7.9. It was found from the cutting tests that washboarding occurs when the tooth passing frequency F, =1122 Hz or 1166 Hz and the bite per tooth is 0.635 mm. It should be noted that the prediction of magnification factors is useful for understanding the mechanism of washboarding though the smooth band model can not give accurate magnification factors of teeth at a high frequency. Since Modes (9, 3) and (10, 4) are close to the tooth passing frequencies, they are the possible modes causing the washboarding. To satisfy the kinematic condition of the existence of washboarding, the dominant vibration of the blade should be at a frequency below the tooth passing frequency. For Mode (9, 3), the solid line stands for the response magnification factor to the excitation at the frequency Ft(l-p). This factor is always larger than other two factors within the tooth passing frequency range of 1080 Hz to 1140 Hz. Consequently, this mode is the washboarding mode. In addition, since the difference between the primary vibration frequency and the tooth passing frequency is equal to the wheel rotation frequency, and the measured vertical pitch is short, the washboarding should be caused by a high-order mode. It means that the damping in this system must be so high that a vibration response consisting of high-order modes along the length appears in this frequency range. If the damping is very light, on the other hand, the magnification factor has several peaks around the natural frequencies. Then the factor is so sensitive to the tooth passing frequency that a small variation of the frequency could cause the washboarding to appear or to disappear. This is not the case when the washboarding occurs during cutting. Consequently, the damping in this system is high and plays an important part in the mechanism of this type of washboarding. Chapter 7 Mechanics of Washboarding 156 0 i , , 1 1 1 — i , , 1 1 850 900 950 1000 1050 1100 1150 1200 1250 1300 Tooth Passing Frequency (Hz) Figure 7.9 Magnification factors of the responses to three excitations (Ro = 36.7 kN, Ki = 60 kN/m, £ = 0.01, /M).0093, Fw0 = Fw, =1 N) According to the kinematics of washboarding, the frequency of the dominant vibration component must be lower than the tooth passing frequency. If the washboarding occurs, the regenerative cutting forces must exist so that the dominant response is at the frequency F,(l-p). In addition, the amplitude of modulation function Fwi in (5.5) at the wheel rotating frequency must reach a certain value so that this response component at the frequency F,(l-p) dominates the vibration of the blade. This is true in the tooth passing frequency range of 1080 Hz to 1130 Hz shown in Figure 7.9. The washboarding pattern caused by the modulated vibration is simulated based on the tooth loci and it is shown in Figure 7.10. It is assumed in the simulation that the wave motion in the blade at an exciting frequency F=Ft(l-p) = 1155 Hz is the same as the wave motion at the frequency F, = 1119 Hz shown in Figure 7.9. The washboarding pitches listed in Table 7.1 are compared with the measured and calculated pitches based on (6.6) and (6.16). It is seen that the simulated and calculated horizontal pitch matches the measured pitch. The simulated and calculated vertical pitches are close to the measured. The difference between them may be caused by the smooth band model in which the teeth, the overhang and the Chapter 7 Mechanics of Washboarding 157 extension of the span due to the guide wearing are not included. If these factors were taken in consideration, the blade would become more flexible. Higher-order modes, such as (10, 3), with shorter wavelengths would exist in this frequency range. The vertical pitches would become shorter. Washboarding Pattern (Fn=1155 Hz, F=1166 Hz, Bite=0.635 mm) P = 67.3 mm Direction Figure 7.10 Simulated washboarding pattern (Mode (9, 3), F = 1155 Hz, F, = 1166 Hz, Bite = 0.635 mm) Table 7.1 A Comparison of Washboard Pitches (Type I) Pitch (mm) Measurement Estimation Simulation Px 66-71 67.3 67.3 Py 30-31 35.5 34.9 _ , ! 1 Estimation is based on Eqns. (6.6) and (6.16). Chapter 7 Mechanics of Washboarding 158 The cutting process of two adjacent teeth is simulated as shown in Figure 7.11. The kerf centers of the two teeth are in wave motion. The locus of the second tooth (dash-dotted line) has a phase-lag to that of the first tooth (dashed line). Due to the large amplitudes of the loci, the washboarding occurs during this cutting. The profile of the first bite (in dashed line) is the envelope of all the first-tooth profiles (in dashed line) in different positions. A series of second-tooth profiles are drawn in the solid lines. Each profile of the second tooth superimposes that of the first one. Their front edges are placed in the same position. Each shaded area with the width of one bite within a tooth profile always cuts the workpiece. Its side cutting forces are balanced and regenerative cutting forces are so small that the system is stable. Consequently, the primary cause for this washboarding is the excessive forced vibration due to the seriously modulated excitations, instead of the instability of the blade. „ „ - 3 Tooth Profiles (F =1155 Hz, F =1166 Hz, Bite=0.635 mm) x 10 v n t .35 0.355 0.36 0.365 0.37 0.375 0.38 0.385 0.39 Position of Tooth (m) 0.395 Figure 7.11 Loci, dynamic profiles and envelopes of two teeth . 159 Chapter 7 Mechanics of Washboarding 7.1.3 Factors Affecting Type I Washboarding — Experimental Results In order to reduce the vibration levels and stop the washboarding, it is necessary to understand the effects of various factors, such as the strain level, the blade speed, the bite per tooth and the depth of cut. I. Strain Level During the cutting tests, two strain levels, 13 klbs and 18 klbs, were applied. With the different strain levels, the saw blade experienced a similar washboarding behavior under the same operating conditions. The existence of washboarding at the lower strain level is the same as that at the higher strain level. This indicates that the washboarding in bandsaws does not greatly depend on the strain levels in the blade because the strain does not greatly affect the high-frequency modes of the blade. This phenomenon matches both the analytical results and previous experiments (Hutton and Zhan, 1989). However, the saw deviations during cutting are strongly dependent on the strain levels. Figure 7.12 shows the extent of washboarding increases with the wheel rotating speed. The extent of washboarding is described by the average depth of patterns. The washboarding occurs at a rotating speed of 600 rpm and a strain level of 13 klbs. It does not occur at the same speed and the higher strain level. This may have been caused by the fact that the guides wore seriously. The vibration of the blade in the cutting span is influenced by the reduction of guide pressure. The decrease of support stiffness and the transmission of vibrations from other spans increase the vibration of the blade in the cutting span. But the vibration energy in this blade may scatter over a wide frequency range and the energy at a specific mode that causes washboarding may be reduced. Therefore this blade does not cut washboards at this speed until the vibration energy at this mode achieves a certain level. Consequently, once washboarding occurs in this case, the pattern should be deeper than the pattern generated by the blade supported with two normal pressure guides. In addition, when a higher strain is Chapter 7 Mechanics of Washboarding 160 applied, the blade has higher stiffness and less deviation and a deeper pattern can be observed. On the other hand, the blade subjected to a lower strain experiences greater deviation and the shallower washboarding pattern is observed. 1.2 E E 0.8 r-f 0.6 g> 0.4 ^ 0.2 0 • T=13 klbs nT=18 klbs 600 625 Rotating Speed (rpm) 650 Figure 7.12 Variation of average depth of washboards with the blade speed II. Blade Speed Figure 7.12 shows that the depth of patterns increases with the wheel rotating speed or the blade speed. The higher the rotating speed, the larger the amplitude of the modulated excitations and the more energy is switched into the blade through the regenerative cutting forces. The vertical pitch does not change with the blade speed. In fact, the small change in the speed does not affect the washboarding mode and the difference between the vibration frequency and the tooth passing frequency. The horizontal pitch may slightly increase with the feed speed that is proportional to the blade speed for a given bite per tooth. III. Feed Speed or Bite It can be seen from Figure 7.13 that if washboarding occurs during cutting, the depth of patterns increases with the feed speed or the bite per tooth. If a small bite is used, no Chapter 7 Mechanics of Washboarding 161 washboarding is observed. Based on the model of regenerative cutting forces, increasing the bite leads to the reduction of unbalanced side-cut forces and the regenerative forces decrease. In this case, however, the forced vibration dominates the transverse motion of the blade. The amplitude of a modulated excitation increases with the blade speed and the bite per tooth. If a big bite is used, the washboarding becomes more serious and the pattern is deeper. 1.6 E £ 1.2 .c £ 0.8 0) 0.4 • T= 13 klbs, n=625 rpm • T= 13 klbs, n=650 rpm sT= 18 klbs, n=650 rpm 0.02 0.028 0.025 Bite (in/tooth) Figure 7.13 Extent of washboarding vs. bite per tooth IV. Depth of Cut It was found in the cutting tests that washboarding only occurred with deep cuts. If the depth of cut was 10 in and the rotating speed was below 625 rpm, no washboarding was observed. For the same bite (0.025 in), when the speed was 650 rpm, a shallow pattern within a small area was found. If the depth of cut increased to 15 in, washboarding occurred at the mentioned speeds. During a deep cut, a high order mode along the length that causes washboarding is more easily excited because its magnification factor is larger than those of other modes. If a shallow washboarding pattern was produced, there was no washboarding pattern close to the top of the lumber. This phenomenon implies that the development of washboarding needs sufficient time to accumulate the vibration energy in the blade teeth. Chapter 7 Mechanics of Washboarding 162 7.2 Washboarding Caused by Self-Excited Vibration A typical washboarding pattern shown in Figure 7.14 was observed during the cutting tests conducted with Blade T2 and 5' Cetec bandmill. This pattern, the horizontal pitch of which is much smaller than the vertical pitch, is called Type II washboarding pattern. Figure 7.15 shows the band-pass filtered vibration signal captured from Blade T2 by an eddy-current probe shown in Figure 7.1. It is clearly seen that the cut starts with a steep increase of vibration amplitude due to a serious impact and ends with a step-down due to the sudden unloading. A visual examination shows that this signal also matches the washboarding pattern on the sawn surface. The washboarding occurs over the first four seconds (except the period marked A) when the vibration displacement is comparatively large. Since the measurement was conducted of the displacement at a point close to the gullet, instead of the tooth tips, the measured displacement is much smaller than the depth of washboarding pattern (about 0.8 ~ 1.2 mm). Figure 7.14 A typical washboarding pattern (Type II) (Blade T2, R0= 18 klbs, F, = 1176 Hz, Fn = 1127 Hz, Bite = 0.397 mm, Depth of Cut = 235 mm, Px = 10 mm, Py= 31 mm) Two main frequency components can be seen from the power spectrum of the signal as shown in Figure 7.16. A series of lower spikes indicate the tooth passing frequencies at different times. The higher spikes are the natural frequencies varying from 1127 Hz to 1135 Hz and they disappear after t = 4 sec. Apparently, this washboarding is caused by self-excited vibration while the saw blade suffers from instability. Chapter 7 Mechanics of Washboarding 163 Displacement at Point 2 (b66092a.600) _ 0.1 E £ 0.05 b -0.1 I i i i I Washboarding' A ' I • L II A..[.. T i ii' I / 1 i i i l ' i ' 'ITIi 1 A i l 1 r J * i l A _ . i — Start ( i )f Cut ! i — End of Cut I [ I I I I I I I 0 1 2 3 4 5 6 7 8 Time (sec) Figure 7.15 The band-pass filtered signal (F, = 1176 Hz) N X ^ 0.2 o- 0 P S D of (b66092a.600) at Probe 2 (n=660 rpm, Bite=0.397 mm) F =1127 -1135 Hz F.= n t Washboardingl 176 Hz d CO CD 23. 0 900 950 1000 1050 1100 1150 1200 1250 1300 Frequency (Hz) Figure 7.16 Spectrum of the captured signal The stability analysis of this blade indicates there are couples of modes whose frequency are close to the tooth passing frequency. Their natural frequencies and wave traveling directions and the vertical pitches of possible washboarding patterns are listed in Table 7.2. The simulation of these patterns is based on the analytical mode shapes and the measured Chapter 7 Mechanics of Washboarding 164 tooth passing frequency and natural frequency. It is found that only Mode (7, 1) can produce the right washboard that matches the sawn pattern and other modes do not follow the kinematics of this washboarding. Table 7.2 Vertical Pitches of Different Modes Mode Frequency Fn (Hz) Traveling Wave —^ Vertical Pitch Py (mm) (6, 2) 1,072 B 40.0 (7,1) 1,101 B 111 (1,5) 1,110 B 43.5 (6, 4) 1,133 F 55.1 (7, 3) 1,140 BIF 46.4 (2, 5) 1,163 B 44.5 * The simulation is based on F, = 1176 Hz and F„ =1127 Hz. ** B = Backward wave, F=Forward wave. The primary instability region of Mode (7, 1) shown in Figure 7.17 varies with the damping ratio £ and the regenerative cutting force coefficient Ki. The primary instability takes place at the natural frequency Fn = 1098 Hz (Point A) for the case C^ =0. For the same regenerative force coefficient AT/=200 kN/m, the onset frequency increases to the frequency at Point C with the damping ratio (^ =0.001. With the same damping ratio, if the force coefficient increases to Ki=250 kN/m, the onset frequency reduces to the frequency at Point B. If the damping ratio is large enough, e.g., (^ =0.0015, the real part of this mode is always negative and the primary unstable region disappears. The mode shape of Mode (7, 1) for the given parameters is shown in Figure 7.18. The washboarding pattern produced by this mode is simulated and shown in Figure 7.19. A comparison of the predicted pitches and the measured can be seen in Table 7.3. Since the simple kinematic model handles the pitches produced by a pure traveling wave, it is unable to provide the vertical pitch of a pattern produced by a group of irregular traveling waves. This is the reason why the simulation can give a better prediction. Chapter 7 Mechanics of Washboarding 15 10 5 t: 0 to D_ ni -5 rr -10 -15 -20 !F =1101 Hz n at F=1176 I Hz : I ^ ^ " ^ w ^ ^ ^ ^ A / B # > > I ^ i ~ 1 g - M / // ^ P . K | = 200 kN/fn, £=0 K,=200 kN/m, ^=0.001 K-250 kN/m, £=0.001 K | = 250 kN/m, £=0.0015 I I I 1 I 1000 1050 1100 1150 1200 Tooth Passing Frequency (Hz) 1250 1300 Figure 7.17 Primary unstable region of Mode (7, 1) Mode (7, 1), F =1127.0 Hz 0.15 0.1 0.05 0 -4 0.2 0.3 Longitude X 0.4 0.5 Mode Shaped Envelope - 2 - 1 0 1 Magnitude W(0.50L, Y, t) Figure 7.18 Mode shape of Mode (7, 1) (i?0=31.67 kN, £,=200 kN/m, £=0.001) Chapter 7 Mechanics of Washboarding 166 Figure 7.19 Simulated washboarding pattern (Type II) Table 7.3 A Comparison of Washboard Pitches (Type II) Pitch (mm) Measurement Estimation* Simulation Px 7-11 9.5-11.4 9.5-11.4 Py 29-32 34.9 - 35.0 32.5 - 32.9 * Estimation is based on Eqns. (6.6) and (6.16). The continuous profiles of two teeth and their envelopes are illustrated in Figure 7.20. Compared with the tooth profiles (Blade Tl) shown in Figure 7.11, these profiles show more serious side cuts. The shaded areas with the width of a bite represent the areas in which the lateral cutting forces for each tooth due to the flank cut are balanced and no lateral resultant appears. Compared the locus of the present tooth (in dash-dotted line) with that of the prior tooth (in dashed line), the second bite has a phase-lag with respect to the first bite due to the fact that the tooth passing frequency is about 50 Hz higher than the natural frequency. During Chapter 7 Mechanics of Washboarding 167 the second bite, one flank of this tooth removes the material inside the envelopes of the first-tooth profiles. The removal of material by one flank results in the lateral cutting force related to the difference between the displacements of this tooth and the prior tooth. Furthermore, the lateral cutting force and the vibration velocity at this moment is in phase. The instability of this blade subjected to such a self-excited vibration could take place during this cut. This washboarding is therefore caused by the self-excited vibration due to the unbalanced side-cutting forces. If the tooth passing frequency is much higher than the excited natural frequency, the phase lag of the locus of the present tooth to the locus of the prior tooth becomes large. If a small bite is used for this cut, the pattern produced by the first tooth could be possibly removed by the rear side-edges of the second tooth. This may be the reason why no washboarding is observed if the tooth passing frequency is much higher than the natural frequency. c <r> E 0) o 03 o_ CO b s CD to 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 „ „ - 3 Tooth Profiles (F =1127 Hz, F,=1176 Hz, Bite=0.397 mm) x 10 v n t Profile of 1st Bite 2nd Tooth 1st Tooth Side Cut Loci of Two Teeth \ • Cutting ^ Direction 1 Bite 1st Bite 2nd Bite 0.305 0.31 0.315 0.32 0.325 0.33 0.335 0.34 0.345 0.35 Position of Tooth (m) Figure 7.20 Loci, dynamic profiles and envelopes of two teeth Chapter 7 Mechanics of Washboarding 168 7.3 Comparison of Two Types of Washboarding Mechanisms Two types of washboarding patterns were observed and studied in the cutting tests. Type I washboarding may be caused by the modulated and forced vibration of the blade due to the displacement excitation from the guides. Type II washboarding results from the self excited vibration of the blade subjected to regenerative cutting forces. The similarities and differences in their mechanisms are summarized in Table 7.4. Table 7.4 Properties of Two Types of Washboarding Patterns Properties Type I Type II Vertical Pitch Py Dependent on mode shape Dependent on mode shape Horizontal Pitch Px Large Small Exciting Forces • Modulated exciting forces • Regenerative cutting forces • Regenerative cutting forces only Side Cutting Involved Serious Vibrations • Modulated zLFt(\-p) • Self-excited at natural • Self-excited (involved) frequency Fn Washboarding Mode Dependent on the response One of modes of high orders along the length Frequencies • Tooth passing frequency Ft • Tooth passing frequency F, Involved • Wheel rotating frequency pF, • Natural frequency Fn Phase-lag between two bites Small Large Damping Effects • Attenuate the amplitude • Attenuate real parts • Modify the wave motion • Reduce the unstable regions Effects of Regenerative Cutting Force • Dependant on the damping & the excitations atF,(l±p) • Equivalent pos/neg damping • Widen the unstable regions • Increase the peak real part • Shift it closer to frequency Fn A 169 Chapter 7 Mechanics of Washboarding Many factors affect the vibration response and stability of a bandsaw blade during sawing. These factors can be classified into two groups. One group mainly affect the vibration amplitude of the blade. These factors are the blade speed, the bite per tooth, the tooth profile, the damping and other factors related to the vibration of the mill frame. The other group may affect the natural frequencies, the instability regions and the frequencies at which the vibration energy is mainly distributed. These factors, for example, include the blade speed, the tooth pitch, the stiffness-related parameters and the damping in this sytem. In order to stop washboarding, one must first determine the main causes for it. After that, one or more of the following measures could be taken: (1) . Check if there are some speed ranges in which washboarding disappears. (2) . Increase tooth stiffness by using thicker blade, shallower cut and shorter span. (3) . Reduce the lateral cutting forces by modifying the tooth profile and using a proper bite per tooth. (4) . Eliminate the excessive vibration of the mill frame and the tooth impacts due to the weldment and dished areas on the blade. (5) . Shift the natural frequencies of the blade by attaching some auxiliary apparatus to the blade in the cutting span. Chapter 8 Conclusions The high frequency vibration characteristics of a traveling plate subjected to internal forces, parametric and modulated excitations have been studied theoretically and experimentally based on the developed smooth band model and finite element model of the plate. The analytical and experimental results on the modes, responses and stability of the plate have been applied to industrial bandsaw blades. The mechanisms of washboarding occurring during the cutting tests have been explained. A kinematic model of washboarding has been used to decode the patterns. Some fundamental findings and the significant characteristics of the traveling plate obtained from this work are summarized below. I. Modal Analysis of a Traveling Band Approximate analytical approaches (Rayleigh Ritz method and FEA) have been developed to conduct the modal analysis of a traveling plate over a wide frequency range. A high frequency mode of a traveling plate is defined based on the orders of the envelope that is the superposition of the time-varying mode shapes. The following factors affect the behavior of different modes of this plate in different ways: (a) . The variation of the span length does not affect the modes of high orders across the width, such as Mode (1, 6). The modes of high orders along the length, such as Modes (9, 1) and (10, 1), are very sensitive to the span length. (b) . Some high frequency modes are insensitive to the blade speed. The natural frequencies of some modes increase with the speed due to mode coupling effects. (c) . The internal stresses, including static tensile stress, tensioning stresses, and stresses induced by the wheel tilt and crown, can either increase or decrease natural frequencies. The influence of these stresses on the lateral and torsional modes depend on the stress distributions that determine the variation of strain energy in a specific mode of the band. 170 Chapter 8 Conclusions 171 (d). The masses and stiffness of teeth can make differences in both natural frequencies and frequency response functions of high frequency modes. The magnitude of a frequency response function is more sensitive to the depth of gullet than is the natural frequency. II. Stability Analysis of A Traveling Band A dynamic model of a smooth band subjected to moving regenerative forces has been built by using the Rayleigh-Ritz method. The primary unstable region occurs where the tooth passing frequency is higher than a natural frequency and below twice the frequency. In this region, the lateral cutting forces and the vibration velocities at the points of application of these forces are in the same direction. The regenerative force coefficient related to the tooth geometry and blade speed affects both the magnitude and location of the crest of the real parts of the eigen-values. Since there are many modes in the frequency range of interest, the primary unstable regions could cover the whole frequency range. III. Finite Element Analysis of a Toothed Blade A finite element model has been built to model the moving regenerative cutting forces applied at the tooth tips of a traveling blade. Three types of traveling plate elements, i.e., bilinear plate element, triangular plate element and variable-domain rectangular plate element have been developed. A toothed blade has lower natural frequencies than a smooth band due to the mass of the teeth. The variation of the depth of gullet does not greatly influence its natural frequencies. The blade with deeper gullets has larger real parts or wider instability regions if the damping is included in this model. IV. Modal Testing of a Traveling Band Modal testing of a stationary smooth band and a traveling band was conducted in order to verify both the continuous and the finite element models. The measured natural frequencies and mode shapes or envelopes are in good agreement with the analytical predictions based on the two models. The moving blade may have some conventional nodes for some modes at a low blade speed or with a short wavelength and some generalized nodes for others at a high blade speed and with a long wavelength. Chapter 8 Conclusions 172 V. Stability and Responses of a Damped Spring-Mass System and a Damped Traveling Plate A deeper look into the vibration behavior of a traveling blade was taken from a model that contains both parametric and modulated excitations as well as damping. The forced vibration response and stability of a single degree-of-freedom spring-mass-dashpot subjected to regenerative and modulated forces were first investigated. This dynamic model was then extended to the study on the vibration characteristics of a traveling blade. (a) . Increasing the damping in a system greatly reduces the real parts of the eigenvalues and the primary unstable regions. The largest unstable regions exist in an undamped system. There is a critical damping ratio above which the unstable regions disappear. (b) . The unstable regions of a damped system can be extended with an increase of the regenerative force coefficient. If this coefficient is small enough, this system is always stable. The locations of regenerative forces determine which modes are dominant in the responses. (c) . Increasing the damping in the traveling plate modifies the shape of the deflected plate. Under heavy damping, a vibration response may contain several modes as if a mode was mixed with its adjacent modes and behaved like a higher order mode. (d) . Among three vibration response components corresponding respectively to three modulated excitations, the maximum component appears at the frequency that is the difference between the tooth passing frequency and the modulating frequency. VI. Kinematics of Washboarding A washboarding pattern contains valuable information on the tooth vibrations. Two kinematic models of washboarding have been developed to predict washboarding patterns by considering the continuous interactions between the tooth profiles that are in wave motion and the work-piece that moves at a feed speed. This pattern, described by a vertical pitch and horizontal pitch and its depth, can be simulated by tracing the loci of teeth in the cutting zone. The vertical pitch depends on the frequency and shape of a washboarding mode and the blade speed. The horizontal pitch is determined based on the feed speed and the phase shift Chapter 8 Conclusions 173 between the loci of two adjacent teeth during two bites that are directly proportional to the difference between the tooth passing frequency and the washboarding frequency. Since different modes might produce the same pattern under certain conditions, a washboarding mode can be identified by analyzing the results from modal analysis and measurements. VII. Cutting Tests and Mechanics of Washboarding Two types of washboarding patterns were observed in the cutting tests. Type I washboarding is possibly caused by the forced vibration due to both the parametric and the modulated excitations produced by the eccentricity of wheels, the weldment and the local dished areas. Type II washboarding results from the instability of the blade subjected to parametric excitations due to unbalanced regenerative cutting forces. An analysis of the dynamic cutting process with two teeth shows that more serious flank cut is involved in Type II washboarding, instead of Type I washboarding. Two types of washboarding patterns are simulated and the predicted pitches match those measured directly from the sawn surfaces. It is recommended that the future work be concentrated on the following aspects: (1) . Modification of Cutting Force Model. The cutting forces should be qualitatively modeled by conducting a fundamental analysis and tests in order to fully understand the effects of the tooth geometry and operating conditions on the cutting forces. (2) . Modification of Finite Element Model. The proposed finite element model should be refined by taking into account factors, such as the guide support stiffness and overhangs, affecting the vibration behavior of the blade. (3) . Analysis of Vibration Response of Teeth. The vibration responses of teeth should be studied by incorporating the modified cutting force model and the estimated damping into the finite element model. The initial conditions for the self-excited vibration must be determined. The effect of geometric non-linearity of the blade is another interesting topic. Bibliography Bathe, K. J., 1996, Finite Element Procedures, Prentice Hall, New Jersey, USA, 796-800. Batoz, J. L., Bath, K. J. and Ho, L. W., 1980, A study of three-node triangular plate bending elements, International Journal for Numerical Methods in Engineering, Vol. 15, 1771-1812. Bazeley, G. P., Cheung, Y. K., Irons, B. M. and Zienkiewicz, O. C , 1965, Triangular elements in plate bending — conforming and non-conforming solutions, Proc. Conf. on Matrix Methods in Structural Mechanics, WPAFB, Ohio, 547-576. Belytschko, T., Stolarski and H., Carpenter, N., 1984, A C° triangular plate element with one-point quadrature, International Journal for Numerical Methods in Engineering, Vol. 20, 787-802. Chung, Y. Y. and Sung, C. K., 1998, Dynamic behavior of the band/wheel mechanical system of industrial band saws, Journal of Vibration and Acoustics, Transaction of ASME, 120(4), 842-847. Clough, R. W. and Tocher, J. L., 1965, Finite element stiffness matrices for analysis of plate bending, Proc. Conf. on Matrix Methods in Structural Mechanics, WPAFB, Ohio, 515-545. Cook, R. D., 1989, Concepts and Applications of Finite Element Analysis, 3 r d edition, John Wiley & sons, 328-332. Cramond, P., 1989, Developments in bandmill design: strain systems, Proceeding of Saw Tech'89, Oakland, California, Oct. 2-3, 1989. D'angelo III, C , Alvarado, N. T., Wang, K. W. and Mote, C. D., Jr., 1985, Current research on circular saw and band saw vibration and stability, Shock and Vibration Digest, 17(11), 11-23. Eschler, A., 1982, Stress and Vibrations in Bandsaw Blades, M.A.Sc. Thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver, B. C , Canada. Foschi, O. F., 1975, The light-gap technique as a tool for measuring residual stresses in bandsaw blade. Wood Science and Technology, 9, 243-255. Hutton, S. G. and Zhan, J., 1989, Causes of washboarding in handsaws, Proceedings of Saw Tech'89, Oakland, California, USA, Oct. 2-3, 1989. 174 Bibliography 175 Hutton, S. G. and Lehman, B. F., 1991, The role of top wheel dynamics in bandmills, Proceedings of the Tenth International Wood Machining Seminar, Berkeley, USA., Oct. 21-23,1991,247-256. Hutton, S. G. and Lehman, B. F., 1993, Modeling of a band saw during cutting, Proceedings of the 11th International Wood Machining Seminar, Honne, Norway, May 25-21, 1993, 441-452. Jeyachandrabose, C , Kirkhope, J. and Ramesh Babu, C , 1985, An alternative explicit formulation for the DKT plate-bending element, International Journal for numerical methods in Engineering, Vol. 21, 1289-1293. Lehman, B. F. and Hutton, S. G., 1991, Factors influencing bandsaw behavior during cutting, Proceedings of the Tenth International Wood Machining Seminar, Berkeley, USA., Oct. 21-23, 1991,257-267. Lehman, B. F., 1993, The Cutting Behavior of Bandsaws, Ph.D. Thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver, B. C , Canada. Lengoc, L. and McCallion, H., 1995a, Wide bandsaw blade under cutting conditions, Part I: Vibration of a plate moving in its plane while subjected to tangential edge loading, Journal of Sound and Vibration, 186(1), 125-142. Lengoc, L. and McCallion, H., 1995b, Wide bandsaw blade under cutting conditions, Part II: Stability of a plate moving in its plane while subjected to parametric excitation, Journal of Sound and Vibration, 186(1), 143-162. Lengoc, L. and McCallion, H., 1995c, Wide bandsaw blade under cutting conditions, Part HI: Stability of a plate moving in its plane while subjected to non-conservative cutting forces, Journal of Sound and Vibration, 186(1), 163-179. Lengoc, L. and McCallion, H., 1999, Self-induced vibration of bandsaw blades during cutting, Proceedings of the Institute of Mechanical Engineers, Part C, 213, 371-380. Lin, C. C , 1997, Stability and vibration characteristics of axially moving plates, International Journal of Solids and Structures, 34(24), 3179-3190. Ling, K., Kimura, S. and Yokochi, H., 1989a, Band saw vibration I: Theoretical analysis of forced vibration during no-load running, Journal of Japan Wood Research Society, 35(4), 293-298. Ling, K., Kimura, S. and Yokochi, H., 1989b, Band saw vibration II: Experimental consideration of forced vibration during no-load running, Journal of Japan Wood Research Society, 35(4), 299-305. Bibliography 176 Ling, K., Kimura, S. and Yokochi, H., 1989c, Band saw vibration HI: Forced vibration during no-load running, Journal of Japan Wood Research Society, 35(8), 710-717. Luo, Z. and Fawcett, J. N., 1996, Detection of incipient defects in a gearbox by demodulation of the vibration signature", Journal of Nanjing University of Aeromautics & Astronautics (English Edition), 13(1), 54-60. MacNeal-Schwendler Corp., 1997, User's Guide, MSCINASTRANfor Windows, Version 3. Mindlin, R. D., 1951, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, ASME, Applied Mechanics, vol. 18, 31-38. Mote, C. D., Jr., 1965a, A study of bandsaw vibrations, Journal of Franklin Institute, 279(6), 430-444. Mote, C. D., Jr., 1965b, Some dynamic characteristics of bandsaws, Forest Production Journal, 15(1), 37-41. Mote, C. D., Jr., and Naguleswaran, S., 1966, Theoretical and Experimental Bandsaw Vibrations, Trans. ASME 88(B), 151-156. Mote, C. D., Jr., 1972, Dynamic stability of axially moving materials, Shock and Vibration Digest, 4(4), 2-11. Mote, C. D., Jr. and Wu, W. Z., 1985, Vibration coupling in continuous belt and band systems, Journal of Sound and Vibration, 102(1), 1-9. Muller, D. E., 1956, A method of solving algebraic equations using an automatic computer, Math Tables and Other Aids to Computation, Vol.10, 208-215. Naguleswaran, S. and Williams, C. J. H., 1968, Lateral vibration of bandsaw blades, pulley belts, and the like, International Journal of Mechanical Science, 10, 239-250. Okai, R., Kimura, S. and Yokochi, H., 1996a, Dynamic characteristics of the candsaw I. Self-excited vibration and washboarding during cutting, Mokuzai Gakkaishi, 42(4), 333-342. Okai, R., Kimura, S. and Yokochi, H., 1996b, Dynamic characteristics of the bandsaw n. Effects of sawdust on the running position of the bandsaw, Mokuzai Gakkaishi, 42(10), 953-960. Okai, R., Kimura, S. and Yokochi, H., 1997, Dynamic characteristics of the bandsaw III. Effects of workpiece thickness and its position from the ground on self-excited vibration and washboarding during sawing, Mokuzai Gakkaishi, 43(7), 551-557. Bibliography 111 Schajer, G. S., 1991, Vibration modes of guided circular saws, The Tenth International Wood Machining Seminar, Univ. of California at Berkeley, USA, Oct. 21-23, 1991, 85 - 99. Simmonds, A., 1980, Wide Bandsaws, The Art of Saw Doctoring, 2 n d Edition, Stobart & Son Ltd, London, 10-11. Stricklin, J. A., Haisler, W. E., Tisdale, P. R. and Gunderson, R., 1969, AIAA Journal, Vol. 7, No. 1, 180-181. Stylianou, M . and Tabarrok, B., 1994, Finite element analysis of an axially moving beam, Part I: Time integration, Journal of Sound and Vibration, 178(4), 433-453. Stylianou, M . and Tabarrok, B., 1994, Finite element analysis of an axially moving beam, Part II: Stability analysis, Journal of Sound and Vibration, 178(4), 455-481. Taylor, J., 1986, The dynamics and Stresses of Bandsaw Blades, M.A.Sc. Thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver, B. C , Canada. Tabarrok, B., Leech, C. M. and Kim, Y. I., 1974, On the dynamics of an axially moving beam, Journal of the Franklin Institute, 297(3), 201-220. Taylor, J., 1993, The Effect of Bandsaw Stresses on Blade Stiffness and Cutting Accuracy, Ph.D. Thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver, B. C , Canada. Tessler, A. and Hughes, T. J. R., 1985, Three-node Mindlin plate element with transverse shear, Computer Methods in Applied Mechanics and Engineering, Vol. 50 71-101. Tian, J., 1998, Self-excited Vibrations of Rotating Discs and Shafts With Applications to Sawing and Milling, Ph. D. thesis, University of British Columbia, Vancouver, B. C , Canada. Tian, J. and Hutton, S. G., 1999, Self-excited vibration in flexible rotating discs subjected to various transverse interactive forces: a general approach, Journal of Applied Mechanics, Trans. ASME, 66(3), 800-805. Ulsoy, A. G., Mote, C. D., Jr. and Szymani, R., 1978a, Principal Developments in Bandsaw Vibration and Stability Research, Holz alsRoh -und Werkstoff, 36(1978), 273-280. Ulsoy, A. G. and Mote, C. D., Jr., 1978b, Band saw vibration and stability, Shock and Vibration Digest, 10(11), 3-15. Bibliography 178 Ulsoy, A. G. and Mote, C. D., Jr., 1982, Vibration of wide band saw blades, Journal of Engineering for Industry, Transactions ofASME, 104(1), 71-78. Ulsoy, A. G., 1986, Coupling between spans in the vibration of axially moving materials, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Trans. ofASME, 108(Oct.), 207-212. Wang, K. W. and Mote, C. D., Jr., 1986, Vibration coupling analysis of band/wheel mechanical systems, Journal of Sound and Vibration, 109(2), 237-258. Wang, K. W. and Mote, C. D., Jr., 1987, Band/wheel system vibration under impulsive boundary excitation, Journal of Sound and Vibration, 115(2), 203-216. Wang, J. and Mote, C. D. Jr., 1994a, Analysis of roller-induced residual stresses in bandsaw plates, Journal of Sound and Vibration, 175(5), 647-659. Wang, J. and Mote, C. D. Jr., 1994b, The effect of roll-tensioning on bandsaw plate vibration and stability, Journal of Sound and Vibration, 175(5), 677-692. Wang, P. K. C. and Wei, J., 1987, Vibration in a moving flexible robotic arm, Journal of Sound and Vibration, 116, 149-160. Wickert, J. A. and Mote, C. D., Jr., 1988, Current research on the vibration and stability of axially moving materials, Shock and Vibration Digest, 20(4), 3-13 Wu, W. Z. and Mote, C. D., Jr., 1986, Parametric excitation of an axially moving band by periodic edge loading, Journal of Sound and Vibration, 110(1), 27-39. Yang, T. Y., 1986, Finite Element Structural Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 439-459. Yuh, J. and Young, T., 1991, dynamic modeling of an axially moving beam in rotation: simulation and experiments, Journal of Dynamics, Measurement and Control, ASME, 113, 34-40. Zhan, J. J., 1990, Dynamics of Bandsaws, M.A.Sc. Thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver, B.C. , Canada. Zienkiewicz, O. C. and Taylor, R. L., 1989, The Finite Element Method, 4 t h edition, McGraw-Hill Book Company, U. K. Appendix A Structure and Parameters of the Bandsaw Table A . l Dimensions and Main Parameters of the Bandmill Symbol Parameter Description Lw 2,410 mm center distance between the wheel axes (Figure A . l (b)) f w 762 mm radius of wheel Bw 228 mm width of wheel rc 64,000 mm radius of the wheel crown (Figure A . l (b)) Bc 2BJ3 distance between the wheel crown center and the wheel rim B, 222 mm distance between the tilting center and the wheel center Be 230 mm width of guide He 55 mm height of guide mw 800 kg mass of wheel Ks> 1.05 MN/m top wheel support stiffness Table A.2 Geometry of the Bandsaw Blades Symbol Smooth Band Blade I Blade II Description Code SI T l T2 mill 5' Cancar 5' Cancar 5' Cetec bandmill used Lh 9,576 mm 9,957 mm 10,268 mm total length L 762 mm 762 mm 502 mm length of the cutting span Bo 256.4 mm 200.0 mm width (back to tooth tip) B 260.4 mm 240.2 mm 182.6 mm width (back to gullet) h 1.45 mm 1.24 mm 1.22 mm thickness P 44.45 mm 44.45 mm tooth pitch E 206.8 GPa 206.8 GPa 206.8 GPa Young's modulus G 80Gpa 80Gpa 80Gpa shear modulus V 0.3 0.3 0.3 Poisson's ratio P 7,800 kg/ m^ 7,800 kg/ m3 7,800 kg/ m" mass density Usage Modal tests Cutting tests Cutting tests 3 blades for different tests 179 Appendix A Structure and Parameters of the Bandsaw i Setworks Square-section Beam (a) Front view without the band and two wheels Figure A . l (1) Structure of the bandmill Appendix A Structure and Parameters of the Bandsaw Blade Crowned Wheel Tilting H4H O I Vertical Column Two Beams Guide Rails Mill Base k o ' ^ o - o o - o ' o ' o . o 0 , 0 . 1 (b) Side view without the band and the bottom wheel Figure A . l (2) Structure of the bandmill Appendix B The System Matrices and the Generalized Force Vector in the Equations of Motion of a Moving Plate The elements of the mass matrix, the gyroscopic matrix, the stiffness matrix and the generalized force vector in (2.12) are given by miJ=phj^ <J>, O y dxdy (B.l) 8ij=2phcjLo Q,0JtXdxdy (B.2) J .L rB + 2(l-v)<J> O 1 + i V O . <E> + N (<£ O +<E> <D ) (B.3) ^ ' i,xy j,xy * x t,x j,x xy\ i,x j,y i,y j,x' v ' + N y ® i , y y ® j , y y y d x d y & ( t ) =f r d x d y (b-4) where /, j =1, 2, ..., m X n and <D, is given by (2.8). 182 Appendix C Shape Functions and Explicit Expressions for MDKT Element Matrices I. Shape Functions The displacement shape functions (Yang, 1986) and the rotation shape functions (Batoz, 1980) are expressed in the following form Vwl =Li+ L2(L2 + L 3) - LX{L2 + L2) (C.l) Ww2 = L2(c3L2-c2L3) + -^(c3-c2)LsL2L3 (C.2) Ww3 = L2 (b2L3 -b 3L 2) + ^ (b2-b3)L,L2L3 (C.3) ^ x ^ - ^ U L 3 - ^ - 2 U L 2 (C.4) b2 +c2 b3+ c3 2h2-c2 2b2-c2 Wx2 = Lx(L, -1) + ^ - 4 - L X L 2 + ^ - 4 - L , L 3 (C.5) b3 + c3 b2 + c2 Wx3 =4^rL,L2 (C.6) x3 b2 + cl ' 2 /? 2+c 2 1 3 T y l " b2+c\ b2+c22LlL3 ( C 7 ) Vy2=Vx3 ' (C.8) 2c 2 - /? 2 2c2-h2 wy3=Ll m - 1 ) + ^ - 4 ^ 2 + # - r - ( c . 9 ) £>3 + c3 e>2 + c2 where the area coordinates are given by Li=^-(ai+bix + ciy) ( C I O ) ai=xjyk-xkyj (C.ll) (C.12) c. = - x ; (C.l3) and L ! + L 2 + L 3 = l (C.14) The expressions for other two sets of shape functions for nodes 2 and 3 can be obtained by cyclically permutating the three node number, i.e. / —> j —> A: —> i and 1 —> 2 —>3 —>1. 183 Appendix C Shape Functions and Explicit Expressions for MDKT Element Matrices II. Explicit Expressions for Matrices The area of element is given by A = (x2y3-X3y2+X3yi-xiy3+xiy2-x2yiy2 (CAS) and A2 = 2A (C16) The following variables can be obtained by cyclically permutating / = 1, 2, 3; j = 2, 3, 1 and k=3, 1,2. bi=yj-yk, Ci=xk-Xj l^bt+cf bi+3=3bici/(4li) ci+3=(2b?-c?)l(4lh ci+6=(2c?-b?)l(4h) ai=Xjyk-xkyj, a,-+.j=c,-//,-, ai+6=bj/li bij=bibj , Cij=CiCj, eij=biCj r3i-i=(ck-cj)l2 , r3i=(brbk)/2 r3i-i,q= bqr3i.i , r3iig= bqr3i (q=2, 3 only) (C.17) (C.18) (C.19) (C.20) (C.21) (C.22) (C.23) (C.24) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -c2 0 0 0 0 0 0 -3 -2c3 2^ 3 h 0 0 0 -4 -3r2 -3r 3 2 2 h h -3 2c2 -2b2 0 0 0 3 c2 -K 2 c3 -h -2 c3 -b3 0 0 0 4 c 3+r 2 r3-b3 -2 c, + c3-r5 -b3-brr6 -2 -R9 4 r2-c2 b2+r3 -2 -rs -h -2 -Cj-c 2-r 8 b{ + b2-r9 2 b2 0 0 0 -2 -c2 b2 (C.25) Appendix C Shape Functions and Explicit Expressions for MDKT Element Matrices 185 N,. 0 -6b2-4b3 -6b3-4b2 6b2 + 4b3 8(b2+b3) 4b2 + 6b3 A2 2(e13 A2) + r23 s23 A2-2el3 r23 s25 2el2 + A2 r22 0 -2bn+r33 2bn+r32 2b -r z.u{3 i33 2(bl3-bt2-r32 r33) -2b -r z.un i32 0 6b2 + 2b3 2b2 -2b3-6b2 -4(b2+b3) -2b2 0 -2e23 + r53 h i 2b23+A2- r5J 2(<?23 + e2l — r52 — r53) -hi 0 2b23+r63 h i -2b23-r63 -2(b2,+b23-r62-r63) -hi 0 2b3 6b3 + 2b2 -2b3 -4(b2+b3) -6b3-2b2 0 2e32 + r82 _ / 83 -2(e3j + e32 — r82 + r83) A2-2(?32-r82 0 r93 -2&32+r92 -r93 2(b3i+b32-r92-r93) 2h -r z.u32 i92 where S23=-2(e12+A2)+r22 and S25=2(ei2-ei3+A2-r23+r22)-(C.26) N„ = 0 -6a6 6a5 6a6 6(a6-a5) -6a5 1 -3 + 4c6 -3 + 4c5 2-4c6 4(l-c5-c6) 2-4c5 0 *K 4b5 -4b6 -4(b5+b6) -4b5 0 6a6 0 -6a6 -6(a4+a6) 0 0 -l + 4c6 0 2-4c6 4(c4-c6) 0 0 *h 0 -4b6 4(*4 A ) 0 0 0 -6a5 0 6(a4 +a5) 6a5 0 0 - l + 4c5 0 4(c4-c5) 2-4c5 0 0 4^5 0 4 A A ) -4b5 (C.27) 0 6a9 -6a 8 -6a9 6(a8-a9) 6as 0 *K 4^5 -4(b5+b6) -4b5 1 -3 + 4c9 -3 + 4c8 2-4cg 4(l-tyc9) 2-4c% 0 -6a 9 0 6ag 6(a1 +a9) 0 0 0 4tf>4 A ) 0 0 - l + 4c9 0 2-4c9 4(c7-c9) 0 0 0 6a8 0 -6(a 7 +a8) -6a 0 0 4^5 0 4 A A ) -4b5 0 0 - l + 4c8 0 4(c7-c8) 2-4c (C.28) Appendix C Shape Functions and Explicit Expressions for MDKT Element Matrices 90 30 30 15 15/2 15 30 15 15/2 9 3 3 A 30 15/2 15 3 3 9 90 15 9 3 6 3/2 1 15/2 3 3 3/2 1 3/2 15 3 9 1 3/2 6 (C.29) M A 8401 840 280 280 140 70 140 84 28 28 84 280 140 70 84 28 28 56 14 28/3 14 280 70 140 28 28 84 14 28/3 14 56 140 84 28 56 14 28/3 40 8 4 4 70 28 28 14 28/3 14 8 4 4 8 140 28 84 28/3 14 56 4 4 8 40 84 56 14 40 8 4 30 5 2 3/2 28 14 28/3 8 4 4 5 2 3/2 2 28 28/3 14 4 4 8 2 3/2 2 5 84 14 56 4 8 40 3/2 2 5 30 210 70 70 35 35/2 35 70 35 35/2 21 7 7 70 35/2 35 7 7 21 35 21 7 14 7/2 7/3 35/2 7 7 7/2 7/3 7/2 35 7 21 7/3 7/2 14 21 14 7/2 10 2 1 7 7/2 7/3 2 1 1 7 7/3 7/2 1 1 2 21 7/2 14 1 2 10 (C.31) 

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