FLUTTER INSTABILITY SPEED OF GUIDED SPLINED DISKS, WITH APPLICATIONS TO SAWING by Ahmad Mohammadpanah M.Sc, Sharif University of Technology, Iran, 2004 M.A.Sc, The University of British Columbia, Canada, 2012 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) February 2015 Β© Ahmad Mohammadpanah, 2015 ii Abstract In this thesis the vibration characteristics of guided splined saws are studied, both analytically and experimentally. Significant insights into the complex dynamic behavior of guided splined saws are presented by analytical investigation of the dynamic behaviour of spinning splined disks and then by conducting idling and cutting experimental tests of guided splined saws. Cutting tests are conducted at different speeds, at critical, supercritical, and post flutter speeds of a guided splined saw. The cutting results are compared to determine the stable operation speeds for guided splined saws. For the analytical studies, the governing linear equations are derived for the transverse motion of a constant speed spinning splined disk. The disk is subjected to lateral constraints and loads. Rigid body translational and tilting degrees of freedom are included in the analysis of total motion of the spinning disk. Also considered in the analyses are applied conservative in-plane edge loads at the outer and inner boundaries. The numerical solution of these equations is used to investigate the effect of the loads and constraints on the natural frequencies, critical speeds, and stability of the spinning disk. The sensitivity of the eigenvalues of the splined spinning disk to the in-plane edge loads is analyzed by taking the derivative of the spinning diskβs eigenvalues with respect to the applied loads. This analysis contains an evaluation of the energy transfer from the applied loads to the disk vibrations and is used to examine the role of critical system components in the development of instability. Experimental results are presented that support the validity of the analysis. The experimental results indicate that flutter instability occurs at speeds when a backward travelling wave of a mode meets a reflected wave of a different mode. The maximum stable operating speed of the rotating splined disk is defined as the initiation of flutter. Flutter instability speeds of splined saws of various sizes were computed and verified experimentally. Then flutter speed charts of splined saws were developed which provides primary practical guide lines for sawmills to choose optimum blade diameter, eye size, blade thickness, and a stable rotation speed. iii Preface Experimental tests results presented in chapter 4 and 5 are based on the tests conducted at FPInnovations, (Vancouver, B.C., Lumber Manufacturing Laboratory). I was responsible for conducting the idling and cutting tests of guided splined saws under the supervision of Professor Stanley G. Hutton. iv Table of Contents Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iii Table of Contents ................................................................................................................... iv List of Tables ......................................................................................................................... vii List of Figures ....................................................................................................................... viii List of Symbols ....................................................................................................................... xi Acknowledgements ............................................................................................................... xii Dedication ............................................................................................................................. xiii Chapter 1: Introduction ......................................................................................................... 1 1.1 Background .................................................................................................................... 1 1.2 Mathematical Modeling of the System .......................................................................... 3 1.3 Variation of Eigenvalues as a Function of Rotation Speed ............................................ 5 1.4 Literature Review ......................................................................................................... 10 1.5 Objective and Scope ..................................................................................................... 19 Chapter 2: Modeling of a Spinning Splined Guided Disk Subjected to In-Plane Edge Loads Applied at the Outer and Inner Boundaries ........................................................... 22 2.1 Introduction .................................................................................................................. 22 2.2 Previous Work on the Effect of Edge Loads on Clamped Disks ................................. 22 2.3 Equations of Motion for a Spinning Splined Guided Disk Subjected to In-Plane Outer and Inner Edge Loads ......................................................................................................... 26 2.4 Sensitivity Analysis of the Eigenvalues of a Spinning Disk to In-plane Edge Loads . 28 2.4.1 Effect of Radial Load ........................................................................................... 29 v 2.4.2 Effect of Tangential Load ..................................................................................... 32 2.5 Numerical Results ........................................................................................................ 33 2.5.1 Effect of In-plane Edge Loads on the Eigenvalues .............................................. 33 2.5.1.1 Special Case (a Rigid Splined Disk) ............................................................. 32 2.5.1.2 General Case (A Guided Splined Flexible Disk) .......................................... 37 2.5.2 Transverse Response of a Disk to In-plane Edge Loads ...................................... 42 2.6 Summary ...................................................................................................................... 44 Chapter 3: Instability Mechanism of a Spinning Splined Guided Disk Subjected to In-plane Edge Loads .................................................................................................................. 46 3.1 Introduction .................................................................................................................. 46 3.2 Work Done by the Edge Loads (Formulation) ............................................................. 46 3.3 Numerical Results of Energy Induced in the Disk by Edge Loads .............................. 49 3.4 Summary ...................................................................................................................... 50 Chapter 4: Experimental Investigations of Idling and Cutting Characteristics of a Splined Guided Saw .............................................................................................................. 52 4.1 Introduction .................................................................................................................. 52 4.2 Experimental Tests Results .......................................................................................... 53 4.2.1 Idling Tests ........................................................................................................... 53 4.2.2 Cutting Tests ......................................................................................................... 56 4.4 Summary ...................................................................................................................... 59 Chapter 5: Flutter Speeds Chart for Guided Splined-Arbor Saws .................................. 60 5.1 Introduction .................................................................................................................. 60 5.2 Experimental Measurements of Idling Response of Guided Splined-Arbor Saws ...... 62 vi 5.3 Numerical Computation of Eigenvalues of Guided Splined-Arbor Saws.................... 66 5.4 Verification of Numerical Results ............................................................................... 68 5.5 The Effect of Geometrical Properties of the System on the Start of Flutter Instability Speeds ................................................................................................................................ 69 5.6 Flutter Instability Speeds Charts .................................................................................. 71 5.7 Conclusion .................................................................................................................... 74 Chapter 6: Conclusions ........................................................................................................ 75 6.1 Summary and Conclusion ............................................................................................ 75 6.2 Suggestions for Further Research ................................................................................ 78 References .............................................................................................................................. 79 Appendices ............................................................................................................................. 84 Appendix A: Stresses Fields in a Disk due to Centrifugal Acceleration .......................... 84 Appendix B: Mathematical Equations for Rigid Body Tilting and Translational Motions .............................................................................................................................. 85 Appendix C: Solution of the Equations of Motion ........................................................... 88 Appendix D: Stresses Fields in a Disk due to In-Plane Edge Loads ................................ 90 Appendix E: The Governing Equations of a Rigid Disk, Subjected to In-plane Edge Loads .................................................................................................................................. 96 Appendix F: Gullet Feed Index and Feed Speed .............................................................. 98 Appendix G: Derivative of Eigenvalues of Spinning Disk with Respect to In-plane Edge Loads .................................................................................................................................. 99 Appendix H: Idling Response of Different Blade Sizes ................................................. 108 Appendix I: Mathematical Calculation of Non-Dimensional Equation of Motion ........ 112 vii List of Tables Table 4.1 Properties of the Blade under Investigation ......................................................... 53 Table 4.2 Cutting Tests Rotation Speed, and Feed Speed ................................................... 57 Table 5.1 List of Blades under Experimental Investigations ............................................... 63 Table 5.2 Properties of the disk under Investigation ............................................................ 66 Table 5.3 Start of Flutter Instability Speeds for the Blades under Investigations ................ 68 Table 5.4 Flutter Speeds for Guided Splined-Arbor Saws ................................................... 72 viii List of Figures Figure 1.1 Clamped Saw vs. Guided Splined Circular Saw .................................................... 2 Figure 1.2 Idealizing the Blade as a Spinning Flat, Thin Disk ................................................ 3 Figure 1.3 Variation of Imaginary and Real Parts of Eigenvalues, (-- dash lines) A Free Clamped Disk, (- solid lines) Disk is Constrained by a Lateral Spring at the Outer radius ..... 6 Figure 1.4 Variation of Imaginary and Real Parts of Eigenvalues, (-- dash lines) Free Splined Disk, (- solid lines) Splined Guided Disk .................................................................... 8 Figure 1.5 Illustrations of Divergence and Flutter Type Instabilities ...................................... 9 Figure 2.1 Schematic of a Clamped Disk, Subjected to a Conservative In-Plane Compressive Edge Load ......................................................................................................... 23 Figure 2.2 Variations of Eigenvalues as a Function of Rotation Speed, (-- dashed lines) Free Spinning Clamped Disk, (- solid lines) Clamped Disk Subjected to the Edge Load F=3.5 D/b ................................................................................................................................................. 25 Figure 2.3 Schematic of a Splined Disk Subjected to In-Plane Edge Loads, and Their Inner Interaction Loads .................................................................................................................... 26 Figure 2.4 Variation of Natural Frequencies as a Function of Rotation Speed, (-- blue dash lines) Free Spinning Splined Rigid Disk, (- blue solid lines) The Rigid Body Motions are Coupled by a Lateral Spring, (-- red dash lines) Free Spinning Splined Rigid Disk, Subjected to a Radial Load, (- red solid lines) Spinning Splined Rigid Disk, Constrained by a Lateral Spring, Subjected to a Radial Load ......................................................................................... 36 Figure 2.5 Variation of eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Splined Disk, (- red solid lines) Free Spinning Splined Disk, Subjected to Edge Loads.............................................................................................................................. 38 ix Figure 2.6 Variation of Imaginary Part of Eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Guided Splined Disk, (- red solid lines) Free Spinning Guided Splined Disk, Subjected to Edge Loads ..................................................................... 40 Figure 2.7 Variation of Real Part of Eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Guided Splined Disk, (- red solid lines) Free Spinning Guided Splined Disk, Subjected to Edge Loads .................................................................................. 41 Figure 2.8 Transverse Vibration of the Disk Computed for Outer Radius at πΌ = 45Β°, Subjected to (a) A Concentrated Radial In-plane Edge Load (b) A Concentrated Tangential In-plane Edge Load (c) Concentrated Radial and tangential In-plane Edge Load, (blue graph) β¦ = 42Hz, (green graph) β¦ = 50Hz, (red graph) β¦ = 53Hz ..................................... 43 Figure 3.1 Rate of Work Done by the Edge Loads, at Rotating Speed (a) 42Hz, (b) 50Hz, (c) 53Hz .................................................................................................................................. 49 Figure 3.2 Rate of Work Done by the Tangential and Radial Edge Loads, Disk Running at a Flutter Instability Speed (β¦ = 53π»π§) ..................................................................................... 50 Figure 4.1 Schematic of the Experimental Setup ................................................................... 53 Figure 4.2 Variation of Excited Frequencies with Rotation Speed for a Saw Blade with no Constraint ................................................................................................................................ 55 Figure 4.3 Variations of Excited Frequencies as a Function of Rotation Speed, Guided Spline Saw .............................................................................................................................. 55 Figure 4.4 Schematic of Cutting Test Setup .......................................................................... 56 Figure 4.5 Cut Profile for Test 1, Cutting at 3200rpm (Critical Speed) ................................ 57 Figure 4.6 Cut Profile for Test 2, Cutting at 3600rpm (A Super Critical Speed) .................. 58 Figure 4.7 Cut Profile for Test 3, Cutting at 4000rpm (A Flutter Speed) .............................. 58 x Figure 5.1 Experimental Idling Tests Setup........................................................................... 62 Figure 5.2 Deflection of Blade 28-8-0.115, during Idling Run-up from 0-3600rpm, Measured by Displacement Probe .......................................................................................... 64 Figure 5.3 Variation of Excited Frequencies of the Disk as a Function of Rotation Speed .. 65 Figure 5.4 Imaginary and Real Parts of Eigenvalues of the Guided Splined Disk ................ 67 Figure 5.5 Non-Dimensionalized Flutter Speeds for a=3in (Eye#3) and a=4in (Eye#4) ...... 70 Figure 5.6 Flutter Instability Speeds Charts, Eye#3 .............................................................. 73 Figure 5.7 Flutter Instability Speeds Charts, Eye#4 .............................................................. 73 xi List of Symbols π Inner radius of the disk π Outer radius of the disk π· Disk rigidity (πΈβ3 12(1 β π2β )) πΈ Youngβs Modulus β Disk thickness π Stiffness of the spring π Number of nodal diameters π Number of nodal circles (π, π, π§) Space-fixed polar coordinate system πππ Eigenvalues for the transverse displacement of the disk π·ππ Eigenfunction π ππ Mode shape in the radial direction for the disk deflection πππ, πΆππ Equilibrium solutions for the amplitude of the π ππ and πππ waves π‘ Time π€ Transverse displacement πΏππ Kronecker delta π Poissonβs ratio π Mass density πππ , πππ Radial and hoop stress due to rotation ππ , ππ Radial and hoop stress due to in-plane edge loads β¦ Rotation speed (rad/sec) xii Acknowledgements I am deeply thankful to my supervisor, Professor Stanley Hutton. I have appreciated his support, guidance and advice during my research. I also would like to thank Professor Mohamed Gadala who has provided me with significant help during my studies. I am thankful to Professor Gary Schajer, Dr. Bruce Lehmann, and Dr. Srikantha Phani for all the great discussions on the subject we have had. I would also like to record my gratitude to Mr. John White, from FPInnovations, for his assistance during all the experimental tests. I am deeply indebted to my wife. Her unlimited love, patience and support have made this work possible. xiii Dedication To My Lovely Wife Chapter 1. Introduction 1 Chapter 1: Introduction 1.1. Background Spinning disks are essential parts of many machines, and find applications in grinding wheels, turbine rotors, brake systems, fans, computer disks memory units, wafer slicing cutters, and circular saw blades. In the wood processing industry circular saws are widely used in the breakdown of logs into boards of varying dimensions. The saws are primarily of two types: a) collared saws; and b) splined arbor saws. It has been found in practice that saws which are not constrained laterally at the inner radius interface, known as βGuided Splined Sawsβ, provide superior cutting performance to βclamped Sawsβ, where the saw is clamped to the arbor (Figure 1). The maximum cutting speed for a collared saw coincides with the lowest critical speed of the rotating disk whereas the splined saw is able to operate at speeds in excess of the lowest critical speed. In splined guided saws, the saw fits loosely on a splined arbor and this arbor provides the driving force to the blade. The clearance between the splined arbour and the matching splined of the disk is in order of 0.25 β 0.5ππ.The lateral location of the blade is determined by the position of guide pads which are supported independently of the blade by two guide arms that are fixed to the saw frame. The clearance between the blade and the guides is in order of 0.05 β0.1ππ. In practice this gap is filled with pressurized water. Chapter 1. Introduction 2 Figure 1.1 Clamped Saw vs. Guided Splined Circular Saw The number of variables that affect cutting performance at high speeds is large and their interaction is complex. Such factors include: blade speed, blade geometry, tooth geometry, blade flatness, depth of cut, wood characteristics, guide size and location, and temperature distribution in the blade. Blade speed is one of the primary factors. At high blade speeds vibrational instability can exist in the saw that leads to poor cutting accuracy. Saw blade vibration directly contributes to production problems such as poor cutting accuracy, and excessive raw material waste. The focus of the current study is on the vibration characteristics of guided splined disks with application to wood sawing. Present understanding is that the critical speed defines the maximum speed at which a circular saw can be run. Experience at sawmills shows that the maximum operation speed for a clamped saw is the lowest critical speed whereas the splined saw is able to operate at speeds in excess of the lowest critical speed. It opens the question of βwhat is the maximum stable operation speed for a guided splined saw?β Chapter 1. Introduction 3 1.2.Mathematical Modeling of the System The first step in investigating the vibration characteristics of the system is to set up a mathematical model that captures the essential physics of the problem. In this regard, the blade is idealized as a flat, thin, circular plate. If there is a lateral constraint, it is modeled as a linear spring. Figure 1.2 shows the idealized representation of a splined guided disk. The transverse displacement of the disk is π€, measured with respect to stationary coordinates (π, π). The rigid body modes of the disk are defined by Z, ππ₯and ππ¦. The lateral constraints (guides in splined saws) are represented by several linear springs of stiffness ππ§ located at (ππ, ππ). Figure 1.2 Idealizing the Blade as a Spinning Flat, Thin Disk The governing equation of motion for a disk, in terms of the transverse displacement π€, with respect to a stationary coordinates π€(π, π), without considering any rigid body motion, is [1]: πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ ββπ(πππππ€,π),π ββπ2ππππ€,ππ = 0 (1.1) Chapter 1. Introduction 4 β, Ο,β¦, D, E and π are thickness, mass density, rotating speed, flexural rigidity, Youngβs modulus and Poissonβs ratio. πππ and πππ are axisymmetric in-plane stresses due to centrifugal acceleration. The closed form solution of πππ and πππ are presented in Appendix A. Consideration of rigid body motions, a space fixed linear spring ππ§ and a space fixed lateral force ππ located at (ππ, ππ) leads to the equation (1.2) [2]: πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ ββπ(πππππ€,π),π ββπ2ππππ€,ππ + πβ?Μ? +πβ(π?Μ?π₯ sin π β π?Μ?π¦ cos π) + πβ(2β¦ π cos π ?Μ?π₯ + 2β¦ π sin π ?Μ?π¦) = βππ§π(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) +πππ (1.2) The equations governing rigid body tilting are [2]: πβπΌ(?Μ?π₯ + 2β¦?Μ?π¦) + β« β« πβ sin π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ = ππ ππ sin ππ + ππππ sin ππ (1.3) πβπΌ(?Μ?π¦ β 2β¦?Μ?π₯) + β« β« πβ cos π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ = βππ ππ cos ππ β ππππ cos ππ (1.4) Where πβπΌ is the moment of inertia for the disk ( πΌπ₯ = πΌπ¦ =π4πβ(π4 β π4) = πβπΌ). The term ππ = ππ§(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) is the spring force. And, the rigid body translational equation in Z is: π?Μ? + β« β« πβ(π€,π‘π‘)ππ2π0πππππ = βππ§(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) + ππ (1.5) Whereas π = πβπ(π2 β π2) is the total mass of the disk. For mathematical details see Appendix B. Chapter 1. Introduction 5 Equations (1.2-1.5) govern the dynamics motion of a splined spinning disk. Solution of the equations can be obtained by application of the Galerkin method. In this solution method the eigenfunction of the stationary disk problem in the polar coordinate system is used as the approximation functions for the Galerkin method. For mathematical details of the solution see Appendix C. 1.3 Variation of Eigenvalues as a Function of Rotation Speed The variation of eigenvalues as function of rotation speed are computed for a disk with physical properties of inner radius π = 75ππ (3ππ), outer radius π = 215ππ(8.5ππ), thickness β = 1.5ππ(0.060ππ), E = 2.03 Γ 1011pa, Ο = 7800 kg m3β , and Ο = 0.29. In this thesis, the term βClamped Diskβ is defined as a disk with clamped inner radius, and βSplined Diskβ as a disk with free inner radius. The term βGuided Splined Diskβ refers to a splined disk which is laterally constrained over a certain area by certain number of lateral linear springs of equal stiffness. Figure 1.3 shows the imaginary and real parts of eigenvalues as a function of rotation speed for a free clamped disk, and a clamped disk subjected to a lateral spring of π = 105ππ at outer radius. Chapter 1. Introduction 6 Figure 1.3 Variation of Imaginary and Real Parts of Eigenvalues, (--dash lines) A Free Clamped Disk, (- solid lines)Disk is Constrained by a Lateral Spring at the Outer radius For each mode there are forward and backward waves travelling around the disk. Forward travelling waves travel in the direction of rotation, and backward waves travel in the opposite direction. The natural frequencies of the forward and backward travelling waves of each Chapter 1. Introduction 7 mode are the same when the disk is stationary. Once the disk spins the natural frequencies of forward travelling waves increase and the natural frequencies of backward travelling waves decrease. Natural frequencies of the modes having more than one nodal diameter decrease until a speed at which the measured natural frequency is zero. This speed is called a βCritical Speedβ. At this speed a constant force can initiate resonance in the disk. Figure 1.4 shows the imaginary and real parts of eigenvalues as a function of rotation speed for a splined disk. The dash line indicates the eigenvalues for a free splined disk, and solid line indicates the eigenvalues of a guided splined disk, constrained by 16 lateral springs each of stiffness π = 105ππ distributed over a 100Γ100 mm square area which represents the guides. Chapter 1. Introduction 8 Figure 1.4 Variation of Imaginary and Real Parts of Eigenvalues, (-- dash lines) Free Splined Disk, (- solid lines) Splined Guided Disk It is noted that with lateral constraints, the movement of the travelling waves become restricted. With the introduction of several springs the curve veering becomes very strong, and the natural frequency curves becomes smoother. Backward and forward travelling wave frequency characteristics are almost completely obliterated. Chapter 1. Introduction 9 The introduction of springs leads to instability. The term βSingle Mode Resonanceβ, also known as βDivergence Instabilityβ is defined when the imaginary part of eigenvalues is zero and the real part positive. The term βMerged-mode Resonanceβ, also known as βFlutter Instabilityβ refers to a case when the imaginary and the real part of eigenvalues are positive. The instability is induced by the presence of springs, and coupling of flexible disk modes. Figure 1.5 illustrates divergence and flutter type instabilities. Figure 1.5 Illustrations of Divergence and Flutter Type Instabilities (a) Divergence Instability (a) Flutter Instability In practice, the maximum operation speed for cutting in a clamped saw coincides with the lowest critical speed of the rotating disk whereas the splined saw is able to operate at speeds in Chapter 1. Introduction 10 excess of the lowest critical speed. The questions arise as to βWhy conducting cutting at supercritical speed by a splined saw is possible?β and βWhat is the maximum stable operation speed for a guided splined saw?β 1.4 Literature Review There exists extensive literature concerning the idling and forced vibration of a spinning disk clamped at the inner radius. The number of studies relating to the dynamics behaviour of splined disks is relatively small; and a large portion of the existing literature relating to the dynamic behaviour of splined disk is concerned with the idling vibration characteristics of spinning disks. Hutton, Chonan, and Lehmann [1] studied dynamic response characteristics of elastically constrained clamped spinning disks, subjected to excitation produced by fixed point loads. They studied the effect of spring stiffness on the frequency characteristics of the spinning clamped disk. In this study, the frequency-speed plots for different values of spring stiffness were presented. They showed the existence of invariant points on the frequency-speed graphs corresponding to modes having a node at the location of a point guide which was represented by a linear spring. They concluded that the introduction of one or two guides did not significantly change the critical speed characteristics of a spinning clamped disk. It was shown that addition of a guide close to the point of force application will reduce the resulting deflection of the disk at the load point. Chen and Body [2] investigated the effect of rigid body tilting on the stability and natural frequency characteristics of a head disk, subjected to a rotating load system. They concluded that to an observer attaching to the tilting frame, only the natural frequencies of one nodal diameter Chapter 1. Introduction 11 modes increase due to the inertial coupling between the rigid-body tilting and the bending modes, while the natural frequencies of the other modes are not affected. They proved that for a spinning disk with rigid-body tilting freedom the centrifugal force is very important in stabilizing the system. In this study they showed that the system of equations of motion is gyroscopic for a spinning disk with rigid-body tilting freedom. Chen and Hsu [3] studied the response of a spinning disk under space-fixed couples analytically. They presented vibration and steady state response of the disk and investigated the effects of the rotation speed and external damping. In their investigation the effect of rotating and space fixed damping was considered. They found that before critical speed, both the rotating and space fixed damping suppress the transient response of a spinning disk. But, at super critical speed, the rotating damping suppresses the transient vibration while the space fixed damping destabilizes the disk. Chen and Wong [4] investigated the effect of evenly space-fixed springs on the divergence instability of a clamped disk with translational degrees of freedom. They proved that divergence instability of the coupled system is induced only when two times the number of nodal diameters of a mode is equal to a multiple of the number of stationary springs. Yang [5] investigated the effect of rigid body degrees of freedom on the stability of splined guided disks. He concluded that, stable operation of the splined disk beyond critical speed is possible. He used modal functions of a stationary disk to compute the elastic displacement of the spinning disk. He proved that the divergence speed is independent of the spring stiffness, however the stable speed range increases with the spring stiffness because of the coupling effects from rigid-body degree of freedom. Chapter 1. Introduction 12 Mote [6] investigated the stability of a collar disk (which slides freely along the axis of symmetry), subjected to a concentrated load moving at uniform speed. In his study he considered both a harmonic transverse load and a load proportional to transverse displacement and velocity. He found that the harmonic loading case leads to a classical critical speed analysis. In his study the proportional loading case represents the transverse guides. He concluded that the number, configuration and mechanical properties of the guides determine the transverse stability of the system. Price [7] studied the effect of a rigid body translational degree of freedom on the dynamic response of rotating disks which are free to translate along the axis of symmetry, with the inner edge constrained to be perpendicular to that axis. He concluded that at super-critical speed interaction of forward and backward frequency loci results in flutter instability. He also concluded that multiple points guides or a sector guide increase the instability range, relative to a single point guide. Khorasany and Hutton [8] investigated the stability characteristics of a spinning disk having rigid body translational degrees of freedom, constrained with only one space fixed spring. They concluded that the interaction of a forward or backward travelling wave with the rigid body translational mode does not induce flutter type instability. They showed that if the stiffness exceeds a certain value, the flutter instability will be induced and never disappears. Tobias and Arnold [9] experimentally investigated the vibration characteristics of imperfect spinning clamped disks. Raman and Mote [10] used experimental results to investigate the vibration response of an imperfect disk. Kang and Raman [11] measured the transverse vibration of a disk coupled to surrounding fluids in an air-filled enclosure. Dβ Angelo, C., Mote, Chapter 1. Introduction 13 C.D. [12] investigated experimentally the aerodynamically excited supercritical disk vibration. Thomas, O. et al [13] investigated experimentally the non-linear forced vibration of circular plates due to large displacements, with the excitation frequency close to the natural frequency of an asymmetric mode. Jana and Raman [14] conducted experiments over a wide range of rotation speeds in the post-flutter region of a flexible disk rotating in an unbounded fluid. Based on their experimental investigations, the existence of a primary instability of a reflected travelling wave of the disk, followed by a secondary instability is confirmed. They observed frequency lock-ins over certain rotation speed ranges. They showed that a non-linear Von-Karman plate theory, coupled with a linear aerodynamic load with the form of a rotating damping is capable of capturing the primary instability of the system. Khorasany and Hutton [15] conducted experimental tests to investigate the effect of large deformations on the frequency characteristics of spinning clamped disks. They present experimental results for vibration behavior of three uniform, thin, disks of different thickness, subjected to a constant lateral force of different amplitude. In particular, they presented the amplitude-speed and frequency-speed characteristic of the disks. Based on the experimental observations, they found that the application of a space fixed lateral force caused separation of the backward and forward travelling waves when the disk is stationary. This was expected due to the lack of symmetry and imperfection in the disks. Khorasany and Hutton [16] numerically investigated the effect of geometrical nonlinear terms on the dynamic characteristics of a spinning clamped disk. They presented an analysis of the amplitude and frequency response of the disk, subjected to geometrical non-linear Chapter 1. Introduction 14 displacements. In a case of a stationary disk, subjected to a stationary load, since the stress distribution is not symmetric, the frequencies of the forward and backward travelling waves do not coincide. In a case of spinning disk, for low levels of non-linearity stationary waves form and collapse within a small speed band. Chen and Wong [17] studied analytically the vibration and stability of a spinning disk with axial spindle displacement in contact with a number of fixed springs. They found the behaviour of divergence instability close to critical speed is different than those of a spinning disk without axial spindle displacement. They concluded divergence instability occurs in a disk with axial displacement if two times the number of nodal diameters is equal to the number of stationary lateral springs. Chen and Bogy [18] studied the effects of different load parameters, such as friction force, transverse mass, damping, and stiffness of a stationary load system in contact with the spinning disk on the stability of the system at sub- and super-critical speeds. They found that only two-mode approximation can exhibit all the features of eigenvalue changes. However, in the case of friction force, at least a four-mode approximation is required. Young et al [19] investigated vibration response of a rotating clamped disk under the constraint of an elastically fixed space oscillating unit. They showed that taking account of the stiffness between the oscillating unit and the disk may result in extra flutter-type instability between the oscillating-unit mode and the dominant reflected disk modes, and these extra unstable regions are much larger than those of the flutter-type instability between flexible modes of the disk. Chapter 1. Introduction 15 Schajer and Wang [20] identified the workpiece interaction with the saw body as an important factor influencing guided circular saw cutting stability. In another work Schajer [21] by using a geometrical model, explained βhunting behaviourβ of guided saw, where the saw blade does not remain perpendicular to the drive shaft but always shifts to one side or the other, as an important factor in analyzing the stability characteristics of guided splined saws. During my master research (M.A.Sc. at the University of British Columbia) [22] I investigated experimentally the effect of guide size and location on stability and cutting performance of a guided splined saw. A comprehensive experimental investigation of idling tests of splined saws with different guide configurations was conducted. The frequencies and amplitudes of the blade vibrations and the mean deflections of the blades were presented. An extensive cutting tests was conducted and the effect of different guide configurations on cutting accuracy was investigated. In the experimental investigations the cutting tests were conducted at different speeds, sub-critical and super-critical speeds for different guide configurations. The cutting results were compared to determine the guide configuration which results in the best cutting accuracy. It was concluded that the guided splined saw with a small pin-guide at the outer radius (blade rim) results in a better performance in compare with a conventional bigger sector guide systems. Although extensive research has been done in this area, the vibration and stability of circular saw blades under cutting conditions is however, not well understood. The papers by Chen J.S. [23-26] appear to be the first thorough investigation, concerning the effect of in-plane edge- loads on the natural frequencies of a spinning clamped disk. He concluded that the effect of the in-plane edge-load on the natural frequencies and stability of the Chapter 1. Introduction 16 rotating disk is through the transverse component of the edge load on the boundary, and not through the membrane stress field it produces inside the disk. He found that, compressive, and conservative in-plane edge-loads decreases the natural frequencies of forward and backward travelling waves, but increases the natural frequencies of the reflected waves. He proved that the compressive edge load induces βdivergence typeβ instability, and βflutter type instabilityβ. To explain the mechanism of divergence and flutter type instability induced by a compressive, conservative, in-plane edge-load (πΉπ), he compared the work done by the edge load over one cycle when the disk is stable, at divergence, and flutter instability situations as follow [23]: The displacement of a disk vibrating in a backward travelling wave at a given radial and angular position of (π, π) can be expressed as [23]: π€ = π ππ(π) cos(ππ + ππππ‘ + πΌππ) Where π ππ(π) is the deflection of the blade as a function of π. πππ is the natural frequency of a mode with π nodal diameters, and π nodal circles, and πΌππ is a constant phase. Then the velocity of a point at the edge of the disk (π = π, π = 0) can be calculated as [23]: ?Μ? = β(ππππ‘ + πβ¦)π ππ(π) sin(ππππ‘ + πΌππ) The disk is subjected to a conservative radial load of πΉπ . The transverse component of the edge load is πΉπ .ππ€ππ ; So, the work done by πΉπ over one cycle can be expressed as[23]: β12πΉπ(ππππ‘ + πβ¦)π ππ(π) ππ ππ(π) ππβ« sin 2(ππππ‘ + πΌππ)ππ‘ ππ¦πππ Chapter 1. Introduction 17 This work is zero when πππ β 0; at a divergence situation (πππ = 0) depends on πΌππ this work might be negative or positive. This can explain why instability occurs when the conservative edge-load is present and the natural frequency of the mode becomes zero. At βflutter type instabilityβ assume that the disk is vibrating in the combination of backward travelling wave π€ππ and a reflected wave π€ππ (πππ, πππ > 0 ): π€ππ = π ππ(π) cos(ππ + ππππ‘ + πΌππ) π€ππ = π ππ(π) cos(ππ + ππππ‘ + πΌππ) The work done by πΉπ .ππ€ππππ and πΉπ .ππ€ππππ over one cycle can be calculated as [23]: β12πΉπ(ππππ‘ + πβ¦)π ππ(π) ππ ππ(π) ππβ« sin((πππ βπππ)π‘ + πΌππ β πΌππ)ππ‘ ππ¦πππ In the situation that πππ β πππ, the work is zero. If πππ = πππ then the work might be negative or positive. This can explain why instability can occurs when the in-plane edge load is present, and two frequency loci merge together. In light of Chenβs work (abovementioned), in an independent study, Shen [27] predicted the stability of a clamped spinning disk, subjected to a stationary, concentrated in-plane edge load. He reached to the same conclusion on the effect of conservative, edge-loads as Chen [23]. He found that conservative edge-loads affect the stability through transverse component of the loads on the boundary. Young [28] extended the work by Chen [23-26] and Shen [27] by considering the rotation speed to be characterized as the sum of a constant speed and a small, periodic perturbation. (His Chapter 1. Introduction 18 motivation for this study was that in practice the spin rates fluctuate within a small interval [28]). He concluded that merge type (flutter) resonance can happen between modes of different nodal diameters if the stationary edge-load is present. He also showed that, when the stationary edge load is uniformly distributed, the lowest few unstable regions, whose maximum width of instability are relatively small, tend to enlarge at first as the load distribution widens and to reach maximum as the load distributed over half a circle. A further increase in the load distribution angle tends to reduce the lowest few unstable regions. In order to understand the wash-boarding mechanism, an analytical model for wood cutting clamped circular saws was developed by Tian and Hutton [29]. In their study, the lateral cutting forces were represented by the product of a time-dependent periodic function and the lateral displacement of the saw teeth. In another work [30] they introduced an approach which predicts the physical instability mechanism that occurs during the interaction of the blade with a space fixed constraint. To provide physical insight into the stability characteristics of the system, they used a physical energy flux equation for the blade. They derived the energy variation βπΈ of the disk over a period of [0,π] for a constant spinning speed as [30]: βπΈ = β«{β¬[π(π€, π, π, π‘)(π€,π‘ +β¦π€,π)] π΄πππππ}ππ‘π0 Where π(π€, π, π, π‘) represents a generalized lateral force, and π€,π‘ + β¦π€,π represents the transverse velocity of a particle, attached to disk, observed by disk-fixed coordinates. Under stable condition βπΈ will be nonzero periodic function of time, but over a complete cycle the change of total energy will be zero. From this equation it is clear that the change of total energy βπΈ will increase when π(π€, π, π, π‘) is in phase with (π€,π‘ + β¦π€,π). In this case driving energy Chapter 1. Introduction 19 required to maintain constant speed will be transferred into transverse vibration and instability will occur [30]. Recently Khorasany, Mohammadpanah, and Hutton [31], conducted experimental tests on the frequency characteristics of a guided splined saw, subjected to large lateral deflection. They found that the blade frequencies show a significant change as a result of the initial lateral displacement imposed by the external force. It was also found that due to the presence of the external force, a stationary wave develops and collapses at a higher speed. For the numerical simulations, they used the nonlinear governing equations based on Von Karman plate theory. They investigated the effect of nonlinearity on the amplitude and frequency response of the guided blade. There is not any available literature or research on the dynamic response of a splined guided disk, subjected to in-plane edge loads. Understanding the effect of in-plane loads on the stability of a splined guided disk is of paramount importance in the current research. Therefore the focus of this work is on the dynamic characteristics of a guided splined disk, subjected to edge loads. 1.5 Objective and Scope The main objective of this research is: βTo provide primary guide lines for sawmills in choosing blade thickness, and stable operating speeds.β To meet this objective, the main task is defined as: βTo investigate the dynamic characteristics of guided splined saws during cutting.β Chapter 1. Introduction 20 In order to fulfill this task, the approach taken in this thesis consists of the following subtasks: - To develop a mathematical model that captures the essential physics of the problem - To study stability characteristics of splined disks, subjected to edge loads - To Conduct Experimental Tests of Guided Splined Saws These tasks are presented in five chapters of this thesis. Chapter 2 is devoted to mathematical modeling of a spinning splined guided disk subjected to in-plane outer and inner edge loads. In this chapter first the previously obtained results on the effect of edge-loads on the characteristics of a clamped disk are independently verified. Then the equations of motion for a guided splined disk subjected to in-plane edge loads on outer boundary and their corresponding interaction on inner boundary of disk are derived. A technique is developed for taking the derivative of eigenvalues of a splined disk with respect to edge loads. Then sensitivity of eigenvalues to edge loads is examined. Numerical solution of the equations of motion for a splined disk is used and the effect of edge loads on the natural frequencies and stability of the system are analyzed. The transverse response of a guided splined disk, subjected to edge loads are computed numerically at sub-critical, critical, super-critical, and flutter speeds of the disk. Chapter 3 is concerned with the mechanism of instability for a spinning splined guided disk subjected to in-plane outer and inner edge loads. In this chapter, an expression for the energy induced into the spinning disk by the in-plane loads, and their interaction at the inner radius, is derived by computation of the rate of work done by the lateral component of the edge Chapter 1. Introduction 21 loads. This chapter provides physical insight into instability mechanism of a guided splined disk subjected to in-plane loads. Chapter 4 presents the experimental results of a guided splined disk. This chapter provides insights into the dynamic behavior of spinning disk by conducting experimental studies. Experimental observations of idling and cutting tests of a splined guided saw are presented in this chapter. First idling tests results are presented for a guided splined saw blade. Then the results of cutting tests for the same blade are presented. Particular interest here is to see how the idling results can be used to predict the stable region of the blade for cutting. In chapter 5 experimental run-up tests of several guided splined saws of different sizes are presented, and the flutter instability zones are identified. Using the equations of motion for guided splined disks, developed in chapter 1 and 2, the flutter instability zones are defined for guided splined saws of various configurations. The experimental and numerical results are compared and discussed. Then the effect of the geometrical parameters of the blade on the start of flutter instability is analyzed. To provide primary practical design guide lines for sawmills the initiation of flutter speeds are computed for different blade diameters. The start of flutter instability speeds were computed and tabulated for different blade thickness. The results are also presented in a chart format. Chapter 6 presents the summary and conclusions. This chapter also provides suggestions for future work on this subject. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 22 Chapter 2: Modeling of a Spinning Splined Guided Disk Subjected to In-Plane Edge Loads Applied at the Outer and Inner Boundaries 2.1 Introduction This chapter presents analytical results of the dynamic behavior of rotating disks subjected to in-plane edge loads. The governing non-damped linear equations of transverse motion of a spinning disk with a splined inner radius and constrained from lateral motion by guide pads are derived. The disk is driven by a matching splined arbor that offers no restraint to the disk in the lateral direction. Rigid body translational and tilting degrees of freedom are included in the analysis of total motion of the spinning disk. The disk is subjected to lateral constraints and loads. Also considered are applied conservative in-plane edge loads at the outer and inner boundaries. The numerical solutions of these equations are used to investigate the effect of the loads and constraints on the natural frequencies, critical speeds, and stability of a spinning disk. 2.2 Previous Work on the Effect of Edge Loads on Clamped Disks Chen J.S. [23] investigated the frequency characteristics of clamped spinning disk, subjected to in-plane concentrated edge loads. He found that compressive edge loads decrease the natural frequencies of the forward and backward travelling waves, but increase the natural frequencies of reflected wave. He also concluded that, the compressive edge loads induce stationary type instability (divergence) before the critical speed and merge-type instability (flutter) after the critical speed when a reflected wave meets a forward or backward wave. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 23 In this section, the previously obtained results by Chen J.S. [23] are independently verified and his conclusions on the effect of edge-loads on the characteristics of a clamped disk are examined. Figure 2.1 Schematic of a Clamped Disk, Subjected to a Conservative In-Plane Compressive Edge Load Figure 2.1 shows a schematic of a clamped disk, subjected to a conservative in-plane compressive edge load. The governing equation of motion for the disk, in terms of the transverse displacement π€, with respect to a space fixed coordinates (π, π), is [23]: πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ + πΏπ€ + ?Μ?π€ = 0 (2.1) L and ?Μ? are the membrane stress field operators due to centrifugal force, and normal edge load, respectively. L and ?Μ? can be calculated by following a procedure described in Chen [23]. The boundary conditions for equation 2.1 are [23]: π€ = 0, π€,π = 0 ππ‘ π = π (2.2) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 24 π΅1π€ + πΉπ΅2π€ = 0, π΅3π€ = 0 ππ‘ π = π (2.3) Whereas π΅1, π΅2 and π΅3 are defined as [23]: π΅1 =πππ(π2ππ2+ππππ+π2π2ππ2) +1βΟ π2(π3ππππ2βπ2πππ2) (2.4) π΅2 =πΏ(π)ππ·πππ (2.5) π΅3 =π2ππ2+Ο π(πππ+π2πππ2) (2.6) The solution of equation 2.1 can be obtained by application of the Galerkin method. A particular solution of the equation is assumed to be: π€(π, π, π‘) = β [πππ(π‘) sinππ + πΆππ(π‘) cosππ]π π(ππππ)βπ,π=0 (2.7) Substituting equation 2.7 into equation of motion 2.1, result in a set of simultaneous partial differential equations in πππ(π‘), πππ πΆππ(π‘), while m, n = 0, 1, 2.... The solution of these equations determines the transverse displacement of the disk (for details see Appendix C). The material and geometric properties of the disk under investigation in [23] are: π = 7840πππ3 , πΈ = 2.03 Γ 1011ππ, Ο = 0.27, π = 101.6ππ, π = 203.2ππ, and β = 1.02ππ Figure 2.2 shows the variation of eigenvalues as a function of speed. The dashed lines are the results for the freely spinning disk, and the solid lines indicate the results for the disk, subjected to πΉ = 3.5π·π= 330π. The results are exactly matched with the results previously obtained by Chen [23]. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 25 Figure 2.2 Variations of Eigenvalues as a Function of Rotation Speed, (--dashed lines) Free Spinning Clamped Disk, (-solid lines) Clamped Disk Subjected to the Edge Load F=3.5 D/b Figure 2.2 illustrates the effect of a conservative in-plane edge load. The graph indicates that the load decreases the natural frequencies of forward and backward waves, and it increases the natural frequencies of reflected waves. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 26 The graph shows that the real part of the eigenvalue for mode (0,3) is positive when the imaginary part is zero in case the disk is subjected to the edge load. Therefore, the load induces a divergence type instability. It also induces a flutter type instability when the backward wave (0,2) intersects the reflected travelling wave of mode (0,3). These findings verified the conclusions of Chen [23] on the effect of in-plane edge load on the dynamic behaviour of a clamped disk. 2.3 Equations of Motion for a Spinning Splined Guided Disk Subjected to In-Plane Outer and Inner Edge Loads Consider an in-plane radial force πΉπ and an in-plane tangential force πΉπ‘ acting on the outer edge of the disk and one possible configuration for the interaction loads on the inner edge of the disk (Figure 2.3): Figure 2.3 Schematic of a Splined Disk Subjected to In-Plane Edge Loads, and Their Inner Interaction Loads Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 27 The governing equation of motion for the disk, in terms of the transverse displacement π€, with respect to a stationary coordinates system (π, π), considering: (π) rigid body translation Z and tilting motions ππ₯and ππ¦; (ππ) a space fixed linear spring ππ§ located at (ππ, ππ); (πππ)a space fixed lateral force ππ located at (ππ , ππ) and (ππ£) Asymmetric membrane stress due to in-plane edge loads is: πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ ββπ(πππππ€,π),π ββπ2ππππ€,ππ + πβ?Μ? + πβ(π?Μ?π₯ sin π βπ?Μ?π¦ cos π) + πβ(2β¦ π cos π ?Μ?π₯ + 2β¦ π sin π ?Μ?π¦) ββπ{(ππππ€,π + ππππ€,π),π + (ππππ€,π +πππ€,π),π} = βππ§π(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) +πππ (2.8) πππ and πππ are axi-symmetric in-plane stresses due to centrifugal acceleration. The closed form solution of πππand πππ for a disk with free inner boundary condition can be found in Appendix A. ππ, ππ and πππ are asymmetric stresses due to the edge loads; and are to be determined from the known values of πΉπ and πΉπ‘. The closed form solution of stresses can be found in Appendix D. The boundary equations at the outer and inner edges respectively are: πΉππΏ(πβπΌ)ππ·π€,π +πΉπ‘πΏ(πβπΌ)π2π·π€,π + (π€,ππ +1ππ€,π +1π2π€,ππ),π +1βππ2((π€,ππ),π β1ππ€,ππ) = 0 (2.9) (π = π) βπΉπ‘(π+π)πΏ(πβπΌ)2π3π·π€,π +πΉπ‘(πβπ)πΏ(π+πβπΌ)2π3π·π€,π + (π€,ππ +1ππ€,π +1π2π€,ππ),π +1βππ2((π€,ππ),π β1ππ€,ππ) βπΉππΏ(πβπΌ)ππ·π€,π = 0 (2.10) (π = π) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 28 Where πΏ(π) is the Dirac delta function. Equations (2.8-2.10) govern the dynamics motion of a splined spinning disk subjected to the edge loads shown in Figure 2.3. 2.4 Sensitivity Analysis of Eigenvalues of Spinning Splined Disk to In-plane Edge Loads To analyze the sensitivity of the eigenvalues of the spinning disk to in-plane edge loads, the derivative of the eigenvalues of the system with respect to the in-plane edge loads is calculated. Adopting a technique used by Chen [23], the derivative of the eigenvalues with respect to the edge loads is derived. The equation of motion of a spinning disk, without considering the membrane stress, can be written as (here for simplicity, we do not consider the membrane stress): πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ = 0 (2.11) Together with the boundary conditions of equations (2.9) and (2.10). Defining the operators: π = πβ πΊ = 2β¦πππ πΎ = β¦2π2ππ2+ π·β4 β4= β2β2= (π2ππ2+1ππππ+1π2π2ππ2)(π2ππ2+1ππππ+1π2π2ππ2) πΏ1 =πππ(π2ππ2+ππππ+π2π2ππ2) +1βΟ π2(π3ππππ2βπ2πππ2) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 29 πΏ2 =πΏ(πβπΌ)π·πππ πΏ3 =πΏ(πβπΌ)π·πππ πΏ4 =πΏ(π+πβπΌ)π·πππ The equation and the boundary conditions become: π?Μ? + πΊ?Μ? + πΎπ€ = 0 (2.12) π2πΏ1π€ + ππΉππΏ2π€ + πΉπ‘πΏ3π€ = 0 at π = π 2π3πΏ1π€ β 2π2πΉππΏ2π€ β (π + π)πΉπ‘πΏ3π€ + (π β π)πΉπ‘πΏ4π€ = 0 at π = π For simplicity we analyze the effect of the radial and tangential loads separately. 2.4.1 Effect of Radial Load Assume a solution in the form: π€ = π€ππ(π, π)πππππ‘ (2.13) Where: π€ππ(π, π) = π π(π)πΒ±πππ (2.14) Substituting the solution (2.13) into the equation (2.12): πππ2 ππ€ππ + ππππΊπ€ππ + πΎπ€ππ = 0 (2.15) Consider πππ0 and π€ππ0 as the eigenvalues and shape functions of free spinning disk. (When πΉπ = 0) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 30 Then the equation (2.15) for a free spinning disk is: (πππ0 )2π π€ππ0 + πππ0 πΊ π€ππ0 + πΎ π€ππ0 = 0 (2.16) The boundary equations become (substitute πΉπ = 0 in the boundary equations) πΏ1 π€ππ0 = 0 at π = π and π = π Taking the derivative of equation (2.15) with respect to πΉπ results in (See the mathematical details in Appendix H): 2πππ0 ππππππΉππ π€ππ0 + (πππ0 )2πππ€ππππΉπ+ππππππΉππΊ π€ππ0 + πππ0 πΊππ€ππππΉπ+ πΎππ€ππππΉπ= 0 (2.17) And the boundary equations become: ππΏ1ππ€ππππΉπ+ πΏ2 π€ππ0 = 0 at π = π ππΏ1ππ€ππππΉπβ πΏ2 π€ππ0 = 0 at π = π Multiplying the conjugate of equation (2.16) by ππ€ππππΉπ results in: (?Μ ?ππ0 )2ππ€ππππΉππ ?Μ ?ππ0 + ?Μ ?ππ0 ππ€ππππΉππΊ ?Μ Μ Μ ?ππ0 +ππ€ππππΉππΎ ?Μ ?ππ0 = 0 (2.18) Multiplying equation (2.17) by ?Μ ?ππ0 , and subtracting (2.18) from it results in: 2πππ0 ππππππΉππ ?Μ ?ππ0 π€ππ0 +ππππππΉπ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ+ ?Μ ?ππ0 πΎππ€ππππΉπββ?Μ ?ππ0 ππ€ππππΉππΊ ?Μ Μ Μ ?ππ0 βππ€ππππΉππΎ ?Μ ?ππ0 = 0 (2.19) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 31 Integrating equation (2.19) over an area of a circular plate ([0 2π] and[π π]), and considering the boundary equations results in: 4ππππππβ(πππ Β± πβ¦)ππππππΉπβ« π π2(π)πππππβ πππ (π π(π)ππ π(π)ππβ π π(π)ππ π(π)ππ) = 0 (2.20) Assume that πππ = πππ + ππππ. For a free spinning disk, the real part of the eigenvalue is zero, therefore, πππ0 = ππππ Rearranging the equation (2.20) results in: ππππππΉπ= βπ(π π(π)ππ π(π)ππβπ π(π)ππ π(π)ππ)4πππππβ(πππΒ±πβ¦)β« π π2(π)πππππ (2.21) For π < π: π π(π)ππ π(π)ππ> π π(π)ππ π(π)ππ and β« π π2(π)πππππ> 0 (See Appendix H for details) Therefore, for forward and backward waves (πππ Β± πβ¦ > 0), equation (2.21) is negative. As a result, the change in πππ is negative. This means that the application of πΉπ decreases the eigenvalues of forward and backward waves. For reflected waves (πππ Β± πβ¦ < 0), equation (2.21) is positive, therefore the change in πππ is positive. This means that the application of πΉπ increases the eigenvalues of reflected waves. The expression (2.21) does not hold when πππ Β± πβ¦ = 0 (critical speeds). Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 32 2.4.2 Effect of Tangential Load Following the same procedure as the radial loads, the derivative of eigenvalues with respect to a tangential load πΉπ‘ , is obtained as: (See Appendix H for complete mathematical calculations) ππππππΉπ‘= βππ(ππ π(π)βππ π(π))4ππβ(πππΒ±πβ¦) β« π π2(π)πππππ (2.22) For forward waves and backward waves (πππ Β± πβ¦ > 0), equation (2.22) is negative, therefore the change in πππ is negative. It means that the application of πΉπ‘ decreases the eigenvalues of forward and backward waves. For reflected waves (πππ Β± πβ¦ < 0), equation (2.22) is positive, therefore the change in πππ is positive. It means that the application of πΉπ‘ increases the eigenvalues of reflected waves. In addition, for π = 0 (modes with no nodal diameter), equation (2.22) becomes zero: ππππππΉπ‘= 0 (2.23) Therefore, tangential loads do not have an effect on modes with no nodal diameter. Equations (2.21) and (2.22) indicate that, for a free spinning splined disk, the derivative of eigenvalues with respect to the radial and tangential loads is purely imaginary. Therefore, in-plane edge loads only affect the imaginary part of eigenvalues. In other words, the effect of in-plane edge loads on the real part of eigenvalues is zero. Since the eigenvalues of a free splined disk are purely imaginary, the real part of the eigenvalues for a free splined disk, subjected to in-plane loads, is zero. It can be concluded that in-plane edge loads do not induce any instability to the splined disk. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 33 2.5 Numerical Results 2.5.1 Effect of In-plane Edge Loads on the Eigenvalues 2.5.1.1 Special Case (a Rigid Splined Disk) We start with the simpler case of a rigid splined disk, subjected to the radial and tangential forces shown in Figure 2.2. The parameters which govern the motion of the disk are the rigid body tilting and translational motion ππ₯, ππ¦and z. Since the applied forces πΉπ and πΉπ are in balance with the inner interaction forces, and the disk is rigid the boundary equations vanish. Consider a rigid disk (π€ = 0), the equations of motion are reduced to (for mathematical details see Appendix E): (?Μ?π₯ sin π β ?Μ?π¦ cos π) + 2β¦ (cos π ?Μ?π₯ + sin π ?Μ?π¦) = 0 (2.24) ?Μ?π₯ + 2β¦?Μ?π¦ +(πβπ) sinπΌπΉππβπΌππ₯ = 0 (2.25) ?Μ?π¦ β 2β¦?Μ?π₯ +(πβπ)cosπΌπΉππβπΌππ¦ = 0 (2.26) Equations (2.25) and (2.26) indicate that πΉπ‘ and the interaction of πΉπ‘ on the inner edge vanish from the equations. Therefore, one initial conclusion is that the tangential load has no effect on the eigenvalues of a splined rigid disk. The solution of this system of second order differential equations can be written as: ππ₯ = ππππππ‘ , ππ¦ = πππππππ‘ (2.27) Where π is a constant. ππ is a real value and can be obtained from the following equation: ππ =2β¦+β4β¦2β2πΎ2 (2.28) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 34 Whereas, πΎ =(πβπ)πβπΌπΉπ and πΌ =π4(π4 β π4) (2.29) Since ππ is a real value, equation (2.28) holds if: πΉπ β€2πβπΌβ¦2(πβπ) (2.30) The term 2πβπΌβ¦2(πβπ) can be defined as the critical load of a free splined rigid disk, subjected to in-plane radial load at a given spinning speed. (Mathematical details can be found in Appendix E). Equations (2.30) suggests that the natural frequency of a free splined spinning rigid disk (πΉπ = 0) is a function of rotation (ππ = 2β¦). The natural frequency of a splined spinning rigid disk, subjected to a radial in-plane force (πΉπ β 0) is a function of rotation speed and magnitude of the load. Taking the derivative of natural frequency with respect to load (Eq. 2.31) indicates that application of πΉπ (for the value below the critical load at any rotation speed) decreases the natural frequency of the system. In addition, the effect of πΉπ decreases as the rotation speed increases. πππππΉπ= βπβππβπΌβ4β¦2β2πΎ (2.31) Now, consider a case where the rigid disk only has rigid body translational movement, and it is constrained by a lateral spring (ππ§). The equation of motion is: πβπ(π2 β π2) ?Μ? = βππ§π, which results in ππ = βππ§πβπ(π2βπ2) (2.32) Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 35 As the equations for the rigid disk indicate the in-plane forces have no effect on the frequencies of the system. Also the in-plane loads provide no coupling between the rigid body tilting and rigid body translational in the absence of lateral springs. Figure 2.4 provides the graphical illustration of above results. The graphs indicate the variation of eigenvalues as a function of rotation speed. The dashed lines are the results for a free spinning splined rigid disk without considering the coupling effect of the spring, and the blue solid lines with considering the coupling effect of the spring. The thick dashed and solid red lines are the results for a same system, subjected to a radial load πΉπ , at outer edge at angular position of πΌ, considering the coupling effect of the spring. The results are obtained for a rigid disk of π = 75ππ, π = 215ππ, β = 1.5ππ, Ο =7800 kg m3β subjected to a lateral spring of ππ§ = 104ππ at outer radius at πΌ = 0Β°, πΉπ = 100π =1660πβπΌπ, at πΌ = 45Β°. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 36 Figure 2.4 Variation of Natural Frequencies as a Function of Rotation Speed, (-- blue dash lines) Free Spinning Splined Rigid Disk, (- blue solid lines) The Rigid Body Motions are Coupled by a Lateral Spring, (-- red dash lines) Free Spinning Splined Rigid Disk, Subjected to a Radial Load, (- red solid lines) Spinning Splined Rigid Disk, Constrained by a Lateral Spring, Subjected to a Radial Load Figure 2.4 illustrates the effect of application of a radial load. For a free spinning rigid disk, the tilting natural frequency is 2β¦; and application of πΉπ = 100π = 1660πβπΌπ decreases the natural frequency after the rotation speed of 23.4Hz. (Before this speed, since the load is higher than the critical load the tilting natural frequency vanishes). For a constrained spinning rigid disk by the lateral spring, application of radial load decreases the natural frequencies. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 37 2.5.1.2 General Case (A Guided Splined Flexible Disk) Consider a flexible disk with the following physical properties: π = 75ππ, π = 215ππ, β = 1.5ππ, E = 2.03 Γ 1011pa, Ο = 7800 kg m3β , and Ο = 0.29. First, to examine the effect of the in-plane edge loads on the bending frequencies of a free splined spinning disk, the eigenvalues are computed numerically for the disk subjected to a radial load of πΉπ = 3.6π·π= 1000π, and a tangential load of πΉπ‘ = 3.6π·π= 1000π. Figure 2.5 shows the variation of eigenvalues as a function of rotation speed for the flexible disk without considering the rigid body motions. The dashed lines are the results for a free spinning splined disk, and the solid lines are the results for the disk, subjected to in-plane loads. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 38 Figure 2.5 Variation of eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Splined Disk, (- red solid lines) Free Spinning Splined Disk, Subjected to Edge Loads Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 39 The results from Figure 2.5 indicate that the edge load decreases the natural frequencies of forward, and backward travelling waves and increases the natural frequencies of reflected waves. However the effect of the loads on the natural frequencies is not significant. The results also indicate that, unlike clamped disks, in-plane edge loads do not have any coupling effect on the bending modes. The in-plane loads thus do not induce any single or merged type resonance in the disk (the real parts are zero). Now, to examine the effect of the in-plane edge loads on the bending frequencies of a guided splined spinning disk (The disk is constrained by 9 lateral springs of stiffness π = 105ππ distributed in a 100Γ100 mm square area which represents the guide) the eigenvalues of the system are computed numerically. Figure 2.6 presents the results for a guided splined disk. The disk is subjected to the edge loads ( πΉπ = πΉπ‘ = 3.6π·π= 1000π, at πΌ = 45Β°). Figure 2.6 indicates the effect of the in-plane loads on the eigenvalues. The results indicate that the loads decrease the natural frequencies of backward waves, but increase the natural frequencies of reflected waves. However, the effect is not significant (less than 0.5% for the 4th frequency path), and for the first two frequency paths is almost zero. Figure 2.7 (which present the real part of the eigenvalues) indicates that there are two minor divergence instabilities at about 42Hz and 59Hz rotation speeds (the real part of eigenvalues is positive, while the imaginary part is zero). The application of edge loads slightly decreases the real part of the eigenvalues at these speeds. It also indicates that the span of rotation speeds at which the real part is positive is smaller for the disk subjected to edge loads. In other word, the edge loads decrease both the level and width of divergence instability zone. But the edge loads, increases the level and width of the flutter instability zone. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 40 Figure 2.6 Variation of Imaginary Part of Eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Guided Splined Disk, (- red solid lines) Free Spinning Guided Splined Disk, Subjected to Edge Loads Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 41 Figure 2.7 Variation of Real Part of Eigenvalues as a Function of Rotation Speed, (--blue dash lines) Free Spinning Guided Splined Disk, (- red solid lines) Free Spinning Guided Splined Disk, Subjected to Edge Loads Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 42 2.5.2 Transverse Response of a Disk to In-plane Edge Loads For further analysis on the effect of the in-plane loads the transverse vibration of the guided splined disk, subjected to the edge loads, is computed in the time domain at different speeds. The results are computed for three different speeds: a) At first minor critical speed (42Hz), b) At a super critical speed (50Hz), c) At a flutter instability speed (53Hz). Figures 2.8 shows the transverse vibration of the outers rim of the disk at πΌ = 45Β°, subjected to a radial load of πΉπ = 3.6π·π= 1000π, and a tangential load of πΉπ‘ = 3.6π·π= 1000π. A small lateral force πΉπ = 5π is applied at the same position as the radial and tangential loads as a source of lateral perturbation for the system. Figure 2.8 shows the transverse response of the guided splined disk subjected to different load configurations at three different speeds. Figure (2.8 a) is the result for the system subjected only to the radial load. Figure (2.8 b) is the result for the system subjected only to the tangential load. And, Figure (2.8 c) is the result for the system subjected to both the radial and tangential loads. Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 43 Figure 2.8 Transverse Vibration of the Disk Computed for Outer Radius at πΆ = ππΒ°, Subjected to (a) A Concentrated Radial In-plane Edge Load (b) A Concentrated Tangential In-plane Edge Load (c) Concentrated Radial and tangential In-plane Edge Load, (blue graph) β¦ = ππππ³, (green graph) β¦ = ππππ³, (red graph) β¦ = ππππ³, For All Cases, the Lateral Load is π π₯ = ππ Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 44 As the results indicate, the disk is unstable only at the flutter instability speed (53Hz). It also can be concluded from the results (Figure 2.8 a, b, and c) that the effect of tangential load is much more pronounced than the radial load. (This issue will be analyzed with more depth in the next chapter through comparing the work done in one cycle by the tangential and radial loads.) 2.6 Summary The effect of in-plane edge loads on the vibration characteristics of a spinning disk was investigated. Analytical investigations show that in-plane edge loads affect the dynamic behaviour of spinning disks through membrane stress and boundary conditions. Analysis indicates that in-plane edge loads tend to decrease the natural frequencies of forward and backward waves, and increase the natural frequencies of reflected waves. Comparing the effect of loads on clamped disks and splined disks reveal that the loads have a weak effect on eigenvalues of a splined disk in comparison with a clamped disk. It is also concluded that in-plane loads tend to merge two frequency paths in the neighborhood of βdegenerate pointsβ (in which two eigenvalues of the freely spinning disk are almost equal) for a clamped disk. However, for a free spinning disk, subjected to the same loads, the frequency loci cross over each other with no coupling. Therefore, the in-plane edge loads do not induce merge-type or flutter instability for a free splined disk, unlike a clamped disk. It was also found that the in-plane loads induce divergence instability in clamped disk, while it is not the case for a free splined disk. Numerical investigations of guided splined disks (a free splined disk is constrained by several springs over a certain area to represents the guides) show that the disk, subjected to edge loads, only suffers the flutter type instability. In other words, the disk is stable at minor critical speeds, and it only goes into instability when a forward wave and reflected wave merge Chapter 2. Mathematical Modeling of a Splined Disk, Subjected to Edge Loads 45 together due to presence of lateral constraints. This will be investigated in more depth in the next chapter by analyzing the energy induced into the disk by the in-plane edge loads. Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 46 Chapter 3: Instability Mechanism of a Spinning Splined Guided Disk Subjected to In-plane Edge Loads 3.1 Introduction In order to provide physical insight into the effect of in-plane edge loads on the stability of a splined spinning disk, the rate of work done by the applied loads is considered. An expression for the rate of work done by the transverse components of the applied in-plane loads, and their interaction at the inner radius, is derived. The energy transfer from the in-plane edge loads to the transverse vibration of the disk is equal to the work done by the transverse components of the in-plane loads. If the total work in one cycle is zero, no energy transfers to the disk and system is stable. If the work done is positive, then since the system is constrained to maintain its constant speed, then the positive energy is transferred into lateral vibration and instability occurs. 3.2 Work Done by the Edge Loads (Formulation) Consider an in-plane radial force πΉπ and an in-plane tangential force πΉπ‘ acting on the outer edge of the disk and their interaction loads on the inner edge of the disk according to the Figure 2.3. If we assume the transverse displacement of the disk with respect to a stationary coordinate system in polar coordinate as π€(π, π, π‘), and the rigid body tilting motions defined by ππ₯ and ππ¦ then the transverse velocity of point (π, π) can be calculated as: π(π, π, π‘) =ππ€ππ‘+ β¦ππ€ππ+ π (πππ₯ππ‘+ β¦ππ₯) β π(πππ¦ππ‘+ β¦ππ¦) (3.1) Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 47 Then the rate of work done by the edge forces in Figure 2.3 can be calculated as: ππΉπ ππ‘ ππ’π‘ππ πΈπππ = πΉπππ€ππ(π, πΌ, π‘). π(π, πΌ, π‘) (3.2) ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ = πΉπ‘ππ€π.ππ(π, πΌ, π‘). π(π, πΌ, π‘) (3.3) Note that the terms πΉπππ€ππ(π, πΌ, π‘) and πΉπ‘ππ€π.ππ(π, πΌ, π‘) define the transverse component of the πΉπ and πΉπ‘ at the point of application (π, πΌ) at time π‘. The rate of work done by the inner edge loads can also be calculated as: ππΉπ ππ‘ πΌππππ πΈπππ = πΉπππ€ππ(π, πΌ, π‘). π(π, πΌ, π‘) (3.4) ππ+π2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ=π+π2ππΉπ‘ππ€π.ππ(π, πΌ, π‘). π(π, πΌ, π‘) (3.5) ππβπ2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ=πβπ2ππΉπ‘ππ€π.ππ(π, πΌ + π, π‘). π(π, πΌ + π, π‘) (3.6) To calculate the work done in one revolution, the integration of the rate of the work is taken in one complete revolution as: ππΉπ ππ‘ ππ’π‘ππ πΈπππ = πΉπ β« (ππ€ππ(π, πΌ, π‘). π(π, πΌ, π‘))ππ‘0πππ πΆππππππ‘π π ππ£πππ’π‘πππ (3.7) ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ = πΉπ‘ β« (ππ€π.ππ(π, πΌ, π‘). π(π, πΌ, π‘))ππ‘0πππ πΆππππππ‘π π ππ£πππ’π‘πππ (3.8) ππΉπ ππ‘ πΌππππ πΈπππ = πΉπ β« (ππ€ππ(π, πΌ, π‘). π(π, πΌ, π‘))ππ‘0πππ πΆππππππ‘π π ππ£πππ’π‘πππ (3.9) Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 48 ππ+π2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ=π+π2ππΉπ‘ β« (ππ€π.ππ(π, πΌ, π‘). π(π, πΌ, π‘))ππ‘0πππ πΆππππππ‘π π ππ£πππ’π‘πππ (3.10) ππβπ2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ=πβπ2ππΉπ‘ β« (ππ€π.ππ(π, πΌ + π, π‘). π(π, πΌ + π, π‘))ππ‘0πππ πΆππππππ‘π π ππ£πππ’π‘πππ (3.11) The total work can be calculated by: π = ππΉπ ππ‘ ππ’π‘ππ πΈπππ + ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ + ππΉπ ππ‘ πΌππππ πΈπππ + ππ+π2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ+ππβπ2ππΉπ‘ ππ‘ ππ’π‘ππ πΈπππ (3.7) The terms πΉπππ€ππ , πΉπ‘ππ€π.ππ , πΉπ‘(π+π)2π2ππ€ππ, and πΉπ‘(πβπ)2π2ππ€ππ are the transverse components of outer and inner edge loads, and π(π, π, π‘) is the lateral velocity of a point on the disk at time t, at position of (π, π). Therefore, π is the rate of work done by the lateral components of the edge loads. If there is no deflection in the disk, then the lateral component of the edge loads are zero and no energy is transferred into the disk. If there is a deflection in the disk, and the lateral components of the loads (including the inner interaction loads) are in phase with π(π, π, π‘) then the work done is positive, and energy will induce into the disk. In this situation, because the disk is spinning at a constant speed, then the positive energy has to be transferred into lateral vibration and instability will occur. If the lateral component of the loads are out of phase with π(π, π, π‘) the work done is negative, and energy will be transferred out the disk. Since the disk is constrained to have constant speed, the disk centers itself and moves to the stable condition. Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 49 3.3 Numerical Results of Energy Induced in the Disk by Edge Loads For the guided splined disk, under investigation, the rate of energy is computed at different speeds. The results are computed for three different speeds: a) At first minor critical speed (42Hz), b) At a super critical speed (e.g. 50Hz), c) At a flutter instability speed (e.g. 53Hz). The disk is subjected to a radial load of πΉπ = 3.6π·π= 1000π, and a tangential load of πΉπ‘ = 3.6π·π= 1000π at = 45Β° , and the inner interaction edge loads. Figure 3.1 shows the rate of work at the three cases ((a), (b), and (c)). And the disk is subjected to both the radial and tangential forces. The results illustrate that at flutter speed the rate of work done by the in-plane edge loads is positive, therefore, instability is induced at flutter speed. Figure 3.1 Rate of Work Done by the Edge Loads, at Rotating Speed (a) 42Hz, (b) 50Hz, (c) 53Hz Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 50 Figure 3.2 shows the rate of work at a flutter instability speed (β¦ = 53π»π§), when the disk is subjected to the radial load, and the tangential load, separately. The results indicate that the rate of work done by the tangential load is higher than the rate of work done by the radial load of the same magnitude. It can be concluded that the effect of the in-plane tangential load on the instability of the disk is greater than the effect of the in-plane radial load. Figure 3.2 Rate of Work Done by the Tangential and Radial Edge Loads, Disk Running at a Flutter Instability Speed (β¦ = 53π»π§) 3.4 Summary In this chapter the rate of work done by the applied loads were formulated and numerically computed. The following main two conclusions were reached: ο· The rate of work done by the in-plane edge loads is positive at flutter speeds for guided splined disks, which induces instability into the system. Chapter 3. Mechanism of Instability of a Splined Disk, Subjected to In-plane Edge Loads 51 ο· The effect of the tangential in-plane edge loads on the instability of guided splined disks is greater than the effect of radial edge loads. Chapter 4. Experimental Investigation 52 Chapter 4: Experimental Investigations of Idling and Cutting Characteristics of a Splined Guided Saw 4.1 Introduction This chapter provides further insights into the dynamic behavior of spinning disks by reporting on the results of experimental studies, conducted by the author. The number of variables that affect cutting performance at high speeds is large and their interaction is complex. Such factors are blade speed, blade geometry, tooth geometry, blade flatness, temperature distribution in the blade, depth of cut, wood characteristics, guide size and location. However, particular interest here is to see how the idling results can be used to predict the stable region of the blade for cutting. Experimental tests facilities were developed which enable conducting idling and cutting tests. Experimental observations of idling and cutting tests for a guided circular saw blade are used to examine the dynamic characteristics of a guided spline saw. First, idling tests results are presented for a guided spline saw blade. Then, the results of cutting tests for the same blade are presented. From idling results the minor critical speed and the start of flutter instability were identified. Then, cutting tests were conducted at about critical speed, a super critical speed, and at a flutter instability speed. Chapter 4. Experimental Investigation 53 The physical properties of the blade under investigation are summarized in Table 4.1: Table 4.1 Properties of the Blade under Investigation Physical Property Value Outer Diameter (Rim) 430ππ (17ππ) Inner Diameter (Eye) 150ππ (6ππ) Thickness 1.5ππ (0.060ππ) Number of teeth 40 Density 7800 (Kgm3) Youngβs Modulus 203(G. Pa) Poissonβs ratio 0.3 4.2 Experimental Tests Results 4.2.1 Idling Tests A schematic of the experimental idling setup is presented in Figure 4.1. In order to measure the blade deflection, a non-contacting inductance probe was used. Electromagnetic excitation was used to provide low level white noise excitation over the frequency range of 0β100 Hz. In order to apply lateral force to the blade during tests an air jet nozzle with a constant pressure were used. Results were obtained by measuring the vibration responses of the blade at the location of the displacement probe as the rotation speed was ramped up from 0 RPM to 4,000 RPM at a constant rate over 600 s. Chapter 4. Experimental Investigation 54 Figure 4.1 Schematic of the Experimental Setup Figures 4.2 and 4.3 show the variation of excited frequencies of the disk as a function of rotation speed for a free spinning spline saw blade and a guided spline saw blade respectively. There are imperfections in the balance and interaction of the components of most practical rotating machines. As a result of these imperfections, rotating parts of machines generate vibration. Common sources of such vibrations arise due to mechanical looseness, mass unbalance, eccentricity, misalignments, bent shaft, pulleys, external forces, and rubbing. These faults usually yield vibrations of 1X, 2X, 3X, and higher order harmonics of the rotation. Further sub-harmonic levels of vibration such as X/2, X/3, β¦ of rotation speed may occur as illustrated in Figures 4.2 and 4.3. Figure 4.2 indicates that the frequency of the first two modes reaches to zero at about 2400rpm and 3200rpm. Chapter 4. Experimental Investigation 55 Figure 4.2. Variation of Excited Frequencies with Rotation Speed for a Saw Blade with no Constraint Figure 4.3 Variations of Excited Frequencies as a Function of Rotation Speed, Guided Spline Saw Chapter 4. Experimental Investigation 56 The experimental idling results of the guided splined saw (Figure 4.3) indicate that the lowest frequency path reaches to zero at about 3200rpm which is identified as the first minor critical speed. The start of flutter for this blade is about 3800rpm. 4.2.2 Cutting Tests Figure 4.4 shows the schematic of the cutting test setup. Cutting tests were conducted at 3200rpm (about the first minor critical speed), 3600rpm (a super critical speed), and 4000rpm (a post-flutter instability speed). The depth of cut was chosen to be 130ππ (stack of 3 Hemlock boards). The feed speeds were chosen to provide the bite per tooth of 1ππ with the gullet feed index of πΊ. πΉ. πΌ. = 0.45 (See Appendix F for details). After each cut, the cutting profile was obtained by laser scanning the surface of the cut 20mm below the upper edge of the board. Each test was repeated 5 times with 5 different cants. Table 4.2 summarizes the rotation and feed speeds. Figure 4.4 Schematic of Cutting Test Setup Chapter 4. Experimental Investigation 57 Table 4.2 Cutting Tests Rotation Speed, and Feed Speed Test No. Rotation Speed (RPM) Feed Speed (m/sec) Idling Condition 1 3200 2.1 First Critical Speed 2 3600 2.4 Super-critical Speed 3 4000 2.6 Flutter Instability Figures 4.5 - 4.7 show the cutting profiles for the cutting tests #1-3 respectively. The graphs indicate that cutting tests at speeds below the flutter instability speed (Test 1 and 2) resulted in an acceptable cutting performance (deviation less than 2ππ). Cutting at a flutter instability speed resulted in large deviations (more than 15ππ). Figure 4.5 Cut Profile for Test 1, Cutting at 3200rpm (Critical Speed) Chapter 4. Experimental Investigation 58 Figure 4.6 Cut Profile for Test 2, Cutting at 3600rpm (A Super Critical Speed) Figure 4.7 Cut Profile for Test 3, Cutting at 4000rpm (A Flutter Speed) Chapter 4. Experimental Investigation 59 4.4 Summary Idling and cutting tests were conducted for a guided circular splined saw. From idling results, the first critical speed and the initiation of flutter instability speeds were obtained. Then, cutting tests were conducted at the critical speed, at a super-critical speed, and at a flutter instability speed. The results indicated that cutting at speeds below the flutter instability, even at the critical speed, resulted in stable operation of the saw blade. Cutting at flutter instability resulted in unstable operation of the saw and cuts with large deviation (about 10 times the thickness of the blade). In conclusion, the idling results can be used to predict the stable operation speeds of a splined saw. Based on the experimental results, obtained in this study the maximum stable operation speed is the flutter speed. This speed may be identified from an idling test.Chapter 5. Flutter Speed Chart 60 Chapter 5: Flutter Speed Chart for Guided Splined-Arbor Saws 5.1 Introduction Choosing an optimum saw blade configuration for primary wood breakdown is a complex problem. There are many interconnected factors which need to be considered. Such factors include: blade diameter, arbour size, blade thickness, teeth number and geometry, gullet area, and a stable operation speed. Blade diameter is constrained by the maximum depth of cut and the arbour size. Two common arbour diameters are 6 inches and 8 inches which correspond to the nominal inner diameter of saw blades. (Also known as βEye#3β and βEye#4β respectively). After determining the blade diameter, and eye size, the next step is to choose optimum blade thickness. Choosing a thick blade results in the reduction of transverse vibrations of blade and increases the maximum stable operation speed. However, this is at the expense of increasing the cutting forces and power consumption. Moreover, a thicker blade means bigger kerf and increased saw dust wastage. The main intention of this chapter is to provide primary practical guide lines for sawmills to choose optimum blade diameter, eye size, blade thickness, and a stable rotation speed. It was shown, both analytically and experimentally, in previous chapters, that cutting at speeds below the flutter instability speed results in stable operation of the saw blade. It was seen that cutting above the flutter instability speed results in unstable operation of the saw and cuts with large deviation. In this chapter, experimental run-up tests of several guided splined disks and saws of different sizes are presented, and the flutter instability zones are identified. The results indicate Chapter 5. Flutter Speed Chart 61 that flutter instability occurs at speeds when a backward travelling wave of a mode meets a reflected wave of a different mode. Sometimes, the system cannot pass a flutter zone, and transverse vibrations of the disk lock into that flutter instability zone. The maximum stable operating speed of the rotating splined disk is defined as the initiation of a flutter which the system cannot pass. Using the equations of motion for guided splined disks (developed in Chapters 1 and 2), the imaginary and real part of eigenvalues is computed and the flutter instability zones are defined. The results show that the mathematical model can accurately predict the flutter instability zones measured in the experimental tests. However, in the case where, in practice, the system cannot pass a flutter zone, the linear model fails to exhibit such behaviour. In other words, while the model indicates an unstable zone, in practice, it may not be possible to pass through the flutter zone by further increasing rotation speed. It should be noted that the mathematical model does not incorporate the effect of stress distributions of the type and magnitude that are caused by roll tensioning of the blade (a universal procedure in primary breakdown sawmills [38]). All the calculations are based on non-tensioned blades. The roll tensioning adds approximately 5-10% to the numerical results of the flutter speeds. This difference leaves users with a safe margin for operation. In this chapter, experimental idling results for several non-tensioned and tensioned blades are presented. Then numerical results of flutter instability speed computation for the same sized blade are presented. The experimental and numerical results are compared and discussed. Then the effect of the geometrical parameters of the blade on the initiation of flutter instability is analyzed. Chapter 5. Flutter Speed Chart 62 5.2 Experimental Measurements of Idling Response of Guided Splined-Arbor Saws Experimental test facilities were developed which enable the measurement of natural frequencies of guided splined saws of various sizes as a function of rotation speed. The experimental setup is presented in Figure 5.1. To measure the disk deflection, a non-contacting inductance probe was used. Electromagnetic excitation was used to provide white noise excitation over frequency range 1-100Hz. To investigate the dynamic characteristics of a rotating disk when subjected to the effect of a stationary lateral constant force, an air jet was used. The magnitude of the air jet force was kept at a constant value by keeping a constant air pressure. Results were obtained by measuring the vibration responses of the disk at the location of the displacement probe as the speed is ramped up from 0 RPM to 4,000 RPM at a constant rate over 600 s. Figure 5.1 Experimental Idling Tests Setup Idling tests were conducted for several blades of different sizes. The experimental tests were conducted for non-tensioned and roll tensioned blades of various sizes. Here as an example the results are present for 5 different blades which are listed in Table 5.1. Chapter 5. Flutter Speed Chart 63 Table 5.1 List of Blades under Experimental Investigations No. Blade Dimension as: βBlade Diameter-Eye Size-Thicknessβ (in) Condition 1 17-6-0.040 Non-Tensioned Disk 2 20-6-0.080 Non-Tensioned Disk 3 28-8-0.115 Non-Tensioned Saw Blade 4 30-8-0.125 Roll Tensioned Saw Blade 5 34-8-0.145 Roll Tensioned Saw Blade Figure 5.2 shows the idling results of the blade 28-8-0.115. (The results for other blades are presented in Appendix H). Figure 5.2 shows the deflection of the disk measured by the displacement probe during run-up from 0-3600rpm at constant acceleration (note that the test had to be stopped at 3600rpm due to very large transverse vibrations of the disk). The variation of the excited frequencies of the disk, as a function of rotation speed, was computed and plotted in a form of frequency-speed color-map (Figure 5.3). Frequency color maps of the power spectrum illustrate the variation of disk frequencies with rotation speed. It illustrates the energy of the signal at each speed and frequency with a color spectrum. From the time domain and color-map two general characteristics of the system can be observed. As the color map indicates, the frequencies (as measured by a space fixed observer) of each mode decrease as the rotation speed increases. At minor critical speeds the frequency of a mode reaches zero. The experimental results show that the blade suffers two minor critical speeds at about 1500rpm, and 2100rpm. It experiences initiation of lock-in flutter vibrations at about 2500rpm. Flutter occurs at super-critical speeds where a reflected wave frequency coincides with Chapter 5. Flutter Speed Chart 64 a backward travelling wave frequency. For example, figure 5.3 illustrates several points where reflected waves of mode 2 and 3 meet the backward waves of mode 2, 3, and 4. The results indicate that by increasing the rotation speed system could pass these flutter zones. But, at a speed about 2500rpm, when the reflected wave of mode 1 reaches the backward wave of mode 4 the system cannot pass the flutter speed zone. We define the term βlocked-in flutter instabilityβ where the system cannot pass a flutter zone. After this speed there are several lock-in frequency paths. The maximum stable operation speed of a guided splined disk is defined as the start of a locked-in flutter. One possible reason is that when system reaches to this flutter, the deflection of the system is large so that system behaves non-linearly. Figure 5.2. Deflection of Blade 28-8-0.115, during Idling Run-up from 0-3600rpm, Measured by Displacement Probe Chapter 5. Flutter Speed Chart 65 Figure 5.3. Variation of Excited Frequencies of the Disk as a Function of Rotation Speed Chapter 5. Flutter Speed Chart 66 5.3 Numerical Computation of Eigenvalues of Guided Splined-Arbor Saws In this section, the imaginary and real parts of the eigenvalues are computed numerically for a disk with physical properties summarized in Table 5.2. The disk is constrained by 16 lateral springs each of stiffness π = 105ππ distributed over a 200Γ200 mm square area which represents the guides. Table 5.2 Properties of the disk under Investigation Physical Property Value Outer Diameter (Rim) 710ππ (28ππ) Inner Diameter (Eye) 203ππ (8ππ) Thickness 2.9ππ (0.115ππ) Density 7800 (Kgm3) Youngβs Modulus 203(G. Pa) Poissonβs ratio 0.3 Figure 5.4 shows the eigenvalues of the disk as a function of rotation speed. Comparing the natural frequency graph predicted by the numerical analysis and those obtained from experimental results (the color map Figure 5.3) indicates that there is a good agreement between the numerical results and experimental results. From the numerical results the start of flutter instability which initiates by the interaction between mode 4 and reflected wave of mode 1 is 2510rpm, which is about the start of lock-in flutter speed obtained from the experimental results during run-up tests. Chapter 5. Flutter Speed Chart 67 Figure 5.4 Imaginary and Real Parts of Eigenvalues of the Guided Splined Disk The color map (Figure 5.3) however illustrates other excited frequencies in the system which analysis did not show. This is due to the fact that there are imperfections in the balance and interaction of the components of most practical rotating machines. As a result of these imperfections, rotating parts of machines generate vibration. Common sources of such vibrations arise due to mechanical looseness, mass unbalance, eccentricity, misalignments, bent shaft, pulleys, external forces, and rubbing. These faults usually yield vibration levels with harmonic of 1β¦, 2 β¦, 3 β¦,., and sub-harmonic levels of vibration such as β¦ /2, β¦ /3, and β¦ of rotation speed which can be seen in the color map. If the frequency of excitation generated by a rotating part coincides with the natural frequencies of the rotating blade, resonant interaction can be expected. Chapter 5. Flutter Speed Chart 68 5.4 Verification of Numerical Results Table 5.3 compares the results of initiation of the lock-in flutter speed computed numerically with that obtained experimentally. Table 5.3 Start of Flutter Instability Speeds for the Blades under Investigations No. Blade Dimension as: βBlade Diameter-Eye Size-Thicknessβ (in) Start of Flutter Difference (%) Measurement (RPM) Numerical Result (RPM) 1 17-6-0.040 2750 2650 -3 2 20-6-0.080 3600 3450 -4 3 28-8-0.115 2500 2510 -1 4 30-8-0.125 (Roll Tensioned) 2450 2175 -11 5 34-8-0.145 (Roll Tensioned) 2150 1910 -11 As may be seen from Table 5.3, the predicted flutter instability speeds are less than measured flutter speeds. This may be due to the existence of tensioning in the blades. Even for the nominally non-tensioned blades under investigation there might be little tensioned in the blade due to hammering and leveling of the blades. However, since it is a common practice in a sawmill to roll-tension blades, the numerical prediction of flutter speed leaves us with about 10% safety margin for stable operation speed. Chapter 5. Flutter Speed Chart 69 5.5 The Effect of Geometrical Properties of the System on the Start of Flutter Instability Speeds A non-Dimensionalized analysis of the speed of a spinning disk indicates that the start of flutter has a linear relationship with thickness (for details see Appendix F): β¦ = βπ·πβπ4 β¦β (5.1) Whereas D is the flexural rigidity: π· =πΈβ312(1βπ2) (5.2) Therefore: β¦ = βπΈ12π(1βπ2) βπ2 β¦β (5.3) β¦β is the non-dimensionalzed rotation speed. β¦β is a function of non-dimensional value of π/π (where π is the inner radius and π is the outer radius of the blade). The Figure 5.5 shows the non-dimensionalzed initiation of flutter speed as a function of π/π for π = 75ππ (3 in) and π = 100ππ (4 in) . Chapter 5. Flutter Speed Chart 70 Figure 5.5 Non-Dimensionalized Flutter Speeds for a=3in (Eye#3) and a=4in (Eye#4) For larger diameter blades (smaller π/π), the effect of eye size is more pronounced. The graph indicates that using the larger eye size increase the start of flutter instability speeds. It should be noted that increasing the eye size is at the expense of reducing the depth of cut. In addition, smaller eye size results in increasing the splined forces and this can increase the stresses in the blade. Therefore, when designing for optimum size blades, we are constrained by the depth of cut, plate thickness, eye size, and maximum operation speed which is the flutter speed. Chapter 5. Flutter Speed Chart 71 5.6 Flutter Instability Speeds Charts To provide primary practical design guide lines for sawmills the initiation of flutter speeds are computed for different blade diameters. In wood primary breakdown industry blades diameter usually varies from 14-34 inches depend on the depth of cut. The start of flutter instability speeds were computed and tabulated for different blade thickness. The results are also plotted in chart format. Table 5.4 summarizes the flutter instability speeds for guided splined-arbor saws. Figure 5.5 and Figure 5.6 is the graphical illustration of the same results. Chapter 5. Flutter Speed Chart 72 Table 5.4 Flutter Speeds for Guided Splined-Arbor Saws Plate Thickness Eye Size Saw Blade Diameter (in) 14 15 16 17 18 19 20 22 24 26 28 30 32 34 0.040 in #3 4100 3440 2940 2550 2250 2000 1760 1460 1210 1010 875 765 665 595 #4 5800 4505 3720 3130 2685 2335 2020 1570 1300 1080 935 810 710 625 0.050 in #3 4960 4160 3560 3090 2720 2410 2120 1740 1450 1210 1045 915 795 710 #4 6800 5295 4385 3695 3100 2730 2410 1895 1560 1295 1125 980 855 755 0.060 in #3 5820 4880 4180 3625 3190 2820 2480 2035 1685 1405 1210 1055 920 815 #4 7780 6130 5080 4280 3625 3190 2815 2220 1820 1510 1310 1135 990 875 0.070 in #3 6690 5620 4810 4165 3660 3240 2850 2330 1920 1610 1385 1205 1050 930 #4 8820 7085 5800 4900 4155 3660 3230 2545 2085 1730 1490 1285 1120 990 0.080 in #3 7575 6360 5440 4710 4140 3660 3230 2620 2165 1820 1560 1355 1180 1045 #4 9880 7860 6535 5520 4695 4130 3640 2875 2355 1950 1670 1445 1255 1105 0.090 in #3 8450 7100 6070 5250 4620 4080 3600 2930 2420 2030 1740 1510 1320 1160 #4 10975 8740 7275 6150 5245 4600 4060 3210 2625 2170 1855 1600 1390 1225 0.100 in #3 9320 7830 6685 5790 5065 4470 3980 3215 2655 2235 1910 1653 1446 1278 #4 12050 9620 7950 6705 5780 5040 4440 3530 2885 2405 2040 1755 1530 1345 0.110 in #3 10180 8550 7320 6340 5570 4920 4350 3540 2920 2450 2100 1820 1590 1400 #4 13135 10480 8750 7400 6315 5550 4890 3870 3165 2625 2235 1925 1670 1470 0.120 in #3 11020 9260 7930 6880 6050 5350 4710 3835 3160 2650 2280 1975 1770 1520 #4 14220 11320 9475 8010 6835 6030 5320 4200 3435 2850 2420 2085 1810 1590 Chapter 5. Flutter Speed Chart 73 Figure 5.6 Flutter Instability Speeds Charts, Eye#3 Figure 5.7 Flutter Instability Speeds Charts, Eye#4 Chapter 5. Flutter Speed Chart 74 5.7 Conclusion Experimental run-up tests of a guided splined disk indicate that flutter instability happens at speeds when a backward wave of a mode meets a reflected wave of another mode at supercritical speeds. As the rotation speed increase system may be able to pass a flutter zone, but there is a speed where system cannot pass a flutter and transverse vibrations of the disk lock into that flutter instability zone. The governing linear equations of transverse motion of a spinning disk with a free inner boundary condition and constrained from lateral motion by linear springs can be used to predict the flutter instability zones. However, this mathematical model fails to predict the behaviour of the system when it locks into a flutter speed. One possible explanation is the geometrical non-linearity of the system at post-flutter speeds due to large transverse deflection of the disk. Therefore, for further analyses of the guided splined disk study of non-linear equations of motion should be undertaken. Flutter speed charts were developed and presented in this chapter. The charts can be used to determine the maximum stable operation speed of a guided splined saw. These results are based upon tests conducted in a laboratory environment. Situations in the mills may not be ideal and this may result in mills not being able to duplicate these results. Therefore, mill trials are necessary in order to see if these results can be duplicated. The results can be used as a measure of how close a mill is to meet optimal conditions. Chapter 6. Conclusion 75 Chapter 6: Conclusion 6.1 Summary and Conclusion The main contributions of this thesis can be summarized as: (π) The effect of in-plane edge loads on the stability and natural frequencies of a spinning disk The governing non-damped linear equations of transverse motion of a spinning splined disk were derived. Rigid body translational and tilting degrees of freedom were included in the analysis. The disk was subjected to conservative in-plane edge loads at the outer and inner boundaries. The numerical solutions of these equations were used to investigate the effect of the loads and constraints on the natural frequencies, critical speeds, and stability of a spinning disk. The conclusions may be summarized as: 1- In-plane edge loads decrease the natural frequency of forward and backward waves, and increase the natural frequency of reflected waves of a spinning splined disk. 2- The effect of in-plane edge loads on the natural frequencies of clamped disks is greater than for splined disks. 3- The edge loads do not couple the bending modes of splined disks, unlike clamped disks. 4- The edge loads do not induce divergence and flutter instability into a free splined disk, unlike clamped disk. 5- The effect of tangential loads on the transverse response of the disk is higher than the effect of radial loads. Chapter 6. Conclusion 76 (ππ) Mechanism of Instability for a Spinning Splined Guided Disk Subjected to In-plane Edge Loads An expression for the rate of work done by the transverse components of the applied in-plane loads, and their interaction at the inner radius, was derived. Then the work done was computed numerically. The results were used to analyze the mechanism of instability at different rotation speeds. It was concluded that: 1- The rate of work done by the in-plane edge loads is positive at flutter speeds for guided splined disks, which induces instability into the system. In other words, application of in-plane edge loads at a flutter instability speed leads to inducing energy to the disk, which transfers to transverse deflection of the disk. 2- The effect of the tangential in-plane edge loads on the instability of guided splined disks is greater than the effect of radial edge loads. (πππ) Experimental Investigations of Idling and Cutting Characteristics of Guided Splined Saws Experimental tests facilities were developed which enables conducting idling and cutting tests. Idling and cutting tests were conducted for a guided splined saw. The conclusions are summarized as follow: 1- From idling run up tests of a guided splined saw, the critical speeds, and flutter instability speeds can be obtained. 2- Cutting at critical speeds, and super critical speeds is possible. They result in stable operational speeds of the blade. 3- Cutting at flutter instability speeds may result in large deviation of the cut. Chapter 6. Conclusion 77 4- Maximum operation speed of a guided splined saw is the initiation of a flutter instability speed. (ππ) Flutter Speed Charts for Guided Splined-Arbor Saws Experimental run-up tests of several guided splined disks and saws of different sizes were presented, and the flutter instability zones were identified. The results indicated that flutter instability occurs at speeds when a backward travelling wave of a mode meets a reflected wave of a different mode. Sometimes, the system cannot pass a flutter zone, and transverse vibrations of the disk lock into that flutter instability zone. The maximum stable operating speed of the rotating splined disk was defined as the start of a flutter which the system cannot pass. Using the equations of motion for guided splined disks the flutter instability zones were identified. The results showed that the mathematical model can predict accurately the flutter instability zones measured in the experimental tests. The effect of the geometrical parameters of the blade on the start of flutter instability was analyzed. The main conclusions can be summarized as: 1- The results show that the mathematical model can predict accurately the flutter instability zones measured in the experimental tests. However, in the case where, in practice, the system cannot pass a flutter zone, the linear model fails to exhibit such behaviour. 2- The initiation of flutter instability has a liner relation with the blade thickness. The non-dimensionalized initiation of flutter speed were computed as a function of a non-dimensionalized parameter π/π, where π and π are the inner and outer radius of the blade respectively. 3- Flutter instability speeds of splined saws of various sizes were computed and verified experimentally. Chapter 6. Conclusion 78 4- Flutter speed charts of splined saws were developed to provide primary guide lines for sawmills in choosing of a blade thickness and stable operation speeds. These results are based upon tests conducted in a laboratory environment. Situations in the mills may not be ideal and this may result in mills not being able to duplicate these results. Therefore, mill trials are necessary in order to see if these results can be duplicated. The results can be used as a measure of how close a mill is to meet optimal conditions. 6.2 Suggestions for Further Research In practice, having a perfect blade with no run-out is impossible. In addition, during cutting, due to friction, heat may induce into the body of the blade. Therefore temperature gradient in the body of the blade is unavoidable. In current research damping has not been included in the analyses. It is possible that there are damping forces (might be negative or positive) in contact between the saw blade and the wood in cutting zone. Therefore, theoretical analysis and experimental investigation of the dynamic behaviour of guided splined saw, considering the following effects, is necessary and need to be undertaken: - The effect of non-flatness - The effect of temperature distribution in the blade - The effect of damping (rotating and non-rotating damping force at the cutting zone on the edge of the blade) 79 References [1] Hutton, S.G., Chonan, S., and Lehmann, B.F., 1987, βDynamic Response of a Guided Circular Saw,β Journal of Sound and Vibration, 112, pp. 527-539. [2] Chen, J.S., and Bogy, D.B., 1993, βNatural Frequencies and Stability of a Flexible Spinning Disk-Stationary Load System With Rigid Body Tilting,β ASME Journal of Applied Mechanics, 60, pp. 470-477. [3] Chen, J.S., and Hsu, C.M. , 1997, βForced Response of a Spinning Disk Under Space-Fixed Couples,β Journal of Sound and Vibration, 206(5), pp. 627-639. [4] Chen, J.S. and Wong, C.C., 1995, βDivergence Instability of a Spinning Disk with Axial Spindle Displacement in Contact with Evenly Spaced Stationary Springs,β Journal of Applied Mechanics, 62, pp. 544-547. [5] Yang, S.M., 1993, βVibration of a Spinning annular Disk with Coupled Rigid-body Motion,β ASME Journal of Vibration and Acoustics, 115, pp. 159-164. [6] Mote, C.D., 1977, βMoving Load Stability of a Circular Plate on a Floating Central Collar,β Journal of Acoustical Society of America, 61, pp. 439-447. [7] Price K.B., βAnalysis of the Dynamics of Guided Rotating Free Centre Platesβ, Ph.D. Dissertation, University of California, Berkeley, 1987. [8] Khorasany, R.M.H., and Hutton, S.G., 2010, βAn Analytical Study on the Effect of Rigid Body Translational Degree of Freedom on the Vibration Characteristics of Elastically 80 Constrained Rotating Disks,β International Journal of Mechanical Sciences, 52, pp. 1186-1192. [9] Tobias, S.A., and Arnold, R.N., 1957, "The Influence of Dynamical Imperfections on the Vibration of Rotating Disks," Institution of Mechanical Engineers, Proceedings 171, pp. 669-690. [10] Raman, A., and Mote, C.D., 2001, βExperimental Studies on the Non-linear Oscillations of Imperfect Circular Disks Spinning Near Critical Speed,β International Journal of Non-Linear Mechanics, 36(2), 2001, pp. 291-305. [11] Kang, N., and Raman, A., 2006, βVibrations and Stability of a Flexible Disk Rotating in a Gas-filled Enclosure-Part 2: Experimental Study,β Journal of Sound and Vibration, 296(4-5), pp. 676-68. [12] DβAngelo, C., Mote, C.D., 1993, βAerodynamically Excited Vibration and Flutter of a Thin Disk Rotating at Supercritical Speed,β Journal of Sound and Vibration, 168, pp. 15-30. [13] Thomas, O., Touze, C., and Chaigne, A., 2003, βAsymmetric Non-linear Forced Vibrations of Free-edge Circular Plates. Part II: Experiments,β Journal of Sound and Vibration, 265(5), pp. 1075-1101. [14] Jana, A., and Raman, A., 2005, βNonlinear Aeroelastic Flutter Phenomena of a Flexible Disk Rotating in an Unbounded Fluid,β Journal of Fluids and Structures, 20(7), pp. 993-1006. 81 [15] Khorasany, R.M.H., and Hutton, S.G., June 2011, βVibration Characteristics of Rotating Thin Disks, Part I: Experimental Results,β ASME Journal of Applied Mechanics. [16] Khorasany, R.M.H., and Hutton, S.G., June 2011, βVibration Characteristics of Rotating Thin Disks, Part II: Analytical Predictions,β ASME Journal of Applied Mechanics. [17] Chen, J.S. and Wong, C.C., 1996, βModal Interaction in a Spinning Disk on a Floating Central Collar and Restrained by Multiple Springs,β Journal of the Chinese Society of Mechanical Engineers, Vol. 17, No.3, pp. 251-259. [18] Chen, J. S., and Bogy, D.B., 1992, βMathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System,β American Society of Mechanical Engineers Journal of Applied mechanics, 59, pp. 390-397. [19] Young, T.H., and Lin, C.Y., 2006, βStability of a Spinning Disk under a Stationary Oscillating Unit,β Journal of Sound and Vibration, 298, pp 307-18. [20] Schajer, G.S., Wang, S.A., 2002, βEffect of Work Piece Interaction on Circular Saw Cutting Satiability,β European Journal of Wood and Wood Products, 02/2002, Volume 60, Number 1, pp.48-54. [21] Schajer G., 1984, βGuided Saw Huntingβ, Forest Products Journal, Vol. 38, No. 4. [22] Mohammadpanah. A., βIdling and Cutting Vibration Characteristics of Guided Circular Sawsβ, January 2012, M.A.Sc. Thesis, The University of British Columbia. [23] Chen J.S., βStability Analysis of a Spinning Elastic Disk under a Stationary Concentrated Edge Loadβ, ASME, Vol. 61, Dec. 1994. 82 [24] Chen J.S., βVibration and Stability of a Spinning Disk under Stationary Distributed Edge Loadsβ, Journal of Applied Mechanics, Vol. 63, June 1996. [25] Chen J.S., βOn the Internal Resonance of a Spinning Disk under Space-Fixed Pulsating Edge Loadsβ, ASME, Vol. 68, November 2001 [26] Chen J.S., βParametric Resonance of a Spinning Disk under Space-Fixed Pulsating Edge Loadsβ, Journal of Applied Mechanics, March 1997, Vol. 64/139-143. [27] Shen I.Y., Song Y., βStability and Vibration of a Rotating Circular Plate Subjected to Stationary In-Plane Edge Loadsβ, Journal of Applied Mechanics, March 1996, Vol. 63/121-127. [28] Young T.H. and Wu M.Y., βDynamic Stability of Disks with Periodically Varying Spin Rates Subjected to Stationary In-Plane Edge Loadsβ, ASME, Vol. 71, July 2004. [29] Tian, J.F., and Hutton, S.G., 2001, βCutting Induced Vibration in Circular Saws,β Journal of Sound and Vibration, 242(5), pp. 907-922. [30] Tian, J.F., and Hutton, S.G., 1999, βSelf Excited Vibration in Flexible Rotating Disks Subjected to Transverse Interaction Forcesβ ASME Journal of Applied Mechanics, 66, pp. 800-805. [31] Khorasany R.M.H., Mohammadpanah A. and Hutton S.G., 2012, βVibration Characteristics of Guided Circular Saws: Experimental and Numerical Analysesβ, J. Vib. Acoust. 134(6), 061004 [32] Timoshenko S. and Goodier J.N. Theory of Elasticity. New York, McGraw-Hill. 83 [33] Meirovitch L., 1997, βPrinciples and Techniques of Vibrationsβ, New Jersey. [34] Carlin J.F., Appl F.C., Bridwel H.C., Dubois R.P., βThe Effect of Tensioning on Buckling and Vibration of Circular Saw Bladesβ ASME Journal of Engineering for Industry, Vol. 2, pp. 37-48. [35] Dubois R.P., "Buckling Loads of Tensioned Circular Plates Subjected to Concentrate In-Plane Loading" A Master Reports Submitted to the Kansas State University 1970. [36] St. Cyr, W.W. "Vibration and Stability of Circular Plates Subjected to Concentrated In-Plane Forces", PhD Dissertation, Kansas State University 1965. [37] Stakgold, Ivar, 1998, βGreen's Functions and Boundary Value Problemsβ, 2nd Edition. [38] Gary S. Schajer, 1983, βAnalysis of Roll Tensioning and its Influence on Circular Saw Stabilityβ Wood Science and Technology, 11/1983,17(4):287-302 [39] Bruce Lehmann, βSaw Tooth Design and Tipping Materialsβ Sr. Engineer, Thin Kerf Technologies Inc. British Columbia, Canada 84 Appendix A Stresses Fields in a Disk due to Centrifugal Acceleration πππ and πππ are axi-symmetric in-plane stresses due to centrifugal acceleration. The close form solution of πππ and πππ are [32]. For a disk with free inner boundary condition: ππ = πβ¦2(3+π8(π2 + π2) β3+π8π2π2π2 β3+π8r2) (A.1) ππ = πβ¦2(3+π8(π2 + π2) +3+π8π2π2π2 β1+3π8π2) (A.2) For a disk with clamped inner boundary condition: ππ = πβ¦2(1+π8(πβ1)π4 β(3+π)π4(πβ1)π2β(1+π)π2+1βπ8π2π2π2(π+1)π2 β(3+π)π2(πβ1)π2β(1+π)π2β3+π8π2) (A.3) ππ = πβ¦2(1+π8(πβ1)π4 β(3+π)π4(πβ1)π2β(1+π)π2β1βπ8π2π2π2(π+1)π2 β(3+π)π2(πβ1)π2β(1+π)π2β1+3π8π2) (A.4) 85 Appendix B Mathematical Equations for Rigid Body Tilting and Translational Motions Consider a point on the disk with respect to the coordinate attach to the disk (Figure B.1). Figure B.1 a) Schematic of Guided Spline Saw, b) Idealizing the Blade as a Spinning Disk πππ ππ‘πππ ππ π πππππ‘ ππ π‘βπ πππ π π€ππ‘β πππ ππππ‘ π‘π ππ‘π‘ππβ πππππ = [π cos ππ sin ππ€] (B.1) Consider O-XYZ as space fixed coordinate (Figure B.1): Position of a point on the disk with respect to inertial frame is: π = [πππ] The following transformation between two coordinates can be used: π = [πππ] = [cos ππ¦ 0 sin ππ¦0 1 0βsin ππ¦ 0 cos ππ¦] [1 0 00 cos ππ₯ βsin ππ₯0 sin ππ₯ cos ππ₯] [π cos ππ sin ππ€] (B.2) 86 The total angular momentum H of the blade, with respect to the origin of the blade O is: π»π = πβ β« β« π Γ ?Μ?ππ2π0 πππππ (B.3) Substituting the approximation (sin ππ₯ β ππ₯ and cos ππ₯ β 1 βππ₯22 and the same for ππ¦) and the moment of inertia for a disk ( πΌπ₯ = πΌπ¦ =π4πβ(π4 β π4) = πβπΌ): ?Μ?ππ₯ = πβπΌ(?Μ?π₯ + 2β¦?Μ?π¦) + β« β« πβ sin π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ (B.4) ?Μ?ππ¦ = πβπΌ(?Μ?π¦ β 2β¦?Μ?π₯) + β« β« πβ cos π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ (B.5) Consider that the disk has rigid body translational in Z direction. Also, Consider there is a spring (ππ§) at position of (ππ, ππ), so the force produce by this spring is: ππ = βππ§(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) (B.6) Also consider there is a force (ππ) at position of (ππ, ππ). By writing the balance equation between the angular momentum, and the torque produced by spring and external force: ?Μ?ππ₯ = ππ ππ sin ππ + ππππ sin ππ (B.7) ?Μ?ππ¦ = βππ ππ cos ππ β ππππ cos ππ (B.8) Substituting from equations (B.4) and (B.5): πβπΌ(?Μ?π₯ + 2β¦?Μ?π¦) + β« β« πβ sin π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ = ππ ππ sin ππ + ππππ sin ππ (B.9) πβπΌ(?Μ?π¦ β 2β¦?Μ?π₯) + β« β« πβ cos π (π€,π‘π‘ + 2β¦π€,π‘π)ππ2π0π2ππππ = βππ ππ cos ππ β ππππ cos ππ 87 (B.10) By adding the translational acceleration term (?Μ?) and angular acceleration (?Μ?π₯ and ?Μ?π¦) to equation (1.1): πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ ββπ(ππππ€,π),π ββπ2πππ€,ππ + πβ?Μ? +πβ(π?Μ?π₯ sin π β π?Μ?π¦ cos π) + πβ(2β¦ π cos π ?Μ?π₯ + 2β¦ π sin π ?Μ?π¦) = βππ§π(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) +πππ (B.11) And, the rigid body translational equation in Z is: π?Μ? + β« β« πβ(π€,π‘π‘)ππ2π0πππππ = βππ§(π€ + π + ππ sin ππ ππ₯ β ππ cos ππ ππ¦) + ππ (B.12) Where, π = π(π2 β π2)πβ is the total mass of the disk. 88 Appendix C Solution of the Equations of Motion For a circular disk the vibration modes can be described by the number of nodal circles (n) and the number of nodal diameters (m). So, the transverse displacement of the disk may be written by a modal expansion as [22]: π(π , π, π‘) = β [πππ(π) sinππ + πΆππ(π) cosππ] π π(ππππ )βπ,π=0 (C.1) First the Non-dimensional parameters are introduced as: π =ππ , π =π€π , π = βπ·πβπ4π‘ , β¦β = βπβπ4π· β¦ (C.2) Substituting of π(π , π, π‘) into equation (2.8) results in[22]: ?Μ?ππ β 2β¦βπ?Μ?ππ + (π4ππ βπ2β¦β2)πππ ββ¦β2π2πβ πππβπ=0 Πππ = πΉππsinππ π π(ππππ ) (C.3) ?Μ?ππ β 2β¦βπ?Μ?ππ + (π4ππ βπ2β¦β2)πΆππ ββ¦β2π2πβ πΆππβπ=0 Πππ = πΉππ cosππ π π(ππππ ) (C.4) Where Πππ = β« (πΆπππ2π π(ππππ )ππ 21ππ+ πΆπππ π(ππππ )ππ βπ2πΆπππ π(ππππ )π π(ππππ )π ππ = 0 (C.5) And 89 πΉππ =βπ3π·β [(ππππ(π) + π?Μ?ππβπ,π=0 (π)) sin ππ + (ππΆππ(π) + π?Μ?ππ(π)) cos ππ]π π( ππππ ) (C.6) The non dimentionalized stationary equation of motion for a plate in polar coordinate system is: β4π = π4π (C.7) Using the boundary conditions we can obtain the eigenvalues πππ associated with the eigenfunctions as: [33] π·ππ = {sinππcosππ}π m(ππππ ) (C.8) π π(ππππ ) = (π1ππ½π(ππππ ) + π2πππ(ππππ ) + π3ππΌπ(ππππ ) + π4ππΎπ(ππππ )) (C.9) Where π½π, ππ, πΌπ, and πΎπare mth order Bessel functions and modified Bessel functions; and π1π , π2π , π3π , and π4π are constants. 90 Appendix D Stresses Fields in a Disk due to In-Plane Edge Loads Following the [34], [35], and [36] procedure the stresses due to the edge loads are calculated (for outer and inner edge): Stress due to the Radial Force (ππ): Consider β as a stress field in a polar coordinates. The resulting stresses are [35]: ππ =1ππβ ππ+1π2π2β ππ2 (1.D) ππ =π2β ππ2 πππ = βπππ(1ππβ ππ) In order to satisfy the compatibility equations β4β = 0 . To satisfy this equation, the general solution is [35]: β = π0ππ(π) + π0π2 + π0π2ππ(π) + π0π2π + πβ²0π + πβ²0π2πππ(π) + π β²0πππ(π) β12π1πππππ π +π1ππππ(π)πππ π + πβ²1ππππ(π)π πππ +12π1πππ πππ + (π1π3 + π β²1πβ1 ++πβ²1πππ(π))πππ π +(π1π3 + π β²1πβ1 ++πβ²1πππ(π))π πππ + β (ππππ + ππππ+2 + π β²ππβπ + πβ²ππβπ+2)πππ ππ +βπ=2β (ππππ + ππππ+2 + π β²ππβπ + πβ²ππβπ+2)π ππ ππβπ=2 (2.D) Using this function the stresses can be calculated as: ππ =1ππβ ππ+1π2π2β ππ2= π0πβ2 + 2π0 + π0 + 2π0π + 2π0ππ(π) + 2πβ²0πππ(π) + πβ²0π + πβ²0ππβ2 +π1ππβ1πππ π + π β²1ππβ1π πππ β 2π1πβ1ππ(π)π πππ + 2π β²1πβ1ππ(π)πππ π + [π1πβ1 + 2π1π β 91 2πβ²1πβ3 + πβ²1πβ1]πππ π + [π1πβ1 + 2π1π β 2πβ²1πβ3 + πβ²1πβ1]π πππ + β (π(1 β π)ππππβ2 +βπ=2(π + 2 β π2)ππππ β π(1 + π)πβ²ππβπβ2 + (2 β π β π2)πβ²ππβπ)πππ ππ + β (π(1 ββπ=2π)ππππβ2 + (π + 2 β π2)ππππ β π(1 + π)π β²ππβπβ2 + (2 β π β π2)πβ²ππβπ)π ππ ππ (3.D) since ππ is an even function due to symmetry about the point of application of the in-plane load, then ππ = ππ = πβ²π = πβ²π = 0 for n=1,2,3,.... Since ππ is a single-valued function of π, then π0 = πβ²0 = πβ²0 = π1 = πβ²1= 0. Substituting these coefficient for ππ: ππ = π0πβ2 + 2π0 + π0 + 2π0ππ(π) + [π1πβ1 + 2π1π β 2πβ²1πβ3 + πβ²1πβ1]πππ π +β (π(1 β π)ππππβ2 + (π + 2 β π2)ππππ β π(1 + π)πβ²ππβπβ2 +βπ=2(2 β π β π2)πβ²ππβπ)πππ ππ (4.D) ππ =π2β ππ2= βπ0πβ2 + 2π0 + 3π0 + 2π0ππ(π) + [6π1π + 2πβ²1πβ3 + πβ²1πβ1]πππ π +β (π(π β 1)ππππβ2 + (π + 2)(π + 1)ππππ + π(1 + π)πβ²ππβπβ2 + (π β 2)(π ββπ=21)πβ²ππβπ)πππ ππ (5.D) πππ = βπππ(1ππβ ππ) = βπβ²0πβ2 + [2π1π β 2πβ²1πβ3 + πβ²1πβ1]π πππ + β (π(π β 1)ππππβ2 +βπ=2π(π + 1)ππππ β π(1 + π)πβ²ππβπβ2 β π(π β 1)πβ²ππβπ)π ππ ππ (6.D) Now, the boundary conditions need to be defined: 92 The stress distribution at the edge of the disk can be found by first considering the Fourier series expansion of a uniformly distributed loading over an arc of length 2π and intensity such that the total load is πΉπ ; then letting π goes to 0. The result of the stress at the outer edge is: ππ(π) =βπΉπππβ[12+ β πππ ππβπ=1 ] (7.D) and similarly for the inner edge: ππ(π) =βπΉπππβ[12+ β πππ ππβπ=1 ] (8.D) At the outer and inner edge the shear stress is zero: πππ(π, π) = πππ(π, π) = 0. So the resulting stress distribution is: ππ =βπΉπππββ [π1π(ππ)πβ2 + π2π(ππ)π + π3π(ππ)βπβ2 + π4π(ππ)βπ]πππ ππβπ=0 (9.D) ππ =βπΉπππββ [π1π(ππ)πβ2 + π2π(ππ)π + π3π(ππ)βπβ2 + π4π(ππ)βπ]πππ ππβπ=0 (10.D) πππ =βπΉπππββ [π1π(ππ)πβ2 + π2π(ππ)π + π3π(ππ)βπβ2 + π4π(ππ)βπ]πππ ππβπ=0 (11.D) And the coefficients are: For n=0, and n=1, the coefficients are: Consider: π =ππ, πΌ =1+Ο 1βΟ , π½ =3βΟ Ο +1, π10 = βπ10 =12π 2(1πΌ+ π 2) π20 = π20 =12πΌ(1πΌ+ π 2) 93 π30 = π40 = π30 = π40 = 0 π10 = π20 = π30 = π40 = 0 π11 = 1 π11 = π11 = 0 π21 = π21 = π21 =3(1βΟ )(Ξ±π 2+1)4(1+π½π 4) π31 = π31 = π31 =π 2(1βΟ )(Ξ±βΞ²π 2)4(1+π½π 4) π41 = π41 = π41 =12(1 β Ο ) which numerical results show we can get sufficient approximation just using the n=0, and n=1. However if we want more accurate results for n=2,3,4,....; the coefficients are: π1π = π1π = π1π =βππ½π β2π+(π2β1+π½2)π 2+π(πβ1)(π2β1)(π βπ β1)2+(π½π βπ β1)2) π2π =(πβ2)(π½π β2πβππ β2+π+1)(π2+1)(π βπ β1)2+(π½π βπ β1)2) π2π =(π+2)(π½π β2πβππ β2+π)4(π½(π π+π βπ)2 π2π =π(π½π β2π+π)(π2β1)(π βπ β1)2+(π½π βπ β1)2) 94 π3π = π3π = π3π =β(π2β1+π½2)π 2βπ(π+1)+ππ½π 2π3π½π 2π+(π2β1)(π βπ β1)2+(π½π βπ β1)2) π4π =(π)(π½π 2πβππ β2βπ+1)(π2β1)(π βπ β1)2+(π½π βπ β1)2) π4π =(π)(π½π 2πβππ β2βπ+1)3(π½(π π)2+(π2β1)(π βπ β1)2+(π½π βπ β1)2) π4π =π(π½π 2πβππ β2βπ)2(π½(π π)2+(π2β1)(π βπ β1)2+(π½π βπ β1)2) Stress due to the Tangential Force (ππ): Using the same procedure by Song [27] the stresses due to tangential loads are: ππ = [β2π1π3+ 2π1π +π1πβπΉπ‘ππβ]π πππ + β ((π β π2)ππππβ2 β (π + π2)πππβπβ2 β (2 + π ββπ=2π2)ππππ + (2 β π β π2)πππβπ)π ππ ππ (12.D) ππ = [2π1π3+ 6π1π +π1π]π πππ + β ((π2 β π)ππππβ2 + (π + π2)πππβπβ2 + (2 + 3π +βπ=2π2)ππππ + (2 β 3π + π2)πππβπ)π ππ ππ (13.D) πππ =πΉπ‘π2πβπ2+ [2π1π3β 2π1π βπ1π]πππ π β β ((π2 β π)ππππβ2 β (π + π2)πππβπβ2 β (π +βπ=2π2)ππππ + (π β π2)πππβπ)π ππ ππ (14.D) and the coefficients are: π1 = πΎ(1 + Ο )b4 + (3 + Ο )b2 95 π1 = πΎ(1 + Ο ) π1 = βπΉπ‘(1βΟ )4πβ whereas K= πΉπ‘(1+Ο )a2β(3+Ο )b28πβ(1+Ο )b4+(3βΟ )a4 And for n=2 π2 = π2 =6πΉπ‘ππβ π2 = π2 =β6πΉπ‘ππ3πβ Numerical results show we can get sufficient accuracy just by using the n=0, and n=1. However if we want more accurate results, we can compute the coefficients for n=2,3,4,.... 96 Appendix E The Governing Equations of a Rigid disk, Subjected to In-plane Edge Loads Consider a rigid splined disk, subjected to the radial and tangential forces according to the Figure 2.3 configuration. The parameters which govern the motion of the disk are just the rigid body tilting and translational motion ππ₯, ππ¦and z. Since the applied forces πΉπ and πΉπ are in balance with the inner interaction forces, and the disk is rigid the boundary equations vanish. Consider a rigid disk (π€ = 0), the equations of motion are: πβπΌ(?Μ?π₯ + 2β¦?Μ?π¦) = βπΉπ(π β π) sin πΌ sin(ππ₯)+(πΉπ‘ sin(ππ₯) (π) βπβπ2ππΉπ‘ sin(ππ₯) π βπ+π2ππΉπ‘ sin(ππ₯) π) sin πΌ (E.1) πβπΌ(?Μ?π¦ β 2β¦?Μ?π₯) = βπΉπ(π β π) cos πΌ sin(ππ¦)+(πΉπ‘ sin(ππ¦) (π) βπβπ2ππΉπ‘ sin(ππ¦) π βπ+π2ππΉπ‘ sin(ππ¦) π) cos πΌ (E.2) Simplifying the equations πΉπ‘ vanishes from the equations. Also, for small oscillation we assume sin(ππ₯) β ππ₯ and sin(ππ¦) β ππ¦ . The equations become: πβπΌ(?Μ?π₯ + 2β¦?Μ?π¦) = βπΉπ(π β π) sin πΌ ππ₯ (E.3) πβπΌ(?Μ?π¦ β 2β¦?Μ?π₯) = βπΉπ(π β π) cos πΌ ππ¦ (E.4) Rearranging the equations: ?Μ?π₯ + 2β¦?Μ?π¦ +(πβπ) sinπΌπΉππβπΌππ₯ = 0 (E.3) ?Μ?π¦ β 2β¦?Μ?π₯ +(πβπ)cosπΌπΉππβπΌππ¦ = 0 (E.4) 97 The solution of this system of second order differential equations is: ππ₯ = ππππππ‘ , ππ¦ = πππππππ‘ (E.5) Whereas π is a constant, and ππ can be obtained from the following equation: ππ =2β¦+β4β¦2β2πΎ2 (E.6) Whereas πΎ =(πβπ)πβπΌπΉπ (E.7) In order for the equations to hold: 4β¦2 β 2(πβπ)πβπΌπΉπ β₯ 0 (E.8) Therefore: πΉπ β€2πβπΌβ¦2(πβπ) (E.9) πππππΉπ= β12(4β¦2 β 2πΎ)β12 (2 (πβππβπΌ)) = βπβππβπΌβ4β¦2β2πΎ (E.10) 98 Appendix F Gullet Feed Index and Feed Speed Gullet Feed Index (GFI) is percentage of tooth blade gullet filled and usually is a number between 0.3 to 0.7 [39]. The maximum feed speed for a given depth of cut might be calculated as [39]: πππ₯πππ’π πΉπππ πππππ =πΊπΉπΌ.π.π΄π.π· π΅ππ‘π ππππ‘ββ =1π(πΉπππ πππππ).60π ππ c = blade rim speed A = Gullet area P=Tooth pitch D= depth of cut N= Number of Teeth 99 Appendix G Derivative of Eigenvalues of Spinning Disk with Respect to In-plane Edge Loads To analyze the sensitivity of eigenvalues of the spinning disk to in-plane edge loads, the derivative of eigenvalues of the system with respect to the in-plane edge loads is calculated. Figure H.1 Schematic of a Splined Disk Subjected to In-Plane Edge Loads, and Their Inner Interaction Loads The equation of motion of spinning disk, without considering the membrane stress, can be written as: πβ(π€,π‘π‘ + 2β¦π€,π‘π +β¦2π€,ππ) + π·β4π€ = 0 (G.1) And the boundary conditions are: πΉππΏ(πβπΌ)ππ·π€,π +πΉπ‘πΏ(πβπΌ)π2π·π€,π + (π€,ππ +1ππ€,π +1π2π€,ππ),π +1βππ2((π€,ππ),π β1ππ€,ππ) = 0 (π = π) 100 βπΉπ‘(π+π)πΏ(πβπΌ)2π3π·π€,π +πΉπ‘(πβπ)πΏ(π+πβπΌ)2π3π·π€,π + (π€,ππ +1ππ€,π +1π2π€,ππ),π +1βππ2((π€,ππ),π β1ππ€,ππ) βπΉππΏ(πβπΌ)ππ·π€,π = 0 (π = π) Defining the operators: π = πβ πΊ = 2β¦πππ πΎ = β¦2π2ππ2+ π·β4 β4= β2β2= (π2ππ2+1ππππ+1π2π2ππ2)(π2ππ2+1ππππ+1π2π2ππ2) πΏ1 =πππ(π2ππ2+ππππ+π2π2ππ2) +1βΟ π2(π3ππππ2βπ2πππ2) πΏ2 =πΏ(πβπΌ)π·πππ πΏ3 =πΏ(πβπΌ)π·πππ πΏ4 =πΏ(π+πβπΌ)π·πππ The equation and the boundary conditions become: π?Μ? + πΊ?Μ? + πΎπ€ = 0 (G.2) π2πΏ1π€ + ππΉππΏ2π€ + πΉπ‘πΏ3π€ = 0 at π = π 2π3πΏ1π€ β 2π2πΉππΏ2π€ β (π + π)πΉπ‘πΏ3π€ + (π β π)πΉπ‘πΏ4π€ = 0 at π = π 101 For simplicity we analyze the effect of the radial and tangential loads separately. Effect of Radial Load Boundary equations become: ππΏ1π€ + πΉππΏ2π€ = 0 at π = π ππΏ1π€ β πΉππΏ2π€ = 0 at π = π Assume a solution by separation of variable method as: π€ = π€ππ(π, π)πππππ‘ (G.3) And assume a separable function for π€ππ(π, π) as: π€ππ(π, π) = π π(π)πΒ±πππ (G.4) Where π π(π) is a real value, and can be a function of Bessel functions as: π π(π) = π1ππ½π(π) + π2ππΌπ(π) (G.5) Where π½π, and πΌπare πth order Bessel functions and modified Bessel functions, and π1π and π2πare constant. Substituting the solution (G.3) into the equation (G.2) [23]: πππ2 ππ€ππ + ππππΊπ€ππ + πΎπ€ππ = 0 (G.6) Consider πππ0 as the eigenvalues of free spinning disk. (πΉπ = 0) and the shape functions are π€ππ0 [23]: (πππ0 )2π π€ππ0 + πππ0 πΊ π€ππ0 + πΎ π€ππ0 = 0 (G.7) 102 The boundary equations become (substitute πΉπ = 0 in the boundary equations) πΏ1 π€ππ0 = 0 at π = π and π = π Taking the derivative of equation (G.6) with respect to πΉπ(prove can be found in [37]): 2πππ0 ππππππΉππ π€ππ0 + (πππ0 )2πππ€ππππΉπ+ππππππΉππΊ π€ππ0 + πππ0 πΊππ€ππππΉπ+ πΎππ€ππππΉπ= 0 (G.8) And the boundary equations become: ππΏ1ππ€ππππΉπ+ πΏ2 π€ππ0 = 0 at π = π ππΏ1ππ€ππππΉπβ πΏ2 π€ππ0 = 0 at π = π The conjugate of equation (G.7) is: (?Μ ?ππ0 )2π ?Μ ?ππ0 + ?Μ ?ππ0 πΊ ?Μ ?ππ0 + πΎ ?Μ ?ππ0 = 0 (G.9) Multiplying the equation (G.9) by ππ€ππππΉπ : (?Μ ?ππ0 )2ππ€ππππΉππ ?Μ ?ππ0 + ?Μ ?ππ0 ππ€ππππΉππΊ ?Μ Μ Μ ?ππ0 +ππ€ππππΉππΎ ?Μ ?ππ0 = 0 (G.10) Multiplying equation (G.8) by ?Μ ?ππ0 : 2πππ0 ππππππΉππ ?Μ ?ππ0 π€ππ0 + (πππ0 )2 ?Μ ?ππ0 πππ€ππππΉπ+ππππππΉπ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ+ ?Μ ?ππ0 πΎππ€ππππΉπ= 0 (G.11) Subtracting (G.10) from (G.11): 103 2πππ0 ππππππΉππ ?Μ ?ππ0 π€ππ0 + (πππ0 )2 ?Μ ?ππ0 πππ€ππππΉπ+ππππππΉπ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ+ ?Μ ?ππ0 πΎππ€ππππΉπβ (?Μ ?ππ0 )2ππ€ππππΉππ ?Μ ?ππ0 β ?Μ ?ππ0 ππ€ππππΉππΊ ?Μ Μ Μ ?ππ0 βππ€ππππΉππΎ ?Μ ?ππ0 = 0 (G.12) It is recognizable that: ππ€ππππΉππ ?Μ ?ππ0 = ?Μ ?ππ0 πππ€ππππΉπ And (?Μ ?ππ0 )2 = (πππ0 )2 Therefore, equation (G.12) becomes: 2πππ0 ππππππΉππ ?Μ ?ππ0 π€ππ0 +ππππππΉπ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ+ ?Μ ?ππ0 πΎππ€ππππΉπββ?Μ ?ππ0 ππ€ππππΉππΊ ?Μ Μ Μ ?ππ0 βππ€ππππΉππΎ ?Μ ?ππ0 = 0 (H.13) And the boundary equations are: ππΏ1ππ€ππππΉπ+ πΏ2 π€ππ0 = 0 and πΏ1 π€ππ0 = 0 at π = π ππΏ1ππ€ππππΉπβ πΏ2 π€ππ0 and πΏ1 π€ππ0 = 0 at π = π Integrating the equation (G.13) over an area of a circular plate([0 2π] and [π π]), and considering the boundary equations result in: 104 4ππππππβ(πππ Β± πβ¦)ππππππΉπβ« π π2(π)πππππβ πππ (π π(π)ππ π(π)ππβ π π(π)ππ π(π)ππ) = 0 (G.14) Note that πππ = πππ + ππππ. For a free spinning disk, the real part of eigenvalues is zero, therefore, πππ0 = ππππ Rearranging the equation (G.14): ππππππΉπ= βπ(π π(π)ππ π(π)ππβπ π(π)ππ π(π)ππ)4πππππβ(πππΒ±πβ¦)β« π π2(π)πππππ (H.15) Considering always π < π, then for π π(π) which is a combination of Bessel functions π½π(π) and πΌπ(π) (equation G.5), the following expression is hold: π π(π)ππ π(π)ππ> π π(π)ππ π(π)ππ (G.16) And β« π π2(π)πππππ> 0 For forward waves and backward waves (πππ Β± πβ¦ > 0), equation (G.15) is negative, therefore the change in πππis negative. It means that the application of πΉπ decreases the eigenvalues of forward and backward waves. For reflected waves (πππ Β± πβ¦ < 0), equation (G.15) is positive, therefore the change in πππis positive. It means that the application of πΉπ increases the eigenvalues of reflected waves. The expression (G.15) does not hold when πππ Β± πβ¦ = 0. In other word the equation (G.15) is not true at critical speeds. 105 Effect of Tangential Load The boundary equations change to: π2πΏ1π€ + πΉπ‘πΏ3π€ = 0 at π = π 2π3πΏ1π€ β (π + π)πΉπ‘πΏ3π€ + (π β π)πΉπ‘πΏ4π€ = 0 at π = π Using the same procedure as for the radial loads, for a free spinning disk (πΉπ‘ = 0)[23] (πππ0 )2π π€ππ0 + πππ0 πΊ π€ππ0 + πΎ π€ππ0 = 0 (G.17) The boundary equations become (substitute πΉπ = 0 in the boundary equations) πΏ1 π€ππ0 = 0 at π = π and π = π Taking the derivative of equation (G.17) with respect to πΉπ‘ [23]: 2πππ0 ππππππΉπ‘π π€ππ0 + (πππ0 )2πππ€ππππΉπ‘+ππππππΉπ‘πΊ π€ππ0 + πππ0 πΊππ€ππππΉπ‘+ πΎππ€ππππΉπ‘= 0 (G.18) And the boundary equations become: π2πΏ1ππ€ππππΉπ‘+ πΏ3 π€ππ0 = 0 at π = π 2π3πΏ1ππ€ππππΉπ‘β (π + π)πΏ3 π€ππ0 + (π β π)πΏ4 π€ππ0 = 0 at π = π Multiplying the conjugate of equation (G.17) by the ππ€ππππΉπ‘ : (?Μ ?ππ0 )2ππ€ππππΉπ‘π ?Μ ?ππ0 + ?Μ ?ππ0 ππ€ππππΉπ‘πΊ ?Μ Μ Μ ?ππ0 +ππ€ππππΉπ‘πΎ ?Μ ?ππ0 = 0 (G.19) 106 Multiplying equation (G.18) by ?Μ ?ππ0 : 2πππ0 ππππππΉπ‘ ?Μ ?ππ0 π π€ππ0 + (πππ0 )2 ?Μ ?ππ0 πππ€ππππΉπ‘+ππππππΉπ‘ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ‘+ ?Μ ?ππ0 πΎππ€ππππΉπ‘= 0 (G.20) Subtract Equation (G.19) from Equation (G.20), results in: 2πππ0 ππππππΉπ‘ ?Μ ?ππ0 π π€ππ0 +ππππππΉπ‘ ?Μ ?ππ0 πΊ π€ππ0 + πππ0 ?Μ ?ππ0 πΊππ€ππππΉπ‘+ ?Μ ?ππ0 πΎππ€ππππΉπ‘β?Μ ?ππ0 ππ€ππππΉπ‘πΊ ?Μ Μ Μ ?ππ0 βππ€ππππΉπ‘πΎ ?Μ ?ππ0 = 0 (G.21) Note that (πππ0 )2 ?Μ ?ππ0 πππ€ππππΉπ‘= (?Μ ?ππ0 )2ππ€ππππΉπ‘π ?Μ ?ππ0 Integration of Equation (G.21) over an area of a disk, and considering the boundary equations, result in: 4ππππππβ(πππ Β± πβ¦)ππππππΉπ‘β« π π2(π)πππππβ ππππ(ππ π(π) β ππ π(π)) = 0 (G.22) Rearranging the equation (G.22): ππππππΉπ‘= βππ(ππ π(π)βππ π(π))4ππβ(πππΒ±πβ¦) β« π π2(π)πππππ (G.23) We can assume that for π < π: ππ π(π) β ππ π(π) > 0 (G.24) 107 And β« π π2(π)πππππ> 0 Then for forward waves and backward waves (πππ Β± πβ¦ > 0), equation (G.23) is negative, therefore the change in πππis negative. It means that the application of πΉπ‘ decreases the eigenvalues of forward and backward waves. For reflected waves (πππ Β± πβ¦ < 0), equation (G.23) is positive, therefore the change in πππis positive. It means that the application of πΉπ‘ increases the eigenvalues of reflected waves. The expression (G.23) does not hold when πππ Β± πβ¦ = 0. In other word the equation (G.23) is not true at critical speeds. In addition, for π = 0 (modes with no nodal diameter), equation (G.23) becomes: ππππππΉπ‘= 0 (G.25) Therefore, tangential loads do not have effect on modes with no nodal diameter. Equations (G.15) and (G.23) indicate that, for a free spinning splined disk, the derivative of eigenvalues with respect to the radial and tangential loads is purely imaginary. Therefore, in-plane edge loads only affect the imaginary part of eigenvalues. In other word, the effect of in-plane edge loads on the real part of eigenvalues is zero. Since the real part of eigenvalues for a splined disk, subjected to in-plane loads is zero, it can be concluded that in-plane edge loads, do not induce any instability to the disk. 108 Appendix H Idling Response of Different Blade Sizes Figure H.1 Idling Response of Disk 17-6-0.040 109 Figure H.2 Idling Response of Disk 20-6-0.080 110 Figure H.3 Idling Response of Blade 30-8-0.125 111 Figure H.4 Idling Response of Blade 34-8-0.145 112 Appendix I Mathematical Calculation of Non-Dimensional Equation of Motion Non-dimensional parameters are introduced as [22]: π =ππ , π =π€π , π = ππ‘ Where π = βπ·πβπ4 Based on the introduced parameters: β¦β = β¦π ππ€ππ‘=π(ππ)π(π πβ )= ππππππ , π2π€ππ‘2= π2ππ2πππ2 , π2π€ππ‘ππ= πππ2πππππ , β4π€ =1π3β4π π2π€ππ2= ππ2πππ2 , ππ€ππ= π(ππ)π(ππ )=ππππ , π2π€ππ2=ππ(ππ )ππ€ππ=1ππ2πππ 2 By substitution of, ππ and ππ into equation of motion, we get: πβ(π€,π‘π‘ + 2β¦π€,π‘π + β¦2π€,ππ) + π·β4π€ ββππβ¦2(π€,π (πΆ1π +πΆ2π+ πΆ3π3)),π β βπβ¦2 (πΆ1π2βπΆ2π4+πΆ4)π€,ππ = π (I.1) By simplification of equation (I.1) the equation can be written in the form of: πβ(π€,π‘π‘ + 2β¦π€,π‘π + β¦2π€,ππ) + π·β4π€ β βπβ¦2 (π€,ππ (πΆ1 +πΆ2π2+ πΆ3π2) + π€,π (πΆ1πβπΆ2π3+ 3πΆ3π)) β βπβ¦2(πΆ1π2βπΆ2π4+ πΆ4)π€,ππ = π (I.2) Here we substitute the non-dimensional parameters into equation (I.2): πβπ2π(π,ππ + 2β¦β2π,ππ + β¦β2π,ππ) +π·π3β4π β 113 πβπ2β¦β2[π,π π π(πΆ1 +πΆ2π 2π2+ πΆ3π 2π2) +π,π (πΆ1π πβπΆ2π 3π3+ 3πΆ3π π)] β πβπ2β¦β2π (πΆ1π 2π2βπΆ2π 4π4+πΆ4)π,ππ = π (I.3) By dividing both side of equation (I.3), the equation may be written as (π,ππ + 2β¦β2π,ππ + β¦β2π,ππ ) +π·πβπ2π4β4π β πβ¦β2ππ[π,π π π(πΆ1 +πΆ2π 2π2+ πΆ3π 2π2) +π,π (πΆ1π πβπΆ2π 3π3+ 3πΆ3π π)] β πβ¦β2π(πΆ1π 2π2βπΆ2π 4π4+ πΆ4)π,ππ =ππβπ2π (I.4) If we put π·πβπ2π4= 1 we get: π = βπ·πβπ4 (I.5) By substituting T in equation (I.4) the equation can be written as π,ππ + 2β¦βπππ + β¦β2π,ππ + β4π ββ¦β2π2(πΆ1 +πΆ2π 2π2+ πΆ3π 2π2)π,π π ββ¦β2π(πΆ1π πβπΆ2π 3π3+3πΆ3π π)π,π ββ¦β21(πΆ1π 2π2βπΆ2π 4π4+ πΆ4)π,ππ =ππ3π· (I.6) This equation may be written in the form of: π,ππ + 2β¦βπ,ππ + β¦β2π,ππ + β4π ββ¦β2π2(πΆπππ,π π + πΆππ,π + πΆπππ,ππ) =ππ3π· (I.7) Where πΆππ = πΆ1 +πΆ2π 2π2+ πΆ3π 2π2 πΆπ = π(πΆ1π πβπΆ2π 3π3+ 3πΆ3π π) πΆππ = π2(πΆ1π 2π2βπΆ2π 4π4+ πΆ4)
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Flutter instability speed of guided splined disks,...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Flutter instability speed of guided splined disks, with applications to sawing Mohammadpanah, Ahmad 2015
pdf
Page Metadata
Item Metadata
Title | Flutter instability speed of guided splined disks, with applications to sawing |
Creator |
Mohammadpanah, Ahmad |
Publisher | University of British Columbia |
Date Issued | 2015 |
Description | In this thesis the vibration characteristics of guided splined saws are studied, both analytically and experimentally. Significant insights into the complex dynamic behavior of guided splined saws are presented by analytical investigation of the dynamic behaviour of spinning splined disks and then by conducting idling and cutting experimental tests of guided splined saws. Cutting tests are conducted at different speeds, at critical, supercritical, and post flutter speeds of a guided splined saw. The cutting results are compared to determine the stable operation speeds for guided splined saws. For the analytical studies, the governing linear equations are derived for the transverse motion of a constant speed spinning splined disk. The disk is subjected to lateral constraints and loads. Rigid body translational and tilting degrees of freedom are included in the analysis of total motion of the spinning disk. Also considered in the analyses are applied conservative in-plane edge loads at the outer and inner boundaries. The numerical solution of these equations is used to investigate the effect of the loads and constraints on the natural frequencies, critical speeds, and stability of the spinning disk. The sensitivity of the eigenvalues of the splined spinning disk to the in-plane edge loads is analyzed by taking the derivative of the spinning diskβs eigenvalues with respect to the applied loads. This analysis contains an evaluation of the energy transfer from the applied loads to the disk vibrations and is used to examine the role of critical system components in the development of instability. Experimental results are presented that support the validity of the analysis. The experimental results indicate that flutter instability occurs at speeds when a backward travelling wave of a mode meets a reflected wave of a different mode. The maximum stable operating speed of the rotating splined disk is defined as the initiation of flutter. Flutter instability speeds of splined saws of various sizes were computed and verified experimentally. Then flutter speed charts of splined saws were developed which provides primary practical guide lines for sawmills to choose optimum blade diameter, eye size, blade thickness, and a stable rotation speed. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2015-02-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0167125 |
URI | http://hdl.handle.net/2429/52060 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2015-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
Download
- Media
- 24-ubc_2015_may_mohammadpanah_ahmad.pdf [ 4.63MB ]
- Metadata
- JSON: 24-1.0167125.json
- JSON-LD: 24-1.0167125-ld.json
- RDF/XML (Pretty): 24-1.0167125-rdf.xml
- RDF/JSON: 24-1.0167125-rdf.json
- Turtle: 24-1.0167125-turtle.txt
- N-Triples: 24-1.0167125-rdf-ntriples.txt
- Original Record: 24-1.0167125-source.json
- Full Text
- 24-1.0167125-fulltext.txt
- Citation
- 24-1.0167125.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0167125/manifest