IDLING AND CUTTING VIBRATION CHARACTERISTICS OF GUIDED CIRCULAR SAWS by Ahmad Mohammadpanah M.Sc, Sharif University of Technology, Iran, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) January 2012 © Ahmad Mohammadpanah, 2012 Abstract In this thesis the vibration characteristics of guided circular saws are studied, both analytically and experimentally. Significant insights into the complex dynamic behavior of guided circular saws are presented first by numerical investigation of rotating disks and then by conducting idling and cutting experimental tests of splined saws with different guide configurations. For the numerical investigations, the governing linear equation of transverse vibration of a rotating disk is used. As a primary interest, the variation of disk natural frequencies with rotation speed and the disk response to applied external force are calculated. Also, the steady state response of the disk at different speeds is calculated. The effects of elastic lateral constraints are investigated in this section. A comprehensive experimental investigation of idling tests of splined saws with different guide configurations is presented. The frequencies and amplitudes of the blade vibrations are documented and the mean deflections of the disks are plotted. The dynamic characteristics of a rotating blade when subjected to the stationary lateral constant force are discussed. Extensive cutting tests are conducted and the effect of different guide configurations on cutting accuracy is presented. Cutting tests are conducted at different speeds, below and above the lowest critical speed for different guide configurations. The cutting results are compared to determine the guide configuration which results in the best cutting accuracy. ii Table of Contents Abstract .................................................................................................................................... ii Table of Contents ................................................................................................................... iii List of Tables .......................................................................................................................... vi List of Figures ........................................................................................................................ vii List of Symbols, and Abbreviations ................................................................................... xiv Acknowledgements .............................................................................................................. xvi Dedication ............................................................................................................................ xvii Chapter 1: Introduction ........................................................................................................ 1 1.1 Background (Clamped Saws vs. Guided Splined Saws) ............................................. 1 1.2 Previous Research on the Vibration Characteristics of Rotating Disks ..................... 3 1.3 Review of Patents on Guided Saws ............................................................................ 9 1.4 Objective and Scope ................................................................................................. 15 Chapter 2: Analytical Investigation of Vibration Characteristics of Disks.................... 18 2.1 Introduction ................................................................................................................ 18 2.2 Linear Equation of Motion ........................................................................................ 18 2.2.1 Formulation ......................................................................................................... 19 2.2.2 Solution ............................................................................................................... 22 2.3 Dynamic Characteristics of Clamped Disks .............................................................. 25 2.3.1 Natural frequencies of Clamped-Free Disks ....................................................... 25 2.3.2 Steady State Forced Response Due to a Space Fixed Constant Force ................ 27 iii 2.3.3 Effect of Transverse Elastic Constraints on the Disk Frequency-Speed Characteristics .............................................................................................................. 29 2.4 Dynamic Characteristics of Free-Free Disks ............................................................. 32 2.4.1 Natural Frequencies of Free-Free Disks ............................................................. 32 2.4.2 Effect of Transverse Elastic Constraints on the Disk Frequency-Speed Characteristics .............................................................................................................. 33 2.5 Dynamic Characteristics of Guided Splined Saws .................................................... 34 2.5.1 Full-Guided Splined Saws .................................................................................. 35 2.5.1.1 Natural Frequencies of Full-Guided Splined Saws ............................................ 36 2.5.1.2 Response of Saw to Applied Forces .................................................................. 37 2.5.2 One-Pin-Eye-Guided Splined Saws .................................................................... 39 2.5.2.1 Natural Frequencies One-Pin-Eye-Guided Splined Saws.................................. 40 2.5.2.2 Response of Saw to Applied Forces .................................................................. 42 2.5.3 One-Pin-Rim-Guided Splined Saws .................................................................. 43 2.5.3.1 Natural Frequencies One-Pin-Rim-Guided Splined Saws ................................. 44 2.5.3.2 Response of Saw to Applied Forces .................................................................. 45 2.5.4 Comparison of Analytical Results of Three Guided Configurations .................. 46 Chapter 3: Experimental Investigations of the Idling Characteristics of Guided Saws ................................................................................................................................................ .50 3.1 Introduction ................................................................................................................ 50 3.2 Experimental Setup and Test Procedure .................................................................... 50 3.3 Experimental Results ................................................................................................. 53 3.3.1 Variations of Blade Deflection with Rotating Speed .......................................... 53 iv 3.3.2 Natural Frequencies ............................................................................................ 59 3.3.2.1 Frequency - Speed Color Map ........................................................................... 59 3.3.2.2 Comparison of Experimental and Analytical Results of Variations of Natural Frequencies ................................................................................................................... 65 3.3.2.3 Frequency - Speed Waterfall Plot ...................................................................... 67 3.3.3 Vibration Response of Saw to Applied External Forces ................................... 73 Chapter 4: Experimental Investigation of the Cutting Characteristics of Guided Saws ................................................................................................................................................. 80 4.1 Introduction ................................................................................................................ 80 4.2 Experimental Setup and Test Procedure .................................................................... 82 4.3 Cutting Test Results ................................................................................................... 84 4.3.1 Standard Deviation of Cut profiles ..................................................................... 85 4.3.2 Cut Profile Results of Different Guide Configurations ...................................... 91 4.4 Cutting Test Results Comparison and Conclusion ................................................... 97 Chapter 5: Conclusions ....................................................................................................... 98 5.1 Summary and Conclusion .......................................................................................... 98 5.2 Recommendations for Future Work......................................................................... 102 References ............................................................................................................................ 104 Appendices ........................................................................................................................... 108 Appendix A Derivation of Non-Dimensional Equation of Motion ................................. 108 Appendix B Mathematical Details of Solution for Linear Equation of Transverse Vibration ........................................................................................................................... 110 Appendix C Gullet Feed Index ........................................................................................ 115 v List of Tables Table 1.1 Different Guided Systems for Splined Saws....................................................... 10 Table 2.1 Physical Properties of the Disk Used for all the Analytical Investigations ........ 25 Table 2.2 First Critical Speed, Three Guided Systems ....................................................... 48 Table 3.1 Properties of the Blade 17-40-60 Used for all the Experimental Tests............... 52 Table 3.2 Speed at Which Impulse Test Was Conducted ................................................... 73 Table 4.1 Rotation and Feed Speeds at Which Cutting Test Were Conducted ................... 82 Table 4.2 Standard Deviation and Mean Value at 2000 RPM for the 17-40-60 Blade, Average of 5 Different Cuts .................................................................................................... 85 Table 4.3 Standard Deviation and Mean Value at 2400 RPM for the 17-40-60 Blade, Average of 5 Different Cuts .................................................................................................... 86 Table 4.4 Standard Deviation and Mean Value at 2800 RPM for the 17-40-60 Blade, Average of 5 Different Cuts .................................................................................................... 87 Table 4.5 Standard Deviation and Mean Value at 3200 RPM for the 17-40-60 Blade, Average of 5 Different Cuts .................................................................................................... 88 Table 4.6 Standard Deviation and Mean Value at 3600 RPM for the 17-40-60 Blade, Average of 5 Different Cuts .................................................................................................... 89 Table 4.7 Top Probe Standard Deviation at Different Speed for the 17-40-60 Blade, Average of 5 Different Cuts for Three Guided Configurations .............................................. 90 vi List of Figures Figure 1.1 Clamped Saw ....................................................................................................... 2 Figure 1.2 Guided Splined Saw ........................................................................................... 2 Figure 1.3 Guided Splined Saw, (a) Full-Guided, (b) Pin-Eye-Guided, and (c) Pin-Rim- Guided ..................................................................................................................................... 16 Figure 2.1 Schematic of Rotating Disk ............................................................................... 19 Figure 2.2 Variations of Natural Frequencies with Disk Rotating Speed, Numerical Results [Clamped Disk, OD17″, ID6″, and Thickness 0.06″] ............................................................. 26 Figure 2.3 Deflection of the Disk as a Function of Angle Around the Rim for a Constant Point Load Applied at the Rim, for Different Rotating Speeds: a) Ω=0, b) Ω=2000rpm, c) Ω=2200rpm, d) Ω=2300rpm, e) Ω=2350rpm, f) Ω=2400rpm ............................................... 28 Figure 2.4 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 5000N/m, Dash Line Unconstrained Clamped-Free Disk) ...................... 29 Figure 2.5 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 10000N/m, Dash Line Unconstrained Clamped-Free Disk) .................... 30 Figure 2.6 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 50000N/m, Dash Line Unconstrained Clamped-Free Disk)…………….30 vii Figure 2.7 Variations of Natural Frequencies with the Disk Rotation Speed, Numerical Results [Free-Free Disk, OD17″, ID6″, and Thickness 0.06″] ............................................... 33 Figure 2.8 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Free-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 10000N/m, Dash Line Unconstrained Free-Free Disk) ................................... 34 Figure 2.9 Full-Guided Splined Saw, all Dimensions in Inch [Blade 17″×6″×0.060″] ...... 36 Figure 2.10 Variations of Natural Frequencies with the Disk Rotation Speed [Full-Guided Splined Saw, Blade17″×6″×0.060″] ...................................................................................... 37 Figure 2.11 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [Full-Guided Spline Saw, Blade 17″×6″×0.060″] ...................... 38 Figure 2.12 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [Full-Guided Splined Saw, Blade 17″×6″×0.060″]…..……..…39 Figure 2.13 One-Pin-Eye-Guided Splined Saw, all Dimensions in Inch [Blade 17″×6″×0.060″] ....................................................................................................................... 40 Figure 2.14 Variations of Natural Frequencies with the Disk Rotation Speed [One-Pin-EyeGuided Splined Saw, Blade 17″×6″×0.060″] ......................................................................... 41 Figure 2.15 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [One-Pin-Eye-Guided Splined Saw, Blade 17″×6″×0.060″] ...... 42 viii Figure 2.16 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [One-Pin-Eye-Guided Splined Saw, Blade 17″×6″×0.060″] ...... 43 Figure 2.17 One-Pin-Rim-Guided Splined Saw, all Dimensions in Inch [Blade 17″×6″×0.060″] ....................................................................................................................... 43 Figure 2.18 Variations of Natural Frequencies with the Disk Rotation Speed [One-Pin-RimGuided Spline Saw, Blade 17″×6″×0.060″] ........................................................................... 44 Figure 2.19 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [One-Pin-Rim-Guided Splined Saw, Blade 17″×6″×0.060″] ..... 45 Figure 2.20 Non-Dimensional Transverse Displacement (Displacement ‘w’ is divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [One-Pin-Rim-Guided Splined Saw, Blade 17″×6″×0.060″] ..... 46 Figure 2.21 Variations of Natural Frequencies, and Real Part of Eigenvalues, Three Configurations......................................................................................................................... 47 Figure 3.1 Schematic of the Experimental Setup ................................................................ 51 Figure 3.2 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [Full-Guided Splined Saw, Blade17-40-60] ........................................................................... 55 Figure 3.3 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [Full-Guided Splined Saw, Blade17-40-60] ........................................................................... 55 Figure 3.4 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] ............................................................. 56 ix Figure 3.5 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] ............................................................. 56 Figure 3.6 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] ............................................................ 57 Figure 3.7 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] ............................................................ 57 Figure 3.8 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [All Configurations, Blade17-40-60] ...................................................................................... 58 Figure 3.9 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [All Configurations, Blade17-40-60] ...................................................................................... 59 Figure 3.10 Variations of Natural Frequencies with Speed, Air Jet OFF [Full-Guided Spline Saw, Blade17-40-60] ................................................................................................... 60 Figure 3.11 Variations of Natural Frequencies with Speed, Air Jet ON [Full-Guided Spline Saw, Blade17-40-60] .............................................................................................................. 60 Figure 3.12 Variations of Natural Frequencies with Speed, Air Jet OFF [One-Pin-EyeGuided Spline Saw, Blade17-40-60] ...................................................................................... 61 Figure 3.13 Variations of Natural Frequencies with Speed, Air Jet ON [One-Pin-EyeGuided Spline Saw, Blade17-40-60] ...................................................................................... 61 Figure 3.14 Variations of Natural Frequencies with Speed, Air Jet OFF [One-Pin-RimGuided Spline Saw, Blade17-40-60] ...................................................................................... 62 Figure 3.15 Variations of Natural Frequencies with Speed, Air Jet ON [One-Pin-Rim- Guided Spline Saw, Blade17-40-60] ...................................................................................... 62 x Figure 3.16 Variations of Natural Frequencies, Experimental and Analytical Results, Three Guided Configurations ............................................................................................................ 66 Figure 3.17 Waterfall Plot, Air Jet OFF [Full-Guided Splined Saw, Blade17-40-60] ....... 68 Figure 3.18 Waterfall Plot, Air Jet ON [Full-Guided Splined Saw, Blade17-40-60] ......... 68 Figure 3.19 Waterfall Plot, Air Jet OFF [One-Pin-Eye-Guided Splined Saw, Blade17-40- 60] ........................................................................................................................................... 69 Figure 3.20 Waterfall Plot, Air Jet ON [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] ................................................................................................................................................. 69 Figure 3.21 Waterfall Plot, Air Jet OFF [One-Pin-Rim-Guided Splined Saw, Blade17-4060] ........................................................................................................................................... 70 Figure 3.22 Waterfall Plot, Air Jet ON [One-Pin-Rim-Guided Splined Saw, Blade17-40- 60] ........................................................................................................................................... 70 Figure 3.23 Impulse Response at 2000 rpm [Full-Guided Spline Saw, Blade 17-40-60] .. 74 Figure 3.24 Impulse Response at 2000 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 74 Figure 3.25 Impulse Response at 2000 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 74 Figure 3.26 Impulse Response at 2400 rpm [Full-Guided Spline Saw, Blade 17-40-60] .. 75 Figure 3.27 Impulse Response at 2400 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 75 Figure 3.28 Impulse Response at 2400 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 75 Figure 3.29 Impulse Response at 2800 rpm [Full-Guided Splined Saw, Blade 17-40-60] 76 xi Figure 3.30 Impulse Response at 2800 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 76 Figure 3.31 Impulse Response at 2800 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 76 Figure 3.32 Impulse Response at 3200 rpm [Full-Guided Splined Saw, Blade 17-40-60] 94 Figure 3.33 Impulse Response at 3200 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 77 Figure 3.34 Impulse Response at 3200 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 77 Figure 3.35 Impulse Response at 3600 rpm [Full-Guided Splined Saw, Blade 17-40-60] 78 Figure 3.36 Impulse Response at 3600 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 78 Figure 3.37 Impulse Response at 3600 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17- 40-60] ...................................................................................................................................... 78 Figure 4.1 Guided Splined saw, (a) Full-Guided, (b) Pin-Eye-Guided, and (c) Pin-Rim- Guided ..................................................................................................................................... 82 Figure 4.2 Cants Cut 3No. 2x10 SPF Boards 8ft long ........................................................ 83 Figure 4.3 X-axis of the Graphs for the Deviations of Cut Surface.................................... 84 Figure 4.4 Averaged Cutting Results at 2000 rpm for the 17-40-60 Blade ........................ 85 Figure 4.5 Averaged Cutting Results at 2400 rpm for the 17-40-60 Blade ........................ 86 Figure 4.6 Averaged Cutting Results at 2800 rpm for the 17-40-60 Blade…………….…87 Figure 4.7 Averaged Cutting Results at 3200 rpm for the 17-40-60 Blade ........................ 88 Figure 4.8 Averaged Cutting Results at 3600 rpm for the 17-40-60 Blade ........................ 89 xii Figure 4.9 Top Probe Standard Deviation at Different Speeds for the 17-40-60 Blade, Average of 5 Different Cuts for Three Guided Configurations .............................................. 90 Figure 4.10 Cut Profile Results at 2000 rpm, for the 17-40-60 Blade ................................ 92 Figure 4.11 Cut Profile Results at 2400 rpm, for the 17-40-60 Blade ................................ 93 Figure 4.12 Cut Profile Results at 2800 rpm, for the 17-40-60 Blade ................................ 94 Figure 4.13 Cut Profile Results at 3200 rpm, for the 17-40-60 Blade ................................ 95 Figure 4.14 Cut Profile Results at 3600 rpm, for the 17-40-60 Blade ................................ 96 xiii List of Symbols, and Abbreviations a Inner radius of the disk b Outer radius of the disk , , , , , Coefficients of the in-plane stress terms Coefficients of the in-plane stresses terms in normalized equation ⁄12 1 D Disk rigidity ( E Young’s modulus F Applied external point forced h Disk thickness i Complex number √ 1 k Stiffness of the spring m Number of nodal diameters M Maximum numbers of nodal diameters in approximation function n Number of nodal circles N Maximum numbers of nodal circles in approximation function , A mode with n and m nodal circles and diameters, respectively , , Space-fixed polar coordinate system Eigenvalues for the transverse displacement of the disk Eigenfunctions Mode shape in the radial direction for the disk deflection , Equilibrium solutions for the amplitude of the and waves of the transverse displacement xiv t Time Displacement in z direction Kronecker delta η The ratio of inner radius to outer radius of the disk (a/b) υ Poisson’s ratio Mode shape in the direction Mass density , Radial and hoop stress due to rotation Ω Rotation speed (rad/sec) FTW Forward travelling wave BTW Backward travelling wave RTW Reflected travelling wave xv 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 grateful to Dr. Ramin M.H. Khorasany, for his help during all the numerical analysis. I would also like to record my gratitude to Mr. John White, from FP innovation center, for his assistance during all the experimental tests. I am deeply indebted to my wife, Saideh. Her unlimited love, patience and support have made this work possible. xvi Dedication To My Lovely Wife, Saideh xvii Chapter 1: Introduction 1.1 Background (Clamped Saws vs. Guided Splined Saws) Rotating disks are a common element in modern rotating machinery such as grinding wheels, turbine rotors, brake systems, fans, computer disks memory units, wafer slicing cutters, and circular saws. In all of these applications the disk may be subjected to severe vibration which can lead to poor output and production of the machine; or if maintained over long periods, may lead to fatigue failure. One particular application of rotating disks is in saw mill industries. Saw blade vibration directly contributes to production problems such as poor cutting accuracy, poor surface quality, and excessive raw material waste. In general, during the specification of a sawing process wood surface finish and dimensional accuracy, feed rate, and amount of sawdust production compete for attention. Hence, an efficient sawing process is sought with the objective to reduce saw blade vibrations. The focus of the current study is on the study of vibration characteristics of circular saws. For primary breakdown, two types of circular saws are widely used in the wood product industry: “clamped saws” and “guided splined saws”. Clamped saws are no longer widely used in North America for primary breakdown. In a clamped saw, the saw is clamped by a central collar to the arbor (Figure 1.1). Splined guided saws have received more attention in the wood milling industry in the past decades and are widely used in North American wood industry. This saw fits loosely (the clearance between the inner spline of the blade and the spline of the arbor is about 0.01inch) on a splined arbor and this arbor provides the driving force to the blade. The location of the blade is determined by the position of guide pads which are supported independently from the blade by two arms that are fixed to the saw frame (Figure 1.2). A combination of water and air is used to provide lubrication and cooling 1 between the blade and the pads. The clearance between the guides and the saw blade is the single most important factor controlling guide lubricant demand, and is more important than guide size. For reduced guide water demand, the guide clearance should be as small as possible compatible with guide setup inaccuracies and saw blade lack of flatness. A suitable range of clearance is 0.002-0.004 in. Figure 1.1 Clamped Saw Figure 1.2 Guided Splined Saw It has been found that saws which are free on the arbor and restrained from lateral motion during cutting by guide-pads, lubricated by air and water, known as “ guided splined saws”, provide superior cutting performance to clamped saws. 2 1.2 Previous Research on the Vibration Characteristics of Rotating Disks There are numerous studies on the dynamic characteristics of rotating disks. A large portion of the existing research in the literature is concerned with the vibration characteristics of clamped circular disks. The number of studies concerned with guided splined saws is limited. Hutton, Chonan, and Lehmann [1] studied analytically the vibration characteristics of elastically constrained clamped spinning disks, subjected to excitation produced by fixed point loads. They examined the dynamic response characteristics of rotating circular disks when subjected to forces produced by stationary point springs. They investigated the effect of spring stiffness, number and location of springs on the frequency speed characteristics of the disk. Chen and Hsu [2] analytically investigated the forced response of a spinning disk under the application of a space fixed couples using the eigenfunction expansion method. They studied the effect of external damping on the transient vibration and the steady state deflection of the spinning disk. They considered a general couple on the disk surface as a superposition of two components. One of the components was pitching, which was in the radial direction, and the other one was a rolling component in the circumferential direction. Chen and Wong [3] examined the instability of a spinning disk with translational degrees of freedom by finite element calculation. They analytically investigated the effect of evenly space-fixed springs on the divergence instability of the disk by using a three-mode eigenfunction approximation of stationary disk. The disk was assumed to have clamped-free boundary conditions. 3 Mote [4] investigated the effect of a collar (which was allowed to move freely on the arbor) on the stability of a guided rotating disk. He studied the stability of the disk, subjected to a concentrated load moving at uniform speed. Yang [5] studied the vibration of a spinning disk with rigid body translational and tilting degrees of freedom. He assumed that the disk has free-free boundary conditions. In particular he investigated the effect of rigid body degrees of freedom on the stability of guided disks. He concluded that due to the coupling effect, stable operation of the disk beyond critical speed is possible, and the disk loses its stability to flutter. Chen and Body [6] investigated the effect of rigid body tilting on the natural frequencies and stability of rotating disks, subjected to a rotating load system. Price [7] studied analytically the dynamic characteristics of a rotating disk that is free to translate while its central clamping collar remains perpendicular to the axis of rotation. He studied the effect of a rigid body translational degree of freedom on the dynamic response of rotating disks. Khorasany and Hutton [8] investigated analytically the stability characteristics of guided spinning disks having rigid body translational degrees of freedom. They used a three mode approximation around the critical speed. They concluded that for a disk that is constrained with one space-fixed spring while it has rigid body translational degrees of freedom, the divergence instability (characterized by having positive real values for the eigenvalues) does not occur. Tobias and Arnold [9] were among the first researchers who experimentally investigated the vibration characteristics of imperfect spinning disks with clamped-free boundary conditions. They recorded the amplitude response of a spinning disk, in the region 4 of its critical speeds, while subjected to a space-fixed external force. They observed that a stationary wave develops in the region of the critical speed of the mode being excited and then collapses at some higher speed whose value is dependent upon the magnitude of the applied load. Raman and Mote [10] conducted experiments on the nonlinear vibrations of a rotating disk close to its critical speed. They used experimentally obtained results to investigate the vibration response of an imperfect disk, induced by point masses bolted to the plate, spinning near to its critical speed. Kang and Raman [11] investigated experimentally the vibration and acoustic oscillations of a rotating disk coupled to surrounding fluids in an air-filled enclosure. They measured the transverse vibration of the disk in a specially designed enclosure. They observed the rotating disk flutter instability of reflected traveling waves at super critical speed. They showed that disk flutter does not occur by mode combination but by a dampinginduced instability leading to the flutter of a single reflected traveling wave. From their experiments they showed that the acoustic modes split into forward and backward traveling waves, as the disk rotates. D’ Angelo, C., Mote, C.D. [12] described experimentally the aerodynamically excited supercritical disk vibration. They showed experimentally that a single super-critical backward wave becomes unstable. They observed at the flutter speed and at higher speeds the fluid and the disk are strongly coupled. Coupling becomes sufficiently strong that the frequency after flutter becomes nearly independent of rotation speed, even though the disk stiffens centrifugally as speed increases. 5 Thomas, O. et al [13] investigated 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. They used an experimental set-up which allows one to measure the vibration amplitudes of the preferential configurations of an asymmetric modal shape. Jana and Raman [14] presented experimental results on the nonlinear aero-elastic flutter phenomena of a flexible disk rotating in an unbounded fluid. They conducted the experiments over a wide range of rotation speeds in the post-flutter region. Based on their experimental results they confirmed the existence of a primary instability of a reflected travelling wave of the disk, followed by a secondary instability. They also observed that at speeds exceeding the onset of flutter instability different phenomena occur such as: complicated nonlinear response, bifurcation points, and frequency lock-ins over certain rotation speed ranges, the presence of higher harmonics of the unstable wave and large frequency and amplitude hysteresis, as the disk speed is ramped up and down across these bifurcations Raman et al [15] investigated experimentally the dynamic response of a thin, flexible disk spinning in a closed air-filled chamber, beyond the beginning of aero-elastic flutter. They concluded that a primary instability leads to the bifurcation of the flat equilibrium to a finite amplitude backward travelling wave. Then a secondary instability causes this travelling wave to jump to a large amplitude frequency locked, traveling wave vibration. Khorasany and Hutton [16] conducted experiments to investigate the effect of large deformations on the frequency characteristics of spinning clamped disks. The disks in their investigations had clamped-free boundary conditions and were under the application of a space-fixed external force provided by an air jet. They noted that depending on the level of 6 nonlinearity, some of the frequency paths around the critical speed level off and maintain a nearly constant level. They also noted that at the speed at which the stationary wave collapses, a sudden change in the measured frequencies is notable. Khorasany and Hutton [17] numerically investigated the effect of geometrical nonlinear terms on the amplitude and frequency characteristics of a spinning disk. To do so, they discretized the governing equations of motion, found the equilibrium solutions, and linearized the governing equations around these solutions. They showed that there is a close correspondence between the numerically and experimentally obtained results. Lee and Kim [18] used the directional frequency response functions in a modal testing method of rotating disks which was proposed for separation of the forward and backward traveling wave modes and identification of the diametrical node numbers associated with modes of interest. They conducted experiments with a rotating disk to verify the theoretical findings and the practicality of the proposed method. Ahn, and Mote Jr. [19] presented a modal testing method which permits identification of the natural frequencies, the number of nodal diameters and the wave motion in a rotating disk. They conducted experimental tests for both symmetric and non-symmetric disks to show the influence of disk rotation speed on the prediction of mode shapes. Tian and Hutton [20] developed an approach to predict the physical instability mechanisms that are involved in the interaction between a rotating flexible disk and a stationary constraining system. They presented instability conditions for various lateral interactive forces using an energy flux analysis. 7 Vogel and Skinner [21] developed the frequency determinants for various combinations of boundary conditions associated with the transverse vibrations of disks. They derived the values of the resonant frequencies for different normal modes. Chen and Wong [22] studied analytically the vibration and stability of a spinning disk with axial spindle displacement in contact with a number of fixed springs. They used a finite element method to calculate the eigenvalues of the coupled system. In their study they focused on the behavior of modal interactions when two modes are almost degenerate. They concluded that when the frequencies of the neighboring modes are away from zero, the behavior of modal interaction is similar to those of a conventional disk without axial translation. Chen and Bogy [23] studied the interaction of the modes at sub-critical and supercritical speeds. They categorized modal interactions to be four and they 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 suband super-critical speeds. They concluded that the interaction between the natural frequencies of backward and reflected travelling waves causes flutter type instability for an elastically constrained disk. When flutter instability happens, the real parts of the eigenvalues have positive parts. Another source of instability for an elastically constrained disk is at the location of the critical speed. This type of instability is called “divergence instability.” When the divergence instability occurs, there is one eigenvalue for the system with a zero imaginary part and a positive real part. 8 Chen et al [24] investigated the possibility of secondary resonance of a spinning disk under space fixed excitations. Deqiang and Suhuan [25] studied the effects of the location, number and stiffness of guide pads on the lowest critical rotational frequencies of the rotating circular saw clamped at the inside and free at the perimeter. They considered the influence of the centrifugal force and the cutting temperature in their analysis. They concluded that the introduction of one guide does not stably increase critical rotational speeds of the circular saw. Chen J.S. [26] investigated the natural frequencies and stability of a spinning elastic disk subjected to a fixed concentrated edge load. He concluded this load decreases the natural frequencies of the forward and backward traveling waves, but it increases the natural frequency of the reflected wave. Tian , and Hutton [27] investigated modal testing of a rotating disk interacting with a stationary restraint. They proposed a practical and effective identification method called the artificial damping method to identify the traveling wave modes for rotating disks based on forced or self-excited resonant responses measured by two displacement probes. Young et al [28] studied the free vibration of a rotating clamped disk under the constraint of an elastically fixed space oscillating unit. The unit he considered was composed of two parallel combinations of springs and dampers attached above and under a mass. He studied the flutter type instabilities imposed by these two units. 1.3 Review of Patents on Guided Saws In the process of shifting from clamped saws to guided splined saws in North America, a number of attempts have been made to improve cutting accuracy and efficiency by restraining the lateral motion of the saw blades using different guide systems. Table 1.1 9 summarizes attempts which have been patented by US Patent Office [29]. In all of these works the blade is either restrained laterally by guides having points of supporting contact proximate to the gullet line (at the rim of the blade or inner tooth edge of the saw blade) or it is restrained by a guide pad which covers a large portion of the blade such as a half or quarter surface of the blade. Table 1.1 Different Guided Systems for Splined Saws US Patent Summary and claims of the No. Figure No. invention A machine setup that allows the blade to float axially to some small degree upon its arbor. The 32853021 1 blade then is restrained laterally by a guide means having points of supporting contact on either side of the blade and proximate the gullet line or inner tooth (1966) edge of the saw blade. In large diameter saws (greater than 24″ diameter) guide means disposed near the top or uppermost peripheral edge of the saw blades to give lateral support above the cut. In the Figure, blade 12 is laterally supported by guide block of 55 having 32853022 2 contact point of A and B and a guide block 56 which provides a contact point C. Point of contact C should be on the forward side of the blade and (1966) preferably close to the gullet line of the blade. 10 US Patent Summary and claims of the No. invention No. Figure This invention is particularly concerned with the problem of mounting circular saws upon an arbor to provide axial floating looseness but yet having a positive drive connection with its supporting arbor that is capable of withstanding the shear forces. In combination with saw guiding apparatus for engaging a saw to its gullet line, an improvement in 3516460 3 mounting a circular saw upon an arbor, comprising: an arbor formed with a plurality of splines arranged (1970) circumferentially in a symmetrical pattern, each spline having a sloped surface relative to a diameter through the geometric center of said spline; and a circular saw formed with a central opening including a plurality of recess complimentary to the periphery of said arbor. A sufficient number of splines must be used to withstand the shear forces which may be expected during use of the saws. The saw guide apparatus has guide arms to be positioned on opposite sides of a saw blade, and spaced slightly from the surfaces of the blade. These arms have surfaces opposed to the blade faces, and provided means for directing a liquid through and 3623520 4 outwardly from these surfaces so as to apply films of the liquid on to the faces of the blade. (1971) 11 US Patent Summary and claims of the No. invention No. Figure The accumulation of sawdust eliminated the clearance between the saw blades and guides, causing a heating of the blades. These undesirable results can be eliminated by providing a plurality of contacts on the supports arms which cooperate in a 3489189 5 manner to inhibit the buildup sawdust. A saw guide comprising a pair of support arms (1970) located on opposite sides of saw blade and having replaceable wear guides supported upon each arm, respectively, the improvement means are provided on each support arm for supporting wear guides as to facilitate the removal of sawdust from the work area. Almost all circular saw guides employed in industrial operation are single guide blocks which support the saw blade either above or below the work piece. The support of the saw blade can be 4977802 6 (1990) further increased by using double or triple guide system. But the alignment of multiple guides has proved to be difficult. The present invention provides a mounting and alignment system for a second guide beyond a first fixed support guide for a circular saw blade. A principal objection of this invention is to provide a saw blade guide and damper unit so as to 4854207 7 reduce blade vibration and in turn provide a smoother cut as well as reduce noise. (1989) 12 US Patent Summary and claims of the No. invention No. Figure An apparatus comprises a blade sits freely on the arbor and a pair of guide, each of the plates have a plurality of spiral grooves formed in its face. The grooves have open loading ends, relative to 3961548 8 direction of rotation of the saw, opening out of edges of the guide plates so that rotation of the saw (1976) blade induces a flow of air into the groove. The grooves are effectively closed at trailing ends so as to develop air cushions between the saw blade and the guide faces and thus stabilize the saw blade between the guide plates. This invention comprises guides near the gullet line of the blade. Each guide comprises a pair of control pads secured in opposed relationship. The 3703915 9 head surfaces of the pads are recessed to form fluid retaining cups which confine a lubricating cushion (1972) of fluid adjacent the sides of the saw blades. In one preferred arrangement, two sets of saw guides are used, the sets being separated between 60º and 120º along the outer periphery of the blades. This invention teaches an improved plug construction with simplified securing means so as to attain nice adjustment of clearance of the plugs from the saw. The inventor has found that dimensionally opposite 3772956 10 (1973) plug pairs supplemented by a third plug pair some 70º from a leading plug pair, gives improved damping and, consequently, improved dimensional control. 13 US Patent Summary and claims of the No. invention No. Figure A pair of saw guides having flat, parallel faces are mounted in opposed relationship to one another with a saw blade sandwiched between them. Near the center of each guide face is a port through which a pressurized fluid, which is preferably a finely atomized oil-air mixture, is introduced into the respective 3918334 11 (1975) gaps, such fluid flowing radially outwardly from the port through the gap toward the periphery of the respective guide face. The predetermined size of each gap is such that, for the particular pressure and flow of fluid involved, regions of sub-atmospheric pressure pursuant to the Bernoulli effect are created on either side of the saw blade in the respective gaps, the suction tending to increase if a respective gap widens and decrease if the gap narrows due to blade deflection. This invention relates to an improved method for guiding saw blades, the improvement in the guides resulting in improved cutting accuracy, surface finish, and reduction of kerfs. The guiding means comprises 3 guides shoes through which the saw blade travels. An external 3674065 force is exerted on one or both of the guide shoes by compressed springs or other equivalent means. A 12 (1972) fluid medium, such as compressed air, water or airwater mixture is introduced under pressure to an interior recess of each of the guide shoes. 14 US Patent Summary and claims of the No. invention No. Figure This invention is a method to extract energy from the lateral motion of the saw blade, similar in a way to the broom handle means, but without any frictional drag. Whereas conventional damping means have attempted to extract vibration energy by enhancing the internal damping of the solid structure of the blade or its support. The present invention is directed toward enhancing the radiation of energy from the sides of the blade by a better 4323145 13 (1982) coupling to the surrounding air medium without making any physical connection with the blade except through the air itself. In this way a strong coupling is achieved between the damping means and the vibration motion of the saw blade and substantially complete decoupling is achieved between the damping means and the circular motion of the saw blade. A large portion of the saw blade must be covered for maximum effectiveness, about ¼ of circumference and ½ of the radius area. 1.4 Objective and Scope Investigation of the vibration characteristics of guided circular saws is of particular interest in the current study. This thesis investigates the dynamic behavior of guided circular saws first by numerical investigation of rotating disks and then by conducting idling and cutting experimental tests of splined saws with different guide configurations. The three main guided configurations which are investigated in the thesis are shown below: 15 Figure 1.3 Guided Splined Saw, (a) Full-Guided, (b) Pin-Eye-Guided, and (c) Pin-Rim-Guided The main objective of this investigation is to analyze the idling and cutting characteristics of a guided splined saw with these three guide configurations. First, the idling characteristics of guided splined saws were studied analytically. Experimental idling tests were also conducted for these guide configurations. Then cutting characteristics of the guide configurations were investigated experimentally. Cutting forces considerably add to the complexity of saw blade behavior in many subtle and inter-related ways. Therefore, studies of guided saws in the past have been primarily confined to the study of idling conditions. In the current work, the cutting characteristics of the abovementioned guide configurations are investigated experimentally. In this regard, a series of cutting tests were conducted with these guided configurations to measure the sawing performance of the different guide systems. This thesis is presented in five chapters. Chapter 2 is devoted to an investigation of the analysis of the dynamic behavior of spinning disk. For numerical investigations, the governing linear equation of transverse vibration for a thin disk is used. The Galerkin method, considering the orthogonality property of the eigenfunction of a stationary disk, is used for solving the equation. As a primary interest, variations of natural frequencies with rotation speed of the disk and disk response to applied external force is calculated. Also the 16 steady state response of the disk at different speeds is calculated. In this part the vibration behavior of a clamp-free disk (clamped at inner radius and free at outer radius) and the vibration behavior of a free-free disk (free at the inner and outer radii) are studied. The effect of lateral elastic constraints is investigated in this part. Then we idealized a saw blade as a perfect disk; and the dynamic behavior of guided saws is investigated analytically. The frequency-speed characteristics of the saw and the effect of the guide and its position are investigated in this chapter. Chapter 3 is concerned with experimental results of guided saws with different guide configurations. An electromagnet was used to generate random white noise excitation. Using inductance displacement probes, the frequencies and amplitudes of the disk vibrations were measured and the mean deflections were calculated. In this chapter the variations of frequencies and their mode shapes as a function of the rotating speed of the saw are presented. The effect of applying an external force at different rotating speeds on the stability of the saw is investigated experimentally. Chapter 4 presents the experimental results of cutting tests for different guide configurations and different feed speeds. A series of cutting tests have been conducted to examine the effect of different guide configurations on cutting accuracy. At different speeds, below and above critical speeds (supercritical speeds), cutting tests have been conducted for different guide configurations. By comparing the cutting results the guide configuration which results in the best cutting accuracy is determined. Chapter 5 presents the summary and conclusions. This chapter also provides suggestions for future work on this subject. 17 Chapter 2: Analytical Investigation of Idling Characteristics of Guided Saws 2.1 Introduction This chapter presents analytical results of the dynamic behavior of rotating disks. The main focus of this chapter is the investigation of idling characteristics of guided saws. In order to study the vibration characteristics of guided saws analytically we idealized a saw blade as a perfect disk. So, in all analysis the saw is idealized as a perfect disk with a circular inner and outer edge, and the teeth are neglected. Hence, first we attempt to understand the vibration characteristics of spinning disk. First the linear equation of motion and its solution for a spinning disk is developed. Then the vibration behavior of a clamped-free disk (clamped at inner radius and free at outer radius) and then the vibration behavior of a free-free disk (free at the inner and outer radii) are studied. Then the effects of a lateral elastic constraint on the vibration characteristics of camped-free and free-free disks are investigated. Considering a saw blade as a disk, the dynamic behavior of guided saws is investigated analytically. The frequency-speed characteristics of a saw and the effect of guide configurations are investigated. 2.2 Linear Equation of Motion Based on the linear theory of vibration (cancelling out higher order terms in the strain-displacement relationship) the following equations of motion can be developed. Consider an axisymmetric disk with an inner radius of ‘a’ and outer radius of ‘b’ and uniform thickness ‘h’ which is small compared to ‘b’ and is rotating with the angular velocity 18 ‘Ω’. The governing linear equations of transverse vibration in terms of lateral displacement are developed with respect to a space fixed coordinate system w(r, θ, t) (Figure 2.1). Figure 2.1 Schematic of Rotating Disk 2.2.1 Formulation The rotation of the blade about its central axis causes in-plane axisymmetric radial and circumferential stresses, and respectively. They are determined by solving the following equations: ⁄ Where ⁄ ⁄ Ω 0 is the mass density, and 0 (2.1) (2.2) and are the in-plane radial and circumferential strains respectively [30]. By solving the above equations the stress distribution in the disk is: Ω Ω (2.3) 19 The constants in equation (2.3) are determined based on the boundary condition of the problem. Therefore for the case of a clamped-free disk, it is assumed that the disk is clamped along the inside while it is free at the outside edge. In this case the constants are calculated as: – – (1.4) And for the case of a disk free at the inner and outer radii, the constants are: (2.5) The governing linear equation of transverse vibration in terms of lateral displacement, with respect to a space fixed coordinate system, may be expressed in the form [1]: , 2Ω , Ω , , , , (2.6) In this equation w(r, θ, t) is the lateral displacement, ‘f’ is the space fixed applied force per unit area that can be either an external applied force or the force that is coming from the 20 guides, h is the thickness, ρ is the mass density, Ω is the rotating speed and D is the flexural rigidity where: E is the Young’s modulus and υ is the Poisson’s ratio. And Non-dimensional parameters are introduced as: , , (2.7) Where Based on the introduced parameters: Ω Using the non-dimensional parameters into equation (2.6), we obtain the non-dimensional equation of motion as: , 2Ω , Ω 2 , 4 Ω 2 2 3 , , , (2.8) Where 3 Appendix A provides mathematical calculations of the non-dimensional equation of motion. 21 2.2.2 Solution Solution of equation (2.8) can be obtained by application of the Galerkin method. In this solution method the eigenfunctions of the stationary disk problem in the polar coordinate system is used as the approximation functions for the Galerkin method. Considering the orthogonality property of the eigenfunctions of a stationary disk simplifies the solution procedure. First we calculate the eigenfunction of a stationary disk. The non dimentionalized stationary equation of motion for a plate in polar coordinate system is: (2.9) Using the boundary conditions we can obtain the eigenvalues associated with the eigenfunctions as: [31] mθ (2.10) Where sin cos , and Where , , , and are mth order Bessel functions and modified Bessel functions. The orthogonality conditions of the eigenfunctions are: , , Where (2.11) is the Kronecker delta. 22 , , , and are constants which can be determined by the normalization condition for the first equation of (2.11) as: 1 (2.12) 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: ∑ , , cos sin , Consider equation (2.8) while 0. Substituting of , , (2.13) from equation (2.13) into equation (2.8) results in: 2Ω 4 2Ω 4 2Ω 2 Ω 2Ω 2 Ω 2 2 2 2 ∑∞ 0 П 0 ∑∞ 0 П 0 (2.14) Where П 0 (2.15) Appendix B presents the mathematical details for obtaining equation (2.14) Equations (2.14) constitute a system of simultaneous differential equations in , , while m, n = 0, 1, 2 . . . the solution of which determines the transverse displacement of the disk via equation (2.13). 23 Now consider that a number of guides constrain the disk laterally. If we model guides with a spring dashpot [1], the spring systems produce a lateral force. If we consider guide (i) in , position , then the force produced by this guide is: (2.16) In this case, the system of simultaneous differential equation (equation 2.14) becomes: 4 2Ω ∑ 2Ω 2 Ω 2Ω 2 Ω 2 2 ∑∞ 0 П ∑∞ 0 П sin 4 2Ω ∑ 2 2 cos (2.17) Where ∑ sin , cos Appendix B provides all the mathematical details for obtaining the equations (2.17) Equations (2.17) constitute a system of simultaneous differential equations in , , while m, n = 0, 1, 2 . . . the solution of which determines the transverse displacement of the guided disk via equation (2.13). MATLAB has been used to solve the equations numerically. 24 2.3 Dynamic Characteristics of Clamped Saws In this section the dynamic behavior of clamped circular saws is investigated analytically. In all analysis the saw is idealized as a perfect disk with a circular inner and outer edge, and the teeth are neglected. So, for the entire analytical investigation we use a perfect disk with the physical properties which are summarized in Table 2.1. The boundary conditions are considered to be clamped at the inside and free at the perimeter. Table 2.1 Physical Properties of the Disk Used for all the Analytical Investigations Property Value Outer Diameter (in) 17 Inner Diameter (in) 6 Thickness (in) 0.060 Density ( ⁄ 7800 Young’s Modulus (G.Pa) 203 Poisson’s ratio 0.3 2.3.1 Natural frequencies of clamped-free disks The frequency response (as measured by a stationary observer) of a rotating disk, as a function of speed is calculated. The solution is assumed to consist of contributions of all modes up to and including 0-3 nodal circles and 0-6 nodal diameters (N, M) = (3, 6). Results are presented for disk response at constant speeds between 0-4000 rpm. Figure 2.2 indicates the variations of natural frequencies of the disk. The modes are shown by , , where ‘n’ refers to the number of nodal circles and ‘m’ refers to the number of nodal diameters. Corresponding to each mode there are two waves travelling around the disk. 25 Forward travelling waves which travel in the direction of rotation, and backward travelling waves which travel in the opposite direction. The Natural frequencies of the backward and forward travelling waves of each mode are the same when the disk is stationary. Once the disks starts to rotate, the natural frequency of forward travelling waves, measured by a stationary observer, increase, and the natural frequency of backward travelling waves decrease. The modes having more than one nodal diameter decrease until a speed at which the measured natural frequency is zero. Figure 2.2 Variations of Natural Frequencies with the Disk Rotation Speed, Numerical Results [Clamped-Free Disk, OD17″, ID6″, and Thickness 0.06″] This speed is called a critical speed of the disk. At this speed a constant force can resonate the disk and the amplitude of motion is only limited by the damping in the system. For this 26 disk, the first three speeds at which the frequency is zero are 3076 rpm, 3485 rpm, and 3645 for modes (0,3), (0,4), and (0,2), respectively. 2.3.2 Steady State Forced Response Due to a Space Fixed Constant Force In this section the steady state response of the disk (the deformation pattern of the disk is time invariant) is calculated. Figure 2.3 illustrates the behavior of a clamped-free disk subjected to a constant lateral force, applied at the rim, as the rotational speed increases. As may be noted the lateral displacement increases drastically as the critical speed is approached. In all cases the applied load was the same (10N). As can be noted, as the first critical speed is approached the deflection is dominated by mode (0, 3). The graphs show the non-dimensional deflection of the disk which is the deflection ‘w’ divided by the disk thickness ‘h’. 27 Figure 2.3 Deflection of the Disk as a Function of Angle Around the Rim for Constant Point Load Applied at the Rim, at 1.5 rad for Different Rotating Speed: a) Ω=0, b) Ω=2000rpm, c) Ω=2800rpm, d) Ω=3000rpm, e) Ω=3050rpm, f) Ω=3076rpm (a) (b) (c) (d) (e) (f) 28 2.3.3 Effect of Transverse Elastic Constraints on the Disk Frequency-Speed Characteristics In this section, the dynamic behavior of a clamped-free disk constrained by a transverse fixed spring placed at the rim is considered. The intent of this study is to investigate the effect of spring stiffness on the frequency speed characteristics of the rotating disk. First, the approximate stiffness of the disk is measured experimentally. Consider the disk with the physical properties in Table 2.1 clamped at the inner radius. We applied a static force at the outer rim, and measured the deflection at the location of the applied force. The result stiffness was 8500N/m which agrees well with the analytical calculated stiffness. Figure 2.4, Figure 2.5, and Figure 2.6 show the variations of the calculated natural frequencies and the real part of the eigenvalues with rotation speed for a clamped disk which is constrained laterally by a linear spring of stiffness 5000, 10000, and 50000N/m respectively. Figure 2.4 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 5000N/m, Dash Line Unconstrained Clamped-Free Disk) 29 Figure 2.5 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 10000N/m, Dash Line Unconstrained Clamped-Free Disk) Figure 2.6 Variations of Natural Frequencies with Disk Speed, and Real Part of the Eigenvalues (Solid Line Clamped-Free Disk with a Lateral Elastic Constraint at the Rim by a Linear Spring 50000N/m, Dash Line Unconstrained Clamped-Free Disk) As Hutton, et al, presented in their study [1] it is evident in all cases that at zero rotational speed the presence of a spring causes the single natural frequency of the unconstrained case to be replaced by two natural frequencies. The lower of these corresponds exactly to the unconstrained case and involves a pure plate mode of vibration with the spring 30 located at a nodal line. The upper frequency corresponds to the case where the spring is aligned in such a position to produce a maximum value for the frequency. It can be noted that unlike the unconstrained case, in which the frequency lines cross over each other, in the constrained case the frequency lines veer away and modal coupling is clear. Frequencies follow the path of the closest mode of the unconstrained case but change direction in favor of an intersecting mode at cross over points. It is evident that for all values of spring stiffness the curves all pass through the intersection point of the frequency lines for the unconstrained disk. From this it may be concluded that the modes corresponding to such points have a node at the spring location, hence the presence of constraint has no effect on the frequency of the system and it is in accord with what Hutton and Lehmann concluded in their study [1]. It is obvious that the presence of one point constraint does not change the critical speed. Therefore, using one point guide for clamped disk does not change the critical speed. This fact is in accord with Rayleigh’s interlacing theorem. Interlacing theorem predicts the changes in natural frequencies of a vibrating structure due to the introduction of a constraint to a structure. According to the theorem: by adding a spring of even infinite stiffness one cannot increase the nth resonant frequency of the new structure beyond the (n+1)th resonant frequency of the original structure[32]. So, in the case of circular saws, introduction of one guide cannot significantly change the critical speed [1]. 31 2.4 Dynamic Characteristics of Free-Free Disks In this section the dynamic behavior of a free-free disk is investigated analytically. The boundary condition is considered to be free at the inside radius and free at the perimeter. 2.4.1 Natural Frequencies of Free-Free Rotating Disks The frequency response (as measured by a stationary observer) of a rotating free-free disk, as a function of speed is calculated. The solution is assumed to consist of contributions of all modes containing up to 0-3 nodal circles and up to 0-6 nodal diameters (N, M) = (3, 6). Results are presented for disk response at constant speeds between 0-4000 rpm. Figure 2.7 shows the variation of natural frequencies of the disk (A free-free disk with the physical properties in Table 2.1). For this disk, the first three speeds at which a frequency is zero are 2350 rpm, 3240rpm, and 3900 rpm. To illustrate this issue, the deflection of the disk when a constant force of F= 10N applied on outer rim has been calculated. At the critical speeds, the displacement result is large (W/h >>10), whence the name critical speeds. 32 Figure 2.7 Variations of Natural Frequencies with the Disk Rotation Speed, Numerical Results [FreeFree Disk, OD17″, ID6″, and Thickness 0.06″] 2.4.2 Effect of Transverse Elastic Constraints on the Disk Frequency-Speed Characteristics Consider a free-free disk in which a spaced fixed linear spring constrains its lateral motion at one point of its outer rim. Figure 2.8 shows the variations of natural frequencies and the real part of the eigenvalues with rotation speed. For the unconstrained case, frequency lines cross over each other. But in the constrained case frequency lines veer away at intersection modes and modal coupling is clear. Frequencies follow the path of the closest mode of the unconstrained case but change direction in favor of an intersecting mode at cross over point. As can be seen, introduction of one point guide for free-free disks does not change the critical speed significantly. 33 Figure 2.8 Variations of Natural Frequencies with the Disk Rotation Speed, and Real Part of the Eigenvalues (Solid Line Free-Free Disk with a Lateral Elastic Constraint at the Rim by a Single Spring 10000N/m, Dash Line Unconstrained Free-Free Disk) 2.5 Idealized Dynamic Characteristics of Guided Splined Saws In this part the idealized dynamic behavior of splined circular saws as used in the sawmilling industry is investigated analytically. In all analysis the saw is idealized as a perfect disk with a circular inner and outer edge, and the teeth are neglected. So, for the entire analytical investigation we use a perfect disk with the physical properties which are summarized in Table 2.1.The boundary conditions is considered to be free at the inside and free at the perimeter. In practice, the location of the blade in guided splined saws 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. A combination of water and air is used to provide lubrication and cooling between the blade and pads. The clearance between the guide pads and the saw blade is in the order of 2-4 thousands of an inch. Guides considerably add to the complexity of dynamic behavior of the saw blade. The guides affect the saw blade in so many subtle and 34 inter related ways. Hence, for analytical investigation of guided splined saws in order to remove a substantial additional source of variability and complexity, we confined this study to models in which the guides are modeled by space fixed lateral linear springs. Therefore, in the area which we intended to study the effect of guide position, and its size, several springs are considered to constrain the lateral motion of the disk. The following parts of this section concern the investigation of natural frequencies and stability of three different guide configurations. 2.5.1 Full-Guided Spline Saws Figure 2.9 represents a schematic of full-guided saws which are used in most of North American sawmill industries. The guide is modeled by several springs (42 springs of 100KN/m) which have the same stiffness and they are distributed equally over the area of the 4.5×5 inches pads. In the real situation, the blade fits loosely on the arbor (The clearance between the inner spline of the disk and the spline of the arbor is about 0.01inch). Therefore there is both a rigid body translational degree of freedom along the axis of rotation, and a rigid body tilting degree of freedom. So, in this study both translational and tilting degrees of freedom have been considered. 35 Figure 2.9 Full-Guided Splined Saw, All Dimensions in Inch [Blade 17″×6″×0.060″] 2.5.1.1 Natural Frequencies of Full-Guided Splined Saws Figure 2.10 shows the analytical frequency response of the full-guided splined saw. It can be noted that we can no longer distinguish individual stationary plate modes due to mode coupling. In this case each frequency path is a combination of forward and backward waves of different stationary modes. From numerical results, the first critical speed is 2507rpm. 36 Figure 2.10 Variations of Natural Frequencies with the Disk Rotation Speed and Real Part of the Eigenvalues [Full-Guided Splined Saw, Blade 17″×6″×0.060″] 2.5.1.2 Response of Full-Guided Spline Saws to Applied Forces Idealizing the full-guided splined saw as a free-free disk with lateral elastic constraints, we used equation (2.13) as the response of the disk to the applied external forces. 37 The solution is assumed to consist of contributions of all modes up to and including 0-3 nodal circles and 0-6 nodal diameters (N, M) = (3, 6). A constant lateral force of 10N, is applied for 0.2 second at outer rim of the disk, at angular position of 90°. Figures 2.11 and Figure 2.12 show the non-dimensional transverse displacement of the disk at outer rim at the position of the applied force. According to the Figures after application of the force, the disk returns to its equilibrium and starts to vibrate about its neutral position. Figure 2.12 indicates that at rotation speed of 3600rpm disk exhibits a sine response with frequency of about 18Hz. In reality, because of damping, the amplitude decreases gradually and the disk settles to its equilibrium position. Figure 2.11 Non-dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [Full-Guided Splined Saw, Blade 17″×6″×0.060″] 38 Figure 2.12 Non-dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [Full-Guided Splined Saw, Blade 17″×6″×0.060″] 2.5.2 One-Pin-Eye-Guided Spline Saws Figure 2.13 shows a schematic when a pair of small guide pads is constraining the blade laterally in an area close to the arbor. We call this configuration a one-pin-eye-guided splined saw. The guide is modeled by several springs (9 springs of 100KN/m) and they are distributed equally over the area of the 1.5×1.5 inches pads. 39 Figure 2.13 One-Pin-Eye-Guided Splined Saw, All Dimensions in Inch [Blade 17″×6″×0.060″] 2.5.2.1 Natural Frequencies of One-Pin-Eye-Guided Splined Saws Figure 2.14 indicates the frequency response of the one-pin-eye-guided splined saw. It can be noted that we can no longer distinguish the fundamental disk modes. In this case each frequency path is a combination of forward and backward waves of different modes. The first critical speed for this configuration is 2383rpm. 40 Figure 2.14 Variations of Natural Frequencies with the Disk Rotation Speed and Real Part of the Eigenvalues [One-Pin-Eye-Guided Splined Saw, Blade 17″×6″×0.060″] 41 2.5.2.2 Response of One-Pin-Eye-Guided Spline Saws to Applied Forces Idealizing the One-pin-eye-guided spline saw as a free-free disk with lateral elastic constraints, we used equation (2.13) as the response of the disk to the applied external forces. The solution is assumed to consist of contributions of all modes up to and including 0-3 nodal circles and 0-6 nodal diameters (N, M) = (3, 6). A constant lateral force of 10N, applied for 0.2 second at outer rim of the disk. Figures 2.15 and Figure 2.16 show the non-dimensional transverse displacement of the disk at outer rim at the position of the applied force. According to the figures, after application of the force, the disk returns to its equilibrium and starts to vibrate about its neutral position. Figure 2.16 indicates that at rotation speed of 3600rpm disk exhibits a sine response with frequency of about 13Hz. In reality, because of damping, the amplitude decreases gradually and the disk settles to its equilibrium position. Figure 2.15 Non-dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [One-Pin-Eye-Guided Splined Saw, Blade 17″×6″×0.060″] 42 Figure 2.16 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [One-Pin-Eye-Guided Splined Saw, Blade 17″×6″×0.060″] 2.5.3 One-Pin-Rim-Guided Spline Saws Figure 2.17 shows a schematic of a guide configuration that consists of a pair of small guide pads located close to the rim of the blade. We called this system of saws a one-pin-rimguided splined saw. The guide is modeled by several springs (9 springs of 100KN/m) distributed equally over the area of the 1.5×1.5 inches pads. Figure 2.17 One-Pin-Rim-Guided Splined Saw, all Dimensions in Inch [Blade 17″×6″×0.060″] 43 2.5.3.1 Natural Frequencies of One-Pin-Rim-Guided Saws Figure 2.18 presents the frequency response of a one-pin-rim-guided splined saw. In this graph each frequency path is a combination of forward and backward waves of different modes. The first critical speed for this configuration is 2368rpm. Figure 2.18 Variations of Natural Frequencies with the Disk Rotation Speed, and the Real Part of Eigenvalues [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] 44 2.5.3.2 Response of Saw to Applied Forces Idealizing the One-pin-eye-guided spline saw as a free-free disk with lateral elastic constraints, we used equation (2.13) as the response of the disk to the applied external forces. The solution is assumed to consist of contributions of all modes up to and including 0-3 nodal circles and 0-6 nodal diameters (N, M) = (3, 6). A constant lateral force of 10N, applied for 0.2 second at outer rim of the disk. Figures 2.19 and Figure 2.20 show the non-dimensional transverse displacement of the disk at outer rim at the position of the applied force. According to the figures, after application of the force, the disk starts to vibrate about its neutral position. Figure 2.20 indicates that at rotation speed of 3600rpm disk exhibits a sine response with frequency of about 2Hz. Figure 2.19 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant Force of 10N, Applied for 0.2 Second, Rotation Speed = 2000rpm [One-Pin-Rim-Guided Splined Saw, Blade 17″×6″×0.060″] 45 Figure 2.20 Non-Dimensional Transverse Displacement (Displacement ‘w’ is Divided by the Disk Thickness ‘h’) of the Disk to a Constant force of 10N, Applied for 0.2 Second, Rotation Speed = 3600rpm [One-Pin-Rim-Guided Splined Saw, Blade 17″×6″×0.060″] 2.5.4 Comparison of Analytical Results of Three Guided Configurations In this section the dynamic characteristics of three guided systems is compared based on the analytical results. First we compare the lowest critical speed, and then we analyze the flutter and divergence type of instabilities in these configurations. For more convenience, the graphs of variations of natural frequencies, and real part of eigenvalues for three guided configurations are repeated in one graph in Figure 2.21. 46 Figure 2.21 Variations of Natural Frequencies, and Real Part of Eigenvalues, Three configurations Full-Guide System: One-Pin-Eye-Guided System: One-Pin-Rim-Guided System: 47 Based on the definition of critical speed, in which the frequency of backward wave is zero, according to analytical results Table 2.2 summarizes the first critical speed for the configurations: Table 2.2 First Critical Speed, Three Guided Systems System Full-Guided Pin-Eye-Guided Pin-Rim-Guided 2507 2383 2368 First Critical Speed (rpm) As shown in the Table 2.2, the pin-rim-guided system has the lowest critical speed, and the full-guided system has the highest first critical speed. Based on the real part of eigenvalues, we can investigate the instability regions of these guided systems. If the real part of eigenvalues has positive values at the location of critical speed the type of instability is named “divergence instability”. According to the Figure 2.21 at the first critical speed, the pin-rim-guided system has the smallest value of real part of eigenvalues (It is close to zero, so it is hardly distinguishable on the graph at the speed around 2368rpm), so the level of divergence instability is the lowest at the first critical speed for this system. Another source of instability for an elastically constrained disk is when real part of eigenvalues is positive while the frequency of backward wave is not zero. As indicated in Figure 2.21, the flutter instabilities occur in several super-critical speeds for the guided configurations. Figure 2.21 shows that the real part of eigenvalues at super-critical speeds for the pin-rim-guided has the highest value compared to other configurations. Therefore, the 48 flutter, and divergence instability is higher in the pin-rim-guided system at super critical speeds. In summary, based on the analytical results, we can conclude that the one-pin-rim-guided system has the lowest first critical speed but it shows the minimum level of divergence instability in this region in compare to other systems. At super-critical speed the one-pin-rimguided has the higher level of divergence and flutter instability. Therefore, it is the most unstable, and the full guided system is the most stable configurations. 49 Chapter 3: Experimental Investigations of the Idling Characteristics of Guided Saws 3.1 Introduction This chapter provides significant insights into the complex behavior of guided circular saws by conducting detailed experimental studies. Guides add considerably to the complexity of saw blade dynamic behavior in many subtle and inter-related ways. Therefore, in most numerical investigations, significant simplifications are made. Experimental results provide more reliable data and a better understanding of the dynamic behavior of guided circular saws. 3.2 Experimental Set up and Test Procedure A schematic of the experimental setup is presented in Figure 3.1. To measure the blade deflection, four non-contacting inductance probes were used. They were located at the outer rim and at the angular positions of 45º, 90º, 180º, and 270º (the reference for measuring the angular position is shown in Figure 3.1). Electromagnetic excitation was used to provide white noise excitation for the frequency range of 0–50 Hz. The electromagnet was located at the outer rim and angular position of 225º. To investigate the dynamic characteristics of rotating saws when subjected to the effect of a stationary lateral constant force, an air jet was used. The air jet was located at the outer rim and on the other side of the blade in front of the 90º probe. It must be noted that the magnitude of the air jet for all tests for each configuration is the same, corresponding to an air pressure of 80psi. Results were obtained by measuring the vibration responses of the blades at the locations of the displacement probes as the speed was ramped up from 0 RPM to 4,000 RPM at a constant rate over 1600 s. 50 Figure 3.1 Schematic of the Experimental Setup Using the National Instruments software package “Signal Express” the data were analyzed to determine mean displacement and also to produce frequency color map of the power spectrum and waterfall plot that illustrate the variations of the blade frequencies with rotation speed, and also show the energy of the signal at each speed and frequency in rms either by color spectrum or waterfall plot. The sampling rate was 2000Hz, and a Gaussian Window with the length of 4096 samples was utilized. The speed resolution was 25 RPM. In order to find the DC level of the disk displacement, the deflection for each probe was averaged over a period of two seconds. DC levels of displacements were evaluated with respect to the neutral position of the disk when it had settled on the arbor. In this regard, the position of the blade for each probe was calculated by averaging the time response of the disk 51 over three revolutions of the blade at a speed of 200rpm. It seems to be rational to consider the speed of 200rpm as a speed when blade has settled on the arbor. So, all the DC data are shifted based on this calculated position. Based on the above procedure the idling tests were conducted for three guide configurations as mentioned in chapter (2) which include: i. Full-guided Splined saw ii. One-Pin-Eye-Guided Splined Saws iii. One-Pin-Rim-Guided Splined Saws The entire tests were conducted with the same blade 17-40-60, whose properties are indicated in Table 3.1. Table 3.1 Properties of the Blade 17-40-60 Used for all the Experimental Tests Property Value Outer Diameter (in) 17 Inner Diameter (in) 6 Thickness (in) 0.060 Number of teeth 40 Density ( ⁄ 7800 Young’s Modulus (G.Pa) 203 Poisson’s ratio 0.3 52 3.3 Experimental Results In this section, a typical set of experimental response data, for the blade 17-40-60 is presented in detail. The first section is concerned with the mean deflection of the blade with rotation speed in both cases of applied air jet and without air jet. As explained in the test procedure the levels of displacements were evaluated with respect to the deflected position of the disk when it becomes settled on the arbor. So, all the DC data are shifted based on the abovementioned procedure. The second section presents the variations of natural frequencies with rotation speed in a color map plot. The last section shows the waterfall plots of the measured power spectrums of transverse blade vibration, during rotation speed sweeps from 0 to 4000rpm. 3.3.1 Variations of Blade Deflection with Rotating Speed Figures 3.2, 3.4, and 3.6 present the variations of the blade deflection with rotation speed while there is no external force applied to the blade. Figures 3.3, Figure 3.5, and Figure 3.7 present the variation of the blade deflection with rotation speed while an external force is applied by the air jet corresponding to an air pressure of 80psi at an angular position of 90º close to the outer rim of the blade. The graphs show the non-dimensional deflection of the blade at the position of applied air jet on the blade. For making a comparison between different configurations, all the graphs were drawn to the same scale. First we compare the case without air jet for three configurations. Figure 3.2 indicates that for the full guided system there is no significant DC deflection during speed ramp up except at speed 3300rpm where probe 180 has a peak of 0.67×60thou, the same situation is repeated for the guided system with one pin guide at the eye where at speed of 3230rpm probe 180 has a peak of 53 0.46×60thou (Figure 3.5). Unlike these two cases the one-pin-rim-guided system presents different behavior. According to Figure 3.7 the levels of DC deflection are significantly larger than the two previous cases, and there is a sudden jump at 3630rpm with magnitude of 2.45×60thou. In the case with air jet ON, for full-guided system (Figure 3.3) there is a noticeable fluctuation for probe 180 over the speed range from 1800 to 2200 rpm; and at speed about 3700rpm probe 180 and probe 90 start sudden increase and decrease respectively. Figure 3.5 indicates small fluctuations about the reference level for all the probes in the one-pin-eyeguided system. The one-pin-rim-guided system represents the largest fluctuations between these three configurations (Figure 3.7). After 2500rpm, Probe 270 starts to increase continuously with some fluctuations. In summary, it may be concluded that based on mean deflection of the blade for these three guide configurations, one-pin-eye-guided system represents the most stable case followed by full-guided system; and one-pin-rim-guided system represents the most unstable case. 54 Figure 3.2 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [Full-Guided Splined Saw, Blade17-40-60] Dc Amplitude/Blade Thickness 5 4 3 2 Probe 45 1 Probe 90 0 Probe 180 ‐1 Probe 270 ‐2 ‐3 0 1000 2000 3000 4000 Speed (RPM) Figure 3.3 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [Full-Guided Dc Amplitude/Blade Thickness Splined Saw, Blade17-40-60] 5 4 3 2 Probe 45 1 Probe 90 0 Probe 180 ‐1 Probe 270 ‐2 ‐3 0 1000 2000 3000 4000 Speed (RPM) 55 Figure 3.4 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [One-Pin-Eye Guided Splined Saw, Blade17-40-60] Dc Amplitude/Blade Thickness 5 4 3 2 Probe 45 1 Probe 90 0 Probe 180 ‐1 Probe 270 ‐2 ‐3 0 500 1000 1500 2000 2500 3000 3500 4000 Speed (RPM) Figure 3.5 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [One-Pin-Eye Guided Splined Saw, Blade17-40-60] Dc Amplitude/Blade Thickness 5 4 3 2 Probe 45 1 Probe 90 Probe 180 0 Probe 270 ‐1 ‐2 ‐3 0 500 1000 1500 2000 2500 3000 3500 4000 Speed (RPM) 56 Figure 3.6 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet OFF [One-Pin-Rim - Dc Amplitude/Blade Thickness Guided Splined Saw, Blade17-40-60] 5 4 3 2 Probe 45 1 Probe 90 0 Probe 180 ‐1 Probe 270 ‐2 ‐3 0 500 1000 1500 2000 2500 3000 3500 4000 Speed (RPM) Figure 3.7 Variations of Mean Deflection of the Blade with Rotation Speed, Air Jet ON [One-Pin-Rim Guided Splined Saw, Blade17-40-60] Dc Amplitude/Blade Thickness 5 4 3 2 Probe 45 1 Probe 90 0 Probe 180 Probe 270 ‐1 ‐2 ‐3 0 500 1000 1500 2000 2500 3000 3500 4000 Speed (RPM) 57 Figure 3.8 and Figure 3.9 present the mean deflection of probe 45 with rotation speed in the three guide configurations while the air jet is OFF, and ON respectively. It should be noted that we chose probe 45 for investigation in this part, because this location is where the first interaction of the blade and work piece happens during the cutting tests. Hence, we are interested in variations of deflection of the blade with rotation speed around this area. Figure 3.8 and Figure 3.9 illustrate that the One-pin-eye-guided system exhibits the most stable idling characteristics and the One-pin-rim-guided system represents the most unstable idling characteristics. Figure 3.8 Variations of Mean Deflection of the Blade at the Position of the Probe 45° with Rotation Speed, Air Jet OFF [All configurations, Blade17-40-60] Dc Amplitude/Blade Thickness 0.4 0.2 0 ‐0.2 ‐0.4 Full-Guided ‐0.6 One-Pin-Eye One-Pin-Rim ‐0.8 ‐1 ‐1.2 0 1000 2000 3000 4000 5000 Rotation Speed (RPM) 58 Figure 3.9 Variations of Mean Deflection of the Blade at the Position of the Probe 45° with Rotation Speed, Air Jet ON [All Configurations, Blade17-40-60] Dc Amplitude/Blade Thickness 0.4 0.2 0 ‐0.2 ‐0.4 Full-Guided ‐0.6 One-Pin-Eye One-Pin-Rim ‐0.8 ‐1 ‐1.2 0 1000 2000 3000 4000 Rotation Speed (RPM) 3.3.2 Natural Frequencies In this section the variations of natural frequencies of the blade with rotation speed is plotted in both the form of a color map plot and in the form of a waterfall plot of the measured power spectrums of transverse blade vibration. 3.3.2.1 Color Map Figures 3.10 – 3.15 show the distribution of excited frequencies of the blade as a function of rotation speed in the form of a color map plot. All the probes exhibit the same fundamental features, so we present here only the data from probe 90º which had a better resolution than the other probes. Frequency color maps of the power spectrum illustrate the variations of the blade frequencies with rotation speed. It also illustrates the energy of the signal at each speed and frequency with a color spectrum (The amplitude has been shown in 59 rms). All the graphs have the same scale so we can compare the graphs for different configurations. Figure 3.10 Variations of Natural Frequencies with Speed, Air Jet OFF [Full-Guided Splined Saw, Blade17-40-60] Figure 3.11 Variations of Natural Frequencies with Speed, Air Jet ON [Full-Guided Splined Saw, Blade17-40-60] 60 Figure 3.12 Variations of Natural Frequencies with Speed, Air Jet OFF [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] Figure 3.13 Variations of Natural Frequencies with Speed, Air Jet ON [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] 61 Figure 3.14 Variations of Natural Frequencies with Speed, Air Jet OFF [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] Figure 3.15 Variations of Natural Frequencies with Speed, Air Jet ON [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] 62 Three main patterns emerge from these graphs: (I) The straight lines which were indicated by 1X, 2X, and so on in the graphs. They are harmonics of the fundamental blade speed that describe the run out or lack of flatness of the blade. We refer to them as first, second and, so on harmonic run-outs. (II) The mode shapes are referred to as first, second and third and so on mode shapes from the lowest to upper frequencies. For example according to the Figure 3.10, and 3.11 the first, second, and so on mode shapes started from 11, 22, 45, and 72 Hz respectively (the fourth mode shapes is hardly distinguishable from noise). Each of these mode shapes might be combination of forward, backward or reflected waves of a different combination of the stationary blade mode shapes. (III) The interaction between harmonic run-outs and mode shapes. Sometimes interaction of mode shapes and harmonic run-outs excite each other and sometimes they just cross over each other without any effect. In this case, it is postulated that mode shapes and harmonic run-outs might be orthogonal to each other, so they cannot excite each other. We can refer to waterfall plots in the next section to observe these features also in a 3D format. In all of the configurations, the red color in the color maps while the air jet is ON is more intense, which indicates the mode shapes and harmonic lines clearer. It shows the increase in amplitude of mode shapes and harmonic run-outs. While the air jet is ON, some harmonic lines such as 0.9 X or 0.6X and so on (Figures 3.11, 3.13, and 3.15) can be detected. They might be non-linear harmonic run-outs due to the application of the air jet producing large deflections in the blade which results in geometric non-linear behavior of the blade. 63 In all cases, the second harmonic run-out (2X) is the strongest signal over the speed range from 0-4000rpm. The first harmonic run-out (1X) triggers after interaction with the third and fourth mode shapes. Interaction of 1X with the first and second mode shapes has no effect. It might be explained by the fact that the blade shape corresponding to the first harmonic is orthogonal to the first and second mode shapes, so their interactions do not generate any excitation. Another noticeable feature of these graphs is frequency lock-in phenomenon. In this phenomenon, the frequency of a mode is almost constant over a range of rotation speed. For example, in the case of full-guided system with air jet ON (Figure 3.11), the frequency of second mode is constant over 3000-4000rpm. For one-pin-eye-guided system with air jet on (Figure 3.13) frequency lock-in is obvious over 1000-2000rpm for the third mode and 32004000rpm for the second mode. It should be mentioned that, electromagnetic excitation was used in the frequency range of 0– 50 Hz, because we are more interested in investigation of the frequency characteristics of the blade in the frequencies less than 50 Hz. Therefore we can see the energy of signals is higher in these range of frequency. 64 3.3.2.2 Comparison of Experimental Results and Analytical Results of Variations of Natural Frequencies For more convenience, Figure 3.16 presents the variations of natural frequencies graphs for the experimental results (without the application of air jet) with analytical results for all the guided configurations. The differences between the numerical and experimental results which might be attributed to the fact that in the analytical model, the effects of lubricant, lack of flatness, and interaction between arbor and blade have not been considered. Moreover, the guide system has been modeled only by linear springs that constrain the lateral motion of the blade. However, numerical results are useful for initial interpretation of the behavior of guided saws with different guide configurations, and for the initial prediction of critical speed. According to the Figure 3.16, at zero speed, the prediction of natural frequencies by numerical analysis is in accord with experimental results. Before 2400rpm, the analytical results general prediction of variations of natural frequencies is in acceptable agreement with experimental results. But after 2400rpm in all the cases, analytical results are not in agreement with experimental results. It might be related to the fact that after 2400rpm, the rotating speed of the blade is in region of super-critical speed, so the dynamic behavior of the blade is no longer linear and the governing linear equation of the motion for the disk are not true for these regions. 65 Figure 3.16 Variations of Natural Frequencies, Experimental and Analytical Results, Three Guided Configurations Full-Guided System Pin-Eye-Guided System Pin-Rim-Guided System 66 3.3.2.3 Waterfall Plots Figure 3.17 – Figure 3.22 show the waterfall plots for the measured power spectrums of transverse blade vibration, during rotation speed sweeps from 0 to 4000rpm, with a speed resolution of 25rpm. All the probes exhibit the same fundamental features, so we present here only the data from probe 90º which had the best resolution. It should be noted that waterfall plots and color maps present the same data in a different format (Color maps show the fluctuation of the power spectrum amplitude in 2D graphs with colors, waterfall plots show them in 3D format). Frequency waterfall plots of the power spectrum illustrate the variations of the blade frequencies with rotation speed, and also shows the root mean square ( ) of the amplitude at each speed and frequency. The amplitude axis is plotted to the same maximum scale of 2 10 and 9 10 for the case of air jet OFF, and ON respectively. So, in this section, we mostly focus on the changes of amplitude of modes especially after intersection with harmonic run-outs. 67 Figure 3.17 Waterfall Plot, Air Jet OFF [Full-Guided Splined Saw, Blade17-40-60] Figure 3.18 Waterfall Plot, Air Jet ON [Full-Guided Splined Saw, Blade17-40-60] 68 Figure 3.19 Waterfall Plot, Air Jet OFF [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] Figure 3.20 Waterfall Plot, Air Jet ON [One-Pin-Eye-Guided Splined Saw, Blade17-40-60] 69 Figure 3.21 Waterfall Plot, Air Jet OFF [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] Figure 3.22 Waterfall Plot, Air Jet ON [One-Pin-Rim-Guided Splined Saw, Blade17-40-60] 70 In general, the level of amplitude in one-pin-rim guided is the largest and for the onepin-eye system it is the smallest. Applying the air jet causes the amplitude to become larger in comparison with the case with no air jet. But a careful inspection of the waterfall plots for these three configurations reveals that application of the air jet has the least effect on the amplitude of the configuration of the one-pin-eye-guided system. In summary, the following are the key response features that are observed to exist for all configurations: (i) Interaction of harmonic run-outs with mode shapes in some cases causes a sudden fluctuation of amplitude along the harmonic run-outs and mode shapes. And in some cases, harmonic run-outs and mode shapes just cross over each other without any effect. This might be explained by the fact that the shape of the harmonic run-out and the mode shape must be orthogonal to each other, so they cannot excite each other. (ii) Amplitude along the 2X line is by far larger than amplitude along 1X line. For example, the amplitudes along 1X are so small that they are hardly distinguishable until they develop after intersection with second and third mode shapes. (iii) It is assumed that the modes with frequencies lower than 10 Hz correspond to the rigid body tilting and translational degrees of freedom (Yang in [5] and Chen in [6] believed that modes with low frequencies corresponds to rigid body tilting motion ). As we can observe in the waterfall graphs, the amplitudes of these modes are the highest in the one-pin-rim system and they are lowest in the one-pin-eye system. It might be postulated that due to the constraints close to the arbor in one-pin-eye, the rigid body motions of the blade are more limited than other configurations. Therefore, we expect the lower amplitudes for the modes with frequencies less than 10Hz in the 71 one-pin-eye-guided, and the highest amplitudes of modes in these regions in the onepin-rim-guided system. (iv) Interactions between 1X and modes are higher in the one-pin-eye-guided system than the other two configurations. It might be postulated that, in the one-pineye-guided system, first harmonic run-outs and guide pads transfer more energy during the interaction of the blade and guide pads than other configurations. (v) Interactions between 2X and modes are higher in the one-pin-rim-guided system than the other two configurations. It might be postulated that, in the one-pinrim-guided system, second harmonic run-outs and guide pads transfer more energy during the interaction of the blade and guide pads than other configurations. (vi) Application of the air jet causes the overall responses to become about 5 times larger than case without air jet, except for the case of the one pin guide at the eye for which the application of the air jet does not results in significant changes in amplitude. (vii) For the experimental conditions used it may be concluded that, constraining the splined blade at the eye has the most stable idling vibration characteristics, and constraining the blade at the rim exhibits the most unstable idling vibration characteristics. 72 3.3.3 Vibration Response of Saws to Applied External Forces To determine saw response characteristics in each of these configurations, an impulse test was conducted at 5 different constant speeds according to Table 3.2. For this test, an impulse force was applied, at the position of 90° probes, by turning ON and OFF very quickly the air jet (about 1 second). Table 3.2 Speed at which Impulse Tests Were Conducted Speed 1 Speed 1 Speed 1 Speed 1 Speed 1 (rpm) (rpm) (rpm) (rpm) (rpm) 2000 2400 2800 3200 3600 Blade 17-40- 60 To compare the response of the blade at these speeds to the impulse force, the DC amplitude of the blade for each of these configurations is compared. In order to find the DC level of the disk displacement, the data documented by each probe was averaged over a period of two seconds. DC levels of displacements were evaluated with respect to the deflected position of the disk when it becomes settled on the arbor. In this regard, the position of the blade for each probe calculated by averaging the time response of the blade over three revolutions of the blade about the speed of 200rpm. At the speed of 200rpm the blade has settled on the arbor. So, all the DC data are shifted based on this calculated position. For comparison between different cases all of the graphs were drawn in the same scale. The following graphs show the impulse response of the blade at different speeds: 73 a) Rotation speed 2000rpm Figure 3.23 Impulse Response at 2000 rpm [Full-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 ‐1 Probe 180 ‐1.5 Probe 270 0 50 100 150 200 Time (Sec.) Figure 3.24 Impulse Response at 2000 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17-40-60] DC Amplitude/ Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 Probe 180 ‐1 Probe 270 ‐1.5 0 20 40 60 80 Time (Sec.) Figure 3.25 Impulse Response at 2000 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1.5 1 0.5 Probe 45 0 Probe 90 ‐0.5 Probe 180 ‐1 ‐1.5 ‐10 Probe 270 40 90 140 Time (Sec.) 74 b) Rotation speed 2400 rpm DC Amplitude / Thickness Figure 3.26 Impulse Response at 2400 rpm [Full-Guided Splined Saw, Blade 17-40-60] 1 0.5 0 Probe 45 ‐0.5 Probe 90 ‐1 Probe 180 ‐1.5 Probe 270 0 100 200 300 Time (Sec.) Figure 3.27 Impulse Response at 2400 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17-40-60] DC Amplitude/ Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 Probe 180 ‐1 Probe 270 ‐1.5 0 20 40 60 80 Time (Sec.) Figure 3.28 Impulse Response at 2400 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1.5 1 0.5 Probe 45 0 Probe 90 ‐0.5 Probe 180 ‐1 ‐1.5 ‐10 Probe 270 40 90 140 Time (Sec.) 75 c) Rotation speed 2800rpm DC Amplitude / Thickness Figure 3.29 Impulse Response at 2800 rpm [Full-Guided Splined Saw, Blade 17-40-60] 1 0.5 0 Probe 45 ‐0.5 Probe 90 ‐1 Probe 180 ‐1.5 Probe 270 0 50 100 150 200 Time (Sec.) Figure 3.30 Impulse Response at 2800 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17-40-60] DC Amplitude/ Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 Probe 180 ‐1 Probe 270 ‐1.5 0 20 40 60 80 100 Time (Sec.) Figure 3.31 Impulse Response at 2800 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1.5 1 0.5 Probe 45 0 Probe 90 ‐0.5 Probe 180 ‐1 Probe 270 ‐1.5 0 50 100 150 Time (Sec.) 76 d) Rotation speed 3200rpm DC Amplitude / Thickness Figure 3.32 Impulse Response at 3200 rpm [Full-Guided Splined Saw, Blade 17-40-60] 1 0.5 0 Probe 45 ‐0.5 Probe 90 ‐1 Probe 180 ‐1.5 Probe 270 0 100 200 300 Time (Sec.) DC Amplitude/ Thickness Figure 3.33 Impulse Response at 3200 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17-40-60] 1 0.5 0 Probe 45 ‐0.5 Probe 90 ‐1 Probe 180 ‐1.5 Probe 270 0 20 40 60 80 100 Time (Sec.) Figure 3.34 Impulse Response at 3200 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1.5 1 0.5 Probe 45 0 Probe 90 ‐0.5 Probe 180 ‐1 Probe 270 ‐1.5 0 50 100 150 Time (Sec.) 77 e) Rotation speed 3600rpm Figure 3.35 Impulse Response at 3600 rpm [Full-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 Probe 180 ‐1 Probe 270 ‐1.5 ‐10 40 90 140 Time (Sec.) Figure 3.36 Impulse Response at 3600 rpm [One-Pin-Eye-Guided Splined Saw, Blade 17-40-60] DC Amplitude/ Thickness 1 0.5 0 Probe 45 ‐0.5 Probe 90 Probe 180 ‐1 Probe 270 ‐1.5 0 20 40 60 80 100 Time (Sec.) Figure 3.37 Impulse Response at 3600 rpm [One-Pin-Rim-Guided Splined Saw, Blade 17-40-60] DC Amplitude / Thickness 1.5 1 0.5 Probe 45 0 Probe 90 ‐0.5 Probe 180 ‐1 Probe 270 ‐1.5 0 50 100 150 Time (Sec.) 78 All the graphs indicate that the blade is stable in its position before applying the external force. After applying the impulse force by air jet no significant unstable behavior can be detected in that the blade recovers its equilibrium position after the force is removed. In general, it can be concluded that the mean deflection in the one-pin-eye-guided system is the smallest and for the one-pin-rim-guided system, it is the largest. Furthermore, a careful inspection of the responses indicates that in the full-guided system, and the one-pin-rimguided system, after applying the impulse force, the blade leans to the direction of force at all probes, but in one-pin-eye-guided system, probe 270° indicates that blade deflects in opposite direction of external force, and at the rest of the probes the deflection is in direction of the force, an indication of a mode shape with one nodal diameter. 79 Chapter 4: Experimental Investigations of the Cutting Characteristics of Guided Saws 4.1 Introduction The cutting characteristics of guided saws are extremely complicated. Interaction between the wood and the blade adds considerably to the complexity of the saw blade behavior. In addition variability of wood characteristics must be considered when investigating the cutting characteristics of guided saws. There are a limited numbers of research studies concerning the cutting characteristics of saws. In order to understand the wash-boarding mechanism, an analytical model for wood cutting of circular saws was developed by Tian and Hutton [33]. In their study, the cutting forces were represented by the product of a time-dependent periodic function and the lateral displacement of the saw teeth. In another work [34] they introduced an approach which predicts the physical instability mechanism that occurs during the interaction of blade with a fixed space constraint. They used a physical energy flux equation for the blade to explain its instability. They also investigated the effect of conservative and non-conservative forces on the stability of the blade. Schajer and Wang [35] described two types of saw-work piece interaction: the interaction between the saw body and the work piece, and the interaction between the saw teeth and the work piece. They explained that the first interaction influences the cutting stability of a guided saw compared with a fixed collar saw, and the second one influences the stability of a climb-cutting saw compared with a counter-cutting saw. In the counter cutting saw the saw applies cutting forces on the work piece in the direction of saw rotation; but in the climb-cutting saw the rotation is reversed. They conducted some cutting tests to explore 80 the effects of the two interaction types in general. They concluded that typically, a guided saw cuts more accurately than a fixed-collar saw, while a climb-cutting saw cuts more accurately than a counter-cutting saw. Superior cutting accuracy of climb-cutting guided saws might be explained by the fact that it cuts the wood in the part of the saw blade that has just left the guide. Climb cutting has become popular in North American sawmills because it typically provides greater cutting accuracy than counter-cutting. In addition the practical advantage of climb-cutting guided saws is that the sawdust they produce is carried away from the guides, thereby greatly reducing the possibility of guide blockage by the sawdust. They presented a geometrical model of the interaction between the body of a guide saw and the work piece. It is the purpose of the present study to make experimental observations of real cutting tests to provide insight into the effect of different guide configurations on cutting accuracy. Another motivation for these investigations was to see how the cutting characteristics and idling characteristics of guided saws can be compared. So, the same saw blade and test equipment which were used in the idling tests has been used in cutting tests. In this study a climb-cutting saw has been used. The same guide configurations as conducted for idling test were used in cutting tests. These guide configurations are repeated in Figure 4.1: 81 Figure 4.1 Guided Splined Saw, (a) Full-Guided, (b) Pin-Eye-Guided, and (c) Pin-Rim-Guided 4.2 Experimental Setup and Test Procedure The same blade as for the idling tests, 17-40-60, was used for cutting tests. The saw was run at different speeds according to Table (4.1), and feed speeds correspond to a Gullet Feed Index (GFI) of 0.45. (Appendix C briefly explains the relation between feed speeds and GFI) Table 4.1 Rotation and Feed Speeds at which Cutting Tests Were Conducted Test (1) Test (2) Test (3) Test (4) Test (5) 2000 2400 2800 3200 3600 259 311 363 414 466 Speed (rpm) Feed Speed (fpm) The cants were made up of 3No. 2x10 SPF boards 8ft long (Fig.4.2). In order to make a better comparison between these 3 guide configurations the cutting tests are conducted in a 82 way such that for all the guide configurations the same set of cants to be used at each speed. The total depth of the cut was 5 in. Figure 4.2 Cants 3No. 2x10 SPF Boards 8ft Long The deviations of the sawn surface were measured using two laser probes focused on spots near the top and near the bottom of the cant. The top laser was focused 3/8in below the top of the cant, and 3/8in above the bottom of the cant. These measured deviations of the cut cant’s profile were taken to represent the cutting accuracy of the sawing parameters used in that test. Five cuts taken off five independent cants were conducted. These results therefore give an indication of the variability of the cut due to differing characteristics of the wood. Then for each cut a graph is drawn in which X-Axis corresponds to length of board (0-8ft) and Yaxis shows the measurements that recorded the distance from the cant to two space fixed laser probes as the cut cant was slowly run back past the probes. As such, the start of the cut corresponds to the end of the graph (8 ft) and zero corresponds to the end of the cut cant (Figure 4.3). 83 For each cut, the deviation for top probe, bottom probe and wedging which is the difference between the top probe and bottom probe reading are tabulated and also are shown as bar graphs. Figure 4.3 X-Axis of the Graphs for the Deviations of Cut Surface 4.3 Cutting Test Results In order to compare the cutting results, the standard deviation and the mean value for each set of recorded data has been calculated. A low standard deviation indicates the data points tend to be close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values. Hence the lower standard deviation of the data, recorded by the laser probes for each cut, indicates the more accurate the cut. 84 4.3.1 Standard Deviation of Cut profiles Table 4.2 to Table 4.6 show the standard deviation of the results recorded by the laser probes for each speed at which the cutting has been conducted. Figures 4.4 to Figure 4.6 show the same results as bar graphs. Table 4.2 Standard Deviation and Mean Value at 2000 RPM for the 17-40-60 Blade, Average of 5 different Cuts System SD_Top SD_Bottom SD_Wedging Mean_Wedging One-Pin-Eye-Guided 35 11 27 18 One-Pin-Rim-Guided 16 6 16 48 Full-Guided 28 10 22 37 Figure 4.4 Averaged Cutting Results at 2000 RPM for the 17-40-60 Blade 85 Table 4.3 Standard Deviation and Mean Value at 2400 RPM for the 17-40-60 Blade, Average of 5 Different Cuts System SD_Top SD_Bottom SD_Wedging Mean_Wedging One-Pin-Eye-Guided 21 8 16 11 One-Pin-Rim-Guided 15 5 13 33 Full-Guided 21 8 18 34 Figure 4.5 Averaged Cutting Results at 2400 RPM for the 17-40-60 Blade 86 Table 4.4 Standard Deviation and Mean Value at 2800 RPM for the 17-40-60 Blade, Average of 5 Different Cuts System SD_Top SD_Bottom SD_Wedging Mean_Wedging One-Pin-Eye-Guided 68 23 50 12 One-Pin-Rim-Guided 14 6 13 27 Full-Guided 23 8 18 14 Figure 4.6 Averaged Cutting Results at 2800 RPM for the 17-40-60 Blade 87 Table 4.5 Standard Deviation and Mean Value at 3200 RPM for the 17-40-60 Blade, Average of 5 Different Cuts System SD_Top SD_Bottom SD_Wedging Mean_Wedging One-Pin-Eye-Guided 98 22 73 30 One-Pin-Rim-Guided 14 5 12 40 Full-Guided 23 8 17 39 Figure 4.7 Averaged Cutting Results at 3200 RPM for the 17-40-60 Blade 88 Table 4.6 Standard Deviation and Mean Value at 3600 RPM for the 17-40-60 Blade, Average of 5 Different cuts System SD_Top SD_Bottom SD_Wedging Mean_Wedging One-Pin-Eye-Guided 156 57 109 -3 One-Pin-Rim-Guided 14 7 13 42 Full-Guided 39 11 33 34 Figure 4.8 Averaged Cutting Results at 3600 RPM for the 17-40-60 Blade 89 Table 4.7 summarizes the standard deviation of recorded data by top laser probes at five conducted speeds for three guided configurations. Figure 4.9 shows the same results as bar graphs. Table 4.7 Top Probe Standard Deviation at Different Speed for the 17-40-60 Blade, Average of 5 Different Cuts for Three Guided Configurations Speed SD_Top Pin-Eye-Guided SD_Top Full-Guided SD_Top Pin-Rim-Guided 2000rpm 35 28 16 2400rpm 21 21 15 2800rpm 68 23 14 3200rpm 98 23 14 3600rpm 156 39 14 Figure 4.9 Top Probe Standard Deviation at Different Speed for the 17-40-60 Blade, Average of 5 Different Cuts for Three Guided Configurations Figure 4.9 indicates that, the sawn surfaces by the one-pin-rim system have the lowest standard deviation for all the conducted cutting speeds. It also has the minimum fluctuation over the different speeds. In comparison, the sawn surfaces by the one-pin-eye-guided system 90 have both the maximum standard deviation and the highest level of fluctuation for different speeds. The full-guided system exhibits the behavior between the eye-guided and the rimguided system in term of cutting accuracy. Based on the standard deviations of the documented data by the laser probes in our experimental cutting tests, we believed that the one-pin-rim-guided splined saw provides the best cutting accuracy, and the one-pin-eye-guided splined saw provides the worst cutting accuracy among the three discussed configurations. 4.3.2 Cut profile Results of Different Guide Configurations To get a sense of cutting accuracy by the three guided configurations, the data recorded by the top and the bottom probes were drawn to the same scale in one graph for all 5 cuts of the same cutting speed. In this way, we can visually compare how the data are spread out over a cutting profile. In the graphs X-Axis is correspond to length of board (08ft) and Y-axis shows the measurements that recorded the distance from the cant to two space fixed laser probes as the cut cant was slowly run back past the probes. As such, the start of the cut corresponds to the end of the graph. Following graphs show the cut profile results of different guide configurations for each speed at which the cutting has been conducted: 91 Figure 4.10 Cut Profile Results at 2000 rpm, for the 17-40-60 Blade 92 Figure 4.11 Cut Profile Results at 2400 rpm, for the 17-40-60 Blade 93 Figure 4.12 Cut Profile Results at 2800 rpm, for the 17-40-60 Blade 94 Figure 4.13 Cut Profile Results at 3200 rpm, for the 17-40-60 Blade 95 Figure 4.14 Cut Profile Results at 3600 rpm, for the 17-40-60 Blade 96 4.4 Comparison and Conclusion on Cutting Test Results Based on the above results, we can conclude that, the guide configuration with one pin at the rim has the most accurate cutting results. The system with one pin guide placed at the eye indicates the worst cutting accuracy. So, it might be concluded that constraining the saw close to the eye results in a weak cutting accuracy. As for the full guided system, the cutting accuracy of full-guided system is better than pin-eye-guided system and is worse than pin-rim-guided system. In sum, using the guide system proximate to the peripheral of the blade (rim) results in better cutting accuracy, and constraining the blade laterally close to the eye results in weak cutting accuracy. 97 Chapter 5: Conclusion 5.1 Summary and Conclusion This thesis made a substantial contribution by providing significant insight into the complex behavior of guided circular saws. It establishes a clear framework for guided saws design to improve sawing accuracy and stability. The main contributions of this study can be summarized as follow: a) Analytical Investigation of Idling Characteristic of Guided Saws The variations of frequencies and mode shapes of clamped saws and guided splined saws with rotation speed have been investigated. The steady state response of the disk (when the deformation pattern of the disk is time invariant) has been calculated. It should be recognized that significant levels of vibration may in practice also exist superimposed on the steady state response. The dynamic behavior of a clamped-free disk and a free-free disk while constrained by a space fixed transverse spring placed at the rim has been studied. The effect of spring stiffness on the frequency speed characteristic of the rotating disk has been investigated. Idealizing the saw as a perfect disk with circular inner and outer edge, frequency-speed characteristics of splined saws with different guide configurations have been calculated. The response of a disk to an impulse force has been calculated. From this study it was found that: (i) Unlike the unconstrained case, in which frequency lines cross over each other, in the constrained case frequency lines veer away and modal coupling is clear. Frequencies follow the path of the closest mode of the unconstrained case but change direction in favor of an intersecting mode at cross over points. 98 (ii) The presence of one point constraint does not change the critical speed, and it is independent of the spring stiffness. Therefore, using one point guide does not change the critical speed of free-free disk noticeably. (iii) Between the pin-guided configurations which have been investigated, one- pin-eye-guided has the highest first critical speed, and one-pin-rim-guided has the lowest critical speed. So, the eye-guided system is the most stable system in term of idling vibration characteristics. b) Experimental Investigations of the Idling Characteristics of Guided Saws A series of concise experimental tests have been conducted to measure the frequency response of rotating saw with different guided configurations. To measure the blade deflection, four space fixed non-contacting inductance probes were used. Electromagnetic excitation was used to provide white noise excitation. An air jet was used to investigate the dynamic characteristics of rotating saw when subjected to the effect of stationary lateral constant force. National Instruments software package “Signal Express” was used for analyzing the experimental data. The data were analyzed to determine mean displacement and also to produce frequency color map and waterfall plot of the power spectrum. They illustrate the variations of blade frequencies with rotation speed, and also show the amplitude at each frequency and speed. To determine saw response characteristics in each of these configurations, an impulse test was conducted at different constant speeds below critical speeds and at super critical speeds. From this investigation it was found that: (i) Intersection of harmonic run-outs with mode shapes in some cases causes a sudden fluctuation of amplitude along the harmonic run-outs and mode shapes. And in some cases, harmonic run-outs and mode shapes just cross over each other without 99 any effect. This might be explained if the harmonic run-out and mode shape are orthogonal to each other, so they cannot excite each other. (ii) The amplitude along the 2X line (Second harmonic run out) is larger than the amplitude along the 1X line (First harmonic run out). For example, the amplitudes along the 1X are so small that they are hardly distinguishable until they develop after intersection with the second and third mode shapes(Refer to Figures 3.17-3.22). (iii) Application of the air jet causes the amplitude to become about 5 times larger than in case without air jet, except for the configuration with one pin guide at the eye, in which application of the air jet does not show significant changes in amplitude. (iv) For the experimental conditions used it may be concluded that, constraining the splined blade at the eye has the most stable idling vibration characteristics, and constraining the blade at the rim exhibits the most unstable idling vibration characteristics. (v) After applying an impulse force by the air jet no significant unstable behavior can be detected. But in general, it might be concluded that the mean deflection in the one-pin-eye-guided system is the smallest and for the one-pin-rim-guided system it is the largest. (vi) Although significant simplifications have been made to obtain the numerical results, there is an acceptable agreement between numerical results and experimental results. There are differences between the numerical and experimental results which might be attributed to the fact that in the analytical model, the effects of lubricant, lack of flatness, interaction between arbor and blade, have not been considered. Moreover, the guide system has been modeled only by linear springs that constrained 100 the lateral motion of the blade. However, numerical results are useful for initial interpretation of the behavior of guided saws with different guide configurations, and for the initial prediction of critical speeds. c) Experimental Investigations of the Cutting Characteristics of Guided Saws Extensive cutting tests have been conducted to provide insight into the effect of different guide configurations on cutting accuracy. The deviations of the sawn surface were measured using two laser probes focused on spots near the top and near the bottom of the cant. These measured deviations of the cut cant’s profile were taken to represent the cutting accuracy of the sawing parameters used in the tests. The saws were run at different speeds below critical speeds and at super critical speeds. At each speed 5 cuts have been done for 5 independent cants each consisted of 3 SPF boards to provide confident results for analyzing. In order to compare the cutting results, the standard deviation and the mean value for each set of data have been calculated. A low standard deviation indicates the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values; hence in our case, lower standard deviation indicates that the cut is more accurate. From this experimental investigation it was found that: (i) The guide configuration with one pin at the rim has the most accurate cutting results (the lowest standard deviation). The system with one pin guide placed at the eye indicates the worst cutting accuracy. (ii) The cutting accuracy of a full-guided system is better than a pin-eye-guided system and is worse than a pin-rim-guided system. 101 (iii) Using the guide system proximate to the peripheral of the blade (rim) results in better cutting accuracy, and constraining the blade laterally close to the eye results in weak cutting accuracy. (iv) The guide system which showed the most stable results in the idling situation (One-pin-eye-guided) presented the worst cutting results in term of accuracy. The one-pin-rim-guided, which gave the worst unstable conditions during the idling tests, provided the best sawing accuracy. (v) There was no correlation between the idling test results and cutting test results. So, any interpretation from the idling tests in term of stability, or level of vibration to use for the cutting would result in wrong conclusions. 5.2 Recommendation for Future Work Further research and investigation in this area should be done on the following aspects: (i) Analytical Investigations: - To model the guided splined saws by considering the mass and stiffness of the guide, and using the system of mass-spring for modeling of the guide - To calculate the effect of different harmonic forces with different frequencies on stability of the blade - To consider imperfections of the blade such as initial run-out of the blade - To consider interaction between the blade and the guide with different guide clearance while the initial run out is considered - To investigate the effect of damping on dynamic behavior of guided saws - To investigate how analysis of cutting can be performed. 102 (ii) Experimental Investigation: - To conduct idling tests by applying different harmonic excitation with different frequencies, in this way we can find how the blade resonates with these excitations. - To conduct cutting tests with different guided system while using a cooling system for blade during the cutting to check the effect of the blade heating on the cutting accuracy. 103 References [1] Hutton, S.G., Chonan, S., and Lehmann, B.F., 1987, “Dynamic response of a guided circular saw,” Journal of Sound and Vibration, 112, pp. 527-539. [2] Chen, J.S., and Hsu, C.M. , 1997, “Forced Response of a Spinning Disk Under SpaceFixed Couples,” Journal of Sound and Vibration, 206(5), pp. 627-639. [3] Chen, J.S. and Wong, C.C., 1995, “Divergence instability of a spinning disk with axial spindle displacement in contact with evenly spaced stationary springs,” Journal of Applied Mechanics, 62, pp. 544-547. [4] Mote, C.D., 1977, “Moving load stability of a circular plate on a floating central collar,” Journal of Acoustical Society of America, 61, pp. 439-447. [5] Yang, S.M., 1993, “Vibration of a spinning annular disk with coupled rigid-body motion,” ASME Journal of Vibration and Acoustics, 115, pp. 159-164. [6] Chen, J.S., and Bogy, D.B., 1993, “Natural Frequencies and Stability of a Flexible Spinning Disk-Stationary Load System With Rigid Body Tilting,” ASME Journal of Applied Mechanics, 60, pp. 470-477. [7] K.B. Price, Analysis of the dynamics of guided rotating free centre plates, Ph.D. Dissertation, University of California, Berkeley, 1987. [8] Khorasany, R.M.H., and Hutton, S.G., 2010, “An Analytical Study on the Effect of Rigid Body Translational Degree of Freedom on the Vibration Characteristics of Elastically Constrained Rotating Disks,” 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. 104 [10] Raman, A., and Mote, C.D. , 2001, “Experimental studies on the non-linear oscillations of imperfect circular disks spinning near critical speed,” International Journal of NonLinear Mechanics, 36(2), 2001, pp. 291-305. [11] Kang, N., and Raman, A., 2006, “Vibrations and stability of a flexible disk rotating in a gas-filled enclosure-Part 2: Experimental study,” Journal of Sound and Vibration, 296(45), pp. 676-68. [12] D’Angelo, C., Mote, C.D., 1993, “Aerodynamically excited vibration and flutter of a thin disk rotating at supercritical speed,” Journal of Sound and Vibration, 168, pp. 15-30. [13] Thomas, O., Touze, C., and Chaigne, A., 2003, “Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: Experiments,” Journal of Sound and Vibration, 265(5), pp. 1075-1101. [14] Jana, A., and Raman, A., 2005, “Nonlinear aeroelastic flutter phenomena of a flexible disk rotating in an unbounded fluid,” Journal of Fluids and Structures, 20(7), pp. 9931006. [15] Raman, A., Hansen, M. H., and Mote, C.D., 2002, “A Note on the Post-Flutter Dynamics of a Rotating Disk,” Journal of Applied Mechanics, 69(6), pp. 864-866. [16] Khorasany, R.M.H., and Hutton, S.G., June 2011, “Vibration Characteristics of Rotating Thin Disks, Part I: Experimental Results,” ASME Journal of Applied Mechanics, in press. [17] Khorasany, R.M.H., and Hutton, S.G., June 2011, “Vibration Characteristics of Rotating Thin Disks, Part II: Analytical Predictions,” ASME Journal of Applied Mechanics, in press. 105 [18] Lee C.-W., and Kim M. -E., January 1995, “Separation and identification of traveling wave modes in rotating disk via directional spectral analysis,” Journal of sound and vibration (1995) 187, pp.851-864 [19] Ahn T.K., and Mote Jr. C.D., December 1998, “Mode identification of a rotating disk,” Experimental Mechanics, v 38, n 4, p 250-254, [20] Tian, J., and Hutton, S.G., 1999, “Self Excited Vibration in Flexible Rotating Discs Subjected to Transverse Interaction Forces – A General Approach,” ASME Journal of Applied Mechanics, 66, pp. 800-805. [21] Vogel, S. M. and Skinner, D. W., 1965, “Natural Frequencies of Transversely Vibrating Uniform Annular Plates,” ASME Journal of Applied Mechanics, 32, pp. 926-931. [22] Chen, J.S. and Wong, C.C., 1996, “Modal Interaction in a Spinning Disk on a Floating Central Collar and Restrained by Multiple Springs,” Journal of the Chinese Society of Mechanical Engineers, Vol. 17, No.3, pp. 251-259. [23] Chen, J. S., and Bogy, D.B., 1992, “Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System,” American Society of Mechanical Engineers Journal of Applied mechanics, 59, pp. 390-397. [24] Chen, J., and Hua, C., 2004, “On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations,” ASME Journal of Vibration and Acoustics, 126, pp. 422-429. [25] Deqiang M., and Suhuan C., 2001, “Effect of the guides on the lowest critical rotational frequencies of circular saw,” Chinese Journal of Mechanical Engineering, Vol. 14, pp. 166170 [26] Chen, J.S., 1994, “Stability Analysis of a Spinning Elastic Disk Under a Stationary Concentrated Edge Load,” ASME Journal of Applied Mechanics, 61, pp. 788-792. 106 [27] Tian, J., and Hutton, S.G., 2001, “Traveling-wave Modal Identification Based on Forced or Self-Excited Resonance of Rotating Discs,” Journal of Vibration and Control , 7, pp. 3-18. [28] Young, T.H., and Lin, C.Y., 2006, “Stability of a Spinning Disk Under a Stationary Oscillating Unit,” Journal of Sound and Vibration, 298, pp 307-18. [29] United States Patent and Trademark Office, official website http://www.uspto.gov/ [30] S. Timoshenko and J.N. Goodier Theory of Elasticity. New York, McGraw-Hill. [31] Meirovitch L., 1997, “Principles and techniques of vibrations”, New Jersey. [32] Rayleigh, Theory of Sound, Volume I, Chapter IV, Section 88 &92 on interlacing of eigenvalues, [33] Tian, J.F., and Hutton, S.G., 2001, “Cutting Induced Vibration in Circular Saws,” Journal of Sound and Vibration, 242(5), pp. 907-922. [34] Tian, J.F., and Hutton, S.G., 1999, “Self Excited Vibration in Flexible Rotating Disks Subjected to Transverse Interaction Forces – A General Approach,” ASME Journal of Applied Mechanics, 66, pp. 800-805. [35] Schajer, G.S., Wang, S.A., 2002, “Effect of work piece interaction on circular saw cutting satiability,” European Journal of Wood and Wood Products, Volume 60, Number 1, pp.48-54. [36] Bruce Lehmann, “saw Tooth Design and Tipping Materials” Sr. Engineer, Thin Kerf Technologies Inc. British Columbia, Canada 107 Appendices Appendix A Mathematical Calculation of Non-Dimensional Equation of Motion Based on the introduced parameters in (2.7) we get: , , By substitution of, 2Ω , , , and , , from equation (2.3) into equation (2.6), we get: Ω Ω Ω , , , , (A.1) By simplification of equation (A.1) the equation may thus be written in the form of: 2Ω , Ω , Ω , , 3 , Ω (A.2) , Here we substitute the non dimensional parameters into equation (A.2): 2Ω , Ω Ω , Ω , , , 3 , (A.3) By dividing both side of equation (A.3), the equation may be written as 108 2Ω , Ω Ω , , , 3 , (A.4) , 1 we get: If we put (A.5) By substituting T in equation (A.4) the equation can be written as , 2Ω Ω , 3 , , (A.6) , This equation may be written in the form of: , 2Ω , Ω 2 , 4 Ω 2 2 3 , , , (A.7) Where 3 109 Appendix B Mathematical Details of Solution for Linear Equation of Transverse Vibration Solution of equation (2.8) can be obtained by application of the Galerkin method. In this solution method the eigenfunctions of a plate problem in the polar coordinate system is used as the approximation function for the Galerkin method. Considering the orthogonality property of the eigenfunctions of a stationary disk makes the solution by far easier. First we calculate the eigenfunctions of a stationary disk. The non dimentionalized stationary equation of motion for a plate in polar coordinate system is (B.1) Using the boundary conditions we can obtain the eigenvalues associated with the eigenfunctions as: [31] mθ (B.2) Where sin cos , and Where , , , are mth order Bessel functions and modified Bessel functions. The orthogonality condition of the eigenfunctions is: , , Where (B.3) is the Kronecker delta. 110 , , , and , are constants which can be determined by the normalization condition for the first equation of (B.3) as 1 (B.4) 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 , , ∑ cos sin , 0. Substituting of Consider equation (2.8) while (B.5) , , from equation (B.5) into equation (2.14) results in: ∑ sin , 2Ω ∑ ∑ , cos cos sin sin , ∑ cos sin , cos ∑ sin , cos 0 By some mathematics simplification we get: 2Ω , Ω Ω sin 111 2Ω , Ω Ω ∑ cos cos sin , 0 (B.6) sin If we multiply equation (B.6) by and then integrate the resultant over the area of the plate ant using the orthogonal property of the eigenfunctions we will have: 2Ω 2 4 2Ω ∑ 0 (B.7) cos If we multiply equation (B.6) by and then integrate the resultant over the area of the plate ant using the orthogonal property of the eigenfunctions we will have: 2Ω 2 4 2Ω ∑ 0 (B.8) Equations (B.7) and (B.8) may be written in the following form: 2Ω 4 2Ω 2 Ω 2 2 ∑∞ 0 П 0 112 2Ω 2 4 2Ω Ω 2 2 ∑∞ 0 П 0 (B.9) Where П 0 (B.10) Equations , (B.10) constitute a system of simultaneous differential equation in , while m, n = 0, 1, 2, . . . the solution of which determines the transverse displacement of the disk via equation (B.5). Now consider number guides constrain the disk laterally. If we model guides with a dashpot spring [1], the spring systems attached to the blade produce a lateral force. If we consider guide (i) in position , , then the force produces by this guide is: (B.11) By Substituting , , from equation (1.19) into equation (1.25) result is: ∑ sin , cos Here, if we substitute (B.12) in equation (2.8) we have: ∑ cos , sin (B.13) 113 Multiplying equation (B.13) by sin and repeating the process with cos ∑ , we will get: 2Ω 2 Ω 2Ω 2 Ω 4 2Ω and integrating over the area of the disk 2 2 ∑∞ 0 П ∑∞ 0 П sin 4 2Ω ∑ 2 2 cos (B.14) Where ∑ sin , cos Equations , (B.14) constitute a system of simultaneous differential equation in , while m, n = 0, 1, 2, . . . the solution of which determines the transverse displacement of the guided disk via equation (B.5). 114 Appendix C Relation between Feed Speeds and GFI The maximum feed speed for a given depth of cut might be calculated as: [36] . . . Where c = blade speed A = Gullet area P=Tooth pitch D= depth of cut GFI = Gullet Feed Index (Which is percentage of gullet filled and usually is a number between 0.3 to 0.7) 115
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Idling and cutting vibration characteristics of guided...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Idling and cutting vibration characteristics of guided circular saws Mohammadpanah Foroutaghe, Ahmad 2012
pdf
Page Metadata
Item Metadata
Title | Idling and cutting vibration characteristics of guided circular saws |
Creator |
Mohammadpanah Foroutaghe, Ahmad |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | In this thesis the vibration characteristics of guided circular saws are studied, both analytically and experimentally. Significant insights into the complex dynamic behavior of guided circular saws are presented first by numerical investigation of rotating disks and then by conducting idling and cutting experimental tests of splined saws with different guide configurations. For the numerical investigations, the governing linear equation of transverse vibration of a rotating disk is used. As a primary interest, the variation of disk natural frequencies with rotation speed and the disk response to applied external force are calculated. Also, the steady state response of the disk at different speeds is calculated. The effects of elastic lateral constraints are investigated in this section. A comprehensive experimental investigation of idling tests of splined saws with different guide configurations is presented. The frequencies and amplitudes of the blade vibrations are documented and the mean deflections of the disks are plotted. The dynamic characteristics of a rotating blade when subjected to the stationary lateral constant force are discussed. Extensive cutting tests are conducted and the effect of different guide configurations on cutting accuracy is presented. Cutting tests are conducted at different speeds, below and above the lowest critical speed for different guide configurations. The cutting results are compared to determine the guide configuration which results in the best cutting accuracy. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-02-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0072569 |
URI | http://hdl.handle.net/2429/40416 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
Download
- Media
- 24-ubc_2012_spring_mohammadpanah_ahmad.pdf [ 7.17MB ]
- Metadata
- JSON: 24-1.0072569.json
- JSON-LD: 24-1.0072569-ld.json
- RDF/XML (Pretty): 24-1.0072569-rdf.xml
- RDF/JSON: 24-1.0072569-rdf.json
- Turtle: 24-1.0072569-turtle.txt
- N-Triples: 24-1.0072569-rdf-ntriples.txt
- Original Record: 24-1.0072569-source.json
- Full Text
- 24-1.0072569-fulltext.txt
- Citation
- 24-1.0072569.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:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0072569/manifest