UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Supercritical speed response of circular saws Yang, Longxiang 1990

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1990_A7 Y37.pdf [ 5.07MB ]
Metadata
JSON: 831-1.0098015.json
JSON-LD: 831-1.0098015-ld.json
RDF/XML (Pretty): 831-1.0098015-rdf.xml
RDF/JSON: 831-1.0098015-rdf.json
Turtle: 831-1.0098015-turtle.txt
N-Triples: 831-1.0098015-rdf-ntriples.txt
Original Record: 831-1.0098015-source.json
Full Text
831-1.0098015-fulltext.txt
Citation
831-1.0098015.ris

Full Text

SUPERCRITICAL SPEED RESPONSE OF CIRCULAR SAWS By Longxiang Yang B.A.Sc. Beijing Institute of Chemical Technology 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 We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1990 © Longxiang Yang, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract This study investigates the dynamics of circular saws at supercritical speeds. A classical governing equation of a circular saw subjected to transverse and membrane forces is derived in both body-fixed and space-fixed coordinate systems. The transverse loads are generalized as spring, damping and out of plane cutting forces, and the mem-brane loads include rotational, thermal and in-plane cutting forces. Galerkin's Method is used to study the natural and forced response of a saw blade, and the effects of spring and damping on the stability at supercritical speed are discussed. An idling experiment is conducted to comprehend the fundamental behavior of cir-cular saws at supercritical speed. The natural frequency, the steady state response, the flexibility and the runout variation of the circular saws are measured. The nonlinear vi-bration of the thin blade is observed. The effects of a spring-damper system and heating on the stability of a circular saw are investigated. A cutting test is conducted to find out the dominant parameters of supercritical speed cutting. At first, the prehminary experiments of the standard blade are conducted. Then, tip side cutting marks are discussed, and finally, the phenomenon of unstable cuttings are presented. The general solution of a rotating string subject to an elastic constraint is developed. Based on this, the discrepancy between theoretical and experimental stability results is analysed for a guided rotating disk. ii Nomenclature o: disk outer radius ao-ct3: coefficients of cutting stresses b: disk inner radius An, Bn, A'n, B'n: coefficients of exact solution of string displacement C: damping coefficient Cn)Sn: time functions of string Cmn, Smn' time functions of disk Di, Z?2, D$, coefficients of centrifugal stresses E: Young's modulus Elmn: coefficient of radial eigenfunction h: disk thickness K: stiffness coefficient p: transverse cutting force P: magnitude of transverse cutting force or string tension q: general transverse force r, 0, z: body-fixed coordinate R,(f>,Z: space-fixed coordinate MrrfMre^erjMgg: bending and twisting moments Nrr , N r e , Ner, Nee '• membrane forces Qr,Qe: transverse shear forces Q: spring and damping forces Rmn'. radial eigenfunction of disk iii S: natural frequency of stationary string t: time T: temperature u: string displacement W: disk displacement aj: thermal expansion coefficient A: eigenvalue fi: Poisson's ratio p: density <rrr)orM»TTit>'- membrane stresses f/Jr) o-^ ,rrr :^ membrane centrifugal stresses arr)°'4><t»TTti>: membrane cutting stresses {rTTiaM»Tr<j>: membrane thermal stresses u/: forcing frequency u>n: natural frequency of string ft: rotation speed iv Table of Contents Abstract ii Nomenclature iii List of Tables ix List of Figures x Acknowledgements xiv 1 Introduction 1 1.1 Background 1 1.2 Review of Previous Work 4 1.2.1 Response of a Flexible Rotating Disk 5 1.2.2 Nonlinear Response 5 1.2.3 Aerodynamic Effects 6 1.2.4 Instability 7 1.2.5 Heating Effects 7 1.3 Purpose and Scope 8 2 Theoretical Background 9 2.1 Introduction 9 2.2 Formulation of Governing EquatioB. - . < 9 2.2.1 Governing Equation in a Body-Fixed Polar Coordinate [13] .... 9 v 2.2.2 Governing Equation in a Space-Fixed Polar Coordinate 13 2.2.3 Transverse and In-Plane Forces in the Governing Equation .... 14 2.2.4 Solution Method 19 2.2.5 Natural Response . 22 2.2.6 Steady State Response 23 3 Numerical Results 24 3.1 Introduction 24 3.2 Single-Mode Assumption 24 3.2.1 Verification of Single-Mode Assumption 24 3.2.2 Effect of Stiffness and Damping on the Stability of a Circular Saw 26 3.3 Multi-Mode Assumption 35 4 Idling Tests 44 4.1 Introduction 44 4.2 Experiment Apparatus 44 4.3 Experiment Set-up 45 4.4 Natural Frequency Measurements 46 4.5 Steady State Response 51 4.6 Flexibility Measurements 56 4.7 Variation in Runout with Rotation Speed 60 4.7.1 Characteristics of Runout 60 4.7.2 Factors Affecting Runout Measurements 62 4.8 Unstable Vibrations of Blade D 65 4.9 Effect of Spring and Damper on the Stability 76 4.9.1 Effect of a Pair of Bearing Supports 76 4.9.2 Effect of a Pair of Pin Guides 76 vi 4.9.3 Effect of Wood 85 4.10 Heating Effect 85 4.11 Effect of Speed on the Static Deflection of a Guided Saw 88 5 Cutting Tests 91 5.1 Introduction 91 5.2 Preliminary Experiments 91 5.3 Investigation of Tip Side Cutting Mark (TSCM) 94 5.3.1 Effect of Parallelness of the Blade to the Track 94 5.3.2 Axial Movement of the Shaft 95 5.3.3 Runout of a Blade 95 5.3.4 Type of Blade 95 5.3.5 Bite Per Tooth 96 5.3.6 Summary of Tip Side Cutting Mark (TSCM) 96 5.4 Experiments on Cutting Stability 97 6 On Discrepancy of Theoretical Stability Results 105 6.1 Introduction 105 6.2 A Guided Rotating Circular String 106 6.2.1 Exact Solution 107 6.2.2 Approximate Solution Ill 6.3 A Guided Rotating Circular Disk 116 7 Conclusions 117 8 Recommendations 120 Bibliography 121 vii Appendices A Differentiation Transformation B Centrifugal Stresses C In-Plane Cutting Stresses D Thermal Stresses E Solution of Elmn F Saw Blade Parameters List of Tables 3.1 Comparison of the Natural Frequency (Hz) and the Critical Speed (rpm) for Blade A (see Appendix F) without a Spring and a Damper 26 5.2 Wood Parameters . 92 5.3 Cutting Results of Preliminary Experiment B 93 5.4 Cutting Results of Prehminary Experiment C 94 5.5 Runout Effect On TSCM 95 5.6 Blade Effect On TSCM 96 5.7 Bite Effect On TSCM At Supercritical Speed 96 5.8 Bite Effect On TSCM At Subcritical Speed 96 ix List of Figures 1.1 Typical Natural Frequency of a Collared Circular Saw 3 2.2 Geometry and Generalized Force System of a Rotating Disk 10 2.3 Equilibrium Element 11 2.4 Transverse Cutting Force and Spring-Damping Loads on the Rotating Disk 15 2.5 In-Plane Cutting Forces 17 3.6 Comparison of Natural Frequency of Blade D. •: Lehmann [43]; —: Yang. 25 3.7 Natural Response of a Blade without a Spring and a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=0, C=0) 27 3.8 Steady State Response of Mode (0,2) 28 3.9 Forced Vibration of a Blade without a Spring and a Damper, (a). Exci-tation Frequency is 6.8 Hz; (b). Excitation Frequency is 26.8 Hz 29 3.10 Natural Response of a Blade with a Spring, (a). Decay Exponent; (b). Natural Frequency. (K=40000, C=0) 31 3.11 Free Vibration of Mode(0,2) at 200 rpm. (K=40000, C=0) 32 3.12 Free Vibration of Mode(0,2) at 800 rpm. (K=40000, C=0) 32 3.13 Free Vibration of Mode(0,2) at 1500 rpm. (K=40000, C=0) 33 3.14 Effects of Stiffness on the Natural Response of a Blade with a Spring, (a). Decay Exponent; (b). Natural Frequency. + : K = 90000, —: K = 40000. 34 3.15 Natural Response of a Blade with a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=0, C=500) 36 3.16 Free Vibration of Mode(0,2) at 400 rpm. (K=0, C=500) 37 x 3.17 Free Vibration of Mode(0,2) at 1400 rpm. (K=0, C=500) 37 3.18 Effects of Damping on the Natural Response of a Blade with a Damper. (a). Decay Exponent; (b). Natural Frequency. + : C = 300, —: C = 500. 38 3.19 Natural Response of a Blade with Both a Spring and a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=40000, C=500) 39 3.20 Natural Frequency of Blade A Guided by Pin Guides 41 3.21 Decay Exponents of Blade A Guided by Pin Guides 42 3.22 Steady State Response of Blade A Subject to a Transverse Static Load . 43 4.23 Experiment Set-up 1 46 4.24 Experiment Set-up 2 . . . 47 4.25 Natural Frequency of Blade A. —: Numerical Results; o : Experimental Results 49 4.26 Natural Frequency of Blade D. —: Numerical Results; o : Experimental Results 50 4.27 Natural Frequency of Blade A Guided by Pin Guides. — : Numerical Results; o : Experimental Results 52 4.28 Steady State Response of Blade A 54 4.29 Steady State Response of Blade D 55 4.30 Flexibility of Blade A 57 4.31 Flexibility of Blade B and C 58 4.32 Flexibility of Blade D 59 4.33 Runout Variation of Blade A, B and C 61 4.34 Runout Variation of Blade D 63 4.35 Effect of Axial Moaor! on Runout 64 4.36 Effect of Probe Support on Runout 66 xi 4.37 Blade D at 500 rpm. (a). RMS Spectrum; (b). Time History 68 4.38 Blade D at 750 rpm. (a). RMS Spectrum; (b). Time History 69 4.39 Blade D at 1200 rpm (Linear State), (a). RMS Spectrum; (b). Time History 70 4.40 Blade D at 1200 rpm (Nonlinear State), (a). RMS Spectrum; (b). Time History 71 4.41 Blade D at 1800 rpm (Nonlinear State), (a). RMS Spectrum; (b). Time History 73 4.42 Effect of Water Damping on the Vibration of Blade D at 900 rpm. (a). Without Water; (b). With Water 74 4.43 Process of Water Damping on the Vibration of Blade D at 900 rpm ... 75 4.44 Blade D Guided by the Bearing Supports at 900 rpm. (a). RMS spectrum; (b). Time History 77 4.45 Blade D Guided by the Pin Guides at 300 rpm. (a). RMS spectrum; (b). Time History 79 4.46 Blade D Guided by the Pin Guides at 800 rpm. (a). RMS spectrum; (b). Time History 80 4.47 Blade D Guided by the Pin Guides at 1100 rpm. (a). RMS spectrum; (b). Time History 81 4.48 Blade D Guided by the Pin Guides at 1200 rpm. (a). RMS spectrum; (b). Time History 82 4.49 Time History of Blade A Guided by the Pin Guides at 3100 rpm 83 4.50 Time History of Blade A Guided by the Pin Guides at 3800 rpm 84 4.51 Blade D Guided by Wood at 900 rpm. (a). RMS Spectrum; (b). Time History 86 4.52 Transverse Impact Response of Blade D Guided by Wood at 900 rpm . . 87 xii 4.53 Heating Effect on The Stationary Natural Frequency of Blade D. - -: Heat-ing, —: Without Heating 89 4.54 Effect of Speed on the Static Deflection of Blade A with the Pin Guides . 90 5.55 Tip Side Cutting Mark 93 5.56 Idling Response at 3700 rpm. (a). RMS Spectrum; (b). Time History . . 99 5.57 Unstable Time History at 3700 rpm 100 5.58 Unstable Time History at 3700 rpm (Enlarged View) 100 5.59 Time History at 3100 rpm 101 5.60 Time History at 3000 rpm 103 5.61 Time History at 2800 rpm. (a). Without Water; (b). With Water .... 104 6.62 A Rotating String Subject to a Spring at a Fixed Point (from Schajer [47]) 106 6.63 Natural Frequencies of a Rotating String for K = 0 (Exact Solution) . . 110 6.64 Natural Frequencies of a Rotating String for K = oo (Exact Solution) . . Ill 6.65 Single-Mode Approximation of a Rotating String for K = 0. (a). Natural Frequencies; (b). Decay Exponents 113 6.66 Single-Mode Approximation of a Rotating String for Finite K. (a). Natural Frequencies; (b). Decay Exponents 114 6.67 Multi-Mode Approximation: Decay Exponents of a Rotating String for Finite K 115 D.68 Temperature Distribution of a Typical Saw Blade 135 xiii Acknowledgements The author would like to express his genuine gratitude to his supervisor, Dr. S.G. Hutton, for his patient guidance and excellent advice in the development of this research work. The author also wishes to thank Mr. B.F. Lehmann and Mr. V. Lee for their useful assistance in both analytical and experimental work. The financial support of the National Science and Engineering Research Council of Canada is greatly appreciated. Dr. G.S. Schajer is acknowleged for his kind suggestions and help for this final draft. The author is indebted to Mrs. M. Bishop for her valuable help in editing the manuscript of this thesis. Finally, special thanks go to his wife, Xin, his son, Yinxiao, and his parents, whose patience and support made this work possible. xiv Chapter 1 Introduction 1.1 Background The circular saw was invented in Holland in the eighteenth century and developed later in England. Since the nineteenth century, it has been widely applied in the wood indus-try as a powerful cutting tool. The special advantages of a circular saw are its simple construction, its convenient operation and its easy maintenance, all of which make it popular in the lumber industry today. Increase in kerf saving of circular saws has been a goal for a long time in sawmills. In 1979 the Weyerhauser Company estimated that the company would save $2 million per annum if 0.01 inch kerf reducing could be obtained for each cut. In the world there are many similar sawmills, therefore a great material saving and economic advantage can be achieved through kerf saving. The cutting accuracy of circular saws is affected by wood properties, operation con-ditions, tooth patterns and lateral stiffness of the blades. Among these effects, lateral stiffness is a controlling factor. A circular saw is subjected to both membrane and trans-verse forces during the cutting. These forces, especially the transverse forces, can cause a lateral deflection of the blade and worsen the cutting surface. Therefore improved lateral stiffness is required in order to decrease the cutting deflection. The dynamic behaviour of a circular saw can be explained the "critical speed" theory. A typical relationship between natural frequency and rotation speed is shown 1 Chapter 1. Introduction 2 in Figure 1.1 for an unguided blade. A pair of numbers (M,N) is used to describe the mode shape of the blade vibration. M represents the number of nodal circles and N is the number of nodal diameters. It can be seen that each mode has only one natural frequency when the blade is stationary. When the blade rotates, each mode will have two natural frequencies except the modes of zero nodal diameter. These two natural frequencies correspond to two travelling waves in the blade. The upward curves in Figure 1.1 correspond to the forward travelling waves which move in the rotation direction. The downward curves correspond to the backward travelling waves which move in the direction opposite to the rotation. It is noted that the natural frequencies vary with the rotation speed. The frequencies of the backward waves become zero at some speeds which are called critical speeds. If a transverse static force, such as the cutting force, is exerted on the blade at a critical speed, its dynamic stiffness is effectively zero. Hence the blade will "snake". It also can be found that an infinite number of critical speeds exist above the fundamental critical speed. All collared circular saws in use at present are operated below their fundamental critical speed. Many measures have been taken to increase the stiffness of a saw blade. These measures are generally classified as two types: internal rigidity improvement and external support. Tensioning of a blade by roll or heating method can increase the critical speed, and hence improve the lateral stiffness for a given speed. Both radial and annular slots can modify in-plane stresses and are of benefit to the cutting process [1]. All these methods can increase the internal stiffness of a blade. As an example of external support, guides are put on the rim of a splined-arbor saw to reduce the vibration amplitude. Although the above methods can increase the stiffness of a blade and improve the cutting performance, the improvement is restricted hy the existing technology. For instance, the maximum increase of the critical speed through roll tensioning is about 40% and this increase is h'mited by the fact that excessive tensioning can cause the blade to be dished [2]. Chapter 1. Introduction 3 Figure 1.1: Typical Natural Frequency of a Collared Circular Saw Chapter 1. Introduction 4 In the last few years, research has been directed on the supercritical speed cutting possibility of circular saws. It is known that the transverse stiffness of a rotating disk is composed of bending and membrane stiffness. The bending stiffness depends on the material and the dimensions of the blade and does not change with rotation speed. However the membrane stiffness will be increased as the rotation speed increases due to the centrifugal forces. At subcritical speed, the bending stiffness is dominant; but at very high speed, the membrane stiffness will govern the dynamics of the blade. Therefore it appears that it may be possible to run a circular saw at supercritical speed. If the supercritical speed cutting of a circular saw can be realized, the following benefits would be achieved. First, increase of arbor speed for a constant feed speed would reduce bite per tooth of the cutting, then a smoother cutting surface should result. In this way, the planing process could be reduced or eliminated providing dimension accuracy can be maintained. The second advantage is that the thickness of saw blades could become thinner and kerf savings could be obtained due to the significant membrane stiffness. Finally, supercritical speed cutting could increase productivity significantly. 1.2 Review of Previous Work Since resonance can occur in a circular saw at critical speeds according to the classical critical speed theory, the rotation speed of all collared circular s a w 6 used now is below their fundamental critical speed. Until now, no systematic study on supercritical speed cutting has been reported. At the University of California in Berkeley an experimental rig has been built and is to be used to study the aerodynamic effects on rotating blades at supercritical speed [3]. The work that is relevant to the behaviour of disks rotating at supercritical speeds is presented in the following section. Chapter 1. Introduction 5 1.2.1 Response of a Flexible Rotating Disk When a circular blade spins at very high speed, the membrane stiffness will be dominant, hence the blade will behave like a membrane. The first analysis of the spinning disk was done by Lamb [4] in which he studied the natural freqencies and mode shapes for a very flexible disk (neglecting bending stiffness), a slowly rotating disk (neglecting membrane stiffness) and a combined case including both bending and membrane stiffness. Sim-monds [5] and Eversman [6, 7] discussed the dynamic response of spinning membranes with different clamping conditions. Analyses of a flexible rotating disk coupled with a read/write head were obtained by Greenberg [8] and Adams [9,10]. Benson [11, 12] stud-ied the transverse deflection due to spatially stationary loads for a near membrane disk. It was shown that a membrane is unable to support an arbitrary lateral load and thus the effect of bending stiffness, no matter how small, had to be included in the analysis in order to get a steady state response of the disk. Cole [13] investigated the effect of imperfection on the transverse deflection of a flexible spinning disk. More recently, using the Galerkin method, Hutton [14] provided the steady state response of a saw blade subjected to a transverse static load. The indication was that the flexibility increased dramatically at lower critical speeds. However, the values between the resonant peaks were much lower than those in the subcritical region, and even the peak values at higher critical speeds, say above twice fundamental critical speed, were approximately equal to that of subcritical speed. Nigh [15] gave similar results using the finite element method. 1.2.2 Nonlinear Response When a circular blade spins at very high speed, it might exhibit a nonlinear response. Richards [16] observed the flexual waves in a rotating disk in 1872. A nonlinear vibration Chapter 1. Introduction 6 of a spinning disk caused by imperfection was noticed by Tobias [17]. Nowinski [18] stud-ied the transverse large-amplitude vibration of an elastic circular disk of small uniform thickness. In his analysis, it was assumed that the vibration was controlled by both Sex-ual stiffness and tension introduced by centrifugal forces. Advani [19] obtained an exact solution for nonlinear waves in a spinning membrane disk. The solution, in the form of harmonic waves, corresponded to a membrane spinning with no nodal circles and two nodal diameters. Advani [20] extended the above investigation to include an exact solu-tion for nonlinear vibration of a spinning membrane. An uncoupled representation of the equation governing free, axisymmetric, large-amplitude, transverse vibration of a spin-ning membrane was also derived by Advani [21]. An exact solution for large-amplitude, asymmetric waves was also obtained for a special case of a membrane spinning with no nodal circles and two nodal diameters [22]. 1.2.3 Aerodynamic Effects Aerodynamics can play a very important role in the supercritical speed cutting of circular saws. However most studies on this topic in the past were concentrated on the noise generated from aerodynamics. Mote [23] indicated that whistling was a self-excitation transverse instability that produced intense noise at a resonant frequency of a circular saw. Leu [24] presented an investigation in which vortices separating from the edges of the cutting teeth were shown to be the dominant source of pressure fluctuation and hence noise in saws. Two types of flow-induced vibration in idling circular saws, random vibration and resonant vibration, were modelled and analyzed by Leu [25]. Stakhiev [26] concluded from experiments that annular slots in a rotating disk could significantly reduce the free vibration caused by aerodynamic forces. Chapter 1. Introduction 7 1.2.4 Instability The instability of a circular saw has drawn much attention for a long time. It was well recognized that the critical speed could cause instability of a spinning blade since a sta-tionary wave developed in the blade at that speed. References [27, 28, 29, 30] discussed this kind of instability in detail. Radcliffe [31] extended the critical speed theory to include the effect of concentrated in-plane edge loads. The theoretical analysis showed that the critical speed was only sensitive to edge load when the resonance mode and the mode of static edge load buckling were identical. Iwan [32] presented the results of research on the effect of a transverse load on the stability of a spinning elastic disk. The load consisted of an elastic restoring force and a damping force. The analysis showed that the disk system was unstable above its fundamental critical speed due to the effects of stiffness and damping of the load. Shen [34] explained physically the instability as-sociated with guided and slotted saws. Both axisymmetric and asymmetric saws were investigated. It was found that for the axisymmetric saws the instability occured only at or above critical speed but for saws with slots the instability occurred below critical speed. Bulkeley [35] analyzed the instability of nonlinear trasverse waves in a spinning membrane. 1.2.5 Heating Effects Excessive heat can be produced in supercritical speed cutting and might significantly affect the blade stability. Cobb [36] and McComas [37] presented methods to calculate temperature distribution in a rotating disk. Mote [38] gave a temperature distribution in circular saws during cutting. Mote also [39, 40] studied thermal stresses and natural frequencies of circular saws. Mote [29] and Nieh [41] discussed the effects of thermal stresses on critical speed. All the above discussion dealt only with rotating saws at Chapter 1. Introduction 8 subcritical speed. 1.3 Purpose and Scope The purpose of this thesis is to study the dynamic behaviour of collared circular saws at supercritical speeds and to conduct experiments to investigate the possibility of super-critical speed cutting for these blades. An analytical model describing the dynamics of a circular saw is presented in which centrifugal stresses, in-plane cutting forces, thermal stresses, and transverse loads such as spring and damping forces are included. Then Galerkin's method is applied to evaluate the effects of spring and damping loads on the stability of supercritical speed cutting. Idling experiments of circular disks at supercritical speeds are conducted so as to un-derstand the idling behaviour of the disks at these speeds. Idling characteristics including flexibility, runout variation in rotation speed, and effects of a spring-damper system and heating on the stability are investigated in the tests. After the idling test, supercritical speed cutting is performed. Different wood pa-rameters and operation conditions axe examined in order to verify the possibility of supercritical speed cutting. Finally, the vibration of a rotating string subject to a spring is investigated. Based on this knowledge, the discrepancy about theoretical stability results is discussed for a guided circular disk at supercritical speed. Chapter 2 Theoretical Background 2.1 Introduction In this chapter, the theoretical analysis of guided circular saws is summarized based on literature [42]. At the beginning, the governing equation of a circular saw subjected to both transverse and membrane forces is developed based on the classical theory of a plate with small deflections. The equation is first derived in a body-fixed coordinate and then transformed into a space-fixed coordinate. At the next stage, the transverse loads are generalized as spring, damping and out of plane cutting forces, and the membrane loads include rotational, thermal and in-plane cutting forces. Finally, Galerkin's Method is used to get a single mode assumption for the equation, and the solutions of the natural and forced response are discussed in detail. 2.2 Formulation of Governing Equation 2.2.1 Governing Equation in a Body-Fixed Polar Coordinate [13] The disk rotating at speed Q, shown in Fig.2.2 is elastic, isotropic and homogeneous. It is subjected to space-fixed time dependent transverse loads q and membrane forces caused by rotation, cutting, and temperature variations of the blade. The thickness of the disk is h, Young's Modulus is E, Poisson's Ratio is fi, and mass density per unit volume is p. The outer radius is a and the inner radius is 6. The body-fixed coordinate is r,8,z. The classical governing equation of a rotating disk can be derived from equilibrating 9 Chapter 2. Theoretical Background 10 Figure 2.2: Geometry and Generalized Force System of a Rotating Disk force components in z direction. An equilibrium element is shown in Fig.2.3. The force components are defined as follows, Nrr, Nre, N$T and Nee'- membrane forces; Qr and Qe'. transverse shear forces; MTT, MTe, Mer and Mee- bending and twisting moments; h p W: inertial force; q: general transverse force. Considering equilibrium in transverse direction yields the following equation FIN + !L8L + + VIIL^L + N 212*1 | N » D W r dr r 86 dr dr r r dr2 r dr 1_ d_Nee_ dW_ Nee_ 8TVV_ 1 dNer dW_ 8W r 2 86 d6 + r 2 862 + r 86 dr r 2 86 [ ' } d2 W I 8Nre 8W N^ 82W _ 82W + r" 8r86~r r dr 86 + r 8T86 + 9 P 8t2 ~ Figure 2.3: Equilibrium Element Chapter 2. Theoretical Background The moment equilibrium in r,9, and z directions yields _ 1 dM66 dMr9 Mr6 Af&. V e r 66 + 6r + r r _ dMrr dM6r A[rr _ Mflfl V r ~ 0r rdB r r For a thin plate with small deflections, the Kirchoff Hypotheses are made: 1. The midplane remains unstretched subsequent to bending; 2. A normal to the midplane remains normal in deflection; 3. The stress normal to the midplane, az, neglegible. Using these hypotheses gives „r/l dW 1 d2Ws d2W, ^A 02W 1 dW. where D is the plate bending stiffness D Substituting Eq.2.3 into 2.1 yields Eh3 12(1 1 d _ Qe = -D-rTeV<W Qr = -I>|-V 2^ C7 7* Substituting Eq. 2.5 into 2.1 and Using Nrr = harr, Nee = hage and JVr$ h Trg, we have r 2 50 r or or r oo h d . dW 1 0W\ . 02W Chapter 2. Theoretical Background 13 where V 4 is the biharmornic operator V 4 = ( 1 1- r K8r2 r 8r r2 862' Eq.2.6 is the governing equation for the vibration of a thin circular disk subjected to the general transverse and in-plane forces in the body-fixed coordinate. 2.2.2 Governing Equation in a Space-Fixed Polar Coordinate It is convenient to solve Eq. 2.6 in a space-fixed coordinate when the transverse loads are fixed in space and the results in such a coordinate system are easily understood by an observer standing on the ground. The space-fixed coordinate system (R, <f>, Z) is shown in Fig. 2.2. The relationship between the body-fixed coordinate and the space-fixed coordinate is given by R = r <t> = e + nt (2.7) Z = z Then the differentiation transformations are (see Appendix A) dmW _ d^W drm ~ 8R™ dmW 8™ W £JlL = £LiL ( m = 1,2) (2.8) 86™ 8(j)m v i / v / 8 8 8t™ K8t ^ 8<f>} Thus, Eq.2.6 becomes r dr Or r 0<p h 8 8W , 18W, h dW t 0 0 82W , n , 0 2 W \ + M _ + 2 n_ +n2_) =gChapter 2. Theoretical Background 14 Equation 2.9 is the governing equation in the space-fixed coordinate system. Since the disk in this study is clamped at the inner radius and free at the outer radius, the boundary conditions can be expressed as aw . Mrr |(.,*o = 0 (2.10) 2.2.3 Transverse and In-Plane Forces in the Governing Equation Transverse Forces In the governing equation, the transverse forces are represented by q which can be gener-alized as cutting, spring and damping forces. The transverse loads are shown in Fig.2.4. The cutting force is caused by the interaction between the teeth and the wood and can be expressed by p(r,<f>) = -P cos utS{r - rp) 6(<f> - <f>p) (2.11) T where P is the amplitude of the load; u> is the frequency of the load; coordinates rp and <j>p indicate the position of the load, and 6( x ) is the Dirac delta function 6{x) = < 1 when x = 0 0 when x ^  0 The spring and damping forces could result from the interaction between the blade and the wood as well as the surrounding air. Both wood and air are elastic materials which can provide the spring force to the blade during the vibration. The air can dissipate the vibration energy of the blade as well. Chapter 2. Theoretical Background 15 Figure 2.4: Transverse Cutting Force and Spring-Damping Loads on the Rotating Disk The spring and damping loads can be written as Q(r,<f>,t) = --KW6{r-rQ)8{<t>-<t>Q) T 8W 1 -C^r-6{T-rq)8{<b-4,Q) (2.12) at T where K is the stiffness coefficient; C is the damping coefficient. For the sake of con-venience, the spring and the damping are considered to act at the same point of the blade. Thus, the total transverse load is q = p + Q = -[P cos wt6(r-rp)6(<l>-<t>p) (2.13) T dW -KW6{T-TQ)6{<j>-<f>q)~ C 6{r - rQ ) 6{4>~ <h)l Chapter 2. Theoretical Background 16 In-Plane Forces The in-plane forces of a circular saw during the cutting are very complicated. In this study, only centrifugal, cutting and thermal stresses are considered. The centrifugal stresses are due to the spinning of the disk. They can be given in the form (see Appendix B) 1 . - 2 o-r„ = Dx + D2 — + D3r 1 T = Dx - D2^ + D4r2 (2.14) rU = 0 where n _1 «'[(! +M)(3 + jQ + (l-M a)(j) 4] 1 " (I + ^ ) + ( I - M ) ( ! ) 2 n _1 a ay[(l-»(3 + M)-(l-M a)(j) a] 2 " 8 ^ (1 + H + ( 1 " M ) ( ! ) 2 D3 = - ^ f i 2 ( 3 + /x) (2.15) DA = -\ptf(l + 3/x) It can be seen that the centrifugal forces are axisymmetric, and increase with the square of the rotation speed. The in-plane cutting forces can be represented as concentrated loads on the edge of the saw blade as shown in Fig.2.5. Thus the membrane stresses caused by the cutting are (see Appendix C) On ,0,1 ^, 2 a\ b\ . , <r = 4 + (~T + 2 0 lr - — 1 + J-) cos^ T r r T , C l , r 2 c ' l dl » • j + (-2a2 - 5fl _ 4&i)cos(2# r Chapter 2. Theoretical Background 17 Figure 2.5: In-Plane Cutting Forces + (-2c' " -it ~ ~ t ) sm(2^) ao , ~. 2 K . <fc = - ^ + (661r + 7 i i + ^)cos^ + ( 6 i i r + ?A + s i n^ (2.16) + (2 a2 + 12 62 r J + -^ 2.) cos(2<£) T 6c7 + (2c2 + 12ri2r2 + — r 2 - ) sin(2<£) r* r* r - ( 2 ^ T" + —) cos^ + 2(a2 + 362r2 - ^ _ *|) s i n ( 2 ^ -2(c 3 + 3<£2r2 _ ^ - ^i) cos(2^ ) where o0 ~ a*2 8 1 6 the coefficients which depend on the boundary conditions. Chapter 2. Theoretical Background 18 It is clear that the cutting loads introduce asymmetric stresses in the blade. At this Btage, the force components PT and Pt are considered as constants. However, they are dependent on the the cutting parameters such as bite per tooth in the actual operation. The thermal stresses are caused by the friction between the blade and the wood. Since — t the heat generated in the cutting process is very complicated and no analytical formulae have been derived, the thermal stresses in this study are assumed to be represented by (see Appendix D) Eon ,r2 - b2 r 2 •(-= „ / Trdr - / Trdr) a2 — b2 Jb Jb „T where aj is the thermal expansion coefficient and T is the temperature distribution which is in the form of It can be seen that the thermal stresses are axisymmetric. Therefore the total in-plane stresses are *** = <>U + °U + (2-19) Tr<t> = T% Substituting Eq. 2.19 into 2.9, the governing equation becomes n „ 4 m , 1 9 . , dW _ h d , 8W _ Jl ?W ~ r d i K T r * d'v J H r 34>] r2 T r * d<f> + + + (,20) Chapter 2. Theoretical Background 19 1 8W 1 -r-KW8(r-rQ)8(<t>-<t>Q)+ C — -6(r-rQ)8(<f>-4>Q) T Ot T = - P cos u>t 8(r — rp) 8(<j> — <j)p) T 2.2.4 Solution Method The governing equation 2.20 is a fourth order partial differential equation, it is not practical to get a closed-form analytical solution. In this study, the Galerkin's method is applied to obtain the approximate solution. For this purpose, the governing equation 2.20 is rewritten as L(W) - q{W) = 0 (2.21) where L = £ V 4 - h-^-[r(ar A + r r ^ A ) ] T or OT r o<p hd_ 8_ 1 j9_ _ h_ 8_ r d<f>[Tr*dr + ***r 8<t>} r*Tr+8<j> + ^ ^ + 2 f i ^ + fi2^> ( 2 ' 2 2 ) + -K6(r-rQ)8(<f>-<t>Q)+C-?-±6(r-rQ)8(<t>-<f>Q) r ot r q = - P cos u; t 8( r — rp) 8( <b — §p ) r An approximate solution of Eq. 2.21 is assumed of the form M N W(r,<t>,t) = £ £[Cm„(i) cos(n^ ) + Smn(t) sin(n^ )] i^ n(r) (2.23) m=0 n=0 where m is the number of nodal circles; n is the number of nodal diameters. Cmn and Smn are unknowns to be determined by the solution. Rmn is the radial eigenfunction which depends on the boundary conditions of the disk. Following the work of Hutton et •al [42], the function Rmn is tabs* ?.s Rnn(r) = Y,Elmnrl+m (2.24) 1=0 Chapter 2. Theoretical Background 20 where Elmn are unknown coefficients. Combining Eq.2.23 and 2.24 into 2.10 and using the normalizing condition Rmn(a) = 1 (2.25) -obtain five linear algebraic equations for solving Elmn (see Appendix E). Use the Galerkin's Method, we have j** £[L{W) - q{W)} coB{t<j>)Rtt{r)Tdrd(j) = 0 jT [L(W) - q{W)} sin(t^ ) Rtt(r) r dr d<f> = 0 (2.26) a = 0,1,2, ...Af t = 0,1,2, ...iV Expanding Eq. 2.26 yields a set of (2 TV + 1) (Af + 1) linear equations. The dynamic characteristics of the system can be investigated by solving these equations. In order to fully understand the stability of a circular saw at supercritical speed, it is necessary to study the effects of different load parameters on the vibration of the disk. A single mode solution is mainly investigated in this study because it i6 much easier to change the load parameters in such a solution than in the multi-mode solution. The single mode solution is assumed by W(r,<p,t) = [Cmn(t) cos(n^ ) + Smn{t) sin(nfl] ^ ( r ) (2.27) m, n, = 0,1,2,3, ... Substituting Eq.2.27 into 2.26 yields the following relationship (1) ^ Cmn . (21 dCmn , (3) dSmn , U) ri 1 (6) c I? v™ ~dl?~ + v™ ~dT v™ IT v™ m n Vm" = 1 ^ + + % + V{:lCmn + r£?STOn = F2 (2.28) Vmn ' vmn rff • -#mn ^ Chapter 2. TheoreticaJ Background 21 where and a P C ( i ) Smn /<») Smn „ ( 1 ) '/ran = „<»> '/mn = Ci2mn(rg)cos2(n^) „<») 'Imn = 2ann41 n + CJZLfo) sin(n^) cos(n^) '/mn = «<2 + KRln(rQ) cos2(n<f>Q) „(«) '/mn = -aC£ 3 n + J f iCfa) sin(n^) cos(n^) „ ( 6 ) 'Imn = Hmi „(T) '/mn = -2B\ln<SX + CR2mn{rQ) sin(n<fo) cos(nfo) Iran = C t ( r « ) « n 2 ( n ^ ) „(•) 'Imn = + i ^ t h ) sin(n^) cos(n^) -(10) 'Imn = ^ d 2 „ + ^ ^ Q ) s i n 2 ( n ^ ) F1 = P Rmn(rP) cos(ra ^ >p) 008(0-1) F2 PRmn(rP) B\n(n<f)p) cos(a;i) 2 ifn = 0 1 ifn ^  0 0 ifn = 0 = < 1 ifn ^  0 = irhp f rR2mndr (2.30) _ T \n (— I— - 1 d 2n* * - Jb *Rmn\Dr{dri + t ^ ^ ^ + ^ ^ ^ ^ 2n2 d 4nJ n 4. D , 0 dR,nn. + -TTr ~ IT + r4" " " h6-r^<Trr-dV) - h ? £ ^ + -hn2RmnO-H> - n2n2hprRmn]dr dq> dr r Chapter 2. Theoretical Background 22 2.2.5 Natural Response —\ If P = 0, Eq. 2.28 will become (1) dtCmn (2) ^ finn (3) dSmn (4) „ (6) Q _ -"dt2 "Si n (6) <^^n (7) ^ mn (8) dSmn (9) ~ (i0) Q _ n /n 01 \ ^ m n 1ft2 ~ A ~~dt ^ron n ,"lB n ~ ^ ' Equation 2.31 describes the free vibration of a circular saw. Let Cmn(t) = CmneXt Smn(t) = 5roneAt (2.32) where C7mn and £ m n are constants; A is the eigenvalue of the free vibration. Substituting Eq. 2.32 into 2.31 yields (*£JA» + ifilA + r/W ) C7 m n + ( £ A + r£) )5 m n = 0 (V{2 A + ) Cmn + (,£> A2 + r£) A + ) Smn = 0 (2.33) For the mode of non-zero nodal diameters (n ^  0), the characteristic equation is „(i) A2 + »<»> A + v{4) n{3) A + i?<8> '/mn ' /mn 1 7mn '/mn " 1 'Iran Q ^ g^ J „(7) A _L „<•) „(6) A2 + (8) A + (10) '/mn i '/mn '/mn ^ '/mn ^ '/mn and for the mode of zero nodal diameter ( n = 0 ), the equation becomes ifijA' + ^ A + flfiJ-O (2.35) The eigenvalue A of the system can be obtained from Eq. 2.34 and 2.35. Chapter 2. Theoretical Background 23 2.2.6 Steady State Response All derivatives with respect to time in Eq. 2.28 become zero in the steady state response (a; = 0). Therefore, for the mode of non-zero nodal diameter(n ^  0), the equation of the steady state response is given by VntlCmn + vUlSmn = P R^fo) cos(n^ p) V{2Cmn + Vm^Smn = 0 (2.36) and for the mode of zero nodal diameter (n = 0), the equation is simplified as vitl Cmo = P Rmo(rP) (2.37) Solving Eq.2.36 and 2.37, we can get coefficients Cmn, and Smn. then, using Eq. 2.27, the steady state response of the disk is obtained. Chapter 3 Numerical Results 3.1 Introduction In this chapter, numerical results are obtained from the above analytical model. Since it is thought that the wood together with the carriage can generate a spring and damping forces on saw blades in the cutting, the effects of spring and damper on the stability of a circular blade at supercritical speeds are discussed in particular. The relationship between the steady state response and the rotation speed is also presented. 3.2 Single-Mode Assumption The single-mode solution is first used to study the dynamics of a guided rotating disk qualitatively. 3.2.1 Verification of Single-Mode Assumption Since the investigation of the stability of a blade is based on the real part of the eigenvalue of the blade vibration, it is necessary to verify the eigenvalue results. Comparison of natural frequency calculations between this study and others is presented in Table 3.1. It can be seen that the maximum error of the stationary frequencies is 5.6% and that of the critical speeds is 4.2%, hence, the numerical results are quite reliable. Fig. 3.6 demonstrates the relationship between the frequency and the speed, which is in agreement with other studies. 24 Chapter 3. Numerical Results Figure 3.6: Comparison of Natural Frequency of Blade D. •: Lehmann [43] Chapter 3. Numerical Results 26 Case Reference Mode (0,0) (0,1) (0,2) (0,3) (0,4) (0,5) Stationary Yang 38.0 38.0 45.0 69.0 113.0 173.0 Leissa [44 36.0 36.0 44.0 N.A. ° N.A. N.A. Lehmann 42 38.0 38.0 45.0 70.0 113.0 170.0 Schajer [50 38.7 38.7 45.7 70.6 113.4 170.7 Critical Speed Yang N.A. N.A. 2120 1800 2050 2400 Lehmann N.A. N.A. 2160 1800 2070 2500 Schajer N.A. N.A. 2199 1850 2101 2457 'Data are not available. Table 3.1: Comparison of the Natural Frequency (Hz) and the Critical Speed (rpm) for Blade A (see Appendix F) without a Spring and a Damper 3.2.2 Effect of Stiffness and Damping on the Stability of a Circular Saw A Blade without a Spring and a Damper When there are no spring or damping loads on a rotating blade, the decay exponents (real parts of the eigenvalue) are zero at any rotation speed. Such results are shown in Fig. 3.7. Since all decay exponents are zero, the free vibration of the blade is stable at any speed. However, when the blade is subjected to a static transverse load, the deflection of the blade becomes significant at the critical speed as shown in Fig. 3.8. Therefore, for a rotating blade without a spring and a damper, the resonance at critical speed is the only mechanism of the instability. A forced response of the backward and the forward waves is shown in Fig. 3.9. The excitation frequencies are 6.8 Hz and 26.8 Hz which respectively correspond to the natural frequency of the backward and the forward waves at 300 rpm in Fig. 3.7 .(b). It can be seen that both backward and forward travelling waves will have a resonance when the excitation frequency is equal to their natural frequency. A Blade with a Spring The natural frequency and decay exponent of a rotating blade mih a spring are plotted in Fig. 3.10. In comparison to Fig. 3.7, the frequency of the forward travelling Chapter 3. Numerical Results 27 10.00 -9.00 -8.00 -7.00 -6.00 -s.oo -4.00 -3.00 -ID 2.00 -z o 1.00 -0. 0.00 -I - 1 .00 -- 2 . 0 0 -o - 3 .00 " - 4 . 0 0 --S .OO --6 .00 --7 .00 -- B O O -- 9 . 0 0 --10 .00 -0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 fThauBands) NOTATION SPEED (rpm) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.6 2 ^Thousands) R T T A T 1 Q N SPEED (rpm) Figure 3.7: Natural Response of a Blade without a Spring and a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=0, C=0) Chapter 3. Numerical Results 28 Figure 3.8: Steady State Response of Mode (0,2) Chapter 3. Numerical Results 29 3 111 O a *> a 630 SCO 400 -300 -200 -100 -0 -100 0.2 — r 0.4 — I — O.S I n r 0.6 (Thousands) ROTATION SPEED (rpm) 1.2 - 1 0 0 -2 111 5 a V) 5 - 2 S O -- 3 0 0 -- 3 5 0 -- 4 0 0 I 0.6 (Thousands) ROTATION SPEED (rpm) r O.S 1.2 Figure 3.9: Forced Vibration of a Blade without a Spring and a Damper, (a). Excitation Frequency is 6.8 Hz; (b). Excitation Frequency is 26.8 Hz. Chapter 3. Numerical Results 30 wave increases, however, the stationary frequency and the critical speed of the backward wave remain unchanged. This occurs because the spring is in an anti-node position of the forward travelling wave and in a node position of the backward wave. The spring in the anti-node position will make the mode stiff but in the node position it cannot change the blade stiffness. In can be seen in Fig. 3.10 that the frequency of the backward wave becomes zero in a region of rotation speed from 700 rpm to 1100 rpm. One of the decay exponents in that region becomes positive, hence the free vibration of the blade will be unbounded. The free vibrations of the blade at different speeds are demonstrated in Fig. 3.11-3.13. The vibration with zero decay exponents such as those at 200 rpm and 1500 rpm speeds remains equal-amplitude oscillation as shown in Fig. 3.11 and Fig. 3.13, while that with the positive decay exponent at 800 rpm is unbounded as shown in Fig. 3.12. This kind of instability is called stiffness instability or zero-frequency instability. The frequencies and the instability region increase as the stiffness of the spring increases as shown in Fig. 3.14. The steady state response of the blade has a resonance at the critical speed, which is similar to that of the blade without a spring. It can be concluded that both resonance and stiffness instability can occur in a blade with a spring. The blade is unstable if its speed is in the instability region. Above the region, the blade becomes stable again. A Blade with a Damper Fig. 3.15 presents the natural frequency and the decay exponent of a rotating blade with a damper. Compared with Fig. 3.7, the stationary natural frequency and the critical speed of the backward wave are not changed, but the frequency of the forward wave decreases. It is assumed that at zero speed the damper is in an anti-node position of the forward wave and in a node position of the backward. But at the critical speed, the damper is in an anti-node position of the backward wave. The oscillating speed of the lateral motion of the blade is expressed as DW/Dt = dW/dt + QdW/d<l>. At Chapter 3. Numerical Results 31 Figure 3.10: Natural Response of a Blade with a Spring, (a). Decay Exponent; (b). Natural Frequency. (K=40000, C=0) Figure 3.12: Free Vibration of Mode(0,2) at 800 rpm. (K=40000, C=0) Chapter 3. Numerical Results 33 U 3 U O 5 Ol TIME (second) Figure 3.13: Free Vibration of Mode(0,2) at 1500 rpm. (K=40000, C=0) stationary, in anti-node position, ft = 0 and 8W/d<f> = 0, then DW/Dt = dWjdt, as a consequence, the dissipating force CdW/dt reduces the frequency of the forward wave; however, in node position, dW/dt = 0 and SldW/d<f> = 0 (ft = 0), hence, the frequency of the backward wave is not changed. When the frequency of the backward wave is zero, dW/dt = 0 and dW/d<f> = 0 in anti-node position, then no damping force acts on the backward wave, so the critical speed remains fixed. It is noted from Fig. 3.15 that all decay exponents are negative below the critical speed and one of them becomes positive above the speed. Therefore, above the critical speed, the blade with a damper is unstable. This type of instability is called damping instability. The free vibration of the blade at both subcritical and supercritical speeds is plotted in Fig. 3.16 and 3.17. It is clear that the vibration below the critical speed is a decayed free oscillation and that at supercritical speed it is unbounded. The effect of the damping on the frequencies and the decay exponents are presented in Fig. 3.18. As the damping is increased, the frequency Chapter 3. Numerical Results 0.2 T 0.4 T 0.6 T — i — i 1 — r 0.8 1 1.2 (Thauaarxds) RDfTATIQN SPEED (rpm) T 0.2 T 0.4 T 0.6 i i — i 1 r o.s 1 1.2 CThausonds) ROTATION SPEED (rpm) Figure 3.14: Effects of Stiffness on the Natural Response of a Blade with a Spring Decay Exponent; (b). Natural Frequency. + : K = 90000, —: K = 40000. Chapter 3. Numerical Results 35 of the backward wave decreases and the characteristics of the decay exponents stay the same although their values are different. At critical speed, the resonance can occur in the blade if it is subjected to a static force as shown before. - From the above analysis it can be seen that the instability of a blade with a damper can be caused by the resonance at the critical speed and also by the damping above the critical speed. A Blade with both a Spring and a Damper Fig. 3.19. shows the natural frequency and the decay exponent of a blade with both a spring and a damper. It can be seen in Fig. 3.19 .(a), that the stiffness and the damping can affect the frequency of the forward wave, but cannot change the frequency at stationary and the critical speed of the backward wave as before. All decay expo-nents are negative below the critical speed and some become positive above the speed in Fig. 3.19 .(b)., therefore, the blade is stable at subcritical speed and unstable at su-percritical speed. In this case, the instability can be caused by resonance, stiffness and damping. 3.3 Multi-Mode Assumption In this section, multi-mode solution is used to reveal some phenomena which cannot be predicted by single-mode assumption. The following results are calculated from computer program GUIDE5 [43]. The natural frequencies of a guided rotating disk are shown in Fig. 3.20. An interest-ing phenomenon presented by the numerical results is the mode crossing. All vibration modes avoid crossing each other at subcritical speed, but some of them will couple to-gether at supercritical speed. As long as two modes cross each other, their frequencies become identical, and the crossing will be maintained in a certain speed region. Beyond Chapter 3. Numerical Results 36 10.00 LI z o 0. $ a a -20 .00 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 (Thousands) ROTATION SPEED (rpm) — i — r 1.80 2.00 120 >-=> a ff ~i—1—71 1 — r 0 . 8 ' 1 1.2 (Thousands) ROTATION SPEED (rpm) r 1.4 T 1.6 1 1.8 Figure 3.15: Natural Response of a Blade with a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=0, C=500) Figure 3.17: Free Vibration of Mode(0,2) at 1400 rpm. (K=0, C=500) Chapter 3. Numerical Results 38 Figure 3.18: Effects of Damping on the Natural Response of a Blade with a Damper, (a). Decay Exponent; (b). Natural Frequency. +:C = 300, —: C = 500. Chapter 3. Numerical Results 39 Ul z o a I 0.8 1 1 2 (Thousands) ROTATION SPEED (rpm) 130 -120 -110 -100 -90 -2 BO --w >• O 70 -z Ul D 60 -o Ul E SO -40 -30 " 20 -10 -0 - T — i — i 1 1 r 0.8 1 1.2 (Thousands) ROTATION SPEED (rpm) Figure 3.19: Natural Response of a Blade with Both a Spring and a Damper, (a). Decay Exponent; (b). Natural Frequency. (K=40000, C=500) Chapter 3. Numerical Results 40 the region, they will seperate again. It should be mentioned that the physics of mode crossing is not quite clear. Leissa [46] and Schajer [47] explained avoided crossing (curve veering), but the mode re-crossing was not discussed. Similar phenomena of avoided crossing and re-crossing can occur in rotor dynamics [48]. Fig. 3.21 shows part of the decay exponents of the above disk at supercritical speed. It is noted that two kinds of mechanism can cause unstable oscillations of the disk. First, the decay exponent of a mode is positive when its natural frequency becomes zero, which has been disclosed by the single-mode assumption. Second, one of the decay exponents corresponding to the crossing modes will be positive when the crossing takes place, which cannot be predicted by the single-mode solution. As long as the decay exponents are positive, the vibration of the blade will become unstable. Fig. 3.22 shows the steady state response of a rotating disk subject to a static trans-verse load. The deflection of the disk is very large at its fundamental critical speeds, and the resonance displacement decreases as the rotation speed increases. Therefore, resonance might not cause cutting instability at higher critical speeds. Figure 3.20: Natural Frequency of Blade A Guided by Pin Guides Chapter 3. Numerical Results 42 Figure 3.21: Decay Exponents of Blade A Guided by Pin Guides Chapter 3. Numerical Results 43 0.02 Figure 3.22: Steady State Response of Blade A Subject to a Transverse Static Load Chapter 4 Idling Tests 4.1 Introduction In this chapter, the idling test of a circular saw at supercritical speeds is discussed. The purpose of the idling experiment is to understand the fundamental behavior of circular saws at supercritical speeds so as to provide guidance for selecting cutting parameters. The experimental apparatus and the set-ups are first introduced. Then the natural frequency, the steady state response, the flexibility and the runout variation of the circular saws are measured at different speeds. As nonlinear vibration occurs in the thin blade at supercritical speed, the phenomenon, as well as possible causes are presented. Finally, the effect of a spring-damper system and heating on the stability of a circular saw at supercritical speeds are investigated. 4.2 Experiment Apparatus All experiments were conducted in the Wood Machining Laboratory of the Department of Mechanical Engineering at the University of British Columbia. The experimental rig of the circular saw is comprised of a rotating shaft and a hydraulic pump.The shaft on which a saw blade is placed is supported by a pair of self-aligning bearings which can allow some axial motion of the shaft. An electric motor of 100 hp drives the hydraulic pump which can rotate from zero to 4000 ?pnr by rieans of changing the angle of a swash plate. In the cutting test, wood is moved by a carriage which is driven by another 44 Chapter 4. Idling Tests 45 hydraulic pump through a cable. The feed speed of the carriage can be varied from zero to 480 fpm. Non-contacting eddy current probes are used to measure the motion of the saw blade. A force transducer is mounted on the back of an electromagnet to detect force applied to the saw. The electromagnet is provided with an electric current by two kinds of signal generator. The random frequency generator produces a white noise signal while the voltage generator supplies direct current. Experimental data are analysed with either Bruel & Kjaer 2034 or Nicolet 660A dual channel spectrum analyser. A TEAC R-71 data recorder with seven channels is used to record experimental data for later analysis. Four circular blades are used in the experiments, and are described in Appendix F. Among these blades, Blade A and D are mainly used to investigate the dynamic characteristics of the circular saws. 4.3 Experiment Set-up Natural frequency, flexibility and steady state response of a circular blade can be mea-sured in the set-up shown in Fig. 4.23. In the set-up connected by solid lines, the electro-magnet excites the blade with random forces which are produced by the random signal generator. The force transducer signal goes to the frequency analyser. The displacement is measured by the eddy current probes and is also sent to the analyser. With these two signals the analyser can determine a transfer function of the blade. The flexibility is derived from the transfer function with frequency span between zero and 10 Hz, while if the frequency span is chosen from zero to 100 Hz, the lower natural frequencies can be obtained. In the set-up corresponding to the dashed lines, a DC current through the electromagnet produces a static force on the blade. A pemanent magnet is also used to generate such static force. The steady state response is measured by the eddy current Chapter 4. Idling Tests 46 Blade Hydraulic Pump C S3 1 Spectrum Analyser Electromagnet Force Transducer 0 Collar Eddy Current Probe Oil Voltage Generator C=7 -4-i . 1 ! Random Frequency I Generator Figure 4.23: Experiment Set-up 1 probe and displayed on the analyser with its time history function. Fig. 4.24 shows set-up 2. The vibration of the blades is measured by the eddy current probes and then recorded by the TEAC tape recorder. After finishing the operation, the signal is analysed by the spectrum analyser. With the time history function of the analyser, the runout variation is investigated, and with the instant spectrum selection, the natural frequency is obtained. 4.4 Natural Frequency Measurements Since Blade A and D are mainly used in the experiments, their natural frequencices are measured in this study. Chapter 4. Idling Tests 47 Hydraulic Pump Bearings < to 1 Spectrum Analyser Blade Collar Eddy Current Probe FM Tape Recorder • rrm Figure 4.24: Experiment Set-up 2 Chapter 4. Idling Tests 48 Both experimental and numerical results of unguided Blade A are shown in Fig. 4.25. The RMS spectrum procedure as shown in Fig. 4.24 is used to measure the frequencies. The numerical results are calculated from the theoretical model given by Hutton [42] and Lehmann [43]. It can be seen that the theoretical curves are quite close to the experimental results except those of mode (0,0) and (0,1). The reason for this is probably that the consideration of roll tensioning effect in the numerical results is not accurate enough and these two modes are more sensitive to the effect. The fundamental critical speed is about 2300 rpm. The speed intervals between two consecutive critical speeds under 2800 rpm are very small, so it is not ideal to conduct a cutting experiment between 2300 rpm and 2800 rpm. Otherwise the resonance instability could occur during the cutting. Because Blade B and C have the same dimensions, the frequency-speed diagram of Fig. 4.25 also can be used as a reference for Blade B and C. The natural frequency variation of Blade D is shown in Fig. 4.26. The fundamental critical speed of the blade is about 650 rpm. The transfer function method is used in the experiment. All experimental data are in agreement with the theoretical results below 900 rpm. Above that speed, however, some lower frequencies in the experiment diverge from the theory. For example, experimental frequencies of mode (0,0) are lower than the theoretical results. And those of mode (0,2), (0,3) and (0,4) do not have a distinguished trace above 900 rpm and seem to be inclined to stay with the backward travelling wave of mode (0,1). The reason for this is due to the nonlinear vibration of the blade above the speed which will be discussed later. Since a stationary spring and/or damper will significantly change the dynamics of circular saws, the natural frequency of Blade A with a pair of pin pad guides is measured. The pad size of the guides is 1 x l | inches and the material is h&vd wood. The guides are put on the rim of the blade, and the stiffness of the arms which hold the guides is about 106N/m. Water is supplied between the blade and the guides in order to reduce Chapter 4. Idling Tests 49 o o u. rx UJ O D UJ Z U >- w X M ^ Figure 4.25: Natural Frequency of Blade A. —: Numerical Results; o : Experimental Results. Chapter 4. Idling Tests o o L C C l J J 0 3 u J Z U > w X . M <—> Figure 4.26: Natural Frequency of Blade D. —: Numerical Results; o : Experimental Results Chapter 4. Idling Tests 51 the heat caused by friction. The transfer function procedure is used in the experiment. The results of the measurement are shown in Fig. 4.27. It can be seen that the experimental results are in agreement with the numerical results below the fundamental critical speed (2300 rpm). However, above that speed the experimental frequencies do not vary with the rotation speed. The physics of the phenomenon shows that the travelling waves in the blade will propagate at constant speeds. The numerical results do not demonstrate this characteristic. One possible cause of the difference may be due to the nonlinearity of the blade and the guides which are not considered in the numerical results. Another possibility is the effect of guide lubricant which is not taken into account in the existing theory either. 4.5 Steady State Response As discussed in Chapter 3, the resonance at critical speed is a major cause of blade instability. The purpose of the experiment is to study the resonance instability at critical speeds. A static force is generated either by an electromagnet or by a permanent magnet shown in set-up 1. This method can simulate the transverse cutting forces, but the simulation is not ideal because the force exerted on the blade is an inverse of the square of the distance between the blade and the magnet. The steady state response of Blade A is measured first. The electromagnetic force is used in the experiment. The voltage generator shown in Fig. 4.23 is used to provide two DC signals of 12 V and 6 V which can excite the blade. The mean static displacement measured by the eddy current probe is analysed by the spectrum analyser. It can be seen in Fig. 4.28 that the responses under these two forces are very similar. The curves in subcritical region are quite flat, ar.xl become larger when approaching fundamental critical speed (2300 rpm). Around this speed the deflection resulting from Figure 4.27: Natural Frequency of Blade A Guided by Pin Guides. Results; o : Experimental Results. — : Numerical Chapter 4. Idling Tests 53 the smaller force becomes significant, while that of the larger force becomes even larger. This indicates that at this speed the blade is at resonance if it is subjected to a static transverse force. Above 2700 rpm, the deflection of the larger force is about 20% of that of subcritical speed, and for the small force, it is about 50%. This shows that the blade stiffness at supercritical speed is much higher than at subcritical speed. The difference between both deflection curves is very small at supercritical speed, which means the blade stiffness increases so significantly so that even the larger force cannot apparently bend the blade. The blade deflects in both directions above the fundamental critical speed because the phase angle changes at resonance. It is known that the electromagnetic force is proportional to the electric current in the coil, therefore, the deflection of the blade is only proportional to the current. Since the current is not proportional to the voltage due to the internal resistance of the generator, hence, the deflection in Fig. 4.28 is not proportional to the voltage. A permanent magnet in Fig. 4.23 is used to produce a static force for measuring the steady state response of Blade D. The response shown in Fig. 4.29 demonstrates a similar characteristic to that of Blade A. Below 400 rpm, the deflection remains almost constant and increases dramatically above the speed. The maximum deflection is around 690 rpm which is a little higher than that of the fundamental critical speed. Then the deflection drops to 0.2 mm which is about one quarter of that of subcritical speed. Since the nonlinear vibration occurs above 900 rpm, and the blade will contact the magnet, the response above the speed cannot be obtained. It can be concluded from the above experiments that resonance definitely occurs around the fundamental critical speed. Hence, the rotation speed for cutting should be kept away from the speed. At higher critical speeds, the deflection is much less than that of the fundamental critical speed as shown in the case of Fig. 4.28. Therefore, the resonance effect on the stability of the blade becomes insignificant. 3.50 3.00 OQ C n to <X3 CO «r>-in o W a 2.50 H ZOO H 1.50 H 1.00 H 0.50 H 0.00 -0.50 a? T 200 400 ROTATION SPEED (rpm) 600 BOO Or Chapter 4. Idling Tests 56 4.6 Flexibility Measurements This experiment is designed to study the flexibility of the blades, compared with the steady state response caused by a magnetic force, the flexibility can provide more accurate results, but its disadvantage is that the environment of the test is not close to the real cutting process. The flexibility of Blade A is illustrated in Fig. 4.30. Compared with Fig. 4.25, it can be seen that speeds with peak value in the plot coincide with critical speeds. This denotes that the blade has a resonance at critical speeds under a static transverse force. However, the flexibility is large only at the fundamental critical speed. Above the speed, the peaks are of the same order as at subcritical speed. In other words, the supercritical resonances are unimportant unless the speed is exactly at a supercritical speed. Note that the flexibilities between critical speeds are approximately 25% of those of subcritical speed. Hence choosing operation speed carefully in these speed intervals may achieve a more accurate cutting. Fig. 4.31 presents the flexibilities of Blade B and C. They demonstrate similar characteristics to Blade A. The flexibility of Blade D shown in Fig. 4.32 possesses a similar tendency as the steady state response of Fig. 4.29. The flexibility at subcritical speeds is almost constant and becomes significant around the fundamental critical speed. Above the speed, the flexi-bility is decreased. Above 900 rpm, the blade becomes unstable due to the aerodynamic interaction effect. The flexibility of the blades as a function of rotation speed exhibits the same char-acteristics as the steady state response. Therefore, the blades have a large deflection at the fundamental critical speed when subjected to a static transverse force, while the resonance may not cause unstable cutting at higher supercritical speeds. Chapter 4. Idling Tests Figure 4.30: Flexibility of Blade A oo Chapter 4. Idling Tests 59 (N/uuu-0 AiTllQIX3Td Figure 4.32: Flexibility of Blade D Chapter 4. Idling Tests 60 4.7 Variation in Runout with Rotation Speed It was found from previous cutting experiments that the deflection of a circular saw at subcritical speed is comprised of both low frequency deflection and runout or initial warping of the blade. In some cases, the amplitude of the runout is greater than that of the low frequency response. Large runout of a saw blade may cause trouble in wood cutting, such as self-excitation and excessive heat. Therefore, reducing runout is of considerable importance. It was expected that the runout variation would decrease as the rotation speed is increased because the centrifugal force should make the blade flatter. The experimental set-up is demonstrated in Fig. 4.24. When a blade rotates at a given speed, the runout of the blade will repeat at multiples of the blade rotation speed. The response signals measured by the probes are recorded with the tape recorder and then the peak-to-peak value of the runout response is obtained with the analyser. 4.7.1 Characteristics of Runout The variation in runout with speed for Blade A, B and C is plotted in Fig. 4.33. The tendencies of the runout variation of these blades are similar below 2200 rpm. Above this speed, they do not follow each other. The amplitude of runout becomes greater as the rotation speed increases although it drops at certain speeds. This result is unexpected for Blade A and B. It should be mentioned that Stakhiev [26] stated that the free vibration of a blade with radial slots will become greater as the rotation speed increases. The experiment of Blade C which has very narrow radial slots is in agreement with his observation. The runout variation of B!?,de D is illustrated in Fig. 4.34. It is found that at subcrit-ical speed the runout decreases as the rotation speed increases and has a minimum value around the fundamental critical speed. Above the speed, the runout increases rapidly. Chapter 4. Idling Tests 62 The measurement of runout above 900 rpm cannot be obtained because the blade is unstable at these speeds. 4.7.2 Factors Affecting Runout Measurements As mentioned before, it was not expected that the runout variation would increase as the rotation speed increases. Large runout of a circular saw can result in poor cutting at supercritical speed. Therefore the mechanism of the runout increase should be under-stood. Among the possible causes, the effects of aerodynamics, the axial movement of the shaft and the motion of the probe support are believed to play an important role. In this study, only the effects of the axial movement of the shaft and the motion of the probe support are examined. Axial Movement of the Shaft If an axial movement of the shaft due to the motion within the bearings is superposed on the runout of the blade, the measurement of the runout will not be accurate. The purpose of the experiment is to investigate the effect of axial movement of the shaft on the runout variation. Two probes are used in the experiment. One is placed near the rim of Blade A as before, the other is located beside the collar (r=3.5 inches). Since the collar is very thick, it does not vibrate transversely, hence the displacement measured by the probe near the collar can represent the shaft axial motion. Results in Fig. 4.35 compare the displacement between the blade and the collar. It is clear that the movement of the collar is much smaller than that of the blade. The difference between the collar and the blade is very close to the displacement of the blade measured directly. Therefore, the axial s.otion of the shaft may change the value of the runout variation a little, but does not change the characteristic of the runout. Chapter 4. lolling Teats r- r- b o o o o o o ( U J U J ) N O I l V i y V A i n o N n y Figure 4.34: Runout Variation of Blade D 05 Chapter 4. Idling Tests 65 Probe Support In the previous experiment, the probe is supported on the frame of the rig which is easily excited by the imperfection of the rotating system, hence the accuracy of the measurement may be affected. In this experiment, another probe is supported on the — i carriage which is isolated from the test rig . Blade A is used in the experiment. Fig. 4.36 illustrates runout measured on the rim of the blade. The difference between the two curves are small and the runout measured from the frame is greater than that from the carriage at some speeds, but smaller at other speeds, so the effect of the frame movement can be neglected. From the above experiments, it can be seen that the effect of the motion of the probe support and the axial movement of the shaft on the increase of runout variation can be neglected. In the tests, the effect of aerodynamics has not been studied. This effect might have considerable influence on the runout response. 4.8 Unstable Vibrations of Blade D The saw blades A, B and C used in the above idling tests can run smoothly up to 4000 rpm and never exhibit large motion. However, the Blade D of 6mall thickness and large diameter has a large vibration above 900 rpm. This experiment is designed to observe the motion in the blade. The experimental set-up is shown in Fig. 4.24. The fundamental critical speed of the blade is about 650 rpm. When the blade spins at subcritical speed, it is always stable. At supercritical speeds below 900 rpm, the blade can be rotated indefinitely. The RMS spectrum and the vibration displacement of the blade at 500 rpm (subcritical) and 750 rpm(supercritical) are shown in Fig. 4.37 and Fig. 4.38 respectively. In the instant spectrum plots (Fig. 4.37 .(a) and Fig. 4.38 .(a)), the peaks at 8.3, 16.6, 25 Hz etc in Fig. 4.37 .(a) and those at 12.5, 25, 37.5 Hz etc Chapter 4. Idling Tests 67 in Fig. 4.38 .(a) comprise the runout spectrum of the blade, and the other significant peaks are natural frequencies of the blade. The natural frequencies do not change in the experimental observation at the above speeds. The amplitudes of the vibration at these two speeds in Fig. 4.37 .(b) and 4.38 .(b) are small and almost the same. However, when the blade runs at or above 900 rpm, it begins to oscillate after a stable period of operation. The stable running time varies with rotation speed. It appears that the time becomes shorter as the speed increases. The vibration at 1200 rpm before wobbling is presented in Fig. 4.39. At this stage, the natural frequency of the blade does not change and the displacement is the same as in previous tests. The stable state lasts only for 30 seconds, and then a travelling wave with gradually increasing amplitude develops in the blade. "It can be observed in the experiment that some frequencies of the blade will shift as the amplitude of the wave increases. The frequency of most apparent change is 14.8 Hz in Fig. 4.39.(a) which becomes smaller. It is known that the natural frequency of linear vibration does not vary with the amplitude, therefore, the vibration in this case must be a nonlinear oscillation. Since the amplitude of the blade vibration increases gradually, a transient process takes place. As the vibration increases, the nonlinear effect of the blade will restrict the increase of the vibration, and maintains a new stable vibration illustrated in Fig. 4.40. The frequency of 14.8 Hz in Fig. 4.39 .(a) shifts to 5.8 Hz in Fig. 4.40 .(a). Note that the amplitude of the largest wave in Fig. 4.40 .(b) is twice as big as that in Fig. 4.39 .(b), and its frequency is about 6 Hz. Since this frequency is coincident with 5.8 Hz in Fig. 4.40 .(a), they must represent the same motion in the blade. When the blade is spinning at 1800 rpm, the stable running time is only about 5 seconds. It is difficult to measure the blade vibration before wobbling -.starts. Fig. 4.41 presents the stable nonlinear vibration of the blade at this speed. The most significant natural frequency in Fig. 4.41 .(a) is 6.8 Hz which is approximately equal to the frequency £to;s!H aun£ 'fa) iranipadg SWH '00 m d l 00S 0. 8PBI9 :Z8t s^Sij 89 Chapter 4. Idling Tests Figure 4.38: Blade D at 750 rpm. (a). RMS Spectrum; (b). Time History Chapter 4. Idling Tests 70 Figure 4.39: Blade D at 1200 rpm (Linear State), (a). RMS Spectrum; (b). Time History Chapter 4. Idling Tests 71 Figure 4.40: Blade D at 1200 rpm (Nonlinear State), (a). RMS Spectrum; (b). Time History Chapter 4. Idling Tests 72 of the large wave in Fig. 4.41 .(b) (6.7 Hz). The frequencies of travelling wave dominating the blade vibration are independent of the rotation speed because they are almost the same at speeds of 1200 rpm and 1800 rpm. The frequency of the nonlinear vibration at 1200 rpm in Fig. 4.40 .(a) is 6 Hz which is in agreement with that of mode (0,1) in Fig. 4.26. Hence the controlling mode in the nonlinear vibration is mode (0,1). When the blade has a nonlinear vibration, the mode shapes are coupled together and the energy can flow from one mode to another. In the case of Fig. 4.41, the energy of lower modes which possess most vibration energy of the blade is probably transfered into mode (0,1) and causes a large amplitude vibration in mode (0,1). It is found that water can damp out the nonlinear vibration significantly. Fig. 4.42 .(a) shows the nonlinear vibration at 900 rpm before water is sprayed on the blade. The amplitude of the motion is about 2.4 mm. Then water is sprayed on one side of the blade near its rim and the amplitude is reduced to 0.7 mm in Fig. 4.42 .(b). Fig. 4.43 presents the time history of the decrease of the nonlinear vibration at 900 rpm. Similar results can be obtained at other rotation speeds. The effect of water damping may be caused by the change of the aerodynamic forces on the blade. Knowledge of nonlinear vibration is very important. Saw blades cannot cut accurately when the motion occurs. The causes of the nonlinear vibration are not clear. It seems that the nonlinear motion is related to the geometric dimension of the blade and aerodynamic effects. The critical speed of blades A, B and C are about 2300 rpm. These blades never exhibit nonlinear motion when they rotate from their fundamental critical speeds up to 4000 rpm. The difference between these blades and blade D is the geometric dimension. Blades A, B and C are of larger thickness and smaller diameter than Blade D, hence, they have larger bending stiffness than Blade D. It is believed that there probably exists a critical value of the bending stiffness for the nonlinear vibration. When the bending Figure 4.41: Blade D at 1800 rpm (Nonlinear State), (a). RMS Spectrum; (b). Time History Chapter 4. Idling Tests 74 Figure 4.42: Effect of Water Damping on the Vibration of Blade D at 900 rpm. (a). Without Water; (b). With Water Chapter 4. Idling Tests 75 Figure 4.43: Process of Water Damping on the Vibration of Blade D at 900 rpm Chapter 4. Idling Tests 76 stiffness is less than the critical value, the aerodynamic forces can cause a nonlinear vibration in the blade. The real causes of the nonlinear vibration must be found out for supercritical speed cutting. 3.9 Effect of Spring and Damper on the Stability The numerical results in Chapter 3 indicate that a spring and a damper can result in instability in a circular blade rotating at supercritical speeds. The purpose of this experiment is to investigate such effects. Blades A and D are used in the experiment. i 4.9.1 Effect of a Pair of Bearing Supports It is very difficult to put a pair of ideal springs on a rotating blade. In this experiment, a pair of cantilever beams with a small ball bearing on their free ends are used. The bearings are pressed on the rim of Blade D, and have a rotating contact with the blade. The stiffness of the supports is about 2.0 x 10BN/m. It can be observed from the experiment that the vibration of the blade is stable at both subcritical and supercritical speeds. Fig. 4.44 presents the dynamic response of the blade at 900 rpm (supercritical speed). The time history as shown in Fig. 4.44 .(b) is finite. Hence, the bearing supports cannot cause the instability of the blade. This device is not suitable to support Blade A because the rotation speed is very high and excess heat can be generated by friction which may damage the bearings. 4.9.2 Effect of a Pair of Pin Guides In this experiment, a pair of pin guides (same as in Section 4.4) are located on each side of the blade rim and water is supplied so each guide so that water film ran be developed in the gap between the guide and the blade when the blade is spinning. It can be imagined Chapter 4. Idling Tests 77 Figure 4.44: Blade D Guided by the Bearing Supports at 900 rpm. (a). RMS spectrum; (b). Time History Chapter 4. Idling Tests 78 that the water film can provide some elastic restoring and dissipating forces besides the stiffness given by the supporting arms. The stiffness of the arms is about 105TV/TO, but the stiffness and the damping caused by the water film are unknown. --, The RMS spectrum and time history of Blade D rotating at 300 rpm are shown in Fig. 4.45. It is clear that the deflection of the blade is very small and quite stable at this speed. All time histories of the blade at subcritical speeds show that no instability occurs in this speed region. When the blade is rotating at 800 rpm, a larger wave of low frequency appears in Fig. 4.46 .(b). The frequency of the wave is about 0.5 Hz and coincides with the frequency of 0.4 Hz in instant spectrum of Fig. 4.46 .(a). The vibration of the blade is not unbounded in this test. The vibration amplitude of the blade at 1100 rpm increases gradually and reaches the level as shown in Fig. 4.47 .(b), however, further observation at this speed shows that the increase will cease and the blade is in a new stable state. The vibration at 1200 rpm shown in Fig. 4.48 is stable and of less fluctuation in comparison with that at 1100 rpm. It can be concluded from the above experiments that the blade with a pair of pin guides does not have unbounded vibration at supercritical speeds. However, these exper-iments do not present a definite trend of the vibration with speed, which can be seen in Fig. 4.46, 4.47, 4.48. Even at the same speed, it can be observed that at the beginning, the blade has a larger vibration and then decreases to a smaller level. In a broad sense, the uncertainty indicates instability. The pin guides are also used to investigate the stability of Blade A. At subcritical speed, the vibration of the blade is always stable. In the supercritical speed region between 2300 rpm and 3800 rpm, the blade is stable as well. The stable time history of the blade at 3100 rpm is shown in Fig. 4.49. However, the blade becomes unstable at 3800 rpm as shown in Fig. 4.50. Since the blade exhibits instability only at this supercritical speed, it cannot be concluded that this instability is caused by the stiffness Chapter 4. Idling Tests 79 Figure 4.45: Blade D Guided by the Pin Guides at 300 rpm. (a). RMS spectrum; (b). Time History Chapter 4. Idling Tests 80 Figure 4.46: Blade D Guided by the Pin Guides at 800 rpm. (a), RMS spectrum; (b). Time History (q) irarupsds SKH *(*) * r a d l 0011 1B sapTO ™d 8H* ^  P^TO d a P B i a :Z*'f " " ^ 18 (q) irampads gWH "(*) ' m d l OUST *B s aPFO u!d 3H» ^ P^FMD d 9P BI9 :8fr'fr Chapter 4. Idling Tests 83 8.0a 2 . 5 a 2 . 0 a 0.5a 0 I , . . . . i • o as t.o t. s 2.0 2.s a.o s.s *. o Time (tecond) Figure 4.49: Time History of Blade A Guided by the Pin Guides at 3100 rpm and the damping of the guides. It may be generated by other factors. One possible cause is the resonance of the blade. It is found that the frequency of the unstable wave in Fig. 4.50 is 4 Hz which is close to the lowest natural frequency of the blade in Fig. 4.27. If the air surrounding the blade or the water between the guides and the blade generates the excitation force of 4 Hz, the blade can develop the motion as shown in Fig. 4.50. The above observations do not agree with the analytical results. One possible reason is that the stiffness and the damping provided by the water film are not the same as those in the theoretical model. The other is that the nonlinearity of the blade can restrict the development of the instability. However, the divergence between theory and experiment migki be due to .^ correct numerical results. A detailed discussion of this is presented in Chapter 6. Chapter 4. Idling Tests 84 8 . 0 a 2.0a 4> 1. . 1 0 . 9 M Figure 4.50: Time History of Blade A Guided by the Pin Guides at 3800 rpm Chapter 4. Idling Tests 85 4.9.3 Effect of Wood As mentioned before, the wood in the cutting process can generate spring and damping forces on saw blades. The purpose of this experiment is to study the effects of the stiffness and the damping generated by the wood on the stability of saw blades. In this experiment, a piece of wood is clamped on the carriage and contacts the rim of a blade. In this way, the real cutting environment can be imitated. At the beginning, the stiffness and the damping coefficient of the wood are measured by the transfer function method. To this end, an impact hammer is used to excite the wood, and an accelerometer is put on different positions along the wood surface facing the blade to pick up the response of the wood. By way of analysing the transfer function, it is found that the stiffness is about 107iV/m, and the damping ( is about 0.01. It should be mentioned that these stiffness and damping include the effect of the carriage system. Strictly speaking, the stiffness and the damping are functions of the position along the wood, and the values given here are the average. The instant spectrum and time history of Blade D at 900 rpm is shown in Fig. 4.51. The results indicate that the vibration is stable at this speed. Even when the blade is subjected to a transverse impact force, its vibration is convergent in Fig. 4.52. The results at other supercritical speeds demonstrate the similar characteristics. Therefore, the stiffness and the damping produced by the wood clamped on the carriage cannot cause the instability of circular blades at supercritical speeds. 4.10 Heating Effect When a circular saw is cutting wood at high speed, friction can cause excessive heat in the blade. This experiment is designed to study the effect of heating on the blade stability. A propane flame generator is used to heat the blade. The heat intensity can be araix '(q) Surtupads SHH '(*) ' m d l 006 V* P 0 0M *q P8P™0 Q 3P*I9 :TST a i n 9 U 98 sjsaj". ffirrrpj f jajden£ Chapter 4. Idling Tests 87 Time (itcond) Figure 4.52: Transverse Impact Response of Blade D Guided by Wood at 900 rpm Chapter 4. Idling Tests 88 adjusted either by the distance between the blade and the generator or by the discharge quantity of propane. Blade D is used in the test. The transfer function of the blade both with and without heating is shown in Fig. 4.53. The plot shows that some natural frequencies in the case of heating are reduced signifi-cantly. Experiments at higher speeds present a similar phenomenon, although the change is not significant. It should be mentioned that the flame can not produce enough heat at high speeds because of the air flow around the blade. So this method cannot be used to observe the effect of heating on the stability at high speeds. When the flame heats the blade at stationary, local warping at the heating position can be observed clearly. This result indicates that if an excessive heat is generated in the blade, a local buckling may occur and decrease the blade stability. 4.11 Effect of Speed on the Static Deflection of a Guided Saw It is found that when a collared saw is guided by the pin guides, it will deflect as the rotation speed changes. Fig. 4.54 presents the effect of the rotation speed on the static deflection of Blade A. It can be seen that below 500 rpm, the blade does not deflect, but above the speed, it begins to deflect. The deflection increases as the rotation speed increases. Similar phenomenon can be observed in Blade D when guided by the pin guides. The deflection is probably caused by the friction between the blade and the guides. We can imagine that a column will deflect when it is subjected to compression forces. Similarly, the friction and the driving force may deflect the blade. If the friction increases as the speed increases, the deflection will become larger. Chapter 4. Idling Tests 89 Figure 4.53: Heating Effect on The Stationary Natural Frequency of Blade D. -—: Without Heating. -: Heating, Chapter 4. Idling Tests 90 BO Time (second) Figure 4.54: Effect of Speed on the Static Deflection of Blade A with the Pin Guides Chapter 5 Cutting Tests 5.1 Introduction In the idling experiments, it has been found that the flexibility of Blade A between supercritical speeds can be reduced considerably and that the vibration of Blade D with a pair of pin guides or bearing supports is bounded at supercritical speeds. Even Blade A with the pin guides can run smoothly up to 3800 rpm. The following cutting experiments are based on the above observations. These experiments are preparatory and are used to find out the parameters which govern the supercritical speed cutting. Since the vibration of the blade cannot be predicted beforehand, eddy current probes are not employed at the beginning of the cutting tests because of possible damage, and the results are examined through visual inspection. At first, prehminary experiments of Blade A are conducted at both subcritical and supercritical speeds in order to obtain some primary information. Then causes of tip side cutting marks in the above tests are discussed in detail and finally, the phenomena of unstable cuttings are investigated. The apparatus and the instrumentation used in the tests are the same as in the idling tests. The parameters of the wood used in the experiments are given in Table 5.2. 5.2 Preliminary Experiments Prehminary experiments are conducted so as to acquire perceptual knowledge of high speed cutting. Blade A is used in the experiment and the displacement of the blade is 91 Chapter 5. Cutting Tests 92 Wood Length(ft.) Depth(in.) Wetness Set 1 6 3 Wet Set 2 i i 1.5 Dry Set 3 3 1.5 Dry Set 4 7 1.5 Dry Table 5.2: Wood Parameters not measured. Experiment A This is the first cutting test. The wood used in the test is set 1 in Table 5.2. The rotation speed is 3000 rpm and the bite per tooth is 0.015 inch. This cutting does not present a satisfactory result. After the blade begins to cut the wood, it is vibrating severely. The blade is stopped after 1.5 feet of cutting has been conducted. It is found that burn marks occurred on both cutting sides, but they are more apparent on the cant side. The surface looks straight although the local area is gouged. The roughness of the surface was caused by marks as shown in Fig. 5.55. Those marks are thought to be made by the sides of the carbide tips and play a very important role in high speed cutting. For the sake of convenience, the marks are called TSCM (Tip Side Cutting Mark) in the following discussion. Experiment B In this experiment, short and dry wood is used to observe whether the above cut-ting instability would occur or not. The wood parameters are set 2 in Table 5.2. The experimental results are shown in Table 5.3. TSCM is more distinct on the board side than on the cant side since the back cutting easily obscures the TSCM pattern on the cant side. It is noted that the forward movement of the carriage in one revolution of the blade is 0.54 inch which is close to TSCM pitch S shown in Fig 5.55. This indicates that the frequency of TSCM variation is approximately equal to the blade rotation speed. It is also noted that the pitch of TSCM becomes larger gradually after the first several Chapter 5. Cutting Tests 93 S Figure 5.55: Tip Side Cutting Mark Speed(rpm) Bite(in.) Stability Burnt Mark TSCM Pitch(in.) 3000 0.015 Yes No 0.57 3200 0.015 Slightly Unstable Slight on Both Sides 0.58 Table 5.3: Cutting Results of Prehminary Experiment B uniform marks occur. At the same time, the depth of TSCM also increases. Experiment C The purpose of the experiment is to confirm the observation of Experiment B. The experiment also includes the subcritical speed cutting for comparison of cutting surfaces. The wood used is the same as in experiment B. Table 5.4 illustrates the test results. It is clear that no instability occurs at supercritical speed. TSCM exists in all cuttings but becomes less indistinct as the rotation speed is decreased. The above prehminary experiments indicate that Blade A is liable to become unstable in supercritical speed cutting. If the instability happens, burnt marks are developed on both cutting surfaces. The causes of instabihty are very complicated, and must be Chapter 5. Cutting Tests 94 Speed(rpm) Bite(in.) Stability Burn Mark TSCM Pitch(in.) 1200 0.015 Yes No 0.054 3000 0.015 Yes No 0.057 3200 0.015 Yes No 0.056 Table 5.4: Cutting Results of Preliminary Experiment C understood for developing successful supercritical speed cutting. TSCM always appears on cutting surfaces in the experiment no matter what speed is employed, and becomes more distinct at higher speed. Burn marks always take place on TSCM. At the end of cutting, TSCM is much greater than at the beginning. This seems to be the sign of commencement of the instability. Therefore, it is necessary to understand the TSCM. 5.3 Investigation of Tip Side Cutting Mark (TSCM) TSCM might be related to the parameters of the operation and the mechanical system. The following experiments are designed to investigate these factors. Wood used in all tests is set 2 in Table 5.2. Blade A is used except where mentioned otherwise. 5.3.1 Effect of Parallelness of the Blade to the Track The parallelness of the blade is measured by a dial gauge which can move along the track. At first the gauge measures one point on the rim of the blade. The blade is then turned 180 degrees and the same point on the blade is measured again. The difference between the two measurements presents the parallelness of the blade. As a result of realignment, the parallelness of the blade is reduced from 0.010 inch to 0.004 inch. The rotation speed is 3200 rpm and the bite is 0.015 inch. The experiment shows that instability does not take place but that TSCM still exists. It can be concluded that the parallelness of the blade has no effect on TSCM. Chapter 5. Cutting Tests 95 Runout (in.) Speed(rpm) Bite(in.) Stability Burn Mark TSCM 0.018 1200 0.015 Yes No Clear 0.005 1200 0.015 Yes No Not As Clear As Above Table 5.5: Runout Effect On TSCM 5.3.2 Axial Movement of the Shaft In this experiment, the shaft is realigned and the axial displacement of the shaft in the bearings is reduced from 0.025 inch to 0.005 inch. The rotation speed is 1200 rpm and the bite is 0.015. The experiment shows that TSCM still exists at 1200 rpm although it is not clear. Based upon the pevious tests, TSCM will occur at higher speeds. Therefore, the axial movement of the shaft does not affect TSCM. 5.3.3 Runout of a Blade The static runout of a blade varies depending upon how the collar is installed, i.e, a different relative position between the blade and the collar may introduce a different static runout. This experiment is used to check the effect of the runout. The results are shown in Table 5.5. In these two cases TSCM always occurs, but, it becomes more clear for 0.018 inch runout. Hence, the runout can influence TSCM. The larger the runout, the more distinct is TSCM. 5.3.4 Type of Blade Because Blade B and C are different from Blade A in tooth pattern or in having slots, they are used to examine the effect of blade types on TSCM. The experiment results are illustrated in Table 5.6. TSCM still exists in the cutting surface. But TSCM produced by Blade B is indistinct compared with those of Blade A and C. Chapter 5. Cutting Tests 96 Blade Speed(rpm) Bite(in.) Runout (in.) Stability Burn Mark TSCM Blade B 1200 0.015 0.005 Yes No Exist But Dim 3200 0.015 Not Available Yes No Exist But Dim Blade C 1200 0.015 0.018 Yes No Exist Table 5.6: Blade Effect 0 n TSCM Bite(in.) 0.015 0.020 0.025 0.030 Speed(rpm) 3200 3200 3200 3200 Stability Yes Yes Yes Yes TSCM Exist Exist Exist Exist And Clear Table 5.7: Bite Effect On TSCM At Supercritical Speed 5.3.5 Bite Per Tooth Different bites are tried in this experiment in order to study the effect of bite per tooth on TSCM. Table 5.7 presents the results for Blade A at 3200 rpm. The experiment shows that TSCM exists and becomes more distinct as the bite increases at this speed. Table 5.8 illustrates the effect of bite at subcritical speed. It is noted that when the bite is 0.040 inch which is used in practical cutting, TSCM disappears. 5.3.6 Summary of Tip Side Cutting Mark (TSCM) Runout, rotation speed and bite can affect TSCM, while other factors such as parallelness of the blade and the axial motion of the shaft have almost no effect on it. Increasing Blade Bite(in.) Speed(rpm) Stability TSCM Blade A 0.015 1200 Yes Exist But Dim 0.040 1200 Yes Disappear Blade C 0.015 1200 Yes Exist But Dim 0.040 | 1200 Yes Disappear Table 5.8: Bite Effect On TSCM At Subcritical Speed Chapter 5. Cutting Tests 97 runout and rotation speed makes TSCM more distinct. At subcritical speed, TSCM becomes indistinct as bite increases, and disappears with 0.040 inch bite. For supercritical speed, the situation is the opposite. It is noted that when conducting supercritical speed cutting, TSCM is uniformly distributed on the first half of the cutting surface, and then its pitch increases and becomes more distinct. This indicates that TSCM is related to the stability of the blade. Instability is liable to take place at supercritical speed cutting. If this happens, the protruding marks cannot contact the blade and be burnt by excessive heat. 5.4 Experiments on Cutting Stability It was found in the prehminary experiments that the pitch of TSCM became larger and its peak was higher at the end of each cut. This might be a sign of the instability of the blade vibration. In the following experiments, a longer piece of wood is cut in order to observe whether the cutting instability happens at supercritical speeds. Blade A is used in the test, and eddy current probes are placed around the blade. At the beginning, wood Set 3 as shown in Table 5.2 is used. The rotation speed is 3700 rpm, and the bite per tooth is 0.015 inch. The idling behavior of the blade before cutting is demonstrated in Fig. 5.56. It can be seen from Fig. 5.56 .(b) that the vibration of the blade is small and stable. The instant spectrum in Fig. 5.56 .(a) shows that the blade speed is 3697 rpm (61.6 Hz) and none of the natural frequencies is zero Hz, so the static resonance can not occur at this speed. At the beginning of the operation, the blade cuts wood smoothly as far as the middle of the wood. After that it begins to vibrate, and its deflection greatly increases. Inspection of the cutting surface on the cant side shows that the TSCM is distributed i:iiformly on the beginning cutting of 16 inches and then becomes larger and deeper. Some are burnt due to the heat caused by friction. On the Chapter 5. Cutting Tests 98 last 4 inches of the wood, the blade is deflected out of the wood. Fig. 5.57 illustrates the whole cutting process. The plot is in agreement with the observation described above. A magnified view of the cutting process is shown in Fig. 5.58. Note that in this process the rotation speed slows down from 3700 rpm at the beginning to 3360 rpm at the end, and the blade passes one of its critical speeds. This experiment indicates that the saw blade is definitely unstable while cutting. As a consequence, a saw is more likely to have instability when cutting longer wood at supercritical speed, because the speed may drop, and pass through a critical speed. Another cut is performed at 3100 rpm. The bite of the cutting is 0.02 inch and wood Set 4 in Table 5.2 is used. It can be seen in Fig. 4.25 that one of the natural frequencies at this speed is zero, therefore the purpose of the experiment is to check the resonance caused by the static transverse force on the stability of the blade. The response of the blade in this cutting is shown in Fig. 5.59. At the beginning of the cutting, the blade is stable, and the straight cutting surface is about 27 inches. Then the blade begins to deflect and jumps out of the cant. Finally, it jumps farther back into the cant again. It is clear that the blade is not stable. The cutting instability may not be caused by the resonance of the static cutting force because the mean level of the deflection does not change at the beginning of the instability as shown in Fig. 5.59. The possible cause of the instability is the mode crossing. It is found in Fig. 5.59 that the frequency of the exponentially increasing wave is about 21 Hz which is close to the natural frequency of the two coupled modes at 3100 rpm in Fig. 4.27. It is known that one decay exponent is positive when two modes cross each other, therefore, the vibration of the blade could become unstable. Although the frequency in Fig. 4.27 is obtained for the blade with the pin guides, the wood ir. a cutting process car*, generate spring and damping forces, so it is possible that the instability of the mode crossing takes place. Fig. 5.60 demonstrates the time history of the cutting at 3000 rpm. In the test, the Chapter 5. Cutting Tests 99 Figure 5.56: Idling Response at 3700 rpm. (a). RMS Spectrum; (b). Time History Chapter 5. Cutting Teats 100 - 5 0 O a 4 S Time (second) Figure 5.57: Unstable Time History at 3700 rpm 9 0 0 a 1 .4 i . e I . B a. D Figure 5.58: Unstable Time History at 3700 rpm (Enlarged View) Chapter 5. Cutting Tests 102 bite is 0.02 inch and the wood parameters are shown in Set 4 of Table 5.2. The blade cuts straight at first 26 inches length, and jumps out of and into the cant the same as at 3100 rpm, so the blade is unstable in the cutting. In Fig. 5.60, the frequency of the larger wave at the beginning of the instability is 5 Hz. Compared with Fig. 4.25 and Fig. 4.27, this frequency is close to one of the lowest natural frequencies of the blade. It is thought that the unstable cutting in the experiment may be due to an excitation at a 5 Hz frequency. Such an excitation force can be generated by the interaction between the wood and the saw in the cutting process. Finally, two cuts are done at 2800 rpm. The bite is 0.03 inch and the wood used is Set 4 in Table 5.2. In this experiment, water is sprayed to the blade for one cut so as to inspect the cooling effect on the stability of the blade. Another purpose of the test is to examine what will happen when the speed is close to the fundamental critical speed. The response of the blade is shown in Fig. 5.61. Although the blade becomes unstable during cutting, the surface is not as bad as those at higher speeds, hence, approaching the fundamental critical speed may not result in a worse cutting. It can be seen that for the cut with water, the stable cutting length is 59 inches, but that of the cut without water is only 40 inches, therefore, reducing the cutting heat can make the blade more stable. Chapter 5. Cutting Tests 103 ( A ) lasmss'BidsiQ Figure 5.60: Time History at 3000 rpm Chapter 5. Cutting Tests 104 as 1.0 1. S 2.0 2. S Time (second) 3 . 0 8 . 3 4. O as t.o t . S 2 . 0 2 . 5 Time (second) 3 . 0 3 . 3 4 . 0 Figure 5.61: Time History at 2800 rpm. (a). Without Water; (b). With Water Chapter 6 On Discrepancy of Theoretical Stability Results 6.1 Introduction In Chapter 3, the numerical results show that the vibration of a rotating disk guided by a pair of springs' is unstable at supercritical speeds. The instability can be divided into two classes. The first type corresponds to a mode of zero natural frequency, and the second one corresponds to the modes which are coupled with each other. Similar analysis was also presented by Iwan [32] and Mote [33]. Shen [34] tried to physically explain the mechanism of the instability of a guided circular saw at supercritical speeds. Although the experimental confirmation on the supercritical speed instability has not been reported in the past, the above conclusion has been widely accepted. However, the experiments conducted in this study do not demonstrate the expected phenomenon. The discrepancy might be caused by the approximate analytical method used. Since Equation 2.9 is a fourth order partial differential equation, it is not possible to get an exact solution for it. In this chapter, a rotating string guided by a spring is studied because such a string bears analogy to a guided rotating disk while its governing equation is just a second order partial differential equation. Schajer [47] investigated a guided rotating string and derived exact solutions for two special cases of the spring stiffness K — 0 and K = oo. A general exact solution for K of any value is developed in this chapter. The results obtained from Galerkin's method are compared with the exact solution. Based on this knowledge, the stability discrepancy of a guided rotating disk is discussed. 105 Chapter 6. On Discrepancy of Theoretical Stability Results 106 Figure 6.62: A Rotating String Subject to a Spring at a Fixed Point (from Schajer [47]) 6.2 A Guided Rotating Circular String A rotating circular string guided by a spring is shown in Fig. 6.62. The radius of the string is r; the density is p\ the rotation speed is fi; the stiffness of the spring is K. The governing equation of the string in space-fixed coordinates [47] is where . P S = — pT2 u is the transverse displacement, and P is the string tension which is assumed independent of the rotation speed. The boundary conditions of the problem [47] are « l(o,t) ~u l(2*,o = 0 (6.39) du . du . r K . "de '(0,t) ~~d~9~ ' ( 2 i r , e ) ~ ~P~ u (0,t) (6.40) Chapter 6. On Discrepancy of Theoretical Stability Results 107 6.2.1 Exact Solution Sack [49] studied the transverse oscillations in a travelling string. The analytical method he proposed to investigate the forced oscillations of the string will be used in this study. The exact solution of Equation 6.38 is assumed as — i u(B,t) = ^{•inWt + ^ M^rinC^J + AcoiC^)] + cosW* + ^ )]Ksin(^) + £ncos(^)]} (6.41) where A n , B n , A'n and B'n are constants which depends on the initial conditions, and b - ~ ~ s ~ V = ^ (6-42) It can be proved that solution 6.41 satisfies governing equation 6.38. The natural frequency of the system can be determined by the following procedure. Substituting solution 6.41 into boundary conditions 6.39 and 6.40 gives „ . „ , . / 27T.. . . 27ru;n 27Tu;n. B n sin u>nt + B n cos uint = sm u>n{t + —)(An sin — — + B n cos + cos wn(t + — )(A'n sin -jr + B'n cos -^) (6.43) fAnWn B'nUns . . , ,A!^n . BnWn x . {~S> jF-)smu,nt + ( — + —)cosu; nt -smu,n(t + T F ) [ { - J , j y " ) c o s _ - ( _ + — ) s m — ] -2TC A'nun B„u;n 27ru;n .AnWn £„u;n . 27ru>ni cos a^t + + ) cos — + (— — ) sm — ] Kr = — (Bnsinu;nt + .Bncosu;nt) (6.44) Let 2l7Wn P _ 2KWn Chapter 6. On Discrepancy of Theoretical Stability Results 108 and decompose the trigonometric terms containing time t, Eq. 6.43 can be rewritten as sinu)nt[Bn — cos ct{Anain(3 + Bn cos/3) -f sin a(A'n sin/? + B'n c o s + cosunt[B'n - sina(Ansin/3 + Bn cos/3) - cos a(A'n sin/3 + B'n c o s = 0(6.45) sinu ; n i { — fT-~PBn ~ cosa[(— — ) c o B f i - ( - ^ - + -^r)smP] + sma[(^- + -^--)cos^-f-(-^ —)sm/?]} + • S' + 0' P n cosu, nt{^  - ^  " (6-46) s i n a [ ( ^ - - ^ ) c o s ^ - ( ^ + -^)sin^] -cos a [ ( ^ + ^ )cos/3 + ( ^ - ^ ) s i n / ? ] } = 0 Equations 6.45 and 6.46 hold only when all coefficients of sinu;nt and cosu>nt become zero. Therefore, we have [ay].x4 {xi}4 = 0 (6.47) where and {x,}r = {An Bn A'n B'n} (6.48) an = — cos a sin /? C 1 2 = 1 — cos a cos 0 O13 = sin a sin f3 du sin a cos /3 021 = — sin a sin /3 a22 = — sin a cos /? Chapter 6. On Discrepancy of Theoretical StabUity Results 109 a23 = — cos a sin B a24 = 1 — cos a cos B (6.49) o-z\ = -~ (1 — cos a cos/?) + — sin a sin/? O it d32 = — cos a sin/? + — sin a cos/3 o' 42' p 0 3 3 = - ^ T sin a cos/? + — cos a sin/? a34 = — — sin a sin/3 — — (1 — cos a cos/3) a4i = — — sin a cos B — — cos a sin 8 0 4 2 = — sinasin/3-f —(1 — cosacos/3) 0 4 3 = —(1 — cosacos/?) + TrrsinasinyS a44 = — cos a sin B + — sin a cos 8 — 0' iV p The determinant |a,j| must be zero if {«,•} are not all zero. Then we get the following characteristic equation 4[l-cos(a-t-/?)][l-cos(a-/?)] (^) 2 + 4sin^(cos/3-cosa)^|^ + sin2/? (Jy)* = 0 (6.50) Solving Eq.6.50 gives the natural frequencies of the system. For the simplicity, two special cases will be examined as follows. Stiffness K = 0 If K = 0, we must have 1 - cos(a + B) = 0 1 - cos(a - B) = 0 Therefore, the natural frequencies are ujn = n(S — n) u>„ = n(S + Q) (6.51) Chapter 6. On Discrepancy of Theoretical Stability Results 110 0.0 0.5 1.0 1.5 2.0 NORMALIZED SPEED Figure 6.63: Natural Frequencies of a Rotating String for K = 0 (Exact Solution) The natural frequencies as a function of rotation speed are shown in Fig.6.63. It can be seen that the frequencies of all backward travelling waves become zero at critical 6peed, and the waves propagate forward above the speed. No instability occurs at either subcritical and supercritical speeds. Stiffness K = oo If K = oo, we must have sin2/? = 0 Then the natural frequency is u>n = 7^(S3 - ft2) (6.52) Fig. 6.64 shows the relationship between the natural frequency and the rotation speed. All frequencies are zero at critical speed and will ascend above the speed. Since Eq. 6.41 only consists of harmonic functions, the vibration of the string is stable at any speed. Chapter 6. On Discrepancy of Theoretical Stability Results 111 0.0 0.5 1.0 1.5 2.0 NORMALIZED SPEED Figure 6.64: Natural Frequencies of a Rotating String for K = oo (Exact Solution) It should be mentioned that these results are exactly the same as those given by Schajer [47]. It can be concluded from Eq. 6.50 that a rotating string subject to an elastic con-straint is always stable at any speed. 6.2.2 Approximate Solution In the following, Galerkin's method is applied to Eq. 6.38 as was done in Chapter 2. To this end, the governing equation 6.38 and the boundary conditions 6.39 and 6.40 are rewritten as U \(0,t) ~ U |(2w,t) = 0 (6'54) Chapter 6. On Discrepancy of Theoretical Stability Results 112 The approximate solution is assumed as where cos n6 + sn(t) sin n8] (6.55) n=0 *n{t) = 5neAt (6.56) and Cn, Sn are constant; A is the eigenvalue. Single-Mode Approximation Following the procedure in Chapter 2, we get (1). n = 0, (2). n ^ 0, TT2A4 + [ 2 7 r V ( f t 2 + S2) + -TT]A 2 + [7rV(ft 2 - 5 2 ) 2 - - 7 r n 2 ( f t 2 - S2)] = 0 (6.58) T r Solving Eq. 6.57 and Eq. 6.58 gives the eigenvalues for the rotating string. The numerical results for the string without a spring are shown in Fig. 6.65. It can be seen that the frequency and the decay exponent are the same as those of the exact solution. Fig. 6.66 presents the results for the string with a spring. In this case, one of the frequencies becomes zero at critical speed and remains zero in a speed region. The corresponding decay exponent in this region becomes positive, hence, the vibration of the string is unstable. Obviously, this result contradicts the conclusions drawn from the exact solution. Multi-Mode Approximation Fig. 6.67 shows the decay exponents of a rotating string subject to a finite spring which are calculated from the multi-mode approximation. It is clear that some eigenvalues are Chapter 6. On Discrepancy of Theoretical Stability Results 113 Figure 6.65: Single-Mode Approximation of a Rotating String for K = 0. (a). Natural Frequencies; (b). Decay Exponents. Chapter 6. On Discrepancy of Theoretical Stability Results 114 NORMALIZED SPEED 0.4-D E 0 . 2 -C A Y E p 0 . 0 -0 N E N T - 0 . 2 --0.4-0.0 T 1 1 1 1 1 1 — 1 1 | 0.5 1-0 T 1 1 1 1 1 1 1 T " 1.5 2.0 NORMALIZED SPEED Figure 6.66: Single-Mode Approximation of a Rotating String for Finite K. (a). Natural Frequencies; (b). Decay Exponents. Chapter 6. On Discrepancy of Theoretical Stability Results 115 0.4 - 0 .2 — 1.0 1.2 1.4 1.6 1.8 2.0 NORMALIZED SPEED Figure 6.67: Multi-Mode Approximation: Decay Exponents of a Rotating String for Finite K positive at supercritical speed so the vibration of the string is unstable. This result is not consistent with the exact solution. The discrepancy might be caused by improper coordinate function 6.55 used to ap-proximate Eq. 6.38. It is known that the coordinate function should satisfy both essential and natural boundary conditions in order to obtain a convergent result when Galerkin's method is applied. Function 6.55 satisfies the essential boundary condition 6.39, how-ever, it does not satisfy the natural boundary condition 6.40, which can be easily verified by substituting 6.55 into them. Because of this, Galerkin's method might present in-correct results. Actually, assumption 6.55 is of the same form as the exact solution for an unguided string. Therefore, the approximate result for the unguided string, even by single-mode calculation, is the same as that of the exact solution. Another interesting point is that the multi-mode approximation can present fair results at subcritical speeds Chapter 6. On Discrepancy of Theoretical Stability Results 116 while it cannot at supercritical speeds. The reason for this is not clear at present. 6.3 A Guided Rotating Circular Disk By way of similar analysis, it can be verified that the approximate function 2.23 used to solve Eq. 2.20 does not satisfy the natural boundary condition produced by a spring. Based on the above discussion, we can deduce that this might be one important cause for the disagreement, although the vibration of a rotating disk is much more complicated than that of a string. Similarly, the approximate solution works well at subcritical speeds, but cannot provide consistent results with the experiments at supercritical speeds. Hence, the coordinate function must be constructed properly in order to achieve a reasonable result at supercritical speeds. The experiment itself also can cause the inconsistency if the experimental condition is not as same as what the theory requires. The spring in the experiments is very difficult to simulate. Both pin-pad guide and bearing guide are used in the tests. The lubricant effect of the pin guide is likely to cause the inconsistency, while the bearing guide is thought to be more similar to a linear spring. In the experiments, neither of them demonstrates the phenomenon predicted by the theory. Therefore, the effect of the experimental condition on the disagreement may not be significant. Chapter 7 Conclusions In this study, analytical and experimental work has been conducted to analyse the be-haviour of collared circular saws at supercritical speeds. The analytical model is based on the theory of a plate with small deflections and solved by Galerkin's method. Previous work [43] shows that the theory can present an accurate analysis for the dynamics of a circular saw at subcritical speed. This study, however, shows that the theory cannot properly predict the behaviour of a circular saw at supercritical speed. The analysis shows that at supercritical speeds a spring which restrains lateral deflec-tion at one point of the blade generates zero-frequency and mode-coupling instabilities, and that a damper always makes the saw unstable above the fundamental critical speed. Iwan [32] and Mote [33] reported similar results. However, experimental results show that no instability occurs. The oscillations of a guided rotating string which bears analogy to a guided rotating disk are discussed. The approximate solution shows that the string is unstable at supercritical speed, but the exact solution indicates that no instability occurs at supercritical speed. The approximate solution fails here because the assumed coordi-nate functions do not satisfy the natural boundary conditions as required by Galerkin's method. In the light of the similarity, it appears that the existing theory is not valid for predicting the stability of a rotating disk subjected to a spring at supercritical speeds. This thesis shows that the analysis cannot provide correct natural frequencies of a guided circular saw at supercritical speeds. The experimental natural frequencies in the 117 Chapter 7. Conclusions 118 supercritical speed region are independent of the rotation speed, whereas the analysis pre-dicts changes in frequencies with speed. The discrepancy might be caused by neglecting the lubricant effect in the analytical model. Experimental results show that a highly flexible circular disk will suddenly have large vibrations above certain supercritical speed. As the rotation speed increases, the transi-tion period between small and large vibrations decreases, and the amplitude of the large vibration increases. The natural frequencies of the vibrations vary with the change of the amplitude, which indicates that the the blade has nonlinear vibrations. It is believed that the aerodynamic forces are the main cause of this motion. Runout variation tends to increase as the rotation speed is increased, although at some speeds it drops. Runout changes are also believed to be closely related to aeroelasticity. These observations are vital to an accurate cutting at supercritical speed, but they have not been considered in the existing analysis. The experiments indicate that the flexibility of a saw blade becomes extremely large at its lowest critical speeds. Above these speeds, flexibility in speed intervals between supercritical speeds is about 25% of that at subcritical speed. Even at supercritical speeds, its value seems to be finite and has the same quantity order as subcritical speed. The theoretical results of flexibility are in agreement with the measurements. This study also proves that for an unguided disk of normal flexibility the numerical results of natural frequency are coincident with the experimental results at supercritical speed. Cutting experiments have been conducted in order to investigate the ability of a conventional circular saw to operate at supercritical speeds. For short lengths (16 inches) of wood, the supercritical speed cutting can be conducted smoothly and no evident instability occurs during the operation. However, tooth marks appear on the cutting surface and their pitch becomes larger at the end of the boards, which indicates the sign Chapter 7. Conclusions 119 of instability. For longer lengths (3, 6 and 7 feet) of wood, the supercritical speed cutting is always unstable and burn marks occur on the surface of the cant. Since excessive heating can cause local buckling of a disk, friction heating generated at supercritical speed is thought to be one of the most important mechanisms to cause cutting instability. Based on the cutting experiments, it seems that the conventional circular saws cannot successfully be used at supercritical speed. Chapter 8 Recommendations 1. Since some experimental phenomena at supercritical speed cannot be explained by the existing theory, modification is needed to improve the analytical model. First, appropriate coordinate functions should be constructed for modeling the dynamics of a guided circular saw when Galerkin's method is applied. The effects of in-plane cutting loads and thermal stresses on the stability of a circular saw at supercritical speed should be studied. Aeroelasticity and nonlinearity should also be considered in the new model. 2. Stability is crucial for the supercritical speed cutting of a circular saw. More idling experiments should be conducted to investigate the possible mechanisms of the instability, such as buckling, heating, resonance, aeroelasticity and dry friction, and to find out the dominant factors causing the instability. Since the burn marks on the cant surface are always associated with the cutting instability, the effect of heating, first of all, should be investigated experimentally. 120 Bibliography [1] Szymani, R., and Mote, C.D.Jr., "Principal Developments in Thin Circular Saw Vibration and Control Research, Part 2: Reduction and Control of Saw Vibration", Holz als Roh-und Werkstoff, 35, 1977, pp 219-225. [2] Schajer, G.S., "Understanding Saw Tensioning", Holz als Roh-und Werkstoff, 42, 1984, pp 425-430. [3] D'Angelo, C.III, and Mote, C.D.Jr., "A Look at Supercritical Speed Saw Phenom-ena", Proceeedings of Nineth Wood Machining Seminar, University of California Forest Products Laboratory, Richmonsd, 1988, pp 457-471. [4] Lamb, H., and Southwell, R.V., "The Vibrations of a Spinning Disk", Proceedings of the Royal Society, Vol. 99, 1921, pp 272-280. [5] Simmonds, J.G., "The Transverse Vibrations of a Flat Spinning Membrane", Journal of the Aeronautical Sciences, Vol. 29, No. 1, 1962, pp 16-18. [6] Eversman, W., "Transverse Vibrations of a Damped Spinning Membrane", AIAA Journal, Vol. 6, No. 7, 1968, pp 1395-1397. [7] Eversman, W., and Dodson, R.O.Jr.,"Free Vibrations of a Centrally Clamped Spin-ning Circular disk", AIAA Journal, Vol. 7, No. 10, 1969, pp 2010-2012. [8] Greenberg, H.J., "Flexible Disk—Read/Write Head Interface", IEEE Trans., Mag-nates, Vol. May-14, No. 5, 1978, pp 336-338. 121 Bibliography 122 [9] Adams, G.G., "Analysis of the Flexible Disk/Head Interface", ASME Trans., Jour-nal of Lubrication Technology, Vol. 102, 1980, pp 86-90. [10] Adams, G.G., "Critical Speeds for a Flexible Spinning Disk", Journal of Mech. Sci., Vol. 29, No. 8, 1987, pp 525-531. [11] Benson, R.C., and Bogy, D.B., "Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load", ASME Trans., Journal of Applied Mechanics, Vol. 45, No. 3, 1978, pp 636-642. [12] Benson, R.C., "Observations on the Steady-State Solution of an Extremely Flexible Spinning Disk with a Transverse Load", ASME Trans., Journal of Applied Mechan-ics, Vol. 50, 1983, pp 525-530. [13] Cole, K.A., "The Effect of Imperfect Geometry on the Transverse Deflection of a Flexible Spinning Disk", Ph.D. Thesis, Department of Mechanical Engineering, University of Rochester, 1986. [14] Hutton, S.G., Lehmann, B.F., and Swanson, J.S., "Circular Saw Behavior at Super-critical Speeds", Forintek Canada Corp., Technical Report, No. 1215K022, 1987. [15] Nigh, G.L., and Olson, M.D., "Finite Element Analysis of Rotating Disk", Journal of Sound and Vibration, Vol. 77, No. 1, 1981, pp 61-78. [16] Richards, J., "Wood-Working Machines", Spon,London, 1872. [17] Tobias, S.A., and Arnold, R.N., "Influence of Dynamical Imperfection on the Vi-bration of Rotating Disks", Proceedings of the Institution of Mechanical Engineers, Vol. 171, 1957, pp 669-690. Bibliography 123 [18] Nowinski, J.L., "Nonlinear Transverse Vibrations of a Spinning Disk", ASME Trans., Journal of Applied Mechanics, Vol. 31, 1964, pp 72-78. [19] Advani, S.H., "Stationary Waves in a Thin Spinning Disk", Intl. Journal of Mech. Set., Vol. 9, 1967, pp 307-313. [20] Advani, S.H., and Bulkeley, P.Z., "Nonliear Transverse Vibrations and Waves in Spinning Membrane Disks", Intl. Journal of Non-Linear Mechanics, Vol. 4, 1969, pp 123-127. [21] Advani, S.H., and Bhattacharjie, A., "Large Amplitude Axisymmetric Transverse Vibrations of Spinning Membrane Disks", Journal of Sound and Vibration, Vol. 9, No. 1, pp 59-64. [22] Advani, S.H., "Large Amplitude Asymmetric Waves in a Spinning Membrane", ASME Trans., Journal of Applied Mechanics, Vol. 34, 1967, pp 1044-1052. [23] Mote, C.D.Jr., and Leu, M.C., "Whistling Instability in Idling Circular Saws", ASME Trans., Journal of Dynamic Systems, Measurement, and Control, Vol. 102, 1980, pp 114-122. [24] Leu, M.C., and Mote, C.D.Jr., "Vortex Shedding: The Source of Noise and Vibration in Idling Circular Saws", ASME Trans., Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol. 106, No. 3, 1984, pp 434-442. [25] Leu, M.C., and Jirapongphan, M., "Modeling and Analysis of Flow-Induced Vibra-tions in Circular Saws", ASME Trans., Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 107, No. 1, 1985, pp 196-202. [26] Stakhiev, Yu.M., "Vibrations in Thin Steel-Disks", Russian Engineering Journal, Vol. 52, No. 8, 1972, pp 14-17. Bibliography 124 [27] Mote, CD.Jr., "Circular Saw Stability—a Theoretical Approach", Forest Product Journal, Vol. 14, No. 6, 1964, pp 244-250. [28] Mote, CD.Jr., "Stabilty of Circular Plates Subjected to Moving Loads", Journal Franklin Inst., Vol. 290, No. 4, 1970, pp 329-344. [29] Mote, C.D.Jr., and Nieh, L.T., "On the Foundation of Circular Saw Stability The-ory", Wood and Fiber, Vol. 5, No. 2, 1973, pp 160-169. [30] Dugdale, D.S., "Stiffness of a Spinning Disk Clamped at its Centre", Journal of Mech. Phys. Solids, Vol. 14, 1966, pp 349-356. [31] Radchffe, C.J.,and Mote, C.D.Jr., "Stability of Stationary and Rotating Disks under Edge Load", Ml Journal Mech. Sci., Vol. 19, 1977, pp 567-574. [32] Iwan, W.D., and Moeller, T.L., "The Stability of a Spinning Elastic Disk with a Transverse Load System", ASME Trans., Journal of Applied Mechanics, Vol. 43, 1976, pp 485-490. [33] Mote, C.D.Jr., "Moving-Load Stability of a Circular Plate on a Floating Central Collar", Journal of the Acoustical Society of America, Vol. 61, No. 2, 1977, pp 439-447. [34] Shen, I.Y., and Mote, C.D.Jr., "On the Mechanisms of Instability in the Circular Saws", Proceedings of Nineth Wood Machining Seminar, University of California Forest Products Laboratory, Richmond, 1988, pp 472-489. [35] Bulkeley, P.Z., "Stability of Transverse Waves in a Spinning Membrane Disk", ASME Trans., Journal of Applied Mechanics, Vol. 40, 1973, pp 133-136. Bibliography 125 [36] Cobb, E.C., and Saunders, O.A., "Heat Transfer from a Rotating Disk", Proceedings of the Royal Society, Vol. 236, No. A, 1956, pp 343-351. [37] McComas, S.T., and Hartnett, J.P., "Temperature Profiles and Heat Transfer Asso-ciated with a Single Disk Rotating in Still Air", 4th Intl. Heat Transfer Conf., Heat Transfer, Vol. 3, 1970, pp 1-11. [38] Mote, C.D.Jr., and Holoyen, S., "The Temperature Distribution in Circular Saws during Cutting", Norsk Treteknisk Institutt Medd., No. 49, Oslo, 1973. [39] Mote, C.D.Jr., "Transient Thermal Stress and Associated Natural Frequency Vibra-tions in Circular Disk Elements", ASME Trans., Vol. 89(B), 1967, pp 265-270. [40] Mote, C.D.Jr., "Theory of Thermal Natural Frequency Vibrations in Disks", Intl. Journal of Mech. Sci., Vol. 8, 1966, pp 547-557. [41] Nieh, L.T., and Mote, C.D.Jr., "Vibration and Stability in Thermally Stressed Ro-tating Disks", Exp. Mech. Vol. 15, No. 7, 1975, pp 258-264. [42] Hutton, S.G., Chonan, S., and Lehmann, B.F., "Dynamic Response of a Guided Circular Saw", Journal of Sound and Vibration, Vol. 112, No. 3, 1987, pp 527-539. [43] Lehmann, B.F., "Dynamics of Guided Circular Saws", Master Thesis, Department of Mechanical Engineering, University of British Columbia, Sept., 1985. [44] Leissa, A.W., "Vibration of Plates", Scientific and Technical Information Division, Office of Technology Utilization, 1969. [45] Timoshenko, S.P., and Goodier, J.N., "Theory of Elasticity", McGraw-Hill, Inc., 1970. Bibliography 126 [46] Leissa, A.W., "On a Curve Veering Aberration", Journal of Applied Mathematics and Physics (ZAMP), Vol. 25, 1974, pp 99-111. [47] Schajer, G.S., "The Vibration of a Rotating Circular String Subject to a Fixed Elastic Restraint", Journal of Sound and Vibration, Vol. 92, No. 1, 1984, pp 11-19. [48] Genta, G., "Whirling of Unsymmetrical Rotors: a Finite Element Approach Baased on Complex Co-ordinates", Journal of Sound and Vibration, Vol. 124, No. 1, 1988, pp 27-53. [49] Sack, R.A., "Transverse Oscillations in Travelling Strings", British Journal of Ap-plied Physics, Vol. 5, 1954, pp 224-226. [50] Schajer, G.S., "CSAW - Guided Circular Saw Vibration and Stability Program". Weyerhauser Technology Center, Tacoma, WA, USA, Feb. 1988. Appendix A Differentiation Transformation Let (r, 0, t) be the variables in the body-fixed coordinate system and (R% <j>, T) in the space-fixed coordinate system. Since R = r <f> = e + nt (A.59) T = t Therefore, similarly, we have dW _ dWdR dWdl , dWdT = dW ( A 6 Q ) ~dr~ ~ dR dr + d<}> dr + dT dr dR and dW _ dW d6 ~ d<f> dW _ dW dW dt ~ dT + d<f> d2W _ &W_ dr2 ~ dR2 d2w _ d2w de2 ~ d<t>2 d2W d2W , n r > d2W 2d2W ~W ~ IF + nid4>dT + d*2 (A.61) (A.62) Eq. A.60 ~ A.62 give the differentiation transformation between the body-fixed coordinate and the space-fixed coordinate. 127 Appendix B Centrifugal Stresses For a rotating disk, the equilibrium equation is [45] 4-(r"rr) - <r<H> + P&r2 = 0 (B.63) dr since E „ E . du u E E u du a+* = T^rui{€^ + tl€rr) = T^ir' + fidr') ( B , 6 4 ) where u is the displacement in r direction. Substituting Eq. B.64 into B.63 gives d?U du 1 — (I2 3 /r> c e \ ' ^ + r * - " = E - " " T ( B - 6 5 ) The boundary conditions are u{b) = 0 <rrr(a) = 0 (B.66) Solve Eq. B.65 with the boundary condition B.66 and then substitute the solution into Eq. B.64, we have 1 <rr% = Dl + D2 -=z + D3 r2 T 1 <rU = Di - D2^ + D< r2 (B.67) 128 Appendix B. Centrifugal Stresses 129 where „ _ 1 n 2 a 2 [ ( l + M)(3 + ^ ) + (l-^)(^) 4 1 1 8P ( i + M) + ( i _ M ) ( J ) i n l l T > , a 2 6 2 [ ( l - M ) ( 3 + / 0 - ( l - M a ) ( ! ) 2 ] 2 " 8 > 0 ( i + „ ) + (! _ „ ) ( { ) . D3 = -\pila(3 + (B.68) o Eq. B.67 and B.68 present the centrifugal stresses of a rotating disk. Appendix C In-Plane Cutting Stresses The stress function for the cutting forces is assumed by [45] ip = aolnr + bor2 + CQ<J> -f- ~r~ v <f> sin <f> -f- ( bi r3 + — + b[r Inr ) cos (f> 2 r ci d - -±r<j> cos(p + (a\r3 + — + d^rlnr) sin^ (C.69) 2 r a1 + (o 2r 2 + 62r4 + -f + o'2) cos(2<£) T + {c2r2 + d2r* + 4 + <*2)sin(2^ ) where a0 ~ ^  a r e constants which depend on the boundary conditions. Since r 5r r2 5^2 0r2 r* drKr d<j>} Substituting Eq. C.69 into C.70 yields '0 r , ai ~. 2 a\ b\ . , + (— + 26xr r + ) cos<^ + (- + 2dtr r + —) T T° r , n 6 a* 4&2 . -(2a 2 + + ^ r ) cos(2^ ) 130 (C.70) Appendix C. In-Plane Cutting Stresses 131 /« 6 cL 4 a", . . ,n .. " ( 2 c > + + -r )8 in(2^) 'o . „, 2 6i . + (66ir + -T" + -) cosc^  I* T Id d' + {Qdir + -±+ -2-)sin^ (C.71) r T + (2a2 + 12 62 r 2 + cos(2<£) r + (2c2 + 12a*2r2 + —j2-) sin(2^ ) ^ M ) = ^ + (26 1r-^-+^-)sin<^ /« , 2 Ci d\ . -(2<fxr r + —) c o s^ + 2(a2 + 362r2 - ^  - ^ |) sin(2 -2(c 2 + Zd2r2 - ^  - ^ )cos(2c^) The stresses on the boundary of the disk shown in Fig. 2.5 are taken as °\rM) = -j^Pr fa) r^fc,^ = r (C.72) crc„(a,<f>) = -\pr6{<f>-<j>0) a n an where — a v T = 2irb*h f Expand Eq. C.72 by Fourier Series and keep the terms with n < 2, we have P PT 2 = — „—TT T7 £t c o s(M>) cos(n^ ) + sin(nc^ o) sin(n^ )] ZTTno irno rUM) = r (C73) Appendix C. In-Plane Cutting Stresses 132 Pr Pr 2 ^rrK^) = _;r~ 7 T~ X][co8(n^o) cos(n0) + sin(n^ o) sin(n^ )] z 7T ft O 7t ft a P P 2 rr*(°)^) = 0 I + —T~ X) [cos(n^ o) cos(n^ ) + sin(n0o) sin(n^ )] z 7r ft a ix ho, n = 1 Constants a0 ~ d'2 can be obtained by comparing the coefficients in Eq. C.72 and C.73. When n = 0, we have 2?+ 21* = - P ' a7 2IT ha Co T = r Solving Eq. C.74 gives o0,6o and Co. When n = 1, we have O i + *i a O i + *i 6 Cl + d[ a Ci + d[ 2a'l Pr + 2 Oi a — = ^ cos 0o ad IT ah 2 a! P + 2 6i b rr1 = {-r COS 0O ft3 7T 0 ft + 2 di a — — sin <po a* IT ah _ , , 2 . . + 2 di 6 YT~ ~ TT S l n ro 6 J ft3 Troft 2 ^ 6 - 2 ^ + ^  = 0 (C.75) 2bxa - + — = r sin0O irah 2ai fc3 6 2a'x K a3 a 2c'i + d[ 63 b 2ci + a3 a 2dib - + = 0 p 2^i a — —7- + — = cos 0o a3 a 7r a ft It is noted that only six equations are independent in C.75, hence, we need to consider the continuity of the displacement. The continuity condition is <n(l - /*)  bi = A Appendix C. In-Plane Cutting Stresses 133 d, = _ c i ( l - / 0 ( 0 > 7 6 ) The constants ai ~ d[ can be solved from Eq. C.75 and C.76. When n = 2, we have —i 6a'2 46', PT , n l . -2o2 = - cos 20o) a4 a2 ir ha 6o'2 46- Pr . - 2 f l 2--6T--6r = - ^ c o s ( 2 ^ ) 6 c, 4d2 _ Pr - 2 c2 — = 7— sm(20oJ a4 oz 7r A a 2a2 + 6b2a2 - - ^  = sin(2#0) a4 a2 7rna „, ,2 6a' 26' 2a2 + 662 62 - - ^ - - ^ = 0 -2c2 - 6d2a2 -f ^  + ^  = ~T- cos(2^ 0) a4 az 7r ft a _ 2 c 2 _ 6 , 2 f c 2 + ^ + ^ = 0 (C.78) Solving Eq. C.77 gives constants o2 ~ d2. Appendix D Thermal Stresses Mote [38] has conducted experiments to investigate the temperature profile in a circular saw during cutting. Fig. D.68 demonstrates the temperature distribution of a typical saw blade in his test. It is clear that the distribution is very complex. For simplicity, the parabolic function is used in this study to approximate the temperature profile in a circular saw, ie, T.= T.£f£ (D.79) where To is the temperature on the rim of the disk. For a circular disk with such an axisymmetric temperature distribution, the equilib-rium equation is * 4 + " *+* = o (D.80) dr r Since the strains of the disk are = - | ( 0 f r - + O.T = i(«L-M«S) + «rr (D.81) where a/ is the thermal expansion coefficient. Solving Eq. D.81, we get o\. — I - ft2 E E [erT + " (1 + 1 - fi2 + fltrr - (1 + li)aiT) (D.82) Since du £rr dr 134 Appendix D. Thermal Stresses 135 5. i A l t X I .f Ul • . - * O J * C CLAMf f »V»*o.i*c IS J * A 2 TEITH J J > J .» .7 J .1 10 TTECU TfMP. frTA TOO * Pl«Wt SAW "WM TESt W i7  T;5'» II HAT :»T3 Figure D.68: Temperature Distribution of a Typical Saw Blade u T Combining Eq. D.82 and D.83 into D.80 gives the following equation d ^  1 d(ru) dr r dr dT dr Integration of Eq. D.84 yields 1 tr C u = a,(l + u)- I Trdr + Cxr + — r Jb r Substituing Eq. D.85 into D.82 gives T aiE r m , . E r „ , „ , x C2 (D.83) (D.84) (D.85) jf r r ^ + ^ l C x C l + M j - ^ C l - M ) ] f Trir + _^_[C7 1(H-/ t) + % ( 1 - M) -a(ET} (D.86) The constants Ci and C% are determined by the following boundary conditions arr(a) = 0 Vrrib) = 0 (D.87) Appendix D. Thermal Stresses 136 Therefore, 1 a2 - b2 Jb Trdr C: 2 Substituting Ci and C2 into D.86 yields T Eon ,r2-b2 r 2 £ A I < -^H^  r + r T r d r ~ T r ' ) (D'89> Eq. D.89 is the thermal stress distribution in the disk. Appendix E Solution of E l The boundary conditions of the disk in the transverse vibration are ^l(M,0 = 0 Q W , Mrr\Mt) = 0 (E.90) where and Rmn(a) = 1 M N W(r,<f>,t) = £ £[G- (0 cos(n0) + 5TOri iin(n^ )] 2L,(r) (E.91) m = 0 n = 0 *™(r) = £it,y (E.92) 1=0 Substitute Eq. E.91 and E.92 into E.90, we have 1=0 1=0 £ ^ Elmn = 1 (E.93) 1=0 £[(ro - 1 + + l) + (i(m + l- n3)]alElmn = 0 1=0 137 Appendix E. Solution of Elmn £ { [ ( m - M ) 3 -n2](m + l-2) -l=o -(l-/x)n 2(m + /-l)}a'£;L Solving these equations gives the coefficients Elmn. Appendix F Saw Blade Parameters Blade A Blade B° Blade Cb Blade D (Standard) (Hi-Earn) (Detenso) (Thin Disk) Tip-To-Tip Diameter (in.) 32 32 32 36 Gullet Depth (in.) 0.875 0.875 0.875 e Eye Diameter (in.) 8.25 8.25 8.25 4.95d Plate Thickness (in.) 0.118 0.118 0.118 0.063 Collar Diameter (in.) 10.5 10.5 10.5 10.5 Tooth Width (in.) 0.202 0.202 0.202 — Tooth Material Carbide Carbide Carbide — Tooth Number 36 35 36 — Tooth Pattern Even Uneven Even — Slot No No Yes — "With an uneven tooth-spacing pattern *With very narrow radial slots 'Data is not available "Pitch diameter 139 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

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>
                        
                    
IIIF logo 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.831.1-0098015/manifest

Comment

Related Items