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Dynamics of guided circular saws Lehmann, Bruce Fredrik 1985

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DYNAMICS OF GUIDED CIRCULAR SAWS By BRUCE FREDRIK LEHMANN B.A.Sc, The University of B r i t i s h Columbia, 1983 A THESIS SUBMITTED AS PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © Bruce Fredrik Lehmann, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s ^understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Bruce F. Lehmann Department of Mechanical Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date September 25, 1985 /Q1 "S A B S T R A C T T h e r e s u l t s o f t h i s w o r k w i l l b e u s e f u l t o t h o s e i n t e r e s t e d i n p r e d i c t i n g o r i m p r o v i n g t h e c u t t i n g a c c u r a c y o f g u i d e d c i r c u l a r s a w s . I n t h i s t h e s i s a n e x p e r i m e n t a l l y v e r i f i e d n u m e r i c a l m o d e l o f t h e g u i d e d f i x e d c o l l a r c i r c u l a r s a w i s p r e s e n t e d . F e a t u r e s o f t h e m o d e l i n c l u d e t h e a b i l i t y t o r e p r e s e n t t h e b l a d e a n d c o l l a r g e o m e t r y , b l a d e r u n o u t , r o t a t i o n a l s t r e s s e s , g u i d e d y n a m i c s , g u i d e p a d s h a p e , g u i d e l u b r i c a n t , a n d g u i d e o r b l a d e m i s a l i g n m e n t . T h e b l a d e i s a s s u m e d t o b e g o v e r n e d b y t h i n p l a t e t h e o r y a n d t h e g u i d e a r m s a r e m o d e l l e d a s a l u m p e d p a r a m e t e r s y s t e m . T h e l u b r i c a t i n g f l u i d i s m o d e l l e d a s a n u m b e r o f m a s s l e s s s p r i n g - d a m p e r s . N u m e r i c a l s o l u t i o n s a r e g i v e n f o r t h e n a t u r a l f r e q u e n c y r e s -p o n s e , t h e f o r c e d r e s p o n s e d u e t o s t a t i c o r h a r m o n i c l a t e r a l l o a d i n g , a n d f o r t h e s e l f - e x c i t e d r e s p o n s e c a u s e d b y t h e i n t e r a c t i o n o f t h e b l a d e r u n o u t w i t h t h e g u i d e s . • T h e b e h a v i o u r o f t h e r u n o u t a s a f u n c t i o n o f b l a d e r o t a t i o n s p e e d a n d t h e c o n d i t i o n s f o r w h i c h a r e s o n a n t c o n d i t i o n i s p r o d u c e d i n t h e g u i d e s a r e a l s o d e t e r m i n e d . E x p e r i m e n t a l r e s u l t s o b t a i n e d f o r t h e n a t u r a l r e s p o n s e , t h e d e f l e c t i o n c a u s e d b y a s t a t i c l o a d , t h e e f f e c t o f s p e e d o n t h e b l a d e r u n o u t , a n d t h e s e l f - e x c i t e d r e s p o n s e c o r r e l a t e w e l l w i t h t h e n u m e r i c a l r e s u l t s . N u m e r i c a l r e s u l t s a r e p r e s e n t e d t o s h o w t h e e f f e c t s o f g u i d e p o s i t i o n , g u i d e s h a p e a n d t h e u s e o f m u l t i p l e g u i d e s o n t h e n a t u r a l a n d f o r c e d r e s p o n s e . i i TABLE OF CONTENTS Page Abstract i i Table of Contents i i i L i s t of Tables v L i s t of Figures v i L i s t of Symbols i x Acknowledgement x i 1. INTRODUCTION 1 Background 1 Purpose and Scope 4 2. APPARATUS 6 3. EXPERIMENTATION: INITIAL INVESTIGATIONS 12 Introduction 12 Natural Frequencies 12 Guide Dynamics 18 Runout 19 Sta t i c Stiffness of Blade Rim and Guide Arm 19 Misalignment 20 Conclusions 22 4. FORMULATION OF THE ANALYTICAL MODEL 23 Introduction 23 Assumptions 24 Description of the Model 25 Governing Equations 31 Approximation of the Blade Deflection 34 Solution Method 35 5. VERIFICATION OF THE ANALYTICAL MODEL 39 Introduction 39 Convergence of the Series Solution 39 Stat i c Deflection of an Unguided Blade 41 Frequency/Speed Relationship for an Unguided Blade 44 Frequency/Speed Relationship for a Guided Blade 44 Stat i c Loading of a Rotating Blade 58 Runout as a Function of Speed 61 Blade and Guide Motion as a Function of Speed 65 i i i TABLE OF CONTENTS (Contd) Page 6. NUMERICAL RESULTS 71 Effect of Guide Position 71 Effect of Guide Shape 74 Effect of Multiple Guides 74 7. CONCLUDING REMARKS 80 Suggestions for Further Work 83 NOTES 84 APPENDICES 86 I Equations of the Anal y t i c a l Model 86 II Solution Method 98 I I I Flowchart of Programs 103 i v LIST OF TABLES Page I Dimensions of the C i r c u l a r Saws 11 II Mode Shapes and Natural Frequencies of an Unguided and a Guided Blade (Blade A). 16 I I I Non-Diraensional Displacement at a Concentrated Load. rQ= 1.0, b/a = 0 .3 , 5 = 1.0, no guides . 40 IV Non-Dimensional Frequencies . b/a = 0 .5 , no guides . 41 v LIST OF FIGURES Page 1. Guided Splined-Arbour Circular Saw. 3 2. General Layout of the Apparatus. 7 3. Experimental Saw Guides. 8 4. Calibration Curve for Displacement Probe. 10 5. Frequency/Speed Plot for Blade A, Unguided. 13 6. Frequency/Speed Plot for Blade A, Guided. 15 7. Receptance of Blade A at 900 r.p.m. ' 17 8. Stiffness of the Rim of Blade A. 21 9. Stiffness of the Guide Arm. 21 10. Model of a Guided Circular Saw. 26 11. Misalignment (a) Shaft, (b) Blade, and (c) Lateral. 28 12. Static Deflection of Blade A (M,N) = (2,8). 42 13. Static Deflection of Blade B (M,N) = (2,8). 43 14. Frequency/Speed Plot for Blade B, Unguided. 45 15. Numerical Solution of the Frequency/Speed Plot for Blade A, Unguided (M,N) = (2,7). 46 16. Numerical Solution of the Frequency/Speed Plot for Blade B, Unguided (M,N) = (2,7). 47 17. Numerical Solution of the Frequency/Speed Plot for Blade A Guided by a Single Spring, (a) ki-j = 22.88 N/mm, 48 (b) kA = 2288.0 N/mm. 49 Guided, Unguided (M,N) = (2,7). 18. Numerically Generated Mode Shapes of a Blade A (Guided) at Zero Speed. k ± i = 2288 N/mm, = 814 N/mm, % = 11.97 Kg, ( r i j . Y i j ) = (1.0,0), (M.NJ = (3,8). 51 19. Contributions of Terms C m n to the Total Displacement for Blade A, (a) k i i = 0.2288 N/mm, 52 (b) kA = 2.288 N/mm. 53 r o f = 1-0, Ynf = 0.0, rU = 1.0, Y i i - 3 A ' 4 ° ' P = 100 N, (MIN) = 2.8).J J v i LIST OF FIGURES (Contd) Page 20. Numerical S o l u t i o n of the Frequency/Speed P l o t f o r Blade A Guided w i t h a Pad Modelled by Four Springs. (a) k i i = 8.14 N/mm, 55 (b) k^i = 81.4 N/mm, 56 (c) k ± j = 814.0 N/mm. (M,N) = (2,7). 57 21. F l e x i b i l i t y of Blade B as a Function of Rotation Speed, (a) Experimental, (b) Numerical. (M,N) = (3,8). 60 22. RMS Spectrum of the Runout of Blade A at 750 r.p.m. 63 23. RMS Runout as a Function of Blade Speed, (a) Sine ( 0 ) component, and (b) Sine (29 ) component. (M,N) = (2,8), Blade A. 64 24. RMS Amplitudes of the S e l f - E x c i t e d Motion of Blade A and the Guide as a Function of Speed, (a) F i r s t Multiple, and 66 (b) Second Multiple of the Blade Speed. 67 25. Numerical Solution of the Self-Excited Motion of Blade A and the Guide; (a) F i r s t M ultiple A Q > 1 = 0.224mm, ( c ) 69 (b) Second Multiple of the Blade Speed, Ao,2 = °' 0 1 7 6 m m « Four-Point Guide, k ± j = 81.4 N/mm. (M,N) = (3,7). 70 26. Blade D e f l e c t i o n at the Load as a Function of the Radial Position of the Guide Relative to the Load. b/a = 0.3,? = 1.0, ^ 0 = 5.0 (M,N) = (1,7), £ = 100.0 . 72 27. Blade D e f l e c t i o n at the Load as a Function of the Angular P o s i t i o n of the Guide R e l a t i v e to the Load. b/a = 0.3, ?= 1.0. (M,N) = (1,7) , £ = 100.0 . 73 28. Guide Pad Shapes. 75 29. E f f e c t of Guide Shape on the D e f l e c t i o n at the Rim Due to a S t a t i c Load; (a) ^ = 2.5, (b) = 5.0. b/a = 0.3. (M,N) = (1,7). 76 30. Frequency/Speed P l o t f o r Blade A w i t h Six Equal l y Spaced Guides. (M,N) = (1,7). 78 31. Frequency/Speed P l o t f o r Blade A w i t h Eight Equal l y Spaced Guides. (M,N) = (1,7). 79 32. Flowchart for Program GUIDE1. 104 33. Flowchart for Program GUIDE2. 105 34. Flowcharts for subroutines MDS, CENST, and POLY. 106 v i i LIST OF FIGURES (Contd) Page 35. Flowcharts for Subroutines STATIC and HARMON. 107 36. Flowchart for Subroutine LOAD. 108 v i i i LIST OF SYMBOLS a = Blade radius b = C o l l a r radius h = Blade thickness p = Density E = Young's modulus D = Eh 3/(12(1-V 2)) u(r,Y) = Blade d e f l e c t i o n i n stationary polar coordinate^system w(r , 8 ) = Blade d e f l e c t i o n i n r o t a t i n g polar coordinate system I = To t a l number of guides J = Total number of spring-dampers per guide 6 = L a t e r a l degree-of-freedom of the i - t h guide arm 6 . , 9 . = Rotational degrees-of-freedom of the i - t h guide arm y i x i M^ = Mass of the i - t h guide arm I .,1 . = Moments of i n e r t i a of the i - t h guide arm X x l (r,Y) = P o s i t i o n of the centre of mass of the i - t h guide ^ i j ' ^ i j = S t i f f n e s s and damping c o e f f i c i e n t of the j - t h spring-damper on the i - t h guide K .,K .,K = Guide arm s t i f f n e s s e s z i y i x i D ,D , ,D .= Guide arm damping c o e f f i c i e n t s z i y i x i t = Time = Blade r o t a t i o n a l speed F = Total number of forces acting on the blade P^,co^ = Magnitude and frequency of the f - t h force A . = L a t e r a l misalignment (c) (s) A ^ »A^ m = Terms for blade misalignment (c) (s) A ,A = Terms for shaft misalignment o o (c) is) A ,AV = Terms for runout ps ps a = In-plane plate stresses ix LIST OF SYMBOLS (Contd) r ' r/a b' b/a A M/pa2h y I/pa uh «U ' d^/^phD hi - K z ia2/D K y ±/D, K x i/D \i D . //phD' Z l D .//pha"D" , D ^/Zpha"^ v i x i T aTft/D/pha 1*' a 0 fi/pha"/D w o f U)f/D/phav A' A /a a' a a2h/D Pfa/D X ACKNOWLEDGEMENT I would l i k e to thank the Science Council of B r i t i s h Columbia and Kockums CanCar Inc. of Surrey, B.C. f o r supporting ray work through the Graduate Research i n Engineering and Applied Technology (GREAT) Award program. I would a l s o l i k e to thank the f o l l o w i n g people f o r t h e i r help: my advisor, Dr. Stanley G. Hutton, Department of Mechanical Engineering, University of B r i t i s h Columbia; Mr. Dave Roper, Manager of Order Engineering, Kockums CanCar Inc; and Dr. S e i j i Chonan, Department of Mechanical Engineering, Tohoku University, Japan, who i s responsible for much of the i n i t i a l development of the a n a l y t i c a l model presented i n t h i s thesis. x i In memory of my father, Fred Lehmann, who had a subtle, but d i r e c t e f f e c t on t h i s work. x i i 1. INTRODUCTION Background C i r c u l a r saws have been used e x t e n s i v e l y by the lumber i n d u s t r y since the middle of the nineteenth century, but i t was not u n t i l the 1950's that systematic research began i n the f i e l d of c i r c u l a r sawing. At the present time blade selection for a particular operation i s based upon past experience and e m p i r i c a l r e l a t i o n s h i p s ^ . The problem of s e l e c t i n g an opti m a l blade i s one of min i m i z i n g the kerf l o s s e s by reducing the plate thickness u n t i l the s t i f f n e s s of the blade decreases to some c r i t i c a l value at which the c u t t i n g accuracy i s no longer acceptable. The most important development i n the a n a l y s i s of the v i b r a t i o n response of c i r c u l a r saws has been c r i t i c a l speed theory, which c o r r e l a t e s poor c u t t i n g accuracy to a resonant c o n d i t i o n i n the blade^. Most of the research on c i r c u l a r saws has been i n the areas of vibration of c i r c u l a r or annular plates and has been directed at finding methods of a l t e r i n g the saw design so that blade resonance i s avoided. The primary method of a l t e r i n g the blade's dynamic c h a r a c t e r i s t i c s , other than by changing the dimensions or m a t e r i a l p r o p e r t i e s of the blade, i s to introduce in-plane stresses i n the plate. I t has been found that the heat generated at the blade rim by the c u t t i n g process produces an adverse s t r e s s d i s t r i b u t i o n i n the blade associated with a compressive hoop stress at the blade rim. I f the rim temperature becomes too great, the blade tends to buckle. To counteract the compressive hoop stresses that are generated by heating, methods of 1 pre-stressing the plate have been developed, through a process of t r i a l and e r r o r , so that the rim i s i n i t i a l l y given a t e n s i l e hoop s t r e s s . Tension, as the p r e - s t r e s s i s c a l l e d , i s given to the blade by c o l d working the blade with special hammers or, more recently, by pressure r o l l i n g of narrow bands around the blade. Hammering i s s t i l l using to remove any warp, or runout, as i t i s c a l l e d by the i n d u s t r y , from the blade. Another method f o r developing tension i s to heat the inner regions of the blade e i t h e r by the use of packings ( f r i c t i o n pads), or by i n d u c t i o n heaters. Tensioning s t i f f e n s the blade so t h a t , i n gen-e r a l , the natural frequencies are increased and the a b i l i t y of the blade to r e s i s t d e f l e c t i o n caused by the l a t e r a l component of the c u t t i n g f o r c e i s a l s o increased. Consequently, t e n s i o n i n g i s the main method for c o n t r o l l i n g the dynamic characteristics of c i r c u l a r saws. However, because t e n s i o n i n g i s s t i l l more of an a r t than a science, the s t r e s s d i s t r i b u t i o n i n a tensioned blade i s not known with any accuracy. Another factor that produces in-plane stresses i n the blade i s the centrifugal force e x i s t i n g i n a rotating blade. The re s u l t i n g stresses are b e n e f i c i a l because they s t i f f e n the blade. A recent innovation i n c i r c u l a r saw design i s the guided splined-arbour saw. In t h i s design the blade i s mounted on a s p l i n e d shaft rath e r than being r i g i d l y f i x e d to the shaft by a c o l l a r arrangement. Because the s p l i n e cannot support the blade, guides must be used to position the blade, see Figure 1. The guides are usually made of babbit and some method of f l u i d l u b r i c a t i o n between the babbit and the blade i s provided. The clearance between the guide and the blade i s t y p i c a l l y 0.001 inches to 0.004 inches per side. Mote 3 and Iwan and S t a h l 4 have 2 Figure 1: Guided Splined-Arbour Circular Saw 3 shown t h a t the support p r o v i d e d by the guides changes the frequency response of the blade quite s i g n i f i c a n t l y , which suggests that through proper design of the guides the blade dynamics can be altered, perhaps i n a way that the c u t t i n g performance of the blade can be improved. To date no a n a l y s i s has been presented of the g e n e r a l f o r c e d r e s -ponse of r o t a t i n g disks, with or without guides although analysis of a r o t a t i n g f l e x i b l e d i s k s u b j e c t e d to a s t a t i c t r a v e r s e l o a d has been s t u d i e d by Benson and Bogy^, Iwan and M o e l l e r ^ , Hutton, Chonan and Lehmann^, and N i g h ^ . An extensive survey of the developments i n t h i n c i r c u l a r saw v i b -r a t i o n and control research has been prepared by Mote and Szymani^. Purpose and Scope The purpose of t h i s work i s to produce an a n a l y s i s of the n a t u r a l and f o r c e d v i b r a t i o n response of guided c i r c u l a r saws s u b j e c t e d to s t a t i c and harmonically varying point loads and to experimentally v e r i f y the r e s u l t s generated by the analysis. At t h i s time the magnitude and frequency content of the a c t u a l f o r c e s t h a t a c t on the blade do not appear to have been measured; however, the a b i l i t y to c a l c u l a t e the f o r c e d response w i l l a l l o w the sawmill industry, through a better understanding of the saw response, to improve and, i n time, p r e d i c t the c u t t i n g c h a r a c t e r i s t i c s of guided c i r c u l a r saws. 4 In t h i s thesis the dynamics of rotating c i r c u l a r disks supported by guides are studied. Because i t i s the i n t e r a c t i o n of the guides w i t h the blade which i s not w e l l understood, t h i s study deals only with a fixed c o l l a r configuration with and without guides. Tensioning i s not included i n the model, but the stresses resulting from the centrifugal forces are included. Because a comparison i s made between a guided and an unguided blade, much i n f o r m a t i o n about the behaviour of unguided blades i s also presented. 5 2. APPARATUS The equipment used f o r the experiments described i n t h i s t h e s i s i s l o c a t e d i n the Wood Sawing L a b o r a t o r y of the U n i v e r s i t y of B r i t i s h Columbia Mechanical Engineering Department. The c i r c u l a r saw was b u i l t in-house and i s configured as a headsaw f o r use w i t h a c a r r i a g e . Blades as l a r g e as 36 inches i n diameter can be mounted. A general layout of the apparatus i s shown i n Figure 2. A one hundred horsepower high pressure h y d r a u l i c system was used to d r i v e the c i r c u l a r saw. The s w a s h - p l a t e type pump was d r i v e n by an e l e c t r i c motor and an e l e c t r o n i c p r o p o r t i o n a l s o l e n o i d was used f o r f l o w c o n t r o l . The h y d r a u l i c motor was connected d i r e c t l y to the saw s h a f t and had a hand-wheel adjustment f o r f i n e c o n t r o l of the r o t a t i o n speed; coarse speed c o n t r o l was obtained w i t h the s o l e n o i d c o n t r o l on the pump. With t h i s c o n f i g u r a t i o n e x c e l l e n t speed c o n t r o l was obtained f o r s h a f t speeds of 30 t o A500 r e v o l u t i o n s per minute. The speed was measured w i t h a tachometer mounted on the sh a f t and was displayed d i g i t a l l y . A d r a w i n g of the saw g u i d e s used i n t h i s study i s shown i n Figure 3. The g u i d e pads were l u b r i c a t e d w i t h water i n t r o d u c e d from i n t e r n a l water l i n e s i n ^ the guide arm. Throughout the experiments the gap between the surface of the guide pad and the blade was kept at l e s s t h a n 0.002 i n c h e s per s i d e . The s i z e and g e n e r a l arrangement of the guides was s i m i l a r to those used i n production saws. To d e t e r m i n e the dynamic c h a r a c t e r i s t i c s of a c i r c u l a r saw the blade must be e x c i t e d and the r e s u l t i n g motion measured. To v i b r a t e the 6 G u i d e s 1 0 0 H P H y d r a u l i c D r i v e D i g i t a l T a c h o m e t e r F o r c e ' T r a n s d u c e r P o w e r A m p l i f i e r F r e q u e n c y G e n e r a t o r o o B l a d e C o l l a r E d d y - c u r r e n t D i s p l a c e m e n t P r o b e D-C h a r g e A m p l i f i e r S p e c t r u m A n a l y s e r I m p a c t H a m m e r a n d A c c e l e r o m e t e r Figure 3: Experimental Saw Guides 8 blade an electromagnet was used. A random frequency s i g n a l or a s i n g l e frequency s i g n a l was provided by a Bru e l & Kjaer 1024 s i g n a l generator. The s i g n a l was a m p l i f i e d by a one hundred watt power a m p l i f i e r connected i n - l i n e between the s i g n a l generator and the electromagnet. The force produced by the e l e c t r o m a g n e t was measured w i t h a B r u e l & K j a e r 8200 f o r c e t r a n s d u c e r , t h e output o f w h i c h was a m p l i f i e d by a K i s t l e r 504D charge a m p l i f i e r . The motion of the blade was measured w i t h Bentley-Nevada n o n - c o n t a c t i n g e d d y - c u r r e n t displacement probes. This type of probe produces a l i n e a r o u t p u t f o r d i s p l a c e m e n t s up to 0.07 i n c h e s (1.8 m i l l i m e t e r s ) . A t y p i c a l c a l i b r a t i o n c u r v e i s shown i n F i g u r e 4. For measuring the dynamic c h a r a c t e r i s t i c s of the guides, as described i n Chapter 3, a PCB P i e z o t r o n i c s impact hammer and accelerometer set was used. Data a n a l y s i s was c a r r i e d out u s i n g a N i c o l e t 660A d u a l c h a n n e l spectrum analyser which provided f r e q u e n c y s p e c t r a and t r a n s f e r f u n c -t i o n s o f the f o r c e and d i s p l a c e m e n t s i g n a l s . Permanent c o p i e s o f the analyses were produced by a Tektronix 4662 d i g i t a l p l o t t e r . Two c i r c u l a r saw blades were used i n the experiments. Blade A has been used i n a s a w m i l l ; consequently, i t has a la r g e amount of t e n s i o n -i n g and was not very f l a t . B l a d e B was a b l a n k d i s k o f a p p r o x i m a t e l y the same s i z e as B l a d e A, but i t had no t e e t h and has been r o l l tensioned only a s m a l l amount so as to keep the disk f l a t . Blade B was l e f t untensioned so that a comparison could be made between experimental r e s u l t s and r e s u l t s g e n e r a t e d by the a n a l y t i c a l model, which does not inc l u d e the e f f e c t s of ten s i o n i n g . Dimensions of the blades are given i n T a b l e I . 9 Figure A: C a l i b r a t i o n Curve f o r Displacement Probe 10 Table I Dimensions of Circular Saws Blade A Blade B Outside Diameter (Tip-to-Tip) 36.0 inches 36.A inches Thickness 0.118 inches 0.121 inches Collar Diameter 10.5 inches 10.5 inches Number and Type of Teeth AO (Rip) 0 Gullet Depth 1.06 inches -11 3. EXPERIMENTATION: INITIAL INVESTIGATIONS I n t r o d u c t i o n The behaviour of guided c i r c u l a r saws i s a complex dynamical prob-lem i n v o l v i n g many pa r a m e t e r s . Thus b e f o r e a t t e m p t i n g to deve l o p a re p r e s e n t a t i v e a n a l y t i c a l model, i t was decided to undertake p r e l i m i n a r y experimental work to i s o l a t e those v a r i a b l e s that need to be considered i n the model. I n t h e s e e x p e r i m e n t s a f i x e d c o l l a r b l a d e i s used even though a vast m a j o r i t y of guided saws used i n s a w m i l l s use a s p l i n e arbour mount-i n g . The f i x e d c o l l a r mounting was used because of e x p e r i m e n t a l convenience i n that i t was found d i f f i c u l t to i s o l a t e the e f f e c t s of the guides from the e f f e c t s caused by having a s p l i n e d arbour. N a t u r a l Frequencies The f i r s t t e s t s c o n s i s t e d of measuring the n a t u r a l frequencies of an unguided f i x e d c o l l a r blade at d i f f e r e n t r o t a t i o n a l speeds. Guides were then mounted on the bla d e and the n a t u r a l f r e q u e n c i e s were a g a i n measured. Comparison of the two f r e q u e n c y / s p e e d p l o t s then i n d i c a t e s the e f f e c t of the guides on the frequency response of the system. F i g u r e 5 shows the f r e q u e n c y / s p e e d p l o t f o r the Bla d e A w i t h no guides. The n o t a t i o n , f o r example, (0,3) i s used to denote a mode w i t h zero nodal c i r c l e s and three nodal diameters. The general form of the r e s u l t s agrees w e l l w i t h developed theory^. For the non-rotating blade (*H) iCouanbajj Figure 5: Frequency/Speed P l o t f o r Blade A, Unguided. 13 there e x i s t s modes of v i b r a t i o n c o n s i s t i n g of n nodal diameters, where n = 0,1,2,... and m n o d a l c i r c l e s , where m = 0,1,2... As the b l a d e speed i s i ncreased the n a t u r a l frequencies s p l i t ; t h i s phenomenon i s explained by r e a l i z i n g t h a t the mode shape r o t a t e s w i t h the b l a d e and t h a t the mode shape i s a standing wave composed of two t r a v e l l i n g waves of equal amplitude but moving i n opposite d i r e c t i o n s around the blade; t h i s can be described mathematically as f o l l o w s : Sin(nA) = 1/2 Sin(nA + c t ) + 1/2 Sin(nA - c t ) The speed o f the wave i s "c". When the bla d e i s r o t a t i n g the frequency of the wave moving i n the same d i r e c t i o n of the r o t a t i o n w i l l appear to i n c r e a s e , as measured by a s t a t i o n a r y o b s e r v e r , w h i l e the frequency of the wave moving opposite to the blade r o t a t i o n w i l l appear to d ecrease. The c u r v e s shown i n F i g u r e 5 a r e concave upward because the c e n t r i f u g a l s t r e s s e s due to r o t a t i o n increase the s t i f f n e s s of the blade, thus i n c r e a s i n g the frequencies as the speed increases. The c r i t i c a l speed of B l a d e A was 1950 rpm. At t h i s speed the b l a d e has a n a t u r a l f r e q u e n c y o f z e r o , so t h a t i f a s t a t i c or low f r e q u e n c y f o r c e i s a p p l i e d , the b l a d e w i l l r e s o n a t e . Mote^ has shown t h a t t h i s r e s o n a n t c o n d i t i o n i s c o r r e l a t e d to poor c u t t i n g a c c u r a c y , i n d i c a t i n g that a s i g n i f i c a n t component of the c u t t i n g f o r c e i s s t a t i c . Note t h a t i t i s the t h r e e n o d a l d i a m e t e r mode w h i c h becomes c r i t i c a l f i r s t . The frequency/speed p l o t f o r the same blade w i t h a guide now mount-ed i s shown i n F i g u r e 6. For a n o n - r o t a t i n g b l a d e the mode shapes and 14 Blade Speed (rpm) the corresponding n a t u r a l frequencies were found by e x c i t i n g the blade w i t h a s i n g l e f r e q u e n c y e x c i t a t i o n and l o c a t i n g the nodes by touch. Nodal r a d i i , r a t h e r t h a n n o d a l d i a m e t e r s , were found, where one n o d a l r a d i u s i s p o s i t i o n e d under the guide. A t y p i c a l r e c e p t a n c e p l o t i s shown i n Figure 7. Table I I i s presented f o r comparing the frequencies and mode shapes of the two arrangements. Table I I Mode Shapes and N a t u r a l Frequencies of a Non-Rotating Unguided and Guided Blade (Blade A). Unguided Guided Frequency Frequency Mode (Hz) Mode (Hz) G 20.75 0 21.75 0 23.0 © 35.0 © 41.25 © 48.0 © 65.25 © 70.25 83.75 For modes w i t h one, two and t h r e e n o d a l d i a m e t e r s the e f f e c t o f adding the guides i s to increase the frequencies of these modes. Note that the s t i f f e n i n g e f f e c t i s greater f o r the higher modes. 16 A few remarks are n e c e s s a r y r e g a r d i n g the r e p e a t a b i l i t y of the r e s u l t s . In general, the day-to-day r e p e a t a b i l i t y of measuring a nat-u r a l frequency was w i t h i n one or two h e r t z : t h i s v a r i a t i o n i s thought to occur because of temperature v a r i a t i o n s i n the room where the e x p e r i -ments were conducted and because of temperature v a r i a t i o n s of the water used f o r the guide l u b r i c a t i o n . Temperature gradients i n the blade are known to a f f e c t the blade dynamics^. Guide Dynamics While measuring the n a t u r a l frequencies of the guided blade (Blade A) a t d i f f e r e n t speeds i t was noted t h a t a t c e r t a i n speeds a very l o u d hammering noise was generated i n the region of the guides. To i n v e s t -i g a t e t h i s problem non-contacting displacement probes were mounted to measure b l a d e motion near the g u i d e and the m o t i o n of the guide. No e x t e r n a l e x c i t a t i o n was p r o v i d e d . I t was found t h a t at speeds of 12.5 cps and 16.5 cps the n o i s e l e v e l would i n c r e a s e and so would the displacement of the guides. At speeds below 12.0 cps the guides and the b l a d e move t o g e t h e r , the m o t i o n b e i n g g e n e r a t e d by the runout of the blade, but at 12.5 and 16.5 cps the guide motion became s i n u s o i d a l w i t h a frequency of 50.0 Hz. This frequency of 50.0 Hz was l a t e r found to be the lowest n a t u r a l frequency of the guide arms. At t h i s time i t was hypothesized that the runout, when represented as a F o u r i e r Trigonometric s e r i e s (see Chapter 4), which has components that appear as m u l t i p l e s of the blade speed when the blade i s r o t a t i n g , 18 was p r o v i d i n g the e x c i t a t i o n . Four t i m e s 12.5 cps and t h r e e t i m e s 16.5 cps are approximately 50.0 Hz, and i t was pred i c t e d that the guides should be resonated by the second m u l t i p l e of the blade speed when the speed i s approximately 25.0 cps: t h i s p r e d i c t i o n proved to be c o r r e c t . Guide r e s o n a n c e s a t speeds of 50/5, 50/6, 50/7 cps , e t c . were not ob-s e r v e d , p o s s i b l y because of damping i n the system. The c o u n t e r p a r t e f f e c t of the b l a d e b e i n g e x c i t e d by the g u i d e / r u n o u t i n t e r a c t i o n was a l s o seen, but no c l e a r resonant c o n d i t i o n s were apparent. Runout From the preceeding s e c t i o n i t i s apparent that the runout, or the i n i t i a l warp of the b l a d e , i s an i m p o r t a n t v a r i a b l e t h a t can a f f e c t c u t t i n g a c c u r a c y : when the gu i d e s are r e s o n a t i n g they do not p r o v i d e much support to the blade during c u t t i n g . The r e s u l t s of these e x p e r i -ments on the b e h a v i o u r o f runout and how i t i n t e r a c t s w i t h the g u i d e s are presented i n Chapter 5. S t a t i c S t i f f n e s s of Blade Rim and Guide Arm The c u t t i n g c h a r a c t e r i s t i c s of a guided saw a r e dependent on the s t i f f n e s s of the b l a d e and of the g u i d e s which s u p p o r t the b l a d e . A l -though t h e s e s t i f f n e s s e s a r e a f f e c t e d by the dynamics of the r o t a t i n g system, the s t i f f n e s s v a l u e s of the s t a t i o n a r y system p r o v i d e some i n d i c a t i o n of the a b i l i t y of the saw to r e s i s t d e f l e c t i o n and the amount of support which the guides can provide. The blade s t i f f n e s s was measured by t y i n g a cord to one of the teeth, running the cord over a pulley, hanging known weights on the end of the cord and measuring the resultant d e f l e c t i o n . A s i m i l a r procedure was used to measure the guide arm s t i f f n e s s , but the cord was attached at the centre of the guide pad. The r e s u l t s are shown i n Figures 8 and 9. The s t i f f n e s s f o r Blade A i s 22.0 N/mm (126 l b s / i n c h ) , and f o r the guide arm, 814 N/mm (4650 lbs/inch). Misalignment One of the problems encountered when mounting the guides on a fi x e d c o l l a r blade i s i n c o r r e c t alignment of the guides. I f the s u r f a c e of the guide pad, a f t e r allowing 0.002 to 0.004 inches f o r the l u b r i c a t i o n gap, i s not i n the same plane as the s u r f a c e of the c o l l a r , one of the guides w i l l cause a s t a t i c p r e - l o a d on the blade when the blade i s clamped i n t o the c o l l a r . As p r e v i o u s l y mentioned, a s t a t i c load can cause a resonant condition i n the blade, therefore, misalignment should be minimized. Misalignment can occur f o r three reasons (see Figure 11): (1) L a t e r a l misalignment: the guide pad and the c o l l a r are para-l l e l but not i n the same plane, 20 Dlsplacaoit (allllaetan) Figure 8: Stiffness of the Rim of Blade A. Slope - 4650 Lbs/in. 2 4 6 Slepleceaeat (0.001 Inches) Figure 9: Stiffness of the Guide Arm. 21 (2) Blade misalignment: the c o l l a r i s not squarely mounted on the shaft , and (3) Shaft misalignment: the shaft i s misa l igned. Conclusions The e x p e r i m e n t a l r e s u l t s d e s c r i b e d i n t h i s c h a p t e r p e r t a i n to f ac tor s which have not been presented i n previous s tudies of sawblade dynamics . The r e s u l t s show t h a t i t i s not s u f f i c i e n t to c o n s i d e r the guide arms as being r i g i d bodies or to assume that the blade i s per fec t -l y f l a t i f the dynamics of guided c i r c u l a r saws are to be a c c u r a t e l y descr ibed . S p e c i f i c f ind ings of the experiments were: (1) The modes o f v i b r a t i o n of gu ided b l a d e s c o n t a i n noda l r a d i i rather than the nodal diameters seen i n unguided blades. (2) One of the nodal r a d i i forms under the guide. (3) The i n t e r a c t i o n of the guide with the blade runout can cause a s e l f - e x c i t e d v i b r a t i o n which occurs when a natura l frequency of the guided system equals a frequency that i s a m u l t i p l e of the blade speed. 22 4. FORMULATION OF THE ANALYTICAL MODEL Introduction As shown in Chapter 3, the following factors appear to be important in determining the dynamic response of guided circular saws: 1. Blade runout or i n i t i a l warp. 2. Guide dynamics. 3. Shaft, blade, and lateral misalignment. 4. Rotational stresses. 5. Thermal and tensioning membrane stresses. For the analysis developed i n th i s thesis a l l of these factors except the membrane stresses w i l l be included. Thermal stresses, as well as the residual stresses resulting from blade tensioning, are not included because of time constraints and because th e i r e f f e c t on the blade i s to shift the natural frequencies, but not to change the general shape of the frequency/speed curves^, H # The method of solution i s to apply Galerkin's method to the equation of the blade, r e s u l t i n g in a set of l i n e a r d i f f e r e n t i a l equations. Another set of equations i s generated for the guides and the solution i s found for the system by solving both sets of equations simultaneously. 23 Assumptions The fo l lowing assumptions are made: 1. The di sk i s homogeneous and i s o t r o p i c . 2. Smal l d e f l e c t i o n t h i n p late theory i s app l i cab le (thus l i m i t -ing the accuracy of the r e s u l t s for displacement l a rger than 0.4 of the plate t h i c k n e s s ^ . 3. The l u b r i c a t i n g f l u i d between the b l ade and the g u i d e s i s m o d e l l e d as a number of raassless s p r i n g - d a m p e r s , where the dampers are i n p a r a l l e l wi th the springs . Th i s l i n e a r model i s a s imple f i r s t approximation for the i n t e r a c t i o n mechanism between the blade and the guides. In prac t i ce the blade makes d i r e c t c o n t a c t w i t h the gu ides as e v i d e n c e d by the p o l i s h e d surface of the blade and the wear of the guides. To proper ly analyse the i n t e r a c t i o n between the blade and the guide, the s p r i n g and damping c o e f f i c i e n t s s h o u l d be f u n c t i o n s of the r e l a t i v e p o s i t i o n s of the b lade and the g u i d e , i . e . a non-l i n e a r model would be required . 4. The guides are modelled as a lumped parameter l i n e a r s p r i n g -mass system with viscous damping. 5. Both gu ide arms are independent of the o t h e r , and are not c o u p l e d by the mot ion of the guide s u p p o r t s t r u c t u r e . T h i s a s s u m p t i o n i s not n e c e s s a r i l y v a l i d f o r a l l saws now i n o p -e ra t ion i n the m i l l s . Desc r ip t ion of the Model F i g u r e 10 shows a b lade of r a d i u s " a " and t h i c k n e s s " h " c lamped i n t o a c o l l a r of r a d i u s " b " , mass d e n s i t y P , and r o t a t i n g w i t h c o n -stant angular v e l o c i t y Q. Two coordinate systems are defined s ince the b l a d e r o t a t e s w h i l e o b s e r v a t i o n s are made from f i x e d p o i n t s i n space. The system r o t a t i n g w i t h the b lade has c o o r d i n a t e s ( r , 0), and the s ta t ionary system has ( r ,y and (x,y). The r a d i a l coordinate i s the same i n each system and the angular coordinates 0 and y are re la ted through the expression Y = 0 + nt ( A . l ) The l a t e r a l d e f l e c t i o n of the blade i s denoted by w(r,e) i n the r o t a t i n g system and by u(r,y) i n the s ta t ionary system. The p o s i t i o n of the c e n t r e of the mass of the i - t h guide i s (j±ty±), i = 1 ,2 ,3 , . . I , where I i s the t o t a l number of gu ides i n the system. Each guide arm i s assumed to have three degrees of freedom: <5 z i s a l a t e r a l motion i n the z - d i r e c t i o n , Qy and 0 X are r o t a t i o n a l motions about axes p a r a l l e l to the y and x axes , r e s p e c t i v e l y . The r i g h t - h a n d r u l e i s used to d e t e r m i n e the d i r e c t i o n of p o s i t i v e r o t a t i o n . Each guide cons i s t s of two guide arms, one on each s ide of the blade and has, therefore , s i x degrees of freedom, 6 1 , © \ © \ 6 2 , 0 2 , and ©J. For each 2 V X Z y X 3 b) Load f and Guide i . Only z d i r e c t i o n degree of freedom shown. c) Degrees of freedom for a guide. Figure 10: Model of a Guided Circular Saw. 26 degree of freedom a s t i f f n e s s K^, a damping c o e f f i c i e n t V±* a n d an i n e r t i a 0 r 1^ are a s s i g n e d . Both guide arms of a g u i d e a r e assumed to have i d e n t i c a l dynamic c h a r a c t e r i s t i c s . To r e p r e s e n t t he f l u i d l u b r i c a t i o n between the g u i d e s u r f a c e and the b l a d e , a number of s p r i n g - d a m p e r s , k i j and d±j a t p o s i t i o n s ( r i j , Y i j ) , j = 1,2,...J, a r e used, where J i s the number of s p r i n g -dampers per g u i d e . By c h o o s i n g ( r i j . Y ^ j ) , t h e shape of the guide pad can be described. The saw blade i s subjected to l a t e r a l loads. These are assumed to be c o n c e n t r a t e d l o a d s at p o s i t i o n ( r 0 f , Y 0 f ) , f = 1,2,3...F, where F i s the t o t a l number of loads. In general, the load has a s t a t i c component as w e l l as a harmonic component of f r e q u e n c y w^, so t h a t t h e t o t a l a p p l i e d load i s assumed to be of the form P f = P s f + pf c°s(u ft) + P*Sin(w ft) (4.2) S h a f t , b l a d e , and l a t e r a l m i s a l i g n m e n t a r e shown i n F i g u r e 11. Shaft misalignment t i l t s the blade i n the s t a t i o n a r y reference frame and can be represented by U s m ( r ' Y , t ) = ( r / a ) { A o C o s ^ ) + A ^ S i n ( Y ) } . (4.3) 27 Figure 11: Misalignment (a) Shaft, (b) Blade, and (c) Lateral. 28 Blade misalignment t i l t s the blade i n the ro t a t i n g reference frame and can be represented by wbm ( r' 0 ) " ( r / a ) K m C o s ( 0 ) + 0 L N ( E ) } (4.4) which can be transformed i n t o the stationary frame using the equation (4.1) so that u b m ( r > Y ' t ) = ('/*)' {A^Cosfr-nt) + A ^ S i n f r - f l t ) } . (4.5) La t e r a l misalignment of the blade i s represented as u l m ( r , y , t ) = A. (4.6) The runout, or more s p e c i f i c a l l y the i n i t i a l warp of the blade, i s approximated by a trigonometric Fourier s e r i e s P S w (r,0) = z I R (r/a) {A* Cos(se) •+ A^ Sin(s0)} (4.7) p=0 s=0 p p s P s i n the r o t a t i n g coordinates, or i n the stationary coordinates as P S u (r.Y.t) = Z Z R (r/a) {A C Cos(s Y-s^t) + A S S i n ( s Y - s ^ t ) } P=0 s=0 p s p s p s (4.8) 29 where Rr . c ( r /a ) i s a f u n c t i o n which i s d e s c r i b e d i n the s e c t i o n on g o v e r n i n g equat ions (equation 4.22). The r o t a t i o n of the blade causes in-plane r a d i a l and c i rcumferen-t i a l s tresses a r a n a * a g » which are determined by so lv ing the f o l l o w i n g equations-'-3 ^ r + V P 9 + pro" = 0 dr r (4.9) e 0 " e r + r ^ 0 = 0 dr where e p and £ Q a re the r a d i a l and c i r c u m f e r e n t i a l s t r a i n s andp the mass d e n s i t y of the b l ade m a t e r i a l . For a b lade c lamped on the i n n e r radius and free at the outside edge the so lu t ions of equations (4.9) are 2 2 a r = pa^fi (A x + A 2 - A 3 ) 2 2 (4.10) °e = P a fl ( A 1 _ A 2 " V where with non-dimensional parameters r ' = r / a and b ' = b /a , A = (l+v)(3+v) + ( l - v ^ Q , ' ) 1 * 1 1+v + ( l - v ) ( b ' ) z A = f h ' / A ' ) 2 ( l - v ) ( 3 + v ) - ( l-v 2)(b') 2  A 2 C b / a > l+v+d-vXb')^ A 3 = (3+vXr') 2 A 4 = ( l + 3 v ) ( r ' ) 2 30 Governing Equations The e q u a t i o n f o r the p l a t e i n the c o o r d i n a t e s of the r o t a t i n g reference frame i s DV\r = q(r,0,t) - ph where D = Eh 3 3 2w b(r,0,t) (4.11) 1 2 ( l - v 2 ) ' 3 ? r 3r p 3 0 ' q = L a t e r a l load per u n i t area w^  = D e f l e c t i o n due to bending only. Transforming t h i s equation into the stationary reference frame by using (4.1) r e s u l t s i n 2 / ^ D V \ ^ ( r f Y , t ) = q ( r , Y , t ) - ph D V * ' ? ' * ' (4.12) Dt 2 where i s the second t o t a l time d e r i v a t i v e and u^r.Y.t) i s the d e f l e c t i o n due only to bending. There are three l a t e r a l loads on the plate: 31 1. The externa l forces as described i n equation (A.2) ff-1 S (4.13) 2. The l a t e r a l component of the in-plane s tresses , which for an a x i a l l y symmetric s t res s d i s t r i b u t i o n , can be wr i t t en as u = The t o . t a l d e f l e c t i o n o f the b l a d e , i n c l u d i n g the e f f e c t of bending, runout and misalignment, = u b + u w + u i m + u s m + U b m The t o t a l d e f l e c t i o n , u , i s used because i n c o n t r a s t to the d e f l e c t i o n due to bending, which i s dependent only on a change i n the c u r v a t u r e o f the p l a t e , the e f f e c t of the i n - p l a n e s tresses i s a l so dependent on the i n i t i a l curvature- 1^. 3. The f o r c e s e x e r t e d on the p l a t e by the s p r i n g - d a m p e r s w h i c h interconnect the blade with the guide arms. See Figure 10. (4.14) where I J Z I (k +d d_)6 ( Y - Y ) « ( r - r ) ( l / r ) i = l j = l 1 3 1 3 d t J J K f i U « k + ^ y i ) t o i J - ( e ^ x i ) A y i r 2 u ( r » Y » t ) ) ( A - 1 9 ) where Ax = I ^ C O S C Y ^ ~ r Cos(if ) A y i j = f ± s i n ^ i ) " r i j S i n ( Y i j ) Therefore, the t o t a l loading on the plate may be written as q = q + q + q, (4.16) M Mp s k As discussed p r e v i o u s l y , each guide has s i x degrees of freedom. The equations of motion for the f i r s t guide arm of the i - t h guide are: Z26l. 36 1. * ± - & + D«±-§r + K z i 6 l Z i = j y k i j + d i j l t - ) ( 1 / r ) 6 ( Y - Y i j ) 6 ( r - r i J ) (4.17) x {u - 6 i ± - + e x l A y i j } Z2Ql . 30 1 + Dy i ^ + K y i 0 y i = j ^ ( k i j + d i j l r : ) ( 1 / r ) 5 ( ^ i J ) 6 ( r - r i j ) (4.18) X (uAx, . -o^.Ax.. - 0 1.(Ax, ,) 2+ e'.Ax Ay. .} i j z i I J y i v ±3 x i 13 J ±3 3 2 0 x i 9 0 x i 1 ^ i " ^ + D x i l f - + K x i 0 x i = j ^ V d i j f e ) ( 1 / r ) 6 ( Y " Y i J > 6 ( r " r i J ) (4.19) xt-uAy^ + fi^Ay^ + B ^ A x ^ A y ^ - ^ ( A y ^ ) * } The equations f o r the second guide arm are i d e n t i c a l to those of the f i r s t , the only d i f f e r e n c e being the change i n s u p e r s c r i p t on the 33 v a r i a b l e from (1) to (2). Approximation for the Blade De f l ec t ion To represent the d e f l e c t i o n of the blade due to bending the f o l l o w -ing approximation i s used: M N u, ( r , Y , t ) = Z Z R (r/a) {C ( t )Cos(n Y ) + S (t)Sin(ny)} ^ m-0 n=0 ™ (4.20) ^mn(t) and S m n ( t ) are unknown functions to be determined. The funct ion R m n ( r / a ) i s chosen to s a t i s f y the boundary condi t ions at the innner and outer r a d i i of the p l a t e , which for the c o l l a r e d blade are : = 0, 9 u b - 0 at r=b 3M (4.21) M = 0, Q - --r1- = 0 at r=a r r r r 3y where M r r -D 3 2 u b + v ( 1 3 u b + 1 * \ ) 7 3 r r 3 r r 2 T y 7 M - D ( l - v ) 1 3 f 3 u b - \ r 3 y \ 3 r r 3 r \ " T r 7 " r 3 r r 2 3 y V ^ ( r / a ) i s assumed to be i n the form of a polynomial i n r / a , R (r/a) = Z E . ( r / a ) * 4 " j _ 1 (4.22) mn mnj 34 Because e q u a t i o n (4.21) has f i v e unknowns, whereas o n l y f o u r boundary condi t ions are provided i n (4.21), an a r t i f i c i a l cond i t ion R - 1.0 at r=a (4.23) mn i s used when s o l v i n g f o r the E m n j . S u b s t i t u t i o n of (4.22) and (4.20) i n t o e q u a t i o n s (4.21) and (4.23) r e s u l t s i n f i v e l i n e a r s i m u l t a n e o u s a lgebra ic equations from which the E m n j are determined. The f u n c t i o n R m n i s the same as R p S , which i s used i n e q u a t i o n (4.7) to descr ibe the runout of the blade. So lu t ion Method The var i ab le s which are to be determined are C m n ( t ) and S m n ( t ) from the approximation of the d e f l e c t i o n due to bending (equation 4.20), and the de f l ec t ions of the guide arms, 6Z, Gy, and 0 x . The method chosen to s o l v e the g o v e r n i n g e q u a t i o n of the p l a t e was to app ly G a l e r k i n ' s Method to the equation. The G a l e r k i n Method i s a weighted r e s i d u a l method that requires the weighting funct ion to co inc ide with the t r i a l funct ion , so that i n t h i s case W l = R g £ ( r / a ) c ° s W Y ) W 2 = R - £ ( r / a > S i n ( A Y > 35 where g = 0,1,2,...,M I = 0,1,2,...,N a n C* 8^5, i s defined by equation (4.22). Applying Galerkin's Method with weighting functions Wj and W 2 to the governing equation for the plate, (4.11), r e s u l t s in two equations, (1.1) and (1.2) which are given i n Appendix I in non-dimensional form. The equations for the guides are also given. The non-dimensional variables are defined i n the L i s t of Symbols. Equations (1.1) and (1.2), combined with the equations (1.3 - 1.7) for the guides constitute a system of simultaneous d i f f e r e n t i a l equations, the solution of which determines the response of the system. These equations can then be arranged into the following matrix equation: [A] d 2{X} + TB] d{X} + [C]{X> = {Q> (4.24 ) dT 2 dT where T {X} = I{ C 0 0' C01 , C02'*''' C10' C11'" ' ' 'Sra , S01' S02' S03'' • • a 1 ^ I T ...S 1 1,S 1 2,...,S M N,6 z,a6 y a9 x) = { { C ™ n } T ' { S « n } • ( G i } } mn mn 1 mn mn a a a are the generalized coordinates of the system, and s u s {Q} = {L} + Z {\P}Cos(sn0T) + {V 2\sin(sn 0T) s=l f Z { f l c o s d j T) + ( f i s i n ( u f T ) . f=l Notice that elements S m n ( t ) where n=0 have been removed from (x(T)}: these elements are the c o e f f i c i e n t s of Sin(O) i n the series approximation (4.20) and must be removed for the matrix solution to be determinate. The matrix [B] i s not proportional to matrices [A] and [C], The matrices are square and of size (M+1)(N+1) - M + 61. The term {L} represents a s t a t i c load on the blade caused by a static external load and/or by the guide when the blade i s misaligned or i f the blade is dished, i.e. the runout has a component with zero nodal diameters. An equal and opposite force i s also applied to the guides. The terms within the summation i n "s" represent the i n t e r a c t i o n forces between the blade and the guides that are generated because the blade has runout. Also included i n these terms i s the e f f e c t of the centrifugal stresses on the runout. The terms within the summation in " f " represent the harmonic components of a number of external loads applied to the blade. The solution of equation (4.24) i s i n two parts: (1) the homo-geneous solution which can be interpreted as the natural response, and (2) the p a r t i c u l a r solution which has a s t a t i c component, a harmonic component of frequency s ^ which can be interpreted as a self-excited response of the system because i t i s generated by the runout, and a harmonic component of frequency w 0f. which can be interpreted as the forced harmonic response of the system. The solution calculations are given in Appendix II. 37 It i s important and interesting to note that the matrices [B] and [C] are dependent on the r o t a t i o n a l speed of the blade, fiQ . S p e c i f i c -ally, [B] has terms in and [C] terms in fiQ2. The result i s that the response of the system i s also dependent on the blade speed. It should also be noted that matrix [B] i s non-zero even i f there i s no damping in the system because i t contains a gyroscopic matrix as well as damping terms. The e f f e c t of the g y r o s c o p i c matrix i s to couple the equations-^. The forced harmonic response of the system i s f a i r l y predictable: the system w i l l resonate when the forcing frequency equals a natural frequency of the system. However, i n the case of the s e l f - e x c i t e d response, resonance w i l l occur when any integer multiple of the blade speed equals a natural frequency. The computer programs used for solving the equations generated by the model were written i n ANSI FORTRAN and executed on a D i g i t a l Equipment Corp. VAX 11/750 mini-computer equipped with a floating-point processor. Two programs were written, one for the natural response and one for the forced and s e l f - e x c i t e d responses. Flow charts for the programs and the subroutines are given in Appendix III. 38 5. VERIFICATION OF THE MODEL Introduct ion Once the model had been developed i t was tested to determine i f i t c o r r e c t l y and a c c u r a t e l y d e s c r i b e d the dynamic re sponse o f a gu ided c i r c u l a r saw. V e r i f i c a t i o n of the model was done by compar ing the r e s u l t s generated by the model to r e s u l t s from experiments. Some of the r e s u l t s can only be compared q u a l i t a t i v e l y because the saw blades tested i n t h i s work had unknown amounts of tens ioning . Besides provid ing a v e r i f i c a t i o n of the model, the r e s u l t s present-ed i n t h i s c h a p t e r are expanded upon so as to p r o v i d e i n s i g h t i n t o the dynamics of guided saws. Convergence of the Ser ies So lu t ion The assumed s o l u t i o n f o r the d e f l e c t i o n of the b l ade i s g i v e n by E q u a t i o n 4.20. W i t h s e r i e s s o l u t i o n s the sum converges to a l i m i t as more terms are added ( i f the s e r i e s i s convergent). Because Ga lerk in ' s Method was used, the na tura l frequencies of the system are bounded and converge from above. T a b l e I I I g i v e s the d e f l e c t i o n a t the l o a d f o r a s t a t i c c o n c e n -t ra ted load located at the rim of the unguided blade for various values of M and N, the upper l i m i t s i n the summations i n (4 .20) . 39 The displacement converges to a value of approximately u/a = 0.1253. For t h i s simple loading and def l e c t i o n a solution with (N,M) = (2,14), i.e. the highest mode has two nodal c i r c l e s and fourteen nodal d i a -meters, i s adequate but for more complex loadings on the blade, es p e c i a l l y where the blade may be expected to have r e l a t i v e l y sharp bends, more terms are required i f the def l e c t i o n i s to be accurately calculated. Table III Non-Diraensional Displacement at a Concentrated Load = 1.0, b/a = 0.3, £ = 1.0, no guides M N 6 8 10 12 14 0 0.1179379 0.1185079 0.1186578 0.1187023 0.1187171 1 0.1200713 0.1211628 0.1215700 0.1217381 0.1218120 2 0.1225869 0.1239152 0.1244761 0.1247408 0.1248743 4 0.1227538 0.1241681 0.1247941 0.1251066 0.1252745 Table IV gives the f i r s t six natural frequencies for d i f f e r e n t assumed values of M;. the solution i s independent of N because for this unguided case the approximation i s an exact solution for the angular component of the mode shape. Also included in Table IV are numerical results for the natural frequencies as calculated by Mote^. Correlation with Mote's results i s very good. 40 Table IV Non-Dimensional N a t u r a l Frequencies b/a = 0.5, no guides M Mode (m,n) (0,0) (0,1) (0,2) (0,3) (0,4) (0,5) 0 1 2 3 4 Mote 13.02789 13.02515 13.02429 13.02426 13.02426 13.02426 13.29885 13.29188 13.28988 13.28979 13.28980 13.28978 14.73197 14.71194 14.70431 14.70384 14.70378 14.70381 18.60499 18.58487 18.56361 18.56211 18.56206 18.56195 25.64368 25.64321 25.59915 25.59613 25.59600 25.59583 35.86873 35.81487 35.73487 35.73070 35.72991 35.73003 S t a t i c D e f l e c t i o n of an Unguided Blade The purpose of t h i s s e c t i o n i s to show how w e l l the model describes the d e f l e c t i o n o f the b l a d e a t a l l p o s i t i o n s around the b l a d e . To do t h i s the case of a blade loaded w i t h a concentrated f o r c e a t the r i m was i n v e s t i g a t e d . F i gure 12 shows both the experimental and the numerical r e s u l t s of the d e f l e c t i o n of the blade r i m f o r Blade A. The d e f l e c t i o n o f t h e b l a d e a t the l o a d i s p r e d i c t e d q u i t e w e l l by the model, but beyond twenty degrees from the load the accuracy d i m i n i s h e s . The l i k e l y cause f o r the d i s c r e p a n c y i s t h e t e n s i o n i n the b l a d e . When B l a d e B, which has very l i t t l e t e n s i o n , was t e s t e d the r e s u l t s were much improved (see Figure 13), however, the e f f e c t of t e n s i o n can s t i l l be seen. 41 ho 00 CO r r C» r t H -o o (T) o r t H -O a o Ml w M 0) Q -tt> 3 S3 K3 00 0.03 o £ 0.021 o •H U (J a) at o 0.0]\ • Experimental Calculated -180 -90 0 90 Position Relative to the Load 180 Frequency/Speed Relationship for an Unguided Blade Figures 5 and 14 show the experimental frequency/speed plots for Blades A and B, respectively, as determined by experiment. Figures 15 and 16 show the frequency/speed plots generated from the model. The difference between the experimental and numerical r e s u l t s , e s p e c i a l l y the upward shift in the c r i t i c a l speed, is due to the tensioning in the blade which tends to s h i f t the natural frequencies. Other than for tensioning the c o r r e l a t i o n between the experimental and numerical r e s u l t s i s good. It was observed that mode (0,0) of Blade A cannot be excited, possibly because Blade A i s dished. Frequency/Speed Relationship for a Guided Blade Because of the many factors which may affect the guided response, the f i r s t numerical results considered were for a blade supported at the rim by a single spring. The e f f e c t s of the dynamics of the guide arm were removed from the analysis by assigning the guide arm a very large stiffness (10^ N/mm) and a small mass. The natural frequencies produced by the model f o r values of the s p r i n g s t i f f n e s s of 22.88, and 2288.0 N/mm are shown in Figures 17 (a) and (b); the dashed l i n e s correspond to the unguided response. For comparison, the spring rate of the rim of the blade, determined numerically, was 22.88 N/mm. The numerically generated mode shapes associated with the f i r s t eight frequencies for a blade with a single point guide aty = 0 degrees 44 o o o o o o o o o o \& m m CM -H ( Z H) Xouanbaaj Figure 14: Frequency/Speed Plot for Blade B, Unguided 45 o o ( Z H ) XouanbaJj Figure 15: Numerical Solution of the Frequency/Speed Plot for Blade A, Unguided (M,N) = (2,7) 46 o o -a-( Z H ) Xouanbajj Figure 16: Numerical Solution of the Frequency/Speed Plot for Blade B, Unguided (M,N) = (2,7) 47 •3 40 o c S 30 500 1000 Blade Soeecl (rpm) 1500 Figure 17: Frequency/Speed P l o t f o r Blade A Guided by a Single Spring, (a) k;.,- = 22.88 N/mm, "Guided, Unguided (M,N), = (2,7). 48 0 500 1000 1500 Blade Speed (rpm) F i g u r e 17: Frequency/Speed P l o t f o r Blade A Guided by a S i n g l e Spring, (b) k - H = 2288.0 N/mm. Guided, Unguided (M,N), = (2,7). 4 9 are shown in Figure 18. The existence of nodal rad i i , determined exper-imentally, i s confirmed. The odd shaped mode at 40.8 Hz i s a mode corresponding to the natural frequency of the guide arm (the guide was assigned a reasonable mass and s t i f f n e s s for t h i s calculation). The modes with an even number (2n) of nodal radii have identical frequencies to the modes i n the unguided blade with n nodal diameters. Note that the mode shapes at 27.0, 30.0 and 46.4 Hz. do not have a true node i n the blade under the guide; instead, the displacement i s a local minimum, but not zero. The shape of the frequency/speed plot of the guided case i s deter-mined by the. c h a r a c t e r i s t i c s of the unguided blade. For k-jj = 22.88 N/mm (Figure 17 (a)) modal coupling i s evident and each frequency curve follows the curve of the unguided blade but changes d i r e c t i o n to follow the frequency of a di f f e r e n t mode at the i n t e r -section points. The frequency curves of the guided case pass through a l l the intersection points of the unguided frequency plot, i.e. through a l l the points at which two modes of the unguided blade have the same natural frequency. The curve veering i s even more pronounced for a very s t i f f guide of k-jj = 2288.0 N/mm (Figure 17 (b)). Notice that the c r i t i c a l speed i s e s s e n t i a l l y unaffected by the presence of a single point guide, regardless of the guide stiffness. The mode shape of the guided blade at these intersection points consists of the two modes which exist at that speed and frequency in the unguided blade. The modal coupling can also be seen in the contributions of the terms C m n to the displacement of a blade with a single guide. Figures 50 51 Figure 19: Contributions of Terras C m n t o the Total Displacement for Blade A, (a) k ± j = 0.2288 N/mm r o f = 1.0, Ynf = 0.0, r> • = 1.0, Y i i = 3 4 - 4 degrees, P = 100 N. (M7N) = (2.8) J 3 52 53 19 (a) and (b) are for the cases corresponding to a blade with guide s t i f f n e s s e s k j j = 0.2288 N/mm and 2.288 N/mm, respectively; a l a t e r a l load of 1.0 N i s applied 34.4 degrees from the guides. For k-y = 0.2288 N/mm the modal contributions are similar to those of an unguided blade. The peak in C 0 >3 at 1380 rpm i s the f i r s t c r i t i c a l speed of the blade, as shown in Figure 15. In the cases of k^- = 2.288 N/mm the magnitude of the terms C r a > n below the c r i t i c a l speed decrease indicating that in this speed region the displacement of every point on the blade i s reduced by the presence of the guide. The next stage i n determining the model parameters required to describe a guided blade was to investigate the eff e c t of the shape of the guide pad and to assign a stiffness to the springs representing the f l u i d lubrication. The shape of the guide pad was represented by four springs, arranged at the corners of the pad. A pad of nine springs, located at the corners, midpoints and centre of the guide pad was also tested but the results were not significantly different from those of a pad of four springs. The stiffness and natural frequency of the guide arms are 814.0 N/mm and 50 Hz, respectively (see Chapter 3): the resultant equivalent mass of the guide arms was 8.24 Kg. To determine the effect of the lubrication stiffness on the system response frequency/speed plots were produced for k i j of 8.14, 81.4 and 814.0 N/mm which are shown i n Figures 20 (a), (b) and (c). The most noticeable effect of increasing k i j i s to straighten, and slightly shift the frequency curves. Comparison of Figure 20 (b) or (c) with the experimental results for a guided blade given in Figure 6 shows that the model i s capable of predicting the natural response of a guided blade i f 54 ( Z H ) Xouanbsaj F i g u r e 20: Numerical S o l u t i o n of the Frequency/Speed P l o t f o r Blade A guided wit h a Pad Modelled by Four Springs, (a) k ± 1 = 8.14 N/mm (M.N) i (2,7). 55 o o o o o o o o r-~ \D m m cs »-i (ZH) Xouanbaaj F i g u r e 20: Numerical S o l u t i o n of the Frequency/Speed P l o t f o r Blade A guided w i t h a Pad Modelled by Four Springs, (b) k±, = 81.4 N/mm (M,N) i (2,7). 56 (ZJI) Xouanbajj Figure 20: Numerical Solution of the Frequency/Speed Plot for Blade A guided with a Pad Modelled by Four Springs, (c) k. • = 814.0 N/mm. (M,N) i (2,7). 57 the lu b r i c a t i o n s t i f f n e s s could be determined beforehand. I t i s s i g n i -f i c a n t , based on a comparison of the experimental and numerical results, that the s t i f f n e s s of the l u b r i c a t i n g f l u i d i s one or two orders of magnitude greater than the s t i f f n e s s of the blade and approximately of the same order as the s t i f f n e s s of the guide arm (814.0 N/mm). Stat i c Loading of a Rotating Blade The e f f e c t of r o t a t i o n on the dynamic s t i f f n e s s of the blade i s presented i n t h i s s e c t i o n . Because a s t a t i c load i s known to e x i s t during c u t t i n g , t h i s s i t u a t i o n i s used to t e s t the forced response generated by the model. To apply a s t a t i c load to the blade an electromagnet, powered by a d i r e c t current power supply, was used. The forc e generated by the electromagnet i s proportional to the square of the current and inversely proportional to the square of the length of the gap between the face of the electromagnet and the blade. A non-contacting displacement probe was used to measure the blade displacement and the change i n the gap length. Since the output from the computer model i s the displacement of the blade caused by a constant force, i t i s convenient for comparison of the experimental and numerical results to calculate the f l e x i b i l i t y of the blade, i.e. the inverse of the blade s t i f f n e s s , I d_ Cdg2 k F I 2 58 where k = blade stiffness (N/mm) d = blade deflection (mm) F = force from electromagnet (N) g = gap length (mm) I = current (amperes) c = proportionality constant dependent on magnet geometry = 7.801(10"3) amp2/N-mm2 Figure 21 (a) i s a plot of the f l e x i b i l i t y of Blade B as a function of blade speed. The load was applied at r' = 0.72, 45 degrees from the centre of the guide pad. The pad shape was modelled by four springs, as was done for Figure 20. As c r i t i c a l speed i s approached the displace-ment increases; recall that the c r i t i c a l speed i s a resonant condition caused by a static load. The numerical results of a blade loaded with a s t a t i c load of 1.0 Newton (N) i s given i n Figure 21 (b): the 1.0 N load was chosen so that the output would be the f l e x i b i l i t y of the blade in mm/N. It was concluded from comparing the experimental and numerical r e s u l t s that the model c o r r e c t l y predicts the displacement of an un-guided blade. The stiffness of a rotating untensioned blade can be considered to have two components; dynamic, or structural stiffness, and stiffness due to rotational stresses. For speeds near c r i t i c a l the dynamic stiffness goes to zero, r e s u l t i n g in the peak i n the f l e x i b i l i t y curve. For speeds below c r i t i c a l the counteracting trends, as the speed i s i n -59 a) Experimental 500 1000 1500 Blade Speed (rpm) Figure 21: F l e x i b i l i t y of Blade B as a Function of Rotation Speed. (M,N) = (3,8). 60 creased, of decreasing dynamic s t i f f n e s s and i n c r e a s i n g r o t a t i o n a l s t i f f n e s s results i n the f l e x i b i l i t y of the blade being insensitive to blade speed. Previous workers 0* have e s t a b l i s h e d from sawing t e s t s that the c u t t i n g accuracy of a c i r c u l a r saw i s poor i f the blade i s operated at speeds above 85% of the c r i t i c a l speed (ft c r ) . Figures 21 (a) and (b) c l e a r l y show that the f l e x i b i l i t y of the blade increases rapidly i n the region above 0.850 ftcr. Also, during the experiments with the electro-magnet at speeds near ft c r i t was observed that when the force was applied or removed the transient response of the blade was a very slow, but l a r g e , o s c i l l a t i o n : the o s c i l l a t i o n was slow because f o r speeds near the lowest natural frequency of the blade i s almost zero. I f a blade o s c i l l a t e s during cutting as was observed i n these tests, i t i s not s u r p r i s i n g that the blade f o l l o w s a meandering path through the lumber, termed "snaking" by the sawmill industry, which produces poor cutting accuracy or possible catastrophic f a i l u r e of the blade. Runout as a Function of Rotation Speed As mentioned i n Chapter 3, the runout of the blade interacts with the guides i n a possibly detrimental manner, but before t h i s interaction can be studied the c h a r a c t e r i s t i c s of runout must be understood. Because of the i n - p l a n e s t r e s s e s due to blade r o t a t i o n , the blade i s being stretched, with the res u l t that i t becomes f l a t t e r as the speed i s increased. 61 As described i n Chapter 4, runout can be written as a trigonometric F o u r i e r s e r i e s (Equation 4.8). This f o r m u l a t i o n i s convenient f o r experimental work because each of the Fourier c o e f f i c i e n t s can be deter-mined with the use of a displacement probe and a spectrum analyser; the frequency of the s-th Fourier c o e f f i c i e n t i s s times the blade speed. A t y p i c a l root-mean-square (RMS) spectrum of the runout of Blade A i s shown i n Figure 22. Figure 23 (a) shows the change i n the F o u r i e r c o e f f i c i e n t which has a frequency equal to the blade speed, Figure 23 (b) f o r the c o e f f i c i e n t having a frequency of t w i c e the blade speed. The numerical results also plotted i n Figure 23 were generated by enter-ing the values of the runout spectra for very low blade speeds, i.e. the s t a t i c or i n i t i a l runout, into the program: the output i s the change i n runout due to bending, which i s subtracted from the i n i t i a l runout to obtain the runout at speed. The correlation between the numerical and the experimental re s u l t s i s quite good, however at certain speeds the experimental results show a resonant behaviour. At 12 Hz (720 rpm) the blade speed equals a natural frequency i n the blade (see Figure 5): presumably the e x c i t a t i o n at t h i s speed was due to r o t a t i o n a l imbalance. At 22.5 Hz (1350 rpm), which i s the c r i t i c a l speed, another peak i s present; p o s s i b l y aero-dynamic forces were responsible. An important consequence of reduced runout with increasing speed i s the r e d u c t i o n i n the width of the kerf cut by the blade as w e l l as the reduction i n the interaction forces betwen the blade and the guides. 62 63 • Experimental — Numerical 10 20 30 • Experimental Numerical (b) 10 20 Blade Speed.(rev/sec) 30 F i g u r e 23: RMS Runout as a Function of Blade Speed, (a) Sine (0) component, and (b) Sine (20) component. (M,N) = ( 2 , 8 ) , Blade A. 64 Blade and Guide Motion as a Function of Rotation Speed As a result of experiments described in Chapter 3 i t was discovered that the interaction of the blade runout with the guides causes a s e l f -excited vibration in the system at speeds where an integer multiple of the blade rotation speed equals the natural frequency of the guide arm. Possible e x c i t a t i o n of the blade by the same mechanism was also men-tioned. To obtain complete information on the self-excited motion, the RMS spectra of the guide motion and the motion of the blade 45 degrees from the guide were measured. The method is the same as described previously for measuring the RMS spectrum of the runout. The RMS amplitudes of the blade and guide motion for the f i r s t and second multiples of the blade rotation speed are shown in Figure 24 (a) and (b) (measured at intervals of 30 rpm). In Figure 24 (a) the peaks i n the blade motion at 750, 1080, 1370, and 1670 rpm a l l correspond, in Figure 6, to speeds at which the blade speed equals a natural frequency of the system. The peaks in Figure 24 (b) at 470, 630, 870, 1150 and 1400 rpm correspond to speeds at which a natural frequency is twice the blade speed. A similar self-e x c i t a t i o n occurs i f blade misalignment, which can be thought of as a once per revolution runout, were to be present. The resonance i n the guides at 1540 rpm i n Figure 24 (b) corres-ponds to the second multiple of the blade speed being equal to the natural frequency of the guide arm. It i s s i g n i f i c a n t that the guide motion due to the s e l f - e x c i t a t i o n for speeds near 1540 rpm, i s i n the 65 o o LO o o o B a. u 13 CD <u p. to a) •V CO 1-1 o o C N o (ram) apn^Txdrav SKR Figure 24: RMS Amplitudes of the Self-Excited Motion of Blade A and the Guide as a Function of Speed; (a) F i r s t Multiple 66 0J CO o) pa (ram) sprinT-rdmy SWH Figure 24: RMS Amplitudes of the Self-Excited Motion of Blade A and the Guide as a Function of Speed; (b) Second Multiple of the Blade Speed. 67 order of 0.05-0.10 mm (0.002-0.004 inches). The r e s u l t s of a numerical a n a l y s i s of the s e l f - e x c i t a t i o n are shown i n Figure 25 (a) and (b); the corresponding frequency/speed p l o t i s shown i n Figure 20 (b). I t can be seen that the model c o r r e c t l y predicts the existence of speeds where large amplitudes of o s c i l l a t i o n i n the blade and the guides occur. 68 i I i LI o o o o C M ° (ram) aprQT-rdrav SKH ° Figure 2 5 : Numerical Solution of the Self-Excited Motion of Blade A and the Guide; , . (a) F i r s t Multiple AQ ^ = 0 . 2 2 4 mm Four-point Guide, k^'- 8 1 . 4 N/mm. (M,N) = ( 3 , 7 ) . 69 0) 73 CO iH 03 Q) •r-t 3 1 I i Lf o CM 0 O O O o" (ram) spn^jxcirav SKH Figure 25: Numerical Solution of the Self-Excited Motion of Blade A and the Guide; (b) Second Mutiple of the Blade Speed, A Q 2 = 0.0176 mm. Four-point Guide, k ± j = 81.4 N/mm. (M,N) = (3,7). 70 6. NUMERICAL RESULTS In t h i s chapter results generated by the model are presented, but not v e r i f i e d . The cases analysed were chosen because of the i r import-ance to the designers of guided c i r c u l a r saws. Although i t i s important to know how the guides a f f e c t the n a t u r a l frequencies of the guided system, the more important i n f o r m a t i o n i s the e f f e c t of the guides on the deflection of the blade i n the cutting region. I t should be remem-bered that the r e s u l t s are f o r a f i x e d c o l l a r blade, and not f o r a splined-arbour blade. Effects of Guide Position To i n v e s t i g a t e how the p o s i t i o n of a guide r e l a t i v e to a load a f f e c t s the d e f l e c t i o n at the load, a point guide was p o s i t i o n e d at various places r e l a t i v e to a s t a t i c l a t e r a l load applied at the rim of the blade. Figure 26 shows the expected r e s u l t that the best r a d i a l p o s i t i o n f o r the guide i s at the blade rim. Figure 27 shows that the best angular p o s i t i o n f o r the guide i s a point as c l o s e as p o s s i b l e to the load. The dashed l i n e s i n Figure 27 are the d e f l e c t i o n s f o r an unguided blade. Simply stated, the best p o l s i t i o n f o r a guide i s as c l o s e as p o s s i b l e to the a p p l i e d load. In f a c t , i t i s apparent trom Figure 27 that the guide i s e f f e c t i v e only i f i t i s within s i x t y degrees of the load. 71 0.003 cfl 3 C o o r-i «W <D Q e •H Pi 0.002 • 0.001 C o l l a r •777777777} 0.2 0.4 0.6 0.8 Radial P o s i t i o n of Guide r/a 1.0 Figure 26: Blade Deflection at the Load as a Function of the Radial Position of the Guide Relative to the Load, b/a = 0.3, ? = l.O.flo = 5.0 (M,N) = (1,7). I = 100.0 72 Position Relative to the Load Figure 27: Blade Deflection at the Load as a Function of the Angular Position of the Guide Relative to the Load, b/a = 0.3, C = 1.0 (M,N) = (1,7). £ = 100.0 73 Effect of Guide Shape Guide shape i s important because the guide can r e s t r a i n how the blade t w i s t s as w e l l as how i t d e f l e c t s l a t e r a l l y . Figure 28 shows three pad shapes used f o r the numerical t e s t s . The s t i f f n e s s and location of the springs k j j are also shown i n Figure 28. In addition, a point guide (modelled as a single spring support) and an unguided blade are also tested. The results are presented i n Figure 29 which shows the shape of the blade rim for the different guide shapes and two rot a t i o n a l speeds, Q Q = 2.5 and 5.0. The c r i t i c a l speed i s approximately ft o • 5.6. The most s t r i k i n g outcome of these t e s t s i s the e f f e c t i v e n e s s at ft = 2.5 of using only a point guide; the maximum blade d e f l e c t i o n i s reduced f o r t y percent from the unguided case and the use of the s m a l l guide (a) reduces the maximum d e f l e c t i o n another t h i r t y percent. At n 0 = 5.0, where the deflections of the blade are greater, the effect of the guide shape i s more important. Although the point guide does r e s t r a i n l a t e r a l d e f l e c t i o n i t does not stop the blade from p i v o t i n g about the point where the support i s applied; the guides that have some area are able to r e s t r a i n the pivoting of the blade with the result that the maximum deflection i s further reduced. Effect of Multiple Guides The c r i t i c a l speed of a blade l i m i t s the maximum speed at which a blade can be operated: higher speed i s desirable because the resulting 74 75 76 cut surface w i l l have s m a l l e r tooth marks. I t was shown i n Chapter 5 that f o r a s i n g l e guide the c r i t i c a l speed i s e s s e n t i a l l y the same as for an unguided blade, however the e f f e c t of mounting s e v e r a l guides should be considered. The number and p o s i t i o n of guides required to produce a s h i f t i n the c r i t i c a l speed depends upon the modes of v i b r a t i o n e x c i t e d at c r i t i c a l speed. Figure 30 shows that the presence of s i x equispaced guides does not have a l a r g e e f f e c t , whereas the presence of eigh t equispaced guides r e s u l t s i n a s i g n i f i c a n t change, as shown i n Figure 31. I t was noted previously that i t i s the (0,3) mode which vibrates at c r i t i c a l speed. The presence of s i x guides a l l o w s the three nodal diameter mode to exist whereas the eight guide configuration forces only the (0,4) mode to e x i s t ; the r e s u l t i n g c r i t i c a l speed can be pre d i c t e d from the (0,4) c r i t i c a l speed of the unguided blade (see Figure 5). 77 500 1000 Blade Speed (rpm) 1500 Figure 30: Frequency/Speed Plot for Blade A with Six Equally Spaced Guides. (M,N) = (1,7). 78 1 2 0 r 0 500 1000 1500 Blade Speed (rpm) Figure 31: Frequency/Speed Plot for Blade A with Eight Equally Spaced Guides. (M,N) = (1,7). 79 7. CONCLUDING REMARKS A model has been presented i n th i s thesis for analysing the dyn-amics of unguided and guided fixed collar circular saws. An important aspect of guided (and unguided) circular saws i s the dependence of the dynamic characteristics on the rotational speed. As a result of this speed dependence, many problems associated with resonance in the guided system can be avoided by changing the speed. The converse also applies: the performance of a saw can become worse as the result of a speed change. The natural response of a guided saw i s dependent primarily on the dynamics of the blade with the c h a r a c t e r i s t i c s of the guide having a secondary e f f e c t . The effect of the guides i s to couple the modes of vibrat i o n i n the blade. For a blade supported by a point guide the natural frequencies at any speed can be determined d i r e c t l y from the frequency/speed plot of the unguided blade: the frequency curves of the guided case pass through a l l the intersection points of the unguided frequency/speed plot. For guide pads that have some area, the modal coupling continues to occur but the frequency curves are shifted slight-l y upward and are straightened. The mode shapes of the guided blade have nodal r a d i i rather than nodal diameters, with one node or l o c a l minimum under the guide, except when the guide resonates, in which case, an anti-node occurs at the guide. The forced response of a guided saw i s required i f cutting accuracy 80 and kerf losses are to be predicted. Solutions for s t a t i c and harmonic concentrated loads are presented, but because a major component of the l a t e r a l cutting force i s s t a t i c , t h i s case was investigated i n d e t a i l . As i n a l l mechanical systems, resonance occurs i n a guided saw when the forcing frequency equals a natural frequency of the system. Because of the dependence of the blade dynamics on r o t a t i o n speed, there i s the p o s s i b i l i t y of resonance o c c u r r i n g when a s t a t i c load i s ap p l i e d . As the speed of a blade subjected to a s t a t i c load i s increased, two counteracting e f f e c t s occur: the dynamic s t i f f n e s s of the blade i s decreasing because a resonance at ft = ftcr i s being approached, but at the same time the st r e s s e s due to r o t a t i o n are s t i f f e n i n g the blade. The result i s that below 0.85ft c r the s t i f f n e s s of the blade i s insens-i t i v e to speed, however, near c r i t i c a l speed the dynamic s t i f f n e s s vanishes and large deflections result. To improve the q u a l i t y of the cut surface of the lumber, the saw designers seek methods of increasing blade speed, but the c r i t i c a l speed l i m i t s the maximum operating speed. A possible method of increasing the c r i t i c a l speed i s to use multiple guides. A single guide has e s s e n t i a l -l y no e f f e c t on the c r i t i c a l speed. I t was found that to s h i f t the c r i t i c a l speed upwards at l e a s t eight equispaced guides are required. Six equispaced guides do not s h i f t the c r i t i c a l speed because they allow the three nodal diameter mode of the blade, which i s the mode that resonates at the f i r s t c r i t i c a l speed, to e x i s t . The eight guide configuration forces only the four nodal diameter mode to exi s t , which has a higher c r i t i c a l speed. Although a single guide does not s i g n i f i c a n t l y change the c r i t i c a l 81 speed of a blade, the guide does re s t r a i n blade deflection. However, i f the guide i s more than s i x t y degrees from the point of a p p l i c a t i o n of the load, the d e f l e c t i o n i s e s s e n t i a l l y the same as f o r an unguided blade. The guide does more than r e s t r a i n l a t e r a l motion, i t a l s o r e -s t r a i n s the blade from p i v o t i n g about the region where the support i s a p p l i e d , w i t h the r e s u l t that the d e f l e c t i o n i s f u r t h e r reduced: the guide shape i s important i n t h i s respect. A self-excited vibration i s generated i n the guided system because of runout and misalignment of the blade r e l a t i v e to the guide. The i n t e r a c t i o n of the runout w i t h the guides causes equal and opposite forces to act on the blade and the guide. This i n t e r a c t i o n force has frequencies that are integer multiples of the blade speed. I f a s e l f -excitation frequency equals a natural frequency of either the blade or the guide arm, a resonant condition i s produced, which results i n large d e f l e c t i o n s of the blade and/or the guide arm. I f the speed i s such that a self-excited resonance occurs, then poor cutting accuracy as wel l as high rates of guide wear can probably be expected. However, at high blade speeds the stresses due to rotation tend to decrease the runout so that the magnitude of the interaction forces i s reduced. Other forms of e x c i t a t i o n are thought to be generated by imbalance or aerodynamic forces. 82 Suggestions for Further Work To make the numerical model more r e a l i s t i c and useful, i t needs to be able to calculate the response of blades with spline-arbour centres. The a b i l i t y to model tensioning and thermal stresses i s also required. The f i n a l aspect of t h i s p r o j e c t w i l l be to c o r r e l a t e blade dynamics, found numerically or from experiments, to cutting performance. Another aspect of the numerical work i s a s t a b i l i t y a n a l y s i s . Mote^ and others^' ^ have shown that a blade guided by a point guide i s unstable for speeds above the f i r s t c r i t i c a l , i.e. the r e a l part of some of the eigenvalues of equation (II.3) are positive. I t should be noted, however, that no i n s t a b i l i t i e s were observed while conducting experi-ments for t h i s thesis (see Figure 6). 83 NOTES 1. Szymani, R., Technological Aspects of Thin-kerf Circular and Band Sawing - A Review of Claassen's System of Saw Selection, Complete  Tree U t i l i z a t i o n of Southern Pine, Symposium Proceedings, New Orleans, LA, Forest Products Research Society, p. 345, 1978. 2. Mote, CD., "Confirmation of the C r i t i c a l Speed Theory for Sym-e t r i c a l C i r c u l a r Saws", Journal of Engineering for Industry, Vol. 97(B), No. 3, p. 1112, 1975. 3. Mote, CD., "Moving Load Stability of a Circular Plate on a Float-ing Central Collar", Journal of the Acoustical Society of America, Vol. 61, No. 2, p. 439, 1977. 4. Iwan, W.D. and Stahl, K.J., "The Response of an E l a s t i c Disk with a Moving Mass System", Journal olf Applied Mechanics, Vol. 40, p.455, 1973. 5. Benson, R.C and Bogy, D.B., "Deflection of a Very F l e x i b l e Spin-ning Disk Due to a Stationary Transverse Load", Journal of Applied  Mechanics. Vol. 45, p. 636, 1978. 6. Iwan, W.D. and Moeller, T.L., The Stability of a Spinning Disk with a Traverse Load System, Journal of Applied Mechanics, Vol. 43, p.485, 1976. 7. Hutton, S.G;, Chonan, S., and Lehmann, B.F., Dynamics of Guided Circular Saws, Submitted to Journal of Sound and Vibration, 1984. 8. Mote, CD. and Szymani, R., Principle Developments in Thin Circular Saw Vibration and Control Research, Holz als Roh und Werkstoff, Vol. 35, pp. 189-196, 219-225, 1977. 9. Prescott, J., Applied E l a s t i c i t y , Dover Publications Inc., New York, p. 587, 1961. 10. Nieh, L.T. and Mote, CD., "Vibration and S t a b i l i t y i n Thermally Stressed Rotating Disks", Experimental Mechanics, Vol. 15, No. 7, p. 258, 1975. 11. Mote, CD., "Free Vibration of I n i t i a l l y Stressed Circular Disks", Journal of Engineering for Industry, Vol. 87(B), No. 2, p. 258, 1965. 12. Timoshenko, S. and Woinowsky-Kreiger, S., Theory of Plates and  Shells, 2nd Ed., McGraw-Hill Book Co. Ltd., Toronto, pp. 407-408, 1959. 13. Timoshenko, S. and Goodier, J.N., Theory of E l a s t i c i t y , 2nd Ed., McGraw-Hill Book Co. Ltd., Toronto, p. 70, 1951. 14. Timoshenko, S. and Woinowsky-Kreiger, S., Theory of Plates and  Shells, 2nd Ed., McGraw-Hill Book Co. Ltd., Toronto, p. 393, 1959. 84 15. Rocard, Y General Dynamics of Vibrations. Crosby Lockwood & Sons L t d . , p.145, 1960. 1 6 * i 9 6 7 ° V i t c h ' L " Analytical Methods in Vibrations. MacMillan, p.422, U ' n 1!?' G , n , > - F i n i t e E l e m e n t Analysis of Rotating Disks. M p s t P r ' thesis, University of British Columbia, 1980. s 85 APPENDIX I  EQUATIONS OF THE ANALYTICAL MODEL Equations for the pl a t e . ..?-{(N<"V + N (f>d + N (* 3 )) 1 C „ + (N (f >d2 + N (f >d + N ( f >) 1 S J glm -gp- gim — g£m ' - ml + I { ( M ( „ 2 ) d + M ( „ 3 ) ) 1 6 1 , + ( M ^ d + M ( ^ } ) 6 1. i-1gZ± dT g U a 2 1 g £ l d f g U y i + (M (">d + M (* 9 )) 6% + (G (* 2>d + G(*3>) 1 6 2 . g U g£i x i gAi — g U 7 - z i + <G<J?>d + G<">> 6% + (G^?>d + G (">) 6%} 8 d f g y 8 d f g 1 x i = L S ? + l± {QflS'Co8(wofT) + Q g f f ) s i n ( w o f T ) } f I {Q (^ 3 )Cos(sft T) + Q ( p 4 ) S i n ( s f t T)} s = 1 9*-s o 3&s o + {Q^ 5 ) C o s ( f t o T ) + Q ^ 6 ) S i n ( f t o T ) } + {Q^ 7 )Cos(£ft oT) + Q^ 8 )Sin(£ft oT)} g = 0,1,2,...,M I = 0,1,2,...,N Equation (1.1) k=l (1.2) k=2 86 E q u a t i o n s f o r t h e g u i d e s . M N Z I { ( N ( k 2 ) d + N ( k 3 ) ) 1 C + ( N ( k ^ d + N ( W ) ) 1 S } m=0 n=0 ^ df m n i ^ 7 mn « i nmi - mn + (M , ( k l ) d + M<k2)d + M, ( k 3 )) 1 &K 1 df" 1 df a + <M<k4)d + M<k5)d + M< k 6 )) 6 1 , 1 df" 1 df 1 y ± + (M , ( k 7 ) d + M<k8)d + M f k 9 ) ) BK 1 dT 1 df + <G<kl>d + G<k2>d + G<k3>) 1 &K 1 df" 1 df ~ Z 1 + <c<k4)d + c<k5>d + Gfk6>) e 2 1 df" 1 df 1 y1 + (G<k7>d + G< k 8 )d + G< k 9 )) BK 1 df" 1 df 1 g - J . 0 0 + I { Q . ( k l ) C o s ( s f t T ) + o j k 2 ) S i n ( s f t T ) > i i i s o i s o + Q ^ k 3 ) C o s ( n Q T ) + Q ^ k 4 ) S i n ( « o T ) i - 1 ,2,3, . . . I E q u a t i o n (1.3) k-3 (1.4) k«*4 (1.5) k-5 (1.6) k«6 (1.7) k=7 (1.8) k=8 87 The fo l lowing terms are de f ined : T $, = ^ v V r , » V r , ) ( r ' ) d ( r , )  T$. - fl< v ( r , ) v ( r , ) ( r ' ) d ( r , ) dim b i»e r a( r . ) " T^+scT^) "cuTT" £ 2 a " V " R m/>( r')> ( r ' ) d ( r ' ) 3 b g l r , - ^ r y a £ 1 2 £ = 0 £ ^ 0 £ = 0 Ax! . r^Cos(Y ± ) - C o s ( Y ± j ) A y ! . = r ! S i n ( Y . ) - r j . S i n ( y . . ) B a bm f-)2 + PS — i 2 r ps -1 = tan l a / IA 1 ^ m . .+ 1 2 ' ^bm -1 = tan / A ( S ) \ ps / 88 Terms for equations (1.1) and (1.2) N ( , U ) = T ( 2 ) TO, glm glm t " Jo J i j l ' W ^ ' w ^ ^ v 0 - ^ ^ ' • I ix Jx 2wr^ ( r«>c° s<^c°B<"v • Jo A jx 2v™(r;j>vr^CosWvsin(nv h$} - j0 i , i , 'w^v^^^-v N ( 2 1 ) - 0 n ^ • I L 3 i 2 n i jv ( r i j ) R ^ < r « ) s i n « Y i J > c o s ( " i ' i J ) • Jo i i A 2 n i jv < r i3 ) v ( r« ) s i n« Y i 3 , s i n < n Y i j ) 89 r ? M N i J N V „ ; = Z E E 2C..R (r!.)R „(r!.)Sin(£y..)Sin(ny..) M3a - -31 V s * < r i J ) C o , " r U > M*-?* - E n..R «(r!.)COS(£Y..) Ay' - j x 5i3v ( ri: , c° s < 1v HSa - -jj Vsi(rij)S1,,WV Msa W r« , s l" wV Mga - W« ) S i" aV ta« M ( 2 6 ) = - E C..R „(r! .)Sixi(£y. •) Ax!. j = 1 ij fl-t i j T i j ' i j M2? = j x T i i j V ( r i J ) s i n ( £ Y i j ) Ayij M g f i } = £ ? i j V ( r i j ) S i n ( £ Y i J ) A y i j 90 L$ = X C s f V ( r o f ) C ° s ( £ Y o f ) I J p - Z . I Z 2? R ( r ' )R ( r ! )B C O S ( £ Y . . ) i = l 1 = i p=o i j 9 1 i j p 0 1 : 1 -£P_ i J J K a I J " ^ ^ S ^ i J ^ ^ V ^ ' + ( r 1 . ) ( A ^ ) C o s ( Y i . ) + A ^ ) S i n ( Y . . ) ) } a a + I 2H T S ) . A ( J ) ' 0 90p _ D O a • I J p - Z Z Z 2? R »(r!.)R _(r!.)B . S i n ( £ Y 4 . ) i = 1 j = 1 p = 0 i j i j pO l j _ D O T i j a " £ ^  2Vs* (V s l n UV { A , + ( ri^ a a (13) P I J Qfl/«i Z Z Z R n / r < - > R <r'->B Cos (Ay..) 9 £ S P=o i = l i = l 9 1 1 3 P s 1 3 -2 s- i J a x { 2 ? i . C o s ( s Y i . - ^ ) ) + 2 s Q A . S i n ( s Y i . - ^ ) ) } (14) P 1 J V . ; ~ 1 E Z R « / ( r < V R C ( r ' - > B Cos(£Y..) p=o i = l i = l 9 1 i j P s U -25. ' i j J a x {2 q . S i n ( s Y i j - ^ s 1 ) ) - 2 s f t o n i . C o s ( S Y . . - ^ ) ) } a 1 3 a x {2Z± .Cos(Y . . - ^ ( 2 ) ) + 2 q p l j S i n ( Y . . - ^ 2 ) ) } ] Q g f } - Sin(m^^  " j j V ( r i J ) C ° S ( £ Y i J ) ( r l j } a J a 9 1 a P a Q™= ^ c ) v ( r o f ) s i n ^ Y o f > • i2S- 4 s )v ( rof ) s i n (^ 0f) (23) I J P a x {2L COS(SY,.-/^) + 2sQ r\. .Sin(sy. .-ii) ( 1 ))} (24) 1 J p Q is  = ~  1 1  Z R n / r ^ ) R n c ^ ! ) B SinU -Y . . ) i=l j=l p=0 3 • l j P s A J -Es i j a x {2C i.Sin(sY i.-^ s 1 )) - 2sft o n..Cos(sY i.-^ s 1 ))} n ( 2 5 ) - T7R T ^ 4 ) C - t ,(2) N a I J ~ jf j . !|n> R g £(r^.) ( r ^ ) Sin(£ Y i j) x ^ . . C o s ^ . - ^ ) + 2fton i.Sin(Y. j-^ ))} 0 (26) _ (4) (2) , a I J " Z Z B KTn R n ^ ( ^ ' - ) <r'-> Sin(£Y ) i=l j = 1 _bH 9* l j i j r i / x {2t: S i n ( Y - ^ 2 ) ) - 2« n..Cos(Y..-^ ( 2 ))} i j i j bm 0 i j ' i j *bm 7 n ( 2 7 ) = y TTR T ( 3 ) c< r , i C D x V P : 0 ^ V P S l n ( , J ;p > (WO) a Q ^ 8 > = 1 !E£ TS C O S ( * P 1 } ) (WO) a 92 Terms for equations (1.3) - (1.8) N ( 3 2 ) = mni j f i n i j W r J j ) C o 8 ( n V N (33) _ mni Z £..R ( r ! .)Cos(ny, .) x ^ i j mn i j • ' i j N ( 3 5 ) -mni Z n . . R ( r ! . ) S i n ( n y . J j = 1 'i;j mnv i j ' N (36) _ mni Z £..R (r!,)Sin(ny..) x s i j mn I J ' ' i j ' N ( 4 2 ) = mni N (A3) mni Z n..R (r!.)Cos(ny..) Ax!. i j mn xj ' i j ' i j J Z £..R (r!.)Cos(ny..) Ax!. j = 1 i j mn i j ' i j i j N (45) mni Z n..R (r!.)Sin(ny,.) Ax!, i j mnv i j ' ' i j ' i j N ( 4 6 ) = mni Z I. .R (r! . ) S i n ( n y J . ) Ax!. ^ i j mnv I J ' ± J i j N ( 5 2 ) -mni Z T). .R ( r ! .)Cos(ny. .) A y ! . ^ i j mn i j ' ' i j ' Jx3 N ( 5 3 ) -mni Z £. .R (r.'.)Cos(ny. .) A y ! . j = 1 ^ i j mnv i j ' i j ' ' i j N ( 5 5 ) -mni Z n..R (r!.)Sin(ny..) A y ! . ^ ' i j mnv i j ' ' i j ' ^ i j N ( 5 6 ) = mni Z K..R (r!.)Sin(ny,.) A y ! . X3 mnv i j ' i j J i j N ( 6 2 ) -mni N (63) = mni N ( 3 2 ) mni N ( 3 3 ) mni N (72) _ „(42) mni mni N ( 6 5 ) = mni N< 6 6 ) = mni N (35) mni N (36) mni mni N ( ? 3 ) = N ( 4 3 ) mm N ( 7 5 ) = N (^> mni mni N ( 7 6 > = N ( 4 6 ) mni mni N ( 8 2 ) = N ( 5 2 ) mni (83) _ N N mni (85) _ = N mni = N N (86) _ mni = N mni (53) mni (55) mni (56) mni 93 (31) _ -(61) i _ G i z i M < 3 2 ) = G < 6 2 ) x i n . + E n . . 2 1 j - i v M M (33) i (34) = G = G (63) m i (64) _ K J + E £. . z i J-1 1 J M ( 3 7 ) = G < 6 ? ) x x M (35) G(65) x E n 4 .Ax! . J-1 ± J 1 J M(36) _ (66) M i " G i E £..Ax!. J-1 1 J 1 J M ° 8 ) - Gf 8 ) x i - E n , . A y ! , j - l i j 1 J M (39) = (69) x i - £ 5, -Ay! . J-1 ± J 1 J M ( 4 D = G(7D . M(47) = G(77) i x x x = 0 M ^ 2 ) x = G (72) _ E nj.Ax!. j - i 1 J 1 J M ( " ) . G (73) x i £ .Ax! . J-1 ± J ± J M(44) = (74) x i y i M (45) = G 1 7 5 > - " y i + £ "l]toij>2 M (46) = G (76) K , + EC..Ax!.Ay!. y i J = 1 i J i J 13 M (48) M (49) = G = G (78) _ (79) _ Z n..Ax'.Ay!. xj i j •'xj j = l J - E £,.Ax!.Ay!. j = 1 i j i j 13 94 (51) _ (81) _ (54) _ (84) _ i ~ G i " M i ~ G i " ° M? 5 5 > . G<85> - - I V 4 y ! . A x - J . i i j = 1 i J i J M ( 5 6 ) , (86) , _ > ^ , j -1 « ' « « M< 5 7 ) - G ( 8 ? > - V x i „<->. or • v + ^ v*i/ For equations (1 .3 ) , (1.4) and (1.5) a l l G ±=0. For equations (1 .6 ) , (1.7) and (1.8) a l l M±=0. L ( 3 ) = L ( 6 ) = J £ . . { A ' + ( r ! , ) ( A ^ c ) C o s ( Y , . ) + A < S > S i n ( Y . , ) ) } i i i = 1 i J i j ° i J ° i J -1 a a J r a L ( 4 ) = (7) = j £ . . { A ' + ( r ! . ) ( A ( C ) C O S ( Y , . ) + A ( s ) S i n ( Y , . ) ) A x ! . i i i = 1 i J i J _9_ i J _ ° _ i J J a a + I Z L . R . ( r ! . ) A ^ ; A x ] . J-1 P=o 1 : 1 p 0 1 J ^ a L i 5 > = L i 8 ) = " ^ C i j { A ' + ( r i j ) ( A ^ C o s ^ ) + A ^ s ) S i n ( Y i : J ) ) A y ^ a a J P . . - Z I „ ( r ! .) A ^ ' Ay! . j -1 p=0 1 J P ° ^ -E9. ^ J r a 95 p Z R (r!.) B ps i j _j>s a x {£..Cos(sY + sQ n ..Sin(sy. i j 13 rps o i j ' 1 3 Tps Z R ( r ! .) B P S i j _ £ S a x {£..Sin(s Y - sft n..Cos(sY. 13 13 rps 0 xj i j ps ( r 1 j> C. . C o s ( Y ± . ) + P o n . Sin(Y < ) > (r!.) B. {£. .Sin(Y. . - ^ 2 ) ) - f t n .Cos(Y. . - i j / 2 ) ) i j bm i j 13 Ybm 0 i j ' i j rbm a P Z R (r!.) B Ax!. ps 13 _ps i j a x { q ^ o s C s y . . - ^ ) + ^ ^ S i n C s Y . . - ^ ^ ) } P Z R ( r ! .) B Ax.'. p= 0 P s _E£ a x { ^ j S i n C s Y ^ - ^ ) - 8 n o n 1 J c o s ( 8 Y L J - ^ ) ) } c l Z R (r!.) B Ay* r a x {£. .Cos(s Y. .-tf/ 1^) + sft n. . S i n ( s Y . . - i l / 1 ^ ) } ^ 1 3 '13 T>s 0 3-3 !J ps Z R (r.' .) B Ay! . F a x {P. . S i n ( s Y . - sfi H . . C O S ( S Y . . - / 1 ) ) } l <=i3 '13 ^ps 0 13 13 rps 96 ( r ! . ) A y ! . B u U Cos ( y , . - ^ 2 ) )+fi^n. .Sin (y , ) > v ±y ij_bm i j i j bin 0 13 i j bm a 97 APPENDIX II SOLUTION METHOD SOLUTION FOR THE NATURAL FREQUENCY RESPONSE The natural response of the system i s found by solving the homogeneous equation [A] d2{X} + IB] d_fX} + [C]{X} = 0 (II.1) dT^ dT Following the method described by Meirovitch l D, equation (II.l) can be manipulated to become where [D] d{p(T)} + [E]{p(T)> = 0 dT (p(T)} = fd{X(T)}/dfl \'{X(T)} J (II.2) [0] [A] [A] [B] [E] = -[A] [0] [0] [C] Assuming a solution of the form {x(T)} = {X}e X T (II.3) 98 where X = a + * i $ = u>/pha4/D' i = /=T $ i s interpreted as the non-dimensional form of the natural frequency, w . Substitution of (II.3) into equation (II.2) r e s u l t s i n the eigen-problem ([D]X + [ E ] ) ^ { X ) \ e X T = 0 (II.4). {X}/ The natural frequencies, $ , were determined numerically from equation (II.4). SOLUTION OF THE STEADY-STATE RESPONSE The steady-state response of the system i s simply the solution of the equation tC]{x) = {L> (II.5) which can be solved numerically for {x}. The displacement of any point on the blade can be calculated by substituting the elements C m n and S m n from {x} in to equation (4.20). 99 SOLUTION OF THE FORCED HARMONIC AND SELF EXCITED RESPONSE The solution procedure for the self-excited and the forced harmonic response i s the same for both cases, the main difference being the change in the frequency from s ^ Q t o to f. Therefore, only the complete derivation of the forced harmonic solution i s shown here. The forced harmonic response i s the solution to the equation [A] d2{X) + [B] d{X}+ [C]{X> = I {^Cos(*.T)-KV^Sin(*.T) (II.6) dr dT f=l for which a solution of "the following form can be assumed: {X(T)} = t {\P}Cos(* T) + {vJ^SinC* T) f=l £ (II.7) Substitution of (II.7) into equation (II.6) r e s u l t s , a f t e r some manipulation, in the matrix equation IC]-»|[A] - » f [ B ] • f [ B ] [C]-*|[A] (II.8) from which {W( 3) fj a n a {w^^f} can be determined numerically. To calculate the motions of the blade and the guides one must f i r s t decompose {w(3)f} and {W^f} into "kcft } 100 as (x(T)} was defined for equation (4.31). The motion of the guides i s then F {G±} = Z { G ^ C o s ^ T ) + {G (^)Sin($ fT) r 6*Cos(<J> T . z i r i " *zif>] ae1.-Cos(3> T y i f f " * y i f ) F E aO1.,Cos(S> T xi f f " *xif> f=l 6 * C o s ( * - T z i r t " *zif> a e y i f C ° s ( V " *yif> , a e x i f C ° s ( V - •*!£>, where (II.9) *zis tan - i , < ^ . > w etc. The motion of the blade, from equation (4.20) i s u(r',Y,T) = 1 1 1 R j n n(r')I((5 3^ fCos(*o f T ) + c f m n fSin ( e>o f T)Cos(nY) a f=l m=0 n=0 — — + ( P f C o s ( * 0 p T ) + ^ f S i n ( $ 0 4 : T ) ) S i n ( n Y ) ] mnf F M N 1 1 Z YBnf<Y'r,CoB(*°fT - ^ (Y ) ) f=l m=0 n=0 (11.10) Y m n f ( T ' r , ) g Rmn(r')[(^ n fCos(nY) + ^ S i n C n y ) ) a a a a + (CfCo,(nY) + Cfsln(ty)2> a a 2i 1/2 = tan -1 S° ,Cos(nY) + ^ - S i n ( t r y ) mnf mnf  i m i i m i l l . P f C o s ( n Y ) + CfSin(nY) 101 From the above solution there i s no d i f f i c u l t y i n seeing that the solution for the self-excited response i s r 6* Cos(sft T z i i o s z -< s=l a6* Cos(sft T yis o r ae 2 Cos(sft T xis o <t>2. ), (11.11) where and zis z i s . / < ^ l s > ( 1 ) > 2 + ( ^ l s > ( 2 ) > 2 tan etc. ( 6 1 . ) z i s (2)\ S M N u(r',y,T) = Z E E Y_ o (r' ,Y) Cos (sQQT - f ( Y) ) a s=l m=0 n=0 1 0 1 1 5 m n s (11.12) where Y (r', Y) = R (r')f(£° Cos(nY) + Sin(n Y)) 2 mns " mn \ mns 1 mns + (cH Cos(ny) + P Sin(ny)) 2) 1 / 2 mns mns J •««oW = tan mns -1 f2> , ^  ' (2^  C Cos(nY) + S Sin(n Y) mns ' mns ' Cos(nY) + S 1 1 Sin(n Y) mns mns 102 APPENDIX III FLOWCHART OF THE PROGRAMS Two programs were developed. The f i r s t , GUIDE1, produces the forced vibration response due to a static or harmonic load and the se l f -excited response. The subroutines HARMON, STATIC and LOAD were part of GUIDE1. The second program, GUIDE2, produces the natural frequencies and mode shapes of the system. The subroutines MDS, POLY and CENST were used by both programs. 103 Program GUIDE1 Execute Subroutines - CENST - KDS - LOAD St a t i c " \ Harmonic Loading' Subroutine STATIC Subroutine HARMON I OMG - OMG + OMGST No Stop Figure 32. Flowchart for Program GUIDE1. 104 Program GUIDE2 Execute Subroutines -CENST -HDS [DA] - 0 . MASS MASS] DAMPj w • -MASS . o ° 1 STIFFJ Solve Eigenproblem (M - D>A3X)W - o Eigenvalues & Eigenvectors and/or Natural Frequencies 6 Mode Shapes | Stop "j Figure 33. Flowchart for Program GUIDE2. 105 Subroutine MDS Start Calculate terns In matrices MASS, DAMP and STIFF based on the equations of the plate and the guides. Return Subroutine CENST Start Calculate terms c1.c2.c3,...,c6 used for describing the centrifugal stresses. Return Subroutine POLY Start Calculate the terms used i n the function R^Cr') Return Figure 34. Flowcharts for Subroutines MDS, CENST, and POLY. 106 Subroutine STATIC Start Solve 6>TIFF][X} - w Calcula Deflec te Blade tion Blade Deflection Beturn Subroutine HARMON STIFF-OHG1 ^ HAS S OHGl*DAMP . -OMGl*DAMP STIFF-OMGl2*MASS. Solve [RM]{X} - {V} 2 2 Amplitude - » l + * 2 -1 Phase " Tan x 2 *1 Amplitude Phase Figure 35. Flowcharts for Subroutines STATIC and HARMON. 107 Subroutine LOAD Start S t a t i c Load Calculate ( V 3 ^ » { V 4 ^ d u e t 0 : - St a t i c External force - Lateral/Shaft Misalignment - Dished Blade Calculate { v l s ] . ^ 2 s } d u e t 0 : - Centrifugal froces - Guide interaction on Runout/Blade Misalignment External Harmonic Load Calculate {v} due to an External Harmonic load Return Figure 36. Flowchart for Subroutine LOAD. 108 

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