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The dynamics and stresses of bandsaw blades Taylor, John 1986

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THE DYNAMICS AND STRESSES OF BANDSAW BLADES by JOHN TAYLOR B . A . S c , The U n i v e r s i t y of B r i t i s h Columbia, 1980 A THESIS SUBMITTED IN 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 paper as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February 1986 © J o h n T a y l o r , 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 26 February 1986 ABSTRACT This study investigates the stresses and dynamics of stationary, idling and cutting bandsaw blades. A method of obtaining an estimate of the stresses i n an i d l i n g bandsaw blade i s presented. The estimate i s determined by measuring the stresses that occur when the blade vibrates in i t s lowest fundamental modes and assuming that the i d l i n g behaviour can be represented by a summation of these modes. The natural frequencies of the bandsaw blade have been measured for various operating conditions and the measured results are compared to existing analytical predictions. A modification to the analysis of torsional motion i s presented that accounts for the i n t e r n a l stress d i s t r i b u t i o n e x i s t i n g i n the blade due to the r o l l tensioning that such blades receive. The displacements and frequency spectra of the bandsaw blade during the cutting process are obtained. The displacements are compared to the surface of the cut lumber, and the frequency spectra are compared to the dynamic response characteristics of the idling blade. The r e s u l t s of t h i s study w i l l be of i n t e r e s t to those wishing to improve their understanding of the stresses and dynamics associated with idling and cutting bandsaw blades and desiring more accurate predictions of blade natural frequencies. i i TABLE OF CONTENTS Page Abstract i i Table of Contents i i i List of Tables v Li s t of Figures v i Nomenclature x Acknowledgements x i i 1. INTRODUCTION 1 1.1 Background 1 1.2 Previous Research 5 1.3 Experimental Aims 7 2. EQUIPMENT AND INSTRUMENTATION 8 2.1 Equipment 8 2.2 Instrumentation 14 2.3 Software 22 3. THEORETICAL CONSIDERATIONS 23 3.1 Theoretical Evaluation of the Strains Due to Vibrational Displacement of the Sawblade 23 3.2 Natural Frequencies of Idling Blade 25 3.2.1 Lateral Natural Frequencies 26 3.2.2 Torsional Natural Frequencies 31 3.2.3 Effect of Non-Linear Stress Distribution on the Torsional Frequencies 34 3.3 Cutting Tests 35 4. EXPERIMENTAL PROCEDURE AND RESULTS 38 4.1 Strain-Mode-Shapes and Strains Due to Vibrational Displacements 38 4.1.1 Strain-Mode-Shapes Procedure 38 4.1.2 Strain-Mode-Shapes Results 40 4.1.3 Strains Due to Forced Displacements -Procedure 45 4.1.4 Strains Due to Forced Displacements -Results 50 i i i Page 4.2 I d l i n g Blade Dynamics 66 4 .2 .1 I d l i n g Blade Dynamics - Procedure 66 A . 2 . 2 I d l i n g Blade Dynamics - Resul ts 69 4 .2 .2 .1 Examples of C o l l e c t e d Data 70 A . 2 . 2 . 2 Comparison of Data with Theory 77 4 .2 .2 .3 M o d i f i c a t i o n of the Theory to Include the E f f e c t of V a r i a b l e In-Plane Stresses 87 4.3 C u t t i n g Tests 93 4 .3 .1 C u t t i n g Tests - Procedure 93 4 .3 .2 C u t t i n g Tests - Resul ts 95 5. CONCLUSIONS 111 5.1 S t r a i n s Due to V i b r a t i o n a l Displacement 111 5.2 I d l i n g Blade Dynamics 112 5.3 C u t t i n g Tests 113 6. REFERENCES 115 APPENDICES 117 I Instrument L i s t 117 II Summary of Computer Programs 118 III Explanation of Notat ion on Graphs from N i c o l e t FFT Frequency Analyser 121 i v LIST OF TABLES Page I Dimensions of the Equipment Used in This Study 21 II Comparison of the Upper and Lower Bound Solutions for the Lateral Blade Frequencies 31 III Average Strain Per Unit Displacement Values 63 IV Theoretical Strain Per Unit Displacement Values for L = 760 mm. 63 V Values of a Obtained Empirically 92 v LIST OF FIGURES Page 2.1 The 5 Foot Bandsaw 9 2.2 Hydraul ic S t r a i n i n g System 10 2.3 D e t a i l s of the C u t t i n g Area 11 2.4 Assumed Stress D i s t r i b u t i o n Due to R o l l - T e n s i o n i n g 12 2.5 S t r a i n Gauge Locat ions Across the Blade 13 2.6 Instrumentation Arrangement 15 2.7 L o a d c e l l C a l i b r a t i o n Curve 16 2.8 Displacement Transducer No. 1, C a l i b r a t i o n Curve 18 2.9 Displacement Transducer No. 2, C a l i b r a t i o n Curve 19 2.10 Displacement Transducer No. 3, C a l i b r a t i o n Curve 20 3.1 Model f o r C a l c u l a t i n g S t r a i n Due to L a t e r a l Displacement 24 3.2 I d e a l i z e d Model of Bandsaw 27 3.3 S t a t i c and Dynamic Components of Blade Tension 29 3.4 Geometry of Blade f o r T o r s i o n a l V i b r a t i o n Model 32 3.5 P a r a b o l i c S t ress D i s t r i b u t i o n i n Blade 32 4.1 S t r a i n Mode Shapes, 11000 l b s S t r a i n 42 4.2 S t r a i n Mode Shapes, 15000 l b s S t r a i n 43 4.3 S t r a i n Mode Shapes, 18500 l b s S t r a i n 44 4.4 S t r a i n Mode Shape Data I n d i c a t i n g Change i n Sign Across Node 46 4.5 Instrument C o n f i g u r a t i o n and Span Lengths f o r S t r a i n Per U n i t Displacement Data 48 4.6 S t r a i n Gauge and Displacement Probe Locat ions f o r S t r a i n Per U n i t Displacement Data 49 4.7 RMS Values f o r S t r a i n and Displacement Data, Instrument/Span C o n f i g u r a t i o n B 51 4.8 T r a n s m i s s i b i l i t y of S t r a i n and Displacement Data, Instrument/Span C o n f i g u r a t i o n B 52 v i Page 4.9 Coherence Between Strain and Displacement Data, Instrument/Span Configuration B 53 4.10 RMS Value for Strain and Displacement Data, Position IB, Instrument/Span Configuration A 54 4.11 RMS Value for Strain and Displacement Data, Position 4B, Instrument/Span Configuration A 55 4.12 RMS Value for Strain and Displacement Data, Position 7B, Instrument/Span Configuration A 56 4.13 Coherence Between Strain and Displacement Data, Position 4B, Instrument/Span Configuration A 57 4.14 Transmissibility of Strain and Displacement Data, Position 4B, Instrument/Span Configuration A 58 4.15 Strain Per Unit Displacement Values for Instrument/Span Configuration A 59 4.16 Strain Per Unit Displacement Values for Instrument/Span Configuration B 60 4.17 Strain Per Unit Displacement Values for Instrument/Span Configuration C 61 4.18 Strain Per Unit Displacement Values for Instrument/Span Configuration D 62 4.19 Displacement Spectrum of the Idling Blade 67 4.20 Transmissibility of Strain and Displacement at Position 7 on the Sawblade 68 4.21 Receptance of Blade @ Zero RPM 71 4.22 Coherence of Blade @ Zero RPM 72 4.23 Receptance of Blade @ 300 RPM 73 4.24 Coherence of Blade @ 300 RPM 74 4.25 Receptance of Blade @ 600 RPM 75 4.26 Coherence of Blade @ 600 RPM 76 4.27 Comparison of Lateral Frequencies with Theory, 10000 lbs Strain (1 of 2) 79 4.28 Comparison of Lateral Frequencies with Theory, 10000 lbs Strain (2 of 2) 80 v i i Page. 4.29 Comparison of Lateral Frequencies with Theory, 16500 lbs Strain (1 of 2) 81 4.30 Comparison of Lateral Frequencies with Theory, 16500 lbs Strain (2 of 2) 82 4.31 Comparison of Torsional Frequencies with Theory, 10000 lbs Strain (1 of 2) 83 4.32 Comparison of Torsional Frequencies with Theory, 10000 lbs Strain (2 of 2) 84 4.33 Comparison of Torsional Frequencies with Theory, 16500 lbs Strain (1 of 2) 85 4.34 Comparison of Torsional Frequencies with Theory, 16500 lbs Strain (2 of 2) 86 4.35 Comparison of Data and Theory with Modified Theory, 10000 lbs Strain (1 of 2) 88 4.36 Comparison of Data and Theory with Modified Theory, 10000 lbs Strain (2 of 2) 89 4.37 Comparison of Data and Theory with Modified Theory, 16500 lbs Strain (1 of 2) 90 4.38 Comparison of Data and Theory with Modified Theory, 16500 lbs Strain (2 of 2) 91 4.39 Experimental Set-Up for Cutting Tests 94 4.40 Sawblade Behaviour During Cutting (78% mfr) 96 4.41 Sawblade Behaviour During Cutting (81% mfr) 97 4.42 Sawblade Behaviour During Cutting (87% mfr) 98 4.43 Sawblade Behaviour During Cutting (94% mfr) 99 4.44 Sawblade Behaviour During Cutting (103% mfr) 100 4.45 Sawblade Behaviour During Cutting (110% mfr) 101 4.46 Displacement Spectrum of Saw Blade During Cutting (78% mfr) 103 4.47 Displacement Spectrum of Saw Blade During Cutting (81% mfr) 104 4.48 Displacement Spectrum of Saw Blade During Cutting (87% mfr) 105 v i i i 4.49 Displacement Spectrum of Saw Blade During C u t t i n g (94% mfr) 4.50 Displacement Spectrum of Saw Blade During C u t t i n g (103% mfr) 4.51 Displacement Spectrum of Saw Blade During C u t t i n g (110% mfr) 4.52 Comparison of Blade Displacement Data with A c t u a l Cut i x NOMENCLATURE A blade cross sectional area Ag gullet area b blade thickness B bite per tooth Bq modified bite per tooth c blade velocity co speed of wave in blade D depth of cut E modulus of elasticity F feed speed of log carriage FL1 f i r s t lateral natural frequency FL2 second lateral natural frequency FT1 f i r s t torsional natural frequency FT2 second torsional natural frequency G bulk modulus GFI gullet feed index h blade width I moment of inertia Ig polar moment of inertia K s top wheel stiffness Kb blade stiffness (AE/L) k non-dimensional top wheel support (1-n.) L span length between guides Lw span length between wheels M bending moment mfr maximum feed rate X P tooth pitch q(t) displacement function Rg static tension in sawblade Rj dynamic tension in sawblade S curved blade length T kinetic energy T g v St. Venant torque U strain energy u(t) displacement function 6j top wheel displacement £ strain e a axial strain ek bending strain H non-dimensional top wheel support 8 angle of twist p mass density o" stress 0Q axial stress Op parabolic stress w frequency wn natural frequency A frequency xi ACKNOWLEDGEMENTS To everyone who helped with t h i s project, thank you. I would p a r t i c u l a r l y l i k e to acknowledge my advisor, Dr. S. G. Hutton, for his continued enthusiasm and encouragement; Bruce Lehraann, for his valued assistance with experiments and knotty problems; Alan Steeves for his support with the computer programs; and finally, my wife, Grace, for her unfailing support and my son, Lucas, for the many weekends and evenings he spent without me. x i i 1 . INTRODUCTION 1.1 Background The handsaw i s one of the most widely used types of saws in the wood cutting industry with duties ranging from primary log breakdown in sawmilling to small dimension work in furniture manufacture. The main advantages of the handsaw are i t s a b i l i t y to handle most log sizes, i t s high cutting speed and i t s relatively thin kerf (thickness of cut). The size of a bandsaw i s described by the diameter of the wheels that support the blade and, for sawmilling, these range from five feet to nine feet i n diameter. The blade i s guided i n the cutting region with pressure guides which displace the blade laterally. The crowned top wheel, supported hydraulically or pneumatically, supplies tension to the blade and can be t i l t e d to control the blade position. The larger size bandmills are used as headrigs, the f i r s t saws in the sawmill production line, which break the logs down into large rectangular cants. The smaller sized handsaws are used as resaws. These break the large cants down into multiples of the required thickness for further reduction to dimensioned lumber, usually by use of circular saws or twin or quad bandmills. The operating d e t a i l s of a bandmill depend on many factors. Of prime importance are the head sawfilers recommendations, these make allowances for: the type of wood being cut; the required quality and accuracy of cut; the volume throughput required; the gauge of blade; the type of tooth; and whether the wood i s frozen. Some average operating d e t a i l s are included at t h i s stage as background information. A nine foot headrig would have a 250 to 300 HP motor and cut logs of up to four feet i n diameter at speeds from 200 to 400 FPM. Larger logs than t h i s 1 could be accommodated but they are becoming scarce. Some headrigs use double cut blades, with teeth on both edges, cutting the log as i t travels in either direction and, although this increases production, the tr a i l i n g edge of teeth tends to spoil cutting accuracy. The smaller sized five foot or six foot diameter resaws, driven with 100 to 150 HP motors, w i l l cut cants up to two feet thick at speeds from 100 to 300 FPM. Resaws are usually run with single cut blades but are often grouped in pairs or quads to improve the lumber throughput. It should be noted that the feed speeds and the horsepowers quoted here are quite general; the feed speed of the lumber w i l l depend on the depth of cut, the speed of the blade and the capacity of the gullet; the horsepowers w i l l depend to a large extent on the size and type of wood. The blades are fabricated from high quality steel with an ultimate tensile strength of 200,000 PSI. The blades range in size from 16 i n . wide by 0.085 to 0.109 i n . thick, for the nine foot bandmills, to 10 i n . wide by .049 to 0.065 in . thick, for the five foot bandmills. The tooth shape, pitch and gullet capacity are usually one of several standard patterns chosen to s u i t the duty of the saw. A side clearance i s allowed to stop the saw binding in the cut and i s created by swaging each tooth the required amount at the t i p . This clearance is very important as i t directly affects the amount of wood lost with each cut, however, side clearance s t i l l tends to vary considerably from mill to m i l l . Side clearances are typically slightly greater than the blade thickness, giving a total cut width of more than twice the blade thick-ness. Carbide and s t e l l i t e tipped circular saws have been used success-fully for many years. However, s t e l l i t e i s gaining in popularity for use on bandsaws due to i t s superior resistance to accidental damage, i t s 2 ease of application and i t s a b i l i t y to be sharpened on t r a d i t i o n a l equipment. Rol l tensioning i s one of the most important processes in blade preparation. It involves pressure rolling narrow bands along the centre region of the blade to plastically extend i t . This introduces compress-ive stresses in the central region of the blade and tensile stresses at the edges. For older, low-strain bandmills, these stresses ensure that the majority of the axial load, applied to the blade by the bandmill, is carried in the edges of the blade, thus keeping the edges taut and stiffening the cutting edge. For modern, high-strain, thin blade band-m i l l s , only a small portion (10 - 15%) of the bandmill s t r a i n i s required to pull out the compressive stresses in the centre of the blade and the remainder of the s t r a i n i s then evenly distributed across the blade. Roll tensioning is also designed to compensate for expansion due to blade heating caused by the cutting action. To maintain optimum performance, frequent maintenance of the saw-blade is required. The standard swaged tooth blades are changed every 2-4 hours for checking and sharpening. The s t e l l i t e tipped blades are changed less frequently because of the reduced wear rate of the teeth. However, care must be taken not to leave the blades cutting or i d l i n g for too long, as fatigue cracks can develop from the extended periods of cyc l i c a l stress due to the blade bending over the wheels. The blades are changed regularly for checking and resharpening and, periodically, this w i l l include additional levelling and re-tensioning work. L e v e l l i n g requires that any bumps due to blade d i s t o r t i o n be beaten out and the re-tensioning work involves checking and correcting the original r o l l tensioning stresses. The s t e l l i t e tipped saws require 3 l e s s frequent maintenance because the teeth remain sharper f o r longer per iods , l e a d i n g to lower c u t t i n g forces and, subsequently, l e s s blade d i s t o r t i o n . Bandmil l performance i s genera l ly estimated from the measured mean and s t a n d a r d d e v i a t i o n s of the lumber p r o d u c e d . However , w i t h the varying economics of the i n d i v i d u a l m i l l s , coupled with the high feed rates t r a d i t i o n a l l y used i n Western Canada, d i f f e r e n t standard d e v i a t i o n v a l u e s w i l l be c o n s i d e r e d a c c e p t a b l e . As a rough g u i d e , a s t a n d a r d d e v i a t i o n of 0.010 i n . to 0.012 i n . would be c o n s i d e r e d v e r y good, w h i l e , f o r a l a r g e h e a d r i g , s t a n d a r d d e v i a t i o n s of up to 0.025 i n . a r e acceptable . Much of the exper t ise i n s e t t i n g up and operat ing handsaws i s based on experience and e m p i r i c a l r e l a t i o n s h i p s and, f o r a bandsaw to operate s u c c e s s f u l l y , w i l l r e l y on the maintenance of a b a l a n c e between the f o l l o w i n g f a c t o r s : the reduct ion of the blade thickness (to minimize kerf ) without i n c r e a s i n g the d e v i a t i o n ; the increase of the a x i a l s t r a i n (to s t i f f e n the blade) without inducing f a t i g u e f a i l u r e ; and the correc t r o l l - t e n s i o n i n g of the blade f o r the p r e v a i l i n g c o n d i t i o n s . R o l l - t e n s i o n i n g , w h i c h tends to be more of an a r t than a s c i e n c e , must be c a r r i e d out c o r r e c t l y to o b t a i n optimum p e r f o r m a n c e and t h i n b l a d e s w i l l t end to e m p h a s i z e any poor workmanship . The s u c c e s s of a bandsaw as an e f f i c i e n t c u t t i n g t o o l can r e l y e n t i r e l y on t h i s one o p e r a t i o n . Some of the problems experienced with handsaws are : blade f a i l u r e s (usual ly from g u l l e t cracks) ; poor surface f i n i s h of the lumber; snaking (weaving f r o m s i d e to s i d e ) of the s a w b l a d e , e s p e c i a l l y a t the h i g h e r feed speeds; poor sawing accuracy due to the i n c o r r e c t d i s t r i b u t i o n of the r o l l - t e n s i o n i n g s t r e s s e s ; and the formation of g u l l e t cracks when 4 i d l i n g . One of the most important developments i n the l a s t two decades has been the advent of the h i g h - s t r a i n b a n d m i l l . The o lder bandmil ls with dead-weight l e v e r s t r a i n i n g mechanisms have been superseded by h y d r a u l i c and pneumatic bandmil ls that provide up to three t imes the s t r a i n . To make use of t h e s e h i g h e r s t r a i n s , t h i n n e r b l a d e s have been used and, a l t h o u g h they have reduced k e r f l o s s e s , they must be c o r r e c t l y r o l l -t e n s i o n e d to o b t a i n good p e r f o r m a n c e , thus e m p h a s i z i n g the need f o r a complete understanding of the e f f e c t s of r o l l - t e n s i o n i n g . In 1981, the Department of Mechanical Engineering at the U n i v e r s i t y of B r i t i s h Columbia, with the ass is tance of the Science C o u n c i l of B.C., set up a wood c u t t i n g laboratory . This study i s a part of the on-going r e s e a r c h a s s o c i a t e d w i t h t h i s l a b o r a t o r y i n an a t t e m p t to more f u l l y understand the parameters governing the c u t t i n g performance of bandsaws. 1.2 Previous Research The s t ress i n bandsaw blades can be separated i n t o two components: permanent s t r e s s , introduced d u r i n g f a b r i c a t i o n by r o l l i n g , s h e a r i n g , t o o t h f o r m a t i o n , r o l l t e n s i o n i n g , and heat t r e a t m e n t ; and t e m p o r a r y s t r e s s , i n t r o d u c e d d u r i n g o p e r a t i o n by b a n d m i l l s t r a i n , t i l t a n g l e , v i b r a t i o n , bending and c u t t i n g . A knowledge of these s t resses and t h e i r d i s t r i b u t i o n i s important i f the dynamic behaviour of the blade i s to be completely understood. Previous research i n t h i s area, aimed at determining the s t resses i n bandsaws, i n c l u d e s the work o f : F o s c h i [ 9 ] , who i n v e s t i g a t e d the e f f e c t i v e n e s s of the ' l i g h t - g a p ' technique, a method used throughout the i n d u s t r y to o b t a i n an e s t i m a t e of the r o l l t e n s i o n i n g s t r e s s e s i n the b l a d e ; A l l e n [ 3 ] , who has p r o v i d e d many u s e f u l methods of c a l c u l a t i n g 5 and estimating the blade stresses in high-strain bandmill systems; and Eschler [8], who investigated the distribution of the stresses in band-saw blades due to band position, axial tension and t i l t angle. One area of research that has created considerable interest has been the generation of a n a l y t i c a l methods for predicting sawblade natural frequencies. Archibald and Emslie [4] investigated the lateral vibrations of a moving string. Mote [14,15] studied the l a t e r a l vib-ration of an axially moving plate with uniform stress distribution and flexural stiffness, including the effect of periodic axial band tension variation. Also included was the dependence of band tension on a x i a l velocity and the pulley mounting system. Alspaugh [2] investigated the torsional vibration of a thin, rectangular, moving strip with uniform stress distribution and torsional stiffness, including the effect of a point load on one edge. Soler [17] studied the combined l a t e r a l and torsional vibration modes of a moving band and the effect of a conserv-ative point load acting on one edge. Anderson [1] studied the lateral vibration of a multiple span moving band. Ulsoy and Mote [20] developed two methods for analyzing the l a t e r a l and tor s i o n a l vibrations of an axially moving plate complete with computer programs for solving them. Wu and Mote [22] investigated the dynamic coupling between the cutting and non-cutting regions of the bandsaw blade. Das [7] experimentally determined the s i g n i f i c a n t blade frequencies and t h e i r mode shapes during the cutting process. For an in-depth review of the available literature associated with bandsaw vibra t i o n and s t a b i l i t y , the reader i s referred to a paper by Ulsoy, Mote and Syzmani [21]. One of the problems associated with the prediction of the band natural frequencies has been the poor correlation between the predicted 6 and experimental torsional frequency values. 1.3 Experimental Aims The aims of t h i s study on the dynamics and stresses of bandsaw blades are threefold: 1.3.1 To measure the strains (and hence the stresses) induced i n the cutting area of a stationary sawblade by forced vibration of the blade. As i t i s not possible to measure the strains induced during cutting, the purpose of t h i s section of the work was to measure the strains induced by exciting the blade in i t s lowest mode shapes and then to use this information to deduce the strains involved during the actual running of the blade (from a knowledge of the spectrum of the measured vibrations). Such information would be of value in attempting to ident-if y the specific factors involved in gullet cracking. 1.3.2 To measure the natural frequencies of the i d l i n g bandsaw blade for various axial prestresses, guide spacings and blade speeds. This knowledge of the dynamic behaviour of the blade i s essential for the validation of the analytical models and the compre-hension of the mechanisms of poor cutting. 1.3.3 To carry out i n i t i a l cutting tests for various blade and feed speeds and measure the frequencies and displacements of the blade during the cutting process and compare the resul t s with the natural frequencies of the blade and the finished surface of the cut lumber. From these resu l t s i t w i l l be possible to investigate the excitation that the blade undergoes during cutting and determine which modes of vibration are most important. 7 2. EQUIPMENT AND INSTRUMENTATION 2.1 Equipment A five foot production bandmill manufactured by Can-Car was used for the experiments (Fig. 2.1). The saw was driven hydraulically via a swash plate type hydraulic pump and 100 hp electric motor. This config-uration was ideal for speed control and the speed could be varied from zero to 700 rpm. The normal operating speed was 600 rpm. The sawblade was strained via a separate hydraulic system (Fig. 2.2) and the pressure was controlled i n two stages. The f i r s t stage loaded the m i l l with a minimal strain of 2000-3000 lbs. A second stage 'surge' was then used to increase the hydraulic pressure up to the pressure r e l i e f valve setting. The maximum setting was 19000 lbs. The blade was guided in the cutting region by two pressure guides and could be lubricated with water jets located above the upper guide (Fig. 2.3). The top wheel could be t i l t e d by an e l e c t r i c motor to a l i g n the running sawblade and the saw could be moved by the hydraulic setworks to adjust the width of the cut lumber. Two sawblades were used for the experiments, a toothed blade and a smooth blade. Both blades had the same basic dimensions and both were roll-tensioned (Fig. 2.4). I n i t i a l l y , the toothed blade was equipped with s t r a i n gauges (Fig. 2.5) and was used for a l l the non-rotating experiments (e.g. data collection for the strain-mode-shapes and for measuring the strains due to vibrational displacement). Later, the strain gauges were removed and the blade was used for the cutting tests. The smooth blade was used for the experiments associated with the idling blade dynamics. For the idling tests, an adjustable guide support frame was manu-factured and attached to the back of the ex i s t i n g guide support arms, 8 s •a O O ai OJ s-3 ' 1 1 • X?\ V 1 1 1 CO LITL Figure 2.2 Hydraulic Straining System Hyd. pump Hyd. mo to r - c a r r i a ge Servo valve Dual r e l i e f s Press, reducing valve Solenoid valve Needle valve Needle valve Check valve Return l i ne f i l t e r Suction s t ra iner Tachometer- c a r r i a g e Elec. motor 25HP I Figure 2.3 Details of the Cutting Area 11 CL CO O-h CTp = s t r e s s due to r o l 1 - t e n s i o n ! ng (assumed p a r a b o l i c ) Figure 2.4 Assumed Stress D i s t r ibut ion due to Roll-Tensioning 12 CL >> h h+g Figure 2.5 Stra in Gauge Locations Across the Blade 13 allowing for incremented positioning of the guides (or guide) between the existing guide locations. For the cutting tests, the standard fixed guides were used and the timber was fed into the saw via a s p e c i a l l y designed log carriage on precision aligned r a i l s . The carriage was driven hydraulically and the feed speeds could be selected as desired from zero to 480 f t . min. 2.2 Instrumentation A l i s t of the instrumentation and equipment used in the experiments i s given in Appendix I and a diagram of the instrument chain i s shown in Fig. 2.6. To measure s t a t i c and dynamic s t r a i n values, seven s t r a i n gauges were attached to the outside of the sawblade and three s t r a i n gauges were attached to the inside of the blade (Fig. 2.5). To measure the axial prestressing force, a link in the hydraulic straining system was equipped with a four arm s t r a i n gauge bridge, referred to as the 'loadcell'. The c a l i b r a t i o n curve for the l o a d c e l l i s shown in Fig. 2.7. The data a c q u i s i t i o n system for the s t r a i n gauges and l o a d c e l l consisted of the following: a Neff 620/300 signal conditioner which provided i n d i v i d u a l excitation voltage, wheatstone bridge completion r e s i s t o r s and bridge balancing for each channel; a Neff 620/100 amplif ier/multiplexer which received the conditioned signals and pro-vided fixed modular and programmable am p l i f i c a t i o n , f i l t e r i n g and analogue to d i g i t a l conversion for each channel; and a Neff 620/500 control unit which provided the necessary i n t e r f a c i n g with a host Vax 750 computer. The system was controlled with a Tektronix 4051 graphics terminal located in the laboratory. Individual strain measure-ments were also taken with a Vishay model P-350A d i g i t a l s t r a i n 14 Amplif ier A/D converter multiplexer Data I/O to computer El ectro-magnet^ Force | g 3 iTransducer> Neff 100 teff 300 I— Signal conditioner -Straingauges -Displacement -4*ansducers Frequency analyser 100 watt ampl i f i e r Figure 2.6 Instrumentation Arrangement o o O O o o Carriage tachometer _ to o o.o I \z = = = = = = = ± — Frequency generator Neff 500 Computer termi nal D ig i ta l p lot ter 1 ( 4 M Depart ment con puter Cant Carriage indicator. Programs written specifically for the Neff data acquisition system, plus packaged graphing routines, enabled the experimental results to be viewed on the terminal and plotted on the Tektronix 4662 plotter. For more information on the Fortran programs developed for the Neff system, see Appendix II. Excitation of the blade was provided by either an electromagnet or a small electromagnetic shaker. The shaker was used where the blade had to be 'tuned i n ' to one of i t s natural frequencies, e.g. the mode shape data was obtained when the blade was 'tuned in' this way. Both magnets were driven by a Bruel & Kjaer No. 1024 frequency generator. The gen-erator signal was amplified with a 100 watt power am p l i f i e r for the large electromagnet or a 10 watt a m p l i f i e r for the electromagnetic shaker. Various methods of mounting the magnets were used. The electro-magnet could be supported independently of the bandsaw with three dimensional positioning on the inside or outside of the blade, or i t could be attached to the bandsaw frame on the inside of the blade, again with three dimensional positioning. The electromagnetic shaker could be mounted on the inside of the blade with a choice of four positions, selected to avoid the nodes of the vibrating blade. The excitation force of the electromagnet was measured with a Bruel & Kjaer piezo-electro force transducer and the signal amplified with a K i s t l e r 504D charge amplifier. The displacement of the sawblade was measured with three non-contacting displacement transducers and matched proximitors, the calibration curves are shown in Figs. 2.8 to 2.10. The data from the force and displacement transducers were analyzed with a Nicolet 660A dual channel FFT frequency analyzer. The analyzer 17 Lead #1 Bentley Nevada Proximitor Model 3106 (no.l) 6 4 0 o o o o Slo pe = 6 5v/i.n (2.57v /mm) Cal i b r a t i on fac tor = 0.391rr m/v 0 1.0 2.0 3. Mil 1ineters Figure 2.8 Displacement Transducer No. l , Ca l ibrat ion Curve Lead #2 Bentley Nevada Proximitor Model 3106 (no.2) 6 Figure 2.9 Displacement Transducer No. 2, Ca l ibrat ion Curve 1.5 V 0 L T S 0.5 Figure 3.0 Displacement Transducer No.3, Ca l ibrat ion Curve TABLE I Dimensions of the Equipment Used in This Study A cross-sectional area of the blade = 0.674 sq. i n . (blank blade) = 0.618 sq. in . (toothed blade) Ag 0.75 sq. in. = gullet area b 0.965 in. = blade thickness D 11.5 in. = depth of cut E 30.0E+6 lbs/sq. i n . = modulus of elas t i c i t y F 273 fpm = log carriage feed speed G 11.5E+6 lbs/sq. i n . = bulk modulus GFI 0.7 = gullet feed index h blade width = 10.375 in. (blank blade) = 9.5 in . (gullet to back) = 10.25 in . (tooth to back) K s 9925 lbs/in. = top wheel stiffness k 0.036 = non-dimensionalized stiffness (1-17 ) L 30 i n . = span length between guides Lw 93.3 in . = distance between wheel centres p 1.75 in. = tooth pitch 21 sampled and stored the information received on each channel and, once calibrated, would calculate and display the receptance, coherence, rms spectrum and the t r a n s m i s s i b i l i t y . Results could be plotted on the Tektronix 4662 plotter. 2.3 Software A l i s t of the computer programs produced to operate the Neff data a c q u i s i t i o n system, with a b r i e f description of each one, i s given i n Appendix II. 22 3. THEORETICAL CONSIDERATIONS 3.1 Theoretical Evaluation of the Strain Due to Vibrational Displace-ment of the Sawblade  The i n i t i a l question here was whether the longitudinal strain from the displacement of the vibrating blade was due to elongation or bending of the blade or a combination of the two. The elongation of the blade due to lateral displacement between the guides was readily obtained from Fig. 3.1 as follows: x = A Sin ^ z ds =^/l + ( x ' ) 2 dz ds = (1 + 1/2 ( x ' ) 2 + ... ) dz L S - f(l + l ^ f ^ ) 2 Cos 2 SI z + ... ) dz •^ o \ L / L /Anrr\2 L S = L + 1/2 j 2 where S = curved length If we assume the f r i c t i o n between the blade and the guides resists elongation of the blade outside the central span, the axial strain i s : e a = S^L = 1 (AnTr) L 4 \ L / If we assume the guides are frictionless, the axial strain i s : 2 £a = S^L = _L_ I AnTT \ Lw ALw V L / The strains due to bending can be obtained from engineering beam theory and are: 23 24 e , A Jl/52T\2 Kax 2 [ L J Where A = blade displacement b = blade thickness Comparing the relative magnitudes of the bending strains with the two cases of axial strain, for the following parameters: h = 1.651 mm Lw = 3L A = 1 mm we find, in the case where elongation of the blade i s resisted by the guides, that: Eb > 3.3 ea and in the case with frictionless guides: Eb > 9.9 Ea In reality, the answer probably l i e s between the two extremes and the ratio gets proportionally larger as A gets smaller. For the experiments associated with this study, the displacements were much less than 1 mm and thus, for comparison with the experimental results, the strains due to vibrational displacement were calculated from beam bending theory. 3.2 Natural Frequencies of Idling Blade In this section the equations of motion, for the prediction of the lateral and torsional natural frequencies, are introduced and solved. The effect of prestressing the sawblade i s also discussed and a mod-if i c a t i o n to the torsional frequency calculations for a blade with uniform stress, to include for the non-linear stress distribution, i s presented. 25 3.2.1 Lateral Natural Frequencies It i s assumed that the section of i n t e r e s t i s the span length between the guides, shown in Figure 3.2. This span i s modeled as a simply supported moving steel band with small amplitude undamped oscillation. The small amplitude equation of motion for the transverse vibration of this span (from Mote [14]) i s : 2 2 2 4 3 x 2c9 x /R_ \ 3 x EI9 x — + - p - n c 2 — + — =o 3t 9z9t \pA / 9z pA9z ... (3.1) Where c = blade speed n, = non-dimensionalized top wheel support stiffness p = mass density A = cross sectional area Rg = static band tension The f i r s t term represents the force due to the lateral acceleration of the blade. The second term is the force associated with the acceler-ation due to the rate of change of the slope (coriolis acceleration). The third term is the force due to the centrifugal acceleration plus the restoring force from the band tension, both associated with the curv-ature, and the fourth term i s the restoring force due to the bending stiffness of the plate. There are a number of factors that influence the band tension 'R?. These are: - the i n i t i a l static tensioning, Rg. - the dynamic tension due to the acceleration of the blade as i t passes around the pulley, R^ . the stiffness of the top wheel support mechanism, Kg, which affects the tension, as follows: 26 Idealised pulley support Figure 3.2 Idealised Model of Bandsaw 27 The top wheel, sensing the loss of downward pressure due to the acceleration of the blade as i t passes around the pulley, moves to maintain the bandmill strain. This movement, 6j say, i s resisted by the top wheel support stiffness, K g6 1. For example, with an in f i n i t e top wheel stiffness the wheel cannot move, the band ten-sion remains constant and, as the speed increases, the dynamic component, R^ , replaces the static component, Rg, until the tension i s virtually a l l dynamic and the blade starts to lose contact with the top wheel. For a frictionless top wheel support the wheel i s free to take up any loss of pressure due to the acceleration of the blade as i t passes around the wheel, and the band tension increase with increasing speed. In this case, the band tension i s the sum of the static tension, Rg, and the dynamic tension, R^ . From Figure 3.3 i t can be seen that by adding the force 2Rd, due to the inertia of the blade as i t moves round the wheel, to the static force balance and allowing the top wheel to move up a distance, 6^ , we obtain the following dynamic force balance: 2 (R g + 6jK b) = (2RS - 5jK s) + 2Rd ... (3.2) rearranging «1 - _ J d h + V 2 The f i n a l band tension i s R = Rs + 6 x l b or R = R s + nRd 28 Stat i c spring balance Stat ic force balance R = Rs + Kb5, 2Rs - Ks5, R = Rs + Kb5, 2Rd Dynamic tension Dynamic force balance Figure 3.3 S tat i c and Dynamic Components of Blade Tension 29 where n - 7751 2AE The band tension is therefore a function of the static tension, Rg, band velocity, c, and the stiffness of the pulley support, Kg. A fixed top pulley would have K g = i n f i n i t y , and a dead weight lever mechanism (DWLM) would have K c = 0, (Fig. 3.2). s Returning to the equation of motion (Equation 3.1), solutions to this equation can be found by using numerical procedures. However, simple, accurate, bounded approximations can be obtained for the natural frequencies. A lower bound can be found by assuming flexural r i g i d i t y is negligible when compared to band tension and assuming the solution to be of the form of Equation 3.3. x = U(t) expUS. (x - c t ) [ f ... (3.3) The resulting, frequency equation (from Mote [14]) i s : 1/2 o mrr /Rs\ (1 - kpAcVR g) O J = L ^ A p J • (1 + npAc 2/R g) 1 / 2 ... (3.4) An upper bound can be obtained by use of Galerkin's method with a two term approximation. The resulting equation (from Mote [14]) i s : 2 4 2 2 4 2 2 2 .,2 2 2 2 /U) L pA - 7T_RGL - TT + kir c L pA\ /oo L pA - 4TT R gL \ EI~ EI EI A EI EI ( 3* 5) - 16TT4 + k 4 T T 2 C 2 L 2 P A \ - (16 L 3p AcoA2 = 0 EI j \3 EI / Table II presents the lower and upper bound frequency values for a bandmill strain of 16500 lbs and a span length of 2.7 f t (the standard span) and compares them to the experimental r e s u l t s (from Section 4.2.2). As can be seen, a l l three values are extremely close for zero blade speed and the experimental value is bracketed between the two bounds for the non-zero blade speeds. 30 TABLE II Comparison of the Upper and Lower Bound Solutions for the Lateral Blade Frequencies (Hz) Bandmill Strain = 16500 lbs Blade Span = 2.7 f t . Blade Speed (fpm) 0 4744 9456 String equation (lower bound) 88.69 86.33 80.23 Experimental value 91.50 88.00 84.00 Galerkin (upper bound) 89.41 89.29 88.91 3.2.2 Torsional Natural Frequencies The model is one of a simply supported thin rectangular strip, translating at constant speed in the longitudinal direction, exhibiting small undamped torsional oscillations. An example of the geometry would be a band running between fixed roller supports, as shown in Figure 3.4. Biot [5] has shown that the effect of uniform axial tension i s to increase the t o r s i o n a l s t i f f n e s s of the band. The r e s u l t i n g expression for torque i s : Torque = l/3hb3G 0 + l/12bh 3a n 0 ... (3.6) L r The f i r s t term i s the torque associated with the torsionally induced shearing stresses and the second term i s the increase in torque assoc-iated with the axial, stress, 0"o. With this expression included in the derivation, the equation of motion for small amplitude torsional vib-ration (from Alspaugh [2]) i s : 31 Figure 3.4 Geometry of Blade for Torsional Vibration Model Figure 3.5 Parabolic Stress D i s t r ibut ion in Blade 32 329 + 2c_9^6_+ ( c 2 - c Q 2 ) 3^ 8 ... (3.7) 2 2 = 0 3t 3z3t 3z Where 9 = angle of twist c = speed of the blade c Q = speed of the wave in the blade c Q i s defined by the expression: C o 2 ' * \h) p p Where p = mass density 0Q = uniform stress due to axial tension. Equation 3.7 is identical in form to that of the transverse vibrations of a moving string and has the same form as the equation governing the lateral vibration of a moving band. The natural frequencies are determined by substituting assumed solutions for 9 (Equation 3.8). ( f 6 = U(t) exp|— (z - ct)J ... (3.8) into Equation 3.7. The resulting equation i s : v 2 , w = c„ I 1 - fc \ | Tun . . . (3.9) E° I1" (if ] Equation 3.9 shows the dependence of the frequency on the velocity ratio, c/c Q, and the axial stress in the blade, 0 Q (which affects c Q ) . Note that as c/c Q approaches unity, U) approaches zero and a standing wave i s produced. This i s known as the c r i t i c a l speed and is to be avoided because of the instability of the blade at this speed. The c r i t i c a l speed i s considerably greater than the maximum speed obtainable with the bandsaw used for this research and i s not invest-igated here. 33 3.2.3 Effect of Non-Linear Stress Distribution on the Torsional Frequencies The roll-tensioning of the blades introduces a non-uniform stress distribution across the sawblade. From Allen [3], we assume this stress distribution to be parabolic (Figure 3.5) and this has the effect of increasing the torsional frequencies. The strain energy relationship, for torsional displacement of a blade with parabolic roll-tensioning stresses, may be shown to be: U = 1/2 o L ri/12bh 3(4) 2G + o 0 + 4/15op)(fY dz ... (3.10) The f i r s t term represents the energy stored by the shearing stresses that provide the torsional resistance of the blade (St. Venant torque), the second term i s the energy stored in torsional stiffness due to uniform axial tension and the third term i s the energy stored in torsional stiffness due to the parabolic component of the axial tension. The f i r s t term i s the expression for torsion associated with the twisting of a thin rectangular bar T = l/3hb3G KdzJ written in terms of strain energy. The second two terms were obtained by substituting an expression for the parabolic roll-tensioning stress 2 a(y) = o a + ay where a = —°^ p h 2 and a a = a Q - l / 3 0 p into the expression for strain energy 'a(y)de 34 Returning to Equation 3.10, i f the parabolic stress compon-ent Op i s set to zero, the expression becomes the same i n form as Alspaugh's equation for strain energy L U = 1/2 j 1/12 bh 3 p^4 j^ b j 2 G + aQj ^39j2dz . . . (3.11) o The expression in parenthesis defines the wave speed in the blade (Section 3 . 2 . 2 ) . Comparing this to Equation 3.10, the wave speed for a blade with parabolic roll-tensioning stresses becomes - 2 . /b\ 2 G a 4a c Q = 4 _ _ + Ho + ™ p W p p 15p and a modified expression for the torsional frequency was obtained by substituting c 0 for c Q into the frequency equation (Equation 3 . 9 ) . This resulted in f = w = £ p f l _ c 2 \ m 2TT 2L \ c^J _ ( 3 U ) Equation 3.12 provides a relationship between the torsional natural frequency of the band and the stress a p (assuming a parabolic distribution of axial stress across the blade). Later, the torsional frequencies are measured and the values of established such that Equation 3.12 predicts the correct frequencies. The results are then compared to the estimate of the parabolic stress distribution obtained by measuring the curvature of the blade. 3.3 Cutting Tests For the cutting tests i t was necessary to know the maximum cutting rate for the sawblade. This i s governed by the capacity of the gullet and i t s a b i l i t y to contain most of the sawdust until the gullet i s free of the cut. Should the capacity of the gullet be exceeded, side s p i l l -35 age occurs. This creates f r i c t i o n between the blade and the lumber, heats up the blade and leads to reduced cutting accuracy. T r i a l and error has shown that the amount of solid wood removed should not exceed 70% of the capacity of the gullet. This factor i s known as the Gullet Feed Index and allows for sawdust expansion less a small amount of side spillage. The blade used for the cutting tests had a g u l l e t area of 9 0.737 in. and a tooth pitch of 1.75 in. The blade speed was 9425 fpm, which corresponds to a bandmill speed of 600 rpm. The depth of cut was 11.5 i n . Prior to c a l c u l a t i n g the feed speed, the following terms are defined: GFI = gullet feed index Ag = gullet area B = bite per tooth c = blade speed P = pitch D = depth of cut F = feed speed mfr = maximum feed rate (see figures) The bite per tooth was obtained from the capacity of the gullet (GFI x A) and the depth of cut (D), i.e. the wood removed by the tooth was equal to the capacity of the gullet. B = GFI x A D The feed speed was controlled by the need to advance the cant a distance 'B' for each tooth and F = B X C P 36 An additional allowance affecting the speed was included to account for the exposed area of gullet which protrudes from the bottom of the cut before the tooth has finished cutting and allows the sawdust to s p i l l out. Making an allowance of 75% of the pitch to account for t h i s (Quelch [16]), the resulting bite per tooth was: B = GFI x A  q D-(.75)P and the f i n a l maximum estimated feed speed was F = B q x c p 37 4. EXPERIMENTAL PROCEDURE AND RESULTS For continuity, the description of the experimental procedures for each section of this study are followed immediately by a discussion of the results. 4.1 Strain-Mode-Shapes and Strains Due to Vibrational Displacement This section has been separated into two subsections. Both of these sections relate to the strains and displacements of the sawblade. However, as the procedures for obtaining the two sets of data were quite different, the two experiments have been kept separate. 4.1.1 Strain-Mode-Shapes Procedure The strain-mode-shapes are plots of the amplitudes of the o s c i l l a t i n g s t r a i n s i n the blade due to the vibration of the blade at each of i t s f i r s t four natural frequencies. As discussed i n Section 3.1, the s t r a i n variations in the vibrating blade were expected to be due to the bending (curvature) of the blade and are, therefore, linearly proportional to the displacement. In order to obtain an accurate picture of the strain distribution, seven s t r a i n gauges were attached to the blade as shown in Figure 2.5. The strain mode shapes were obtained by exciting the blade with the electro-magnet at one of i t s natural frequencies and measuring the oscillations in the longitudinal strains at seven points across the blade. A plot of the magnitude of the strain variation vs. position on the blade was then generated to obtain the strain-mode-shape of the blade for each frequency. The displacements at positions 1 and 7 and the bandmill strain were also monitored for each data run. The s t r a i n mode shapes were obtained for three d i f f e r e n t s t r a i n l e v e l s ; 11000 lbs, 15000 lbs, and 18000 lbs; and for each of the 38 f i r s t four natural frequencies. In the industry today, 15000 lbs is considered an upper level of strain for this type of bandmill and gauge of blade. The signals from the loadcell and strain gauges were fed into the Neff 300 signal conditioner. The excitation voltage (9.85v) and bridge balancing are contained in the unit and both are manually adjusted. The bridge balancing for the loadcell was completed when the straining system supported the top wheel only (no axial loading in the blade). The reading was then readily converted to axial load in the blade. Strain gauges 1 to 7 were attached to the Neff 300 in a 1/4 bridge arrangement. The three bridge completion resistors were mounted on a plug-in mode card (one per channel) inside the unit. The card also contained the manual adjustments for excitation voltage (9.85v) and bridge balancing. The displacement probes were connected directly to the Neff 100. Axial s t r a i n was introduced into the blade with the hydraulic straining system shown in Figure 2.2. The level of blade strain was controlled by a spring loaded pressure r e l i e f valve which was set manually. Once strained, the blade was excited using the electro-magnet shaker attached to the blade. An oscilloscope was connected to one of the strain gauge channels to monitor the signal. The conditioned signals from the Neff 300 ( which were the wheatstone bridge outputs) were fed into the Neff 100. The f i l t e r s and amplifiers for the experiment were "500 Hz low pass" and "1000 gain" respectively. As well as the fixed amplification i t was possible to set additional programmable gains (2, 4, 8, 16 and 32) for each channel. 39 These ensured the res u l t i n g signal was of suitable magnitude to make f u l l use of the range of the Neff 100, which w i l l transmit a signal of up to lO.Ov f u l l scale. The sampling rate of the Neff was set so that 360 samples, at 1800 samples a second, were obtained for each channel. A l l channels were sampled simultaneously. The data collection was triggered by running the main pro-gram ca l l e d "MODE" in conjunction with the data f i l e generated by "SCANLIST". Once a complete set of data had been obtained, i t was either displayed on the terminal screen using "BREAK" and "EZGRAF" or presented in tabular form using "CONVERT". The mode shape could then be plotted either by hand directly from the tabulated values or by feeding the points into the "EZGRAF" plotting routine. The step-by-step procedure for obtaining the data was as follows: The blade was strained to the preset axial load and vibrated at the f i r s t natural frequency, FL1. The signal was carefully monitored on the oscilloscope for shape and amplitude and when assessed to be at the natural frequency three sets of data were taken. A further three sets of data were then taken, without excitation, to check on the back-ground noise of the instrumentation. This was repeated for the remaining three frequencies, FL2, FT1 and FT2. This procedure was followed for each of the three bandmill strain levels. 4.1.2 Strain-Mode-Shapes Results It should be noted that the strain-mode-shapes were obtained from a blade undergoing point force exc i t a t i o n at a natural frequency and, as such, w i l l not s t r i c t l y be the exact mode shapes. Having obtained the strain variations (in the longitudinal direction) at seven locations across the blade for each of the f i r s t 40 four natural frequencies, the magnitude of these strain variations were plotted against position on the blade to indicate the strain-mode-shapes. The r e s u l t s are presented i n Figures 4.1 to 4.3 for the three strain levels. Blade excitation levels were 1/2 to 3/4 of the blade thick-ness (0.065 in.) for the fundamental frequencies. This exceeded the blade operational vibration levels while providing signals with a min-imal axial strain content. Severe blade excitation at 1-1/2 to 3 times the blade thickness, well above operational l e v e l s , were found to i n -crease the a x i a l s t r a i n component to a s i g n i f i c a n t level,as expected from the theory presented i n Section 3.1. The l a t e r a l strain-mode-shapes were almost the same for a l l modes and s t r a i n l e v e l s and were seen to be a function of the roll-tensioning stresses. The portions of the blade carrying the most load had the least lateral displacement and consequently reduced l e v e l s of s t r a i n . This i s very clear i n the l a t e r a l strain-mode-shapes where the tight tooth side shows up quite d i s t i n c t l y , as does the lesser stressed centre section and the s l i g h t tightening of the back edge. The small reduction at position four indicates this blade was "tight centred", an expression indicating that the roll-tensioning stresses were not evenly distributed. The torsional strain modes did not compare quite as well as the l a t e r a l . The shape associated with the f i r s t t o r s i o n a l frequency varied slightly for each level of bandmill strain. However, the shape associated with the second torsional frequency compared very well for a l l three l e v e l s of s t r a i n . The node position was consistent for a l l t o r s i o n a l modes and coincided with the displacement node position located on the blade by touch. 41 II ro O IS) Figure 4.1 Stra in Mode Shapes, 11000 Lbs. Stra in 42 II CD fO (J OO FT2 Stra in Gauge Posit ion Figure 4.3 Stra in Mode Shapes, 18500 Lbs. Stra in 44 Figure 4.4 i s an example of the data obtained from strain gauges 1, 2, 3 and 4 in the second torsional mode, with a bandmill strain of 10,000 lbs. The change in phase between strain gauge signals 3 and 4 indicates the location of the node. In Section 3.1 the strains were shown to be linearly prop-o r t i o n a l to displacement and the experiments of the next section (Section 4.1.4) corroborated this. The displacement mode shapes w i l l , therefore, be proportional to the strain mode shapes and a reasonable estimate of the physical shape of the blade can be obtained from the strain mode shapes. One of the aims of this section of the work was to establish the stresses induced in the cutting area of the blade due to vibration. The stresses, at vibration amplitudes considerably higher than those recorded for the idling band, were measured and found to be at the most 900 psi. This i s only 1-2% of the maximum working stress and not likely to cause any of the gullet cracking experienced with high strain band-saws. (It has been noticed that an idling bandsaw can develop fatigue cracks more rapidly than one used for cutting, Claassen [6].) 4.1.3 Strains Due to Forced Displacements - Procedure The object of this section of the study was to find the factor that associated the strains in the vibrating blade with blade displacements. This would allow an estimate of the stresses in the idling band to be obtained from a knowledge of the displacements. It would also ensure that the strain shapes measured in the previous sec-tion were, in fact, independent of displacement. To obtain these objectives, a comparison was made between the change i n st r a i n and the change in displacement due to blade oscillations, at each of the f i r s t four natural frequencies. The 45 Figure 4.4 Strain-Mode-Shape Data Indicating Change in Sign Across Node 46 results were obtained with the blade undergoing random frequency excit-ation and are presented as strain per unit displacement. Strain and displacement measurements were taken at positions 1, 4 and 7 (Fig. 2.5) and the str a i n s on the inside and the outside of the blade were compared separately to the displacements. Two different span lengths were used and the readings taken at two positions within each span (Fig. 4.5). The blade a x i a l s t r a i n was set at 15000 lbs. •Strain gauges 1, lb, 4, 4b, 7 and 7b were connected to the Nicolet via the Neff data a c q u i s i t i o n system to take advantage of the amplification (1000 gain) in the Neff 100 unit. The three displacement probes were positioned immediately below strain gauges 1, 4 and 7. This configuration (Fig. 4.6) enabled the displacements and the strains on either side of the blade to be obtained. To acquire the data, the signals from one displacement transducer and an adjacent strain gauge were fed into the frequency analyser and one hundred averages were taken. The bandmill s t r a i n was also recorded. Using the i n - b u i l t functions of the analyser, both RMS spectrums were displayed on the screen and the f i r s t four natural frequencies, the average maximum displacements, the average maximum strains and the strains per unit displacement were a l l recorded. The "RMS" values of the two signals, the "transmissibility" and the "coherence", were displayed and recorded with the Tektronix plotter. The power to the electromagnetic shaker was then changed to obtain a d i f f e r e n t amplitude of o s c i l l a t i o n and the data run repeated (another one hundred averages taken) and the numerical values of the strains, the displacements and the ratio of the two were again recorded. The latter figure was then compared to that from the f i r s t run to check 47 SGS .E Guide _S£! D F Guide F = Position of electromagnetic shaker SGS = Position of strain gauges and displacement probe Figure 4.5 Instrument Configuration and Span Lengths for Strain per Unit Displacement Data 48 - t f f -t • 41 , 4 ,7 ) -Strain gauges C1B.4B.7B) Probe i Figure 4.6 Strain Gauge and Displacement Probe Locations for Strain per Unit Displacement Data 49 the linearity of the results. This procedure was repeated for a l l six strain gauges. The configuration of the instruments and guides were then changed and the process repeated. 4.1.4 Strains Due to Forced Displacements - Results The change in strain per unit of lateral displacement was measured at six points across the blade, these were at strain gauge positions 1, 4 and 7 and IB, 4B and 7B. Examples of the actual data collected are presented in Figures 4.7 to 4.14. A description of the notation on these figures i s given i n Appendix I I I . The combined results of a l l the data were plotted as a bar chart and are presented in Figures 4.15 to 4.18. The average values for the four modes for each of the four instrument configurations are presented in Table III. The theoretical values are presented in Table IV for comparison. The process for obtaining the strain per unit displacement values using configuration "B" (Fig. 4.5) and the data for position 1, was as follows: (a) From the displacement and strain spectra (Fig. 4.7), the f i r s t four natural frequencies were located at 60 hz, 79 hz, 120 hz and 153 hz respectively. (b) The transmissibility (Fig. 4.8) is the ratio of the two signals and provides the value of the strain per unit displacement. At the four frequencies of interest the values were 11.5, 11.9, 65.6 and 59 micro-strain/mm. These were the values recorded in Fig. 4.16 for strain gauge 1. (c) The coherence (Fig. 4.9) i s a measure of the linear relationship between the two signals and i s one, or very close to i t , for the frequencies of interest. 50 Figure 4.7 RMS Values for S t ra in and Displacement Data, Instrument/Span Conf igurat ion B 51 Figure 4.8 T r a n s m i s s i b i l i t y of S t ra in and Displacement Data, Instrument/Span Conf igurat ion B 52 Figure 4.9 Coherence Between S t ra in and Displacement Data, Instrument/Span Conf igurat ion B 53 CD _ l > LU L d 1 00 G> I + • © © © © • < QQ L d LU \ \ > > CO 00 © © I I • CM 00 © O) • 00 LO CM N X 00 \ < + © < LO CD Figure 4.10 RMS Values for Stra in and Displacement Data, Posit ion IB, Instrument/Span Configuration A 54 < 51 D O V) CD, 21 CO 6) LY Figure 4.11 RHS Values for Stra in and Displacement Data, Posit ion 4B, Instrument/Span Configuration A 55 CD - J > LU LU 00 <S> I + < Q J LU L d \ \ > > <S> 00 © G> + I c\i LO G> CM LO H 1 \ CM N X 00 < < < Figure 4.12 RMS Values for Stra in and Displacement Data PosHion 7B, Instrument/Span Configuration A 56 o OJ LO H 1 1 1 1 1 1 1 \ LY => © X (J) © o - o Figure 4.13 Coherence Between Strain and Displacement Data, Posit ion 4B, Instrument/Span Configuration A © © CM N X 00 \ < 57 58 o co o s-4-> to O s-o o O LO Theory 56.42 4-> a; E o o |C0 i f-cn • C Q I C Q -1' C Q •3" •co ir~-FL2 FT2 Figure 4.15 Stra in per Unit Displacement Values for Instrument/Span Configuration A 59 ra s_ 4-> 1/1 O s-o c oi E <D O ra a. s-cu CL S-80 70 -60 . 50 40 30 20 -10 . Theory 56.42-y Figure 4.16 Stra in per Unit Displacement Values for Instrument/Span Configuration B 60 20 <0 s_ +-> to o S-o 15 . Theory 14.11-2 10. o. CO 4-> s-o. s-+-> 0 0 5. Theory 3.53-I I I C O I _! FL1 13 cr. cr FT1 FL2 FT2 Figure 4.17 Stra in per Unit Displacement Values for Instrument/Span Configuration C 61 20 to s-o S-o c E d) (J n3 CL to • r -Q S-S-15 10 . Theory 3.53 I CO FL1 Theory I 4 - 1 1 FT1 . co 7 1 CO a. FL2 FT2 Figure 4.18 Stra in per Unit Displacement Values for Instrument/Span Configuration D 62 TABLE III Average Strain Per Unit Displacement Values Configuration FL1 FT1 FL2 FT2 Span Posit] Lon P+SG's Force A 13.07 15.45 54.31 59.29 L 1/6 1/3 B 13.37 14.88 61.36 60.31 L 1/3 1/3 C 3.47 3.23 13.77 14.33 2L 1/3 1/6 D 3.44 3.42 13.68 14.44 2L 1/5 1/6 TABLE IV Theoretical Strain Per Unit Displacement Values for L = 760 mm Configuration FL1 FT1 FL2 FT2 A 14.11 14.11 56.42 56.42 B 14.11 14.11 56.42 56.42 C 3.53 3.53 14.11 14.11 D 3.53 3.53 14.11 14.11 63 (d) Figure 4.16 presents the data for a l l six locations for instrument configuration "B". The values from (b) above are the f i r s t values i n each of the four columns. From the transmissibility of Figure 4.8, note two distinct l e v e l s of s t r a i n per unit displacement. The lower l e v e l i s over the range of the fundamental lateral and torsional frequencies, FL1 and FT1, and i s the s t r a i n per unit displacement value for single curvature of the blade over the span length. The second l e v e l i s over the range of the second l a t e r a l and t o r s i o n a l frequencies, FL2 and FT2, and i s the s t r a i n per unit displacement value for double curvature of the blade over the span length. The experimental values of the strains per unit displacement for a l l four instrument/span configurations are presented in Table III. The theoretical values of strain per unit displacement are presented in Table IV and the correlation i s extremely good except where the strain gauge was located close to a node. Where t h i s occurred, the values of strain per unit displacement have been marked with an "N" (Figs. 4.15 to 4.18) and in most cases errant values were located close to a node. To further analyse the errant values, the actual strain and displacement data were investigated. Figures 4.10 to 4.14 present the data c o l l e c t e d for configuration "A" at positions IB, 4B and 7B. The data include the RMS spectra for the displacement probe and strain gauge signals at a l l three positions and the coherence and t r a n s m i s s i b i l i t y for position 4B, as this was the location of the dominant errant value. Examination of the frequency spectrum for position 4B (Fig. 4.11) revealed a discontinuity i n the s t r a i n and displacement traces at the f i r s t and second torsional frequencies. To explain this, i t was recog-nized that the strain/displacement data, for at least one of the three 64 positions of the blade, experienced a 180 degree phase s h i f t between each natural frequency. On closely inspecting the strain/displacement spectra for a l l three positions (Figures 4.10, 4.11 and 4.12), the following blade behaviour pattern emerged: - at FL1 (60 Hz) a l l three positions (IB, 4B and 7B) were in phase; - between FL1 and FT1, position IB maintained a smooth transition while position 7B went through a discontinuity at approximately 70 Hz, and position 4B went through a discontinuity at FT1 (approximately 80 Hz); - from this information i t was concluded that position IB maintained the same phase while position 7B switched to being 180 degrees out of phase at 70 Hz, and position 4B at approximately 80 Hz right at the formation of FT1; - f i n a l l y , i t was concluded that the phase change, occurring at a natural frequency (which was where the data was sampled), caused the discontinuity in the data. The effects of the 180 degree phase shift were particularly severe at the centre of the blade where the t o r s i o n a l vibration amp-litudes were minimal compared to the lateral and the displacements went abruptly to zero very close to the t o r s i o n a l frequencies (Fig. 4.11). This caused the loss of coherence (Fig. 4.13), the subsequent non-linear peak in the transmissibility (Fig. 4.14), and the large errant values in the torsional frequency data from position 4B. The data for a l l of the errant values were investigated and in every case, at that position, the 180 degree phase shift occurred very close to the natural frequency of int e r e s t . The difference between back-to-back s t r a i n gauge readings (i.e. 1 and IB) was noted and, although several affects were considered, no explanation could be found. The effect of the axial strain component 65 (assumed to be small in the formulation of the theory) was given careful consideration. The plots of t r a n s r a i s s i b i l i t y are of p a r t i c u l a r i n t e r e s t when considering the o r i g i n a l aim of being able to deduce the strains associated with the running blade. By measuring the displacement spec-trum at a selected position on the running blade and knowing the strains per unit displacements at several positions across the blade, there was enough information to calculate the most s i g n i f i c a n t s t r a i n s i n the running blade. Figure 4.19 i s a plot of the displacement spectrum at position 7. Comparing t h i s to the t r a n s m i s s i b i l i t y plot at the same location (Fig. 4.20), i t was possible to estimate the strains i n the running blade from the two plots. The re s u l t of th i s c a l c u l a t i o n has been added to Figure 4.19 and, for the worst combination of mode shapes ( a l l additive), was less than two microstrain. 4.2 Idling Blade Dynamics 4.2.1 Idling Blade Dynamics - Procedure The dynamics of the idling blade were investigated by excit-ing the blade between the guides and measuring the applied excitation force and the resulting blade displacement. These values were then used to generate frequency response functions from which the natural frequencies could be obtained. The f i r s t four natural frequencies were investigated for five guide spacings, two axial loadings and five blade speeds. Blade e x c i t a t i o n was provided by the electromagnet driven with the signal generator and power amplifier. At times, maximum output was required to overcome the noise i n the data caused by the s e l f -excited vibrations of the running blade. The exci t a t i o n force was measured with a force transducer built into the magnet support bracket 66 \ 1 1 1 1 1 1 1 1 . CD CD 00 — H Figure 4.19 Displacement Spectrum of the Id l ing Blade 67 Figure 4.20 Transmiss ib i l i ty of Stra in and Displacement at Posit ion 7 on the Sawblade 68 and the displacement of the blade was measured with one of the displace-ment probes. Both the magnet and probe were positioned 1/3 of the span up from the bottom guide and 1/6 of the blade width from the gullet line (Fig. 4.5). The probe was on the opposite side of the blade to magnet. The signals from the force transducer (on the magnet) and the displacement probe were fed into the frequency analyser and from f i f t y to one hundred samples were taken for each frequency response function to completely stabilize the data. The "receptance" (the ratio of the displacement response to the applied force) and "coherence" were displayed on the analyser screen and copies obtained from the Tektronix plotter. During each data run the bandmill strain was monitored and recorded. This indicated the i n i t i a l static axial loading and the change due to the dynamic axial loading caused by the blade rotation around the wheels. 4.2.2 Idling Blade Dynamics - Results This section investigates the effect of blade speed on the natural frequencies of the blade. The investigation was completed for two levels of axial loading and five different guide spacings. The data has been presented in three stages. F i r s t , typical examples of the data collected are presented (4.2.2.1). The second stage is the collation and plotting of a l l the data collected and the comparison of this to theory (4.2.2.2). The third stage compares the data and known theory with the modified theory and investigates the sensitivity of the modified theory (4.2.2.3). The guide spacings are referred to by the following code and the spacing i s the ins i d e - t o - i n s i d e distance between the guides (Fig. 2.3). For calculating the natural frequencies a more accurate 69 span length was required and this was obtained by tapping the sawblade over the surface of the guide to locate the contact point where the blade span ended. In this manner, r e a l i s t i c and accurate span lengths were obtained and these are also listed. Guide Spacing Distance (mm) Span Length (mm) (Inside-to-Inside) A 415 487 B 520 584 C (standard) 760 822 D 1636 1689 E 2368 2400 Axial Loading Upper level 16500 lbs Lower level 10000 lbs Speed Variation RPM FT/MIN 0 0 150 2356 300 4713 450 7069 600 9425 4.2.2.1 Examples of Collected Data Typical examples of the data collected, for an axial s t r a i n of 16500 lbs and guide spacing C, are presented i n Figures 4.21 to 4.26 i n the form of receptance (displacement/force) and coherence plots. Figure 4.21 shows the zero rpm receptance and the f i r s t four 70 T ' f T " 1 1 1 H f -(0 LO h-Figure 4.21 Receptance of Blade @ Zero RPM 71 V) LO O O Figure 4.22 Coherence of Blade @ Zero RPM 72 Figure 4.23 Receptance of Blade @ 300 RPM 73 CY D O I (0 LO O U Figure 4.24 Coherence of Blade @ 300 RPM 74 if) LO h-Figure 4.25 Receptance of Slade @ 600 RPM 75 76 natural frequencies are easily discernible and have been identified. The upper trace "P" i s the phase angle of the displacement with respect to the excitation force and at each natural frequency passes through 90 degrees, indicating a frequency ratio of unity (u/co(n) = 1). Figures 4.23 and 4.25 show the receptance at 300 and 600 rpm and a l l four frequencies can be seen to decrease with increasing blade speed as we would expect from the theory. Figures 4.22, 4.24 and 4.26 show the coherence plots for the zero, 300 and 600 rpm data and the excellent coherence of the zero rpm conditions can be seen to rapidly deteriorate once the bandsaw i s set in motion. This i s due to blade excitation from sources other than the electromagnet. The loss of coherence at each of the resonant frequencies i s due to the frequency resolution of the analyzer and i s known as bias error. The resolution, in this case 1/2 Hz for the zero to 200 Hz range, was too large to describe the rapidly changing functions that were encountered near resonance on the lightly damped blade. 4.2.2.2 Comparison of Data with Theory The fundamental l a t e r a l and t o r s i o n a l natural frequencies for each of the guide spacings are compared to the theoretically predicted values and the results are plotted against blade speed. Figures 4.27 to 4.30 indicate the lateral frequency comparison and Figures 4.31 to 4.34, the torsional. For the purpose of cla r i t y , the five spans have been s p l i t between two figures, e.g. Figure 4.27 shows the data for spans A, C and E and figure 4.28 shows the data for spans B and D. It should be noted that the theoretical results of Mote and Alspaugh are based on the assumption that there i s constant stress distribution across the blade. 77 The lateral natural frequencies compare very well with theory, which is the string equation, particularly for the longer span lengths D and E. The shorter span lengths, A and B, show some discrepancy especially for the lower (10,000 lb) strain level. This i s probably due to the boundary conditions, which were modeled as simple supports, not being an exact representation of the end conditions and this would tend to have a greater affect on the shorter span lengths. It should also be noted that spans A and B are somewhat shorter than the standard span which showed excellent correlation with the predicted frequencies. In contrast to the l a t e r a l frequencies, the torsional frequencies compared very poorly with the theory (Alspaugh [ 2 ] ) , p a r t i c u l a r l y for the shorter span lengths A, B and C. The difference i s thought to be due primarily to the model and the blade having different stress distributions. The additional stress in the edges of the sawblade due to the parabolic stress distribution would be expected to have a strong e f f e c t on the to r s i o n a l frequencies, especially for the shorter span lengths, hence the greater loss of accuracy in predicting them. Another factor, not included in Alspaugh's theory, i s the increase in stress due to the rotation of the blade around the wheels, sometimes called dynamic tension, which increases the axial strain and hence the frequency with increasing speed. In a l l cases the experimental results were much higher than the theoretically predicted values indicating additional stiffness in the blade. 78 120 Theory Mote [ l * ] 100 o o o Experiment 5* 80 J c <1> =3 cr <D S-Span A « 60 40 20 Blade Speed (Ft/s) Figure 4.27 Comparison of Lateral Frequencies with Theory, 10000 Lbs. Stra in CI of 2) 79 120 Theory Mote [l4] ioo . Q Q Q Experiment 5 80 . >> (_> c: 1 CD c r £ 60 -o o _ Span B U- o -o o 5 40 . Span D 20 - — n -o 0 ( ) 40 80 Blade Speed (Ft/s ) 120 160 Figure 4.28 Comparison of Lateral with Theory, 10000 Lbs (2 of 2) Frequencies . Stra in 80 120 100 J Theory Mote [14] o o o Experiment 80 >> o sz <u cr <D s-<D ra 60 40 20 3 lib" no Blade Speed (Ft/5) Figure 4.29 Comparison of Theory, 16500 Lateral Frequencies with Lbs. Stra in (1 o f 2) 81 120 . Theory Mote [14) ^100-o o o Experiment < > > o o . o _ Span B S 80. a i ^> u . c u ™ 60 CQ 40 . Span D »_ o — n 20-0 « 0 40 8d 120 160 Blade Speed (Ft/s) Figure 4.30 Comparison of Lateral Frequencies with Theory, 16500 Lbs. Strain C2 of 2) 82 140 -0 0 0 Theory Alspaugh Experiment [2] 120 . > o 0 N n: 100 0 to CD O c Freqi 80 - Span A ra c o Q o 60 . o -tl o 1— 40 -Span C 2 0 : O o ° Span E o" 0 • 1 40 80 120 160 Blade Speed (Ft/s) Figure 4.31 Comparison of Torsional Frequencies with Theory, 10000 Lbs. Stra in (1 of 2) 83 •2120 - Theory AT spaugh [2] •— 0 0 0 Experiment I/) QJ o c §L00 4 CT <V S-u_ O 0 0 § 80 -•r— o cn s_ o h- Span B 60 • 40 -> o o ° Span D ° 20 • n 40 80 120 160 Blade Speed (Fb/s) Figure 4.32 Comparison of Torsional with Theory, 10000 Lbs. Frequencies Strain ( 2 o f 2 ) 84 Figure 4.33 Comparison of Torsional Frequencies with Theory, 16500 Lbs. Stra in (1 of 2) 85 Theory A l spaugh [2] Figure 4.34 Comparison of Torsional Frequencies with Theory, 16500 Lbs. Stra in (2 of 2) 86 A.2.2.3 Modification of the Theory to Include the Effect of Variable In-Plane Stresses The results of the previous section indicate that the model for torsional vibration i s unable to predict the frequencies accurately. This i s probably due to the inadequate modelling of the stress distribution which has not taken into account the effects of roll-tensioning and centrifugal forces. In this section, these effects are included and the results analyzed. As presented in Section 3.2, the equations of motion can be modified to include for a parabolic stress distribution across the blade as an approximation of the roll-tensioning effects. As the actual magnitude of the rol l i n g stresses are unknown, this introduced an unknown stress level, Op, into the equation of motion. Op i s the maximum value of the assumed parabolic stress (Fig. 3.5). In this work, the theoretical value of the fundamental torsional frequency was made to agree exactly with the data obtained, at zero rpm, by choosing an appropriate value of o_. The value of C was then used in equation 3.12 to predict the blade frequencies for the non-zero rpm condition (Section 3.2.3). The modified theoretical curves for several different sets of parameters, including the dynamic tension effects, were then compared to the original theory and to the data, Figures A.35 to A.38. For standard span lengths and longer (C, D and E) the correlation between the data and the modified theory was excellent. However, the data for spans A and B at the 10,000 lbs strain level and span B at the 16,500 lb strain level exhibit a significant offset for a l l the non-zero blade speed points for which no explanation could be found. 87 140 Theory Alspaugh [2] o o o Experiment - - - Modified Theory ( Eqn . 3.12) 120!' 100-o c OJ 3 cr <L> J -80-60. co n3 40-Span C 20- . -o i 160 — i — 40 — i — 80 — i — 120 Blade Speed (Ft/s) Figure 4.35 Comparison of Data and Theory with Modified Theory, 10000 Lbs. Stra in CI of 2) 88 140 Theory Alspaugh [2] Modified Theory •(•E.qn. 3.12) 0 0 0 Experiment >, 120-•• c cr cu s--o 80 -60 Span B 40-20. Span D 40 80 120 Blade Speed (Ft/5) 160 Figure 4.36 Comparison of Data and Theory with Modified Theory, 10000 Lbs. Strain (2 of 2) 89 Theory A l spaugh [2] Modified Theory (Eqn.3 .12) 0 0 0 Experiment 140-0 J, - i — — — 1 1 »-40 80 120 160 Blade Speed (Ft/s) Figure 4.37 Comparison of Data and Theory with Modified Theory, 16500 Lbs. Stra in (1 of 2) 90 Theory Alspaugh [2] o o o Experiment - - - M o d i f i e d Theory 140. ( Eqn . 3.12) o 120 • Span B _ 80. zn >~> o t— I 60-u. - 40, " —'•—• Span D 20-0 40 80 120 160 Blade Speed (Ft /s) Figure 4.38 Comparison of Data and Theory with Modified Theory, 16500 Lbs. Stra in (2 of 2) 91 Introducing an assumed parabolic stress distribution (o"p) into the torsional frequency equation, and assuming the difference between the theoretical (Alspaugh [2]) and experimental results was due entirely to this stress distribution, enabled an empirical value for to be obtained. Using the methods of Allen [3], the stress (Op) due to roll-tensioning was estimated to be in the order of 20,000 psi. The values of o^, obtained empirically from the torsional frequency data, are shown in Table V. For a bandmill strain of 10,000 lbs the values averaged 23,504 psi with a standard deviation of 1269 psi (about 5%). The values obtained for a bandmill strain of 16,500 lbs were also very consistent with an average value of 23,986 psi and a standard deviation of 751 psi (3%). The average of both sets of data was 23,745 psi with a standard deviation of 1015 psi (4.3%). The results are reasonably close to the estimated value of 20,000 psi for this sawblade, indicating the the error in the torsional frequency prediction was due primarily to the stress distribution from roll-tensioning. TABLE V Values of a Obtained Empirically Span 10,000 lbs 16,5000 lbs Op psi Op psi A 24910 23310 B 24300 23590 C 22060 24000 D 23980 25250 E 22270 23780 92 4.3 Cutting Tests 4.3.1 Cutting Tests - Procedure The c u t t i n g t e s t s were intended as a p r e l i m i n a r y i n v e s t i g a t i o n into blade displacement and modes of vibration while cutting at various feed speeds. It should be emphasized that this was not intended to be an in-depth study of the blade behaviour during cutting. The experiments were completed to pull together the work of blade stresses and dynamics and to prepare a starting point for the next stage of investigation. The cutting test data were obtained by recording the displacements of the front and back edges of the blade, during the actual cutting process, for various cutting speeds. These ranged from 78% to 110% of the maximum recommended cutting capacity of the sawblade (see Section 3.3). The bandmill rpm, cant feed speed and the bandmill strain were also recorded during each of the cutting tests. Details of the experimental set-up are shown in Figure 4.39. The variations in cutting rate were obtained by setting the log carriage feed speed to the recommended maximum for a bandmill speed of 600 rpm and then varying the bandmill wheel speed to obtain the desired result. From Section 3.3, the maximum feed speed was calculated to be 273 fpm for a bandmill speed of 600 rpm. The maximum feed speed of the log carriage was obtained by recording the output of a d.c. generator attached to the carriage drive system. Seasoned hemlock was used for the cutting tests and, to ensure comparable r e s u l t s , a l l the cutting rate data were obtained from the same cant. The signals from the displacement probes and the d.c. gen-erator were fed into the Neff 100 data a c q u i s i t i o n system and the Nicolet frequency analyzer. The axial strain value was set to 16500 lbs 93 -6ft x 2ft x 1ft cant c = Blade Speed F = Cant Feed Speed D i sp lacement r—i probes ~IZJ~ 1" set T ' f Guide 10" ,12%" - Q l Guide 12" Figure 4.39 Experimental Set-up for Cutting Tests and the bandmill speed was monitored at the beginning of each run. The data for the cutting tests were obtained using both the Nicolet and the Neff. The Nicolet was set to be triggered by the d.c. generator signal and the Neff was activated by running the program "CUT.FOR" on the computer terminal. This program prompted the user for the name of the data f i l e (previously generated by using the program "SCANLIST") and set up the Neff data a c q u i s i t i o n system i n a s e l f -t r i g g e r i n g mode which continuously sampled channel 9 for a non zero voltage. The log carriage was set i n motion to carry the cant toward the bandsaw at the preset speed. Just before the cant reached the saw, the log carriage tripped a microswitch which made contact between the d.c. generator output and the input to channel 9 of the Neff 100 and one channel of the Nicolet. The Neff 100, on receiving t h i s voltage on channel 9, immediately sampled a l l three of the input signals (two probes and d.c. generator) at the sp e c i f i e d sampling rate u n t i l the storage buffer i n the Neff 500 computer interface unit was f u l l (4096 samples). The program then prompted the user for the name of the stor-age f i l e for the data. The Nicolet, triggered by the same signal, sampled the data from the probe at the front of the blade. Having completed the run, the data captured by the Nicolet was plotted using the Tektronix plotter. 4.3.2 Cutting Tests - Results The behaviour of the sawblade during the actual cutting process was investigated and the results compared to the known natural dynamic behaviour of the blade and to the cut surface in the cant. Six data collection runs were made for cutting rates of 78%, 81%, 87%, 94%, 103% and 110% of the maximum cutting rate (see Section 3.3) for the sawblade. The plots of the behaviour of the blade during 95 LO 00 I >-I cu X> O S-a. +-> c o S-cu X) o S-o to CO in I CM co CU XI o S-Q. to CU x : o to O) S-+-> c (O o CU LO CM I CO CO CS> CO Q 2 O c_> LO CO v_/ LU C\J c\j CO CM CO o CS> CO I z: 2: Figure 4.40 Sawblade Behaviour during Cutting (78% mfr.) 96 Figure 4.41 Sawblade Behaviour during Cutting (81% mfr) 97 Figure 4.42 Sawblade Behaviour during Cutting (87% mfr) 98 Figure 4.43 Sawblade Behaviour during Cutting (94% mfr) 99 100 Q LO LO CM LO O 1 • I • I 03 — CM I I I Figure 4.45 Sawblade Behaviour during Cutting (110% mfr) 101 the cut are presented i n Figures A.40 to A.A5. The abcissa has been marked with the times that the leading edge of the cant reached the front and back probes (FP and BP) and, for the preset feed speed of 273 fpm, this occurred at .06 seconds and .22 seconds respectively. The time taken to complete the cut was approximately 1.3 seconds. Returning to the blade displacements, the large amplitude idling oscillations can be seen to rapidly decrease as the cant reached each probe i n turn. From this point on, the displacement of the blade was composed of small high frequency o s c i l l a t i o n s superimposed on a very low frequency oscillation. I n i t i a l l y , the magnitude of the low frequency o s c i l l a t i o n appeared relatively insensitive to the increase in cutting rate (78% to 87%), however, this changed very rapidly as the estimated 100% value was approached and the plot of the sawblade path disappeared from the graph (or, in fact, exceeded the sensitivity range of the probes) for the 110% value. This was expected and reinforced confidence in the methods used for estimating maximum cutting rates based on gullet capacity. The instantaneous spectrum of the blade displacements during each cut, obtained from the probe at the front of the blade, are pre-sented i n Figures A.A6 to 4.51. An examination of the displacement spectrum for each of the six data runs shows the large low frequency component and also two other noticeable "peaks" at approximately 120 Hz and 140 Hz. To compare the frequency spectrum with the i d l i n g blade behaviour, the natural frequencies for the sawblade at 600 rpm were estimated from the blank blade data (Section 4.2.2) and are as follows: 102 CD _ J O > > CD CD + (0 00 CD —-c • r - C \ J •4-> h -+ J L l _ 3 C_> cr cu S- co 4 - c o S o +-> i — to CU i — cn - i -s- o to CO _ J o o to +J o Z ) CD 00 — 00 H Figure 4.46 Displacement Spectrum of Sawblade During Cutting (78% mfr) 103 3 . 1 6+00 V VLG C A M SU 1 6 IS-tt Large low freq, Osc i l lat ions Wheel Rotation 10.9 HZ 10.9 HZ Multiples IA A I il 11 n 5.0A - 5 . 0 D B/16 FT2 (Cutting) FT2 ( Idl ing) 200 Figure 4.48 Displacement Spectrum of Sawblade During Cutting (87% mfr) 105 Figure 4.49 Displacement Spectrum of Sawblade During Cutting (94% mfr) 106 Eigure 4.50 Displacement Spectrum of Sawblade During Cutting (103% mfr) 107 -J fD to -o ai o n> 3 rt> 3 c+ 0 0 •o fD o r+ -s c 3 GO 0) O. ro a tz -5 — i . 3 ca o c r+ c+ — i . 3 CO O 3 -h -S - 0 . 0 4 9 7 D L T A 1 0 0 . - 0 3 V V L G C A M S U 1 6 I S + Large low freq. Oscillations FL2 Wheel Rotation 8.5 HZ FT2 (Cutting) FT2 (Idling) 5 . 0 A - 5 . 0 D B / 1 6 H Z 2 0 0 FL1 = 61 Hz FT1 = 81 Hz FL2 = 123 Hz FT2 = 160 Hz Soler [17] showed that the torsional frequencies decreased with an edge load while the lateral frequencies were virtually unaffected and, i f we take this into consideration, the two "peaks" are seen to be the second lateral and torsional frequencies of the idling sawblade. The lateral frequency remained v i r t u a l l y unchanged at 120 Hz but the t o r s i o n a l frequency was reduced from 160 Hz to approximately 140 Hz. The wheel rotation frequencies (11.6 Hz in Fig. 4.46) and the corresponding mult-iples were clearly visible in a l l of the displacement spectrum data and were responsible for most of the remaining peaks on the graphs. A comparison of the recorded blade displacement with the finished surface of the cant i s presented in Figure 4.52. The correlation between the low frequency oscillation of the blade and the oscillation in the cut surface of the cant was very good. However, the magnitudes of the displacements i n the cut surface were larger than those measured for the blade. This was possibly due to the probe being positioned 2 i n . behind the cutting l i n e of the teeth, 2.25 in. above the cant and, consequently, 6.75 i n . above the l i n e of measurement of the cant surface. 109 Figure 4.52 Comparison of Blade Displacement Data with Actual Cut 110 5. CONCLUSIONS The purpose of the work was threefold: to obtain an estimate of the stresses induced in an idling blade, with a view to identifying the specific factors involved in gullet cracking; to measure the natural frequencies of the idling blade for validation of the analytical models and for comparison with the dynamic behaviour of the blade during cutting; and to examine the behaviour of the blade during the cutting process, for comparison with the known dynamic characteristics of the blade and with the finished profile of the cut lumber. The conclusions based on the findings are as follows: 5.1 Strains Due to Vibrational Displacement The strains associated with small amplitude vibrational blade dis-placement are due primarily to bending of the blade and not to the change in axial length. The data shows that the strain i s a linear function of displacement, as predicted by bending theory, and the values are inversely proportional to the square of the span length. By measuring the displacement spectrum of the running blade and knowing the strain per unit displacement values for the various blade frequencies, i t was possible to obtain an estimate of the strain in the running blade. The estimate indicated that the stresses due to idling vibration were small compared to the normal operating stresses, estimat-ed using the methods of Allen [3], and not likely to cause any of the gullet cracking problems experienced in idling handsaws. The strain-mode-shapes (the distribution of strain across the blade due to vibration in a fundamental mode) were found to be independent of a x i a l loading for both the l a t e r a l and tor s i o n a l modes and were a function of the stress distribution in the blade due to roll-tensioning. For example, the portions of the blade with high axial strain experience 111 reduced v i b r a t i o n a l displacement leading to a reduction in the strain mode shape at this position. The magnitude of stress, due to forced e x c i t a t i o n of the blade at amplitudes at least ten times greater than those exhibited by the idling blade, was at most 1-2% of the t o t a l estimated stresses i n the i d l i n g band and reinforced the conclusion that gullet cracking i s unlikely to be caused by idling blade vibrations. Due to the lin e a r r e l a t i o n s h i p between the strains and displace-ments, coupled with the resu l t s of the s t r a i n mode shape data, a good estimate of the displacement mode shapes of the blade can be obtained from the strain mode shape results. 5.2 Idling Blade Dynamics The natural frequencies vs. blade speed were obtained for each of five different span lengths and two different axial prestresses. The experimental l a t e r a l band natural frequencies exhibited excellent c o r r e l a t i o n with theory (which i n th i s case was the s t r i n g equation as the plate bending effects had been shown to be negligible) providing accurate span lengths were used. The method of tapping the guide face area to locate the end of the span worked well. The experimental torsional band natural frequencies exhibited poor c o r r e l a t i o n with theory (which assumed constant axial stress d i s t r i b -ution) e s p e c i a l l y for the shorter span lengths. This was largely attributed to the roll-tensioning stresses in the blade and the dynamic tension effects. Modifying the torsional frequency equations to include for a para-b o l i c r o l l - t e n s i o n i n g stress d i s t r i b u t i o n across the blade enabled a constant empirical value of the parabolic stress to be obtained (for a 112 combination of span lengths and a x i a l loadings). The value obtained compared well with available theory, indicating that the error in the torsional frequency prediction was due primarily to inadequate modelling of the in-plane stress distribution caused by roll-tensioning. Use of the modified stress distribution in the torsional frequency equations, plus the e f f e c t s of dynamic tension due to blade rotation around the wheels, gave a much improved prediction of the t o r s i o n a l frequencies. 5.3 Cutting Tests Due to the preliminary nature of the cutting tests, the results are far from conclusive, however, certain events occurred frequently enough for the following observations to be made. From the displacement graphs of the cutting blade, i t can be seen that the major inaccuracies were due to the low frequency oscillations of the blade. These o s c i l l a t i o n s occurred even when the cutting rate was well below the estimated maximum and were more likely a function of blade stiffness than vibration. The higher frequency components of the cutting blade were more l i k e l y to a f f e c t the kerf width and surface quality and were, in t h i s case, composed of the second l a t e r a l and t o r s i o n a l frequencies and the wheel speed frequency, plus a l l i t s multiples. From these r e s u l t s i t i s apparent that improving blade s t i f f n e s s i s going to have the most s i g n i f i c a n t e f f e c t on cutting accuracy. Controlling blade vibrations w i l l help reduce kerf width and improve surface quality, but the major improvements w i l l be due to the reduction of the low frequency oscillations of the blade. Correlation between the low frequency o s c i l l a t i o n s i n the blade displacement data and the cut surface of the lumber was very good, with a l l the major displacements of the blade e a s i l y discernible in the 113 finished surface. The magnitude of the displacements in the cut were g e n e r a l l y l a r g e r than those recorded f o r the blade. T h i s was attributed, i n part, to the l a t e r a l f l e x i b i l i t y of the teeth being greater than that of the sawblade at the probe location and, in part, to the separation of the grain "tear out" in the cutting process. 114 6. REFERENCES [I] Anderson, D.L., "Natural Frequency of Lateral Vibrations of a Multiple Span Moving Band Saw". Research Report for the Forestry Directorate, Environment Canada, Western Forest Products Laboratory (now Forintek Canada corp.), 6620 N.W. Marine Drive, Vancouver, B.C., V6T 1X3, January 1974. [2] Alspaugh, D.W., "Torsional Vibrations of a Moving Band". J. Franklin Institute, Volume 283(4): 328-338, 1967. [3] Allen, F.E., "High Strain Theory and Application". Proceedings of the 8th Wood Machining Seminar, University of California, Forest Products Lab., Richmond, California, October 1985. [4] Archibald, F.R., Emslie, A.G., "The Vibration of a String Having a Uniform Motion Along Its Length". Journal of Applied Mechanics, American Society of Mechanical Engineers, Paper No. 58-APM 7, 1957. [5] Biot, M.A., "Increase of Torsional S t i f f n e s s of a P r i s m a t i c a l Bar Due to Axial Torsion". Journal of Applied Physics, Vol. 10, No. 12, pp.860-864, December 1939. [6] Claassen, L., "Determination of the Feasibility of Increasing the Band Speed of High Strain, Thin Kerf Bandsaws". Research Report prepared for Hawker Siddeley Canada Ltd., Canadian Car (P a c i f i c ) Division (now Kockums Cancar Inc.), P.O. Box 4200, Vancouver, B.C., V6B 4K6, 25p, July 1975. [7] Das, A.K., "Analysis of Dynamic S t a b i l i t y of Bandsawing Systems". Proceedings of the 7th Wood Machining Seminar, University of C a l i f o r n i a , Forest Products Lab., Richmond, C a l i f o r n i a , October 1982. [8] Eschler, A., "Stresses and Vibrations i n Bandsaw Blades", M.A.Sc. Thesis, Dept. of Mechanical Engineering, University of B r i t i s h Columbia, Vancouver, V6T 1Z2, 1982. [9] Foschi, R.O., "The Light Gap Technique as a Tool for Measuring Residual Stresses i n Bandsaw Blades". Wood Science & Technology 9:243-255, 1975. [10] G a r l i c k i , A.M., Mirza, S., "The Mechanics of Bandsaw Blades". Department of the Environment, Eastern Forest Products Laboratory (now Forintek Canada Corp.), 800 Montreal Road, Ottawa, Ont. K1G 3Z5, 1972. [II] G a r l i c k i , A.M., Mirza, S., "Lateral S t a b i l i t y of Wide Band Saws". Proceedings of the 4th Symposium on Engineering Applications of Solid Mechanics, held Ontario Research Foundation, 25-26 September, 1978, V2:273-287. 115 [12] Kirbach, E., Bonac, T., "The Effect of Tensioning and Wheel T i l t i n g on the Torsional and Lateral Fundamental Frequencies of Bandsaw Blades". Society of Wood Science and Technology, Wood and Fibre, 9(4) 1978, pp.245-251. [13] Kirbach, E., Bonac, T., "Experimental Study on the Lateral Natural Frequencies of Bandsaw Blades". Society of Wood Science and Tech-nology, Wood and Fibre, 10(1) 1978, pp.19-27. [14] Mote, CD., "Some Dynamic Ch a r a c t e r i s t i c s of Bandsaws". Forest Products Journal, Vol. XV, No. 1, January 1965A. [15] Mote, CD., "A Study of Bandsaw Vibrations". J. Franklin Inst-i t u t e , Vol. 279, pp.430-444, 1965. [16] Quelch, P.S., "Sawmill Feed and Speeds". Armstrong Mfg. Co., Portland, Oregon, 1964. [17] Soler, D.I., "Vibrations and S t a b i l i t y of a Moving Band". J. Franklin Institute, Vol. 286, No. 4, pp.295-307, October 1968. [18] Tanaka, C, Shiota, A., "Experimental Studies on Band Saw Blade Vibration". Wood Science and Technology 15, pp.145-159, 1981. [19] Timoshenko, S., Woinowksy-Kreiger, A., "Theory of Plates and Shells". McGraw-Hill, 1979, Second Ed. [20] Ulsoy, A.G., Mote, CD., "Analysis of Bandsaw Vibration". Wood Science, Vol. 13, No. 1, pp.1-10, July 1980. [21] Ulsoy, A.G., Mote, CD., Syzmani, R., "Pr i n c i p l e Developments i n Bandsaw Vibration and S t a b i l i t y Research". Holz a l s Roh-und Werkstoff, 36 (1978), 273-280. [22] Wu, W.Z., Mote, CD., "Analysis of Vibration i n a Band Saw System". 7th Wood Machining Seminar, University of California, Forest Pro-ducts Lab., Richmond, California, October 1982. 116 APPENDIX I INSTRUMENT LIST 1. Loadcell Strain Gauges, EA-06-125AD-120, K=2.065, 120 Ohms. 2. Bruel & Kjaer Piezo-Electric Loadcell. 3. 9 No. Strain Gauges, Kiowa KFC-5-C1.11, K=2.10. 4. 3 No. Strain Gauges, M-M EP-08-250BG-120. 5. 2 No. Bentley Nevada Non-Contacting Displacement Probes and Proximitors. 6. Electro-Magnet. 7. Bruel & Kjaer Electromagnetic Shaker. 8. Neff 620/300 Signal Conditioner. 9. Neff 620/100 Amplifier and A/D Converter. 10. Neff 620/500 Computer Interface and Data Storage Unit. 11. PDP 11/34 Computer. 12. Vax 11/750 Computer. 13. Tektronix 4051 Terminal. 14. Tektronix 4662 Digital Plotter. 15. Bruel & Kjaer 1024 Signal Generator. 16. Nicolet 660A Dual Channel FFT Frequency Analyser. 17. Kistler 504D Charge amplifier. 18. 1 No. Kamen Non-Contacting Displacement Probe. 19. 1 No. Kamen Oscillator Demodulator Unit. 20. Vishay P-350A Digital Strain Indicator. 21. 10 Watt Power Amplifier. 22. 100 Watt Power Amplifier. 117 APPENDIX II SUMMARY OF COMPUTER PROGRAMS The following is a l i s t of the computer programs produced to operate the Neff data a c q u i s i t i o n system, with a br i e f description of the i r function. SCANLIST This program int e r a c t s with the user to name and build a f i l e of basic information required to run the Neff. MODE This program, when supplied with the name of the f i l e generated by using SCANLIST, runs the Neff and stores the data i n a f i l e of the user's choice. The data i s stored in a single column of values in the following order (example for three channels): Channel No. Data Point 1 1 2 1 3 1 1 2 2 2 3 2 The values are s t i l l subject to the fixed and programmable gains applied during the sampling, have been multiplied by 32768 (2 to power 15) and are displayed as integers. 118 BREAK When supplied with the name of the data f i l e generated by MODE, this program w i l l interact with the user to convert a maximum of four sets of data to the correct values (remove the gains, etc.) and store them in four f i l e s named SET l.DAT to SET 4.DAT. It also generates a command f i l e c a l l e d GRAF.DAT that takes most of the work out of operating EZGRAF, the packaged graphing routine in the computer. Having run BREAK, i t i s only necessary to run EZGRAF then run GRAF and the f i r s t set of data i s plotted on the terminal screen. Adjusting the range of the y coordinate w i l l enable the other sets to be plotted, either singly or overlaid, depending on the user's range selection. CONVERT When supplied with the name of the data f i l e generated by MODE, this program w i l l interact with the user to tabulate the results of each channel. For a constant input signal, the average value of a l l the readings i s given. For a sinusoidal input signal the average maximum and minimum values are given. NEFFLIB For the program MODE to work, several subroutines are required. Some of them are listed in this f i l e , the remainder are listed below: LENGTH, FREQ, MSAMP These subroutines are required to run SCANLIST, MODE, BREAK and CONVERT. SORT This subroutine i s required to run CONVERT and, as the name implies, sorts a set of values into increasing order. 119 CUT This program i s used to c o l l e c t the data from the cutting tests. The program i s set up to run when triggered by a voltage on channel 9. It w i l l then sample the data as directed by the data f i l e generated using SCANLIST. It should be noted that once the program has been set to run, i t continuously samples channel 9 un t i l a voltage i s detected. There i s approximately 30 ms delay between the detection of the voltage and the capture of the f i r s t sample. 120 APPENDIX III EXPLANATION OF THE NOTATION ON GRAPHS  FROM NICOLET FFT FREQUENCY ANALYZER 121 

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