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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.Sc,  The U n i v e r s i t y o f 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 in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING  We a c c e p t t h i s paper as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA February ©John  1986  Taylor,  1986  In p r e s e n t i n g  this thesis  r e q u i r e m e n t s f o r an of  British  it  freely available  agree that  in partial  advanced degree a t  Columbia, I agree that for reference  permission  understood for  by  that  h i s or  be  her  s h a l l not  s h a l l make  and  study.  I  V6T  Date  1Y3  26 February  1986  of  further this  Columbia  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  Mechanical Engineering  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada  the  representatives.  permission.  Department o f  University  Library  g r a n t e d by  be  the  the  copying or p u b l i c a t i o n  f i n a n c i a l gain  the  for extensive copying of  f o r s c h o l a r l y p u r p o s e s may department or  f u l f i l m e n t of  written  ABSTRACT  T h i s study i n v e s t i g a t e s the s t r e s s e s and dynamics of stationary, i d l i n g and cutting bandsaw blades. A method of o b t a i n i n g an e s t i m a t e of the s t r e s s e s i n an bandsaw blade i s presented.  idling  The estimate i s determined by measuring the  stresses that occur when the blade vibrates i n i t s lowest fundamental modes and assuming t h a t 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 f o r v a r i o u s o p e r a t i n g c o n d i t i o n s and the measured r e s u l t s are compared to existing a n a l y t i c a l predictions.  A modification  to the analysis of t o r s i o n a l motion i s presented that accounts f o r the i n t e r n a l s t r e s s 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  roll  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 c h a r a c t e r i s t i c s of the i d l i n g 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 w i s h i n g to improve their understanding of the stresses and dynamics associated with i d l i n g and cutting bandsaw blades and desiring more accurate predictions of blade natural frequencies.  ii  TABLE OF CONTENTS Page Abstract Table of Contents L i s t of Tables L i s t of Figures Nomenclature Acknowledgements  1.  2.  3.  INTRODUCTION  1  1.1  Background  1  1.2  Previous Research  5  1.3  Experimental Aims  7  EQUIPMENT AND INSTRUMENTATION  8  2.1  Equipment  8  2.2  Instrumentation  14  2.3  Software  22  THEORETICAL CONSIDERATIONS 3.1  3.2  3.3 4.  i i i i i v vi x xii  23  Theoretical Evaluation of the Strains Due to Vibrational Displacement of the Sawblade  23  Natural Frequencies of Idling Blade  25  3.2.1 3.2.2 3.2.3  26 31  Lateral Natural Frequencies Torsional Natural Frequencies E f f e c t of Non-Linear Stress D i s t r i b u t i o n on the Torsional Frequencies  Cutting Tests  35  EXPERIMENTAL PROCEDURE AND RESULTS 4.1  34  Strain-Mode-Shapes and Strains Due to Vibrational Displacements 4.1.1 Strain-Mode-Shapes Procedure 4.1.2 Strain-Mode-Shapes Results 4.1.3 Strains Due to Forced Displacements Procedure 4.1.4 Strains Due to Forced Displacements Results iii  38 38 38 40 45 50  Page 4.2  I d l i n g Blade Dynamics  66  4.2.1 A.2.2  I d l i n g Blade Dynamics - Procedure I d l i n g Blade Dynamics - R e s u l t s  66 69  4.2.2.1 A.2.2.2 4.2.2.3  70 77  4.3  5.  6.  Examples o f C o l l e c t e d Data Comparison of Data w i t h Theory M o d i f i c a t i o n o f the Theory t o I n c l u d e the E f f e c t o f V a r i a b l e I n - P l a n e Stresses  87  Cutting Tests  93  4.3.1  C u t t i n g T e s t s - Procedure  93  4.3.2  Cutting Tests - Results  95  CONCLUSIONS  111  5.1  S t r a i n s Due t o 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  Cutting Tests  113  REFERENCES  115  APPENDICES  117  I  Instrument L i s t  117  II  Summary o f Computer Programs  118  III  Explanation Frequency  o f N o t a t i o n on Graphs from N i c o l e t FFT  Analyser  121  iv  LIST OF TABLES Page  I  Dimensions of the Equipment Used i n This Study  21  II  Comparison of the Upper and Lower Bound for the L a t e r a l Blade Frequencies  31  Solutions  III  Average Strain Per Unit Displacement Values  63  IV  Theoretical Strain Per Unit Displacement Values for L = 760 mm.  63  Values of a  92  V  Obtained Empirically  v  LIST OF FIGURES Page 2.1  The 5 Foot Bandsaw  9  2.2  H y d r a u l i c S t r a i n i n g System  10  2.3  D e t a i l s o f the C u t t i n g Area  11  2.4  Assumed S t r e s s 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 L o c a t i o n s  13  2.6  I n s t r u m e n t a t i o n 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  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 T e n s i o n  29  3.4  Geometry o f 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  Parabolic  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  A c r o s s the Blade  Displacement  S t r e s s D i s t r i b u t i o n i n Blade  S i g n A c r o s s 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 L o c a t i o n s f o r S t r a i n Per U n i t Displacement Data  49  RMS V a l u e s f o r S t r a i n and Displacement Instrument/Span C o n f i g u r a t i o n B  51  4.7  4.8  Data,  T r a n s m i s s i b i l i t y o f S t r a i n and Displacement Instrument/Span C o n f i g u r a t i o n B  vi  Data, 52  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 f o r 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 T r a n s m i s s i b i l i t y 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 f o r Instrument/Span Configuration D 4.19 Displacement Spectrum of the I d l i n g Blade  62 67  4.20 T r a n s m i s s i b i l i t y 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) 4.28 Comparison of Lateral Frequencies with Theory, 10000 lbs Strain (2 of 2)  79  vii  80  Page.  4.29 Comparison of Lateral Frequencies with Theory, 16500 l b s Strain (1 of 2)  81  4.30 Comparison of L a t e r a l Frequencies with Theory, 16500 l b s Strain (2 of 2)  82  4.31 Comparison of Torsional Frequencies with Theory, 10000 l b s Strain (1 of 2)  83  4.32 Comparison of Torsional Frequencies with Theory, 10000 l b s S t r a i n (2 of 2)  84  4.33 Comparison of Torsional Frequencies with Theory, 16500 l b s Strain (1 of 2)  85  4.34 Comparison of Torsional Frequencies with Theory, 16500 l b s Strain (2 of 2)  86  4.35 Comparison of Data and Theory with Modified Theory, 10000 l b s Strain (1 of 2)  88  4.36 Comparison of Data and Theory with Modified Theory, 10000 l b s Strain (2 of 2)  89  4.37 Comparison of Data and Theory with Modified Theory, 16500 l b s Strain (1 of 2)  90  4.38 Comparison of Data and Theory with Modified Theory, 16500 l b s S t r a i n (2 of 2)  91  4.39 Experimental Set-Up f o r 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) 4.47 Displacement Spectrum of Saw Blade During Cutting (81% mfr) 4.48 Displacement Spectrum of Saw Blade During Cutting (87% mfr)  viii  103 104  105  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 w i t h A c t u a l Cut  ix  NOMENCLATURE A  blade cross sectional area  Ag  g u l l e t area  b  blade thickness  B  bite per tooth  Bq  modified bite per tooth  c  blade v e l o c i t y  c  o  speed of wave i n blade  D  depth of cut  E  modulus of e l a s t i c i t y  F  feed speed of log carriage  FL1  f i r s t l a t e r a l natural frequency  FL2  second l a t e r a l natural frequency  FT1  f i r s t t o r s i o n a l natural frequency  FT2  second t o r s i o n a l natural frequency  G  bulk modulus  GFI  g u l l e t feed index  h  blade width  I  moment of i n e r t i a  Ig  polar moment of i n e r t i a  K  s  top wheel s t i f f n e s s  K  b  blade s t i f f n e s s (AE/L)  k  non-dimensional  L  span length between guides  Lw  span length between wheels  M  bending moment  mfr  maximum feed rate  top wheel support (1-n.)  X  P  tooth  pitch  q(t) displacement function R  g  s t a t i c tension i n sawblade  Rj  dynamic tension i n sawblade  S  curved blade length  T  k i n e t i c energy  T  g v  U  St. Venant torque s t r a i n energy  u(t) displacement function 6j  top wheel displacement  £  strain  e e  a  k  axial strain bending  strain  H  non-dimensional top wheel support  8  angle of twist  p  mass density  o"  stress  0  a x i a l stress  Op  parabolic stress  w  frequency  w  natural frequency  A  frequency  Q  n  xi  ACKNOWLEDGEMENTS  To everyone who helped w i t h t h i s p r o j e c t , thank you. I would p a r t i c u l a r l y l i k e to acknowledge my a d v i s o r , Dr. S. G. Hutton, f o r h i s continued enthusiasm  and encouragement; Bruce Lehraann,  assistance with experiments and knotty problems;  f o r h i s valued  Alan Steeves for h i s  support with the computer programs; and f i n a l l y , my wife, Grace, for her u n f a i l i n g support and my son, Lucas, for the many weekends and evenings he spent without me.  xii  1.  INTRODUCTION  1.1  Background The handsaw i s one of the most widely used types of saws i n the  wood cutting industry with duties ranging from primary log breakdown i n sawmilling to small dimension  work i n 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 r e l a t i v e l y thin kerf (thickness of cut). The s i z e of a bandsaw i s d e s c r i b e d by the diameter of the wheels that support the blade and,  for sawmilling, these range from f i v e feet  to nine f e e t i n diameter.  The blade i s guided i n the c u t t i n g r e g i o n  with pressure guides which displace the blade l a t e r a l l y .  The crowned  top wheel, supported h y d r a u l i c a l l y or pneumatically, supplies tension to the blade and can be t i l t e d to c o n t r o l the blade p o s i t i o n .  The l a r g e r  s i z e b a n d m i l l s are used as h e a d r i g s , the f i r s t saws i n the s a w m i l l production l i n e , which break the logs down into large rectangular cants. The s m a l l e r s i z e d 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 c i r c u l a r saws or twin  or quad bandmills. The o p e r a t i n g d e t a i l s of a b a n d m i l l depend on many f a c t o r s .  Of  prime importance are the head s a w f i l e r s recommendations, these make allowances f o r :  the type of wood being cut; the r e q u i r e d q u a l i t y 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 i n c l u d e d at t h i s stage as background i n f o r m a t i o n .  A nine  foot headrig would have a 250 to 300 HP motor and cut logs of up to four f e e t i n diameter at speeds from 200 to 400 FPM.  1  L a r g e r l o g s than t h i s  could be accommodated but they are becoming scarce. double  cut blades, with  teeth on  both  Some headrigs use  edges, cutting  the  log as i t  travels i n either direction and, although this increases production, the trailing  edge of teeth tends  to s p o i l  sized five foot or six foot diameter motors, w i l l 300 FPM.  cut cants up to two  cutting accuracy.  The smaller  resaws, driven with 100 to 150 f e e t t h i c k at speeds from  100  HP to  Resaws are usually run with single cut blades but are often  grouped i n 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 the speed  of the lumber w i l l depend on the depth of cut,  of the blade and the capacity of the g u l l e t ; 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 t e n s i l e strength of 200,000 PSI. wide by 0.085 to 0.109  The blades range i n size from 16 i n .  i n . thick, for the nine foot bandmills, to 10 i n .  wide by .049 to 0.065 i n . thick, for the f i v e foot bandmills. shape, pitch and  gullet  capacity are usually  p a t t e r n s chosen to s u i t allowed  to stop the saw  each tooth the  the duty  one  The tooth  of several standard  of the saw.  A side clearance i s  binding i n the cut and  i s created by swaging  required amount at the  tip.  This clearance i s very  important as i t d i r e c t l y a f f e c t s the amount of wood l o s t with each cut, however, side clearance s t i l l mill.  tends to vary considerably from m i l l  Side clearances are t y p i c a l l y  slightly  greater than  the  to  blade  thickness, giving a t o t a l cut width of more than twice the blade thickness. Carbide and fully  s t e l l i t e tipped c i r c u l a r saws have been used  for many years.  success-  However, s t e l l i t e i s gaining i n popularity for  use on bandsaws due to i t s superior resistance to accidental damage, i t s 2  ease of a p p l i c a t i o n and  its ability  to be sharpened on  traditional  equipment. R o l l t e n s i o n i n g i s one of the most important processes i n blade preparation.  I t involves pressure r o l l i n g narrow bands along the centre  region of the blade to p l a s t i c a l l y extend i t .  This introduces compress-  ive stresses i n the central region of the blade and the edges.  For older, low-strain bandmills,  t e n s i l e stresses at  these stresses ensure that  the majority of the a x i a l load, applied to the blade by the bandmill, i s carried  i n the edges of the blade,  s t i f f e n i n g the cutting edge. mills,  only  a small  thus keeping the edges taut  and  For modern, high-strain, thin blade band-  p o r t i o n (10  - 15%)  of the  bandmill  strain is  required to p u l l out the compressive stresses i n the centre of the blade and the remainder of the s t r a i n i s then evenly d i s t r i b u t e d a c r o s s blade.  the  R o l l tensioning i s also designed to compensate for expansion due  to blade heating caused by the cutting action. To maintain optimum performance, frequent blade i s required. 2-4  The  standard  hours for checking and  maintenance of the  saw-  swaged tooth blades are changed every  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 c u t t i n g or i d l i n g for too long, as fatigue cracks can develop from the extended periods of c y c 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, work.  t h i s w i l l include additional l e v e l l i n g and  L e v e l l i n g r e q u i r e s that any  beaten out and  the re-tensioning  bumps due  re-tensioning  to blade d i s t o r t i o n be  work involves checking and  the o r i g i n a l r o l l tensioning stresses.  correcting  The s t e l l i t e tipped saws require  3  less  frequent  periods,  maintenance  l e a d i n g to  lower  because  the  teeth  cutting  forces  remain  and,  sharper  for  subsequently,  longer  less  blade  distortion. B a n d m i l l performance and s t a n d a r d varying  deviations  economics  of  the  i s generally of  the  estimated  from the measured mean  lumber produced.  individual  mills,  However,  coupled w i t h the  r a t e s 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 values  will  deviation while,  be c o n s i d e r e d  of  0.010  for a large  in.  to  acceptable. 0.012  headrig,  in.  As a r o u g h g u i d e ,  would  be  considered  with high  the feed  deviation  a  standard  very  good,  s t a n d a r d d e v i a t i o n s o f up t o 0.025 i n . a r e  acceptable. Much of the e x p e r t i s e i n s e t t i n g on e x p e r i e n c e  and e m p i r i c a l r e l a t i o n s h i p s and,  successfully, following kerf) (to  up and o p e r a t i n g handsaws i s  will  factors:  r e l y on t h e the  maintenance  reduction  of  the  w i t h o u t i n c r e a s i n g the d e v i a t i o n ;  s t i f f e n the blade)  roll-tensioning  of  the  f o r a bandsaw to of a balance  blade  thickness  the i n c r e a s e  without inducing f a t i g u e  must be c a r r i e d o u t c o r r e c t l y  (to  the  minimize  and the  strain correct  an e f f i c i e n t  than a  science,  t o o b t a i n o p t i m u m 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 e n d t o e m p h a s i z e any p o o r w o r k m a n s h i p . as  between  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 t e n d s t o be more o f an a r t  bandsaw  operate  of the a x i a l  failure;  based  cutting  tool  can  rely  The s u c c e s s o f a  entirely  on t h i s  one  operation. Some of the problems experienced ( u s u a l l y from g u l l e t c r a c k s ) ;  w i t h handsaws a r e :  poor s u r f a c e  blade  f i n i s h of the lumber;  (weaving from s i d e to s i d e ) of the sawblade,  e s p e c i a l l y at the  feed speeds;  incorrect  the  poor sawing accuracy  roll-tensioning  stresses;  and  due to the  the  formation  4  of  failures snaking higher  distribution  gullet  of  c r a c k s when  idling. One of  the most i m p o r t a n t developments i n the l a s t two decades has  been the advent dead-weight  high-strain  bandmill.  The o l d e r  bandmills with  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 make u s e  of the  b a n d m i l l s t h a t p r o v i d e up to  of these higher s t r a i n s ,  three  times  the  strain.  To  t h i n n e r b l a d e s h a v e been u s e d a n d ,  a l t h o u g h t h e y have r e d u c e d k e r f l o s s e s , t e n s i o n e d t o o b t a i n good p e r f o r m a n c e ,  t h e y must be c o r r e c t l y  roll-  t h u s e m p h a s i z i n g t h e need f o r a  complete u n d e r s t a n d i n g 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 M e c h a n i c a l E n g i n e e r i n g at the U n i v e r s i t y  of B r i t i s h C o l u m b i a , set  w i t h the a s s i s t a n c e of the  up a wood c u t t i n g  research associated  laboratory.  Science  Council  T h i s study i s a p a r t of the o n - g o i n g  with this laboratory  i n an a t t e m p t t o more  understand the parameters governing the c u t t i n g performance of 1.2  Previous  tooth  stress,  formation,  stress,  introduced d u r i n g f a b r i c a t i o n  bandsaws.  roll  i n t o two  components:  by r o l l i n g ,  shearing,  t e n s i o n i n g , and h e a t t r e a t m e n t ; and  temporary  introduced during operation  vibration,  fully  Research  The s t r e s s i n bandsaw b l a d e s can be separated permanent  of B.C.,  bending and c u t t i n g .  by b a n d m i l l s t r a i n ,  tilt  angle,  A knowledge of these s t r e s s e s and t h e i r  d i s t r i b u t i o n i s i m p o r t a n t i f the dynamic behaviour of the blade i s to be completely  understood.  Previous  research  in this  area,  i n bandsaws,  i n c l u d e s t h e work o f :  effectiveness  of  the  aimed at  F o s c h i [9],  ' l i g h t - g a p ' technique,  i n d u s t r y t o o b t a i n an e s t i m a t e o f t h e r o l l blade; A l l e n [3],  d e t e r m i n i n g the  who i n v e s t i g a t e d  the  a method used throughout  the  tensioning stresses in  the  who has p r o v i d e d many u s e f u l methods o f  5  stresses  calculating  and  estimating  Eschler  the blade stresses i n high-strain bandmill systems; and  [8], who investigated the d i s t r i b u t i o n of the stresses i n band-  saw blades due to band position, a x i a l tension and t i l t angle. One area of r e s e a r c h  that has created  considerable  been the g e n e r a t i o n of a n a l y t i c a l methods natural frequencies.  i n t e r e s t has  for predicting  sawblade  Archibald and Emslie [4] investigated the l a t e r a l  v i b r a t i o n s of a moving s t r i n g .  Mote [14,15] s t u d i e d the l a t e r a l v i b -  ration of an a x i a l l y moving plate with uniform stress d i s t r i b u t i o n and f l e x u r a l s t i f f n e s s , including the e f f e c t of periodic a x i a l band tension variation.  Also i n c l u d e d was the dependence of band t e n s i o n on a x i a l  velocity and the pulley mounting system.  Alspaugh [2] investigated the  torsional vibration of a thin, rectangular,  moving s t r i p with uniform  stress d i s t r i b u t i o n and t o r s i o n a l s t i f f n e s s , including the e f f e c t of a p o i n t load on one edge.  S o l e r [17] s t u d i e d the combined l a t e r a l and  t o r s i o n a l vibration modes of a moving band and the e f f e c t of a conservative point  load acting on one edge.  Anderson [1] studied  vibration of a multiple span moving band. two methods f o r a n a l y z i n g  the l a t e r a l  Ulsoy and Mote [20] developed  the l a t e r a l and t o r s i o n a l v i b r a t i o n s of an  a x i a l l y moving plate complete with computer programs f o r solving them. Wu and Mote [22] investigated the dynamic coupling and  between the cutting  non-cutting regions of the bandsaw blade.  Das [7] experimentally  determined the s i g n i f i c a n t blade f r e q u e n c i e s  and t h e i r mode shapes  during  the cutting process.  For an in-depth review of the available l i t e r a t u r e associated  with  bandsaw v i b r a t i o n and s t a b i l i t y , the reader i s r e f e r r e d to a paper by Ulsoy, Mote and Syzmani [21]. One of the problems a s s o c i a t e d natural frequencies  w i t h the p r e d i c t i o n of the band  has been the poor c o r r e l a t i o n 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 s t r e s s e s of bandsaw  blades are threefold: 1.3.1  To measure the s t r a i n s (and hence the s t r e s s e s ) induced i n  the c u t t i n g area of a s t a t i o n a r y sawblade by f o r c e d v i b r a t i o n of the blade. As i t i s not possible to measure the strains induced during c u t t i n g , the purpose of t h i s s e c t i o n of the work was to measure the strains induced by exciting the blade i n i t s lowest mode shapes and then to use t h i s 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 i n attempting to ident-  i f y the s p e c i f i c factors involved i n g u l l e t cracking. 1.3.2  To measure the n a t u r a l f r e q u e n c i e s of the i d l i n g bandsaw  blade for various a x i a l prestresses, guide spacings and blade speeds. T h i s knowledge of the dynamic behaviour of the blade i s essential f o r the v a l i d a t i o n of the a n a l y t i c a l models and the comprehension of the mechanisms of poor cutting. 1.3.3  To c a r r y out i n i t i a l c u t t i n g t e s t s f o r v a r i o u s blade and  feed speeds and measure the frequencies and displacements of the blade during the c u t t i n g process and compare the r e s u l t s w i t h the n a t u r a l frequencies of the blade and the finished surface of the cut lumber. From these r e s u l t s i t w i l l be p o s s i b l e to i n v e s t i g a t e the e x c i t a t i o n that the blade undergoes during cutting and determine which modes of vibration are most important.  7  2.  EQUIPMENT AND INSTRUMENTATION  2.1  Equipment A f i v e foot production bandmill manufactured  for the experiments  (Fig. 2.1).  The saw  was  by Can-Car was  driven hydraulically v i a a  swash plate type hydraulic pump and 100 hp e l e c t r i c motor. uration was  This config-  i d e a l for speed control and the speed could be varied  zero to 700 rpm. The  The normal operating speed was 600  sawblade  was  strained  (Fig. 2.2) and the pressure was  via a  separate  stage 'surge' was  from  rpm. hydraulic  c o n t r o l l e d i n two stages.  system  The  stage loaded the m i l l with a minimal s t r a i n of 2000-3000 lbs.  first  A second  then used to increase the hydraulic pressure up to the  pressure r e l i e f valve setting. blade was  used  The maximum setting was  19000 lbs.  The  guided i n 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 c o u l d 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 a d j u s t the width of the cut lumber.  Two  sawblades were used f o r the  experiments, a toothed blade and a smooth blade. same basic dimensions Initially,  and both were roll-tensioned (Fig. 2.4).  the toothed blade was  (Fig. 2.5) and was  Both blades had the  equipped  with s t r a i n  used for a l l the non-rotating experiments  c o l l e c t i o n f o r the strain-mode-shapes to v i b r a t i o n a l displacement).  and for measuring  gauges  (e.g. data  the strains due  Later, the s t r a i n gauges were removed and  the blade was used for the cutting tests.  The smooth blade was used for  the experiments associated with the i d l i n g blade dynamics. For the i d l i n g tests, an adjustable guide support frame was  manu-  f a c t u r e d and a t t a c h e d to the back of the e x i s t i n g guide support arms,  8  s •a  O O  ai  OJ  s3  Hyd. pump Hyd. m o t o r - c a r r i a g e Servo valve Dual  reliefs  Press, reducing valve Solenoid valve Needle valve Needle valve Check valve Return l i n e f i l t e r Suction s t r a i n e r Tachometer-  carriage  Elec. motor 25HP  I '  1  1  • X?\ V  CO  1 1  1  LITL Figure 2.2  Hydraulic S t r a i n i n g System  Figure 2.3  Details of the Cutting Area  11  CL CO  O-  h  CTp = s t r e s s  Figure 2.4  due t o r o l 1 - t e n s i o n ! ng (assumed  parabolic)  Assumed Stress D i s t r i b u t i o n due to Roll-Tensioning  12  CL  >>  h h+g  Figure 2.5  S t r a i n Gauge Locations Across the Blade  13  a l l o w i n g f o r incremented p o s i t i o n i n g 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 i n t o the saw v i a a s p e c i a l l y designed  precision aligned r a i l s .  The carriage was  l o g c a r r i a g e on  driven h y d r a u l i c a l l y and  feed speeds could be selected as desired from zero to 480 f t . 2.2  the  min.  Instrumentation A l i s t of the instrumentation and equipment used i n the experiments  i s given i n Appendix I and a diagram of the instrument chain i s shown i n 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 o u t s i d e of the sawblade and three s t r a i n gauges were attached  to the i n s i d e of the blade ( F i g . 2.5).  To measure the  a x i a l prestressing force, a l i n k i n the hydraulic straining system  was  equipped  the  with  'loadcell'. Fig.  a four arm  The  strain  calibration  gauge bridge,  curve  for  the  r e f e r r e d to as  loadcell  i s shown i n  2.7. The  data a c q u i s i t i o n system f o r the s t r a i n gauges and  c o n s i s t e d of the f o l l o w i n g : provided  a Neff 620/300 s i g n a l c o n d i t i o n e r which  i n d i v i d u a l e x c i t a t i o n v o l t a g e , wheatstone bridge  resistors  and  bridge  loadcell  balancing  f o r each channel;  completion  a Neff  620/100  a m p l i f i e r / m u l t i p l e x e r which received the conditioned signals and provided f i x e d modular and  programmable a m p l i f i c a t i o n ,  filtering  and  analogue to d i g i t a l c o n v e r s i o n f o r each channel; and a Neff 620/500 c o n t r o l u n i t which provided the necessary Vax 750  computer.  The  system was  with  a Vishay  14  a host  c o n t r o l l e d with a Tektronix  graphics terminal located i n the laboratory. ments were a l s o taken  i n t e r f a c i n g with  4051  Individual s t r a i n measure-  model P-350A d i g i t a l  strain  Amplifier A/D converter multiplexer  Data I/O to computer  Neff 100 teff 300  Neff 500 I—  Signal conditioner  Computer termi nal  Frequency analyser El e c t r o magnet^ Force |g3 iTransducer>  (4  _ \z t o = o= =  o o  100 watt ampl i f i e r  O  Carriage tachometer  o.o ± I—  = = = =  o O  Depart ment con puter  Frequency generator  o  Cant Carriage  Figure 2.6  Instrumentation Arrangement  M  1  -Straingauges -Displacement -4*ansducers  Digital plotter  indicator. Programs written s p e c i f i c a l l y for the Neff data a c q u i s i t i o n system, plus packaged graphing routines, enabled the experimental r e s u l t s to be viewed on the terminal and plotted on the Tektronix 4662 plotter. more information on the Fortran programs developed  For  for the Neff system,  see Appendix I I . Excitation of the blade was a small electromagnetic shaker.  provided by either an electromagnet or The shaker was used where the blade had  to be 'tuned i n ' to one of i t s n a t u r a l f r e q u e n c i e s , e.g. the mode shape data was obtained when the blade was  'tuned i n ' t h i s way.  Both magnets  were d r i v e n by a B r u e l & Kjaer No. 1024 frequency generator.  The gen-  e r a t o r s i g n a l was a m p l i f i e d with a 100 watt power a m p l i f i e r f o r the l a r g e electromagnet  or a 10 watt a m p l i f i e r f o r the e l e c t r o m a g n e t i c  shaker. Various methods of mounting the magnets were used. magnet could be supported  independently  The  electro-  of the bandsaw w i t h three  d i m e n s i o n a l p o s i t i o n i n g on the i n s i d e or o u t s i d e 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 i n s i d e of the blade w i t h a c h o i c e of four p o s i t i o n s , 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 a m p l i f i e r .  The displacement of the sawblade  was  measured with three non-contacting displacement transducers and matched proximitors, the c a l i b r a t i o n curves are shown i n 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. 17  The analyzer  Lead #1 Bentley Nevada Proximitor Model 3106 (no.l) 6 o o o o 4  Slo pe = 6 5v/i.n (2.57v /mm)  Cal i b r a t i on fac t o r = 0.391rr m/v  0 0  1.0  2.0  Mil 1ineters  Figure 2.8 Displacement Transducer N o . l , C a l i b r a t i o n Curve  3.  Lead #2 Bentley Nevada Proximitor Model 3106 (no.2) 6  Figure 2.9  Displacement Transducer No. 2, C a l i b r a t i o n Curve  1.5  V 0 L T S 0.5  3.0  Figure  Displacement Transducer No.3, C a l i b r a t i o n Curve  TABLE I Dimensions of the Equipment Used i n This Study  A  cross-sectional area of the blade =  0.674 sq. i n . (blank blade)  =  0.618 sq. i n . (toothed blade)  Ag  0.75 sq. i n . = g u l l e t area  b  0.965 i n . = blade thickness  D  11.5 i n . = depth of cut  E  30.0E+6 lbs/sq. i n . = modulus of e l a s 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 =  K  s  10.375 i n . (blank blade)  =  9.5 i n . (gullet to back)  =  10.25 i n . (tooth to back)  9925 l b s / i n . = top wheel s t i f f n e s s  k  0.036 = non-dimensionalized  L  30 i n . = span length between guides  Lw  93.3 i n . = distance between wheel centres  p  1.75 i n . = tooth pitch  21  s t i f f n e s s (1-17  )  sampled  and  calibrated,  stored the information received on each channel and, would calculate and display the receptance,  spectrum and the t r a n s m i s s i b i l i t y .  coherence,  once rms  R e s u l t s c o u l d be p l o t t e d on the  Tektronix 4662 plotter. 2.3  Software A list  of the computer programs produced to operate the Neff data  a c q u i s i t i o n system, w i t h a b r i e f d e s c r i p t i o n of each one, i s given i n Appendix II.  22  3.  THEORETICAL CONSIDERATIONS  3.1  Theoretical Evaluation of the Strain Due ment of the Sawblade  to Vibrational Displace-  The i n i t i a l question here was whether the longitudinal s t r a i n 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 l a t e r a l displacement between the guides was readily obtained from F i g . 3.1 as follows: x = A Sin ^ z  ds =^/l + ( x ' ) dz 2  ds = (1 + 1/2 ( x ' ) + ... ) dz 2  L  f(l  S -  l^f^)  +  2  \L /  •^o /Anrr\  L  j  2  2  S = L + 1/2  Cos SI z + ... ) dz L 2  where S = curved length If we assume the f r i c t i o n between the blade and the guides r e s i s t s elongation of the blade outside the central span, the a x i a l s t r a i n i s : e  a  =  S^L L  1 4  =  (AnTr) \ L /  If we assume the guides are f r i c t i o n l e s s , the a x i a l s t r a i n i s : 2 £a =  S^L Lw  =  _L_ AL  w  I AnTT \  V / L  The strains due to bending can be obtained from engineering beam theory and are:  23  24  e  ,  Kax  A  Jl/52T\2 2 [ LJ  Where A = blade displacement b = blade thickness Comparing the r e l a t i v e magnitudes of the bending cases of a x i a l s t r a i n , f o r the following h  strains with the two  parameters:  = 1.651 mm  Lw = 3L A  = 1 mm  we f i n d ,  i n the case where elongation of the blade i s resisted by the  guides, that: Eb ea  > 3.3  and i n the case with f r i c t i o n l e s s guides: Eb Ea  > 9.9  In r e a l i t y ,  the answer probably l i e s between the two extremes and the  r a t i o gets proportionally larger as A gets smaller.  For the experiments  associated with this study, the displacements were much less than 1 mm and thus, f o r comparison with the experimental r e s u l t s , the strains due to v i b r a t i o n a l displacement were calculated from beam bending theory. 3.2  Natural Frequencies of Idling Blade In t h i s section the equations of motion, f o r the prediction of the  lateral The  and t o r s i o n a l  effect  ification  natural  of prestressing to the torsional  uniform stress,  frequencies, are introduced and solved.  the sawblade i s also discussed and a modfrequency  calculations  for a  blade  with  to include f o r the non-linear stress d i s t r i b u t i o n , i s  presented.  25  3.2.1  Lateral Natural Frequencies It  i s assumed that  the  s e c t i o n of i n t e r e s t i s the  length between the guides, shown i n Figure 3.2. a  simply  supported  oscillation.  The  moving  steel  band  with  span  This span i s modeled as small  amplitude  small amplitude equation of motion for the  undamped transverse  vibration of t h i s span (from Mote [14]) i s : 2 3 x — 3t  2 2c9 x  /R_ - p 9z9t \pA  +  -nc  2  2 \ 3x — / 9z  4 EI9 x —  +  =o pA9z ...  Where c  (3.1)  = blade speed  n, = non-dimensionalized top wheel support s t i f f n e s s p  = mass density  A  = cross sectional area  R  = s t a t i c band tension  g  The f i r s t term represents the force due to the l a t e r a l acceleration of the blade. ation due  The second term i s the force associated with the acceler-  to the rate of change of the slope  ( c o r i o l i s acceleration).  The t h i r d term i s the force due to the c e n t r i f u g a l acceleration plus the restoring ature,  force from the  and  the  fourth  band tension,  both associated  with the  term i s the restoring force due  to the  curv-  bending  s t i f f n e s s 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 s t a t i c tensioning,  -  the  dynamic  tension  due  passes around the pulley,  to  R.  the  g  acceleration of  the  blade as i t  R^.  the s t i f f n e s s of the top wheel support mechanism, K , g  the tension, as follows: 26  which a f f e c t s  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 by  the bandmill  strain.  the top wheel support  stiffness,  i n f i n i t e top wheel s t i f f n e s s sion  remains  constant  This movement, 6j say, i s resisted K 6 . g  1  For example, with an  the wheel cannot move, the band ten-  and, as the speed  increases,  the dynamic  component, R^, replaces the s t a t i c component, R , u n t i l the tension g  i s v i r t u a l l y a l l dynamic and the blade s t a r t s to lose contact with the top wheel.  For a f r i c t i o n l e s s 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 t h i s case, the band tension i s the sum  of the s t a t i c tension, R , and the dynamic tension, R^. g  From Figure 3.3 i t can be seen that by adding the force 2R , due to the d  i n e r t i a of the blade as i t moves round the wheel, to the s t a t i c force balance and allowing the top wheel to move up a distance, 6^, we obtain the following dynamic force balance: 2 (R + 6jK ) = (2R - 5jK ) + 2R g  b  S  s  rearranging «1 - _ J d  h  +V  2  The f i n a l band tension i s R = R  s  + 6 l x  b  or R = R  s  + nR  d  28  d  ... (3.2)  S t a t i c spring balance  S t a t i c force balance  R = Rs + Kb5,  2Rs - Ks5,  2Rd Dynamic tension  R = Rs + Kb5, Dynamic force balance  Figure 3.3  S t a t i c and Dynamic Components of Blade Tension  29  where  7751  n -  2AE  The band tension i s therefore a function of the s t a t i c tension, R , g  band v e l o c i t y , c, and the s t i f f n e s s of the pulley support, K . g  top pulley would have K (DWLM) would have K  A fixed  = i n f i n i t y , and a dead weight lever mechanism  g  = 0, ( F i g . 3.2).  c  s  Returning this  equation  to the equation of motion (Equation 3.1), can  be  found  by  using numerical  solutions to  procedures.  However,  simple, accurate, bounded approximations can be obtained for the natural frequencies.  A lower bound can be found by assuming f l e x u r a l  rigidity  i s n e g l i g i b l e 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)  ...  (3.4)  The resulting, frequency equation (from Mote [14]) i s : mrr /Rs\ =  OJ  L  1/2  o (1 - kpAcVR ) g  •  ^ApJ  (1 + n p A c / R ) 2  1 / 2  g  An upper bound can be obtained by use of Galerkin's method with a two term approximation. 24 2 2 /U) L  \  - 7T_R L  pA  -  G  EI~  - 16TT  4  The resulting equation (from Mote [14]) i s : 4 2 2 2 .,2 2 2 2 TT  +  kir c L  EI +  k  A  EI 4TT C L PA\ 2  2  EI  2  j  -  /oo L  pA\  L p AcoA EI /  (16 \3  3  pA  -  4TT  RL g  EI 2  EI  (*) 3  5  = 0  Table II presents the lower and upper bound frequency values for a bandmill s t r a i n of 16500 lbs and a span length of 2.7 f t (the standard span) and 4.2.2). blade  compares them to the experimental r e s u l t s  (from S e c t i o n  As can be seen, a l l three values are extremely close for zero  speed  and  the  experimental  value  bounds for the non-zero blade speeds. 30  i s bracketed  between the  two  TABLE II Comparison of the Upper and Lower Bound Solutions for the Lateral Blade Frequencies (Hz) Bandmill Strain = 16500 l b s 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  strip,  model  translating  exhibiting  at  i s one  of  constant  small undamped  a  simply  speed  torsional  supported  i n the  thin rectangular  longitudinal  oscillations.  An  direction,  example of the  geometry would be a band running between fixed r o l l e r supports, as shown i n Figure 3.4. Biot [5] has shown that the e f f e c t of uniform a x i a l tension i s to  i n c r e a s e the  torsional  stiffness  of  the  band.  The  resulting  expression f o r torque i s : Torque = l/3hb G 0 + l/12bh a L 3  The  first  term  n  ... (3.6)  i s the torque associated with the t o r s i o n a l l y  shearing stresses and the second  induced  term i s the increase i n torque assoc-  iated with the a x i a l , stress, 0".  With t h i s expression included i n the  o  derivation,  0 r  3  the equation of motion  ration (from Alspaugh [2]) i s :  31  for small amplitude  torsional  vib-  Figure 3.4  Geometry of Blade for Torsional V i b r a t i o n Model  Figure 3.5  Parabolic Stress D i s t r i b u t i o n i n Blade  32  3 9 + 2c_9^6_+ ( c - c 2 3t 3z3t 2  2  2 Q  ) 3^8 2 3z  ... (3.7) 0  =  Where 9 = angle of twist c = speed of the blade c c  Q  = speed of the wave i n the blade  Q  i s defined by the expression: C o  2  ' *  \h)  p  p  Where p = mass density 0  Q  = uniform stress due to a x i a l tension. Equation 3.7 i s i d e n t i c a l i n form to that of the transverse  vibrations  of a moving s t r i n g  and has the same form  as the equation  governing the l a t e r a l vibration of a moving band. The  n a t u r a l f r e q u e n c i e s are determined  by s u b s t i t u t i n g  assumed solutions f o r 9 (Equation 3.8). 6 = U(t) e x p | — (z - ct)J  ... (3.8)  (f  into Equation 3.7.  I" (if ] 1  c„ I 1 -  w =  E°  The resulting equation i s : fc  Equation velocity  ratio,  affects c ) . Q  v \  2  ,  | Tun  3.9 shows the dependence of the frequency  c / c , and the a x i a l Q  The  be avoided critical  stress  i n the blade,  0  Q  (3.9)  on the (which  Note that as c / c approaches unity, U) approaches zero and Q  a standing wave i s produced. to  ...  This i s known as the c r i t i c a l speed and i s  because of the i n s t a b i l i t y speed  of the blade at t h i s speed.  i s c o n s i d e r a b l y g r e a t e r than the maximum  obtainable with the bandsaw used  speed  f o r t h i s research and i s not invest-  igated here.  33  3.2.3  E f f e c t of Non-Linear Stress D i s t r i b u t i o n on the Torsional Frequencies The  r o l l - t e n s i o n i n g of the blades introduces a non-uniform  stress d i s t r i b u t i o n across the sawblade. stress d i s t r i b u t i o n to be parabolic of increasing  From Allen [3], we assume t h i s  (Figure 3.5) and t h i s has the e f f e c t  the t o r s i o n a l frequencies.  The  s t r a i n energy relationship,  f o r torsional displacement  of a blade with parabolic r o l l - t e n s i o n i n g stresses, may be shown to be:  U = 1/2  L r  i/12bh (4) G 3  2  +  o  0 +  4  /15o )(fY p  dz ... (3.10)  o The stresses  f i r s t term represents the energy stored  that provide the t o r s i o n a l resistance  torque), the second term i s the energy stored to  uniform  by the shearing  of the blade (St. Venant i n t o r s i o n a l s t i f f n e s s due  a x i a l tension and the t h i r d term i s the energy stored i n  t o r s i o n a l s t i f f n e s s due to the parabolic component of the a x i a l The  tension.  f i r s t term i s the expression f o r torsion associated with  the twisting of a thin rectangular bar T = l/3hb G 3  KdzJ  written  i n terms of s t r a i n energy. The  second  two terms were obtained  expression f o r the parabolic  r o l l - t e n s i o n i n g stress  2 a  (y) = o  a  + ay  where a = —°^p h 2  and  a  a  = a  Q  -  l/30  p  into the expression f o r s t r a i n energy 'a(y)de  34  by s u b s t i t u t i n g  an  Returning to Equation 3.10,  i f the parabolic stress compon-  ent Op i s s e t to zero, the e x p r e s s i o n becomes the same i n form as Alspaugh's equation for s t r a i n energy L U = 1/2 j  1/12 b h p^4 j^b j 3  2  G  a j ^39j dz  ...  2  +  Q  (3.11)  o The  expression i n parenthesis defines the wave speed i n the  blade (Section 3 . 2 . 2 ) .  Comparing t h i s to Equation 3.10,  the wave speed  for a blade with parabolic r o l l - t e n s i o n i n g stresses becomes c  2 Q  . /b\ G = 4 _ _ 2  a 4a Ho + ™ p  p  W and  +  p  15p  a modified expression f o r the torsional frequency  substituting c resulted i n f = w  0  for c  = £  2TT  p  f  2L  \  Q  into the frequency equation (Equation 3 . 9 ) .  l _ c  Equation 3.12 natural frequency  2  \  This  m  c^J  _  (  3  U  )  provides a relationship between the torsional  of the band and the stress a  d i s t r i b u t i o n of a x i a l stress across the blade). frequencies are measured and the values Equation 3.12  was obtained by  of  predicts the correct frequencies.  p  (assuming a parabolic Later, the torsional established such  that  The results are then  compared to the estimate of the parabolic stress d i s t r i b u t i o n 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 f o r 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 u n t i l the g u l l e t 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  friction  between the blade and  the  lumber,  heats up the blade and leads to reduced cutting accuracy. Trial should  and  not  error has  exceed 70%  shown that the amount of s o l i d  of the capacity of the g u l l e t .  wood removed  This factor i s  known as the Gullet Feed Index and allows for sawdust expansion less a small amount of side s p i l l a g e . The  blade  used  for  the  cutting  t e s t s had  a  gullet  area  of  9 0.737 i n .  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  f o l l o w i n g terms  are  11.5 i n . Prior  to c a l c u l a t i n g  the  feed  speed, the  defined: GFI  = g u l l e t feed index  Ag  = g u l l e t 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 g u l l e t . 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 g u l l e t which protrudes from the bottom of the cut  before the tooth has finished out.  Making  cutting  and allows the sawdust to s p i l l  an allowance of 75% of the p i t c h to account f o r t h i s  (Quelch [16]), the resulting bite per tooth was: B  = GFI x A D-(.75)P  q  and the f i n a l maximum estimated feed speed was F = q B  x  c  p  37  4.  EXPERIMENTAL PROCEDURE AND RESULTS  For continuity, the description of the experimental procedures f o r each section of t h i s study are followed immediately by a discussion of the r e s u l t s . 4.1  Strain-Mode-Shapes  and Strains Due to Vibrational Displacement  T h i s s e c t i o n has been separated i n t o two s u b s e c t i o n s .  Both of  these sections r e l a t e 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 p l o t s 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 v i b r a t i o n of the blade at each of i t s f i r s t four natural frequencies. As d i s c u s s e d i n S e c t i o n 3.1, the s t r a i n v a r i a t i o n s i n the v i b r a t i n g blade were expected to be due to the bending ( c u r v a t u r e ) of the blade and are, therefore, l i n e a r l y proportional to the displacement. In order to obtain an accurate picture of the s t r a i n d i s t r i b u t i o n , seven s t r a i n gauges were attached to the blade as shown i n F i g u r e 2.5. The s t r a i n mode shapes were obtained by exciting the blade with the electromagnet at one of i t s natural frequencies and measuring  the o s c i l l a t i o n s  in the longitudinal strains at seven points across the blade.  A plot of  the magnitude of the s t r a i n variation vs. position on the blade was then generated frequency.  to o b t a i n the strain-mode-shape  of the blade  f o r each  The d i s p l a c e m e n t s a t p o s i t i o n s 1 and 7 and the b a n d m i l l  s t r a i n were also monitored f o r each data run. The s t r a i n mode shapes were obtained f o r three d i f f e r e n t s t r a i n l e v e l s ; 11000 l b s , 15000 l b s , and 18000 l b s ; and f o r each of the 38  first  four natural frequencies.  In the industry today,  15000 lbs i s  considered an upper l e v e l of s t r a i n for this type of bandmill and gauge of blade. The  signals  into the Neff 300 and  from  the  l o a d c e l l and  signal conditioner.  bridge balancing are contained  adjusted.  gauges were fed  excitation voltage (9.85v)  i n the unit and  both are manually  The bridge balancing for the l o a d c e l l was  completed when the  straining system supported blade).  The  strain  The  reading was  the top wheel only (no a x i a l loading i n the then readily  converted  to a x i a l load i n the  blade. Strain gauges 1 to 7 were attached to the Neff 300 i n a bridge arrangement.  The three bridge completion r e s i s t o r s were mounted  on a plug-in mode card (one per channel) inside the u n i t . contained  the  manual adjustments  bridge balancing. the Neff  1/4  The  for excitation  displacement  The card also  voltage  (9.85v)  probes were connected  directly  and to  100. Axial  hydraulic  strain  straining  system  was  introduced  shown i n Figure  into  the  2.2.  blade  The  with  level  of  the blade  s t r a i n was controlled by a spring loaded pressure r e l i e f valve which was set manually.  Once strained, the blade was  magnet shaker attached to the blade.  excited using the e l e c t r o -  An oscilloscope was connected  to  one of the s t r a i n gauge channels to monitor the s i g n a l . The  conditioned signals from the Neff 300  wheatstone bridge outputs) were fed into the Neff 100. amplifiers  f o r the experiment were "500  respectively.  Hz  ( which were the The f i l t e r s and  low pass" and  "1000  gain"  As well as the fixed amplification i t was possible to set  additional programmable gains  (2, 4,  39  8,  16 and  32)  for each  channel.  These ensured the r e s u l t i n g s i g n a l was of s u i t a b l e magnitude to make f u l l use of the range of the Neff 100, which w i l l t r a n s m i t a s i g n a l of up to lO.Ov f u l l s c a l e .  The sampling r a t e of the Neff was set so that  360 samples, at 1800 samples a second,  were obtained f o r each channel.  A l l channels were sampled simultaneously. The data c o l l e c t i o n was triggered by running the main program c a l l e d "MODE" i n c o n j u n c t i o n w i t h the data f i l e "SCANLIST".  generated by  Once a complete s e t of data had been obtained, i t was  e i t h e r d i s p l a y e d on the t e r m i n a l screen using "BREAK" and "EZGRAF" or presented i n tabular form using "CONVERT".  The mode shape could then be  plotted either by hand d i r e c t l y from the tabulated values or by feeding the points into the "EZGRAF" p l o t t i n g routine. The step-by-step procedure f o r o b t a i n i n g the data was as follows: The blade was strained to the preset a x i a l load and vibrated at the f i r s t natural frequency, FL1. The signal was c a r e f u l l y  monitored  on the oscilloscope f o r shape and amplitude and when assessed to be at the n a t u r a l frequency three s e t s of data were taken.  A f u r t h e r three  sets of data were then taken, without excitation, to check on the background  n o i s e of the i n s t r u m e n t a t i o n .  remaining  T h i s was repeated  for  three f r e q u e n c i e s , FL2, FT1 and FT2. T h i s procedure  the was  followed for each of the three bandmill s t r a i n levels. 4.1.2  Strain-Mode-Shapes Results It should be noted that the strain-mode-shapes  were obtained  from a blade undergoing p o i n t f o r c e e x c i t a t i o n at a n a t u r a l 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 s t r a i n v a r i a t i o n s ( i n the l o n g i t u d i n a l d i r e c t i o n ) a t seven l o c a t i o n s a c r o s s the blade f o r each of the f i r s t 40  four natural frequencies, the magnitude of these s t r a i n variations were p l o t t e d a g a i n s t p o s i t i o n on the blade to i n d i c a t e the shapes.  strain-mode-  The r e s u l t s are presented i n F i g u r e s 4.1 to 4.3 f o r the three  strain levels. Blade excitation l e v e l s were 1/2 to 3/4 of the blade thickness (0.065 in.) f o r the fundamental f r e q u e n c i e s .  T h i s exceeded the  blade operational vibration l e v e l s while providing signals with a minimal a x i a l s t r a i n content.  Severe blade excitation at 1-1/2  to 3 times  the blade t h i c k n e s s , w e l l above o p e r a t i o n a l 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 l e v e l , a s expected from the theory presented i n S e c t i o n 3.1.  The l a t e r a l  strain-mode-  shapes were almost the same f o r 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 r o l l - t e n s i o n i n g stresses.  The portions of  the blade carrying the most load had the least l a t e r a l displacement and consequently  reduced  l e v e l s of s t r a i n .  l a t e r a l strain-mode-shapes  T h i s i s very c l e a r i n the  where the t i g h t tooth s i d e shows up q u i t e  d i s t i n c t l y , as does the l e s s e r s t r e s s e d c e n t r e s e c t i o n and the s l i g h t t i g h t e n i n g of the back edge.  The  s m a l l r e d u c t i o n at p o s i t i o n four  indicates this blade was "tight centred", an expression indicating that the r o l l - t e n s i o n i n g stresses were not evenly distributed. The t o r s i o n a l s t r a i n modes did not compare quite as well as the l a t e r a l .  The shape a s s o c i a t e d w i t h the f i r s t t o r s i o n a l frequency  varied s l i g h t l y  for each l e v e l of bandmill strain.  associated with the second a l l three l e v e l s of s t r a i n .  However, the shape  t o r s i o n a l frequency compared very well for The node p o s i t i o n was c o n s i s t e n t f o r a l l  t o r s i o n a l modes and c o i n c i d e d w i t h the displacement node p o s i t i o n located on the blade by touch.  41  II ro  O  IS)  Figure 4.1  S t r a i n Mode Shapes, 11000 Lbs. S t r a i n  42  II CD  FT2  fO (J OO  S t r a i n Gauge P o s i t i o n  Figure 4.3  S t r a i n Mode Shapes, 18500 Lbs. S t r a i n  44  Figure 4.4 i s an example of the data obtained from gauges 1, 2, 3 and 4 i n the second s t r a i n of 10,000 l b s .  torsional  strain  mode, with a bandmill  The change i n phase between s t r a i n gauge signals  3 and 4 indicates the location of the node. In Section 3.1 the strains were shown to be l i n e a r l y ortional  to displacement  and the experiments  (Section 4.1.4) corroborated t h i s .  of the next  prop-  section  The displacement mode shapes w i l l ,  therefore, be proportional to the s t r a i n mode shapes and a reasonable estimate of the physical shape of the blade can be obtained from the s t r a i n mode shapes. One of the aims of t h i s section of the work was to establish the stresses induced i n the cutting area of the blade due to v i b r a t i o n . The  stresses,  at vibration  amplitudes  considerably higher than  those  recorded for the i d l i n g band, were measured and found to be at the most 900 p s i . This i s only 1-2% of the maximum working stress and not l i k e l y to cause any of the gullet cracking experienced with high s t r a i n bandsaws. ( I t has been noticed that an i d l i n g  bandsaw can develop fatigue  cracks more rapidly than one used for cutting, Claassen [6].) 4.1.3  Strains Due to Forced Displacements - Procedure The  factor  that  section  of the study was to find the  associated the strains i n the vibrating  displacements. idling  object of this  blade with blade  This would allow an estimate of the stresses i n the  band to be obtained from a knowledge of the displacements.  It  would also ensure that the s t r a i n shapes measured i n the previous section were, i n fact, independent of displacement. To  obtain these objectives, a comparison  the change i n s t r a i n oscillations,  at each  was made between  and the change i n displacement of the f i r s t 45  four  natural  due to blade  frequencies.  The  Figure 4.4  Strain-Mode-Shape Data I n d i c a t i n g Change i n Sign Across Node  46  r e s u l t s were obtained with the blade undergoing random frequency e x c i t ation and are presented as s t r a i n per unit  displacement.  Strain and displacement measurements were taken at positions 1, 4 and 7 ( F i g . 2.5) and the s t r a i n s on the i n s i d e and the o u t s i d e of the blade were compared separately to the displacements.  Two d i f f e r e n t  span l e n g t h s were used and the readings taken at two p o s i t i o n s w i t h i n each span (Fig. 4.5).  The blade a x i a l s t r a i n was s e t at 15000 l b s .  • S t r a i n gauges 1, l b , 4, 4b, 7 and 7b were connected to the N i c o l e t v i a the Neff data a c q u i s i t i o n system to take advantage of the a m p l i f i c a t i o n (1000 gain) i n the Neff 100 unit. probes were positioned immediately c o n f i g u r a t i o n (Fig.  below s t r a i n gauges 1, 4 and 7. This  4.6) enabled the displacements and the s t r a i n s on  e i t h e r s i d e of the blade to be obtained. signals from  The three displacement  one displacement  To a c q u i r e the data, the  transducer and an adjacent s t r a i n gauge  were f e d i n t o the frequency a n a l y s e r and one hundred averages taken.  The b a n d m i l l s t r a i n was a l s o recorded.  were  Using the i n - b u i l t  f u n c t i o n s of the a n a l y s e r , both RMS spectrums were d i s p l a y e d on the screen and the f i r s t  four n a t u r a l f r e q u e n c i e s , the average  d i s p l a c e m e n t s , the average  maximum  maximum s t r a i n s and the s t r a i n s per u n i t  displacement were a l l recorded.  The "RMS" v a l u e s of the two s i g n a l s ,  the " t r a n s m i s s i b i l i t y " and the "coherence",  were displayed and recorded  with the Tektronix plotter. The power to the electromagnetic shaker was then changed to o b t a i n 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 n u m e r i c a l values of the strains, the displacements and the r a t i o of the two were again recorded. The l a t t e r figure was then compared to that from the f i r s t run to check  47  .E  Guide  SG  S  _S£!  F  Guide D F = Position of electromagnetic shaker SG = Position of strain gauges and displacement probe S  Figure 4.5  Instrument Configuration and Span Lengths for Strain per Unit Displacement Data  48  Strain gauges  t•  41,4,7)-  - t f f -  C1B.4B.7B)  Probe  i  Figure 4.6 Strain Gauge and Displacement Probe Locations for Strain per Unit Displacement Data  49  the l i n e a r i t y of the r e s u l t s . strain  gauges.  The  This procedure was repeated for a l l six  configuration of the instruments and  guides were  then changed and the process repeated. 4.1.4  Strains Due to Forced Displacements - Results The  change i n s t r a i n  per unit of l a t e r a l displacement  was  measured at six points across the blade, these were at s t r a i n gauge positions 1, 4 and collected  7 and  are presented  IB, 4B and  7B.  i n Figures 4.7  Examples of the actual data  to 4.14.  A description of the  n o t a t i o n on these f i g u r e s 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 i n Figures 4.15 the  to 4.18.  four instrument  The average values for the four modes for each of configurations are presented  i n Table  III.  The  theoretical values are presented i n Table IV for comparison. The  process for obtaining the s t r a i n per unit  values using configuration "B"  ( F i g . 4.5)  displacement  and the data for position 1,  was as follows: (a)  From the displacement and s t r a i n spectra ( F i g . 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 t r a n s m i s s i b i l i t y ( F i g . 4.8) i s the r a t i o of the two signals and  provides the value of the s t r a i n  per unit displacement.  frequencies of interest the values were 11.5,  11.9,  strain/mm.  i n F i g . 4.16  These were the  values recorded  65.6  At the four and 59 microfor s t r a i n  gauge 1. (c)  The  coherence ( F i g . 4.9)  between the two  s i g n a l s and  i s a measure of the linear i s one,  frequencies of i n t e r e s t .  50  or very c l o s e  relationship  to i t , f o r the  F i g u r e 4.7  RMS Values f o r S t r a i n and Displacement Data, Instrument/Span  51  Configuration B  F i g u r e 4.8  T r a n s m i s s i b i l i t y o f 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  F i g u r e 4.9  Coherence Between S t r a i n and Displacement Data, Instrument/Span  53  Configuration B  CM  CD _l >  LU L d 00  1  G>  N X  I +  • ©  © © ©  •  00  \ <  < QQ L d LU \  \  > >  CO 00  © ©  +©  00 ©  < LO  I I • CM  O) • 00 LO  Figure 4.10  CD  RMS Values for S t r a i n and Displacement Data, P o s i t i o n IB, Instrument/Span  54  Configuration A  < 51  Figure 4.11  DO CO 6)  V)  LY  CD, 21  RHS Values for S t r a i n and Displacement Data, P o s i t i o n 4B, Instrument/Span Configuration A  55  H  1\ CM  CD -J >  LU LU 00  <S>  N X  I +  00 <  QJ  \  \  <  LU L d  > >  <  <S> 00 © G> + I c\i  <  LO G> CM LO  Figure 4.12  RMS Values f o r S t r a i n and Displacement Data PosHion 7B, Instrument/Span Configuration A  56  © © CM o  N X  00  \ <  OJ LO  H  LY  Figure 4.13  1 1 1 1 1 1 1 \ => © (J)  © -  X  o o  Coherence Between S t r a i n and Displacement Data, P o s i t i o n 4B, Instrument/Span Configuration A  57  58  o co  o  o  Theory 56.42  s-  4-> to O  so  4->  O LO  o  a; E  o  |C0  if-  cn  • CQ  CQ  ICQ  FL2  Figure 4.15  S t r a i n per Unit Displacement Values for Instrument/Span Configuration A  59  •3"  -1'  FT2  •co ir~-  80  70  -  60  . Theory 56.42-y  ra  s_  4->  1/1  O  so  50  c oi E <D O  40  ra  a.  30  scu  CL  20  -  10  .  S-  Figure 4.16  S t r a i n per Unit Displacement Values for Instrument/Span Configuration B  60  20  <0  s_ +-> to o  15 .  S-  Theory 14.11-  o  2  o.  10.  CO  4->  so. s+-> 00  5. Theory 3.53-  I I I CO  13  cr  cr.  I  _! FL1  Figure 4.17  FT1  FL2  S t r a i n per Unit Displacement Values for Instrument/Span Configuration C  61  FT2  20  to s-  o  S-  15  Theory  o  I 4  11  7  c E d) (J  n3  CL to  •r  10 .  -  Q  S-  S-  Theory  I  3.53  . co  CO  FL1  Figure 4.18  FT1  FL2  1  FT2  S t r a i n per Unit Displacement Values f o r Instrument/Span Configuration D  62  CO  a.  TABLE I I I Average Strain Per Unit Displacement Values  FL1  FT1  FL2  FT2  Span  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  Configuration  Posit]Lon Force P+SG's  TABLE IV Theoretical Strain Per Unit Displacement Values f o r L = 760 mm  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  Configuration  63  (d)  Figure 4.16  presents the data for a l l six locations for instrument  c o n f i g u r a t i o n "B".  The values from (b) above are the f i r s t values i n  each of the four columns. From the t r a n s m i s s i b i l i t y of Figure 4.8, l e v e l s of s t r a i n per u n i t displacement.  note two  distinct  The lower l e v e l i s over the  range of the fundamental l a t e r a l and t o r s i o n a l frequencies, FL1 and FT1, and i s the s t r a i n per u n i t displacement value f o r s i n g l e c u r v a t u r e of the blade over the span l e n g t h .  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 f r e q u e n c i e s , FL2 and FT2, and i s the s t r a i n per u n i t displacement value f o r double c u r v a t u r e 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 i n Table III.  The t h e o r e t i c a l values of s t r a i n per unit displacement are presented i n Table IV and the c o r r e l a t i o n i s extremely good except where the s t r a i n gauge was l o c a t e d c l o s e to a node.  Where t h i s occurred, the values of  s t r a i n per unit displacement have been marked with an "N" (Figs. 4.15 to 4.18) and i n most cases errant values were located close to a node. To further analyse the errant values, the actual s t r a i n and displacement  data were investigated.  Figures 4.10  to 4.14  present the  data c o l l e c t e d f o r c o n f i g u r a t i o n "A" at p o s i t i o n s IB, 4B and 7B. data include the RMS  spectra for the displacement probe and s t r a i n gauge  signals at a l l three positions and the coherence for position 4B, Examination  The  as t h i s was  and  transmissibility  the location of the dominant errant value.  of the frequency  spectrum  f o r p o s i t i o n 4B  (Fig.  4.11)  r e v e a l e d a d i s c o n t i n u i t y i n the s t r a i n and displacement t r a c e s at the f i r s t and second t o r s i o n a l frequencies.  To explain t h i s , i t was  recog-  nized that the strain/displacement data, for at least one of the three  64  p o s i t i o n s of the blade, experienced a 180 degree phase s h i f t between each natural frequency. spectra  On closely  inspecting the strain/displacement  f o r a l l three p o s i t i o n s ( F i g u r e s 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 i n phase; - between FL1 and FT1, position IB maintained a smooth t r a n s i t i o n while position 7B went through a discontinuity at approximately 70 Hz,  and  position 4B went through a discontinuity at FT1 (approximately 80 Hz); - from t h i s information i t was concluded that position IB maintained the same phase w h i l e p o s i t i o n 7B switched to being 180 degrees out of phase at 70 Hz, and p o s i t i o n 4B at approximately 80 Hz r i g h t at the formation of FT1; - finally,  i t was  concluded  natural frequency (which was  that the phase change, o c c u r r i n g at a where the data was  sampled),  caused  the  discontinuity i n the data. The e f f e c t s of the 180 degree phase s h i f t were p a r t i c u l a r l y severe at the c e n t r e of the blade where the t o r s i o n a l v i b r a t i o n amplitudes were minimal compared to the l a t e r a l and the displacements went a b r u p t l y to zero very c l o s e to the t o r s i o n a l f r e q u e n c i e s ( F i g . 4.11). This caused the loss of coherence (Fig. 4.13), the subsequent non-linear peak i n the t r a n s m i s s i b i l i t y (Fig. 4.14), and the large errant values i n the t o r s i o n a l frequency data from position 4B.  The data for a l l of the  errant values were investigated and i n every case, at that position, the 180 degree phase s h i f t occurred very close to the natural frequency of interest.  The d i f f e r e n c e between back-to-back s t r a i n gauge readings  (i.e. 1 and IB) was  noted and, although several a f f e c t s were considered,  no explanation could be found.  The e f f e c t of the a x i a l s t r a i n component  65  (assumed to be small i n the formulation of the theory) was given careful consideration. The p l o t s of t r a n s r a i s s i b i l i t y a r e o f p a r t i c u l a r i n t e r e s t when c o n s i d e r i n g the o r i g i n a l aim of being a b l e to deduce the s t r a i n s 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 i n f o r m a t i o n to c a l c u l a t e 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. p o s i t i o n 7.  F i g u r e 4.19 i s a p l o t of the d i s p l a c e m e n t spectrum a t  Comparing t h i s to the t r a n s m i s s i b i l i t y p l o t a t the same  l o c a t i o n ( F i g . 4.20), i t was p o s s i b l e to e s t i m a t e the s t r a i n s i n the running blade from the two p l o t s .  The r e s u l t of t h i s c a l c u l a t i o n has  been added to Figure 4.19 and, f o r the worst combination of mode shapes ( a l l additive), was l e s s than two microstrain. 4.2  Idling Blade Dynamics 4.2.1  I d l i n g Blade Dynamics - Procedure The dynamics of the i d l i n g blade were investigated by e x c i t -  ing the blade between the guides and measuring the applied force and the r e s u l t i n g blade displacement. to  generate  frequency  response  frequencies could be obtained.  excitation  These values were then used  f u n c t i o n s from  which  the n a t u r a l  The f i r s t four natural frequencies were  investigated f o r f i v e guide spacings, two a x i a l loadings and f i v e 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 r e q u i r e d t o overcome the n o i s e i n the data caused by the s e l f e x c i t e d v i b r a t i o n s of the running blade.  The e x c i t a t i o n f o r c e was  measured with a force transducer b u i l t into the magnet support bracket 66  \  1 1 1 1 1 1 1 1  00 —  Figure 4.19  H  Displacement Spectrum of the I d l i n g Blade  67  . CD CD  Figure 4.20  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 at P o s i t i o n 7 on the Sawblade  68  and the displacement of the blade was measured with one of the displacement 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 g u l l e t l i n e (Fig.  4.5).  The probe was on the opposite side of the blade to magnet. The  signals  the displacement fifty  to one  from  the force transducer (on the magnet) and  probe were fed into the frequency  hundred  samples were taken  function to completely s t a b i l i z e the data. of  for each  analyser and frequency  from  response  The "receptance" (the r a t i o  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 s t r a i n was recorded.  This indicated  the  change due  to the dynamic a x i a l  initial  static  axial  loading caused  monitored  and  loading and  the  by the blade  rotation  around the wheels. 4.2.2  Idling Blade Dynamics - Results This section investigates the e f f e c t of blade speed on the  natural frequencies of the blade.  The investigation was  completed for  two l e v e l s of a x i a l loading and f i v e d i f f e r e n t guide spacings. The data has been presented i n three stages. examples  of the  data  collected  stage i s the c o l l a t i o n and comparison data and  are  presented  First, typical  (4.2.2.1).  The  second  p l o t t i n g of a l l the data collected and  of t h i s to theory (4.2.2.2). known theory with  The  the modified  the  t h i r d stage compares the  theory  and  investigates  the  s e n s i t i v i t y of the modified theory (4.2.2.3). The guide spacings are referred to by the following code and the (Fig.  spacing 2.3).  i s the For  inside-to-inside  calculating  d i s t a n c e between  the  guides  the natural frequencies a more accurate 69  span length was required and t h i s was obtained by tapping the sawblade over the s u r f a c e of the guide to l o c a t e the c o n t a c t p o i n t where the blade span ended.  In t h i s manner, r e a l i s t i c and accurate span lengths  were obtained and these are also l i s t e d .  Guide Spacing  Distance (mm) (Inside-to-Inside)  Span Length  A  415  487  B  520  584  C (standard)  760  822  D  1636  1689  E  2368  2400  (mm)  Axial Loading Upper l e v e l 16500 l b s Lower l e v e l 10000 l b s 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, f o r an a x i a l  s t r a i n of 16500 l b s and guide spacing C, are presented i n F i g u r e s 4.21 to 4.26 i n the form of receptance ( d i s p l a c e m e n t / f o r c e ) and coherence plots.  F i g u r e 4.21 shows the zero rpm receptance and the f i r s t four  70  T'fT"  1  1  (0 LO Figure 4.21  1 H f h-  Receptance of Blade @ Zero RPM 71  V) LO Figure 4.22  O O  Coherence of Blade @ Zero RPM 72  Figure 4.23  Receptance of Blade @ 300 RPM 73  CY Figure 4.24  D O (0 LO  I O U  Coherence of Blade @ 300 RPM 74  if) LO Figure 4.25  h-  Receptance of Slade @ 600 RPM 75  76  natural  frequencies are e a s i l y  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 r a t i o of unity 4.23  and  4.25  (u/co(n) = 1).  show the receptance at 300 and 600  rpm  Figures  and a l l four  frequencies can be seen to decrease with increasing blade speed as we would  expect  coherence  from  the theory.  Figures 4.22,  plots f o r the zero, 300 and  4.24  600 rpm  and  data and  4.26  show the  the excellent  coherence of the zero rpm conditions can be seen to rapidly deteriorate once the bandsaw i s set i n motion.  This i s due to blade e x c i t a t i o n from  sources other than the electromagnet. the  resonant  analyzer 1/2 Hz  frequencies i s due  and  i s known as  to  The loss of coherence at each of the  bias  error.  for the zero to 200 Hz  range,  frequency The  was  resolution  resolution, too large  of  i n this  the case  to describe the  rapidly changing functions that were encountered near resonance on the l i g h t l y damped blade. 4.2.2.2  Comparison of Data with Theory The  frequencies  fundamental  f o r each  of  the  guide  lateral  and  spacings  torsional  are  natural  compared  to  the  t h e o r e t i c a l l y predicted values and the results are plotted against blade speed.  Figures 4.27  and Figures 4.31 the f i v e  spans  to 4.30  to 4.34,  indicate the l a t e r a l frequency  the t o r s i o n a l .  have been s p l i t  For the purpose of c l a r i t y ,  between two  figures, e.g. Figure  shows the data f o r spans A, C and E and figure 4.28 spans B and D.  comparison  4.27  shows the data for  I t should be noted that the t h e o r e t i c a l r e s u l t s of Mote  and Alspaugh are based on the assumption  that there i s constant stress  d i s t r i b u t i o n across the blade.  77  The  lateral  natural frequencies  with theory, which i s the s t r i n g equation, span lengths discrepancy probably  D and  The  p a r t i c u l a r l y for the  shorter span lengths, A and  longer  This i s  to the boundary conditions, which were modeled as  not  being  t h i s would tend  an exact  representation of the end  well  B, show some  especially for the lower (10,000 lb) s t r a i n l e v e l .  due  supports,  E.  compare very  simple  conditions and  to have a greater a f f e c t 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 torsional [2]),  frequencies  particularly  difference having  contrast compared  f o r the  to  especially  have  a  for the  i s the  strong  shorter  increase  lateral  poorly  shorter  with  frequencies,  the  span lengths  theory A,  The  (Alspaugh  B and  primarily to the model and  the  C.  The  the  blade  additional stress i n the  to the parabolic stress d i s t r i b u t i o n would be  accuracy i n predicting them. theory,  the  stress d i s t r i b u t i o n s .  edges of the sawblade due expected  very  i s thought to be due  different  to  effect span  on  the  lengths,  torsional  hence the  frequencies,  greater  loss of  Another factor, not included i n Alspaugh's  i n stress due  to  the  rotation of  the  blade  around the wheels, sometimes called dynamic tension, which increases the axial  strain  and  hence the  cases  the experimental  frequency  with  increasing speed.  results were much higher than the  theoretically  predicted values indicating additional s t i f f n e s s i n the blade.  78  In a l l  Theory Mote  120 o o o  [l*]  Experiment  100  5* c  80 J  <1> =3  Span A  cr <D S-  «  60  40  20  Blade Speed (Ft/s)  Figure 4.27  Comparison of Lateral  Frequencies  with Theory, 10000 Lbs. S t r a i n CI of 2)  79  120 Theory Mote [l4] Q Q Q Experiment  ioo .  5  80 .  >> (_> c: CD  1  o  cr  £  U-  60 -  o  _  Span B o o  -o  5  40 .  Span D 20 -  —n-  o  0 ()  40  80  120  Blade Speed (Ft/s )  Figure 4.28  Comparison of Lateral Frequencies with Theory, 10000 Lbs . S t r a i n ( 2 of 2)  80  160  120 Theory Mote o o o  [14]  Experiment  100 J  80 >>  o sz <u cr <D s-  60  <D ra  40  20 3  no  lib" Blade Speed (Ft/5)  Figure 4.29  Comparison of Lateral Frequencies with Theory, 16500 Lbs. S t r a i n (1  81  of  2)  120 . Theory Mote [14) o o o  ^100<  o  >> o  S  .  o  —  n  Experiment  _  Span B  80.  a i ^> u . cu  ™ 60 CQ  40 . Span D  »_  o  «  0  20-  0 40  8d Blade Speed  Figure  4.30  120 (Ft/s)  Comparison of Lateral  Frequencies  with Theory, 16500 Lbs. Strain C2 of 2)  82  160  140  -  Theory Alspaugh Experiment  0 0 0  120  .  >  o 0  N n: 100  0  to CD O  Freqi  c  80 -  o  ra c  o tl o  Span A  60  .  Q  o  -  1—  Span C 40 -  20:  0  O  o  • 40  1 80  °  Span E  120  Blade Speed (Ft/s)  Figure 4.31  Comparison of Torsional Frequencies with Theory, 10000 Lbs. S t r a i n (1  83  of  2)  o"  160  [2]  •2120 •— I/) QJ o  0 0 0  Theory AT spaugh [2] Experiment  c  §L00  4  CT <V Su_  O  0  0 o  § 80 s_ o h-  •r— cn  Span B  60 •  40 >  o  o  °  Span D °  20 •  n 40  80  120  160  Blade Speed (Fb/s)  Figure 4.32  Comparison of Torsional Frequencies with Theory, 10000 Lbs. S t r a i n 84  ( 2  o f  2  )  Figure 4.33  Comparison of Torsional Frequencies with Theory, 16500 Lbs. S t r a i n (1 of 2)  85  Theory Alspaugh  Figure 4.34  Comparison of Torsional Frequencies with Theory, 16500 Lbs. S t r a i n (2  86  of  2)  [2]  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 f o r t o r s i o n a l vibration i s unable to predict the frequencies accurately. stress  This i s probably  distribution  due to the inadequate  which has not taken  r o l l - t e n s i o n i n g and c e n t r i f u g a l forces.  into  modelling of the  account  the effects of  In t h i s section, these e f f e c t s  are included and the r e s u l t s analyzed. As presented  i n Section 3.2, the equations of motion  can be modified to include f o r a parabolic stress d i s t r i b u t i o n across the blade as an approximation  of the r o l l - t e n s i o n i n g e f f e c t s .  As the  actual magnitude of the r o l l i n g stresses are unknown, t h i s introduced an unknown stress l e v e l , Op, into the equation of motion. Op i s the maximum value of the assumed parabolic stress ( F i g . 3.5). In this work, the theoretical agree  value  of the fundamental t o r s i o n a l  e x a c t l y with  frequency  was made to  the data o b t a i n e d , at zero rpm, by choosing an  appropriate value of o_. The value of C  was then used i n equation 3.12  to predict the blade frequencies for the non-zero rpm condition (Section 3.2.3).  The modified t h e o r e t i c a l curves f o r several d i f f e r e n t sets of  parameters,  including the dynamic tension e f f e c t s , were then compared to  the o r i g i n a l 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 f o r spans A and B at the 10,000 lbs s t r a i n l e v e l and span B at the 16,500 l b s t r a i n l e v e l exhibit a s i g n i f i c a n t o f f s e t f o r all  the non-zero blade speed  points f o r which no explanation could be  found.  87  Theory Alspaugh 140  ooo  Experiment  - - -  Modified Theory (Eqn.  [2]  3.12)  120!'  100-  o c OJ 3  80-  cr  <L> J-  60.  co n3  Span C 40-  -o  20-.  — i —  — i —  — i —  i  40  80  120  160  Blade Speed (Ft/s)  Figure 4.35  Comparison of Data and Theory with Modified Theory, 10000 Lbs. S t r a i n CI of 2)  88  Theory Alspaugh 140  0  0  0  [2]  Modified Theory •(•E.qn. Experiment  3.12)  >, 120-••  c  cr cu s-o  80 Span B  60  40Span D 20.  40  80  160  120  Blade Speed (Ft/5)  Figure 4.36  Comparison of Data and Theory with Modified Theory, 89  10000  Lbs. S t r a i n  (2  of  2)  Theory  Alspaugh  [2]  Modified Theory ( E q n . 3 . 1 2 ) 0  0  Experiment  0  140-  0  J,  -  i  —  40  — —  1  1  80  120  Blade Speed (Ft/s) Figure 4.37  Comparison of Data and Theory with Modified Theory, 16500 Lbs. S t r a i n (1  90  o f 2)  »-  160  Theory Alspaugh [2] o o o -  Experiment  - - Modified  140.  (Eqn.  Theory  3.12)  o  120 • Span B  _  80.  zn >~>  o  t—  I  60-  u.  -  40, "  —'•—•  Span D  20-  0 40  80  120  Blade Speed ( F t / s )  Figure 4.38  Comparison of Data and Theory with Modified Theory, 16500 Lbs. S t r a i n (2 of 2)  91  160  Introducing an assumed parabolic  stress  distribution  (o"p) into the t o r s i o n a l frequency equation, and assuming the difference between the t h e o r e t i c a l (Alspaugh [2]) and experimental r e s u l t s was due e n t i r e l y to t h i s stress d i s t r i b u t i o n , enabled an empirical to be obtained.  Using the methods of Allen [3], the stress ( O p ) due to  roll-tensioning values of o^,  value for  was estimated  to be i n the order of 20,000 p s i .  obtained empirically  are shown i n Table V.  from the t o r s i o n a l frequency  The data,  For a bandmill s t r a i n of 10,000 l b s the values  averaged 23,504 p s i with a standard deviation  of 1269 p s i (about 5%).  The values obtained f o r a bandmill s t r a i n of 16,500 l b s were also very consistent  with an average value of 23,986 p s i and a standard  of 751 p s i (3%).  deviation  The average of both sets of data was 23,745 p s i with a  standard deviation of 1015 p s i (4.3%).  The r e s u l t s are reasonably close  to the estimated value of 20,000 p s i f o r t h i s sawblade, indicating the the error i n the torsional frequency prediction was due primarily to the stress d i s t r i b u t i o n from r o l l - t e n s i o n i n g .  TABLE V Values of a  Span  Obtained  Empirically  10,000 l b s Op p s i  16,5000 l b s Op p s i  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  cutting  tests  were  i n v e s t i g a t i o n i n t o blade displacement cutting at various feed speeds.  intended  as a p r e l i m i n a r y  and modes of v i b r a t i o n  I t should be emphasized that t h i s was  not intended to be an i n - d e p t h study of the blade behaviour cutting.  The experiments  while  were completed  during  to p u l l together the work of  blade stresses and dynamics and to prepare a s t a r t i n g point f o r the next stage of investigation. The displacements  cutting  test  data  were obtained  by r e c o r d i n g the  of the f r o n t and back edges of the blade, during the  actual cutting process, f o r 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  s t r a i n were also recorded during each of the cutting tests.  Details of  the experimental set-up are shown i n Figure 4.39. The variations i n cutting rate were obtained by setting the log carriage feed speed to the recommended maximum f o r a bandmill speed of 600 rpm and then v a r y i n g the b a n d m i l l wheel speed desired result.  to o b t a i n the  From Section 3.3, the maximum feed speed was calculated  to be 273 fpm f o r a bandmill speed of 600 rpm.  The maximum feed speed  of the l o g c a r r i a g e was obtained by r e c o r d i n g the output of generator attached to the carriage drive system.  a d.c.  Seasoned hemlock was  used f o r the c u t t i n g t e s t s 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 s i g n a l s from the displacement probes and the d.c. gene r a t o r were f e d i n t o the Neff 100 data a c q u i s i t i o n system and the Nicolet frequency analyzer.  The a x i a l s t r a i n value was set to 16500 l b s 93  -6ft x 2 f t x 1ft cant  Guide  T'f 10" Displacement probes  c = Blade Speed  ,12%"  r—i ~IZJ~  F = Cant Feed Speed  12"  1" set - Q l  Figure 4.39  Guide  Experimental Set-up for Cutting Tests  and the bandmill speed was monitored at the beginning of each run. The data f o r the cutting tests were obtained using both the N i c o l e t and the Neff.  The N i c o l e t was set to be t r i g g e r e d by the d.c.  generator s i g n a l and the Neff was a c t i v a t e d by running the program "CUT.FOR" on the computer terminal.  This program prompted the user f o r  the name of the data f i l e ( p r e v i o u s l y generated by u s i n g the program "SCANLIST") and s e t 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 c o n t i n u o u s l y sampled channel 9 f o r a non zero voltage.  The l o g c a r r i a g e was set i n motion to c a r r y the cant toward  the bandsaw at the preset speed.  Just before the cant reached the saw,  the l o g 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 N i c o l e t . channel 9, immediately  The Neff 100, on r e c e i v i n g t h i s v o l t a g e on sampled a l l three of the input s i g n a l s (two  probes and d.c. generator) a t the s p e c i f i e d sampling  r a t e u n t i l the  storage b u f f e r i n the Neff 500 computer i n t e r f a c e u n i t was f u l l (4096 samples).  The program then prompted the user for the name of the s t o r -  age f i l e f o r the data. sampled the data from completed  The N i c o l e t ,  t r i g g e r e d by the same s i g n a l ,  the probe a t the f r o n t of the blade.  Having  the run, the data captured by the N i c o l e t was p l o t t e d u s i n g  the Tektronix plotter. 4.3.2  Cutting Tests - Results The  behaviour  of the sawblade d u r i n g the a c t u a l  cutting  process was investigated and the r e s u l t s compared to the known natural dynamic behaviour of the blade and to the cut surface i n the cant. Six data c o l l e c t i o n runs were made for cutting rates of 78%, 81%, 87%, 94%, 103% and 110% of the maximum c u t t i n g r a t e (see S e c t i o n 3.3) f o r the sawblade.  The p l o t s of the behaviour of the blade during 95  CM  cu X> O S-  X) o S-  +-> c o S-  CO  a.  cu  o to  CO  LO 00  co CU XI o SQ. to CU  I  >-  I  CO CS>  x: o to  O) S+-> c (O o CU C\J  c\j CM  CO  CO o  CS> in  LO  I  CM I  z: 2: Figure 4.40  Sawblade Behaviour during Cutting (78% mfr.)  96  CO I  CO Q 2 O c_> LO CO v_/ LU  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%  99  mfr)  100  Q  LO 03 I  1  Figure 4.45  LO • — I  CM I  LO • CM I  Sawblade Behaviour during Cutting (110% mfr)  101  O I  the cut are presented i n F i g u r e s A.40 to A.A5.  The a b c i s s a has been  marked w i t h the times t h a t the l e a d i n g edge of the cant reached the f r o n t and back probes (FP and BP) and, f o r the p r e s e t feed speed of 273 fpm, t h i s occurred at .06 seconds and .22 seconds respectively. time taken to complete the cut was approximately 1.3 seconds.  The  Returning  to the blade displacements, the large amplitude i d l i n g o s c i l l a t i o n s can be seen to r a p i d l y decrease as the cant reached each probe i n turn. From t h i s point on, the displacement of the blade was composed of small high  frequency  oscillations  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  r e l a t i v e l y insensitive to the increase i n cutting rate (78% to  87%), however, t h i s changed very rapidly as the estimated 100% value was approached and the plot of the sawblade path disappeared from the graph (or, i n fact, exceeded the s e n s i t i v i t y range of the probes) for the 110% value.  This was  expected and reinforced confidence i n the methods used  for estimating maximum cutting rates based on g u l l e t capacity. The instantaneous spectrum of the blade displacements during each cut, obtained from the probe at the f r o n t of the blade, are p r e sented i n F i g u r e s A.A6  to 4.51.  An examination of the displacement  spectrum f o r each of the s i x data runs shows the l a r g e low frequency component and also two other noticeable "peaks" at approximately 120 Hz and 140 Hz.  To compare the frequency spectrum w i t h the i d l i n g blade  behaviour, the n a t u r a l f r e q u e n c i e s f o r the sawblade a t 600 rpm were estimated from the blank blade data (Section 4.2.2) and are as follows:  102  CD _J >  O CD — -  c  •r•4-> +J 3 C_>  C\J h Ll_  > CD CD  + (0  00  o  cr cu  to  +J  S- co 4- c o  o  S  o +-> i — to CU  i—  cn - i s- o  to CO o  _ J  Z ) CD 00 — Figure 4.46  00 H  Displacement Spectrum of Sawblade During Cutting (78% mfr)  103  3 . 1 6+00  V  VLG C  A M SU 16  Large low freq, Oscillations  IS-tt  FT2 (Cutting) FT2 ( I d l i n g ) Wheel Rotation 10.9 HZ 10.9 HZ Multiples  5.0A  IA A I il 11 n  -5.0D  B/16  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  - 0 . 0 4 9 7  -o ai o n> 3 rt> 3  c+ 00  •o fD  O.  ro  V L G  M Large low freq. Oscillations  -s c  0)  V  A  r+  GO  1 0 0 . - 0 3  C  o  3  D L T A  SU 1 6  FT2 (Cutting) FL2  I S  +  FT2  Wheel Rotation 8.5 HZ  (Idling)  a tz  5 —i. 3  ca  o c  r+ c+ —i.  3  CO  O 3  -h -S  5.0A  - 5 . 0 D  B / 1 6  HZ  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 l a t e r a l frequencies were v i r t u a l l y unaffected and,  i f we  take this into consideration, the two "peaks" are seen to be the second l a t e r a l and t o r s i o n a l frequencies of the i d l i n g sawblade. frequency remained frequency was  The  lateral  v i r t u a l l y unchanged at 120 Hz but the t o r s i o n a l  reduced from  The  wheel  rotation frequencies (11.6 Hz i n Fig. 4.46) and the corresponding  mult-  i p l e s were c l e a r l y  160 Hz to approximately  140 Hz.  v i s i b l e i n a l l of the displacement spectrum  data and  were responsible for most of the remaining peaks on the graphs. A comparison finished  surface  of  of the recorded blade displacement with the  the  cant  i s presented  i n F i g u r e 4.52.  correlation between the low frequency o s c i l l a t i o n  The  of the blade and the  o s c i l l a t i o n i n the cut surface of the cant was very good.  However, the  magnitudes of the d i s p l a c e m e n t s i n the cut s u r f a c e were l a r g e r than those measured f o r the blade.  This was  possibly due to the probe being  p o s i t i o n e d 2 i n . behind the c u t t i n g l i n e of the t e e t h , 2.25 the cant and, consequently, 6.75  i n . above  i n . above the l i n e of measurement of  the cant surface.  109  F i g u r e 4.52  Comparison o f Blade Displacement  110  Data with Actual  Cut  5.  CONCLUSIONS  The  purpose  of the work was  threefold:  to obtain an estimate of  the stresses induced i n an i d l i n g blade, with a view to i d e n t i f y i n g the specific  factors  involved i n g u l l e t  cracking; to measure the  natural  frequencies of the i d l i n g blade for validation of the a n a l y t i c a l models and  f o r comparison  cutting; and  with the dynamic behaviour of the blade d u r i n g  to examine the behaviour of the blade during the cutting  process, for comparison  with the known dynamic c h a r a c t e r i s t i c s of the  blade and with the finished p r o f i l e 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 v i b r a t i o n a l blade d i s placement  are due  change i n a x i a l  primarily  length.  to bending  The  of  the  data shows that  blade and the s t r a i n  not  to the  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 s t r a i n  per unit displacement  frequencies, i t was  possible to obtain an estimate of the s t r a i n i n the  running blade.  values for the various blade  The estimate indicated that the stresses due to i d l i n g  vibration were small compared to the normal operating stresses, estimated using the methods of Allen  [3], and not l i k e l y to cause any of the  gullet cracking problems experienced in i d l i n g handsaws. The strain-mode-shapes  (the d i s t r i b u t i o n of s t r a i n across the blade  due to vibration i n a fundamental axial  mode) were found to be independent of  l o a d i n g f o r both the l a t e r a l  and  t o r s i o n a l modes and were a  function of the stress d i s t r i b u t i o n i n the blade due to r o l l - t e n s i o n i n g . For example, the portions of the blade with high a x i a l s t r a i n experience 111  reduced  v i b r a t i o n a l displacement  l e a d i n g to a reduction i n the s t r a i n  mode shape at t h i s position. The magnitude of s t r e s s , due to f o r c e d 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 i d l i n g blade, was a t most 1-2% of the t o t a l e s t i m a t e d s t r e s s e s i n the i d l i n g band and reinforced the conclusion that g u l l e t cracking i s unlikely to be caused by i d l i n g blade vibrations. Due to the l i n e a r r e l a t i o n s h i p between the s t r a i n s and d i s p l a c e ments, coupled w i t h the r e s u l t s of the s t r a i n mode shape data, a good e s t i m a t e of the displacement mode shapes of the blade can be obtained from the s t r a i n mode shape results. 5.2  Idling Blade Dynamics The natural frequencies vs. blade speed  were obtained for each of  f i v e d i f f e r e n t span lengths and two d i f f e r e n t a x i a l prestresses. The  experimental  lateral  band  natural frequencies exhibited  e x c e l l e n t c o r r e l a t i o n w i t h theory (which i n t h i s case was the s t r i n g equation as the plate bending  e f f e c t s had been shown to be negligible)  p r o v i d i n g a c c u r a t e span l e n g t h s 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 w i t h theory (which assumed constant a x i a l stress d i s t r i b ution) e s p e c i a l l y  f o r the s h o r t e r span l e n g t h s .  T h i s was l a r g e l y  attributed to the r o l l - t e n s i o n i n g stresses i n the blade and the dynamic tension e f f e c t s . Modifying the t o r s i o n a l frequency equations to include for a parab o l i c r o l l - t e n s i o n i n g s t r e s s d i s t r i b u t i o n a c r o s s the blade enabled a constant empirical value of the parabolic stress to be obtained (for a  112  combination of span l e n g t h s and a x i a l l o a d i n g s ) .  The value obtained  compared well with available theory, indicating that the error i n the t o r s i o n a l frequency prediction was due primarily to inadequate modelling of the in-plane stress d i s t r i b u t i o n caused by roll-tensioning. Use of the modified stress d i s t r i b u t i o n i n the t o r s i o n a l frequency equations, p l u s the e f f e c t s of dynamic t e n s i o n due to blade r o t a t i o n around the wheels, gave a much improved  p r e d i c t i o n 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, c e r t a i n 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 o s c i l l a t i o n s of the blade.  These o s c i l l a t i o n s occurred even when the c u t t i n g r a t e  was well below the estimated maximum and were more l i k e l y a function of blade s t i f f n e s s than vibration.  The higher frequency components of the  c u t t i n g blade were more l i k e l y to a f f e c t the k e r f width and s u r f a c e q u a l i t y and were, i n t h i s case, composed of the second torsional multiples.  f r e q u e n c i e s and  the wheel speed  frequency,  lateral  plus a l l i t s  From these r e s u l t s i t i s apparent t h a t improving  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 accuracy.  and  blade  cutting  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 o s c i l l a t i o n s of the blade. C o r r e l a t i o n 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 all  the major d i s p l a c e m e n t s of the blade e a s i l y d i s c e r n i b l e i n the  113  f i n i s h e d surface. generally attributed,  The magnitude of the d i s p l a c e m e n t s i n the cut were  l a r g e r than those recorded  f o r the  i n p a r t , to the l a t e r a l f l e x i b i l i t y  blade.  This  of the t e e t h  was being  greater than that of the sawblade at the probe location and, i n part, to the separation of the grain "tear out" i n the cutting process.  114  6.  REFERENCES  [I]  Anderson, D.L., "Natural Frequency of L a t e r a l V i b r a t i o n s of a Multiple Span Moving Band Saw". Research Report for the Forestry Directorate, Environment Canada, Western Forest Products Laboratory (now F o r i n t e k Canada corp.), 6620 N.W. Marine D r i v e , Vancouver, B.C., V6T 1X3, January 1974.  [2]  Alspaugh, D.W., " T o r s i o n a l V i b r a t i o n s of a Moving Franklin I n s t i t u t e , Volume 283(4): 328-338, 1967.  [3]  Allen, F.E., "High Strain Theory and Application". Proceedings of the 8th 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 1985.  [4]  A r c h i b a l d , F.R., E m s l i e , A.G., "The V i b r a t i o n of a S t r i n g Having a Uniform Motion Along I t s Length". J o u r n a l of A p p l i e d Mechanics, American Society of Mechanical Engineers, Paper No. 58-APM 7, 1957.  [5]  B i o t , M.A., "Increase of T o r s i o n a l 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 A x i a l T o r s i o n " . J o u r n a l of A p p l i e d P h y s i c s , V o l . 10, No. 12, pp.860-864, December 1939.  [6]  Claassen, L., "Determination of the F e a s i b i l i t y of Increasing the Band Speed of High S t r a i n , T h i n K e r f Bandsaws". Research Report prepared f o r Hawker S i d d e l e y 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, J u l y 1975.  [7]  Das, A.K., " A n a l y s i s of Dynamic S t a b i l i t y of Bandsawing Systems". Proceedings of the 7th Wood Machining Seminar, U n i v e r s i t y of C a l i f o r n i a , F o r e s t Products Lab., Richmond, C a l i f o r n i a , October 1982.  [8]  E s c h l e r , A., " S t r e s s e s and V i b r a t i o n s i n Bandsaw Blades", M.A.Sc. T h e s i s , Dept. of M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, V6T 1Z2, 1982.  [9]  F o s c h i , R.O., "The L i g h t Gap Technique as a T o o l f o r Measuring R e s i d u a l S t r e s s e s i n Bandsaw Blades". Wood Science & Technology 9:243-255, 1975.  Band".  J.  [10] G a r l i c k i , A.M., M i r z a , S., "The Mechanics of Bandsaw Blades". Department of the Environment, Eastern Forest Products Laboratory (now F o r i n t e k Canada Corp.), 800 M o n t r e a l Road, Ottawa, Ont. K1G 3Z5, 1972. [II] G a r l i c k i , A.M., M i r z a , S., " L a t e r a l S t a b i l i t y of Wide Band Saws". Proceedings of the 4th Symposium on E n g i n e e r i n g A p p l i c a t i o n s of Solid Mechanics, held Ontario Research Foundation, 25-26 September, 1978, V2:273-287.  115  [12] Kirbach, E., Bonac, T., "The E f f e c t of Tensioning and Wheel T i l t i n g on the T o r s i o n a l and L a t e r a l 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 L a t e r a l Natural Frequencies of Bandsaw Blades". Society of Wood Science and Technology, Wood and Fibre, 10(1) 1978, pp.19-27. [14] Mote, CD., "Some Dynamic C h a r a c t e r i s t i c s of Bandsaws". Products Journal, Vol. XV, No. 1, January 1965A. [15] Mote, CD., "A Study of Bandsaw V i b r a t i o n s " . i t u t e , V o l . 279, pp.430-444, 1965. [16] Quelch, P.S., "Sawmill Feed and Speeds". Portland, Oregon, 1964.  J. Franklin  Armstrong  Forest Inst-  Mfg. Co.,  [17] S o l e r , D.I., " V i b r a t i o n s and S t a b i l i t y of a Moving Band". Franklin Institute, Vol. 286, No. 4, pp.295-307, October 1968.  J.  [18] Tanaka, C , S h i o t a , A., "Experimental S t u d i e s on Band Saw Blade Vibration". Wood Science and Technology 15, pp.145-159, 1981. [19] Timoshenko, S., Woinowksy-Kreiger, A., "Theory Shells". McGraw-Hill, 1979, Second Ed.  of P l a t e s and  [20] Ulsoy, A.G., Mote, CD., " A n a l y s i s of Bandsaw V i b r a t i o n " . Science, V o l . 13, No. 1, pp.1-10, J u l y 1980.  Wood  [21] Ulsoy, A.G., Mote, CD., Syzmani, R., " P r i n c i p l e Developments i n Bandsaw V i b r a t i o n 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., " A n a l y s i s of V i b r a t i o n i n a Band Saw System". 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.  116  APPENDIX I INSTRUMENT LIST  1.  Loadcell Strain Gauges, EA-06-125AD-120, K=2.065, 120 Ohms.  2.  Bruel & Kjaer P i e z o - E l e c t r i c 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  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 D i g i t a l P l o t t e r .  15.  Bruel & Kjaer 1024 Signal Generator.  16.  Nicolet 660A Dual Channel FFT Frequency Analyser.  17.  K i s t l e r 504D Charge amplifier.  18.  1 No. Kamen Non-Contacting  19.  1 No. Kamen O s c i l l a t o r Demodulator Unit.  20.  Vishay P-350A D i g i t a l Strain Indicator.  21.  10 Watt Power Amplifier.  22.  100 Watt Power Amplifier.  Displacement  117  Probe.  Probes and  APPENDIX I I SUMMARY OF COMPUTER PROGRAMS The following i s 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, w i t h a b r i e f d e s c r i p t i o n of t h e i r function.  SCANLIST T h i s program i n t e r a c t s w i t h the user to name and b u i l d a f i l e of b a s i c 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 s t o r e s the data i n a f i l e choice.  of the user's  The data i s stored i n a single column of values i n the following  order (example f o r three channels): Channel No.  Data Point  1 2 3 1 2 3  1 1 1 2 2 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, t h i s  program w i l l interact with the user to convert a maximum of four sets of data to the correct four  f i l e s named SET  file  called  EZGRAF,  the  values (remove the gains, etc.) and l.DAT to SET  4.DAT.  store them i n  It also generates a command  GRAF.DAT t h a t takes most of the work out of o p e r a t i n g packaged  graphing  routine i n the  computer.  Having  BREAK, i t i s only necessary to run EZGRAF then run GRAF and the set  of data i s plotted on the terminal screen.  run first  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 s e l e c t i o n .  CONVERT When supplied with the name of the data f i l e program w i l l channel.  interact  with  For a constant  readings i s given.  the  user  generated  by MODE, t h i s  to tabulate the results of each  input s i g n a l ,  the average value of a l l the  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 l i s t e d i n t h i s f i l e , the remainder are l i s t e d 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,  sorts a set of values into increasing order. 119  as the name implies,  CUT T h i s program i s used to c o l l e c t the data from the c u t t i n g t e s t s . program i s set up to run when t r i g g e r e d by a v o l t a g e on channel 9.  The 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 u n t i l a voltage i s detected.  There i s  approximately 30 ms delay between the detection of the voltage and capture of the f i r s t  sample.  120  the  APPENDIX III EXPLANATION OF THE NOTATION ON GRAPHS FROM NICOLET FFT FREQUENCY ANALYZER  121  

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