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
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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
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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
N X
I +
00 <
QJ
\
\
<
LU L d
> >
<
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
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
«
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
60
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
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
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