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An experimental investigation of circular saw vibration via a thin plate model
Int. J. Mach. Tools Manufact. Vol. 34. No. 7. pp. 893-905. 1994 Copyright© 1994ElsevierScienceLtd Printed in Great Britain. All rights reserved 0890-6955/9457.00 + .00
Pergamon
0890-6955(93)E0001-L AN EXPERIMENTAL VIBRATION
INVESTIGATION
OF CIRCULAR
SAW
VIA A THIN PLATE MODEL
E . M A R U I , t S. EMA~ a n d R . MIYACHIt
(Received 1 June 1993; in final form 1 October 1993)
Abstract--This paper deals with important practical problems related to vibration reduction in a circular saw used for wood cut-offs. A n investigation is made of the static and dynamic characteristics of thin plate structures as a model of a circular saw, to obtain fundamental knowledge to improve circular saw performance. The circular saw is represented experimentally by a thin rectangular plate clamped by two shims. The bending rigidity and damping capacity of a structure are measured for six different clamping methods. The results are evaluated by a coefficient, which is introduced through Merritt's regenerative chatter vibration theory. The most effective thin plate structure is found to improve vibration-proof capacity. This structure is easily applicable to an actual circular saw.
1.
INTRODUCTION
WHEN A FLEXURALvibration is induced in a thin disc-type cutter such as a circular saw for wood cut-offs at high spindle speeds, the operating efficiency is lowered and wood resources are greatly wasted. Many studies have reported on vibration problems involving circular saws for wood cut-offs or thin discs and their remedies. Representative reports deal with natural frequency and principal mode of vibration [1-4], forced vibration [5], unstable or self-excited vibration [6-9], increase in dynamic stability [10-14], stiffness [15] and noise [16-18]. An increase in system damping capacity is the key technology for preventing selfexcited or forced vibrations of a circular saw during cutting operation. The contact condition between the circular saw blade and the flange of the main spindle of the circular saw machine has a great influence on the initiation of the system damping capacity of the circular saw. However, to our knowledge, no reported study to date has treated the interrelationship between the system damping capacity and the contact condition. In this paper, the relationships between the static bending rigidity, the damping capacity and the contact condition at the clamping part are measured for a thin plate structure, which is a simple model of the circular saw blade and the clamping flange of the circular saw itself. A fundamental device for improving the cutting efficiency and the stability of the circular saw is clarified. Partly, the effectiveness is ascertained at the thin disc structure model. 2.
E X P E R I M E N T A L A P P A R A T U S AND P R O C E D U R E S
Figure 1 shows an experimental apparatus for measuring thin plate structures, which are used in experiments on bending rigidity and damping capacity. The experimental apparatus was mounted on a surface plate. The hatched parts in the figure indicate the thin plate and the shims for clamping it. These correspond to the circular saw blade and the flange at the main spindle of the circular saw, respectively. They were made of carbon tool steel. Some of the shims were made of cast iron. The thin plate and two shims were clamped by a vice through load cells. The clamping load was 4 kN per load cell. A wire was attached to the free end of the thin plate by a hook. A weight was hung at the other end of the wire. The deflection at the free end of the
tFaculty of Engineering, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan. ~tFaculty of Education, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan. 893
894
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gap detector
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pass f i l t e r H A n p l i f i e r
j
FIG. 1. Experimental apparatus for measuring dynamic characteristics of thin plate structures. thin plate was read by a dial gauge, and the bending stiffness of the plate was calculated by linear regression analysis between the deflection at the free end of the plate and the bending load caused by the weight. Then, the hook was removed instantaneously from the free end of the plate and a step function input applied to the plate. The plate underwent a damped free vibration which was detected by an eddy current displacement sensor. The detected signal was resolved into harmonics by an FFT analyzer, and the frequency of the principal mode of vibration obtained. The logarithmic decrement was calculated from the damping vibration signal by micro-computer. A band pass filter was used to cut the low frequency vibration of the surface plate and some high frequency disturbance from the environment.
3. CONSTRUCTION OF THIN PLATE Figure 2 shows the basic thin plate structure used in this experiment. The thin plate was made from carbon tool steel of 54 mm width, 130 mm length and 2 mm thickness. This plate was clamped by two shims of 2 mm thick and made of the same materials. In some cases, shims of different materials and sizes were used. The length of the 130
40
/
Carbon
tool steel \
FIG. 2. Fundamental thin plate structure.
895
Experimental Investigation of Circular Saw Vibration
j
\Carbon
Epoxy r e s i n
tool s t e e l
FIG. 3. Bonded thin plate structure.
TABLE 1. BONDED THIN PLATESTRUCTURE Plate number B~ B2 B3 B4 B5
Thickness of adhesive bond (mm) 0.097 0.487 0.926 1.171 1.395
tape
'Carb0n
to01 s t e e l
FIG. 4. Adhesive tape thin plate structure.
TABLE 2. ADHESIVETAPETHIN PLATESTRUCTURE Plate number T~ T2 T3 T4
Number of adhesive tape strips 1 2 3 4
shims was 40 mm. Six types of clamping were incorporated and 23 kinds of structures were tested.
3.1. Bonded thin plate structure In this structure, the thin plate and two shims were bonded by an epoxy-based adhesive, as shown in Fig. 3. The adhesive hardens at room temperature. Five adhesive thicknesses were used in the experiment, as shown in Table 1. Adhesive tape thin plate structure The basic structure is given in Fig. 4. The thin plate and two shims were attached to one another with double-faced industrial adhesive tape. The tape thickness was 0.43 mm when no load was applied. This tape has good therma 1 resistance and sufficient bonding strength is maintained up to 150°C. As shown in Table 2, only one strip of adhesive tape was used in the structure T~, and two, three and four strips are applied in layers in structures T2, T3 and T4, respectively. 3.2.
3.3. Wire thin plate structure This type of structure was manufactured as follows: one strip of adhesive tape was applied to the thin plate and piano wires of 0.3 mm diameter were spaced equidistantly
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U
U
U
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i
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o '
U
l~Adhesive
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~tmmmEmmmmmm~d Wira
Ws, W,o, W~s
WS,o
Fio. 5. Wire thin plate structure.
on it. Then, two shims were pressed on to the thin plate as seen in Fig. 5. The piano wires were arranged in parallel with the thin plate axis (structures Ws, Wlo, W,5) or perpendicularly to the axis (structure WSlo). The former arrangement corresponded to the radial direction of the circular saw blade, and the latter to the blade circumferential direction. Table 3 gives the number of piano wires and their pitch P.
3.4.
Projection plate thin plate structure
This type of thin plate was manufactured by the process shown in Fig. 6. A strip of adhesive tape was pre-attached to the thin plate. Rugged shims were pressed on it. Projections on the shims were trimmed by a milling cutter. The projections were arranged at 2.54 mm intervals in a lattice. The size of the projections was varied by changing the depth of cut of the milling cutter, on three levels. The projection height Q is shown in Table 4. The projection of a large Q value has a small contact area at its tip, and becomes sharp. The vertical angle of the projections was 90° , as seen in Fig. 6. 3.5.
Non-bonded thin plate structure
This type of structure was prepared to compare the damping capacity of the preceding structures. The thin plate contacted the two shims directly, as shown in Fig. 7. Shims of different thickness and material were used. The detailed structure is shown in TABLE3. WIRE THIN PLATESTRUCTURE Plate number W5 W,o W,~ WS,o
Number of piano wires
Wire pitch P (mm)
5 10 15 10
10 5 3.5 3.5
Remarks Parallel Parallel Parallel Perpendicular
897
Experimental Investigation of Circular Saw Vibration r.O
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~
~
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~
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e
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i
v
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e
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..................2 ""'""""'"1 It H
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FI6.6. Projection plate thin plate structure. (a) Structure. (b) Configuration of projections.
TABLE4. PROJECTIONPLATET H I N PLATESTRUCTURE Plate number P15 P2. P2_s
Height of projection Q (ram) 0.38 0.51 0.64
Flat or projecting
FIG. 7. Non-bonded thin plate structure.
898
E. MARUi et al. TABLE 5. NON-BONDED THIN PLATE STRUCTURE
Plate number
No Nz N6 NFC NPt5 NP:s
Metal sheet thickness (mm) 0 2 6 2 6 6
I
I I
Material
Type
Carbon tool steels Carbon tool steels Spheroidal graphite iron castings Carbon tool steels Carbon tool steels
Carbon
tool steel
Flat Plate Flat Plate Flat Plate Projected plate Projected plate
I
I
FIG. 8. Solid thin plate structure.
Table 5. The thin plate contacted the vice directly in the structure No, as the shims were not used here. In structures N2 and N6, the carbon tool steel shims, measuring 2 mm and 6 mm thick, respectively, were pressed directly onto the thin plate. Cast iron shims of 2 mm thick were used in the structure NFC. The shims of structures NPIs and NP2s were made of carbon tool steel, and those in structures NP15 and NP2s were similar to the projection plate thin plate structures PlS and P2s. 3.6.
Solid thin plate structure
This structure incorporated the thin plate and two shims, as shown in Fig. 8. The entire structure was made of carbon tool steel. The thickness of the incorporated part was 6 mm. 4.
SIMULATION MODEL OF THIN PLATE STRUCTURE AND PERFORMANCE COEFFICIENT FOR VIBRATION PROOF CAPACITY
Figure 9 is a simulation model of the thin plate structure. The vibratory system of Fig. 9 is a one degree of freedom system, whose mass, viscous damping coefficient and spring rigidity are represented by m, c and kt, respectively. Cutting force does not act in the axial direction of the circular saw blade in stable cutting operation. However, when the bending vibration occurs in the circular saw blade, the moving direction of the blade deviates slightly from its rotational direction. The inclination of the cutting edge relative to the workpiece surface varies at every moment. As a result, the axial component of the cutting force is generated and the circular saw blade is affected by the regenerative effect during bending vibration. Then, the bending vibration of the circular saw blade is equivalent to the vibratory system of one degree of freedom in Fig. 9.
"//////////,~
FIG. 9. Transformation of thin plate structure into a lumped mass systcr.
Experimental Investigation of Circular Saw Vibration
899
An external force F = kcu acts as the cutting force in the axial direction. The symbol kc is the cutting rigidity, and u is the depth of cut as a function of the axial bending of circular saw blade. The equation of motion of the system shown in Fig. 9 can be written as follows: d2x
m~
dx
(1)
+ c - ~ + ktx = kcu,
where x is the bending displacement of the thin plate structure equivalent to the axial bending of the circular saw blade. The Laplace transformation of equation (1) is ( m s 2 + c s + k t ) X = k~ U,
(2)
where the symbols X and U are the Laplace transformations of x and u. By putting s = jo~ in equation (2), we obtain equation (3): X
kc = ( k t - mto 2) + j c w "
(3)
Cutting wood with a circular saw takes place in a condition of regenerative chatter vibration. According to Merritt's [19] theory of regenerative chatter vibration, a vibratory system is stable and regenerative chatter vibration cannot be induced, when the real part of equation (3) is larger than -0.5 [19]. The larger the difference between the magnitude of the real part of equation (3) and -0.5, the larger the margin for chatter initiation, and the system correspondingly becomes more stable. The real part of equation (3) can be expressed as
mo > Re
= (kt_~)~
o~2.
(4)
The minimum value of the negative real part is obtained from equation (4) as follows: X] - kc Re ~ mi, - 4 k, (8 + 82) '
(5)
where, 8 is a damping ratio and can be written as follows: c
8 - 2 X/~k,
(6)
Then, the system having a larger magnitude of h in equation (7) is more capable of preventing unstable vibration, such as chatter vibration. This h is used as the performance coefficient to evaluate the vibration-proof capacity of the thin plate structure x = k, (8 + 82).
(7) 5.
EXPERIMENTAL RESULTS AND DISCUSSION
Figure 10 represents the logarithmic decrement 80 (= ~rS) and bending rigidity kt of various thin plate structures in bar graphs. Logarithmic decrement is plotted on a log scale. Broken lines in the figure indicate the results for the structure No, which is quite similar to the case of a conventional circular saw in the clamping condition. The structures, whose logarithmic decrement or bending rigidity are greater than the results for the structure No, have excellent damping capacity and static rigidity. However, the structure, which has a good damping capacity, has a poor static rigidity. In either the
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B2 B3
B3
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Bs
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TI T2 T3 T4,
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No N2 N6 NFC NPI5 NP2s
NO N2
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NPIs NP~5 i I
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0.01
J
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0.1
logarithmic decrement 6 0
0
I
I
10
20
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I;
30
40
Bending r i g i d i t y k, kN/m
FI~. 10. Experimental results of logarithmic decrement and bending rigidity.
bonded or solid structures, static rigidity is high but damping capacity is conversely low. The damping capacities of the structures with adhesive tape, wire and projection plate are excellent. It is necessary to assess the practical vibration-proof capacity of the thin plate structure by the performance coefficient h, given by equation (7). The assessed result is shown in Fig. 11. In the figure, the ordinate is the logarithmic decrement ~o, and the abscissa is the bending stiffness kt. Moreover, the fine solid lines in the same figure are the performance curves having the same magnitude of performance coefficient. The thick solid line indicates the performance reference line, whose h magnitude is equal to that of the structure No (h = 0.131). The larger the performance coefficient, the higher the stability of the structure. For the structure having a large bending rigidity kt and small logarithmic decrement ~o, the plotted point locates on the lower right region of the figure, and the performance coefficient is small. In contrast, in the case of a large logarithmic decrement and somewhat small bending rigidity, the plotted point locates on the upper left region of the figure, and the performance coefficient is large. As a result, the practical assessment of the vibration-proof capacity of each thin plate structure can be carried out using Fig. 11. The projection plate, wire and adhesive tape structures are superior in vibration proof capacity. In the adhesive tape structure T4, the bending rigidity decreases about 15%, and the logarithmic decrement increases 330%, compared with the reference structure No. As a result, the performance coefficient is 3.7 times as large as the reference structure No. Thus, T4 is an excellent vibration-proof structure. The solid structure I has the largest bending rigidity of all structures tested, exceeding that of the reference structure No by 25%. However, the logarithmic decrement is 90% smaller than that of the reference structure No and the performance coefficient for this type of structure is the smallest of all.
Experimental Investigation of Circular Saw Vibration
1
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I
o Bonded type o Adhesive Tape type 6 | i r e type • Projection Plate type • non-Bonded type x Solid type
0.5
0.05
I
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•
o.
~--~___.______
-0. l a l
0.02
Performance coefficient
,NFcl
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-----
o.
"--'---- ------" • MPIs
O. I
o
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. , ~
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OBt
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3p.4.............
o. 005
xl "-----------
..-.--_._.__. _____..__.
O. 002
0.01 0.001 24
26
28
30
32
Bending rigidity
34
36
k,
38
40
kN/m
FIG. 11. Estimation of vibration-proof capacity by performance coefficient.
In Fig. 12, the ordinate indicates the performance coefficient in log scale, and the vibration-proof capacity of each thin plate structure is arranged schematically. The higher the position of the thin plate structure is in the figure, the larger is its performance coefficient and the better its vibration-proof capacity. The arrow lines connecting two different type thin plate structures indicate the mechanism improving vibration-proof capacity. For example, the vibration-proof capacity of the bonded thin plate structure is improved by the absorption of vibratory energy accompanying the shear deformation of bonding agent layers, compared with the solid thin plate structure. Also, the vibration proof capacity of the non-bonded thin plate structure is improved by the consumption of vibratory energy by the friction acting on the contact surface between the thin plate and two shims, compared with the solid thin plate structure. It is clear in the same figure that the improvement owing to the friction loss of vibratory energy at contact surfaces is larger than the improvement owing to the consumption of vibratory energy by the shearing deformation of bonding agent layers. In the thin plate structures N6, NP~ and NP25, which have shims 6 mm thick, the vibration-proof capacities of the projection plate thin plate structures, NP15 and NP25, are lower than that of the flat plate N6. The reason for this is that the vibratory energy consumption differs in terms of the apparent contact areas. The vibration-proof capacity of the thinner shim structure is greater than for the thicker shim structure as seen in the comparison between the results for the structures N6 and N2, because the energy consumption is large in the case of the thinner shim structure. It is also generally recognized that the vibrationproof capacity is vastly improved by friction loss accompanied by interposition of the wire, and by the constrained contact area at the top of projections. 6.
CONFIRMATION OF DAMPING CAPACITY IMPROVEMENT AT THIN DISC STRUCTURE
Damping capacity improvement was experimentally examined at a thin disc structure of adhesive tape type, which was remarkably effective in the vibration-proof capacity improvement of a cantilevered thin plate structure.
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Adhesive tape type
P~*
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P,6 Pto
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Tz Ill
\
0.3
tire type !
|ll
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®
0.2
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i N,
1
o
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o o to o c
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!
0.1 Bz
g
Bi
0.05
NPzs
Bt B,
.
Bs
.
.
.
]]
_-7"___-_, I
Non-bonded type O. 03
Solid type 0.02
0.01 FIG. 12. Vibration-proof mechanism.
The outline of the experimental apparatus is shown in Fig. 13. A disc, which is equivalent to a circular saw blade, was clamped on an arbor by a circular nut. Thickness and diameter of the disc were 3 and 255 mm, respectively. A detailed description of the clamping part is given in Fig. 14. The flange diameter of the disc was 100 mm, as recognized in Fig. 14. Strain gauges were pasted on the arbor, to detect a clamping load of disc. The disc was made of carbon tool steel, and the arbor and the nut were made of carbon steel for machine structure. This arbor was set at the main spindle of a lathe by an independent chuck. An impulse was applied uniformly at the periphery of the disc by a cupped hammer, which can slide smoothly along a guide bar on a tail stock of the lathe. Then, damped free vibration of fundamental mode with zero nodal circle and zero nodal diameter was generated in the disc. The vibration was detected by an eddy current displacement
Experimental Investigationof Circular Saw Vibration
i
. ~l.
St.rsin
903
Oisk\
~ulex ~
~
/Guide
Eddy:current gap oetector
bar
I J
IA, plifierl--]Band pass filterl~FFT
analyzer I
FIG. 13. Experimentalapparatus for obtaining dampingcapacity of disc structure. t3
Arbor
~
---ii:
Straingauge
~ ....
/ Adhesive tape
Circular nut 1
Fi6. 14. Detailed description of disc clampingpart at arbor. sensor set at the opposite side to the cupped hammer. The logarithmic decrement was calculated from the damping vibration signal by micro-computer. A band pass filter was used to cut the low frequency vibration of the lathe and some high frequency disturbance from the environment. The measurement was carried out varying the clamping load, for the direct contact between the disc and the arbor or the circular nut, and also for the indirect contact via three sheets of adhesive tape. The experimental result of the logarithmic decrement of the disc is given in Fig. 15. The logarithmic decrement decreased slightly with the increase of the clamping load for both contact situations. The logarithmic decrement in the case of the indirect contact via adhesive tape was about 2.9 times larger than that in the case of the direct contact. This improvement of the damping capacity was inferior to the improvement at the cantilevered thin plate structure in Fig. 4. However, the effectiveness of the adhesive tape structure for improving the damping capacity was also confirmed at the thin disc structure. 7. CONCLUSION To enhance the vibration-proof capacity of a thin-bladed tool such as a circular saw for wood-cutting applications, we experimentally investigated the effects on damping
904
E. MARUIet al. 0.10
---a.. 0.08 o
to
With adhesive tape 0.06
Direct contact
o
O. 04
S*
""'"--0 O. 02
10
20
30 Clamping load
40
50
60
F kN
Fro. 15. Experimental result for logarithmic decrement of disc structure. force of the c l a m p i n g c o n d i t i o n o f a thin p l a t e structure. A thin p l a t e structure o f cantilever t y p e s e r v e d as the m o d e l . A d h e s i v e t a p e , wire a n d the like w e r e i n s e r t e d b e t w e e n the plate a n d the shims, the v i b r a t i o n e n e r g y a d s o r p t i o n o r c o n s u m p t i o n effects w e r e i n c r e a s e d , o r the shim was given p r o j e c t i o n s in o r d e r to increase friction loss, as a m e a n s of vastly i m p r o v i n g the d a m p i n g p e r f o r m a n c e of the thin p l a t e structure system. T h e o p t i m u m i m p r o v e m e n t r e s u l t e d f r o m the insertion of strips of a d h e s i v e t a p e o r the p r o j e c t i o n p l a t e a r r a n g e m e n t . A d h e s i v e t a p e s t r u c t u r e , which is r e m a r k a b l y effective in i m p r o v i n g the d a m p i n g capacity o f a thin p l a t e s t r u c t u r e of c a n t i l e v e r t y p e , was a p p l i e d to a thin disc structure. A g o o d result was o b t a i n e d in i m p r o v i n g the d a m p i n g c a p a c i t y of the disc structure. A thin disc structure having g o o d v i b r a t i o n - p r o o f c a p a c i t y is e x p e c t e d by the c l a m p i n g m e t h o d s shown in this p a p e r . REFERENCES [1] S. G. HUTTON, Dynamic response of a guided circular saw, J. Sound Vibr. 112, 527-539 (1987). [2] G. S. SCHAJER, Simple formulas for natural frequencies and critical speeds of circular saws, Forest Products J. 36, 37-43 (1986). [3] C. D. MOTE Jr, Free vibration of initially stressed circular discs, Trans. A S M E J. Engng Ind. 87, 258--265 (1965). [4] W. H. LIu, Flexural vibrations of an annular saw, Theory Mach. Mech. 2, 763-766 (1987). [5] H. REISMANN,Forced vibration of a circular plate, A S M E J. Appl. Mech. 81,526--527 (1959). [6] D. S. DU6DALE, Self-excited vibration of circular saws cutting aluminium, Proc. Int. Mach. Tool Des. Res. Conf. 15, 279-285 (1975). [7] C. D. MoTE Jr, Stability of circular plates subjected to moving loads, J. The Franklin Institute 290, 329-344 (1970). [8] B. F. LEHMANNand S. G. HurroN, Self-excitation in guided circular saws, A S M E J. Vibr. Acoustics, Stress Reliab. Des. llO, 338-344 (1989). [9] C. D. MOTEJR and M. C. LEO, Whistling instability in idling circular saws, A S M E J. dyn. Syst. Meas. Control 102, 114--122 (1980). [10] G. S. SCnAJER, Why are guided circular saws more stable than unguided saws, Holz als Roh--und Werkstoff 44, 465--469 (1986). [ 11] C. D. MOTEJr, et al., Circular saw vibration control by induction of thermal membrane stresses, A S M E J. Engng Ind. 103, 81-89 (1981). [ 12] R. C. Yu and C. D. MOTEJr, Vibration of circular saws containing slots, Holz als Roh--und Werkstoff 45, 155-160 (1987). [13] C. D. MOTE Jr, Stability control analysis of rotating plates by finite element; emphasis on slots and holes, A S M E J. dyn. Syst. Meas. Control 94, 64-70 (1972). [14] A. TROCmDIS,Vibration damping of circular saws, Acoustica 69, 270-275 (t989).
Experimental Investigation of Circular Saw Vibration
905
[15] D. S. DUGDALE, Stiffness of a spinning disc clamped at its centre, J. Mech. Phys. Solids 14, 349-356 (1966). [16] D. S. DUGDALE,Discrete frequency noise from free running circular saws, J. Sound Vibr. 10, 296--304 (1969). [17] T. WANGet al., The vibration and sound response of the metal circular saw plate, Proc. Int. Noise 87, 211-214 (1987). [18] M. NAKAZAWAet al., Lateral vibration of free running circular saw and emitted whistling noise, Proc. VI Int. Congr. Expl. Mech. Portland, 1179-1184 (1988). [19] H. E. MERRITT, Theory of self-excited machine tool chatter, contribution to roaching-tool chatter, Tram. A S M E J. Engng Ind. 87, 447---452 (1965).
Pergamon
0890-6955(93)E0001-L AN EXPERIMENTAL VIBRATION
INVESTIGATION
OF CIRCULAR
SAW
VIA A THIN PLATE MODEL
E . M A R U I , t S. EMA~ a n d R . MIYACHIt
(Received 1 June 1993; in final form 1 October 1993)
Abstract--This paper deals with important practical problems related to vibration reduction in a circular saw used for wood cut-offs. A n investigation is made of the static and dynamic characteristics of thin plate structures as a model of a circular saw, to obtain fundamental knowledge to improve circular saw performance. The circular saw is represented experimentally by a thin rectangular plate clamped by two shims. The bending rigidity and damping capacity of a structure are measured for six different clamping methods. The results are evaluated by a coefficient, which is introduced through Merritt's regenerative chatter vibration theory. The most effective thin plate structure is found to improve vibration-proof capacity. This structure is easily applicable to an actual circular saw.
1.
INTRODUCTION
WHEN A FLEXURALvibration is induced in a thin disc-type cutter such as a circular saw for wood cut-offs at high spindle speeds, the operating efficiency is lowered and wood resources are greatly wasted. Many studies have reported on vibration problems involving circular saws for wood cut-offs or thin discs and their remedies. Representative reports deal with natural frequency and principal mode of vibration [1-4], forced vibration [5], unstable or self-excited vibration [6-9], increase in dynamic stability [10-14], stiffness [15] and noise [16-18]. An increase in system damping capacity is the key technology for preventing selfexcited or forced vibrations of a circular saw during cutting operation. The contact condition between the circular saw blade and the flange of the main spindle of the circular saw machine has a great influence on the initiation of the system damping capacity of the circular saw. However, to our knowledge, no reported study to date has treated the interrelationship between the system damping capacity and the contact condition. In this paper, the relationships between the static bending rigidity, the damping capacity and the contact condition at the clamping part are measured for a thin plate structure, which is a simple model of the circular saw blade and the clamping flange of the circular saw itself. A fundamental device for improving the cutting efficiency and the stability of the circular saw is clarified. Partly, the effectiveness is ascertained at the thin disc structure model. 2.
E X P E R I M E N T A L A P P A R A T U S AND P R O C E D U R E S
Figure 1 shows an experimental apparatus for measuring thin plate structures, which are used in experiments on bending rigidity and damping capacity. The experimental apparatus was mounted on a surface plate. The hatched parts in the figure indicate the thin plate and the shims for clamping it. These correspond to the circular saw blade and the flange at the main spindle of the circular saw, respectively. They were made of carbon tool steel. Some of the shims were made of cast iron. The thin plate and two shims were clamped by a vice through load cells. The clamping load was 4 kN per load cell. A wire was attached to the free end of the thin plate by a hook. A weight was hung at the other end of the wire. The deflection at the free end of the
tFaculty of Engineering, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan. ~tFaculty of Education, Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan. 893
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cell
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i{ook~c=~rr.l,v~£ id Y- c u r r e n t
gap detector
[FFT a n a l y z e r ~ B a n d
pass f i l t e r H A n p l i f i e r
j
FIG. 1. Experimental apparatus for measuring dynamic characteristics of thin plate structures. thin plate was read by a dial gauge, and the bending stiffness of the plate was calculated by linear regression analysis between the deflection at the free end of the plate and the bending load caused by the weight. Then, the hook was removed instantaneously from the free end of the plate and a step function input applied to the plate. The plate underwent a damped free vibration which was detected by an eddy current displacement sensor. The detected signal was resolved into harmonics by an FFT analyzer, and the frequency of the principal mode of vibration obtained. The logarithmic decrement was calculated from the damping vibration signal by micro-computer. A band pass filter was used to cut the low frequency vibration of the surface plate and some high frequency disturbance from the environment.
3. CONSTRUCTION OF THIN PLATE Figure 2 shows the basic thin plate structure used in this experiment. The thin plate was made from carbon tool steel of 54 mm width, 130 mm length and 2 mm thickness. This plate was clamped by two shims of 2 mm thick and made of the same materials. In some cases, shims of different materials and sizes were used. The length of the 130
40
/
Carbon
tool steel \
FIG. 2. Fundamental thin plate structure.
895
Experimental Investigation of Circular Saw Vibration
j
\Carbon
Epoxy r e s i n
tool s t e e l
FIG. 3. Bonded thin plate structure.
TABLE 1. BONDED THIN PLATESTRUCTURE Plate number B~ B2 B3 B4 B5
Thickness of adhesive bond (mm) 0.097 0.487 0.926 1.171 1.395
tape
'Carb0n
to01 s t e e l
FIG. 4. Adhesive tape thin plate structure.
TABLE 2. ADHESIVETAPETHIN PLATESTRUCTURE Plate number T~ T2 T3 T4
Number of adhesive tape strips 1 2 3 4
shims was 40 mm. Six types of clamping were incorporated and 23 kinds of structures were tested.
3.1. Bonded thin plate structure In this structure, the thin plate and two shims were bonded by an epoxy-based adhesive, as shown in Fig. 3. The adhesive hardens at room temperature. Five adhesive thicknesses were used in the experiment, as shown in Table 1. Adhesive tape thin plate structure The basic structure is given in Fig. 4. The thin plate and two shims were attached to one another with double-faced industrial adhesive tape. The tape thickness was 0.43 mm when no load was applied. This tape has good therma 1 resistance and sufficient bonding strength is maintained up to 150°C. As shown in Table 2, only one strip of adhesive tape was used in the structure T~, and two, three and four strips are applied in layers in structures T2, T3 and T4, respectively. 3.2.
3.3. Wire thin plate structure This type of structure was manufactured as follows: one strip of adhesive tape was applied to the thin plate and piano wires of 0.3 mm diameter were spaced equidistantly
896
E. MARU!et al.
'ooooo IQ
ot o o o
Q Q U U Q U
U
U
U
~
i
pr!ss
o '
U
l~Adhesive
\ ~ ' ~ Ca r bon
tool
tape steel
,1,
~tmmmEmmmmmm~d Wira
Ws, W,o, W~s
WS,o
Fio. 5. Wire thin plate structure.
on it. Then, two shims were pressed on to the thin plate as seen in Fig. 5. The piano wires were arranged in parallel with the thin plate axis (structures Ws, Wlo, W,5) or perpendicularly to the axis (structure WSlo). The former arrangement corresponded to the radial direction of the circular saw blade, and the latter to the blade circumferential direction. Table 3 gives the number of piano wires and their pitch P.
3.4.
Projection plate thin plate structure
This type of thin plate was manufactured by the process shown in Fig. 6. A strip of adhesive tape was pre-attached to the thin plate. Rugged shims were pressed on it. Projections on the shims were trimmed by a milling cutter. The projections were arranged at 2.54 mm intervals in a lattice. The size of the projections was varied by changing the depth of cut of the milling cutter, on three levels. The projection height Q is shown in Table 4. The projection of a large Q value has a small contact area at its tip, and becomes sharp. The vertical angle of the projections was 90° , as seen in Fig. 6. 3.5.
Non-bonded thin plate structure
This type of structure was prepared to compare the damping capacity of the preceding structures. The thin plate contacted the two shims directly, as shown in Fig. 7. Shims of different thickness and material were used. The detailed structure is shown in TABLE3. WIRE THIN PLATESTRUCTURE Plate number W5 W,o W,~ WS,o
Number of piano wires
Wire pitch P (mm)
5 10 15 10
10 5 3.5 3.5
Remarks Parallel Parallel Parallel Perpendicular
897
Experimental Investigation of Circular Saw Vibration r.O
~Plate .
t.
~
~
"
~
A press
d
h
e
s
,ith projections
i
v
'~Carbon
e
tape
tool steel
..................2 ""'""""'"1 It H
XXX 2.54 am
90"
FI6.6. Projection plate thin plate structure. (a) Structure. (b) Configuration of projections.
TABLE4. PROJECTIONPLATET H I N PLATESTRUCTURE Plate number P15 P2. P2_s
Height of projection Q (ram) 0.38 0.51 0.64
Flat or projecting
FIG. 7. Non-bonded thin plate structure.
898
E. MARUi et al. TABLE 5. NON-BONDED THIN PLATE STRUCTURE
Plate number
No Nz N6 NFC NPt5 NP:s
Metal sheet thickness (mm) 0 2 6 2 6 6
I
I I
Material
Type
Carbon tool steels Carbon tool steels Spheroidal graphite iron castings Carbon tool steels Carbon tool steels
Carbon
tool steel
Flat Plate Flat Plate Flat Plate Projected plate Projected plate
I
I
FIG. 8. Solid thin plate structure.
Table 5. The thin plate contacted the vice directly in the structure No, as the shims were not used here. In structures N2 and N6, the carbon tool steel shims, measuring 2 mm and 6 mm thick, respectively, were pressed directly onto the thin plate. Cast iron shims of 2 mm thick were used in the structure NFC. The shims of structures NPIs and NP2s were made of carbon tool steel, and those in structures NP15 and NP2s were similar to the projection plate thin plate structures PlS and P2s. 3.6.
Solid thin plate structure
This structure incorporated the thin plate and two shims, as shown in Fig. 8. The entire structure was made of carbon tool steel. The thickness of the incorporated part was 6 mm. 4.
SIMULATION MODEL OF THIN PLATE STRUCTURE AND PERFORMANCE COEFFICIENT FOR VIBRATION PROOF CAPACITY
Figure 9 is a simulation model of the thin plate structure. The vibratory system of Fig. 9 is a one degree of freedom system, whose mass, viscous damping coefficient and spring rigidity are represented by m, c and kt, respectively. Cutting force does not act in the axial direction of the circular saw blade in stable cutting operation. However, when the bending vibration occurs in the circular saw blade, the moving direction of the blade deviates slightly from its rotational direction. The inclination of the cutting edge relative to the workpiece surface varies at every moment. As a result, the axial component of the cutting force is generated and the circular saw blade is affected by the regenerative effect during bending vibration. Then, the bending vibration of the circular saw blade is equivalent to the vibratory system of one degree of freedom in Fig. 9.
"//////////,~
FIG. 9. Transformation of thin plate structure into a lumped mass systcr.
Experimental Investigation of Circular Saw Vibration
899
An external force F = kcu acts as the cutting force in the axial direction. The symbol kc is the cutting rigidity, and u is the depth of cut as a function of the axial bending of circular saw blade. The equation of motion of the system shown in Fig. 9 can be written as follows: d2x
m~
dx
(1)
+ c - ~ + ktx = kcu,
where x is the bending displacement of the thin plate structure equivalent to the axial bending of the circular saw blade. The Laplace transformation of equation (1) is ( m s 2 + c s + k t ) X = k~ U,
(2)
where the symbols X and U are the Laplace transformations of x and u. By putting s = jo~ in equation (2), we obtain equation (3): X
kc = ( k t - mto 2) + j c w "
(3)
Cutting wood with a circular saw takes place in a condition of regenerative chatter vibration. According to Merritt's [19] theory of regenerative chatter vibration, a vibratory system is stable and regenerative chatter vibration cannot be induced, when the real part of equation (3) is larger than -0.5 [19]. The larger the difference between the magnitude of the real part of equation (3) and -0.5, the larger the margin for chatter initiation, and the system correspondingly becomes more stable. The real part of equation (3) can be expressed as
mo > Re
= (kt_~)~
o~2.
(4)
The minimum value of the negative real part is obtained from equation (4) as follows: X] - kc Re ~ mi, - 4 k, (8 + 82) '
(5)
where, 8 is a damping ratio and can be written as follows: c
8 - 2 X/~k,
(6)
Then, the system having a larger magnitude of h in equation (7) is more capable of preventing unstable vibration, such as chatter vibration. This h is used as the performance coefficient to evaluate the vibration-proof capacity of the thin plate structure x = k, (8 + 82).
(7) 5.
EXPERIMENTAL RESULTS AND DISCUSSION
Figure 10 represents the logarithmic decrement 80 (= ~rS) and bending rigidity kt of various thin plate structures in bar graphs. Logarithmic decrement is plotted on a log scale. Broken lines in the figure indicate the results for the structure No, which is quite similar to the case of a conventional circular saw in the clamping condition. The structures, whose logarithmic decrement or bending rigidity are greater than the results for the structure No, have excellent damping capacity and static rigidity. However, the structure, which has a good damping capacity, has a poor static rigidity. In either the
900
E. MARUI et al.
I
I]1
BI
B2 B3
B3
114 Bs
Bs
TI T2 T3 1"4
TI T2 T3 T4,
Ws Wio WI5
WSIo
15 ~10 1115 |$1o
Pls P2o P25
Pts P2o P~s
No N2 N6 NFC NPI5 NP2s
NO N2
I
i
'
I
N6
NFC
NPIs NP~5 i I
0.001
0.01
J
I
0.1
logarithmic decrement 6 0
0
I
I
10
20
"
I;
30
40
Bending r i g i d i t y k, kN/m
FI~. 10. Experimental results of logarithmic decrement and bending rigidity.
bonded or solid structures, static rigidity is high but damping capacity is conversely low. The damping capacities of the structures with adhesive tape, wire and projection plate are excellent. It is necessary to assess the practical vibration-proof capacity of the thin plate structure by the performance coefficient h, given by equation (7). The assessed result is shown in Fig. 11. In the figure, the ordinate is the logarithmic decrement ~o, and the abscissa is the bending stiffness kt. Moreover, the fine solid lines in the same figure are the performance curves having the same magnitude of performance coefficient. The thick solid line indicates the performance reference line, whose h magnitude is equal to that of the structure No (h = 0.131). The larger the performance coefficient, the higher the stability of the structure. For the structure having a large bending rigidity kt and small logarithmic decrement ~o, the plotted point locates on the lower right region of the figure, and the performance coefficient is small. In contrast, in the case of a large logarithmic decrement and somewhat small bending rigidity, the plotted point locates on the upper left region of the figure, and the performance coefficient is large. As a result, the practical assessment of the vibration-proof capacity of each thin plate structure can be carried out using Fig. 11. The projection plate, wire and adhesive tape structures are superior in vibration proof capacity. In the adhesive tape structure T4, the bending rigidity decreases about 15%, and the logarithmic decrement increases 330%, compared with the reference structure No. As a result, the performance coefficient is 3.7 times as large as the reference structure No. Thus, T4 is an excellent vibration-proof structure. The solid structure I has the largest bending rigidity of all structures tested, exceeding that of the reference structure No by 25%. However, the logarithmic decrement is 90% smaller than that of the reference structure No and the performance coefficient for this type of structure is the smallest of all.
Experimental Investigation of Circular Saw Vibration
1
|
I
]
901
I
o Bonded type o Adhesive Tape type 6 | i r e type • Projection Plate type • non-Bonded type x Solid type
0.5
0.05
I
........
.~
•
o.
~--~___.______
-0. l a l
0.02
Performance coefficient
,NFcl
-----
-----
o.
"--'---- ------" • MPIs
O. I
o
O. O1
. , ~
*NP25
- --_..~ ~ t3Bs
OBt
-------
0.05
3p.4.............
o. 005
xl "-----------
..-.--_._.__. _____..__.
O. 002
0.01 0.001 24
26
28
30
32
Bending rigidity
34
36
k,
38
40
kN/m
FIG. 11. Estimation of vibration-proof capacity by performance coefficient.
In Fig. 12, the ordinate indicates the performance coefficient in log scale, and the vibration-proof capacity of each thin plate structure is arranged schematically. The higher the position of the thin plate structure is in the figure, the larger is its performance coefficient and the better its vibration-proof capacity. The arrow lines connecting two different type thin plate structures indicate the mechanism improving vibration-proof capacity. For example, the vibration-proof capacity of the bonded thin plate structure is improved by the absorption of vibratory energy accompanying the shear deformation of bonding agent layers, compared with the solid thin plate structure. Also, the vibration proof capacity of the non-bonded thin plate structure is improved by the consumption of vibratory energy by the friction acting on the contact surface between the thin plate and two shims, compared with the solid thin plate structure. It is clear in the same figure that the improvement owing to the friction loss of vibratory energy at contact surfaces is larger than the improvement owing to the consumption of vibratory energy by the shearing deformation of bonding agent layers. In the thin plate structures N6, NP~ and NP25, which have shims 6 mm thick, the vibration-proof capacities of the projection plate thin plate structures, NP15 and NP25, are lower than that of the flat plate N6. The reason for this is that the vibratory energy consumption differs in terms of the apparent contact areas. The vibration-proof capacity of the thinner shim structure is greater than for the thicker shim structure as seen in the comparison between the results for the structures N6 and N2, because the energy consumption is large in the case of the thinner shim structure. It is also generally recognized that the vibrationproof capacity is vastly improved by friction loss accompanied by interposition of the wire, and by the constrained contact area at the top of projections. 6.
CONFIRMATION OF DAMPING CAPACITY IMPROVEMENT AT THIN DISC STRUCTURE
Damping capacity improvement was experimentally examined at a thin disc structure of adhesive tape type, which was remarkably effective in the vibration-proof capacity improvement of a cantilevered thin plate structure.
902
E. MARUIet
al.
1
Projection plate type
Adhesive tape type
P~*
0.5
®
P,6 Pto
Tt
,,
Tz Ill
\
0.3
tire type !
|ll
N:
®
0.2
NFC
I I I
@
[ I
i N,
1
o
I I
iiii!lliiill
Bonded type o
o o to o c
Ne
!
0.1 Bz
g
Bi
0.05
NPzs
Bt B,
.
Bs
.
.
.
]]
_-7"___-_, I
Non-bonded type O. 03
Solid type 0.02
0.01 FIG. 12. Vibration-proof mechanism.
The outline of the experimental apparatus is shown in Fig. 13. A disc, which is equivalent to a circular saw blade, was clamped on an arbor by a circular nut. Thickness and diameter of the disc were 3 and 255 mm, respectively. A detailed description of the clamping part is given in Fig. 14. The flange diameter of the disc was 100 mm, as recognized in Fig. 14. Strain gauges were pasted on the arbor, to detect a clamping load of disc. The disc was made of carbon tool steel, and the arbor and the nut were made of carbon steel for machine structure. This arbor was set at the main spindle of a lathe by an independent chuck. An impulse was applied uniformly at the periphery of the disc by a cupped hammer, which can slide smoothly along a guide bar on a tail stock of the lathe. Then, damped free vibration of fundamental mode with zero nodal circle and zero nodal diameter was generated in the disc. The vibration was detected by an eddy current displacement
Experimental Investigationof Circular Saw Vibration
i
. ~l.
St.rsin
903
Oisk\
~ulex ~
~
/Guide
Eddy:current gap oetector
bar
I J
IA, plifierl--]Band pass filterl~FFT
analyzer I
FIG. 13. Experimentalapparatus for obtaining dampingcapacity of disc structure. t3
Arbor
~
---ii:
Straingauge
~ ....
/ Adhesive tape
Circular nut 1
Fi6. 14. Detailed description of disc clampingpart at arbor. sensor set at the opposite side to the cupped hammer. The logarithmic decrement was calculated from the damping vibration signal by micro-computer. A band pass filter was used to cut the low frequency vibration of the lathe and some high frequency disturbance from the environment. The measurement was carried out varying the clamping load, for the direct contact between the disc and the arbor or the circular nut, and also for the indirect contact via three sheets of adhesive tape. The experimental result of the logarithmic decrement of the disc is given in Fig. 15. The logarithmic decrement decreased slightly with the increase of the clamping load for both contact situations. The logarithmic decrement in the case of the indirect contact via adhesive tape was about 2.9 times larger than that in the case of the direct contact. This improvement of the damping capacity was inferior to the improvement at the cantilevered thin plate structure in Fig. 4. However, the effectiveness of the adhesive tape structure for improving the damping capacity was also confirmed at the thin disc structure. 7. CONCLUSION To enhance the vibration-proof capacity of a thin-bladed tool such as a circular saw for wood-cutting applications, we experimentally investigated the effects on damping
904
E. MARUIet al. 0.10
---a.. 0.08 o
to
With adhesive tape 0.06
Direct contact
o
O. 04
S*
""'"--0 O. 02
10
20
30 Clamping load
40
50
60
F kN
Fro. 15. Experimental result for logarithmic decrement of disc structure. force of the c l a m p i n g c o n d i t i o n o f a thin p l a t e structure. A thin p l a t e structure o f cantilever t y p e s e r v e d as the m o d e l . A d h e s i v e t a p e , wire a n d the like w e r e i n s e r t e d b e t w e e n the plate a n d the shims, the v i b r a t i o n e n e r g y a d s o r p t i o n o r c o n s u m p t i o n effects w e r e i n c r e a s e d , o r the shim was given p r o j e c t i o n s in o r d e r to increase friction loss, as a m e a n s of vastly i m p r o v i n g the d a m p i n g p e r f o r m a n c e of the thin p l a t e structure system. T h e o p t i m u m i m p r o v e m e n t r e s u l t e d f r o m the insertion of strips of a d h e s i v e t a p e o r the p r o j e c t i o n p l a t e a r r a n g e m e n t . A d h e s i v e t a p e s t r u c t u r e , which is r e m a r k a b l y effective in i m p r o v i n g the d a m p i n g capacity o f a thin p l a t e s t r u c t u r e of c a n t i l e v e r t y p e , was a p p l i e d to a thin disc structure. A g o o d result was o b t a i n e d in i m p r o v i n g the d a m p i n g c a p a c i t y of the disc structure. A thin disc structure having g o o d v i b r a t i o n - p r o o f c a p a c i t y is e x p e c t e d by the c l a m p i n g m e t h o d s shown in this p a p e r . REFERENCES [1] S. G. HUTTON, Dynamic response of a guided circular saw, J. Sound Vibr. 112, 527-539 (1987). [2] G. S. SCHAJER, Simple formulas for natural frequencies and critical speeds of circular saws, Forest Products J. 36, 37-43 (1986). [3] C. D. MOTE Jr, Free vibration of initially stressed circular discs, Trans. A S M E J. Engng Ind. 87, 258--265 (1965). [4] W. H. LIu, Flexural vibrations of an annular saw, Theory Mach. Mech. 2, 763-766 (1987). [5] H. REISMANN,Forced vibration of a circular plate, A S M E J. Appl. Mech. 81,526--527 (1959). [6] D. S. DU6DALE, Self-excited vibration of circular saws cutting aluminium, Proc. Int. Mach. Tool Des. Res. Conf. 15, 279-285 (1975). [7] C. D. MoTE Jr, Stability of circular plates subjected to moving loads, J. The Franklin Institute 290, 329-344 (1970). [8] B. F. LEHMANNand S. G. HurroN, Self-excitation in guided circular saws, A S M E J. Vibr. Acoustics, Stress Reliab. Des. llO, 338-344 (1989). [9] C. D. MOTEJR and M. C. LEO, Whistling instability in idling circular saws, A S M E J. dyn. Syst. Meas. Control 102, 114--122 (1980). [10] G. S. SCnAJER, Why are guided circular saws more stable than unguided saws, Holz als Roh--und Werkstoff 44, 465--469 (1986). [ 11] C. D. MOTEJr, et al., Circular saw vibration control by induction of thermal membrane stresses, A S M E J. Engng Ind. 103, 81-89 (1981). [ 12] R. C. Yu and C. D. MOTEJr, Vibration of circular saws containing slots, Holz als Roh--und Werkstoff 45, 155-160 (1987). [13] C. D. MOTE Jr, Stability control analysis of rotating plates by finite element; emphasis on slots and holes, A S M E J. dyn. Syst. Meas. Control 94, 64-70 (1972). [14] A. TROCmDIS,Vibration damping of circular saws, Acoustica 69, 270-275 (t989).
Experimental Investigation of Circular Saw Vibration
905
[15] D. S. DUGDALE, Stiffness of a spinning disc clamped at its centre, J. Mech. Phys. Solids 14, 349-356 (1966). [16] D. S. DUGDALE,Discrete frequency noise from free running circular saws, J. Sound Vibr. 10, 296--304 (1969). [17] T. WANGet al., The vibration and sound response of the metal circular saw plate, Proc. Int. Noise 87, 211-214 (1987). [18] M. NAKAZAWAet al., Lateral vibration of free running circular saw and emitted whistling noise, Proc. VI Int. Congr. Expl. Mech. Portland, 1179-1184 (1988). [19] H. E. MERRITT, Theory of self-excited machine tool chatter, contribution to roaching-tool chatter, Tram. A S M E J. Engng Ind. 87, 447---452 (1965).