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Analysis of the cutting process of a cylindrical workpiece clamped by a four-jaw chuck
Journal of Mechanical Working Technology, 19 (1989) 73-84
73
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
ANALYSIS OF THE CUTTING PROCESS OF A CYLINDRICAL W ORKPIEC E CLAMPED BY A F O U R - J A W CHUCK
M. RAHMAN
Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge (Singapore 0511) (Received November 7, 1988; accepted in revised form November 24, 1988)
Industrial Summary A stability analysis of the cutting process of a solid cylindrical rotating workpiece clamped by a four-jaw chuck with balanced pressure at the jaws has been carried out. The net reaction balancing the cutting force is found to consist of an infinite number of force components oscillating at frequencies 4(0, 8(0, 12(o..... etc., where o9 is the angular speed of rotation of the workpiece, and a stationary force term 2Fo where Fo is the static balancing force impressed along each jaw-line. The stationary force term tends to give a uniform depth of cut. However the harmonic force terms tend to spoil the uniformity in the depth of cut. It is predicted that under the resonant condition of the fourth-harmonic force, the minimum depth of cut will occur at the mid positions between two jaws and the maximum at the jaw-positions. Chatter vibration of the system will be observed at angular spindle speeds approaching O)o/4, o90/8, O9o/12..... etc., where too is the angular natural frequency of the system. A stability chart has been drawn to suggest the safe operating regions.
Notation (o
angular speed of rotation of the workpiece
Fo static balancing force impressed along each jaw-line F'o circumferential force that revolves the workpiece (Or angular speed of rotation at resonance (OO angular natural-frequency
P ll 12
an integer length of overhang of the workpiece clamped length of the workpiece (jaw length) [(t) resultant cutting-force f r ( t ) radial component of the cutting force a, b Fourier coefficients angle between the resultant cutting-force and the radial component of the cutting force r any radial distance inside the workpiece radius of the workpiece R 0378-3804/89/$03.50
© 1989 Elsevier Science Publishers B.V.
74
M B K Zn ,On
Y,~ x
mass of the system damping coefficient of the system stiffness of the system impedance of the system phase angle of the system admittance of the system radial depth of cut
1. I n t r o d u c t i o n
In turning processes, a deviation of shape called out-of-roundness is commonly observed [1,2]. This deviation of shape is found to vary with the type of clamping, the level of the cutting force, the overhang of the workpiece, the workpiece stiffness, etc. [ 1,3 ], all of which are related directly to the performance of the machining process. For example, it has been observed that if a cylindrical workpiece is turned using a three-jaw chuck, a deviation in shape as shown in Fig. 1 (a) is developed [ 1,3 ]. On the other hand, if it is turned using a four-jaw chuck, the shape deviation observed is as shown in Fig. 1 (b). These shape deviations become distinct at the onset of chatter and beyond [ 3,4 ]. The cause of the shape deviation in the case of a three-jaw chuck has been clarified already [5]. Currently, self-centering type four-jaw chucks are used quite commonly also for turning cylindrical workpieces. However, no theoretical analysis has been carried out so far regarding the deviation of shape of a machined workpiece J
a
b
J Fig. 1. Deviation in shape observed by the author after the onset of chatter of a workpiece turned in: (a) a three-jaw chuck (after Ref. [3 ] ); (b) a four-jaw chuck. (J: jaw position; length of overhang: 145 mm; spindle speed: 450 rpm; depth of cut: 3 mm; diameter of workpiece: 60 mm. )
75
clamped by a four-jaw chuck. In this study, an analysis of the cutting process of a solid cylindrical workpiece clamped by a four-jaw chuck with balanced pressure at the jaws has been carried out to find the cause of shape deviation in such a cutting process. 2. A n a l y s i s o f t h e e u t t i n g chuck
process for a workpiececlampedby
a four-jaw
Under resonant conditions it would be expected that the profile of the radial depth of cut on a cylindrical workpiece clamped by a four-jaw chuck would be as sketched in Fig. 2. The maximum depth of cut is expected along the lines AC and BD and the minimum depth of cut is expected at 45 ° on both sides of the lines AC and BD. The minimum depth of cut is attributed to the maximum reaction Fo which in terms of the forces F~ applied at the jaws A', B', C' and D' can be obtained by taking the moment about the clamped end of the workpiece, where, for equilibrium, Fo = F 'olff (/1 + 12). The direction of Fo is radially outwards, as indicated. The expected reaction diagram will be in compliance with the expected cutting profile of Fig. 2. Such reaction profile can be determined by Fourier analysis of the cutting force. Let fr (t) be the radial component of the cutting force Fo
iA
/
,EXPECTED CUTTING PROFILE
A'
I
CYLINDRICAL WORXPIECE
B'D' - ~ I I D - -
•
fz (t)
TOOL j
l, - JAW CHUCK
c' \\ I
1 17
Fig. 2. Expected cutting profile. The jaw positions are A', B', C' and D' with clamping force F ~.
76 at point P at time t developed for balancing the cutting force exciting the system, expressed as fr ( t ) =]CrA + / r B
+frC +/rD
(1)
where, RrA, frB, frC and rid are the reactional forces from points A, B, C and D. Using a Fourier series expansion with the abbreviation 0--cot, where co is the angular speed of the spindle/workpiece, oo
/ra = ~ + ~__ [am cosmO+bmsinmO]
(2)
--1
frB=2+
~ m=l
frc=~+Z Z
(3)
[amcosm(0+180°)+bmsinm(0+180°)]
(4)
[am c o s t a ( 0 + 2 7 0 ° ) + b m s i n m ( 0 + 2 7 0 °)]
(5)
m =--1
ao ~ f~D=~-+~ ~
[amcosm(O+90°)+bmsinm(O+90°)]
m ==1
In the above the constant term ao/2 and the cosine harmonics arise from the cosine-components and the sine harmonics arise from the sine-components of the reactions at A, B, C and D. Considering the reaction from A, oo
F. cos 0 = 2 +
~ am cos mO
(6)
trt=l
and Fosin0= ~ bmsinm0
(7)
m=l
it is tempting to simplify the analysis by using the cosine harmonics only. However, some higher harmonics of the component Fo sin 0 are found to be in the radial direction: this will be clear in the following analysis. As the net reaction is minimum at the mid-position between any two jaws, by integrating between these mid-positions the Fourier coeffcients am and bm are given by
n/4 am =-
7g
Fo cos mO dO --~/4
-F°[sin(m+l)~/4'sin(m-1)zc/417~ m+l m-
(8)
77 n/4
bm=l
7r
f
Fosin0sinm0d0
--n/4
- F °I s i n n
(m-1)n/4ml_
sin (m+ l )n/4
(9)
The average value is obtained by setting m = 0 in eqn. (8), i.e.,
ao o[ - 2n
•.
sin n / 4 - sin ( - n/4)
----nsin n/4
(10) (11)
ao=0.45Fo
Substituting eqns. (2)- (5) into eqn. (1),
fi(t)=2ao+ ~ (amCm(O)+bmSm(O))
(12)
m=l
where
Cm(0)=cosmO+cosm(O+90°)+cosm(O+180°)+cosm(O+270 °)
(13)
Sm(O)=sinmO+sinm(O+90°)+sinm(O+180°)+sinm(O+270 °)
(14)
These summations will vanish except for m = 4 , 8, 12, ..., 4n etc. The nonvanishing terms are
C4(0)=4cos40
$4(0)=4 sin40
C8(0) = 4 cos 80
$8(0) = 4 sin 80
C12(0) = 4 cos 120
$12(0)= 4 sin 120
(15)
Hence by using the relations of eqn. (15) the series eqn. (12) can be written as
[r(t) = 2 a o + 4 ~ [atn cos(4nO)+b4n sin(4n0)] n=l
(16)
T h u s both cosine and sine harmonics are involved in the radial reaction satisfying the condition It(t) =Fo for 0=0 ° , 90 °, 180 ° and 270 ° . A plot of this function versus 0 gives the expected reaction profile. T h e radial c o m p o n e n t of the cutting force will be balanced by this reaction i.e.
f(t) cos fl=2ao + 4 ~ [a4ncos(4no)t) q-bnnsin(4nrot) ] rt~l
(17)
78 It is interesting to observe that although the spindle rotates at an angular speed w, the cutting force oscillates at frequency 40), 80), 12o9, ..., etc. This means that reactions from the jaws are transmitted at speeds faster than the spindle speed. Incorporating the radial variation of the reaction, the cutting force impressed at any radial distance r inside the workpiece can be expressed as: 4n
f(t'r) c°sfl=2a°(R)+4 (R)n=l ~
[a4~cos(nn0)t)+bnnsin(4n0)t)]
(18)
3. Analysis of the radial depth of cut
The radial depth of cut is a solution of the differential equation
Md'~x BdX+Kx=Fx(f) dt 2 ~- dt 4n
=2a°(R)+4
(~R ) n = l
[a4ncos(4n0)t)+b4nsin(4n0)t)]
(19)
where M, B and K are respectively the mass, damping and stiffness factors involved in the dynamic compliance of the system. Consider the depth of cut when r--,R. As 40), 80), 12o9..... 4n0), etc. are the frequencies involved in the cutting force, the time harmonic solution is to be obtained by setting d d r - (j4n0))
( 20 )
d2 dt ~ - (j4n0)) 2
(21)
where j = \ / - 1 ~<7r/2 (in polar form). Using eqns. (20) and (21) and r--.R, the solution of eqn. (19) is given by
x ( t , R ) =2ao - - ~ 4. . ~ a4n cos(4n0)t- ½7c-On) +b4~sin(4n0)t- ~~ -On) =1 4neoZ~
(22)
For r < R, the solution is
x(t,r) = ~2 a- ° ( R ) +4 ~
=1
a4ncos(4n0)t-½Jr-~)+b4nsin(4n0)t-½~-On) 4n0)Zn
(23)
79 The effective depth of cut is given by the difference Ax(t) =x(t,R) -x(t,r) or Ax(t)
1-
+ ~ F.
[ r~4n-][-a4nsin(4n0)t-O,)-b4ncos(4n0)t-On)-]
,,z'_-,L
JL
j
(24)
where Z,, and 0,, are the impedence and phase angle of the system respectively:
Z,, =x,/B2 + ( 4n0)M- K/ 4n0) ) 2 •
0n =can
_J4n0)M-K/4n0)~ ~
~
(25)
j
(26)
It is worth noting that the system impedance Z, and phase angle 0n are sensitive to the oscillating frequencies 4n0). Under resonant conditions, 0~ = 0, i.e. the system is in phase with the cutting force so that 4n0)M-K/4n0)--0 and Zn = B. Hence the resonant spindle speeds are given by i
(Or = ~
I(K)
~
1
(27)
= 4--~0)o
where 0)o = (K/M) ~ is the angular natural frequency of the system. The occurrence of resonance in the system with these spindle speeds is explained in the following manner. Suppose that the spindle speed is set to 0)r = 0)o/4. Then from eqn. (24), the resulting depth is given by
2a°(l-R)-~a4sin(40)rt)-b4c°s(40)rt) [ (R)4]
Ax (t)re~ ----~-\
-~
0)rB
2-0)rZ2
I--
112
-~a12sin(120)rt-dP3)--b~2c°s(120)rt--O3)[40)r Z,~ + ...etc.
1- (R)
] (28)
Evidently the second term will resonate, vibrating the system at frequency 40)~= 0)o. This means that corresponding to the spindle speed 0)~=0)o/4 the system will experience resonant vibration at the natural frequency. This is just an unstable situation. The third term yields non-resonant vibration at twice the natural frequency, adding to the instability. Similarly the fourth term gives non-resonant vibration at three times the natural frequency and so on. Thus
80
the spindle speed o)r= o)0/4 sets the system to an unstable situation, which will be manifested in the form of severe chattering in the system. In such a case the resonant depth of cut will dominate over the non-resonant harmonic cuts so that the depth of cut is given approximately by 4
2ao r t. (atsin4o)rt-b4cos4o)rt)wrB I I - ( R ) Ax(t)res______~_(l__~)_
I
(29)
This is minimum when O---o)rt=O°, 90 °, 180 ° and 270 ° and m a x i m u m when o)~t=45 ° , 135 ° , 225 ° and 315 ° . The minimum depth of cut is given by Ax (t)res =Xo --Xl
(30)
where Xo= (2ao/K) ( 1 - r / R ) , depth of cut is given by
Xx= (4b,/o)oB) [ 1 - (r/R)4]. The maximum
AX ( t ) r e s = X 0 "F X 1
The results are sketched in a polar diagram in Fig. 3. This profile will be distorted, however, with minute cut marks at different angular locations aris-
I /
\
/
\
/I
\
,1
,
--.
//
-J- -t-- "
/
i
xo.xl
~
(NAX. OEPIH
o
I
\
OF CUT)
I
\
*Y
xo
x
\
" ~
/
Xo-X I (NIN. OEPTHOF CUT)
li"
I I
Fig. 3. Schematic radial depth of cut. Xo is the targetted uniform depth of cut at a radius r. R is the radius of the worlq~ieee before cutting.
81
ing from the non-resonant higher harmonic vibrations. Next consider that the spindle is set to wr = o)0/8. In this case the 8th harmonic force will resonate, vibrating the system at the natural frequency. The 4th harmonic will be nonresonant at o)o/2, the 12th harmonic will be non-resonant at 30)0/2, the 16th harmonic at 20)o and so on. This is another stage of the occurrence of chattering. In this case, since the spindle speed is lower than before, the resulting depth of cut will increase but the amplitude of chattering will be less, because the resonance is now with the 8th harmonic, which is much lower in amplitude compared with the 4th harmonic. Thus chattering of lower amplitudes will be observed at spindle speeds o)r = O)o/12, o)o/16,...,o)o4P, etc. Now every resonant phenomenon is perceived over a certain range of frequencies. In the above, although the resonating spindle speeds for the onset of chatter can be identified, in practice the chattering effect will be observed over a certain range or "bandwidth" of spindle speed around the resonant speed. These speed ranges can be evaluated from a study of the admittance of the system, where the latter is given by
~-I [B2+(4no)M- 1 ~2~-~ Y"-~,4no)K] J
(31)
The harmonic depth of cut is directly proportional to the admittance, which latter controls the storage and dissipation of energy supplied to the system. Maximum energy will be dissipated under resonant conditions i.e. when Y, = B or at o)~=o)o/4P, where P = 1, 2, 3,...,n etc. Suppose that appreciable perception of chattering starts when the energy dissipated in the cutting process is half of that at resonance. T h e n the corresponding spindle speeds can be obtained from the relation B2+
4PO)M-~
(32)
or
Rearranging,
K =+_x/~B, i.e. 16P2O)2M~4x/3PO)B-K=O 4Po)M- 4Po)
(33)
This is a quadratic equation in o). Solving for o9, o)-
+4v/3PB+x/48P2B2+64p2MK +x//3B+o)o ~ / 3B2 32P2M - - 8PM - 4P 14 4MK
(34)
82 Since M K > > B 2 and (90 = ( K / M ) ½ (9= + "/'3B+(9° - 8 P M - 4-P
i.e.
co= +
(35)
- 8 P M +- (gr
The minus sign on the second term will give a negative value of (9, which is impracticable. Taking the positive sign on the second term, therefore, + x~ B . (36)
( 9 = __ 8 P M t (.Or
This indicates that for each value of P there are two values of (9 at which the energy dissipation is half of that at resonance. Let these angular spindle speeds be (9P1 and (gp~ such that 09pl = (9 r - 8 P M = (gr - A ( g p
(37)
x/~ S
(9.~ ~---(gr"~ 8-PM=(gr +h(gp
(38)
where, AOJp --
8PM-
(39)
8PK
In terms of the revolution per second of the spindle, the angular resonant speed can be designated as (or-- 27r N p , where Np is the Pth resonant speed. Hence
Npl = N p
(40)
-ANp
This is the lower limiting spindle speed for the onset of chatter. Again
Np~--Np-I-ANp
(41)
This is the upper limit of the spindle speed where chatter is feeble. The resulting speed range or bandwidth for chattering is given, therefore, by 2 A N p = Np.~ - N p , -
2A(_op x//3B(9~ 2re - 8 7 c P K
(42)
With increase of the integer number P this bandwidth decreases, which means that the range of spindle speeds for low-frequency chatter will be narrower than that for high-frequency chatter. All these phenomena can be summarized in a sketch shown in Fig. 4 drawn as depth of cut versus spindle speed. The chattering resonance occurring over the largest speed range is observed when the spindle speed is (No/4), where No = (9o/27r is the natural frequency of the system in r.p.s. The bandwidth of chattering gradually decreases at lower spin-
83
!
UNSTABLE
%...A t 2.,,. N3
II
Xo
IN
.
J-.-. _'..J.-.~.
I ~
[
No/l~ No/l~
No/~
N%
p=l.
P:2
P:I
N/,, ~ 3 1 1
e=3
I
REGION
I I
IN21 I I
I N1 ]
I
1 I
SPINDLE SPEEO No= ~° ,NAIURAL FREQUENCY OF VIBRAIION ZT'C
Fig. 4. Expected stability of a machine, indicating the chattering speed ranges. The spikes are resonances occurring at spindle speeds ~Oo/4P. At the spikes for P=2, 3, 4, etc., the differences between the solid line and the dotted lines are contributions from non-resonant vibrations.
dle speeds. It is interesting to note that quite stable and uniform depths of cut can be achieved if it is possible to raise the spindle speed above (No/4) + AN1, where the depth of cut gradually approaches the asymptote of a stable uniform depth of cut. 5. C o n c l u s i o n
In this analysis for a cylindrical workpiece clamped by a 4-jaw chuck it is observed that the cutting force is balanced by a stationary reaction and harmonic reaction supplied by the jaws at speeds 40), 8o), 120), ..., etc., where 0) is the spindle speed. Resonant vibrations or chatter vibrations will be observed at the frequencies 0)0/4, 0)0/8, 0)0/12,.., etc., where 0)0 is the angular natural frequency of the system. For stable operation of the system it is advisable to keep the spindle speed above o)o/4. The chattering ranges of spindle speeds are also given in terms of system parameters. In order to avoid chattering at low spindle speeds, the system parameters e.g. mass, stiffness and damping, must be adjusted or, alternatively, the chucking mechanisms should be modified.
84
References 1 C.H. Kahng, H.W. Lord and T.L. Davis, The effect of chucking methods on roundness error in boring process, Trans. ASME (Ser. B), 98 (1) (1976) 233-238. 2 K. Miyao and T. Iwaki, Deflection of circular plate clamped in a three-jaw chuck, Trans. JSME, 35 (170) (1969) 313-317. 3 M. Rahman and Y. Ito, Machining accuracy of a cylindrical workpiece held by a three-jaw chuck, Bull. Jpn. Soc. Proc. Eng., 13(1) (1979) 7-12. 4 M. Rahman, A study of dynamic performance test for lathes, Dr. Eng. Thesis, Tokyo Institute of Technology, Japan, 1979. 5 M.A. Matin and M. Rahman, Analysis of the cutting process of a cylindrical workpiece clamped by a three-jaw chuck, Trans. ASME, J. Eng. Ind., (accepted for publication).
73
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
ANALYSIS OF THE CUTTING PROCESS OF A CYLINDRICAL W ORKPIEC E CLAMPED BY A F O U R - J A W CHUCK
M. RAHMAN
Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge (Singapore 0511) (Received November 7, 1988; accepted in revised form November 24, 1988)
Industrial Summary A stability analysis of the cutting process of a solid cylindrical rotating workpiece clamped by a four-jaw chuck with balanced pressure at the jaws has been carried out. The net reaction balancing the cutting force is found to consist of an infinite number of force components oscillating at frequencies 4(0, 8(0, 12(o..... etc., where o9 is the angular speed of rotation of the workpiece, and a stationary force term 2Fo where Fo is the static balancing force impressed along each jaw-line. The stationary force term tends to give a uniform depth of cut. However the harmonic force terms tend to spoil the uniformity in the depth of cut. It is predicted that under the resonant condition of the fourth-harmonic force, the minimum depth of cut will occur at the mid positions between two jaws and the maximum at the jaw-positions. Chatter vibration of the system will be observed at angular spindle speeds approaching O)o/4, o90/8, O9o/12..... etc., where too is the angular natural frequency of the system. A stability chart has been drawn to suggest the safe operating regions.
Notation (o
angular speed of rotation of the workpiece
Fo static balancing force impressed along each jaw-line F'o circumferential force that revolves the workpiece (Or angular speed of rotation at resonance (OO angular natural-frequency
P ll 12
an integer length of overhang of the workpiece clamped length of the workpiece (jaw length) [(t) resultant cutting-force f r ( t ) radial component of the cutting force a, b Fourier coefficients angle between the resultant cutting-force and the radial component of the cutting force r any radial distance inside the workpiece radius of the workpiece R 0378-3804/89/$03.50
© 1989 Elsevier Science Publishers B.V.
74
M B K Zn ,On
Y,~ x
mass of the system damping coefficient of the system stiffness of the system impedance of the system phase angle of the system admittance of the system radial depth of cut
1. I n t r o d u c t i o n
In turning processes, a deviation of shape called out-of-roundness is commonly observed [1,2]. This deviation of shape is found to vary with the type of clamping, the level of the cutting force, the overhang of the workpiece, the workpiece stiffness, etc. [ 1,3 ], all of which are related directly to the performance of the machining process. For example, it has been observed that if a cylindrical workpiece is turned using a three-jaw chuck, a deviation in shape as shown in Fig. 1 (a) is developed [ 1,3 ]. On the other hand, if it is turned using a four-jaw chuck, the shape deviation observed is as shown in Fig. 1 (b). These shape deviations become distinct at the onset of chatter and beyond [ 3,4 ]. The cause of the shape deviation in the case of a three-jaw chuck has been clarified already [5]. Currently, self-centering type four-jaw chucks are used quite commonly also for turning cylindrical workpieces. However, no theoretical analysis has been carried out so far regarding the deviation of shape of a machined workpiece J
a
b
J Fig. 1. Deviation in shape observed by the author after the onset of chatter of a workpiece turned in: (a) a three-jaw chuck (after Ref. [3 ] ); (b) a four-jaw chuck. (J: jaw position; length of overhang: 145 mm; spindle speed: 450 rpm; depth of cut: 3 mm; diameter of workpiece: 60 mm. )
75
clamped by a four-jaw chuck. In this study, an analysis of the cutting process of a solid cylindrical workpiece clamped by a four-jaw chuck with balanced pressure at the jaws has been carried out to find the cause of shape deviation in such a cutting process. 2. A n a l y s i s o f t h e e u t t i n g chuck
process for a workpiececlampedby
a four-jaw
Under resonant conditions it would be expected that the profile of the radial depth of cut on a cylindrical workpiece clamped by a four-jaw chuck would be as sketched in Fig. 2. The maximum depth of cut is expected along the lines AC and BD and the minimum depth of cut is expected at 45 ° on both sides of the lines AC and BD. The minimum depth of cut is attributed to the maximum reaction Fo which in terms of the forces F~ applied at the jaws A', B', C' and D' can be obtained by taking the moment about the clamped end of the workpiece, where, for equilibrium, Fo = F 'olff (/1 + 12). The direction of Fo is radially outwards, as indicated. The expected reaction diagram will be in compliance with the expected cutting profile of Fig. 2. Such reaction profile can be determined by Fourier analysis of the cutting force. Let fr (t) be the radial component of the cutting force Fo
iA
/
,EXPECTED CUTTING PROFILE
A'
I
CYLINDRICAL WORXPIECE
B'D' - ~ I I D - -
•
fz (t)
TOOL j
l, - JAW CHUCK
c' \\ I
1 17
Fig. 2. Expected cutting profile. The jaw positions are A', B', C' and D' with clamping force F ~.
76 at point P at time t developed for balancing the cutting force exciting the system, expressed as fr ( t ) =]CrA + / r B
+frC +/rD
(1)
where, RrA, frB, frC and rid are the reactional forces from points A, B, C and D. Using a Fourier series expansion with the abbreviation 0--cot, where co is the angular speed of the spindle/workpiece, oo
/ra = ~ + ~__ [am cosmO+bmsinmO]
(2)
--1
frB=2+
~ m=l
frc=~+Z Z
(3)
[amcosm(0+180°)+bmsinm(0+180°)]
(4)
[am c o s t a ( 0 + 2 7 0 ° ) + b m s i n m ( 0 + 2 7 0 °)]
(5)
m =--1
ao ~ f~D=~-+~ ~
[amcosm(O+90°)+bmsinm(O+90°)]
m ==1
In the above the constant term ao/2 and the cosine harmonics arise from the cosine-components and the sine harmonics arise from the sine-components of the reactions at A, B, C and D. Considering the reaction from A, oo
F. cos 0 = 2 +
~ am cos mO
(6)
trt=l
and Fosin0= ~ bmsinm0
(7)
m=l
it is tempting to simplify the analysis by using the cosine harmonics only. However, some higher harmonics of the component Fo sin 0 are found to be in the radial direction: this will be clear in the following analysis. As the net reaction is minimum at the mid-position between any two jaws, by integrating between these mid-positions the Fourier coeffcients am and bm are given by
n/4 am =-
7g
Fo cos mO dO --~/4
-F°[sin(m+l)~/4'sin(m-1)zc/417~ m+l m-
(8)
77 n/4
bm=l
7r
f
Fosin0sinm0d0
--n/4
- F °I s i n n
(m-1)n/4ml_
sin (m+ l )n/4
(9)
The average value is obtained by setting m = 0 in eqn. (8), i.e.,
ao o[ - 2n
•.
sin n / 4 - sin ( - n/4)
----nsin n/4
(10) (11)
ao=0.45Fo
Substituting eqns. (2)- (5) into eqn. (1),
fi(t)=2ao+ ~ (amCm(O)+bmSm(O))
(12)
m=l
where
Cm(0)=cosmO+cosm(O+90°)+cosm(O+180°)+cosm(O+270 °)
(13)
Sm(O)=sinmO+sinm(O+90°)+sinm(O+180°)+sinm(O+270 °)
(14)
These summations will vanish except for m = 4 , 8, 12, ..., 4n etc. The nonvanishing terms are
C4(0)=4cos40
$4(0)=4 sin40
C8(0) = 4 cos 80
$8(0) = 4 sin 80
C12(0) = 4 cos 120
$12(0)= 4 sin 120
(15)
Hence by using the relations of eqn. (15) the series eqn. (12) can be written as
[r(t) = 2 a o + 4 ~ [atn cos(4nO)+b4n sin(4n0)] n=l
(16)
T h u s both cosine and sine harmonics are involved in the radial reaction satisfying the condition It(t) =Fo for 0=0 ° , 90 °, 180 ° and 270 ° . A plot of this function versus 0 gives the expected reaction profile. T h e radial c o m p o n e n t of the cutting force will be balanced by this reaction i.e.
f(t) cos fl=2ao + 4 ~ [a4ncos(4no)t) q-bnnsin(4nrot) ] rt~l
(17)
78 It is interesting to observe that although the spindle rotates at an angular speed w, the cutting force oscillates at frequency 40), 80), 12o9, ..., etc. This means that reactions from the jaws are transmitted at speeds faster than the spindle speed. Incorporating the radial variation of the reaction, the cutting force impressed at any radial distance r inside the workpiece can be expressed as: 4n
f(t'r) c°sfl=2a°(R)+4 (R)n=l ~
[a4~cos(nn0)t)+bnnsin(4n0)t)]
(18)
3. Analysis of the radial depth of cut
The radial depth of cut is a solution of the differential equation
Md'~x BdX+Kx=Fx(f) dt 2 ~- dt 4n
=2a°(R)+4
(~R ) n = l
[a4ncos(4n0)t)+b4nsin(4n0)t)]
(19)
where M, B and K are respectively the mass, damping and stiffness factors involved in the dynamic compliance of the system. Consider the depth of cut when r--,R. As 40), 80), 12o9..... 4n0), etc. are the frequencies involved in the cutting force, the time harmonic solution is to be obtained by setting d d r - (j4n0))
( 20 )
d2 dt ~ - (j4n0)) 2
(21)
where j = \ / - 1 ~<7r/2 (in polar form). Using eqns. (20) and (21) and r--.R, the solution of eqn. (19) is given by
x ( t , R ) =2ao - - ~ 4. . ~ a4n cos(4n0)t- ½7c-On) +b4~sin(4n0)t- ~~ -On) =1 4neoZ~
(22)
For r < R, the solution is
x(t,r) = ~2 a- ° ( R ) +4 ~
=1
a4ncos(4n0)t-½Jr-~)+b4nsin(4n0)t-½~-On) 4n0)Zn
(23)
79 The effective depth of cut is given by the difference Ax(t) =x(t,R) -x(t,r) or Ax(t)
1-
+ ~ F.
[ r~4n-][-a4nsin(4n0)t-O,)-b4ncos(4n0)t-On)-]
,,z'_-,L
JL
j
(24)
where Z,, and 0,, are the impedence and phase angle of the system respectively:
Z,, =x,/B2 + ( 4n0)M- K/ 4n0) ) 2 •
0n =can
_J4n0)M-K/4n0)~ ~
~
(25)
j
(26)
It is worth noting that the system impedance Z, and phase angle 0n are sensitive to the oscillating frequencies 4n0). Under resonant conditions, 0~ = 0, i.e. the system is in phase with the cutting force so that 4n0)M-K/4n0)--0 and Zn = B. Hence the resonant spindle speeds are given by i
(Or = ~
I(K)
~
1
(27)
= 4--~0)o
where 0)o = (K/M) ~ is the angular natural frequency of the system. The occurrence of resonance in the system with these spindle speeds is explained in the following manner. Suppose that the spindle speed is set to 0)r = 0)o/4. Then from eqn. (24), the resulting depth is given by
2a°(l-R)-~a4sin(40)rt)-b4c°s(40)rt) [ (R)4]
Ax (t)re~ ----~-\
-~
0)rB
2-0)rZ2
I--
112
-~a12sin(120)rt-dP3)--b~2c°s(120)rt--O3)[40)r Z,~ + ...etc.
1- (R)
] (28)
Evidently the second term will resonate, vibrating the system at frequency 40)~= 0)o. This means that corresponding to the spindle speed 0)~=0)o/4 the system will experience resonant vibration at the natural frequency. This is just an unstable situation. The third term yields non-resonant vibration at twice the natural frequency, adding to the instability. Similarly the fourth term gives non-resonant vibration at three times the natural frequency and so on. Thus
80
the spindle speed o)r= o)0/4 sets the system to an unstable situation, which will be manifested in the form of severe chattering in the system. In such a case the resonant depth of cut will dominate over the non-resonant harmonic cuts so that the depth of cut is given approximately by 4
2ao r t. (atsin4o)rt-b4cos4o)rt)wrB I I - ( R ) Ax(t)res______~_(l__~)_
I
(29)
This is minimum when O---o)rt=O°, 90 °, 180 ° and 270 ° and m a x i m u m when o)~t=45 ° , 135 ° , 225 ° and 315 ° . The minimum depth of cut is given by Ax (t)res =Xo --Xl
(30)
where Xo= (2ao/K) ( 1 - r / R ) , depth of cut is given by
Xx= (4b,/o)oB) [ 1 - (r/R)4]. The maximum
AX ( t ) r e s = X 0 "F X 1
The results are sketched in a polar diagram in Fig. 3. This profile will be distorted, however, with minute cut marks at different angular locations aris-
I /
\
/
\
/I
\
,1
,
--.
//
-J- -t-- "
/
i
xo.xl
~
(NAX. OEPIH
o
I
\
OF CUT)
I
\
*Y
xo
x
\
" ~
/
Xo-X I (NIN. OEPTHOF CUT)
li"
I I
Fig. 3. Schematic radial depth of cut. Xo is the targetted uniform depth of cut at a radius r. R is the radius of the worlq~ieee before cutting.
81
ing from the non-resonant higher harmonic vibrations. Next consider that the spindle is set to wr = o)0/8. In this case the 8th harmonic force will resonate, vibrating the system at the natural frequency. The 4th harmonic will be nonresonant at o)o/2, the 12th harmonic will be non-resonant at 30)0/2, the 16th harmonic at 20)o and so on. This is another stage of the occurrence of chattering. In this case, since the spindle speed is lower than before, the resulting depth of cut will increase but the amplitude of chattering will be less, because the resonance is now with the 8th harmonic, which is much lower in amplitude compared with the 4th harmonic. Thus chattering of lower amplitudes will be observed at spindle speeds o)r = O)o/12, o)o/16,...,o)o4P, etc. Now every resonant phenomenon is perceived over a certain range of frequencies. In the above, although the resonating spindle speeds for the onset of chatter can be identified, in practice the chattering effect will be observed over a certain range or "bandwidth" of spindle speed around the resonant speed. These speed ranges can be evaluated from a study of the admittance of the system, where the latter is given by
~-I [B2+(4no)M- 1 ~2~-~ Y"-~,4no)K] J
(31)
The harmonic depth of cut is directly proportional to the admittance, which latter controls the storage and dissipation of energy supplied to the system. Maximum energy will be dissipated under resonant conditions i.e. when Y, = B or at o)~=o)o/4P, where P = 1, 2, 3,...,n etc. Suppose that appreciable perception of chattering starts when the energy dissipated in the cutting process is half of that at resonance. T h e n the corresponding spindle speeds can be obtained from the relation B2+
4PO)M-~
(32)
or
Rearranging,
K =+_x/~B, i.e. 16P2O)2M~4x/3PO)B-K=O 4Po)M- 4Po)
(33)
This is a quadratic equation in o). Solving for o9, o)-
+4v/3PB+x/48P2B2+64p2MK +x//3B+o)o ~ / 3B2 32P2M - - 8PM - 4P 14 4MK
(34)
82 Since M K > > B 2 and (90 = ( K / M ) ½ (9= + "/'3B+(9° - 8 P M - 4-P
i.e.
co= +
(35)
- 8 P M +- (gr
The minus sign on the second term will give a negative value of (9, which is impracticable. Taking the positive sign on the second term, therefore, + x~ B . (36)
( 9 = __ 8 P M t (.Or
This indicates that for each value of P there are two values of (9 at which the energy dissipation is half of that at resonance. Let these angular spindle speeds be (9P1 and (gp~ such that 09pl = (9 r - 8 P M = (gr - A ( g p
(37)
x/~ S
(9.~ ~---(gr"~ 8-PM=(gr +h(gp
(38)
where, AOJp --
8PM-
(39)
8PK
In terms of the revolution per second of the spindle, the angular resonant speed can be designated as (or-- 27r N p , where Np is the Pth resonant speed. Hence
Npl = N p
(40)
-ANp
This is the lower limiting spindle speed for the onset of chatter. Again
Np~--Np-I-ANp
(41)
This is the upper limit of the spindle speed where chatter is feeble. The resulting speed range or bandwidth for chattering is given, therefore, by 2 A N p = Np.~ - N p , -
2A(_op x//3B(9~ 2re - 8 7 c P K
(42)
With increase of the integer number P this bandwidth decreases, which means that the range of spindle speeds for low-frequency chatter will be narrower than that for high-frequency chatter. All these phenomena can be summarized in a sketch shown in Fig. 4 drawn as depth of cut versus spindle speed. The chattering resonance occurring over the largest speed range is observed when the spindle speed is (No/4), where No = (9o/27r is the natural frequency of the system in r.p.s. The bandwidth of chattering gradually decreases at lower spin-
83
!
UNSTABLE
%...A t 2.,,. N3
II
Xo
IN
.
J-.-. _'..J.-.~.
I ~
[
No/l~ No/l~
No/~
N%
p=l.
P:2
P:I
N/,, ~ 3 1 1
e=3
I
REGION
I I
IN21 I I
I N1 ]
I
1 I
SPINDLE SPEEO No= ~° ,NAIURAL FREQUENCY OF VIBRAIION ZT'C
Fig. 4. Expected stability of a machine, indicating the chattering speed ranges. The spikes are resonances occurring at spindle speeds ~Oo/4P. At the spikes for P=2, 3, 4, etc., the differences between the solid line and the dotted lines are contributions from non-resonant vibrations.
dle speeds. It is interesting to note that quite stable and uniform depths of cut can be achieved if it is possible to raise the spindle speed above (No/4) + AN1, where the depth of cut gradually approaches the asymptote of a stable uniform depth of cut. 5. C o n c l u s i o n
In this analysis for a cylindrical workpiece clamped by a 4-jaw chuck it is observed that the cutting force is balanced by a stationary reaction and harmonic reaction supplied by the jaws at speeds 40), 8o), 120), ..., etc., where 0) is the spindle speed. Resonant vibrations or chatter vibrations will be observed at the frequencies 0)0/4, 0)0/8, 0)0/12,.., etc., where 0)0 is the angular natural frequency of the system. For stable operation of the system it is advisable to keep the spindle speed above o)o/4. The chattering ranges of spindle speeds are also given in terms of system parameters. In order to avoid chattering at low spindle speeds, the system parameters e.g. mass, stiffness and damping, must be adjusted or, alternatively, the chucking mechanisms should be modified.
84
References 1 C.H. Kahng, H.W. Lord and T.L. Davis, The effect of chucking methods on roundness error in boring process, Trans. ASME (Ser. B), 98 (1) (1976) 233-238. 2 K. Miyao and T. Iwaki, Deflection of circular plate clamped in a three-jaw chuck, Trans. JSME, 35 (170) (1969) 313-317. 3 M. Rahman and Y. Ito, Machining accuracy of a cylindrical workpiece held by a three-jaw chuck, Bull. Jpn. Soc. Proc. Eng., 13(1) (1979) 7-12. 4 M. Rahman, A study of dynamic performance test for lathes, Dr. Eng. Thesis, Tokyo Institute of Technology, Japan, 1979. 5 M.A. Matin and M. Rahman, Analysis of the cutting process of a cylindrical workpiece clamped by a three-jaw chuck, Trans. ASME, J. Eng. Ind., (accepted for publication).