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Stability analysis of chatter vibration for a thin-wall cylindrical workpiece
t
Int. J. Math. Toob Manufact. VoL 35. No. 3. pp. 431--444,1995 CopyrightC 1994ElsevierScienceLid Printed in Great Britain. All rights n~rved
Pergmon
0890-695519559.50 + .00
STABILITY ANALYSIS OF CHATFER VIBRATION THIN-WALL CYLINDRICAL WORKPIECE
FOR A
G. J. LAIt~ and J. Y. CHANG? (Received 1 September 1993; in final form 10 January 1994)
AlmtnctmTbe vibration characteristics and the vibration direction angle between the beam mode and shell mode of the three-jaw clamped thin.wall cylindrical workpiece will change accordin8 to the relative position of the cutting tool to the chucking jaws. This will induce parametric vibration and cause the chatter characteristics of a thin-wall cylindrical workpiece to become more complicated than those of a solid workpi¢ce. In this study, the three-jaw damped thin-wall cylindrical workpiece was considered as a parametrically excited vibration system. Initially, the relationship between the parametric vibration and the workpiece system structure was investigated, and then the equivalent stability cutting depth of the thin-wall cylindrical workpiece was analyzed by a numerical calculation method. When the three-jaw clamped thinwall cylindrical workpiece was considered as a parametric vibration system, the analytical results were consistent with the experimental results.
NOMENCLATURE
C1,C12,C~ Di
F
f
Gc
G . , G~, h
Ko K. Kl,KI2,Ke M,M~ m-1 n s
T U
Uo Y
Y Ybd, Y.d Zbd, Z~d
B~, 812 0.1) O'12
k~, k12
6, ~1, tt
tO tOl, tOl2
damping coefficients of modes of vibration of structure workpiece inner diameter resultant cutting force or vector force exciting the structure frequency normalized cutting process dynamics normalized dynamic compfiance of structure workpiece thickness static directional cutting stiffness (slope of curve of F versus U) static directional stiffness of structure (slope of curve F versus y) stiffness coefficients of modes of vibration of structure equivalent masses of structure number of nodes in axial mode shape number of circumferential waves Laplace operator delay time between the primary and secondary chatter instantaneous uncut chip-thickness average or steady state uncut chip-thickness relative displacement between tool and workpiece normal to the machined surface radial direction eigenvectors on the direction y for beam and shell deflections, respectively eigenvectors on the direction ~ for beam and shell deflections, respectively angles of modes of vibration of the structure relative to a line normal to the machined surface angle between the resultant cutting force and a fine normal to the machined surface damping ratios of modes of vibration of structure vibration direction angles of vibration modes stiffness factors angle factors overlap factor =2=f, radian frequency undamped natural frequencies of vibration modes of structure I.
INTRODUCTION
BASED on a previous study [1], it was found that a three-jaw clamped thin-wall cylindrical workpiece was easily excited. The vibration of a thin-wall cylindrical workpiece consists of the characteristics of both beam and shell vibration modes, and the
tDepartment of Mechanical Engineering, Tatung Institute of Technology, 40 Chung-Shan N. Road, Sec. 3, Taipei, Taiwan 10451, R.O.C. ~To whom correspondence should be addressed. 431
432
G.J. LAi and J. Y, CHASe
stability cutting depth for a thin-wall and a solid cylindrical workpiece with the same moment of inertia will be quite different. The three-jaw chuck is the most practical and generally used support for the turning process. Due to the non-continuity of the three-jaw clamping condition, the stiffness coefficient, the damping coefficient, the equivalent mass and the vibration direction angle between the beam mode and the shell mode will vary according to the relative position of the cutting tool to the chucking jaw. This variation will induce the parametric vibration and cause the chatter characteristics of this three-jaw turning system to become very complicated. Rahman and Ito [2] have discussed in detail the chatter characteristics of the threejaw clamped solid cylindrical workpiece, but did not study the thin-wall cylindrical workpiece. Hence, the purpose of this study was to investigate the relationship between the vibration pattern of a three-jaw clamped thin-wall cylindrical workpiece and the stability cutting depth. In this study, the three-jaw clamped thin-wall cylindrical workpiece was considered as a parametric vibration system and its equilibrium equation was established. Initially, a numerical method was applied to analyze the effect of the relative position of the cutting tool to the chucking jaws on the variations of the stiffness coefficient and the vibration direction angle between the beam mode and the shell mode for a thin-wall cylindrical workpiece. Then, the parametric vibration induced by the variations of the stiffness coefficient and the vibration direction angle was analyzed. Finally a series of cutting experiments were carried out and the stability cutting depth of the thin-wall cylindrical workpiece was established. 2. THE VIBRATION PROPERTIES OF A THIN-WALL CYLINDRICALWORKPIECE The effect of the relative position of the cutting tool to the chucking jaws on the shell mode shape was obtained by a numerical method and the results are shown in Fig. 1 (the numerical method will be discussed in a later section). In Fig. l(a), the solid line represents the thin-wall cylindrical workpiece clamped by a three-jaw chuck. The positions of these three jaws are Jo, J1 and J2. The points A1 and A2 represent the different positions of point A. The dashed line and the dotted line represent the 12 f
1
IJ 2
" q - -.....
l ~(~
" ~
I
i<
TZ,d
-I Z~d
" _ /
I
(b) e = 3 o °
(o) e = o °
A2 '
I
I
1
"'] ~
A2i
dz
I
•
J
d o\"-'F - ~ / IJo
I
(~) e=6o °
(d)
e=9o °
r (e) e =1200
FIG. 1. The effect of the relative position of the cutting tool to the chucking jaw on the m mode shapes.
=
1, n
=
2 shell
Stability Analysis of Chatter Vibration
433
Pn = 1, n = 2 shell mode shape of a thin-wall cylindrical workpiece [1]. m and n are the subscript indices of the vibration modes for all the vibration properties in the following discussion. In Fig. l(a), the jaw Jo and the cutting tool are located on the same side and the jaws J1 and J 2 are located symmetrically about the y axis. At that moment, point A will oscillate between point A1 and point A2 when the shell vibration mode occurs. When the workpiece (or chucking jaws) as shown in Fig. l(a) rotates counter-clockwise 30, 60, 90 and 120° , the relative position of the cutting tool to the chucking jaws will change, and the shell mode shapes of the workpiece vibration will be as shown in Fig. l(b), (c), (d) and (e), respectively. During the cutting process, the motion locus of point A changes according to the relative position of the cutting tool to the chucking jaws. In Fig. l(b), Y~d and Z~dStand for the eigenvectors of point A in direction ~ and direction 7, respectively, and vrt2 represents the vibration direction angle of the m = 1, n = 2 shell mode and Crl2 = tan - t ( ~ , ~ / ~ ) ; when the workpiece rotates through one revolution, the variation in their values will be as shown in Fig. 2, which shows the periodic change according to the relative position of the cutting tool to the chucking jaws. Figure 2 shows that when the angle 0 between the cutting tool and a chucking jaw is equal to 60, 180 or 300°, the value of ~ shows a maximum value, expressed as 2 ~ m ~ . When the angle 0 is equal to 0, 120 or 240°, the value of ~ shows a minimum value denoted by Ysdmin. As the workpiece rotates through one revolution and the angle 0 is equal to 30, 150 or 270 °, the value of ~t2 shows a maximum value denoted by O'12max. y ~ is inversely proportional to the shell mode stiffness coefficient Ke and crt2 represents the shell mode vibration direction angle, and both are functions of the angle Ysdrno~x
'
I
'
,
,
I
,
I
,
I
I
,
t
I
,
L
,
I
L
L
I
I
7--sd
i ,w
+ v e~ 0
I
/ 0"12max
o
/
.
I
L,,,I,,, 0°
60°
I,
i i lllLI
120 °
180°
i t ill 240 °
300 °
ill 360 °
0 Fic. 2. The effect of variation of the angle of 0 between the cutting tool and the chucking jaw on y~, zs, and cry2.
434
G . J . LAI and J. Y. CHANG
0 between the cutting tool and a chucking jaw. From a previous study [1], it has been shown that the shell mode natural frequency t~12 and the damping ratio 812 are the same for all positions of the cutting tool relative to the chucking jaw. The equivalent mass Me was equal to Ke/to22 and the damping coefficient Ce was equal to 2512(MeKe) i. Therefore, Ce, M¢ and K~ can be defined as functions of the angle 0. In the threejaw clamped thin-wall cylindrical workpiece system, the vibration characteristics must include both the shell mode and the beam mode. The eigenvector Ybd, vibration direction angle 0"1, damping coefficient and equivalent mass for m = 1 beam mode are also functions of the angle 0 between the cutting tool and the chucking jaws. It is known from shell vibration theory that the vibration properties of the shell mode and the beam mode of a thin-wall cylindrical workpiece are dominated by the ratio of the inner diameter to wall thickness Di/h. For convenience in discussing the effect of Di/h o n Yba,Y~d, trl and tr12 on the rotary workpiece, the following four definitions are used: Y~dmax-- Y~=i,, (1) the stiffness factor for shell mode, h12 = Ysdmax + Ysdmin
2 (2) the stiffness factor for beam mode, hi
----- Ybdraax -- Ybdmin
Ybdmax "l- Ybdmin ; 2 (3) the direction angle variation for shell mode, Atr12 = O'12max; (4) the direction angle variation for beam mode, Atrl = trlm~x. The three-jaw clamped workpiece was considered as a parametric vibration system and the effect of the parametric vibration on the stability cutting depth in the turning process of the thin-wall cylindrical workpiece was investigated. 3.
3.1.
ANALYSIS FOR THE STABILITY CUTTING DEPTH IN THE TURNING PROCESS OF THE THIN-WALL CYLINDRICAL WORKPIECE
Derivation of thin-wall workpiece equilibrium equation
To calculate the stability cutting depth, it was convenient to use the chatter theory proposed by Merritt [3]• Merritt's method was used and whereas the parametric vibration of the three-jaw clamped thin-wall cylindrical workpiece may have both linear and nonlinear components, it was assumed here to be a linear vibration. For the stability analysis of the thin-wall cylindrical workpiece, three equations, the uncut chipthickness relationship, the cutting force relationship and the equilibrium equation, must be established. The first two equations can be derived from the results of previous studies [3, 4]. It can be seen from the schematic diagram of Fig. 3 that the uncut chip-thickness equation [3] can be written as
Tool (rlgldty mount;eoD FIG. 3. Uncut chip-thickness variation.
435
Stability Analysis of Chatter Vibration
U(t) = Uo(t) - y(t) + ~ y(t - T),
(1)
where Uo(t) is the average or steady state uncut chip-thickness, y(t) is the relative displacement between tool and workpiece normal to the machined surface, T is the delay time between the primary and secondary chatter, and IXis the regenerative effect or overlap factor. The Laplace transform of equation (1) is U(s) = Uo(s) - y ( s ) + Ix e-sTy(s).
(2)
If dynamic vibration within the cutting process is neglected, the cutting force equation [4] is F(t) = K= U(t),
(3)
where Kc is the static directional cutting stiffness and is a constant for a fixed cutting depth. The Laplace transform of equation (3) is F(s) = K= U(s).
(4)
In the following section, the equilibrium equation of the parametric vibration will be derived. In the preceding section it was stated that the vibration pattern of the three-jaw clamped thin-wall cylindrical workpiece simultaneously includes both the beam and shell parametric vibration modes. In the stability analysis of the thin-wall cylindrical workpiece, the beam and shell modes of the thin-wall cylindrical workpiece are considered as a vibration system with two degrees of freedom (Fig. 4). The beam mode and the shell mode are connected in series. Initially, only the easily excited shell mode was considered and the equilibrium equation of the one degree of freedom parametric vibration was derived. This equilibrium equation was then extended to two degrees of freedom, composed of the beam and shell modes. In this study, the effect of the two degrees of freedom parametric vibration on the stability cutting depth is discussed. 3.1.1. The equilibrium equation for one degree o f freedom. It was assumed that the workpiece had only one degree of freedom and its equilibrium equation is then Mc (d21dF)[y(t)]lcos oq2(t) + Cc (d/dr) [y(t)l/cos ot,2(t)
(5)
+ K~ [y(t)]/cos oq2(t) = F(t)cos[oh2(t) - f~],
I
beam
\ \ \ \ \ \ \ \ \ \ \
I
FIG. 4. Structure with two degrees of freedom for the thin-wall cylindrical workpiece.
436
G . J . LAI and J. Y. CnAr4C
where Me, Ce, and Ke represent the equivalent mass, damping coefficient, and stiffness coefficient, respectively, of the m = 1, n = 2 shell mode. a~2(t) is the vibration angle of the structure relative to a line normal to the machined surface, and 13 is the angle between the resultant cutting force and a line normal to the machined surface. From the previous results, it is known that the stiffness coefficient Ke, damping coefficient Ce, and equivalent mass Me vary cyclically three times when the three-jaw clamped thin-wall cylindrical workpiece rotates once. Considering the linear vibration and the effect of the variations of these vibration properties on the equilibrium equation, the equation now takes the following form: (1-h~2sin 3to~t)M (d2/dt2)[y(t)]/cos a~2(t) + (1-h12sin 3tOrt)Clz (d/d/) [y(t)]/cos a~2(t) + (1-k~2sin 3to#)K~2 [y(t)]/cos a~2(/) = F(/)cos [a12(t)-13],
(6)
where M, C12, and K~e represent the mean value of the equivalent mass, damping coefficient, and stiffness coefficient, respectively, of the m = 1, n = 2 shell mode. tOr is the rotational speed of the workpiece. Rearranging equation (6) one can obtain
M y"(t) + C~2y'(t) + K12y(t) = F(t)(1-h12sin 30Jrt)-' g12(t),
(7)
where
y"(t) = (d2/dt2)[y(t)] y'(t) = (d/dt)[y(t)] g12(t) = cos[a12(t) - 13]cos[alz(t)].
(8)
These results show that the vibration direction angle o12 will vary according to the relative position of the cutting tool to the chucking jaws. This means that the vibration angle otlz(t ) has a frequency variation of three times that of the rotating workpiece. That is Ot12(/) = ~12 -- A0.1zsin(3to~t),
(9)
where ~12 is the mean value of the vibration angle of the structure relative to a line normal to the machined surface. Substituting a12(t) of equation (9) into equation (8) and expanding the right hand side of equation (8), one can obtain g12(t) = cos~12 cos(~12-13) cos2 [A0.12sin(3tOrt)] + COS(~12-- 13) sin~12 cos[A0.12sin(3oJd) ] sin[A0.12sin(3ort) ] + cos~12 sin(~12 -13) cos[A0.12sin(3o#)] sin[A0.12sin(3ort)] + sin~12 sin(ax2 -13) sin2[A0.~2sin(3oJ#)].
(10)
When the value of A0.12is small, g12(t) becomes approximately g12(t) = cosal2 cos(al2 - 13) + A0.I2[COS(~12-13) sin~12 + c0s~12 sin(~12-13)] sin3oJ~t = g12 (l+~12sin3tort), where
(11)
Stability Analysis of Chatter Vibration
437
g12 = COS~12 coS(~12 -- ~) Ao'12sin(2~12 --~) 612 = COS~12 coS(~12 -- [~)"
612 is called the angle factor, where the values of K12 and 13 are determined from the tool geometry, workpiece material and cutting conditions, and the value of Atr12 is a function of Dt/h. Substituting g12(t) of equation (11) into equation (7), expanding the right hand side of equation (7), and assuming that higher orders can be neglected, the equilibrium equation of the three-jaw clamped thin-wall cylindrical workpieee can be expressed as (12)
M y"(t) + C12 y'(t)+ K12 y(t) = F(/)g12 (1 + M2 sin3oJrt + 612 simo~t).
Expanding the sine function of equation (12) and considering only the first two terms, yields
MY"(t)+C12Y'(t)+KI2y(t) = g12 [F(/) + 3F(t)kt2tOrt + 3F(t)612tOrt - 9/2F(t)k12to3t 3 - 912F(t)612o3t3 + ...].
(13)
If the stability analysis of the chatter loop for a step response is considered, assuming that a kind of trigger such as a sudden change of system parameter causes the chatter onset, then the Laplace transform of equation (13) is
y(s)
g12
=
(M$3 + ~
+
K12)
(3k12~Or 3612fOr 27kt2to3 27612to3 1
+
s
+ --$
$3
--$3
+
) ...
F(s). (14)
The dynamic compliance is then derived as
y(s)
d12($)
F(s) Km (~12 + 2812$(I)12"[-1)
(15) ~
where d12($) = 1 + 3k12tOr -t 3612tOt 27k12¢03 276120~r3 S
$
$3
$3
+ "'" '
1~Kin = g12/K12, Km is the directional static stiffness of the structure, to22 = K12/M, and 812 = C12/2(Kt2M) t. 3.1.2. The equilibrium equation for two degrees of freedom. In Fig. 4, the beam and the shell mode are now connected in series, and the equilibrium equation for two degrees of freedom can be obtained. This will be Y($)
F(s)
=
gI2dl2(s)
(so~2+ 28t2s + 1) K12 0012
where
gldl($)
-# K1
(o~__~t 281s +
' + 1 Iil1
)'
(16)
438
G.J. LAI and J. Y. Cn^r~o d12(s ) = 1 +
dl(S) = 1 +
3h12tOr 3~12tOr 27h120~r3 + - -S S $3
3hltOr S
+
3~ltor 27kltOr 3 $
27~12t03 t- ... $,.3
27~1o3
$3
S3
~- "'"
g12 = COS~12 C0S(~12--13), gl = COS~I COS (~1--13)
Act12 sin(2~12 -13) Act1 sin(2~l --13) ~12 = c0s~12 cos(~l 2 _13), ~1 = cos~ 1 cos(~l _13) K12 K1 0)22 = ~ - , {~02= " ~ ,
~12 = C12/2( K12M) i, 81 = C1/2( K1M)k The right hand side of equation (16) can be written with a common denominator, and y(s) F(s)
d(s) Km
+
+1 (1112
(17) +
+ 1 ¢.01
where
Km
K12
K1
and
Klg12dlx(S)
d(s) =
+ o) 1
+ 1 + K12gldl(s)
+
0,112
+ 1
g12K1 + glK12
It is convenient to write the dynamic compliance using the normalized compliance Gm(s), and thus equation (17) takes the following form: y(s) 1 F(s) - Km Gin(s).
(18)
From equations (16) and (17), it is seen that Gin(s) is a function of the stiffness factors (hi, h12) and the angle factors (~1, ~12). The parametric degree is determined from the totality of the stiffness factors (hi, h12) and the angle factors (~1, ~12). Hence, the parametric degree will change the value of Gm(s) and influence the chatter stability cutting depth. When the vibration properties of the structural system are constant, namely when hx = ~k12 : 0 and ~x = ~12 = 0 , then Gin(s) is consistent with the expression for the general solid workpiece [3]. 3.1.3. The effect o f Ddh on the su'ffness factor and angle factor. From the expressions for the angle factor in the preceding section, it will be seen that the angle factor is a function of ~1 o r ~ 1 2 , AOrl or Aort2 and 13, where the values of or1, 0/-12, and 13 are determined from the tool geometry, the workpiece material and the cutting conditions. The values of Act1 and A f t 1 2 a r e determined from Di/h. The main purpose of the present study was to investigate the effect of Di/h on the chatter vibration of the thin-wall workpiece. In all the theoretical analyses and cutting experiments, the value of Di/h was taken as the only variable, and all the other parameters were kept
Stability Analysis of Chatter Vibration
439
constant. Hence, the angle factors (El, ~x2) were determined only from the values of Dl/h. Moreover, from the definitions of the stiffness factor, it is seen that the stiffness factors (kl, kt2) are functions of Di/h. In determining the effect of Dt/h on the stabifity cutting depth of the thin-wall cylindrical workpieee, a numerical method was used to obtain the values of the stiffness factors and the angle factors. In the previous study [1], as the value of Di/h was decreased, the thin-wall workpiece showed the characteristics of the beam and shell vibration modes. When the value of Dt/h was large, the dynamics of the shell vibration mode were more evident than those of the beam vibration mode. To obtain the general characteristics of the thin-wall workpiece, a workpiece with a low value of Di/h which showed the beam and shell modes' vibration simultaneously was initially chosen for analysis. The overhang length L and the inner diameter D~ were chosen as 100 mm and 30.5 mm, respectively, while the different values of Di/h were obtained by changing the outer diameter. To compare the difference between a solid and a thin-wall workpiece, a thin-wall workpiece (Di/h = 10.17) and a solid workpiece with Do = 30.9 mm which both had the same moment of inertia were chosen. For this solid workpiece it was considered that Di/h was equal to two for comparison purposes. The commercial finite element package (NISA II) was used for the numerical analysis. The thin-wall cylindrical workpiece was divided into 240 3-D shell elements. Each element had four nodes and each node had six degrees of freedom. The solid workpiece was divided into 480 3-D beam elements. Each element had eight nodes and each node had six degrees of freedom. In applying the boundary conditions it was assumed that at the chucking jaw positions the displacements were regarded as zero. The relationships between Di/h and the stiffness factors (kl, k12) and the angle factors (~1, ~12) were solved and the results are shown in Fig. 5(a) and (b). Figure 5(a) shows that kl is equal to 0.003 for the solid workpiece. The values of kt and k12for the thin-wall workpiece increase with an increase of Dt/h. Furthermore, the value of k12 is much larger than that of k~. The effect of DJh on the angle factor
,,
0.240.20
,,/,:
--
•
: fix" t h e boom mode of
•
: for thw bl~W'l mode of
t h e solid i m r l ~ l l c e
t.<
0.16 -o
t h e th|n woll workplace
0.12 "
0 : for t h t Ihe~ mode of
0.08
--
t h e thin woll workplece
0.04- - -
-
o -~'
,
10
O2
(o)
20
I
40
D i/h
0.42
j,
I
30
!
0.35
• -
: foe t h e boom mode of t h e solid wortq=lecal
0.28 •
: for t h e boom mode of the thIn wall worlq~lece
0.21
0 : for t h e shell mode of
0.14 e-o
the thin won workpkK:e
0.07!
0
-~'O~
10
(b)
20
30
4-0
Di/h
FIG. 5. The effect of Di/h on kl, Xl2, ~1 and ~,2.
440
G . J . Lpa and J. Y. CHANG
is shown in Fig. 5(b), where ~1 = ~12 = 45 °, and 13 = 15°. It is for this reason that the side cutting edge angle is equal to 45 ° and the chatter vibration direction of the shell mode and the beam mode is in the radial direction, so the mean values of the vibration angle of the structure relative to a line normal to the machined surface, ~1 and ti12, are equal to 45 °. The angle between the resultant force direction and the feed direction is equal to 60 °, so the value of the angle between the resultant cutting force and a line normal to the machined surface, 13, is equal to 15°. Figure 5(b) shows that ~1 is equal to 0 for the solid workpiece. The values of 61 and ~12 for the thin-wall workpiece increase with an increase of Di/h, and furthermore, the value of ~2 is much larger than that of ~1. From the above analyses, it is seen that for the thin-wall workpiece the extent of parametric vibration of the shell vibration mode is larger than that of the beam vibration mode. Furthermore, the extent of parametric vibration of the shell vibration mode increases with an increase of Di/h. Also, when the thin-wall workpiece and the solid workpiece have the same moment of inertia, the extent of parametric vibration of the thin-wall workpiece is larger than that of the solid workpiece.
3.2. Derivation of the chatter loop On combining equations (2), (4) and (18), one can obtain the equation for the block diagram of the chatter loop as shown in Fig. 6. The transfer function relating U(s) and Uo(s) can be obtained from the block diagram of the chatter loop as
U(s) 1 Uo(s) - 1 + ( 1 - p . e -st) KJKm Gin(s)"
(19)
From equation (19), the characteristic equation of the chatter loop is 1 + ( 1 - p . E sT) KJKm Gin(s)
=
(20)
0.
For the chatter loop of Fig. 6, the transfer function relating y(s) and U(s) can therefore be obtained as
y(s) _F(s) y(s) =K~Gm(s). U(s) U(s)F(s) Km
(21)
Cutting process
Structure
dynamics
dynamics
y(s)
U0(s)
I
Primary feedback path
Regenerative feedback path
Fro. 6. Block diagram of chatter loop.
Stability Analysisof Chatter Vibration
441
The gain-phase plot of y/U can be obtained by substituting s = ]to into equation (21), i.e.
Y0"to)
rc -
CmUto).
(22)
In normal cutting, the chatter vibration direction of the thin-wall cylindrical workpiece is in the radial direction (direction ~), and the angle between the chatter vibration direction and the chip-thickness variation direction is equal to 45 °. When the effect of this angle is considered, equation (22) will become y(jto) gc Gm(]to) U(jto) -- Km ~7~ '
(23)
where 1 -y(jto) Km G~(jto) = F(jto) ' which is the dynamic radial equilibrium equation. The plot of the critical locus of G~p can also be obtained from the derivation by Merritt, where _ Gcp
-1 - Y(Jto) (l_p. ~/,oT) U(jto) "
(24)
From the characteristic equation of the chatter loop of equation (20), it will be seen that when the gain-phase plot of equation (23) and the critical locus Gcpof equation (24) intersect, chatter vibration occurs. Kc of equation (23) is a function of cutting depth. Because an increase of cutting depth will increase the value of Kc and cause the chatter occurrence, it can be said that the value of Kc/Km is the value of the equivalent stability cutting depth. From equations (16) and (23), it is known that the extent of parametric vibration and the structure dynamics will influence the value of the equivalent stability cutting depth (Kc/Km), and this will be investigated in the following section. 3.3. Analysis of stability cutting depth From the preceding investigation, it is known that the extent of parametric vibration of the three-jaw clamped thin-wall cylindrical workpiece is a function of Di/h. Equations (16) and (23) reveal that the extent of parametric vibration will influence the value of y/U, and that the values of y/U and the equivalent stability cutting depth (Kc/Km) have a close relationship. Hence, the extent of parametric vibration is the dominant factor in determining the equivalent stability cutting depth for the three-jaw clamped thin-wall cylindrical workpiece. For the general thin-wall cylindrical workpiece, the extent of parametric vibration of the shell vibration mode is larger than that of the beam vibration mode. Hence, initially, the effect of the parametric vibration of the shell vibration mode on the value of the y/U for the three-jaw clamped thin-wall cylindrical workpiece was investigated. Although the values of the stiffness factor and the angle factor for the shell vibration mode were different from those for the beam vibration, the extent of parametric vibration of the shell vibration mode was determined by these two factors. In investigating the effect of the extent of parametric vibration of the shell vibration mode on the y/U gain-phase plot, the angle factor of the shell vibration mode was, for convenience, assumed to be zero. The characteristics of a thin-wall cylindrical workpiece with HTI4 35:3-F
442
G . J . LAI and J. Y. CttANG
Di/h = 10.17 had been obtained previously [1], and this workpiece was investigated further. The results for various stiffness factors kt2 and an overlap factor of W = 1 are shown in Fig. 7. The reason for choosing Ix = 1 was that an overlap factor of unity would be the most critical value for the commencement of chatter. Figure 7(a) indicates that when the stiffness factor k12 is equal to 0.0, parametric vibration does not happen, and that the chatter vibration occurs when Kc/Km is equal to 0.092. When the value of ~,12is increased, the y/U gain-phase plot will move leftward considerably, as shown in Fig. 7(b), (c) and (d). Hence, when the effect of the stiffness factor of the shell vibration mode does exist, chatter will occur at a smaller value of KdKm. That is, the extent of parametric vibration is the dominant factor in determining the value of the equivalent stability cutting depth (KdKm). Figure 8 shows the effect of the stiffness factor k12 and the angle factor ~12 of the shell vibration mode on the equivalent stability cutting depth (KdK=) of the thin-wall cylindrical workpiece. Figure 8(a) and (b) show that when the stiffness factor A12 and the angle factor ~x2 of the shell vibration mode increase, KdKm quickly decreases. Also from Fig. 8(a) and (b) it is seen that the stiffness factor and the angle factor of the shell vibration mode have the same effect on KdKm. From the previous study [1], it was shown that like the stiffness factor and the angle factor, the structural vibration is also a function of Di/h. To understand the effect of all the above three factors on the chatter stability of the three-jaw clamped thin-wall workpiece, Di/h was considered as the only variable in numerical calculations to assess the effect of these three factors on the equivalent stability cutting depth. A series of cutting experiments were also conducted (cutting speed V = 125 m/min, feed rate +4 10 (a)
Xle = 0.0 /~= [ critical
+4 10 (b)
;~le "= 0 . 0 0 1 /~= 1 critical +2 10
-2 10
-2 10
--4 10
-~,eO° -~1~° -240° - I ~ '~ - I ~ "
-6~
o•
• _~lSOo
,
.
_ ~ o
i .
.
2~o
i
.
_1~u
J _120o
.
.., _60o
--4 10
i
oo
-I-4 10 (c)
Ate0.01 /~.- 1 ctltlcol
+4 10 (d)
+2
;~m = OA /~', 1 critical +2 10
1 -2 10
•
1
.
z
.
a
.
,
-360" -300e -240° -180° -120 °
.
-4 10
t
-60 °
o"
-2 10
I
,
I
-aao" - 3 o o ° -24o"
~
I
, ,
J
-~ao ° -12o °
-4 10
* -eo °
o•
FIG. 7. G a i n - p h a s e plot of critical loci with plot of y/U superimposed for the different values of X~2 (DJh = 10.17; Kc/K= = 0.092, here KdK~ is equivalent stability cutting depth without considering the parametric vibration),
Stability Analysis of Chatter Vibration
443
0.12 .c_
0.10
3 (.3
~J~ 0.08 0.06 0.04 ll}
-o
0.02 0 0
---A\
I-3 ' '
10
....
,J
......
-2
-I
10
10
(O) stiffness factor o~ ._
0.12
\\
I
........
i
........
0.10
D
_ ~
0.08
g 0.06 c--
-~ -~ r (3-
0.04
~
0.02
g
0 0
"-~
~-3 ........ ~- 2 ....... - I 10 10 10
(b) angle factor ~12
FIG. 8. The effect of stiffness factor X12and angle factor ~12 on the equivalent stability cutting depth
(K¢/Km).
S 0.1 mm/rev) to verify the effect of the parametric vibration on the stability cutting depth, and the results are shown in Fig. 9. In Fig. 9, the ordinate is the value of Di/h and the abscissa is the equivalent stability cutting depth (KJKm) obtained from the numerical calculations and the stability cutting depth obtained from the cutting experiment. Symbols [] and © represent the numerical results for the solid workpiece (Di/h = 2) and thin-wall workpieces, respectively, and symbols • and • represent the experimental stability cutting depth for the solid =
0.18 '
v
'
0.15
'
'
I
. . . .
I
. . . .
I
'
"
'
13, • : solid workpiece o, •
Q. ~3
12
'
"
10 E
E
Thin wall workpiece
B, 0 : computed result
8 N0 .
Ca
0.12
0
0.09
6
0.06
4-
>,
0.03
2
~
.E_.
~ 0
O)
>o G}
0 -.,,,
0
I .... 5
I .-l~---~..
10
15
0 20
Di/h FIG. 9. The effect of
DJh on the equivalent stability cutting depth and stability cutting depth (cutting speed V = 125 mlmin, feed rate S = 0.I mmlrev).
444
G.J. L^l and J. Y. CHANG
workpiece and thin-wall workpieces. Figure 9 shows that the equivalent stability cutting depth (Kc/Km) and the stability cutting depth show the same tendency, as they both decrease quickly with an increase in Di/h. Even though the thin-wall (Di/h = 10.17) and the solid cylindrical workpiece (Di/h = 2) have the same value of moment of inertia, their equivalent stability cutting depths (Kc/Km) and their stability cutting depths are quite different. It is for this reason that the extent of parametric vibration and the dynamic compliance increase with an increase of Di/h. Figure 9 shows that when the three-jaw clamped thin-wall cylindrical workpiece is considered as a parametric vibration system, the analytical results agree quite well with the experimental results. From all the above analyses, it has been shown that the parametric vibration will influence the stability cutting depth considerably and cause the three-jaw clamped thinwall cylindrical workpiece to be easily excited. 4. CONCLUSIONS The beam mode and shell mode vibration properties (vibration characteristics and vibration direction angle) of the three-jaw clamped thin-waU cylindrical workpiece are functions of the relative position of the cutting tool to the chucking jaw, which will induce the parametric vibration. The stability analysis of the chatter vibration for the thin-wall cylindrical workpiece and the solid workpiece was investigated considering the parametric vibration by numerical calculations and experimental cutting. The following conclusions can be drawn. 1. For the three-jaw clamped thin-wall cylindrical workpiece, not only the stiffness coefficient of beam mode but also the vibration direction angle of beam mode, the stiffness coefficient and the vibration direction angle of shell mode will change according to the relative position of the cutting tool to the chucking jaw. This will induce the parametric vibration and cause the chatter characteristics of a thin-wall cylindrical workpiece to become more complicated than those of a solid workpiece. 2. For the thin-wall cylindrical workpiece, the values of the stiffness factor and the angle factor of the shell vibration mode are larger than those for the beam vibration mode. Hence, the extent of parametric vibration of the shell vibration mode is increased. This causes the thin-wall cylindrical workpiece and the solid workpiece with the same moment of inertia to show different stability cutting depths. 3. When the ratio of inner diameter to wall thickness Di/h increases, the extent of parametric vibration and the dynamic compliance will increase. These cause the stability cutting depth of the three-jaw clamped thin-wall cylindrical workpiece to become very small. Acknowledgements--The authors would like to acknowledgethe financial support from the National Science Council of the Republic of China under Grant No. NSC-81-0401-E036-04,and from the Tatung Company.
[1] [2] [3] [4]
REFERENCES J. Y. CnAr~6, G. J. LAI and M. F. CuEs, Int. J. MTDR (in press). M. RAHMANand Y. Iro, J. Sound Vibr. 102(4), 515 (1985). H. E. MEvatrrr, Trans. ASME ii87, 447 (1985). S. A. TOalASand W. FISHWlCK,Trans. ASME 1079 (1958).
Int. J. Math. Toob Manufact. VoL 35. No. 3. pp. 431--444,1995 CopyrightC 1994ElsevierScienceLid Printed in Great Britain. All rights n~rved
Pergmon
0890-695519559.50 + .00
STABILITY ANALYSIS OF CHATFER VIBRATION THIN-WALL CYLINDRICAL WORKPIECE
FOR A
G. J. LAIt~ and J. Y. CHANG? (Received 1 September 1993; in final form 10 January 1994)
AlmtnctmTbe vibration characteristics and the vibration direction angle between the beam mode and shell mode of the three-jaw clamped thin.wall cylindrical workpiece will change accordin8 to the relative position of the cutting tool to the chucking jaws. This will induce parametric vibration and cause the chatter characteristics of a thin-wall cylindrical workpiece to become more complicated than those of a solid workpi¢ce. In this study, the three-jaw damped thin-wall cylindrical workpiece was considered as a parametrically excited vibration system. Initially, the relationship between the parametric vibration and the workpiece system structure was investigated, and then the equivalent stability cutting depth of the thin-wall cylindrical workpiece was analyzed by a numerical calculation method. When the three-jaw clamped thinwall cylindrical workpiece was considered as a parametric vibration system, the analytical results were consistent with the experimental results.
NOMENCLATURE
C1,C12,C~ Di
F
f
Gc
G . , G~, h
Ko K. Kl,KI2,Ke M,M~ m-1 n s
T U
Uo Y
Y Ybd, Y.d Zbd, Z~d
B~, 812 0.1) O'12
k~, k12
6, ~1, tt
tO tOl, tOl2
damping coefficients of modes of vibration of structure workpiece inner diameter resultant cutting force or vector force exciting the structure frequency normalized cutting process dynamics normalized dynamic compfiance of structure workpiece thickness static directional cutting stiffness (slope of curve of F versus U) static directional stiffness of structure (slope of curve F versus y) stiffness coefficients of modes of vibration of structure equivalent masses of structure number of nodes in axial mode shape number of circumferential waves Laplace operator delay time between the primary and secondary chatter instantaneous uncut chip-thickness average or steady state uncut chip-thickness relative displacement between tool and workpiece normal to the machined surface radial direction eigenvectors on the direction y for beam and shell deflections, respectively eigenvectors on the direction ~ for beam and shell deflections, respectively angles of modes of vibration of the structure relative to a line normal to the machined surface angle between the resultant cutting force and a fine normal to the machined surface damping ratios of modes of vibration of structure vibration direction angles of vibration modes stiffness factors angle factors overlap factor =2=f, radian frequency undamped natural frequencies of vibration modes of structure I.
INTRODUCTION
BASED on a previous study [1], it was found that a three-jaw clamped thin-wall cylindrical workpiece was easily excited. The vibration of a thin-wall cylindrical workpiece consists of the characteristics of both beam and shell vibration modes, and the
tDepartment of Mechanical Engineering, Tatung Institute of Technology, 40 Chung-Shan N. Road, Sec. 3, Taipei, Taiwan 10451, R.O.C. ~To whom correspondence should be addressed. 431
432
G.J. LAi and J. Y, CHASe
stability cutting depth for a thin-wall and a solid cylindrical workpiece with the same moment of inertia will be quite different. The three-jaw chuck is the most practical and generally used support for the turning process. Due to the non-continuity of the three-jaw clamping condition, the stiffness coefficient, the damping coefficient, the equivalent mass and the vibration direction angle between the beam mode and the shell mode will vary according to the relative position of the cutting tool to the chucking jaw. This variation will induce the parametric vibration and cause the chatter characteristics of this three-jaw turning system to become very complicated. Rahman and Ito [2] have discussed in detail the chatter characteristics of the threejaw clamped solid cylindrical workpiece, but did not study the thin-wall cylindrical workpiece. Hence, the purpose of this study was to investigate the relationship between the vibration pattern of a three-jaw clamped thin-wall cylindrical workpiece and the stability cutting depth. In this study, the three-jaw clamped thin-wall cylindrical workpiece was considered as a parametric vibration system and its equilibrium equation was established. Initially, a numerical method was applied to analyze the effect of the relative position of the cutting tool to the chucking jaws on the variations of the stiffness coefficient and the vibration direction angle between the beam mode and the shell mode for a thin-wall cylindrical workpiece. Then, the parametric vibration induced by the variations of the stiffness coefficient and the vibration direction angle was analyzed. Finally a series of cutting experiments were carried out and the stability cutting depth of the thin-wall cylindrical workpiece was established. 2. THE VIBRATION PROPERTIES OF A THIN-WALL CYLINDRICALWORKPIECE The effect of the relative position of the cutting tool to the chucking jaws on the shell mode shape was obtained by a numerical method and the results are shown in Fig. 1 (the numerical method will be discussed in a later section). In Fig. l(a), the solid line represents the thin-wall cylindrical workpiece clamped by a three-jaw chuck. The positions of these three jaws are Jo, J1 and J2. The points A1 and A2 represent the different positions of point A. The dashed line and the dotted line represent the 12 f
1
IJ 2
" q - -.....
l ~(~
" ~
I
i<
TZ,d
-I Z~d
" _ /
I
(b) e = 3 o °
(o) e = o °
A2 '
I
I
1
"'] ~
A2i
dz
I
•
J
d o\"-'F - ~ / IJo
I
(~) e=6o °
(d)
e=9o °
r (e) e =1200
FIG. 1. The effect of the relative position of the cutting tool to the chucking jaw on the m mode shapes.
=
1, n
=
2 shell
Stability Analysis of Chatter Vibration
433
Pn = 1, n = 2 shell mode shape of a thin-wall cylindrical workpiece [1]. m and n are the subscript indices of the vibration modes for all the vibration properties in the following discussion. In Fig. l(a), the jaw Jo and the cutting tool are located on the same side and the jaws J1 and J 2 are located symmetrically about the y axis. At that moment, point A will oscillate between point A1 and point A2 when the shell vibration mode occurs. When the workpiece (or chucking jaws) as shown in Fig. l(a) rotates counter-clockwise 30, 60, 90 and 120° , the relative position of the cutting tool to the chucking jaws will change, and the shell mode shapes of the workpiece vibration will be as shown in Fig. l(b), (c), (d) and (e), respectively. During the cutting process, the motion locus of point A changes according to the relative position of the cutting tool to the chucking jaws. In Fig. l(b), Y~d and Z~dStand for the eigenvectors of point A in direction ~ and direction 7, respectively, and vrt2 represents the vibration direction angle of the m = 1, n = 2 shell mode and Crl2 = tan - t ( ~ , ~ / ~ ) ; when the workpiece rotates through one revolution, the variation in their values will be as shown in Fig. 2, which shows the periodic change according to the relative position of the cutting tool to the chucking jaws. Figure 2 shows that when the angle 0 between the cutting tool and a chucking jaw is equal to 60, 180 or 300°, the value of ~ shows a maximum value, expressed as 2 ~ m ~ . When the angle 0 is equal to 0, 120 or 240°, the value of ~ shows a minimum value denoted by Ysdmin. As the workpiece rotates through one revolution and the angle 0 is equal to 30, 150 or 270 °, the value of ~t2 shows a maximum value denoted by O'12max. y ~ is inversely proportional to the shell mode stiffness coefficient Ke and crt2 represents the shell mode vibration direction angle, and both are functions of the angle Ysdrno~x
'
I
'
,
,
I
,
I
,
I
I
,
t
I
,
L
,
I
L
L
I
I
7--sd
i ,w
+ v e~ 0
I
/ 0"12max
o
/
.
I
L,,,I,,, 0°
60°
I,
i i lllLI
120 °
180°
i t ill 240 °
300 °
ill 360 °
0 Fic. 2. The effect of variation of the angle of 0 between the cutting tool and the chucking jaw on y~, zs, and cry2.
434
G . J . LAI and J. Y. CHANG
0 between the cutting tool and a chucking jaw. From a previous study [1], it has been shown that the shell mode natural frequency t~12 and the damping ratio 812 are the same for all positions of the cutting tool relative to the chucking jaw. The equivalent mass Me was equal to Ke/to22 and the damping coefficient Ce was equal to 2512(MeKe) i. Therefore, Ce, M¢ and K~ can be defined as functions of the angle 0. In the threejaw clamped thin-wall cylindrical workpiece system, the vibration characteristics must include both the shell mode and the beam mode. The eigenvector Ybd, vibration direction angle 0"1, damping coefficient and equivalent mass for m = 1 beam mode are also functions of the angle 0 between the cutting tool and the chucking jaws. It is known from shell vibration theory that the vibration properties of the shell mode and the beam mode of a thin-wall cylindrical workpiece are dominated by the ratio of the inner diameter to wall thickness Di/h. For convenience in discussing the effect of Di/h o n Yba,Y~d, trl and tr12 on the rotary workpiece, the following four definitions are used: Y~dmax-- Y~=i,, (1) the stiffness factor for shell mode, h12 = Ysdmax + Ysdmin
2 (2) the stiffness factor for beam mode, hi
----- Ybdraax -- Ybdmin
Ybdmax "l- Ybdmin ; 2 (3) the direction angle variation for shell mode, Atr12 = O'12max; (4) the direction angle variation for beam mode, Atrl = trlm~x. The three-jaw clamped workpiece was considered as a parametric vibration system and the effect of the parametric vibration on the stability cutting depth in the turning process of the thin-wall cylindrical workpiece was investigated. 3.
3.1.
ANALYSIS FOR THE STABILITY CUTTING DEPTH IN THE TURNING PROCESS OF THE THIN-WALL CYLINDRICAL WORKPIECE
Derivation of thin-wall workpiece equilibrium equation
To calculate the stability cutting depth, it was convenient to use the chatter theory proposed by Merritt [3]• Merritt's method was used and whereas the parametric vibration of the three-jaw clamped thin-wall cylindrical workpiece may have both linear and nonlinear components, it was assumed here to be a linear vibration. For the stability analysis of the thin-wall cylindrical workpiece, three equations, the uncut chipthickness relationship, the cutting force relationship and the equilibrium equation, must be established. The first two equations can be derived from the results of previous studies [3, 4]. It can be seen from the schematic diagram of Fig. 3 that the uncut chip-thickness equation [3] can be written as
Tool (rlgldty mount;eoD FIG. 3. Uncut chip-thickness variation.
435
Stability Analysis of Chatter Vibration
U(t) = Uo(t) - y(t) + ~ y(t - T),
(1)
where Uo(t) is the average or steady state uncut chip-thickness, y(t) is the relative displacement between tool and workpiece normal to the machined surface, T is the delay time between the primary and secondary chatter, and IXis the regenerative effect or overlap factor. The Laplace transform of equation (1) is U(s) = Uo(s) - y ( s ) + Ix e-sTy(s).
(2)
If dynamic vibration within the cutting process is neglected, the cutting force equation [4] is F(t) = K= U(t),
(3)
where Kc is the static directional cutting stiffness and is a constant for a fixed cutting depth. The Laplace transform of equation (3) is F(s) = K= U(s).
(4)
In the following section, the equilibrium equation of the parametric vibration will be derived. In the preceding section it was stated that the vibration pattern of the three-jaw clamped thin-wall cylindrical workpiece simultaneously includes both the beam and shell parametric vibration modes. In the stability analysis of the thin-wall cylindrical workpiece, the beam and shell modes of the thin-wall cylindrical workpiece are considered as a vibration system with two degrees of freedom (Fig. 4). The beam mode and the shell mode are connected in series. Initially, only the easily excited shell mode was considered and the equilibrium equation of the one degree of freedom parametric vibration was derived. This equilibrium equation was then extended to two degrees of freedom, composed of the beam and shell modes. In this study, the effect of the two degrees of freedom parametric vibration on the stability cutting depth is discussed. 3.1.1. The equilibrium equation for one degree o f freedom. It was assumed that the workpiece had only one degree of freedom and its equilibrium equation is then Mc (d21dF)[y(t)]lcos oq2(t) + Cc (d/dr) [y(t)l/cos ot,2(t)
(5)
+ K~ [y(t)]/cos oq2(t) = F(t)cos[oh2(t) - f~],
I
beam
\ \ \ \ \ \ \ \ \ \ \
I
FIG. 4. Structure with two degrees of freedom for the thin-wall cylindrical workpiece.
436
G . J . LAI and J. Y. CnAr4C
where Me, Ce, and Ke represent the equivalent mass, damping coefficient, and stiffness coefficient, respectively, of the m = 1, n = 2 shell mode. a~2(t) is the vibration angle of the structure relative to a line normal to the machined surface, and 13 is the angle between the resultant cutting force and a line normal to the machined surface. From the previous results, it is known that the stiffness coefficient Ke, damping coefficient Ce, and equivalent mass Me vary cyclically three times when the three-jaw clamped thin-wall cylindrical workpiece rotates once. Considering the linear vibration and the effect of the variations of these vibration properties on the equilibrium equation, the equation now takes the following form: (1-h~2sin 3to~t)M (d2/dt2)[y(t)]/cos a~2(t) + (1-h12sin 3tOrt)Clz (d/d/) [y(t)]/cos a~2(t) + (1-k~2sin 3to#)K~2 [y(t)]/cos a~2(/) = F(/)cos [a12(t)-13],
(6)
where M, C12, and K~e represent the mean value of the equivalent mass, damping coefficient, and stiffness coefficient, respectively, of the m = 1, n = 2 shell mode. tOr is the rotational speed of the workpiece. Rearranging equation (6) one can obtain
M y"(t) + C~2y'(t) + K12y(t) = F(t)(1-h12sin 30Jrt)-' g12(t),
(7)
where
y"(t) = (d2/dt2)[y(t)] y'(t) = (d/dt)[y(t)] g12(t) = cos[a12(t) - 13]cos[alz(t)].
(8)
These results show that the vibration direction angle o12 will vary according to the relative position of the cutting tool to the chucking jaws. This means that the vibration angle otlz(t ) has a frequency variation of three times that of the rotating workpiece. That is Ot12(/) = ~12 -- A0.1zsin(3to~t),
(9)
where ~12 is the mean value of the vibration angle of the structure relative to a line normal to the machined surface. Substituting a12(t) of equation (9) into equation (8) and expanding the right hand side of equation (8), one can obtain g12(t) = cos~12 cos(~12-13) cos2 [A0.12sin(3tOrt)] + COS(~12-- 13) sin~12 cos[A0.12sin(3oJd) ] sin[A0.12sin(3ort) ] + cos~12 sin(~12 -13) cos[A0.12sin(3o#)] sin[A0.12sin(3ort)] + sin~12 sin(ax2 -13) sin2[A0.~2sin(3oJ#)].
(10)
When the value of A0.12is small, g12(t) becomes approximately g12(t) = cosal2 cos(al2 - 13) + A0.I2[COS(~12-13) sin~12 + c0s~12 sin(~12-13)] sin3oJ~t = g12 (l+~12sin3tort), where
(11)
Stability Analysis of Chatter Vibration
437
g12 = COS~12 coS(~12 -- ~) Ao'12sin(2~12 --~) 612 = COS~12 coS(~12 -- [~)"
612 is called the angle factor, where the values of K12 and 13 are determined from the tool geometry, workpiece material and cutting conditions, and the value of Atr12 is a function of Dt/h. Substituting g12(t) of equation (11) into equation (7), expanding the right hand side of equation (7), and assuming that higher orders can be neglected, the equilibrium equation of the three-jaw clamped thin-wall cylindrical workpieee can be expressed as (12)
M y"(t) + C12 y'(t)+ K12 y(t) = F(/)g12 (1 + M2 sin3oJrt + 612 simo~t).
Expanding the sine function of equation (12) and considering only the first two terms, yields
MY"(t)+C12Y'(t)+KI2y(t) = g12 [F(/) + 3F(t)kt2tOrt + 3F(t)612tOrt - 9/2F(t)k12to3t 3 - 912F(t)612o3t3 + ...].
(13)
If the stability analysis of the chatter loop for a step response is considered, assuming that a kind of trigger such as a sudden change of system parameter causes the chatter onset, then the Laplace transform of equation (13) is
y(s)
g12
=
(M$3 + ~
+
K12)
(3k12~Or 3612fOr 27kt2to3 27612to3 1
+
s
+ --$
$3
--$3
+
) ...
F(s). (14)
The dynamic compliance is then derived as
y(s)
d12($)
F(s) Km (~12 + 2812$(I)12"[-1)
(15) ~
where d12($) = 1 + 3k12tOr -t 3612tOt 27k12¢03 276120~r3 S
$
$3
$3
+ "'" '
1~Kin = g12/K12, Km is the directional static stiffness of the structure, to22 = K12/M, and 812 = C12/2(Kt2M) t. 3.1.2. The equilibrium equation for two degrees of freedom. In Fig. 4, the beam and the shell mode are now connected in series, and the equilibrium equation for two degrees of freedom can be obtained. This will be Y($)
F(s)
=
gI2dl2(s)
(so~2+ 28t2s + 1) K12 0012
where
gldl($)
-# K1
(o~__~t 281s +
' + 1 Iil1
)'
(16)
438
G.J. LAI and J. Y. Cn^r~o d12(s ) = 1 +
dl(S) = 1 +
3h12tOr 3~12tOr 27h120~r3 + - -S S $3
3hltOr S
+
3~ltor 27kltOr 3 $
27~12t03 t- ... $,.3
27~1o3
$3
S3
~- "'"
g12 = COS~12 C0S(~12--13), gl = COS~I COS (~1--13)
Act12 sin(2~12 -13) Act1 sin(2~l --13) ~12 = c0s~12 cos(~l 2 _13), ~1 = cos~ 1 cos(~l _13) K12 K1 0)22 = ~ - , {~02= " ~ ,
~12 = C12/2( K12M) i, 81 = C1/2( K1M)k The right hand side of equation (16) can be written with a common denominator, and y(s) F(s)
d(s) Km
+
+1 (1112
(17) +
+ 1 ¢.01
where
Km
K12
K1
and
Klg12dlx(S)
d(s) =
+ o) 1
+ 1 + K12gldl(s)
+
0,112
+ 1
g12K1 + glK12
It is convenient to write the dynamic compliance using the normalized compliance Gm(s), and thus equation (17) takes the following form: y(s) 1 F(s) - Km Gin(s).
(18)
From equations (16) and (17), it is seen that Gin(s) is a function of the stiffness factors (hi, h12) and the angle factors (~1, ~12). The parametric degree is determined from the totality of the stiffness factors (hi, h12) and the angle factors (~1, ~12). Hence, the parametric degree will change the value of Gm(s) and influence the chatter stability cutting depth. When the vibration properties of the structural system are constant, namely when hx = ~k12 : 0 and ~x = ~12 = 0 , then Gin(s) is consistent with the expression for the general solid workpiece [3]. 3.1.3. The effect o f Ddh on the su'ffness factor and angle factor. From the expressions for the angle factor in the preceding section, it will be seen that the angle factor is a function of ~1 o r ~ 1 2 , AOrl or Aort2 and 13, where the values of or1, 0/-12, and 13 are determined from the tool geometry, the workpiece material and the cutting conditions. The values of Act1 and A f t 1 2 a r e determined from Di/h. The main purpose of the present study was to investigate the effect of Di/h on the chatter vibration of the thin-wall workpiece. In all the theoretical analyses and cutting experiments, the value of Di/h was taken as the only variable, and all the other parameters were kept
Stability Analysis of Chatter Vibration
439
constant. Hence, the angle factors (El, ~x2) were determined only from the values of Dl/h. Moreover, from the definitions of the stiffness factor, it is seen that the stiffness factors (kl, kt2) are functions of Di/h. In determining the effect of Dt/h on the stabifity cutting depth of the thin-wall cylindrical workpieee, a numerical method was used to obtain the values of the stiffness factors and the angle factors. In the previous study [1], as the value of Di/h was decreased, the thin-wall workpiece showed the characteristics of the beam and shell vibration modes. When the value of Dt/h was large, the dynamics of the shell vibration mode were more evident than those of the beam vibration mode. To obtain the general characteristics of the thin-wall workpiece, a workpiece with a low value of Di/h which showed the beam and shell modes' vibration simultaneously was initially chosen for analysis. The overhang length L and the inner diameter D~ were chosen as 100 mm and 30.5 mm, respectively, while the different values of Di/h were obtained by changing the outer diameter. To compare the difference between a solid and a thin-wall workpiece, a thin-wall workpiece (Di/h = 10.17) and a solid workpiece with Do = 30.9 mm which both had the same moment of inertia were chosen. For this solid workpiece it was considered that Di/h was equal to two for comparison purposes. The commercial finite element package (NISA II) was used for the numerical analysis. The thin-wall cylindrical workpiece was divided into 240 3-D shell elements. Each element had four nodes and each node had six degrees of freedom. The solid workpiece was divided into 480 3-D beam elements. Each element had eight nodes and each node had six degrees of freedom. In applying the boundary conditions it was assumed that at the chucking jaw positions the displacements were regarded as zero. The relationships between Di/h and the stiffness factors (kl, k12) and the angle factors (~1, ~12) were solved and the results are shown in Fig. 5(a) and (b). Figure 5(a) shows that kl is equal to 0.003 for the solid workpiece. The values of kt and k12for the thin-wall workpiece increase with an increase of Dt/h. Furthermore, the value of k12 is much larger than that of k~. The effect of DJh on the angle factor
,,
0.240.20
,,/,:
--
•
: fix" t h e boom mode of
•
: for thw bl~W'l mode of
t h e solid i m r l ~ l l c e
t.<
0.16 -o
t h e th|n woll workplace
0.12 "
0 : for t h t Ihe~ mode of
0.08
--
t h e thin woll workplece
0.04- - -
-
o -~'
,
10
O2
(o)
20
I
40
D i/h
0.42
j,
I
30
!
0.35
• -
: foe t h e boom mode of t h e solid wortq=lecal
0.28 •
: for t h e boom mode of the thIn wall worlq~lece
0.21
0 : for t h e shell mode of
0.14 e-o
the thin won workpkK:e
0.07!
0
-~'O~
10
(b)
20
30
4-0
Di/h
FIG. 5. The effect of Di/h on kl, Xl2, ~1 and ~,2.
440
G . J . Lpa and J. Y. CHANG
is shown in Fig. 5(b), where ~1 = ~12 = 45 °, and 13 = 15°. It is for this reason that the side cutting edge angle is equal to 45 ° and the chatter vibration direction of the shell mode and the beam mode is in the radial direction, so the mean values of the vibration angle of the structure relative to a line normal to the machined surface, ~1 and ti12, are equal to 45 °. The angle between the resultant force direction and the feed direction is equal to 60 °, so the value of the angle between the resultant cutting force and a line normal to the machined surface, 13, is equal to 15°. Figure 5(b) shows that ~1 is equal to 0 for the solid workpiece. The values of 61 and ~12 for the thin-wall workpiece increase with an increase of Di/h, and furthermore, the value of ~2 is much larger than that of ~1. From the above analyses, it is seen that for the thin-wall workpiece the extent of parametric vibration of the shell vibration mode is larger than that of the beam vibration mode. Furthermore, the extent of parametric vibration of the shell vibration mode increases with an increase of Di/h. Also, when the thin-wall workpiece and the solid workpiece have the same moment of inertia, the extent of parametric vibration of the thin-wall workpiece is larger than that of the solid workpiece.
3.2. Derivation of the chatter loop On combining equations (2), (4) and (18), one can obtain the equation for the block diagram of the chatter loop as shown in Fig. 6. The transfer function relating U(s) and Uo(s) can be obtained from the block diagram of the chatter loop as
U(s) 1 Uo(s) - 1 + ( 1 - p . e -st) KJKm Gin(s)"
(19)
From equation (19), the characteristic equation of the chatter loop is 1 + ( 1 - p . E sT) KJKm Gin(s)
=
(20)
0.
For the chatter loop of Fig. 6, the transfer function relating y(s) and U(s) can therefore be obtained as
y(s) _F(s) y(s) =K~Gm(s). U(s) U(s)F(s) Km
(21)
Cutting process
Structure
dynamics
dynamics
y(s)
U0(s)
I
Primary feedback path
Regenerative feedback path
Fro. 6. Block diagram of chatter loop.
Stability Analysisof Chatter Vibration
441
The gain-phase plot of y/U can be obtained by substituting s = ]to into equation (21), i.e.
Y0"to)
rc -
CmUto).
(22)
In normal cutting, the chatter vibration direction of the thin-wall cylindrical workpiece is in the radial direction (direction ~), and the angle between the chatter vibration direction and the chip-thickness variation direction is equal to 45 °. When the effect of this angle is considered, equation (22) will become y(jto) gc Gm(]to) U(jto) -- Km ~7~ '
(23)
where 1 -y(jto) Km G~(jto) = F(jto) ' which is the dynamic radial equilibrium equation. The plot of the critical locus of G~p can also be obtained from the derivation by Merritt, where _ Gcp
-1 - Y(Jto) (l_p. ~/,oT) U(jto) "
(24)
From the characteristic equation of the chatter loop of equation (20), it will be seen that when the gain-phase plot of equation (23) and the critical locus Gcpof equation (24) intersect, chatter vibration occurs. Kc of equation (23) is a function of cutting depth. Because an increase of cutting depth will increase the value of Kc and cause the chatter occurrence, it can be said that the value of Kc/Km is the value of the equivalent stability cutting depth. From equations (16) and (23), it is known that the extent of parametric vibration and the structure dynamics will influence the value of the equivalent stability cutting depth (Kc/Km), and this will be investigated in the following section. 3.3. Analysis of stability cutting depth From the preceding investigation, it is known that the extent of parametric vibration of the three-jaw clamped thin-wall cylindrical workpiece is a function of Di/h. Equations (16) and (23) reveal that the extent of parametric vibration will influence the value of y/U, and that the values of y/U and the equivalent stability cutting depth (Kc/Km) have a close relationship. Hence, the extent of parametric vibration is the dominant factor in determining the equivalent stability cutting depth for the three-jaw clamped thin-wall cylindrical workpiece. For the general thin-wall cylindrical workpiece, the extent of parametric vibration of the shell vibration mode is larger than that of the beam vibration mode. Hence, initially, the effect of the parametric vibration of the shell vibration mode on the value of the y/U for the three-jaw clamped thin-wall cylindrical workpiece was investigated. Although the values of the stiffness factor and the angle factor for the shell vibration mode were different from those for the beam vibration, the extent of parametric vibration of the shell vibration mode was determined by these two factors. In investigating the effect of the extent of parametric vibration of the shell vibration mode on the y/U gain-phase plot, the angle factor of the shell vibration mode was, for convenience, assumed to be zero. The characteristics of a thin-wall cylindrical workpiece with HTI4 35:3-F
442
G . J . LAI and J. Y. CttANG
Di/h = 10.17 had been obtained previously [1], and this workpiece was investigated further. The results for various stiffness factors kt2 and an overlap factor of W = 1 are shown in Fig. 7. The reason for choosing Ix = 1 was that an overlap factor of unity would be the most critical value for the commencement of chatter. Figure 7(a) indicates that when the stiffness factor k12 is equal to 0.0, parametric vibration does not happen, and that the chatter vibration occurs when Kc/Km is equal to 0.092. When the value of ~,12is increased, the y/U gain-phase plot will move leftward considerably, as shown in Fig. 7(b), (c) and (d). Hence, when the effect of the stiffness factor of the shell vibration mode does exist, chatter will occur at a smaller value of KdKm. That is, the extent of parametric vibration is the dominant factor in determining the value of the equivalent stability cutting depth (KdKm). Figure 8 shows the effect of the stiffness factor k12 and the angle factor ~12 of the shell vibration mode on the equivalent stability cutting depth (KdK=) of the thin-wall cylindrical workpiece. Figure 8(a) and (b) show that when the stiffness factor A12 and the angle factor ~x2 of the shell vibration mode increase, KdKm quickly decreases. Also from Fig. 8(a) and (b) it is seen that the stiffness factor and the angle factor of the shell vibration mode have the same effect on KdKm. From the previous study [1], it was shown that like the stiffness factor and the angle factor, the structural vibration is also a function of Di/h. To understand the effect of all the above three factors on the chatter stability of the three-jaw clamped thin-wall workpiece, Di/h was considered as the only variable in numerical calculations to assess the effect of these three factors on the equivalent stability cutting depth. A series of cutting experiments were also conducted (cutting speed V = 125 m/min, feed rate +4 10 (a)
Xle = 0.0 /~= [ critical
+4 10 (b)
;~le "= 0 . 0 0 1 /~= 1 critical +2 10
-2 10
-2 10
--4 10
-~,eO° -~1~° -240° - I ~ '~ - I ~ "
-6~
o•
• _~lSOo
,
.
_ ~ o
i .
.
2~o
i
.
_1~u
J _120o
.
.., _60o
--4 10
i
oo
-I-4 10 (c)
Ate0.01 /~.- 1 ctltlcol
+4 10 (d)
+2
;~m = OA /~', 1 critical +2 10
1 -2 10
•
1
.
z
.
a
.
,
-360" -300e -240° -180° -120 °
.
-4 10
t
-60 °
o"
-2 10
I
,
I
-aao" - 3 o o ° -24o"
~
I
, ,
J
-~ao ° -12o °
-4 10
* -eo °
o•
FIG. 7. G a i n - p h a s e plot of critical loci with plot of y/U superimposed for the different values of X~2 (DJh = 10.17; Kc/K= = 0.092, here KdK~ is equivalent stability cutting depth without considering the parametric vibration),
Stability Analysis of Chatter Vibration
443
0.12 .c_
0.10
3 (.3
~J~ 0.08 0.06 0.04 ll}
-o
0.02 0 0
---A\
I-3 ' '
10
....
,J
......
-2
-I
10
10
(O) stiffness factor o~ ._
0.12
\\
I
........
i
........
0.10
D
_ ~
0.08
g 0.06 c--
-~ -~ r (3-
0.04
~
0.02
g
0 0
"-~
~-3 ........ ~- 2 ....... - I 10 10 10
(b) angle factor ~12
FIG. 8. The effect of stiffness factor X12and angle factor ~12 on the equivalent stability cutting depth
(K¢/Km).
S 0.1 mm/rev) to verify the effect of the parametric vibration on the stability cutting depth, and the results are shown in Fig. 9. In Fig. 9, the ordinate is the value of Di/h and the abscissa is the equivalent stability cutting depth (KJKm) obtained from the numerical calculations and the stability cutting depth obtained from the cutting experiment. Symbols [] and © represent the numerical results for the solid workpiece (Di/h = 2) and thin-wall workpieces, respectively, and symbols • and • represent the experimental stability cutting depth for the solid =
0.18 '
v
'
0.15
'
'
I
. . . .
I
. . . .
I
'
"
'
13, • : solid workpiece o, •
Q. ~3
12
'
"
10 E
E
Thin wall workpiece
B, 0 : computed result
8 N0 .
Ca
0.12
0
0.09
6
0.06
4-
>,
0.03
2
~
.E_.
~ 0
O)
>o G}
0 -.,,,
0
I .... 5
I .-l~---~..
10
15
0 20
Di/h FIG. 9. The effect of
DJh on the equivalent stability cutting depth and stability cutting depth (cutting speed V = 125 mlmin, feed rate S = 0.I mmlrev).
444
G.J. L^l and J. Y. CHANG
workpiece and thin-wall workpieces. Figure 9 shows that the equivalent stability cutting depth (Kc/Km) and the stability cutting depth show the same tendency, as they both decrease quickly with an increase in Di/h. Even though the thin-wall (Di/h = 10.17) and the solid cylindrical workpiece (Di/h = 2) have the same value of moment of inertia, their equivalent stability cutting depths (Kc/Km) and their stability cutting depths are quite different. It is for this reason that the extent of parametric vibration and the dynamic compliance increase with an increase of Di/h. Figure 9 shows that when the three-jaw clamped thin-wall cylindrical workpiece is considered as a parametric vibration system, the analytical results agree quite well with the experimental results. From all the above analyses, it has been shown that the parametric vibration will influence the stability cutting depth considerably and cause the three-jaw clamped thinwall cylindrical workpiece to be easily excited. 4. CONCLUSIONS The beam mode and shell mode vibration properties (vibration characteristics and vibration direction angle) of the three-jaw clamped thin-waU cylindrical workpiece are functions of the relative position of the cutting tool to the chucking jaw, which will induce the parametric vibration. The stability analysis of the chatter vibration for the thin-wall cylindrical workpiece and the solid workpiece was investigated considering the parametric vibration by numerical calculations and experimental cutting. The following conclusions can be drawn. 1. For the three-jaw clamped thin-wall cylindrical workpiece, not only the stiffness coefficient of beam mode but also the vibration direction angle of beam mode, the stiffness coefficient and the vibration direction angle of shell mode will change according to the relative position of the cutting tool to the chucking jaw. This will induce the parametric vibration and cause the chatter characteristics of a thin-wall cylindrical workpiece to become more complicated than those of a solid workpiece. 2. For the thin-wall cylindrical workpiece, the values of the stiffness factor and the angle factor of the shell vibration mode are larger than those for the beam vibration mode. Hence, the extent of parametric vibration of the shell vibration mode is increased. This causes the thin-wall cylindrical workpiece and the solid workpiece with the same moment of inertia to show different stability cutting depths. 3. When the ratio of inner diameter to wall thickness Di/h increases, the extent of parametric vibration and the dynamic compliance will increase. These cause the stability cutting depth of the three-jaw clamped thin-wall cylindrical workpiece to become very small. Acknowledgements--The authors would like to acknowledgethe financial support from the National Science Council of the Republic of China under Grant No. NSC-81-0401-E036-04,and from the Tatung Company.
[1] [2] [3] [4]
REFERENCES J. Y. CnAr~6, G. J. LAI and M. F. CuEs, Int. J. MTDR (in press). M. RAHMANand Y. Iro, J. Sound Vibr. 102(4), 515 (1985). H. E. MEvatrrr, Trans. ASME ii87, 447 (1985). S. A. TOalASand W. FISHWlCK,Trans. ASME 1079 (1958).