Int. J. Mach. Tool Des. Res.
Vol. 11, pp. 145-175.
Pergamon Press 1971.
Printed in Great Britain
VIBRATION ANALYSIS A N D W O R K - R O U N D I N G MECHANISM IN CENTERLESS GRINDING* YuJI FURUKAWA~, MASAKAZU MIYASH1TA~"and SUSUMU SHIOZAKlJ~
(Received 30 August 1970) Abstract--Several studies on the work-rounding mechanism in centerless grinding have been made so far, but it is not solved systematically analyzing the various complicated vibratory behaviors generated in the grinding system as a result of relative influences between the dynamic characteristics of grinding machine, the geometrical work supporting condition and the disturbances associated with machining. From this point of view, the following are solved analytically and experimentally in the present paper. 1. Work-rounding mechanism in the stable region. Work-surface waviness caused by the static deformation of machine tool, the periodic forced vibration and the aperiodic forced vibration. 2. Work-rounding mechanism in the unstable region. Instability causea by the geometrical work supporting condition, the static digging-in, the work-regenerative chatter vibration and the wheel-regenerative chatter vibration. And then several methods to make the out-of-roundness of work as small as possible avoiding the generation of vibrations are given. 1. I N T R O D U C T I O N THE GENERATION mechanism of work-surface waviness in centerless grinding is studied in the past from the standpoint of the geometrical work-supporting condition or from the side of vibration generated while machining, which can be summarized as follows: Geometrical work-rounding mechanism. The waviness is formed under the relative relation between the geometrical arrangement of work-wheel-blade and the primarily existed waviness [1], the periodic disturbance [2] and the aperiodic one [3, 4]. These are studied in detail theoretically and experimentally and the optimum work-supporting condition to make the out-of-roundness of work minimum is proposed [3, 5]. Geometrical instability. Under some conditions of the geometrical work-rounding mechanism, the waviness builds up unrelated to the machine's vibration [6-9]. Regenerative chatter theory. Applied, the chatter theory in turning to the centerless grinding system and the existence of unstable grinding regions is theoretically pointed out [7, 10], but the practical chatter behaviors cannot be explained yet, like on those points of most practical operations must be in unstable regions or the theoretically assumed number of waviness is not practical and so on. These studies are carefully developed in each point but pick up one of the complicated work-rounding mechanisms, furthermore, cannot fully explain the whole aspects of practical behaviors in centerless grinding. Consequently, each shape of waviness formed on the * This paper was presented at the llth International Machine Tool Design and Research Conference, Birmingham, 14-18 September 1970. t Mechanical Engineering, Faculty of Technology, Tokyo Metropolitan University, 2-1-1 Fukazawa, Setagaya-ku, Tokyo, Japan. 11 145
146
YuJJ FURUKAWA,MASAKAZUMIYASHITAand SOSUMUSHIOZAKI
ground work-surface does not correspond to several kinds of vibrations generated while grinding. Therefore, it is difficult to diminish the out-of-roundness by changing the grinding condition and avoiding the causes to generate the waviness. The present paper puts importance on the work-regenerative chatter vibration among the above summarized work-rounding mechanisms, especially on its generation and growing-up process, as well as the influences of the elastic deformation of machine, the digging in type of instability and the wheel regeneration on the work-rounding mechanism. And it is aimed to solve the work-rounding mechanism systematically, furthermore, to get the grinding condition so as to minimize the influence of vibration on the generation of waviness. 2. G E N E R A L T H E O R Y OF W O R K - R O U N D I N G IN C E N T E R L E S S G R I N D I N G
MECHANISM
2. I. Centerless grinding system The geometrical work-arrangement and nomenclature are shown in Fig. 1. It is represented by the center height h in the past but the center height angle 7 seems to be more X
grinding "''~(1~ ~ T-'~. 0 . w h ~ ~ ] / . ' , ' . .
7
1'~1~/-: ~.W,..~ ] regulating /
,::.::;
ge
i..Z.
/i///f/I'///f/i
FIG. 1. Geometrical work supporting condition. convenient to show the relative relation of work-wheel-blade arrangement. Laplace transforming the equations with respect to the time coordinate ¢, which means the work revoluting angle, the centerless grinding system in Fig. 1 can be shown by the block diagram form in Fig. 2 (Appendix 1). The fundamental work-rounding mechanisms on this block diagram are consisted of
those of force vibration and those of self-excited one generated under the instability of machining system. Work-rounding mechanism in the stable grinding region. The work-surface waviness caused by, the static deformation of machine tool, the periodic forced vibration and the aperiodic forced vibration.
Vibration Analysisand Work-RoundingMechanismin CenterlessGrinding RW waviness
~ I
V-n J
J
I " R W.wear
,
k
I F-7-1LR.w.
.r---7
Wheel-regeneralive effect
G,W.wa~iness
r-- 1
Wheel-wear stiffness
~ I
"1" G W. wear
E
e " Rw(S) work out-of-roundness -rYo~i~l'~tricalw,urk7 ----amg rnucnonmm _ +
,1" L
_I
Ciasod loop A
I
o ==
l
G~ (S), ~
147
| J
Regenerative effecl"
s L
i
Grinding stiffness
J
e-ZTrs ~
• . • Ym(S}, dynamic stiffness of grind er r
depth of cut
~)
I I
a~,
L
I JI
L_
D(S) (l;]b -y-~)
disturbance
I B,ado
,~': Influence of unit work wovi~less at contoc~ point B(fig.I) an the depth of cut (I-~): Likewise, at point R
FIG. 2. Block d i a g r a m of centerless grinding system.
Work-rounding mechanism in the unstable grinding region. Instability caused by, the geometrical work-supporting condition, the static digging-in, the work-regenerative chatter vibration and the wheel-regenerative chatter vibration. On the analysis of these surface forming processes, the geometrical work-supporting condition and the grinding stiffness are given by the grinding conditions but the dynamic characteristics of grinder or the input disturbance are specific for each machine tool. Therefore it is necessary to examine these for the given centerless grinder.
2.2. Dynamic characteristics of grinder The mass of centerless grinder's wheel is so big as its vibratory model can be considered to be the lumped coefficient system, which makes its shape of vibration mode clear. Furthermore, the relative natural frequencies between the grinding wheel (G.W.) and the regulating wheel (R.W.) are much lower than those of other types of grinding machine and this is one of the important reasons why self-excited vibrations are more apt to be generated in this machine. The natural frequencies between G.W. and R.W. are calculated for the given grinder from the equivalent value of mass and spring constant which is statically measured while driving the machine. At the same time, the natural frequencies are measured from the transient response curve obtained by giving the impulse on the wheels.
148
YuJi FURUKAWA, MASAKAZU MIYASHITA and S u s u i t r SHIOZAKI
o~
E y J j 2 r r f ) vector locus where, -
j2
ym62~f)- c-33~ y~ + = 120 Hz -jl
Ks +
Kr
Fzo. 3. Dynamic stiffness between G.W. and R.W.
The calculated and the measured values almost coincide and the relative dynamic stiffness between G.W. and R.W. has such a response curve as shown in Fig. 3, The first mode of vibration is almost decided by the G.W. system and has the natural frequency at about 90 Hz. The second is about 150 Hz which is mainly decided by the R.W. system. Generally the natural frequency of first mode seems to be existed in the neighbourhood of 100 Hz for the 16 in. type of centerless grinder [11]. 2.3. Input disturbance to act on the system
Whenever the waviness is formed, its nucleus' generation depends upon the input disturbance to act on the machining system whether in the forced or the self-excited vibration. The kinds of disturbances transferred inside and/or outside of machine tool are as follows: I unbalances of rotating parts; A.C. Motor, G.W., R.W., shape of work; key way, Periodic ( hydraulic noise; (number of vanes) × (number of rev) Hz, | driving mechanism; gear, chain, belt, J irregularity of G.W. quality, Aperiodic l irregularity of infeed rate; stic slip of slider, | noises generated by the grinding action itself, I, ground vibration. These disturbances can be classified to the considerably periodic and the aperiodic one. Each of them has the next response relation with the waviness through the grinding system. periodic forced disturbance frequency response--waviness aperiodic forced disturbance--statistical response--waviness The disturbing deflection between the G.W. and the R.W. bearings is measured while driving the machine with no load for the given grinder. Its statistically treated result is shown in Fig. 4. The peak of 25 Hz is caused by the G.W. unbalance and those of 85 and
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
149
l/ 1.5
I/ /|[
The auto correlation function was standardized by puttng
E. •~ I.O
£ O.
0
25
50
75
I00 125 Frequency f, 14z
150
;75
200
FIG. 4. Power spectrum density for the disturbing deflection between G.W. and R.W. 165 Hz seem to be influenced by the resonance of machine tool. Therefore, the disturbing force D to act on the system may be considered to be the band limited white noise distributing till about 100 Hz except the 25 Hz peak. 3. E X P E R I M E N T A L D E V I C E A N D
METHOD
Given grinder, 16 in. (G.W. 4054, × 150) centerless grinder, infeed wet grinding, G.W. surface speed; 30 m/sec, G.W. dressing condition; 2/100q~ -- 2/100q~ -- 0, The dressing is made for every one work grinding in order to keep the G.W. surface to the same condition. Deflection pick-up for wheel waviness
20 Hz 20kH~z
•: Accelerometer
S~;/p~'~goge strain
I : II i~
Strain-meter ~ l
Data-recorder, I:/
Low pass f,,ter l i[
[
Synchro- scope
]
]
FIG. 5. Set up of measurement.
Pen writer
J q
150
YuJI FURUKAWA,MASAKAZUMIYASHITAand SUSUMOSHIOZAKI
Work primary out-of-roundness; 0.5-1.0/~ with no periodic waviness. Set up of measurement. The through feeding force and the blade's acceleration are measured by the device in Fig. 5. At the same time the work and the wheel wavinesses are measured. 4. W O R K - R O U N D I N G MECHANISM IN THE GRINDING REGION
STABLE
In the centerless grinding operation, it has been a problem why the Gleichdicke or the periodic waviness are formed on the work-surface even under the stable grinding condition with no self-excited vibration. The generation mechanism of these wavinesses are due to the relative relation between the geometrical work-rounding action and the input disturbance.
4.1. Waviness caused by the static elastic deformation of machine The irregularity of infeed rate Si(s) comes from stic slip of slider "~. step-like infeed of slider; unskilled operator f Si(s) = 1Is shortage of spark out grinding after steady infeed ramp-like infeed of slider; Si(s)= 1/s 2.
(
I+x s
Calcuiated Experimented
(cl) Step-like infeed response
7=2 ° (S,(O): 5/.t)
//
Calculated
~ ......... Experimented f X = 2°
(b) Romp-like infeed response
~
~,Si=41.t/revJ
FIG. 6. Waviness caused by the elastic deformation of machine.
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding In the case of the center height angle change of the depth of cut and leave the W h e n ~, -- 0, 1/Gg(s) = 1 -t- e -as. The waviness Rw(s) becomes as follows, when is considered:
-- Rs~ (s) =
ag(s)
151
~, is small, these irregularities invite the sudden discontinuity on the work-surface. response between the infeed rate Si(s) and the only the static stiffness of machine Ym(O) = k m
1
1 + Gg(s)(l -- e-Zas)k o Ym(s) __ 1 - - x s {(1 - - e - a s + e 1 +Ks
-2as-e
-3a~+
... )
+ Ks(1 + Ks e -as -+- K~ e - z a ' -~- . . . )}
(1)
Where Ks = ko/(km -~- kg); rate of static cutting residue. Step response: substituting S i ( s ) = 1Is into equation (1) and the waviness Rw(qb) can be solved by Laplace inverse transformation. The result obtained is shown in Fig. 6(a). -- Rw(~)
=
1 9 5 s_ [{u(~b) -- u(q~ -- 7r) A- u(~b -- 2zr) . . . 1 +xs
}
÷ Ks{u(q~) + Ksu(4, -- zr) + K~u(4, -- 27r) ÷
. . . }1
(2)
where u(4) = L -1 (l/s); unit step function. R a m p response: stopping the operation suddenly after the steady ramp-like infeed, then the waviness shown in Fig. 6(b) can be got
-- Rw(qb) -- 1 -- Ks [{h(~) -- h(4, -- ~r) + h(~ -- 2zr) . . . } 1 +Ks
q- Ks{h(q~) + ,~sh(gp-- 7r) + K~h(/p -- 27r) +
. . . }1
(3)
where h(~) = L-a(1/s2); unit r a m p function. I m p r o v e m e n t of grinding conditions: Set the center height angle ~, = 7 °. Proper time of spark out grinding. W a r m i n g up of slider.
4.2. Waviness caused by the periodic forced disturbance A m o n g the disturbances associated with machining, are the unbalance of wheel, the shape of work like a noise. These disturbances have the frequency response where nl means the forced vibration frequency for one
those to have a strong periodicity key way and the hydraulic p u m p with the waviness. Putting s ~-jns work revolution
Rw(jni) _ Go(jnl) D(jn/) Ym(jn:) -J- kg. Gu(jn/). (1 -- e-2"J"/)"
(4)
As an example, if the change of depth of cut caused by the G.W. unbalance is existed, there appears the fluctuation of grinding force to have the frequency f / = 25 H z as shown in Fig. 7 and the n u m b e r of waviness n / = f / / N w ~- 25/5.2 -- 5 coincides with that of measured. I m p r o v e m e n t . Detect the source of forced vibration and take it off. Set the n u m b e r of work revolution Nw so as the forced vibrating frequency does not coincide with the
152
YuJI FURUKAWA, MASAKAZU MIYASHITA and SusuMu SHIOZAKI
Blade's acceleration
Through feed grinding force
Grinding condition, y = 7 5 ° Work, $40 C 19-4't'xco .5
Nw= .5 2 rev/sec Total depth of cut, 30%m
FIG. 7. Forced vibration (G.W. unbalance, 25 Hz) and work-surface waviness.
geometrical specific number of waviness ns which is explained in the following section,
Nw :~ fl/ns. 4.3. Waviness caused by the aperiodic forced disturbance The band limited white noise is always acting on the system even when the periodic forced vibration is taken off. This noise has a statistical response relation with the waviness and for the round number of waviness nr there exists the next spectrum relation.
SRw(nr) So(nr)
l =
where Saw(nr) and D respectively.
Gg(jnr) . . . . I2 Ym(jnr) + kgGg(jnr)(1 -- e-2"Jnr) Ga [2 -~,m (jn r)
(5)
SD(nr) show the power spectrum density of waviness Rw and disturbance
Vibration Analysis and W o r k - R o u n d i n g Mechanism in Centerless Grinding
153
r : i _¢,e-~i% (1_ ~)e-¢zi, %~jn) 7 =7°3C', E'=0.146,(I-~)=0.940 d?t = 0'317'%
~2~ C ' 9 4 :T
n:14 8 b2
I0
Real 1
7
II
FIG. 8(a). 1/Gg(]n) vector locus. Consequently those round number of waviness nr as Gg(jnr) or 1/Ym(jnr) becomes to have a peak value may be much more amplified than other components and formed on the work-surface. Especially, the number nr = ns to make IGg(jns) l maximum is important and apt to be generated in a practical operation. This waviness is called the geometrical specific waviness as it is decided only by the geometrical condition. As an example, the vector locus of 1/Gg(jnr) is shown in Fig. 8(a) for the center height angle 7 = 7.5 ° and I Go(Jnr) I becomes to be maximum at nr = ns ~---24 when the locus passes nearest to the origin. The waviness to have the number ns = 24 is formed practically as shown in Fig. 8(b), when there is almost no periodic variation in the grinding force. In order to know the gain characteristic of Go(jn) for the center height angle ~,, Gg(jn) is approximated by equation (6) 1
--
1 -- E,e-¢lJ~ Jr- (1 -- ¢) e -¢~n
Gg(jn) -- 1 q- e-C--r)J"
(6)
At this time, the locus of 1/Gg(jn) is shown by the unit circle and the gain characteristic I Go(Jn) ] can be expressed by the angle 0 or the number of waviness n as shown in Fig. 9. Such a waviness as [ Gg(jn) I has a peak is formed and its number is specifically decided by the center height angle 7'. Its tendency is almost as follows: y < 7 ° : n s 0 = 3 , 5, 7 . . . Gleichdicke 7 > 7°: n,p = [~h']~, [2~/~,]0,
[3~h,], . . . .
However, the waviness more than [27r/y]0 are not generated in a practical case. It is proposed [3, 5] to set y = 7 ° in order to minimize the amplitude of waviness while
stable grinding.
154
YuJl FURUKAWA,MASAKAZUMIYASHITAand Sus1JMU SHIOZAKI
Blade's acceleration ~t
Tl%oug h feed grinding force
Grinding condition, y = 7 5 ° Nw=5.2 rev/sec Work~ $40C 19 4dPx40 Total depth of cut, 30~u.m
FIG. 8(b). Aperiodic forced vibration and work-surface waviness.
There is a possibility that those components of wavinesses except the geometrical specific one may be formed to a certain level by disturbance but not so remarkable and especially the waviness to situate in 0 ° ~ 0 < 120 ° and 240 ° < 0 =< 360 ° of Fig. 9 are hard to be formed by the aperiodic disturbance because I Gg(jn) l becomes smaller than unit for such a number of waviness. Consequently these wavinesses are difficult to become the nucleus o f chatter generation, which is developed in Section 5.2. 5. W O R K - R O U N D I N G M E C H A N I S M IN T H E U N S T A B L E GRINDING REGION When the machining system is unstable, the work-rounding mechanism cannot be solved by the geometrical consideration in the previous section. On the block diagram in Fig. 2 it is possible for these to be unstable, namely, the geometric instability, the static digging-in and the work and the wheel regenerative chatter virbation. The work-regenerative chatter vibration seems to be most important among these because the vibration builds up violently and the finishing accuracy becomes extremely bad in this case [12].
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
n
\
7r
"
f
'
155
/Oor I .-Real
n
"3,r
3
i
i
I 0o
I
6O"
120°
8
180 °
I
I
I
I
nI
n2
n3
n4
240 ° I
n5
300
°
I
n6
FIG. 9. Approximated geometrical work-rounding mechanism. 5.1. Geometrical instability
If the geometrical work-supporting condition satisfies ~bl = n e y 2 = (ne - - 1)7,
where n~ and ne are even numbers.
(7)
The geometrical work-rounding mechanism Gg(jne) for the number of waviness ne becomes Ga(jne) = 1/{1 -- ,' e-OUne + (1 -- ~) e -o2-~ne} = 1/(, -- ,') < 0
(8) (Appendix 2)
Therefore, the system is positively fed back and becomes unstable unrelated to the machine's vibration. Equation (7) is seldom satisfied in the usual centerless grinder and the geometrical instability does not almost occur in a practical operation. Figure 10 shows the change of out-of-roundness with respect to the positioning angle of front shoe 4,z under the shoe-type centerless grinder. In the neighbourhood of 41 = 66 ° as to satisfy equation (7), the out-of-roundness becomes to be maximum and it diverges with continued grinding [13]. Preferable finishing accuracy can be got i f to set 7 - - 7° and 4 1 : (he" - - 1)7, when the out-of-roundness becomes to be minimum.
156
Y u J I F U R U K A W A , MASAKAZU MIYASH1TA
_ •
and SUSUMU SHIOZAKI
Ng/t
Grinding
condition
N w= 12O rev/min
S~ = O.25/.z/rev 7= I1" total depth of cut
toff¢ work 5 5 ~ X IO turn
Front shoe
[
2.0 ¸
~L
E O
IO
i
It 0
,
g
50 °
,
0o
70 o
shoe position angle,
Front
16~
FIG. 10. Geometrical instability.
5.2. Static
digging-in type of instability The equivalent spring constant km with regard to the depth of cut is 1
1
~gr _ c~g~
(9)
If km becomes smaller than zero, the true depth of cut will be over the set depth of cut, which invites the static digging-in of wheel on work. At this time, there appears the intermittent fiat mark on the work-surface as shown in Fig. 11. The causes of the static digging-in come from
"~
The center height is too high The top angle of blade is too l a r g e . , ~gb becomes large. Lack of static stiffness of blade Lack of tightening of bolts
"1> kb becomes small.
J
Even if these causes are taken off, there is the case when the digging-in occurs according to the increase of grinding force associated with dulling of grains. Because kb has the
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
157
Ground w o r k - s u r f a c e
TALYSURF
i ....
....
i
~
-0
Iv i/; i/i
t
"-20 - -40
- -60
i
I
i i,,,
7
~m
MeQsured wQviness
FiG. 11. Digging-in type of instability. linear spring characteristic for the wide range of load but ks and kr have usually the hard spring characteristics. Therefore km is possible to be smaller than zero once the grinding force will be over a certain limit. 5.3. Work-regenerative chatter vibration 5.3.1. Stability criteria in the steady state. As explained in the next section, the selfexciting vibration in centerless grinding is not generated as soon as the operation takes place but gradually grows up with the continued grinding. However, it is regarded in the present section whether the vibration fully grown after the enough continued grinding is kept or not, that is, the stability criteria for the machining system in the steady state are made. The characteristic equation for the closed loop centerless grinding system A on the block diagram in Fig. 2 is Qo(s) = Ym(S) _~_ (I -- e-2"s)Gg(s) kg
(10)
It is well known in the automatic control theory that the machining system is just situated on the stability limit if Qg(jn) = O. Therefore, the stability equation becomes as follows: (1 -- e-2"s)Gg(s) ~ -- Ym(s) kg
(11)
158
Y u J I FURUKAWA, MASAKAZU MIYASHITA
and SustlMU SmOZAKI
Here, Gg(jn) is decided only by the geometrical work-supporting conditions, especially by the center height angle 9' (Fig. 8 and 9) and Ym(jn) is the specific function for the given grinder as shown in Fig. 3. The solution of equation (11) n ----na, which means the number of waviness generated by the self-excited chatter, can be got from the cross point of two vector loci for the right and the left term of equation (11) [12]. In Fig. 12(a), the example of the stability criteria diagram for the given grinder is shown against the center height angle 9' ----7.5 ° and the grinding stiffnessk~ ---- 1 kg/t~. Here Ym(j2zrf) and (1 -- e-2"J")Gg(jn) are described with respect to the f r e q u e n c y f Hz and the number of waviness n respectively. Therefore, if the number of work revolution N w is so chosen as to be N w = fe/na (fc and na are the frequency and the number of waviness at the cross point), equation (11) is satisfied and the system is just on the stability limit. For example, set N w = 4.0, 4.7, 5-5 rev/sec, the chatter mark to have the number na = 20, 18, 16 are formed by the selfexcited vibration. TABLE 1. CHARACTERISTICS OF WORK-REGENERATIVE CHATTER VIBRATION IN THE CASE OF THE DAMPING COEFFICIENT OF GRINDER IS NEGLIGIBLY SMALL
Chatter frequency Phase lag of grinder Geometrical depth of cut I II
fi, Hz
Z_{1/Ym(j2~rf)}
fel ~ fnl, fn2 fcl ~ fn2
- - 9 0 °, - - 2 7 0 ° -- 360 °
/(1
7.25
--
Phase differenceof waviness
e-2~rJn)Gg(jn)
/(Rwn/Rwn_l)
-- 90 ° 0°
-- 0 ° A b o u t -- 180 °
5.25
q. (j .2=f) kg
F I G . 12(a). G r a p h i c a l
solution of the number
of regenerative chatter waviness
rid.
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
159
B]ode~s acceler(3tion
Through feed grinding force
Grinding condition, 7"= 7.5 ° Work, $ 4 0 C 19.4~x95
Nw` 5.2 rev/sec Total depth of cut, 50'/'p.m
FIG. 12(b). Regenerative chatter vibration and work-surface waviness. The characteristics of stability limit are classified into two groups as shown in Table 1 (I and II). In the type 1I stability limit, the phase difference of waviness is in the neighbourhood of -- 180 °, which interrupts the generation of nucleus of, and the growth of vibration in a practical operation. The stability criteria are made as well as for the other center height angles and the characteristics of work-rounding mechanism under the generation of self-excited vibration are summarized as follows: The self-exciting frequency is in the neighbourhood o f the natural frequency o f grinder. Referring to Fig. 8(a) and Fig. 12(a), the number o f self-excited waviness na does not coincide with that o f geometrical specific waviness ns and na exists in the first quadrant o f 1/Gg locus, na = nsp -- n* (n* small even number, 2, 4, 6, . . . ).
Figure 12(b) shows one of the practical case when the distinct vibration is generated. At this time, the heated chattering sound can be heard in accordance with continued grinding and there exists the chatter component of the frequency fc = 90 Hz among the grinding force. The finishing accuracy becomes extremely bad by the generation of waviness na = f c / N w ---- 95/5.2 = 18, which coincides with the measured value. When these clear
160
YusI
FURUKAWA,
MASAKAZU
MIYASH1TA
and
SUSUMU
SHIOZAKI
vibrations being generated, those frequencies fc distribute in the neighbourhood of natural frequencies between the G.W. and the R.W. and the phase difference of waviness is usually 30 ° phase lag as shown in Fig. 13, which proves this kind of vibration is due to the selfexcitation based on the work-surface regenerative effect shown in Table I (I). Figure 14 shows the number of waviness n a generated under the same geometrical work-supporting condition as in Fig. 12(a). The theoretical value is calculated from Fig. 12(a), when the number of waviness is rounded by neglecting the phase difference, therefore, it is almost unrelated to the grinding stiffness. ~._~iWork,
o--- Lw=65 o--- 85
4oc2o , .^
. ..... "6
150 l - ~ -
o ~t
o{ --#-
i I00
°'-7
fnr
/
t~
N a t uf r eaql u e n c y
.
£3 1" J
0
~ i
Number
t
[
I
I
I
i
t
i
I
I
[
5 I0 of work revolution, Nw r e v / s e c
~ ,~J
~ I
L I
J i
i
1 I ~
Work, S40C 19qSx95 through f e e d S p e e d phose
/-- ~n_,/-
.....
,,,
l!
;=~;i=
~ ', ', ', I
Nw= 4-8 0.58
rev/sec
mm/rev
difference~
-~r~v ~ - 3 0 °
FtG. 13. Chatter frequency~and phase difference.
Results obtained from the above analyses and experiments are: The generation of practical self-excited vibration can be explained by the steady state work-regenerative chatter theory. The characteristics of this vibration can be approximated by those of at the first stability limit. Most of the practical operations must be situated in the unstable region according to the steady state stability criteria because the vector locus o f
--
Ym(j2rrf)/kg
and
that of ( 1 - - e-2~n)Gg(jn) crosses each other under most grinding conditions. This contradiction can be explained by the transient characteristic of vibration which is to be solved in the following section.
Vibration Analysis and Work-Rounding Mechanismin CenterlessGrinding o---L 20 18 e~ t 6 - -- l ~
E]._O
-Q E Z
:65ram
m---
75
U---
85
e---
95
I~' -- 7 ° 3 0 '
w
--o
'*o
I
1
10
e
--O- --C,j ~ Theoretical value by fig 12(a)
6-
-
4 -
•
0
4
First mode of chatter Second mode of c h a t t e r - -
,'5
G
7
8
161
9
j
/ --//' J "/"
i0
II
12
N u m b e r o f w o r k revolutions,
13
14
15
N~, r e v / s e c
FIG. 14. Number of waviness by regenerative chatter vibration.
5.3.2. Generation and growing up process of work-regenerative chatter vibration. The steady state stability criteria in the previous section can be applied to the problem whether the already existed waviness, which corresponds to the self-excited vibration, is damped or diverges. Consequently, in most practical operations where the primarily existed waviness does not have any periodicity, the time history of the self-excited vibration seems to be consisted of the following process [14]. 1. At the beginning of grinding, the frequency response by the periodic disturbance or the statistical response by the aperiodic one take the main part of work-rounding mechanism, when the machining system is always stable. 2. The primary waviness Rwo, as the nucleus of self-excitation, whose number of waviness satisfies the unstable condition in the steady state, is formed to a certain level on the process (1) and after then the self-excited vibration grows up according to the regeneration begins to effect and the machining system transits to be unstable. 1.
SRw(n) 1Gg SD(n) = [ Ym (in)
2
(12) 1
2.
Rwn (in) = Rwo
kg Ym(jn) ~g(jn) 1
1 + (1 -- e-2"~)kg Ym(jn) Gg(jn) kg
l
Ym + kg 1 Gg
kv Ym
G--~+ kg
where ko
Ym
Gg + kg 12
- - Kd
. e 2,J,,
(13)
162
YuJI FtmUKAWA, MASAKAZU MIYASHITA and SUSUMU SHIOZAKI"
coincides with the rate of dynamic cutting residue [15]. •
""
Rwn
Rw~o(Jn)=Ka(1 + K a e - 2 " j n + K ~ e - a - ~ n + K ~ e - 6 - ~ +
nt e2nt~rJn
... +K a
... )
(14)
When the periodic disturbance is not acting on the system, the generation process of the primary waviness Rwo depends upon the geometrical work-rounding action as considered in Section 4.3. Therefore, the number of waviness n which situates 0 ° ~ O < 120 ° or 240 ° < 0 g 360 ° in Fig. 9 is difficult to become the nucleus of chatter generation even though it theoretically satisfies the unstable condition in the steady state. Once the primary waviness Rwo is formed, the fluctuation of grinding force by the disturbance becomes to be negligible compared with that of Rwo being ground and the waviness builds up with work-revolution according to equation (14) like Rwo, xaRwo, K~Rwo. . . . Rw,, = ~;Rwo (]5) If xa = 1, the waviness does not increase nor decrease and the machining system repeats the constant amplitude of vibration, which means the system is just on the stability limit explained in the previous section (Appendix 3). Only in the case of ~ca > 1, the self-excited vibration grows up so as the finishing accuracy becomes to be a problem. Examining the stability criteria for one grinding condition carefully, it turns out to be clear that several instabilities with different values of xa and over the stability limit are satisfied simultaneously for one work-revolution Nw and one grinding stiffness. But in practice, only the self-excited vibration which is a little over the first stability limit is most apt to be generated by the effects of resonance of grinder and equation (12) [14, 15]. The theoretical time history of vibration can be proved by the experimental result in Fig. 15. The growing up process of vibration from the grinding start point until finished
g
[OOC 5C0
Through feed grinding force
Grinding condition 7" = 7 ° , Nw= 5 rev/sec Si = 0 4¢ f f m / r e v ¢ Work, $ 4 0 C 2 5 x 9 0
Blades acceleration
I
2
3
4
5
Confinued grindingtirne~
6
•
8
TconSec
FIG. 15(a). Growing up process of self-excited vibration.
9
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding T h r o u g h feed g r i n d i n g force 0
--
----
.
.
.
Work-surface
163
waviness
. 400g
2~m
5 i_,°
~2 E
.E -
-
llV
Fro. 15(b). Waviness generation process. is continuously measured. At the beginning of grinding, there is not any remarkable vibratory component and the machining system is stable but it transits to be unstable gradually with the growth of vibratory component whose frequency almost coincides with the natural one of machine. Corresponding the fluctuating component of grinding force with the work surface waviness as in Fig. 15(b), the primary waviness Rwo to have a considerable order of periodicity is formed while the continued time ~-eon-~- 10-15 sec, after then the vibration grows up violently as the regenerative effect begins to help the self-excitation. These theoretical and experimental commentaries result in:
Even under the unstable grinding conditions according to the steady state stability criteria, the machining system is stable on the transient process at the beginning of grinding. Therefore, if finish the operation for one work within the transient process, the self-excited vibration may not be generated. There are many cases where the through feeding of a workpiece is performed within several seconds and, furthermore, Gg(s) is changeable according to the through feeding position, therefore, all of the operation is finished on the transient process and the vibration does not occur at all. If the gain of geometrical work-rounding mechanism I Go(jna) I is so small, the primary waviness to become the nucleus of self-excitation is difficult to be formed. The growing up process of waviness is decided by the rate of dynamic cutting residue ,~a and under the conditions of ~ca is equaled to or a little larger than unit, the influence of self-excited vibration on the finishing accuracy can be neglected in practice. 5.3.3. Graphical solution of the rate of dynamic cutting residue and the experimental result. It is necessary to calculate the rate of dynamic cutting residue in order to solve the growing up process of self-excited chatter vibration. kg a -- V~
~-'~"(in) +
kg
(16)
In order to know the approximate characteristic of Ka, let assume as follows: The self-exciting frequencies can be represented by the natural frequency. This assumption seems to be reasonable from the experimental result in Section 5.3.1.
164
YuJI FLrRUKAWA, MASAKAZU MIYASHITA a n d S u s u M u SHIOZAKI
E
k/ ~ ~ ~ N n
- k0
7-
%n(J2~fn)
Real Ym(0)=k~
0J n 4
(e) Approximdted gruphicet solution of K d
.F
1.25 1-20
-o 2 E
/,4"o= i ;~ + " ~
t15
\
/
E
105 t¢
IO0
3o.
(b)
60.
9o.
,20.
=o.
xdwith respect tokg ond 0
OJ
l i ReQI
~// e
I (c)
\'~_
/_
ha=
Stobte waviness ndson I/%(jn) locus
FIG. 16. Graphical solution o f Kd.
The geometrical work-rounding mechanism G~(s) can be reduced to equation (6) and the vector locus of 1/Gg(jn) is shown by a unit circle in Fig. 9. Therefore, Ka at the natural frequency can be solved graphically as shown in Fig. 16(a). Where, 2
0
(17)
When the damping factor I of machine is sufficiently small, =a takes the maximum value at 0 -- 90 ° as shown in Fig. 16(b). The number o f n to satisfy Ka > 1, the vibration grows up, situates in the first quadrant of I/Gg(jn) locus. This coincides with the result obtained in Section 5.3.1. The primary waviness which is to become the nucleus o f self-excitation situates in 120 ° _< 0 --< 240 ° (result in Section 5.3.2).
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
165
The range where Ka becomes larger than unit is a little influenced by the value of grinding stiffness but is approximately shown by, about 0°~< 0 <~ 180 ° (result o f Fig. 16(b)). The waviness in 45 ° < 0 < 120 ° is difficult to become the primary waviness because o f I Gg(jn) I < 1, but Ka > 1 for that range of 0. Therefore, even the slight amplitude o f primary waviness is formed, it will grow to the self-excited vibration. From these commentaries and the experimental results described in the following, the stable and the unstable range of 0 and n can be approximately shown as follows : The stable range of 0 and the number of waviness has 0 ° - < 0 ~ < 4 5 °, 0 ~ nas(even) ~ [rr/4y]e,
180 ° - < 0<--360 ° 1 % nas(odd) <= [~r/~']o
(18)
The unstable range of 0 and the number of waviness na 45 ° < 0 < 180 ° [Tr/47]e + 2 <~ ha(even) ~ [Tr/y]e -- 2
-~ m
04
0:5
o o
o
No. 5 - - a w =69 fc =97 n d = 14
No ~r4=1.09 N re~/~sed fc* Hz • nd •
n2
/
m
/
.c o
~
(19)
Work, $ 4 0 C hardened 2 5 ~ x 7 0 y = 7 5 ° , S = 7 3 ¢ /.z/sec
2 =47---K%=108 =94 ~
I
= 20
i
Z~No.4--[ *. ~ /d IN =bO
7f.
K~ =107 qex
=96
. . . . . NO 3 - - - K d = I "U'e •, , ~ e, •/ mw=o'a
No - - - K d e ~= 0 N =4.2n=20.~ w fc = 8 3 ~rr d
~.~
0~
,~
5
!C
Continued
grinding time,
u
//
"t~ ~
104
'~
102
/
/
~o, s e c
xperirnenta I
T
~
~2 Nol
I
No.2
No5
No 4
No5
~----~/I
-,
¥~(j z rre3)
"i%84'1o
.......... t'-2'I
Fia. 17. Influence of ne upon the growing-up process of vibration.
I !
166
YuJI FURUKAWA, MASAKAZU MIYASHITA a n d SUSUMU SHIOZAKI
Especially those n~ near to [~r/7]e -- 2 are more apt to be generated. In equation (18) and (19), those even number of waviness to pass on I/Gg(jn) locus first time were regarded. Because except the center height angle 9' is so large as more than 9 ° , the odd number of nd to pass on 1/Gg locus second time are not formed in practice [12]. In Fig. 17, the growing up process of vibration is shown for the several number of work revolution. The experimented value of Xdex differs for the number n~. The grinding stiffness kg is known from the other experiment about kg -- 4.5 kg/F during
Nw = 4" 2-6" 9 rev/sec. Therefore the theoretical value of Knth can be graphically calculated for kg ---- 4.5 kg//z,
fc = 95 H z and na ---- 14--20 and the theoretical value well coincides with the experimental one. Like this example, the growing-up process of vibration can be represented by the rate of dynamic cutting residue. Likewise, when the grinding stiffness or the chattering frequency vary, the growing up process can be shown by Ka in the same way [15]. In Fig. 18, the growing up process is shown against the center height angle ~,. In case of 7----3.5 ° the system was stable during 2 minutes' continued grinding; however, in other center heights, the self-excited vibration to have almost equal value of na and fe Work, $40C hardened 25¢x 70 Nw = 6.7rev/sec fc = 9 4 H z
40O y =5°30 '
2
~
300
K~cf
i. 08"---~
nd = 14
&
_ " f =7030 ' ~de/106
o
S
~oo
~S 3 2S I--~-
~oc
y = 9°30'/0 Kde~ [ 0 ~
/ ~" =3°50 ' with no ~ . ~ chatter vibra tiol~,,W -~ 5
i
i
i
I
I
:o
t5
20
25
50
Continued grinding time,
Io8
-cc~n
~
sec
Theoretical
, o6
1o4
~/Experimen~al 3o50 ,
~°30'
~
'~
7°30 '
9o30'
)" Ym(j2~'94)
FIG. 18. Influence of y upon the growing-up process of vibration.
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
167
was generated. Assuming that the same vibration occurs in case of), = 3.5 °, the theoretical value of tCdthbecomes Ka(jl4) -- 1.10 y
~
3.5 °
and the vibration must be developed. But actually, the vibration is not induced because the primary waviness is not formed as I Gg(]14) ~, : =
3,5 o
is enough small. When the center height angle ~, is small, the number of waviness which is possible to be generated by the self-excited vibration becomes to be considerably large from equation (19). In order these wavinesses are generated under the grinder to have the natural frequency at about a hundred and so Hz, the number of work revolutions must be very low. Therefore, under the usual operating condition, the chatter vibration is not induced ([ setting the center height angle y : 2°-3 °. In Fig. 19, the region where the vibrations are generated is shown. It is experimentally proved that the system is stable if the center height angle ~, is set on a small value. ,2
<
~7 I,
~G g
I
'
'\ t \
5
z F
"ql S~ = 4 2 , 5
E
O-I
I
q
t
I I < ,%
4
"
1 la
/
? 8,7.5 ~./see
,~
I wcrk, S40C 25 x T O
Stable
I- Q - - Firsl rnode of c h a t t e r v i b r a t i o n •A ' - - Second mode of c h a r i e r v~bla1'ion 3°30 '
5°30
7°50 '
Center height angle,
Z) 50'
I°30 '
y
FIG. 19. Regions where chatter vibration generated.
5.3.4. Work-regenerative chatter vibration, sub-conclusion. 1. The characteristics of fully grown self-excited vibration can be explained by the application of the regenerative chatter theory to the centerless grinding system. 2. On the generation and the growing up process of the vibration, the geometrical work-rounding mechanism takes the main part of waviness formation at the beginning of grinding and gradually the regenerative chatter replaces that part in accordance with the formation of primary waviness which is to become the nucleus of selfexcitation. The growing-up speed of vibration is decided by the rate of dynamic cutting residue. 3. If the gain of geometrical work-rounding mechanism I Gg(jn) l is much smaller than unit or the rate of dynamic cutting residue is smaller or a little larger than unit, the self-excited vibration does not so grow as to effect on the finishing accuracy. The practical grinding conditions to avoid the vibration are considered in Section 6.
168
YuJ[ FURUKAWA, MASAKAZU MIYASHITA and SUSUMU SHIOZAKI
5.4. Wheel-regenerative chatter vibration The influence of wheel waviness on the finishing accuracy are studied on the cylindrical or the surface grinder [16]. However, on the centerless grinding whose natural frequency is usually much lower than those, the generation process of wheel waviness may also be different. Even under the stable grinding, the finishing accuracy decreases in accordance with continued grinding, which seems to be caused by the dulling of wheel surface. As an example, on the through-feed grinding of rollers, the work regenerative chatter is not induced but the circular roughness increases with the total number of ground rollers and the partial waviness can be recognized on the wheel surface as shown in Fig. 20. At this Work, roller 15¢xl5 H&=62 Through feed time, 2.5 sec Total depth of cut, 0.2¢mm Entrance center height angle, 5* Exit center height angle, 7.5 °
25
8 ['5
0 o
o
8 0
O
I0
_ ~ "
8 "00
~o
~ / "
o
E 2-0 :L
3:
0 . ~ s
8
0
O
0
0
0
p*---
One revolution of G.W
~ "~ • ; 'Jlt;'~. t'lt
~
.
,-;~ll~i.', ~ '
!,:'
a{
j~L
o
i) O5
GW (WA60 LV) waviness offer ground 1500 rollers
•
.
500
I
1000
_
_
i0 JSO
I
2000
Total N u m b e r of g r o u n d rollers
FIG. 20. Through feed grinding of rollers. time, any periodic waviness is not generated on the work and the regulating wheel surface. Therefore, on the analysis of the grinding wheel waviness, Gg(jn). (1 -- e -2-j') may be approximately put to unit in Fig. 2. Consequently, the stability equation for the grinding wheel's surface becomes as follows:
-,+ kg
1 ,
= kas(1 -- e-2"Ns ~")
(20)
Ym (jn)
Here, kas is the mean wear stiffness of G.W. including the falling-off, the fracture and the wear of grains. The characteristic of kas is discussed carefully [17], but here let consider kas to be a linear coefficient. Describing equation (20) on the complex plain and Fig. 21 can be got, where two vector loci cross each other always as the wear stiffness of G.W. kas is generally takes a large value. In the neighbourhood of the natural frequency
kas >~ Ym(j2rrfn), therefore, one of the unstable roots coincides with the natural frequency. Practically the infeed grinding is performed for a long period, under the condition where the work-regeneration is not induced, against a different grain size and grade. There
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
_
~
0
~-~___.._
•
Real I
kff k m
~g+~m(jn)
%' kas.f 1 - e ~rrNlslrl)
Flo. 21. Graphical solution of the stability[limit for the grinding-wheel waviness.
50
25
22 20
o~ c
J5
10
05
5
i0
20
Total n u m b e r of g ' o u n d works
FIG.
22(a). Wheel characteristics and finishing accuracy.
169
170
YuJI FURUKAWA,MASAKAZUMIYASHITAand SUSUMUSHIOZAKI One revolution of G.W.
t
WA 4 6 LV
WA 6 0 LV
WA 8 0 UV
WA 8 0 LV
WA 8 0 NV
WA I00 LV FIG. 22(b). Waviness on the grinding wheel surface (after ground twenty pieces of works). appears the case when the waviness is formed on G.W. and the finishing accuracy decreases as shown in Fig. 22 (a) and (b). The waviness to have the frequency of about 100--300 Hz seems to be formed on those wheels to have lower grain size and grade of bond. Such a wheel is usually considered that the grains are apt to be fallen off. The higher frequency waviness with 3 kHz, which corresponds to the blade's natural frequency, is partially recognized on those wheels with higher grain size and strong grade of bond as well as lower and weaker ones, but it does not become a problem on accuracy. Figure 23 shows the statistically treated result for the waviness of WA46LV. There are peaks at about 90 and 150 Hz both of which coincide with the first and the second
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
171
%-
g Auto
correlation WA46LV
o
function
wheel
for
waviness
05
o
,2
.
v
.
.
.
2o
J ¢0 :>,
~o
13-
{I
*
'
'
'
t ' IOO
I
I
I , 150
I
Frequency
I
~ J , 200 f,
,
,
,
I , 250
,
i
i
I 500
Hz
FIG. 23. Statistically treated result of wheel waviness.
natural frequency of given grinder. Therefore, the generation mechanism of this kind of waviness seems to come from the wheel-regenerative effect. 6. S E T T I N G OF T H E M A C H I N I N G C O N D I T I O N I N O R D E R TO A V O I D T H E G E N E R A T I O N OF V I B R A T I O N 1. Grinding wheel selection. It is a difficult problem which can not be easily decided. Because the selection of wheel is greatly affected by the matching with the workmaterial. Under the consideration of wheel waviness formation, the wheel to have the grain size nos. 80-100 and the grade of bond L - N seems to fit to grind the usual hardened steel. 2. Continued grinding time for one workpiece. Like in the through-feed grinding, if the continued grinding time for one workpiece is so short as to be several seconds, the operation is always performed on the transient process and the work-regenerative chatter is not induced. In this case, the work-rounding mechanism is due to the setting of the center height angle. 3. Geometrical work-arrangement, center height angle ~,. When the grinding is finished on the transient process or the work-regenerative chatter can be avoided by the selection of work revolution, the center height angle ~, must be set y -- 7 ° in order to minimize the out-of-roundness. In Fig. 24 the calculation chart of center height h when ~,----7 ° is shown. In the case when the machining efficiency is much more important than the finishing accuracy, set ~, = 2o-3 °. At this time, the Gleichdicke is formed on the work-surface.
172
YUJI FURUKAWA, MASAKAZU MIYASHITA a n d S u s u M u SHIOZAKI
500
400
E E
'
3o0
JI q \
+
200 ~
~2"25
",,,
\\\
IO0
"/'20
'~"
\x/(5 \\
0
lOG
:500
200 r. + rw,
mm
FIG. 24. Calculation diagram of center height when 7 = 7 °. 4. N u m b e r of work revolution to avoid chatter under 7' = 7°. Low-speed stable region; f r o m equation (19) the m a x i m u m n u m b e r of waviness by chatter namax is //O~max =
[7"r/7']e - - 2
Therefore to avoid the chatter generation,
namaxNw ~ f c -- fn (natural frequency). Example: for 7' = 7 °, [rr/7']e = 26.
.'.
namax =
24
iffn = 90 H z then, Nw ~fn/namax = 90/24 = 3" 75 rev/sec. Practically the self-excited chatter vibration is not induced when Nw = 3 rev/see as shown in Fig. 19. High-speed stable region; from equation (19) the m i n i m u m n u m b e r of waviness by chatter namin is Ramin ~--- [~r/47]e + 2. Therefore,
fc ~ ndminNW. Example: for 7, = 7 °, r/dmin ~- 8 Nw ~ f n / r l d m i n ~ 90/8 -- 11 rev/sec. Practically the generation of first mode of chatter can be avoided by this method but simultaneously the second m o d e of chatter is induced. It is necessary to separate the first and the second natural frequency by a proper a m o u n t in order to diminish the chatter by this method [15].
Vibration Analysis and Work-Rounding Mechanism in Centerless Grinding
7. G E N E R A L
173
CONCLUSIONS
In the present paper, the complicated work-rounding mechanism in centerless grinding is solved systematically based on the block diagram of machining system. Consequently, each shape of wavinesses can be corresponded to its generative cause and several method to take off that cause are proposed. Especially the work-regenerative chatter vibration is fully developed theoretically and experimentally, where the transient phenomena, which are important when to consider the method to avoid the generation of vibration, are carefully examined as well as the stability criteria in the steady state. From these results obtained, the condition to minimize the influence of vibration on the out-of-roundness is proposed. Acknowledgements--The authors wish to express their appreciation to President Chuji Fukushima and
Factory Manager Tadamasa Fukushima of the Ohmiya Seisakusho Co., Ltd. for the support given throughout the course of research.
REFERENCES [1] S. YONETSU, Trans. J S M E 19, 53, 59 (1953). [2] M. SUI~A,BMI. Tokyo Inst. Tech. 25, 1 (1963). [3] K. OGAWA and M. MIYASHITA,Trans. JSPE, 24, 89 (1958); 24, 279 (1958); 26, 169 (1960); 27, 421 (1961). [4] W. B. ROWE, M. M. BARASHand F. KOEN1GSBERGER,Int. J. Mach. Tool Des. Res. 5, 203 (1965). [5l W. B. ROWE, Mach. Prod. Engng 3, 984 (1965). [6] M. MIYASHITA,Mere. Fac. Tech. Tokyo Metropolitan Univ. 15, 1177 (1965). 17] W. B. ROWE and F. KOENIGSBERGER,Int. J. Mach. Tool Des. Res. 4, 175 0964). [8] D. REEKA, Ind. Anz. 90, 461 (1968). [9] E. A. BECKER, Ind. Anz. 87, 327 (1965). [10] J. P. GtmNEY, Trans. A S M E , Ser. B, 86, 163 (1964). [11] D. J. PmKINGTONand E. R. AUSTIN,Adv. Mach. Tool Des. Res. 7, 477 (1967). [12] Y. FURUKAWA,M. MIYASHITAand S. SHIOZAKLTrans. JSME, 36, 328 (1970). [13] M. MIYASHITA,Trans. JSPE 35, 54 (1969). [14] S. SHIOZAKI,M. MIYASI-nTAand Y. FURUKAWA,Trans. JSME, 36, 143 (1970). [15] Y. FURUKAWA,M. MIYASmTAand S. SHIOZAKI,Lect. JSME, No. 214, 105 (1969). [16] R. SNOEYS, CIRP (1967). [171 P. D. SINGHAt.and H. KALISZER,Adv. Mach. Tool Des. Res. 6, 629 (1965). APPENDIX
1. If the work surface waviness can be shown by rw(¢), the depth of cut tw(¢) turns out to be as follows: rw(¢) = rw(¢ - - 21r) - - e'rw(¢ - - ¢1) + (1 - -
¢)r~.(6
- - Cz)
+
si(¢)
(1)
Where s,(¢) means the relative distance between the work and the grinding wheel. And e' means the ratio how much the unit wave from the work mean circle rwo at contact point B (see Fig. l) influences on the depth of cut. Likewise, (1 -- E) at contact point R. ~' = sin (c~ + f l ) / c o s (0 - - c0 (1 - - ~) = c o s (0 q- f l ) / c o s (0 - - ct)
At the same time, the present work form rw(4,) can be shown, r~(¢)=~(¢
- 2~)-
t~(¢)
(2)
174
Yusi FURUKAWA, MASAKAZU MIYASHITA and SusuMu SHIOZAKI
Supposing that the radial grinding force is proportioned to the depth of cut and its proportional coefficient is called the grinding stiffness.
Pa(4) = katw(4)
(3)
Once the grinding force acts, the machine tool has the elastic deformation according to its dynamic stiffness. x(4) = Pg(4)/ym(4) (4) Laplace transforming equations (1)-(4) with respect to the work revoluting angle 4, they become to be as follows:
Rw(s) 1 Si(s) -- I -- e'e-¢l' + (1 -- ¢)e-$2s
(5)
Tw(s) _ 1 -- e -2~s
(6)
Po(s) Tw(s) = kg
(7)
Rw(s)
X(s)_ 1 Co(s) rr.(s)
(8)
Likewise, the wheel regeneration can be considered in the same way. 2. 1 -- E'e-OlJn + (1 -- ¢)e-O,J" = {1 -- cos 41n + (1 -- ,) cos 4zn} + j { ¢ ' sin 41n -- (1 -- E) sin 42n}
(9)
When the imaginary part is zero, the real part becomes, R.P.=realpart=
1 - - ¢' cos 41n ~ ( 1 - - ¢)
1-- \1--
E] sin42n
-- 1 -- E' cos 41/'/::[:: (1 - - E)
(I0)
and has the minimum value (~ -- d) when cos 41n = 1, that is, 41n = n'ezr. The number of waviness n becomes the even number ne = ~r/~,; therefore, the condition that the real part takes the minimum value can be arranged as follows:
z=rr--y=(ne--
1)y
(11)
On the other hand, ~' + (1 -- E) becomes from the geometrical consideration,
sin(~+fl)+cos(O+fi) e' + ( 1
c°s (41 2 7)
- - ,) =
cos ( 4 1 f ~1
cos ( 0 - ~) Generally, 0 < 41 < rr and ~, is small, therefore,
O<
< 4~ + ~, < 2
2
Vibration Analysis and W o r k - R o u n d i n g Mechanism in Centerless Grinding
J+(1--
E)>I
and ~ - - ~ ' < 0 .
3. From equation (11), the steady state stability limit is (1 -- e-2~J'~)Gg(.jn) = -- Ym(jn)/ko. this can be transformed like follows: kg
Kd - ---
Gg
(.in)
- - e 2~jn +
kg
l e 2~jn ] ~
1.
175