Wear, 55 (1979) 59 - 70 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
NON-UNIFORMLY VARI~LE TURNING OPERATION A. F. RASHED,
H. R. EL-SAYED
E~ineering Alexandria (Egypt}
Production
(Received
April 27,1978;
ERRORS I~ER~NT
59
IN THE
and T. EL-WARDANY
Department,
Faculty of Engineering, Aiex~~dri~ University,
in final form November
2,1978)
Variations of the depth of cut during the turning operation result in the generation of dynamic cutting forces which act upon the processing system to introduce non-uniformly variable errors. These are in turn transferred to the turned workpiece. Equations were derived to describe the dynamic forces generated owing to initial errors in the blank. Evaluation of the dynamic forces generated and the corresponding transferred error is presented.
1. Introduction Owing to roundness and straightness errors characterizing blanks for turning, variations in the pre-set depth of cut take place. These in turn generate dynamic cutting forces which introduce non-uniformly variable errors [l].Determination of the function describing this type of error makes it possible to control its effects. Determinations of the magnitude of this error under steady state conditions have been reported [ 1 -61. The ~dividu~ effects of dynamic cutting force components have also been investigated [ 7,8] . No solution considering the actual conditions of the turning process has been published. The present work was undertaken to obtain a solution.
2. Conditions governing the de~ation consideration
of the error function under
Complicated vibratory structures can be reduced to simple elements with one degree of freedom. The processing system will be analysed in the
60
Fig. 1. Incremental
variation of Bi.
same way. The receptance of each of its components
L
R=
is therefore given by
(1)
S-Mw2
The roundness and straightness errors change the radius of the workpiece both along the turning axis and within the same section. Therefore the radius of the workpiece at any section x is assumed to be rx
=r
- Ql +
mx
=r mx f ax
(2)
sin&e
The variations in the workpiece radius change the pre-set depth of cut and consequently make the cutting tool oscillate during the turning operation. The instantaneous amplitude of oscillation can be determined in terms of the steady state radial deflection of the tool and its carrying structure. The equations defining the steady state force components, the radial deflection of the tool and the tool support system have been derived elsewhere [l] . From these equations the resultant radial deflection of the tool and its support can be written in the form 6 st = kd2
(3)
Therefore the dynamic radial deflection is 6d
= 6,,
’
s(6,,)
= 6,, * (2kd6d + d26k)
(4)
Using the same data the [ 1 J the magnitude of K is found to be of the order 10d6; hence 6 K can be neglected. The instantaneous amplitude ati of the tool oscillation is therefore given by at, =+ 2Kdtid
(5)
61
Vi
Fig. 2. Incremental
variation of (Y,.
Based on this analysis and the conditions previously assumed [l] , the non-uniformly variable error function is derived.
3. The dynamic cutting forces Owing to the errors characterizing the workpiece, incremental variations take place in some of the parameters included in the equations of the steady state force components. 3.1. Incremental variation of the shearing stress The incremental variation of the shearing stress of the workpiece material is given by [9] AS, = 0.0356e
(6)
3.2. Incremental variation of the inclination angle 8 i Owing to tool oscillation the direction of the cutting velocity v varies and consequently the value of 8i varies as shown in Fig. 1. From this figure 6Bie
Aat,
AX .
=-atx V
3.3. Incremental variation of the normal rake angle (IL,
Figure 2 shows the general position of the cutting tool and the corresponding variation in (Y,. Hence
(7)
Fig. 3. Incremental
variation of u.
(8) 3.4. Incremental variation of the normal shear plane angle @,, Nigm [9] reported that
(9) where 6 J, is the variation of the free surface slope relative to the instantaneous cutting direction. In the case considered here
3.5. Incremental variation of the angle of approach x Koenigsberger [3] showed the effects of the angle of approach x on the components of the cutting force. Owing to the variation in the depth of cut, the incremental variation of the angle of approach is
3.6. Incremental variation of the cutting velocity The instantaneous relative cutting velocity V, between the tool and workpiece is the vectorial sum of the cutting velocity v and the rate of tool oscillation. From Fig. 3 Vi = V+Ci*68i and therefore
63
Fig. 4. Incremental
6”
-
variation of d.
LJ2
(12)
u
3.7. Incremental variation of the depth of cut Figure 4 shows the variation of the depth of cut due to tool oscillation and the nature of the workpiece surface. For small incremental variations of the depth of cut the shear plane changes its nosition but keens the same inclination to the mean cutting velocity. From Fig. 4 it folio-ws that d2 -dl
=atx - 2al + da 19~cot 4,
Therefore 6d =ati -2a1+-
i,d v
cot 9,
(13)
3.8. The dynamic cutting force components The magnitude of any component of the dynamic cutting force can be expressed by Fd = F,, f SF,
(14)
where Fd and FSt represent the dynamic and steady state values of the component respectively and SFSt represents the amount of change in the steady state component considered due to incremental variations of the influencing parameters. Therefore the changes in the tangential, axial and radial components Ft, F, and F, of the steady state cutting force are
64
+
w
(15)
Sd
mi
SF,=-
ei
ass 6ss + ae
a-c i
sei +
al;g
aF, a& -6X+-----
84,
ax
----~hl+
pGl+ n
au
a&
(16)
+adSd 6F,=S
&_&
a&.
+
a8i
s
6v-t
&&
:
aFr
San
aan
+%u+
+aF,*& +aFr6x ah
ax
au
a&
(17)
+izd
The equations defining the dynamic force components can be derived by using the equations for the steady state cutting force derived in ref. 1 and eqns. (6) - (17).
4. Errors arising owing to deflections of the components of the processing system Dynamic deflections of the individual components of the processing system under the effect of each of the dynamic cutting force components can be expressed by & = FdR = (F,, + 6 F,,)R =6,
+ A6,
(18)
where &d and 6, represent the dynamic and mean deflections of the component under consideration in the direction of the acting component, Figure 5 shows deflections of the tool-carrying structure under the effect of the three components of dynamic force. The minimum and maximum radii of the turned workpiece can be derived from this figure and are expressed by rtcd = [&z
+ (Lrm * A&AI2
+ @tctm 4 A&m )211’2
(1%
The corresponding errors induced in the turned workpiece will be E ted
= 2hcd
-
bz)
(20)
By proceeding as described previously [ l] the induced and transferred error equations of the processing system and its components can be derived.
65
Fig. 5. Tool-carrying structure deflection.
5. Numerical evaluation of the force and error equations The general equations we have derived are valid for any surface configuration of the workpiece. For the purpose of numerical evaluation a particular cross-sectional configuration (a, = O.ld, k, = 8) was chosen and assumed to extend uniformly along the workpiece axis. The masses (kg) of the different parts of the lathe were estimated (bed, 0.94; carriage, 0.306; headstock, 0.735; tailstock, 0.061). Using these and previously presented data [l] the anticipated cutting force components and the corresponding induced and transferred errors were evaluated for turned brass and cast iron workpieces. The results obtained are shown in Figs. 6 - 11.
66 1
’
’
’
’
’
--components 30 y”
components 28
5 5
26
16 14 12 10
8
.__ \
.
. . .-_
.___
. _.
-.
6
\
.
\// ‘.
\
/
/
Material brass 0, z 30mm
I’
d N 4%
0
= .75 mm z 720rpm 10
I
I
I
I
I
I
I
I
,
1
2
3
&
5
6
7
0
9 time
Fig. 6. Dynamic
10
set
cutting force components.
6. Discussion and conclusions Figures 6 - 9 show interrelated results and illustrate the dynamic force components generated, the induced and transferred errors due to deflections of the individual components of the processing system and the transferred error of the processing system which occurs when a brass workpiece is turned under defined cutting conditions. Figures 10 and 11 show the dynamic force components generated and the transferred error of the corresponding processing system which occurs when a cast iron workpiece is turned. Figures 6 and 10 show that the components of the cutting force be-
67
urn
w
0
25
urn urn
03
o!iTd
urn
095um
urn
Ok!7
urn
Id
o-22 Fig. 7. Individual
induced errors: (a) Ew
urn
0.75 um
;(b) E,,,d ;(c)Et Ed;(d) E,.
have similarly. Their magnitude depends, for constant depth of cut, on the shearing stress of the material being turned. Figures 7 and 8 show that the ratios of transferred to induced errors of the different components of the processing system follow the same trend as those of the steady state errors.
{;)
ofifs )J
Fig. 8. Individual
07.57 12.5JJ transferred errors: (a) Ew;
(b) E,d;
d.5
II
(c) Et ldr;(d) E,dr.
Therefore the same reasoning is valid [l].The system transferred errors (Figs. 9 and 11) are greater than those under steady state conditions. From the results the following conclusions are drawn: (1) roundness and straightness errors characterizing the workpiece generate a dynamic cutting force;
69
07r-r-Lu
x=O*lL
I
:
0
9
I 29.75
u
50.75
x=0.5 L
0 5.?25
20.2Su
x= I_
Fig. 9. System transferred error E, dr: brass; Di = 30 mm; d a 0.75 mm; N =: 720 rev mine’; L/Di = 10.
components
1
2
3
Fig. 10. Dynamic cutting force components.
5
6
7
8
9 Time
set
31
70
0525
55
010 40 77.5
tt-+025
Fig. ll._!ystem transferred error Es&: cast iron; Di = 30 mm; d = 0.75 mm; N = 205 rev min ; L/Di = 10.
(2) the final shape of a turned workpiece depends to a great extent on the shape of the initial errors of the blank; (3) the magnitude of the transferred error depends upon the ratio of the tangential to the radial receptance of the processing system. Nomenclature
ax d Etcds Etcdr
amplitude of blank error at any section x, mm depth of cut, mm induced and transferred dynamic errors due to carrying structure, E.trn induced and transferred dynamic errors due to piece-carrying structure, pm induced and transferred dynamic errors due to tool, pm induced and transferred dynamic errors due to piece, pm mean radius at any section x, mm set-to-size radius, mm
deflection of the tooldeflection of the workdeflection of the cutting deflection of the work-
References H. R. El-Sayed and T. El-Wardany, Steady state errors transferred to the turned workpiece, Wear, 55 (1979) 41 - 58. S. M. Said, Static compliance of the centre lathe processing system, Znt. J. Mach. Tool Des. Res., 4 (1966) 223 - 241. F. Koenigsberger, Machine tool design, Prod. Eng., 43 (11) (Nov. 1964) 534 - 542. R. L. Murthy, Interaction of machine tool and workpiece rigidities, Znt. J. Mach. Tool Des. Res., 10 (2) (June 1970) 317 - 325. S. Bajpai, Optimization of workpiece for turning accurate cylindrical parts, Znt. J. Mach. Tool Des. Res.. 12 (3) (Sept. 1972) 221- 228. R. Umbach, Problems of stiffness and accuracy of large size machine tool, Proc. 6th Znt. Machine Tool Design and Research Conf., Pergamon Press, Oxford, 1965, pp. 95 - 121. V. Kovan, Fundamentals of Process Engineering, MIR Publications, Moscow, 1970. F. Koenigsberger and J. Tlusty, Machine Tool Structure, Vol. 1, Pergamon Press, Oxford, 1970. M. M. Nigm, Prediction of dynamic cutting coefficients from steady state cutting data, Proc. Znt. Machine Tool Design and Research Conf., Pergamon Press, Oxford, September 1972.
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