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The Effect of Time Varying Parameters on the Build-Up of Chatter in Turning
The Effect of Time Varying Parameters on the Build-Up of Chatter in Turning B . J. Stone (1)
In man? dynamic performance tests on machine tools the cutting situation IS not one with constant parameters, whereas analyses of chatter assume this is the case. It is t h e r e f o r e important to determine the effect if any of the malor parameters v a r y , which i s often the case in practice. The ilarameters investisated are those o f width of cut and daniping and it is shown that it is possible to obtain performance test values which are significantly in error. compared to stradv state conditions. The main reasons f o r this are shown to be the rate of Srowth of vibration and the stsbilising effect of varyins width with time. most
INTRODUCTION FOK some time there has been an interest in determining the dynamic performance of machine tools in addition to their accuracy. This has resulted in various dynamic performance tests being proposed'>'. The main objective of these tests has been to determine the limit of stability so that a comparison can be made between different makes and thus a purchaser will have this additional and important information available. As the information will be used in such a critical way there has inevitably arisen the question of both the relevance and reliability of the information obtained from such tests. The major question to be answered concerns the decision whether to use cutting tests o r vibration excitation tests, that may be used for a comparison. The advocates of the former argue that the average person is more likely to purchase on the basis of a cutting test rather than an indirect test, the theoretical basis of which he does not comprehend. The latter argue that cutting tests involve so many other variables, that are difficult to control that a more reliable result is obtained by the indirect test. The results of these conflicting views resulted in attempts being made to reduce the variability of results obtained from cutting tests. For example one of th major variables is the workpiece material; attempts were made' to compare the tendency to chatter of nominally identical materials supplied from different sources. This work used a test structure which it was expected would give consistent vibration characteristics and was not sensitive to small changes in the cutting force direction. In the event it was found to be difficult to obtain consistent results between the different institutions involved in the This may have seemed to collaborative research exercise. provide good support f o r the advocates of vibration tests, but results were being obtained which cast doubts on the reliability of that approach. It is self evident that the final confirmation of the validity of a vibration test will have to be a cutting test and a comparison of many vibration tests and cutting tests4 did not show a good correlation. The major difficulty of vibration tests is that the characteristics obtained should be those that would have been effective under working conditions, ie. 'the operative response o r receptance' as it has been designated. This has been extremely difficult to achieve because of the requirements to have all the relevant components in motion and under preload, whilst at the same time applying equal and opposite forces to the tool and workpiece. These difficulties suggested to some workers the possibility of obtaining the required characteristics whilst cutting; this would involve the subsequent removal of the effect of the cutting process from the results. To the authors knowledge noone has claimed success using this method. This is mainly because of the difficulty of measuring the relevant forces and displacements on a working machine and also the lack of knowledge of the cutting process forces. The solution would therefore appear to lie with reliable cutting tests if they can be achieved. This would seem to require test workpieces made from the same material. Such tests were proposed some time ago for lathes and the repeatability of In general the results results was examined at some length. obtained could be repeated to within 15% and gave the width at which chatter commenced. Since that time however various papers have been publish d which indicate the significance of time varying parametersaBb and of course during a cutting test it is For example the inevitable that some parameters will vary. contact position between the tool and workpiece must change and this may cause the operative structural characteristics to change. It has therefore been considered necessary to investigate the effect of some time varying parameters on the onset of chatter and the method chosen has been to use This has the advantage of simulation on a digital computer. removing the variability already discussed but has the
disadvantage of requiring a model of the machining operation which will be somewhat simplified. However it is argued that any major effects found would be present to some extent in the real situation The time varying parameters that will be discussed are those of width and structural response.
.
PREVIOUS WORK
The main interest in time varying parameters has centred on the possible deliberate variation of such parameters to provide an increase in stability. The parameter which has received the Thus by greatest attention would appear to be speed. continuously changing speed, ie cyclically, so mprovements have been obtained. These investigations6sr8?8.6 indicated improvements for turning of the order of 30% and there were indications that greater improv ments might be obtained in grinding. Also a report was made' of the succes8 in grinding of eliminating chatter in a roll grinding operation by having centres of different stiffness so that there waa a continually varying response. These results indicate that the onset of chatter can be affected by time varying parameters and thus any dynamic performance test which involves ti= varying parameters may not give a true indication of performance.
TXEORY Most of the theoretical examination of the onset of chatter has used the condition at the stability boundary as the basis for any analysis. This is. however, a condition which is not achievable in practice but nevertheless is useful in recognising the major parameters and does indicate the width of cut at which the process will begin to go unstable. The actual manner in which the instability grows depends on the initial disturbance and the extent to which the limiting width la exceeded. It is therefore necessary with a time varying situation to look at the transient behaviour of the process as a steady state solution is not available. The method used in this paper is similar to that described i paper on the build up and decay of vibration in grindingq0: However, as grinding is the most complex operation to analyse as it involves feedback on both the grinding wheel and workpiece and because the dynamic performance testing of lathes is an important subject. the build up of vibration in turning wag examined. The model used is shown in Figure 1, and fndicatea that the representation of the structure was with the conventional spring, mass, damper model. This is a great simplification but satisfactorily represents a structure with a predominant rode.
k
Figure 1. Dynamic model of lathe structure
Annals of the ClRP Vol. 34/1/1985
371
The derivation of the relevant equations and the method of digital simulation will first be described for constant parameters and then subsequently be modified for the inclusion of ti= varying effects. The equation of motion resulting from the instantaneous cutting force P(t) is given by
+
mu 1 (t)
+ kul(t)
cGl(t)
-
(1)
-P(t)
The cutting force depends on the instantaneous depth of cut so that for a constant width of cut, h P(t)
-
Rb(ul(t-r)
-
(2)
ul(t))
R is the cutting force coefficient, assumed constant and ul(t-T) indicates the surface left one revolution earlier, ie, at time (t-T) where T is the time for one revolution of the workpiece. It is useful to non-dimensionalise the equations by dividing dk/m and 5 cl2dmk. Thus throughout by k and substituting wn
-
-
and
(4) If a time increment At i s considered from the situation at some time t then it is necessary to find ul(t+At), uI(t+At) and P(t+At) knowing ul(t), ul(t) and P(t). This is achieved as follows. Consider Figure 2 which shows the value of P at t and t+At. Over the time interval At the structure is subjected to a force which is trapezoidal in form and has. at the start of this time increment an initial displacement ul(t) and an initial velocity ul(t).
Figure 2.
Trapezoidal force/time eler&ent
It can be shown that the effect of the initial conditions ul(t) and ul(t) is to produce u,(t+AC)
-
(il(t)/u"+bl(r))
-Cu,At e
unbC +
[U,(t)EOS/l-S2
."At]
(5)
onbt]
(6)
s i n f I-f2
1'1-2
and (?+At)
-
( Ci (t ) /u,+U
-WnAt e
[ul(t) e 0 s 4 l - C ~ u2At
+
l(c 1)
sidl-6'
4 1-2
At the same t i m e the trapezoidal force produces
ul(t+At)
+
+
-
(P(t+At)-P(t))
25 {1--+------wnAt
k
(zr2-1) sinf 14,'~ f1-5 L2sinfl-52
A]]
+
{l
-e
e-SwnAt [25 cosfl-S2 wnAt w At
-5w
At
[cosil-E'
wnAt
(7)
w 2 At]!
f 1-5
The method of storing the values of ul(t-T),
ie, the amplitudes the previous revolution, was to use a specific number of points around the workpiece periphery and remember the current revolution values for use on the next revolution. For the first revolution it was assumed that there had been no vibration previously so that all the previous revolution values were set to zero. The vibration vas commenced by taking as initial The latter value is arbitrary as values ul(0)-O and u l ( 0 ) = l O . the process is linear in the sense that doubling ul(0) simply doubled all subsequent values of ul(t). As an example of what was obtained consider Figure 3 which shows the situation for w T 50 and i = 0.05 and Rb/k 0.1, 0.125 and 0.2. The lgtter correspond to stable, boundary of stability and unstable machining. on
-
-
It was found to be helpful in examining the build up of vibration to plot the maximum amplitude on each workpiece revolution against time and this is shown, in Figure 4 for the It should be noted that this curve three examples of Figure 3 need not be smooth because of beating effects and therefore may exhibit some degree of oscillation itself.
.
In order to investigate time varying parameters the particular parameter(s) are changed as required with each time increment At. Thus for example when the width of cut was increased linearly with rime the value of Rb/k was given by Rblk- (Rb/k)o
+
taper*At
(11)
TEST PROGRANHE The parameters investigated were width and structural characteristics. One of the dynamic performance tests (2) was designed around tapered workpieces so that the width of cut was continuously increased until chatter was detected by the The need to ascertain increases in the level of vibration. whether the rate of increase of width can lead to either a stabilising effect or give a false estimate of stability is therefore apparent. A l s o the increasing width occurs with a corresponding approach of the cutting tool towards the chuck so that the effective static stiffness will be increasing and there will be an increase in the preload applied to the Structure. The increase in preload is likely to further change the structural characteristics and this has been modelled by an increase in damping. The change in stiffness will, as is the case for the width, change the non-dimensional parameter Rb/k The main problem with the investigation concerned the difficulty of determing whether a varying parameter was affecting the onset of stability by simply the change in magnitude andlor was having a stabilising effect because of being time varying. The method adopted was to determine the limit of stability that is predicted from conventional theory and to check that for constant parameters this was the limit of stability, eg, as Then for example when one parameter wan shown in Figure 3. varied it was known at what time the process would be expected to become unstable and comparisons could be made.
As an example of this approa'ch the width parameter Rb/k was increased linearly with time for a value of rotational speed 150 and with 5 * 0.05. This value was corresponding to wnT chosen as being typical of a turning operation and avoiding the possibility of being in a speed range where there was the danger of moving between stability lobes. The rate of increase of Rb/k was chosen to give a reasonably rapid increase of width with workpiece revolutions. The starting value was chosen to be 0.11, which is slightly above the unstable width f o r this speed 0.108). The results obtained are shown in Figure 5 as (Rblk the variation of Rb/k and maximum amplitude on each revolution against workpiece revolutions. The initial conditions were again taken to be zero displacement but with an initial velocity. This example shows that before the amplitude of vibration begins to grow to any significant extent the width parameter Rb/k has increased to a value far in excess of the unstable width for constant width machining.
-
-
and
It is thus possible to calculate by superposition the values of As this ul(t+At) and ul(t+At) provided that P(t+At) is known. is not known an iterative method is used to find P(t+At) by making an initial estimate of ul(t+At) by the simple extrapolation ul(t+At)-
ul(t)
+ Ll(t)
At
This in turn allows an initial estimate of P(t+At) P(t+At)
-
Rb/k(ul(t+At-r)
-
ul(t+At))
(9)
since (10)
As a result it is possible to calculate a second estimate of This in turn allows a ul(t+At) using equations (5) and (7). second estimate of P(t+At) to be made using equation (10). It was found that four such iterations were sufficient to give stable convergence in the computation.
37 2
This example indicates either that the rate of increase of vibration is intrinsically slow or that varying the width parameter inhibits the growth of vibration. In Figure 4 a rapid growth rate was achieved for a value of Rb/k exceeding the Thus it was unstable width, ie, 0.2 as opposed to 0.125. thought likely that the varying width was having a stabilising effect. To confirm this a much more rapid increase of width was investigated, which would be extremely unlikely in practice, but would make any time varying effect more apparent. Figure 6 shows the values obtained for the same values of speed and damping. The width was varied so that over eight workpiece revolutions the value of Rb/k increased from zero to 2.5, ie approximately twenty five times the unstable width for constant width cutting. The initial conditions of displacement and velocity were kept the same for all the tests. The results obtained indicate that no instability would have been observed The dotted portion of the curve up to a value of Rb/k of 2.5. indicates the onset of numerical instability in the program. This could be delayed by using a smaller time interval, but it was considered that the effect had been established. It should be noted that the variation in Rb/k attributed to width could also have been the result of a variation in k but without a corresponding change in natural frequency.
t - =
Pigure 3.
Build up o f vibration for
I
i =
50 and
5
=
0.2
0.05 f o r varicus values of Rb/k
_." k
1c
0.14
0.13
0.12
0.11
Figure 5 .
Number of revs Figure 4 .
Variation of maximum applitude on successive revolutions
24
2.5
2C
2.0
16
1.5
12
1
8
0.5
4
ul(t)
0
2
4
6
8
10
Variation of maximum amplitude f o r rapidly increasing width
I 10
20
30
40
50
60
Xurnber o f revs
Number o f revs Figure 6.
Variation o f Rblk and maximum amplitude on successive revolutions
Figure 7.
Variation of Rb/k, 5 and maximum amplitude for successive revolutions
313
The other major parameter that was investigated was damping a5 it is possible for the structural characteristics to change because of changing cutting position and also changing preload resulting from increasing the width. Initially constant values of Rb/k and damping were investigated to observe the rate of increase of vibration. For each value of damping the value of Rb/k was found at the stability boundary and the rate of growth observed for a value exceeding this by a constant percentage. It was found that the growth rate was lower for the structures This can therefore result in an with the higher damping. overestimate of the stability boundary for increasing width tests when the structural damping is high because of the slow growth rate of the vibration.
In order to investigate this further for both varying width and damping it was arranged for the ratio of (Rb/k)/S to be increasing at a constant rate but with the rate of increaseldecrease of Rb/k being different. The value of (Rb/k)/& at the stability boundary had been found to be Thus the rate of approximately constant for a range of 5. change with respect t a the stability boundary was the same for each test but with a different rate of variation of damping. The results are shown in Figure 7. Each result was for the same speed and initial conditions. It can be seen that the rate of increase of vibration is significantly different for each case. It is also noticeable that the lower damping situation results in the more rapid increase in vibration. It should also be noted that on the thirteenth revolution when the values of Rb/k and 5. are the same for each test the boundary of stability has been exceeded by 20% and that by the time the vibration would have risen to noticeable levels the value of Rb/k would have a significant error compared to the stability boundary at that level of damping. It remains to determine whether the variation of damping in itself provides a stabilising effect. This is illustrated in Figure 8. A test was run for constant values of Rb/k = 0.11 and 5 0.03. This resulted in a rapid rise in vibration levels after the second revolution. It was then arranged for the damping to decay linearly and very rapidly from a value of 0.05 for the same constant value of Rblk. The result was that after six and one half revolutions the damping was at 0.03 as for the non varying case and comparisons could be made. The results show (Figure 8) that there is only a modest stabilising effect as the rate of growth is similar in both cases.
-
also there is a stabilising effect due to the variation with time. The main parameter which has not been investigated but which could be significant in practice is that of frequency. This is likely to have a similar effect to varying apeed, which is already known to have a stabilising effect when varied. The main conclusion to be drawn is that dynamic performance teats should be conducted with great care and confirmation of results should be obtained if possible under conditions where the main parameters are constant. Also It would appear that there are even more ways yet to be found for etabiliaing machining processes by the uae of time varying parameters. Finally the use of digital simulation of instability in machining processes allows an examination of effects which until now had been left alone. Even with the assumptions made in modelling the major effects which are thus determined and likely to be present to the same extent in the real situation. REFERENCES 1.
'Specifications and tests of metal cutting machine tools', M I S T report to Ministry of Technology, Conf. W E T , Feb. 1970
2.
Stone, B. J., 'The development of a dynamic performance test for lathes', 12th MTDR Conf. 1971
3.
CIRP collaborative project on the determination of dynamic cutting force coefficient.
4.
Lombard, J.. Mirski, F. and Beeaon, 8. D., 'Dynamic Some findings from Performance Tests for Machine Tools the experiences of CERHO and MTIRA', Annals of C.I.R.P., Vol 25/1/1976.
5.
Sexton, J. S. and Stone, B. J., 'A meyhod of reducing chatter in roll grinding', 1.C.M.E Conf. Melbourne 1980.
6.
Sexton, J. S. and Stone, B. J., 'The stability of machining with continuously vary spindle speed', Annals of C.I.R.P., Vol 271 I / 1978.
7.
Takemura, T. et al., 'Active suppression of chatter by programmed variation of spindle speed. I , Annals of C.I.R.P.. Yo1 2 3 / 1 / 1 9 7 4 .
8.
Inamura, T. and Sata, T., 'Stability analysis of cutting under varying spindle speed.', Annals of C.I.R.P., Vol
-
231 1/ 1974.
't
0.05
T. et al., 'Study for practical application of fluctuating speed cutting for regenerative chatter control.', Annal's of C.I.R.P., Vol 25/1/1977.
9.
Hoshi,
10.
Baylis, R. J. and Stone, B. J., 'The build up and decay of vibration during grinding', Annals of C.I.R.P., Vol 321 1/ 1983.
Number o f revs
Figure 8.
Variation of amplitude w i t h varyino, damping
CONCLUSIONS An examination has been made of the effect of some time varying parameters on the rate of build up of vibration durlng turning. It has been shown that varying the width of cut may have a considerable stabilising effect so that the width of cut at which vibration is observed to build up may be far in excess of the width. if kept constant at which the process would have become unstable. Further it has been shown that the higher the level of damping is then proportionately the slower the rate of build up. The effect of varying stiffness but not frequency, as is the case when the overhang in turning varies, is also likely to have a stabilising effect on two accounts. Increasing stiffness results in an increase in the stability boundary and
374
In man? dynamic performance tests on machine tools the cutting situation IS not one with constant parameters, whereas analyses of chatter assume this is the case. It is t h e r e f o r e important to determine the effect if any of the malor parameters v a r y , which i s often the case in practice. The ilarameters investisated are those o f width of cut and daniping and it is shown that it is possible to obtain performance test values which are significantly in error. compared to stradv state conditions. The main reasons f o r this are shown to be the rate of Srowth of vibration and the stsbilising effect of varyins width with time. most
INTRODUCTION FOK some time there has been an interest in determining the dynamic performance of machine tools in addition to their accuracy. This has resulted in various dynamic performance tests being proposed'>'. The main objective of these tests has been to determine the limit of stability so that a comparison can be made between different makes and thus a purchaser will have this additional and important information available. As the information will be used in such a critical way there has inevitably arisen the question of both the relevance and reliability of the information obtained from such tests. The major question to be answered concerns the decision whether to use cutting tests o r vibration excitation tests, that may be used for a comparison. The advocates of the former argue that the average person is more likely to purchase on the basis of a cutting test rather than an indirect test, the theoretical basis of which he does not comprehend. The latter argue that cutting tests involve so many other variables, that are difficult to control that a more reliable result is obtained by the indirect test. The results of these conflicting views resulted in attempts being made to reduce the variability of results obtained from cutting tests. For example one of th major variables is the workpiece material; attempts were made' to compare the tendency to chatter of nominally identical materials supplied from different sources. This work used a test structure which it was expected would give consistent vibration characteristics and was not sensitive to small changes in the cutting force direction. In the event it was found to be difficult to obtain consistent results between the different institutions involved in the This may have seemed to collaborative research exercise. provide good support f o r the advocates of vibration tests, but results were being obtained which cast doubts on the reliability of that approach. It is self evident that the final confirmation of the validity of a vibration test will have to be a cutting test and a comparison of many vibration tests and cutting tests4 did not show a good correlation. The major difficulty of vibration tests is that the characteristics obtained should be those that would have been effective under working conditions, ie. 'the operative response o r receptance' as it has been designated. This has been extremely difficult to achieve because of the requirements to have all the relevant components in motion and under preload, whilst at the same time applying equal and opposite forces to the tool and workpiece. These difficulties suggested to some workers the possibility of obtaining the required characteristics whilst cutting; this would involve the subsequent removal of the effect of the cutting process from the results. To the authors knowledge noone has claimed success using this method. This is mainly because of the difficulty of measuring the relevant forces and displacements on a working machine and also the lack of knowledge of the cutting process forces. The solution would therefore appear to lie with reliable cutting tests if they can be achieved. This would seem to require test workpieces made from the same material. Such tests were proposed some time ago for lathes and the repeatability of In general the results results was examined at some length. obtained could be repeated to within 15% and gave the width at which chatter commenced. Since that time however various papers have been publish d which indicate the significance of time varying parametersaBb and of course during a cutting test it is For example the inevitable that some parameters will vary. contact position between the tool and workpiece must change and this may cause the operative structural characteristics to change. It has therefore been considered necessary to investigate the effect of some time varying parameters on the onset of chatter and the method chosen has been to use This has the advantage of simulation on a digital computer. removing the variability already discussed but has the
disadvantage of requiring a model of the machining operation which will be somewhat simplified. However it is argued that any major effects found would be present to some extent in the real situation The time varying parameters that will be discussed are those of width and structural response.
.
PREVIOUS WORK
The main interest in time varying parameters has centred on the possible deliberate variation of such parameters to provide an increase in stability. The parameter which has received the Thus by greatest attention would appear to be speed. continuously changing speed, ie cyclically, so mprovements have been obtained. These investigations6sr8?8.6 indicated improvements for turning of the order of 30% and there were indications that greater improv ments might be obtained in grinding. Also a report was made' of the succes8 in grinding of eliminating chatter in a roll grinding operation by having centres of different stiffness so that there waa a continually varying response. These results indicate that the onset of chatter can be affected by time varying parameters and thus any dynamic performance test which involves ti= varying parameters may not give a true indication of performance.
TXEORY Most of the theoretical examination of the onset of chatter has used the condition at the stability boundary as the basis for any analysis. This is. however, a condition which is not achievable in practice but nevertheless is useful in recognising the major parameters and does indicate the width of cut at which the process will begin to go unstable. The actual manner in which the instability grows depends on the initial disturbance and the extent to which the limiting width la exceeded. It is therefore necessary with a time varying situation to look at the transient behaviour of the process as a steady state solution is not available. The method used in this paper is similar to that described i paper on the build up and decay of vibration in grindingq0: However, as grinding is the most complex operation to analyse as it involves feedback on both the grinding wheel and workpiece and because the dynamic performance testing of lathes is an important subject. the build up of vibration in turning wag examined. The model used is shown in Figure 1, and fndicatea that the representation of the structure was with the conventional spring, mass, damper model. This is a great simplification but satisfactorily represents a structure with a predominant rode.
k
Figure 1. Dynamic model of lathe structure
Annals of the ClRP Vol. 34/1/1985
371
The derivation of the relevant equations and the method of digital simulation will first be described for constant parameters and then subsequently be modified for the inclusion of ti= varying effects. The equation of motion resulting from the instantaneous cutting force P(t) is given by
+
mu 1 (t)
+ kul(t)
cGl(t)
-
(1)
-P(t)
The cutting force depends on the instantaneous depth of cut so that for a constant width of cut, h P(t)
-
Rb(ul(t-r)
-
(2)
ul(t))
R is the cutting force coefficient, assumed constant and ul(t-T) indicates the surface left one revolution earlier, ie, at time (t-T) where T is the time for one revolution of the workpiece. It is useful to non-dimensionalise the equations by dividing dk/m and 5 cl2dmk. Thus throughout by k and substituting wn
-
-
and
(4) If a time increment At i s considered from the situation at some time t then it is necessary to find ul(t+At), uI(t+At) and P(t+At) knowing ul(t), ul(t) and P(t). This is achieved as follows. Consider Figure 2 which shows the value of P at t and t+At. Over the time interval At the structure is subjected to a force which is trapezoidal in form and has. at the start of this time increment an initial displacement ul(t) and an initial velocity ul(t).
Figure 2.
Trapezoidal force/time eler&ent
It can be shown that the effect of the initial conditions ul(t) and ul(t) is to produce u,(t+AC)
-
(il(t)/u"+bl(r))
-Cu,At e
unbC +
[U,(t)EOS/l-S2
."At]
(5)
onbt]
(6)
s i n f I-f2
1'1-2
and (?+At)
-
( Ci (t ) /u,+U
-WnAt e
[ul(t) e 0 s 4 l - C ~ u2At
+
l(c 1)
sidl-6'
4 1-2
At the same t i m e the trapezoidal force produces
ul(t+At)
+
+
-
(P(t+At)-P(t))
25 {1--+------wnAt
k
(zr2-1) sinf 14,'~ f1-5 L2sinfl-52
A]]
+
{l
-e
e-SwnAt [25 cosfl-S2 wnAt w At
-5w
At
[cosil-E'
wnAt
(7)
w 2 At]!
f 1-5
The method of storing the values of ul(t-T),
ie, the amplitudes the previous revolution, was to use a specific number of points around the workpiece periphery and remember the current revolution values for use on the next revolution. For the first revolution it was assumed that there had been no vibration previously so that all the previous revolution values were set to zero. The vibration vas commenced by taking as initial The latter value is arbitrary as values ul(0)-O and u l ( 0 ) = l O . the process is linear in the sense that doubling ul(0) simply doubled all subsequent values of ul(t). As an example of what was obtained consider Figure 3 which shows the situation for w T 50 and i = 0.05 and Rb/k 0.1, 0.125 and 0.2. The lgtter correspond to stable, boundary of stability and unstable machining. on
-
-
It was found to be helpful in examining the build up of vibration to plot the maximum amplitude on each workpiece revolution against time and this is shown, in Figure 4 for the It should be noted that this curve three examples of Figure 3 need not be smooth because of beating effects and therefore may exhibit some degree of oscillation itself.
.
In order to investigate time varying parameters the particular parameter(s) are changed as required with each time increment At. Thus for example when the width of cut was increased linearly with rime the value of Rb/k was given by Rblk- (Rb/k)o
+
taper*At
(11)
TEST PROGRANHE The parameters investigated were width and structural characteristics. One of the dynamic performance tests (2) was designed around tapered workpieces so that the width of cut was continuously increased until chatter was detected by the The need to ascertain increases in the level of vibration. whether the rate of increase of width can lead to either a stabilising effect or give a false estimate of stability is therefore apparent. A l s o the increasing width occurs with a corresponding approach of the cutting tool towards the chuck so that the effective static stiffness will be increasing and there will be an increase in the preload applied to the Structure. The increase in preload is likely to further change the structural characteristics and this has been modelled by an increase in damping. The change in stiffness will, as is the case for the width, change the non-dimensional parameter Rb/k The main problem with the investigation concerned the difficulty of determing whether a varying parameter was affecting the onset of stability by simply the change in magnitude andlor was having a stabilising effect because of being time varying. The method adopted was to determine the limit of stability that is predicted from conventional theory and to check that for constant parameters this was the limit of stability, eg, as Then for example when one parameter wan shown in Figure 3. varied it was known at what time the process would be expected to become unstable and comparisons could be made.
As an example of this approa'ch the width parameter Rb/k was increased linearly with time for a value of rotational speed 150 and with 5 * 0.05. This value was corresponding to wnT chosen as being typical of a turning operation and avoiding the possibility of being in a speed range where there was the danger of moving between stability lobes. The rate of increase of Rb/k was chosen to give a reasonably rapid increase of width with workpiece revolutions. The starting value was chosen to be 0.11, which is slightly above the unstable width f o r this speed 0.108). The results obtained are shown in Figure 5 as (Rblk the variation of Rb/k and maximum amplitude on each revolution against workpiece revolutions. The initial conditions were again taken to be zero displacement but with an initial velocity. This example shows that before the amplitude of vibration begins to grow to any significant extent the width parameter Rb/k has increased to a value far in excess of the unstable width for constant width machining.
-
-
and
It is thus possible to calculate by superposition the values of As this ul(t+At) and ul(t+At) provided that P(t+At) is known. is not known an iterative method is used to find P(t+At) by making an initial estimate of ul(t+At) by the simple extrapolation ul(t+At)-
ul(t)
+ Ll(t)
At
This in turn allows an initial estimate of P(t+At) P(t+At)
-
Rb/k(ul(t+At-r)
-
ul(t+At))
(9)
since (10)
As a result it is possible to calculate a second estimate of This in turn allows a ul(t+At) using equations (5) and (7). second estimate of P(t+At) to be made using equation (10). It was found that four such iterations were sufficient to give stable convergence in the computation.
37 2
This example indicates either that the rate of increase of vibration is intrinsically slow or that varying the width parameter inhibits the growth of vibration. In Figure 4 a rapid growth rate was achieved for a value of Rb/k exceeding the Thus it was unstable width, ie, 0.2 as opposed to 0.125. thought likely that the varying width was having a stabilising effect. To confirm this a much more rapid increase of width was investigated, which would be extremely unlikely in practice, but would make any time varying effect more apparent. Figure 6 shows the values obtained for the same values of speed and damping. The width was varied so that over eight workpiece revolutions the value of Rb/k increased from zero to 2.5, ie approximately twenty five times the unstable width for constant width cutting. The initial conditions of displacement and velocity were kept the same for all the tests. The results obtained indicate that no instability would have been observed The dotted portion of the curve up to a value of Rb/k of 2.5. indicates the onset of numerical instability in the program. This could be delayed by using a smaller time interval, but it was considered that the effect had been established. It should be noted that the variation in Rb/k attributed to width could also have been the result of a variation in k but without a corresponding change in natural frequency.
t - =
Pigure 3.
Build up o f vibration for
I
i =
50 and
5
=
0.2
0.05 f o r varicus values of Rb/k
_." k
1c
0.14
0.13
0.12
0.11
Figure 5 .
Number of revs Figure 4 .
Variation of maximum applitude on successive revolutions
24
2.5
2C
2.0
16
1.5
12
1
8
0.5
4
ul(t)
0
2
4
6
8
10
Variation of maximum amplitude f o r rapidly increasing width
I 10
20
30
40
50
60
Xurnber o f revs
Number o f revs Figure 6.
Variation o f Rblk and maximum amplitude on successive revolutions
Figure 7.
Variation of Rb/k, 5 and maximum amplitude for successive revolutions
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The other major parameter that was investigated was damping a5 it is possible for the structural characteristics to change because of changing cutting position and also changing preload resulting from increasing the width. Initially constant values of Rb/k and damping were investigated to observe the rate of increase of vibration. For each value of damping the value of Rb/k was found at the stability boundary and the rate of growth observed for a value exceeding this by a constant percentage. It was found that the growth rate was lower for the structures This can therefore result in an with the higher damping. overestimate of the stability boundary for increasing width tests when the structural damping is high because of the slow growth rate of the vibration.
In order to investigate this further for both varying width and damping it was arranged for the ratio of (Rb/k)/S to be increasing at a constant rate but with the rate of increaseldecrease of Rb/k being different. The value of (Rb/k)/& at the stability boundary had been found to be Thus the rate of approximately constant for a range of 5. change with respect t a the stability boundary was the same for each test but with a different rate of variation of damping. The results are shown in Figure 7. Each result was for the same speed and initial conditions. It can be seen that the rate of increase of vibration is significantly different for each case. It is also noticeable that the lower damping situation results in the more rapid increase in vibration. It should also be noted that on the thirteenth revolution when the values of Rb/k and 5. are the same for each test the boundary of stability has been exceeded by 20% and that by the time the vibration would have risen to noticeable levels the value of Rb/k would have a significant error compared to the stability boundary at that level of damping. It remains to determine whether the variation of damping in itself provides a stabilising effect. This is illustrated in Figure 8. A test was run for constant values of Rb/k = 0.11 and 5 0.03. This resulted in a rapid rise in vibration levels after the second revolution. It was then arranged for the damping to decay linearly and very rapidly from a value of 0.05 for the same constant value of Rblk. The result was that after six and one half revolutions the damping was at 0.03 as for the non varying case and comparisons could be made. The results show (Figure 8) that there is only a modest stabilising effect as the rate of growth is similar in both cases.
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also there is a stabilising effect due to the variation with time. The main parameter which has not been investigated but which could be significant in practice is that of frequency. This is likely to have a similar effect to varying apeed, which is already known to have a stabilising effect when varied. The main conclusion to be drawn is that dynamic performance teats should be conducted with great care and confirmation of results should be obtained if possible under conditions where the main parameters are constant. Also It would appear that there are even more ways yet to be found for etabiliaing machining processes by the uae of time varying parameters. Finally the use of digital simulation of instability in machining processes allows an examination of effects which until now had been left alone. Even with the assumptions made in modelling the major effects which are thus determined and likely to be present to the same extent in the real situation. REFERENCES 1.
'Specifications and tests of metal cutting machine tools', M I S T report to Ministry of Technology, Conf. W E T , Feb. 1970
2.
Stone, B. J., 'The development of a dynamic performance test for lathes', 12th MTDR Conf. 1971
3.
CIRP collaborative project on the determination of dynamic cutting force coefficient.
4.
Lombard, J.. Mirski, F. and Beeaon, 8. D., 'Dynamic Some findings from Performance Tests for Machine Tools the experiences of CERHO and MTIRA', Annals of C.I.R.P., Vol 25/1/1976.
5.
Sexton, J. S. and Stone, B. J., 'A meyhod of reducing chatter in roll grinding', 1.C.M.E Conf. Melbourne 1980.
6.
Sexton, J. S. and Stone, B. J., 'The stability of machining with continuously vary spindle speed', Annals of C.I.R.P., Vol 271 I / 1978.
7.
Takemura, T. et al., 'Active suppression of chatter by programmed variation of spindle speed. I , Annals of C.I.R.P.. Yo1 2 3 / 1 / 1 9 7 4 .
8.
Inamura, T. and Sata, T., 'Stability analysis of cutting under varying spindle speed.', Annals of C.I.R.P., Vol
-
231 1/ 1974.
't
0.05
T. et al., 'Study for practical application of fluctuating speed cutting for regenerative chatter control.', Annal's of C.I.R.P., Vol 25/1/1977.
9.
Hoshi,
10.
Baylis, R. J. and Stone, B. J., 'The build up and decay of vibration during grinding', Annals of C.I.R.P., Vol 321 1/ 1983.
Number o f revs
Figure 8.
Variation of amplitude w i t h varyino, damping
CONCLUSIONS An examination has been made of the effect of some time varying parameters on the rate of build up of vibration durlng turning. It has been shown that varying the width of cut may have a considerable stabilising effect so that the width of cut at which vibration is observed to build up may be far in excess of the width. if kept constant at which the process would have become unstable. Further it has been shown that the higher the level of damping is then proportionately the slower the rate of build up. The effect of varying stiffness but not frequency, as is the case when the overhang in turning varies, is also likely to have a stabilising effect on two accounts. Increasing stiffness results in an increase in the stability boundary and
374