When the tool is not in contact with the workpiece, i.e. on the time interval (4), Eq. (2) applies; there is a well-known exact solution, and this solution is clearly stable, because the oscillations are exponentially damped. The situation is not so simple on the time interval (3) when Eq. (1) applies. What we have done is to approximate the motion on the time interval (3) in the case that p << I. Here, we will summarize the results; for more details, see Davies, et al. 12•13
One of the key ideas in our analysis is to borrow a technique from nonlinear dynamics theory, and treat the time evolution of the interrupted system by means of a two- dimensional discrete mapping, that takes the position and velocity ( x i-P v i-J ) of the tool at time ti-J = (i -1)r to the new position and velocity (xi, vi) at time ti = ir. We do this by employing a two-step "subcycling" approach. Using the exact solution of (2), we compute the
- E
- Es •
mapping
HIGHLY INTERRUPTED MACHINING
- E E 5
-lit
, / "
.•. / ' ' /
,...,, .. \ _,...., ... ,,,. .... ãã '-..
0+-~1ãã-=ã=ã '---~~~~~~~~~~~~~~~-.---~~~
25 50 75
O(krpm)
Figare 5. Stability lobe plot for p=I and p=0.05.
100
157
evolution of the system from time ti-I = (i - l)r to the time just before impact, f;-p = (i - p )r, where the position and velocity of the tool are now given by (xi-p, V;-p ).
The next step is to approximate the evolution of the system according to Eq. (1), assuming that pô l. Intuitively, because the impact of the tool and the workpiece takes place over a very small time interval, we would expect to leading order in p that the position of the tool does not change, but that there is a discontinuous jump in the tool velocity. If we integrate Eq. (1) over the time interval (3), and make the assumption that pw = 71, a constant similarity parameter, as p-!. 0, then we get the following result.
KrT/
Xi =Xi-p• Vi =Vi-p - mQ (j + X; -Xi-1).
Because (x;-p, vi-p) are known from the first "subcycle", we may conclude that there is a matrix B such that the discrete time evolution of the system is detemined by the linear mapping
(5)
158 M.A. DA VIES ET AL.
It can be shown that the matrix 1-B has an inverse, where I is the identity matrix. As a consequence, the mapping (5) has the fixed point
( :)' ~-(1 - B) -'[ K:~ f) (6)
The stability of the cutting operation thus reduces to the study of the stability of the mapping (5) with respect to the fixed point (6), which is a simple two-dimensional eigenvalue problem. From this, it is straightforward to determine the stability lobes corresponding to the neutral stability boundaries in the w vs. fl plane. The analysis shows that, in addition to most stable cutting speeds that occur when the period of the spindle (or of the tool tooth engagement) is an integer multiple of the period of oscillation of the tool (2), new local most stable cutting speeds also occur when the period of the spindle is equal to an odd multiple of one half of the tool oscillation period; see Fig. 6. Comparing this figure with Fig. 5, we see that our approximate analysis of the model gives results that are in agreement with the numerical simulations.
4 3.5 3
2.5
e 2
i E 1.5
1 0.5
5 10
fl(krpm)
15
Figure 6. Analytically determined curves of neutral stability for very low radial immersion model.
20
DISCUSSION We have pr immersion milling.
machining, the most frequency Ct>d of the flexible mode of the same location at the time spent in the cut same location at the milling that COIT
O=mdl(Nn), n=
addition, however, rotation period, and location at spindle oscillation period of the lobe splitting thllt speeds at which rel ã vibrations. Recent found in the paper by
REFERENCES
l.
2.
3.
4.
5.
6.
7.
8.
9.
IO.
11.
12.
HIGHLY INTERRUPTED MACHINING 159
DISCUSSION AND CONCLUDING REMARKS
We have presented an outline of a new theory of the stability of very low radial immersion milling. Intuitively, the results can be explained as follows. For full immersion machining, the most stable cutting speeds correspond to integer fractions of the natural frequency (J) d of the damped harmonic oscillator (2) that models the dynamics of the most flexible mode of the machine-tool structure, because at these speeds the tool returns to the same location at the start of each tool tooth engagement. For intermittent machining, as the time spent in the cut per revolution of the tool becomes very small, the tool can return to the same location at the start of each cut in exactly the same way, i.e., at cutting speeds for milling that correspond to integral multiples of the natural tool oscillation period, 0 = md l(Nn), n = 1,2,3, ... , where N denotes the number of flutes on the tool. In addition, however, because the tool is not in contact with the workpiece for most of each rotation period, and thus acts as an impact oscillator, it can also return to the same cutting location at spindle speeds that correspond to odd half-integer multiples of the natural damped oscillation period of the tool, i.e. 0 = 2m d I[ N(2n -1 )], n = 1,2,3, .... This accounts for the lobe splitting that occurs asp ~ 0 , and thus also for the appearance of new spindle speeds at which relatively large depths of cut can be used without introducing unstable chatter vibrations. Recent experimental results that support the conclusions presented here may be found in the paper by Davies et al. 13
REFERENCES
I. S.A. Tobias, Machine-Tool Vibration (Blackie & Son, Glasgow, 1965).
2. J. Tlusty, Dynamics of high-speed milling, J. Eng.for Industry 108, 59.(,7 (1986).
3. F. C. Moon, editor, Dynamics and Chaos in Manufacturing Processes (John Wiley & Sons, Inc., New York, 1998).
4. F. C. Moon and M. A Johnson, in: Dynamics and chaos in manufacturing processes, edited by F. C. Moon (John Wiley & Sons, Inc., New York, 1998), pp. 3-32.
5. G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Functions, Pitman Research Notes in Mathematics, vol. 210 (Longman Scientific and Technical, London, 1989).
6. J. Tlusty, S. Smith, and W. Winfough, Techniques for use of long slender endmills in high speed machining, CIRP Annals 45(1), 583-589 (1996).
7. M. A. Davies, J. Pratt, A. J. Schaul, and J. Bryan, On the dynamics of high-speed milling with long, slender endmills, CJRP Annals 41( I), 55.()0 ( 1998).
8. S. Smith, W. Winfough, and J. Halley, The effect of tool length on stable metal removal rate in high speed milling, CIRP Annals 47( I), 307-310 ( 1998).
9. J. Halley, A. Helvey, S. Smith, and W. Winfough, The impact of high-speed machining on the design and filbrication of aircraft components. Proceedings of the 1999 ASME Design Engineering and Technical Conferences, Las Vegas, Nevada, September 12-16, 1999, DETC99-VIB8057.
10. F. W. Taylor, On the art of cutting metals, Trans. ASME 28, 31-350 ( 1907).
11. S. Smith, T. Jacobs, and J. Halley, The effect of drawbar force on metal removal rate in milling. CIRP Annals 48( I), 293-296 (1999).
12. M. A. Davies, J. Pratt, B. S. Dutterer, and T. J. Burns, The stability of low radial immersion milling, CIRP Annals 49(1), 37-40 (2000).
160 M.A. DAVIES ET AL.
13. M. A. Davies, J. Pratt, B. S. Dutterer, and T. J. Bums, Stability prediction for low radial immersion milling, submitted to J. Man. Sci. and Eng. (2000).
14. T. lnsperger and G. Stepan, Vibration frequencies in high-speed milling processes, submitted to J. Man. Sci.
and Eng. (2000).
Janez Gradi8ek, Edvard Klocke*
1. INTRODUCTION