Solid State Phenomena Vol. 164 (2010) pp 285-290 © (2010) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.164.285
Simulation and Experiments of High Speed Machining Vibration Monitoring with Variable Spindle Velocity Krzysztof J. Kalinski1, Marek Galewski1 1
Gdansk University of Technology, Faculty of Mechanical Engineering, G. Narutowicza 11/12, 80-233 Gdansk, Poland
[email protected],
[email protected]
Keywords: High Speed Machining, ball-end milling, vibration control, simulation, experiments, chatter
Abstract. The paper is devoted to vibration monitoring of rotating tools in modern milling machines. Dynamic analysis of slender ball-end milling process was performed and dynamics of the controlled structure was described. Instantaneous change in the spindle speed is applied in order to reduce vibration level. The method of vibration monitoring by means of spindle speed optimal-linear control was developed and implemented with success. Vibration monitoring during high speed milling was performed on the basis of results of computer simulation. These results were verified during experimental investigation on the Alcera Gambin 120CR milling machine. Introduction Contour milling using slender ball-end mills due to the technological reasons is a frequent case of milling in contemporary production centers. It is usually finishing work so the depth of cut can be very small and it cannot be further reduced. Then tool-workpiece relative vibration plays a principal role. Due to existence of certain conditions, it may lead to a loss of stability and generate selfexcited chatter vibration [1]. There are many different methods for the chatter vibration monitoring which purpose is to reduce vibration level. Amongst them, the methods that utilize spindle speed variation appear to be successful [2-4]. In this paper the former method of vibration monitoring by using spindle speed optimal-linear control [5] was developed. The novelty is that the simulation results of milling of only one material allow us to predict real behavior concerned with the milling of various materials. Former applications required tuning-up model parameters for every material. Results of simulations and experimental investigation performed on the Alcera Gambin 120CR milling machine demonstrated good efficiency of the vibration monitoring. Cutting Process Dynamics and Tool-workpiece Vibration Monitoring Using Spindle Speed Optimal-linear Control Dynamic analysis of a slender ball-end milling process was performed based on the following assumptions [5, 6]. - The spindle together with the tool fixed in the holder and the table with the workpiece are separated from the machine tool whole structure. - Only flexibility of the tool is considered. - Position of symmetry axis of the tool, relatively to the feed speed vf, suits to the pulling milling, in order to prevent the top-ball-end milling. - Coupling elements (CEs) are applied for modeling cutting process dynamics [5, 6]. The CEs are defined by transfer functions between geometry and kinematic variables, and instantaneous cutting forces. Tool edges, having contact with a material, are called “active”. - The effect of the first pass of the edge along cutting layer is idealized as proportional to feedback, and the effect of multiple passes is idealized additionally as delayed feedback. Existence of the latter is a main reason of the chatter vibration development.
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The reasons above imply a simplified model of the milling process (Figs. 1-2) being purposed for computer simulation.
Fig. 1. A scheme of ball-end milling
Fig. 2. Discrete model of the process in which the ball-end mill has two cutting edges
For conventional contact point of tool edge and workpiece (i.e. CE no. l), proportional model of the cutting dynamics is included [1, 5, 6]. Thus, we get: k a [h (t ) − ∆hl (t ) + ∆hl (t - τ l )] Fyl1 (t ) = dl p Dl 0 µ k a [h (t ) − ∆hl (t ) + ∆hl (t - τ l )] Fyl 2 (t ) = l dl p Dl 0 Fyl 3 (t ) ≡ 0,
for
hDl (t ) − ∆hl (t ) + ∆hl (t - τ l ) > 0,
for
hDl (t ) − ∆hl (t ) + ∆hl (t - τ l ) ≤ 0,
for
hDl (t ) − ∆hl (t ) + ∆hl (t - τ l ) > 0,
for
hDl (t ) − ∆hl (t ) + ∆hl (t - τ l ) ≤ 0,
(1) (2) (3)
where: kdl is an average dynamic specific cutting pressure, hDl(t) is desired cutting layer thickness; hDl (t) ≅ fz cosϕl(t) ∆hl(.) is dynamic change in cutting layer thickness, µl is cutting force ratio (a quotient of forces Fyl2 and Fyl1), τl is time-delay between the same angular position (ϕl) of CE no. l and CE no. l-1, fz is feed per tooth. Zero values of the components concern an effect of the tool-workpiece contact loss. Let us consider a milling process being performed at time-varying spindle speed n=n(t). In this case, time-delay τl for edge no. l becomes function of instantaneous spindle speed n, i.e. τl = τl(n). It is shown [5, 6], that after transformation of displacements to co-ordinate system x1e, x2e, x3e, milling at changing spindle speed can be described by dynamic equation of controlled system, i.e.:
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+ Lq + K * (t )q = f * + B u u , Mq
(4)
where: il
K (t ) = K + ∑ TlT (t )D Pl Tl (t ) , *
l =1
il
il
f * = ∑ TlT (t )Fl0 (t ) + ∑ TlT (t )DOl ∆w l (t − τ l ) , l =1
l =1
∆t l (t − τ l ) ⋅ − Bu (t ) = ∑ TlT (t ) DOl ∆w , ( ) n ϕ l =1 l q is vector of generalized displacements of the system, M, L, K are matrices of inertia, damping and stiffness of decoupled system (i.e. the slender ball-end mill fixed in the holder, but without cutting process consideration), Bu, u are control matrix and control command of a complete system, 0 Fl (t ) is vector of desired forces of CE no. l. The desired forces refer only to kinematic and geometric conditions, without consideration of dynamic interaction; DPl, DOl are matrices of proportional and delayed feedback of CE no. l, ∆w l (t − τ l ) is vector of deflections of CE no. l for time-instant t-τl, il is number of “active” CEs. The matrix of transformation Tl(t) is time-dependent because cutting edges change their positions due to tool rotation and workpiece feed. Now we define energy performance index, which considers time-varying kinetic and potential energy, with respect to trajectory of desired motion (determined by vector of generalized displacements q and velocities q ), and also – instantaneous energy of control command [5, 6]: il
J (t ) =
1 (q − q )T Q1M (q − q ) + 1 (q − q )T Q 2K * (q − q ) + 1 uT Ru , 2 2 2
(5)
where: Q1, Q2, R are matrices of dimensionless weighing coefficients selected arbitrary. Application of the variational calculus allows us to determine optimal control command [5, 6], which minimizes the energy performance index (5) in a time-domain, i.e.: t
[
]
u = −R −1 ∫ BTu (τ ) 0 ΦT (t ,τ )dτ t0
{T M Q q + 12 T (K Q T 1
T
T 1
T 2
*
T 2
)(
+ Q 2 K *T q − K *−1f 0
)} ,
(6)
where:
Φ(t, t 0 ) is fundamental matrix, which is solution to homogeneous equation x = A(t )x, x(t0 ) = I , − M −1L − M −1K * (t ) is state matrix of the system, A(t ) = , 0 I T x is state vector, x = [q T qT ] ; x(t0) is initial state vector, T1, T2 are matrices of transformation, T1 = [I 0] , T2 = [0 I ] . Due to identity u ≡ δn , the optimal control command is an instantaneous change δn in the spindle speed n(t).
A(t)
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Direct application of relationship (6) causes that instantaneous values of the spindle speed are monotonously growing to infinity. Thus, the optimal control command (6) cannot provide any way to reduce spindle speed, and no alternative criterion of optimality is selected at the moment. In order to avoid this problem, an idea of the optimal-linear spindle speed control has been incorporated. During the spindle speed rise period the optimal control is applied. After reaching allowed maximum value nmax, the speed is linearly reduced to nominal value n0, in accordance with the following relationship: t − tj (7) n(t ) = nmax − (nmax − n0 ) ⋅ , t j ≤ t ≤ t j + TO , TF where: TF is falling time, tj is time instant of the beginning of fall no. j. Vibration Monitoring Procedure Vibration monitoring procedure consists of two main stages. At the first stage, parameters of simulation model are tuned-up, but as opposed to previous approaches – for milling of only one material. Experimental data for milling at constant speed is used as a reference for simulation results. Once simulation results are satisfactory, (i.e. comparable values of displacement RMS and of maximum amplitude qch in the chatter spectrum, between simulation and experimental data), one can proceed to the second stage. In a series of simulations a set of control parameters like Q1, Q2, R and TF is chosen. As a result a program, which implementation is intended to reduce the chatter vibration during milling of various materials, is generated. A scheme of a complete procedure is illustrated in Fig. 3.
Fig. 3. A scheme of vibration monitoring procedure. Experimental Investigation In order to perform experimental investigation, a special stand was built which allows us to measure tool rotation speed and displacement of the rotating tool in two orthogonal axes, lateral to the tool. Experiments were conducted on the Alcera Gambin 120CR milling machine equipped with the S2M electrospindle. High speed machining of aluminium alloy EN AW-2017A, carbon steel C45, and bronze CC331G were performed using slender ball-end mill (160×16mm). Simulations demonstrated that variable spindle speed that switches every 0.3 – 0.5 s allows us to reduce chatter vibration significantly. Two variable spindle speed programs were chosen for performance: V04 –
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with TF = 0.4 s and V05 – with TF = 0.5 s. Speed values were varying between 15000 and 16500 rot/min. Experiments for constant speed 15000 and 16500 rot/min were performed as well. Some results for milling aluminium alloy are presented. In case of constant spindle speed 16500 rot/min, although vibration decreased after 2 seconds (because of shorter length of the workpiece, Fig. 4a), milling with depth ap=0.3 mm yields a strong chatter vibration (Fig. 4b). The benefit of variable spindle speed program is that reduction of maximum amplitude of chatter vibration in the spectrum was about 62% in comparison with milling at constant spindle speed 16500 rot/min (Figs 4b and 5c). The RMS value of tool displacement was reduced by about 14% (Figs 4a and 5b). Fig. 6 shows that similar results were obtained for bronze and steel. Summary The method of vibration monitoring by using spindle speed optimal-linear control was developed with success. Vibration monitoring was performed by means of the proposed method on the basis of results of computer simulation. The latter consisted in generation of suitable spindle speed program. Results of experimental investigation during high speed milling on machine Alcera Gambin 120CR confirmed a very good efficiency of the vibration monitoring. Peak-values of resonant chatter vibration were significantly reduced. The RMS values are decreased as well. It should be noted that successful results with respect to chatter suppression are expected due to the small investment. Except for experimental instrumentation, only the standard control system of the CNC milling machine is employed. This low-cost approach can be easily applied in industrial environment if the machine control system is capable of changing spindle speed during milling process. Acknowledgments The research was supported by The Polish Ministry of Science and Education, Grant No. 5 T07C 037 25 and subjective subvention (No. 155/E-359/SPB/Cooperation with FP UE/DIE 485/2004). Experimental verification was performed thanks to Ecole Nationale d’Ingenieurs de Metz, France.
Fig. 4. Aluminium alloy (EN AW-20017A) face milling with constant spindle speed, n = 16’500 rev/min, ap = 0.3 mm; (a) tool displacement along the xe2 axis and (b) relevant amplitude spectrum. Time period for computing RMS and FFT marked as a light line at (a).
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Fig. 5. Aluminium alloy (EN AW-20017A) face milling with variable spindle speed programme V04, ap = 0.3 mm; (a) spindle speed, (b) tool displacement along the xe2 axis and (c) relevant amplitude spectrum. Time period for computing RMS and FFT marked as a light line at (b).
Fig. 6. Efficiency of surveillance for a face ball-end milling, (a) the RMS of displacement and (b) maximum chatter amplitude in spectrum. 100% is a reference level for chosen constant spindle speed programme. References [1] J. Tomkow: Vibrostability of machine tools, The Scientific and Technical Publications; Warsaw (1997) [2] K. Jemielniak, A. Widota, Int. J. Mach. Tool Des. and Res., vol. 24/3 (1984), p. 207–214 [3] S. C .Lin, R. E. DeVor, S. G. Kapoor: Trans. of the ASME Journal of Engineering for Industry, Vol. 112 (1990), p. 1–11 [4] E. Soliman, F. Ismail: Int. J. Mach. Tools Manufact., Vol. 37/3 (1997), p. 355–369 [5] M. Galewski, K. Kalinski: Vibration surveillance during high speed slender milling at variable spindle speed, The Publication of GUT, Gdansk, Poland (2009) (in Polish) [6] K. Kalinski: Vibration surveillance of mechanical systems which are idealised discretely. The Publication of GUT, Gdansk (2001) (in Polish)
