Active in-process chatter control E.J.J. Doppenberg*, R.P.H. Faassen**, N. van de Wouw**, J.A.J. Oosterling*, H. Nijmeijer**, *TNO Science and Industry, P.O. Box 155, 2600 AD Delft, The Netherlands 1 **Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Ed.doppenberg@tno.nl Abstract: A control stategy for high speed milling has been developed to prevent the onset of chatter, and at the same time to increase machining efficiency and surface finish quality. The main objective of this approach is to detect the onset and adjust the spindle speed in such a manner, that the machining process is forced into a region of stable operation. In the paper, this approach - active in-process chatter control - is described and illustrated in simulations. Keywords: Active chatter control, time-evolving regenerative-process-database, machining simulation. 1. INTRODUCTION To some degree, all milling tools exhibit vibrations as their teeth cut into the work piece, leaving a wavy surface. If the waves of the succeeding cuts match, the chip thickness is constant and the milling process is stable. When the waves are out of synchronization, the chip thickness varies, and the start of the minute regenerative vibrations can grow to undesired high vibration levels, which can excess to the phenomenon chatter. This results in heavy vibrations of the entire cutting system - from the cutting edge geometry to the machine tool spindle - which degrades the machining process. Besides the degradation of for instance the surface quality, life time of the tool, the process reliability and the life span of the machine tool spindle will be negatively affected. The key to prevent chatter is to maintain the synchronization of succeeding wavy surfaces, keeping the chip thickness constant. Machine designers use passive strategies to prevent regenerative vibrations by absorbing the vibration energy, or by redirecting the vibration energy [Semercigil and Chen, 2002;Tarng et al, 2000]. A trend in the design approach to control chatter behaviour, is to optimize the machine s dynamic behaviour at the design process, maximizing stiffness and optimizing damping of the entire cutting system [Zhang and Sims, 2005; Kyung and Lee, 2003]. The objective of the abovementioned strategies is to minimize the energy feedback of the unwanted regenerative vibrations to the cutting process for a vast range of process parameters. In a schematic diagram, depicted in Figure 1, the ideal, but unfeasible situation would be the total elimination of the feedback path to the 1 This research is supported in part by Jabro Tools, Philips ETG and Somatech. http://www.tno.nl/chattercontrol.
process, in which case the dynamic chip thickness (h dyn (t)) would be zero. From a control-engineering viewpoint, this design strategy focuses on the control (indirectly) of the properties of the regenerative vibration process, by tuning the machine dynamics at the design process [Altintas and Cao, 2005]. On the other hand, high-speed milling increasingly demands a very high response, high feed rates, which requires machines with relatively short response time, and with constrained energy properties. Therefore the trend in high speed milling machines is to design a relatively lightweight cutting tool system with increased cutting power, which will constrain the machine developer in the design process even more. Thus, the set of requirements for robust cutting without chatter and high response are becoming more opposed, making machine designing a time consuming, critical path in the machine development. To ease the inevitable design trade offs in the ever-demanding machine development, the indirect control (at the design process) of the regenerative vibration process (to prevent chatter) will increasingly require complementary control approaches. One of these complementary alternatives is the active control strategy of the regenerative vibration process itself during the machining process to prevent the onset of chatter. The main objective of this approach is to detect the onset of chatter and to adjust the spindle speed in such a manner, that the machining process is forced into a region of stable operation. This approach - active in-process control of chatter - is presented in the paper. The paper is organized as follows: In Section 2, a brief mathematical model of the regenerative vibration process is presented. In Section 3, the alternative control approach is described. In Section 4, the simulation set up is presented. In Section 5, the simulation results are discussed. The conclusions are drawn up in Section 6. Delay hstat + h dyn (t) + Cutting F(t) Machine + Trigonometric functions Figure 1; Block diagram of the milling process.
2. THE MILLING PROCESS In Figure 1, a block diagram of the milling process is shown. The static thickness h stat is a result of the pre-defined motion of the tool with respect to the work piece. This chip thickness results in the force F(t) that acts on the tool via the cutting process (block Cutting). Interaction of this force with the spindle and tool dynamics (block Machine), results in a dynamic displacement of the tool v(t), which is superimposed on the predefined tool motion. The block Delay is due to the regenerative effect (see e.g. [Tlusty and Polacek, 1963; Altintas and Budak, 1995; Stépán, 2001;Faassen et al., 2003]). Via trigonometric relations, the tool motion results in a dynamic chip thickness h dyn (t), which is added to the static chip thickness. In the milling process, the static chip thickness is periodic. The movement of the cutter v(t) can therefore be described by the movement v p (t), which is periodic with 1 60 period time T = τ = =. Here, τ is the delay as mentioned in the block Delay, z is ft zω the number of teeth on the cutter, and Ω the spindle speed in rpm. A perturbation on that periodic movement is denoted by v u (t). If no chatter occurs, the periodic movement v p (t) is an asymptotically stable solution of the set of delay differential equations describing the milling process, and the perturbation motion v u (t) tends to zero asymptotically. When the periodic solution loses its stability (e.g. with an increasing axial depth-of-cut), in most cases a secondary Hopf bifurcation occurs, and in other cases a periodic doubling bifurcation occurs [Insperger et al., 2003]. This means that the original periodic solution v p (t) becomes unstable, and a new periodic motion v u (t) with a different frequency f c is superimposed on the original periodic solution. This new periodic motion v u (t) is strongly correlated with the dynamic chip thickness h dyn (t) and can be used as a measure for h dyn (t). In the remainder of this paper f c is referred to as the basic chatter frequency. 3. THE COMPLEMENTARY CONTROL STRATEGY The main objective of the alternative control strategy is to enhance machining performance, and at the same time to guarantee robustness of performance. This means that chatter has to be inhibited. There are a number of secondary objectives that the control strategy has to fulfil: 1. The control should not alter essential machining process parameters: e.g. surface properties of the end products should maintain their quality; 2. The control is complementary in the sense that the imposed control action on the cutting system is an enhancement, not a vital system to sustain the machining process. The main objective, inhibiting chatter, can be defined as to synchronize the wavy surfaces of each succeeding cut, minimizing the variation of the dynamic chip thickness
h dyn (t). The objective of the controller can be re-defined as the optimization of cost function J(t), which is rudimentarily defined as: J ( t) = E{ h ( t) 2 dyn }. (1) The perturbation motion v u (t) can be used as a measure of chip thickness h dyn (t). Then an alternative cost function becomes: J ( t) E{ v ( t) 2 v = u }. (2) To meet the first secondary control objective, the machining parameters (depth-of-cut, radial depth-of-cut, and chip load) must remain unchanged. To meet the second secondary objective, the automated machining process (CNC software program) should not be altered (from a practical viewpoint the altering of the CNC program during the milling process is not an option). From the entire set of machining parameters, the spindle speed Ω is a suitable candidate with which to manipulate the regenerative vibration process during cutting, if the chip load is kept constant [Smith and Delio, 1992]. It is expected, that when altering the spindle speed, it will have a negligible impact on the performance of the milling process. Therefore, the use of the spindle system as a control actuator will not violate the secondary objectives. However, unlimited changes of the spindle speed do endanger machining processing and machine safety. To limit the control effort (large speed changes), the demanded power level of the spindle system to adjust to the new speed must be guarded, and should not exceed the maximum power level. It is preferable to have the spindle system power handled by the control system rather than to leave the handling to the spindle motor amplifier system, since the latter would introduce undesirable nonlinear behaviour in the control actuator system. Augmenting the cost function (2) to J vp (t) will incorporate the spindle power handling by the control system. The augmented cost function is: 2 J v = E{ Avu + B P( t, }. (3) p Here, P(t, the spindle power is required to adjust to the new spindle speed. The parameters A and B are tuneable weighting parameters, which will influence the performance and robustness of the control system. Summarizing, the control strategy will constitute four main functionalities: 1. Optimization of the cost function to realize the main control objective; 2. Sensing of the cutting process. In particular the perturbation motion v u (t); 3. The controller. A description of the controller input and output properties will be presented; 4. Actuation of the cutting process. In particular the actuation of the regenerative process by changing the spindle speed Ω. 3.1. Optimizing of the cost function To maintain the objective of enhancing machining performance, and at the same time to guarantee robustness of performance, the defined cost function (3) is optimized according to the following rule: min{ J vp }, subject to v u δ 0 and Ω min < Ω < Ω max (4) Here δ 0 is a parameter, that determines the maximum allowable value of the perturbation v u (t). The parameters Ω are the restrictive bounds of spindle min,ω max
speed. All parameters are determined from the model of the cutting process [Faassen et al., 2003], from the properties of the spindle system, and by the selected machining working point. To estimate the cost function, the perturbation motion v u (t) is estimated from the movement of the cutter v(t), and the required spindle power of the spindle system is estimated from the measured spindle motor torque. 3.2. Sensing of the perturbation motion Detection of the perturbation motion v u (t, from the cutter movement v(t) is performed by a dedicated method, which is described in [Faassen et al., 2006]. To demonstrate the performance of this method, the measured, and detected perturbation motion v u,xy (t) in x- and y-direction (the normal direction) of the tool is reconstructed real time, and depicted in Figure 2. The perturbation motion distinctively grows at t > 2 s. Figure 2; Reconstruction of the perturbation motion v u,xy (t) of the cutter. - Using eddy current sensors; - The spindle speed = 42000 rpm, axial Depth-Of-Cut (DOC) = 3 ->5 mm. For the control system only one sensor signal in one direction is used to detect the perturbation motion v u (t,. Simulations and experiments [Faassen et al., 2006] have revealed that the measured acceleration in the normal direction near the lower spindle bearing is a suitable signal for detection purposes of the onset of chatter. In Figure 3, the perturbation motion v u (t, is detected real time from the measured acceleration at the spindle bearing and is presented. At time = 1.65 s the tool enters the cut. At t = 2.2 s the onset of chatter is (visually) detectable. Full-blown chatter starts at t = 2.3 s, which can be clearly seen.
Figure 3; Detected v u (t) from the measured acceleration at the spindle bearing. - Axial Depth-Of-Cut (DOC) = 3 ->5 mm. 3.3 The controller The design of the controller depends on the optimization of the cost function (4). The properties of the cost function are estimated real time from the perturbation motion v u (t,. More precisely, the perturbation motion v u (t, is modelled with a parametric model and is estimated at discrete time instances, discrete spindle speeds, and is denoted by V u (f,t,. The function V u (f,t, can be considered as a four-dimensional database, which describes the properties of the regenerative vibration process for four domains: frequency, time, magnitude and spindle speed. The database innovates as the machine process proceeds, and will therefore evolve with time. From the database different types of queries and combination of queries can be made. To optimize the cost function (4) the spindle speed is adjusted. However, to adjust the spindle speed while at the same time the constraint not to exceed the maximum allowable spindle power is preserved, the adjustment is performed via an optimal speed trajectory. This trajectory is generated using a gradient-based adaptation scheme [Haykin, 2001]. The trajectory will be dependent of the evolving function V u (f,t,, of the estimated spindle power and of the selected weighting parameters A and B. Specifically, a recursive method is used to implement the adaptation scheme to determine the trajectory. The trajectory will therefore vary and evolve with the time due to the nature of the adaptation scheme. To have a better understanding of the relation between the control design and the evolving function V u (f,t,, the function is estimated in a simulation for a spindle speed sweep. The start and end values of the spindle speed are selected such, that the left and right boundaries of stable/unstable cut of the stability lobe diagram are deliberately exceeded. The axial DOC is kept constant at 3.5 mm. Two queries are made. The first query is the function f cd (, the dominant chatter frequency with which the function f onset ( = kf rpm ( - f cd ( is derived. The second query is the function V( = 20log( V u (f,t, ) at f = f cd (.
Figure 4; The frequency-domain decomposition of the perturbation motion v u (t) for a spindle speed sweep. Here kf rpm ( is the higher harmonic of the spindle speed frequency where the dominant chatter frequency f cd ( is manifest (k is the harmonic number). The basic chatter frequency equals: f c = f onset ( at full blown chatter, which occurs at approximately Ω min < 32 10 3 rpm and at Ω max > 36 10 3 rpm. In the graph, depicted in Figure 4, both functions are displayed as function of the spindle speed. In the graph a rectangle boundary box is drawn which intersects with the function V( at the level of 60 db. This is a graphical representation of the optimization rule v u δ0. The control action can now be graphically interpreted. To optimize the cost function the spindle speed is forced to the middle of the stability lobe close to (but outside) the red boundary box at V( 60 db. The direction of the approach is determined by the sign of f onset (. The speed of adjustment is determined by the time evolving trajectory. The onset of chatter will be prevented by the controller without exceeding the maximum allowable spindle power. 4. THE SIMULATION SETUP The defined controller approach has been simulated and tested using the Simulink software package. The set up is presented in Figure 5. The model consists of the following functions: 1. The cutting system [Faassen et al., 2003]; 2. Detection of the regenerative process [Faassen et al., 2006]; 3. Estimation of the regenerative process;
4. The control design and action of the controller; 5. The spindle system (Mikron HSM 700) The simulations are performed with the following settings: 1. Sampling frequency f s = 20 khz. 2. Spindle speed is 30000 rpm. 3. The axial depth-of-cut (DOC) varies from 3 to 5 mm in 1 second 4. The feed rate is 0.15 mm/teeth, full immersion. 5. The maximum allowed relative perturbation δ 0 = 0.510-3 m/s 2. 6. The maximum available incremental spindle power = 1 kw. 5. SIMULATION RESULTS The results depicted in Figure 6 and 7 show that the control system minimizes the perturbation motion V u (f,t, in the event that the machining process tends to become unstable. There where the spindle system has unlimited available power to adapt to the new spindle speed, the control system reacts immediately and maintains stable processing. The repeated experiment with limited spindle system power, depicted in Figure 7, indicates that the response of the system is relatively slow, thereby affecting the performance. The perturbation motion exceeds at DOC = 4.2 mm the desired performance of δ 0 = 0.510-3 m/s 2 (unscaled). This is also due to the inaccurate modelling of the spindle system dynamics. Despite a relatively soft control action the model of the dynamics is excited unrealistically. Upcoming experiments with the Mikron HSM 700 milling machine will be used to validate the simulations and most importantly, the relation between control performance and the quality of machining will be analyzed. rpm desired Spindle system rpm(t) Cutting system a(t) Onset detection (t) rpm adj Torque(t) rpm(t) Controller v u V u ( f, t, Onset estimation rpm(t) Figure 5; Model of the active in-process chatter control.
Figure 6; Control of chatter, with unlimited available spindle power. Figure 7; Control of chatter, with limited spindle power. 6. THE CONCLUSIONS In this paper an alternative control approach to prevent chatter is presented. The control approach is considered as a complementary tool or an add-on tool to give answer to the growing demand for higher process efficiency and to ease the inevitable design tradeoffs in the ever-demanding machine development at the design process. The approach aims to in-process control the regenerative vibration process, which is the underlying phenomenon of onset of chatter during machining. The controller uses a sophisticated detection and estimation method to determine this regenerative vibration process and to construct and maintain the time-evolving regenerative-process-database. The controller calculates real-time the optimal spindle speed trajectory to prevent the onset of chatter from this database and from the measured spindle system power. The simulations that are performed with Simulink reveal that a spindle system with limited power will deteriorate the performance of the control. With the new results of the upcoming experiments with Mikron HSM 700 milling machine, the relation between control performance, and the quality of machining will be analyzed.
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