Chatter Suppression in NC Machine Tools by Cooperative Control of Spindle and Stage Motors

Transcription

1 IEEJ International Workshop on Sensing, Actuation, and Motion Control Chatter Suppression in NC Machine Tools by Cooperative Control of Spindle and Stage Motors Satoshi Fukagawa a) Student Member, Hiroshi Fujimoto Senior Member Shinji Ishii Non-member, Yuki Terada Non-member The chatter vibration is well known as an undesirable phenomenon, so various methods for the chatter suppression have been proposed. To vary the spindle speed in the cutting process is one of the methods for the chatter suppression. However, the chatter vibration like as triangular velocity is not suppressed completely by the method of variable spindle speed. This remained chatter is caused by that the spindle speed cannot follow the spindle speed reference. Moreover, the triangular spindle speed variation need large torque and good response system of the spindle motor. This paper proposes the method of chatter suppression by a cooperative control of the spindle and the stage motors to decrease the spindle motor torque. The trajectory and the cooperative control are evaluated by simulations and experiments. Furthermore, as a next work, the spindle speed and the stage speed trajectory to suppress the chatter vibration more than only the triangular spindle speed variation is proposed. Keywords: NC Machine tools, Endmilling, Self excited chatter vibration, Cooperative control. Introduction. Background A machining is exceedingly important process to produce any product because it is adaptive for various materials, high precise and much effective. Until today, a considerable amount of study has been made on the NC machine tools to attain higher accuracy and high efficiently in machining () (). Chatter vibration is one of the causes to be worse the products quality. Chatter vibration generalizes undesired vibration in cutting process. Thus, recently the chatter suppression has been focused on and many papers has been published. Particularly, the tools and works which are low stiffness and low damping induces chatter vibration. Chatter vibration is classified as self-excited chatter vibration or forced chatter vibration. Especially, self-excited chatter vibration is the most undesirable, being caused by the interaction between the tools and works. If the chatter vibration becomes large enough, the system becomes uncontrollable. The methods to suppress self-excited chatter is divided into four types (3). Two of these types analyze the conditions under self-excited chatter does not occur, and cuts under these conditions (4) (5). However, these approaches require parameters for the tools, and the measurement of these parameters requires labor and specified skills. Furthermore, due to tools being worn down and being deformed by heat, these parameters are constantly changing, making accurate modeling of the system is difficult. The third a) Correspondence to: fukagawa4@hflab.k.u-tokyo.ac.jp The University of Tokyo 5--5, Kashiwanoha, Kashiwa, Chiba, Japan DMG MORI SEIKI CO., LTD Meieki, Nakamura-ku, Nagoya City, Aichi 45-, Japan Fig.. Schematic view of cutting process. The cutting thickness is determined by the difference between previous and present tool position. method is passive chatter suppression like using the variable pitch cutting tools or the dumper (6). However, this method is not sufficient about robustness for variable parameter. The fourth method is active self-excited chatter suppression (8) (7). For example, spindle motor velocity is varied in cutting process.. The Purpose of Study In (7), the proposed method to suppress chatter vibration is to use the higher amplitude of the spindle speed variation. In contrast, the author s research group proposed a high frequency variation speed control of the spindle to suppress self-excited chatter vibration and demonstrated its effectiveness. The method proposed in (9), was shown to be effective in suppressing chatter by having the spindle speed follow a triangle wave speed trajectory. However, in (9), there are two problems: c 5 The Institute of Electrical Engineers of Japan.

2 Fig.. Block diagram of self-excited chatter vibration. thickness to actual chip thickness is h(s) h (s) = + ( e τs. (4) )K f ag(s) Here, G(s) is the second order lag system shown in Fig. and compliance transfer function describes how the machine constructions vibrate. When the cutting situation is critical state, the critical cutting width is defined as a lim,and chatter frequency is defined ω c. Then, Eq. (5) is described as + ( e jωcτ )K f a lim (Re[G(ω c )] + jim[g(ω c )]) =, (5) ( ) The triangle wave speed trajectory requires a discontinuous change in motor torque. ( ) The chatter occurs in the point that the spindle speed cannot follow the triangle wave speed trajectory Therefore in addition to variable spindle speed control, a cooperative variable feed speed control is implemented to solve these problems above in this paper. The following methods is proposed: Proposed method A velocity trajectory of the spindle and the stage are sinusoidal wave. This method archives to suppress the chatter vibration and to decrease the required motor torque.. The Mechanism of Self-Excited Chatter Vibration (9) () Fig. shows the schematic view of cutting process. Here, the variation in relative position of a work and a tool induces variation in cutting depth. The variation in cutting thickness determines the quality of cutting surface. The cutting thickness is the difference between the previous tool position and the present tool position, so the variation in previous cutting thickness affects the present cutting thickness. This phenomenon is called regeneration. Because of the variation in cutting thickness induced by regeneration, the cutting force changes. So this variation in cutting force induces variation in cutting thickness. This process consists of unstable closed loop, therefore the self-excited chatter vibration occurs. Fig. shows the block diagram of self-excited chatter vibration (9) (). Here, a is the width of a tool. Nominal chip thickness h is described as below: π h = v s. () ω Here, ω is the spindle speed. Displacement of tool y(t) is excited by cutting force. As Fig. shows, h(s) is described as h(s) = h (s) + y(s)e τs y(s), () where, τ is a period of the spindle turning. Then, the thrust cutting force F n is proportional to the specific cutting force K f and described as F n = ak f h(s). (3) This cutting force vibrates a work and cases displacement y(s). Therefore, the transfer function from nominal chip where, Re[G(ω c )], Im[G(ω c )] are real part of G(ω c ) and imaginary part of G(ω lim ). Thus, both sides of real part and imaginary part are equal. Critical cutting width a lim is compute as: a lim =. (6) K f Re[G(ω c )] Equation. (6) shows that compliance of the machine structure Re[G(ω c )] is in inverse proportion to critical cutting width a lim. 3. Methods of Chatter Suppression As shown in the previous section, the mechanism of the self-excited chatter vibration is that the periodic change of cutting thickness induces the periodic change of cutting force. In a previous study, it was proposed that the variable spindle speed suppressed the chatter vibration. Moreover, in the authors research group, the method of high frequency spindle speed variation was proposed and the effect for chatter suppression was confirmed (9). In this paper, this method is regarded as a conventional method. And in this section, the conventional method is mentioned and then the proposed method is mentioned. 3. Chatter suppression by High Frequency Spindle Speed Variation(conventional method) In this subsection, the method of chatter suppression by high frequency spindle speed variation (9) is introduced. To generalize amplitude and frequency of spindle speed variation, the parameters A RVA and f RVF are defined as follows A RVA = N A, (7) N av f RVF = π N av T. (8) Here, T is the period of variation in spindle speed, N av is the average of spindle speed, and N A is the maximum value of spindle speed. The variation in the spindle speed changes the wave number on cutting surface caused by vibration of the tool in cutting of each teeth. Accordingly, the periodic variation in cutting thickness is disturbed by change in the spindle speed. Thus, chatter vibration is suppressed. Moreover, it is mentioned that the triangular velocity trajectory is effective for chatter suppression in (9) because to change the wave number on cutting surface constantly is important to suppress chatter vibration. 3. Chatter Suppression by the Sinusoidal Velocity Trajectory Cooperative Control (proposed method) The triangular velocity trajectory (9) of spindle needs discon-

3 Fig. 3. The definition of nominal cutting thickness. C s and P s are the stage controller and plant. C ω and P ω are the spindle controller and plant. The nominal cutting thickness is defined by feed speed v s and ω Spindle, stage velocity Spindle Time Feed Fig. 4. Sinusoidal velocity trajectory cooperative control (proposed method). The spindle and the stage velocity are sinusoidal. There is phase difference between the spindle speed and the feed speed. tinuous variation in motor torque, so that the bigger power motor is required. The bigger motors need more expense. Also it is difficult to track the triangular velocity trajectory, so synchronous motor was used as the spindle motor in (9). Although synchronous motors good in tracking control, usually induction motors are used for most of machine tools. Therefore, this paper proposes cooperative control of spindle and stage and these speed trajectory are sinusoidal. Fig. 4 shows the conceptual diagram. The spindle and the feed speed is sinusoidal and additionally, there is phase difference between the spindle speed and the feed speed. This proposed method does not requires discontinuous variation in motor torque of spindle. Moreover, when the ratio of spindle speed variation is small, the feed speed changes and then varies cutting thickness. 4. Simulation of Chatter Suppression To simulate the suppression of chatter vibration, cutting thickness is defined as below and the block diagram is shown Fig. 3. The nominal cutting thickness h (t) is h (t) = v s (t) π, (9) ω(t) the spindle speed ω(t) is ω(t) = ω av + A spindle ω av S spindle ( f spindle, t), () and the feed speed v s (t) is v s (t) = v s av + A stage v av S stage ( f stage, t). () Table. Table. Reference parameter for simulation. ω av 6 rad/s A spindle.3 f spindle.7 v s av mm/s A stage. Model parameter for simulation. Parameter Value Width of cut a 5 mm Specific cutting force K f 3 Mpa Dynamic mass M Ns /m Mechanical impedance B Ns/m Dynamic rigidity K 5 5 N/m Here, ω av is the mean value of spindle speed, A spindle is the amplitude coefficient of the spindle speed variation defined in Eq. (7), S spindle ( f spindle, t) is the shape function of spindle speed trajectory and f spindle is the frequency coefficient of spindle speed variation defined in Eq. (8). For example, S spindle ( f spindle, t) in Eq. (8) is trigonometric function or triangle wave. In the same way, v s av is the average of feed speed, A stage is the amplitude coefficient of the feed speed variation, S stage ( f spindle, t) is the shape function of stage speed trajectory and f stage is the frequency coefficient of stage speed variation. Fig. also is used for simulation, and Eq. (9) () are used to compute the valuable nominal cutting thickness and Eq. () is used to compute τ. The τ is a period of the spindle turning. Therefore, in Fig., τ is changed approximately depending on time. 4. Simulation by Sinusoidal Velocity Trajectory Cooperative Control (Proposed Method) This proposed method used sinusoidal velocity trajectory. The spindle and the feed speed are defined as below: ω(t) = ω av + A spindle ω av sin ( f spindle ω av t), () v s (t) = v av + A stage v av sin ( f stage ω av t ϕ). (3) Here, ϕ is phase difference between the spindle speed and the stage speed and the simulation parameter is Table.. The optimal point of f stage, ϕ are decided by simulation of the each condition f stage from to and ϕ from to π. The f stage, ϕ 3

4 Fig. 8. Experimental setup. Fig. 5. Chatter vibration simulation dependence of f stage and phase shift by sinusoidal trajectory when the spindle RVF =.7. Chatter vibration tends to suppress around phase shift = π/ and RVF =.7 3 x time t [s] Fig. 6. Chatter vibration simulation by sinusoidal trajectory (time dependent). Some peaks of chatter vibration are suppressed. 3.5 x Frequency [Hz] Fig. 7. Chatter vibration simulation by sinusoidal trajectory (FFT). The chatter vibration around Hz is suppressed. dependence of chatter vibration given by the simulation is Fig. 5. This result explains that the chatter vibration tends to be suppressed around ϕ = π/ and f stage =.7. Moreover, in this optimal parameters of ϕ = π/ and f stage =.7, the time dependent simulation and FFT results are shown in Fig 6, 7. The reference parameters of this simulation are shown in Table.. Table. is Model parameter used in this simulation. In Fig. 7, the vibration around 3 Hz are caused by the variation in the speed of spindle and the feed. Now, the resonant frequency of the plant model used for simulation is about Table 3. Table 4. The reference parameters for experiment. ω av 9 rad/s A spindle. f spindle.5 v s av.5 mm/s A stage. f stage.5 The spindle and the stage parameters. J spindle kg m D spindle Nm s K t spindle.47 NmA J stage kg m D stage K t stage Table 5..4 Nm s.75 Nm/A Cutting condition. End mill flutes End mill ϕ mm Radial depth of cut 5 mm Axial depth of cut mm Work piece C Hz. It is known that the chatter vibration occurs around resonant frequency of the plant model (). The chatter vibration frequency around Hz is suppressed by proposed method. 5. Controlled System Fig. 8 shows the experimental equipment. In this paper, the spindle and the stage used velocity controller. The spindle and the stage models are described as transfer function below G(s) = K t. (4) Js + D The parameters of the spindle and the stage are shown in Table. 4. Here, subscript spindle means the parameters of the 4

5 Reference Spindle, stage velocity Spindle Tspindle Response Tstage Feed vs_av Astage Time Fig.. Triangular velocity trajectory cooperative control (proposed method). The spindle and the stage velocity are triangular. When the spindle speed does not change, the stage is moved. Fig. 9. Experimental result FFT. The vibration from 58 Hz to 7 Hz is suppressed. 3 x time t [s] Fig.. Chatter vibration by triangular trajectory (time dependent). Some peaks of chatter vibration are suppressed. Fig.. The experimental spindle speed of proposed method in (9). The response cannot track the reference at top of triangle spindle speed trajectory. spindle and stage means the parameters of the stage. The PI controllers are used in both systems, and the pole assignment of the spindle is 5 Hz and the stage is 4 Hz. Then, the stage system is semi closed loop control. 6. Experimental Results In this section, the proposed method is evaluated by experiments. The efficacy of the proposed method is evaluated by cutting experiment. The reference parameters are shown in Table. 3 and the cutting conditions are shown in Table. 5. The efficacy is evaluated by using the acceleration sensor. The the result of the cutting experiment is shown in Fig. 9. In Fig. 9, the waves shows FFT of acceleration. The pink and broken line shows the acceleration of the spindle with the spindle speed and the stage speed are constant. The green and broken line shows the acceleration with the spindle speed is sinusoidal trajectory and stage speed is constant. The blue line shows the acceleration with proposed method. The spindle and the stage have the phase difference of π/. As shown in Fig. 9, the chatter vibration appears from 58 Hz to 7 Hz. Then, compared between the proposed method and constant, the chatter vibration are much suppressed by proposed method. Furthermore, compared between the proposed method and the conventional, the chatter vibration are suppressed especially from 6 Hz to 7 Hz. 3.5 x Frequency [Hz] Fig. 3. Chatter vibration by triangular trajectory (FFT). The chatter vibration around Hz is suppressed. 7. Another Trajectory as Next Works In this section, another trajectory of the stage and the spindle is also proposed and simulated as next works. 7. Triangular Velocity Trajectory Cooperative Control (Next Proposed Method) The chatter vibration is not completely suppressed in (9). The reason is the spindle speed cannot follow the triangle wave speed trajectory. The remained chatter occurs in the point that the spindle speed cannot follow the triangular spindle speed reference. The experimental wave of spindle speed is shown in Fig.. At the top of triangular wave, the rate of the change in the spindle speed is zero, so that the wave number on cutting surface is not varied in each cut and that induces periodic vibration in 5

6 Table 6. Characters of each method. Only spindle variation Sinusoidal cooperation (Proposed) Triangle cooperation (Next proposed) Chatter suppression Good Good Very good Motor torque Normal Small Normal cutting thickness. Thus, the chatter vibration occurs. Suppressing chatter vibration, it is essential to vary cutting thickness constantly. Hence, At the point that the spindle speed cannot follow the triangle wave speed trajectory, the feed speed is changed to suppress chatter vibration more than speed variation. This conceptual diagram is shown Fig.. Here, a broken and red line shows the spindle speed and blue one shows the feed speed. As shown Fig., the feed speed is changed when the rate of the variation in spindle speed is zero. The aim of this method is to change cutting thickness irregularly at all times. 7. Simulation by Triangular Velocity Trajectory Cooperative Control (Next Proposed Method) The simulation model is Fig.. Reference parameters are shown in Table. and Table.. Additionally, T spindle =.7 s, T stage =.34 s. Fig. shows the simulation result of time dependent chatter vibration by the triangular velocity trajectory. The red and broken line shows the chatter vibration h with the variation in just spindle. The blue one is with the variation in both the spindle and the stage. As shown in Fig. 3 which is the FFT graph of Fig., the chatter vibrations around Hz witch is resonant frequency are more suppressed by the sinusoidal velocity trajectory cooperative control. Here, the vibrations around 3 Hz are caused by active variation in the spindle and the feed speed. 8. Conclusion In this paper, the cooperative control between the spindle speed and stage speed for self excited chatter suppression is proposed. The proposed method using sinusoidal velocity trajectory is evaluated by simulations and experiments. It is effective to suppress the chatter vibration and decrease the discontinuous variation in motor torque. Moreover, it is possible that induction motors, which usually used for products of machine tools can track the sinusoidal reference. This proposed method archives to save cost of machine tools. The method of velocity trajectory using triangle trajectory is also evaluated as next works. This method evaluated by simulation. The simulation shows effectiveness for chatter suppression more than the spindle velocity triangular variation and cooperative sinusoidal velocity trajectory. The characters of the proposed methods are shown in Table. 6. If to decrease motor torques is required, the sinusoidal velocity trajectory cooperative variation is selected. If more high precision products and decreasing chatter more than sinusoidal speed variation, the triangular cooperative variation is chosen. As the future works, triangular velocity trajectory cooperative control is going to be experiment. Moreover, it is necessary to optimize the spindle speed and the feed speed trajectory. The frequency response is going to analyzed and the variation in velocity will optimized. Optimization of the spindle and feed speed variation is going to make more chatter suppression. References ( ) A. Matsubara, & S. Ibaraki Proposal of fast and high-precision control for ball-screw-driven stage by explicitly considering elastic deformation, Advanced Motion Control (AMC), (4) ( ) Z. Hongzhong, & H. Fujimoto, Monitoring and Control of Cutting Forces in Machining Processes: A Review, Int. J. of Automation Technology, Vol. 3, No. 4 (9) ( 3 ) G. Quintana, & J. Ciurana, Chatter in machining processes: A review, International Journal of Machine Tools and Manufacture 5.5, pp () ( 4 ) Altintas. Y, & E. Budak, Analytical prediction of stability lobes in milling, CIRP Annals-Manufacturing Technology 44., pp (995) ( 5 ) H. Cao, B. Li, & Z. He, Chatter stability of milling with speed-varying dynamics of spindles, International Journal of Machine Tools and Manufacture 5., pp () ( 6 ) J. Munoa, I. Mancisidor, N. Loix, L. G. Uriarte, R. Barcena, & M. Zatarain, Chatter suppression in ram type travelling column milling machines using a biaxial inertial actuator, CIRP Annals-Manufacturing Technology, 6(), pp (3) ( 7 ) S. Seguy, T. Insperger, L. Arnaud, G. Dessein & G. Peigne, SUPPRESSION OF PERIOD DOUBLING CHATTER IN HIGH SPEED MILLING BY SPINDLE SPEED VARIATION, Journal of Machining Science and Technology, Vol. 5, pp.53 7() ( 8 ) Y. Kakinuma, K. Enomoto, T. Hirano, & K. Ohnishi, Active chatter suppression in turning by band-limited force control, CIRP Annals-Manufacturing Technology, 63(), pp (4) ( 9 ) T. Ishibashi, H. Fujimoto, S. Ishi, K. Yamamoto, & Y. Terada, High frequency variation speed control of spindle motor for chatter vibration suppression in NC machine tools, In American Control Conference (ACC), pp7 77 (4) ( ) E. Shamoto, Mechanism and Suppression of Chatter VIbrations in Cutting, Electric Furnace Steel, Vol. 8, No., pp ()(in Japanese) 6