| 1. IntroductionModel a two-dimension oscillator model to express machine tool cutting system . The two oscillators are coupling. By computing maximal Lyapunov exponent, we discover there is chaotic vibration in the cutting chatter process. The current paper simulates the control on the chaotic chatter based on the model. Simulate the cutting process by chaotic fluctuating speed and do examinations to validate the conclusion.2. Chaotic Chatter Control AnalysisTwo-dimension vibration model of machine tool cutting system is showed in Fig.1 Fig.1 two dimension vibration model of machine tool cutting systemAssume cutting force is and the displacement of cutting blade in can be expressed by . Then , the system can be expressed : (1)The two equivalent masses in two directions are equal; k and r are damping and stiffness coefficient of the main vibration system, respectively. For sake of simplifying analysis, we transform variables as: (2)Then, the cutting dynamics model can be expressed by two normal coupling constrain vibration nonlinear differential equations such as: (3) The meanings of the parameters are as follows (4) (5) (6) (7) (8)where and are unit step function and sign function, respectively , that is (9)We do some control simulation to two-dimension oscillator model by using the following three ways: adding stochastic noise to control chaos chatter, adding periodic perturbation to control chaos chatter and delaying feedback to control chaos chatter. By comparison and analysis, we find that add delay on both x and y at the same time is a better control method to control chaos chatter. Fig. 2 are time domain charts of x and y without controls on the model respectively. And figure 3 are time domain charts of x and y with delay feedback control and their part amplification charts on both x and y. We can see that the effect after the control is more acceptable . Fig. 2 The time domain charts of x and y without controls on the model Fig. 3 (a): The time domain chart of x Fig. 3 (b): the time domain chart of y with delay with delay feedback on x and y feedback on x and y Fig.3 (c): The part amplification of time domain chart of x Fig.3 (d): The part amplification of time domain chart with delay feedback control on both x and y of y with delay feedback control on both x and y.3 .Chaotic fluctuating speed ResearchAssume that the mode of machine tool mechanics can be expressed by an equivalently single degree of freedom system. Fig. 4 is the regeneration chatter dynamic model. Fig. 4 the regeneration chatter dynamic model.According to Fig. 4, we have the following equation about y (10)Where is the equivalent mass of principal vibration, is equivalent damping, is equivalent stiffness, dynamic cutting force , the instantaneous cutting thickness of dynamic cutting , is nominal cutting thickness, is the vibratory veins trajectory of this cutting, is the vibratory veins trajectory that has been left by the last cutting, is the time of work piece every turn, is superposition coefficient.We do some fluctuating spindle speed analysis, which is based on the dynamic model of Fig. 4. The vibratory response under cutting with fluctuating spindle speed belongs to the instantaneous response under frequency conversion excitation, which is much smaller than that under constant frequency excitation. This is the nature theory of vibration mechanism, by which fluctuating spindle speed cutting can do the remarkable absorption of vibration. Chaos signals are analogous stochastic signal and aren't periodic signals. The difference between the turning speed of current signal and that of the next time may be very small or very large, that is uncertainty. So chaos signals can overcome the inherent weakness of periodic s... |
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