MODEL REFERENCE ESTIMATION AND CONTROL OF TOOL CHATTER Wai i h Kanso Assistant Professsor Mechanical Engineering Gmerican University of Beirut Beirut Lebanon Abstract In this paper a mathematical model i s derived t.0 predict tool chatter in milling processes. The mathematical model parameters are estimated under changing machining conditions by an on-line model reference series-parallel estimation strategy. By applying the proposed method, variations in the machining process are compensated +or by subsequent changes in both the feed velocity and the spindle speed. Changes of feed velocity and spindle speed are carried out by a model reference parallel-. parallel adaptive control strategy. The spindle speed and feed velocity are modified to minimize tool Lhatter and maintain stability of the whole system. The sttategy of estimation and adaptation has been implemented on a computer controlled milling machine. Introduction Adaptive control systems i n machining processes are intended to eliminate the effects of distur- bances, and maintain performance at a desired level despite changes i n the controlled systemC1,21. The main problem of estimation lies in selecting an effective estimation strategy that gives accurate parameters. Inaccurate estimation would cause false feedback signal. Another important characteristics of estimation i s the speed of the process, the le55 the computation required for parameter estimation the fast the controller response. This i s specifically critical when controlling tool chatter, where the process frequency i s high and requires fast computer control. In this work we intend to estimate and control tool chatter. The feedback control loop i s composed of an estimation process to estimate parameters under varying machining conditions, and a control algorithm that updates it's decision based on estimated parame- ters. Control signal i s carried out by the spindle speed and the feed velocity. The fact that machining performance is a function of both parameters, simul- taneous change of the feed velocity and the spindle speed i s necessary. Next, the mathematical model that represents tool chatter will be presented, the estimation and adapta- tion strategy will be derived, and experimental results of the proposed model will be shown. Mathematical Model of Tool Chatter Machining of materials i s often accompanied by a violent relative vibration between work-piece and the tool. This damaging vibration i s known as chatter. This phenomenon i s undesired in metal cutting because of it's adverse effects on surface finish, machining accuracy, tool l i f e , and machine l i f e C3,43. As demonstrated in figure 1, the tool chatter component Y i s developed along a direction perpendic- ular to the feed direction (Y-axis). Since we are considering a two axis of milling, it i s reasonable to assume that the vertical component of tool chatter i s negligible. (Horizontal plane) 4Y r\ Fr Fy Fig.1 Model of Dynamic Cutting Process Chatter 15 caused by an excitation force. The machine tool and i t ' s support may be regarded as a simple spring-mass system with damping. If the system i s regarded as a 5imple single degree of freedom then the equation of motion of tool chatter in the Y-direction can be given a5 CS,bl: M,Y +c,Y+ K,Y- F, (1) Where: M , is the equivalent mass of tool system, Cd i s the structural damping coefficient, K . is the structural stiffness, and F , i s the cutting force exerted on machine tool in the direction of chatter. As suggested by Tobias and Kegg C5,61, the force F , normal to the feed direction can be given as: Where: K1 and M are functions of cutting parame- ters siich as width of cut and chip thickness, V is the feed velocity, and n i s the spindle speed. The proposed dynamical model i n equations (1) and (Z), will be used a5 an estimation model to estimate the machining parameters Ed, K , , M, K1, and M. Farameters will be estimated under varying machining conditions. For adaptation, the first order model presented i n equation 2 will be used, where the spindle speed V and the feed velocity n will be modified to compensate f o r subsequent changes of the estimated parameters K1 and M. The use of the first order model provides faster control loop than the use of the second order model. Estimation Algorithm Estimation or identification comes i n when little information i s known about the dynamics of the machining process. The series-parallel estimation procedure will be used, due to the fact that the estimation algorithm does not require any prior information about the parameters of the machining process. The measured values of Y, y, and FY will be sufficient for estimating the machining parameters. The parameters involved i n the mathematical model i n equations (1) and (2) are: the structural stiff- ness K . , the damping factor E , , , the equivalent mass Me, and the parameters K1 and M. The second order model given by equation (11, can be used in a series-parallel estimator in state space form a5 fOllOW5 C7,81:
В этом варианте мы изобразили букву “А” вырезанную из формы кубика. Добавочные элементы, создают эффект матрицы, а так же добавляют знаку динамики.Знак может существовать в лайновом исполнении, так и в цветном.
