Evaluation of the influence of the reaction rate of the thermodynamic subsystem on the dynamics of the cutting process in metalworking

OBRABOTKAMETALLOV Vol. 23 No. 2 2021 TECHNOLOGY where  0 is the coef fi cient that characterizes the chip pressure on the front face of the tool at the standard temperature of the experiment (kg/mm 2 ). To synthesize a model of the system of tool deformation movements’ dynamics, the following system of equations is taken: 2 11 12 13 11 12 13 1 2 2 21 22 23 21 22 23 2 2 2 31 32 33 31 32 33 3 2 , d x dx dy dz m h h h c x c y c z F dt dt dt dt d y dx dy dz m h h h c x c y c z F dt dt dt dt d z dx dy dz m h h h c x c y c z F dt dt dt dt                                   (6) where c1, c2, c3 are the coef fi cients that take into account the decomposition of the cutting force on the axis of tool deformation. Thus, a mathematical model of the cutting system, described by a set of equations 1–6 is obtained. To conduct an experiment with the resulting model, several programs in the Matlab and Matlab/Simulink environments are developed. The initial data for these models are obtained based on analyses of experiments conducted earlier and published in [32, 33]. For all experiments, it is denoted that the system of equations of instrument motion is described by the following parameters: 0,0065 0 0 0 0, 0065 0 0 0 0, 0065 m            kg  s 2 /mm, 0, 844 0, 39 0, 37 0, 39 0, 77 0, 36 0, 37 0, 36 0, 75 h            kg  s/mm, 1390 190 165 190 795 150 165 150 970 ñ            kg/mm. Coef fi cients of expansion of the cutting force on the tool deformation axis:  x = 0,3369,  2 = 0,48,  3 = 0,81. Parameters of the process mode: depth t p = 2 mm, feed S = 0.1 mm, spindle speed n = 1000 rpm,  = 400 kg/mm 2 , radius of the work piece R = 50 mm. Results and discussion The results of the experiments carried out in the Matlab/Simulink environment are shown below in a series of pictures, the fi rst of which is to consider the dynamics of the cutting system at a time constant of the thermodynamic subsystem of the cutting system equal to 0.7 seconds (see Figure 3). As it can be seen from fi gure 3, after the tool deformations increase by 0.1 seconds of the experiment, there is a certain stabilization of the deformation coordinates and even a subsequent decrease; this is due to the in fl uence of the operator being introduced in expression 5, which displays the dependence of the cutting force on the temperature in the cutting zone. Taking into account the time constant of the thermodynamic subsystem of the cutting system introduced in the experiment of 0.7 seconds, the reaction of the system to the change in the cutting force is approximately 2/3 of this time, that is, the process of temperature stabilization of the change in the deformation coordinates takes about 1.05 seconds. To assess how the state coordinates of the deformation subsystem of the cutting system react to the increase in temperature during cutting, consider the phase trajectories of the deformation coordinates shown in fi gure 4. As it can be seen from fi gure 4, the tool deformation coordinates in the x direction are contracted from a maximum value of 0.0075 mm to 0.005 mm, in the y direction from 0.034 to 0.024 mm, and in the z direction from 0.059 to 0.04 mm. As it was pointed out earlier, this is due to a drop in the cutting force with