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 As it can be seen from fi gure 2, during the cutting process, a heat transfer zone is formed, due to the formed worn place along the back face; the temperature from the previous processing stages is transmitted through it. Due to this temperature transmission channel, a thermodynamic connection is formed, which affects the force response from the cutting process to the shaping movements of the tool. The force itself, which prevents the shaping movements, can be determined because of the hypothesis of the proportionality of the cutting force to the area of the cut layer (see expression 1): )[ ] t p x t dx F Q t y V dt dt             , (1) where  ( Q ) is the coef fi cient that characterizes the chip pressure on the front face of the tool (kg/mm 2 ), Q is the temperature in the cutting zone (  C), t p is the feed per revolution (mm) recorded in the CNC program, T is the rotation time of the spindle with the part fi xed in it (c –1 ), V x is the tool feed rate (mm/s), dx dt is the tool vibration rate in the feed direction (mm/s), y is the tool deformation in the radial direction (mm). The rotation time of the spindle with the part fi xed in it also depends on the coordinates of the deformations of the tool, based on the relationship of the rotation time with the rotation frequency 2     (  – rad/s), where the rotation frequency can be described in terms of the cutting speed z V R   (mm/s), we get the following dependence describing the rotation period of the spindle: , z R dz V dt     (2) where dz dt is the tool deformation rate in the cutting direction (mm/s), R is the radius of the work piece (mm). Thus, a mathematical model that describes the cutting force in the coordinates of the tool deformations is obtained, but here (see expression 1), there is a temperature in the cutting zone, taking into account our previous studies published in [33-34], the dependence of the temperature in the cutting zone on the power of irreversible transformations can be represented by the following expression: 0 ( ) , Q dQ T Q Q kN dt    (3) where T Q is the time constant of the thermodynamic subsystem (c –-1 ), Q is the current temperature in the cutting zone (  C), Q 0 is the temperature of the part to be processed before the start of processing (  C), k – the coef fi cient of conversion of the power allocated in the contact zone of the tool and the processed part into temperature Ñ ñ Í ìì          , N – the power of irreversible transformations in the cutting zone ( )  Í ìì . The power of irreversible transformations, taking into account the entered coordinates of tool deformations (see Figure 1), is described by the following expression: 2 2 2 , z dz dx dy N F V V x dt dt dt                  (4) Taking into account the accepted model of forming the temperature in the cutting zone (see expressions 3, 4), as well as on the basis of the metalworking dependences of the tensile strength of the metal on the temperature of the experiment, the following dependence ) Q  is assumed: ) , Q Q e           (5)