Full-factor matrix model of accuracy of dimensions performed on CNC multipurpose machines

OBRABOTKAMETALLOV Vol. 23 No. 4 2021 TECHNOLOGY    1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P P P x x x x P P P y y y y P P P z z z z x y z P x y z t P x y z P C t S p C t S C t S - - - æ ö÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ çè ø ;    2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 P P P y y y y P P P x x x x P P P z z z z x y z P x y z t P x y z P C t S p C t S C t S - - - æ ö÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç= ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ ç ÷ çè ø ; (2)     1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P P P x x x x x P P P y y y y y P P P z z z z z x y z P P x y z t P P x y z P P x t C S p x t C S x t C S - - - æ ö÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ çè ø ;     2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 P P P y y y y y P P P x x x x x P P P z z z z z x y z P P x y z t P P x y z P P x C t S p x C t S x C t S - - - æ ö÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ ç= ÷ ç ÷ ç ÷ ç ÷ ç ÷÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ çè ø . (3) The value of the cutting force is calculated according to the well-known formula 1 i i i x y z i p c t s v = i = x; y; z. depending on the parameter t [11]. The model of the scattering fi eld of the performed dimensions on the transverse carriage in the opposite adjustment is similar. In non-opposite adjustment, there is no mutual in fl uence of cutting forces. Therefore, scattering interval of distortion has a single positive position. The maximum of this interval occurs with large depths of cut on both carriages, maximum workpiece hardness and minimum fl exibility of the technological system, and the minimum is obtained with reverse position. As a result, for the scattering fi eld of the performed size on the longitudinal carriage in the non-opposite adjustment, we respectively obtain:           1 1 0 0 0 0 1 2 1 2 1 1 0 0 1 0 0 2 1 01 1 0 2 01 1 0 2 1 0 1 0 2 O O O O O O ( ) t t t t t t w e t p e t p e t p e t p a a a a t p a a t p é ù é ù é ù = + + + + - + - + ê ú ê ú ê ú ê ú ê ú ê ú ë û ë û ë û ( )        1 1 0 0 0 0 1 1 0 0 1 0 0 2 1 0 1 0 2 O O O O O ; t t O a a a a t p a a t p é ù + - + - ê ú ê ú ë û (4) In the non-opposite adjustment, the model of the scattering fi eld of the performed size on the transverse carriage is similar. Most real multi-tool adjustments are not uniform in the direction of cutting forces. For this reason, there is no single calculation scheme for the scattering fi eld for non-uniform directional adjustments. The calculated scheme of the scattering fi eld is determined separately for each coordinate direction, so one adjustment can be opposite in the direction of one dimension to be performed, non-opposite to the direction of another dimension to be performed. If the adjustment in the considered direction is opposite, then some of the developed models (for example, model (1) for the longitudinal carriage) are applied in the direction of the considered dimension, and for non-opposite adjustments, some of the developed models are used in the direction of the considered dimension (for example, model (4) for the longitudinal carriage). During double-tool machining on multipurpose CNC machines, as a result of the development of full- factor matrix mathematical models of errors in the dimensions performed using the appropriate input parameters for solving various problems of practical importance, some parameters should be determined experimentally. With the determination of the complex characteristics of the compliance of the technological system for each subsystem consisting of two matrix complexes – the elasticity of the subsystems along the coordinate axes and the coordinate matrix of compliance e , characterizing its mutual in fl uence, and showing the resistance of the subsystems to its rotations around the coordinate axes and the angular matrix of elasticity ξ , which characterizes its mutual in fl uence, we de fi ne a complex characteristic of the compliance of subsystems. Using this characteristic, it is possible to calculate the total distortion of the performed dimensions for a given adjustment [27].