Determination of the optimal metal processing mode when analyzing the dynamics of cutting control systems

OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology The system (12) v jT e− ω is like a lagging argument, where v T is the revolution period of the workpiece. For the subsequent analysis of the control system, let’s move on to the operator form of the system (12), it means, let’s implement the Laplace transform, assuming that the initial conditions are zero (p = d/dt), the following formulas are obtained. 2 11 12 13 11 1 3 2 12 1 3 13 1 1 2 21 22 23 21 2 ( ) ( ) ( ) ( ) ( ) (1 )( ) cos( ) ( )[ ( ) cos( ) ] ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) (1 )( ) v v jT p p p p jT mp x p h px p h py p h pz p x p c e h t y p c S h c z p Q p t S mp y p h px p h py p h pz p x p c e − ω − ω + + + +   + + χ − ρ + ρµ + ϕ σ α +   + + χ ρ + ρµ + ϕ σ + + χ ρµα = + + + + + + χ − ρ + ρµ 3 2 22 2 3 23 2 1 2 31 32 33 31 3 0 3 2 32 3 0 3 sin( ) ( ) ( ) sin( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) (1 )( ) ( ) ( ) ( ) ( ) v p p p p jT t t p p t t h t y p c S h c z p Q p t S mp z p h px p h py p h pz p x p c e k k h t y p c S k k h c − ω   + ϕ σ α +   + + χ ρ + ρµ + ϕ σ + + χ ρµα =     + + + +   + + χ − ρ + ρµ + + ∆ σ α +   + + χ ρ + ρµ + + ∆ σ +     33 3 1 1 2 3 2 1 2 1 2 0 3 0 0 3 1 3 1 0 3 3 0 0 ( ) ( ) ( ) 0, 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1 )( ) ( v jTv t p p f f p p p t t p jT p p t t p c t z p k Q p t S t h T T p Q p T T pQ p pz p k t S k k k h t pz p k t S k k k h t e x p e V k k k − ω − ω + ∆   χ ρµα + + α − α σ =     + + + χ ρ + ρµ + + ∆ σ +     + χ ρ + ρµ + + ∆ σ +     + χ − ρ + ρµ + + ∆ 3 2 0 3 1 0 3 2 1 3 0 3 3 1 0 1 3 3 1 1 2 ) ( ) (1 )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) 2 v v v t p c jT jT c t t p c p c t t c jT p c t t c t p p c f f h t V k x p e V k k k h t V k e y p S kV k k kV h y p S k V k k k V h e k Q p t S kV t − ω − ω − ω   σ α +       + χ − ρ + ρµ + + ∆ σ α +   + χ ρ + ρµ + + ∆ σ +     + χ ρ + ρµ + + ∆ σ +     ∆ + + χ ρµα + α − α 3 3 1 1 1 2 3 1 ( ) ( ) 0. 2 v p c jT t p p c f f p c h kV k Q p e t S k V t h k V − ω                                                 σ +        ∆    + χ ρµα + α − α σ =       (13) It is convenient to consider the system (13) in matrix-vector form: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) ( ) 0, ( ) ( ) a a a a x p a a a a y p a a a a z p a a a a Q p             =                 (14)

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