OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 ( ) ( ) ( ) 2 11 1 0 3 11 0 12 1 0 0 3 12 0 13 1 0 1 0 13 3 21 2 0 3 21 0 2 22 2 0 ( ) (1 ) cos( ) ( ) cos( ) ( )( ) ( ) cos( ) 0, ( ) (1 ) sin( ) ( ) c v c c v jT V p h p V h p h Q p jT V p h p x p mp h p e t K h t c y p h p S h c z p h p p t S c Q p k h t x p h p e t K h t c y p mp h p − ω − ω + + χ − ρ + ϕ σ + + + + χ ρ + ϕ σ + + + − χ ρ µα + − ϕ = + χ − ρ + ϕ σ + + + + + χ ρ ( ) ( ) ( ) ( ) 0 0 3 22 23 2 0 1 0 23 3 31 3 31 0 3 0 0 2 32 3 0 32 0 3 0 33 3 0 1 0 23 0 3 sin( ) ( )( ) ( ) sin( ) 0, ( ) (1 ) ( ) ( ) ( ) ( ) ( ) c c v c V h p h Q p jT V p h p t t V t t p h h Q p S h c z p h p p t S c Q p k h t x p h p e t c K h t k k y p h p S c h k k z p mp h p p t S c Q p k h t − ω + ϕ σ + + + − χ ρ µα + − ϕ = + χ − ρ + + σ + ∆ + + + χ ρ + + σ + ∆ + + − χ ρ µα + − − ( ) 0 2 1 0 3 3 0 0 3 0 3 0 0 0 3 0 3 0 0 0 3 0 3 0 1 0 ( ) ( ) 0, ( ) (1 ) ( ) ( ) ( ) ( ) ( ) v v c v v c v v v t t f f t p jT jT V jT p h p t t jT V jT t t V jT jT p t t p k k K K k h t x p k e e t V K h t k k e V y p k e S V h k k e V z p kp S h t k k e e t S − ω − ω − ω − ω − ω − ω − ω + ∆ + − ∆ σ = χ − ρ + σ + ∆ + + χ ρ + σ + ∆ + + χ ρ + σ + ∆ − χ ρ µα c ñ ñ ñ ( ) 2 1 2 1 2 3 0 2 1 0 3 ( ) ( ) 1 ( ) ( ) 0. v h jT h Q p t t f f t p V Q p T T p T T p k k h t k k K K k h t e V − ω + + + + + − − + ∆ + − ∆ σ = ñ ñ (30) where p – the Laplace transform operator, which is equal to d p dt = under zero initial conditions The system (30) in matrix-vector form is given below: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0. h h h h a p x p a p y p a p z p a p Q p a p x p a p y p a p z p a p Q p a p x p a p y p a p z p a p Q p a p x p a p y p a p z p a p Q p + + + = + + + = + + + = + + + = (31) where the coefficients of the matrix are: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a p a p a p a p a p a p a p a p A a p a p a p a p a p a p a p a p = . (32)
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