OBRABOTKAMETALLOV Vol. 25 No. 1 2023 technology ij a , i = 1...4, j = 1…4, are represented by the following expressions: 2 11 11 1 0 3 11 0 12 12 1 0 0 3 12 0 13 13 1 0 1 0 13 14 3 21 21 2 0 3 21 0 22 ( ) (1 ) cos( ) , ( ) cos( ) , ( ) ), ( ) cos( ) , ( ) (1 ) sin( ) , ( ) c v c c v jT V p h p V p h Q p jT V p h p a p mp h p e t K h t c a p h p S h c a p h p p t S c a p k h t a p h p e t K h t c a p mp − ω − ω = + + χ − ρ + ϕ σ + = + χ ρ + ϕ σ + = − χ ρ µα + = ϕ = + χ − ρ + ϕ σ + = 2 22 2 0 0 3 22 0 23 23 2 0 1 0 23 24 3 31 31 3 31 0 3 0 0 32 32 3 0 32 0 3 0 0 2 33 33 3 0 1 sin( ) , ( ) , ( ) sin( ) , ( ) (1 ) ( ), ( ) ( ), ( ) c c v c V p h Q p jT V p h p t t V t t p h p S h c a p h p p t S c a p k h t a p h p e t c K h t k k a p h p S c h k k a p mp h p p t S − ω + + χ ρ + ϕ σ + = − χ ρ µα + = ϕ = + χ − ρ + + σ + ∆ = + χ ρ + + σ + ∆ = + − χ ρ µα ( ) 0 23 34 3 0 2 1 0 3 41 3 0 0 3 0 42 3 0 0 0 3 0 43 3 0 0 0 3 0 , ( ) ( ) ( ) , ( ) (1 ) ( ) , ( ) ( ) , ( ) ( ) v v v v c v h Q p t t f f t p jT jT V jT p h p t t jT V jT ñ t t V j p t t c a p k h t k k K K k h t a p k e e t V K h t k k e V a p k e S V h k k e V a p kp S h t k k e − ω − ω − ω − ω − ω − + = + ∆ + − ∆ σ = χ − ρ + σ + ∆ = χ ρ + σ + ∆ = χ ρ + σ + ∆ c c ñ ñ ( ) 3 0 1 0 2 44 1 2 1 2 3 0 2 1 0 3 , ( ) ( ) 1 ( ) ( ) . v v v T jT p h Q p t t jT f f t p e t S V a p T T p T T p k k h t k k K K k h t e V ω − ω − ω − χ ρ µα = + + + − + ∆ + + − ∆ σ ñ ñ (33) Similarly to the reasoning of the previous section, in the future it is necessary to move to the time domain by replacing p = jω, and the characteristic polynomial of the control system is nothing more than the determinant of the matrix A, what is seen in the following equation: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) det( ( )) . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a j a j a j a j a j a j a j a j D j A j a j a j a j a j a j a j a j a j ω ω ω ω ω ω ω ω ω = ω = ω ω ω ω ω ω ω ω (34) Thus, the expression (34) is a Mikhailov vector that should be followed for behavior on the complex plane when the frequency ω changes from zero to infinity.
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