OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 where the coefficients have the following values: 2 11 11 11 1 3 2 12 12 12 1 3 13 13 13 14 1 1 21 21 21 2 3 2 2 22 22 22 2 3 23 (1 )( ) cos( ) , ( ) cos( ) , , , (1 )( ) sin( ) , ( ) sin( ) , v v jT p p p p jT p p a mp h p c e h t a h p c S h a h p c a t S a h p c e h t a mp h p c S h a − ω − ω = + + + χ − ρ + ρµ + ϕ σ α = + + χ ρ + ρµ + ϕ σ = + = χ ρµα = + + χ − ρ + ρµ + ϕ σ α = + + + χ ρ + ρµ + ϕ σ = 23 23 24 2 1 31 31 31 3 0 3 2 32 32 32 3 0 3 2 33 33 33 34 3 1 1 2 3 41 3 0 0 3 , , (1 )( ) ( ) , ( ) ( ) , , ( ) , 2 (1 )( ) ( ) v jTv p p jT t t p p t t t p p f f p c t t h p c a t S a h p c e k k h t a h p c S k k h a mp h p c k a t S t h a e V k k k h − ω − ω + = χ ρµα = + + χ − ρ + ρµ + + ∆ σ α = + + χ ρ + ρµ + + ∆ σ = + + ∆ = χ ρµα + α − α σ = χ − ρ + ρµ + + ∆ σ 2 0 3 1 0 3 2 1 42 3 0 0 0 3 3 1 0 1 3 43 0 3 0 0 3 1 3 (1 )( ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) ( ) v v v p c jT jT c t t p c p c t t c jT p c t t c p p t t p p p t V k e V k k k h t V k e a S k V k k k V h S k V k k k V h e a p k t S k k k h t p k t S − ω − ω − ω α + + χ − ρ + ρµ + + ∆ σ α = χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ + + ∆ σ = χ ρ + ρµ + + ∆ σ + + χ ρ + ρµ 1 0 3 2 44 1 2 1 2 3 1 0 1 2 3 0 3 1 1 1 2 3 1 ( ) , ( ) ( ) 1 ( ) 2 ( ) . 2 v v jT t t p t p p c f f p c jT t p p c f f p c k k k h t e k a T T p T T p t S k V t h k V k e t S k V t h k V − ω − ω + + ∆ σ ∆ = + + + + χ ρµα + α − α σ + ∆ + χ ρµα + α − α σ (15) Subsequently, 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 presented in equation (14). 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ( ) det( ) . a a a a a a a a D j A a a a a a a a a ω = = (16) Thus, the equation (16) is the characteristic polynomial of the control system that needs to be researched for behavior on the complex plane when the frequency of ω changes from zero to infinity.
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