OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 2 4 From (4) we calculate the autospectra of oscillations 1 1 , ( ) x x S ω , 2 2 , ( ) x x S ω , 3 3 , ( ) x x S ω . For example, the spectrum of deformations X1 responsible for the height irregularities of the topology C (L, X) is 1 1 , ( ) ( ) ( ) X X p j S W p W p = ω ω = - (5) where 1 ( ) ( ) / ( ) X W p p p = ∆ ∆ ; 1 2,1 2,1 3,1 3,1 1 2 2,2 2,2 3,2 3,2 2 2 2,3 2,3 3,3 3,3 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) X h p c h p c p mp h p c h p c h p c mp h p c + + -χ ∆ = + + + -χ + + + -χ ; 3 3 2 1,1 1,1 2,1 2,1 3,1 3,1 1 2 1,2 1,2 2,2 2,2 3,2 3,2 2 2 1,3 1,3 2,3 2,3 3,3 3,3 3 (0) (0) (0) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [1 ] [1 ] 1 V V P P P P mp h p c h p c h p c h p c mp h p c h p c p h p c h p c mp h p c e S e t t S pT -ζ -ζ ∗ ∗ + + + + -χ + + + + -χ ∆ = + + + + -χ ρ + µ ρ + µ -ρµζ + ; (0) 1 P P t t X ∗ ∗ = - . Experiments show that a dynamical system under real conditions is always perturbed. If the equilibrium is stable, then small perturbations correspond to a sequence satisfying the hypotheses of stationary randomness ( ) 1 ( ) U X t . Let the measured signal (U)( ) 1 X t is a sequence )( ) u t = ( 1 X { } ( ) ( ) ( ) ( ) 1 1 1 1 (0), ( ), (2 ), ... ( ) T U U U U X X t X t X s t = ∆ ∆ ∆ . Here 1 ( ) t - ∆ is the Nyquist frequency. It is determined by an order of magnitude higher than the upper natural frequency of the oscillating circuits. The sequence (U)( ) 1 X t allows to calculate the autocorrelation function and its Fourier image, i.e. spectrum ( ) ( ) 1 1 , ( ) U U X X S ω ( ) ( ) 1 1 ( ) ( ) , ( ) ( ) ( ) . U U U U p j X X S W p W p = ω ω = - (6) To assess the quality of model (4), we can introduce a proximity estimator { } ( ) ( ) 2 ( ) ( ) , , , 1 ( ) ( ) ( ) , 0, , 1, 2; 1, 2,3, U U S S S S S S i i X X X X X X S S i s t ℘ ω = ω - ω ω∈ = = ∆ (7) where ù -Äù ( ) ( ) , , 1 ( ) ( ) S S S S i i X X X X S S d ω ω = ω ω ∆ω ∫ ; ù -Äù ( ) ( ) ( ) ( ) , , 1 ( ) ( ) U U U U S S S S X X X X S S d ω ω = ω ω ∆ω ∫ – moving averages in the frequency window ∆ω; spectrum (1) , ( ) S S X X S ω is calculated by the formula (5); spectrum (2) , ( ) S S X X S ω refers to the time sequence derived from the transformed “white” noise; frequency window ∆ω is selected substantially less than the bandwidth of oscillating circuits. Finally, the adequacy analysis used amplitude-frequency characteristics measured directly on model (2) when the system is excited by forces φ0 sin(ωt) with slowly varying frequency ω. The obtained frequency response corresponds to (1) 2 , ( ) ( ) S S S X X S A ω = ω (fig. 4). Here A is the ratio of the amplitude at the output to the amplitude at the input. We can also consider the dispersion estimation in the frequency domain
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