OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 7 No. 3 2025 ( ) ( ) ( ) + + + + + + + + + = ∈ + ∆ = ∆ = − ∈ + ∆ + ∆ + ∆ = ∆ = ∈ + ∆ + ∆ ( , ) ( , ) ( , ) ( , ) 1 1 1 0, 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0, 2 1 1 1 1 0, 1 ( , ) ( , ) ( , ) 1 1 1 ( ) , , , / ; ( ) , , , / ; ( ) 0, , . N k f N k f N k N k i i i i i i N k N k f N k N k N k f N k N k i i i i i i i i i N k N k N k i i i i i f t k t t t t T k f T f t f k t t t T t T t k f t f t t t T t t (8) The sequence represents a set of standard pulses (8) ( , )( ) N k i f t = ( ) ( ) ( ) { } ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 1 1 2 1 ( ), 0, ; ( ), , ; ...; ( ), , N k N k N k N k N k N k N k N k s i i i s f t t T f t t T T f t t T T − = ∈ ∈ ∈ . The tangential components of elementary forces fi (T,k)(t) have a similar structure. In the following, the work is limited to considering the impulse sequence fi (N,k)(t), which has two stages. In the first stage, there is an accumulation of potential energy (time interval + ∆ ( , ) 1 N k i T ). Here, elastic displacement of microcontacts occurs up to values at which the bonds break. This time depends on the cutting speed. In the second stage, the energy is released (time interval + ∆ ( , ) 1 N k i t ). The release of energy is accompanied by an impulse of irreversible energy transformations, which generates heat, but also causes other physical interaction effects. The release of energy also forms an elastic wave impulse. The standard sequence is characterized by the following parameters: the distance between pulses Ti (N,k), rise time ΔTi (N,k), fall time Δt i (N,k), and height ( , )) ,0 N k i f (Fig. 6, d). This impulse can be caused by elasticplastic interaction or the destruction of adhesive and other bonds. First, the frequency-domain representations of the sequence are analyzed (Fig. 7). In the illustration, the upper two curves represent the sequence in the time domain, but on different scales. The lower two diagrams correspond to their spectral representations, also shown on two scales. Note the following key properties of these representations. 1. If the distance between pulses is constant Ti (N,k) = 0.02 s = const, then the spectrum consists of a discrete set of frequencies depending on the pulse duration relative to the distance ΔTi (N,k). If the pulse width equals the distance between pulses, then in the frequency domain we observe frequency bursts corresponding to a Fourier series expansion. In this case, the spectrum contains constant components: the first and third harmonics (Fig. 7, a). When ΔTi (N,k) is reduced, additional frequencies appear (Fig. 7, b), which eventually transform into a continuous spectrum (Fig. 7, c). As ΔTi (N,k)→0, the spectrum approaches “white” noise. 2. If uncertainty is introduced in ΔTi (N,k), then the discrete spectrum shown in Fig. 7, b transforms into a continuous spectrum (Fig. 7, d). A broadening of the spectral lines is observed. The frequency of the spectral maximum corresponds to the mathematical expectation of the distance between pulses, and the variability (spread) of the intervals between pulses is reflected in the broadening of the spectral line. 3. Changes in the ratio between the rising and falling stages of the pulses also affect the spectrum, but modeling shows that these changes are insignificant. Regarding amplitude uncertainty, the spectrum amplifies the components located between the main bursts. If a narrowband filter selects a signal in the frequency window Δω, an increase in amplitude uncertainty corresponds to an increase in amplitude modulation of the signal within this frequency window Δω. In the contact region between the flank face of the tool and the workpiece, numerous interactions occur (Fig. 6, c), which generate the emission signal: = = = = = ∑ ∑ ( , ) ( ) 1 1 ( ) ( ), i n k m N k N i i k F t f t (9) where k and i are the numbers of rows and contacts per row on the interaction surface, respectively. It is necessary to formulate additional hypotheses regarding the influence of the probability distribution of these contacts on the contact surface in order to determine the spectral properties of the set of random sequences.
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