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Определение трехмерной слоисто-неоднородной скоростной модели среды по данным метода отраженных волн.

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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ɫɬɪ.95-112
Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ
ɦɨɞɟɥɢ ɫɪɟɞɵ ɩɨ ɞɚɧɧɵɦ ɦɟɬɨɞɚ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ
ɗ.Ⱥ. Ȼɥɹɫ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ
Ⱥɧɧɨɬɚɰɢɹ. Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɬɪɟɯɦɟɪɧɵɯ ɫɤɨɪɨɫɬɧɵɯ ɦɨɞɟɥɟɣ ɩɨ ɞɚɧɧɵɦ
ɫɟɣɫɦɨɪɚɡɜɟɞɤɢ ɦɟɬɨɞɨɦ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ. Ⱦɚɟɬɫɹ ɨɩɢɫɚɧɢɟ ɨɫɧɨɜɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ,
ɞɟɥɚɟɬɫɹ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ, ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɪɟɢɦɭɳɟɫɬɜɚ ɢ ɧɟɞɨɫɬɚɬɤɢ ɤɚɠɞɨɝɨ ɦɟɬɨɞɚ.
ɉɪɢɜɨɞɢɬɫɹ ɧɨɜɵɣ ɢɬɟɪɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ ɨɩɪɟɞɟɥɟɧɢɹ ɦɨɞɟɥɢ ɫɪɟɞɵ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ
ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ ɫɥɨɹɦɢ, ɧɟ ɬɪɟɛɭɸɳɢɣ ɩɟɪɟɫɱɟɬɚ ɤɪɢɜɢɡɧ ɮɪɨɧɬɚ ɜɨɥɧɵ. ɉɨɤɚɡɚɧɨ, ɱɬɨ ɞɥɹ
ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɧɟɨɞɧɨɪɨɞɧɵɯ ɫɥɨɟɜ ɞɨɫɬɚɬɨɱɧɨ ɭɦɟɬɶ ɪɟɲɚɬɶ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ ɜ ɪɚɦɤɚɯ ɦɨɞɟɥɢ
ɫ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ. ɉɪɢɜɨɞɢɬɫɹ ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ, ɧɟ ɬɪɟɛɭɸɳɢɣ ɦɧɨɝɨɤɪɚɬɧɵɯ
ɪɟɲɟɧɢɣ ɩɪɹɦɵɯ ɡɚɞɚɱ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɫɨɤɪɚɬɢɬɶ ɜɪɟɦɹ ɫɱɟɬɚ.
Abstract. The given paper deals with the problem of 3-D velocity layered model definition from reflection
seismic data. It gives the description of basic methods for the named problem, gives the compare analysis,
discusses advantages and disadvantages of these methods. It demonstrates a new iteration algorithm for the
determination of the medium model with inhomogeneous curve-linear layers which does not require the wave
front curvature. It is evident from the work that in order to define velocities in the inhomogeneous strata one has
to solve an inverse problem for the model with locally-homogeneous layers. It demonstrates the optimization
algorithm that does not require multiple solution of the direct problem and reduces time-consumption
significantly.
1. ȼɜɟɞɟɧɢɟ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨɩɪɟɞɟɥɟɧɢɹ ɬɪɟɯɦɟɪɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ɫ ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ
ɝɨɪɢɡɨɧɬɚɥɶɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ ɩɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ. Ⱦɥɹ ɟɟ
ɪɟɲɟɧɢɹ ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ, ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɦɵɟ ɩɪɢ ɪɟɲɟɧɢɢ ɨɛɪɚɬɧɵɯ
ɡɚɞɚɱ ɝɟɨɮɢɡɢɤɢ (Ƚɨɥɶɰɦɚɧ, 1971; Ƚɨɥɶɞɢɧ, 1979; Ƚɨɥɶɞɢɧ ɢ ɞɪ., 1993; əɧɨɜɫɤɚɹ, ɉɨɪɨɯɨɜɚ, 1983). ȼ
ɪɚɛɨɬɚɯ ɋ.ȼ.Ƚɨɥɶɞɢɧɚ ɩɨɞɪɨɛɧɨ ɪɚɫɫɦɨɬɪɟɧɵ ɪɚɡɥɢɱɧɵɟ ɦɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ, ɩɪɢɦɟɧɹɟɦɵɟ ɩɪɢ
ɪɟɲɟɧɢɢ ɨɛɪɚɬɧɵɯ ɞɜɭɦɟɪɧɵɯ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ. Ɉɧɢ ɨɫɧɨɜɚɧɵ ɧɚ ɦɧɨɝɨɤɪɚɬɧɨɦ (ɫɨɬɧɢ ɪɚɡ)
ɪɟɲɟɧɢɢ ɩɪɹɦɨɣ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɢ ɦɢɧɢɦɢɡɚɰɢɢ ɪɚɫɯɨɠɞɟɧɢɹ ɦɟɠɞɭ ɧɚɛɥɸɞɚɟɦɵɦɢ ɢ
ɪɚɫɫɱɢɬɚɧɧɵɦɢ ɜɪɟɦɟɧɧɵɦɢ ɩɨɥɹɦɢ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ. ȼ ɩɪɢɧɰɢɩɟ ɷɬɢ ɦɟɬɨɞɵ ɩɨɡɜɨɥɹɸɬ ɭɱɟɫɬɶ
ɤɪɢɜɢɡɧɭ ɥɭɱɟɣ ɜ ɧɟɨɞɧɨɪɨɞɧɵɯ ɫɥɨɹɯ, ɬɨ ɟɫɬɶ ɧɚɣɬɢ ɫɤɨɪɨɫɬɧɭɸ ɦɨɞɟɥɶ ɜ ɤɥɚɫɫɟ ɦɨɞɟɥɟɣ ɫ
ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ, ɧɨ ɞɥɹ ɢɯ ɩɪɢɦɟɧɟɧɢɹ ɬɪɟɛɭɟɬɫɹ ɯɨɪɨɲɟɟ ɧɚɱɚɥɶɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ. Ʉɪɨɦɟ ɬɨɝɨ,
ɩɪɢ ɧɚɥɢɱɢɢ ɫɢɥɶɧɵɯ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɟɣ ɪɟɲɟɧɢɟ ɩɪɹɦɨɣ ɡɚɞɚɱɢ ɜ ɫɪɟɞɟ ɫ
ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ ɬɪɟɛɭɟɬ ɛɨɥɶɲɢɯ ɡɚɬɪɚɬ ɦɚɲɢɧɧɨɝɨ ɜɪɟɦɟɧɢ ɞɚɠɟ ɜ ɞɜɭɦɟɪɧɨɦ
ɫɥɭɱɚɟ. Ⱦɥɹ ɬɪɟɯɦɟɪɧɵɯ ɦɨɞɟɥɟɣ ɫɪɟɞ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɧɟɫɨɝɥɚɫɧɵɦɢ ɫɥɨɹɦɢ ɡɚɬɪɚɬɵ
ɦɚɲɢɧɧɨɝɨ ɜɪɟɦɟɧɢ ɫɭɳɟɫɬɜɟɧɧɨ ɜɨɡɪɚɫɬɚɸɬ, ɧɨ, ɱɬɨ ɟɳɟ ɛɨɥɟɟ ɜɚɠɧɨ, ɞɥɹ ɨɩɢɫɚɧɢɹ ɩɨɜɟɪɯɧɨɫɬɟɣ ɫɥɨɟɜ
ɢ ɩɥɚɫɬɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɦɨɠɟɬ ɩɨɧɚɞɨɛɢɬɶɫɹ ɛɨɥɶɲɨɟ ɱɢɫɥɨ ɛɚɡɢɫɧɵɯ ɮɭɧɤɰɢɣ, ɱɬɨ ɭɦɟɧɶɲɚɟɬ
ɭɫɬɨɣɱɢɜɨɫɬɶ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ. Ʉɪɨɦɟ ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɯ ɚɥɝɨɪɢɬɦɨɜ, ɞɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ
ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ɩɨ ɷɮɮɟɤɬɢɜɧɵɦ ɩɚɪɚɦɟɬɪɚɦ ɦɟɬɨɞɚ ɈȽɌ ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɢɬɟɪɚɰɢɨɧɧɵɟ
ɢɧɜɟɪɫɧɵɟ ɚɥɝɨɪɢɬɦɵ (R-ɚɥɝɨɪɢɬɦɵ ɩɨ ɬɟɪɦɢɧɨɥɨɝɢɢ ɋ.ȼ.Ƚɨɥɶɞɢɧɚ), ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɩɪɢɛɥɢɠɟɧɧɨɦ
ɪɟɲɟɧɢɢ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ (Ƚɥɨɝɨɜɫɤɢɣ, Ƚɨɝɨɧɟɧɤɨɜ, 1978; Ƚɥɨɝɨɜɫɤɢɣ, 1985; Sattleger, 1975; Ƚɨɥɶɞɢɧ,
1986). ɋɯɨɞɢɦɨɫɬɶ ɬɚɤɨɝɨ ɚɥɝɨɪɢɬɦɚ ɞɥɹ ɞɜɭɦɟɪɧɵɯ ɦɨɞɟɥɟɣ ɫɥɨɢɫɬɨ-ɨɞɧɨɪɨɞɧɵɯ ɫɪɟɞ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ
ɜ (Ȼɥɹɫ, Ʌɟɜɢɬ, 1989), ɝɞɟ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɵɯ ɫɥɭɱɚɟɜ ɨɧ ɫɯɨɞɢɬɫɹ ɡɚ 2 - 4
ɲɚɝɚ. ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ R-ɚɥɝɨɪɢɬɦɨɜ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɥɨɢ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɥɨɤɚɥɶɧɨɨɞɧɨɪɨɞɧɵɦɢ, ɬɚɤ ɤɚɤ ɬɨɥɶɤɨ ɜ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɨɦ ɫɥɨɟ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɢɧɜɟɪɫɧɵɣ ɦɟɬɨɞ
ɨɩɪɟɞɟɥɟɧɢɹ ɟɝɨ ɫɤɨɪɨɫɬɢ. ɇɚɩɨɦɧɢɦ, ɱɬɨ ɫɥɨɣ ɹɜɥɹɟɬɫɹ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦ, ɟɫɥɢ ɜ ɩɪɟɞɟɥɚɯ
ɨɬɞɟɥɶɧɨɝɨ ɥɭɱɚ ɟɝɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ, ɬɨ ɟɫɬɶ ɩɪɟɧɟɛɪɟɱɶ ɤɪɢɜɢɡɧɨɣ ɥɭɱɚ ɢ ɫɜɹɡɚɧɧɵɦ ɫ ɧɟɣ
ɢɡɦɟɧɟɧɢɟɦ ɜɪɟɦɟɧɢ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɯɨɬɹ ɢɬɟɪɚɰɢɨɧɧɵɟ R-ɚɥɝɨɪɢɬɦɵ ɛɵɫɬɪɨ ɫɯɨɞɹɬɫɹ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɧɢɯ
ɢɧɜɟɪɫɧɵɟ ɦɟɬɨɞɵ ɪɚɡɪɚɛɨɬɚɧɵ ɬɨɥɶɤɨ ɞɥɹ ɫɪɟɞɵ ɫ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ,
ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɩɨɡɜɨɥɹɸɬ ɜɨɫɫɬɚɧɚɜɥɢɜɚɬɶ ɫɤɨɪɨɫɬɢ ɫ ɭɱɟɬɨɦ ɥɨɤɚɥɶɧɨɣ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ
ɫɥɨɟɜ, ɧɨ ɬɪɟɛɭɸɬ ɛɨɥɶɲɢɯ ɡɚɬɪɚɬ ɦɚɲɢɧɧɨɝɨ ɜɪɟɦɟɧɢ. ȼ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɚɥɝɨɪɢɬɦɵ
95
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
ɨɛɨɢɯ ɜɢɞɨɜ. ɇɟɞɨɫɬɚɬɤɨɦ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ ɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɦɧɨɝɨɤɪɚɬɧɨɝɨ (ɫɨɬɧɢ ɪɚɡ)
ɪɟɲɟɧɢɹ ɩɪɹɦɨɣ ɡɚɞɚɱɢ, ɤɨɬɨɪɚɹ ɞɥɹ ɪɟɚɥɶɧɵɯ ɦɧɨɝɨɤɪɚɬɧɵɯ ɫɯɟɦ ɧɚɛɥɸɞɟɧɢɣ ɫɜɨɞɢɬɫɹ ɤ ɪɚɫɱɟɬɭ
ɞɟɫɹɬɤɨɜ ɬɵɫɹɱ ɥɭɱɟɣ. ȼ ɫɪɟɞɚɯ ɫ ɫɢɥɶɧɵɦɢ ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ ɩɪɟɥɨɦɥɹɸɳɢɦɢ ɝɪɚɧɢɰɚɦɢ ɜ ɜɟɪɯɧɟɣ
ɱɚɫɬɢ ɪɚɡɪɟɡɚ ɧɚ ɷɮɮɟɤɬɢɜɧɭɸ ɫɤɨɪɨɫɬɶ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɨɤɚɡɵɜɚɸɬ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɷɬɢɯ
ɝɪɚɧɢɰ, ɩɪɢɱɟɦ ɷɬɨ ɜɥɢɹɧɢɟ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɪɨɫɬɨɦ ɝɥɭɛɢɧɵ ɡɚɥɟɝɚɧɢɹ ɜɨɫɫɬɚɧɚɜɥɢɜɚɟɦɨɝɨ ɫɥɨɹ. ɗɬɨ
ɡɧɚɱɢɬ, ɱɬɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɝɥɭɛɨɤɨ ɡɚɥɟɝɚɸɳɢɯ ɫɥɨɟɜ ɧɟɨɛɯɨɞɢɦɨ ɜɟɪɯɧɢɟ ɪɟɡɤɢɟ ɝɪɚɧɢɰɵ
ɜɨɫɫɬɚɧɚɜɥɢɜɚɬɶ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɜɬɨɪɵɯ ɩɪɨɢɡɜɨɞɧɵɯ (Ȼɥɹɫ, 1988, 1991). Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɨɬɪɚɠɚɸɳɢɟ
ɝɪɚɧɢɰɵ ɩɨ ɪɟɚɥɶɧɵɦ ɞɚɧɧɵɦ ɜɨɫɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɬɨɥɶɤɨ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɟɪɜɵɯ ɩɪɨɢɡɜɨɞɧɵɯ (ɟɫɥɢ
ɭɱɢɬɵɜɚɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɩɨɞɯɨɞɚ ɧɨɪɦɚɥɶɧɨɝɨ ɥɭɱɚ), ɩɨɷɬɨɦɭ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɝɥɭɛɢɧɵ ɩɨɝɪɟɲɧɨɫɬɢ
ɩɨɫɬɪɨɟɧɢɣ ɪɚɫɬɭɬ. ɂɫɯɨɞɹ ɢɡ ɷɬɨɝɨ, ɦɨɠɧɨ ɩɪɟɞɥɨɠɢɬɶ ɫɥɟɞɭɸɳɢɣ ɩɨɞɯɨɞ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɫɤɨɪɨɫɬɧɵɯ
ɦɨɞɟɥɟɣ ɜ ɫɪɟɞɚɯ ɫ ɫɢɥɶɧɵɦɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɦɢ ɧɟɨɞɧɨɪɨɞɧɨɫɬɹɦɢ ɜ ɩɨɤɪɵɜɚɸɳɟɣ ɬɨɥɳɟ. Ɂɚɞɚɱɚ
ɪɟɲɚɟɬɫɹ ɩɨɫɥɨɣɧɨ ɫɜɟɪɯɭ ɜɧɢɡ. ɋɧɚɱɚɥɚ ɩɪɢɦɟɧɹɟɬɫɹ R-ɚɥɝɨɪɢɬɦ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɬɟɤɭɳɟɣ ɝɪɚɧɢɰɵ, ɚ
ɡɚɬɟɦ ɩɪɢɦɟɧɹɟɬɫɹ ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɣ ɦɟɬɨɞ, ɩɪɢ ɤɨɬɨɪɨɦ ɭɬɨɱɧɹɸɬɫɹ ɜɫɟ ɧɚɣɞɟɧɧɵɟ ɪɚɧɟɟ ɫɥɨɢ.
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɚɩɪɢɨɪɧɵɯ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɩɚɪɚɦɟɬɪɵ ɨɩɪɟɞɟɥɹɟɦɵɯ ɫɤɨɪɨɫɬɟɣ ɢ ɝɪɚɧɢɰ ɩɪɢ
ɦɢɧɢɦɢɡɚɰɢɢ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɫɪɚɡɭ ɩɨ ɩɚɪɚɦɟɬɪɚɦ ɜɫɟɯ ɫɥɨɟɜ ɩɨɡɜɨɥɹɟɬ ɭɦɟɧɶɲɢɬɶ ɪɟɝɭɥɹɪɧɵɟ
ɩɨɝɪɟɲɧɨɫɬɢ ɜɨɫɫɬɚɧɚɜɥɢɜɚɟɦɵɯ ɝɥɭɛɨɤɢɯ ɫɥɨɟɜ ɡɚ ɫɱɟɬ ɜɥɢɹɧɢɹ ɤɪɢɜɢɡɧɵ ɫɢɥɶɧɵɯ ɩɪɟɥɨɦɥɹɸɳɢɯ
ɝɪɚɧɢɰ ɜ ɩɨɤɪɵɜɚɸɳɟɣ ɬɨɥɳɟ, ɬɚɤ ɤɚɤ ɜ ɬɚɤɨɦ ɩɪɨɰɟɫɫɟ ɭɬɨɱɧɹɸɬɫɹ ɧɟ ɬɨɥɶɤɨ ɨɬɪɚɠɚɸɳɢɟ ɝɪɚɧɢɰɵ ɢ
ɫɤɨɪɨɫɬɢ ɜ ɩɨɤɪɵɜɚɸɳɢɯ ɢɯ ɩɥɚɫɬɚɯ, ɧɨ ɢ ɩɪɟɥɨɦɥɹɸɳɢɟ ɝɪɚɧɢɰɵ, ɨɩɪɟɞɟɥɟɧɧɵɟ ɪɚɧɟɟ. Ɂɞɟɫɶ
ɧɟɨɛɯɨɞɢɦɨ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɨɩɬɢɦɢɡɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɩɪɢɦɟɧɹɸɬɫɹ ɬɚɤɠɟ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɫɤɨɪɨɫɬɧɨɣ
ɦɨɞɟɥɢ ɫɪɟɞɵ ɩɨ ɞɚɧɧɵɦ ɦɟɬɨɞɚ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɫɟɣɫɦɢɱɟɫɤɨɝɨ ɩɪɨɮɢɥɢɪɨɜɚɧɢɹ (ȼɋɉ), ɤɨɝɞɚ ɩɪɢɟɦɧɢɤɢ
ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɜ ɫɤɜɚɠɢɧɟ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɧɹɬɢɟ ɫɤɨɪɨɫɬɢ ɫɭɦɦɢɪɨɜɚɧɢɹ (ɚɧɚɥɨɝɢɱɧɨ ɫɤɨɪɨɫɬɹɦ
ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɪɢ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɧɚɛɥɸɞɟɧɢɹɯ) ɧɟɩɪɢɦɟɧɢɦɨ, ɩɨɷɬɨɦɭ ɨɩɬɢɦɢɡɚɰɢɹ ɹɜɥɹɟɬɫɹ
ɩɪɚɤɬɢɱɟɫɤɢ ɟɞɢɧɫɬɜɟɧɧɵɦ ɦɟɬɨɞɨɦ ɭɬɨɱɧɟɧɢɹ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɨɤɨɥɨɫɤɜɚɠɢɧɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ.
2. ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ
ɉɭɫɬɶ Fk(x,y) – ɮɭɧɤɰɢɢ, ɨɩɢɫɵɜɚɸɳɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɫɥɨɟɜ, vk(x,y) – ɩɥɚɫɬɨɜɵɟ ɫɤɨɪɨɫɬɢ, k –
ɧɨɦɟɪ ɫɥɨɹ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɦɟɟɬɫɹ ɧɟɫɤɨɥɶɤɨ ɫɟɣɫɦɢɱɟɫɤɢɯ ɩɪɨɮɢɥɟɣ, ɩɨɤɪɵɜɚɸɳɢɯ ɢɫɫɥɟɞɭɟɦɭɸ
ɩɥɨɳɚɞɶ. ɇɚ ɤɚɠɞɨɦ ɩɪɨɮɢɥɟ ɢɡɜɟɫɬɧɵ ɥɢɧɢɢ t0 – ɜɪɟɦɟɧɚ ɩɪɨɛɟɝɚ ɜɨɥɧ ɫ ɧɭɥɟɜɵɦ ɪɚɫɫɬɨɹɧɢɟɦ ɜɡɪɵɜɩɪɢɟɦ, ɷɬɢ ɜɪɟɦɟɧɚ ɢɡɜɟɫɬɧɵ ɞɥɹ ɜɫɟɯ ɨɬɪɚɠɚɸɳɢɯ ɝɪɚɧɢɰ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɩɨɫɥɟ
ɩɪɨɜɟɞɟɧɢɹ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɢɡɜɟɫɬɧɵ ɫɤɨɪɨɫɬɢ ɫɭɦɦɢɪɨɜɚɧɢɹ vɨɝɬ. ɏɨɬɹ ɫɤɨɪɨɫɬɢ ɫɭɦɦɢɪɨɜɚɧɢɹ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɫɟɣɫɦɨɝɪɚɦɦɚɦ ɢ ɡɚɜɢɫɹɬ ɨɬ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ, ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɷɬɚ
ɫɤɨɪɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɩɚɪɚɦɟɬɪ ɝɢɩɟɪɛɨɥɵ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɝɨɞɨɝɪɚɮ ɈȽɌ tCDP(l) ɩɨ ɦɟɬɨɞɭ
ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɧɚ ɛɚɡɟ ɧɚɛɥɸɞɟɧɢɣ, l – ɪɚɫɫɬɨɹɧɢɟ ɜɡɪɵɜ-ɩɪɢɟɦ (ɉɭɡɵɪɟɜ, 1979). ɋɤɨɪɨɫɬɢ
ɫɭɦɦɢɪɨɜɚɧɢɹ ɧɚɯɨɞɹɬɫɹ ɞɥɹ ɜɫɟɯ ɨɬɪɚɠɚɸɳɢɯ ɝɪɚɧɢɰ. Ɏɚɤɬɢɱɟɫɤɢ ɤɨɥɢɱɟɫɬɜɨ ɪɟɝɭɥɹɪɧɵɯ ɨɬɪɚɠɟɧɢɣ
ɜɦɟɫɬɟ ɫɨ ɫɤɨɪɨɫɬɹɦɢ ɈȽɌ (ɫɭɦɦɢɪɨɜɚɧɢɹ ɩɪɢ ɩɨɥɭɱɟɧɢɢ ɪɚɡɪɟɡɨɜ ɈȽɌ) ɢ ɨɩɪɟɞɟɥɹɟɬ ɤɨɥɢɱɟɫɬɜɨ ɫɥɨɟɜ
ɜ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚ ɤɚɠɞɨɦ ɢɡ ɩɪɨɮɢɥɟɣ ɢɦɟɟɬɫɹ 2n ɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ: t0k, vCDP(k),
k=1,2,...,n. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɥɚɫɬɨɜɵɟ ɫɤɨɪɨɫɬɢ vk(x,y) ɢ ɨɬɪɚɠɚɸɳɢɟ ɝɪɚɧɢɰɵ Fk(x,y).
ɉɪɟɠɞɟ ɱɟɦ ɪɚɫɫɦɨɬɪɟɬɶ ɢɬɟɪɚɰɢɨɧɧɵɣ R-ɚɥɝɨɪɢɬɦ, ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɦɨɬɪɟɬɶ ɢɧɜɟɪɫɧɵɣ ɦɟɬɨɞ
ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ, ɩɪɢɦɟɧɟɧɢɟ ɤɨɬɨɪɨɝɨ ɹɜɥɹɟɬɫɹ ɫɨɫɬɚɜɧɨɣ ɱɚɫɬɶɸ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ.
3. ɂɧɜɟɪɫɧɵɣ ɚɥɝɨɪɢɬɦ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ
Ɂɚɞɚɱɭ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɛɭɞɟɦ ɪɟɲɚɬɶ ɩɨɫɥɨɣɧɨ ɫɜɟɪɯɭ ɜɧɢɡ. ɋɧɚɱɚɥɚ ɧɚɯɨɞɢɦ
ɫɤɨɪɨɫɬɶ ɢ ɦɨɳɧɨɫɬɶ ɩɟɪɜɨɝɨ ɫɥɨɹ ɩɨ ɢɡɜɟɫɬɧɵɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɩɚɪɚɦɟɬɪɚɦ ɜɨɥɧ, ɨɬɪɚɠɟɧɧɵɯ ɨɬ ɟɝɨ
ɩɨɞɨɲɜɵ. ɉɨɫɥɟ ɷɬɨɝɨ, ɫɱɢɬɚɹ ɩɚɪɚɦɟɬɪɵ ɩɟɪɜɨɝɨ ɫɥɨɹ ɢɡɜɟɫɬɧɵɦɢ, ɧɚɯɨɞɢɦ ɩɨɞɨɲɜɭ ɢ ɫɤɨɪɨɫɬɶ
ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧ ɜɨ ɜɬɨɪɨɦ ɫɥɨɟ ɢ ɬ.ɞ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɧɜɟɪɫɧɵɣ ɚɥɝɨɪɢɬɦ ɫɬɪɨɢɬɫɹ ɜ
ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɧɟɢɡɜɟɫɬɧɵɦɢ ɹɜɥɹɸɬɫɹ ɩɚɪɚɦɟɬɪɵ ɬɨɥɶɤɨ ɨɞɧɨɝɨ ɫɥɨɹ, ɚ ɢɦɟɧɧɨ, ɬɨɝɨ, ɨɬ ɩɨɞɨɲɜɵ
ɤɨɬɨɪɨɝɨ ɜɵɞɟɥɟɧɵ ɨɬɪɚɠɟɧɧɵɟ ɜɨɥɧɵ. Ɉɛɪɚɬɧɚɹ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɩɨ ɷɮɮɟɤɬɢɜɧɵɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ
ɩɚɪɚɦɟɬɪɚɦ ɷɬɢɯ ɜɨɥɧ.
ɂɬɚɤ, ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɡɜɟɫɬɧɵ ɩɚɪɚɦɟɬɪɵ ɩɟɪɜɵɯ (n-1) ɫɥɨɟɜ, ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ
v(x,y) ɢ ɩɨɞɨɲɜɭ Fn(x,y) n-ɝɨ ɫɥɨɹ ɩɨ ɢɡɜɟɫɬɧɵɦ ɥɢɧɢɹɦ to(p), vo(p), grad to(p) ɩɨ ɧɟɤɨɬɨɪɨɣ ɫɢɫɬɟɦɟ
ɩɪɨɮɢɥɟɣ, p – ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ ɧɚ ɩɪɨɮɢɥɟ. Ⱦɥɹ ɤɚɠɞɨɣ ɬɨɱɤɢ ɩɪɨɮɢɥɹ ɛɭɞɟɬ ɧɚɣɞɟɧɚ ɫɤɨɪɨɫɬɶ vn ɜ
ɨɤɪɟɫɬɧɨɫɬɢ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ, ɫɜɹɡɚɧɧɨɝɨ ɫ ɷɬɨɣ ɬɨɱɤɨɣ, ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɚ ɬɨɱɤɚ ɨɬɪɚɠɟɧɢɹ ɷɬɨɝɨ ɥɭɱɚ
ɨɬ ɩɨɞɨɲɜɵ ɫɥɨɹ. ɂɦɟɹ ɞɨɫɬɚɬɨɱɧɨ ɩɥɨɬɧɭɸ ɫɢɫɬɟɦɭ ɩɪɨɮɢɥɟɣ, ɦɨɠɧɨ ɡɚɬɟɦ ɫɝɥɚɠɢɜɚɧɢɟɦ
ɜɨɫɫɬɚɧɨɜɢɬɶ ɨɬɪɚɠɚɸɳɭɸ ɩɨɜɟɪɯɧɨɫɬɶ Fn(x,y) ɢ ɫɤɨɪɨɫɬɶ vn(x,y) n-ɝɨ ɫɥɨɹ. Ⱦɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ
ɨɞɧɨɪɨɞɧɵɯ ɫɥɨɟɜ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɜɨ ɦɧɨɝɢɯ ɪɚɛɨɬɚɯ (ɑɟɪɧɹɤ, Ƚɪɢɰɟɧɤɨ, 1979; Ƚɨɥɶɞɢɧ, 1993), ɡɞɟɫɶ ɦɵ
ɩɪɢɜɨɞɢɦ ɞɪɭɝɨɣ ɚɥɝɨɪɢɬɦ, ɧɟ ɫɜɹɡɚɧɧɵɣ ɫ ɩɟɪɟɫɱɟɬɨɦ ɤɪɢɜɢɡɧ ɮɪɨɧɬɚ ɜɨɥɧɵ.
ɉɭɫɬɶ (X,Y) - ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɈȽɌ ɧɚ ɧɟɤɨɬɨɪɨɦ ɩɪɨɮɢɥɟ. Ɂɧɚɹ grad to , ɦɨɠɧɨ, ɢɫɩɨɥɶɡɭɹ
ɡɚɤɨɧ Ȼɟɧɧɞɨɪɮɚ (Ƚɨɥɶɞɢɧ, 1979), ɜɨɫɫɬɚɧɨɜɢɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɩɨɞɯɨɞɚ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ ɤ ɬɨɱɤɟ (X,Y) ɢ
96
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ɫɬɪ.95-112
ɩɪɨɬɪɚɫɫɢɪɨɜɚɬɶ ɥɭɱ ɞɨ (n-1)-ɨɣ ɝɪɚɧɢɰɵ – ɞɨ ɤɪɨɜɥɢ n-ɝɨ ɫɥɨɹ; ɷɬɭ ɝɪɚɧɢɰɭ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɱɟɪɟɡ
F(x,y), ɨɬɛɪɚɫɵɜɚɹ ɢɧɞɟɤɫ "n-1" ɞɥɹ ɫɨɤɪɚɳɟɧɢɹ ɡɚɩɢɫɢ. Ɍɨɱɤɭ ɩɟɪɟɫɟɱɟɧɢɹ ɞɚɧɧɨɝɨ ɥɭɱɚ ɫ (n-1)-ɨɣ
ɝɪɚɧɢɰɟɣ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ ([0,K0,F0), F0 = F([0,K0). Ⱦɚɥɟɟ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɪɚɜɟɧɫɬɜɨɦ, ɜɩɟɪɜɵɟ ɫɬɪɨɝɨ
ɞɨɤɚɡɚɧɧɵɦ ɋ.Ⱥ.Ƚɪɢɰɟɧɤɨ (ɑɟɪɧɹɤ, Ƚɪɢɰɟɧɤɨ, 1979; Ƚɨɥɶɞɢɧ, 1986)
2 /(tove2) = d 2W /ds 2 .
(1)
Ɂɞɟɫɶ W(s) – ɷɣɤɨɧɚɥ ɥɭɱɟɣ, ɜɵɲɟɞɲɢɯ ɢɡ (ɢɫɤɨɦɨɣ) ɬɨɱɤɢ M([n,Kn,Fn) ɨɬɪɚɠɟɧɢɹ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ ɢ
ɩɪɢɲɟɞɲɢɯ ɜ ɬɨɱɤɭ ɩɪɨɮɢɥɹ, ɭɞɚɥɟɧɧɭɸ (ɫ ɭɱɟɬɨɦ ɡɧɚɤɚ) ɧɚ ɪɚɫɫɬɨɹɧɢɟ s ɨɬ ɬɨɱɤɢ Ao(X,Y) (ɪɢɫ.1); ve –
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ, ɧɚɣɞɟɧɧɚɹ ɩɨ ɩɪɨɮɢɥɶɧɨɦɭ ɝɨɞɨɝɪɚɮɭ ɈȽɌ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ Ao(X,Y), L –
ɩɪɹɦɚɹ, ɧɚ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɨɠɟɧɵ ɢɫɬɨɱɧɢɤɢ ɢ ɩɪɢɟɦɧɢɤɢ (ɩɪɨɮɢɥɶ ɧɚɛɥɸɞɟɧɢɣ). ɑɟɪɟɡ N([,K,F([,K))
ɨɛɨɡɧɚɱɢɦ ɬɨɱɤɭ ɩɟɪɟɫɟɱɟɧɢɹ ɫ (n-1)-ɨɣ ɝɪɚɧɢɰɟɣ ɥɭɱɚ, ɜɵɲɟɞɲɟɝɨ ɢɡ ɬɨɱɤɢ Ɇ ɢ ɩɪɢɲɟɞɲɟɝɨ ɜ ɬɨɱɤɭ
ɩɪɨɮɢɥɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ s ɨɬ ɬɨɱɤɢ Ao(X,Y), ɬɨ ɟɫɬɶ N(s) – ɬɨɱɤɚ ɩɟɪɟɫɟɱɟɧɢɹ ɫ (n-1)-ɨɣ ɝɪɚɧɢɰɟɣ ɥɭɱɚ,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɷɣɤɨɧɚɥɭ W(s). Ɍɨɝɞɚ W(s) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
W = T1(s,[,K) + tn([,K) = T(s,[,K).
(2)
Ɂɞɟɫɶ T1 – ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɢɡ ɬɨɱɤɢ N([,K,F([,K)) ɞɨ ɬɨɱɤɢ A ɧɚ ɩɪɨɮɢɥɟ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ s ɨɬ Ao, tn – ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧ n-ɨɦ ɫɥɨɟ. Ɍɚɤ ɤɚɤ ɨɩɟɪɚɬɨɪ M ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ
ɫɬɪɨɢɬɫɹ ɞɥɹ ɫɪɟɞɵ ɫ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦ n-ɵɦ ɫɥɨɟɦ, ɬɨ ɜ ɷɬɨɦ ɫɥɨɟ ɥɭɱ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
ɩɪɹɦɨɥɢɧɟɣɧɵɣ ɨɬɪɟɡɨɤ, ɢ ɞɥɹ tn ɫɩɪɚɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ
tn = [(Fn-F([,K))2 + ([n-[) + (Kn– K)2]2 / vn.
(3)
Ɋɢɫ.1. ɋɯɟɦɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɥɭɱɟɣ ɢɡ ɬɨɱɤɢ ɨɬɪɚɠɟɧɢɹ ɜ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨɣ ɫɪɟɞɟ.
ɂɡ ɩɪɢɧɰɢɩɚ Ɏɟɪɦɚ (ɫɬɚɰɢɨɧɚɪɧɨɫɬɢ ɬɪɚɟɤɬɨɪɢɢ) ɫɥɟɞɭɟɬ, ɱɬɨ ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ
ɪɚɜɟɧɫɬɜɚ:
wT /w[ = wT1(s,[,K) /w[ + w tn([,K) /w[ = 0,
(4)
wT /wK = wT1(s,[,K) /wK + w tn([,K) /wK = 0.
ɉɨɞɫɬɚɜɥɹɹ ɜ ɞɚɧɧɵɟ ɪɚɜɟɧɫɬɜɚ tn ɢɡ (3), ɩɨɥɭɱɢɦ, ɱɬɨ
[([ - [n ) + (F - Fn) (wF/wx)] / [vn [(F - Fn)2 + ([ - [n )2 + (K - Kn)2]1/2] = - wT /w[ ,
(5)
[(K - Kn) + (F - Fn) (wF/wy)] / [vn [(F - Fn)2 + ([ - [n )2 + (K - Kn)2]1/2] = - wT /wK.
97
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
ɉɭɫɬɶ G = F - Fn – ɧɟɢɡɜɟɫɬɧɚɹ ɜɟɥɢɱɢɧɚ. Ɍɨɝɞɚ ɢɡ (3) ɢ (5) ɩɨɥɭɱɚɟɦ, ɱɬɨ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ
[(wF/w[) G + (wT1 /w[) vn 't]2 + [(wF/wK)G + (wT1 /wK) vn 't]2 + G2 = (vn 't)2.
(6)
Ɂɞɟɫɶ ɢ ɜɫɸɞɭ ɧɢɠɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɣ T1, F ɛɟɪɭɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɬɨɱɤɟ (0,[(0),K(0)) ɢ ([(0),K(0))
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɬɨɱɤɟ ɩɟɪɟɫɟɱɟɧɢɹ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ ɫ (n-1)-ɨɣ ɝɪɚɧɢɰɟɣ, 't = t0/2 - T1 – ɢɡɜɟɫɬɧɚɹ
ɜɟɥɢɱɢɧɚ. Ʉɨɨɪɞɢɧɚɬɵ [ ,K ɩɪɨɟɤɰɢɢ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɥɭɱɚ, ɜɵɯɨɞɹɳɟɝɨ ɢɡ ɬɨɱɤɢ Ɇ, ɡɚɜɢɫɹɬ ɨɬ s, ɬɨ
ɟɫɬɶ [ ,K ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɩɟɪɟɦɟɧɧɨɣ s, ɩɨɷɬɨɦɭ [(0), K(0) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɮɭɧɤɰɢɢ [ ,K ɛɟɪɭɬɫɹ ɜ
ɬɨɱɤɟ s = 0, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɰɟɧɬɪɚɥɶɧɨɦɭ ɥɭɱɭ.
Ɋɚɜɟɧɫɬɜɨ (6) ɩɨɫɥɟ ɧɟɫɥɨɠɧɵɯ ɬɨɠɞɟɫɬɜɟɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
B1 G2 + B2 G vn2 + B3 vn4 + B4 vn2 = 0,
(7)
ɝɞɟ
B1 = 1 + (wF/wx)2 + (wF/wy)2,
B2 = 2(wF/wx wT1 /w[ + wF/wy wT1 /wK) 'tn ,
B3 = ('tn)2 [(wT1 /w[ )2 + (wT1 /wK)2],
B4 = - ('tn)2.
ȼ ɭɪɚɜɧɟɧɢɢ (7) ɧɟɢɡɜɟɫɬɧɵɦɢ ɹɜɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ vn ɢ G, ɞɥɹ ɢɯ ɧɚɯɨɠɞɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɢɦɟɬɶ ɟɳɟ
ɨɞɧɨ ɭɪɚɜɧɟɧɢɟ. ȿɝɨ ɩɨɥɭɱɢɦ ɢɡ ɪɚɜɟɧɫɬɜɚ (1). Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ d 2W / ds2 ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɩɨɫɨɛɨɦ,
ɜɩɟɪɜɵɟ ɢɡɥɨɠɟɧɧɵɦ ɜ (Ȼɥɹɫ, 1985), ɚ ɢɦɟɧɧɨ, ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭɟɦ (2) ɞɜɚ ɪɚɡɚ ɩɨ s ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ [,
K ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ s. ɂɫɩɨɥɶɡɭɹ ɪɚɜɟɧɫɬɜɚ (4), ɩɨɥɭɱɢɦ
dW /ds = wT /ws ,
d 2W /w 2s = w 2T1 /ws2 + (w 2T1 /ws w[)( d[ /ds) + (w 2T1 /ws w[)( dK /ds).
(8)
Ⱦɚɥɟɟ, ɞɢɮɮɟɪɟɧɰɢɪɭɹ (4) ɩɨ s, ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ d[ /ds, dK /ds:
(w 2T /w[ 2) (d[ /ds) + (w 2T /w[wK) (dK /ds) = - w 2T1 /w[ws,
(w 2T /w[wK)(d[ /ds) + (w 2T /w 2K) (dK /ds) = - w 2T1 /wKws.
Ɋɟɲɚɹ ɞɚɧɧɭɸ ɫɢɫɬɟɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨ d[ /ds, dK /ds ɢ ɩɨɞɫɬɚɜɥɹɹ ɢɯ ɜɨ ɜɬɨɪɨɟ ɪɚɜɟɧɫɬɜɨ (8), ɧɚɣɞɟɦ
d 2W /ds2. ɉɨɞɫɬɚɜɥɹɹ ɧɚɣɞɟɧɧɨɟ ɬɚɤɢɦ ɨɛɪɚɡɨɦ d 2W /ds2 ɜ ɪɚɜɟɧɫɬɜɨ (1), ɩɨɥɭɱɢɦ
2 /(t0 ve2) = (w 2T1 /ws2) – {(w 2T1 /w[ws) [(w 2T1 /w[ws)(w 2T/wK2) – (w 2T1 /wKws) (w 2T/w[wK)] +
+(w 2T1 /wKws) [(w 2T1 /wKws)((w 2T/w[ 2) – (w 2T1 /w[ws) (w 2T/w[wK)] }/ [(w 2T/w[2)(w 2T/wK2) – (w 2T/w[wK)2].
ɉɨɞɫɬɚɜɥɹɹ ɜ ɷɬɨ ɪɚɜɟɧɫɬɜɨ Ɍ ɢɡ (2) ɢ ɜɵɱɢɫɥɹɹ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɜɟɥɢɱɢɧɵ tn, ɨɩɪɟɞɟɥɟɧɧɨɣ
ɪɚɜɟɧɫɬɜɨɦ (3), ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɨ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
A1 G2 + A2G + A3Gvn2 + A4 vn2 + A5 vn4 = 0.
(9)
Ɂɞɟɫɶ
('tn) A1 = [(w 2T1 /ws2) – 2 /(t0 ve2)] [(w 2F/wx2)(w 2F/wy2) – (w 2F/wxwy)2] ,
('tn)2A2 = - [(w 2T1 /ws2) – 2 /(t0 ve2)][(w 2F/wx2)(1+Fy2) + (w 2F/wy2)(1+Fx2) – 2(w 2F/wxwy)FxFy] ,
'tn A3 = (w 2T1 /w[ws)2 (w 2F/wy2) + (w 2T1 /wKws)2 (w 2F/wx2) – 2 (w 2T1 /w[ws)(w 2T1 /wKws)(w 2F/wxwy) –
– [(w 2T1 /w 2s) – 1 / (t0 ve2)] [(w 2F/wx2)D2 + (w 2F/wy2)D1 – 2 (w 2F/wywx) D3 ] ,
(10)
'tn A4 = - (w 2T1 /w[ws )2 (1+Fy2) + (w 2T1 /wKws)2 (1+Fx2) - 2(w 2T1 /w[ws)(w 2T1 /wKws)(wF/wx)(wF/wy)- [(w 2T1 /w 2s) - 1 /(t0ve2)] [(1+Fx2)D2 + (1+Fy2)D1 - 2 Fx Fy D3)] ,
'tn A5 = (w 2T1 /w[ws)D2+(w 2T1 /wKws)D1 - 2(w 2T1 /w[ws)(w 2T1 /wKws)D3 - [(w 2T1 /ws2) – /(t0 ve2)](D1D2- D32) ,
ɝɞɟ
98
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
D1 = (w 2T1 /w[ 2) - (wT1 /w[)2 ('tn ) -1,
ɫɬɪ.95-112
D2 = (w 2T1 /wK2) - (wT1 /wK )2 ('tn )-1,
D3 = (w 2T1 /w[wK) - (wT1 /w[) (wT1 /wK) ('tn )-1.
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɣ T1, F ɛɟɪɭɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɬɨɱɤɚɯ (0,[0,K0) ɢ ([0,K0),
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɬɨɱɤɟ ɩɟɪɟɫɟɱɟɧɢɹ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ ɫ (n-1)-ɨɣ ɝɪɚɧɢɰɟɣ z = F(x,y). ɂɬɚɤ, ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ G, vn ɢɦɟɟɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ (7), (9). ɉɭɫɬɶ D = G / vn2, ɬɨɝɞɚ ɞɚɧɧɵɟ ɭɪɚɜɧɟɧɢɹ
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
B1 D2 vn2 + B2 D vn2 + B3 vn2 + B4 = 0,
(11)
A1 D2 vn2 + A2 D + A3 D vn2 + A4 + A5 vn2 = 0.
ɂɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ (11) ɧɚɯɨɞɢɦ vn2 = - B4 /(B1 D2 + B2 D + B3). ɉɨɞɫɬɚɜɥɹɹ ɨɬɫɸɞɚ vn ɜɨ ɜɬɨɪɨɟ
ɪɚɜɟɧɫɬɜɨ (11), ɩɨɥɭɱɚɟɦ ɤɭɛɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ D:
B4 (A1D 2 + A3D + A5) = (A2D + A4)(B1D 2 + B2D + B3).
(12)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɯɨɠɞɟɧɢɟ ɩɥɚɫɬɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ ɫɜɨɞɢɬɫɹ ɤ ɪɟɲɟɧɢɸ
ɤɭɛɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ (12) ɢ ɧɚɯɨɠɞɟɧɢɸ vn ɩɨ ɮɨɪɦɭɥɟ
vn= [- B4 /(B3+B2D+B3D2)] 1/2.
Ʉɚɤ ɚɧɚɥɢɬɢɱɟɫɤɢ ɩɨɤɚɡɚɧɨ ɚɜɬɨɪɨɦ (Ȼɥɹɫ, 1991), ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɝɥɭɛɢɧɵ ɡɚɥɟɝɚɧɢɹ
ɩɪɟɥɨɦɥɹɸɳɟɣ ɝɪɚɧɢɰɵ ɜɥɢɹɧɢɟ ɟɟ ɜɬɨɪɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɧɚ ve ɭɦɟɧɶɲɚɟɬɫɹ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɞɥɹ
ɩɨɥɭɱɟɧɢɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɩɥɚɫɬɨɜɨɣ ɫɤɨɪɨɫɬɢ ɦɨɠɧɨ ɧɟ ɭɱɢɬɵɜɚɬɶ ɜɥɢɹɧɢɟ ɜɬɨɪɵɯ
ɩɪɨɢɡɜɨɞɧɵɯ ɤɪɨɜɥɢ ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ, ɬɨ ɟɫɬɶ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɨɩɟɪɚɬɨɪɚ M ɩɨɥɚɝɚɬɶ
w 2F/wx 2=w 2F/wxwy =w 2F/wy 2 = 0. Ɍɨɝɞɚ ɢɡ (10) ɜɢɞɧɨ, ɱɬɨ Ⱥ1=Ⱥ2=Ⱥ3=0 ɢ ɢɡ ɭɪɚɜɧɟɧɢɹ (9) ɫɪɚɡɭ ɧɚɯɨɞɢɦ
vn:
vn = (- A4 /A5) 1/2.
ȼ ɮɨɪɦɭɥɵ (10) ɜɯɨɞɹɬ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɜɪɟɦɟɧɧɨɝɨ ɩɨɥɹ T1(s,[,K) ɜɞɨɥɶ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ,
ɷɬɢ ɩɪɨɢɡɜɨɞɧɵɟ ɦɨɠɧɨ ɧɚɣɬɢ ɦɟɬɨɞɨɦ, ɢɡɥɨɠɟɧɧɵɦ ɜ (Ȼɥɹɫ, 1985). ɂɬɚɤ, ɞɚɧɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɡɜɨɥɹɟɬ
ɧɚɣɬɢ ɫɤɨɪɨɫɬɶ ɜɞɨɥɶ ɰɟɧɬɪɚɥɶɧɨɝɨ ɥɭɱɚ, ɚ ɡɧɚɱɢɬ, ɢ ɬɨɱɤɭ ɨɬɪɚɠɟɧɢɹ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɥɨɤɚɥɶɧɨɣ
ɨɞɧɨɪɨɞɧɨɫɬɢ ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɵɲɟ ɡɚɥɟɝɚɸɳɢɟ ɫɥɨɢ ɧɟ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɥɨɤɚɥɶɧɨ
ɨɞɧɨɪɨɞɧɵɦɢ, ɬɚɤ ɤɚɤ ɞɨ ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ ɰɟɧɬɪɚɥɶɧɵɣ ɥɭɱ ɢ ɩɪɨɢɡɜɨɞɧɵɟ ɜɞɨɥɶ ɧɟɝɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ
ɫ ɭɱɟɬɨɦ ɤɪɢɜɢɡɧɵ ɥɭɱɟɣ (ɉɪɢɥɨɠɟɧɢɟ 2).
4. ɂɬɟɪɚɰɢɨɧɧɵɣ R-ɚɥɝɨɪɢɬɦ
Ɋɚɫɫɦɨɬɪɢɦ ɢɬɟɪɚɰɢɨɧɧɵɣ R-ɚɥɝɨɪɢɬɦ, ɨɩɢɪɚɸɳɢɣɫɹ ɧɚ ɢɡɥɨɠɟɧɧɵɣ ɢɧɜɟɪɫɧɵɣ ɦɟɬɨɞ
ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɚɫɬɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɢ ɨɬɪɚɠɚɸɳɢɯ ɝɪɚɧɢɰ ɜ ɬɪɟɯɦɟɪɧɵɯ ɫɥɨɢɫɬɵɯ ɫɪɟɞɚɯ. Ⱦɚɧɧɵɣ
ɚɥɝɨɪɢɬɦ ɬɚɤɠɟ ɪɚɛɨɬɚɟɬ ɩɨɫɥɨɣɧɨ, ɩɨɷɬɨɦɭ ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɡɜɟɫɬɧɵ (ɪɚɧɟɟ ɧɚɣɞɟɧɵ) ɩɚɪɚɦɟɬɪɵ
ɩɟɪɜɵɯ (n-1) ɫɥɨɟɜ – ɮɭɧɤɰɢɢ Fk(x,y), vk(x,y), k=1,2,..., n-1. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ vn(x,y) ɢ
ɨɬɪɚɠɚɸɳɭɸ ɩɨɜɟɪɯɧɨɫɬɶ z=Fn(x,y) n-ɝɨ ɫɥɨɹ ɩɨ ɢɡɜɟɫɬɧɵɦ ɮɭɧɤɰɢɹɦ t0, vcdp ɜɨɥɧ, ɨɬɪɚɠɟɧɧɵɯ ɨɬ ɷɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɩɨɥɭɱɟɧɧɵɯ ɩɨ ɩɨɥɟɜɵɦ ɫɟɣɫɦɨɝɪɚɦɦɚɦ. Ⱦɚɧɧɭɸ ɡɚɞɚɱɭ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɡɚɞɚɱɭ
ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɹ ɨɩɟɪɚɬɨɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
f(S)= d,
d = (t0,vCDP).
(13)
Ɂɞɟɫɶ S – ɢɫɤɨɦɚɹ ɦɨɞɟɥɶ, ɬɨ ɟɫɬɶ ɩɨɞɨɲɜɚ Fn(x,y) ɢ ɫɤɨɪɨɫɬɶ vn(x,y) ɜ ɫɥɨɟ: S = (Fn(x,y), vn(x,y)), f –
ɨɩɟɪɚɬɨɪ ɪɟɲɟɧɢɹ ɩɪɹɦɨɣ ɡɚɞɚɱɢ, ɞɟɣɫɬɜɢɟ ɤɨɬɨɪɨɝɨ ɫɜɨɞɢɬɫɹ ɤ ɫɥɟɞɭɸɳɢɦ ɨɩɟɪɚɰɢɹɦ. ɋɧɚɱɚɥɚ
ɱɢɫɥɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ ɞɥɹ ɡɚɞɚɧɧɨɣ ɦɨɞɟɥɢ S = (Fn(x,y), vn(x,y)) ɫɥɨɹ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɝɨɞɨɝɪɚɮɵ ɈȽɌ
ɞɥɹ ɡɚɞɚɧɧɨɝɨ ɧɚɛɨɪɚ ɩɪɨɮɢɥɟɣ. ɉɨɫɥɟ ɷɬɨɝɨ ɧɚɯɨɞɢɬɫɹ ɢɧɬɟɝɪɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ vCDP ɤɚɤ ɩɚɪɚɦɟɬɪ
ɝɢɩɟɪɛɨɥɵ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɝɨɞɨɝɪɚɮ ɈȽɌ tCDP(l) ɩɨ ɦɟɬɨɞɭ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ, ɬɨ ɟɫɬɶ ɢɡ
ɭɫɥɨɜɢɹ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ
99
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
I
Ɏ(vCDP) = ¦ [tCDP(li) - (t02 + li2/vCDP2)1/2]2,
(14)
i=1
ɝɞɟ li - ɪɚɫɫɬɨɹɧɢɟ ɜɡɪɵɜ-ɩɪɢɟɦ ɧɚ i-ɨɣ ɬɪɚɫɫɟ ɫɟɣɫɦɨɝɪɚɦɦɵ ɈȽɌ, I - ɤɨɥɢɱɟɫɬɜɨ ɬɨɱɟɤ ɜ ɝɨɞɨɝɪɚɮɟ ɈȽɌ.
ɍɪɚɜɧɟɧɢɟ (13) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɬɚɤɢɟ ɡɧɚɱɟɧɢɹ ɩɥɚɫɬɨɜɨɣ ɫɤɨɪɨɫɬɢ vn(x,y) ɢ
ɝɪɚɧɢɰɵ Fn(x,y), ɞɥɹ ɤɨɬɨɪɵɯ ɜɟɥɢɱɢɧɵ t0 , vCDP, ɪɚɫɫɱɢɬɚɧɧɵɟ ɱɢɫɥɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ, ɫɨɜɩɚɞɚɸɬ ɫ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ t0 , vCDP .
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɦɵ ɭɦɟɟɦ ɩɪɢɛɥɢɠɟɧɧɨ ɪɟɲɚɬɶ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ, ɬɨ ɟɫɬɶ ɭɦɟɟɦ ɫɬɪɨɢɬɶ
(ɱɢɫɥɟɧɧɨ ɢɥɢ ɚɧɚɥɢɬɢɱɟɫɤɢ) ɬɚɤɨɣ ɨɩɟɪɚɬɨɪ M, ɱɬɨ S | M(d), ɨɬɤɭɞɚ M | f -1, ɝɞɟ f -1 – ɨɩɟɪɚɬɨɪ, ɨɛɪɚɬɧɵɣ
ɨɩɟɪɚɬɨɪɭ f. Ɋɚɧɟɟ ɬɚɤɨɣ ɨɩɟɪɚɬɨɪ ɛɵɥ ɩɨɫɬɪɨɟɧ. Ɉɧ ɹɜɥɹɟɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦ, ɬɚɤ ɤɚɤ ɧɟ ɭɱɢɬɵɜɚɟɬ
ɤɪɢɜɢɡɧɭ ɥɭɱɚ ɜ ɨɰɟɧɢɜɚɟɦɨɦ ɫɥɨɟ ɢ ɨɬɥɢɱɢɟ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP ɨɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ve,
ɤɨɬɨɪɚɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɢɧɬɟɝɪɚɥɶɧɨɣ ɩɪɢ ɧɭɥɟɜɨɣ ɛɚɡɟ (ɉɭɡɵɪɟɜ, 1989). Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɨɩɟɪɚɬɨɪ M
ɩɨɫɬɪɨɟɧ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɡɧɚɱɟɧɢɹ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP ɨɩɪɟɞɟɥɟɧɵ ɩɪɢ ɦɚɥɵɯ ɛɚɡɚɯ
ɝɨɞɨɝɪɚɮɨɜ ɈȽɌ, ɚ ɬɚɤɠɟ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɥɨɤɚɥɶɧɨɣ ɨɞɧɨɪɨɞɧɨɫɬɢ ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɩɨɞɪɨɛɧɟɟ ɩɨɧɹɬɢɟ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP ɢ ɫɜɹɡɚɧɧɨɟ ɫ ɧɢɦ
ɩɨɧɹɬɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɲɚɝ 'l = li+1 - li ɞɨɫɬɚɬɨɱɧɨ ɦɚɥ, ɬɚɤ ɱɬɨ ɫɭɦɦɭ
ɩɨ i ɜ (14) ɦɨɠɧɨ ɡɚɦɟɧɢɬɶ ɧɚ ɢɧɬɟɝɪɚɥ (ɧɚ ɩɪɚɤɬɢɤɟ ɷɬɨ ɭɫɥɨɜɢɟ ɨɛɵɱɧɨ ɜɵɩɨɥɧɹɟɬɫɹ) (Ɇɟɲɛɟɣ, 1985). ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ɫɤɨɪɨɫɬɶ vCDP, ɦɢɧɢɦɢɡɢɪɭɸɳɚɹ ɮɭɧɤɰɢɸ Ɏ, ɡɚɜɢɫɢɬ ɨɬ L1, L2, ɝɞɟ L1, L2 – ɦɢɧɢɦɚɥɶɧɨɟ ɢ
ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɪɚɫɫɬɨɹɧɢɣ li ɜɡɪɵɜ-ɩɪɢɟɦ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ vCDP = vCDP(L1, L2). ȿɫɥɢ L1, L2
ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ, ɬɨ vCDP(L1, L2) ɫɬɪɟɦɢɬɫɹ ɤ ɧɟɤɨɬɨɪɨɦɭ ɡɧɚɱɟɧɢɸ, ɤɨɬɨɪɨɟ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɱɟɪɟɡ ve , ɢ
ɤɨɬɨɪɨɟ ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɟɥɶɧɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɶɸ. ȿɫɥɢ ɝɨɞɨɝɪɚɮ ɈȽɌ ɹɜɥɹɟɬɫɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦ
(ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ ɫ ɩɥɨɫɤɨɣ ɨɬɪɚɠɚɸɳɟɣ ɝɪɚɧɢɰɟɣ), ɬɨ vCDP ɧɟ ɡɚɜɢɫɢɬ ɨɬ L1, L2, ɢ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, vCDP = ve. ȼ ɛɨɥɟɟ ɫɥɨɠɧɵɯ ɦɨɞɟɥɹɯ ɪɟɚɥɶɧɵɯ ɫɪɟɞ (ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨɫɥɨɢɫɬɨɣ ɦɨɞɟɥɢ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ) vCDP ɨɬɥɢɱɚɟɬɫɹ ɨɬ ve.
ɍɦɟɧɶɲɟɧɢɟ ɛɚɡɵ ɝɨɞɨɝɪɚɮɚ ɩɪɢ ɩɪɨɜɟɞɟɧɢɢ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɩɪɢɜɨɞɢɬ ɤ ɫɢɥɶɧɨɦɭ
ɭɦɟɧɶɲɟɧɢɸ ɭɫɬɨɣɱɢɜɨɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ, ɧɨ, ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɩɪɢ ɷɬɨɦ ɜ ɛɨɥɶɲɢɧɫɬɜɟ
ɫɥɭɱɚɟɜ ɝɨɞɨɝɪɚɮ ɈȽɌ ɛɥɢɡɨɤ ɤ ɝɢɩɟɪɛɨɥɟ, ɩɨɷɬɨɦɭ ɨɬɥɢɱɢɟ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɨɬ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɧɟ ɨɱɟɧɶ ɜɟɥɢɤɨ. ȼɥɢɹɧɢɟ ɧɟɭɱɟɬɚ ɤɪɢɜɢɡɧɵ ɥɭɱɟɣ ɜ ɫɚɦɨɦ ɨɰɟɧɢɜɚɟɦɨɦ ɫɥɨɟ ɩɪɢ
ɨɩɪɟɞɟɥɟɧɢɢ ɟɝɨ ɫɤɨɪɨɫɬɢ ɢ ɦɨɳɧɨɫɬɢ ɬɚɤɠɟ ɧɟɜɟɥɢɤɨ (Ȼɥɹɫ, 1991). ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɧɟɭɱɟɬ ɨɬɥɢɱɢɹ
ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP ɨɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ve ɢ ɧɟɭɱɟɬ ɤɪɢɜɢɡɧɵ ɥɭɱɚ ɜ ɨɰɟɧɢɜɚɟɦɨɦ (!) ɫɥɨɟ ɧɟ
ɩɪɢɜɨɞɢɬ ɤ ɛɨɥɶɲɢɦ ɩɨɝɪɟɲɧɨɫɬɹɦ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɩɥɚɫɬɨɜɨɣ ɫɤɨɪɨɫɬɢ ɢ ɦɨɳɧɨɫɬɢ, ɩɨɷɬɨɦɭ ɨɩɟɪɚɬɨɪ M
ɜ ɫɚɦɨɦ ɞɟɥɟ ɞɚɟɬ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ, ɬɨ ɟɫɬɶ M | f -1. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɟɫɥɢ ɞɥɹ
ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ ɩɪɢɦɟɧɹɬɶ ɬɨɥɶɤɨ ɨɩɟɪɚɬɨɪ M, ɬɨ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɚɬɢɱɟɫɤɢɟ ɩɨɝɪɟɲɧɨɫɬɢ,
ɤɨɬɨɪɵɟ ɯɨɬɹ ɢ ɧɟɜɟɥɢɤɢ, ɧɨ ɦɨɝɭɬ ɫɭɳɟɫɬɜɟɧɧɨ ɩɨɜɥɢɹɬɶ ɧɚ ɪɟɡɭɥɶɬɚɬɵ ɝɟɨɥɨɝɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ,
ɨɫɨɛɟɧɧɨ ɜ ɪɚɣɨɧɚɯ ɫ ɩɥɚɬɮɨɪɦɟɧɧɵɦ ɡɚɥɟɝɚɧɢɟɦ ɩɨɪɨɞ ɢ ɦɚɥɨɚɦɩɥɢɬɭɞɧɵɦɢ ɫɬɪɭɤɬɭɪɚɦɢ.
Ⱦɥɹ ɭɱɟɬɚ ɨɬɥɢɱɢɹ vCDP ɨɬ ve ɪɚɫɫɦɨɬɪɢɦ ɫɥɟɞɭɸɳɢɣ ɢɬɟɪɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ:
Sm+1 = Sm - M ( f (Sm)) + M (d ),
S1 = M(d),
m = 1,2,... .
(15)
Ⱦɟɣɫɬɜɢɟ ɞɚɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ ɫɜɨɞɢɬɫɹ ɤ ɫɥɟɞɭɸɳɟɦɭ. ɇɚ ɩɟɪɜɨɦ ɲɚɝɟ ɨɩɟɪɚɬɨɪ M ɩɪɢɦɟɧɹɟɬɫɹ ɤ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɞɚɧɧɵɦ d (ɤ ɡɧɚɱɟɧɢɹɦ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP ɢ ɜɪɟɦɟɧɢ t0 ɩɨ ɩɪɨɮɢɥɹɦ ɈȽɌ),
ɟɝɨ ɩɪɢɦɟɧɟɧɢɟ ɞɚɟɬ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ – ɩɨɥɭɱɚɟɦ S1 = M (d ). Ɍɚɤ ɤɚɤ ɨɩɟɪɚɬɨɪ M
ɞɚɟɬ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɬɨɥɶɤɨ ɞɥɹ ɩɪɟɞɟɥɶɧɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ ɢ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɨɝɨ
ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ, ɬɨ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ S1 ɛɭɞɟɬ ɩɪɢɛɥɢɠɟɧɧɵɦ. Ʉ ɩɨɥɭɱɟɧɧɨɣ ɧɚ ɩɟɪɜɨɦ ɲɚɝɟ
ɦɨɞɟɥɢ S1 ɩɪɢɦɟɧɹɟɬɫɹ ɨɩɟɪɚɬɨɪ f ɪɟɲɟɧɢɹ ɩɪɹɦɨɣ ɡɚɞɚɱɢ – ɞɥɹ ɞɚɧɧɨɣ ɦɨɞɟɥɢ ɱɢɫɥɟɧɧɨ
ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɝɨɞɨɝɪɚɮɵ ɈȽɌ tCDP( l ), ɤɨɬɨɪɵɟ ɚɩɩɪɨɤɫɢɦɢɪɭɸɬɫɹ ɝɢɩɟɪɛɨɥɚɦɢ. ɉɚɪɚɦɟɬɪɵ
ɝɢɩɟɪɛɨɥɵ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɢɡ ɭɫɥɨɜɢɹ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ (14). Ʉ ɩɨɥɭɱɟɧɧɵɦ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɡɧɚɱɟɧɢɹɦ
(ɱɢɫɥɟɧɧɨ ɪɚɫɫɱɢɬɚɧɧɵɦ ɞɥɹ ɦɨɞɟɥɢ S1, t02, vCDP) ɩɪɢɦɟɧɹɟɬɫɹ ɨɩɟɪɚɬɨɪ M, ɤɨɬɨɪɵɣ ɞɚɟɬ ɦɨɞɟɥɶ
S1c = M ( f (S1)). Ɂɚɬɟɦ ɦɨɞɟɥɶ S1 ɤɨɪɪɟɤɬɢɪɭɟɬɫɹ ɞɨɛɚɜɥɟɧɢɟɦ ɜɟɥɢɱɢɧɵ 'S1 = S1 - S1c: S2 = S1 + 'S1. ɇɚ mɨɦ ɲɚɝɟ ɧɚɯɨɞɢɦ Smc = M ( f (Sm)), 'Sm = Sm - Smc ɢ ɧɨɜɭɸ ɦɨɞɟɥɶ Sm+1 = Sm + 'Sm.
ȿɫɥɢ M = f -1 (ɬɨ ɟɫɬɶ ɧɚ ɩɟɪɜɨɦ ɲɚɝɟ ɩɨɥɭɱɚɟɦ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ S1 = S* ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ), ɬɨ
S1c = M ( f (S1)) = S* ɢ 'S1 = S1 - S1c = 0. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɫɟ ɦɨɞɟɥɢ Sm ɫɨɜɩɚɞɚɸɬ, ɢ ɩɟɪɜɚɹ ɢɬɟɪɚɰɢɹ ɞɚɟɬ
ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ. ȿɫɥɢ M z f -1, ɬɨ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɚɥɝɨɪɢɬɦɚ ɩɨɥɭɱɚɟɦ ɪɚɡɥɢɱɧɵɟ ɦɨɞɟɥɢ
S1, S2,..., Sm,... .
ȼɨɡɧɢɤɚɟɬ ɜɨɩɪɨɫ ɨ ɫɯɨɞɢɦɨɫɬɢ ɞɚɧɧɨɝɨ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ ɢ ɨ ɦɨɞɟɥɢ, ɤ ɤɨɬɨɪɨɣ ɨɧ
ɫɯɨɞɢɬɫɹ. ɂɫɫɥɟɞɭɟɦ ɫɯɨɞɢɦɨɫɬɶ ɚɥɝɨɪɢɬɦɚ. Ɋɚɜɟɧɫɬɜɨ (15) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
100
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
Sm+1 = Ⱥ(Sm),
ɫɬɪ.95-112
Ⱥ(S) = S - M ( f (S)) + M (d),
m = 1,2,.. .
(16)
ɇɚɣɞɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɨɩɟɪɚɬɨɪɚ Ⱥ. ɂɡ ɪɚɜɟɧɫɬɜɚ (16) ɫɥɟɞɭɟɬ, ɱɬɨ Ac = E - Mc fc, ɝɞɟ ȿ –
ɟɞɢɧɢɱɧɵɣ ɨɩɟɪɚɬɨɪ. Ɍɚɤ ɤɚɤ M | f--1, ɬɨ M c | (f c)-1, ɨɬɤɭɞɚ E | M cf c ɢ ||Ac|| = ||E - M cf c|| | 0, ɝɞɟ ||Ⱥ|| – ɧɨɪɦɚ
ɨɩɟɪɚɬɨɪɚ Ⱥ. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɨɩɟɪɚɬɨɪ Ⱥ ɹɜɥɹɟɬɫɹ ɫɠɢɦɚɸɳɢɦ (ɟɝɨ ɧɨɪɦɚ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ), ɢ, ɩɨ
ɬɟɨɪɟɦɟ ɨɛ ɨɩɟɪɚɬɨɪɟ ɫɠɚɬɢɹ, ɢɬɟɪɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ (16) ɫɯɨɞɢɬɫɹ (Ʉɨɥɦɨɝɨɪɨɜ, Ɏɨɦɢɧ, 1968).
ɋɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɡɚɜɢɫɢɬ ɨɬ ||Ⱥc|| ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
||Sm - S*|| d ||Ac||m ||A(S1) - S1|| (1- ||Ac|| ) -1 ,
ɝɞɟ S* – ɦɨɞɟɥɶ ɫɪɟɞɵ, ɤ ɤɨɬɨɪɨɣ ɫɯɨɞɢɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ Sn ɩɪɢ n o f. ɉɟɪɟɯɨɞɹ ɜ ɪɚɜɟɧɫɬɜɟ (15) ɤ
ɩɪɟɞɟɥɭ ɩɪɢ n o f, ɩɨɥɭɱɢɦ, ɱɬɨ S* ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ
M ( f (S)) = M (d).
ȿɫɥɢ ɨɩɟɪɚɬɨɪ M ɢɦɟɟɬ ɨɛɪɚɬɧɵɣ, ɬɨ ɢɡ ɩɨɫɥɟɞɧɟɝɨ ɪɚɜɟɧɫɬɜɚ ɫɥɟɞɭɟɬ, ɱɬɨ S* ɭɞɨɜɥɟɬɜɨɪɹɟɬ
ɭɪɚɜɧɟɧɢɸ (13), ɬɨ ɟɫɬɶ S* ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ
ɜɵɛɨɪɚ ɨɩɟɪɚɬɨɪɚ M. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɜɚɠɧɵɣ ɜɵɜɨɞ: ɨɩɟɪɚɬɨɪ M ɦɨɠɧɨ ɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɜ ɛɨɥɟɟ ɩɪɨɫɬɨɣ
ɦɨɞɟɥɢ ɫɪɟɞɵ (ɢɥɢ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɫɯɟɦɟ ɧɚɛɥɸɞɟɧɢɣ) ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɧɚɦ ɢɡɜɟɫɬɧɵ ɢɫɯɨɞɧɵɟ
ɞɚɧɧɵɟ, ɤɨɬɨɪɵɟ ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɩɨ ɪɟɚɥɶɧɵɦ ɫɟɣɫɦɨɝɪɚɦɦɚɦ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɵ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ
ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ (ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (13)) ɧɟɨɛɯɨɞɢɦɨ ɜ ɨɩɟɪɚɬɨɪɟ f ɪɟɲɟɧɢɹ ɩɪɹɦɨɣ ɡɚɞɚɱɢ
ɭɱɢɬɵɜɚɬɶ ɛɨɥɟɟ ɫɥɨɠɧɭɸ ɦɨɞɟɥɶ, ɱɟɦ ɬɚ, ɤɨɬɨɪɚɹ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɨɩɟɪɚɬɨɪɚ M. ȼ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɪɟɲɟɧɢɢ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ (ɩɨɫɬɪɨɟɧɢɢ ɨɩɟɪɚɬɨɪɚ M)
ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɨɞɟɥɶ ɫ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦ ɨɰɟɧɢɜɚɟɦɵɦ ɫɥɨɟɦ (ɞɥɹ ɤɨɬɨɪɨɝɨ ɬɨɥɶɤɨ ɢ ɦɨɠɧɨ
ɩɨɫɬɪɨɢɬɶ ɤɨɧɫɬɪɭɤɬɢɜɧɵɣ ɚɥɝɨɪɢɬɦ ɨɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɦɨɳɧɨɫɬɢ) ɢ ɫɱɢɬɚɬɶ, ɱɬɨ ɢɡɜɟɫɬɧɚ
ɩɪɟɞɟɥɶɧɚɹ ɷɮɮɟɤɬɢɜɧɚɹ ɫɤɨɪɨɫɬɶ (ɯɨɬɹ ɧɚ ɩɪɚɤɬɢɤɟ ɨɧɚ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɭɫɬɨɣɱɢɜɨ ɨɩɪɟɞɟɥɟɧɚ ɩɨ
ɝɨɞɨɝɪɚɮɭ, ɬɚɤ ɤɚɤ ɟɟ ɧɟɫɦɟɳɟɧɧɚɹ ɨɰɟɧɤɚ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɭɸ ɞɢɫɩɟɪɫɢɸ). ɉɪɢ ɪɚɫɱɟɬɟ ɠɟ ɬɪɚɟɤɬɨɪɢɣ ɢ
ɜɪɟɦɟɧ ɜ ɩɪɹɦɨɣ ɡɚɞɚɱɟ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɤɪɢɜɢɡɧɭ ɥɭɱɟɣ, ɚ ɡɚɬɟɦ ɪɚɫɫɱɢɬɵɜɚɬɶ ɢɧɬɟɝɪɚɥɶɧɭɸ
ɫɤɨɪɨɫɬɶ ɩɨ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɝɨɞɨɝɪɚɮɨɜ ɈȽɌ.
ɂɬɚɤ, ɞɚɧɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɧɭɸ ɦɨɞɟɥɶ ɫ ɭɱɟɬɨɦ ɤɪɢɜɢɡɧɵ ɥɭɱɟɣ ɜ
ɝɨɪɢɡɨɧɬɚɥɶɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɯ ɫɥɨɹɯ ɧɟɫɦɨɬɪɹ ɧɚ ɬɨ, ɱɬɨ ɢɧɜɟɪɫɧɵɣ ɦɟɬɨɞ ɩɪɟɞɩɨɥɚɝɚɟɬ ɥɨɤɚɥɶɧɭɸ
ɨɞɧɨɪɨɞɧɨɫɬɶ ɨɰɟɧɢɜɚɟɦɨɝɨ ɫɥɨɹ.
5. Ɉɩɬɢɦɢɡɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ɩɨ
ɝɨɞɨɝɪɚɮɚɦ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ. Ɍɚɤ ɤɚɤ ɧɚ ɩɪɚɤɬɢɤɟ ɩɪɢ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɧɚɛɥɸɞɟɧɢɹɯ ɜ ɛɨɥɶɲɢɧɫɬɜɟ
ɫɥɭɱɚɟɜ ɨɬɪɚɠɟɧɧɵɟ ɜɨɥɧɵ ɜɵɞɟɥɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ, ɚ ɧɟ ɢɯ ɩɨɬɪɚɫɫɧɨɣ
ɤɨɪɪɟɥɹɰɢɟɣ, ɬɨ ɧɚɦ ɢɡɜɟɫɬɧɵ ɧɟ ɫɚɦɢ ɝɨɞɨɝɪɚɮɵ, ɚ ɢɯ ɚɩɩɪɨɤɫɢɦɚɰɢɢ, ɱɚɳɟ ɜɫɟɝɨ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɟ
(Ɇɟɲɛɟɣ, 1985). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨ ɜɵɞɟɥɟɧɧɵɦ ɝɢɩɟɪɛɨɥɚɦ ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɝɨɞɨɝɪɚɮɵ ɢ ɫɱɢɬɚɬɶ, ɱɬɨ
ɧɚɦ ɢɡɜɟɫɬɧɵ ɜɪɟɦɟɧɚ ɩɪɢɯɨɞɚ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɩɪɢ ɫɤɜɚɠɢɧɧɵɯ ɫɟɣɫɦɢɱɟɫɤɢɯ
ɧɚɛɥɸɞɟɧɢɹɯ ɨɛɵɱɧɨ ɢɡɜɟɫɬɧɵ ɫɚɦɢ ɝɨɞɨɝɪɚɮɵ.
ɉɭɫɬɶ, ɤɚɤ ɢ ɪɚɧɟɟ, Fk(x,y), vk(x,y,z) – ɮɭɧɤɰɢɢ, ɨɩɢɫɵɜɚɸɳɢɟ ɝɪɚɧɢɰɵ ɫɥɨɟɜ ɢ ɩɥɚɫɬɨɜɵɟ
ɫɤɨɪɨɫɬɢ ɫɪɟɞɵ, ɫɤɨɪɨɫɬɧɭɸ ɦɨɞɟɥɶ ɤɨɬɨɪɨɣ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ S. ɑɟɪɟɡ i ɨɛɨɡɧɚɱɢɦ ɧɨɦɟɪ ɬɨɱɤɢ
ɧɚɛɥɸɞɟɧɢɹ, i=1,2,..., I; p – ɧɨɦɟɪ ɜɨɥɧɵ ɜ ɦɧɨɠɟɫɬɜɟ ɜɵɞɟɥɟɧɧɵɯ ɜɨɥɧ, p=1,2,...,P. ɉɨɞ i-ɨɣ ɬɨɱɤɨɣ
ɧɚɛɥɸɞɟɧɢɹ ɩɨɧɢɦɚɟɬɫɹ ɩɚɪɚ ɢɫɬɨɱɧɢɤ-ɩɪɢɟɦɧɢɤ, ɢɦɟɸɳɚɹ ɧɨɦɟɪ i ɜ ɨɛɳɟɣ ɧɭɦɟɪɚɰɢɢ. ɑɟɪɟɡ tip
ɨɛɨɡɧɚɱɢɦ ɢɡɜɟɫɬɧɨɟ ɢɡ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɜɪɟɦɹ ɩɪɨɛɟɝɚ p-ɨɣ ɜɨɥɧɵ ɜ i-ɨɣ ɬɨɱɤɟ ɧɚɛɥɸɞɟɧɢɹ,
tip(S') – "ɬɟɨɪɟɬɢɱɟɫɤɨɟ" ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ, ɪɚɫɫɱɢɬɚɧɧɨɟ ɞɥɹ ɦɨɞɟɥɢ Sc. Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ
ɬɪɟɯɦɟɪɧɵɟ ɦɨɞɟɥɢ ɫɪɟɞ, ɜ ɤɨɬɨɪɵɯ ɝɪɚɧɢɰɵ ɫɥɨɟɜ Fk(x,y) ɢ ɩɥɚɫɬɨɜɵɟ ɦɟɞɥɟɧɧɨɫɬɢ vk-1(x,y,z)
ɨɩɢɫɵɜɚɸɬɫɹ ɥɢɧɟɣɧɵɦɢ ɤɨɦɛɢɧɚɰɢɹɦɢ ɡɚɪɚɧɟɟ ɜɵɛɪɚɧɧɵɯ ɛɚɡɢɫɧɵɯ ɮɭɧɤɰɢɣ:
vk-1(x,y,z)= Dj(k) Mj(x,y,z),
k=1,...,K
(17)
Fk(x,y)= Ej(k) \j(x,y,z).
Ɇɟɬɨɞ ɨɩɬɢɦɢɡɚɰɢɢ ɫɜɨɞɢɬ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ ɤ ɧɚɯɨɠɞɟɧɢɸ ɤɨɷɮɮɢɰɢɟɧɬɨɜ Dj(k), Ej(k) ɢɡ ɭɫɥɨɜɢɹ
ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɨɧɚɥɚ Ɏ, ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɪɚɜɟɧɫɬɜɨɦ
101
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
Ɏ(D,E) = ¦ (tip - tip(D,E))2qip ,
(18)
i,p
ɝɞɟ qip – ɜɟɫɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɩɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ tip, D, E – ɜɟɤɬɨɪɵ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ,
ɫɨɫɬɨɹɳɢɦɢ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɣ (17):
D = (D1(1), D2(1),...DJv(1), D1(2),..., DJv(2),..., D1(K), D2(K),..., DJv(K)),
(19)
(1)
(1)
E = (E1 , E2 ,..., E
(1)
JF ,
(2)
E1 ,..., E
(2)
JF ,...,
(K)
(K)
E1 , E2 ,..., E
(K)
JF ).
ɑɟɪɟɡ Pn([n(i,p), Kn(i,p), Fn([n(i,p),Kn(i,p))) ɨɛɨɡɧɚɱɢɦ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ p-ɨɣ ɜɨɥɧɵ ɫ ɝɪɚɧɢɰɚɦɢ
ɫɥɨɟɜ, ɩɟɪɟɧɭɦɟɪɨɜɚɧɧɵɯ ɜ ɩɨɪɹɞɤɟ ɜɫɬɪɟɱɢ ɫ ɥɭɱɨɦ, n=1,...,N. ȼɪɟɦɹ Wip(D,E) ɩɪɨɛɟɝɚ p-ɨɣ ɜɨɥɧɵ ɜ i-ɨɣ
ɬɨɱɤɟ ɧɚɛɥɸɞɟɧɢɹ ɜ ɦɨɞɟɥɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
N
Wip(D,E) = ¦ tn([n-1(i,p), Kn-1(i,p), [n(i,p), Kn(i,p)) = T(ip)(D,E,[(i,p),K(i,p)),
(20)
n=1
ɝɞɟ tn([n-1(i,p), Kn-1(i,p), [n(i,p), Kn(i,p)) – ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɦɟɠɞɭ ɬɨɱɤɚɦɢ Pn-1, Pn, ɬɨ ɟɫɬɶ ɜɪɟɦɹ ɩɪɨɛɟɝɚ
ɜɨɥɧɵ ɜ n-ɨɦ ɫɥɨɟ, [(i,p) = ([1(i,p), [2(i,p),..., [N(i,p)), K(i,p)=(K1(i,p), K2(i,p),..., KN(i,p)) – ɜɟɤɬɨɪɵ, ɨɩɢɫɵɜɚɸɳɢɟ
ɬɪɚɟɤɬɨɪɢɸ p-ɨɣ ɜɨɥɧɵ ɜ i-ɨɣ ɬɨɱɤɟ ɧɚɛɥɸɞɟɧɢɹ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɬɚɰɢɨɧɚɪɧɨɫɬɢ
wT(i,p)/w[n(i,p)=0, wT(i,p)/wKn(i,p) = 0,
n=1,...,N.
(21)
ɂɬɟɪɚɰɢɨɧɧɵɣ ɚɥɝɨɪɢɬɦ ɧɚɯɨɠɞɟɧɢɹ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ Ɏ, ɤɨɬɨɪɚɹ ɨɩɢɫɵɜɚɟɬ ɤɜɚɞɪɚɬɢɱɧɨɟ
ɨɬɤɥɨɧɟɧɢɟ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɜɪɟɦɟɧ ɨɬ ɪɟɚɥɶɧɵɯ, ɫɯɨɞɢɬɫɹ ɡɚ ɧɟɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɢɬɟɪɚɰɢɣ. ɇɚ ɤɚɠɞɨɣ
ɢɬɟɪɚɰɢɢ ɜɵɩɨɥɧɹɟɬɫɹ ɥɢɧɟɚɪɢɡɚɰɢɹ ɮɭɧɤɰɢɨɧɚɥɚ Ɏ ɢ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɬɪɚɟɤɬɨɪɢɢ (ɱɢɫɥɨ ɪɚɫɱɟɬɨɜ
ɬɪɚɟɤɬɨɪɢɣ ɫɨɜɩɚɞɚɟɬ ɫ ɱɢɫɥɨɦ ɢɬɟɪɚɰɢɣ), ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɦɢɧɢɦɭɦɚ ɥɢɧɟɚɪɢɡɨɜɚɧɧɨɝɨ ɮɭɧɤɰɢɨɧɚɥɚ
ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɦɟɬɨɞɵ ɤɜɚɞɪɚɬɢɱɧɨɣ ɦɢɧɢɦɢɡɚɰɢɢ, ɤɨɬɨɪɵɟ ɫɯɨɞɹɬɫɹ ɡɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɲɚɝɨɜ.
Ɋɚɫɫɦɨɬɪɢɦ m-ɵɣ ɲɚɝ ɚɥɝɨɪɢɬɦɚ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɨɧɚɥɚ Ɏ. ɉɭɫɬɶ S0 = (D0,E0) – ɦɨɞɟɥɶ,
ɩɨɥɭɱɟɧɧɚɹ ɧɚ m-ɨɣ ɢɬɟɪɚɰɢɢ. ɑɟɪɟɡ {[0(i,p),K0(i,p)} ɨɛɨɡɧɚɱɢɦ ɬɪɚɟɤɬɨɪɢɸ ɪ-ɨɣ ɜɨɥɧɵ ɜ ɦɨɞɟɥɢ S.
Ɋɚɡɥɨɠɢɦ ɮɭɧɤɰɢɸ Tip(D, E, [(i,p), K(i,p)) ɜ ɪɹɞ Ɍɷɣɥɨɪɚ ɜ ɬɨɱɤɟ (D0,E0, [0(i,p),K0(i,p)). ɋ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ
[=[(D,E), K=K(D,E) ɢ ɪɚɜɟɧɫɬɜ (21) ɩɨɥɭɱɢɦ, ɱɬɨ
T(i,p) = T0(i,p) + wT0(i,p)/wD (D-D0) + wT0(i,p)/wE (E-E0) + R(D, E),
(22)
ɝɞɟ R(D,E) – ɜɟɥɢɱɢɧɚ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ||D-D0||, ||E-E0||; wT0( i,p)/wD, wT0( i,p)/wE –
ɜɟɤɬɨɪɵ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ
wT0 ( i,p)/wD = (wT0 ( i,p)/wD1 (1), wT0 ( i,p)/wD2 (1),..., wT0 ( i,p)/wDJv(1), wT0 ( i,p)/wD1(2) ,..., wT0 ( i,p)/wDJ u(1)),
wT0 ( i,p)/wE = (wT0 ( i,p)/wE1 (1), wT0 ( i,p)/wE2 (1),..., wT0 ( i,p)/wEJv(1), wT0 ( i,p)/wE1(2) ,..., wT0 ( i,p)/wEJ u(1)),
“0” ɜ T(i,p) ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɦɢ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɨɧɢ ɛɟɪɭɬɫɹ ɜ ɬɨɱɤɟ (D0,E0,[0(i,p),K0(i,p)).
ɉɨɞɫɬɚɜɢɦ (22) ɜ (18) ɢ ɨɝɪɚɧɢɱɢɦɫɹ ɭɱɟɬɨɦ ɥɢɧɟɣɧɵɯ ɱɥɟɧɨɜ ɩɨ D-D0, E-E0. Ɍɨɝɞɚ ɩɨɥɭɱɢɦ, ɱɬɨ
Ɏ=¦ [tip - T(i,p) - wT0 ( i,p)/wD (D-D0) - wT0 ( i,p)/wE (E - E0)]2qip= Ɏ(D,E).
(23)
i,p
Ɍɚɤ ɤɚɤ ɮɭɧɤɰɢɹ T( i,p) ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ wT0 ( i,p)/wD, wT0 ( i,p)/wE ɜɵɱɢɫɥɹɸɬɫɹ ɜ ɢɡɜɟɫɬɧɨɣ ɦɨɞɟɥɢ S0, ɬɨ
ɧɟɢɡɜɟɫɬɧɵɟ D, E ɜɯɨɞɹɬ ɜ Ɏ ɤɜɚɞɪɚɬɢɱɧɨ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɡɚɞɚɱɚ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ Ɏ ɫɜɨɞɢɬɫɹ ɤ
ɦɢɧɢɦɢɡɚɰɢɢ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɭɧɤɰɢɢ ɩɨ ɤɨɨɪɞɢɧɚɬɚɦ ɜɟɤɬɨɪɨɜ D-D0, E-E0 . ɗɬɚ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɡɚ
ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɲɚɝɨɜ, ɧɚɩɪɢɦɟɪ, ɦɟɬɨɞɨɦ ɫɨɩɪɹɠɟɧɧɵɯ ɝɪɚɞɢɟɧɬɨɜ.
ɉɭɫɬɶ 'tip = tip-T0(i,p) – ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɧɚɛɥɸɞɚɟɦɵɦɢ ɜɪɟɦɟɧɚɦɢ ti,p ɢ ɜɪɟɦɟɧɚɦɢ T(i,p),
ɪɚɫɫɱɢɬɚɧɧɵɦɢ ɞɥɹ ɦɨɞɟɥɢ S0, 'D =D - D0, 'E =E - E0 – ɢɫɤɨɦɵɟ ɞɨɛɚɜɤɢ ɤ ɡɧɚɱɟɧɢɹɦ D, E ɦɨɞɟɥɢ S0,
ɩɨɥɭɱɟɧɧɨɣ ɧɚ ɩɪɟɞɵɞɭɳɟɣ ɢɬɟɪɚɰɢɢ. Ɍɨɝɞɚ ɮɭɧɤɰɢɨɧɚɥ Ɏ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
Ɏ = ¦('ti,p - wT0 ( i,p)/wD 'D - wT0 ( i,p)/wE 'E)2qi,p.
i,p
102
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ɫɬɪ.95-112
ȼ Ɏ ɜɯɨɞɹɬ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ T(i,p) ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦ Dj(k), Ej(k) ɪɚɡɥɨɠɟɧɢɣ ɩɥɚɫɬɨɜɵɯ
ɫɤɨɪɨɫɬɟɣ ɢ ɝɪɚɧɢɰ ɫɥɨɟɜ. Ⱦɥɹ i-ɨɣ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ ɷɬɢ ɩɪɨɢɡɜɨɞɧɵɟ ɫɱɢɬɚɸɬɫɹ ɜɞɨɥɶ ɬɪɚɟɤɬɨɪɢɢ,
ɫɨɟɞɢɧɹɸɳɟɣ ɢɫɬɨɱɧɢɤ ɢ ɩɪɢɟɦɧɢɤ, k – ɧɨɦɟɪ ɫɥɨɹ. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɷɬɢɯ ɩɪɨɢɡɜɨɞɧɵɯ ɞɨɫɬɚɬɨɱɧɨ
ɪɟɲɢɬɶ ɨɞɢɧ ɪɚɡ ɩɪɹɦɭɸ ɡɚɞɚɱɭ – ɧɚɣɬɢ ɬɪɚɟɤɬɨɪɢɢ ɜ ɦɨɞɟɥɢ S0 = (D0,E0), ɚ ɡɚɬɟɦ ɹɜɧɵɦ
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɮɭɧɤɰɢɢ (20) ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɵɟ ɜɪɟɦɟɧɢ Ɍ(i,p) ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦ Dj(k), Ej(k)
ɪɚɡɥɨɠɟɧɢɣ (17) ɝɪɚɧɢɰ ɢ ɫɤɨɪɨɫɬɟɣ.
ȼ ɮɭɧɤɰɢɸ Ɍ(i,p) ɜɯɨɞɹɬ ɜɪɟɦɟɧɚ tk ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜɨ ɜɫɟɯ ɩɟɪɟɫɟɤɚɟɦɵɯ ɫɥɨɹɯ, k = 1,2,...,n. ɉɪɢ
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɢ ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦ Dj(k) ɫɤɨɪɨɫɬɢ k-ɝɨ ɫɥɨɹ ɨɫɬɚɧɭɬɫɹ ɬɨɥɶɤɨ ɜɪɟɦɟɧɚ ɩɪɨɛɟɝɚ ɜ ɷɬɨɦ
ɫɥɨɟ, ɟɫɥɢ ɠɟ ɦɵ ɧɚɯɨɞɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦ Ej(k) n-ɨɣ ɝɪɚɧɢɰɵ, ɬɨ ɨɫɬɚɸɬɫɹ ɩɪɨɢɡɜɨɞɧɵɟ
ɜɪɟɦɟɧ ɜ ɫɥɨɹɯ, ɪɚɡɞɟɥɹɟɦɵɯ ɷɬɨɣ ɝɪɚɧɢɰɟɣ, ɬɨ ɟɫɬɶ ɜ ɫɥɨɹɯ ɫ ɧɨɦɟɪɚɦɢ k-1 ɢ k.
ȿɫɥɢ ɫɥɨɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɥɨɤɚɥɶɧɨ-ɨɞɧɨɪɨɞɧɵɦɢ, ɬɨ ɜɪɟɦɹ tk ɜ k-ɨɦ ɫɥɨɟ ɦɨɠɧɨ ɧɚɣɬɢ ɩɨ ɹɜɧɨɣ
ɮɨɪɦɭɥɟ
tk = {[Fk([k,Kk) – Fk-1([k-1,Kk-1)]2 + ([k–[k-1)2 + (Kk–Kk-1)2}2 mk ,
(24)
ɝɞɟ mk=vk-1(([k-1+[k)/2,(Kk-1+Kk)/2,(Fk-1+Fk)/2) – ɡɧɚɱɟɧɢɟ ɦɟɞɥɟɧɧɨɫɬɢ ɜ ɫɪɟɞɧɟɣ ɬɨɱɤɟ ɨɬɪɟɡɤɚ [Pk-1, Pk].
ɉɨɞɫɬɚɜɥɹɹ ɜ (24) ɪɚɜɟɧɫɬɜɚ (17), ɩɨɥɭɱɢɦ ɹɜɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɜɪɟɦɟɧɢ tk ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɨɜ Dj(k), Ej(k-1),
Ej(k) ɩɨ ɤɨɬɨɪɨɣ ɥɟɝɤɨ ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɵɟ. Ⱦɥɹ ɫɪɟɞɵ ɫɨ ɫɥɚɛɨ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ ɜ ɉɪɢɥɨɠɟɧɢɢ 2
ɩɨɥɭɱɟɧɚ ɹɜɧɚɹ ɮɨɪɦɭɥɚ ɞɥɹ ɜɪɟɦɟɧɢ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜ ɫɥɨɟ, ɩɨ ɤɨɬɨɪɨɣ ɬɚɤɠɟ ɦɨɠɧɨ ɧɚɣɬɢ
ɩɪɨɢɡɜɨɞɧɵɟ, ɟɫɥɢ ɜ ɧɟɟ ɩɨɞɫɬɚɜɢɬɶ ɪɚɜɟɧɫɬɜɚ (17). ȿɫɥɢ ɠɟ ɫɥɨɣ ɫɭɳɟɫɬɜɟɧɧɨ ɧɟɨɞɧɨɪɨɞɧɵɣ, ɢ
ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɜ ɧɟɦ ɧɟɥɶɡɹ ɭɱɟɫɬɶ ɦɟɬɨɞɨɦ ɜɨɡɦɭɳɟɧɢɣ, ɬɨ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɣ ɧɚɞɨ
ɩɪɢɦɟɧɹɬɶ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɚ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ
ɩɪɨɢɡɜɨɞɧɵɯ ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɦɟɬɨɞ, ɩɪɟɞɥɨɠɟɧɧɵɣ ɚɜɬɨɪɨɦ (Ȼɥɹɫ, 1985).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɯɨɠɞɟɧɢɟ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɨɧɚɥɚ Ɏ ɫɜɨɞɢɬɫɹ ɤ ɧɚɯɨɠɞɟɧɢɸ ɦɢɧɢɦɭɦɚ
ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɭɧɤɰɢɢ Ɏ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɚɥɝɨɪɢɬɦɚ. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɷɬɨɣ
ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɭɧɤɰɢɢ ɧɟɨɛɯɨɞɢɦɨ ɨɞɢɧ ɪɚɡ ɪɟɲɢɬɶ ɩɪɹɦɭɸ ɡɚɞɚɱɭ – ɪɚɫɫɱɢɬɚɬɶ ɬɪɚɟɤɬɨɪɢɢ ɜɨɥɧ ɞɥɹ
ɡɚɞɚɧɧɨɣ ɫɯɟɦɵ ɧɚɛɥɸɞɟɧɢɣ, ɩɨɫɥɟ ɱɟɝɨ ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ T0 ( i,p), ɜɯɨɞɹɳɢɟ ɜ (23).
ɇɚɯɨɠɞɟɧɢɟ ɷɬɢɯ ɩɪɨɢɡɜɨɞɧɵɯ ɫɜɨɞɢɬɫɹ ɤ ɧɚɯɨɠɞɟɧɢɸ ɩɪɨɢɡɜɨɞɧɵɯ ɜɪɟɦɟɧ ɩɪɨɛɟɝɚ ɜ ɨɬɞɟɥɶɧɵɯ ɫɥɨɹɯ.
ɑɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɟɫɥɢ ɩɥɚɫɬɨɜɵɟ ɫɤɨɪɨɫɬɢ ɦɨɞɟɥɢ ɧɭɥɟɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ
(ɚɩɪɢɨɪɧɨɣ ɦɨɞɟɥɢ) ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɢɫɬɢɧɧɵɯ ɫɤɨɪɨɫɬɟɣ ɧɟ ɛɨɥɟɟ ɱɟɦ ɧɚ 15 - 20%, ɬɨ ɚɥɝɨɪɢɬɦ ɫɯɨɞɢɬɫɹ
ɡɚ 3 - 5 ɢɬɟɪɚɰɢɣ.
ɉɪɢɥɨɠɟɧɢɟ 1. ɗɮɮɟɤɬɢɜɧɵɟ ɫɤɨɪɨɫɬɢ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɩɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɢ ɞɢɧɚɦɢɱɟɫɤɢɦ
ɝɨɞɨɝɪɚɮɚɦ
ȼ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ, ɤɚɫɚɸɳɢɯɫɹ ɢɧɬɟɝɪɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ vCDP, ɨɛɵɱɧɨ
ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɷɬɚ ɫɤɨɪɨɫɬɶ ɧɚɣɞɟɧɚ ɩɨ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɝɨɞɨɝɪɚɮɚ tCDP(l) ɈȽɌ ɩɨ
ɦɟɬɨɞɭ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ (ɉɭɡɵɪɟɜ, 1979; Ƚɨɥɶɞɢɧ, 1993). Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, vCDP ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ
ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ v, ɩɪɢ ɤɨɬɨɪɨɦ ɮɭɧɤɰɢɹ Ɏ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɪɚɜɟɧɫɬɜɨɦ
N
Ɏ(v) = ¦ [ W(li) - (t02 + li2/v2)1/2 ]2,
(ɉ1.1)
i=1
ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɧɚɢɦɟɧɶɲɟɝɨ ɡɧɚɱɟɧɢɹ (li – ɭɞɚɥɟɧɢɟ ɜɡɪɵɜ-ɩɪɢɟɦ ɧɚ i-ɨɦ ɩɪɢɟɦɧɢɤɟ). ȼ ɩɪɚɤɬɢɤɟ
ɫɟɣɫɦɨɪɚɡɜɟɞɨɱɧɵɯ ɪɚɛɨɬ ɦɟɬɨɞɨɦ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ ɫɚɦɢ ɝɨɞɨɝɪɚɮɵ ɈȽɌ ɨɛɵɱɧɨ ɧɟɢɡɜɟɫɬɧɵ, ɢ vCDP
ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɫɟɣɫɦɨɝɪɚɦɦ ɈȽɌ (Ɇɟɲɛɟɣ, 1985; ɍɪɭɩɨɜ, Ʌɟɜɢɧ, 1985).
Ɇɟɬɨɞ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɫɨɫɬɨɢɬ ɜ ɧɚɤɚɩɥɢɜɚɧɢɢ (ɜ ɬɨɦ ɢɥɢ ɢɧɨɦ ɜɢɞɟ) ɜ ɧɟɤɨɬɨɪɨɦ ɫɤɨɥɶɡɹɳɟɦ
ɜɪɟɦɟɧɧɨɦ ɨɤɧɟ ɫɟɣɫɦɢɱɟɫɤɢɯ ɬɪɚɫɫ ui(t) ɫ ɧɟɤɨɬɨɪɵɦɢ ɩɟɪɟɛɢɪɚɟɦɵɦɢ ɩɨɞɜɢɠɤɚɦɢ Ti (i – ɧɨɦɟɪ ɬɪɚɫɫɵ
ɫɟɣɫɦɨɝɪɚɦɦɵ ɈȽɌ). Ɉɛɵɱɧɨ Ti = (t02 + li2 /vCDP2) 1/2 – t0, ɧɨ ɢɧɨɝɞɚ,
ɜ ɫɥɭɱɚɟ ɫɥɨɠɧɨɝɨ ɫɬɪɨɟɧɢɹ
ɩɨɤɪɵɜɚɸɳɟɣ ɬɨɥɳɢ, ɤɨɝɞɚ ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɨɬɥɢɱɢɟ ɝɨɞɨɝɪɚɮɚ ɈȽɌ ɨɬ ɝɢɩɟɪɛɨɥɵ, ɩɪɢɦɟɧɹɸɬɫɹ
ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɚɩɩɪɨɤɫɢɦɚɰɢɢ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɬɪɟɯ ɩɟɪɟɛɢɪɚɟɦɵɯ ɩɚɪɚɦɟɬɪɨɜ (Ƚɨɥɶɞɢɧ, 1993). Ɋɟɡɭɥɶɬɚɬ
ɧɚɤɚɩɥɢɜɚɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɱɢɫɥɨɜɨɦ ɜɢɞɟ, ɬɨ ɟɫɬɶ ɞɥɹ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɜɪɟɦɟɧɧɨɝɨ ɨɤɧɚ ɤɚɠɞɨɦɭ
ɧɚɛɨɪɭ ɡɚɞɟɪɠɟɤ (ɜ ɫɥɭɱɚɟ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ – ɤɚɠɞɨɦɭ ɡɧɚɱɟɧɢɸ, ɨɩɪɟɞɟɥɹɸɳɟɦɭ ɷɬɢ
ɡɚɞɟɪɠɤɢ) ɫɬɚɜɢɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɧɟɤɨɬɨɪɨɟ ɱɢɫɥɨ E (ɪɢɫ.2ɛ), ɞɚɸɳɟɟ ɨɰɟɧɤɭ ɤɨɝɟɪɟɧɬɧɨɫɬɢ
(ɫɢɧɮɚɡɧɨɫɬɢ) ɤɨɥɟɛɚɧɢɣ ɜɞɨɥɶ ɤɪɢɜɨɣ Ti, (ɪɢɫ.2ɚ). Ɉɩɟɪɚɬɨɪ, ɩɟɪɟɜɨɞɹɳɢɣ ɪɟɡɭɥɶɬɚɬ ɧɚɤɚɩɥɢɜɚɧɢɹ (ɢɥɢ
ɧɟɤɨɬɨɪɨɣ ɩɪɨɰɟɞɭɪɵ ɨɰɟɧɤɢ ɤɨɝɟɪɟɧɬɧɨɫɬɢ ɬɪɚɫɫ ɜ ɡɚɞɚɧɧɨɦ ɨɤɧɟ, ɜɤɥɸɱɚɸɳɟɣ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɮɨɪɦɟ
ɫɭɦɦɢɪɨɜɚɧɢɟ ɬɪɚɫɫ) ɜ ɱɢɫɥɨ, ɧɚɡɵɜɚɟɬɫɹ ɨɩɟɪɚɬɨɪɨɦ ɪɟɝɭɥɢɪɭɟɦɨɝɨ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɚɧɚɥɢɡɚ (ɊɇȺ).
Ɂɧɚɱɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ɊɇȺ, ɩɨɥɭɱɟɧɧɵɟ ɞɥɹ ɧɟɤɨɬɨɪɨɝɨ ɦɧɨɠɟɫɬɜɚ ɡɧɚɱɟɧɢɣ vCDP, ɧɚɡɵɜɚɸɬɫɹ ɫɤɨɪɨɫɬɧɵɦ
ɫɩɟɤɬɪɨɦ, ɪɢɫ.2ɛ (ɍɪɭɩɨɜ, Ʌɟɜɢɧ, 1985; Ɇɚɥɨɜɢɱɤɨ, 1990; Ʉɚɪɚɫɢɤ, 1993). Ɍɚɤ ɤɚɤ ɨɩɟɪɚɬɨɪ ɫɤɨɪɨɫɬɧɨɝɨ
103
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
ɚɧɚɥɢɡɚ ɡɚɜɢɫɢɬ ɬɚɤɠɟ ɨɬ ɨɤɧɚ ɫɭɦɦɢɪɨɜɚɧɢɹ, ɬɨ ɜɟɥɢɱɢɧɚ ȿ ɡɚɜɢɫɢɬ ɨɬ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ: ɨɬ ɫɤɨɪɨɫɬɢ
ɫɭɦɦɢɪɨɜɚɧɢɹ ɢ ɜɪɟɦɟɧɢ t0 – ɰɟɧɬɪɚ ɨɤɧɚ ɫɭɦɦɢɪɨɜɚɧɢɹ (ɪɢɫ.2ɜ).
ȿɫɥɢ ɜ ɨɤɧɟ ɚɧɚɥɢɡɚ ɫɟɣɫɦɨɝɪɚɦɦɵ ɢɦɟɟɬɫɹ ɨɞɧɚ ɪɟɝɭɥɹɪɧɚɹ ɜɨɥɧɚ ɫ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦ ɢɥɢ
ɛɥɢɡɤɢɦ ɤ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɦɭ ɝɨɞɨɝɪɚɮɨɦ, ɬɨ ɩɪɢ ɫɨɜɩɚɞɟɧɢɢ ɤɪɢɜɨɣ ɫɭɦɦɢɪɨɜɚɧɢɹ (t02 + li2/vCDP2)1/2 ɫ
ɝɨɞɨɝɪɚɮɨɦ ɷɬɨɣ ɜɨɥɧɵ ɩɪɨɢɫɯɨɞɢɬ ɫɢɧɮɚɡɧɨɟ ɧɚɤɚɩɥɢɜɚɧɢɟ, ɢ ɨɩɟɪɚɬɨɪ ɊɇȺ ɩɪɢɧɢɦɚɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ
ɡɧɚɱɟɧɢɟ. ɗɬɨɦɭ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ vCDP, ɩɨɥɭɱɚɟɦɨɟ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ. ɗɬɨ ɡɧɚɱɟɧɢɟ vCDP ɜ ɞɚɥɶɧɟɣɲɟɦ ɢɫɩɨɥɶɡɭɟɬɫɹ ɤɚɤ ɞɥɹ ɩɨɥɭɱɟɧɢɹ
ɫɭɦɦɚɪɧɵɯ ɪɚɡɪɟɡɨɜ ɈȽɌ, ɬɚɤ ɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɚɫɬɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ
ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɬɟɦ ɢɥɢ ɢɧɵɦ ɦɟɬɨɞɨɦ.
Ɋɢɫ.2. ɋɤɨɪɨɫɬɧɨɣ ɚɧɚɥɢɡ ɫɟɣɫɦɨɝɪɚɦɦɵ.
ɇɟɫɦɨɬɪɹ ɧɚ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɪɚɡɥɢɱɧɵɯ ɨɩɟɪɚɬɨɪɨɜ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ, ɨɰɟɧɤɢ vCDP,
ɩɨɥɭɱɚɟɦɵɟ ɷɬɢɦɢ ɨɩɟɪɚɬɨɪɚɦɢ, ɛɥɢɡɤɢ ɞɪɭɝ ɤ ɞɪɭɝɭ, ɚ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɩɨɦɟɯ (ɤɚɤ ɪɟɝɭɥɹɪɧɵɯ, ɬɚɤ ɢ
ɫɥɭɱɚɣɧɵɯ ɲɭɦɨɜ) ɜɨɨɛɳɟ ɫɨɜɩɚɞɚɸɬ. Ɏɚɤɬɢɱɟɫɤɢ ɜɫɟ ɷɬɢ ɨɩɟɪɚɬɨɪɵ ɹɜɥɹɸɬɫɹ ɨɩɟɪɚɬɨɪɚɦɢ ɜɬɨɪɨɝɨ
ɩɨɪɹɞɤɚ ɢ ɜɵɪɚɠɚɸɬɫɹ ɱɟɪɟɡ ɨɞɧɢ ɢ ɬɟ ɠɟ ɜɟɥɢɱɢɧɵ, ɫɜɹɡɚɧɧɵɟ ɫ ɫɭɦɦɨɣ ɬɪɚɫɫ ɜ ɡɚɞɚɧɧɨɦ ɨɤɧɟ.
ɉɨɞɪɨɛɧɨ ɷɬɢ ɜɨɩɪɨɫɵ ɪɚɫɫɦɨɬɪɟɧɵ ɜɨ ɦɧɨɝɢɯ ɪɚɛɨɬɚɯ, ɨɛɡɨɪ ɷɬɢɯ ɪɚɛɨɬ ɢ ɢɯ ɚɧɚɥɢɡ ɞɚɧ Ɇɚɥɨɜɢɱɤɨ
(Ɇɚɥɨɜɢɱɤɨ, 1993). Ɂɞɟɫɶ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ ȿ, ɞɚɸɳɢɣ ɷɧɟɪɝɢɸ ɫɭɦɦɨɬɪɚɫɫɵ.
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɞɟɣɫɬɜɢɟ ɞɪɭɝɢɯ ɨɩɟɪɚɬɨɪɨɜ.
ɉɭɫɬɶ ui(t) – i-ɚɹ ɬɪɚɫɫɚ ɫɟɣɫɦɨɝɪɚɦɦɵ ɈȽɌ, W(v)=(t02+ li2 /vCDP2)1/2 – ɤɪɢɜɚɹ ɫɭɦɦɢɪɨɜɚɧɢɹ.
ȼɦɟɫɬɨ ɤɨɷɮɮɢɰɢɟɧɬɚ v ɭɞɨɛɧɟɟ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɚɪɚɦɟɬɪ b = 1/v2, ɜɯɨɞɹɳɢɣ ɜ ɩɨɞɤɨɪɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ
ɥɢɧɟɣɧɨ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɫɚɦɚ ɫɤɨɪɨɫɬɶ ɫɭɦɦɢɪɨɜɚɧɢɹ ɜɯɨɞɢɬ ɜ ɩɨɞɤɨɪɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɧɟɥɢɧɟɣɧɨ.
Ʉɪɨɦɟ ɬɨɝɨ, ɜɜɟɞɟɧɢɟ ɩɚɪɚɦɟɬɪɚ b ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɢ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɫɥɭɱɚɣ ɩɟɪɟɜɨɪɚɱɢɜɚɧɢɹ
ɝɨɞɨɝɪɚɮɨɜ ɈȽɌ, ɜɵɡɜɚɧɧɨɝɨ ɪɟɡɤɢɦɢ ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ ɩɪɟɥɨɦɥɹɸɳɢɦɢ ɝɪɚɧɢɰɚɦɢ ɜ ɩɨɤɪɵɜɚɸɳɟɣ
ɫɪɟɞɟ (Ȼɥɹɫ, 1991). Ⱦɥɹ ɩɚɪɚɦɟɬɪɚ b ɤɪɢɜɚɹ ɫɭɦɦɢɪɨɜɚɧɢɹ W(b) ɢɦɟɟɬ ɜɢɞ W(b) = (t02 + bli2)1/2, ɢ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ ȿ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
T
N
E(b) = ³ [ ¦ u (t + (t02 + bli2)1/2)]dt.
0
(ɉ1.2)
i=1
Ɂɞɟɫɶ ɞɥɹ ɭɞɨɛɫɬɜɚ ɢɫɫɥɟɞɨɜɚɧɢɣ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɲɚɝ ɜɪɟɦɟɧɧɨɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ 't ɞɨɫɬɚɬɨɱɧɨ ɦɚɥ,
ɢ ɜɦɟɫɬɨ ɫɭɦɦɵ ɩɨ t ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɷɧɟɪɝɢɢ ɫɭɦɦɨɬɪɚɫɫɵ ɛɟɪɟɬɫɹ ɢɧɬɟɝɪɚɥ. ɇɚ ɩɪɚɤɬɢɤɟ ɷɬɨ ɭɫɥɨɜɢɟ ɜ
ɩɨɞɚɜɥɹɸɳɟɦ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɚɤ ɤɚɤ ɪɚɛɨɬɵ ɦɟɬɨɞɨɦ ɈȽɌ ɩɪɨɜɨɞɹɬɫɹ ɩɪɢɟɦɧɵɦɢ
ɭɫɬɪɨɣɫɬɜɚɦɢ ɫ ɲɚɝɨɦ ɞɢɫɤɪɟɬɢɡɚɰɢɢ 1 ɦɫ, ɱɬɨ ɫɨɫɬɚɜɥɹɟɬ ɫɨɬɵɟ ɞɨɥɢ ɨɬ ɩɟɪɢɨɞɚ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ. ɉɪɢ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɨɥɟɡɧɚɹ ɜɨɥɧɚ ɩɨɥɧɨɫɬɶɸ ɩɨɩɚɞɚɟɬ ɜ ɢɧɬɟɪɜɚɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜ ɨɤɧɨ ɚɧɚɥɢɡɚ ɩɨɩɚɞɚɸɬ ɞɜɟ ɪɟɝɭɥɹɪɧɵɟ ɢɧɬɟɪɮɟɪɢɪɭɸɳɢɟ ɜɨɥɧɵ, ɬɨ ɟɫɬɶ,
ɱɬɨ
104
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ui(t) = f(t - Wf (li)) + g(t - Wï(li)),
ɫɬɪ.95-112
i=1,2,...,n.
(ɉ1.3)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɩɪɨɢɡɜɨɥɶɧɨɟ ɱɢɫɥɨ ɢɧɬɟɪɮɟɪɢɪɭɸɳɢɯ ɜɨɥɧ, ɚ ɬɚɤɠɟ
ɭɱɟɫɬɶ ɧɚɥɢɱɢɟ ɧɟɪɟɝɭɥɹɪɧɨɝɨ (ɧɟɤɨɪɪɟɥɢɪɭɟɦɨɝɨ ɩɨ ɬɪɚɫɫɚɦ) ɲɭɦɚ, ɟɫɥɢ ɩɨɞ ɫɥɚɝɚɟɦɵɦ g ɩɨɧɢɦɚɬɶ ɧɟ
ɨɞɧɭ ɜɨɥɧɭ, ɚ ɫɭɦɦɭ ɧɟɫɤɨɥɶɤɢɯ ɪɟɝɭɥɹɪɧɵɯ ɜɨɥɧ ɢ ɫɥɭɱɚɣɧɨɝɨ ɲɭɦɚ. Ⱦɥɹ ɫɨɤɪɚɳɟɧɢɹ ɡɚɩɢɫɢ ɦɵ
ɨɝɪɚɧɢɱɢɜɚɟɦɫɹ ɭɱɟɬɨɦ ɢɧɬɟɪɮɟɪɟɧɰɢɢ ɞɜɭɯ ɪɟɝɭɥɹɪɧɵɯ ɜɨɥɧ.
ɉɭɫɬɶ f (t) – ɫɢɝɧɚɥ ɩɨɥɟɡɧɨɣ ɨɞɧɨɤɪɚɬɧɨ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ, ɱɟɣ ɝɨɞɨɝɪɚɮ ɨɩɢɫɵɜɚɟɬɫɹ
ɮɭɧɤɰɢɟɣ
W (l) = [c0 + c1 l 2 + c2 l 4] 1/2.
(ɉ1.4)
Ɂɞɟɫɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɪɟɚɥɶɧɵɣ ɝɨɞɨɝɪɚɮ ɨɩɢɫɵɜɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ (ɉ1.4), ɬɨ ɟɫɬɶ ɛɨɥɟɟ ɫɥɨɠɧɨɣ
ɤɪɢɜɨɣ, ɱɟɦ ɝɢɩɟɪɛɨɥɚ. Ⱥɜɬɨɪɨɦ (Ȼɥɹɫ, 1988) ɩɨɤɚɡɚɧɨ, ɱɬɨ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɝɨɞɨɝɪɚɮ ɈȽɌ ɫ ɜɵɫɨɤɨɣ
ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɮɨɪɦɭɥɨɣ (ɉ1.4). Ɍɚɤ ɤɚɤ ɦɵ ɢɫɫɥɟɞɭɟɦ ɪɟɡɭɥɶɬɚɬɵ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ
ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ, ɬɨ ɡɞɟɫɶ ɭɦɟɫɬɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɟɫɥɢ ɝɨɞɨɝɪɚɮ ɈȽɌ ɫɢɥɶɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ
ɝɢɩɟɪɛɨɥɵ, ɬɨ ɜ ɨɩɟɪɚɬɨɪɟ ɊɇȺ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɷɬɨ ɨɬɥɢɱɢɟ, ɢɧɚɱɟ ɦɚɤɫɢɦɭɦɵ ɫɩɟɤɬɪɨɜ
ɫɤɨɪɨɫɬɟɣ ɛɭɞɭɬ ɫɭɳɟɫɬɜɟɧɧɨ ɢɫɤɚɠɟɧɵ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɤɨɪɨɫɬɢ ɫɭɦɦɢɪɨɜɚɧɢɹ ɧɟɜɨɡɦɨɠɧɨ ɛɭɞɟɬ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ ɞɚɥɶɧɟɣɲɟɦ ɤɚɤ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ, ɬɚɤ ɢ ɞɥɹ ɫɭɦɦɢɪɨɜɚɧɢɹ ɈȽɌ.
ɉɨɞɫɬɚɜɢɜ (ɉ1.3) ɜ (ɉ1.2), ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɭ, ɞɚɸɳɭɸ ɹɜɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɨɩɟɪɚɬɨɪɚ ȿ ɨɬ
ɩɚɪɚɦɟɬɪɚ b:
T
N
E(b) = ³{¦ [ f (t - Wf (li) + (t02 + bli2)1/2) + g(t - Wg(li) + (t02 + bli2)1/2)]2}dt.
0
(ɉ1.5)
i=1
ɇɚɫ ɢɧɬɟɪɟɫɭɟɬ ɬɨɱɤɚ ɦɚɤɫɢɦɭɦɚ ɷɬɨɣ ɮɭɧɤɰɢɢ, ɬɨ ɟɫɬɶ ɬɚɤɨɟ ɡɧɚɱɟɧɢɟ b* ɩɚɪɚɦɟɬɪɚ b, ɜ ɤɨɬɨɪɨɣ
dE/db = 0. ɉɨɞɫɬɚɜɥɹɹ ɜ ɷɬɨ ɪɚɜɟɧɫɬɜɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ (ɉ1.5), ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɟɟ ɭɪɚɜɧɟɧɢɟ, ɤɨɬɨɪɨɦɭ
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɬɨɱɤɚ b* ɦɚɤɫɢɦɭɦɚ ɨɩɟɪɚɬɨɪɚ ȿ:
T
N
³{[ ¦ f(t - Wf(li) + (t0
N
2
0
N
u
+
)+ ¦ g(t - Wg(li) + (t02 + bli2)1/2) ] u
bli2)1/2
i=1
i=1
(ɉ1.6)
N
[ ¦ f c(t - Wf(li) + (t02 + bli2)1/2)+ ¦ g c(t - Wg(li) + (t02 + bli2)1/2) ] } [li2/(t02 + bli2)] dt.
i=1
i=1
Ɍɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɢɬɶ ɧɟ ɭɞɚɟɬɫɹ, ɧɨ, ɢɫɩɨɥɶɡɭɹ ɦɟɬɨɞ
ɜɨɡɦɭɳɟɧɢɣ, ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɹɜɧɨɟ ɪɟɲɟɧɢɟ ɫ ɥɸɛɨɣ ɬɨɱɧɨɫɬɶɸ. Ⱥɧɚɥɢɡ ɞɚɧɧɨɝɨ ɪɟɲɟɧɢɹ ɩɨɡɜɨɥɹɟɬ
ɢɫɫɥɟɞɨɜɚɬɶ ɨɬɥɢɱɢɟ vCDP, ɧɚɣɞɟɧɧɨɝɨ ɩɨ ɝɨɞɨɝɪɚɮɭ ɈȽɌ ɦɟɬɨɞɨɦ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ (ɬɨ ɟɫɬɶ ɱɢɫɬɨ
ɤɢɧɟɦɚɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɶɧɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ, ɩɨɥɭɱɟɧɧɨɟ ɩɪɢ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ
Ɏ), ɨɬ ɡɧɚɱɟɧɢɹ vCDP, ɩɨɥɭɱɟɧɧɨɝɨ ɩɪɢ ɫɤɨɪɨɫɬɧɨɦ ɚɧɚɥɢɡɟ ɫɟɣɫɦɨɝɪɚɦɦ. ɂɧɬɟɝɪɚɥɶɧɭɸ ɫɤɨɪɨɫɬɶ,
ɨɩɪɟɞɟɥɟɧɧɭɸ ɩɭɬɟɦ ɦɢɧɢɦɢɡɚɰɢɢ ɮɭɧɤɰɢɢ Ɏ, ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɶɸ
ɢ ɨɛɨɡɧɚɱɚɬɶ ɟɟ ɱɟɪɟɡ wCDP, ɮɨɪɦɭɥɚ ɞɥɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ ɛɭɞɟɬ ɩɨɥɭɱɟɧɚ ɧɢɠɟ.
ɗɮɮɟɤɬɢɜɧɭɸ ɫɤɨɪɨɫɬɶ, ɧɚɣɞɟɧɧɭɸ ɩɪɢ ɫɤɨɪɨɫɬɧɨɦ ɚɧɚɥɢɡɟ ɫɟɣɫɦɨɝɪɚɦɦɵ ɈȽɌ, ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ
ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɤɨɪɨɫɬɶɸ (ɢɥɢ ɫɤɨɪɨɫɬɶɸ ɫɭɦɦɢɪɨɜɚɧɢɹ) ɢ ɩɨ-ɩɪɟɠɧɟɦɭ ɨɛɨɡɧɚɱɚɬɶ vCDP. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɫɤɨɪɨɫɬɶ wCDP ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɝɨɞɨɝɪɚɮɚ ɈȽɌ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ (ɟɫɥɢ ɨɧ ɨɬɥɢɱɟɧ ɨɬ
ɝɢɩɟɪɛɨɥɵ) ɢ ɨɬ ɛɚɡɵ ɧɚɛɥɸɞɟɧɢɹ (ɨɬ ɩɪɨɦɟɠɭɬɤɚ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɪɟɚɥɶɧɨɝɨ ɝɨɞɨɝɪɚɮɚ ɝɢɩɟɪɛɨɥɨɣ), ɜ
ɬɨ ɜɪɟɦɹ ɤɚɤ ɞɢɧɚɦɢɱɟɫɤɚɹ ɫɤɨɪɨɫɬɶ vCDP ɡɚɜɢɫɢɬ ɬɚɤɠɟ ɨɬ ɞɢɧɚɦɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɜɨɥɧ – ɮɨɪɦɵ ɢɯ
ɫɢɝɧɚɥɨɜ f(t), g(t) ɢ ɨɬ ɯɚɪɚɤɬɟɪɚ ɢɧɬɟɪɮɟɪɟɧɰɢɢ.
ɉɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɜɨɡɦɭɳɟɧɢɣ ɧɚɤɥɚɞɵɜɚɟɬ ɧɟɤɨɬɨɪɵɟ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɦɨɞɟɥɶ ɫɟɣɫɦɨɝɪɚɦɦɵ,
ɚ ɢɦɟɧɧɨ – ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɦɨɞɟɥɶ ɞɨɥɠɧɚ “ɧɟ ɨɱɟɧɶ ɫɢɥɶɧɨ” ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɛɚɡɨɜɨɣ ɦɨɞɟɥɢ (ɦɨɞɟɥɢ
ɧɭɥɟɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ), ɞɥɹ ɤɨɬɨɪɨɣ ɪɟɲɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɭɪɚɜɧɟɧɢɹ ɥɟɝɤɨ ɧɚɯɨɞɢɬɫɹ. ɗɬɨ
ɩɪɢɜɨɞɢɬ ɤ ɫɥɟɞɭɸɳɢɦ ɩɪɟɞɩɨɥɨɠɟɧɢɹɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɨɞɟɥɢ ɫɟɣɫɦɨɝɪɚɦɦɵ, ɤɨɬɨɪɵɟ ɧɚ ɩɪɚɤɬɢɤɟ
ɱɚɫɬɨ ɜɵɩɨɥɧɹɸɬɫɹ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɞɥɹ ɯɨɪɨɲɨ ɜɵɪɚɠɟɧɧɵɯ ɨɬɪɚɠɟɧɢɣ ɢ ɩɥɚɬɮɨɪɦɟɧɧɵɯ ɭɫɥɨɜɢɣ
ɡɚɥɟɝɚɧɢɹ ɩɨɪɨɞ. ȼɨ-ɩɟɪɜɵɯ, ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɜɵɩɨɥɧɹɟɬɫɹ c2 l 4 « t0+c1 l 2, ɬɨ ɟɫɬɶ ɜɥɢɹɧɢɟ ɬɪɟɬɶɟɝɨ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɫ2 ɧɚ ɝɨɞɨɝɪɚɮ ɈȽɌ ɧɚɦɧɨɝɨ ɦɟɧɶɲɟ, ɱɟɦ ɜɥɢɹɧɢɟ ɫɭɦɦɵ ɩɟɪɜɵɯ ɞɜɭɯ. ɗɬɨ ɭɫɥɨɜɢɟ ɧɚ
ɩɪɚɤɬɢɤɟ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɜɵɩɨɥɧɹɟɬɫɹ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɩɪɢ ɭɞɚɥɟɧɢɹɯ, ɧɟ ɩɪɟɜɵɲɚɸɳɢɯ
ɝɥɭɛɢɧɵ ɨɬɪɚɠɚɸɳɟɣ ɝɪɚɧɢɰɵ (Ȼɥɹɫ, 1988). ɋɬɪɨɝɨ ɝɨɜɨɪɹ, ɩɨɧɹɬɢɟ ɦɚɥɨɫɬɢ ɨɬɤɥɨɧɟɧɢɹ ɝɨɞɨɝɪɚɮɚ ɨɬ
ɟɝɨ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɩɪɟɨɛɥɚɞɚɸɳɟɣ ɱɚɫɬɨɬɵ ɫɢɝɧɚɥɚ, ɫɚɦɨ ɨɬɤɥɨɧɟɧɢɟ
105
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
ɞɨɥɠɧɨ ɜɵɪɚɠɚɬɶɫɹ ɜ ɞɨɥɹɯ ɩɟɪɢɨɞɚ ɫɢɝɧɚɥɚ, ɩɨɞɪɨɛɧɟɟ ɷɬɨɬ ɜɨɩɪɨɫ ɛɭɞɟɬ ɪɚɫɫɦɨɬɪɟɧ ɜ ɤɨɧɰɟ
ɩɪɢɥɨɠɟɧɢɹ. ȼɨ-ɜɬɨɪɵɯ, ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɪɢ b=ɫ1, ɬɨ ɟɫɬɶ ɩɪɢ ɫɭɦɦɢɪɨɜɚɧɢɢ ɩɨ ɤɪɢɜɨɣ, ɛɥɢɡɤɨɣ ɤ
ɝɨɞɨɝɪɚɮɭ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ, ɷɧɟɪɝɢɹ ɫɭɦɦɚɪɧɨɝɨ ɫɢɝɧɚɥɚ ɛɨɥɶɲɟ ɷɧɟɪɝɢɢ ɩɨɦɟɯɢ. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ,
ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ ɝɢɩɟɪɛɨɥɟ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɝɨɞɨɝɪɚɮ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ,
ɩɨɡɜɨɥɹɟɬ ɜɵɞɟɥɢɬɶ ɷɬɭ ɜɨɥɧɭ ɧɚ ɮɨɧɟ ɩɨɦɟɯɢ, ɬɨ ɟɫɬɶ ɪɚɡɪɚɫɬɚɧɢɟ ɧɚ ɫɤɨɪɨɫɬɧɨɦ ɫɩɟɤɬɪɟ ɜɵɡɜɚɧɨ
ɩɨɥɟɡɧɨɣ ɜɨɥɧɨɣ. ɑɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ ɧɚ ɦɨɞɟɥɹɯ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ, ɟɫɥɢ ɨɬɥɢɱɢɟ ɝɨɞɨɝɪɚɮɨɜ Wf ɢ Wg ɧɚ
ɩɨɫɥɟɞɧɢɯ ɤɚɧɚɥɚɯ (ɩɪɢ ɦɚɤɫɢɦɚɥɶɧɨɦ ɪɚɫɫɬɨɹɧɢɢ ɜɡɪɵɜ-ɩɪɢɟɦ) ɞɨɫɬɢɝɚɟɬ ɜɟɥɢɱɢɧɵ ɩɨɪɹɞɤɚ ɩɟɪɢɨɞɚ
ɫɢɝɧɚɥɚ, ɬɨ ɩɪɢ ɪɚɜɧɵɯ ɚɦɩɥɢɬɭɞɚɯ ɷɬɨ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɹɟɬɫɹ.
ɋɨɝɥɚɫɧɨ ɦɟɬɨɞɭ ɜɨɡɦɭɳɟɧɢɣ, ɦɚɥɨɫɬɶ ɨɬɤɥɨɧɟɧɢɣ ɝɨɞɨɝɪɚɮɚ ɈȽɌ Wf ɨɬ ɝɢɩɟɪɛɨɥɵ ɢ ɦɚɥɨɫɬɶ
ɫɭɦɦɚɪɧɨɣ ɜɨɥɧɵ ɩɨɦɟɯɢ (ɦɚɥɨɫɬɶ ɩɨɦɟɯɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɢɝɧɚɥɨɦ ɧɚ ɫɭɦɦɨɬɪɚɫɫɟ) ɩɟɪɟɧɟɫɟɦ ɜ ɦɚɥɵɣ
ɛɟɡɪɚɡɦɟɪɧɵɣ ɩɚɪɚɦɟɬɪ H, ɞɥɹ ɱɟɝɨ ɜɦɟɫɬɟ ɫ ɢɫɯɨɞɧɵɦ ɭɪɚɜɧɟɧɢɟɦ (ɉ1.6) ɪɚɫɫɦɨɬɪɢɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ
T
N
N
³{[¦ f(t - Wf(li,H) + (t0
2
0
+
i=1
i=1
N
u
) + H ¦ g(t - Wg(li) + (t02 + bli2)1/2) ] u
bli2)1/2
(ɉ1.7)
N
[¦ f c(t - Wf (li,H) + (t02 + bli2)1/2) + H ¦ gc(t - Wg(li) + (t02 + bli2)1/2)]} [li2/(t02 + bli2)] dt ,
i=1
i=1
2
4 1/2
ɝɞɟ Wf(li,H) = [c0 + c1 l + H c2 l ] .
ɉɪɟɠɞɟ ɜɫɟɝɨ ɩɨɤɚɠɟɦ, ɱɬɨ ɩɪɢ H=0, ɬɨ ɟɫɬɶ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɜɨɥɧɵ-ɩɨɦɟɯɢ ɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɦ
ɝɨɞɨɝɪɚɮɟ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ, ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɨ ɝɨɞɨɝɪɚɮɭ ɜɨɥɧɵ ɩɪɢɜɨɞɢɬ ɤ ɦɚɤɫɢɦɭɦɭ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ
ɨɩɟɪɚɬɨɪɚ, ɬɨ ɟɫɬɶ ɤ ɦɚɤɫɢɦɭɦɭ ɫɩɟɤɬɪɚ ɫɤɨɪɨɫɬɟɣ. ȼɨ ɜɪɟɦɟɧɧɨɣ ɨɛɥɚɫɬɢ ɷɬɨɬ ɮɚɤɬ ɨɛɨɫɧɨɜɵɜɚɟɬɫɹ ɫ
ɩɨɦɨɳɶɸ ɧɟɪɚɜɟɧɫɬɜɚ Ʉɨɲɢ – Ȼɭɧɹɤɨɜɫɤɨɝɨ (Ʉɨɥɦɨɝɨɪɨɜ, Ɏɨɦɢɧ, 1968): ɞɥɹ ɥɸɛɵɯ ɱɢɫɟɥ xi, yi,
i=1,2,...,n, ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
n
n
n
( ¦ xi yi )2 d ¦ xi ¦ yi ,
i=1
i=1
i=1
ɩɪɢɱɟɦ ɪɚɜɟɧɫɬɜɨ ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɥɶɤɨ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ xi= yi ɞɥɹ ɜɫɟɯ i=1,2,...,n. ɉɨɞɫɬɚɜɥɹɹ ɜ ɞɚɧɧɨɟ
ɪɚɜɟɧɫɬɜɨ xi = f(t - Ti ), ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ ³ f 2(t - Ti ) dt ɧɟ ɡɚɜɢɫɢɬ ɨɬ i, ɩɨɥɭɱɢɦ, ɱɬɨ
T
N
T
³ ( ¦ f ( t - Tic))2 dt d N2 ³ f 2(t)dt,
0
0
i=1
0
2 1/2
ɝɞɟ T ci= (t +bl ) – Wf (li,0). ɋɬɪɨɝɨɟ ɪɚɜɟɧɫɬɜɨ ɜɵɩɨɥɧɹɟɬɫɹ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ f(t-Ti) ɩɪɢɧɢɦɚɸɬ ɨɞɢɧɚɤɨɜɵɟ
ɡɧɚɱɟɧɢɹ ɞɥɹ ɜɫɟɯ i, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɢɧɮɚɡɧɨɦɭ ɫɭɦɦɢɪɨɜɚɧɢɸ.
ɂɬɚɤ, ɩɪɢ H =0 ɭɪɚɜɧɟɧɢɟ (ɉ1.7) ɢɦɟɟɬ ɪɟɲɟɧɢɟ b0=c1, ɩɨɷɬɨɦɭ ɫɨɝɥɚɫɧɨ ɦɟɬɨɞɭ ɜɨɡɦɭɳɟɧɢɣ
ɦɨɠɧɨ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ H :
b0 = c1 + H b1 + H 2b2 + ... .
(ɉ1.8)
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ bi, i=1,2,... ɪɹɞɚ (ɉ1.8) ɩɨɞɫɬɚɜɢɦ ɟɝɨ ɜ ɭɪɚɜɧɟɧɢɟ (ɉ1.7) ɢ ɪɚɡɥɨɠɢɦ
ɥɟɜɭɸ ɱɚɫɬɶ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ H. Ɍɚɤ ɤɚɤ ɩɪɚɜɚɹ ɱɚɫɬɶ ɩɨɥɭɱɢɜɲɟɝɨɫɹ ɬɨɠɞɟɫɬɜɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ H, ɬɨ ɜɫɟ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɫɬɟɩɟɧɹɯ H ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɬɚɤɠɟ ɞɨɥɠɧɵ ɪɚɜɧɹɬɶɫɹ ɧɭɥɸ. ɉɪɢɪɚɜɧɢɜɚɹ ɢɯ ɤ ɧɭɥɸ ɢ
ɭɱɢɬɵɜɚɹ, ɱɬɨ ɫ1 – ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (ɉ1.7) ɩɪɢ H=0, ɩɨɥɭɱɢɦ ɥɢɧɟɣɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ
ɤɨɷɮɮɢɰɢɟɧɬɨɜ bi, i=2,3,... . ȼ ɤɚɠɞɨɟ ɬɚɤɨɟ ɭɪɚɜɧɟɧɢɟ, ɩɨɥɭɱɚɸɳɟɟɫɹ ɩɪɢɪɚɜɧɢɜɚɧɢɟɦ ɤ ɧɭɥɸ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɪɢ Hi, ɜɯɨɞɹɬ ɤɨɷɮɮɢɰɢɟɧɬɵ b1, b2, ..., bi-1, ɢ ɥɢɧɟɣɧɨ bi. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɪɟɲɚɹ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨɥɭɱɚɸɳɢɟɫɹ ɭɪɚɜɧɟɧɢɹ, ɦɨɠɧɨ ɧɚɣɬɢ ɫɤɨɥɶɤɨ ɭɝɨɞɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɹɞɚ (ɉ1.8).
ȿɫɥɢ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɢɝɧɚɥɵ ɨɩɢɫɵɜɚɸɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɮɭɧɤɰɢɹɦɢ, ɬɨ ɩɨ ɬɟɨɪɟɦɟ ɨɛ ɚɧɚɥɢɬɢɱɧɨɫɬɢ
ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ (ɌɎɄɉ) ɪɹɞ (ɉ1.8) ɬɚɤɠɟ ɨɩɢɫɵɜɚɟɬ ɚɧɚɥɢɬɢɱɟɫɤɭɸ ɮɭɧɤɰɢɸ ɜ ɧɟɤɨɬɨɪɨɣ ɨɤɪɟɫɬɧɨɫɬɢ
ɧɭɥɹ, ɬɨ ɟɫɬɶ ɹɜɥɹɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ ɪɹɞɨɦ. ɏɨɬɹ ɜ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɧɚɣɬɢ ɫɤɨɥɶɤɨ ɭɝɨɞɧɨ ɱɥɟɧɨɜ ɪɹɞɚ
(ɉ1.8), ɞɥɹ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɞɨɫɬɚɬɨɱɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɩɟɪɜɵɦɢ ɞɜɭɦɹ ɫɥɚɝɚɟɦɵɦɢ.
ɑɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɨɧɢ ɞɚɸɬ ɡɧɚɱɟɧɢɟ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ ɫ ɞɨɫɬɚɬɨɱɧɨ ɜɵɫɨɤɨɣ
ɬɨɱɧɨɫɬɶɸ.
Ɉɝɪɚɧɢɱɢɜɚɹɫɶ ɭɱɟɬɨɦ ɥɢɧɟɣɧɵɯ ɱɥɟɧɨɜ ɩɨ H ɢ ɩɟɪɟɯɨɞɹ ɨɬ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (ɉ1.7) ɤ
ɢɫɯɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ (ɉ1.6) (ɤɨɬɨɪɨɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ H), ɩɨɥɭɱɢɦ, ɱɬɨ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ ȿ
ɞɨɫɬɢɝɚɟɬ ɦɚɤɫɢɦɭɦɚ ɩɪɢ ɡɧɚɱɟɧɢɢ bCDP, ɤɨɬɨɪɨɟ ɞɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ
106
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
N
bCDP = c1 + [( ¦
N
li2
i=1
/
Di1/2
) –N¦
2
N
li4
N
{[¦
-1
/ Di ]
i=1
T
ɫɬɪ.95-112
(li4
/
i=1
T
Di1/2) (
i=1
N
N
¦
li2
/
Di1/2)
- N ( ¦ li6 / Di )] c2 –
i=1
(ɉ1.9)
T
– [2 / ³ f c2(t) dt] [ ³ f c(t) G(t) dt ¦ li2 / Di1/2 + N ³ f(t) G1(t) dt ]},
0
0
i=1
0
ɝɞɟ
Di = c0 + c12l i,
G(t) = ¦ gi (t - Wg(li) + (c0 + c1 li2)1/2),
(ɉ1.10)
G1 (t) = ¦ gci (t - Wg(li) + (c0 +
2
c1li2)1/2
)
li2
/
Di1/2.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɞɚɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɥɟɞɭɸɳɢɦ ɩɪɢɟɦɨɦ, ɩɨɡɜɨɥɹɸɳɢɦ ɩɟɪɟɣɬɢ
ɨɬ ɫɭɦɦ ɤ ɢɧɬɟɝɪɚɥɚɦ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɲɚɝ 'l = li+1 - li ɦɟɠɞɭ ɬɪɚɫɫɚɦɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥ, ɢ ɫɭɦɦɵ ɩɨ i
ɦɨɠɧɨ ɡɚɦɟɧɢɬɶ ɧɚ ɢɧɬɟɝɪɚɥɵ ɜ ɩɪɟɞɟɥɚɯ ɛɚɡɵ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ, ɧɚɩɪɢɦɟɪ,
N
L
¦ [li2/(c0 + c li2)] | (N/L) ³ [li2dl/ (c0 + c1l2)] .
i=1
0
ɇɚ ɩɪɚɤɬɢɤɟ ɷɬɨ ɭɫɥɨɜɢɟ ɜ ɩɨɞɚɜɥɹɸɳɟɦ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɚɤ ɤɚɤ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ
ɪɚɛɨɬɵ ɦɟɬɨɞɨɦ ɈȽɌ ɩɪɨɜɨɞɹɬɫɹ ɫ ɩɪɢɟɦɧɵɦɢ ɭɫɬɪɨɣɫɬɜɚɦɢ, ɜ ɤɨɬɨɪɵɯ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɩɪɢɟɦɧɢɤɚɦɢ
ɫɨɫɬɚɜɥɹɟɬ ɫɨɬɵɟ ɢ ɞɚɠɟ ɬɵɫɹɱɧɵɟ ɞɨɥɢ ɨɬ ɞɥɢɧɵ ɪɚɫɫɬɚɧɨɜɤɢ.
Ⱦɚɥɟɟ, ɜɜɟɞɟɦ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɛɟɡɪɚɡɦɟɪɧɭɸ ɜɟɥɢɱɢɧɭ J L ɩɨ ɮɨɪɦɭɥɟ J L = c1L2 / c0. Ⱦɥɹ
ɝɨɪɢɡɨɧɬɚɥɶɧɨ-ɫɥɨɢɫɬɨɣ ɫɪɟɞɵ ɫ ɦɨɳɧɨɫɬɹɦɢ ɢ ɩɥɚɫɬɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ JL=(1/4) (L / He)2, ɝɞɟ
N
N
He = ( ¦ hkvk ¦ hk /vk )1/2
k=1
k=1
– ɷɮɮɟɤɬɢɜɧɚɹ ɝɥɭɛɢɧɚ, ɛɥɢɡɤɚɹ ɤ ɝɥɭɛɢɧɟ ɇ ɨɬɪɚɠɚɸɳɟɣ ɝɪɚɧɢɰɵ. Ⱦɥɹ ɫɪɟɞ ɫ ɧɟɛɨɥɶɲɢɦɢ ɭɝɥɚɦɢ
ɧɚɤɥɨɧɚ (ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɩɥɚɬɮɨɪɦɟɧɧɵɦ ɭɫɥɨɜɢɹɦ ɨɫɚɞɤɨɧɚɤɨɩɥɟɧɢɹ) ɜɟɥɢɱɢɧɚ J L ɬɚɤɠɟ ɩɪɢɦɟɪɧɨ
ɪɚɜɧɚ ɨɞɧɨɣ ɱɟɬɜɟɪɬɨɣ ɤɜɚɞɪɚɬɚ ɨɬɧɨɲɟɧɢɹ ɞɥɢɧɵ ɪɚɫɫɬɚɧɨɜɤɢ ɤ ɝɥɭɛɢɧɟ ɨɬɪɚɠɚɸɳɟɣ ɝɪɚɧɢɰɵ,
ɮɨɪɦɢɪɭɸɳɟɣ ɩɨɥɟɡɧɭɸ ɜɨɥɧɭ, ɬɨ ɟɫɬɶ ɦɧɨɝɨ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ.
Ȼɟɪɹ ɢɧɬɟɝɪɚɥɵ ɜ ɹɜɧɨɦ ɜɢɞɟ, ɪɚɡɥɚɝɚɹ ɩɨɥɭɱɚɸɳɢɟɫɹ ɜɵɪɚɠɟɧɢɹ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ J ɢ
ɨɝɪɚɧɢɱɢɜɚɹɫɶ ɭɱɟɬɨɦ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ J, ɪɚɜɟɧɫɬɜɨ (ɉ1.9) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɛɨɥɟɟ ɩɪɨɫɬɨɦ ɜɢɞɟ:
T
bCDP = c1 + (6 / 7)c2 L + (45 c0 / 2NL ) (1+ 6Jl / 7) / [ ³ f c2 (t) dt] u
2
4
0
T
u
(ɉ1.11)
T
[L2 (1 – 3J L / 10) / (3t0) ³ f c(t) G(t) dt + ³ f(t) G1(t) dt ].
0
0
Ⱦɚɧɧɚɹ ɮɨɪɦɭɥɚ ɞɚɟɬ ɹɜɧɨɟ (ɩɪɢɛɥɢɠɟɧɧɨɟ) ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɬɨɱɤɢ ɦɚɤɫɢɦɭɦɚ ɫɤɨɪɨɫɬɧɨɝɨ ɫɩɟɤɬɪɚ ɜ
ɫɥɭɱɚɟ ɧɟɛɨɥɶɲɢɯ ɨɬɥɢɱɢɣ ɝɨɞɨɝɪɚɮɚ ɈȽɌ ɨɬ ɝɢɩɟɪɛɨɥɵ ɢ ɧɟɛɨɥɶɲɨɣ ɷɧɟɪɝɢɢ ɜɨɥɧ-ɩɨɦɟɯ ɧɚ
ɫɭɦɦɨɬɪɚɫɫɟ ɈȽɌ. ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɞɚɟɬ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ b, ɫɨɜɩɚɞɚɸɳɟɟ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɫ1, ɬɨ
ɟɫɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɡɧɚɱɟɧɢɟɦ, ɩɨɥɭɱɚɸɳɢɦɫɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɛɚɡɟ ɚɧɚɥɢɡɚ ɩɪɢ
ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɦ ɝɨɞɨɝɪɚɮɟ ɢ ɨɬɫɭɬɫɬɜɢɢ ɩɨɦɟɯ. ȼɬɨɪɨɟ ɢ ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɵɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɮɨɪɦɭɥɵ
(ɉ1.11) ɞɚɸɬ ɫɦɟɳɟɧɢɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɚ bCDP ɨɬɧɨɫɢɬɟɥɶɧɨ ɫ1=1/ve2. ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɨɩɢɫɵɜɚɟɬ
ɫɦɟɳɟɧɢɟ, ɜɵɡɜɚɧɧɨɟ ɨɬɥɢɱɢɟɦ ɝɨɞɨɝɪɚɮɚ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ ɨɬ ɝɢɩɟɪɛɨɥɵ, ɬɪɟɬɶɟ ɞɚɟɬ ɜɥɢɹɧɢɟ ɜɨɥɧɵɩɨɦɟɯɢ, ɤɨɬɨɪɚɹ ɢɫɤɚɠɚɟɬ ɨɰɟɧɤɭ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ.
ȿɫɥɢ ɜɨɥɧɚ-ɩɨɦɟɯɚ ɨɬɫɭɬɫɬɜɭɟɬ, ɬɨ ɢɡ (ɉ1.11) ɩɨɥɭɱɚɟɦ
bCDP = 1 / v2CDP = c1 + (6 / 7 )L2c2.
(ɉ1.12)
ɉɪɟɠɞɟ ɜɫɟɝɨ ɨɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɩɨɦɟɯɢ ɜɟɥɢɱɢɧɚ bCDP, ɚ ɜɦɟɫɬɟ ɫ ɧɟɣ ɢ vCDP=(1/c1)1/2 ɧɟ
ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɫɢɝɧɚɥɚ, ɚ ɬɨɥɶɤɨ ɨɬ ɛɚɡɵ ɫɤɨɪɨɫɬɧɨɝɨ ɚɧɚɥɢɡɚ ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɫ2, ɨɩɢɫɵɜɚɸɳɟɝɨ
ɨɬɥɢɱɢɟ ɝɨɞɨɝɪɚɮɚ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ ɨɬ ɝɢɩɟɪɛɨɥɵ.
Ⱦɥɹ ɚɧɚɥɢɡɚ ɨɬɥɢɱɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɢ ɞɢɧɚɦɢɱɟɫɤɨɣ ɷɮɮɟɤɬɢɜɧɵɯ ɫɤɨɪɨɫɬɟɣ ɧɚɣɞɟɦ ɡɧɚɱɟɧɢɟ
ɤɨɷɮɮɢɰɢɟɧɬɚ b, ɦɢɧɢɦɢɡɢɪɭɸɳɟɝɨ ɮɭɧɤɰɢɸ Ɏ, ɨɩɪɟɞɟɥɟɧɧɭɸ ɪɚɜɟɧɫɬɜɨɦ (ɉ1.1). Ʉɚɤ ɢ ɜ ɫɥɭɱɚɟ
ɞɢɧɚɦɢɱɟɫɤɨɣ ɷɮɮɟɤɬɢɜɧɨɣ ɫɤɨɪɨɫɬɢ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɝɨɞɨɝɪɚɮ W(l) ɨɩɢɫɵɜɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ (ɉ1.4).
107
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɹɜɧɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ b ɬɚɤɠɟ ɩɪɢɦɟɧɢɦ ɦɟɬɨɞ ɜɨɡɦɭɳɟɧɢɣ, ɞɥɹ ɱɟɝɨ
ɜɦɟɫɬɟ ɫ ɢɫɯɨɞɧɨɣ ɮɭɧɤɰɢɟɣ Ɏ ɪɚɫɫɦɨɬɪɢɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɮɭɧɤɰɢɸ Ɏ, ɡɚɜɢɫɹɳɭɸ ɨɬ ɦɚɥɨɝɨ
ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ H ɢ ɨɩɪɟɞɟɥɟɧɧɭɸ ɪɚɜɟɧɫɬɜɨɦ
N
Ɏ(v) = ¦ [(c0 + c1li2 + H c2li4 )1/2 – (c0 + dli2)1/2 ]2.
(ɉ1.13)
i=1
Ɍɨɱɤɚ ɦɢɧɢɦɭɦɚ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ
N
N
wɎ/w b = ¦ [(c0 + c1 l 2 + c2 l 4)1/2 /(c0 + dl2)1/2]l2 = ¦ l2 .
i=1
(ɉ1.14)
i=1
ɉɪɢ H = 0 ɨɧɨ ɢɦɟɟɬ ɨɱɟɜɢɞɧɨɟ ɪɟɲɟɧɢɟ d = c1, ɩɨɷɬɨɦɭ ɛɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɞɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ H :
d = c1 + H d1 + H 2d2 + ... .
ɉɨɞɫɬɚɜɢɦ ɞɚɧɧɵɣ ɪɹɞ ɜ ɭɪɚɜɧɟɧɢɟ (ɉ1.14), ɪɚɡɥɨɠɢɦ ɥɟɜɭɸ ɱɚɫɬɶ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ H ɢ
ɩɪɢɪɚɜɧɹɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɫɬɟɩɟɧɹɯ H ɤ ɧɭɥɸ. ɉɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ, ɪɟɲɚɹ ɤɨɬɨɪɵɟ, ɦɨɠɧɨ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɣɬɢ ɫɤɨɥɶɤɨ ɭɝɨɞɧɨ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɹɞɚ (ɉ1.8). Ɉɝɪɚɧɢɱɢɜɚɹɫɶ ɭɱɟɬɨɦ ɩɟɪɜɨɣ
ɫɬɟɩɟɧɢ H, ɩɨɥɭɱɢɦ
N
N
6
dCDP = c1 + c2 ¦ li /(c0 +
c1 li2
) / ¦ li4/(c0 + c1 li2).
i=1
i=1
Ʉɚɤ ɢ ɪɚɧɟɟ, ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɞɚɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ ɡɚɦɟɧɢɦ ɫɭɦɦɵ ɧɚ ɢɧɬɟɝɪɚɥɵ ɢ ɨɝɪɚɧɢɱɢɦɫɹ ɭɱɟɬɨɦ
ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɩɚɪɚɦɟɬɪɚ J. Ɍɨɝɞɚ ɞɚɧɧɚɹ ɮɨɪɦɭɥɚ ɭɩɪɨɳɚɟɬɫɹ ɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
dCDP = 1 / w2CDP = c1 + (5 / 7)L2c2.
(ɉ1.15)
ɂɡ ɪɚɜɟɧɫɬɜ (ɉ1.15) ɢ (ɉ1.12) ɫɥɟɞɭɟɬ, ɱɬɨ ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɢ ɞɢɧɚɦɢɱɟɫɤɚɹ ɫɤɨɪɨɫɬɢ ɞɚɠɟ ɜ
ɨɬɫɭɬɫɬɜɢɟ ɩɨɦɟɯ ɨɬɥɢɱɚɸɬɫɹ, ɧɨ ɷɬɨ ɨɬɥɢɱɢɟ ɧɟɜɟɥɢɤɨ, ɢ ɥɟɝɤɨ ɦɨɠɟɬ ɛɵɬɶ ɭɱɬɟɧɨ ɩɪɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ
ɢɫɫɥɟɞɨɜɚɧɢɹɯ – ɞɥɹ ɷɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ ɜɨ ɜɫɟɯ ɮɨɪɦɭɥɚɯ, ɤɭɞɚ ɜɯɨɞɢɬ ɢɧɬɟɝɪɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ,
ɜɦɟɫɬɨ L ɩɨɞɫɬɚɜɢɬɶ (6 / 5)1/2 L .
Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɜɚɠɧɵɣ ɞɥɹ ɩɪɚɤɬɢɤɢ ɜɨɩɪɨɫ ɨ “ɦɚɥɨɫɬɢ” ɤɨɷɮɮɢɰɢɟɧɬɚ ɫ2, ɩɪɢ ɤɨɬɨɪɨɣ
ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɪɚɡɥɨɠɟɧɢɟɦ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ H ɜ ɭɪɚɜɧɟɧɢɢ (ɉ1.7) ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɜ ɪɹɞ
ɩɨ ɫɬɟɩɟɧɹɦ ɫ2. Ʉɚɱɟɫɬɜɟɧɧɨ ɹɫɧɨ, ɱɬɨ ɩɨɧɹɬɢɟ ɦɚɥɨɫɬɢ ɫ2, ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɦɚɥɨɫɬɢ ɨɬɥɢɱɢɹ
ɝɨɞɨɝɪɚɮɚ ɩɨɥɟɡɧɨɣ ɜɨɥɧɵ ɨɬ ɝɢɩɟɪɛɨɥɵ, ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ ɫɢɝɧɚɥɚ – ɱɟɦ ɜɵɲɟ ɟɝɨ ɱɚɫɬɨɬɚ, ɬɟɦ
ɫɢɥɶɧɟɟ ɜɥɢɹɧɢɟ ɧɟɝɢɩɟɪɛɨɥɢɱɧɨɫɬɢ ɝɨɞɨɝɪɚɮɚ ɧɚ ɫɭɦɦɚɪɧɵɣ ɫɢɝɧɚɥ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɤɨɥɢɱɟɫɬɜɟɧɧɵɯ
ɨɰɟɧɨɤ ɪɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
\(t) = ¦ f (t - (c0 + c1l 2 + c2l 4)1/2 + (c0 + c1l 2)1/2)
ɜ ɫɩɟɤɬɪɚɥɶɧɨɣ ɨɛɥɚɫɬɢ. ɉɭɫɬɶ F(Z) - ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f(t), ɬɨ ɟɫɬɶ
f
-iZt
F(Z) = ³ f(t) e
dt .
0
Ɍɨɝɞɚ ɢɡ ɫɜɨɣɫɬɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɏɭɪɶɟ ɫɥɟɞɭɟɬ, ɱɬɨ
-iZWk
< (Z) = F(Z) ¦ e
,
ɝɞɟ < (Z) – ɫɩɟɤɬɪ ɮɭɧɤɰɢɢ \(t), Wk = (c0 + c1l2)1/2 - (c0 + c1 l2 + c2 l4)1/2. ɋ ɬɨɱɧɨɫɬɶɸ ɞɨ Ɉ(ɫ22) ɢɦɟɟɦ
ɪɚɜɟɧɫɬɜɨ
Wk = - ( 1 / 2 )c2 l 4 / (c0 + c1l 2)1/2,
-iZW
k
ɩɨɷɬɨɦɭ ɪɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ ɫ2 ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɫ22 ɪɚɜɧɨɫɢɥɶɧɨ ɪɚɡɥɨɠɟɧɢɸ e
ɜ ɪɹɞ ɩɨ
2
ɫɬɟɩɟɧɹɦ W ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ (iZWk) . Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɜɨɡɦɨɠɧɨɫɬɶ ɭɱɟɬɚ ɬɨɥɶɤɨ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫ2 (ɫ
ɨɬɛɪɚɫɵɜɚɧɢɟɦ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɫɬɟɩɟɧɟɣ) ɮɚɤɬɢɱɟɫɤɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɨɡɦɨɠɧɨɫɬɢ ɨɬɛɪɚɫɵɜɚɧɢɹ ɫɬɟɩɟɧɟɣ
108
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ɫɬɪ.95-112
(iZW) ɜɵɲɟ ɩɟɪɜɨɣ ɩɪɢ ɪɚɡɥɨɠɟɧɢɢ ɮɭɧɤɰɢɢ ɟiZW ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ iZW. Ⱦɥɹ ɮɭɧɤɰɢɢ ɟZ ɫɩɪɚɜɟɞɥɢɜɨ
ɪɚɡɥɨɠɟɧɢɟ
ez = 1 + z/1! + z2/2! + z3/3! + ... + zn/n! + ... .
Ɇɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɟz | 1+z ɩɪɢ ~ɯ~< 0,5, ɬɨ ɟɫɬɶ ɜ ɪɚɡɥɨɠɟɧɢɢ ɮɭɧɤɰɢɢ eiZW ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
iZW ɦɨɠɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɭɱɟɬɨɦ ɬɨɥɶɤɨ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ iZW, ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ ~ZW~< 0,5.
Ⱦɚɥɟɟ, ɩɭɫɬɶ Zɩ – ɩɪɟɨɛɥɚɞɚɸɳɚɹ ɱɚɫɬɨɬɚ ɫɢɝɧɚɥɚ, Ɍɩ – ɩɟɪɢɨɞ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɷɬɨɣ ɱɚɫɬɨɬɟ,
ɬɨ ɟɫɬɶ Ɍɩ = 2S/Z. Ɍɨɝɞɚ ɭɫɥɨɜɢɟ ɜɨɡɦɨɠɧɨɫɬɢ ɭɱɟɬɚ ɬɨɥɶɤɨ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫ2 ɩɪɢ ɪɚɡɥɨɠɟɧɢɢ ɜ ɪɹɞ
ɩɪɢɧɢɦɚɟɬ ɜɢɞ 2SW /Ɍɩ < 0,5, ɢɥɢ ɩɪɢɛɥɢɠɟɧɧɨ W < Ɍ/10. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɜɨɡɦɭɳɟɧɢɣ
ɞɥɹ ɨɩɢɫɚɧɢɹ ɨɬɥɢɱɢɹ ɝɨɞɨɝɪɚɮɚ Wf(l) ɨɬ ɝɢɩɟɪɛɨɥɵ ɩɪɢ ɚɧɚɥɢɡɟ ɫɩɟɤɬɪɚ ɫɤɨɪɨɫɬɟɣ ɜɨɡɦɨɠɧɨ, ɟɫɥɢ ɷɬɨ
ɨɬɥɢɱɢɟ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ 1/10 ɜɢɞɢɦɨɝɨ ɩɟɪɢɨɞɚ. ȼ ɩɪɚɤɬɢɤɟ ɦɨɪɫɤɢɯ ɫɟɣɫɦɨɪɚɡɜɟɞɨɱɧɵɯ ɪɚɛɨɬ
ɩɪɟɨɛɥɚɞɚɸɳɢɣ ɩɟɪɢɨɞ ɩɨɥɟɡɧɵɯ (ɨɞɧɨɤɪɚɬɧɨ ɨɬɪɚɠɟɧɧɵɯ ɩɪɨɞɨɥɶɧɵɯ) ɜɨɥɧ ɞɥɹ ɝɥɭɛɢɧ ɩɨɪɹɞɤɚ 2 - 3
ɤɦ ɫɨɫɬɚɜɥɹɟɬ 30 - 40 ɦɫ, ɩɨɷɬɨɦɭ ɨɬɥɢɱɢɟ ɝɨɞɨɝɪɚɮɚ ɨɬ ɝɢɩɟɪɛɨɥɵ ɧɟ ɞɨɥɠɧɨ ɩɪɟɜɨɫɯɨɞɢɬɶ 3 - 4 ɦɫ, ɱɬɨ,
ɤɚɤ ɩɨɤɚɡɵɜɚɸɬ ɱɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ, ɜɵɩɨɥɧɹɟɬɫɹ ɞɥɹ ɛɨɥɶɲɢɧɫɬɜɚ ɧɟɮɬɟɧɨɫɧɵɯ ɪɚɣɨɧɨɜ (ɉɭɡɵɪɟɜ,
1979; Ɇɚɥɨɜɢɱɤɨ, 1990).
ɉɪɢɥɨɠɟɧɢɟ 2. ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɢ ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜ ɧɟɨɞɧɨɪɨɞɧɨɦ ɫɥɨɟ
ɉɭɫɬɶ ɜ ɧɟɨɞɧɨɪɨɞɧɨɦ ɫɥɨɟ ɫɨ ɫɤɨɪɨɫɬɶɸ v=v(x,y,z) ɡɚɞɚɧɵ ɞɜɟ ɬɨɱɤɢ Q1([1,K1,]1) ɢ Q2([2,K2,]2).
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɪɚɟɤɬɨɪɢɸ J : x=x(z), y=y(z), ɫɨɟɞɢɧɹɸɳɭɸ ɷɬɢ ɬɨɱɤɢ, ɢ ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ. ɋɧɚɱɚɥɚ
ɩɨɥɭɱɢɦ ɩɪɢɛɥɢɠɟɧɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ, ɚ ɡɚɬɟɦ ɧɚɣɞɟɦ ɜɪɟɦɹ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ ɦɚɥɨɝɨ
ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ. Ɏɭɧɤɰɢɢ x=x(z), y=y(z), ɨɩɢɫɵɜɚɸɳɢɟ ɬɪɚɟɤɬɨɪɢɸ J, ɭɞɨɜɥɟɬɜɨɪɹɸɬ
ɭɪɚɜɧɟɧɢɹɦ ɗɣɥɟɪɚ
w F/wx - d(wF/wxc)/ dz = 0,
w F/wy - d(wF/wyc)/ dz = 0,
(ɉ2.1)
ɝɞɟ
F(x,y,z,xc,yc) = (1 + xc 2 + yc 2)1/2 / v(x,y,z) ,
(ɉ2.2)
ɢ ɮɭɧɤɰɢɢ x(z), y(z), ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ
x(]1) = [1,
y(]1) = K1.
(ɉ2.3)
ɉɨɞɫɬɚɜɥɹɹ (ɉ2.2) ɜ (ɉ2.1) ɢ ɪɚɫɤɪɵɜɚɹ ɩɪɨɢɡɜɨɞɧɵɟ ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ x ɢ ɭ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ z,
ɩɨɥɭɱɢɦ, ɱɬɨ ɮɭɧɤɰɢɢ x(z), y(z) ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɢɫɬɟɦɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
(w 2F /wxc 2)xcc + (w 2F/wxcwyc ) ycc + (w 2F/wxcwx) xc + (w 2F/wxcwy) yc + (w 2F /wxcwz) - w F/wx = 0 ,
(ɉ2.4)
(w 2F /wxcwyc )xcc + (w 2F/wyc 2) ycc + (w 2F/wycwx) xc + (w 2F/wycwy) yc + (w 2F /wycwz) - w F/wy = 0 .
Ⱦɥɹ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɵ, ɤɨɝɞɚ ɫɤɨɪɨɫɬɶ v ɧɟ ɡɚɜɢɫɢɬ ɨɬ x, y, z, ɫɢɫɬɟɦa (ɉ2.4) ɩɟɪɟɯɨɞɢɬ ɜ ɭɪɚɜɧɟɧɢɹ
(w 2F/wxc 2)xcc + (w 2F /wxcwyc) ycc = 0 ,
(w 2F /wxcwyc )xcc + (w 2F /wyc 2)ycc = 0 ,
ɨɬɤɭɞɚ x''(z)=0, y''(z)=0. Ɍɨɝɞɚ c ɭɱɟɬɨɦ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ (ɉ2.3) ɩɨɥɭɱɚɟɦ, ɱɬɨ
x(z)=[1 + [([2-[1)/(]2-]1)](z-]1),
(ɉ2.5)
y(z)=K1 + [(K2-K1)/(]2-]1)](z-]1),
ɢ ɬɪɚɟɤɬɨɪɢɹ J ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɹɦɭɸ. ȿɫɥɢ ɫɥɨɣ ɧɟɨɞɧɨɪɨɞɧɵɣ, ɬɨ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ (ɉ2.4)
ɭɞɚɟɬɫɹ ɧɚɣɬɢ ɬɨɥɶɤɨ ɞɥɹ ɫɪɟɞɵ ɫ ɥɢɧɟɣɧɵɦ ɢɡɦɟɧɟɧɢɟɦ ɫɤɨɪɨɫɬɢ (ɉɭɡɵɪɟɜ, 1979). ȼ ɬɨ ɠɟ ɜɪɟɦɹ, ɤɚɤ
ɩɨɤɚɡɚɧɨ ɜ (Ȼɥɹɫ, 1991), ɜ ɜɟɪɯɧɟɣ ɱɚɫɬɢ ɪɚɡɪɟɡɚ ɨɫɧɨɜɧɭɸ ɪɨɥɶ ɢɝɪɚɸɬ ɧɟɥɢɧɟɣɧɵɟ (ɤɜɚɞɪɚɬɢɱɧɵɟ)
ɢɡɦɟɧɟɧɢɹ ɩɥɚɫɬɨɜɵɯ ɫɤɨɪɨɫɬɟɣ, ɩɨɷɬɨɦɭ ɜɚɠɧɨ ɢɦɟɬɶ ɹɜɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜɪɟɦɟɧɢ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜ
ɫɥɨɹɯ ɫ ɧɟɥɢɧɟɣɧɵɦ ɢɡɦɟɧɟɧɢɟɦ ɫɤɨɪɨɫɬɟɣ.
109
Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
ɉɭɫɬɶ n(x,y,z) – ɦɟɞɥɟɧɧɨɫɬɶ, ɬɨ ɟɫɬɶ ɜɟɥɢɱɢɧɚ, ɨɛɪɚɬɧɚɹ ɫɤɨɪɨɫɬɢ: n(x,y,z) = 1 / v(x,y,z). Ɍɨɝɞɚ
ɭɪɚɜɧɟɧɢɹ (ɉ2.4) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ
nx''=(1+x' 2+y' 2)(wn/wx - wn/wz),
(ɉ2.6)
ny''=(1+x' 2+y' 2)(wn/wy - wn/wz).
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɹɜɧɨɣ ɮɨɪɦɭɥɵ ɞɥɹ ɬɪɚɟɤɬɨɪɢɢ ɜɨɥɧɵ ɢ ɟɟ ɜɪɟɦɟɧɢ ɩɪɨɛɟɝɚ ɜɦɟɫɬɟ ɫ ɢɫɯɨɞɧɨɣ ɦɨɞɟɥɶɸ
ɦɟɞɥɟɧɧɨɫɬɢ, ɨɩɢɫɵɜɚɟɦɨɣ ɮɭɧɤɰɢɟɣ n(x,y,z), ɪɚɫɫɦɨɬɪɢɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɦɨɞɟɥɶ ɫ ɦɟɞɥɟɧɧɨɫɬɶɸ,
ɡɚɜɢɫɹɳɟɣ ɨɬ ɦɚɥɨɝɨ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ H:
n(x,y,z,H) = n0 + H m(x,y,z).
(ɉ2.7)
Ⱦɥɹ ɦɨɞɟɥɢ ɫ ɦɟɞɥɟɧɧɨɫɬɶɸ n(x,y,z,H) ɫɢɫɬɟɦɚ (ɉ2.7) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
(n0 + H m(x,y,z)) x'' = H (1+x' 2+y' 2)(wm/wx - wm/wz),
(ɉ2.8)
(n0 + H m(x,y,z)) y'' = H (1+x' 2+y' 2)(wm/wy - wm/wz).
ɋɨɝɥɚɫɧɨ ɦɟɬɨɞɭ ɜɨɡɦɭɳɟɧɢɣ, ɪɟɲɟɧɢɟ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (ɉ2.3), (ɉ2.8) ɢɳɟɦ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ H:
x (z,H) = x0(z) +H x1(z) + H 2x2(z) + ... ,
(ɉ2.9)
y (z,H) = y0(z) + H y1(z) + H 2y2(z) + ... .
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ xi(z), yi(z) ɪɹɞɨɜ (ɉ2.9) ɩɨɞɫɬɚɜɢɦ ɢɯ ɜ ɭɪɚɜɧɟɧɢɹ (ɉ2.8),
ɪɚɡɥɨɠɢɦ ɥɟɜɵɟ ɢ ɩɪɚɜɵɟ ɱɚɫɬɢ ɜ ɪɹɞɵ ɩɨ ɫɬɟɩɟɧɹɦ H ɢ ɩɪɢɪɚɜɧɹɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ ɥɟɜɨɣ ɢ ɩɪɚɜɨɣ
ɱɚɫɬɹɯ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ H. ɉɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ xi(z), yi(z),
i=0,1,2,... . Ⱦɥɹ ɯ0, ɭ0 ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɨɞɧɨɪɨɞɧɨɦɭ ɫɥɨɸ, ɟɟ ɪɟɲɟɧɢɟ ɞɚɟɬɫɹ
ɪɚɜɟɧɫɬɜɚɦɢ (ɉ2.5). Ⱦɥɹ ɫɥɟɞɭɸɳɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ xi(z), yi(z) (i=1,2,3,...) ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ
ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ
xi(z) = fi(x0, y0, x1, y1, x2, y2, ..., xi-1, yi-1) ,
i=1,2,...
(ɉ2.10)
yi(z) = gi(x0, y0, x1, y1, x2, y2, ..., xi-1, yi-1) ,
ɝɞɟ fi, gi – ɮɭɧɤɰɢɢ, ɤɨɬɨɪɵɟ ɧɚɯɨɞɹɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɜɵɩɨɥɧɟɧɢɹ ɨɩɢɫɚɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ. Ɍɚɤ ɤɚɤ x0(z),
y0(z) ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ (ɉ2.3), ɬɨ ɢɡ (ɉ2.9) ɫɥɟɞɭɟɬ, ɱɬɨ ɮɭɧɤɰɢɢ xi(z), yi(z) (i=1,2,3...)
ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɧɭɥɟɜɵɦ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ
xi(]1) = yi(]1) = 0, i = 1, 2, 3,... .
(ɉ2.11)
ɂɡ ɭɪɚɜɧɟɧɢɣ (ɉ2.10) ɫɥɟɞɭɟɬ, ɱɬɨ xi(z), yi(z), i=1,2,3... ɦɨɠɧɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɣɬɢ ɩɪɨɫɬɵɦ
ɞɜɭɤɪɚɬɧɵɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɪɚɜɵɯ ɱɚɫɬɟɣ (ɜ ɤɨɬɨɪɵɯ ɩɪɟɞɵɞɭɳɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɚɧɟɟ ɧɚɣɞɟɧɵ) ɢ
ɨɩɪɟɞɟɥɟɧɢɟɦ ɩɨɥɭɱɚɸɳɢɯɫɹ ɞɜɭɯ ɩɨɫɬɨɹɧɧɵɯ ɢɡ ɭɫɥɨɜɢɣ (ɉ2.11). Ⱦɥɹ x1(z), y1(z) ɩɨɥɭɱɢɦ
]2
z
x1(z) = (1+b +c )/no [ ³ Mx(s)ds – (z – ]1) / (]2 – ]1) ³ Mx(s)ds],
2
2
]1
]1
]2
z
y1(z) = (1+b2+c2)/no [ ³ My(s)ds – (z – ]1) / (]2 – ]1) ³ My(s)ds],
]1
]1
ɝɞɟ
b = ([2 – [1) / (]2 – ]1),
c = (K2 – K1) / (]2 – ]1),
z
z
Mx(z) = ³(w m /wx)ds,
My(z) = ³(w m /wy)ds.
]1
]1
110
(ɉ2.12)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 1, ʋ2, 1998 ɝ.
ɫɬɪ.95-112
ȼ ɷɬɢɯ ɪɚɜɟɧɫɬɜɚɯ ɚɪɝɭɦɟɧɬɨɦ ɮɭɧɤɰɢɢ m ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɯ ɹɜɥɹɟɬɫɹ (xo(s), yo(s), s). Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɢɥɭ
ɩɪɢɧɰɢɩɚ ɫɬɚɰɢɨɧɚɪɧɨɫɬɢ ɧɚɯɨɠɞɟɧɢɟ ɜɪɟɦɟɧɢ t ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ H i ɬɪɟɛɭɟɬ ɧɚɯɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ
(ɮɭɧɤɰɢɣ x(z), y(z)) ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ H i-1. Ɉɬɫɸɞɚ, ɜ ɱɚɫɬɧɨɫɬɢ, ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɪɟɦɟɧɢ t ɫ
ɬɨɱɧɨɫɬɶɸ ɞɨ o(H 2) ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ xi(z), yi(z) ɞɥɹ i=0,1.
ȼɪɟɦɹ t(Q1,Q) ɩɪɨɛɟɝɚ ɜɨɥɧɚ ɦɟɠɞɭ ɬɨɱɤɚɦɢ Q1([1,K1,]1) ɢ Q2([2,K2,]2) ɞɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ
]2
t(Q1,Q) = ³ [1 + xc 2(z) + yc 2(z)]2 n(x(z), y(z), z) dz.
]1
ɉɨɞɫɬɚɜɥɹɹ ɜ ɞɚɧɧɨɟ ɪɚɜɟɧɫɬɜɨ x(z), y(z), ɧɚɣɞɟɧɧɵɟ ɩɨ ɮɨɪɦɭɥɚɦ (ɉ2.9) ɢ ɪɚɡɥɚɝɚɹ ɟɝɨ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
H, ɩɨɥɭɱɢɦ ɹɜɧɨɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜɪɟɦɟɧɢ t ɜ ɜɢɞɟ
t(Q1, Q2) = t0(Q1, Q2) + t1(Q1, Q2)H + t2(Q1, Q2)H2 + ... .
ɋ ɬɨɱɧɨɫɬɶɸ ɞɨ o(H 2) ɨɧɨ ɢɦɟɟɬ ɜɢɞ
]2
]2
t(Q1, Q2) = n0D + HD/(]2 – ]1) ³ mdz + e {[D/(]2 – ]1)] ³ [(w m /wx)x1 + (w m /wy)y1)]dz +
2
]1
(ɉ2.13)
]1
]2
]2
+ [(]2 – ]1)/D] ³ (bx1'+cy1c )mdz + (1/2)no(]2 – ]1)3/D3 ³ [x1' 2+y1' 2 + (by1'- cx1')2]dz},
]2
]2
ɝɞɟ D = (([2 – [1)2 + (K2 – K1)2 + (z2 – z1)2)1/2 – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ Q1, Q2.
Ɏɨɪɦɭɥɚ (ɉ2.13) ɞɚɟɬ ɹɜɧɨɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜɪɟɦɟɧɢ t ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜ
ɧɟɨɞɧɨɪɨɞɧɨɦ ɫɥɨɟ ɢɡ ɬɨɱɤɢ Q1 ɜ ɬɨɱɤɭ Q2 ɞɥɹ ɦɨɞɟɥɢ ɫ ɦɟɞɥɟɧɧɨɫɬɶɸ (ɉ2.7). Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɢɡ
ɞɚɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ ɩɨɥɭɱɢɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɞɥɹ ɜɪɟɦɟɧɢ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɜ ɫɪɟɞɟ ɫ ɦɟɞɥɟɧɧɨɫɬɶɸ n(x,y,z),
ɞɨɫɬɚɬɨɱɧɨ ɜ ɧɟɦ ɡɚɦɟɧɢɬɶ H m(x,y,z) ɧɚ n(x,y,z)- no. ɑɢɫɥɟɧɧɵɟ ɪɚɫɱɟɬɵ ɩɨɤɚɡɚɥɢ ɜɵɫɨɤɭɸ ɬɨɱɧɨɫɬɶ
ɞɚɧɧɨɣ ɮɨɪɦɭɥɵ ɜɨ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɵɯ ɫɥɭɱɚɹɯ. ɉɪɢ ɢɡɦɟɧɟɧɢɢ ɫɤɨɪɨɫɬɢ ɧɚ 10y15% ɜ ɨɛɥɚɫɬɢ,
ɫɨɞɟɪɠɚɳɟɣ ɬɨɱɤɢ Q1, Q2, ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ 0,1y0,15%.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɭɱɟɬ ɱɥɟɧɨɜ ɩɨɪɹɞɤɚ H ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɯɨɠɞɟɧɢɸ ɜɪɟɦɟɧɢ t ɜɞɨɥɶ ɩɪɹɦɨɣ,
ɫɨɟɞɢɧɹɸɳɟɣ ɬɨɱɤɢ Q1, Q2, ɬɨ ɟɫɬɶ ɧɚɯɨɠɞɟɧɢɸ ɜɪɟɦɟɧɢ ɜɞɨɥɶ ɬɪɚɟɤɬɨɪɢɢ, ɩɨɥɭɱɟɧɧɨɣ ɜ ɫɪɟɞɟ ɧɭɥɟɜɨɝɨ
ɩɪɢɛɥɢɠɟɧɢɹ – ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ. ȼ ɫɚɦɨɦ ɞɟɥɟ,
]2
]2
noD + H D/(z2-z1) ³ m dz = ³ (1+x'o2 + y'o2)1/2 (no + H m)dz.
]2
]2
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɟɪɜɵɟ ɞɜɚ ɫɥɚɝɚɟɦɵɟ ɜ (ɉ2.13) ɞɚɸɬ ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɛɟɡ ɭɱɟɬɚ ɤɪɢɜɢɡɧɵ ɥɭɱɚ ɜ
ɧɟɨɞɧɨɪɨɞɧɨɦ ɫɥɨɟ, ɚ ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ (ɱɥɟɧ ɩɨɪɹɞɤɚ H 2) ɩɨɡɜɨɥɹɟɬ ɭɱɟɫɬɶ ɜɥɢɹɧɢɟ ɤɪɢɜɢɡɧɵ ɥɭɱɚ ɧɚ
ɜɪɟɦɹ ɩɪɨɛɟɝɚ ɜɨɥɧɵ.
ȼ ɮɨɪɦɭɥɭ (ɉ6.7) ɜɯɨɞɹɬ ɢɧɬɟɝɪɚɥɵ ɨɬ ɦɟɞɥɟɧɧɨɫɬɢ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɯ, ɩɪɢ ɱɢɫɥɟɧɧɵɯ ɠɟ
ɪɚɫɱɟɬɚɯ ɞɥɹ ɤɨɧɤɪɟɬɧɵɯ ɦɨɞɟɥɟɣ ɠɟɥɚɬɟɥɶɧɨ ɢɦɟɬɶ ɹɜɧɵɟ ɮɨɪɦɭɥɵ, ɧɟ ɫɨɞɟɪɠɚɳɢɟ ɢɧɬɟɝɪɚɥɵ. ȿɫɥɢ
ɦɟɞɥɟɧɧɨɫɬɶ ɨɩɢɫɵɜɚɟɬɫɹ ɩɨɥɢɧɨɦɨɦ (ɢɥɢ ɫɩɥɚɣɧɨɦ), ɡɚɜɢɫɹɳɢɦ ɨɬ ɩɟɪɟɦɟɧɧɵɯ x, y, z, ɬɨ ɢɧɬɟɝɪɚɥɵ,
ɜɯɨɞɹɳɢɟ ɜ ɪɚɜɟɧɫɬɜɨ (ɉ6.7), ɛɟɪɭɬɫɹ ɜ ɹɜɧɨɦ ɜɢɞɟ.
Ʌɢɬɟɪɚɬɭɪɚ
Goldin S.V. Seismic travel time inversion. Tulsa, 364p., 1986.
Sattleger I.W. A method of computing true interval velocities from expanding spread data in the case of
arbitrary long spread and arbitrary dipping plane interfaces. Geophysical Prospecting, v.13, ʋ2, p.306318, 1975.
Ȼɥɹɫ ɗ.Ⱥ. ȼɪɟɦɟɧɧɵɟ ɩɨɥɹ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ ɜ ɬɪɟɯɦɟɪɧɵɯ ɫɥɨɢɫɬɵɯ ɫɨ ɫɥɚɛɨ ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ
ɝɪɚɧɢɰɚɦɢ ɪɚɡɞɟɥɚ ɢ ɥɚɬɟɪɚɥɶɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɫɥɨɹɦɢ. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɩɪɨɛɥɟɦɵ
ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɞɚɧɧɵɯ ɫɟɣɫɦɨɪɚɡɜɟɞɤɢ. ɇɨɜɨɫɢɛɢɪɫɤ, ɇɚɭɤɚ, ɫ.98-128, 1988.
Ȼɥɹɫ ɗ.Ⱥ. Ʉɢɧɟɦɚɬɢɤɚ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ ɜ ɬɪɟɯɦɟɪɧɵɯ ɩɨɩɟɪɟɱɧɨ-ɢɡɨɬɪɨɩɧɵɯ ɝɨɪɢɡɨɧɬɚɥɶɧɨɧɟɨɞɧɨɪɨɞɧɵɯ ɫɪɟɞɚɯ. Ɇɟɬɨɞɵ ɪɚɫɱɟɬɚ ɢ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɫɟɣɫɦɢɱɟɫɤɢɯ ɜɨɥɧɨɜɵɯ ɩɨɥɟɣ.
ɇɨɜɨɫɢɛɢɪɫɤ, ɇɚɭɤɚ, ɫ.148-188, 1991.
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Ȼɥɹɫ. ɗ.Ⱥ. Ɉɩɪɟɞɟɥɟɧɢɟ ɬɪɟɯɦɟɪɧɨɣ ɫɥɨɢɫɬɨ-ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɤɨɪɨɫɬɧɨɣ ɦɨɞɟɥɢ ɫɪɟɞɵ ...
Ȼɥɹɫ ɗ.Ⱥ. ɉɪɢɛɥɢɠɟɧɧɵɣ ɫɩɨɫɨɛ ɧɚɯɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɣ ɥɭɱɟɣ ɜ ɬɪɟɯɦɟɪɧɵɯ ɫɥɨɢɫɬɨ-ɨɞɧɨɪɨɞɧɵɯ
ɫɪɟɞɚɯ. Ƚɟɨɥɨɝɢɹ ɢ ɝɟɨɮɢɡɢɤɚ, ʋ12, ɫ.80-87, 1985.
Ȼɥɹɫ ɗ.Ⱥ., Ʌɟɜɢɬ Ⱥ.ɇ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɚɫɬɨɜɨɣ
ɦɨɞɟɥɢ ɩɨ ɞɚɧɧɵɦ ɦɟɬɨɞɚ ɈȽɌ. Ƚɟɨɥɨɝɢɹ ɢ ɝɟɨɮɢɡɢɤɚ, ʋ3, ɫ.112-120, 1989.
Ƚɥɨɝɨɜɫɤɢɣ ȼ.Ɇ. Ⱥɧɚɥɢɡ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɨɣ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɆɈȼ ɜ ɧɟɨɞɧɨɪɨɞɧɵɯ
ɫɪɟɞɚɯ. Ɇ., Ɍɪɭɞɵ ɏɏɏ Ɇɟɠɞɭɧɚɪɨɞɧɨɝɨ ɝɟɨɮɢɡɢɱɟɫɤɨɝɨ ɫɢɦɩɨɡɢɭɦɚ, ɬ.2, ɫ.106-116, 1985.
Ƚɥɨɝɨɜɫɤɢɣ ȼ.Ɇ., Ƚɨɝɨɧɟɧɤɨɜ Ƚ.ɇ. ɋɯɨɞɢɦɨɫɬɶ ɢɬɟɪɚɬɢɜɧɨɝɨ ɦɟɬɨɞɚ ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɚɫɬɨɜɵɯ ɫɤɨɪɨɫɬɟɣ
ɩɨ ɫɟɣɫɦɢɱɟɫɤɢɦ ɞɚɧɧɵɦ. Ɇ., ɇɟɞɪɚ, ɉɪɢɤɥɚɞɧɚɹ ɝɟɨɮɢɡɢɤɚ, ɜɵɩ.92, ɫ.65-78, 1978.
Ƚɨɥɶɞɢɧ ɋ.ȼ. Ⱦɜɭɦɟɪɧɚɹ ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɫɟɣɫɦɨɝɪɚɦɦ ɜ ɫɥɨɢɫɬɵɯ ɫɪɟɞɚɯ. ɇɨɜɨɫɢɛɢɪɫɤ,
ɇɚɭɤɚ, 208ɫ., 1993.
Ƚɨɥɶɞɢɧ ɋ.ȼ. ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɞɚɧɧɵɯ ɫɟɣɫɦɢɱɟɫɤɨɝɨ ɦɟɬɨɞɚ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ. Ɇ., ɇɟɞɪɚ, 344ɫ., 1979.
Ƚɨɥɶɰɦɚɧ Ɏ.Ɇ. ɋɬɚɬɢɫɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɢɧɬɟɪɩɪɟɬɚɰɢɢ. Ɇ., ɇɟɞɪɚ, 328ɫ., 1971.
Ʉɚɪɚɫɢɤ ȼ.Ɇ. ɂɡɭɱɟɧɢɟ ɫɤɨɪɨɫɬɟɣ ɫɟɣɫɦɢɱɟɫɤɢɯ ɜɨɥɧ ɤɨɦɩɥɟɤɫɨɦ ɦɟɬɨɞɨɜ. Ɇ., ɇɟɞɪɚ, 224ɫ., 1993.
Ʉɨɥɦɨɝɨɪɨɜ Ⱥ.ɇ., Ɏɨɦɢɧ ɋ.ȼ. ɗɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɮɭɧɤɰɢɣ ɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɚɧɚɥɢɡɚ. Ɇ., ɇɚɭɤɚ, 324ɫ.,
1968.
Ɇɚɥɨɜɢɱɤɨ Ⱥ.Ⱥ. Ʉɢɧɟɦɚɬɢɱɟɫɤɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɞɚɧɧɵɯ ɰɢɮɪɨɜɨɣ ɫɟɣɫɦɨɪɚɡɜɟɞɤɢ ɜ ɭɫɥɨɜɢɹɯ
ɜɟɪɬɢɤɚɥɶɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɯ ɫɪɟɞ. ɋɜɟɪɞɥɨɜɫɤ, ɍɪɈ Ⱥɇ ɋɋɋɊ, 270ɫ., 1990.
Ɇɟɲɛɟɣ ȼ.ɂ. Ɇɟɬɨɞɢɤɚ ɦɧɨɝɨɤɪɚɬɧɵɯ ɩɟɪɟɤɪɵɬɢɣ ɜ ɫɟɣɫɦɨɪɚɡɜɟɞɤɟ. Ɇ., ɇɟɞɪɚ, 264ɫ., 1985.
ɉɭɡɵɪɟɜ ɇ.ɇ. ȼɪɟɦɟɧɧɵɟ ɩɨɥɹ ɨɬɪɚɠɟɧɧɵɯ ɜɨɥɧ ɢ ɦɟɬɨɞ ɷɮɮɟɤɬɢɜɧɵɯ ɩɚɪɚɦɟɬɪɨɜ. ɇɨɜɨɫɢɛɢɪɫɤ,
ɇɚɭɤɚ, 294ɫ., 1979.
ɍɪɭɩɨɜ Ⱥ.Ʉ., Ʌɟɜɢɧ Ⱥ.ɇ. Ɉɩɪɟɞɟɥɟɧɢɟ ɢ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɫɤɨɪɨɫɬɟɣ. Ɇ., ɇɟɞɪɚ, 290ɫ., 1985.
ɑɟɪɧɹɤ ȼ.ɋ., Ƚɪɢɰɟɧɤɨ ɋ.Ⱥ. ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɷɮɮɟɤɬɢɜɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɈȽɌ ɞɥɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ
ɫɢɫɬɟɦɵ ɨɞɧɨɪɨɞɧɵɯ ɫɥɨɟɜ ɫ ɤɪɢɜɨɥɢɧɟɣɧɵɦɢ ɝɪɚɧɢɰɚɦɢ. Ƚɟɨɥɨɝɢɹ ɢ ɝɟɨɮɢɡɢɤɚ, ʋ12, ɫ.112-119,
1979.
əɧɨɜɫɤɚɹ Ɍ.Ȼ., ɉɨɪɨɯɨɜɚ Ʌ.ɇ. Ɉɛɪɚɬɧɵɟ ɡɚɞɚɱɢ ɝɟɨɮɢɡɢɤɢ. Ʌ., ɂɡɞ-ɜɨ ɅȽɍ, 212ɫ., 1983.
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