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# О построении матриц жесткости трехмерного дискретно-континуального конечного элемента с четырехугольным поперечным сечением методом базисных вариаций.

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```4/2011
ВЕСТНИК
МГСУ
Ɉ ɉɈɋɌɊɈȿɇɂɂ ɆȺɌɊɂɐ ɀȿɋɌɄɈɋɌɂ ɌɊȿɏɆȿɊɇɈȽɈ
ȾɂɋɄɊȿɌɇɈ-ɄɈɇɌɂɇɍȺɅɖɇɈȽɈ ɄɈɇȿɑɇɈȽɈ ɗɅȿɆȿɇɌȺ
ɋ ɑȿɌɕɊȿɏɍȽɈɅɖɇɕɆ ɉɈɉȿɊȿɑɇɕɆ ɋȿɑȿɇɂȿɆ
ɆȿɌɈȾɈɆ ȻȺɁɂɋɇɕɏ ȼȺɊɂȺɐɂɃ
CONSTRUCTION OF STIFFNESS MATRICES
OF THREE-DIMENSIONAL DISCRETE-CONTINUAL FINITE
BY METHOD OF BASIC VARIATIONS
ɉ.Ⱥ. Ⱥɤɢɦɨɜ, Ɇ.Ʌ. Ɇɨɡɝɚɥɟɜɚ, ȼ.ɇ. ɋɢɞɨɪɨɜ
Pavel A. Akimov, Marina L. Mozgaleva, Vladimir N. Sidorov
ȽɈɍ ȼɉɈ ɆȽɋɍ
Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɨɪɪɟɤɬɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɫɬɪɨɟɧɢɹ ɦɚɬɪɢɰ ɠɟɫɬɤɨɫɬɢ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ ɫ ɱɟɬɵɪɟɯɭɝɨɥɶɧɵɦ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ,
ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɟɬɨɞɚ ɛɚɡɢɫɧɵɯ ɜɚɪɢɚɰɢɣ.
Correct algorithm of construction of stiffness matrices of three-dimensional discretecontinual finite element with quadrangular cross-section by method of basic variations are
under consideration in the distinctive paper.
ȼɜɟɞɟɧɢɟ.
ȼ ɧɚɫɬɨɹɳɟɣ ɫɬɚɬɶɟ ɨɩɢɫɵɜɚɟɬɫɹ ɚɥɝɨɪɢɬɦ ɩɨɫɬɪɨɟɧɢɹ ɦɚɬɪɢɰ ɠɟɫɬɤɨɫɬɢ ɬɪɟɯɦɟɪɧɨɝɨ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ ɫ ɱɟɬɵɪɟɯɭɝɨɥɶɧɵɦ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɟɬɨɞɚ ɛɚɡɢɫɧɵɯ ɜɚɪɢɚɰɢɣ [1, 2, 4, 5],
ɤɨɬɨɪɵɣ ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɜɵɫɨɤɨɣ ɫɬɟɩɟɧɶɸ ɚɥɝɨɪɢɬɦɢɱɧɨɫɬɢ ɢ ɭɧɢɜɟɪɫɚɥɶɧɨɫɬɢ, ɚ ɫ ɞɪɭɝɨɣ – ɨɬɧɨɫɢɬɟɥɶɧɨ ɛɨɥɶɲɢɦ ɨɛɴɟɦɨɦ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɪɚɛɨɬɵ
(ɷɬɨ, ɜɩɪɨɱɟɦ, ɧɟ ɤɪɢɬɢɱɧɨ ɧɚ ɫɨɜɪɟɦɟɧɧɨɦ ɷɬɚɩɟ ɪɚɡɜɢɬɢɹ ɤɨɦɩɶɸɬɟɪɧɨɣ ɬɟɯɧɢɤɢ).
1. Ⱦɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɚɹ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɚɹ ɦɨɞɟɥɶ ɤɨɧɫɬɪɭɤɰɢɢ.
ɉɭɫɬɶ x3 – ɩɟɪɟɦɟɧɧɚɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɧɚɩɪɚɜɥɟɧɢɸ ɪɟɝɭɥɹɪɧɨɫɬɢ ɮɢɡɢɤɨɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɤɨɧɫɬɪɭɤɰɢɢ (ɨɫɧɨɜɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ),
ɩɪɢɱɟɦ ɩɭɫɬɶ ɜɞɨɥɶ x3 ɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɤɨɧɫɬɪɭɤɰɢɢ ɢɡɦɟɧɹɸɬɫɹ
ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɨ (ɡɚɦɟɬɢɦ, ɱɬɨ ɩɨ ɩɟɪɟɦɟɧɧɵɦ x1 , x2 ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɚɪɚɦɟɬɪɵ
ɦɨɝɭɬ ɢɡɦɟɧɹɬɶɫɹ ɩɪɨɢɡɜɨɥɶɧɨ); : k , k = 1, 2, ..., nk 1 – ɩɨɞɨɛɥɚɫɬɢ ɩɨɫɬɨɹɧɫɬɜɚ ɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɤɨɧɫɬɪɭɤɰɢɢ ɫ ɩɨɩɟɪɟɱɧɵɦ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ) ɫɟɱɟɧɢɟɦ S:(k ) ; Zk , k = 1, 2, ..., nk 1 – ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɪɚɫɲɢɪɟɧɧɵɟ ɨɛɥɚɫɬɢ, ɨɤɚɣɦɥɹɸɳɢɟ ɢɫɯɨɞɧɵɟ : k , k = 1, 2, ..., nk 1 , ɩɪɢɱɟɦ : k  Zk .
ɉɪɢɧɢɦɚɟɬɫɹ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɚɹ ɦɨɞɟɥɶ ɨɛɴɟɤɬɚ ɫɥɟɞɭɸɳɟɝɨ ɬɢɩɚ: ɩɨ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ (ɜɞɨɥɶ ɨɫɢ Ox3 ) ɤɨɧɫɬɪɭɤɰɢɢ ɡɚɞɚɱɚ ɨɫɬɚɟɬɫɹ ɤɨɧɬɢɧɭɚɥɶɧɨɣ, ɩɨ ɩɨɩɟɪɟɱɧɵɦ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ) ɧɚɩɪɚɜɥɟɧɢɹɦ (ɜɞɨɥɶ ɨɫɟɣ Ox1 ɢ
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Ox2 ) ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɟɬɨɱɧɚɹ ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɫɬɚɧɞɚɪɬɧɨɣ ɬɟɯɧɢɤɢ ɦɟɬɨɞɚ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ [3, 6].
Ɋɚɫɲɢɪɟɧɧɵɟ ɨɛɥɚɫɬɢ Zk , k = 1, 2, ..., nk 1 ɜɵɛɢɪɚɸɬɫɹ ɫɬɚɧɞɚɪɬɧɵɦɢ ɜ ɜɢɞɟ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɨɜ. Ⱥɩɩɪɨɤɫɢɦɚɰɢɹ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɚɧɞɚɪɬɧɨɣ ɨɛɥɚɫɬɢ ɫɨɫɬɨɢɬ ɜ
ɡɚɞɚɧɢɢ ɫɟɬɤɢ, ɬɨɩɨɥɨɝɢɱɟɫɤɢ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɨɧɚ
ɤɚɤ ɦɨɠɧɨ ɥɭɱɲɟ ɫɨɨɬɜɟɬɫɬɜɨɜɚɥɚ ɨɱɟɪɬɚɧɢɹɦ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɤɨɧɫɬɪɭɤɰɢɢ (ɪɢɫ.
1.1, 1.2). ɉɨɧɹɬɢɟ ɬɨɩɨɥɨɝɢɱɟɫɤɨɣ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɨɧɚ
ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ ɢɡ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɟɬɤɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɧɟɤɨɬɨɪɨɣ ɧɟɜɵɪɨɠɞɟɧɧɨɣ
ɞɟɮɨɪɦɚɰɢɢ ɹɱɟɟɤ ɩɨɫɥɟɞɧɟɣ ɛɟɡ ɢɯ «ɩɟɪɟɤɪɭɱɢɜɚɧɢɹ». ȼɵɛɨɪ ɬɚɤɨɝɨ ɤɥɚɫɫɚ ɫɟɬɨɤ, ɫ
ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ, ɚ ɫ ɞɪɭɝɨɣ – ɩɨɡɜɨɥɹɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɨɫɬɭɸ ɪɟɝɭɥɹɪɧɭɸ ɧɭɦɟɪɚɰɢɸ
ɭɡɥɨɜ (ɞɜɭɯɢɧɞɟɤɫɧɭɸ), ɱɬɨ ɩɪɢɜɨɞɢɬ ɜ ɞɚɥɶɧɟɣɲɟɦ ɤ ɭɞɨɛɧɵɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɮɨɪɦɭɥɚɦ, ɷɮɮɟɤɬɢɜɧɵɦ ɜɵɱɢɫɥɢɬɟɥɶɧɵɦ ɫɯɟɦɚɦ ɢ ɚɥɝɨɪɢɬɦɚɦ, ɚ ɬɚɤɠɟ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɟɬ ɫɛɨɪ ɢɫɯɨɞɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɢ ɜɵɜɨɞ ɪɟɡɭɥɶɬɚɬɨɜ. ɉɟɪɟɯɨɞ ɤ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɟɬɤɟ ɫ
ɟɞɢɧɢɱɧɵɦ ɲɚɝɨɦ ɥɟɝɤɨ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɥɨɤɚɥɶɧɨɣ ɡɚɦɟɧɨɣ ɤɨɨɪɞɢɧɚɬ ɜɧɭɬɪɢ ɫɟɬɨɱɧɨɣ
ɹɱɟɣɤɢ. ɉɪɢ «ɜɵɩɪɹɦɥɟɧɢɢ» ɫɟɬɤɢ ɨɛɳɢɣ ɜɢɞ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɭɪɚɜɧɟɧɢɣ ɫɨɯɪɚɧɹɟɬɫɹ
– ɦɟɧɹɸɬɫɹ ɥɢɲɶ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɨ
ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɷɮɮɟɤɬɢɜɧɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɩɪɹɦɨɭɝɨɥɶɧɵɯ ɫɟɬɨɤ, ɩɨɫɤɨɥɶɤɭ ɷɬɨ ɡɧɚɱɢɬɟɥɶɧɨ ɭɩɪɨɳɚɟɬ ɢ ɭɫɤɨɪɹɟɬ ɚɥɝɨɪɢɬɦɵ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɡɜɨɥɹɟɬ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɭɡɥɨɜ, ɱɬɨ ɩɪɢɜɨɞɢɬ ɜ ɢɬɨɝɟ ɤ ɭɜɟɥɢɱɟɧɢɸ ɬɨɱɧɨɫɬɢ.
Ɋɢɫ 1.1. ɉɪɢɦɟɪ ɜɵɛɨɪɚ ɫɟɬɤɢ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ «ɩɨɩɟɪɟɱɧɨɟ» ɩɨ ɨɬɧɨɲɟɧɢɸ
ɤ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ ɫɟɱɟɧɢɟ ɤɨɧɫɬɪɭɤɰɢɢ
Ɋɢɫ. 1.2. Ɉɛɳɢɣ ɜɢɞ ȾɄɄɗ ɫ ɱɟɬɵɪɟɯɭɝɨɥɶɧɵɦ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ
2. ɇɟɤɨɬɨɪɵɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɨɛɨɡɧɚɱɟɧɢɹ.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ: (i, j ) – ɞɜɭɯɢɧɞɟɤɫɧɵɣ ɧɨɦɟɪ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɞɢɫɤɪɟɬɧɨɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ (ȾɄɄɗ) ɫ ɱɟɬɵɪɟɯɭɝɨɥɶɧɵɦ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ; Zi, j – ɨɛɥɚɫɬɶ, ɡɚɧɢɦɚɟɦɚɹ ȾɄɄɗ; SZi , j – ɨɛɥɚɫɬɶ, ɡɚɧɢɦɚɟɦɚɹ ɩɨɩɟɪɟɱɧɵɦ (ɩɨ ɨɬ-
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ɧɨɲɟɧɢɸ ɤ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ) ɫɟɱɟɧɢɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ȾɄɄɗ (ɩɨɫɬɨɹɧɧɚ
ɜɞɨɥɶ x3 ); l3 – ɞɥɢɧɚ ȾɄɄɗ ɩɨ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ ( x3  [0, l3 ] ); x3b,k , k = 1, ..., nk
– ɤɨɨɪɞɢɧɚɬɵ ɫɟɱɟɧɢɣ, ɜ ɤɨɬɨɪɵɯ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ «ɫɤɚɱɤɨɨɛɪɚɡɧɨɟ» (ɪɚɡɪɵɜɵ ɩɟɪɜɨɝɨ ɪɨɞɚ) ɢɡɦɟɧɟɧɢɟ ɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɤɨɧɫɬɪɭɤɰɢɢ).
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ȾɄɄɗ Zi, j (ɪɢɫ. 1.2) ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ:
nk 1
Zi , j
Z
k ,i , j
, ɝɞɟ Zk ,i , j
{( x1 , x2 , x3 ) : ( x1 , x2 )  SZi , j ; x3  [ x3b,k , x3b,k 1 ]} .
(2.1)
k 1
3. Ʌɨɤɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ. ȼɨɫɩɨɥɧɟɧɢɟ ɧɟɢɡɜɟɫɬɧɵɯ ɧɚ ɷɥɟɦɟɧɬɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɢɡɜɨɥɶɧɵɣ (i, j ) -ɣ ȾɄɄɗ. ɉɭɫɬɶ ( x1( i , j ) , x2( i , j ) ) – ɤɨɨɪɞɢɧɚɬɵ (i, j ) ɝɨ ɭɡɥɚ ɫɟɬɤɢ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɯ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɯ
ɤɨɧɫɬɪɭɤɰɢɸ. Ɍɨɝɞɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ȾɄɄɗ ɢɦɟɟɬ ɭɡɥɵ (ɭɡɥɨɜɵɟ ɥɢɧɢɢ) ɫɨ ɫɥɟɞɭɸɳɢɦɢ ɧɨɦɟɪɚɦɢ: ( x1( i , j ) , x2( i , j ) ) , ( x1( i 1, j ) , x2( i1, j ) ) , ( x1( i , j 1) , x2( i , j 1) ) ɢ ( x1( i 1, j 1) , x2( i 1, j 1) ) .
ȼ ɩɨɩɟɪɟɱɧɨɦ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɫɧɨɜɧɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ) ɫɟɱɟɧɢɢ ȾɄɄɗ ɜɜɨɞɢɬɫɹ ɥɨɤɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ Ot1 ɢ Ot2 (ɪɢɫ. 3.1), ɩɪɢ ɷɬɨɦ t1  [0, 1]; t 2  [0, 1] .
Ɋɢɫ. 3.1. ɉɟɪɟɯɨɞ ɤ ɥɨɤɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɧɚ ȾɄɄɗ
ɂɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɜɟɬɫɬɜɢɟ ɝɥɨɛɚɥɶɧɵɯ ɢ ɥɨɤɚɥɶɧɵɯ ɤɨɨɪɞɢɧɚɬ ɭɡɥɨɜ ɷɥɟɦɟɧɬɚ:
( x1( i , j ) , x2( i , j ) )  (0, 0) ; ( x1( i 1, j ) , x2( i 1, j ) )  (1, 0) ;
(3.1)
( x1( i , j 1) , x2( i , j 1) )  (0, 1) ; ( x1( i 1, j 1) , x2( i 1, j 1) )  (1, 1) .
T
T
ɉɭɫɬɶ t [ t1 t 2 ] ɢ x [ x1 x2 ] – ɜɟɤɬɨɪɵ ɤɨɨɪɞɢɧɚɬ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ ɜ ɥɨɤɚɥɶɧɨɣ ɢ ɢɫɯɨɞɧɨɣ ɝɥɨɛɚɥɶɧɨɣ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ. Ɏɨɪɦɭɥɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɧɚ ɷɥɟɦɟɧɬɟ (ɪɢɫ. 3.1):
(3.2)
x (t1 , t 2 ) xn( i , j ) t1'1 x t 2 ' 2 x t1t 2 '12 x ,
ɝɞɟ '1 x ( i , j ) xn( i1, j ) xn( i , j ) ; ' 2 x ( i , j ) xn( i , j 1) xn( i , j ) ; '12 x ( i , j ) xn( i1, j 1) xn( i1, j ) ' 2 x . (3.3)
Ɂɞɟɫɶ xn( i , j ) , tn( i , j ) – ɜɟɤɬɨɪɵ ɤɨɨɪɞɢɧɚɬ (i, j ) -ɝɨ ɭɡɥɚ ɷɥɟɦɟɧɬɚ ɜ ɝɥɨɛɚɥɶɧɨɣ ɢ ɥɨɤɚɥɶɧɨɣ
ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɡɚɦɟɧɵ ɩɟɪɟɦɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɟ ɩɪɨɢɡɜɨɞɧɵɯ ɩɨ x1 ɢ x2 ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɫɥɨɠɧɨɣ ɮɭɧɤɰɢɢ:
2
wM
wM w t k
, s 1, 2 ,
(3.4)
¦
t k wx s
wx s
w
k 1
ɝɞɟ M – ɧɟɤɨɬɨɪɚɹ ɮɭɧɤɰɢɹ.
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ȼɟɥɢɱɢɧɵ w t k / wxs ɢɡ (3.4) ɨɛɪɚɡɭɸɬ ɹɤɨɛɢɚɧ ɫɢɫɬɟɦɵ. Ɇɚɬɪɢɰɚ əɤɨɛɢ:
Dt
Dx
w tk
w t1 / w x2 º ªD1(,i1, j ) (t1 , t 2 ) D1(,i2, j ) (t1 , t 2 )º
ª w t1 / w x1
» «D (i , j ) (t , t ) D (i , j ) (t , t )» / i , j (t1 , t 2 ) , ɝɞɟ D k ,s wx , (3.5)
« w t /w x
t
/
x
w
w
1
2
2 ¼
2, 2
1 2 ¼
¬ 2
¬ 2,1 1 2
s
ɗɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɨɩɪɟɞɟɥɢɬɟɥɹ E Dx / Dt ɧɚɯɨɞɹɬɫɹ ɤɚɤ
ª E 1(,1i , j ) (t1 , t 2 ) E 1(,i2, j ) (t1 , t 2 )º
« E ( i , j ) (t , t ) E ( i , j ) (t , t )» ,
2, 2
1
2 ¼
¬ 2 ,1 1 2
' q x (pi , j ) t3q '12 x (pi , j ) , p 1, 2; q 1, 2 ;
ȼ (t1 , t 2 )
E p( i,,qj ) (t1 , t 2 )
ɝɞɟ
'1 x
(i , j )
p
x
( i 1, j )
p
x
(i , j )
p
, p 1, 2 ; ' 2 x
'12 x
(i , j )
p
x
( i 1, j 1)
p
(i , j )
p
x
x
( i 1, j )
p
( i , j 1)
p
'2 x
x
(i , j )
p
(i, j )
p
(3.6)
(3.7)
, p 1, 2 ;
, p 1, 2 .
(3.8)
Ɏɨɪɦɭɥɚ ɜɵɱɢɫɥɟɧɢɹ ɨɩɪɟɞɟɥɢɬɟɥɹ ɦɚɬɪɢɰɵ ȼ(t1 , t 2 ) :
J i , j (t1 , t 2 ) det[/ i , j (t1 , t 2 )] E1(,1i , j ) (t1 , t 2 ) E 2(,i2, j ) (t1 , t 2 ) E1(,i2, j ) (t1 , t 2 ) E 2(,i1, j ) (t1 , t 2 ) .
Ɇɚɬɪɢɰɚ əɤɨɛɢ / (t1 , t 2 ) ɢɦɟɟɬ ɜɢɞ:
ªD ( i , j ) (t , t )
/ i , j (t1 , t 2 ) ȼi, 1j (t1 , t 2 ) « 1(,i1, j ) 1 2
¬D 2 ,1 (t1 , t 2 )
ɝɞɟ D1(,i1, j )
E 2(,i2, j ) (t1 , t 2 )
D1(,i1, j ) ( x1 , x2 )
J i , j (t1 , t 2 )
; D1(,i2, j )
D 2( i,1, j ) D 2( i,1, j ) ( x1 , x2 ) D 1(,i2, j ) (t1 , t 2 )º
,
D 2( i, 2, j ) (t1 , t 2 )»¼
D1(,i2, j ) ( x1 , x2 ) E 2(,i1, j ) (t1 , t 2 )
J i , j (t1 , t 2 )
; D 2( i, 2, j )
(3.9)
(3.10)
E1(,i2, j ) (t1 , t 2 )
J i , j (t1 , t 2 )
D 2(i, 2, j ) ( x1 , x2 )
;
E1(,i1, j ) (t1 , t 2 )
J i , j (t1 , t 2 )
. (3.11)
ȼ ɤɚɱɟɫɬɜɟ ɨɫɧɨɜɧɵɯ ɧɟɢɡɜɟɫɬɧɵɯ ɜ ɭɡɥɚɯ ɩɪɢɧɢɦɚɸɬɫɹ ɫɨɫɬɚɜɥɹɸɳɢɟ ɩɟɪɟɦɟɳɟɧɢɣ u1( k ) , u2( k ) , u3( k ) ɢ ɢɯ ɩɪɨɢɡɜɨɞɧɵɟ v1( k ) , v2( k ) , v3( k ) ɩɨ ɩɟɪɟɦɟɧɧɨɣ x3 , ɬ.ɟ. ɞɥɹ ( p, q) -ɝɨ
ɭɡɥɚ ɷɬɨ u1( k , p ,q ) , u2( k , p ,q ) , u3( k , p ,q ) , v1( k , p ,q ) , v2( k , p ,q ) , v3( k , p ,q ) ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɟɤɬɨɪɵ ɧɟɢɡɜɟɫɬɧɵɯ
ɝɞɟ
u ( k , p,q )
u ( k , p , q ) ( x3 )
ªu1( k , p , q ) º
«u ( k , p , q ) »; v ( k , p , q )
« 2( k , p , q ) »
«¬u 3
»¼
v ( k , p ,q ) ( x3 )
ªv1( k , p , q ) º
«v ( k , p , q ) » ,
« 2( k , p , q ) »
«¬v3
»¼
(3.12)
Ɏɨɪɦɭɥɚ ɜɨɫɩɨɥɧɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɣ ɧɚ ɷɥɟɦɟɧɬɟ:
(3.13)
u ( k ) (t1 , t 2 ) u ( k ,i , j ) t1'1u ( k ,i , j ) t 2 ' 2u ( k ,i , j ) t1t 2 '12u ( k ,i , j ) ,
( k ,i , j )
( k ,i 1, j )
( k ,i , j )
( k ,i , j )
( k ,i , j 1)
( k ,i , j )
u
u
u
u
; ' 2u
;
ɝɞɟ '1u
'12u ( k ,i , j ) u ( k ,i 1, j 1) u ( k ,i 1, j ) ' 2u ( k ,i , j ) .
(3.14)
Ɏɨɪɦɭɥɚ ɜɨɫɩɨɥɧɟɧɢɹ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɩɟɪɟɦɟɳɟɧɢɣ ɩɨ x3 ɧɚ ɷɥɟɦɟɧɬɟ:
(3.15)
v ( k ) (t1 , t 2 ) v ( k ,i , j ) t1'1v ( k ,i , j ) t 2 ' 2 v ( k ,i , j ) t1t 2 '12 v ( k ,i , j ) ,
( k ,i , j )
( k ,i 1, j )
v
v ( k ,i , j ) ; ' 2 v ( k ,i , j ) v ( k ,i , j 1) v ( k ,i , j ) ;
ɝɞɟ '1v
'12 v ( k ,i , j ) v ( k ,i 1, j 1) v ( k ,i1, j ) ' 2v ( k ,i , j ) .
(3.16)
4. ȼɵɱɢɫɥɟɧɢɟ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɩɟɪɟɦɟɳɟɧɢɣ, ɞɟɮɨɪɦɚɰɢɣ ɢ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɷɥɟɦɟɧɬɟ.
ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɩɟɪɟɦɟɳɟɧɢɣ ɩɨ t1 ɢ t 2 ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:
ª w (k ) º
u »
(t1 , t2 ) ' p u ( k ,i , j ) t3 p '12u ( k ,i , j ) , p 1, 2 ;
(4.1)
«
w
t
¬« p
¼» ( x , x )Z
1
24
2
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Ɏɨɪɦɭɥɵ ɨɩɪɟɞɟɥɟɧɢɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɩɟɪɟɦɟɳɟɧɢɣ ɩɨ x1 , x2 ɢ x3 :
ª w (k ) º
u »
«
»¼ ( x , x )Z
¬« wx p
1
2
2
ª w
¦ «« wt
(t1 , t 2 )
q 1
k .i . j
¬
q
º
u (k ) »
¼» ( x , x )Z
1
2
(t1 , t 2 )D q( i, ,pj ) (t1 , t 2 ), p 1, 2 ;
k .i . j
ª w (k ) º
u »
(t1 , t2 ) v ( k ) ( x , x )Z .
«
w
x
¬ 3
¼ ( x , x )Z
Ɏɨɪɦɭɥɚ ɞɥɹ ɤɨɦɩɨɧɟɧɬ ɬɟɧɡɨɪɚ ɞɟɮɨɪɦɚɰɢɣ ɧɚ ɷɥɟɦɟɧɬɟ:
§
·
ª w (k ) º
1 ¨ ª w (k ) º
¸
u
(
t
,
t
)
u
(
t
,
t
)
,
H p( k,q,i , j ) (t1 , t 2 )
«
»
«
»
1 2
1 2 ¸
q
p
w
2 ¨¨ ¬« wxq
x
¸
» ( x , x )Z
« p
» ( x , x )Z
¼
¬
¼
¹
p 1, 2, 3; q 1, 2, 3.
Ɏɨɪɦɭɥɚ ɞɥɹ ɨɛɴɟɦɧɨɣ ɞɟɮɨɪɦɚɰɢɢ:
1
1
2
(4.2)
(4.3)
k .i . j
2
k .i . j
1
2
k .i . j
1
2
(4.4)
k .i . j
3
H ( k ,i , j ) (t1 , t 2 )
¦H
( k ,i , j )
p, p
(t1 , t 2 ) .
(4.5)
Ɏɨɪɦɭɥɚ ɞɥɹ ɤɨɦɩɨɧɟɧɬ ɬɟɧɡɨɪɚ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɷɥɟɦɟɧɬɟ:
V p( k,q,i , j ) (t1 , t2 ) G p ,q Oi , jH ( k ,i , j ) (t1 , t2 ) 2Pi , j H p( k,q,i , j ) (t1 , t 2 ), p 1, 2, 3; q 1, 2, 3 ,
(4.6)
p 1
ɝɞɟ G p,q – ɫɢɦɜɨɥ Ʉɪɨɧɟɤɟɪɚ [4, 5]; O k ,i , j
T k ,i , j O ; P k ,i , j T k ,i , j P ; T k ,i , j – ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ.
5. ȼɵɱɢɫɥɟɧɢɟ ɤɜɚɞɪɚɬɢɱɧɨɣ ɱɚɫɬɢ ɮɭɧɤɰɢɨɧɚɥɚ ɷɧɟɪɝɢɢ ɧɚ ɷɥɟɦɟɧɬɟ.
Ʉɜɚɞɪɚɬɢɱɧɚɹ ɱɚɫɬɶ ɮɭɧɤɰɢɨɧɚɥɚ ɷɧɟɪɝɢɢ ɧɚ (i, j ) -ɦ ɷɥɟɦɟɧɬɟ ɧɚ ɮɪɚɝɦɟɧɬɟ ɤɨɧɫɬɪɭɤɰɢɢ Zk ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
1 1 M M 3 3 ( k ,i , j ) ( r ) ( s )
~
Ɏ ( k ,i , j )
(5.1)
¦¦¦¦ f ([1 , [ 2 ) ,
2 M 1M 2 r 1 s 1 p 1 q 1
1
3
f ( k ,i , j ) (t1 , t 2 )
ɝɞɟ
2
3
¦¦V
(k )
p ,q
(t1 , t 2 )H p( k,q) (t1 , t 2 ) ;
(5.2)
p 1 q 1
M 1 ɢ M 2 – ɤɨɥɢɱɟɫɬɜɨ ɬɨɱɟɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜɞɨɥɶ ɨɫɢ Ot1 ɢ ɨɫɢ Ot2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
(ɤɚɤ ɩɪɚɜɢɥɨ, M 1 M 2 ); [1( r ) ɢ [ 2( s ) – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ,
1 §
1·
1 §
1·
(s)
(5.3)
[1( r )
¨ r ¸, r 1, 2, ..., M 1 ; [ 2
¨ s ¸, s 1, 2, ..., M 2 ;
2¹
2¹
ɤɨɦɩɨɧɟɧɬɵ V p( k,q,i , j ) (t1 , t 2 ) ɢ H p( k,q,i , j ) (t1 , t2 ) ɬɟɧɡɨɪɨɜ ɧɚɩɪɹɠɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (4.6) ɢ (4.4) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
6. ɋɜɹɡɶ ɥɨɤɚɥɶɧɨɣ ɢ ɝɥɨɛɚɥɶɧɨɣ ɢɧɞɟɤɫɚɰɢɢ.
ȼɟɤɬɨɪ ɧɟɢɡɜɟɫɬɧɵɯ ɧɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ (i, j ) -ɦ ȾɄɄɗ ɢɦɟɟɬ ɜɢɞ:
ªu~ ( k ,i , j ) º
~
~
U ( k ,i , j ) U ( k ,i , j ) ( x3 ) « ~ ( k ,i , j ) » ,
¬v
¼
ɝɞɟ
u~ ( k ,i , j )
u~ ( k ,i , j ) ( x3 )
ªu1( k ,i , j ) º
«u ( k ,i , j ) » ;
« 2( k ,i , j ) »
«¬u3 »¼
v~ ( k ,i , j )
v~ ( k ,i , j ) ( x3 )
ªv1( k ,i , j ) º
«v ( k ,i , j ) » ;
« 2( k ,i , j ) »
«¬v3 »¼
(6.1)
(6.2)
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u p( k ,i , j )
ª u (pk ,i , j ) º
« u ( k ,i1, j ) »
p
u p( k ,i , j ) ( x3 ) « ( k ,i , j 1) »; v p( k ,i , j )
« up
»
«u ( k ,i1, j 1) »
¬ p
¼
ª v (pk ,i , j ) º
« v ( k ,i1, j ) »
p
v p( k ,i , j ) ( x3 ) « ( k ,i , j 1) » ,
« vp
»
«v ( k ,i1, j 1) »
¬ p
¼
p 1, 2, 3 .
(6.3)
ȼɜɟɞɟɦ ɫɩɥɨɲɧɭɸ ɧɭɦɟɪɚɰɢɸ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɥɨɤɚɥɶɧɨɟ ɩɟɪɟɨɛɨɡɧɚɱɟɧɢɟ
ɷɥɟɦɟɧɬɨɜ ɜɟɤɬɨɪɚ ɧɟɢɡɜɟɫɬɧɵɯ
~
(6.4)
U ( k ,i , j ) ( x3 )  Y ( x3 ) , ɝɞɟ Y ( x3 ) [ y1 y2 ... y24 ]T .
ɉɭɫɬɶ ig – ɝɥɨɛɚɥɶɧɵɣ ɢɧɞɟɤɫ ɷɥɟɦɟɧɬɚ ɜɟɤɬɨɪɚ (6.4). ɉɨɫɬɚɜɢɦ ɟɦɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɥɨɤɚɥɶɧɵɟ ɢɧɞɟɤɫɵ.
ɇɨɦɟɪ jn ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɭɡɥɚ (ɭɫɥɨɜɧɨ ɜɞɨɥɶ ɨɫɢ Ox2 ):
ª ig 1 º
jn j «
»,
¬ 12 ¼
(6.5)
ɝɞɟ ɡɞɟɫɶ ɢ ɞɚɥɟɟ ɡɚɩɢɫɶ ɬɢɩɚ [n] ɨɛɨɡɧɚɱɚɟɬ ɰɟɥɭɸ ɱɚɫɬɶ ɱɢɫɥɚ n .
ɇɨɦɟɪ in ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɭɡɥɚ (ɭɫɥɨɜɧɨ ɜɞɨɥɶ ɨɫɢ Ox1 ):
in
ª i 12( jn j ) 1º
i« g
».
6
¬
¼
(6.6)
ɂɧɞɢɤɚɬɨɪ ind ɬɢɩɚ ɧɟɢɡɜɟɫɬɧɨɣ ( ind 1 – ɧɟɢɡɜɟɫɬɧɚɹ ɤɨɦɩɨɧɟɧɬɚ ɩɟɪɟɦɟɳɟɧɢɹ; ind 2 – ɧɟɢɡɜɟɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɨɦɩɨɧɟɧɬɵ ɩɟɪɟɦɟɳɟɧɢɹ ɩɨ x3 ):
ind
ª i 12( jn j ) 6(in i ) 1 º
1 « g
».
3
¼
¬
(6.7)
ɇɨɦɟɪ p ɧɟɢɡɜɟɫɬɧɨɣ ɤɨɦɩɨɧɟɧɬɵ
p ig 12( jn j ) 6(in i ) 3(ind 1) .
(6.8)
Ɉɱɟɜɢɞɧɵɦ ɫɥɟɞɫɬɜɢɟɦ ɮɨɪɦɭɥɵ (6.8) ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ:
ig 12( jn j ) 6(in i ) 3(ind 1) p .
(6.9)
7. Ⱥɥɝɨɪɢɬɦ ɮɨɪɦɢɪɨɜɚɧɢɹ ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɨɝɨ
ɤɨɧɟɱɧɨɝɨ ɷɥɟɦɟɧɬɚ.
Ɏɨɪɦɢɪɨɜɚɧɢɟ ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ ȾɄɄɗ ɩɪɨɢɡɜɨɞɢɬɫɹ ɦɟɬɨɞɨɦ ɛɚɡɢɫɧɵɯ ɜɚɪɢɚɰɢɣ [4, 5]. Ɏɨɪɦɭɥɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ ɢɦɟɟɬ ɜɢɞ:
~
~
~
~
( K k( i , j ) ) ig , jg Ɏ ( k ,i , j ) (eig e jg ) Ɏ ( k ,i , j ) (eig ) Ɏ ( k ,i , j ) (e jg ) Ɏ ( k ,i , j ) (e0 ),
(7.1)
ig 1, 2, ..., 24; j g 1, 2, ..., 24,
ɝɞɟ e p , e0 – 24-ɯ ɦɟɪɧɵɟ ɜɟɤɬɨɪɵ, ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
(e p ) q
G p ,q ; (e0 ) q
0, q 1, 2, ..., 24 ;
(7.2)
G p,q – ɫɢɦɜɨɥ Ʉɪɨɧɟɤɟɪɚ [4, 5].
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɨɩɢɫɚɧɧɵɣ ɧɢɠɟ ɚɥɝɨɪɢɬɦ ɮɨɪɦɢɪɨɜɚɧɢɹ
ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ ȾɄɄɗ.
1. ɇɚ ɨɫɧɨɜɚɧɢɢ ɜɵɲɟɩɪɢɜɟɞɟɧɧɵɯ ɮɨɪɦɭɥ (3.9), (3.11), (4.1)-(4.6), (6.5)-(6.8), (7.2)
~
ɜɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ Ɏ ( k ,i , j ) (e0 ) .
2. ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɟɪɟɛɢɪɚɸɬɫɹ ig 1, 2, ..., 24 . Ⱦɥɹ ɤɚɠɞɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ig ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɞɟɣɫɬɜɢɹ:
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~
2.1. ȼɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ Ɏ ( k ,i , j ) (ei ) ɩɨ ɮɨɪɦɭɥɚɦ (3.9), (3.11), (4.1)-(4.6), (6.5)g
(6.8), (7.2);
2.2. ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɟɪɟɛɢɪɚɸɬɫɹ j g
1, 2, ..., 24 . Ⱦɥɹ ɤɚɠɞɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ
ɡɧɚɱɟɧɢɹ jg ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɞɟɣɫɬɜɢɹ:
~
2.2.1. ȼɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ Ɏ ( k ,i , j ) (ei e j ) ɩɨ ɮɨɪɦɭɥɚɦ (3.9), (3.11), (4.1)g
g
(4.6), (6.5)-(6.8), (7.2);
~
2.2.2. ȼɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ Ɏ ( k ,i , j ) (e j ) ɩɨ ɮɨɪɦɭɥɚɦ (3.9), (3.11), (4.1)-(4.6),
g
(6.5)-(6.8), (7.2);
2.2.3. ȼɵɱɢɫɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟ ɷɥɟɦɟɧɬɚ ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ ȾɄɄɗ ( K k( i , j ) ) i
g , jg
ɩɨ
ɮɨɪɦɭɥɟ (7.1).
ɋɬɪɭɤɬɭɪɚ ɩɨɥɭɱɚɟɦɨɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ K k( i , j ) ȾɄɄɗ ɫɥɟɞɭɸɳɚɹ:
ª K k1,,1uu
« K 1,1
« k2,,vu1
« K k ,uu
« K k2,,vu1
« 3,1
« K k3,,uu1
« K k ,vu
« K 4 ,1
« k4,,uu1
«¬ K k ,vu
K k1,,1uv
K k1,,1vv
K k2,,uv1
K k2,,vv1
K k3,,uv1
K k3,,vv1
K k4,,uv1
K k4,,vv1
K k1,,uu2
K k1,,vu2
K k2,,uu2
K k2,,vu2
K k3,,uu2
K k3,,vu2
K k4,,uu2
K k4,,vu2
K k1,,uv2
K k1,,vv2
K k2,,uv2
K k2,,vv2
K k3,,uv2
K k3,,vu2
K k4,,uv2
K k4,,vu2
K k1,,uu3
K k1,,3vu
K k2,,uu3
K k2,,vu3
K k3,,uu3
K k3,,vu3
K k4,,uu3
K k4,,vu3
K k1,,uv3
K k1,,vv3
K k2,,uv3
K k2,,vv3
K k3,,uv3
K k3,,vv3
K k4,,uv3
K k4,,vv3
K k1,,uu4
K k1,,vu4
K k2,,uu4
K k2,,vu4
K k3,,uu4
K k3,,vu4
K k4,,uu4
K k4,,vu4
K k1,,uv4 º
K k1,,vv4 »
»
K k2,,uv4 »
K k2,,vv4 »
K k( i , j )
».
K k3,,uv4 »
K k3,,vv4 »
K k4,,uv4 »
»
K k4,,vv4 »¼
Ɂɞɟɫɶ K kl,,muu , K kl ,,muv , K kl ,,mvu , K kl ,,mvv , l , m 1, 2, 3, 4 – ɦɚɬɪɢɰɵ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ.
(7.3)
ɇɚ ɨɫɧɨɜɟ ɦɚɬɪɢɰɵ ɠɟɫɬɤɨɫɬɢ ȾɄɄɗ (7.3) ɮɨɪɦɢɪɭɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɨɷɥɟɦɟɧɬɧɵɟ ɦɚɬɪɢɰɵ K k( i,uu, j ) , K k( i,uv, j ) , K k( i,vu, j ) ɢ K k( i,vv, j ) 12-ɝɨ ɩɨɪɹɞɤɚ:
K k( i,uu, j )
ª K k1,,1uu
« K 2 ,1
« k3,,uu1
« K k ,uu
« K k4,,uu1
¬
K k1,,uu2
K k2,,uu2
K k3,,uu2
K k4,,uu2
K k1,,uu3
K k2,,uu3
K k3,,uu3
K k4,,uu3
ª K k1,,1vu
« K 2 ,1
« k3,,vu1
« K k ,vu
« K k4,,vu1
¬
K k1,,vu2
K k2,,vu2
K k3,,vu2
K k4,,vu2
K k1,,vu3
K k2,,vu3
K k3,,vu3
K k4,,vu3
K k1,,uu4 º
K k2,,uu4 »
» ; K k( i,uv, j )
K k3,,uu4 »
K k4,,uu4 »¼
ª K k1,,1uv
« K 2 ,1
« k3,,uv1
« K k ,uv
« K k4,,uv1
¬
K k1,,uv2
K k2,,uv2
K k3,,uv2
K k4,,uv2
K k1,,uv3
K k2,,uv3
K k3,,uv3
K k4,,uv3
K k1,,uv4 º
K k2,,uv4 »
» ; (7.4)
K k3,,uv4 »
K k4,,uv4 »¼
ª K k1,,1vv K k1,,vv2
K k1,,vu4 º
K k1,,vv3
K k1,,vv4 º
2, 4 »
2
,
1
2
,
2
2
,
3
«K
K k ,vu
K k ,vv K k ,vv K k2,,vv4 »
(i , j )
k ,vv
»
» . (7.5)
«
K k( i,vu, j )
K
;
3 ,1
3, 2
3, 3
3, 4
k ,vv
K k3,,vu4 »
« K k ,vv K k ,vv K k ,vv K k ,vv »
« K k4,,vv1 K k4,,vv2 K k4,,vv3 K k4,,vv4 »
K k4,,vu4 »¼
¼
¬
Ɂɚɦɟɱɚɧɢɹ. ɂɫɫɥɟɞɨɜɚɧɢɹ ɩɪɨɜɨɞɢɥɢɫɶ ɜ ɪɚɦɤɚɯ ɫɥɟɞɭɸɳɢɯ ɪɚɛɨɬ:
1. Ƚɪɚɧɬ ɇɒ-8684.2010.8 ɉɪɟɡɢɞɟɧɬɚ Ɋɨɫɫɢɣɫɤɨɣ Ɏɟɞɟɪɚɰɢɢ ɞɥɹ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɣ
ɩɨɞɞɟɪɠɤɢ ɜɟɞɭɳɢɯ ɧɚɭɱɧɵɯ ɲɤɨɥ Ɋɨɫɫɢɣɫɤɨɣ Ɏɟɞɟɪɚɰɢɢ «Ɇɧɨɝɨɭɪɨɜɧɟɜɵɟ ɱɢɫɥɟɧɧɵɟ, ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɪɨɱɧɨɫɬɢ ɡɞɚɧɢɣ ɢ
ɫɨɨɪɭɠɟɧɢɣ ɫ ɭɱɟɬɨɦ ɤɨɧɫɬɪɭɤɬɢɜɧɵɯ ɢ ɮɢɡɢɱɟɫɤɢɯ ɨɫɨɛɟɧɧɨɫɬɟɣ» ɧɚ 2010-2011 ɝɝ.
2. Ƚɪɚɧɬ 2.3.9 Ɋɨɫɫɢɣɫɤɨɣ ɚɤɚɞɟɦɢɢ ɚɪɯɢɬɟɤɬɭɪɵ ɢ ɫɬɪɨɢɬɟɥɶɧɵɯ ɧɚɭɤ «Ɋɚɡɪɚɛɨɬɤɚ ɢ ɢɫɫɥɟɞɨɜɚɧɢɟ ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɯ ɦɟɬɨɞɨɜ ɞɥɹ ɪɚɫɱɟɬɚ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ ɫ ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɵɦɢ ɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɩɨ ɨɞɧɨɦɭ ɢɡ
ɧɚɩɪɚɜɥɟɧɢɣ» ɧɚ 2011-2013 ɝɝ.
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3. ɇɂɊ «Ɋɚɡɪɚɛɨɬɤɚ ɬɟɨɪɢɢ ɢ ɚɥɝɨɪɢɬɦɨɜ ɩɨɫɬɪɨɟɧɢɹ ɤɨɪɪɟɤɬɧɵɯ ɚɧɚɥɢɬɢɱɟɫɤɢɯ
ɪɟɲɚɬɟɥɟɣ ɦɧɨɝɨɬɨɱɟɱɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɚɫɱɟɬɚɦ ɫɬɪɨɢɬɟɥɶɧɵɯ
ɤɨɧɫɬɪɭɤɰɢɣ», ɜɵɩɨɥɧɹɟɦɨɣ ɩɨ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɜɟɞɨɦɫɬɜɟɧɧɨɣ ɰɟɥɟɜɨɣ ɩɪɨɝɪɚɦɦɟ
«Ɋɚɡɜɢɬɢɟ ɧɚɭɱɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɜɵɫɲɟɣ ɲɤɨɥɵ (2009-2011 ɝɨɞɵ)» (ɩɪɨɟɤɬ 2.1.2/12148).
Ʌɢɬɟɪɚɬɭɪɚ
1. Ⱥɤɢɦɨɜ ɉ.Ⱥ. Ʉɨɪɪɟɤɬɧɵɣ ɦɟɬɨɞ ɬɨɱɧɨɝɨ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ ɦɧɨɝɨɬɨɱɟɱɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɪɚɫɱɟɬɚ ɤɨɧɫɬɪɭɤɰɢɣ ɞɥɹ ɫɢɫɬɟɦ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ ɫ ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. // ȼɟɫɬɧɢɤ ɆȽɋɍ, ʋ1, 2011, ɫ. 11-16.
2. Ⱥɤɢɦɨɜ ɉ.Ⱥ. Ʉɨɪɪɟɤɬɧɵɣ ɦɟɬɨɞ ɬɨɱɧɨɝɨ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ ɦɧɨɝɨɬɨɱɟɱɧɵɯ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɪɚɫɱɟɬɚ ɤɨɧɫɬɪɭɤɰɢɣ ɞɥɹ ɫɢɫɬɟɦ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɬɨɪɨɝɨ
ɩɨɪɹɞɤɚ ɫ ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. // ȼɟɫɬɧɢɤ ɆȽɋɍ, ʋ4, ɬ. 1, 2010, ɫ. 24-28.
3. Ɂɟɧɤɟɜɢɱ Ɉ. Ɇɟɬɨɞ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɜ ɬɟɯɧɢɤɟ.– Ɇ.: Ɇɢɪ, 1975. – 511 ɫ.
4. Ɂɨɥɨɬɨɜ Ⱥ.Ȼ., Ⱥɤɢɦɨɜ ɉ.Ⱥ., ɋɢɞɨɪɨɜ ȼ.ɇ., Ɇɨɡɝɚɥɟɜɚ Ɇ.Ʌ. Ⱦɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɟ
ɦɟɬɨɞɵ ɪɚɫɱɟɬɚ ɫɨɨɪɭɠɟɧɢɣ. – Ɇ.: ɂɡɞɚɬɟɥɶɫɬɜɨ «Ⱥɪɯɢɬɟɤɬɭɪɚ – ɋ», 2010. – 336 ɫ.
5. Ɂɨɥɨɬɨɜ Ⱥ.Ȼ., Ⱥɤɢɦɨɜ ɉ.Ⱥ., ɋɢɞɨɪɨɜ ȼ.ɇ., Ɇɨɡɝɚɥɟɜɚ Ɇ.Ʌ. Ⱦɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɣ
ɦɟɬɨɞ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ. ɉɪɢɥɨɠɟɧɢɹ ɜ ɫɬɪɨɢɬɟɥɶɫɬɜɟ. – Ɇ.: ɂɡɞɚɬɟɥɶɫɬɜɨ Ⱥɋȼ, 2010. – 336 ɫ.
6. ɋɟɤɭɥɨɜɢɱ Ɇ. Ɇɟɬɨɞ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ. – Ɇ.: ɋɬɪɨɣɢɡɞɚɬ, 1993. – 664 ɫ.
References
1. Akimov P.A. Correct Analytical Solution of Multipoint Boundary Problems of Structural
Analysis for Set of First-order Differential Equations with Piecewise-constant Coefficients. // Bulletin
ɆSUSE, #1, 2011, pp. 11-16 (in Russian).
2. Akimov P.A. Correct Analytical Solution of Multipoint Boundary Problems of Structural
Analysis for Set of Second-order Differential Equations with Piecewise-constant Coefficients. // Bulletin
ɆSUSE, #4, vol. 1, 2010, pp. 24-28 (in Russian).
3. Zienkiewicz O.C. The Finite Element Method in Engineering Science, London, Mc.Graw-Hill,
1971, 521 pages.
4. Zolotov A.B., Akimov P.A., Sidorov V.N., Mozgaleva M.L. Discrete-continual methods of
structural analysis. Moscow, “Arkhitectura – S”, 2010, 336 pages (in Russian).
5. Zolotov A.B., Akimov P.A., Sidorov V.N., Mozgaleva M.L. Discrete-continual finite element
method. Applications in Construction. Moscow, “ASV”, 2010, 336 pages (in Russian).
6. Sekulowicz M. The Finite Element Method. Moscow, “Stroyizdat”, 1993, 664 pages.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɦɧɨɝɨɬɨɱɟɱɧɚɹ ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ, ɫɢɫɬɟɦɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ, ɤɭɫɨɱɧɨ-ɩɨɫɬɨɹɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɬɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ, ɞɢɫɤɪɟɬɧɨ-ɤɨɧɬɢɧɭɚɥɶɧɵɟ ɦɟɬɨɞɵ, ɪɚɫɱɟɬɵ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ.
Key words: multipoint boundary problem, set of first-order differential equations, piecewiseconstant coefficients, correct analytical solution, discrete-continual methods, structural analysis.
Ⱥɜɬɨɪɵ:
1. Ⱥɤɢɦɨɜ ɉɚɜɟɥ Ⱥɥɟɤɫɟɟɜɢɱ, ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɱɥɟɧ-ɤɨɪɪɟɫɩɨɧɞɟɧɬ
ɊȺȺɋɇ (ȽɈɍ ȼɉɈ ɆȽɋɍ); 129337, Ɋɨɫɫɢɹ, ɝ. Ɇɨɫɤɜɚ, əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ, ɞɨɦ 26; ɬɟɥ./ɮɚɤɫ:
+7(499) 183-59-94; e-mail: pavel.akimov@gmail.com.
2. Ɇɨɡɝɚɥɟɜɚ Ɇɚɪɢɧɚ Ʌɟɨɧɢɞɨɜɧɚ, ɤɚɧɞɢɞɚɬ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ, ɩɪɨɮɟɫɫɨɪ (ȽɈɍ
ȼɉɈ ɆȽɋɍ); 129337, Ɋɨɫɫɢɹ, ɝ. Ɇɨɫɤɜɚ, əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ, ɞɨɦ 26; ɬɟɥ./ɮɚɤɫ: +7(499) 183-5994; e-mail: marina.mozgaleva@gmail.com.
3. ɋɢɞɨɪɨɜ ȼɥɚɞɢɦɢɪ ɇɢɤɨɥɚɟɜɢɱ, ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɫɨɜɟɬɧɢɤ ɊȺȺɋɇ,
ɡɚɜɟɞɭɸɳɢɣ ɤɚɮɟɞɪɨɣ (ȽɈɍ ȼɉɈ ɆȽɋɍ); 129337, Ɋɨɫɫɢɹ, ɝ. Ɇɨɫɤɜɚ, əɪɨɫɥɚɜɫɤɨɟ ɲɨɫɫɟ, ɞɨɦ
26; ɬɟɥ./ɮɚɤɫ: +7(499) 183-59-94; e-mail: sidorov.vladimir@gmail.com.
Ɋɟɰɟɧɡɟɧɬ: Ȼɟɥɨɫɬɨɰɤɢɣ Ⱥ.Ɇ., ɩɪɨɮɟɫɫɨɪ, ɞ.ɬ.ɧ., ɝɟɧɟɪɚɥɶɧɵɣ ɞɢɪɟɤɬɨɪ ɁȺɈ «ɇɚɭɱɧɨɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ ɰɟɧɬɪ ɋɬɚȾɢɈ»
28
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