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Абстрактные изометрические отображения геометрических фигур определяющие путь к реальному допустимому преобразованию в области упругих деформаций некоторых строительных элементов.

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ВЕСТНИК
МГСУ
4/2010
ȺȻɋɌɊȺɄɌɇɕȿ ɂɁɈɆȿɌɊɂɑȿɋɄɂȿ ɈɌɈȻɊȺɀȿɇɂə
ȽȿɈɆȿɌɊɂɑȿɋɄɂɏ ɎɂȽɍɊ, ɈɉɊȿȾȿɅəɘɓɂȿ ɉɍɌɖ Ʉ
ɊȿȺɅɖɇɈɆɍ ȾɈɉɍɋɌɂɆɈɆɍ ɉɊȿɈȻɊȺɁɈȼȺɇɂɘ ȼ
ɈȻɅȺɋɌɂ ɍɉɊɍȽɂɏ ȾȿɎɈɊɆȺɐɂɃ ɇȿɄɈɌɈɊɕɏ
ɋɌɊɈɂɌȿɅɖɇɕɏ ɗɅȿɆȿɇɌɈȼ
ABSTRACT ISOMETRIC MAPS OF GEOMETRIC FIGURES
DETERMINING THE WAY TO A REAL ADMISSIBLE
TRANSFORMATION IN THE FIELD OF ELASTIC
DEFORMATIONS OF SOME STRUCTURAL ELEMENTS
ȼ.ɋ. Ʌɟɧɟɜ
V.S. Lenev
ɆȽɋɍ
ȼ ɫɬɚɬɶɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɚɛɫɬɪɚɤɬɧɵɟ ɬɨɩɨɥɨɝɢɱɟɫɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɤɨɬɨɪɵɟ ɜ ɢɡɦɟɧɺɧɧɨɦ ɜɢɞɟ ɩɪɢɦɟɧɹɸɬɫɹ ɤ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ ɪɟɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɚ ɜ ɨɛɥɚɫɬɢ ɭɩɪɭɝɢɯ ɞɟɮɨɪɦɚɰɢɣ, ɝɞɟ ɭɱɢɬɵɜɚɟɬɫɹ ɬɨɥɳɢɧɚ ɫɬɟɧɨɤ ɢ ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ
ɦɚɬɟɪɢɚɥɚ.
The author bases himself upon the abstract topological transformations and describes
the transformations of a real cylinder in the field of elastic deformations, where the wall
thickness value and the elasticity modulus of the material are considered.
1. ɉɨɹɫɧɢɦ ɨɛɳɭɸ ɫɢɬɭɚɰɢɸ. ȼ ɫɬɚɬɶɟ ɞɟɥɚɟɬɫɹ ɩɨɩɵɬɤɚ ɩɟɪɟɧɟɫɬɢ ɬɟɯɧɨɥɨɝɢɸ
ɪɚɛɨɬɵ ɫ ɧɟɩɪɟɪɵɜɧɵɦɢ ɢɡɨɦɟɬɪɢɱɟɫɤɢɦɢ ɨɬɨɛɪɚɠɟɧɢɹɦɢ ɨɝɪɚɧɢɱɟɧɧɵɯ ɡɚɦɤɧɭɬɵɯ
ɤɭɫɤɨɜ ɪɚɡɜɺɪɬɵɜɚɸɳɢɯɫɹ ɩɨɜɟɪɯɧɨɫɬɟɣ (ɫɦ. [2]), ɤɚɤ ɧɚɩɪɢɦɟɪ, ɚɛɫɬɪɚɤɬɧɵɯ ɰɢɥɢɧɞɪɨɜ ɫ ɪɚɡɦɟɪɚɦɢ H, R, ɧɚ ɪɟɚɥɶɧɵɟ – ɰɢɥɢɧɞɪɵ ɫ ɨɩɪɟɞɟɥɺɧɧɨɣ ɦɚɪɤɢ ɫɬɚɥɢ ɫ ɬɨɥɳɢɧɨɣ ɫɬɟɧɨɤ G . ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɜ ɨɛɥɚɫɬɢ ɭɩɪɭɝɢɯ ɞɟɮɨɪɦɚɰɢɣ.
ɉɪɨɛɥɟɦɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɭɫɬɚɧɨɜɢɬɶ ɤɪɢɬɢɱɟɫɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɪɚɡɦɟɪɚɦɢ R, L ɢ h, ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɰɢɥɢɧɞɪ ɜɵɜɟɪɧɭɬɶ ɧɚɢɡɧɚɧɤɭ. ȼ
ɪɚɛɨɬɚɯ [2], [4] ɛɵɥɨ ɩɨɥɭɱɟɧɨ ɬɚɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɞɥɹ ɚɛɫɬɪɚɤɬɧɨɝɨ ɰɢɥɢɧɞɪɚ ɐ (ɫɦ.
ɪɢɫ. 1) ɫ ɧɭɥɟɜɨɣ ɬɨɥɳɢɧɨɣ ɫɬɟɧɨɤ ɢ ɚɛɫɨɥɸɬɧɨ ɝɢɛɤɨɝɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɥɸɛɨɣ ɟɝɨ ɬɨɱɤɢ:
L ! Sh 2h, L
2SR
(1)
ɐɟɥɶ ɞɚɧɧɨɣ ɪɚɛɨɬɵ ɫɨɫɬɨɢɬ ɜ ɩɨɥɭɱɟɧɢɢ ɩɨɞɨɛɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɪɟɚɥɶɧɨɝɨ
ɰɢɥɢɧɞɪɚ.
2. ɉɪɢɜɟɞɺɦ ɩɨɷɬɚɩɧɭɸ ɬɟɯɧɨɥɨɝɢɸ, ɩɨɡɜɨɥɹɸɳɭɸ ɩɪɢɞɬɢ ɤ ɫɨɨɬɧɨɲɟɧɢɸ (1), ɩɨ
ɚɧɚɥɨɝɢɢ ɫ ɤɨɬɨɪɨɣ ɫɨɝɥɚɫɧɨ ɬɟɦ ɠɟ ɩɪɢɧɰɢɩɚɦ ɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦ ɲɚɝɚɦ, ɧɨ ɩɪɢɦɟɧɺɧɧɵɦ ɤ ɪɟɚɥɶɧɨɦɭ ɨɛɴɟɤɬɭ, ɪɟɲɚɟɬɫɹ ɩɨɫɬɚɜɥɟɧɧɚɹ ɜɵɲɟ ɡɚɞɚɱɚ.
26
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ВЕСТНИК
МГСУ
ɂɬɚɤ, ɬɟɨɪɟɦɚ 1. ɉɭɫɬɶ ɢɦɟɟɦ ɨɞɧɨɪɨɞɧɭɸ ɝɢɛɤɭɸ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɩɥɚɫɬɢɧɭ ɉ:
ABCD (l – ɞɥɢɧɚ, 2h – ɜɵɫɨɬɚ/ɲɢɪɢɧɚ) (ɫɦ. ɪɢɫ. 2), ɫ ɨɫɶɸ ɫɢɦɦɟɬɪɢɢ OO1 , ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɟɪɟɞɢɧɭ ɫɬɨɪɨɧ AB ɢ CD. Ɍɨɝɞɚ ɦɚɤɫɢɦɚɥɶɧɵɣ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɫɜɨɛɨɞɧɨɣ
ɫɬɨɪɨɧɵ AB ɜɨɤɪɭɝ ɨɫɢ OO1 ɩɨ ɩɪɢɧɹɬɨɦɭ ɭɫɥɨɜɢɸ ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ ɛɟɡ ɪɚɫɬɹɠɟɧɢɣ, ɫɦɹɬɢɣ ɢ ɪɚɡɪɵɜɨɜ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ ɛɭɞɟɬ ɪɚɜɟɧ
l
h
\ max
(2)
ɋɥɟɞɫɬɜɢɟ. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨɣ ɞɥɢɧɟ l-ɩɨɥɨɫɵ (h=const), ɭɝɨɥ ɩɨɜɨɪɨɬɚ \ max ɦɨɠɟɬ
ɛɵɬɶ, ɧɚɩɪɢɦɟɪ, ɪɚɜɟɧ 180º. Ɍɚɤɭɸ ɩɨɥɨɫɭ, ɨɩɪɟɞɟɥɺɧɧɵɦ ɨɛɪɚɡɨɦ ɫɤɥɚɞɵɜɚɹ, ɦɨɠɧɨ
ɭɥɨɠɢɬɶ ɧɚ ɩɥɨɫɤɨɫɬɶ P, ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ CD ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ OO1 . Ɍɨɝɞɚ ɬɨɱɤɚ
A, ɩɨɜɟɪɧɭɜɲɢɫɶ ɧɚ ɷɬɨɬ ɭɝɨɥ, ɫɨɜɩɚɞɺɬ ɫ ɬɨɱɤɨɣ D.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɋɚɡɨɛɶɟɦ ɩɨɥɨɫɤɭ ɉ ɧɚ 2n ɲɬɭɤ ɨɞɢɧɚɤɨɜɵɯ ɜɟɪɬɢɤɚɥɶɧɵɯ ɩɨɥɨɫɨɤ ɲɢɪɢɧɨɣ H, ɬɚɤ, ɱɬɨ l 2nH . ȼ ɤɚɠɞɨɦ ɨɛɪɚɡɨɜɚɜɲɟɦɫɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤɟ ɩɪɨɜɟɞɺɦ ɨɞɧɭ ɞɢɚɝɨɧɚɥɶ, ɱɬɨɛɵ ɜɫɟ ɜɦɟɫɬɟ ɞɢɚɝɨɧɚɥɢ ɫɨɫɬɚɜɢɥɢ 2n-ɡɜɟɧɧɭɸ ɥɨɦɚɧɭɸ, ɫɨɟɞɢɧɹɸɳɭɸ ɬɨɱɤɢ B ɢ D. Ⱦɥɹ ɧɚɫ ɡɜɟɧɶɹ ɛɭɞɭɬ ɹɜɥɹɬɶɫɹ ɨɫɹɦɢ, ɜɨɤɪɭɝ ɤɨɬɨɪɵɯ ɛɭɞɟɦ
ɩɨɜɨɪɚɱɢɜɚɬɶ ɩɨɥɨɫɤɭ ɉ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɝɢɛɚɹ ɟɺ ɢ ɭɤɥɚɞɵɜɚɹ ɟɺ ɧɚ ɩɥɨɫɤɨɫɬɶ
Pɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɨɜɟɪɧɭɜ ɩɨɥɨɫɤɭ ɉ ɜɨɤɪɭɝ CD. ɗɬɨ ɧɚɩɨɦɢɧɚɟɬ ɫɩɟɰɢɚɥɶɧɭɸ
ɩɥɢɫɫɢɪɨɜɤɭ ɨɞɟɠɞɵ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨɣ ɞɥɢɧɟ l (ɫɦ. ɪɢɫ. 3).
ɉɨɞɪɨɛɧɨ ɷɬɨ ɞɟɥɚɟɬɫɹ ɬɚɤ. ɉɟɪɜɵɣ ɩɨɜɨɪɨɬ ɜɨɤɪɭɝ CDc ɫ ɭɤɥɚɞɵɜɚɧɢɟɦ ɩɨɥɨɫɤɢ ɉ ɧɚɥɟɜɨ, ɜɬɨɪɨɣ – ɜɨɤɪɭɝ C cDc ɫ ɭɤɥɚɞɵɜɚɧɢɟɦ ɩɨɥɨɫɤɢ ɉ ɧɚɩɪɚɜɨ. ɂ ɬɚɤ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨɜɨɪɚɱɢɜɚɹ ɩɨɥɨɫɤɭ ɧɚ 180º ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɩɨɥɨɫɤɚ ɉ ɧɟ ɩɪɟɜɪɚɬɢɬɫɹ ɜ
ɫɥɨɠɟɧɧɭɸ ɮɢɝɭɪɭ, ɩɨɯɨɠɭɸ ɧɚ ɞɜɚ ɞɢɚɦɟɬɪɚɥɶɧɨ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɫɟɤɬɨɪɚ ɩɪɚɜɢɥɶɧɨɝɨ n-ɭɝɨɥɶɧɢɤɚ, ɫɨɫɬɨɹɳɭɸ ɢɡ ɞɜɭɯ ɫɟɪɢɣ ɩɨ n ɪɚɜɧɨɛɟɞɪɟɧɧɵɯ ɬɪɟɭɝɨɥɶɧɢɤɨɜ
' , ɫɱɢɬɚɹ ɞɜɟ ɤɨɧɰɟɜɵɟ ɩɨɥɨɜɢɧɤɢ ɡɚ ɨɞɢɧ. (ɫɦ. ɪɢɫ. 3) Ɉɛɨɡɧɚɱɢɦ ɜ ɤɚɠɞɨɦ ɬɪɟɭɝɨɥɶɧɢɤɟ ɭɝɨɥ ɩɪɢ ɜɟɪɲɢɧɟ Dn ɢ ɜɵɫɨɬɭ h n . ɋɭɦɦɚ ɞɥɢɧ ɨɫɧɨɜɚɧɢɣ ɬɪɟɭɝɨɥɶɧɢɤɨɜ ɜ
ɷɬɨɣ ɫɟɪɢɢ ɪɚɜɧɚ l=2nH. ɉɨɞɫɱɢɬɚɟɦ, ɤɚɤ ɜɵɪɚɠɚɟɬɫɹ ɫɭɦɦɚ ɭɝɥɨɜ ɩɪɢ ɜɟɪɲɢɧɚɯ ɷɬɢɯ
ɬɪɟɭɝɨɥɶɧɢɤɨɜ. ȼɨɡɶɦɺɦ ɬɪɟɭɝɨɥɶɧɢɤ
ɧɨ
ɦɚɥɵɯ
n
¦D
i 1
i
|
ɭɝɥɚɯ
Dn
l
2H
˜ n = , ɬ.ɤ.
hn
hn
ɢɦɟɟɦ
H=
' (ɫɦ. ɪɢɫ. 3). ȼ ɧɺɦ tg
Dn
2
|
H
hn
o Dn |
Dn
2H
hn
2
=
H
hn
ɢ,
. ɉɪɢ ɞɨɫɬɚɬɨɱɫɥɟɞɨɜɚɬɟɥɶɧɨ,
l
. ɉɟɪɟɯɨɞɹ ɤ ɩɪɟɞɟɥɭ ɩɪɢ n o f ɜɢɞɢɦ, ɱɬɨ
2n
hn o h ɢ ɮɢɝɭɪɚ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɞɜɚ ɫɟɤɬɨɪɚ ɨɤɪɭɠɧɨɫɬɢ ɤɚɠɞɵɣ ɫ ɭɝl
ɥɨɦ D =\ max = . ɑ. ɬ. ɞ.
h
Ɂɚɤɥɸɱɟɧɢɟ ɧɚ ɫɥɭɱɚɣ ɝɥɚɞɤɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ: ɟɫɥɢ ɩɪɢ ɬɚɤɨɣ ɬɟɯɧɨɥɨɝɢɢ ɫɥɨɠɟɧɢɹ ɩɨɥɨɫɤɢ (ɜ ɩɪɨɰɟɫɫɟ ɩɨɜɨɪɨɬɚ ɫɬɨɪɨɧɵ AB) ɜ ɞɨɩɪɟɞɟɥɶɧɭɸ ɮɢɝɭɪɭ ɞɨɩɭɫɤɚɬɶ
ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɶ ɧɚ ɫɤɥɚɞɤɟ (ɝɥɚɞɤɨɟ ɝɨɦɨɬɨɩɢɱɟɫɤɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɩɨɥɨɫɤɢ ɉ), ɬɨ ɩɨɥɭɱɚɟɦ ɝɥɚɞɤɨɟ ɩɥɚɜɧɨɟ ɫɨɩɪɹɠɟɧɢɟ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɫɤɥɚɞɨɤ ɫ
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ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɦ ɪɚɞɢɭɫɨɦ r= G >0 ɢ ɝɥɚɞɤɭɸ (ɛɟɡ ɫɤɥɚɞɨɤ) ɮɢɝɭɪɭ ɫ ɭɩɨɦɹɧɭɬɵɦ
l
, ɱɬɨ ɞɚɫɬ ɩɪɢɜɟɞɺɧɧɭɸ ɧɢɠɟ ɮɨɪɦɭɥɭ
h
l
(3)
\d
h
ɭɝɥɨɦ ɩɨɜɨɪɨɬɚ \ , ɫɤɨɥɶ ɭɝɨɞɧɨ ɛɥɢɡɤɢɦ ɤ
Ɍɚɤ ɤɚɤ ɱɚɫɬɶ ɞɥɢɧɵ l ɬɪɚɬɢɬɫɹ ɧɚ ɢɡɝɢɛ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨ ɪɚɞɢɭɫɭ
U
ɜ ɦɟɫɬɟ ɩɪɹ-
ɦɨɥɢɧɟɣɧɨɝɨ ɢɡɝɢɛɚ (ɧɚɩɪɢɦɟɪ, A1 B , ɪɢɫ. 2).
3. ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ ɧɚ ɨɫɧɨɜɚɧɢɢ ɜɵɲɟɢɡɥɨɠɟɧɧɵɯ ɩɨɫɬɪɨɟɧɢɣ ɩɪɢɦɟɧɢɦ ɚɧɚɥɨɝɢɱɧɵɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɞɭɪɵ ɢɡɝɢɛɚɧɢɹ ɫɬɟɧɨɤ ɞɥɹ ɪɟɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɚ ɐ
(ɬɚɤɠɟ ɫɯɟɦɚɬɢɱɟɫɤɢ ɢɡɨɛɪɚɠɺɧɧɨɝɨ ɧɚ ɪɢɫ. 1), ɩɪɢɜɨɞɹɳɢɟ ɤ ɜɵɜɨɪɚɱɢɜɚɧɢɸ ɐ ɧɚɢɡɧɚɧɤɭ, ɟɫɥɢ ɩɨɡɜɨɥɹɬ ɪɚɡɦɟɪɵ ɰɢɥɢɧɞɪɚ L ɢ R, ɬɨɥɳɢɧɚ ɫɬɟɧɤɢ G ɢ ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ
E (ɧɚɩɪɢɦɟɪ, ɫɬɚɥɶ ɨɩɪɟɞɟɥɺɧɧɨɣ ɦɚɪɤɢ). Ⱦɥɹ ɩɪɟɞɥɨɠɟɧɧɨɝɨ ɫɩɨɫɨɛɚ ɜɵɜɺɪɬɵɜɚɧɢɹ
ɧɟɨɛɯɨɞɢɦɨ, ɜɨ-ɩɟɪɜɵɯ, ɜɵɱɢɫɥɢɬɶ ɦɢɧɢɦɚɥɶɧɵɣ ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɢɱɟɫɤɨɝɨ ɢɡɝɢɛɚ U
ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨ ɮɨɪɦɭɥɟ:
V ɞɨɩ
E
U
˜yoU
E
V ɞɨɩ
˜ y ɮɨɪɦɭɥɚ
(4)
(ɫɦ. ɩɨɞɪɨɛɧɟɟ ɜ [3])
Ɋɢɫ. 1
ɋɦɵɫɥ ɜɵɜɺɪɬɵɜɚɧɢɹ ɰɢɥɢɧɞɪɚ ɐ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɫɨɛɪɚɜ ɱɚɫɬɶ ɛɨɤɨɜɨɣ
ɫɬɟɧɤɢ ɐ ɜ ɧɟɤɨɬɨɪɵɟ ɝɥɚɞɤɢɟ ɫɤɥɚɞɤɢ (ɜ ɩɥɢɫɫɢɪɨɜɤɭ), ɩɨɜɟɪɧɭɬɶ ɢɯ ɜɧɭɬɪɶ ɰɢɥɢɧɞɪɚ
ɧɚ ɭɝɨɥ S , ɡɚɬɪɚɬɢɜ ɩɪɢ ɷɬɨɦ ɨɩɪɟɞɟɥɺɧɧɭɸ ɱɚɫɬɶ ɞɥɢɧɵ L (L – ɞɥɢɧɚ ɨɤɪɭɠɧɨɫɬɢ
ɨɫɧɨɜɚɧɢɹ). Ɂɚɞɚɱɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɨɰɟɧɢɬɶ ɬɭ ɦɢɧɢɦɚɥɶɧɭɸ ɞɥɢɧɭ L, ɩɨɡɜɨɥɹɸɳɭɸ ɩɪɢ ɞɚɧɧɵɯ h, ' , E ɢ ɩɪɢɧɹɬɨɣ ɬɟɯɧɨɥɨɝɢɢ ɜɵɜɺɪɬɵɜɚɧɢɹ ɜɵɜɟɪɧɭɬɶ ɰɢɥɢɧɞɪ ɧɚɢɡɧɚɧɤɭ. ɇɚ ɪɢɫɭɧɤɟ 2 ɢɡɨɛɪɚɠɟɧɵ ɩɭɧɤɬɢɪɨɦ ɥɢɧɢɢ, ɩɨ ɤɨɬɨɪɵɦ ɧɚɞɨ ɫɤɥɚɞɵɜɚɬɶ ɚɛɫɬɪɚɤɬɧɭɸ ɩɨɥɨɫɤɭ ɫ ' 0 , ɱɬɨɛɵ ɩɨɜɟɪɧɭɬɶ ɟɺ ɨɬɪɟɡɨɤ AB ɧɚ 180º. Ⱦɥɹ
ɮɢɡɢɱɟɫɤɨɣ ɩɥɚɫɬɢɧɤɢ – ɫɬɟɧɤɢ ɰɢɥɢɧɞɪɚ – ɢɡɝɢɛɚɧɢɟ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɞɨ ɨɩɪɟɞɟɥɺɧɧɨɣ ɤɪɢɜɢɡɧɵ ɩɨɜɟɪɯɧɨɫɬɢ, ɨɩɪɟɞɟɥɹɟɦɨɣ ɪɚɞɢɭɫɨɦ U . ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɨɛɫɭɞɢɦ ɧɨɜɭɸ ɫɯɟɦɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɪɟɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɚ ɐ, ɨɩɢɪɚɹɫɶ ɜ ɪɚɫɫɭɠɞɟɧɢɹɯ ɧɚ ɪɢɫɭɧɨɤ 4.
ɑɬɨɛɵ ɩɨɜɟɪɧɭɬɶ AB ɜɧɭɬɪɶ ɰɢɥɢɧɞɪɚ, ɧɚɞɨ ɫɠɚɬɶ ɟɝɨ ɫɬɟɧɤɢ ɫɩɟɰɢɚɥɶɧɨɣ ɭɩɪɭɝɨɣ ɩɥɢɫɫɢɪɨɜɤɨɣ. ȿɫɥɢ ɪɚɧɧɟɟ (ɪɢɫ. 2) ɩɥɢɫɫɢɪɨɜɤɚ ɲɥɚ ɩɨ ɥɢɧɢɹɦ AB, A1 B , A1 B1 ,
A1 B2 ɢ ɬ.ɞ., ɬɨ ɬɟɩɟɪɶ ɧɟɨɛɯɨɞɢɦɨ ɷɬɢ ɥɢɧɢɢ ɩɪɟɜɪɚɬɢɬɶ ɜ ɩɨɥɭɰɢɥɢɧɞɪɵ ɫ ɪɚɞɢɭɫɨɦ
U (ɫɦ. ɪɢɫ. 4; ɬɨɱɤɚ ɨɡɧɚɱɚɟɬ ɜɵɝɢɛɚɧɢɟ ɧɚ ɫɟɛɹ, ɩɥɸɫ – ɨɬ ɫɟɛɹ). ɉɪɢ ɬɚɤɨɦ ɢɡɝɢɛɚɧɢɢ
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ɧɚɞɨ ɡɧɚɬɶ ɧɚ ɤɚɤɨɣ ɭɝɨɥ M ɤɚɠɞɵɣ ɪɚɡ ɩɨɜɨɪɚɱɢɜɚɟɬɫɹ ɨɬɪɟɡɨɤ AB. ɉɪɢɜɟɞɺɦ ɪɚɫɱɺɬ
ɭɝɥɚ M ɢ ɱɢɫɥɚ ɩɨɜɨɪɨɬɨɜ n, ɧɟɨɛɯɨɞɢɦɵɯ ɧɚ ɩɨɜɨɪɨɬ AB ɢ 180º (ɩɨɥɶɡɭɹɫɶ ɪɢɫ. 4).
ɗɬɢ ɩɚɪɚɦɟɬɪɵ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɟ:
­M ˜ n S ;
°
°x S ˜ U ;
sin M
°
°
h
®y
;
°
tgM
°
°y
h2 y 2
°
x
¯h
(5)
Ɋɢɫ. 2
Ɋɢɫ. 3
Ɋɢɫ. 4
ȼɵɱɢɫɥɟɧɢɟ ɡɧɚɱɟɧɢɣ M , n, x, y ɨɩɪɟɞɟɥɹɟɬ ɪɚɡɛɢɟɧɢɟ (ɪɢɫ. 4) ɢ ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ
ɧɭɠɧɭɸ ɞɥɢɧɭ l ɞɥɹ ɤɨɧɫɬɪɭɤɰɢɢ ɜɵɜɺɪɬɵɜɚɧɢɹ. ɗɬɚ ɞɥɢɧɚ ɫɨɫɬɨɢɬ ɢɡ:
1) ɡɚɬɪɚɬɵ ɞɥɢɧɵ L ɩɪɢ ɩɥɢɫɫɢɪɨɜɤɟ, ɬ.ɟ. l1 (SU x y ) ˜ n ;
2) ɡɚɬɪɚɬɵ L ɧɚ ɦɢɧɢɦɚɥɶɧɭɸ ɲɢɪɢɧɭ ɩɥɢɫɫɢɪɨɜɤɢ -
l2
§
SU ·
¨¨ 2 U ¸˜n;
2 sin M ¸¹
©
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3) ɡɚɬɪɚɬɵ L ɧɚ ɩɨɜɨɪɨɬ ɜɧɭɬɪɶ ɐ (ɨɰɟɧɢɜɚɟɬɫɹ ɤɚɫɚɧɢɟ ɫ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɷɥɟɦɟɧɬɨɦ ɜɵɜɺɪɬɵɜɚɧɢɹ): l3 U y s , ɝɞɟ s SU ˜ cos M ;
l
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚ ɜɟɫɶ ɩɨɜɨɪɨɬ ɜ 180º ɬɪɚɬɢɬɫɹ ɞɥɢɧɵ L ɜ ɤɨɥɢɱɟɫɬɜɟ
l1 l2 l3 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɜɨɡɦɨɠɧɨɫɬɢ ɝɥɚɞɤɨɝɨ ɜɵɜɺɪɬɵɜɚɧɢɹ ɜ ɞɚɧɧɨɣ ɬɟɯ-
ɧɨɥɨɝɢɢ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɟɧɢɟ ɫɥɟɞɭɸɳɟɝɨ ɭɫɥɨɜɢɹ:
§
SU ·
L ! (SU x y ) ˜ n ¨¨ 2 U ¸˜n U y s
2 sin M ¸¹
©
ɂɡ ɚɧɚɥɢɡɚ ɩɪɢɜɟɞɺɧɧɵɯ ɭɬɜɟɪɠɞɟɧɢɣ ɧɚɩɪɚɲɢɜɚɟɬɫɹ ɜɵɜɨɞ, ɱɬɨ ɦɨɝɭɬ ɜɨɡɧɢɤɧɭɬɶ ɫɢɬɭɚɰɢɢ, ɜ ɤɨɬɨɪɵɯ ɧɚ ɧɟɤɨɬɨɪɵɣ ɨɛɴɟɤɬ ɞɟɣɫɬɜɭɟɬ ɬɚɤɨɟ ɫɨɱɟɬɚɧɢɟ ɫɢɥ (ɩɚɪɚɦɟɬɪɨɜ), ɱɬɨ ɨɧ ɦɨɠɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɢɬɶɫɹ, ɧɟ ɬɟɪɹɹ ɫɜɨɢɯ ɩɪɨɱɧɨɫɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ, ɢ ɩɪɨɣɞɹ ɫɜɨɸ ɡɨɧɭ ɛɢɮɭɪɤɚɰɢɢ, ɦɨɠɟɬ ɡɧɚɱɢɬɟɥɶɧɨ ɩɨɦɟɧɹɬɶ ɫɜɨɸ ɤɨɧɮɢɝɭɪɚɰɢɸ. Ɍɚɤɢɟ ɫɢɬɭɚɰɢɢ ɫɜɹɡɚɧɵ ɫ ɬɟɨɪɢɟɣ ɤɚɬɚɫɬɪɨɮ [1]. ɉɨɷɬɨɦɭ ɩɪɢ ɪɚɫɱɺɬɟ ɫɨɨɪɭɠɟɧɢɣ ɧɚɞɨ ɩɨɦɧɢɬɶ ɨ ɫɢɫɬɟɦɧɨɦ ɩɨɞɯɨɞɟ ɜ ɩɪɨɰɟɫɫɟ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɮɨɪɦɭɥ ɢ ɞɟɬɟɪɦɢɧɢɫɬɢɱɟɫɤɢɯ ɩɨɥɨɠɟɧɢɣ.
Ʌɢɬɟɪɚɬɭɪɚ
1. Ⱥɪɧɨɥɶɞ ȼ. ɂ. Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ. Ɇ., ɇɚɭɤɚ, 1990
2. Ʌɟɧɺɜ ȼ.ɋ. ɉɪɟɞɟɥɶɧɵɟ ɩɨɥɨɠɟɧɢɹ ɧɟɬɪɢɜɢɚɥɶɧɵɯ ɨɝɪɚɧɢɱɟɧɧɵɯ ɤɨɦɩɚɤɬɨɜ ɡɚɦɤɧɭɬɵɯ
ɪɚɡɜɺɪɬɵɜɚɸɳɢɯɫɹ ɩɨɜɟɪɯɧɨɫɬɟɣ ɜ ɧɟɤɨɬɨɪɵɯ ɫɜɨɢɯ ɝɨɦɨɬɨɩɧɵɯ ɫɟɦɟɣɫɬɜɚɯ.
// ȼɨɩɪɨɫɵ ɦɚɬɟɦɚɬɢɤɢ, ɦɟɯɚɧɢɤɢ ɫɩɥɨɲɧɵɯ ɫɪɟɞ ɢ ɩɪɢɦɟɧɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɜ
ɫɬɪɨɢɬɟɥɶɫɬɜɟ: ɋɛ. ɬɪ. Ɇ.:ɆȽɋɍ(Ɇɂɋɂ), 1999.
3. ɋɦɢɪɧɨɜ Ⱥ. Ɏ. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ. Ɇ., ȼɵɫɲɚɹ ɲɤɨɥɚ, 1975.
The literature
4. Halpern B., Weaver C. Inverting a cylinder through isometric immersions and isometric embeddings // transaction of the American Mathematical Society. 1977. V. 230
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɢɡɨɦɟɬɪɢɱɟɫɤɢɟ ɨɬɨɛɪɚɠɟɧɢɹ, ɞɨɩɭɫɬɢɦɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɨɪɢɟɧɬɚɰɢɹ
ɩɨɜɟɪɯɧɨɫɬɟɣ, ɝɥɚɞɤɚɹ ɞɟɮɨɪɦɚɰɢɹ ɩɨɜɟɪɯɧɨɫɬɟɣ, ɫɦɹɬɢɹ ɢ ɫɤɥɚɞɤɢ, ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜ ɨɛɥɚɫɬɢ
ɭɩɪɭɝɢɯ ɞɟɮɨɪɦɚɰɢɣ, ɫɦɟɧɚ ɨɪɢɟɧɬɚɰɢɢ, ɬɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ, ɛɢɮɭɪɤɚɰɢɹ.
Key words: isometric maps, admissible transformations, orientation of surfaces, smooth deformation of surfaces, crushing and folding, transformation in the field of elastic deformations, changing of
orientation, catastrophe theory, bifurcation.
Ɍɟɥ. 3290412, 8 915 109507; e-mail: vladlenev@rambler.ru
Ɋɟɰɟɧɡɟɧɬ: ȼɚɪɚɩɚɟɜ ȼɥɚɞɢɦɢɪ ɇɢɤɨɥɚɟɜɢɱ, ɞɨɤɬɨɪ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ,
ɩɪɨɮɟɫɫɨɪ ɤɚɮɟɞɪɵ ɦɚɬɟɦɚɬɢɤɢ ɋȽȺ (ɋɨɜɪɟɦɟɧɧɚɹ ɝɭɦɚɧɢɬɚɪɧɚɹ ɚɤɚɞɟɦɢɹ)
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