Насыров А.М. Волновые процессы. Электромагнитные волны в линиях передач

ɄȺɁȺɇɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ

ɤɚɮɟɞɪɚ ɪɚɞɢɨɷɥɟɤɬɪɨɧɢɤɢ

Ⱥ.Ɇ. ɇɚɫɵɪɨɜ

ȼɈɅɇɈȼɕȿ ɉɊɈɐȿɋɋɕ

ɑȺɋɌɖ 10.

ɗɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜ ɥɢɧɢɹɯ ɩɟɪɟɞɚɱ.

ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ

Ʉɚɡɚɧɶ - 2010

2

ɍȾɄ 538.86+537.86

ɉɟɱɚɬɚɟɬɫɹ ɩɨ ɪɟɲɟɧɢɸ

ɪɟɞɚɤɰɢɨɧɧɨ-ɢɡɞɚɬɟɥɶɫɤɨɝɨ ɫɨɜɟɬɚ

ɮɢɡɢɱɟɫɤɨɝɨ ɮɚɤɭɥɶɬɟɬɚ

Ʉɚɡɚɧɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ

Ɋɟɰɟɧɡɟɧɬɵ:

ɉɪɨɮɟɫɫɨɪ ɤɚɮɟɞɪɵ ɪɚɞɢɨɮɢɡɢɤɢ Ʉɚɪɩɨɜ Ⱥ.ȼ..

ɇɚɫɵɪɨɜ Ⱥ.Ɇ.

ȼɨɥɧɨɜɵɟ ɩɪɨɰɟɫɫɵ. ɑɚɫɬɶ 10. ɗɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜ ɥɢɧɢɹɯ

ɩɟɪɟɞɚɱ. ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ. ñ Ʉɚɡɚɧɶ: ɄȽɍ, 2010. ñ49 ɫɬɪ.

Ⱥɧɧɨɬɚɰɢɹ

ɂɡɥɚɝɚɸɬɫɹ ɨɫɧɨɜɵ ɬɟɨɪɢɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ,

ɦɚɝɧɢɬɧɵɯ ɢ ɷɥɟɤɬɪɢɱɟɫɤɢɯ ɜɨɥɧ ɜ ɜɨɥɧɨɜɨɞɚɯ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɢ ɤɪɭɝɥɨɝɨ

ɫɟɱɟɧɢɹ, ɜ ɤɨɚɤɫɢɚɥɶɧɵɯ ɥɢɧɢɹɯ, ɜ ɞɜɭɯɩɪɨɜɨɞɧɵɯ ɥɢɧɢɹɯ. Ɋɚɫɫɦɨɬɪɟɧɵ

ɩɪɢɧɰɢɩɵ ɜɨɡɛɭɠɞɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɜ ɪɟɡɨɧɚɬɨɪɚɯ. ɉɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ

ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɪɚɞɢɨɮɢɡɢɤɨɜ.

ɭɧɢɜɟɪɫɢɬɟɬ, 2010

Ʉɚɡɚɧɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ

ɭɧɢɜɟɪɫɢɬɟɬɚ, 2010

3

ɋɈȾȿɊɀȺɇɂȿ

1. ɄɅȺɋɋɂɎɂɄȺɐɂə ȼɈɅɇ 4

1.1. ɋɜɨɛɨɞɧɵɟ ɢ ɧɚɩɪɚɜɥɟɧɧɵɟ ɜɨɥɧɵ 4

1.2. Ʉɨɧɰɟɩɰɢɹ ɩɚɪɰɢɚɥɶɧɵɯ ɜɨɥɧ 4

1.3. ɍɫɥɨɜɢɹ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɜɨɥɧ ɪɚɡɥɢɱɧɨɝɨ ɬɢɩɚ 6

1.4. Ɋɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɜ ɥɢɧɢɹɯ ɩɟɪɟɞɚɱ 7

1.5. ɉɨɩɟɪɟɱɧɵɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ 8

1.6. ɗɥɟɤɬɪɢɱɟɫɤɢɟ ɜɨɥɧɵ

()

0,0 =&

zz

HE

9

1.7. Ɇɚɝɧɢɬɧɵɟ ɜɨɥɧɵ

()

0,0 &=

zz

HE

12

2. ɋɬɪɭɤɬɭɪɚ ɩɨɥɹ ɜ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɩɨɥɵɯ ɜɨɥɧɨɜɨɞɚɯ

ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ

12

2.1. ȼɨɥɧɵ ɬɢɩɚ ȿ ɜ ɜɨɥɧɨɜɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ 12

2.2. ȼɨɥɧɵ ɬɢɩɚ ɇ ɜ ɜɨɥɧɨɜɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ 15

3. ɋɨɛɫɬɜɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɨɛɴɟɦɧɵɯ ɪɟɡɨɧɚɬɨɪɨɜ 18

4. ȼɨɥɧɵ ɬɢɩɨɜ ȿ ɢ ɇ ɜ ɜɨɥɧɨɜɨɞɚɯ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ 22

4.1. ȼɨɥɧɵ ɬɢɩɚ ȿ ɜ ɜɨɥɧɨɜɨɞɚɯ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ 23

4.2. ȼɨɥɧɵ ɬɢɩɚ ɇ ɜ ɜɨɥɧɨɜɨɞɚɯ ɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ 26

5. ɗɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜ ɤɨɚɤɫɢɚɥɶɧɨɣ ɥɢɧɢɢ 28

5.1. ȼɨɥɧɵ ɬɢɩɚ ɌȿɆ ɜ ɤɨɚɤɫɢɚɥɶɧɨɣ ɥɢɧɢɢ 29

5.2. ȼɨɥɧɵ ɬɢɩɨɜ ȿ ɢ ɇ ɜ ɤɨɚɤɫɢɚɥɶɧɨɣ ɥɢɧɢɢ 32

6. ɗɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɜ ɨɞɧɨɪɨɞɧɵɯ ɥɢɧɢɹɯ 36

6.1. Ɍɟɥɟɝɪɚɮɧɵɟ ɭɪɚɜɧɟɧɢɹ 36

6.2. Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ, ɤɨɷɮɮɢɰɢɟɧɬ ɮɚɡɵ ɢ ɮɚɡɨɜɚɹ

ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ

40

6.3. Ʉɨɷɮɮɢɰɢɟɧɬ ɨɬɪɚɠɟɧɢɹ 41

6.4. Ʉɨɷɮɮɢɰɢɟɧɬ ɛɟɝɭɳɟɣ ɜɨɥɧɵ 42

ɄɈɇɌɊɈɅɖɇɕȿ ȼɈɉɊɈɋɕ

ɁȺȾȺɑɂ

43

45

ɅɂɌȿɊȺɌɍɊȺ 49

4

1. ɄɅȺɋɋɂɎɂɄȺɐɂə ȼɈɅɇ

1.1. ɋɜɨɛɨɞɧɵɟ ɢ ɧɚɩɪɚɜɥɟɧɧɵɟ ɜɨɥɧɵ

ɋɜɨɛɨɞɧɵɦɢ

ɧɚɡɵɜɚɸɬ ɜɨɥɧɵ, ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɢɟɫɹ ɜ ɫɜɨɛɨɞɧɨɦ

ɩɪɨɫɬɪɚɧɫɬɜɟ (ɬ.ɟ. ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɨɞɧɨɪɨɞɧɨɣ ɢɡɨɬɪɨɩɧɨɣ ɫɪɟɞɟ) ɢ

ɩɨɬɟɪɹɜɲɢɟ ɜ ɩɪɨɰɟɫɫɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɫɜɹɡɶ ɫɨ ɫɜɨɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ.

ɇɚɩɪɚɜɥɟɧɧɵɦɢ

ɧɚɡɵɜɚɸɬ ɜɨɥɧɵ, ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɢɟɫɹ ɜɞɨɥɶ ɤɚɤɢɯ-

ɥɢɛɨ ɬɟɥ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɢɯ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ. ɇɚɩɪɢɦɟɪ: ɨɬɤɪɵɬɵɟ

ɦɟɬɚɥɥɢɱɟɫɤɢɟ, ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɟ ɢɥɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜɵɟ ɩɨɜɟɪɯɧɨɫɬɢ,

ɨɞɧɨɩɪɨɜɨɞɧɵɟ, ɦɧɨɝɨɩɪɨɜɨɞɧɵɟ ɢ ɤɨɚɤɫɢɚɥɶɧɵɟ ɥɢɧɢɢ, ɜɨɥɧɨɜɨɞɵ ɢɡ

ɩɨɥɵɯ ɬɪɭɛ, ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɟ ɫɬɟɪɠɧɢ ɢ ɞɪ.

ȼ ɩɨɫɨɛɢɢ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɨɥɶɤɨ ɧɚɩɪɚɜɥɟɧɧɵɟ ɜɨɥɧɵ. Ɉɧɢ

ɞɟɥɹɬɫɹ ɧɚ ɩɨɩɟɪɟɱɧɵɟ (ɨɛɨɡɧɚɱɚɸɬɫɹ ɌȿɆ), ɷɥɟɤɬɪɢɱɟɫɤɢɟ (ɨɛɨɡɧɚɱɚɸɬɫɹ

ȿ ɢɥɢ ɌɆ) ɢ ɦɚɝɧɢɬɧɵɟ (ɨɛɨɡɧɚɱɚɸɬɫɹ ɇ ɢɥɢ Ɍȿ). ȼɟɤɬɨɪɵ ɧɚɩɪɹɠɟɧɧɨɫɬɟɣ

ɷɥɟɤɬɪɢɱɟɫɤɢɯ ɢ ɦɚɝɧɢɬɧɵɯ ɩɨɥɟɣ ɜɨɥɧ ɌȿɆ ɪɚɫɩɨɥɨɠɟɧɵ ɜ ɩɥɨɫɤɨɫɬɢ,

ɨɪɬɨɝɨɧɚɥɶɧɨɣ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ.

ɗɥɟɤɬɪɢɱɟɫɤɢɦɢ

ɧɚɡɵɜɚɸɬ ɜɨɥɧɵ, ɭ ɤɨɬɨɪɵɯ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɧɨɫɬɢ

ȿ

r

ɩɨɦɢɦɨ ɩɨɩɟɪɟɱɧɨɣ, ɢɦɟɟɬ ɩɪɨɞɨɥɶɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ (ɬ.ɟ.

ɫɨɫɬɚɜɥɹɸɳɭɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ), ɜɟɤɬɨɪ

ɧɚɩɪɹɠɟɧɧɨɫɬɢ

ɇ

r

ɢɦɟɟɬ ɬɨɥɶɤɨ ɩɨɩɟɪɟɱɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ.

ɗɥɟɤɬɪɢɱɟɫɤɢɟ ɜɨɥɧɵ ɧɚɡɵɜɚɸɬ ɬɚɤɠɟ ɩɨɩɟɪɟɱɧɨ-ɦɚɝɧɢɬɧɵɦɢ

ɜɨɥɧɚɦɢ.

Ɇɚɝɧɢɬɧɵɦɢ

ɧɚɡɵɜɚɸɬ ɜɨɥɧɵ, ɭ ɤɨɬɨɪɵɯ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɧɨɫɬɢ

ɇ

r

ɢɦɟɟɬ ɩɨɩɟɪɟɱɧɭɸ ɢ ɩɪɨɞɨɥɶɧɭɸ ɫɨɫɬɚɜɥɹɸɳɢɟ. Ⱥ ɜɟɤɬɨɪ ɧɚɩɪɹɠɟɧɧɨɫɬɢ

ȿ

r

ɢɦɟɟɬ ɬɨɥɶɤɨ ɩɨɩɟɪɟɱɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ. Ɇɚɝɧɢɬɧɵɟ ɜɨɥɧɵ ɧɚɡɵɜɚɸɬ

ɩɨɩɟɪɟɱɧɨ-ɷɥɟɤɬɪɢɱɟɫɤɢɦɢ

ɜɨɥɧɚɦɢ.

1.2. Ʉɨɧɰɟɩɰɢɹ ɩɚɪɰɢɚɥɶɧɵɯ ɜɨɥɧ [2]

Ɉɫɨɛɟɧɧɨɫɬɢ ɫɬɪɭɤɬɭɪɵ ɪɚɡɥɢɱɧɵɯ ɬɢɩɨɜ ɧɚɩɪɚɜɥɹɟɦɵɯ ɜɨɥɧ ɦɨɠɧɨ

ɨɛɴɹɫɧɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜɨɥɧɵ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ

ɜɞɨɥɶ ɨɫɢ Z. Ɋɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɷɧɟɪɝɢɢ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ

ɜɨɥɧɵ ɜ ɞɚɥɶɧɟɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ

ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɜɟɤɬɨɪɨɦ ɍɦɨɜɚ-ɉɨɣɧɬɢɧɝɚ

ɉ

r

[

]

HEɉ

rrr

= ;

H

E

ɉ

rrr

 (1.1)

ɂɡ (1.1) ɫɥɟɞɭɟɬ, ɱɬɨ ɜɟɤɬɨɪ

ɉ

r

ɧɚɩɪɚɜɥɟɧ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ

ɪɚɫɩɨɥɨɠɟɧɢɹ ɜɟɤɬɨɪɨɜ

ȿ

v

ɢ

ɇ

r

(Ɋɢɫ. 1).

5

Ɋɢɫ 1

Ɋɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɷɧɟɪɝɢɢ ɜɞɨɥɶ ɨɫɢ

z ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɥɢɛɨ ɩɨ

ɩɚɪɚɥɥɟɥɶɧɵɦ ɨɫɢ ɥɢɧɢɹɦ (Ɋɢɫ.2ɚ), ɥɢɛɨ ɩɨ ɥɨɦɚɧɵɦ ɥɢɧɢɹɦ- (ɤɪɢɜɵɦ,

Ɋɢɫ.2ɛ) ɩɪɢ ɨɛɳɟɦ ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɷɧɟɪɝɢɢ ɜɞɨɥɶ ɨɫɢ

z.

Ɋɢɫ 2ɚ Ɋɢɫ 2ɛ

ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ (1.1) ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɜɟɤɬɨɪɵ

ȿ

v

ɢ

ɇ

r

ɞɨɥɠɧɵ ɧɚɯɨɞɢɬɶɫɹ

ɜ ɩɥɨɫɤɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ

z. Ɍ.ɟ ɦɵ ɢɦɟɟɦ ɜɨɥɧɭ ɬɢɩɚ ɌȿɆ. ȼɨ

ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɜɟɤɬɨɪɵ

ȿ

v

ɢ

ɇ

r

ɞɨɥɠɧɵ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɥɨɫɤɨɫɬɢ,

ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɭɱɚɫɬɤɭ ɥɨɦɚɧɨɣ (ɤɪɢɜɨɣ) ɥɢɧɢɢ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɯɨɬɹ

ɛɵ ɨɞɢɧ ɢɡ ɜɟɤɬɨɪɨɜ

ȿ

v

ɢɥɢ

ɇ

r

ɧɟ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɨɫɢ z (Ɋɢɫ 3 ɢ Ɋɢɫ 4). Ɍ.ɟ.

ɥɢɛɨ

ȿ

v

, ɥɢɛɨ

ɇ

r

, ɥɢɛɨ ɨɛɚ ɜɟɤɬɨɪɚ ɢɦɟɸɬ ɩɪɨɞɨɥɶɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɬɨɪɨɣ ɫɥɭɱɚɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɨɥɧɚɦ ɬɢɩɚ

ȿ

ɢɥɢ

ɇ

, ɢɥɢ

ɨɞɧɨɜɪɟɦɟɧɧɨɦɭ ɫɭɳɟɫɬɜɨɜɚɧɢɸ ɜɨɥɧ ɬɢɩɨɜ

ȿ

ɢ

ɇ

.

6

Ɋɢɫ.3. Ɋɢɫ.4.

1.3. ɍɫɥɨɜɢɹ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɜɨɥɧ ɪɚɡɥɢɱɧɨɝɨ ɬɢɩɚ

ȼ ɨɞɧɨɪɨɞɧɨɣ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɫɪɟɞɟ ɜɨɥɧɵ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ

ɩɪɹɦɨɥɢɧɟɣɧɨ. ɂɡɝɢɛ ɩɭɬɢ ɜɨɡɦɨɠɟɧ ɜ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ. Ɉɬɤɥɨɧɟɧɢɟ ɨɬ

ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɡɦɨɠɧɨ ɜ ɧɟɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ, ɚ ɬɚɤɠɟ ɜ

ɫɥɭɱɚɟ, ɟɫɥɢ ɫɪɟɞɚ ɝɪɚɧɢɱɢɬ ɫ ɞɪɭɝɨɣ ɫɪɟɞɨɣ (ɢɥɢ ɨɤɪɭɠɟɧɚ), ɧɚ ɝɪɚɧɢɰɟ

ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɨɬɪɚɠɟɧɢɟ ɜɨɥɧ. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɜɨɥɧɵ ɬɢɩɨɜ

ȿ

ɢ

ɇ

ɜɨɡɦɨɠɧɵ ɜ ɩɨɥɵɯ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɬɪɭɛɚɯ, ɜ ɤɨɚɤɫɢɚɥɶɧɵɯ ɥɢɧɢɹɯ,

ɦɟɠɞɭ ɞɜɭɦɹ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɩɥɚɫɬɢɧɚɦɢ, ɜ ɞɪɭɝɢɯ ɩɨɞɨɛɧɵɯ ɫɢɫɬɟɦɚɯ, ɜ

ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɯ ɫɬɟɪɠɧɹɯ. ȼ [4] ɩɪɢɜɟɞɟɧ ɩɪɢɦɟɪ ɩɨɹɜɥɟɧɢɹ ɩɪɨɞɨɥɶɧɨɣ

ɫɨɫɬɚɜɥɹɸɳɟɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɜ ɚɧɢɡɚɬɪɨɩɧɨɣ ɩɥɚɡɦɟ.

ɉɪɢɦɟɪɚɦɢ ɥɢɧɢɢ, ɜɞɨɥɶ ɤɨɬɨɪɨɣ ɧɟɜɨɡɦɨɠɧɨ

ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɜɨɥɧ

ȿ ɢɥɢ ɇ ɬɢɩɨɜ ɹɜɥɹɟɬɫɹ ɨɞɧɨɩɪɨɜɨɞɧɚɹ ɥɢɧɢɹ ɢɞɟɚɥɶɧɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ,

ɧɚɯɨɞɹɳɚɹɫɹ ɜ ɫɜɨɛɨɞɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. Ɉɞɧɨɪɨɞɧɨɫɬɶ ɢ ɛɟɡɝɪɚɧɢɱɧɨɫɬɶ

ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɢɫɤɥɸɱɚɸɬ ɜɨɡɦɨɠɧɨɫɬɶ ɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ (ɢɥɢ

ɤɪɢɜɨɥɢɧɟɣɧɨɝɨ) ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧ, ɢɫɤɥɸɱɚɹ ɜɨɡɦɨɠɧɨɫɬɶ

ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɜɨɥɧ ɬɢɩɨɜ ȿ ɢ ɇ. ɇɚɥɢɱɢɟ ɩɨɬɟɪɶ ɜ ɨɞɧɨɩɪɨɜɨɞɧɵɯ

ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɥɢɧɢɹɯ ɩɪɢɜɨɞɢɬ ɤ ɩɪɨɧɢɤɧɨɜɟɧɢɸ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ

ɜɧɭɬɪɶ ɦɟɬɚɥɥɢɱɟɫɤɨɝɨ ɩɪɨɜɨɞɧɢɤɚ. ɉɪɢ ɷɬɨɦ ɫɨɡɞɚɸɬɫɹ ɭɫɥɨɜɢɹ ɞɥɹ

ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧ ȿ ɢɥɢ ɇ ɬɢɩɨɜ ɜɞɨɥɶ ɥɢɧɢɢ.

ȼ ɩɨɥɵɯ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɬɪɭɛɚɯ (ɜɨɥɧɨɜɨɞɚɯ) ɧɟɜɨɡɦɨɠɧɨ

ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɜɨɥɧ ɬɢɩɚ ɌȿɆ. ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.

Ɇɚɝɧɢɬɧɵɟ ɫɢɥɨɜɵɟ ɥɢɧɢɢ ɞɨɥɠɧɵ ɨɛɪɚɡɨɜɵɜɚɬɶ ɡɚɦɤɧɭɬɵɟ ɤɨɧɬɭɪɵ,

ɤɨɬɨɪɵɟ ɞɥɹ ɜɨɥɧ ɌȿɆ ɞɨɥɠɧɵ ɧɚɯɨɞɢɬɶɫɹ ɜ ɩɨɩɟɪɟɱɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɉɞɧɚɤɨ,

ɷɬɨ ɧɟɜɨɡɦɨɠɧɨ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɩɪɨɞɨɥɶɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɜɟɤɬɨɪɚ

ȿ

v

. ɉɪɢ

ɨɬɫɭɬɫɬɜɢɢ ɷɬɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɨɬɫɭɬɫɬɜɭɟɬ ɩɪɨɞɨɥɶɧɵɣ ɬɨɤ ɫɦɟɳɟɧɢɹ, ɜ

ɫɥɟɞɫɬɜɢɟ ɱɟɝɨ ɦɚɝɧɢɬɧɵɟ ɫɢɥɨɜɵɟ ɥɢɧɢɢ ɜ ɩɨɩɟɪɟɱɧɨɣ ɩɥɨɫɤɨɫɬɢ

ɨɬɫɭɬɫɬɜɭɸɬ. ȼ ɫɥɭɱɚɟ ɤɨɚɤɫɢɚɥɶɧɨɣ ɥɢɧɢɢ ɩɨ ɜɧɭɬɪɟɧɧɟɦɭ ɩɪɨɜɨɞɧɢɤɭ

ɦɨɠɟɬ ɬɟɱɶ ɬɨɤ ɩɪɨɜɨɞɢɦɨɫɬɢ. ɉɨɷɬɨɦɭ ɜ ɬɚɤɨɣ ɥɢɧɢɢ ɜɨɡɦɨɠɧɨ

ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɜɨɥɧ ɬɢɩɚ ɌȿɆ, ɬɚɤɠɟ ɤɚɤ ɢ ɜɨɥɧ ɇ ɢ ȿ.

7

1.4. Ɋɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɜ ɥɢɧɢɹɯ ɩɟɪɟɞɚɱ.

ȼɜɟɞɟɦ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɨɫɶɸ

z, ɧɚɩɪɚɜɥɟɧɧɨɣ

ɜɞɨɥɶ ɨɫɢ ɜɨɥɧɨɜɨɞɚ. ɋɨɫɬɚɜɥɹɸɳɢɟ

z

ȿ ɢ

z

ɇ ɜɞɨɥɶ ɨɫɢ z, ɭɞɨɜɥɟɬɜɨɪɹɸɬ

ɜɨɥɧɨɜɵɦ ɭɪɚɜɧɟɧɢɹɦ. Ɉɧɢ ɫɜɨɞɹɬɫɹ ɤ ɭɪɚɜɧɟɧɢɹɦ Ƚɟɥɶɦɝɨɥɶɰɚ ɞɥɹ

ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɩɨɥɟɣ. Ⱦɥɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɩɨɥɟɣ ɭɪɚɜɧɟɧɢɹ Ɇɚɤɫɜɟɥɥɚ

ɢɦɟɸɬ ɜɢɞ:

HiErot

a

r

tµ

<= (1.2)

EiHrot

r

¡t

~

= (1.3)

ɝɞɟ

aa

µ¡

, - ɚɛɫɨɥɸɬɧɚɹ ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɢ ɦɚɝɧɢɬɧɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɢ,

)1(

~

t¡

m

¡¡

i

a

<= - ɤɨɦɩɥɟɤɫɧɚɹ ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ. ɂɡ (1.2),

(1.3) ɫɥɟɞɭɸɬ ɜɨɥɧɨɜɵɟ ɭɪɚɜɧɟɧɢɹ :

0

~

2

=+6 EE

a

r

µ¡t

(1.4)

0

~

2

=+6 HH

a

r

µ¡t

(1.5)

ɡɞɟɫɶ

6 -Ʌɚɩɥɚɫɢɚɧ.

ɉɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɜɨɥɧɵ ɜɞɨɥɶ ɨɫɢ

z ɡɚɜɢɫɢɦɨɫɬɶ ɜɫɟɯ

ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɨɥɹ ɨɬ ɤɨɨɪɞɢɧɚɬɵ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

z

eNN

a

<

=

0

(1.6)

ɡɞɟɫɶ N - ɥɸɛɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɜɟɤɬɨɪɨɜ

E

r

ɢ

a

,H

r

- ɤɨɷɮɮɢɰɢɟɧɬ

ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ, ɪɚɜɧɵɣ :

_

a

+= ik ,

ɡɞɟɫɶ

h

/

/2=k - ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ,

h

- ɞɥɢɧɚ ɜɨɥɧɵ,

_

-ɤɨɷɮɮɢɰɢɟɧɬ

ɡɚɬɭɯɚɧɢɹ. ȼ [4] ɛɵɥɨ ɜɜɟɞɟɧɨ ɤɨɦɩɥɟɤɫɧɨɟ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ

_

ikk <=

v

,

ɫɥɟɞɨɜɚɬɟɥɶɧɨ ki

v

=

a

. ȼ ɫɪɟɞɚɯ ɛɟɡ ɩɨɬɟɪɶ ik=

a

. Ɂɧɚɱɟɧɢɹ k ɢ

_

ɨɩɪɟɞɟɥɹɸɬɫɹ ɡɚɜɢɫɢɦɨɫɬɹɦɢ [4]:

11

2

1)(1

2

22

++=++=

b

¡µ

tt¡m

¡µ

t

tgk

aa

a

aa

(1.7)

11

2

1)(1

2

22

<+=<+=

b

¡µ

tt¡m

¡µ

t_

tg

aa

a

aa

(1.8)

8

ȼ (1.7) ɢ (1.8):

m

-ɭɞɟɥɶɧɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ,

t

-ɰɢɤɥɢɱɟɫɤɚɹ ɱɚɫɬɨɬɚ ɩɨɥɹ,

aa

µ¡

, -ɚɛɫɨɥɸɬɧɵɟ ɷɥɟɤɬɪɢɱɟɫɤɢɟ ɢ ɦɚɝɧɢɬɧɵɟ ɩɪɨɧɢɰɚɟɦɨɫɬɢ,

)(

a

tg

t¡mb

= -ɬɚɧɝɟɧɫ ɭɝɥɚ ɞɢɷɥɟɤɬɪɢɱɟɫɤɢɯ ɩɨɬɟɪɶ.

ɉɪɨɢɡɜɨɞɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɨɥɹ ɩɨ ɤɨɨɪɞɢɧɚɬɟ

z ɪɚɜɧɵ:

N

z

N

a

<=

,

(1.9)

N

z

N

2

a

=

,

(1.10)

ɉɨɞɫɬɚɜɢɜ ɜ ɭɪɚɜɧɟɧɢɟ Ɇɚɤɫɜɟɥɥɚ (1.2) ɢ (1.3) ɜɵɪɚɠɟɧɢɟ (1.9) ɢ, ɢɫɩɨɥɶɡɭɹ

ɩɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɜɵɪɚɡɢɦ

yxyx

HHEE ,,, ɱɟɪɟɡ

z

E ɢ

z

H :

¸

¹

·

¨

©

§

,

+

,

+

<=

y

H

i

x

E

z

a

z

x

tµa

`a

2

1

2

1

(1.11)

¸

¹

·

¨

©

§

,

+

,

<

+

=

x

H

i

y

E

z

a

z

y

tµa

`a

2

1

2

1

(1.12)

¸

¹

·

¨

©

§

,

+

,

<

+

=

y

E

i

x

H

zz

x

¡ta

`a

~

1

2

1

2

(1.13)

¸

¹

·

¨

©

§

,

+

,

+

<=

x

E

i

y

H

zz

y

¡ta

`a

~

1

2

1

2

(1.14)

ȼ ɩɪɢɜɟɞɟɧɧɵɯ ɮɨɪɦɭɥɚɯ

a

µ¡t`

~

1

= . Ⱥɧɚɥɨɝɢɱɧɵɟ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ

ɩɨɥɭɱɢɬɶ ɢ ɜ ɞɪɭɝɢɯ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ.

1.5. ɉɨɩɟɪɟɱɧɵɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɜɨɥɧɵ

Ɉɩɪɟɞɟɥɢɦ ɤɨɷɮɮɢɰɢɟɧɬ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ, ɮɚɡɨɜɭɸ ɫɤɨɪɨɫɬɶ ɢ

ɜɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ. ɋɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɹɦ (1.11)-(1.14) ɩɪɢ

0==

zz

HE ɜɫɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɪɚɜɧɵ ɧɭɥɸ, ɟɫɥɢ

0

~

22

&+=

ac

k

µ¡ta

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɫɥɨɜɢɟɦ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɜɨɥɧ ɬɢɩɚ

ɌȿɆ, ɞɥɹ ɤɨɬɨɪɵɯ

0==

zz

HE , ɹɜɥɹɟɬɫɹ ɜɵɩɨɥɧɟɧɢɟ ɪɚɜɟɧɫɬɜɚ

0

~

22

=+

a

µ¡ta

(1.15)

Ɉɬɫɸɞɚ

a

i

µ¡ta

~

±= (1.16)

9

ɂɫɩɨɥɶɡɭɹ (1.7) ɞɥɹ ɮɚɡɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɨɥɧɵ ɢɦɟɟɦ:

b

t

2

11

2

tg

c

k

v

++

==

(1.17)

ɝɞɟ

ɫ- ɫɤɨɪɨɫɬɶ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɜ ɜɚɤɭɭɦɟ.

ȼɨɥɧɨɜɨɟ (ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ) ɫɨɩɪɨɬɢɜɥɟɧɢɟ

c

z ɪɚɜɧɨ ɨɬɧɨɲɟɧɢɸ

ɩɨɩɟɪɟɱɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɜɟɤɬɨɪɨɜ

E

r

ɢ

H

r

:

¡t

a

~

22

=

+

=

yx

c

HH

EE

z

(1.18)

ȼ ɫɪɟɞɚɯ ɛɟɡ ɩɨɬɟɪɶ (ɬ.ɟ. ɩɪɢ ɭɞɟɥɶɧɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ

0=

m

)

¡

=

~

. ȼ ɷɬɨɦ

ɫɥɭɱɚɟ:

aaaa

cvikk

µ¡a_µ¡t

/1,,0, ===== (1.19)

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɪɚɜɧɚ ɮɚɡɨɜɨɣ ɫɤɨɪɨɫɬɢ.

ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɩɨɬɟɪɶ v ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ ɞɥɹ ɜɨɥɧɵ ɬɢɩɚ ɌȿɆ. ɉɪɢ

ɷɬɨɦ ɞɥɢɧɚ ɜɨɥɧɵ ɌȿɆ ɪɚɜɧɚ

kfɫ

ɌȿɆ

/2/

/hh

=== , f ñ ɥɢɧɟɣɧɚɹ ɱɚɫɬɨɬɚ. (1.20)

ȼ ɫɪɟɞɚɯ ɛɟɡ ɩɨɬɟɪɶ (ɞɢɷɥɟɤɬɪɢɤɢ).

aabɌȿɆc

HEzz

¡µ

// === (1.21)

1.6. ɗɥɟɤɬɪɢɱɟɫɤɢɟ ɜɨɥɧɵ

()

0,0 =&

zz

HE

ȼ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɭɪɚɜɧɟɧɢɟ Ƚɟɥɶɦɝɨɥɶɰɚ ɞɥɹ

ɫɨɫɬɚɜɥɹɸɳɟɣ

z

E ɫɥɟɞɭɟɬ ɢɡ (1.4):

()

0

~

22

2

=++

,

+

,

za

zz

E

y

E

x

E

µ¡ta

(1.22)

Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɩɪɢɦɟɦ, ɱɬɨ ɫɪɟɞɚ ɧɟ ɨɛɥɚɞɚɟɬ ɩɪɨɜɨɞɢɦɨɫɬɶɸ

(

0=

m

), ɬɨɝɞɚ

a

¡¡

=

~

. ȼ ɷɬɨɦ ɫɥɭɱɚɟ:

()

2

222

/2

h/aµ¡ta

+=+=

aac

k

22

kk

c

<±=

a

(1.23)

ɂɡ (1.23) ɫɥɟɞɭɟɬ, ɱɬɨ

a

ɹɜɥɹɟɬɫɹ ɜɟɳɟɫɬɜɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɟɫɥɢ

kk

c

> , ɢ ɦɧɢɦɨɣ , ɟɫɥɢ kk

c

< . ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɩɨɥɟ ɜ ɧɚɩɪɚɜɥɟɧɢɢ z

10

ɡɚɬɭɯɚɟɬ, ɬ.ɟ. ɩɟɪɟɧɨɫ ɷɧɟɪɝɢɢ ɨɬɫɭɬɫɬɜɭɟɬ. ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɷɧɟɪɝɢɹ

ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɛɟɡ ɡɚɬɭɯɚɧɢɹ. Ʉɪɢɬɢɱɟɫɤɚɹ ɱɚɫɬɨɬɚ

ɤɪ

f ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ

ɭɫɥɨɜɢɹ

()

ɤɪc

fɫk /2

/

= . Ɉɬɫɸɞɚ ɤɪɢɬɢɱɟɫɤɚɹ ɱɚɫɬɨɬɚ:

aa

c

ɤɪ

k

f

µ¡/

2

=

(1.24ɚ)

Ʉɪɢɬɢɱɟɫɤɚɹ ɞɥɢɧɚ ɜɨɥɧɵ

c

ɤɪ

k

f

ɫ

/

h

2

== (1.24ɛ)

ȼɨɥɧɵ ɬɢɩɚ ȿ ɦɨɝɭɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɬɨɥɶɤɨ ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɣ

ɤɪɢɬɢɱɟɫɤɨɣ ɱɚɫɬɨɬɵ, ɨɩɪɟɞɟɥɹɟɦɨɣ (1.24ɚ). Ʉɨɷɮɮɢɰɢɟɧɬ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ

()

2

/1

ɤɪ

ik

hha

<= (1.25)

ȼ ɨɛɥɚɫɬɢ ɱɚɫɬɨɬ ɜɵɲɟ ɤɪɢɬɢɱɟɫɤɨɣ ɞɥɹ ɜɨɥɧɵ ɬɢɩɚ ȿ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ

()

2

/1

ɤɪE

kk

hh

<= (1.26)

Ɏɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ:

()

2

/1

ɤɪ

E

c

k

v

hh

t

<

== (1.27)

Ɂɞɟɫɶ

ɫñ ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ ɜ ɜɚɤɭɭɦɟ.

h

ñ ɞɥɢɧɚ ɜɨɥɧɵ ɌȿɆ ɜ ɫɪɟɞɟ ɛɟɡ

ɩɨɬɟɪɶ.

ɋɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ:

()

2

/1

ɤɪɷ

ɫ

v

ɫ

v

hh

<== (1.28)

ɂɡ (1.27) , (1.28) ɫɥɟɞɭɟɬ, ɱɬɨ ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢ ɫɤɨɪɨɫɬɶ

ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɡɚɜɢɫɹɬ ɨɬ ɱɚɫɬɨɬɵ ɩɨɥɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ

ɫɬɪɭɤɬɭɪɚ ɩɨɥɹ ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ. ɉɪɢ

ɤɪ

ff = ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɪɚɜɧɚ

ɛɟɫɤɨɧɟɱɧɨɫɬɢ. ɗɬɨ ɦɨɠɧɨ ɩɨɹɫɧɢɬɶ ɫ ɩɨɦɨɳɶɸ ɪɢɫɭɧɤɚ, ɧɚ ɤɨɬɨɪɨɦ

ɩɪɢɜɟɞɟɧɚ ɞɢɫɩɟɪɫɢɨɧɧɚɹ ɤɪɢɜɚɹ ɞɥɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ ɜ ɜɨɥɧɨɜɨɞɟ.

Ɏɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɚɧɝɟɧɫɨɦ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤ ɩɪɹɦɨɣ,

ɩɪɨɜɟɞɟɧɧɨɣ ɢɡ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɜ ɬɨɱɤɭ (

0,

ɤɪ

t

). ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ

'==

0

90tgtg

_

.

11

Ɋɢɫ.5

ȼ ɩɪɟɞɟɥɟ ɩɪɢ

'Af ɮɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɪɚɜɧɚ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ

aa

cv

µ¡

1

== . ɇɚ Ɋɢɫ. 5 ɷɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɢɫɩɟɪɫɢɨɧɧɵɦ ɤɪɢɜɵɦ,

ɨɛɨɡɧɚɱɟɧɧɵɦ ɩɭɧɤɬɢɪɨɦ, ɩɪɨɜɟɞɟɧɧɵɦ ɩɨɞ ɭɝɥɨɦ 45

0

[3]. ɋɤɨɪɨɫɬɶ

ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɩɪɢ

ɤɪ

ff = ɪɚɜɧɚ ɧɭɥɸ. ɗɬɨ ɜɢɞɧɨ ɢɡ Ɋɢɫ.5., ɧɚ

ɤɨɬɨɪɨɦ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɚɧɝɟɧɫɨɦ ɭɝɥɚ

ɧɚɤɥɨɧɚ ɩɪɹɦɨɣ, ɤɚɫɚɬɟɥɶɧɨɣ ɤ ɞɢɫɩɟɪɫɢɨɧɧɨɣ ɤɪɢɜɨɣ ɜ ɬɨɱɤɟ (

0,

ɤɪ

t

) . ȼ

ɷɬɨɦ ɫɥɭɱɚɟ

00

0

== tgtg

_

.

Ⱦɥɢɧɚ ɜɨɥɧɵ ɬɢɩɚ ȿ ɪɚɜɧɚ :

()

2

/1

2

ɤɪ

E

ȿ

k

hh

h

/

h

<

==

(1.29)

ȼɢɞɧɨ, ɱɬɨ

ȿ

h

ɛɨɥɶɲɟ ɞɥɢɧɵ ɜɨɥɧɵ

h

, ɫɜɨɛɨɞɧɨ ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɟɣɫɹ ɜ

ɨɞɧɨɪɨɞɧɨɣ ɢɡɨɬɪɨɩɧɨɣ ɫɪɟɞɟ ɛɟɡ ɩɨɬɟɪɶ.

Ɉɩɪɟɞɟɥɢɦ ɜɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɪɟɞɵ ɞɥɹ ɜɨɥɧɵ ɬɢɩɚ ȿ.

ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɹɦɢ (1.11)-(1.14), ɩɨɞɫɬɚɜɢɜ ɜ ɧɢɯ

0=

z

H ,

a

¡¡

=

~

. ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɧɚɯɨɞɢɦ:

y

a

x

H

i

E

t¡

a

<= (1.30)

x

a

y

H

i

E

t¡

a

= (1.31)

Ɉɬɫɸɞɚ:

a

yx

c

i

HH

EE

z

t¡

a

=

+

=

22

(1.32)

12

ɍɱɢɬɵɜɚɹ, ɱɬɨ

()

2

/1

ɤɪ

i

hh`a

<= , ɩɨɥɭɱɚɟɦ:

()

2

CE

/1z

ɤɪTEM

z

hh

+= (1.33)

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɨɛɥɚɫɬɢ ɱɚɫɬɨɬ ɜɵɲɟ ɤɪɢɬɢɱɟɫɤɨɣ ɜɨɥɧɨɜɨɟ

ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɨɥɧɵ ɬɢɩɚ E ɦɟɧɶɲɟ ɜɨɥɧɨɜɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ

aaTEM

z

¡µ

/= ɜɨɥɧɵ ɬɢɩɚ ɌȿɆ. ɉɪɢ 0==

CEɤɪ

zff .

ɉɪɢ ɢɡɦɟɧɟɧɢɢ ɱɚɫɬɨɬɵ ɨɬ

ɤɪ

f ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɜɨɥɧɨɜɨɟ

ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɭɜɟɥɢɱɢɜɚɟɬɫɹ, ɫɬɪɟɦɹɫɶ ɤ

aaTEM

z

¡µ

= . ȼ ɨɛɥɚɫɬɢ ɱɚɫɬɨɬ

ɧɢɠɟ ɤɪɢɬɢɱɟɫɤɨɣ ɜɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ

CE

z ɹɜɥɹɟɬɫɹ ɦɧɢɦɨɣ

ɜɟɥɢɱɢɧɨɣ. Ⱥɧɚɥɨɝɢɱɧɨɟ ɹɜɥɟɧɢɟ ɢɦɟɟɬ ɦɟɫɬɨ ɜɧɟ ɩɨɥɨɫɵ ɩɪɨɡɪɚɱɧɨɫɬɢ ɜ

ɩɨɥɨɫɨɜɵɯ ɮɢɥɶɬɪɚɯ.

1.7. Ɇɚɝɧɢɬɧɵɟ ɜɨɥɧɵ

()

0,0 =&

zz

EH

Ⱥɧɚɥɢɡ ɜɨɥɧɵ ɬɢɩɚ ɇ ɩɪɨɜɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɚɧɚɥɢɡɭ ɜɨɥɧ ɬɢɩɚ ȿ.

ȼɨɥɧɵ ɬɢɩɚ ɇ ɬɚɤɠɟ ɢɦɟɸɬ ɤɪɢɬɢɱɟɫɤɢɟ ɱɚɫɬɨɬɵ. ɇɚ ɱɚɫɬɨɬɚɯ

ɤɪ

ff > ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɷɧɟɪɝɢɢ ɩɪɨɢɫɯɨɞɢɬ ɛɟɡ ɡɚɬɭɯɚɧɢɹ, ɟɫɥɢ ɜ ɫɪɟɞɟ

ɨɬɫɭɬɫɬɜɭɸɬ ɩɨɬɟɪɢ. ɇɚ ɱɚɫɬɨɬɚɯ

ɤɪ

ff < ɩɨɥɟ ɜɞɨɥɶ ɨɫɢ z ɡɚɬɭɯɚɟɬ.

Ʉɪɢɬɢɱɟɫɤɚɹ ɱɚɫɬɨɬɚ ɪɚɜɧɚ:

/

2

c

ɤɪ

ɫk

f = , ɤɪɢɬɢɱɟɫɤɚɹ ɞɥɢɧɚ ɜɨɥɧɵ

ɤɪc

ɤɪ

f

ɫ

k

==

/

h

2

ȼɟɥɢɱɢɧɵ

ɷ

vv ,,

a

ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɚɦɢ:

()

2

/1

ɤɪ

i

hh`a

<=

()

2

/1

ɤɪ

c

v

hh

`

t

<

==

()

2

/1

ɤɪɷ

c

v

c

v

hh

<==

ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ

0=

z

E ɜ ɭɪɚɜɧɟɧɢɹ (1.11) - (1.14) ɩɨɥɭɱɚɟɦ:

y

a

x

H

i

E

a

t

µ

=

x

a

y

H

i

E

a

tµ

<=

(1.34)

13

ȼɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ:

a

tµ

a

yx

CH

i

HH

EE

z =

+

=

22

. Ɉɬɫɸɞɚ

()

2

CH

/1

z

ɤɪ

TEM

z

hh

<

=

(1.35)

ȼɢɞɧɨ, ɱɬɨ ɜɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɨɥɧɵ ɬɢɩɚ ɇ ɛɨɥɶɲɟ

aaTEM

zz

¡µ

/

0

= . ɉɪɢ

ɤɪ

ff = , ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɪɟɞɵ ɫɬɚɧɨɜɢɬɫɹ

ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɦ. ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ

f ɨɬ

ɤɪ

f , ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɜɨɥɧɨɜɨɟ

ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɭɦɟɧɶɲɚɟɬɫɹ, ɫɬɪɟɦɹɫɶ ɤ

TEM

z .

2. ɋɬɪɭɤɬɭɪɚ ɩɨɥɹ ɜ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɩɨɥɵɯ ɜɨɥɧɨɜɨɞɚɯ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ

ɫɟɱɟɧɢɹ

2.1. ȼɨɥɧɵ ɬɢɩɚ ȿ ɜ ɜɨɥɧɨɜɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ

ɋɯɟɦɚ ɜɨɥɧɨɜɨɞɚ ɩɪɢɜɟɞɟɧɚ ɧɚ Ɋɢɫ.6.

Ɋɢɫ.6.

Ɋɚɫɫɦɨɬɪɢɦ ɩɨɥɟ ɜɨɥɧɵ ɬɢɩɚ ȿ

()

0,0 &=

zz

EH ɜ ɜɨɥɧɨɜɨɞɟ

ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ. ɉɨɩɟɪɟɱɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɜɟɤɬɨɪɨɜ

ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ:

x

E

k

E

z

x

,

u

+

<=

22

a

(2.1)

14

y

E

k

E

z

y

,

u

+

<=

22

a

(2.2)

y

E

k

i

H

z

a

x

,

u

+

=

22

a

t¡

(2.3)

x

E

k

i

H

z

a

y

,

u

+

<=

22

a

t¡

(2.4)

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ

z

E ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ Ƚɟɥɶɦɝɨɥɶɰɚ:

0

2

=+

,

+

,

zc

zz

Ek

y

E

x

E

(2.5)

ɇɚɩɨɦɧɢɦ, ɱɬɨ

aɫ

k

µ¡ta

~

222

+= .

ɍɪɚɜɧɟɧɢɟ (2.5) ɦɨɠɧɨ ɪɟɲɢɬɶ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ:

a

z

XYeE

<

= (2.6)

ɝɞɟ X ñɮɭɧɤɰɢɹ ɬɨɥɶɤɨ ɯ, Y ñ ɮɭɧɤɰɢɹ ɬɨɥɶɤɨ y. ɉɨɞɫɬɚɜɢɜ (2.6) ɜ (2.5)

ɢɦɟɟɦ:

XYkXYYX

c

2

<=

vv

<

vv

ɢɥɢ

2

c

k

X

Y

X

<=

vv

<

vv

(2.7)

Ɉɬɫɸɞɚ:

2

x

k

X

<=

vv

,

2

y

k

Y

<=

vv

, ɝɞɟ

222

cyx

kkk =+ (2.8)

ɍɪɚɜɧɟɧɢɹ (2.8) ɢɦɟɸɬ ɪɟɲɟɧɢɹ:

(

)

(

)

() ()

YkDSinykCCosY

xkBSinxkACosX

yy

xx

+=

(2.9)

ɉɨɞɫɬɚɜɥɹɹ (2.9) ɜ (2.6) ɢɦɟɟɦ:

[]

[

]

a

z

yyxxz

eykDSinykCCosxkBSinxkACosE

<

+u+= )()()()( (2.10)

Ȼɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɫɬɟɧɤɢ ɜɨɥɧɨɜɨɞɚ ɹɜɥɹɸɬɫɹ ɢɞɟɚɥɶɧɨ

ɩɪɨɜɨɞɹɳɢɦɢ (

0=

m

). ȼ ɷɬɨɦ ɫɥɭɱɚɟ 0,0

00

==

== y

z

x

z

EE , ɱɬɨ

ɜɨɡɦɨɠɧɨ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɭɫɥɨɜɢɹ Ⱥ=ɋ=0. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɭɱɚɟɦ:

a

z

yxz

eykSinxkSinEE

<

= )()(

0

, (2.11)

ɝɞɟ

DȼE u=

0

.

ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɲɟ ɫɨɨɬɧɨɲɟɧɢɹ, ɩɨɥɭɱɚɟɦ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɢɯ

ɩɨɥɹ:

15

()

a

h

hh

z

yx

ɤɪ

xy

ɤɪ

x

eykSinxkCosE

kk

iE

<

u

<

<=

)()(

/1

0

2

(2.12)

=

y

E

()

a

h

hh

z

yx

ɤɪ

xy

ɤɪ

eykCosxkSinE

kk

i

<

u

<

)()(

/1

0

2

(2.13)

a

h

z

yx

ɤɪ

yxB

CE

y

x

eykCosxkSinE

kkz

i

z

E

H

<

+

=<= )()(

)/(1

0

2

(2.14)

a

h

z

yx

ɤɪ

xyB

CE

x

y

eykSinxkCosE

kkz

i

z

E

H

<

+

<

== )()(

)/(1

0

2

(2.15)

Ɉɩɪɟɞɟɥɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ

x

k ɢ

y

k .

ɂɡ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ

0=

=by

z

E ɫɥɟɞɭɟɬ:

abkSin

y

=)(

/

nbk

y

= , (2.16)

ɨɬɫɸɞɚ

bnk

y

/

= . (2.17)

ɂɡ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ

0=

=ax

z

E ɫɥɟɞɭɟɬ

amk

x

/

= . (2.18)

ȼ (2.17) ɢ (2.18) m ɢ n ñɥɸɛɵɟ ɰɟɥɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ.

ɂɡ (2.17) ɢ (2.18) ɩɨɥɭɱɚɟɦ:

()()

22

// bnamkkk

yxc

//

+=+= (2.19).

2.2. ȼɨɥɧɵ ɬɢɩɚ ɇ ɜ ɜɨɥɧɨɜɨɞɟ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɡ (1.11)-(1.14) ɫɥɟɞɭɸɬ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɢɯ

ɩɨɥɹ:

y

H

k

i

E

z

x

,

u

+

<=

22

a

tµ

(2.20)

x

H

k

i

E

z

y

,

u

+

=

22

a

tµ

(2.21)

16

x

H

k

H

z

x

,

u

+

<=

22

a

(2.22)

y

H

k

H

z

y

,

u

+

<=

22

a

(2.23)

ɋɨɫɬɚɜɥɹɸɳɚɹ

z

H ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɪɚɜɧɟɧɢɹ Ƚɟɥɶɦɝɨɥɶɰɚ:

0

2

=+

,

+

,

zc

zz

Hk

y

H

x

H

(2.24)

ɂɫɩɨɥɶɡɭɹ ɦɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ (ɬɚɤ ɠɟ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ

ɜɵɲɟ) ɩɨɥɭɱɚɟɦ:

[]

[

]

a

z

yyxxz

eykSinDykCosCxkSinBxkCosAH

<

+u+= )()()()(

1111

(2.25)

ɂɡ (2.20) ɫɥɟɞɭɟɬ: ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ

0

=

=y

x

E

ɬɪɟɛɭɟɬɫɹ, ɱɬɨɛɵ

0

=

,

=y

z

y

H

.

ɂɡ ɷɬɨɝɨ ɫɥɟɞɭɟɬ ɬɪɟɛɨɜɚɧɢɟ 0

1

=D .

ɂɡ ɭɪɚɜɧɟɧɢɹ (2.21) ɫɥɟɞɭɟɬ: ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɹ

0

=

=x

y

E ɬɪɟɛɭɟɬɫɹ,

ɱɬɨɛɵ

0

=

,

=x

z

x

H

. Ɉɬɫɸɞɚ ɜɨɡɧɢɤɚɟɬ ɬɪɟɛɨɜɚɧɢɟ 0

1

=B . Ɍɚɤɢɦ

ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɭɪɚɜɧɟɧɢɸ:

a

z

yxz

eykCosxkCosHH

<

= )()(

0

,

110

CAH = (2.26)

ɉɨɞɫɬɚɜɥɹɹ (2.26) ɜ ɭɪɚɜɧɟɧɢɹ (2.20)-(2.23) ɩɨɥɭɱɚɟɦ:

()

a

h

z

yx

ɤɪ

yx

B

x

eykSinxkCosH

kk

z

iE

<

u

+

= )()(

/1

0

2

(2.27)

()

a

h

z

yx

ɤɪ

yx

B

y

eykCosxkSinH

kk

z

iE

<

u

+

<= )()(

/1

0

2

(2.28)

()

a

h

hh

z

yx

ɤɪ

xy

ɤɪ

CH

y

x

eykCosxkSinH

kk

i

z

E

H

<

+

<

<=<= )()(

)/(1

/1

0

2

(2.29)

17

()

a

h

hh

z

yx

ɤɪ

xy

ɤɪ

CH

x

y

eykSinxkCosH

kk

i

z

E

H

<

+

<

== )()(

)/(1

/1

0

2

(2.30)

Ɂɧɚɱɟɧɢɹ

x

k ɢ

y

k ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɚɦɢ (2.17) ɢ (2.18). ȼ ɨɬɥɢɱɢɟ ɨɬ

ɜɨɥɧ ɬɢɩɚ ȿ ɨɞɧɨ ɢɡ ɱɢɫɟɥ m ɢɥɢ n ɜ ɜɨɥɧɚɯ ɬɢɩɚ ɇ ɦɨɠɟɬ ɪɚɜɧɹɬɶɫɹ ɧɭɥɸ.

ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɵ (2.12)-(2.15) ɢ (2.26)-(2.30) ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɤɚɪɬɢɧɭ

ɫɢɥɨɜɵɯ ɥɢɧɢɣ

ȿ

v

ɢ

ɇ

r

ɞɥɹ ɥɸɛɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ. Ⱦɥɹ

ɩɪɢɦɟɪɚ ɧɚ Ɋɢɫ.7 ɩɪɢɜɟɞɟɧɚ ɫɬɪɭɤɬɭɪɚ ɜɨɥɧ ɬɢɩɚ ȿ ɢ ɇ ɜ ɜɨɥɧɨɜɨɞɟ

ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɫɟɱɟɧɢɹ.

ɉɨɞɫɬɚɜɥɹɹ ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɤɪɢɬɢɱɟɫɤɨɣ ɱɚɫɬɨɬɵ

¡µ/

2

c

ɤɪ

k

f =

ɡɧɚɱɟɧɢɹ

c

k , ɩɨɥɭɱɢɦ ɡɧɚɱɟɧɢɹ ɤɪɢɬɢɱɟɫɤɨɣ ɱɚɫɬɨɬɵ ɢ ɤɪɢɬɢɱɟɫɤɨɣ ɞɥɢɧɵ

ɜɨɥɧɵ ɜ ɩɪɹɦɨɭɝɨɥɶɧɨɦ ɜɨɥɧɨɜɨɞɟ

22

2

1

¸

¹

·

¨

©

§

+

¸

¹

·

¨

©

§

=

b

n

a

m

f

aa

ɤɪ

µ¡

(2.31)

() ()

22

2

namb

ab

ɤɪ

+

=

h

(2.32)

Ɇɚɤɫɢɦɚɥɶɧɭɸ ɞɥɢɧɭ ɦɨɠɟɬ ɢɦɟɬɶ ɜɨɥɧɚ ɬɢɩɚ ɇ ɩɪɢ m=0 (ɜɨɥɧɚ

10

ɇ ) ɢɥɢ

ɩɪɢ n=0 (ɜɨɥɧɚ ɬɢɩɚ

01

ɇ ). ɍ ɜɨɥɧɵ ɚɇ

ɤɪ

2

10

=

h

, ɭ ɜɨɥɧɵ

ɜɇ

ɤɪ

2

01

=

h

. ȼɫɸɞɭ ɜ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɨɥɚɝɚɬɶ, ɱɬɨ ɜɚ * .

18

Ɋɢɫ.7.

ȼ ɬɟɯɧɢɤɟ ɋȼɑ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɨɥɧɚ ɬɢɩɚ

10

ɇ . ɉɨɞɫɬɚɜɥɹɹ ɜ

ɭɪɚɜɧɟɧɢɹ (2.26)-(2.30) ɡɧɚɱɟɧɢɹ

2

,0,1,0

/

====

xy

kkmn

ɉɨɥɭɱɢɦ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɜɟɤɬɨɪɨɜ ɩɨɥɹ ɜɨɥɧɵ

10

ɇ :

10

0

a

/

z

e

a

x

CosHɇ

<

¸

¹

·

¨

©

§

=

(2.33)

10

0

2

a

/

h

z

By

e

a

x

SinzH

a

iE

<

¸

¹

·

¨

©

§

u<=

(2.34)

()

10

0

2

2/1

a

/

h

z

x

e

a

x

SinH

a

aiH

<

¸

¹

·

¨

©

§

u<=

(2.35)

0==

yx

HE (2.36)

()

2

1010

)2/(1

2

/1 aiikik

ɤɪ

h

/

hha

<=<== (2.37)

Ɂɞɟɫɶ

1010

,k

a

- ɤɨɷɮɮɢɰɢɟɧɬ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɢ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ ɜɨɥɧɵ ɬɢɩɚ

10

ɇ .

Ɏɚɡɨɜɚɹ ɫɤɨɪɨɫɬɶ ɬɚɤɨɣ ɜɨɥɧɵ:

19

2

10

)2/(1 a

c

v

h

<

=

(2.38)

Ⱦɥɢɧɚ ɜɨɥɧɵ ɜ ɜɨɥɧɨɜɨɞɟ:

2

10

)2/(1

1

a

c

v

ȼ

h

<

==

(2.39)

ȼɨɥɧɨɜɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ:

2

)2/(1

10

a

z

B

CH

h

<

=

(2.40)

ɋɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ:

2

10

)2/(1 aɫv

ɗ

h

<= (2.41)

ȼɫɸɞɭ ɜɵɲɟ

ɫɌȿɆB

zz ==

¡µ

/

3. ɋɨɛɫɬɜɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɨɛɴɟɦɧɵɯ ɪɟɡɨɧɚɬɨɪɨɜ

Ɉɛɴɟɦɧɵɦɢ ɪɟɡɨɧɚɬɨɪɚɦɢ ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɨɝɪɚɧɢɱɟɧɧɚɹ

ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɫɬɟɧɤɨɣ. ɇɚ ɜɵɫɨɤɢɯ ɱɚɫɬɨɬɚɯ ɨɧɢ ɢɦɟɸɬ ɫɜɨɣɫɬɜɚ

ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɤɨɧɬɭɪɚ ɫ ɜɵɫɨɤɨɣ ɞɨɛɪɨɬɧɨɫɬɶɸ. Ɋɚɫɫɦɨɬɪɢɦ ɫɨɛɫɬɜɟɧɧɵɟ

(ɬ.ɟ. ɫɜɨɛɨɞɧɵɟ) ɤɨɥɟɛɚɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɮɨɪɦɵ.

ɉɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɫɬɟɧɤɢ ɪɟɡɨɧɚɬɨɪɚ ɢɦɟɸɬ ɢɞɟɚɥɶɧɭɸ ɩɪɨɜɨɞɢɦɨɫɬɶ (ɬ.ɟ.

0=

m

), ɚ ɡɧɚɱɟɧɢɹ 1==

µ

¡

. ɋɯɟɦɚ ɪɟɡɨɧɚɬɨɪɚ ɩɪɢɜɟɞɟɧɚ ɧɚ Ɋɢɫ.8.

Ɋɢɫ.8.

ȼ ɪɟɡɨɧɚɬɨɪɟ ɜɵɩɨɥɧɟɧɵ ɭɫɥɨɜɢɹ:

lzbyaɯ )))))) 00,0 (3.1)

ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɧɚ ɫɬɟɧɤɚɯ ɩɨɥɨɫɬɢ ɞɨɥɠɧɨ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ

ɭɫɥɨɜɢɹɦ:

axɢxɩɪɢEE

zy

==== 00

byɢyɩɪɢEE

xz

==== 00 (3.2)

20

lzɢzɩɪɢEE

yx

==== 00

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɩɨɬɟɪɶ ɜ ɩɨɥɨɫɬɢ (ɪɟɡɨɧɚɬɨɪɟ) ɧɟɬ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ

ɤɚɠɞɨɟ ɢɡ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɛɭɞɟɬ ɧɟɡɚɬɭɯɚɸɳɢɦ. ɉɭɫɬɶ

t

- ɤɪɭɝɨɜɚɹ

ɱɚɫɬɨɬɚ ɨɞɧɨɝɨ ɢɡ ɬɚɤɢɯ ɤɨɥɟɛɚɧɢɣ,

k ñ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ ɜ ɩɭɫɬɨɬɟ. ɑɬɨɛɵ

ɧɚɣɬɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɟ ɩɨɥɟ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɜɨɫɩɨɥɶɡɭɟɦɫɹ

ɜɨɥɧɨɜɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ.

0,0

2

==+6 EdivEkE

rrr

(3.3)

Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (3.3) ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ

ɩɟɪɟɦɟɧɧɵɯ. ɉɨɥɭɱɢɦ:

l

zp

Sin

b

yn

Sin

a

xm

ACosE

x

/

=

l

zp

Sin

b

yn

Cos

a

xm

BSinE

y

/

= (3.4)

l

zp

Cos

b

yn

Sin

a

xm

CSinE

z

/

=

Ɂɞɟɫɶ Ⱥ, ȼ, ɋ ñ ɤɨɦɩɥɟɤɫɧɵɟ ɩɨɫɬɨɹɧɧɵɟ m, n, p ñɰɟɥɵɟ ɱɢɫɥɚ 0, 1, 2, Ö

ȼ ɭɪɚɜɧɟɧɢɹɯ (3.4) ɫɢɧɭɫɵ ɨɛɟɫɩɟɱɢɜɚɸɬ ɜɵɩɨɥɧɟɧɢɟ ɝɪɚɧɢɱɧɵɯ

ɭɫɥɨɜɢɣ (3.2), ɚ ɤɨɫɢɧɭɫɵ ɨɛɟɫɩɟɱɢɜɚɸɬ ɩɭɬɟɦ ɩɨɞɛɨɪɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ

Ⱥ, ȼ, ɋ ɜɵɩɨɥɧɟɧɢɟ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ (3.3). ɉɨɞɫɬɚɜɥɹɹ (3.4) ɜ ɜɨɥɧɨɜɨɟ

ɭɪɚɜɧɟɧɢɟ (3.3), ɧɚɯɨɞɢɦ ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ:

222

¸

¹

·

¨

©

§

+

¸

¹

·

¨

©

§

+

¸

¹

·

¨

©

§

=

l

p

b

n

a

m

k

/

(3.5)

Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɩɨɥɭɱɚɟɦ ɢɡ ɭɪɚɜɧɟɧɢɹ

Ɇɚɤɫɜɟɥɥɚ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɩɨɥɹ.

Erot

ik

H

rr

1

= (3.6)

ɂɡ (3.6) ɫɥɟɞɭɟɬ:

l

zp

Cos

b

yn

Cos

a

xm

SinB

l

p

C

b

n

ik

H

x

////

¸

¹

·

¨

©

§

<=

l

zp

Cos

b

yn

Sin

a

xm

CosC

a

m

B

l

p

ik

H

y

////

¸

¹

·

¨

©

§

<=

(3.7)

l

zp

Sin

b

yn

Cos

a

xm

CosA

b

n

B

a

m

ik

H

z

////

¸

¹

·

¨

©

§

<=

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