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Файл:Расчет переходных процессов в электроэнергетических системах методом интеграла Дюамеля. Учебное пособие
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Ɂɚɜɢɫɢɦɨɫɬɶ iɋ ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
1
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɬɨɤ ɜ ɟɦɤɨɫɬɢ ɰɟɩɢ, ɚ
ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫɧɵɣ ɬɨɤ
௧
= ݄
ᇱ
ሼݐሽ
1
ି
݁
ሼݐሽ
݇
= െ
ି
= ݁
௧
ோమ
ήܪሺݐሻ+ Ɂሺݐ
ோమ
ήܪሺݐሻ൨
ᇱ
=
ሻ
.
(1.65)
ܴଶܥ
Ⱦɚɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ
«1/ɫɟɤ».
Ɂɚɜɢɫɢɦɨɫɬɶ i2 ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɬɨɤ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ R2
ɰɟɩɢ, ɚ ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫ ɧɚɩɪɹɠɟɧɢɹ
ܪሺݐሻ൨
ᇱ
=
(1.66)
ି
ோ
ήܪሺݐ
௧
ʬ
൰
ሻ
.
= ݄
௨
ᇱ
ሼݐሽ
=
ሼݐሽ
݇
௨
=
ሺ
ܴଵ+ ܴ
ܴଵ+ ܴ
1
ሻ
ܴʬܥ
ଶ
1
1 െ݁
൬
ଶ
௧
ି
ோʬ
݁
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «ɋɦ/ɫɟɤ».
Ɂɚɜɢɫɢɦɨɫɬɶ i2 ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
1
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɬɨɤ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ R2
ɰɟɩɢ, ɚ ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫɧɵɣ ɬɨɤ
31
௧
௧
ோమ
ି
ோ
ήܪሺݐ
మ
ሻ
ήܪሺݐሻ൨
൰
.
= ݄
ᇱ
ሼݐሽ
=
= ൬1 െ݁
1
ି
݁
ܴ
ܥ
ଶ
ሼݐሽ
݇
ᇱ
=
(1.67)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «1/ɫɟɤ».
Ɂɚɜɢɫɢɦɨɫɬɶ uC ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ u
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɟɦɤɨɫɬɢ
ɰɟɩɢ, ɚ ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫ ɧɚɩɪɹɠɟɧɢɹ
ܪሺݐሻ൨
ᇱ
=
(1.68)
ܴ
ሼݐሽ
݇
௨௨
= ݄
௨௨
ᇱ
ሼݐሽ
=
=
ܴଵ+ ܴ
1
݁
ି
ଶ
௧
ோʬ
൬
ଶ
ήܪሺݐ
1 െ݁
௧
ି
ோʬ
൰
ሻ
.
ܴଵܥ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «1/ɫɟɤ».
Ɂɚɜɢɫɢɦɨɫɬɶ uC ɨɬ ɞɟɣɫɬɜɭɸɳɟɝɨ ɧɚ ɜɯɨɞɟ i
1
– ɤɨɝɞɚ ɩɨɞ ɪɟɚɤɰɢɟɣ ɩɨɧɢɦɚɟɬɫɹ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɟɦɤɨɫɬɢ
ɰɟɩɢ, ɚ ɧɚ ɜɯɨɞɟ ɞɟɣɫɬɜɭɟɬ ɟɞɢɧɢɱɧɵɣ ɢɦɩɭɥɶɫɧɵɣ ɬɨɤ
ܪሺݐሻ൨
ᇱ
=
(1.69)
௧
= ݄
௨
ᇱ
ሼݐሽ
=
= ܴ
1
݁
൬
ଶ
௧
ି
ோమ
ሼݐሽ
݇
௨
1 െ݁
ήܪሺݐ
ି
ோమ
൰
ሻ
.
ܥ
32
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
݂
݂
ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ «Ɉɦ/ɫɟɤ».
ȼɨɡɦɨɠɧɵɟ ɜɢɞɵ ɢɦɩɭɥɶɫɧɵɯ ɩɟɪɟɯɨɞɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɞɥɹ ɡɚɜɢɫɢɦɨɫɬɟɣ ɜɟɥɢɱɢɧ ɬɨɤɚ ɢ ɧɚɩɪɹɠɟɧɢɹ ɢ ɢɯ ɪɚɡɦɟɪɧɨɫɬɢ ɫɜɟɞɟɧɵ ɜ ɬɚɛɥ. 1.2.
Ɍɚɛɥɢɰɚ 1.2
ɂɦɩɭɥɶɫɧɚɹ ɩɟɪɟɯɨɞɧɚɹ
Ɍɢɩ
ɜɨɡɞɟɣɫɬɜɢɹ
u i kui{t} ɋɦ/ɫɟɤ
u u kuu{t} 1/ɫɟɤ
i i kii{t} 1/ɫɟɤ
i u kiu{t} Ɉɦ/ɫɟɤ
Ɍɢɩ
ɪɟɚɤɰɢɢ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ
Ɉɛɨɡɧɚɱɟɧɢɟ Ɋɚɡɦɟɪɧɨɫɬɶ
1.4. Ɏɨɪɦɵ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɩɪɢɧɰɢɩɚ ɧɚɥɨɠɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɟɬ, ɱɬɨ
ɪɟɚɤɰɢɟɣ ɧɚ ɩɨɥɧɨɟ ɜɨɡɞɟɣɫɬɜɢɟ ɹɜɥɹɟɬɫɹ ɫɭɦɦɚ ɪɟɚɤɰɢɣ ɧɚ
ɨɬɞɟɥɶɧɵɟ ɟɝɨ ɫɨɫɬɚɜɥɹɸɳɢɟ. Ɍɨɝɞɚ ɩɪɟɞɫɬɚɜɢɦ ɜɨɡɞɟɣɫɬɜɢɟ f (t) ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɷɥɟɦɟɧɬɚɪɧɵɯ ɞɟɥɶɬɚɨɛɪɚɡɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ, ɜɨɡɧɢɤɚɸɳɢɯ ɜɨ ɜɫɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɜ ɢɧɬɟɪɜɚɥɟ 0…t (ɪɢɫ. 1.14).
Ⱦɥɹ ɢɦɩɭɥɶɫɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɜɨɡɞɟɣɫɬɜɢɹ ¨f {t} ɜ
ɬɨɱɤɟ t = IJ ɫ ɩɥɨɳɚɞɶɸ ɢɦɩɭɥɶɫɚ f (IJ)dIJ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ
ሼݐሽ
ο
=
ሺɒሻ
݀ɒɁሺݐെɒ
33
ሻ
.
(1.70)
݂
݂
݂
Ɋɢɫ. 1.14
ɋɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɟɣ ɫɨɫɬɚɜɥɹɸɳɚɹ ɪɟɚɤɰɢɢ ɤ ɦɨɦɟɧɬɭ t
ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
ሽ
οݔሼݐ
=
ሺɒሻ
݀ɒ݇ሼݐെɒ
ሽ
,
(1.71)
ɝɞɟ ɚɪɝɭɦɟɧɬ t – IJ y ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ
ɩɨɤɚɡɵɜɚɟɬ ɜɪɟɦɹ ɞɟɣɫɬɜɢɹ ɧɚ ɜɯɨɞɟ ɢɦɩɭɥɶɫɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɞɨ ɦɨɦɟɧɬɚ ɧɚɛɥɸɞɟɧɢɹ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɚɤɰɢɹ x{t} ɧɚ ɩɨɥɧɨɟ ɜɨɡɞɟɣɫɬɜɢɟ, ɭɱɢɬɵɜɚɸɳɚɹ ɤ ɦɨɦɟɧɬɭ ɧɚɛɥɸɞɟɧɢɹ ɜɫɟ ɢɦɩɭɥɶɫɧɵɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɜɨɡɞɟɣɫɬɜɢɹ ɜ ɢɧɬɟɪɜɚɥɟ 0…t, ɜɵɪɚɠɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ
௧
௧
ݔሺݐሻ= න
ሺɒሻ
݀ɒ݇ሼݐെɒ
ሽ
= න
ሺ
ݐെɒሻ ݇ሼɒሽ݀ɒ
(1.72)
.
Ɋɚɫɤɪɵɜɚɹ ɡɧɚɱɟɧɢɟ ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ k{t}, ɩɨɥɭɱɚɟɦ
34
݂
௧
݂
݂
݂
ሺ
ݐെɒሻ ݇ሺɒሻ ܪሺɒሻ݀ɒ+
௧
ష
ሺ
ݐെɒሻ ݄
ሺ0ሻ Ɂሺɒሻ
݀ɒ
(1.73)
.
ݔሺݐሻ= න
+ න
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɨ ɜɬɨɪɨɦ ɢɧɬɟɝɪɚɥɟ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ
ɧɢɠɧɢɣ ɩɪɟɞɟɥ ɫɦɟɳɟɧ ɤ ɬɨɱɤɟ 0
, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɭɱɟɫɬɶ ɡɧɚ-
–
ɱɟɧɢɟ ɢɦɩɭɥɶɫɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɜ ɬɨɱɤɟ t = 0. ȼ ɫɥɭɱɚɟ ɠɟ
ɫ ɧɭɥɟɜɵɦ ɧɢɠɧɢɦ ɩɪɟɞɟɥɨɦ ɢɧɬɟɝɪɚɥ ɧɟ ɢɦɟɟɬ ɫɦɵɫɥɚ.
ȼ ɫɢɥɭ ɮɢɥɶɬɪɭɸɳɟɝɨ ɞɟɣɫɬɜɢɹ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɜɬɨɪɨɣ
ɢɧɬɟɝɪɚɥ ɫɨɨɬɧɨɲɟɧɢɹ (1.73) ɭɩɪɨɳɚɟɬɫɹ ɞɨ h(0) f
(t), ɚ ɜ
ɩɟɪɜɨɦ ɢɧɬɟɝɪɚɥɟ ɮɭɧɤɰɢɹ H(IJ) = 1, ɬ. ɤ. ɜ ɩɪɟɞɟɥɚɯ ɜɫɟɝɨ
ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ IJ > 0. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɟɦ
ɧɚɩɢɫɚɬɶ
௧
ݔሺݐሻ= ݄
ሺ0ሻ݂ሺݐሻ
+ න
ሺ
ݐെɒሻ ݇ሺɒሻ݀ɒ
, (1.74)
ɢɥɢ
௧
ݔሺݐሻ= ݄
ሺ0ሻ݂ሺݐሻ
+ න
ሺɒሻ ݇ሺ
ݐെɒሻ݀ɒ
.
(1.75)
ɉɨɥɭɱɟɧɧɵɟ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɜɵɪɚɠɟɧɢɹ (1.74) ɢ (1.75)
ɧɚɡɵɜɚɸɬɫɹ ɬɪɟɬɶɟɣ ɢ ɱɟɬɜɟɪɬɨɣ ɮɨɪɦɚɦɢ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ
Ⱦɸɚɦɟɥɹ. ɋ ɩɨɦɨɳɶɸ ɧɢɯ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɪɟɚɤɰɢɹ ɥɢɧɟɣɧɨɣ ɰɟɩɢ ɧɚ ɩɪɨɢɡɜɨɥɶɧɨɟ ɜɨɡɞɟɣɫɬɜɢɟ f {t} ɩɨ ɡɚɞɚɧɧɨɣ
35
ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ. ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨ-
݂
݂
ɥɚɝɚɟɬɫɹ, ɱɬɨ ɮɭɧɤɰɢɹ f (t), ɜɯɨɞɹɳɚɹ ɜ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ,
ɜɵɪɚɠɚɟɬ ɡɧɚɱɟɧɢɹ ɜɨɡɞɟɣɫɬɜɢɹ ɜ ɦɨɦɟɧɬ ɧɚɛɥɸɞɟɧɢɹ t.
Ɉɬɦɟɬɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɤɭɫɨɱɧɨ-ɝɥɚɞɤɨɟ ɜɨɡɞɟɣɫɬɜɢɟ f{IJ} ɧɚ ɢɧɬɟɪɜɚɥɟ 0…t ɩɪɟɬɟɪɩɟɜɚɟɬ ɪɚɡɪɵɜɵ ɥɢɛɨ
ɧɚɪɭɲɟɧɢɹ ɝɥɚɞɤɨɫɬɢ ɜ ɬɨɱɤɚɯ t
(Ȟ = 1, 2, … q), ɬɨ ɢɧɬɟɝɪɚɥɵ
Ȟ
(1.74) ɢ (1.75) ɪɚɡɛɢɜɚɸɬɫɹ ɧɚ q + 1 ɢɧɬɟɝɪɚɥɨɜ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɩɪɟɞɟɥɚɦɢ
௧
௧
௧
భ
మ
௧
௧
න=
න+
න+
௧
భ
… න+
௧
షభ
න.
௧
ɉɪɢɦɟɧɢɜ ɜ ɮɨɪɦɭɥɟ (1.75) ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ,
ɩɨɥɭɱɢɦ
௧
ሺɒሻ
ᇱ
ሺ
݄
ݐെɒሻ݀ɒ
ሺ0ሻ ݄ሺݐሻ
+ ݂
ݐെɒሻ݀ɒ
ሻ|
=
௧
+
(1.77)
+
.
= ݄
ሺ0ሻ݂ሺݐሻ
= ݄ሺݐ
ሺ0ሻ݂ሺݐሻ
ሺ0ሻ݂ሺݐሻ
௧
+ න݂
െ݂ሺݐሻ ݄
௧
+ න݂
ሻ݂ሺ0ሻ
+ න
െ݂ሺɒሻ ݄ሺݐെɒ
ᇱ
ሺɒሻ ݄ሺ
ݐെɒሻ݀ɒ=
ሺ0ሻ
ᇱ
ሺɒሻ ݄ሺ
ݐെɒሻ݀ɒ=
௧
ᇱ
+ න
ሺɒሻ ݄ሺ
ݔሺݐሻ= ݄
= ݄
(1.76)
36
ȼ ɩɪɢɜɟɞɟɧɧɨɦ ɜɵɲɟ ɜɵɪɚɠɟɧɢɢ ɩɪɨɢɡɜɨɞɧɚɹ f '(IJ) ɜɨ
݂
݂
ɜɫɟɯ ɬɨɱɤɚɯ ɪɚɡɪɵɜɚ t
ɫɨɞɟɪɠɚɬɶ ɢɦɩɭɥɶɫɧɭɸ ɫɨɫɬɚɜɥɹɸɳɭɸ ¨f(t
(Ȟ = 1, 2, … q) ɧɚ ɢɧɬɟɪɜɚɥɟ 0…t ɛɭɞɟɬ
Ȟ
į(IJ – tȞ). ȼ ɷɬɨɦ
Ȟ)
ɫɥɭɱɚɟ ɢɡ ɢɧɬɟɝɪɚɥɚ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɢɡɜɥɟɤɭɬɫɹ ɫɥɚɝɚɟɦɵɟ ɜɢɞɚ
ሺ
ሻ ݄ሺ
ݐ
ο
ݐെݐ
ሻ
.
(1.78)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɤ ɧɢɦ ɨɬɧɨɫɢɬɫɹ ɢ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ (1.77)
ɩɪɢ t = 0.
ɉɨɤɚɠɟɦ ɷɬɨ, ɜɵɩɨɥɧɢɜ ɞɚɥɶɧɟɣɲɟɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
(1.77). Ⱦɥɹ ɷɬɨɝɨ ɧɚɣɞɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɚɯ
ɪɚɡɪɵɜɚ tȞ (Ȟ = 1, 2, … q), ɝɞɟ ɬɚ ɜɵɪɚɠɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟɦ
ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ f (t) ɧɚ ɟɞɢɧɢɱɧɭɸ H(t)
௧
ݔሺݐሻ= ݄ሺݐ
= ݄ሺݐሻ݂
ሻ݂ሺ0ሻ
ሺ0ሻ
௧
+ න
௧
+ න݂
ሾ݂ሺɒሻܪሺɒሻሿ
ᇱ
ሺɒሻܪሺɒሻ ݄ሺ
ᇱ
݄ሺݐെɒሻ ݀ɒ
ݐെɒሻ ݀ɒ
+
=
(1.79)
+ න
ሺɒሻߜሺɒሻ ݄ሺ
ݐെݐ
ሻ
݀ɒ
.
ȼ ɩɪɟɞɟɥɚɯ ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ tȞ > 0 ɜ ɫɨɨɬɧɨɲɟɧɢɢ (1.79) ɮɭɧɤɰɢɹ H(tȞ) = 1, ɚ ɩɨɫɥɟɞɧɢɣ ɢɧɬɟɝɪɚɥ ɜ ɫɢɥɭ
ɮɢɥɶɬɪɭɸɳɟɝɨ ɫɜɨɣɫɬɜɚ ɭɩɪɨɳɚɟɬɫɹ. Ɍɨɝɞɚ ɪɟɚɤɰɢɹ x(t)
ɩɪɢɧɢɦɚɟɬ ɜɢɞ
37
݂
ሺ0ሻ
݂
݂
݂
݂
݂
݂
+ න
ሻ
௧
ᇱ
ሺɒሻ ݄ሺ
ݔሺݐሻ= ݄ሺݐ
ሺ
+
ݐെɒሻ ݀ɒ
ݐ
ሻ ݄ሺ
ݐെݐ
.
ሻ
+
(1.80)
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ ɮɨɪɦɵ
ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ
ݔሺݐሻ= ο
ఔୀ
ሺ
ݐ
ఔ
ሻ
݄ሺݐെݐ
ఔ
ሻ
+ න
௧
ᇱ
ሺɒሻ ݄ሺ
ݐെɒሻ݀ɒ
,
ɢɥɢ
(1.81)
ݔሺݐሻ= ο
ఔୀ
ሺ
ݐ
ఔ
ሻ
݄ሺݐെݐ
ఔ
ሻ
+ න
௧
ᇱ
ሺ
ݐെɒሻ ݄ሺɒሻ݀ɒ
(1.82)
.
ɋ ɩɨɦɨɳɶɸ ɧɢɯ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɪɟɚɤɰɢɹ ɥɢɧɟɣɧɨɣ ɰɟɩɢ
ɧɚ ɩɪɨɢɡɜɨɥɶɧɨɟ ɜɨɡɞɟɣɫɬɜɢɟ f {t} ɩɨ ɡɚɞɚɧɧɨɣ ɫɬɭɩɟɧɱɚɬɨɣ
ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ.
ɋɥɚɝɚɟɦɵɟ ɫɭɦɦ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɜɵɪɚɠɟɧɢɣ (1.81) ɢ (1.82)
ɭɱɢɬɵɜɚɸɬ ɪɚɡɪɵɜɵ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ, ɩɪɟɞɲɟɫɬɜɭɸɳɢɯ ɦɨɦɟɧɬɭ ɧɚɛɥɸɞɟɧɢɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ
ɩɪɨɰɟɫɫ ɜ ɢɧɬɟɪɜɚɥɟ t1…t2 (ɪɢɫ. 1.15), ɬɨ ɫɭɦɦɚ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɫɥɚɝɚɟɦɨɟ ɬɨɥɶɤɨ ɜ ɬɨɱɤɟ t = 0, ɟcɥɢ ɜ ɢɧɬɟɪɜɚɥɟ t2…t3,
ɬɨ ɫɥɚɝɚɟɦɵɟ ɜ ɬɨɱɤɚɯ t = 0 ɢ t = t2. ȼ ɬɨɱɤɟ t1 ɧɚɪɭɲɚɟɬɫɹ
ɝɥɚɞɤɨɫɬɶ ɮɭɧɤɰɢɢ, ɧɨ ɪɚɡɪɵɜ ɧɟ ɢɦɟɟɬ ɦɟɫɬɚ, ɬɚɤ ɱɬɨ ɩɪɨɢɡɜɨɞɧɚɹ ɧɟ ɩɪɢɨɛɪɟɬɚɟɬ ɧɟɨɝɪɚɧɢɱɟɧɧɵɯ ɡɧɚɱɟɧɢɣ.
38
݂
݂
Ɋɢɫ. 1.15
ɂɧɬɟɝɪɚɥɵ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɜɵɪɚɠɟɧɢɣ (1.81) ɢ (1.82) ɪɚɡɛɢɜɚɸɬɫɹ ɧɚ ɫɭɦɦɵ ɢɧɬɟɝɪɚɥɨɜ, ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɞɧɨɦɭ ɢɡ ɢɧɬɟɪɜɚɥɨɜ ɝɥɚɞɤɨɫɬɢ ɜɨɡɞɟɣɫɬɜɢɹ.
ɉɪɢ ɷɬɨɦ ɩɪɨɢɡɜɨɞɧɚɹ, ɫɬɨɹɳɚɹ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ, ɧɟ
ɫɨɞɟɪɠɢɬ ɢɦɩɭɥɶɫɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ.
ɉɹɬɚɹ ɢ ɲɟɫɬɚɹ ɮɨɪɦɵ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ ɩɨɥɭɱɚɸɬɫɹ ɩɪɢ ɩɨɦɨɳɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɨɩɪɟɞɟɥɟɧɧɨɝɨ
ɢɧɬɟɝɪɚɥɚ ɩɨ ɩɚɪɚɦɟɬɪɭ
௧
ݔሺݐ
ሻ
=
ሺ
න
ݐെɒሻ ݄ሺɒሻ݀ɒ
;
݀ݐ
௧
݀
ݔሺݐ
݀
ሻ
=
න
ሺɒሻ ݄ሺ
ݐെɒሻ݀ɒ
.
(1.83)
(1.84)
݀ݐ
ɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɫɨɤɪɚɳɟɧɧɭɸ ɡɚɩɢɫɶ ɩɟɪɜɨɣ ɢɥɢ ɜɬɨɪɨɣ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɬɪɟɬɶɟɣ ɢɥɢ ɱɟɬɜɟɪɬɨɣ ɮɨɪɦ.
39
ȼɵɛɨɪ ɬɨɣ ɢɥɢ ɢɧɨɣ ɮɨɪɦɵ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ
ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɫɯɨɞɹ ɢɡ ɭɞɨɛɫɬɜɚ ɢ ɩɪɨɫɬɨɬɵ ɜɵɩɨɥɧɟɧɢɹ
ɜɵɱɢɫɥɟɧɢɣ. ɉɪɢ ɷɬɨɦ ɩɪɟɞɩɨɱɬɟɧɢɟ ɫɥɟɞɭɟɬ ɨɬɞɚɜɚɬɶ ɬɨɣ
ɮɨɪɦɟ, ɞɥɹ ɤɨɬɨɪɨɣ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɨɤɚɠɟɬɫɹ
ɩɪɨɳɟ. ɉɨɫɥɟɞɧɟɟ ɡɚɜɢɫɢɬ ɨɬ ɭɫɥɨɜɢɣ ɤɨɧɤɪɟɬɧɨɣ ɡɚɞɚɱɢ, ɚ
ɢɦɟɧɧɨ ɨɬ ɜɢɞɚ ɮɭɧɤɰɢɢ ɜɨɡɞɟɣɫɬɜɢɹ ɢ ɜɢɞɚ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ.
1.5. Ⱥɥɝɨɪɢɬɦ ɪɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɵɯ ɩɪɨɰɟɫɫɨɜ
ɦɟɬɨɞɨɦ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ
Ⱥɥɝɨɪɢɬɦ ɪɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɵɯ ɩɪɨɰɟɫɫɨɜ ɦɟɬɨɞɨɦ ɢɧɬɟ-
ɝɪɚɥɚ Ⱦɸɚɦɟɥɹ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɫɥɟɞɭɸɳɢɦ ɷɬɚɩɚɦ:
1. ɇɚ ɩɟɪɜɨɦ ɷɬɚɩɟ ɡɚɩɢɫɵɜɚɸɬ ɢɧɬɟɝɪɚɥ Ⱦɸɚɦɟɥɹ ɜ
ɮɨɪɦɟ ɧɚɢɛɨɥɟɟ ɭɞɨɛɧɨɣ ɞɥɹ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɢ ɫ ɦɨɦɟɧɬɚ
t = 0 ɞɨ ɦɨɦɟɧɬɚ ɧɚɛɥɸɞɟɧɢɹ.
2. ɇɚ ɜɬɨɪɨɦ ɷɬɚɩɟ ɧɚɯɨɞɹɬ ɩɟɪɟɯɨɞɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɜɵɛɪɚɧɧɨɣ ɮɨɪɦɟ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ
Ⱦɸɚɦɟɥɹ ɤɥɚɫɫɢɱɟɫɤɢɦ ɢɥɢ ɨɩɟɪɚɬɨɪɧɵɦ ɦɟɬɨɞɨɦ.
3. Ɂɚɬɟɦ, ɜ ɫɥɭɱɚɟ ɩɟɪɜɨɣ ɢɥɢ ɜɬɨɪɨɣ ɮɨɪɦ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ ɜɵɱɢɫɥɹɸɬ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ
ɮɭɧɤɰɢɢ. Ⱦɥɹ ɷɬɨɝɨ ɫɧɚɱɚɥɚ ɨɩɪɟɞɟɥɹɸɬ ɩɪɨɢɡɜɨɞɧɭɸ ɜɨ
ɜɪɟɦɟɧɢ t, ɚ ɡɚɬɟɦ ɡɚɦɟɧɹɸɬ ɩɟɪɟɦɟɧɧɨɣ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ IJ
ɢɥɢ (t – IJ).
4. Ⱦɚɥɟɟ ɭɱɢɬɵɜɚɸɬ ɪɚɡɪɵɜɵ ɜ ɧɚɱɚɥɟ ɢ ɤɨɧɰɟ ɤɚɠɞɨɝɨ
ɢɧɬɟɪɜɚɥɚ, ɩɪɟɞɲɟɫɬɜɭɸɳɟɝɨ ɦɨɦɟɧɬɭ ɧɚɛɥɸɞɟɧɢɹ. ȼ ɫɥɭɱɚɟ ɠɟ ɬɪɟɬɶɟɣ ɢɥɢ ɱɟɬɜɟɪɬɨɣ ɮɨɪɦ ɡɚɩɢɫɢ ɢɧɬɟɝɪɚɥɚ
Ⱦɸɚɦɟɥɹ ɭɱɢɬɵɜɚɟɬɫɹ ɬɨɥɶɤɨ ɡɧɚɱɟɧɢɟ ɢɦɩɭɥɶɫɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɜ ɬɨɱɤɟ t = 0.
40
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