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ȁDzǸ dzǯǼȟȜȘȖțȎ
ȟȠȝȞȓȝ
DzǮǼȟȪȘȖț
ȘȎțȒȠȓȣțțȎȡȘȒȜȤ
ǮǮDzȩȒȎ
ȒȞȠȓȣțțȎȡȘȝȞȜȢ
ȺȾȺɉɌɂȼɇȺəɂȾȿɇɌɂɎɂɄȺɐɂəɉȺɊȺɆȿɌɊɈȼɋɍȾɇȺ
ɇȺɈɋɇɈȼȿɉɊɈɋɌɕɏɆɈȾȿɅȿɃ
6,03/(02'/(6%$6('1$'$37,9($,'(17,),&$7,21
2)6+,33$5$0(7(56
ȼ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɚɹ ɡɚɞɚɱɚ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ
ɫɭɞɧɚɪɟɲɟɧɢɟɤɨɬɨɪɨɣɞɟɦɨɧɫɬɪɢɪɭɟɬɫɹɧɚɩɪɢɦɟɪɟɩɪɨɫɬɟɣɲɢɯɦɨɞɟɥɟɣɝɨɩɨɪɹɞɤɚɥɢɧɟɣɧɨɣɦɨɞɟɥɢ
ɇɨɦɨɬɨɢɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɪɛɢɧɚȾɥɹɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜɢɫɩɨɥɶɡɨɜɚɧɦɟɬɨɞɫɤɨɪɨɫɬɧɨɝɨ
ɝɪɚɞɢɟɧɬɚɫɭɬɶɤɨɬɨɪɨɝɨɡɚɤɥɸɱɚɟɬɫɹɜɫɨɫɬɚɜɥɟɧɢɢɢɧɬɟɝɪɚɥɶɧɨɝɨɢɥɢɥɨɤɚɥɶɧɨɝɨɰɟɥɟɜɨɝɨɮɭɧɤɰɢɨɧɚɥɚɜɵɪɚɠɚɸɳɟɝɨɫɬɟɩɟɧɶɛɥɢɡɨɫɬɢɢɞɟɧɬɢɮɢɰɢɪɭɟɦɨɝɨɨɛɴɟɤɬɚɫɭɞɧɚɢɟɝɨɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢ
ɉɨɥɭɱɟɧɵɚɥɝɨɪɢɬɦɵɚɞɚɩɬɢɜɧɨɣɧɚɫɬɪɨɣɤɢɨɰɟɧɨɤɩɚɪɚɦɟɬɪɨɜɢɞɟɧɬɢɮɢɰɢɪɭɟɦɨɝɨɨɛɴɟɤɬɚȺɥɝɨɪɢɬɦɵɨɫɧɨɜɚɧɵɧɚɜɵɱɢɫɥɟɧɢɢɝɪɚɞɢɟɧɬɚɩɪɨɢɡɜɨɞɧɨɣɩɨɜɪɟɦɟɧɢɰɟɥɟɜɨɝɨɮɭɧɤɰɢɨɧɚɥɚɜɵɱɢɫɥɟɧɧɨɝɨɜɫɢɥɭ
ɭɪɚɜɧɟɧɢɣɫɢɫɬɟɦɵɑɢɫɥɟɧɧɵɟɷɤɫɩɟɪɢɦɟɧɬɵɩɨɞɬɜɟɪɞɢɥɢɪɚɛɨɬɨɫɩɨɫɨɛɧɨɫɬɶɩɨɥɭɱɟɧɧɵɯɚɥɝɨɪɢɬɦɨɜ
ɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɊɚɫɫɦɨɬɪɟɧɧɵɣɩɨɞɯɨɞɦɨɠɟɬɛɵɬɶɪɚɫɩɪɨɫɬɪɚɧɟɧɧɚɪɟɲɟɧɢɟɡɚɞɚɱɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɫɢɫɩɨɥɶɡɨɜɚɧɢɟɦɥɢɧɟɣɧɵɯɢɧɟɥɢɧɟɣɧɵɯɦɨɞɟɥɟɣɛɨɥɟɟɜɵɫɨɤɨɝɨɩɨɪɹɞɤɚ
7KH SDSHU LV GHYRWHG WR WKH SUREOHP RI VKLS SDUDPHWHU LGHQWL¿FDWLRQ 7KH VROYLQJ RI WKLV SUREOHP LV
GHPRQVWUDWHGRQWKHEDVLVRIVLPSOHVKLSPRGHOVRI¿UVWRUGHUQDPHO\1RPRWR¶VOLQHDUPRGHODQG1RUELQ¶VQRQOLQHDU
PRGHO7RLGHQWLI\WKHSDUDPHWHUVWKHVSHHGJUDGLHQWPHWKRGLVDSSOLHG7KHPDLQLGHDRIWKHPHWKRGFRQVLVWVLQ
VXFKDQDGMXVWPHQWRIFRQWUROOHGREMHFWPRGHOSDUDPHWHUVWKDWDGLIIHUHQFHEHWZHHQG\QDPLFVRIDQREMHFWDQGLWV
PRGHOWHQGVWR]HUR$OJRULWKPVRIDGDSWLYHDGMXVWPHQWIRUHVWLPDWHVRILGHQWL¿HGREMHFWSDUDPHWHUVDUHGHULYHG
1XPHULFDOVLPXODWLRQVKDGFRQ¿UPHGDQHIIHFWLYHQHVVRIFRQVLGHUHGDOJRULWKPVRIDGDSWLYHLGHQWL¿FDWLRQ
Выпуск 2
Ʉɥɸɱɟɜɵɟɫɥɨɜɚɭɩɪɚɜɥɟɧɢɟɫɭɞɧɨɦɦɨɞɟɥɶɇɨɦɨɬɨɦɨɞɟɥɶɇɨɪɛɢɧɚɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹɢɞɟɧɬɢɮɢɤɚɰɢɹɦɟɬɨɞɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚɚɥɝɨɪɢɬɦɚɞɚɩɬɚɰɢɢ
.H\ZRUGVVKLSFRQWURO1RPRWR¶VPRGHO1RUELQ¶VPRGHOSDUDPHWHULGHQWL¿FDWLRQVSHHGJUDGLHQWPHWKRGW
DGDSWDWLRQDOJRULWKP
24
ȼɜɟɞɟɧɢɟ
Ɂɚɞɚɱɢɭɩɪɚɜɥɟɧɢɹɫɥɨɠɧɵɦɢɞɢɧɚɦɢɱɟɫɤɢɦɢɫɢɫɬɟɦɚɦɢɜɬɨɦɱɢɫɥɟɦɨɪɫɤɢɦɢɩɨɞɜɢɠ
ɧɵɦɢɨɛɴɟɤɬɚɦɢɆɉɈɩɪɟɞɩɨɥɚɝɚɸɬɨɛɟɫɩɟɱɟɧɢɟɬɪɟɛɭɟɦɨɣɪɟɚɤɰɢɢɫɢɫɬɟɦɵɜɫɥɟɞɫɬɜɢɟɫɮɨɪ
ɦɢɪɨɜɚɧɧɵɯɭɩɪɚɜɥɹɸɳɢɯɫɢɝɧɚɥɨɜɄɚɤɩɪɚɜɢɥɨɧɟɫɬɚɛɢɥɶɧɨɫɬɶɧɟɥɢɧɟɣɧɨɫɬɶɢɧɟɨɩɪɟɞɟɥɺɧ
ɧɚɹɞɢɧɚɦɢɤɚɪɟɚɝɢɪɨɜɚɧɢɹɨɛɴɟɤɬɚɨɛɭɫɥɚɜɥɢɜɚɸɬɫɥɨɠɧɨɫɬɶɡɚɞɚɱɭɩɪɚɜɥɟɧɢɹɌɪɚɞɢɰɢɨɧɧɨ
ɞɥɹɫɢɧɬɟɡɚɤɚɱɟɫɬɜɟɧɧɨɣɫɢɫɬɟɦɵɭɩɪɚɜɥɟɧɢɹɬɪɟɛɭɟɬɫɹɡɧɚɧɢɟɩɚɪɚɦɟɬɪɨɜɭɩɪɚɜɥɹɟɦɨɝɨɨɛɴɟɤ
ɬɚɬɟɪɟɲɟɧɢɟɡɚɞɚɱɢɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɋɩɟɰɢɮɢɤɨɣɆɉɈɹɜɥɹɟɬɫɹɢɯɫɥɨɠɧɨɟ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɫɜɧɟɲɧɟɣɫɪɟɞɨɣɜɵɪɚɠɚɸɳɟɟɫɹɧɚɩɪɢɦɟɪɜɧɚɥɢɱɢɢɩɪɢɫɨɟɞɢɧɺɧɧɵɯɦɚɫɫɢ
ɦɨɦɟɧɬɨɜɢɧɟɪɰɢɢɜɥɢɹɧɢɢɜɟɬɪɨɜɨɥɧɨɜɵɯɜɨɡɞɟɣɫɬɜɢɣɢɢɡɦɟɧɟɧɢɢɯɚɪɚɤɬɟɪɢɫɬɢɤɩɨɜɟɪɯɧɨɫɬɢ
ɫɭɞɧɚɟɝɨɡɚɝɪɭɡɤɢɢɞɪ>@±>@ȼɫɨɜɪɟɦɟɧɧɨɣɬɟɨɪɢɢɭɩɪɚɜɥɟɧɢɹɪɚɡɪɚɛɨɬɚɧɨɞɨɫɬɚɬɨɱɧɨɦɧɨ
ɝɨɦɟɬɨɞɨɜɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜɞɢɧɚɦɢɱɟɫɤɢɯɨɛɴɟɤɬɨɜɫɪɟɞɢɤɨɬɨɪɵɯɤɱɢɫɥɭɧɚɢɛɨɥɟɟ
ɩɟɪɫɩɟɤɬɢɜɧɵɯɦɨɠɧɨɨɬɧɟɫɬɢɦɟɬɨɞɵɨɫɧɨɜɚɧɧɵɟɧɚɚɞɚɩɬɢɜɧɨɦɩɨɞɯɨɞɟȼɧɚɫɬɨɹɳɟɣɪɚɛɨɬɟ
ɪɟɲɚɟɬɫɹɡɚɞɚɱɚɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɞɥɹɩɪɨɫɬɵɯɦɨɞɟɥɟɣɫɭɞɨɜ>@±>@ɨɫɧɨɜɚɧɧɚɹɧɚ
ɦɟɬɨɞɟɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚ>@>@
Ɉɫɧɨɜɧɚɹɱɚɫɬɶ
Ɇɨɞɟɥɶɧɚɞɜɨɞɧɨɝɨɜɨɞɨɢɡɦɟɳɚɸɳɟɝɨɫɭɞɧɚɜɩɪɨɰɟɫɫɟɪɟɲɟɧɢɹɡɚɞɚɱɢɭɩɪɚɜɥɟɧɢɹɤɭɪɫɨɦ
ɩɪɢɧɟɤɨɬɨɪɵɯɭɩɪɨɳɚɸɳɢɯɞɨɩɭɳɟɧɢɹɯ>@ɦɨɠɟɬɛɵɬɶɩɪɟɞɫɬɚɜɥɟɧɚɜɜɢɞɟɫɥɟɞɭɸɳɢɯɞɢɮɮɟ
ɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣ
ϕ = ω
= kδ − cω
Jω
ɝɞɟij²ɤɭɪɫȦ²ɭɝɥɨɜɚɹɫɤɨɪɨɫɬɶɫɤɨɪɨɫɬɶɪɵɫɤɚɧɶɹį²ɭɝɨɥɨɬɤɥɨɧɟɧɢɹɪɭɥɹNɢɫ²ɤɨ
ɷɮɮɢɰɢɟɧɬɵɦɨɦɟɧɬɚɫɢɥɵɢɜɹɡɤɨɝɨɫɨɩɪɨɬɢɜɥɟɧɢɹɫɨɨɬɜɟɬɫɬɜɟɧɧɨ-²ɨɛɳɢɣɦɨɦɟɧɬɢɧɟɪɰɢɢ
ɫɭɱɺɬɨɦɩɪɢɫɨɟɞɢɧɺɧɧɵɯɦɚɫɫɜɨɞɵ
ɉɪɢɜɟɞɺɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɨɫɬɟɣɲɭɸ ɦɨɞɟɥɶ ɢɡ
ɜɟɫɬɧɭɸɤɚɤɦɨɞɟɥɶɇɨɦɨɬɨɝɨɩɨɪɹɞɤɚ>@ɋɨɨɬɜɟɬɫɬɜɭɸɳɚɹɟɣɩɟɪɟɞɚɬɨɱɧɚɹɮɭɧɤɰɢɹɢɦɟɟɬɜɢɞ
W (s) =
ɝɞɟ k p
k
T
c
kp
s (Ts + 1) )
J
c
Ɇɨɞɟɥɶɇɨɪɛɢɧɚ>@ɭɱɢɬɵɜɚɟɬɧɟɥɢɧɟɣɧɭɸɡɚɜɢɫɢɦɨɫɬɶɦɨɦɟɧɬɚɜɹɡɤɨɝɨɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɨɬɫɤɨɪɨɫɬɢɪɵɫɤɚɧɶɹɢɜɩɪɨɫɬɟɣɲɟɦɫɥɭɱɚɟɦɨɠɟɬɛɵɬɶɡɚɩɢɫɚɧɚɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
ϕ = ω = k2 δ − c2 ω3
Jω
ɇɟɨɛɯɨɞɢɦɨɨɬɦɟɬɢɬɶɱɬɨɩɚɪɚɦɟɬɪɵɩɪɢɜɟɞɺɧɧɵɯɦɨɞɟɥɟɣɦɨɝɭɬɜɚɪɶɢɪɨɜɚɬɶɫɹɜɞɨɫɬɚ
ɬɨɱɧɨɲɢɪɨɤɢɯɩɪɟɞɟɥɚɯɜɡɚɜɢɫɢɦɨɫɬɢɨɬɫɤɨɪɨɫɬɢɞɜɢɠɟɧɢɹɡɚɝɪɭɡɤɢɫɭɞɧɚɢɞɪɭɝɢɯɮɚɤɬɨɪɨɜ
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɟɬɨɞɚ ɫɤɨɪɨɫɬɧɨɝɨ ɝɪɚɞɢɟɧɬɚ ɞɥɹ ɫɢɧɬɟɡɚ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ ɞɜɢ
ɠɟɧɢɟɦ ɫɭɞɧɚ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɩɚɪɚɦɟɬɪɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɫɭɞɧɚ ɉɪɢ ɪɟɲɟɧɢɢ ɷɬɨɣ
ɡɚɞɚɱɢɜɞɚɧɧɨɣɪɚɛɨɬɟɢɫɩɨɥɶɡɭɟɬɫɹɦɟɬɨɞɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚɪɚɡɪɚɛɨɬɚɧɧɵɣɢɨɩɢɫɚɧɧɵɣ
Ⱥ Ʌ Ɏɪɚɞɤɨɜɵɦ ɤɨɬɨɪɵɣ ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ ɱɬɨ ɞɢɧɚɦɢɱɟɫɤɢɣ ɨɛɴɟɤɬ
ɭɩɪɚɜɥɟɧɢɹɨɩɪɟɞɟɥɺɧɭɪɚɜɧɟɧɢɟɦ
x = F ( x p t ) ɝɞɟ[²ɜɟɤɬɨɪɫɨɫɬɨɹɧɢɹɭɩɪɚɜɥɹɟɦɨɝɨɨɛɴɟɤɬɚɢS²ɜɟɤɬɨɪɟɝɨɩɚɪɚɦɟɬɪɨɜ
ɐɟɥɶɸɭɩɪɚɜɥɟɧɢɹɹɜɥɹɟɬɫɹɦɢɧɢɦɢɡɚɰɢɹɮɭɧɤɰɢɢ
Q = Q ( x, t ) → min ɇɚɩɟɪɜɨɦɲɚɝɟɚɥɝɨɪɢɬɦɚɦɟɬɨɞɚɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚɜɵɱɢɫɥɹɟɬɫɹɫɤɨɪɨɫɬɶɢɡɦɟɧɟɧɢɹ
Q[SWɜɞɨɥɶɬɪɚɟɤɬɨɪɢɢɨɩɪɟɞɟɥɹɟɦɨɣɭɪɚɜɧɟɧɢɟɦ
∂Q n ∂Q
Q ( x, t ) =
+∑
Fi ( x, p, t ) ∂t i =1 ∂xi
Выпуск 2
ɋɥɟɞɭɸɳɢɣɲɚɝ²ɧɚɯɨɠɞɟɧɢɟɝɪɚɞɢɟɧɬɚɞɥɹɩɨɥɭɱɟɧɧɨɣɩɪɨɢɡɜɨɞɧɨɣɩɨɧɚɫɬɪɚɢɜɚɟɦɵɦ
ɩɚɪɚɦɟɬɪɚɦ
∂Q ( x, t )  ∂Q ( x, t ) ∂Q ( x, t ) 
= 
…

∂p
∂pn 
 ∂p1
ȼɡɚɤɥɸɱɟɧɢɟɞɥɹɦɢɧɢɦɢɡɚɰɢɢɰɟɥɟɜɨɣɮɭɧɤɰɢɢQ[SWɜɵɩɨɥɧɹɟɬɫɹɧɚɫɬɪɨɣɤɚɜɟɤɬɨɪɚ
ɢɡɦɟɧɹɟɦɵɯɩɚɪɚɦɟɬɪɨɜɜɧɚɩɪɚɜɥɟɧɢɢɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦɫɤɨɪɨɫɬɧɨɦɭɝɪɚɞɢɟɧɬɭ
∂Q ( x, t )
dp
= −Ƚ
dt
∂p
25
ɝɞɟȽ²ɫɢɦɦɟɬɪɢɱɧɚɹɩɨɥɨɠɢɬɟɥɶɧɨɨɩɪɟɞɟɥɺɧɧɚɹɧɚɩɪɢɦɟɪɞɢɚɝɨɧɚɥɶɧɚɹɦɚɬɪɢɰɚ
0 0 
 γ1 0
0 γ
0 0 
2
Ƚ 
 0 0 … 0


 0 0 0 γn 
ɤɨɷɮɮɢɰɢɟɧɬɵȖiɨɛɭɫɥɚɜɥɢɜɚɸɬɫɤɨɪɨɫɬɶɚɞɚɩɬɚɰɢɢɧɚɫɬɪɚɢɜɚɟɦɵɯɩɚɪɚɦɟɬɪɨɜ
ɊɚɫɫɦɨɬɪɢɦɚɞɚɩɬɢɜɧɭɸɢɞɟɧɬɢɮɢɤɚɰɢɸɩɚɪɚɦɟɬɪɨɜɆɉɈ
Ɇɨɞɟɥɶɇɨɦɨɬɨ. ɉɪɢɦɟɧɢɦɦɟɬɨɞɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚɞɥɹɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟ
ɬɪɨɜɦɨɞɟɥɟɣɆɉɈɅɢɧɟɣɧɭɸɦɨɞɟɥɶɇɨɦɨɬɨɝɨɩɨɪɹɞɤɚɜɬɨɪɨɟɭɪɚɜɧɟɧɢɟɦɨɠɧɨɡɚɩɢ
ɫɚɬɶɜɜɢɞɟ
= aω + bδ ω
ɝɞɟa = −
kp
1
;b = T
T
ȼɵɛɟɪɟɦɧɚɫɬɪɚɢɜɚɟɦɭɸɢɞɟɧɬɢɮɢɰɢɪɭɸɳɭɸɦɨɞɟɥɶɜɜɢɞɟ
m = Am ω + Bδ + v ω
ɝɞɟaPEP²ɧɚɫɬɪɚɢɜɚɟɦɵɟɩɚɪɚɦɟɬɪɵɚY²ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɣɫɢɝɧɚɥ
ȼɜɟɞɺɦɞɨɩɨɥɧɢɬɟɥɶɧɭɸɩɟɪɟɦɟɧɧɭɸ
6 Z±ZP
ɢɨɩɪɟɞɟɥɢɦɰɟɥɟɜɭɸɮɭɧɤɰɢɸ
Q 1 s 2 2
ȼɵɱɢɫɥɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɩɨ ɜɪɟɦɟɧɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɭɪɚɜɧɟɧɢɹɦɢ
ɢ
−ω
m ) = s ( aω + bδ − Aω − Bm δ − v )
Q = ss = s ( ω
ȼɵɩɨɥɧɢɦɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɩɨɩɚɪɚɦɟɬɪɚɦɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢ
∂Q
∂Q
= − sω = − sδ ∂Am
∂Bm
ȼɪɟɡɭɥɶɬɚɬɟɚɥɝɨɪɢɬɦɧɚɫɬɪɨɣɤɢɩɚɪɚɦɟɬɪɨɜɦɨɞɟɥɢɦɨɠɟɬɛɵɬɶɡɚɩɢɫɚɧɜɜɢɞɟ
A m = γsω Выпуск 2
B m = γsδ ν = ν 0sign ( V )
26
ɜɨɡɦɨɠɧɵɪɚɡɥɢɱɧɵɟɜɚɪɢɚɧɬɵɜɵɛɨɪɚɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨɫɢɝɧɚɥɚ
Ʉɚɤɩɨɤɚɡɚɧɨɜ>@ɩɪɢɨɩɪɟɞɟɥɺɧɧɵɯɭɫɥɨɜɢɹɯɢɦɟɟɬɦɟɫɬɨɫɥɟɞɭɸɳɟɟɢɞɟɧɬɢɮɢɤɚɰɢɨɧɧɨɟ
ɫɜɨɣɫɬɜɨɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢ
aPĺa
EPĺE
Ɇɨɞɟɥɶɇɨɪɛɢɧɚ.Ⱦɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɪɛɢɧɚɝɨɩɨɪɹɞɤɚɛɭɞɟɦɫɱɢɬɚɬɶɱɬɨɦɨɦɟɧɬ
ɫɢɥɵɜɹɡɤɨɝɨɫɨɩɪɨɬɢɜɥɟɧɢɹɢɦɟɟɬɯɚɪɚɤɬɟɪɤɭɛɢɱɟɫɤɨɣɡɚɜɢɫɢɦɨɫɬɢɨɬɫɤɨɪɨɫɬɢɪɵɫɤɚɧɶɹ
=−
ω
ɩɪɢɧɹɜ a =
kp
J
b = −
k
kδ
δ + p ω J
J
kδ
ɩɨɥɭɱɢɦ
J
= aω + bδ
ω
m = Am ω + Bm δ + ν ω
ɋɭɱɺɬɨɦɜɵɱɢɫɥɢɦɩɪɨɢɡɜɨɞɧɭɸɰɟɥɟɜɨɣɮɭɧɤɰɢɢɩɨɜɪɟɦɟɧɢ
−ω
m ) = s ( aω + bδ − Am ω − Bm δ − ν ) Q = ss = s ( ω
ɬɚɬɵ
Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɩɨɩɚɪɚɦɟɬɪɚɦɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢɞɚɺɬɫɥɟɞɭɸɳɢɟɪɟɡɭɥɶ
∂Q
= − sω ∂Am
∂Q
= − sδ ∂Bm
Y V
Ɍɚɤɢɦɨɛɪɚɡɨɦɚɥɝɨɪɢɬɦɧɚɫɬɪɨɣɤɢɧɟɥɢɧɟɣɧɨɣɢɞɟɧɬɢɮɢɰɢɪɭɸɳɟɣɦɨɞɟɥɢɦɨɠɟɬɛɵɬɶ
ɡɚɩɢɫɚɧɜɜɢɞɟ
A m = γsω
B m = γsδ ν = ν 0sign ( V ) Am ≈ a = −0, 3; Bm ≈ b = 0, 05.
Выпуск 2
Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɪɚɛɨɬɨɫɩɨɫɨɛɧɨɫɬɢ ɩɨɥɭɱɟɧɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɚɞɚɩɬɢɜɧɨɣ ɢɞɟɧɬɢɮɢɤɚɰɢɢ
ɧɚɨɫɧɨɜɟɩɪɨɫɬɵɯɦɨɞɟɥɟɣɇɨɦɨɬɨɢɇɨɪɛɢɧɚɛɵɥɨɩɪɨɜɟɞɟɧɨɦɨɞɟɥɢɪɨɜɚɧɢɟɜɫɪɟɞɟ0$7/$%
6LPXOLQNɋɢɫɬɟɦɚɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜɞɥɹɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɦɨɬɨɝɨɩɨ
ɪɹɞɤɚɩɪɢɜɟɞɟɧɚɧɚɪɢɫɇɚɜɯɨɞɨɛɴɟɤɬɚɭɩɪɚɜɥɟɧɢɹɛɵɥɩɨɞɚɧɡɚɞɚɸɳɢɣɫɢɝɧɚɥɢɡɞɜɭɯɝɚɪ
ɦɨɧɢɤɪɚɡɥɢɱɚɸɳɢɯɫɹɩɨɚɦɩɥɢɬɭɞɟɢɱɚɫɬɨɬɟ
ɉɟɪɟɦɟɧɧɚɹVɪɢɫɨɬɪɚɠɚɟɬɨɬɤɥɨɧɟɧɢɟɫɢɝɧɚɥɚɧɚɜɵɯɨɞɟɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢȦPɨɬ
ɫɢɝɧɚɥɚɧɚɜɵɯɨɞɟɦɨɞɟɥɢɪɭɟɦɨɝɨɨɛɴɟɤɬɚȦ
Ɋɟɡɭɥɶɬɚɬɵɦɨɞɟɥɢɪɨɜɚɧɢɹɩɨɞɬɜɟɪɞɢɥɢɱɬɨɧɚɫɬɪɚɢɜɚɟɦɵɟɩɚɪɚɦɟɬɪɵ$Pɢ%Pɪɢɫɫɨ
ɝɥɚɫɧɨɢɞɟɧɬɢɮɢɤɚɰɢɨɧɧɨɦɭɫɜɨɣɫɬɜɭɦɨɞɟɥɢɩɨɜɟɥɢɱɢɧɟɫɬɪɟɦɹɬɫɹɤɚɧɚɥɨɝɢɱɧɵɦɤɨɷɮɮɢɰɢ
ɟɧɬɚɦɡɚɞɚɧɧɵɦɞɥɹɨɛɴɟɤɬɚɭɩɪɚɜɥɟɧɢɹ Am ≈ a = −0, 8; Bm ≈ b = 0, 05.
ɇɚɪɢɫɩɪɢɜɟɞɟɧɚɫɢɫɬɟɦɚɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜɞɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟ
ɥɢɇɨɪɛɢɧɚɝɨɩɨɪɹɞɤɚ
ɉɪɨɰɟɫɫɵɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟɚɞɚɩɬɢɜɧɭɸɢɞɟɧɬɢɮɢɤɚɰɢɸɞɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɩɪɢɜɟ
ɞɟɧɵɧɚɪɢɫɢ
Ɍɚɤɠɟɤɚɤɢɞɥɹɥɢɧɟɣɧɨɣɫɢɫɬɟɦɵɚɞɚɩɬɢɜɧɚɹɢɞɟɧɬɢɮɢɤɚɰɢɹɞɥɹɧɟɥɢɧɟɣɧɨɝɨɨɛɴɟɤɬɚ
ɨɛɟɫɩɟɱɢɜɚɟɬɫɨɨɬɜɟɬɫɬɜɢɟɧɚɫɬɪɚɢɜɚɟɦɵɯɩɚɪɚɦɟɬɪɨɜ$Pɢ%Pɪɢɫɢɯɢɫɬɢɧɧɵɦɡɧɚɱɟɧɢɹɦ
27
Выпуск 2
Ɋɢɫɋɢɫɬɟɦɚɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜ
ɞɥɹɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɦɨɬɨɝɨɩɨɪɹɞɤɚ
28
ɊɢɫɈɬɤɥɨɧɟɧɢɟɫɢɝɧɚɥɚɧɚɜɵɯɨɞɟɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢɨɬɫɢɝɧɚɥɚ
ɧɚɜɵɯɨɞɟɨɛɴɟɤɬɚɞɥɹɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɦɨɬɨɝɨɩɨɪɹɞɤɚ
Ɋɢɫ$Pɢ%Pɞɥɹɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɦɨɬɨɝɨɩɨɪɹɞɤɚ
Выпуск 2
29
Ɋɢɫɋɢɫɬɟɦɚɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚɪɚɦɟɬɪɨɜ
ɞɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɪɛɢɧɚɝɨɩɨɪɹɞɤɚ
ɊɢɫɈɬɤɥɨɧɟɧɢɟɫɢɝɧɚɥɚɧɚɜɵɯɨɞɟɧɚɫɬɪɚɢɜɚɟɦɨɣɦɨɞɟɥɢɨɬɫɢɝɧɚɥɚ
ɧɚɜɵɯɨɞɟɨɛɴɟɤɬɚɞɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɪɛɢɧɚ
Выпуск 2
Ɋɢɫ$Pɢ%Pɞɥɹɧɟɥɢɧɟɣɧɨɣɦɨɞɟɥɢɇɨɪɛɢɧɚ
30
ȼɵɜɨɞɵ
Ɇɟɬɨɞɫɤɨɪɨɫɬɧɨɝɨɝɪɚɞɢɟɧɬɚɩɨɡɜɨɥɢɥɩɨɫɬɪɨɢɬɶɫɯɟɦɵɚɞɚɩɬɢɜɧɨɣɢɞɟɧɬɢɮɢɤɚɰɢɢɩɚ
ɪɚɦɟɬɪɨɜɥɢɧɟɣɧɨɝɨɢɧɟɥɢɧɟɣɧɨɝɨɞɢɧɚɦɢɱɟɫɤɢɯɨɛɴɟɤɬɨɜ
ɉɨɥɭɱɟɧɧɵɟɪɟɡɭɥɶɬɚɬɵɦɨɞɟɥɢɪɨɜɚɧɢɹɩɨɡɜɨɥɹɸɬɩɪɟɞɩɨɥɨɠɢɬɶɱɬɨɢɫɩɨɥɶɡɨɜɚɧɧɵɣ
ɩɨɞɯɨɞɦɨɠɟɬɛɵɬɶɪɚɫɩɪɨɫɬɪɚɧɺɧɧɚɛɨɥɟɟɲɢɪɨɤɢɣɤɥɚɫɫɧɟɥɢɧɟɣɧɵɯɞɢɧɚɦɢɱɟɫɤɢɯɨɛɴɟɤɬɨɜ
ɜɱɚɫɬɧɨɫɬɢɧɚɫɢɫɬɟɦɵɫɮɭɧɤɰɢɨɧɚɥɶɧɨɣɧɟɨɩɪɟɞɟɥɺɧɧɨɫɬɶɸȺɤɬɭɚɥɶɧɨɫɬɶɪɟɲɟɧɢɹɩɨɞɨɛɧɨɣ
ɡɚɞɚɱɢɧɚɩɪɢɦɟɪɞɥɹɆɉɈɨɛɴɹɫɧɹɟɬɫɹɚɩɪɢɨɪɧɨɧɟɨɩɪɟɞɟɥɺɧɧɵɦɧɟɥɢɧɟɣɧɵɦɜɡɚɢɦɨɞɟɣɫɬɜɢ
ɟɦɫɜɧɟɲɧɟɣɫɪɟɞɨɣ
ɋɩɢɫɨɤɥɢɬɟɪɚɬɭɪɵ
ɅɭɤɨɦɫɤɢɣɘȺɇɚɜɢɝɚɰɢɹɢɭɩɪɚɜɥɟɧɢɟɞɜɢɠɟɧɢɟɦɫɭɞɨɜɭɱɟɛɧɢɤɘȺɅɭɤɨɦɫɤɢɣȼȽɉɟ
ɲɟɯɨɧɨɜȾȺɋɤɨɪɨɯɨɞɨɜ²ɋɉɛɗɥɦɨɪ²ɫ
0DQXHO +DUR &DVDGR 5HFXUVLYH LGHQWL¿FDWLRQ SURFHGXUH RI WKH QRQOLQHDU PRGHO VKLS EDVHG RQ WKH
WXUQLQJWHVWPDQRHXYULQJ0DQXHO+DUR&DVDGR$)HUQDQGH]$PHDO&$06,)$&&RQIHUHQFHRQ&RQWURO
$SSOLFDWLRQVLQ0DULQH6\VWHPV²$QFRQD,WDO\²3±
ȼɚɝɭɳɟɧɤɨ Ʌ Ʌ ɋɢɫɬɟɦɵ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɟɦ ɫɭɞɧɚ Ʌ Ʌ ȼɚɝɭɳɟɧɤɨ
ɇɇɐɵɦɛɚɥ²ɈɞɟɫɫɚɅɚɬɫɬɚɪ²ɫ
ɘɞɢɧɘɂɂɫɩɨɥɶɡɨɜɚɧɢɟɢɞɟɧɬɢɮɢɰɢɪɨɜɚɧɧɵɯɦɚɬɟɦɚɬɢɱɟɫɤɢɯɦɨɞɟɥɟɣɫɭɞɧɚɞɥɹɨɛɟɫɩɟɱɟ
ɧɢɹɛɟɡɨɩɚɫɧɨɫɬɢɫɭɞɨɜɨɠɞɟɧɢɹɘɂɘɞɢɧȺȽɋɬɟɩɚɯɧɨȺɇȽɨɥɨɥɨɛɨɜȼɟɫɬɧɢɤɆȽɌɍ²²
Ɍ²ʋ²&±
ɊɨɦɚɧɨɜȺȼɋɬɪɭɤɬɭɪɧɚɹɢɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹɢɞɟɧɬɢɮɢɤɚɰɢɹɦɚɬɟɦɚɬɢɱɟɫɤɨɣɦɨɞɟɥɢɜɨɞɨɢɡɦɟ
ɳɚɸɳɢɯɫɭɞɨɜȺȼɊɨɦɚɧɨɜȼɟɫɬɧɢɤɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨɭɧɢɜɟɪɫɢɬɟɬɚɦɨɪɫɤɨɝɨɢɪɟɱɧɨɝɨɮɥɨɬɚɢɦɟɧɢ
ɚɞɦɢɪɚɥɚɋɈɆɚɤɚɪɨɜɚ²²ʋ²ɋɚ±
ɀɚɛɤɨɇȺɉɚɪɚɦɟɬɪɢɱɟɫɤɚɹɢɞɟɧɬɢɮɢɤɚɰɢɹɞɢɧɚɦɢɱɟɫɤɢɯɦɨɞɟɥɟɣɦɨɪɫɤɢɯɫɭɞɨɜɇȺɀɚɛ
ɤɨ ȼɟɫɬɧɢɤȼɨɪɨɧɟɠɫɤɨɝɨɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨɬɟɯɧɢɱɟɫɤɨɝɨɭɧɢɜɟɪɫɢɬɟɬɚ²Ɍ²ʋ²ɋ±
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Ɋɚɡɪɚɛɨɬɚɧɚɬɟɯɧɨɥɨɝɢɹɦɨɞɟɥɢɪɨɜɚɧɢɹɞɢɧɚɦɢɱɟɫɤɨɝɨɜɨɡɞɟɣɫɬɜɢɹ©ɜɨɥɧɵɭɛɢɣɰɵªɜɵɫɨɬɨɣɦ
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