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Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
)ɛ)
w
½
1
°
°
w
°
°
2
¾
®
w
3
°
°
°
°
w
¯
¿
4
04,8
½
°
°
79,5
°
°
®
48,8
°
°
37,11
¯
3
ɦ10
¾
°
°
¿
04,8
½
°
°
79,5
°
°
®
48,8
°
°
37,11
¯
.ɦɦ
¾
°
°
¿
ɑɢɫɥɨɜɨɟ ɩɨɥɟ ɩɪɨɝɢɛɨɜ w ɜ ɭɡɥɚɯ ɫɟɬɤɢ ɩɪɟɞɫɬɚɜɥɟɧɨ ɧɚ
ɪɢɫ 2.6, ɚɉɨɜɟɪɯɧɨɫɬɶɢɡɨɝɧɭɬɨɣɩɥɢɬɵɢɡɨɛɪɚɠɟɧɚɧɚɪɢɫ 2.6, ɛ.
ɚ
0 0 0 0 0
0 5,79 8,04 5,79 0
0 8,48 11,37 8,48 0
0 5,79 8,04 5,79 0
0 0 0 0 0
y
w, ɦɦ
Ɋɢɫ 2.6. Ɂɧɚɱɟɧɢɹɩɪɨɝɢɛɨɜɜɭɡɥɚɯɫɟɬɤɢ
x
8,0
5,8
8,5
5,8
5,8
8,5
5,8
8,0
11,4
ɋɩɨɦɨɳɶɸɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯɨɩɟɪɚɬɨɪɨɜɜɧɭɬɪɟɧɧɢɯɭɫɢ
ɥɢɣ ɫɦ ɪɢɫ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡɝɢɛɚɸɳɢɟ ɢ ɤɪɭɬɹɳɢɟ
ɦɨɦɟɧɬɵɜɭɡɥɚɯɩɨɫɟɱɟɧɢɹɦɩɥɢɬɵ
ȼɵɱɢɫɥɢɦɡɧɚɱɟɧɢɹɦɧɨɠɢɬɟɥɟɣɜɷɬɢɯɨɩɟɪɚɬɨɪɚɯ
3
1082,21
2
'
2
2
)1(
Q
D
4
''
yx
3
DD
;ɤɇɦ1046,5
2
'
yx
3
)15,01(1082,21
5,124
3
1082,21
2
5,1
3
3
;ɤɇɦ107,9
,ɤɇɦ1055,1
ɢɡɚɩɢɲɟɦɨɩɟɪɚɬɨɪɵɫɦɪɢɫ ɜɱɢɫɥɨɜɨɦɜɢɞɟɪɢɫ 2.7).
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɤɥɚɞɵɜɚɟɦ ɷɬɢ ɨɩɟɪɚɬɨɪɵ ɧɚ ɭɡɥɵ ɫɟɬɤɢ
ɞɥɹɫɟɱɟɧɢɣ ɜ ɤɨɬɨɪɵɯɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ
ɉɪɢɜɟɞɟɦ ɜɵɱɢɫɥɟɧɢɟ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ M
, M
x
ɜ ɭɡɥɟ
y
ɢɤɪɭɬɹɳɟɝɨ ɦɨɦɟɧɬɚ H ɜɭɡɥɚɯ O, P ɢPƍ ɫɦɪɢɫɚ).
51
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
M
x
k
–0,267
M
y
k
–1
–1–1 2,533
3
–1
–1
0
îÂ
¨
y
¨
x
–0,267
îÂ
1
H
k
–1 1
Ɋɢɫ 2.7. Ɉɩɟɪɚɬɨɪɵɞɥɹɜɵɱɢɫɥɟɧɢɹɦɨɦɟɧɬɨɜ
Mx1= 5,46 Â103(2,533 Âw1– 1(w2+ w2) – 0,267 (wO+ w4) =
= 5,46 Â10
= Â
3
(2,533 Â8,04 – 1 Â2 Â5,79 – 0,267 Â11,37) = ɤɇ Â ɦɦ
M
= 9,7 Â103(2,169 Âw1– 0,084 (w2+ w2) – 1(wO+ w4) =
y1
3
(2,169 Â8,04 – 0,084 Â2 Â5,79 – 1 Â11,37) = ɤɇ Â ɦɦ
H
= 1,55 Â103(w2–w2+ w2ƍ–w2ƍ) = 0.
O
H
= 1,55 Â103(w1–wS–w1ƍ) =1,55 (8,04 + 8,04) =ɤɇ Â ɦɦ
P
–0,084–0,084 2,169
3
îÂ
3
H
= 1,55 Â103(wSƍ–w1+ w1ƍ) = 1,55 Â2(–8,04) =–ɤɇ Â ɦɦ
Pƍ
Ⱥɧɚɥɨɝɢɱɧɨ ɜɵɱɢɫɥɹɸɬɫɹ ɢɡɝɢɛɚɸɳɢɟ ɢ ɤɪɭɬɹɳɢɟ ɦɨɦɟɧɬɵ
ɜ ɨɫɬɚɥɶɧɵɯ ɭɡɥɚɯ ɫɟɬɤɢ ɑɢɫɥɨɜɵɟ ɩɨɥɹ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ M
M
, H ɩɪɟɞɫɬɚɜɥɟɧɵɧɚɪɢɫ 2.8.
y
Ɂɧɚɱɟɧɢɹ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ ɜ ɭɡɥɚɯ ɩɨɡɚɤɪɟɩɥɟɧɧɵɦ ɫɬɨɪɨ
ɧɚɦɩɥɢɬɵɦɨɠɧɨɧɚɣɬɢɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶɭɪɚɜɧɟɧɢɟɦɜɜɢ
ɞɟ
4
k
''
yx
kk
,)/(
DVqw
ɨɬɫɸɞɚ
k
4
qwDV
k
Ɍɨ
.)(
''
yx
ɝɞɚɪɚɫɩɪɟɞɟɥɟɧɢɟɪɟɚɤɰɢɣɩɨɤɪɚɹɦɩɚɪɚɥɥɟɥɶɧɵɦɨɫɹɦx ɢy:
44
qwDVVqwDVV
k
x
k
y
k
k
52
y
kxky
k
' ' ' '
k
,
x
.)(;)(
x
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Mx, ɤɇ Â ɦ/ɦ
0 0 0 0 0
–63,1 23,8 31,4 23,8 –63,1
–92,5 38,3 41,3 38,3 –92,5
–63,1 23,8 31,4 23,8 –63,1
0 0 0 0 0
y
H, ɤɇ Â ɦ/ɦ
8,9 24,8 0 –24,8 –8,9
0 17,6 0 –17,6 0
0 17,6 0 –17,6 0
0 –17,6 0 17,6 0
–8,9 –24,8 0 24,8 8,9
y
My, ɤɇ Â ɦ/ɦ
0 0 0 0 0
x
–9,5 32,9 49,3 32,9 –9,5
–13,9 56,7 69,4 56,7 –13,9
–9,5 32,9 49,3 32,9 –9,5
0 0 0 0 0
y
x
x
Ɋɢɫ 2.8. Ɂɧɚɱɟɧɢɹɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣɜɭɡɥɚɯɫɟɬɤɢ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɧɚɤɥɚɞɵɜɚɹɨɩɟɪɚɬɨɪɪɢɫɛɧɚɭɡɥɵO,
P, R ɜɵɱɢɫɥɹɟɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɩɨɪɧɨɣ ɪɟɚɤɰɢɢ V
ɩɨ ɜɟɪɯɧɟɦɭ
y
ɲɚɪɧɢɪɧɨɨɩɟɪɬɨɦɭɤɪɚɸ ɩɥɢɬɵɋɭɱɟɬɨɦ ɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ ɧɚ
ɷɬɨɦɤɪɚɟw1ƍ=–w1; w2ƍ=–w
V
= VyP= (0 –q/2) ¨y = –15 Â1,5 = –22,5 ɤɇ/ɦ;
yO
V
= (0 –q/4) ¨y = –7,5 Â1,5 = –11,25 ɤɇ/ɦ.
yR
ɩɨɥɭɱɚɟɦ
2
4
w
,,
RPO
ɬɨɝɞɚ
0
Ɂɧɚɤɦɢɧɭɫ ɝɨɜɨɪɢɬ ɨɬɨɦ ɱɬɨɷɬɚ ɪɟɚɤɰɢɹɤɚɤ ɜɧɟɲɧɹɹ ɞɥɹ
ɩɥɢɬɵɫɢɥɚɧɚɩɪɚɜɥɟɧɚɩɪɨɬɢɜɨɫɢzɬ ɟɜɜɟɪɯ
Ɋɟɡɭɥɶɬɚɬɵɪɚɫɱɟɬɚɚɢɦɟɧɧɨɷɩɸɪɵɩɪɨɝɢɛɨɜɢɢɡɝɢɛɚɸɳɢɯ
ɦɨɦɟɧɬɨɜM
, M
ɞɥɹɫɪɟɞɧɢɯɫɟɱɟɧɢɣɩɥɢɬɵɷɩɸɪɚɤɪɭɬɹɳɢɯɦɨ
x
y
53
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɦɟɧɬɨɜH ɜ ɫɟɱɟɧɢɢɩɨɜɟɪɯɧɟɣ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɤɪɨɦɤɟ ɩɥɢɬɵ
ɚɬɚɤɠɟɪɚɫɩɪɟɞɟɥɟɧɢɟɨɩɨɪɧɵɯɪɟɚɤɰɢɣV
ɩɨɷɬɨɦɭɤɪɚɸ, ɩɪɢɜɨ
y
ɞɹɬɫɹɧɚɪɢɫ 2.9.
Ɉɛɪɚɬɢɦɜɧɢɦɚɧɢɟ ɧɚ ɬɨɱɬɨɟɫɥɢ ɷɩɸɪɵɩɪɨɝɢɛɨɜɪɟɚɤɰɢɣ
ɢɢɡɝɢɛɚɸɳɢɯɦɨɦɟɧɬɨɜɞɥɹɫɢɦɦɟɬɪɢɱɧɨ ɧɚɝɪɭɠɟɧɧɨɣɩɥɢɬɵɹɜ
ɥɹɸɬɫɹ ɫɢɦɦɟɬɪɢɱɧɵɦɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ ɫɢɦɦɟɬɪɢɢ ɬɨ ɷɩɸɪɵ
ɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵ
8,0
x
11,4
8,0
31,4
41,3
31,4
R P O Pƍ R ƍ
S 2 1 2
Sƍ 2 1 2
y
F
T 3 4 3
a/2 a/2
F
b/2
b/2
w, ɦɦ Mx, ɤɇ Â ɦ/ɦ My, ɤɇ Âɦɦ
00
8,5
11,4
8,5
w, ɦɦ
92,592,5
Mx, ɤɇ Â ɦɦ
38,3 41,3 38,3
13,9
8,5
24,8
69,4
56,856,8
0
24,8
13,9
My, ɤɇ Â ɦɦ
H, ɤɇ Â ɦɦ
8,5
Vy, ɤɇɦ
11,25
22,5 22,5
11,25
Ɋɢɫ Ɋɟɡɭɥɶɬɚɬɵɪɚɫɱɟɬɚɩɥɢɬɵ
ɆɟɬɨɞȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ
ɈɫɧɨɜɧɵɟɩɨɥɨɠɟɧɢɹɦɟɬɨɞɚȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ
49,3
69,4
49,3
Ɇɟɬɨɞ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɆȻȽ ɨɬɧɨɫɢɬɫɹ ɤ ɱɢɫɥɟɧɧɨɚɧɚɥɢɬɢɱɟɫɤɢɦ ɩɪɨɟɤɰɢɨɧɧɵɦ ɦɟɬɨɞɚɦ ɪɟɲɟɧɢɹ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɗɬɨɬ ɦɟɬɨɞ ɛɵɥ ɩɪɟɞɥɨɠɟɧ
ɜɝ ɂ ȽȻɭɛɧɨɜɵɦɢɧɟɡɚɜɢɫɢɦɨɨɬ ɧɟɝɨɜ ɝȻȽȽɚ
ɥɺɪɤɢɧɵɦ
54
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ȼ ɨɫɧɨɜɟ ɦɟɬɨɞɚ ɥɟɠɢɬ ɩɨɧɹɬɢɟ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɭɧɤɰɢɣ
Ⱦɜɟ ɮɭɧɤɰɢɢ:
ɬɟɪɜɚɥɟ l
d ɯd l
1
)( xf
ɢ
(ɚɧɚɥɨɝɢɱɧɨɢ ɜɤɚɤɨɣ-ɥɢɛɨɦɧɨɝɨɦɟɪɧɨɣɨɛɥɚ
2
– ɧɚɡɵɜɚɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɜ ɢɧ
)(xM
ɫɬɢɟɫɥɢɜɵɩɨɥɧɹɟɬɫɹɭɫɥɨɜɢɟ
l
2
.0)()(
M
³
l
1
ȿɫɥɢ ɨɞɧɚ ɢɡ ɮɭɧɤɰɢɣ ɧɚɩɪɢɦɟɪ
ɧɭɥɸɬɨ ɨɧɚ ɛɭɞɟɬ ɨɪɬɨɝɨɧɚɥɶɧɚɤ ɥɸɛɨɣ ɮɭɧɤɰɢɢ
ɦɟɪ ɮɭɧɤɰɢɹ
IV
qEIwxf
dxxxf
ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɚ
)( xf
ɇɚɩɪɢ
)( xiM
ɹɜɥɹɸɳɚɹɫɹ ɥɟɜɨɣ ɱɚɫɬɶɸ ɞɢɮɮɟ
,)(
ɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɡɨɝɧɭɬɨɣ ɨɫɢ ɛɚɥɤɢ ɞɥɢɧɨɣ l ɬɨɠɞɟ
ɫɬɜɟɧɧɨɪɚɜɧɚɧɭɥɸɩɪɢɜɫɟɯɡɧɚɱɟɧɢɹɯɯɩɨɷɬɨɦɭ
IV
³
l
i
M
dxxqEIw .0)()(
ȿɫɥɢ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ
ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ
N
¦
N
i
1
ɩɪɢɛɥɢɠɟɧɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)( xw
, ɬɨ ɜɵɪɚɠɟɧɢɟ
MD
xw
)(
ii
IV
)(
qEIwN
ɭɠɟ ɧɟ ɛɭɞɟɬ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɨ ɧɭɥɸ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟ ɛɭɞɟɬ
ɨɪɬɨɝɨɧɚɥɶɧɨ ɥɸɛɨɣ ɮɭɧɤɰɢɢ Ɍɨɝɞɚɩɨɬɪɟɛɭɟɦɱɬɨɛɵɨɧɨɛɵɥɨ
ɨɪɬɨɝɨɧɚɥɶɧɨɩɨɤɪɚɣɧɟɣɦɟɪɟɤɤɚɠɞɨɣɢɡɮɭɧɤɰɢɣ
ɥɹɸɳɢɯ
, ɬ ɟɱɬɨɛɵɜɵɩɨɥɧɹɥɢɫɶɭɫɥɨɜɢɹ
w
N
IV
N
³
l
i
M
NidxxqEIw
.,...,2,1,0)()(
ɫɨɫɬɚɜ
)( xiM
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ N ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ
ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ N ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɜɩɪɢɛɥɢɠɟɧɧɨɟɪɟɲɟɧɢɟ
Ɏɭɧɤɰɢɢ
)( xiM
w
.
N
ɧɚɡɵɜɚɸɬɫɹɛɚɡɢɫɧɵɦɢ, ɢɥɢ ɤɨɨɪɞɢɧɚɬɧɵɦɢ,
ɜɯɨɞɹɳɢɯ
D
i
ɮɭɧɤɰɢɹɦɢ Ɉɧɢ ɞɨɥɠɧɵ ɛɵɬɶ ɡɚɞɚɧɵ ɬɚɤɢɦ ɨɛɪɚɡɨɦɱɬɨɛɵ ɭɞɨ
ɜɥɟɬɜɨɪɹɬɶɭɫɥɨɜɢɹɦɧɚɝɪɚɧɢɰɚɯɨɛɥɚɫɬɢɄɪɨɦɟɬɨɝɨ ɫɢɫɬɟɦɚɛɚ
ɡɢɫɧɵɯɮɭɧɤɰɢɣ
i
ɫɢɦɨɣ ɬ ɟ ɧɢ ɨɞɧɭ ɢɡ ɮɭɧɤɰɢɣ
, ɞɨɥɠɧɚɛɵɬɶɥɢɧɟɣɧɨɧɟɡɚɜɢ
Nix
,...,2,1),( M
ɧɟɥɶɡɹ ɜɵɪɚɡɢɬɶ ɥɢɧɟɣɧɨɣ
)( xiM
ɤɨɦɛɢɧɚɰɢɟɣɞɪɭɝɢɯɮɭɧɤɰɢɣɷɬɨɣɠɟɫɢɫɬɟɦɵ
55
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɉɪɢɜɟɞɟɧɧɵɟ ɪɚɫɫɭɠɞɟɧɢɹ ɩɪɢɦɟɧɢɦɵ ɢ ɤ ɮɭɧɤɰɢɹɦ ɞɜɭɯ
ɢɛɨɥɟɟ ɩɟɪɟɦɟɧɧɵɯ ɍɫɥɨɜɢɟɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɭɧɤɰɢɣ
ɢ
ɜɞɜɭɦɟɪɧɨɣɨɛɥɚɫɬɢȍɡɚɩɢɲɟɬɫɹɬɚɤ
),( yxM
),( yxf
³³
dydxyxɭxf
:
.0),(),( M
ɉɪɟɞɫɬɚɜɢɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ
ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
q
wL
(2.14)
.0)(
D
Ɂɞɟɫɶɫɢɦɜɨɥɨɦ L ɨɛɨɡɧɚɱɟɧɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣɨɩɟɪɚɬɨɪɭɪɚɜɧɟ
ɧɢɹɢɡɝɢɛɚɬ ɟ
.
wwL4)(
Ȼɭɞɟɦɢɫɤɚɬɶɩɪɢɛɥɢɠɟɧɧɨɟɪɟɲɟɧɢɟɭɪɚɜɧɟɧɢɹɜɜɢɞɟ
N
(2.15)
,),(
yxw
jj
ɝɞɟ
M
j
N
í ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɞɥɟɠɚɳɢɟ ɨɩɪɟɞɟɥɟɧɢɸ
D
j
í ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɟ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ ɜɵɛɪɚɧɧɵɟ
),( yx
MD
1¦
j
ɬɚɤɢɦɨɛɪɚɡɨɦɱɬɨɛɵ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶ ɩɪɟɞɩɨɥɚɝɚɟɦɵɟ ɩɟɪɟɦɟ
ɳɟɧɢɹɜɧɚɩɪɚɜɥɟɧɢɢɨɫɢ z ɢɭɞɨɜɥɟɬɜɨɪɹɬɶɜɫɟɦɤɢɧɟɦɚɬɢɱɟɫɤɢɦ
ɢɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ>1].
ɉɨɞɫɬɚɜɢɜɜɭɪɚɜɧɟɧɢɟɩɨɥɭɱɢɦ
¦
j
N
L
1
q
jj
D
(2.16)
.0])([
MD
ɉɨɬɪɟɛɭɟɦɱɬɨɛɵɥɟɜɚɹɱɚɫɬɶɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ
(2.16) ɛɵɥɚ ɨɪɬɨɝɨɧɚɥɶɧɚ ɤɚɠɞɨɣ ɢɡ ɛɚɡɢɫɧɵɯ ɮɭɧɤɰɢɣ ɪɹɞɚ
. ɗɬɢ ɭɫɥɨɜɢɹ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ
Nix
i
,...,2,1),( M
ɪɚɡɦɟɪɚɦɢ a×b (0 d ɯd a,0d y d b) ɩɪɢɜɨɞɹɬɤɫɥɟɞɭɸɳɟɣɫɢɫɬɟ
ɦɟ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɫɤɨɦɵɯ
ɧɟɢɡɜɟɫɬɧɵɯ
N
¦
j
:
D
j
ab
dydxL
)(
1
ijj
ab
1
][
³³³³
D
0000
.,...,2,1
Ni
56
dydxyxq
MMMD
i
,0),(
(2.17)
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Ʉɚɠɞɨɟ i-ɟɭɪɚɜɧɟɧɢɟɷɬɨɣɫɢɫɬɟɦɵɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɜɢɞɟ
N
¦
j
ɢɥɢ
BA
D
i
1
jji
i
2211
BAAA
DDD
,...
NN
iii
ɝɞɟɤɨɷɮɮɢɰɢɟɧɬɵɩɪɢɧɟɢɡɜɟɫɬɧɵɯ
ab
jiji
;)(
ɢɫɜɨɛɨɞɧɵɟɱɥɟɧɵ
D
j
ab
1
BdydxLA
D
M MM
ii
³³³³
0000
.),(
dydxyxq
ȿɫɥɢɧɚɩɥɚɫɬɢɧɭ ɩɨɦɢɦɨɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢ ɞɟɣɫɬɜɭ
ɟɬɫɢɫɬɟɦɚɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯɫɢɥ
ab
BBB
1
³³
D
00
k
),(),(
iFiqii
dydxyxyxq
ɬɨ
nkF
,...,2,1,
F
n
F
1
D
¦
k
MM
1
.),(
yxF
kkik
ȼɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵɭɪɚɜɧɟɧɢɣ ɧɚɯɨɞɢɦ ɡɧɚɱɟɧɢɹ
ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɛɨɜ
),( yxw
. ȿɫɥɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɥɢɧɟɣ
N
ɧɨɟ ɬɨ ɢ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɢ ɬɟɦ ɫɚɦɵɦ ɩɪɢɛɥɢɠɟɧɧɭɸ ɮɭɧɤɰɢɸɩɪɨɝɢ
D
j
ɛɭɞɟɬ ɬɚɤɠɟ ɧɟɥɢ
D
j
ɧɟɣɧɨɣ
ȼɫɥɭɱɚɟ ɟɫɥɢ ɞɥɹɡɚɞɚɧɢɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɢɫɩɨɥɶ
ɡɭɟɬɫɹɨɞɧɚ ɛɚɡɢɫɧɚɹ ɮɭɧɤɰɢɹ ɬ ɟ
;)(
,AB D
ɝɞɟ
BdydxLA
abab
1
³³³³
D
0000
ɷɮɮɢɰɢɟɧɬ
dydxyxq
),(
ɬɨ ɢɫɤɨɦɵɣ ɤɨ
),( yxw MD
n
F
1
D
¦
k
MM MM
1
yxF
.),(
kkk
ɏɨɬɹ ɨɩɢɫɚɧɢɟ ɦɟɬɨɞɚ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɩɪɢɜɟɞɟɧɨ ɞɥɹ
ɨɞɧɨ- ɢ ɞɜɭɦɟɪɧɨɣ ɨɛɥɚɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɨɧ ɟɫɬɟɫɬɜɟɧɧɨ ɩɪɢ
ɦɟɧɢɦ ɢ ɞɥɹ ɬɪɟɯɦɟɪɧɵɯ ɡɚɞɚɱ ɗɬɨɬ ɦɟɬɨɞ ɩɪɢɦɟɧɢɦ ɬɚɤɠɟ
ɢɤɫɢɫɬɟɦɚɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
ȼɷɬɨɣɫɜɹɡɢɫɥɟɞɭɟɬɭɩɨɦɹɧɭɬɶɨɦɟɬɨɞɚɯɪɨɞɫɬɜɟɧɧɵɯ
ɆȻȽɗɬɨɨɛɨɛɳɟɧɧɵɣɦɟɬɨɞȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚɦɟɬɨɞɉɟɬɪɨɜɚ –
Ƚɚɥɺɪɤɢɧɚ ɦɟɬɨɞ Ʉɚɧɬɨɪɨɜɢɱɚ – ȼɥɚɫɨɜɚ ɦɟɬɨɞɵ ɤɨɥɥɨɤɚɰɢɣ
ɜ ɪɚɡɥɢɱɧɵɯ ɜɢɞɚɯ ɜɧɭɬɪɟɧɧɟɣ ɝɪɚɧɢɱɧɨɣ ɤɨɦɛɢɧɢɪɨɜɚɧɧɨɣ
ɤɨɥɥɨɤɚɰɢɢɦɟɬɨɞɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜɢɬɞ
57
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
y
(
)
(
)
()(
)
Ɍɚɤɜ ɦɟɬɨɞɟ ɉɟɬɪɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ
ɡɚɞɚɟɬɫɹɱɟɪɟɡɨɞɧɭɫɢɫɬɟɦɭɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣɚɭɫɥɨɜɢɹɨɪɬɨɝɨ
ɧɚɥɶɧɨɫɬɢɡɚɩɢɫɵɜɚɸɬɫɹɩɨɨɬɧɨɲɟɧɢɸɤɞɪɭɝɨɣɛɚɡɢɫɧɨɣɫɢɫɬɟɦɟ
ȿɫɥɢɞɜɟɷɬɢɫɢɫɬɟɦɵɫɜɹɡɚɧɵɦɟɠɞɭɫɨɛɨɣɩɨɫɪɟɞɫɬɜɨɦɥɢɧɟɣɧɨɝɨ
ɨɩɟɪɚɬɨɪɚɬɨɞɚɧɧɵɣɦɟɬɨɞɧɚɡɵɜɚɟɬɫɹɦɟɬɨɞɨɦɦɨɦɟɧɬɨɜ.
ɜɪɚɡ
D
j
ɥɨɠɟɧɢɢɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɢɳɭɬɫɹɢɡɭɫɥɨɜɢɣɱɬɨɛɵ
ɷɬɨ ɩɪɢɛɥɢɠɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɥɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ɜɡɚɞɚɧɧɵɯɬɨɱɤɚɯk (x
, y
ɧɚɡɵɜɚɟɦɵɯɭɡɥɚɦɢɤɨɥɥɨɤɚɰɢɢ,
k
k
ɩɪɢɷɬɨɦɫɯɨɞɢɦɨɫɬɶɦɟɬɨɞɚɡɚɜɢɫɢɬɨɬɜɵɛɨɪɚɷɬɢɯɭɡɥɨɜ
ɋɭɬɶɦɟɬɨɞɚɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ ɡɚɤɥɸɱɚɟɬɫɹɜɧɚɯɨɠɞɟ
ɧɢɢ ɢɫɤɨɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɢɡ ɭɫɥɨɜɢɹ ɦɢɧɢɦɢɡɚɰɢɢ ɜ ɡɚ
D
j
ɞɚɧɧɵɯ ɬɨɱɤɚɯ k ɧɟɜɹɡɤɢ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɜ ɮɨɪɦɟ
ɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ
n
N
k
§
¨
¦¦
¨
11
kj
©
L
jj
2
·
q
¸
MD D)
o
¸
D
¹
,min])([)(
ɝɞɟnk– ɱɢɫɥɨɬɨɱɟɤɤɨɥɥɨɤɚɰɢɢnk N).
ɆɟɬɨɞɄɚɧɬɨɪɨɜɢɱɚ – ȼɥɚɫɨɜɚ ɩɨɡɜɨɥɹɟɬ ɫɜɟɫɬɢ ɞɜɭɦɟɪɧɭɸ
ɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢ ɬɪɟɯɦɟɪɧɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɞɥɹ ɞɢɮɮɟɪɟɧ
ɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣɜɱɚɫɬɧɵɯɩɪɨɢɡɜɨɞɧɵɯɤɤɪɚɟɜɨɣɡɚɞɚɱɟ ɞɥɹ
ɫɢɫɬɟɦɵɨɛɵɤɧɨɜɟɧɧɵɯɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣȾɥɹɪɟɲɟ
ɧɢɹ ɬɚɤɢɯ ɫɢɫɬɟɦ ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɟ ɫɭ
ɳɟɫɬɜɭɟɬɞɨɫɬɚɬɨɱɧɨɷɮɮɟɤɬɢɜɧɵɯɦɟɬɨɞɨɜ
ɂɦɟɸɬɫɹɢ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɜ ɤɨɬɨɪɵɯɡɚɞɚɱɚ ɛɨɥɟɟ ɜɵɫɨɤɨɣ
ɪɚɡɦɟɪɧɨɫɬɢɫɜɨɞɢɬɫɹ ɤ ɡɚɞɚɱɟ ɦɟɧɶɲɟɣɪɚɡɦɟɪɧɨɫɬɢ ɩɪɚɜɞɚ ɤɚɤ
ɩɪɚɜɢɥɨɡɚɫɱɟɬɭɜɟɥɢɱɟɧɢɹɱɢɫɥɚɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣ
ɌɪɭɞɧɨɫɬɢɜɩɪɢɦɟɧɟɧɢɢɆȻȽɢɞɪɭɝɢɯɩɪɨɟɤɰɢɨɧɧɵɯɩɨɥɭ
ɚɧɚɥɢɬɢɱɟɫɤɢɯɦɟɬɨɞɨɜ ɜɨɫɧɨɜɧɨɦ ɫɜɹɡɚɧɵ ɫɩɨɞɛɨɪɨɦ ɛɚɡɢɫɧɵɯ
ɮɭɧɤɰɢɣ
ɬɚɤ ɤɚɤ ɩɨɫɥɟɞɧɢɟ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɧɟ ɬɨɥɶɤɨ
M
i
ɜɫɟɦɤɢɧɟɦɚɬɢɱɟɫɤɢɦɧɨɢɩɨɜɨɡɦɨɠɧɨɫɬɢ) ɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱ
ɧɵɦɭɫɥɨɜɢɹɦ
2.2.2. Ȼɚɡɢɫɧɵɟɮɭɧɤɰɢɢ
Ȼɚɡɢɫɧɵɟɮɭɧɤɰɢɢɤɚɤɭɠɟɝɨɜɨɪɢɥɨɫɶɞɨɥɠɧɵɩɪɟɞɫɬɚɜɥɹɬɶ
ɫɨɛɨɣ ɩɪɟɞɩɨɥɚɝɚɟɦɵɟ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ ɢ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɤɪɚɟ
58
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
y
(0)
(b)
(a)
ɜɵɦɭɫɥɨɜɢɹɦ ɡɚɞɚɱɢȼ ɤɚɱɟɫɬɜɟɛɚɡɢɫɧɵɯ ɮɭɧɤɰɢɣ ɨɛɵɱɧɨ ɩɪɢ
ɦɟɧɹɸɬɫɹɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɢɫɬɟɩɟɧɧɵɟ ɮɭɧɤɰɢɢ
Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɪɚɡɦɟɪɚɦɢ a×b ɜɵɛɨɪ ɛɚɡɢɫ
ɧɵɯɮɭɧɤɰɢɣ ɭɩɪɨɳɚɟɬɫɹ ɬɟɦ ɱɬɨ ɨɧɢ ɦɨɝɭɬ ɛɵɬɶɡɚɞɚɧɵ ɜ ɜɢɞɟ
ɩɪɨɢɡɜɟɞɟɧɢɹɞɜɭɯɮɭɧɤɰɢɣɨɬɨɞɧɨɝɨɚɪɝɭɦɟɧɬɚ:
,
)()(),(
yxyx MM M
21
ɝɞɟ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢy ɩɪɢɯ 0; a);
ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɧɚ ɤɪɚɹɯ
)(
xM
1
í ɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɧɚ
)(
yM
2
ɤɪɚɹɯɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢɯ ɩɪɢy=0; b).
Ɋɚɫɫɦɨɬɪɢɦɩɥɚɫɬɢɧɭɪɢɫ ɡɚɳɟɦɥɟɧɧɭɸ ɩɨɫɬɨɪɨɧɚɦ
ɩɚɪɚɥɥɟɥɶɧɵɦɨɫɢy (ɜɷɬɨɦɫɥɭɱɚɟ ɡɚɞɚɧɵɬɨɥɶɤɨɤɢɧɟɦɚɬɢɱɟɫɤɢɟ
ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɢ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɭɸ ɩɨ ɫɬɨɪɨɧɚɦ ɩɚɪɚɥ
ɥɟɥɶɧɵɦɨɫɢx ɫɦɟɲɚɧɧɵɟɝɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹɉɨɞɛɟɪɟɦɞɥɹɧɟɟ
ɛɚɡɢɫɧɵɟɮɭɧɤɰɢɢɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟɜɩɟɪɜɭɸɨɱɟɪɟɞɶɤɢɧɟɦɚɬɢ
ɱɟɫɤɢɦɢɩɨɜɨɡɦɨɠɧɨɫɬɢɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦ
0
a
y
w (0) = 0
wƍ(0) = 0
Ɋɢɫ 2.10ɉɪɹɦɨɭɝɨɥɶɧɚɹɩɥɚɫɬɢɧɚɫɝɪɚɧɢɱɧɵɦɢɭɫɥɨɜɢɹɦɢ
ɉɨɞɛɟɪɟɦ ɫɧɚɱɚɥɚɮɭɧɤɰɢɸ
w (a) = 0
wƍ
1
x
b
x
= 0
ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸɡɚɳɟɦ
)(
xM
w (0) =0
wƎ
w (b) =0
wƎ
= 0
= 0
ɥɟɧɢɸ ɩɨɤɪɚɹɦ ɯ 0, aɁɞɟɫɶɩɨɞɨɣɞɟɬɧɚɩɪɢɦɟɪɬɪɢɝɨɧɨɦɟɬɪɢ
ɱɟɫɤɚɹɮɭɧɤɰɢɹɜɢɞɚ
2
x
S
§
M
¨
©
·
mx
a
c
¸
11
¹
59
m
S
)(;2cos1)(
x
M
a
x
S
,2sin
m
a
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɝɞɟ m – ɰɟɥɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɩɨɞɫɬɚɜɢɜ
ɜɷɬɭ ɮɭɧɤɰɢɸɢɟɟɩɪɨɢɡɜɨɞɧɭɸɡɧɚɱɟɧɢɹ ɯ= 0 ɢ ɯ=a, ɩɨɥɭɱɢɦ
m
S
2
1
M
S M
c
M
)0(;0)11(0cos1)0(
1
c
11
a
S
m
2
M
ama
)(;0)11(2cos1)(
a
;00sin
S
m
.02sin
Ɍɟɩɟɪɶ ɩɨɞɛɟɪɟɦ ɮɭɧɤɰɢɸ
ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɲɚɪ
)(
yM
2
ɧɢɪɧɨɦɭɨɩɢɪɚɧɢɸɩɨɤɪɚɹɦ y=0; b. ȼ ɷɬɨɦɫɥɭɱɚɟ ɧɚɢɛɨɥɟɟɩɨɞ
ɯɨɞɹɳɟɣɹɜɥɹɟɬɫɹɮɭɧɤɰɢɹɫɢɧɭɫɚ
y
S
my
M
b
cc
y
22
22
SS
m
M
2
b
y
.sin)(;sin)(
m
b
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɩɪɢ y=0; b:
S
m
c
M
M
z
b
m
c
S M
22
S
M
bmb
zS
b
cc
222
cc
M
bm
2
22
m
S
M
2
b
22
m
S
2
b
;00sin)0(;00cos)0(;00sin)0(
S
m
ȼ ɢɬɨɝɟ ɛɚɡɢɫɧɚɹ ɮɭɧɤɰɢɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɝɨ ɜɢɞɚ ɞɥɹ
ɩɪɢɜɟɞɟɧɧɵɯɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɛɭɞɟɬɢɦɟɬɶ ɜɢɞ
x
S
§
M
¨
©
·
myx
¸
a
¹
y
S
.sin2cos1),(
m
b
ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ ɩɪɢɛɥɢɠɟɧɧɚɹ ɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ
ɩɪɢ ɤɨɧɟɱɧɨɦ ɱɢɫɥɟ ɱɥɟɧɨɜ ɜ ɪɚɡɥɨɠɟɧɢɢ ɦɨɠɟɬ
),( yxw
ɛɵɬɶɡɚɞɚɧɚɫɥɟɞɭɸɳɢɦɨɛɪɚɡɨɦ
NN
ª
§
D MD
m
mm
¦¦
m
¨
m
«
©
¬
11
S
x
·
myxyxw
¸
a
¹
S
y
º
.sin2cos1),(),(
m
»
b
¼
ɇɚɩɪɢɦɟɪɩɪɢɞɜɭɯɱɥɟɧɚɯɫɭɦɦɵɜɪɚɡɥɨɠɟɧɢɢ
.0sin)(;0cos)(;0sin)(
x
§
yxw
¨
©
S
a
y
S
60
§
¨
b
©
·
¸
¹
x
S
·
¸
a
¹
y
S
.2sin4cos1sin2cos1),(
b
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