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ɩɪɢ ɧɚɱɚɥɟ ɞɟɣɫɬɜɢɹ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ, ɭɠɟ ɩɪɚɤɬɢɱɟɫɤɢ ɡɚɬɭɯɥɢ, ɢ ɩɪɨɰɟɫɫ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɪɢɧɹɥ ɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɯɚɪɚɤɬɟɪ.
ɉɟɪɟɦɟɳɟɧɢɟ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɹɯ
ɉɪɟɞɫɬɚɜɢɦ ɧɚɣɞɟɧɧɨɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɟ ɬɨɱɟɱɧɨɣ
ɦɚɫɫɵ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
t
ɜɜɢɞɟ:
)()(
ɫɬ
tutu E ,
(1.89)
ɝɞɟ
G
Z
o
o
P
m
P
u
2
ɫɬ
;
(1.90)
WWZTZ E
WH
³
dtett
t
t
)(sinsin)(
)(
0
= )sin(
D
TE
t
.
(1.91)
Ɂɞɟɫɶ:
x ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɞɚɬɥɢɜɨɫɬɢ
2
/
1
Z
G m
ɢɡɜɟɫɬɟɧ ɢɡ (1.13);
x
ɫɬ
u ɹɜɥɹɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢɦ ɩɪɨɝɢɛɨɦ ɛɚɥɤɢ ɜ ɦɟɫɬɟ ɪɚɫɩɨɥɨɠɟɧɢɹ ɦɚɫɫɵ
ɨɬ ɫɬɚɬɢɱɟɫɤɢ ɩɪɢɥɨɠɟɧɧɨɣ ɫɢɥɵ
o
P ;
x )(
t
E ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɧɚɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ (ɢɫɩɨɥɶɡɭɟɬɫɹ
ɬɚɤɠɟ ɧɚɡɜɚɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɧɚɦɢɱɧɨɫɬɢ), ɤɨɬɨɪɵɣ ɯɚɪɚɤɬɟɪɢɡɭɟɬ
ɨɬɧɨɲɟɧɢɟ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ )(
t
u (1.89) ɤ ɫɬɚɬɢɱɟɫɤɨɦɭ
ɩɟɪɟɦɟɳɟɧɢɸ
ɫɬ
u (1.90).
ɉɨɞɫɬɚɜɢɜ (1.91) ɜ (1.89), ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɞɢɧɚɦɢɱɟɫɤɨɝɨ
ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɜ ɜɢɞɟ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ:
)sin()(
DT
t
A
t
u ,
(1.92)
ɝɞɟ
A
ɹɜɥɹɟɬɫɹ ɚɦɩɥɢɬɭɞɨɣ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɣ, ɪɚɜɧɨɣ
ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ )(
t
u , ɤɨɝɞɚ
1)sin(
r MT
t
:
E
ɫɬ
)(max utuA
t
.
(1.93)
ɉɪɢ ɷɬɨɦ
)(max t
t
E E =
4
2
22
2
2
4)1(
1
Z
T
H
Z
T
(1.94)
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɧɚɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ, ɪɚɜɧɵɣ ɦɚɤɫɢɦɚɥɶɧɨɦɭ
ɡɧɚɱɟɧɢɸ ɜɟɥɢɱɢɧɵ )(
t
E (1.91).
ɉɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɡɚɞɚɱɢ ɨ ɫɜɨɛɨɞɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɡɚɬɭɯɚɸɳɢɯ
ɤɨɥɟɛɚɧɢɹɯ ɫ ɱɚɫɬɨɬɨɣ
Z|Z
D
ɤɨɷɮɮɢɰɢɟɧɬ H (1.63), ɩɨɥɭɱɟɧɧɵɣ ɩɪɢ
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ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɝɢɩɨɬɟɡɵ «ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ» Ɏɨɣɝɬɚ, ɛɵɥ ɫɤɨɪɪɟɤɬɢɪɨɜɚɧ ɢ
ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ (1.80) ɫ ɭɱɟɬɨɦ (1.81).
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜɵɪɚɠɟɧɢɣ (1.80) ɢ (1.81) ɜɦɟɫɬɨ (1.94) ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ:
2
2
2
ɧ
2
2
2
2
2
22
2
2
4
2
22
2
2
)1(
1
4)1(
1
4)1(
1
Z
T
J
Z
T
Z
T
]
Z
T
Z
T
H
Z
T
E
.
(1.95)
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɜɵɪɚɠɟɧɢɹ (1.92), ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ
ɤɨɥɟɛɚɧɢɹɯ ɢɡɦɟɧɟɧɢɟ ɩɟɪɟɦɟɳɟɧɢɹ )(
t
u ɜɨ ɜɪɟɦɟɧɢ ɩɪɨɢɫɯɨɞɹɬ ɫ ɱɚɫɬɨɬɨɣ T
ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ tPtP T sin)(
o
, ɧɨ ɢɦɟɸɬ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɟɣ ɫɞɜɢɝ ɩɨ
ɮɚɡɟ ɧɚ ɜɟɥɢɱɢɧɭ D .
ɂɧɨɝɞɚ ɜɵɪɚɠɟɧɢɟ (1.92) ɩɪɟɞɫɬɚɜɥɹɸɬ ɜ ɜɢɞɟ:
tAtAtAtu T
cc
T
c
DT cossin)sin()(,
(1.96)
ɝɞɟ D
cc
D
c
sin;cos
AAAA – ɫɨɫɬɚɜɥɹɸɳɢɟ ɩɨɥɧɨɣ ɚɦɩɥɢɬɭɞɵ
A
ɜɵɧɭɠɞɟɧɧɵɯ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ. ȼɫɜɹɡɢɫɷɬɢɦ
ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɪɚɫɫɦɨɬɪɢɦ ɜɨɩɪɨɫ ɨ ɩɨɥɭɱɟɧɢɢ ɮɨɪɦɭɥ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ
D
ccc
ɢ,,
AAA .
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɜ ɜɢɞɟ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (1.96) ɢ
ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɢɥɵ ɜ ɜɢɞɟ tPtP
o
T sin)( ɩɨɞɫɬɚɜɥɹɸɬ ɜ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ
ɤɨɥɟɛɚɧɢɣ (1.86). Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (1.86) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
ttmPtNtL
o
TT TT cos0sin)/(cossin ,
ɝɞɟ
AANAAL
cc
TZ
c
HT
cc
HT
c
TZ )(2;2)(
2222
.
ɉɪɢɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɫɢɧɭɫɟ ɢ ɤɨɫɢɧɭɫɟ ɩɪɟɨɛɪɚɡɨɜɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ,
ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
A
A
ccc
ɢ
:
.0)(2
);/(2)(
22
o
22
cc
TZ
c
TH
cc
TH
c
TZ
AA
mPAA
Ɉɬɫɸɞɚ
.4)(
;/2;/)(
22222
o
22
o
THTZ '
'TH
cc
'TZ
c
mPAmPA
(1.97)
ɉɨ ɚɧɚɥɨɝɢɢ ɫ ɬɟɦ, ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɹɯ (ɫɦ. (1.17)
ɢ ɪɢɫ. 1.3), ɜɵɪɚɠɟɧɢɟ (1.96) ɫɜɹɡɵɜɚɸɬ ɫ ɜɪɚɳɟɧɢɟɦ ɫ ɪɚɜɧɨɦɟɪɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ
T
ɜɟɤɬɨɪɚ D
c
cosAA ɢ ɨɬɫɬɚɸɳɟɝɨ ɨɬ ɧɟɝɨ ɧɚ ɭɝɨɥ 2/S ɜɟɤɬɨɪɚ D
cc
sinAA (ɪɢɫ. 1.19)
ɜɨɤɪɭɝ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ 0 [14].
ɇɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ
A
cc
ɜɵɛɪɚɧɨ ɬɚɤɢɦ, ɱɬɨɛɵ ɭɱɟɫɬɶ ɡɧɚɤ ɦɢɧɭɫ ɜ (1.96). Ɍɨɝɞɚ,
ɟɫɥɢ ɫɩɪɨɟɤɬɢɪɨɜɚɬɶ ɜɟɤɬɨɪɵ
AAA
ccc
,, ɧɚ ɨɫɶ X (ɫɦ. ɪɢɫ. 1.19), ɩɨɥɭɱɚɬɫɹ ɥɟɜɚɹ ɢ ɩɪɚɜɚɹ
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ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (1.96).
Ɋɢɫ. 1.19
ɋɭɦɦɚɪɧɚɹ ɚɦɩɥɢɬɭɞɚ A ɢ ɬɚɧɝɟɧɫ ɭɝɥɚ
D ɬɟɩɟɪɶ ɨɩɪɟɞɟɥɹɬɫɹ (ɫɦ. ɪɢɫ. 1.19) ɢɡ
ɮɨɪɦɭɥ:
¸
¸
¹
·
¨
¨
©
§
TZ
TH
D
TZ
TH
c
cc
DE
cc
c
)(
2
arctg;
)(
2
tg;)()(
2222
ɫɬ
22
A
A
uAAA
,
(1.98)
ɝɞɟ ɮɨɪɦɭɥɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɫɬ
u ɢ E ɢɦɟɸɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɢɞ (1.90) ɢ (1.95).
ɉɨɞɫɬɚɜɥɹɹ ɜ (1.98) ɜɵɪɚɠɟɧɢɟ (1.80) ɞɥɹ
H ɫ ɭɱɟɬɨɦ (1.81) ɮɨɪɦɭɥɵ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ
D ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:
2
2
ɧ
2
2
2
2
2
1
tgarc
1
2
tgarc
1
2
tgarc
Z
T
Z
T
J
Z
T
Z
T
]
Z
T
Z
T
H
D .
(1.99)
ȼɬɨɪɚɹ ɢ ɬɪɟɬɶɹ ɮɨɪɦɭɥɵ ɜ (1.95) ɢ (1.99) ɩɨɥɭɱɢɥɢɫɶ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɞɥɹ
ɤɨɪɪɟɤɬɢɪɨɜɤɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ
H
ɜ ɭɪɚɜɧɟɧɢɢ (1.86) ɜɵɪɚɠɟɧɢɹ (1.80). ȼ
ɪɹɞɟ ɪɚɛɨɬ, ɫɩɢɫɨɤ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɩɨɫɦɨɬɪɟɬɶ ɜ ɪɚɛɨɬɚɯ [5, 11], ɩɨɤɚɡɚɧɨ, ɱɬɨ ɜɟɥɢɱɢɧɚ
H
ɩɪɢ ɜɵɧɭɠɞɟɧɧɵɯ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɹɯ ɫ ɱɚɫɬɨɬɨɣ
T ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ ɡɚɜɢɫɢɬ ɧɟ
ɬɨɥɶɤɨ ɨɬ ɜɟɥɢɱɢɧɵ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɵ
Z, ɧɨ ɢ ɨɬ ɱɚɫɬɨɬɵ T ɢ ɩɪɢ ɤɨɪɪɟɤɬɢɪɨɜɤɟ
ɜɟɥɢɱɢɧɵ
H
ɜɦɟɫɬɨ ɮɨɪɦɭɥɵ (1.80) ɞɨɥɠɧɚ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɚ:
)/(
2
TZ] H
.
(1.100)
ɉɪɢ ɷɬɨɦ ɮɨɪɦɭɥɵ (1.95) ɢ (1.99) ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɧɟɭɩɪɭɝɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɧ
J
ɛɭɞɭɬ ɢɦɟɬɶ ɜɢɞ:
;
)1(
1
2
ɧ
2
2
2
J
Z
T
E
2
2
ɧ
1
tgarc
Z
T
J
D .
(1.101)
Ⱥɧɚɥɢɡ ɢɡɦɟɧɟɧɢɹ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ E ɢ ɚɦɩɥɢɬɭɞɵ
A
.
əɜɥɟɧɢɟ ɪɟɡɨɧɚɧɫɚ
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɚ ɜɚɪɢɚɧɬɚ:
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1. ɇɟɭɩɪɭɝɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɤɨɥɟɛɚɧɢɹɦ ɨɬɫɭɬɫɬɜɭɟɬ (ɧɚɩɪɢɦɟɪ, ɜ (1.95), (1.99) ɢ (1.101)
0
ɧ
J
);
2. ɂɦɟɟɬɫɹ ɪɟɚɥɶɧɨɟ ɧɟɭɩɪɭɝɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɤɨɥɟɛɚɧɢɹɦ (ɡɚɞɚɧɧɨɟ, ɧɚɩɪɢɦɟɪ, ɜ ɜɢɞɟ
ɤɨɷɮɮɢɰɢɟɧɬɚ
ɧ
J
).
ȼɚɪɢɚɧɬ 1 ( 0
ɧ
J
) . Ɍɨɝɞɚ (ɫɦ. (1.91), (1.99), (1.101))
tAtu T sin)(,
(1.102)
))/(1(
1
2
ZT
E
.
(1.103)
ɉɪɢ ɷɬɨɦ (ɫɦ. (1.93)) E
ɫɬ
uA .
ɉɪɨɫɥɟɞɢɦ ɜ ɷɬɨɦ ɜɚɪɢɚɧɬɟ ɢɡɦɟɧɟɧɢɟ ɚɛɫɨɥɸɬɧɵɯ ɡɧɚɱɟɧɢɣ (ɦɨɞɭɥɟɣ)
E (1.103) ɢ
A
(1.93) ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɢɡɦɟɧɟɧɢɹ ɨɬɧɨɲɟɧɢɹ
Z
T/ (ɪɢɫ. 1.20) [8].
Ƚɪɚɮɢɤɭ ɢɡɦɟɧɟɧɢɹ ɡɧɚɱɟɧɢɣ ɦɨɞɭɥɹ E ɩɪɢ
0
ɧ
J
(1.93) ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɫɥɟɞɭɸɳɢɟ
ɨɪɞɢɧɚɬɵ (ɫɦ. ɪɢɫ.1.20):
ZT/
0.25 0.5 0.75 0.875 1.25
2
1.5 1.75 2 2.5
E
1.07 1.33 2.29 4.29 3.76 1 0.8 0.48 0.33 0.19
ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɨɬɧɨɲɟɧɢɹ ZT/ ɨɬ 0 ɞɨ 1 ɩɪɨɢɫɯɨɞɢɬ ɭɜɟɥɢɱɟɧɢɟ ɦɨɞɭɥɹ ɞɢɧɚɦɢɱɟɫɤɨɝɨ
ɤɨɷɮɮɢɰɢɟɧɬɚ E . ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɡɪɚɫɬɚɧɢɹ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɩɪɢɛɥɢɠɟɧɢɟɦ ɨɬɧɨɲɟɧɢɹ
ZT/
ɤ 1. Ⱥɧɚɥɨɝɢɱɧɨ ɢɡɦɟɧɹɟɬɫɹ ɢ ɚɦɩɥɢɬɭɞɚ A (1.93).
Ɋɢɫ. 1.20
ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɨɬɧɨɲɟɧɢɹ
ZT/ ɨɬ 1 ɞɨ f ɩɪɨɢɫɯɨɞɢɬ ɭɦɟɧɶɲɟɧɢɟ ɡɧɚɱɟɧɢɣ ɦɨɞɭɥɹ
ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ E ɢ ɚɦɩɥɢɬɭɞɵ
A ɨɬ
f
ɞɨ 0 (ɫɧɚɱɚɥɚ ɛɵɫɬɪɨɟ, ɡɚɬɟɦ
44
ɦɟɞɥɟɧɧɨɟ).
Ɉɬɦɟɬɢɦ ɜ ɬɚɛɥɢɰɟ, ɩɪɢɜɟɞɟɧɧɨɣ ɧɢɠɟ ɞɚɧɧɨɝɨ ɚɛɡɚɰɚ, ɯɚɪɚɤɬɟɪɧɵɟ ɡɧɚɱɟɧɢɹ ɦɨɞɭɥɟɣ E
ɢ
A ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɯɚɪɚɤɬɟɪɧɵɯ ɨɬɧɨɲɟɧɢɹɯ ZT/ . Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɩɪɢ
ɨɬɧɨɲɟɧɢɢ
41.12/ ZT ɡɧɚɱɟɧɢɟ ɦɨɞɭɥɹ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɜɧɨ ɟɞɢɧɢɰɟ, ɚ
ɩɪɢ
2/ !ZT
ɦɨɞɭɥɶ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ E ɛɭɞɟɬ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ, ɢɦɨɞɭɥɶ
ɚɦɩɥɢɬɭɞɵ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɣ E
ɫɬ
uA ɛɭɞɟɬ ɦɟɧɶɲɟ ɦɨɞɭɥɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɩɪɨɝɢɛɚ
ɫɬ
u ɨɬ ɫɢɥɵ
o
P .
1
0/ ZT
1 E
ɫɬ
uA
2
1/ ZT
f E
f A
3
41.12/ ZT
1 E
ɫɬ
uA
4
f ZT/
0 E
0 A
ȼ ɪɟɚɥɶɧɵɯ ɫɨɨɪɭɠɟɧɢɹɯ ɡɧɚɱɟɧɢɣ f E ɢ f
A ɛɵɬɶ ɧɟ ɦɨɠɟɬ. Ɍɚɤ ɱɬɨ ɜɚɪɢɚɧɬ 1
(
0
ɧ
J ) ɹɜɥɹɟɬɫɹ ɱɢɫɬɨ ɬɟɨɪɟɬɢɱɟɫɤɢɦ. ɇɨ ɞɚɠɟ, ɟɫɥɢ 0
ɧ
J , ɜɟɥɢɱɢɧɵ AɢE ɦɨɝɭɬ ɫɬɚɬɶ
ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɦɢ ɬɨɥɶɤɨ ɡɚ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɨɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ. ɗɬɨ ɩɨɤɚɡɚɧɨ ɜ
ɪɚɛɨɬɟ [11], ɝɞɟ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɜɚɪɢɚɧɬɚ ɩɪɢɜɟɞɟɧɨ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ,
ɤɨɝɞɚ
1/ ZT :
tt
m
P
tu Z
Z
sin
2
)(
o
.
(1.104)
ɉɪɢ ɧɚɲɢɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɢɦɟɟɬ ɜɢɞ (1.89), ɝɞɟ
ttt
Z
Z E sin5.0)( .
(1.105)
Ƚɪɚɮɢɱɟɫɤɢ ɢɡɦɟɧɟɧɢɟ
)(tE
(1.105), ɚ ɡɧɚɱɢɬ ɢ )(
t
u (1.89), ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ
ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 1.21.
Ɋɢɫ. 1.21
Ʉɚɤ ɜɢɞɧɨ, ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɜɪɟɦɟɧɢ
t
ɩɪɨɢɫɯɨɞɢɬ ɧɟɨɝɪɚɧɢɱɟɧɧɨɟ ɜɨɡɪɚɫɬɚɧɢɟ
ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɢ ɚɦɩɥɢɬɭɞ ɤɨɥɟɛɚɧɢɣ, ɚ ɫɚɦ ɩɪɨɰɟɫɫ ɤɨɥɟɛɚɧɢɣ, ɤɨɝɞɚ
1/ ZT , ɢɦɟɟɬ ɜɢɞ ɩɨɫɬɟɩɟɧɧɨɣ ɪɚɫɤɚɱɤɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɭɩɪɭɝɨɣ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ
ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ. ɇɢɠɟ ɩɪɢɜɟɞɟɧɵ ɡɧɚɱɟɧɢɹ ɚɦɩɥɢɬɭɞ ɤɨɥɟɛɚɧɢɣ, ɩɨɥɭɱɟɧɧɵɟ ɩɨ ɮɨɪɦɭɥɟ
(1.104) ɩɪɢ ɭɤɚɡɚɧɧɵɯ ɜ ɬɚɛɥɢɰɟ ɡɧɚɱɟɧɢɹɯ
t
:
45
t
ZS 2/ ZS 2/3 ZS 2/5 ZS 2/7
tutA Z
ɫɬ
5.0)( 4/
ɫɬ
uS 4/3
ɫɬ
uS 4/5
ɫɬ
uS 4/7
ɫɬ
uS
ɋɨɜɩɚɞɟɧɢɟ ɱɚɫɬɨɬɵ T ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ ɫ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ
Z
ɧɟɡɚɬɭɯɚɸɳɢɯ
ɤɨɥɟɛɚɧɢɣ ɫɢɫɬɟɦɵ ɢ ɜɨɡɧɢɤɚɸɳɟɟ ɩɪɢ ɷɬɨɦ ɹɜɥɟɧɢɟ ɪɚɫɤɚɱɤɢ ɢ ɜɨɡɪɚɫɬɚɧɢɹ ɚɦɩɥɢɬɭɞ
ɤɨɥɟɛɚɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɪɟɡɨɧɚɧɫɨɦ.
ɉɪɢ ɨɱɟɧɶ ɧɟɛɨɥɶɲɨɦ ɪɚɡɥɢɱɢɢ ɜ ɜɟɥɢɱɢɧɚɯ
T ɢ Z ɜɨɡɧɢɤɚɸɬ ɤɨɥɟɛɚɧɢɹ ɫ ɹɜɥɟɧɢɟɦ
ɛɢɟɧɢɹ [11, 15]. ɗɬɨ ɫɨɫɬɨɹɧɢɟ ɫɨɨɪɭɠɟɧɢɹ ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɪɟɡɨɧɚɧɫɧɵɦ.
Ɂɨɧɚ ɡɧɚɱɟɧɢɣ ɨɬɧɨɲɟɧɢɹ
ZT/ , ɜ ɤɨɬɨɪɨɣ ɞɚɠɟ ɧɟɛɨɥɶɲɨɟ ɢɡɦɟɧɟɧɢɟ ɨɬɧɨɲɟɧɢɹ ZT /
ɩɪɢɜɨɞɢɬ ɤ ɡɧɚɱɢɬɟɥɶɧɨɦɭ ɢɡɦɟɧɟɧɢɸ ɜɟɥɢɱɢɧ AɢE , ɧɚɡɵɜɚɟɬɫɹ ɪɟɡɨɧɚɧɫɧɨɣ ɡɨɧɨɣ.
ɒɢɪɢɧɚ ɪɟɡɨɧɚɧɫɧɨɣ ɡɨɧɵ (ɩɨ ɨɫɢ ɡɧɚɱɟɧɢɣ
ZT/ ) ɭɫɥɨɜɧɚ ɢ ɡɚɜɢɫɢɬ ɨɬ ɤɪɢɬɟɪɢɟɜ,
ɩɪɢɧɢɦɚɟɦɵɯ ɩɪɢ ɟɟ ɨɩɪɟɞɟɥɟɧɢɢ (ɧɚɩɪɢɦɟɪ, ɤɪɢɬɟɪɢɹ ɨ ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɹɯ ɜɟɥɢɱɢɧ
AɢE ɢɥɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɦ ɜɟɥɢɱɢɧ ɩɟɪɟɦɟɳɟɧɢɣ, ɭɫɢɥɢɣ ɢ ɧɚɩɪɹɠɟɧɢɣ ɩɪɢ ɪɚɫɱɟɬɟ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɭɩɪɭɝɨɝɨ ɫɨɨɪɭɠɟɧɢɹ ɧɚ ɠɟɫɬɤɨɫɬɶ ɢ ɩɪɨɱɧɨɫɬɶ). Ɋɚɡɥɢɱɧɵɟ
ɩɪɢɛɥɢɠɟɧɧɵɟ ɨɰɟɧɤɢ ɲɢɪɢɧɵ ɷɬɨɣ ɡɨɧɵ ɞɚɧɵ, ɧɚɩɪɢɦɟɪ, ɜ ɭɱɟɛɧɵɯ ɩɨɫɨɛɢɹɯ [9, 11].
ɂɡ ɫɨɨɛɪɚɠɟɧɢɣ ɞɨɩɭɫɬɢɦɨɫɬɢ ɡɧɚɱɟɧɢɣ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɦɨɠɟɬ ɛɵɬɶ ɭɫɬɚɧɨɜɥɟɧɨ ɢ
ɜɪɟɦɹ
ɞɨɩ
t ɛɟɡɨɩɚɫɧɨɣ ɞɥɹ ɫɨɨɪɭɠɟɧɢɹ ɪɚɫɤɚɱɤɢ ɩɪɢ ɪɟɡɨɧɚɧɫɟ.
ȼɚɪɢɚɧɬ 2 ( 01
ɧ
!J!! ). ɉɪɢ ɧɚɥɢɱɢɢ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɚ ɫɨɨɪɭɠɟɧɢɹ
(ɩɪɢ ɤɨɧɤɪɟɬɧɨɦ ɡɧɚɱɟɧɢɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɡɚɬɭɯɚɧɢɹ) ɢ ɩɪɢ ɨɬɧɨɲɟɧɢɢ
1/ ZT
ɞɢɧɚɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ E ɢ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ A ɭɠɟ ɧɟ ɛɭɞɭɬ ɛɟɫɤɨɧɟɱɧɨ
ɛɨɥɶɲɢɦɢ. ɂɯ ɤɨɧɟɱɧɵɟ ɡɧɚɱɟɧɢɹ ɨɩɪɟɞɟɥɹɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢɡ ɜɵɪɚɠɟɧɢɣ (1.106) ɩɨɫɥɟ
ɩɨɞɫɬɚɧɨɜɤɢ ɜ ɧɢɯ ɡɚɞɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ
ɧ
J
.
ɧ
2
ɧɧ
ɫɬ
ɧ
1
;
1
J
Z
J
G
J
J
E
m
PPu
A
oo
(1.106)
Ʉɚɤ ɜɢɞɧɨ (ɫɦ. ɪɢɫ. 1.20), ɢɡɦɟɧɟɧɢɟ ɡɧɚɱɟɧɢɹ
ɧ
J ɜ ɜɵɪɚɠɟɧɢɢ
E
(1.101) ɡɚɦɟɬɧɨ ɜɥɢɹɟɬ
ɧɚ ɢɡɦɟɧɟɧɢɟ E ɢ A ɬɨɥɶɤɨ ɜ ɪɟɡɨɧɚɧɫɧɨɣ ɡɨɧɟ, ɬ.ɟ. ɩɪɢ ɡɧɚɱɟɧɢɹɯ ɨɬɧɨɲɟɧɢɹ
ZT/
ɛɥɢɡɤɨɦ
ɤ
1/ ZT
.
ȼɧɟ ɪɟɡɨɧɚɧɫɧɨɣ ɡɨɧɵ, ɬɨɱɧɟɟ ɩɪɢ ɭɞɚɥɟɧɢɢ ɨɬɧɨɲɟɧɢɹ
ZT/ (ɜɥɟɜɨ ɢ ɜɩɪɚɜɨ ɧɚ ɪɢɫ. 1.20)
ɨɬ ɡɧɚɱɟɧɢɹ
1/ ZT , ɢɡɦɟɧɟɧɢɟ ɡɧɚɱɟɧɢɹ
ɧ
J ɜ ɜɵɪɚɠɟɧɢɢ
E
ɦɚɥɨ ɜɥɢɹɟɬ ɧɚ ɢɡɦɟɧɟɧɢɟ
ɡɧɚɱɟɧɢɟ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ.
Ⱥɧɚɥɢɡ ɢɡɦɟɧɟɧɢɹ ɜɟɥɢɱɢɧɵ ɫɞɜɢɝɚ ɩɨ ɮɚɡɟ D ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɟ )(tP
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɤɨɥɟɛɚɧɢɹɦ, ɤɨɝɞɚ, ɧɚɩɪɢɦɟɪ,
0
ɧ
J , ɡɧɚɱɟɧɢɟ 0 D (1.99). ɋɢɥɚ
)(tP
ɢ ɩɪɨɝɢɛ
)(tu
ɜ ɷɬɨɦ ɜɚɪɢɚɧɬɟ ɢɡɦɟɧɹɸɬɫɹ
ɫɢɧɯɪɨɧɧɨ ɩɨ ɡɚɤɨɧɭ tTsin ɛɟɡ ɫɞɜɢɝɚ ɩɨ ɮɚɡɟ.
ɉɪɢ
1/ ZT ɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ 0
ɧ
!J ɩɨɥɭɱɢɦ 2/S
D . Ɍɚɤɨɣ ɫɞɜɢɝ ɩɨ ɮɚɡɟ ɦɟɠɞɭ
ɫɢɥɨɣ ɢ ɩɟɪɟɦɟɳɟɧɢɟɦ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɪɢ ɞɨɫɬɢɠɟɧɢɢ ɫɢɥɨɣ tPtP T sin)(
o
:
x ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ
o
P (ɧɚɩɪɢɦɟɪ, ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ 4/
o
Tt , ɤɨɝɞɚ
46
1)
4
2
sin(sin
S
T
o
o
T
T
t ) ɩɟɪɟɦɟɳɟɧɢɟ ɞɜɢɠɭɳɟɣɫɹ ɦɚɫɫɵ ɛɭɞɟɬ ɪɚɜɧɨ ɧɭɥɸ
0)
24
2
sin()(
o
S
S
T
T
Atu
o
;
x ɧɭɥɟɜɨɝɨ ɡɧɚɱɟɧɢɹ (ɧɚɩɪɢɦɟɪ, ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ 2/
o
Tt , ɤɨɝɞɚ
0)
2
2
sin(sin
o
S
T
T
T
t
o
) ɩɟɪɟɦɟɳɟɧɢɟ ɞɜɢɠɭɳɟɣɫɹ ɦɚɫɫɵ ɛɭɞɟɬ ɪɚɜɧɨ
ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸ
A
T
T
Atu
S
S
)
22
2
sin()(
o
o
.
ȼɨ ɜɫɟɯ ɷɬɢɯ ɜɵɪɚɠɟɧɢɹɯ ɜɟɥɢɱɢɧɚ TS /2
o
T ɹɜɥɹɟɬɫɹ ɩɟɪɢɨɞɨɦ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ
ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɫ ɱɚɫɬɨɬɨɣ
T ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ.
ɉɪɢ
1/ ZT ( 1/ !ZT ) ɢ ɥɸɛɵɯ ɡɧɚɱɟɧɢɹɯ 0
ɧ
!J ɩɨɥɭɱɢɦ 2/SD ( 2/S!D ).
ɏɚɪɚɤɬɟɪ ɢɡɦɟɧɟɧɢɹ ɜɟɥɢɱɢɧɵ
D
ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɢɡɦɟɧɟɧɢɹ ɨɬɧɨɲɟɧɢɹ
ZT/
ɩɪɢ
ɧɟɤɨɬɨɪɵɯ ɡɚɞɚɧɧɵɯ ɩɨɫɬɨɹɧɧɵɯ ɡɧɚɱɟɧɢɹɯ
0
ɧ
!J ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ. 1.22.
ɑɟɦ ɦɟɧɶɲɟ ɡɧɚɱɟɧɢɟ
ɧ
J , ɬɟɦ ɫɢɥɶɧɟɟ ɢɡɦɟɧɹɟɬɫɹ ɮɭɧɤɰɢɹ D ɜ ɡɨɧɟ ɪɟɡɨɧɚɧɫɚ. ȼ ɩɪɟɞɟɥɟ
(ɩɪɢ
0
ɧ
J ɢ ɩɪɢ
1/ ZT
) ɩɪɨɢɫɯɨɞɢɬ ɫɤɚɱɤɨɨɛɪɚɡɧɨɟ ɢɡɦɟɧɟɧɢɟ
D
: ɨɬ ɧɭɥɟɜɨɝɨ ɡɧɚɱɟɧɢɹ
ɩɪɢ
1/ ZT
ɞɨ
S
(ɲɬɪɢɯɨɜɚɹ ɥɢɧɢɹ ɧɚ ɪɢɫ. 1. 22).
ɉɪɢ
S D
ɜɨɡɦɭɳɚɸɳɚɹ ɫɢɥɚ
tPtP T sin)(
o
ɢ ɩɟɪɟɦɟɳɟɧɢɟ ɦɚɫɫɵ
)
2
sin()(
o
D
S
t
T
Atu ɧɚɯɨɞɹɬɫɹ ɜ ɩɪɨɬɢɜɨɮɚɡɟ. ɗɬɨ ɛɭɞɟɬ, ɧɚɩɪɢɦɟɪ, ɤɨɝɞɚ 4/
o
Tt , ɬ. ɤ. ɜ
ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
o
o
o
)
4
2
sin()( P
T
T
PtP
o
S
ɢ A
T
T
Atu
o
S
S
)
4
2
sin()(
o
.
Ɋɢɫ. 1.22
ɋɭɦɦɚɪɧɚɹ ɫɢɥɚ )(
t
S
ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ
ɢ ɟɟ ɫɢɧɭɫɨɢɞɚɥɶɧɚɹ ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɢɟ
ɉɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ (1.92) ɭɪɚɜɧɟɧɢɹ (1.86) ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɚɝɪɭɡɤɢ
47
tPtP
o
T sin)( ɫɨɩɨɫɬɚɜɢɦ ɫ ɭɪɚɜɧɟɧɢɟɦ ɞɜɢɠɟɧɢɹ (1.84), ɤɨɬɨɪɨɟ, ɜɜɟɞɹ
ɨɛɨɡɧɚɱɟɧɢɟ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ
)()()()(
ɧ
tRtJtPtS
, ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ (1.84)
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɪɚɜɟɧɫɬɜɨ
)sin()(
o
DT tStS
,
(1.107)
ɝɞɟ
E
GE
G
GE G
ooɫɬo
/)(/)(/ PPuAS
(1.108)
ɹɜɥɹɟɬɫɹ ɚɦɩɥɢɬɭɞɨɣ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ )(
t
S
.
Ɉɩɪɟɞɟɥɢɜ ɚɦɩɥɢɬɭɞɭ
o
S ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ
ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɪɨɝɢɛ ɛɚɥɤɢ
2
o
o
Z
G
m
S
SA
(1.109)
ɢ ɨɩɪɟɞɟɥɢɬɶ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ ɛɚɥɤɢ ɢ ɭɫɢɥɢɹ ɜ ɟɟ ɫɟɱɟɧɢɹɯ.
ɂɧɨɝɞɚ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨɛ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ
ɤɨɥɟɛɚɧɢɹɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟ ɩɨɥɧɚɹ ɚɦɩɥɢɬɭɞɚ
o
S ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ )(
t
S
, ɚ
ɚɦɩɥɢɬɭɞɵ
S
S
ccc
ɢ
ɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɢɯ
ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɟɟ ɜ ɜɢɞɟ
tStStStS
o
T
c
c
T
c
DT cossin)sin()(.
(1.110)
Ɏɨɪɦɭɥɵ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɚɦɩɥɢɬɭɞ
S
S
cc
c
ɢ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɢɥ, ɚ ɬɚɤɠɟ
ɩɨɥɧɨɣ ɚɦɩɥɢɬɭɞɵ
o
S , ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢɡ ɢɦɟɸɳɢɯɫɹ ɮɨɪɦɭɥ (1.97) ɢ (1.98)
ɞɥɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɦ ɚɦɩɥɢɬɭɞ
A
A
ccc
ɢ ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɟɪɟɦɟɳɟɧɢɣ, ɚ
ɬɚɤɠɟ ɩɨɥɧɨɣ ɚɦɩɥɢɬɭɞɵ
A
, ɫ ɭɱɟɬɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɟɲɟɧɢɹ (1.92) ɜ ɜɢɞɟ
(1.96):
tAtAtAtu T
cc
T
c
DT cossin)sin()(.
ɉɪɟɨɛɪɚɡɭɟɦ ɩɟɪɜɭɸ ɮɨɪɦɭɥɭ (1.101) ɢ ɮɨɪɦɭɥɵ (1.97), ɩɪɟɞɫɬɚɜɢɜ ɢɯ ɫ
ɭɱɟɬɨɦ (1.96) ɢ (1.110) ɜ ɜɢɞɟ:
2
ɧ
2
1
JF
E
, )1(
2
2
Z
T
F
.
G
c
Z
c
FE
Z
c
S
m
S
m
P
A
o
2
2
2
; G
cc
Z
cc
JE
Z
cc
S
m
S
m
P
A
o
2
ɧ
2
2
.
ɝɞɟ
FE
c
Z
c
22
o
PAmS ;
ɧ
22
JE
cc
Z
cc
o
PAmS .
(1.111)
ȼ ɩɨɫɥɟɞɧɢɯ ɜɵɪɚɠɟɧɢɹɯ ɞɥɹ
A
c
ɢ
A
cc
ɭɱɬɟɧɨ, ɱɬɨ G Z )/1(
2
m .
ɉɨɥɧɚɹ ɚɦɩɥɢɬɭɞɚ
A
ɩɟɪɟɦɟɳɟɧɢɣ )(
t
u (1.96) (ɫɦ. ɪɢɫ. 1.18,ɚ) ɩɪɢ
ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ ɩɨ ɮɨɪɦɭɥɟ
48
(1.98) ɚ ɬɚɤ ɠɟ ɩɨ ɮɨɪɦɭɥɟ (1.109), ɝɞɟ
22
o
)()( SSS
cc
c
.
(1.112)
ɇɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ ɚɦɩɥɢɬɭɞɚ
o
S
ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ ɢɦɟɟɬ, ɤɨɝɞɚ ɩɪɢ
ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ ɫɢɥɚ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ (ɩɪɢɧɢɦɚɟɬɫɹ
ɭɫɥɨɜɢɟ, ɱɬɨ
0
ɧ
J
). Ɍɨɝɞɚ ɫɢɥɵ )(ɢ)(
t
J
t
P
ɞɟɣɫɬɜɭɸɬ ɫɢɧɯɪɨɧɧɨ (ɫɢɧɮɚɡɧɨ),
ɛɟɡ ɫɞɜɢɝɚ ɩɨ ɮɚɡɟ (
0 D ), ɢ ɫɢɥɚ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɨɬɫɭɬɫɬɜɭɟɬ.
ɍɫɢɥɢɹ ɢ ɧɚɩɪɹɠɟɧɢɹ ɜ ɛɚɥɤɟ ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɹɯ
ȼɚɪɢɚɧɬ 1
( 0
ɧ
J
). ɉɨɫɤɨɥɶɤɭ ɜ ɷɬɨɦ ɜɚɪɢɚɧɬɟ ɢɡɦɟɧɟɧɢɟ ɜɨ ɜɪɟɦɟɧɢ
ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ ɩɪɨɢɫɯɨɞɢɬ ɫɢɧɯɪɨɧɧɨ, ɬɨ ɦɚɤɫɢɦɚɥɶɧɵɯ
ɡɧɚɱɟɧɢɣ ɷɬɢ ɫɢɥɵ ɢ ɫɭɦɦɚɪɧɚɹ ɫɢɥɚ (1.110) ɞɨɫɬɢɝɚɸɬ ɜ ɨɞɢɧɚɤɨɜɵɟ ɦɨɦɟɧɬɵ
ɜɪɟɦɟɧɢ, ɧɚɩɪɢɦɟɪ, ɤɨɝɞɚ
4/
o
Tt (ɩɪɢ ɷɬɨɦ 1sin T
t
).
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɜɨ ɜɪɟɦɟɧɢ ɥɸɛɵɯ
ɞɢɧɚɦɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɛɟɡ
ɭɱɟɬɚ ɫɢɥ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɬɚɤɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɪɚɫɱɟɬɚ:
1. ȼɵɩɨɥɧɹɟɬɫɹ ɪɚɫɱɟɬ ɛɚɥɤɢ ɨɬ ɞɟɣɫɬɜɢɹ ɫɬɚɬɢɱɟɫɤɨɣ ɫɢɥɵ
o
P
(ɪɢɫ. 1.23, ɚ): ɨɩɪɟɞɟɥɹɸɬɫɹ ɨɩɨɪɧɵɟ ɪɟɚɤɰɢɢ 2/
o
PVV
o
A
o
A
, ɩɪɨɝɢɛ ɛɚɥɤɢ
G
oɫɬ
Pu ; ɫɬɪɨɹɬɫɹ ɷɩɸɪɵ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɢ ɩɨɩɟɪɟɱɧɵɯ ɫɢɥ;
ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɨɪɦɚɥɶɧɵɟ ɢ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɢɧɬɟɪɟɫɭɸɳɢɯ
ɪɚɫɱɟɬɱɢɤɚ ɫɟɱɟɧɢɹɯ ɛɚɥɤɢ.
ɇɚ ɪɢɫ. 1.23 ɩɪɢɜɟɞɟɧɵ:
ɚ) ɷɩɸɪɚ
ɫɬ
o
MM ɫ ɭɤɚɡɚɧɢɟɦ ɦɚɤɫɢɦɚɥɶɧɨɣ
ɨɪɞɢɧɚɬɵ ɜ ɫɪɟɞɧɟɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ, ɤɨɝɞɚ ɫɢɥɚ
o
P
ɧɚɩɪɚɜɥɟɧɚ ɜɧɢɡ; ɛ) ɷɩɸɪɚ
ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ
o
V ɫ ɭɤɚɡɚɧɢɟɦ ɨɪɞɢɧɚɬ WM /
o
max
o
2,1
r V ɧɚ ɧɢɠɧɟɣ
(1) ɢ ɜɟɪɯɧɟɣ (2) ɫɬɨɪɨɧɚɯ ɫɟɱɟɧɢɹ ɛɚɥɤɢ ɫ ɦɨɦɟɧɬɨɦ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
W .
Ɋɢɫ. 1.23
49
2. ɉɨɞɫɱɢɬɵɜɚɟɬɫɹ ɞɢɧɚɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ E , ɧɚɩɪɢɦɟɪ ɩɨ ɮɨɪɦɭɥɟ
(1.103).
3. Ɉɩɪɟɞɟɥɹɸɬɫɹ ɚɦɩɥɢɬɭɞɧɵɟ ɡɧɚɱɟɧɢɹ ɫɥɟɞɭɸɳɢɯ ɜɟɥɢɱɢɧ ɜ ɫɪɟɞɧɟɦ
ɫɟɱɟɧɢɢ ɛɚɥɤɢ:
– ɩɟɪɟɦɟɳɟɧɢɣ ɦɚɫɫɵ
2
oɫɬ
/)( ZE E mPuA ;
– ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ
E
oooo
PJPS
, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɛɚɥɤɭ ɜ ɦɟɫɬɟ
ɪɚɫɩɨɥɨɠɟɧɢɹ ɦɚɫɫɵ.
– ɨɪɞɢɧɚɬɵ ɷɩɸɪɵ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ
E
ɫɬɞ
MM , ɝɞɟ
ɫɬ
M –
ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɨɪɞɢɧɚɬɚ ɷɩɸɪɵ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɨɬ ɞɟɣɫɬɜɢɹ ɫɢɥɵ
o
P
(ɫɦ. ɪɢɫ.1.23,
ɚ);
– ɨɪɞɢɧɚɬɵ ɷɩɸɪɵ ɧɨɪɦɚɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɜ ɬɨɱɤɚɯ 1 ɢ 2 ɩɨɩɟɪɟɱɧɨɝɨ
ɫɟɱɟɧɢɹ ɛɚɥɤɢ )(
o
2,12,1
EVr V , ɝɞɟ
o
2,1
V –ɦɚɤɫɢɦɚɥɶɧɚɹ ɨɪɞɢɧɚɬɚ ɷɩɸɪɵ
ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɜ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɨɬ ɞɟɣɫɬɜɢɹ ɫɢɥɵ
o
P
(ɫɦ. ɪɢɫ. 1.23,
ɛ).
ȼɚɪɢɚɧɬ 2 (01
ɧ
!J!! ). ȼ ɷɬɨɦ ɜɚɪɢɚɧɬɟ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɚɦɩɥɢɬɭɞɧɵɯ
ɡɧɚɱɟɧɢɣ ɭɫɢɥɢɣ ɢ ɧɚɩɪɹɠɟɧɢɣ ɨɛɵɱɧɨ ɩɨɫɬɭɩɚɸɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
1. Ɉɩɪɟɞɟɥɹɟɦ ɞɢɧɚɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ
E , ɧɚɩɪɢɦɟɪ, ɩɨ ɮɨɪɦɭɥɟ
(1.101).
2. Ɂɚɬɟɦ ɩɨ ɮɨɪɦɭɥɚɦ (1.111) ɨɩɪɟɞɟɥɹɟɦ ɚɦɩɥɢɬɭɞɵ
S
S
ccc
ɢ
ɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ.
3. ɉɨ ɮɨɪɦɭɥɚɦ (1.97) ɨɩɪɟɞɟɥɹɟɦ ɚɦɩɥɢɬɭɞɵ
A
A
ccc
ɢ
ɩɟɪɟɦɟɳɟɧɢɣ
ɦɚɫɫɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɞɟɣɫɬɜɢɸ ɧɚ ɛɚɥɤɭ ɫɢɥ
S
S
ccc
ɢ
.
4. Ɉɬɞɟɥɶɧɨ ɨɬ ɞɟɣɫɬɜɢɹ ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ
S
S
ccc
ɢ
ɫɢɧɭɫɨɢɞɚɥɶɧɨɣ
ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɭɦɦɚɪɧɨɣ ɧɚɝɪɭɡɤɢ ɫɬɪɨɢɦ ɞɥɹ ɛɚɥɤɢ
ɧɟɨɛɯɨɞɢɦɵɟ ɷɩɸɪɵ ɭɫɢɥɢɣ. Ɉɝɪɚɧɢɱɢɜɲɢɫɶ ɜ ɛɚɥɤɟ ɪɚɫɫɦɨɬɪɟɧɢɟɦ ɬɨɥɶɤɨ
ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ, ɩɨɫɬɪɨɢɦ ɷɩɸɪɵ
M
M
c
c
c
ɢ ɢ ɨɩɪɟɞɟɥɢɦ ɦɚɤɫɢɦɚɥɶɧɵɟ
ɚɦɩɥɢɬɭɞɵ ɦɨɦɟɧɬɨɜ ɜ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɩɨɞ ɫɢɥɨɣ:
maxmax
ɢ MM
ccc
.
5. ɉɨɫɤɨɥɶɤɭ ɚɦɩɥɢɬɭɞɧɵɟ ɡɧɚɱɟɧɢɹ ɫɢɥ
S
S
ccc
ɢ ɧɟ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ
ɞɪɭɝɨɦ ɜɪɟɦɟɧɧɵɦ ɮɚɤɬɨɪɨɦ, ɜɫɟ ɫɭɦɦɚɪɧɵɟ ɚɦɩɥɢɬɭɞɧɵɟ ɡɧɚɱɟɧɢɹ ɢɫɤɨɦɵɯ
ɜɟɥɢɱɢɧ (
o
S – ɩɨɥɧɨɣ ɚɦɩɥɢɬɭɞɵ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ )(
t
S
,
A
– ɩɨɥɧɨɣ
ɚɦɩɥɢɬɭɞɵ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ )(
t
u ,
max
M – ɩɨɥɧɨɣ ɚɦɩɥɢɬɭɞɵ ɢɡɝɢɛɚɸɳɟɝɨ
ɦɨɦɟɧɬɚ ɜ ɫɪɟɞɧɟɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ ɢ ɬ. ɞ.) ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ ɜɢɞɚ
(1.98) ɞɥɹ ɚɦɩɥɢɬɭɞɵ ɩɟɪɟɦɟɳɟɧɢɹ ɢ (1.112) ɞɥɹ ɚɦɩɥɢɬɭɞɵ ɫɭɦɦɚɪɧɨɣ ɫɢɥɵ.
ɉɨɷɬɨɦɭ ɩɨɥɧɚɹ ɚɦɩɥɢɬɭɞɚ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɜ ɫɪɟɞɧɟɦ ɫɟɱɟɧɢɢ ɛɚɥɤɢ
ɨɩɪɟɞɟɥɢɬɫɹ ɩɨ ɮɨɪɦɭɥɟ: