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16
x
cx
cd
P
x
g
P
EX
. (2)
ȼ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɤ ɝɪɭɡɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ
:
Ɋ
–
ɟɝɨ ɜɟɫ
,
ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɧɢɡ
,
ɫɬɚɬɢɱɟɫɤɚɹ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ
F
ɫɬ
=cd
,
ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ
.
Ɍɚɤ ɤɚɤ ɝɪɭɡ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ
,
ɬɨ
0
cd
P
.
(3)
ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
(2)
ɜ ɜɢɞɟ
0
2
2
x
k
x
n
x
,
(4)
ɝɞɟ
P
g
n
P
cg
k
x
x
2
,
,
E
X
,
ɩɨɥɭɱɚɟɦ
k = 10 c
-1
, n = 8
ɫ
-1
,
ɬɚɤɢɦ
ɨɛɪɚɡɨɦ
,
n < k
.
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
(4):
.
,
0
2
2
2
2
2
2
,
1
2
2
n
k
i
n
k
n
n
k
n
r
r
O
O
O
ɋɥɟɞɨɜɚɬɟɥɶɧɨ
,
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ
))
sin(
)
cos(
(
2
2
2
2
2
1
t
n
k
c
t
n
k
c
e
x
nt
. (5)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
,
ɩɨɥɭɱɚɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
2
2
0
0
2
0
1
,
n
k
nx
x
c
x
c
.
ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ
,
ɨɩɪɟɞɟɥɹɹ
D
D
cos
,
sin
2
2
0
0
0
A
n
k
nx
x
A
x
.
(6)
Ɍɟɩɟɪɶ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
)
sin(
2
2
D
t
n
k
Ae
x
nt
.
(7)
Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɡɚɬɭɯɚɸɳɢɦ
(
ɬ
.
ɤ
.
ɩɪɢ
t
ĺ
x
ĺ
0)
ɫ ɤɪɭɝɨɜɨɣ
ɱɚɫɬɨɬɨɣ
2
2
n
k
k
c
.
ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɨɪɦɭɥɵ
,
ɧɚɯɨɞɢɦ
Ⱥ
=7,2
ɫɦ
,
Į
=0,59
ɪɚɞ
,
k
c
=6
ɫ
-1
.
ɂɬɚɤ
,
ɝɪɭɡ ɫɨɜɟɪɲɚɟɬ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ
ɫɦ
t
e
x
t
)
59
,
0
6
sin(
2
,
7
8
. (8)
ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ
ɫ
k
T
c
c
05
,
1
2
S
.
Ɂɚɞɚɱɚ ʋ
2.
Ɋɟɲɢɬɶ ɩɪɟɞɵɞɭɳɭɸ ɡɚɞɚɱɭ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ
,
ɱɬɨ ɫɢɥɚ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɪɚɜɧɚ
R=
ȕȣ
,
ɝɞɟ ȕ
=5,2 H
ɫ
/
ɫɦ
.
ȼ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɝɪɭɡ ɛɵɥ ɫɦɟɳɟɧ ɢɡ ɩɨɥɨɠɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɧɚ
4
ɫɦ
,
ɢ ɟɦɭ ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɜɟɪɯ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ȣ
0
=240
ɫɦ
/
ɫ
.
Ɋɟɲɟɧɢɟ
.
ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ ɩɪɭɠɢɧɟ
,
ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ
ɜɨɡɶɦɟɦ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
.
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
17
ɧɚɱɚɥɶɧɵɟ
ɭɫɥɨɜɢɹ
ɞɜɢɠɟɧɢɹ
ɝɪɭɡɚ
ɢɦɟɸɬ
ɜɢɞ
0
0
4
ɩɪɢ
0
.
240
/
x
x
ɫɦ
t
x
x
ɫɦ ɫ
®
¯
ɋɥɟɞɭɹ ɪɟɲɟɧɢɸ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ
,
ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ
0
2
kx
x
n
x
,
ɝɞɟ
.
2
,
,
P
g
n
P
cg
k
x
x
E
X
ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
,
ɩɨɥɭɱɚɟɦ
k=10 c
-1
, n=26
ɫ
-1
,
ɬɚɤɢɦ
ɨɛɪɚɡɨɦ
,
n>k
(
ɫɥɭɱɚɣ ɛɨɥɶɲɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
).
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
,
0
2
2
2
k
n
O
O
ɟɝɨ ɤɨɪɧɢ
ɪɚɜɧɵ
.
,
2
2
2
2
2
1
n
k
n
k
n
n
O
O
Ɍɚɤ ɤɚɤ
n > k
,
ɬɨ ɤɨɪɧɢ Ȝ
1
ɢ Ȝ
2
ɹɜɥɹɸɬɫɹ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɢ
ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ
.
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ
t
t
e
c
e
ɫ
x
2
1
2
1
O
O
.
(9)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
,
ɧɚɣɞɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
:
2
1
0
0
1
2
2
1
0
0
2
1
,
O
O
O
O
O
O
x
x
c
x
x
c
.
ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
(1)
ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
:
>
@
t
t
e
x
x
e
x
x
x
1
2
)
(
)
(
1
0
0
2
0
0
1
2
1
O
O
O
O
O
O
. (10)
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɧɚɱɟɧɢɹɦɢ Ȝ
1
ɢ Ȝ
2
ɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ
ɮɭɧɤɰɢɹɦɢ
,
ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
(2)
ɜ ɜɢɞɟ
>
@
t
k
n
ch
k
n
x
k
n
sh
nx
x
k
n
e
x
nt
2
2
2
2
0
2
2
0
0
2
2
)
(
. (11)
Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɚɩɟɪɢɨɞɢɱɟɫɤɢɦ ɢ ɩɪɢɬɨɦ ɡɚɬɭɯɚɸɳɢɦ
,
ɬ
.
ɤ
.
ɩɪɢ
t
ĺ
x
ĺ
0.
ɉɨɞɫɬɚɜɢɦ ɜ
(3)
ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
,
ɩɨɥɭɱɢɦ
)
5
29
(
6
1
24
24
26
t
t
t
e
e
e
x
ɢɥɢ
)
24
17
24
12
(
3
1
26
t
sh
t
ch
e
x
t
ȼɵɹɫɧɢɦ
,
ɩɟɪɟɯɨɞɢɬ ɥɢ ɝɪɭɡ
ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
:
0
)
5
29
(
6
1
24
24
26
t
t
t
e
e
e
.
ȼɵɱɢɫɥɹɹ
,
ɩɨɥɭɱɚɟɦ
t
1
=0,037
ɫ
, t
2
=
.
18
Ɂɧɚɱɟɧɢɟ
t
1
ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɩɟɪɟɯɨɞɭ
ɝɪɭɡɚ
ɱɟɪɟɡ
ɩɨɥɨɠɟɧɢɟ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
, t
2
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɚɬɭɯɚɧɢɸ ɞɜɢɠɟɧɢɹ
.
ɂɬɚɤ
,
ɜ
ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɝɪɭɡ ɩɪɨɯɨɞɢɬ ɨɞɢɧ ɪɚɡ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ
ɪɚɜɧɨɜɟɫɢɹ ɢ ɡɚɬɟɦ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɤ ɧɟɦɭ ɩɪɢɛɥɢɠɚɟɬɫɹ ɫ ɞɪɭɝɨɣ
ɫɬɨɪɨɧɵ
.
§5.
ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ
Ɂɚɞɚɱɚ ʋ
1.
ɇɚ
ɪɢɫɭɧɤɟ
ɢɡɨɛɪɚɠɟɧɚ ɫɯɟɦɚ ɩɪɢɛɨɪɚ ɞɥɹ
ɢɡɦɟɪɟɧɢɹ ɞɚɜɥɟɧɢɹ
.
Ʉ ɩɨɥɡɭɧɭ Ⱥ
ɜɟɫɨɦ
Ɋ
=196
Ƚ
ɩɪɢɤɪɟɩɥɟɧɚ
ɫɬɪɟɥɤɚ
ȼ
,
ɨɬɦɟɱɚɸɳɚɹ
ɩɨɤɚɡɚɧɢɹ
ɧɚ
ɧɟɩɨɞɜɢɠɧɨɣ
ɲɤɚɥɟ
ɋ
.
ɉɨɥɡɭɧ
Ⱥ
,
ɩɪɢɤɪɟɩɥɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ
D,
ɩɟɪɟɦɟɳɚɟɬɫɹ
ɩɨ
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ
ɢɞɟɚɥɶɧɨ
ɝɥɚɞɤɨɣ ɩɥɨɫɤɨɫɬɢ
.
Ʉ ɩɨɥɡɭɧɭ
ɩɪɢɥɨɠɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ
S = H·sin(pt),
ɝɞɟ ɇ
= 1,6
ɤȽ
,
ɪ
= 60
ɫ
-1
.
Ʉɨɷɮɮɢɰɢɟɧɬ
ɭɩɪɭɝɨɫɬɢ ɪɚɜɟɧ ɫ
= 2
ɤȽ
/
ɫɦ
.
ȼ
ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ
ɩɨɥɡɭɧ
ɧɚɯɨɞɢɥɫɹ ɜ ɩɨɤɨɟ
,
ɜ ɩɨɥɨɠɟɧɢɢ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
.
Ɉɩɪɟɞɟɥɢɬɶ
:
1)
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɜ ɫɥɭɱɚɟ ɨɬɫɭɬɫɬɜɢɹ ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
;
2)
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɩɪɢ
ɧɚɥɢɱɢɢ
ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
,
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ
ɩɟɪɜɨɣ
ɫɬɟɩɟɧɢ
ɫɤɨɪɨɫɬɢ
ɩɨɥɡɭɧɚ
R=
ȕȣ
,
ɝɞɟ ȕ
=25,6
Ƚ ɫ
/
ɫɦ
.
Ɋɟɲɟɧɢɟ
.
ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɜɩɪɚɜɨ
,
ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ
ɩɨɥɨɠɟɧɢɢ ɩɨɥɡɭɧɚ
,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɟ
.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ
:
0
,
0
0
ɩɪɢ
x
x
t
.
ɂɡɨɛɪɚɡɢɦ ɩɨɥɡɭɧ ɫɦɟɳɟɧɧɵɦ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɩɪɚɜɨ ɧɚ ɯ
.
ɉɪɢ ɷɬɨɦ ɩɪɭɠɢɧɚ ɪɚɫɬɹɧɟɬɫɹ ɧɚ
D =
ɯ
.
Ʉ ɩɨɥɡɭɧɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ
:
Ɋ
–
ɜɟɫ ɩɨɥɡɭɧɚ
, N –
ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ
,
ɫɢɥɚ
S,
ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɪɚɫɬɹɧɭɬɨɣ
ɩɪɭɠɢɧɵ
F.
ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ
ɯ
:
x
x
F
S
x
m
ɢɥɢ
x
P
cg
pt
P
Hg
x
sin
,
ɨɬɤɭɞɚ
,
sin
2
pt
k
x
k
x
(1)
ɝɞɟ
P
hg
h
P
cg
k
,
.
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
k=100
ɫ
-1
, h=8000
ɫɦ
/
ɫ
-2
.
Ɋɟɲɚɹ
(1),
ɧɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
19
)
sin(
)
cos(
2
1
1
kt
c
kt
c
x
. (2)
ɑɚɫɬɧɨɟ
ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ
(1)
ɩɪɢɦɟɦ
ɜ
ɜɢɞɟ
)
cos(
)
sin(
2
pt
B
pt
A
x
,
ɬɨɝɞɚ
)
sin(
2
2
2
pt
p
k
h
x
. (3)
Ɂɚɩɢɲɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ
(1),
ɫɥɨɠɢɜ
(2)
ɢ
(3):
)
sin(
)
sin(
)
cos(
2
2
2
1
pt
p
k
h
kt
c
kt
c
x
.
(4)
ɂɫɩɨɥɶɡɭɹ
ɧɚɱɚɥɶɧɵɟ
ɭɫɥɨɜɢɹ
,
ɨɩɪɟɞɟɥɢɦ
ɩɨɫɬɨɹɧɧɵɟ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
:
2
2
2
1
,
0
p
k
h
k
p
c
c
.
ɂɬɚɤ
,
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɟ ɫɬɪɟɥɤɢ
)
sin(
)
sin(
2
2
2
2
pt
p
k
h
kt
p
k
h
k
p
x
. (5)
ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
,
ɩɨɥɭɱɢɦ
c
ɦ
t
t
x
))
60
sin(
25
,
1
)
100
sin(
75
,
0
(
. (6)
Ɋɟɲɢɦ ɷɬɭ ɡɚɞɚɱɭ ɫ ɭɱɟɬɨɦ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ
.
Ʉ ɫɢɥɚɦ
,
ɪɚɧɟɟ ɩɪɢɥɨɠɟɧɧɵɦ ɤ ɩɨɥɡɭɧɭ
,
ɞɨɛɚɜɥɹɟɬɫɹ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ
R,
ɧɚɩɪɚɜɥɟɧɧɚɹ ɜ ɫɬɨɪɨɧɭ
,
ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ
.
x
x
x
R
F
S
x
m
ɢɥɢ
x
P
g
x
P
cg
pt
P
Hg
x
E
sin
,
ɨɬɤɭɞɚ
,
sin
2
2
pt
k
x
k
x
n
x
(7)
ɝɞɟ
P
hg
h
P
g
n
P
cg
k
,
2
,
E
.
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
k = 100
ɫ
-1
, h = 80
ɫɦ ɫ
-2
,
n = 64
ɫ
-1
,
ɪ
= 60
ɫ
-1
.
ɂɬɚɤ
,
n < k
ɢ
p < k
.
8
,
0
4
)
(
8
,
76
2
2
2
2
2
2
2
p
n
p
k
h
a
n
k
87
,
0
2
2
2
p
k
np
arctg
H
.
Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ
,
ɜɵɛɢɪɚɹ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ
)
sin(
2
H
pt
a
x
,
ɩɨɥɭɱɢɦ
).
87
,
0
60
sin(
8
,
0
)
8
,
76
sin
8
,
76
cos
(
2
1
64
t
t
c
t
c
e
x
t
(8)
ɂɫɩɨɥɶɡɭɹ
ɧɚɱɚɥɶɧɵɟ
ɭɫɥɨɜɢɹ
,
ɨɩɪɟɞɟɥɢɦ
ɩɨɫɬɨɹɧɧɵɟ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
:
ɫ
1
= 0,62;
ɫ
2
= 0,12.
ɉɟɪɟɩɢɲɟɦ ɮɨɪɦɭɥɭ
(9):
).
87
,
0
60
sin(
8
,
0
)
8
,
76
sin
12
,
0
8
,
76
cos
62
,
0
(
64
t
t
t
e
x
t
(9)
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
:
0,62=bsin
Į
;
0,12=bcos
Į
,
ɩɨɥɭɱɢɦ
b=0,63,
Į
=1,74
.
ɂɬɚɤ
,
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ Ⱥ ɢ ɫɬɪɟɥɤɢ ȼ ɢɦɟɟɬ ɜɢɞ
:
>
@
.
)
87
,
0
60
sin(
8
,
0
)
74
,
1
8
,
76
sin(
63
,
0
64
ɫɦ
t
t
e
x
t
(10)
20
ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɹ ɨɩɪɟɞɟɥɹɬ ɤɨɥɟɛɚɧɢɟ ɫɬɪɟɥɤɢ ɫ ɱɚɫɬɨɬɨɣ
ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ
,
ɤɨɬɨɪɵɟ ɛɵɫɬɪɨ ɡɚɬɭɯɚɸɬ ɛɥɚɝɨɞɚɪɹ ɧɚɥɢɱɢɸ
ɦɧɨɠɢɬɟɥɹ
e
-64t
.
ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬ ɜɵɧɭɠɞɟɧɧɵɟ
ɤɨɥɟɛɚɧɢɹ ɫɬɪɟɥɤɢ ȼ
.
§6.
Ɋɟɡɨɧɚɧɫ
Ɂɚɞɚɱɚ
ʋ
1.
Ɉɩɪɟɞɟɥɢɬɶ
ɭɪɚɜɧɟɧɢɟ
ɞɜɢɠɟɧɢɹ
ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ Ɇ ɜɟɫɨɦ
Ɋ
=196
Ƚ
,
ɞɜɢɠɭɳɟɣɫɹ ɜɞɨɥɶ ɨɫɢ ɯ
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ
F
ɢ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ
S.
ɉɪɨɟɤɰɢɢ
ɷɬɢɯ ɫɢɥ ɧɚ ɨɫɶ ɯ ɪɚɜɧɵ
: F
x
= –cx,
S
x
= H·sin(pt),
ɝɞɟ
ɫ
= 2
ɤȽ
/
ɫɦ
,
ɇ
= 1,6
ɤȽ
,
ɪ
= 101
ɫ
-1
.
ȼ
ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ
ɬɨɱɤɚ
ɧɚɯɨɞɢɥɚɫɶ ɜ ɩɨɤɨɟ ɜ ɧɚɱɚɥɟ
ɨɬɫɱɟɬɚ
ɨɫɢ
ɯ
.
ɋɢɥɨɣ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɞɜɢɠɟɧɢɸ
ɩɪɟɧɟɛɪɟɱɶ
.
Ɋɟɲɟɧɢɟ
.
Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ Ɇ ɜ
ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ
x
x
S
F
x
m
ɢɥɢ
pt
H
cx
x
m
sin
pt
h
x
k
x
sin
2
,
(1)
ɝɞɟ
2
1
1
2
10000
,
101
,
8000
/ .
c
H
k
ɫ
p
ɫ
h
ɫɦ ɫ
m
m
.
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɤɨɥɟɛɚɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜɛɥɢɡɢ ɪɟɡɨɧɚɧɫɚ
(
ɪɟɡɨɧɚɧɫ
ɢɦɟɟɬ ɦɟɫɬɨ ɩɪɢ
p=k)
ɜ ɡɨɧɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɛɨɥɶɲɨɣ ɱɚɫɬɨɬɵ
(p>k).
Ɂɧɚɱɢɬ
,
ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
pt
p
k
h
kt
p
k
h
k
p
x
sin
sin
2
2
2
2
. (2)
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ
,
ɢɫɤɨɦɨɟ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɪɟɡɭɥɶɬɚɬɨɦ
ɧɚɥɨɠɟɧɢɹ ɞɜɭɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ
,
ɩɪɨɢɫɯɨɞɹɳɢɯ ɫ ɩɨɱɬɢ
ɪɚɜɧɵɦɢ ɤɪɭɝɨɜɵɦɢ ɱɚɫɬɨɬɚɦɢ ɫɜɨɛɨɞɧɵɯ
k
ɢ ɜɵɧɭɠɞɟɧɧɵɯ
ɪ
ɤɨɥɟɛɚɧɢɣ
.
Ɍ
.
ɤ
.
k
§
p
,
ɬɨ ɛɭɞɟɦ ɫɱɢɬɚɬɶ
.
2
2
,
1
p
k
k
p
k
p
|
|
|
(3)
ɂɫɩɨɥɶɡɭɹ ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
(3),
ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
(2),
)
sin
(sin
)
)(
(
kt
pt
p
k
p
k
h
x
|
.
ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
(3),
ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
:
)
sin
(sin
)
(
2
kt
pt
p
k
k
h
x
. (4)
ɉɪɟɨɛɪɚɡɨɜɚɜ ɜɵɪɚɠɟɧɢɟ
,
ɩɨɥɭɱɢɦ
pt
t
a
x
cos
)
(
, (5)