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46
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ
M
5
ɛɥɨɤɚ
Ʉ
ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ
Ɇ
,
ɩɪɢɧɹɜ
ɡɚ
ɩɨɥɸɫ
ɬɨɱɤɭ
N
:
v
v
v
Mx
Nx
MN
x
(
)
,
ɬ
.
ɟ
.
(
)
v
v
v
s
s
MN
x
Mx
Nx
1
2
.
Ɍɚɤ ɤɚɤ
(
)
v
MN
r
MN
x
M
M
5
5
5
2
,
ɬɨ
2
5 5
1
2
r
s
s
M
,
ɨɬɤɭɞɚ ɭɝɥɨɜɚɹ
ɫɤɨɪɨɫɬɶ ɛɥɨɤɚ
Ʉ
:
M
5
1
2
5
2
s
s
r
. (11)
Ʉɢɧɟɬɢɱɟɫɤɢɟ ɷɧɟɪɝɢɢ ɝɪɭɡɨɜ
Ⱥ
ɢ
ȼ
,
ɫɨɜɟɪɲɚɸɳɢɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ
ɞɜɢɠɟɧɢɟ
,
ɢɦɟɸɬ ɜɢɞ
T
P
g
s
( )
1
1
1
2
1
2
,
T
P
g
s
( )
2
2
2
2
1
2
.
(12)
ȼɵɱɢɫɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɨɜ
D
ɢ
ȿ
,
ɜɪɚɳɚɸɳɢɯɫɹ ɜɨɤɪɭɝ
ɧɟɩɨɞɜɢɠɧɵɯ ɨɫɟɣ
:
2
3
5
)
3
(
2
1
M
O
I
T
,
2
4
4
)
3
(
2
1
M
O
I
T
.
ɉɨɞɫɬɚɜɢɜ
ɡɧɚɱɟɧɢɹ
ɦɨɦɟɧɬɨɜ
ɢɧɟɪɰɢɢ
ɛɥɨɤɨɜ
¸¸
¹
·
¨¨
©
§
g
r
P
I
g
r
P
I
O
O
2
,
2
2
4
4
4
2
3
3
5
ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ
(9),
ɧɚɯɨɞɢɦ
T
P s
g
( )
3
3 1
2
4
,
T
P s
g
( )
4
4 2
2
4
.
(13)
Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɚ
Ʉ
,
ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ
,
ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
2
5
5
2
5
5
)
5
(
2
1
2
1
M
O
O
I
v
g
P
T
.
ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɛɥɨɤɚ
Ʉ
¸¸
¹
·
¨¨
©
§
g
r
P
I
O
2
2
5
5
5
ɢ
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ
(10)
ɢ
(11),
ɩɨɥɭɱɢɦ
:
T
P
g
s
s
P
g
s s
( )
5
5
1
2
2
2
5
1 2
3
16
1
8
.
(14)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ
L,
ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ
,
ɢɦɟɟɬ ɜɢɞ
T
P
g
v
O
( )
6
6
2
1
2
5
.
ɉɪɢɦɟɧɢɜ ɮɨɪɦɭɥɭ
(10),
ɨɩɪɟɞɟɥɹɟɦ
47
T
P
g
s
s
( )
6
6
1
2
2
1
8
. (15)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ
ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ
ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɮɨɪɦɭɥɭ
(8)
ɜɵɪɚɠɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɢɡ ɮɨɪɦɭɥ
(12)–(15).
ȼ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ
:
2
2
2
2
3
1
2
4
1
2
1
2
2
2
2
5
5
6
1
2
1 2
1
2
1
1
1
1
2
2
4
4
3
1
1
(
)
(
) ,
16
8
8
P
P
P
P
T
s
s
s
s
g
g
g
g
P
P
P
s
s
s s
s
s
g
g
g
ɬ
.
ɟ
.
T
P
P
P
P
g
s
P
P
P
P
g
s
P
P
g
s s
8
4
3
2
16
8
4
3
2
16
2
8
1
3
5
6
1
2
2
4
5
6
2
2
5
6
1 2
. (16)
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫɥɟɞɭɟɬ
ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ɍ
ɩɨ ɨɛɨɛɳɟɧɧɵɦ
ɫɤɨɪɨɫɬɹɦ
s
1
ɢ
s
2
1
6
5
2
6
5
4
2
2
2
6
5
1
6
5
3
1
1
8
2
8
2
3
4
8
,
8
2
8
2
3
4
8
s
g
P
P
s
g
P
P
P
P
s
T
s
g
P
P
s
g
P
P
P
P
s
T
w
w
w
w
ɢ ɜɡɹɬɶ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɩɨ ɜɪɟɦɟɧɢ
°
°
¿
°°
¾
½
1
6
5
2
6
5
4
2
2
2
6
5
1
6
5
3
1
1
8
2
8
2
3
4
8
8
2
8
2
3
4
8
s
g
P
P
s
g
P
P
P
P
s
T
dt
d
s
g
P
P
s
g
P
P
P
P
s
T
dt
d
w
w
w
w
.
(17)
ɍɱɢɬɵɜɚɹ
,
ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ
Ɍ
,
ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɨɪɦɭɥɨɣ
(16)
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ
s
1
ɢ
s
2
,
ɢɦɟɟɦ
0
1
s
T
w
w
,
0
2
s
T
w
w
.
(18)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɮɨɪɦɭɥ
(4), (7), (17)
ɢ
(18)
ɜ ɭɪɚɜɧɟɧɢɟ
(1)
ɩɨɥɭɱɢɦ
ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ
s
1
ɢ
s
2
:
48
8
4
3
2
8
2
8
1
2
8
4
3
2
8
2
8
1
2
1
3
5
6
1
5
6
2
1
5
6
2
4
5
6
2
5
6
1
2
5
6
P
P
P
P
g
s
P
P
g
s
P
P
P
P
P
P
P
g
s
P
P
g
s
P
P
P
sin
,
sin
.
D
E
Ɋɟɲɚɹ ɷɬɭ ɫɢɫɬɟɦɭ
,
ɩɨɥɭɱɢɦ
:
(
)
sin
sin
s
g
D C
B
C P
B P
AB
C
1
2
1
2
8
E
D
,
(
)
sin
sin
s
g
D C
A
A P
C P
AB
C
2
2
1
2
8
E
D
,
2
1
2
5
sin
)
(
sin
)
(
)
2
(
4
C
AB
P
C
B
P
C
A
B
A
C
D
g
w
O
D
E
,
ɝɞɟ
8
4
3
2
1
3
5
6
P
P
P
P
A
,
8
4
3
2
2
4
5
6
P
P
P
P
B
,
P
P
C
5
6
2
ɢ
P
P
D
5
6
2
.
§12.
ɋɩɢɫɨɤ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ
[2]
26.1, 26.5, 26.6, 26.9.
27.1, 27.3, 27.12, 27.13, 27.34, 27.42, 27.53, 27.56.
28.1, 28.6, 28.7, 28.15, 28.17, 28.21.
29.7, 29.14.
30.4, 30.6, 30.11, 30.12, 30.15, 30.19, 30.22.
32.1, 32.2, 32.5, 32.16, 32.24, 32.26, 32.37, 32.53, 32.55, 32.57, 32.68,
32.70, 32.78, 32.79, 32.86, 32.93, 32.94, 32.96, 32.98, 32.99.
34.2, 34.3, 34.9, 34.11, 34.12, 34.15, 34.21.
35.3, 35.4, 35.10, 35.13, 35.14, 35.19.
49
36.3, 36.6, 36.9.
37.1, 37.3, 37.9, 37.14, 37.34, 37.39, 37.43, 37.46, 37.53, 37.56.
38.2, 38.4, 38.9, 38.20, 38.24, 38.27, 38.30, 38.40, 38.42, 38.44, 38.46,
38.47, 38.50.
48.5, 48.6, 48.12, 48.13, 48.28, 48.29, 48.30, 48.35, 48.44.
§13.
Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɧɚɦɢɤɢ
ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ
:
F
w
m
,
ɝɞɟ
m
–
ɢɧɟɪɬɧɚɹ ɦɚɫɫɚ ɬɨɱɤɢ
,
w
–
ɭɫɤɨɪɟɧɢɟ
,
F
–
ɫɢɥɚ
.
°
°
°
°
¯
°
°
°
°
®
¸
¹
·
¨
©
§
M
M
¸¸
¹
·
¨¨
©
§
¸
¹
·
¨
©
§
M
M
¸¸
¹
·
¨¨
©
§
M
M
¸
¹
·
¨
©
§
M
M
¸
¸
¹
·
¨
¨
©
§
¸
¹
·
¨
©
§ M
M
dt
dz
dt
d
dt
dr
z
r
t
F
dt
z
d
m
dt
dz
dt
d
dt
dr
z
r
t
F
dt
d
dt
dr
dt
d
r
m
dt
dz
dt
d
dt
dr
z
r
t
F
dt
d
r
dt
r
d
m
z
r
,
,
,
,
,
,
,
,
,
,
,
,
2
,
,
,
,
,
,
2
2
2
2
2
2
2
.
°
°
°
¯
°
°
°
®
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
dt
dS
S
t
F
dt
dS
S
t
F
dt
dS
m
dt
dS
S
t
F
dt
S
d
m
n
,
,
0
,
,
1
,
,
2
2
W
W
U
.
( , , , , , , )
( , , , , , , ) .
( , , , , , , )
x
y
z
mx
F t x y z x y z
my
F t x y z x y z
mz
F t x y z x y z
°
®
°
¯
50
¦
N
k
k
k
z
r
m
J
1
2
.
³
³
³
)
(
2
)
(
2
)
(
2
)
,
,
(
V
V
M
z
dv
r
V
M
dv
r
z
y
x
dm
r
J
U
.
2
1
Md
J
J
Zc
Z
.
2
MR
J
z
.
¦
n
k
e
k
F
w
M
1
c
=
.
¦
n
i
e
i
F
Q
dt
d
1
)
(
;
³
W
0
0
1
)
(
dt
t
F
Q
Q
.
)
(
1
e
k
n
k
Z
Z
F
mom
dt
dL
¦
.
¦
)
(
10
0
1
F
A
T
T
.
T
m v
k
k
k
n
¦
2
1
2
.
T
Mv
C
2
2
,
ɝɞɟ
M –
ɦɚɫɫɚ ɬɟɥɚ
;
C
v
–
ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
.
2
2
Z
Z
J
T
,
ɝɞɟ
Z
J
–
ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ
,
Z
–
ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ
.
T
Mv
J
C
C
2
2
2
2
Z
.
G
W
A
F dr
Fds
F
F dx
F dy
F dz
x
y
z
cos( , )
.
A
Fdr
F ds
F dx
F dy
F dz
L
L
x
y
z
L
³
³
³
W
(
)
.
A
Fs
F s
cos( , )
.
A
P
P z
z
C
C
12
1
2
( )
(
)
.