Яковенко Г.Н. Краткий курс теоретической механики
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W
B
B
W
O
O ωωω
ωω
εεε
εε
i
1
i
2
i
3
e
1
e
2
e
3
e
3
i
3
e
3
= i
3
= const. (6.1)
e
1
e
2
e
3
= i
3
e
1
O
e
2
j(t)
i
2
i
1
e
1
e
2
i
1
i
2
ϕ i
1
e
1
e
1
= i
1
cos ϕ + i
2
sin ϕ,
e
2
= −i
1
sin ϕ + i
2
cos ϕ.
(6.2)
t
˙
e
1
= (−i
1
sin ϕ + i
2
cos ϕ) ˙ϕ = e
2
˙ϕ,
˙
e
2
= (−i
1
cos ϕ + i
2
sin ϕ) ˙ϕ = −e
1
˙ϕ.
(6.3)
ωωω
ωω
ωωω
ωω
=
1
2
3
X
i=1
[e
i
,
˙
e
i
] =
1
2
([e
1
,
˙
e
1
] + [e
2
,
˙
e
2
]) =
1
2
([e
1
, e
2
] − [e
2
, e
1
]) ˙ϕ = ˙ϕe
3
.
ωωω
ωω
= ˙ϕe
3
= ωe
3
(6.4)
e
3
= i
3
ω = ˙ϕ t
εεε
εε
εεε
εε
= ˙ω˙ω˙ω
˙ω˙ω
= ¨ϕe
3
= ˙ωe
3
, (6.5)
e
3
= i
3
ε = ¨ϕ = ˙ω
B
C
H
A
a
O
h
F
w
e
2
e
3
e
1
e
r
B
A
V
A
= 0 O
V
O
≡ 0 B
V
B
= [ωωω
ωω
, r] r = OB A
V
A
= [ωωω
ωω
, r] = 0 r = OA
ωωω
ωω
ωωω
ωω
= ωe
2
e
2
C V
C
ω ω = V
C
/h h C
ωωω
ωω
εεε
εε
F ωωω
ωω
εεε
εε
˙ω˙ω˙ω
˙ω˙ω
V
F
F {e
2
, e
3
} e
3
V
F
V
F
ΩΩΩ
ΩΩ
ωωω
ωω
ΩΩΩ
ΩΩ
Ωe
3
{e
2
, e
3
} Ω = V
C
/H
ωωω
ωω
OF εεε
εε
= εe
1
{e
2
, e
3
}
ε = Ωω = V
2
C
/Hh = (V
2
C
/h
2
) tg α = ω
2
tg α α
B
i
1
r
B
r
O
e
1
e
2
e
3
O
i
3
i
2
r
r
0
(t) e
1
(t) e
2
(t) e
3
(t) B
y
k
(t)
ρρρ
ρρ
=
3
X
k=1
y
k
e
k
, (7.1)
ρρρ
ρρ
OB e
1
e
2
e
3
B
r = r
0
+ ρρρ
ρρ
= r
0
+
3
X
k=1
y
k
e
k
, (7.2)
r
0
(t) e
1
(t) e
2
(t) e
3
(t)
y
1
(t) y
2
(t)
y
3
(t) B
ρρρ
ρρ
t
ρρρ
ρρ
V
W
V =
˙
r
0
+
3
X
k=1
y
k
˙
e
k
, (7.3)
W =
¨
r
0
+
3
X
k=1
y
k
¨
e
k
. (7.4)
r
0
e
1
e
2
e
3
t
V W
V =
3
X
k=1
˙y
k
e
k
, (7.5)
W =
3
X
k=1
¨y
k
e
k
. (7.6)
V
e
W
e
V
r
W
r
r
0
(t) e
1
(t) e
2
(t) e
3
(t)
y
1
(t) y
2
(t) y
3
(t)
V
V = V + V . (7.7)
¤ t
V =
˙
r
0
+
3
X
k=1
y
k
˙
e
k
| {z }
V
+
3
X
k=1
˙y
k
e
k
| {z }
V
= V + V . (7.8)
¥
W
W = 2[ωωω
ωω
, V ], (7.9)
ωωω
ωω
W = W + W + W . (7.10)
¤ t
W
W =
¨
r
0
+
3
X
k=1
y
k
¨
e
k
| {z }
W
+
3
X
k=1
¨y
k
e
k
| {z }
W
+ 2
3
X
k=1
˙y
k
˙
e
k
| {z }
W
= W + W + W .
W
2
3
X
k=1
˙y
k
˙
e
k
= 2
3
X
k=1
˙y
k
[ωωω
ωω
, e
k
] = 2[ωωω
ωω
,
3
X
k=1
˙y
k
e
k
] = 2[ωωω
ωω
, V ].
¥
V
a
W
a
V W
W
c
P O
i
1
P
r = OP ϕ r = OP
O
e
1
e
2
P
j
r
r
i
1
e
1
r = OP r = re
1
e
2
e
1
e
3
e
1
e
2
e
3
e
3
ωωω
ωω
= ˙ϕe
3
εεε
εε
= ¨ϕe
3
O
e
1
e
2
P
V
îòí
V
ïåð
i
1
P
ϕ
r
V = [ωωω
ωω
, r] = ˙ϕr[e
3
, e
1
] = ˙ϕre
2
. (8.1)
V = ˙re
1
. (8.2)
V = V = V + V = V
ϕ
+ V
r
(8.3)
V
ϕ
= V = ˙ϕr V
r
= V = ˙r
V
2
= ˙r
2
+ ˙ϕ
2
r
2
. (8.4)
O
e
1
e
2
P
W
î ò í
W
ï å ð
t
W
n
å ðï
W
êî ð
i
1
W
τ
W
n
W
τ
= [εεε
εε
, r] = ¨ϕr[e
3
, e
1
] = ¨ϕre
2
,
W
n
= [ωωω
ωω
, V ] = ˙ϕ
2
r[e
3
, e
2
] = − ˙ϕ
2
re
1
.
W
W = ¨re
1
.
W
W = 2[ωωω
ωω
, V ] = 2 ˙ϕ ˙r[e
3
, e
1
] = 2 ˙ϕ ˙re
2
.
W = W = W + W
n
+ W
τ
+ W =
= (¨r − ˙ϕ
2
r)e
1
+ ( ¨ϕr + 2 ˙ϕ ˙r)e
2
= W
r
+ W
ϕ
.
(8.5)
W
r
W
ϕ
W
r
= ¨r − ˙ϕ
2
r, W
ϕ
= ¨ϕr + 2 ˙ϕ ˙r (8.6)