Беляк О.А., Дрезин С.В. Динамика материальной точки
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∆
i
= ∆
i
+ a
i
ϕ, i = 1, 2, 3,
F
i
= c
i
∆
i
, i = 1, 2, 3.
O F
i
(i = 1, 2, 3)
F = c∆ = c(∆ + bϕ)
b O
F b =
3
X
i=1
F
i
a
i
.
c(∆ + bϕ)b =
3
X
i=1
c
i
(∆
i
+ a
i
ϕ)a
i
.
ϕ = 0
c∆ b =
3
X
i=1
c
i
∆
i
a
i
.
c
cb
2
ϕ =
3
X
i=1
c
i
ϕa
2
i
, c =
c
1
a
2
1
+ c
2
a
2
2
+ c
3
a
2
3
b
2
.
M
b O mb ¨ϕ + cbϕ = 0
k =
r
c
m
c = 19,6
x
0
= 4
∆
∗
> 0
Ox O
∆
1
= ∆
∗
+ x, ∆
2
= ∆
∗
− x.
Ox
m¨x = −F
1
+ F
2
.
m¨x = −c(∆
∗
+ x) + c(∆
∗
− x), m¨x = −2cx
x(0) = 4 ˙x(0) = 0
k =
r
2c
m
=
r
2 · 1960
10
−1
≈ 1 9, 8
−1
,
c = 1960
C
1
= x
0
, C
2
= v
0
/k = 0
N
x
F
1
∆
∗
x
∆
∗
O
F
2
mgmg
x(t) = x
0
cos
r
2c
m
t = 4 cos(19, 8t) ,
v(t) = ˙x(t) = −x
0
k sin kt,
T =
2π
k
= π
r
2m
c
≈ 3 , 14
r
2 · 10
1960
≈ 0,3172
v
max
= |x
0
k| = 4 · 19,8 = 79, 2 .
Q m AB = l
S
A
a
x
α β
Q Ox
|S
1
| = |S
2
|
m¨x = −S sin α − S sin β + mg.
l −a
a
α
β
mg
S
1
S
2
Q
x
O
BAA
sin α sin β x
AOQ OQB
tg α =
x
a
, tg β =
x
l − a
.
sin α =
tg α
p
1 + tg
2
α
=
x/a
p
1 + (x/a)
2
=
x
a
−
x
3
2a
3
+ O
x
a
5
,
sin β =
x/(l − a)
p
1 + (x/(l −a))
2
=
x
l − a
−
x
3
2(l − a)
3
+ O
x
l −a
5
!
,
sin α, sin β
m¨x + x
Sl
a(l − a)
= mg,
k =
r
Sl
ma(l −a)
Q m
l c
c
1
= 3EJ/l
3
(E−
J−
c
1
Q
l
Q
c
1
c
mg
F
mgmg
c
∗
c
∗
=
cc
1
c + c
1
=
3cEJ
3EJ + cl
3
k =
r
c
∗
m
=
s
3EJc
m(3EJ + cl
3
)
.
Q m
c
1
c
2
l T
J E
l/2
QQ
l/2
Q
c
1
c
2
c
2
c
c
1
c
1
c
∗
1
y
c
yc = mg y =
mgl
3
48EJ
E J
c =
48EJ
l
3
.
Q
c c
2
c
1
c c
2
c
∗
1
c
∗
1
=
cc
2
c + c
2
.
Q c
∗
1
c
1
˜c
˜c = c
∗
1
+ c
1
=
cc
2
c + c
2
+ c
1
=
cc
2
+ cc
1
+ c
1
c
2
c + c
2
,
k =
r
˜c
m
.
T =
2π
k
= 2π
s
(c + c
2
)m
cc
2
+ cc
1
+ c
1
c
2
.
c T
T
2
(cc
2
+ cc
1
+ c
1
c
2
) = 4π
2
(c + c
2
)m, c =
4π
2
c
2
m − T
2
c
1
c
2
T
2
(c
1
+ c
2
) − 4π
2
m
.
l
Q T
l =
3
r
48EJ
c
=
3
s
48EJ(T
2
(c
1
+ c
2
) − 4π
2
m)
4π
2
c
2
m − T
2
c
1
c
2
.
A
B
A B
x
−x
v
x
−x
x
x
A
A
B
B
O
N
N
F
F
mg
F
mg
F
y
y
v
A T
A x
Oy O
A mg
F F N
F = fN,
Oy
0 = N − mg, N = mg.
A Ox
m¨x = −cx −fmg
˙x
|˙x|
.
c = 2 , 45 = 245
T =
π
k
= π
r
m
c
≈ 3, 14
r
0, 5
245
c ≈ 0 , 142 c.
x =
fmg
c
=
0, 2 · 0, 5 · 9, 81
245
≈ 0 , 004 = 0, 4 .
x(0) = x
0
= 3 x
0
> 0 ˙x(0) = 0
(2i − 1)x 6 x
0
6 (2i + 1)x .
−0, 4 + 0, 8i 6 3 6 0, 4 + 0, 8i, i = 4.
x
01
= 3
x
11
= 2x −x
01
= 2 · 0, 4 − 3 = −2, 2 ,
l
1
= x
01
− x
11
= 5, 2
x
02
= x
11
= −2, 2 , x
02
< 0,
x
12
= −2x − x
02
= −2 · 0, 4 + 2, 2 = 1, 4 ,
l
2
= x
12
− x
02
= 3, 6 .
i = 2 i = 4i = 3i = 1
˙x
x
01
x
11
x
02
x
12
x
03
x
13
x
04
x
14
−7x −5x −3x
9x7x5x3x
−x
x
O
xx
x
03
= x
12
= 1, 4 , x
13
= 2x −x
03
= 0, 8 − 1, 4 = −0, 6 ,
l
3
= x
03
− x
13
= 2 .
x
04
= x
13
= −0, 6 , x
14
= −0, 8 + 0, 6 = −0, 2 ,
l
4
= −0, 2 + 0, 6 = 0, 4 .
x
14
= −0, 2
−0, 4 6 x 6 0, 4
A
x
13
x
12
x
11
x
14
x
01
T
2T
3T 4T
t
x
−x
x
O
c = 19, 6
R = αv α = 3, 5 ·
x
0
= 1
R
mg
O
x
FF
Ox
O
mg F R,
R = αv = α ˙x Ox
h =
α
2m
=
3, 5
2 · 0, 1
−1
= 17, 5
−1
k =
r
c
m
=
r
19, 6
0, 1
−1
= 14
−1
λ
1
= −7, λ
2
= −28.
x
0
x(0) = 0, 01 , ˙x(0) = 0.
x(0) = C
1
+ C
2
,
˙x(0) = C
1
λ
1
+ C
2
λ
2
.
C
1
= 0, 132 , C
2
= −0, 033 .
x(t) =
0, 01 33e
−7t
− 0, 0033e
−28t
.
F (t) = 156, 8 sin 4t
Ox O
mg
F F
F (t) = Q
0
sin pt, Q
0
= 156,8 p = 4
−1
Ox h = 0, k =
r
c
m
=
r
392
24, 5
−1
= 4
−1
, P =
Q
0
m
= 6,4
2