Колесников А.А. Синергетические методы управления сложными системами: механические и электромеханические системы
Подождите немного. Документ загружается.
ψ =0
mg
mAω
2
g(t)
|g| g
0
ml
J
g
u
1
= −
Jωx
1
x
2
ml
A
2
− x
2
1
sin x
1
−
Jψ
1
x
2
1
mlT
1
sin x
1
.
ψ
1
=0 u
1
u
1ψ
= −
Jω
2
x
1ψ
ml sin x
1ψ
,
˙x
1ψ
(t)=x
2ψ
;
˙x
2ψ
(t)=−ω
2
x
1ψ
+
mlg
J
sin x
1ψ
.
Jω
2
>mlg −π<x
1ψ
<π.
J = ml
2
lω
2
>g.
A
2
ω
2
> 2gl,
l =0, 3 ω =50 A =1 T =0, 02
x
1
x
2
x
1
x
2
u
1
ψ
1
=0
sin x
1ψ
≈ x
1ψ
−
1
6
x
3
1ψ
.
u
1ψ
J = ml
2
u
1ψ
∼
=
−
6lω
2
6 − x
2
1ψ
,
˙x
1ψ
(t)=x
2ψ
,
˙x
2ψ
(t)=
1
l
g −
6lω
2
6 − x
2
1ψ
sin x
1ψ
.
x
1ψ
=0
lω
2
>g x
2
1
< 6,
ψ
1
=0
u
1ψ
u
1ψinf
= lim
x
1ψ
→0
$
$
$
$
lω
2
x
1ψ
sin x
1ψ
$
$
$
$
= lω
2
,
x
1ψ
(t) A
u
1ψ
u
1ψsup
= −
lω
2
A
sin A
A<π .
u
1ψ
B = A
u
=0, 5lω
2
A
sin A
− 1
,A<π
.
B = A
u
∼
=
0, 5lω
2
A
2
6 − A
2
,A<
√
6 .
ψ
1
=0
u
1ψ
ψ
2
=0, 5x
2
2
+0, 5ω
2
0
x
2
1
− E
0
.
ψ
2
T
2
˙
ψ
2
(t)+x
2
2
sin x
2
1
ψ
2
=0
ml
J
u
2
= −
ml
J
g −
ω
2
0
x
1
sin x
1
−
ψ
2
x
2
sin x
1
T
2
.
ml
J
ml
J
u
2
= −
ω
2
0
x
1
sin x
1
−
ψ
2
x
2
sin x
1
T
2
.
˙x
1
(t)=x
2
;
˙x
2
(t)=−
mgl
J
sin x
1
− ω
2
0
x
1
−
ψ
2
x
2
sin
2
x
1
T
2
.
u
2
ψ
2
=0
Jω
2
0
>mgl lω
2
0
>g,
ψ
2
=0
l =0, 3 ω
0
=10
T
2
=0, 2 E
0
=2 g =9, 81
2
g =0
l =0, 3 ω
0
=10 T
2
=0, 2 E
0
=7 g =0
ψ
3
=0, 5x
2
2
+ ω
2
0
(1 − cos x
1
) − E
0
.
ψ
3
T
3
˙
ψ
3
(t)+x
2
2
sin x
2
1
ψ
3
=0
ml
J
u
3
= −
ml
J
g − ω
2
0
−
x
2
sin x
1
T
3
ψ
3
.
ml
J
g
ml
J
u
3
= −ω
2
0
−
x
2
sin x
1
T
3
ψ
3
.
˙x
1
(t)=x
2
;
˙x
2
(t)=−
mgl
J
sin x
1
− ω
2
0
sin x
1
−
x
2
sin
2
x
1
T
3
ψ
3
.
Jω
2
0
>mgl
l =0, 3 ω
0
=
7
T
3
=0, 2 E
0
=1 g =9, 81
2
g =0
ψ
2
=0 ψ
3
=0
u
2
u
3
E
0
u = −γ(H −H
∗
)x
2
sin x
1
,
H =0, 5Jx
2
2
+mgl(1−cos x
1
)
H
∗
ml
2
ψ
3
= H − H
∗
lω
2
0
=2g
−
ω
2
0
x
1
sin x
1
−ω
2
0
u t →∞
u
u
3
−ω
2
0
u
2
x
1
→ 0 −ω
2
0
u
2
u
3
x
1
x
2
u(x
1
,x
2
)
u(x
1
,x
2
)
x
10
x
20
u
1ψ
u(t)
J
¨
θ(t) − ml(g + aω
2
sin ωt)sinθ =0.
u(t)=−aω
2
sign sin ωt,
u
1ψ
(t)=−
lω
2
A sin(ωt + ϕ
0
)
sin
A sin(ωt + ϕ
0
)
.
u
1ψ
(t) A
u
u
1ψ
u
1ψ
(t)=−
6lω
2
6 − A
2
sin
2
(ωt + ϕ
0
)
sin
2
(ωt + ϕ
0
)=0, 5 − 0, 5 cos 2(ωt + ϕ
0
)
u
1ψ
(t)=−
12lω
2
12 − A
2
+ A
2
cos 2(ωt + ϕ
0
)
.
u
1ψ
(t)
A
u
ω
1
=2ω T
∼
=
π
ω
u
1ψ
(x
1
)
∼
=
−lω
2
1+
1
6
x
2
1ψ
+
7
360
x
4
1ψ
+ ...
.
x
1ψ
u
1ψ
(t) ≈−lω
2
1+
A
2
12
−
A
2
12
cos 2(ωt + ϕ
0
)
.
ω
1
=2ω t = π/ω
ω
1
: ω =2
u
1ψ
u
1ψ
(t)
u
1ψ
(t)
u
1ψ
(t)
x
1
x
2
u
1ψ
A
u
ω
1
∼
=
2ω
T
∼
=
π
ω
lω
2
>g lω
2
1
> 4g
u
1ψ
(t)
ω
1
: ω =2
ω
1
=2ω
T =
π
ω
u
x
1
x
2
x
10
,x
20
ψ =0
A ω u
1
ψ =0
u
1ψ
(t)
u(t)=aω
2
sin ωt
a
2
ω
2
> 2lg
a ω
u(t)=aω
2
sin ωt
u
ω A
¨x(t) − µ
1
(1 − α
1
x
2
)˙x(t)+ω
2
x =0,
¨y(t) − µ
2
(1 − α
2
˙y
2
)˙y(t)+ω
2
y =0,
˙x
1
(t)=− ωx
2
+ x
1
(A
2
− x
2
1
− x
2
2
),
˙x
2
(t)=+ωx
1
+ x
2
(A
2
− x
2
1
− x
2
2
).
x =˙y(t)
x ˙x(t)
y ˙y(t)
µ
i
α
i
µ
i
µ
i
x
1
= r cos ϕ
x
2
= r sin ϕ
˙r(t)=r(A
2
− r
2
) ˙ϕ(t)=ω.
r(t)=A
r
2
0
r
2
0
+(A −r
2
0
)e
−2A
2
t
; ϕ(t)=ωt + ϕ
0
.
A ω
x
1s
(t)=A cos(ωt + ϕ
0
); x
2s
(t)=A sin(ωt + ϕ
0
).
ψ = A
2
− x
2
1
− x
2
2
,
A
ψ =0
˙x
1ψ
(t)=−ωx
2ψ
, ˙x
2ψ
(t)=−ωx
1ψ
¨x
1ψ
(t)+ω
2
x
1ψ
=0.
ψ =0
˙x
1ψ
(t)=±
A
2
− ω
2
x
2
1ψ
.
ψ =0
div
˙
x(t)=
∂ ˙x
1
∂x
1
+
∂ ˙x
2
∂x
2
=2(A − x
2
1
− x
2
2
) − 2x
2
1
− 2x
2
2
< 0.
ψ =0
ψ =0