Учебники / [Филяев К.Ю.] Математические задачи энергетики
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ò Mdδ = A .
. 2.4. ,
101
14 ( -
)
M = − MII = TJ ddtω .
TJ – , ( -
); d ω/dt = d2 /dt2 = – ; –
, ω0.
-
:
MII = f (δ, ω .
, MII = MIImsin (
. 2.4, ). -
T d2δ |
= |
M , |
(2.22 ) |
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J |
dt2 |
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, = , |
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T |
d2δ |
= |
P |
(2.22 ) |
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J |
dt2 |
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M = 0 − MmII sin δ , |
(2.22 ) |
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P = P |
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− PII sin δ. |
(2.22 ) |
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0 |
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m |
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, -
0, 0 Pm
.
ω , -
.
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14 « -
» « ».
102
2.4.2.
,
ω(t) (t) -
, (2.22 ). , -
( . 2.4, , ).
(2.22 ), . = f(t), -
15 .
, (2.22 ) -
. .
,
-
– . [1.8, §4.6], . 2.4.
, -
I II , -
. -
, . . ,
,
, ( ).
, -
v m F, -
, , -
. , b1 b2
, ,
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15 , -
( , , -
, ).
103
mv |
2 b 2 |
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= òFdx = A . |
(2.23) |
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2 |
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b1 |
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F = f(x) , -
, .
( . 2.3)
-
ω0, (2.23)
δ0II |
T |
ω2 |
. |
(2.24) |
A1 = ò Mdδ = |
J |
2 |
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δ0I |
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1 abca . 2.4, .
(2.24), -
ω = |
2 |
ò Pdδ . |
(2.25) |
T |
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J δ |
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ω = f ) . 2.4, , = ϕ(t) ω = ψ(t)
– 2.4, .
, , ,
δ0II
ò Pdδ
δIo
1 . 2.4, . ,
òPdδ 2. 1 2 -
δIIo
, -
: |
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1 = 2 |
(2.26 ) |
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104
ò Pdδ = 0. |
(2.26 ) |
δ |
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[1.8, §4.6].
. 2.4, , -
, . , -
, -
. -
, , -
.
2.4.3.
.
, -
,
= f(t) P = ϕ(t).
-
. (2.22 ).
, [1.8, §7.1]:
;
2),
;
3), , -
.
, ,
105
( ). -
.
[1.8, §7.2]
( . 2.3). -
,
, (2.22 ) :
d2d |
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360f0DP |
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a = dt2 |
= |
TJ |
. |
(2.27) |
,
, ,
. : [t] = [TJ] = , [ ] = ; [a] = 2, [f0] = .
(2.27) ( = ).
, , -
( . 2.4, , , ). -
t
.
( . 2.5, ), ,
. 2.5, ). -
, ,
, .
. ,
.
t = 0,02…0,05 .
,
a0 ( . . 2.5, ) -
.
Dd |
(1) |
= 0,5a |
0 |
Dt2 |
= 0,5 × 360f |
0 |
(DP |
T Dt2 . |
(2.28) |
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( ) |
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(0 ) |
J |
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106
.
, t, – (1),
(1) , , .
: a(1) = (DP1 ( T) J 360f0 .
w(1), -
, a(1), -
(1). -
Dd |
(2 ) |
= Dw |
1 |
Dt + 0,5a |
1 |
Dt2 . |
) |
(2.29) |
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( ) |
( |
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. -
:
Dw(1 ) = 0,5(a 0 (+ )a 1 Dt( .)
(2.29),
Dd(2 ) = 0,5a 0 D( t)2 + a 1 Dt2( )
, (2.28),
Dd(2 ) = Dd 1 (+) a 1 Dt(2).
,
. -
( ) , -
,
ìDd(1) = K0,5DP(0); |
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ï |
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= Dd(1) + KDP(1); |
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ïDd(2) |
(2.30) |
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í |
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ïL |
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ïDd |
(n) |
= Dd |
(n −1) |
+ KDP |
, |
î |
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(n −1) |
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K = (360f0/TJ t2.
107
. 2.5.
:
1 –
M = ψ , t)
, 2 – ; 0 – α = α(0) = const, 0 –
( (1))
, , -
(n–1), (n–1) ( . 2.6), -
n
δ |
(n ) |
= δ |
n −(1 |
+ K0,5( |
P |
n −1 |
+ |
( |
P' |
. |
( ) |
(2.31) |
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) |
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n)−1 |
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108
. 2.6
: ( . 2.4) -
, – ,
.
,
. -
-
.
[1.3, . VIII, §7] -
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dδ |
= F(t,δ ) |
(2.32) |
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dt |
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(t0) = 0 k+1- |
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δk +1 = δk + (k1 + 2k2 + 2k3 + k4 )/6 , |
(2.33) |
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k = 0, ..., n–1; |
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k1 == F(tk , δk ) t ; |
(2.33 ) |
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109
k2 |
= F(tk + |
t/2,δk + k1/2) |
t ; |
(2.33 ) |
k3 |
= F(tk + |
t/2, δk + k2/2) |
t ; |
(2.33 ) |
k4 |
= F(tk + |
t,δk + k3 ) t . |
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(2.33 ) |
(2.22 ) .
.
2.4.4.2.
16
.
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dyi = F (t, y ), |
(2.34) |
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dt |
i |
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i = 1,..., n; y = (y1, y2, …, yn); y(t0) = y(0) = (y(0)1, …, y(0)n).
k+1-
yi(k+1)=yi(k)+(k1(i)+2k2(i)+2k3(i)+k4(i))/6, |
(2.35) |
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k1(i)=Fi(x(k),y(k))h; |
(2.35 ) |
k2(i)=Fi(x(k)+h/2,y(k)+k1(i)/2)h; |
(2.35 ) |
k3(i)=Fi(x(k)+h/2,y(k)+k2(i)/2)h; |
(2.35 ) |
k4(i)=Fi(x(k+1),y(k)+k3(i))h. |
(2.35 ) |
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16 – ,
, P(x,y)dx + Q(x,y)dy = 0,
F(t,x, ,…,x(n)). ,
, , -
, .
110