Надежность технических систем 5-230-22198-4
Изложены основные понятия, определения и критерии, используемые в теории надежности. Рассмотрены общие методы расчета на
181 105 1005KB
Russian Pages 113 Year 2002
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(
)
. .
2002
-2681.51 – 192 (075.8) 32.965 – 6 6.5
:
. .
. .
– ., 2002 . –
113 .
ISBN 5–230–22198–4 ,
,
. , . IV
,
2101 –
«
». , .
ISBN 5–230–22198–4 681.51 – 192 (075.8) 32.965 – 6 6.5 ©
. ., 2002.
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1.2.
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-8К . . , ,
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1.3.
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2
3
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-91.3. (
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1
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2
3 , -
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-10Э
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1.3. . .Э .
–
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Э (
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-11( (
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– .
К
–
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1.4. 1.4.1.
, ,
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. . . : ,
,
,
( ( .
1.
). –
) –
,
-12-
, . . t.
, , ,
t
. , :
p(t ) = P { > t }, t ≥ 0. , p(t) , p(∞)=0, . .
(1.1) , . . p(0)=1, 1
0 ( t→+∞
. 1.1.). .
. 1.1.
. p(t)
t 2.
(
)–
,
. q(t) , –
,
q(t ) = P { ≤ t } = 1 − p(t ), 3.
, t ( ≤t). >t,
:
t ≥ 0.
(1.2)
. q(t) ω( t ) =
dq(t ) dp(t ) =− dt dt
, : (1.3)
-13-
. 1.2.
.
(1.3)
,
p(t ) = ∫ ω(t )dt ∞
(0,t) t ∞(
):
t
(1.2)
(1.3):
p(t ) = 1 − ∫ ω(t )dt . t
(1.4) t0
t0
,
=0 → (0)=ω(0), . .
-16→∞
,
( )≈−
( )=−
,
(1.17)
ω ′( ) d = − ln ω( ). ω( ) d
:
.
d ln p( ). d
(1.19) (1.19)
∫
(1.18) p(t) :
(1.16)
p(0)=1, t
, ,
0
t,
,
:
( )d = − ln p(t ).
(1.20)
0
:
⎧⎪ t p(t ) = exp⎨− ∫ ( )d ( ⎪⎩ 0 ,
⎫⎪ )⎬. ⎪⎭
(1.21) ,
,
, . ()
∫ t
.
,
( )d =0.
, . .:
0
1.4.
.
ω(t ) dq(t ) dt
q(t ) q(t )
∫ ω(t )dt t
ω(t )
(t )
1 − ∫ ω(t )dt 0
1 − p(t )
⎞ ⎛ t 1 − exp⎜ − ∫ (t )dt ⎟ ⎟ ⎜ ⎠ ⎝ 0 =const,
1 − q (t ) t
0
p(t )
p(t )
. . :
−
dp(t ) dt
⎞ ⎛ t (t ) ⋅ exp⎜ − ∫ (t )dt ⎟ ⎟ ⎜ ⎠ ⎝ 0
⎞ ⎛ t exp⎜ − ∫ (t )dt ⎟ ⎟ ⎜ ⎠ ⎝ 0
(t ) 1 dq(t ) ⋅ 1 − q(t ) dt ω(t ) 1 − ∫ ω(t )dt t
−
0
1 dp(t ) ⋅ p(t ) dt
-17= ∫ p(t )dt = ∫ e − t dt = − t
T
t
0
1
e−
t
∞ 0
0
=
1
.
(1.22)
1.4
, .
. p(t), , ω( t ) = 2 e
− t
(t)
T ,
:
(1 − e − t ).
. 1.
(1.4).
⎡t
⎤ p(t ) = 1 − ∫ ω(t )dt = 1 − 2 ⋅ ⎢ ∫ e − t dt − ∫ e −2 t dt ⎥ = 2e − t − e −2 t . ⎥⎦ 0 0 ⎣⎢ 0 2. (1.17): t
ω(t ) ⋅ (1 − e − t ) (t ) = = . p(t ) (1 − 0,5e − t ) 3. : T
[
]
= ∫ p(t )dt = ∫ 2e − t − e −2 t dt = ∞
∞
0
0
1.4.2.
:
t
.
(1.8)
3 . 2
,
,
. . η( ( )
. 1. , ( . 1.4). p (t ) = P {η ≤ t }, t ≥ 0
). , t.
(1.23)
-18-
. 1.4.
. p (t)
(
η. Э
0(
t→∞). 2. , . . , q (t ) = p{η > t } = 1 − p (t ).
( :
η>t,
t=0)
1
) (1.24)
3. ω (t ) =
: dp (t ) , dt
t ≥ 0.
(1.25) .
ω (t),
p (t) ,
.
4.
.
mk = ∫ t k ω (t )dt = k ∫ t k −1q (t )dt . ∞
∞
0
0
(1.26)
(1.26) dp (t ) dq (t ) =− ω (t ) = dt dt :
. ∞
m = ∫ t ω (t )dt = − ∫ t dq (t ) = − t ⋅ q (t ) + ∫ q (t )dt = k ∫ t k −1q (t )dt , 123 ∞
∞
k
0
k
k
→0
0
0
∞
∞
k
0
0
. (
) T
:
T = ∫ q (t )dt = ∫ [1 − p (t )]dt . ∞
∞
0
0
(1.27)
,
-19-
,
T (
.1.4). :
= 2 ∫ t [1 − p (t )]dt − T 2 . ∞
(1.28)
0
5. , ,
p (t/ ), t,
. .
,
, : p (t / ) = [ p ( + t ) − p ( )] /[1 − p ( )].
(1.29)
t, q (t / ) = 1 − p (t / ) = q ( + t ) / q ( ).
: (1.30)
p (t/ )
′ p (t / ) p ( ) = t →0 t 1− p ( )
t→0, . . , . ( ),
,
:
( ) = lim
, ω ( ) , ( )=− 1− p ( )
dp (t ) ′ p (t ) = = ω (t ) dt
0
0
: (1.32)
(1.31)
,
∫
t→0,
> 0.
d ( ) = − ln[1 − p ( )] d
t
(1.31)
: (1.33) (1.33)
p (0)=0.
t:
( )d = − ∫ d ln[1 − p ( )] =− ln[1 − p ( )] 0 = − ln[1 − p (t )]. t
t
0
,
⎫⎪ ⎬. ⎪⎭ : ⎫⎪ ( )d ⎬. ⎪⎭
⎧⎪ t 1 − p (t ) = exp⎨− ∫ ( )d ⎪⎩ 0 ⎧⎪ t p (t ) = 1 − exp⎨− ∫ ⎪⎩ 0 ,
:
(1.34)
(1.35) .
,
-20-
, . , , 1.5
. , .
1.5.
.
q (t )
ω (t )
1 − p (t )
dp (t ) dt
p (t ) p (t ) 1 − q (t )
q (t )
∫ ω (t )dt
1 − ∫ ω (t )dt
⎛ ⎞ 1 − exp⎜ − ∫ (t )dt ⎟ ⎜ ⎟ ⎠ ⎝ 0
⎞ ⎛ exp⎜ − ∫ (t )dt ⎟ ⎜ ⎟ ⎝ 0 ⎠
ω (t )
−
dq (t ) dt
t
1 dq (t ) ⋅ q (t ) dt
ω (t )
1 − ∫ ω (t )dt t
0
0
t
dp (t ) 1 ⋅ 1 − p (t ) dt
t
t
(t )
−
(t )
⎛ ⎞ (t ) ⋅ exp⎜ − ∫ (t )dt ⎟ ⎜ ⎟ ⎝ 0 ⎠
0
t
1.4.3.
(
) , ,
. . t=0
t. (0,t)
n Fn (t ) = P {Vt ≥ n}. P {Vt
(1.36)
= n} = P {Vt n
Vt . ,
Fn(t) – , . .:
≥ n} − P {Vt
, Vt –
.
(0,t) (1.36)
≥ n + 1} = Fn (t ) − Fn +1(t ). (0,t):
(1.37)
1. H (t ) = m1{Vt }.
(0,t). Э .
H(t). (1.38)
-21-
,
{
}
,
{ }
{ }
m1 Vt2 − Vt1 = m1 Vt2 − m1 Vt1 = H (t 2 ) − H (t1 ). (
H (t ) = ∑ nP {Vt = n}.
)
∞
(t1,t2)
: (1.39)
:
n =0
(1.40)
H (t ) = ∑ nFn (t ) − ∑ nFn +1(t ). (1.37)
∞
n =0
H(t)
(0,t).
(1.40)
∞
,
:
n =0
(1.41) (1.41)
n=0
.
m=n+1.
H (t ) = ∑ nFn (t ) − ∑ (m − 1)Fm (t ). (1.41)
:
∞
∞
n =1
m =1
,
H (t ) = ∑ Fn (t ). :
∞
n =1
(1.42)
2.
[H (t 2 ) − H (t1 )]. (t 2 − t1 ) ω ω
ω
(t1,t2), (1.39)
(t1–t2).
:
H (t + Δt ) − H (t ) dH (t ) = . Δt →0 Δt dt (1.43) (1.42) ,
(t ) = lim
(1.43)
(t ) = ∑
:
∞
dFn (t ) . dt n =1
(1.44)
3. Fn(t),
. . , . .
. : 1. . 2.
. k
k-
.
:
-22ς=
k
ς[
−
k −1,
(1.45) ]–
,
ξk.
ηk
.1.5. 0
= η1 = 0; n-
1
= ∑ςk .
= ς1 = ,
1.
,
:
n
n
k =1
,
(1.46) (0,t) n-
,
n
, –
,
: Fn (t ) = P {Vt ≥ n} = P { n < t }
t.
⎧n ⎫ Fn (t ) = P ⎨∑ ς k < t ⎬ ⎩ k =1 ⎭
ς1 – ςn
(1.47) ,
Fn(t) . . ως k (t ) –
Θ ς k (iv )
ςk.
, . .:
Θ ς k ( −iv ) = ∫ ως k (t )e ivt dt . ∞
(1.48)
0
ω ς k (t ) =
1 2π
∫ Θς
:
∞
−∞
k
(iv )e −ivt dt . (1.49)
(1.48)
, e ivς k .
ςk
∑n = ∑ ς k ,
:
n
k =1
{ }
⎧⎪ iv ∑n ς k ⎫⎪ n Θ ∑n (iv ) = m1 ⎨e k =1 ⎬ = ∏ m1 e ivς k , ⎪⎭ k =1 ⎪⎩
-23Θ ∑n (iv ) = ∏ Θ ς k (iv ), n
:
k =1
(1.50)
. . . W∑ n ( t ) =
1 2π
:
∫ ∏ Θς
∞
n
−∞ k =1
k
( iv )e ivt dv
(1.51)
ςk = ηk+ξk Θ ς k (iv ) = Θ ηk (iv ) ⋅ Θ k (iv ), Θηk ( iv )
Θξ k ( iv )
,
: (1.52)
–
, . ωηk (t ) ω k (t ) :
Θ ηk (iv ) = ∫ ωηk (t )e ivt dt , ∞
(1.53)
Θ k (iv ) = ∫ ω k (t )e ivt dt . 0
∞
(1.54)
0
(1.47), (1.49), (1.51)÷(1.54) Fn(t) :
Fn (t ) = ∫ W∑n (t )dt = t
=
1 2π
∞ ⎡∞ ⎤ ivx ( ) ⋅ ω x e d x ω k ( y )e ivy d y ⎥ ⋅ e −ivt dvdt . ⎢ ∫ ηk ∫∫∏ ∫ ⎥⎦ 0 −∞ k =1 ⎢ 0 ⎣0 0 t ∞
n
ω(t) (1.55) 1 Fn (t ) = 2π : 1 Fn (t ) = 2π 4.
ω (t)
. n-
,
, . .:
∞ ⎡∞ ⎤ ivx ivy −ivt ( ) ( ) ω x e d x ω y e d y ⋅ ⎢ ⎥ ⋅ e dvdt . ∫ ∫ ⎢∫ ∫ ⎥⎦ 0 −∞ ⎣ 0 0 , t ∞
n
⎡∞ ⎤ ivy −ivt ( ) ω y e d y ⎢ ⎥ ⋅ e dvdt . ∫ ∫ ⎢∫ 0 −∞ ⎣ 0 ⎦⎥ t ∞
(1.55)
(1.56) ω (t)= (x),
n
(1.57)
,
-24-
Ω(t)dt
,
.
(t,t+dt)
,
, . . .
, (t,t+dt) dt,
, ,
An
dt
. (t,t+dt)
,
n-
. ,
n-
dFn (t ) , dt
P {An } = dFn (t ). , Ω(t)dt, , n: ∞ ⎧ ⎫ Ω(t )dt = P ⎨U An ⎬. ⎩n =1 ⎭
An
: ,
(1.58) Ak
(1.58)
(t,t+dt) An
Ar
k≠r
,
∞ ⎫ ∞ ⎧∞ Ω(t )dt = P ⎨U An ⎬ = ∑ P {An } =∑ dFn (t ). n =1 ⎩n =1 ⎭ n =1 Ω(t)
,
Ω(t ) = ∑
:
(1.59) .
(1.59)
∞
:
dFn (t ) . dt n =1
(1.60) ,
(1.42.),
: dH (t ) Ω(t ) = =ω dt . .
(t ). ω
(1.61) Ω(t) (t)
!
,
Ω(t)
, . (1.61)
H(0)=0,
t
0
tc
,
:
H (t ) = ∫ Ω(t )dt . t
(1.62) Ω(t)
0
, ω(t)
ω(t).
Ω(t) :
-25-
Ω(t ) = ω(t ) + ∫ Ω( )ω(t − )d , t
(1.63)
0
.
ω(t),
(1.63)
, . ,
(1.63) ,
–
, ,
. . .
Ω(t) (1.63)
. Ω( p ) = Ω( p ) ⋅ ω( p ) + ω( p ). (1.63) ω( p ) , Ω( p ) = 1 − ω( p )
ω( p ) =
Ω( p ) . 1 + Ω( p ) (1.64)
,
: : (1.64)
(1.65) (1.65)
, (1.64)
ω(p) (1.65).
:
lim Ω(t ) = lim pΩ( p ).
t →∞
Ω(p)
p →0
p ∫ ω(t )e − pt dt ∞
lim pΩ( p ) = lim
p →0
p →0
1 − ∫ ω(t )e − pt dt 0 ∞
.
0
∫ ω(t )dt
,
∞
lim pΩ( p ) = lim
p →0
∫ tω(t )dt
0 p →0 ∞
=
:
1 . T
0
lim Ω(t ) =
t →∞
. .
: 1 , T , , Ω(t)
Ω(t) . :
t→∞,
-26-
1. ω(t)
2.
Ω(t)
1/T ;
t→∞ 3.
) Ω(t)>ω(t);
(
(t) – (t) –
,
4.
, (t)>Ω(t)>ω(t); Ω(t)> (t)>ω(t); , . .
(t)= =const,
(t ) ≠ ∑ Ω i (t ),
Ω
:
N
i =1
Ω(t) = (t) = . Ω(t),
. , :
ω(t ) =
2
te − t . .
ω( p ) = ∫ ω(t )e ∞
0
Ω( p ) =
Ω(p).
− pt
dt = ∫
∞
5.
2
te −(
+ p )t
dt =
0
ω( p ) = . 1 − ω( p ) p(p + 2 ) Ω(t)
(
. 2 + p) (1.64), 2
:
2
:
p1 = 0; p2 = −2 .
Ω(t ) =
(1.64),
ω(t):
(
2⎡
)
1 e −2 t ⎤ −2 t . ⎢ − ⎥ = 1− e 2 2 2 ⎣ ⎦ G(t) (
)
. , g ( t ) = 1 − G( t )
t
,
(
): (1.66)
G(t) , . (0,t)
g(t) – N
Δ.
-27-
.1.6. ,
Bk –
,
Δ
(t – k – Δ k). Ω( k ) ⋅ Δ ⋅ p(t − k ) .
,
,
: (
.
),
G(t), . . t:
⎧N ⎫ G(t ) = p(t ) + lim P ⎨U Bk ⎬ max Δt k →0 ⎩k =1 ⎭ Bi Bj i≠j ⎧N ⎫ N P ⎨U Bk ⎬ = ∑ p(t − k ) ⋅ Ω( k ) ⋅ Δ k . ⎩k =1 ⎭ k =1
,
:
(1.67)
max Δ k→0,
:
G(t ) = p(t ) + ∫ p(t − ) ⋅ Ω( )d . t
(1.68)
0
, . :
g (t ) = 1 − G(t ) = 1 − p(t ) − ∫ p(t − ) ⋅ Ω( )d = t
= g (t ) − ∫ p(t − ) ⋅ Ω( )d .
0
t
(1.69)
0
,
,
.1.7.
(
.
. 1.7):
-28-
. . A1, A2, … An . PAi(B) –
(
)
B
,
B→P(B) Ai. : P (B ) = P ( A1 ) ⋅ PA1 (B ) + P ( A2 ) ⋅ PA2 (B ) + K + P ( Ai ) ⋅ PAi (B ) + K + P ( An ) ⋅ PAn (B ). 6. К
(К ) G(t)
(
t
. 1.7), . . K = lim G(t ). t →∞
K = 1−
(1.70) (1.68)
К t→∞:
T T = . T +T T +T
(1.71)
,К
К
К .
К
.
t. KП = 1− KГ =
,
:
Tв T + Tв
(1.72)
К
К t→∞.
t, G( t ) − K
< ,
(1.73)
–
. , t , K
:
= K ⋅ p(t
). К
(1.74) ,
(t)
(t) .
const, ,
(p(0)=1)
p (t ) =
+
+
. .
t=0 +
e −(
+ )t
: ,
-29-
p (t ) = K + (1 − K )e
=
=
1 ; T
−
,
K =
1 ; T
. .
t K t
(1.75) T T
+T
. К
(1.75)
t. (1.75)
p (t)→К
,
,К
t→∞, ,
. ,
. ,
= 0,02 1/ =const, t =10 . .
(1.22):
=
T
K =
1
= 50 .
T T
=
+T
(1.71)
50 = 0,83. 50 + 10
G(t ) = p (t ) = K + (1 − K )e
−
:
(1.75): t K t
= 0,83 + 0,17e −0,12t .
1.4.4.
1. , –
. T ),
(
( t
T ): = m1[T ]
(1.76) -
t,
:
P {T
>t }=
100.
(1.77) –
, t , (
).
. , .
-30-
, .
Э
-
. . .
2. , . , Э
. . . .
____________________________________________ : 1. . 2. « » . 3. . 4. 5. 6. , 7. ? 8. ?
,
.
Ω(t). .
-31-
II. 2.1.
,
, (
«
»
)
, (
. 2.1).
. 2.1.
.
I.
. Э
. , .
II.
.
. Э .
,
,
,
. III.
. Э
. ,
.
,
. ,
,
(
)
, , . , ,
, «
, »
( .
, ,
. 2.1), ,
.
. , ,
, . .,
-322.2.
, . ( (0,t) ( . 2.2): p(t ) = exp(− t ), t ≥ 0; > 0; > 0.
. 2.2.
(2.1)
p(t)
.
(2.1) ω(t ) = − p ′(t ) = = ∫ e − t dt = ∞
T
(1.3) . 2.3): t −1 exp( − t ). (
,
(
−
1
(1 +
1
)
),
(2.3)
0
( )–
-
.
. -
:
( ) = ∫ t x −1e −t dt . ∞ 0
: ( ) = ( x − 1) ( x − 1) = ( x − 1)( x − 2) ( x − 2) = ... : (4,7)=3,7·2,7·1,7· (1,7) (1,7)=0,9086 – 0;
2
)−
2
(1 +
(1.11): 1
)].
(2.4)
(1.17)
t exp(− t ) = exp( − t ) −1
> 0. , – (t)
,
:
, (2.5) ,
=1 – =const,
. 2.3).
. 2.3.
ω(t)
.
T
, –4 =10 1/
)
−1
t
1 – (t)
( (2.2),
(2.1)
(t). (t) =1,5;
t=100 . . (2.3)
T =
T
−
: 1
1 (1 + ) = (10 −4 ) −0,67 ⋅ (1,67). -
T ≈418 .
,
(1,67)=0,9033
: (2.5)
(100 ) = ⋅ ⋅ (100 )
−1
= 1,5 ⋅ 10 −3 1/ .
, t=100 :
2.3.
Э =1.
(Э ) :
-34-
p(t ) = e − t , t ≥ 0; > 0. (2.5)
(2.6) ,
=1
(t)≡ . , . :
, ,
,
,
Э
. – T ,
= ∫ e − t dt = ∞
T
– 1/ , . .: 1
. (2.7) =1, . . (2)=1.
0
(2.7) p(t ) = e p( :
−
(2.6) t ≥ 0;
1
)=e
; −1
(2.3) :
1/T ,
> 0.
Э
t=T
≅ 0,368.
(
. 2.4):
Э
ω(t ) = − p ′(t ) = e − t .
(2.8) ,
– T .
,
, ,
, 2 T
T . ⎛ 1⎞ = 2 ∫ te − t dt − ⎜ ⎟ ⎝ ⎠ 0 ∞
2
2
∞
−x
dx −
1 2
=
2 2
−
1 2
=
1 2
.
0
t = x; t =
. 2.4. Э
xe 2 ∫
= {
:
x
.
.
-35-
∫ xe
ax
dx =
∫ xe dx =
∞
−x
0
lim
x →∞
2 T
x +1 e
=
:
x
1 2
(ax − 1),
e ax a2
∞
(− x − 1) (− 1)2 0 e −x
= lim
x →∞
1 ex
=
x +1
→ 0.
ex
∞
= 1,
0
,
=T 2
: (2.9) , t, . :
p(t + ) e − ( t + p(t ) = = p( ) e−
)
= e− .
(2.10) : ,
. Э
( . .
≠const,
). . .
.
=10–4 1/ .
t=2000 , . (2.6)
p(2000 ) = e −10 (2.8)
−4
⋅2000
:
= 0,819. :
ω(t ) = 10 −4 ⋅ e −10 ⋅2000 = 8,19 ⋅ 10 −6 1/ . (2.7) 1 T = = 10 4 .
2.4.
(0,t)
(
2.5. ):
−4
p(t) T .
:
ω(t)
-36-
⎛ t2 ⎞ ⎟, p(t ) = exp⎜⎜ − 2 ⎟ 2 ⎝ ⎠ : –
(2.11) , . ω(t). m,
, ω(t ) = − p ′(t ) =
(t ) =
t 2
exp( −
t2 2
2
,
).
(2.12) (
ω(t ) t = 2. p(t )
. 2.5. ): (2.13)
–T .( :
= ∫ tω(t)dt = ∞
T
0
2 T
π⎞ ⎛ = ⎜2 − ⎟ 2⎠ ⎝
2
∫
∞
t2 2
e
−
t2 2
)
dt =
0
= 0,4292
= 1,253 .
π 2
(2.14) :
2
. (2.15)
. 2.5.
2.5.
p( m). ( . 2.5. ):
.
-
:
ω(t ) =
2 0
(r ) (r) –
r=1,
t r −1 exp(−
> 1,
( 0t
)
), -
(2.16) . .
r. r –
-37-
,
0
=
: 1
.
T
0
-
r
Э
. (0,t)
2.6. ):
( 0 t )i . p(t ) = exp( − 0 t ) ∑ i! i =0
(
.
rm =1
ω(t ) = (t ) =
=
T
0
( 0 t ) r −1 exp( − 0 t ). (r − 1)! r −1
=
(2.19) : r 2 0
0
t0
= rT ,
r– 2 T0
. 2.6.
=r
.
: 2 T.
-
) )
. 2.6. ):
.
2 T
,
(2.17) . 2.6. ): (2.18)
(
( 0 t )i (r − 1)! ∑ i! i =0 0 ( 0t ) r −1
r
(
:
; ) .
; ,
r
.
. .
-
.
-382.6.
Э (t , t ),
t
t –
. (Δω(t)
1. (t) (
. 2.7. );
2. 3.
p(t) (
. 2.7.
. 2.7. );
:
ω(t)
) ) ω(t ) = h,
):
(t);
p(t). 1 h(t − t ) = 1. 2
: 2(t − t ) ⎧ ⎪ (t − t )(t − t ) ⎪ ω(t ) = ⎨ 2(t − t ) ⎪ ⎪⎩ (t − t )(t − t )
ω(t)
t ≤t ≤t , t ≤t ≤t . (2.20)
p(t)
⎧ (t − t ) 2 ⎪1 − ⎪ (t − t )(t − t ) p(t ) = ⎨ (t − t ) 2 ⎪ ⎪ (t − t )(t − t ) ⎩
t ≤t ≤t ,
:
t ≤t ≤t . (t)
2(t − t ) ⎧ ⎪ 2 ⎪ (t − t )(t − t ) − (t − t ) (t ) = ⎨ ⎪ 2 t ≤t ≤t . ⎪⎩ t − t
(2.21) :
t ≤t ≤t ,
(2.22)
-39-
Δ:
1
=
2
=
t
=
h t −t )=
p(t
t
2 (t − t )(t − t )
Δ-
(2.23) :
1 . 2
=t −
:
1 2(t − t )(t − t ). 2
= ∫ tω(t )dt = t
T
⎫ t ≤t ≤t ⎪ ⎪ ⎬ t ≤t ≤t ⎪ ⎪⎭
h 2 = t −t (t − t )(t − t )
t
(2.24) T :
1 (t + t + t ). 3
(2.25)
ΔJ=
t −t
.
:
. t −t t = t ⇒ J = 0, t = t ⇒ J = 0. : T −t t −t t −t J = ; J = ; T = . t −t t −t t −t (2.24) (2.25) 1 J = 1− 2(1 − J ), 2 1 T = (1 + J ). 3 p(t ) = 1 −
J=
t −t
t −t
ΦΔ(J)
:
(2.26) :
Δ ( J ),
⎧J 2 0≤J ≤J , ⎪ ⎪J Δ (J ) = ⎨ ⎪J + (J − J )(2 − J − J ) ⎪⎩ 1− J (2.28) – ΦΔ(J)
(2.27)
J ≤ J ≤ 1.
(2.28) ,
. ΦΔ(J)
, .
,
t,
-40-
t, t , t ΦΔ(J) ( . 2.8).
J
J ,
Δ-
. 2.6.
2.7.
(
ΦΔ(J).
)
,
,
. :
ω(t ) = c1ω1(t ) + c 2 ω 2 (t ), : ω1(t) ω2(t) – , c2 – c1 , c1+c2=1.
.
ω(t ) = c1 1e p(t ) = c1e (t ) =
− 1t
− 1t
+ c2
+ c 2e
− 2t − 1t
ω(t ) c1 1e = p(t ) c1e − t
t→∞
2e
,
1t
− 2t
(2.29) , , : . :
. + c2
2e − 2t
+ c 2e e– 1t
(1.8)
− 2t
:
.
e–
t
1
2
– t
e
2
(t)≈ (t)→ 1.
1 1+ 2 2
(
. 2.9).
-41-
. 2.9. 2 >
(t)
1.
T
=
1
:
+
2
1
.
2
2.8.
«
» >1 . (
ω( t ) =
Э
1 2π
2
. 2.10): ⎛ (t − T ) 2 ⎞ ⎟. exp⎜ − 2 ⎜ ⎟ 2 T ⎝ ⎠
2 T
.
(2.30) : .
2.10. .
T
-42-
p(t ) = ∫ ω( )d = ∫ ∞
∞
t
=
1 2π
−
1 2π
t
2 T
( −T )
:
2
2
e
2 T
d =
∫ 2π t −T
∞
1
∫
e
1 ⎛ −T − ⋅⎜⎜ 2⎝ T
⎞ ⎟ ⎟ ⎠
2
⎛ −T d ⎜⎜ T ⎝
⎞ ⎟= ⎟ ⎠
T
x2 ∞ − e 2 dx.
t −T
(2.31) ) :
T
q(t ) = 1 − p(t ) =
t −T
F(x) =
2π
∫
T
1
−∞
2π
⎛ t −T . . q(t ) = F ⎜⎜ T ⎝ 1
(0,t) (
⎞ ⎟, ⎟ ⎠
∫e x
−∞
−
e
−
x2 2 dx,
(2.32) (2.33)
2
u 2
du.
(2.34)
(2.34) –
, . , , . .
(2.30) 0
Э
t> ω(t)
,
T,
. . t>
. ,
T
. ω(t)
. (
. 2.10) t T, T
e
2
2 T
(2.41)
. T =8000 , t=4000 . . (2.38)
:
T
=2000
.
-45⎛ 4000 − 8000 ⎞ F⎜ ⎟ 2000 ⎝ ⎠ = F ( −2 ) = 1 − F ( −2 ) . P ( 4000 ) = F ( 4) F ( 4) ⎛ 8000 ⎞ F⎜ ⎟ ⎝ 2000 ⎠
F(x) (
): F(2)=0,97725; F(4)=1; 1 − 0,97725 P ( 4000 ) = = 0,2275. 1 (2.37) ω(t ) =
F (T
. . F (T
ω(t ) =
)⋅
1
) = F (4) = 1, T
T
φ( x )
2π
T
,
⋅e
−
.
T
: φ( x ) =
1 2π
φ(x) :
x=
t −T
:
t −T
⋅e
−x
2
2.
.
.
T
, : ⎛ 4000 − 8000 ⎞ φ⎜ ⎟ 2000 ⎠ = φ( −2) = φ(2) = 0.05399 = 2,7 ⋅ 10 −5 1/ . ⎝ ω( 4000 ) = 2000 2000 2000 T (1.17): (t ) =
(T )
ω(t ) 2,7 ⋅ 10 −5 = = 11,87 ⋅ 10 −4 1/ . p(t ) 0,02275 (2.41) =T
+
F (T
T T
)⋅
−
2π
e
T
2
2
2 T
:
= 8000 +
2000 ⋅ e −8
F ( 4) ⋅ 2π
= 8000,26 .
2.9.
. p (t ) = 1 − e , t
ω ( t ) = p ′ (t ) = e
t ≥ 0, − t
,
p (t)
> 0. t ≥ 0,
: (2.42) :
> 0.
(2.43) :
-46(t ) =
ω (t ) = . 1 − p (t ) ,
(2.44) . :
1 T = ∫ [1 − p (t )]dt = ∫ e − t dt = . ∞
∞
0
p (t ) = 1 − e
:
(2.45)
0
−
(2.42),
1/T t T
t ≥ 0, T > 0
,
= 2∫ t [1 − p (t )]dt − T = 2 ∫ te − t dt − ∞
2
:
∞
2
0
(2.46)
1 2
=
1 2
= T 2. (2.47)
0
. , ,
, .
2.10.
,
,
. ,
, ,
,
. . 1. 0,1,2…,n. m n
:
Pn (m ) = C nm p m q n −m [m] = np; p– q=1–p. 2.
0,1,2…,n.
2 T [m ]
(2.48) :
= npq,
(2.49) ;
m
:
-47-
Pm =
⋅ e− .
m
m!
[m ] = ; – 3. Pm = pq
[m ] =
2 T [m]
(2.50) :
= ,
(2.51) . 0,1,2…,n.
m −1
1 ; p
.
(2.52) 2 T [m]
=
:
q p2
, (2.53)
p– q=1–p. ____________________________________________ : 1. « » . 2. « » ? 3. ( ) « » . 4. « » 5. , 6. .
;
? .
-48-
III.
(
)
3.1
, ( ,
,
,
) ,
. ,
,
: ,
-
(
)
;
( ); (
-
,
,
,
,
); . .
,
,
, . , . , ,
, ,
(
,
. .).
,
, ,
.
, . .
, ,
,
, . ,
,
,
,
.
, : -
Э (Э
,
Э
). ,
: ,
,
Э
; . , . .
3.2.
,
N Э . .
. Э
-49-
: p(t ), ω(t ), – 1.
(t ), T Э
p(t ) → p(t ) = N– n(t) –
q (t ) =
N − n( t ) , N Э
:
: (3.1) ;
Э
t;
n( t ) . N
(3.2)
2.
Э
Э
, n( Δt ) , N ⋅ Δt n(Δt) – ⎛ Δt ⎞ ⎜t − ⎟ 2⎠ ⎝
:
ω(t ) =
3.
(t ) → (t ) =
⎛ Δt ⎞ ⎜ t + ⎟. 2⎠ ⎝
Э
n( Δt ) ; N ⋅ Δt
:
N + N i +1 = i 2 –
N
(3.3)
Э
(3.4)
Э
Ni – Ni+1 –
Э
Δt; Δt.
Э
4.
∑ ti
:
N
T
=
i =1
,
N
ti –
Э
i-
(3.5) . T
Э
(3.5)
,
.
,
Э
i-
∑ ni t
:
m
T
≈
i =1
N t
i
=
i
,
t i −1 + t i ; 2
m=
(3.6) tk ; Δt
;
-50-
ti-1 – ti – tk – , ni – Δt = ti – ti-1 –
i-
;
i-
.
; Э
Э
;
. 1000 Э
.
3000
3000÷4000 3000
;
i-
3000÷4000
50 Э
80 Э .
,
Э
.
. (3.1) (3.2) : N − n(t ) 1000 − 80 = P (3000 ) = = 0,92; N 1000 n(t ) 80 q(3000 ) = = = 0,08; N 1000 N − n(3500 ) 1000 − 105 P (3500 ) = = 0,895, = N 1000 N i + N i +1 920 + 870 = 1000 − = 1000 − 895 = 105 2 2 – Э t=3500 ( ). (3.3) (3.4) : n( Δt ) 80 ω(3000 ) = = ≈ 2,67 ⋅ 10 −5 1/ ; N ⋅ Δt 1000 ⋅ 3000 50 ω(3500 ) = = 5 ⋅ 10 −5 1/ ; 1000 ⋅ 1000 80 n( Δt ) (3000 ) = = ≈ 2,98 ⋅ 10 −5 1/ ; N ⋅ Δt 895 ⋅ 3000 n(3500 ) = N − N
(3500 ) =
=N−
50 ≈ 5,59 ⋅ 10 −5 1/ . 895 ⋅ 1000
3.3.
N Э ( ,
.
Э ). ,
-51-
. , Ω(t),
-
t .
1. Э
, :
Э
, : n( Δt ) Ω (t ) = , N ⋅ Δt n(Δt) – ⎛ Δt ⎞ ⎜t − ⎟ 2⎠ ⎝ N– Δt – (3.7) 2.
Э
(3.7) ⎛ Δt ⎞ ⎜ t + ⎟; 2⎠ ⎝
Э
;
.
Ω(t). .
Э
∑ ti :
n
=
t
i =1
n ti – n–
,
∑∑ t ij N
=
t
(i-1)-
i-
,
t. (3.8) Э .
t,
(3.8)
Э ,
N Э :
nj
j =1 i =1 N
∑nj
,
j =1
tij –
Э
j;
nj –
j-
Э
Ω (t )
(3.9) (i-1)-
t. t
,
.
.
,
, . K ,
.
i-
-52-
K =
: t
t +t
, (3.10)
tp – t –
,
t = ∑t i ; n
i =1
tpi – t i– n–
t = ∑t i ,
,
n
i =1
, )Э
(
(i-1)i-
(3.11) ,
i,
.
(3.10) tp K =
, t t
+t
.
K. t
:
, (3.12)
t – t –
, . K ,
K
=
. t t +t
(3.13)
, K K K
=
, . . t t
+t
:
. (3.14)
K = 1− K .
(1.72): K
K =
(1.71): T T
Э : t =T ,
+T
. , (3.15)
t – T –
; .
. . 6
, 11
. 181
,
– 329
– 245
.
8
-53-
. . t Σ = ∑∑ t ij = 181 + 329 + 245 = 755
:
nj
N
j =1 i =1
n Σ = ∑ n j = 6 + 11 + 8 = 25
.
:
N
j =1
.
(3.9)
∑∑ t ij N
t
=
:
nj
j =1 i =1 N
∑nj j =1
=
tΣ 755 = = 30,2 . nΣ 25
____________________________________________ : 1. ? 2.
Э
? 3. .
-54-
ё
IV.
4.1.
.
:
1. 2. 3. 4.
. . . .
1. : 1. 2. 3.
. . . ,
. .
. (
). . . .
, , . , .
2. , (
. 4.1.
)
.
.
-55-
, 1
2(
. 4.1). tp
= ∫ P (t )dt
P1(t)>P2(t). T
∞ 0
(
P(t))) , . .T
(
1 < T ,
,
1.
)
,
. , . . 1968
«
, »,
.:
,
, 1968. , . ,
1972
« », Э
.:
, 1972.
.
, . .
:
1.
, ,
, .
2.
. : )
,
, ;
)
, ;
)
,
, . . η,
– η– (
:
, η )
. : ~ T,
T, T.
-56-
, :
=
~ (T ,T ~ η = η (T ,T
,
) ⎫⎪ ⎬ )⎪⎭
(4.1)
(
η
(4.1) ):
m =Э=
) ⎫⎪ ⎬ mη = W = η (T ,T )⎪⎭ , Э– W– , T , T – . , , (T ,T
(4.2) ,
, T K,
T .
,
φ, . .
,
(tj, tj+1). – . ( )
,
. ,
. ,
, : 1)
–
P(t) ω(t),
(t). 2) P(t1,t2) Ω(t).
(t1,t2), 3) . ,
, G(t). ,
, P(t1,t2).
, (
),
: 1.
: ) )
i;
T ;
-57-
)
P(Δt3)
Δt3.
2. (
,
): ) T ; ; ) ) ) P(t1), P(t2) – (0,t1) 3.
(0,t2).
,
:
- P(t) (t); Ω(t) – -
; (
).
4.2.
. . , . , (
.
) . , . :
1) 2) 3) 4) 5) 6)
; ; ; (
,
,
. .);
; .
1.
. (
,
. .). . , .
–
-58-
(
y)
( (
x) (
.4.2).
. 4.2) . y=a+bx. .
. 4.2. J = ∑ [a + bx i − y i ] −2 = min.
:
K
i =1
(4.3)
∂J = ∑ [a + bx i − y i ] = 0, ∂a i =1
a
b
:
K
K ∂J = ∑ [a + bx i − y i ] x i = 0. ∂b i =1
(4.4) x1,…,xn,
(a+b1x11+…+bnxni–y1)
a,b1,…,bn. , .
. . . . P(t) : xi yi Wi,
⎫ ⎪ ⎪ ⎬ Pi (t ) ⎪ = P (t ) ⎪⎭ , W – =
W.
Wi W
(4.5) , i-
;
-59-
Pi(t), P (t) – .
,
i.
. 4.3. , (
)
. ( ) K ,
. Э . 2.
,
,
,
, . ,
,
.
.
, . . . 4.4).
(
, K , .
. : 1) 2)
; , . . .
-60-
. 4.4.
.
. . 3. . . K . . : 1. 2.
. . .
, .
3. 4.
, ,
. .
Э ,
.
.
. Э
– K
, . K K
. ,
«
(
,
)
»
r (t ) = − ln P (t ) = ∫ ( )d
. :
t
0
.
(4.6)
-61-
, r
. Э
, ,
. .
(
,
. .).
Ω(t).
(t) . . (
. 4.5), (
)
. , ,
. .
Э
=
U− W−
. 4.5.
U U
; = .
W W
⎫ ,⎪ ⎪ ⎪⎪ ⎬ ,⎪ ⎪ ⎪ ⎪⎭
:
(4.7)
-62-
.
,
(t) 10°C. , . (t)
Ki =
i *
, (4.8) .
*– , (t). : 1)
(t ) = f (
:
: 0 ,t
0
, 1,
2 ... n ),
(4.9)
0 – 0 t – 1,…, n, –
, , .
2)
: ,
-
, (t)
.
ó
.
. : 1;
-
2.
. = f(
, 0 ,t
0
, 1,
2 ).
= (1 + C1 + C 2 )
: 0
,
(4.10)
C1 – C2 –
1; 2
t0;
: , 1 = U /U W + W0 + W ; 2 = W +W W , W0, W – W , W0 –
, .
C1 (
.4.6).
C2
1,
2
t0
,
-63-
.4.6. . , : 1 U
: , J
2 3 4 5
: : : :
6
:
,
6
U
J W
, W . U J < 75% U U J < 90% U U J < 90% U
J J J U,
; ; ; J
W < 50% W W < 75% W W < 90% W W
50% 25% . . . 90%
. U, J
W
100%
. . (t)
.4.7.
(
.4.7).
100%
-64,
, . .
4. Э , .
Э :
,
,
. (Э ) 1. 2. 3.
: ;
; -
.
, . ,
,
, . .
-
,
. , . , (
)
. .
,
Э(t ) = −(
1
+
:
2)+
t,
1 – 2 – –
(4.11) ;
,
; ;
t– Э = −(
1
. +
(
2)+
)
T ,
(4.12)
T –
. . , T
К
=
. T
.
(4.13)
(0,T ,
K).
a
:
-65−
=
2
+
T
T
.
К
[
]
⎞ ⎛ ⎜ S0 = 1 − exp(− ℵt ) ⎟ ℵ ⎠ ⎝ − 2+ T Э =− 1+ 1 − expℵT ℵT К =−
1+
ℵ –
(
⎛ 2 ⎜ − ℵ ⎜⎝ T К
(4.14) :
(
К
)=
⎞ ⎟ 1 − expℵT ⎟ ⎠
К
),
(4.15) ,
T = 8760 ; ℵ = 13 ⋅ 10 −6 1/ .
E = 0,12; .
Э .
ΔЭ
i
:
=Э
i
−Э
Э0 Э
: 0
(4.16)
– ; i
– ,
i.
ΔЭ i. ,
.
, .
, . , , .
4.3.
( ) . , . : 1. 2. 3.
. . .
-664.
, . . 1. . P(t)=0,98
t =2000 .
(t) .
P (t ) = [P P (t ) = e −
t
]3 ;
−1 T
;
P (t ) = e ≈ 1− t : 1 − 0,98 = = 10 −5 1/ . 2000
≤
−
.
=3
. (t )
=e
.
;
T
1 = T 3
;
= 0,98.
t
10 −5 = 3,3 ⋅ 10 −6 1/ . 3 2.
1. 2. 3.
,
A, B, C. t1=100 .
P (t1)=0,97 , –4
A0=10
A, B, C, –4
1/ ;
B0=8·10
A1, B1, C1 ⇒
–4
1/ ; A1,
C0=3·10 B1,
: 1/ ;
C1.
. 1.
, -
Kj =
j
j-
, ,
j
–
j-
. Kj Kj =
∑
, :
j0
,
n
i =1
ji 0
n–
. :
K = 0
+
0 0
+
= 0
10 −4
(1 + 8 + 3) ⋅ 10
−4
=
1 ; 12
-67K = K =
0
+
0
+
0 0 0 0
=
+
0
+
=
0
2.
8 ⋅ 10 −4
(1 + 8 + 3) ⋅ 10 3 ⋅ 10 −4
(1 + 8 + 3) ⋅ 10 −4 (t)
P (t 1 ) = 1 − 3. 1
=K
1
=K
1
=K
−4
= =
=
t1 = 0,97 ;
2 ; 3 1 . 4
1 − 0,97 = 3 ⋅ 10 − 4 1/ . 100 :
=
1 1 ⋅ 3 ⋅ 10 − 4 = ⋅ 10 −4 = 2,5 ⋅ 10 −5 1/ ; 4 12 2 = ⋅ 3 ⋅ 10 −4 = 2 ⋅ 10 −4 1/ ; 3 3 1 = ⋅ 3 ⋅ 10 −4 = ⋅ 10 − 4 = 7,5 ⋅ 10 −5 1/ . 4 4 3.
1. A1
B1.
2.
P(t)=0,97 t1=100 .
3. 4.
–2007 . 1992÷2002 ,
exp[− 92 – L–
=
(L − 1992 )] ,
92
:
A1
B1,
: ,
A0:
92
B0:
92
= 1,4 ⋅ 10
−4
. 1/ ;
1992
= 0,034 1/
;
= 0,14 1/
= 28 ⋅ 10 −4 1/ ;
. A
A1
.
1. Э 07 07
2. K K
1
1
= 1,4 ⋅ 10
−4
B1.
⋅ exp[− 0,034 ⋅ (2007 − 1992 )] = 8,4 ⋅ 10
−5
2007 .: 1/ ;
= 28 ⋅ 10 −4 ⋅ exp[− 0,14 ⋅ (2007 − 1992 )] = 34 ⋅ 10 −5 1/ 2 :
= = =
=
+
07
+
07
07
07
07
=
8,4 ⋅ 10 −5
(8,4 + 34 ) ⋅ 10 −5 34 ⋅ 10 −5
(8,4 + 34) ⋅ 10 −5
= 0,2;
= 0,8
1 − P (t ) 1 − 0,98 = = 2 ⋅ 10 −4 1/ ; t1 100 07
B
;
.
Kj
;
-68-
1 1
=K
=K
= 0,4 ⋅ 10 −4 1/ ;
= 1,6 ⋅ 10 −4 1/ .
1 1
4. 1. 2. 3.
1, 2, 3, 4.
ΩC=10–5 1/ ; =5 t =20
4.
(1 =
j
K
) j-
+
Ωj
j
,
. :
0j,
Ωj –
j-
;
–
0j
,
; .⋅
:
K
1
= 1,6 ⋅ 10 −4
K
2
=K
01
=
3
=K
=
02
03
;
2
4
=
.⋅
= 3 ⋅ 10 −4 04
=0
;
2
( ).
5. = K Эj ⋅ Ω j + : :
j
K Э1 = 4 ⋅ 10 6 01
=
02
=
03
=
0j,
./ 04
;
= 0.
K Э 2 = K Э 3 = K Э 4 = 1,7 ⋅ 10 6
6.
./
;
, =∑
:
n
j,
j =1
:
Cj – n–
j-
, .
Ωj. . – . : Эj
=
0j
=
:
ℵ j
ℵ
[1 − exp(− ℵt p )] =
0j
[1 − exp(− ℵt p )] ;
0j
+
jΩ j,
-69=
j
[1 − exp(− ℵt p )] .
K Эj ℵ
nj
=
n
=
[exp(ℵ
ℵ j
1 ⎛⎜ ℵ ⎜⎝
0j
:
α0 j = =
n
:
ℵ K nj
0j
+
) − 1] .
:
K nj ⎞ α ⎟[exp(ℵ ) − 1] = α 0 j + j , Ω j ⎟⎠ Ωj
(4.17)
[exp(ℵ ) − 1] ;
[exp(ℵ ) − 1] .
ℵ
= ∑(
,
+ α0 j )+ ∑
n
j =1
n
j =1
0j
αj
Ωj
+∑
jΩ j.
n
j =1
(4.18)
. Ωi
∑ Ω j − Ω i = 0.
Ωj
:
n
j =1
Ωj,
(4.19) . .
)= ∑
Φ (Ω 1,K Ω n
n
j =1
αj
Ωj
+∑ n
j =1
⎟ , ⎜ j Ω j + ∑ Ω j − ΩC ⎛ n ⎜ j =1 ⎝ .
– Ω1,...,Ωn
⎞ ⎟ ⎠
:
:
α1 ⎧ ∂Φ ⎪ ∂Ω = − 2 + 1 + = 0 Ω1 ⎪ 1 ⎪ ⎨KKKKKKKKKKK ⎪ ∂Φ α ⎪ = − n2 + n + = 0 Ωn ⎪⎩ ∂Ω n =
α1
Ω 12
−
1
=
α2
Ω 22 :
−
2
=K=
αn
Ω n2
−
: n.
(4.19)
-70-
Ωj
=
αj
α1
Ω 12
−
1
.
+
j
(4.20) (4.20)
n ⎛α Ω 1 + ∑ α j ⎜⎜ 12 − j =2 ⎝ Ω1
1
+
j
Э A(Ω1 ) = B(Ω1 ) , :
A(Ω 1 ) = Ω C − Ω 1 − B( Ω 1 ) = ∑
A(Ω1)
hj =
.4.8.
(
.4.9).
,
:
⎫ ⎪ ⎞⎪ ⎟ 1+ 2⎟⎪ ⎠⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (4.21)
1
+
j
⎞ ⎟ ⎟ ⎠
(4.21)
B(Ω1).
Ω1
,
Ωj ⎛Ωj ⎞ ⎟ ⎜ ⎜Ω ⎟ ⎝ 1⎠
⎛ α1 ⎜ ⎜Ω2 − ⎝ 1
⎛ α1 ⎜ ⎜Ω2 − ⎝ 1
j =3
⎞ ⎟ − Ω C = 0. ⎟ ⎠
:
α2
αj
n
(4.19),
(4.20),
. (4.20)
=
⋅
αj α1
j
−
α1
1
1 , 1+ h j
⋅ Ω 12 .
:
.
-71-
.4.9. 1 (1 + h j ), α j α1.
(
. 4.9). A(Ω1) B(Ω1) ⎛ α2 1 ⋅ A(Ω 1 ) = Ω C − Ω 1 ⎜1 + ⎜ α1 1 + h2 ⎝
B( Ω 1 ) = Ω 1 ∑ n
αj
j =3
α1
⋅
:
⎞ ⎟ , ⎟ ⎠
1 . 1+ h j
(4.22)
Ω1,
(4.22). ℵ = 13 ⋅ 10 α1 =
−6
1/ ;
1,6 ⋅ 10 −4 13 ⋅ 10
[exp(13 ⋅ 10
−6
α2 = α3 = α4 = 1
= =
4 ⋅ 10 6
13 ⋅ 10 =
:
−6
3 ⋅ 10 −5
13 ⋅ 10
−6
−6
[ (
1,7 ⋅ 10 5
) ]
⋅ exp 13 ⋅ 10 −6 ⋅ 8760 ⋅ 5 − 1 = 1,78;
[1 − exp(− 13 ⋅ 10
=
) ]
⋅ 8760 ⋅ 5 − 1 = 0,955;
−6
)]
⋅ 8760 ⋅ 20 = 2,76 ⋅ 1011
[1 − exp(− 13 ⋅ 10
−6
3
4
)]
;
⋅ 8760 ⋅ 20 = 1,17 ⋅ 1010
13 ⋅ 10 −6 1,17 ⋅ 1010 − 2,76 ⋅ 1011 h2 = ⋅ Ω 12 = −2,77 ⋅ 1011 ⋅ Ω 12 . 0,955 (4.22) A(Ω1) Ω1 = 1,43·10–6 1/ . : Ω2 = Ω3 = Ω4 = 2,86·10–6 1/ . 2
.⋅
B(Ω1).
.⋅
;
-724.4.
,
, (
)
.
: 1. 2. 3. 4.
. .
Э .
.
,
, .
(
). ,
:
1) 2) 3)
; ; , . ,
,
. , .
.
, .
,
, .
К . .
Э
,
. ,
,
. : . . . .
, , ,
. K ,
. :
-73=
t
t
, K t – K –
; .
4.5.
. . «
» (
)
. . . , (
)
. (
), .
. .
, , .
Э
.
, (
) (
). ( 1.
). (
)
, .
. 4.10.
, T
(
)
.
:
≅ min(T j ), j = 1,2,..., n ,
(4.23)
n–
.
P (t ) = ∏ P j (t ), n
j =1
Pj(t) –
(4.24) j-
. n
:
-74=∑ n
j =1
T
(
i
j
= const ). (4.25) :
=
∑ 1/ T 1
,
n
j =1
T
j
(4.26)
j –
j-
.
(1.21)
(4.24)
:
n ⎧⎪ n t ⎫⎪ ⎧⎪ t P (t ) = ∏ exp⎨− ∫ j (t )dt ⎬ = exp⎨− ∑ ∫ ⎪⎩ j =1 0 ⎪⎭ ⎪⎩ 0 j =1 , ( =const) :
P (t ) = e −
= ∑nj
t
=e
⎫⎪
j (t )dt ⎬.
⎪⎭
(4.27)
− t
,
,
(4.25) :
(4.26).
r
j =1
j,
(4.28)
nj – r–
j-
; .
P (t)
.
–
e
·t