Прикладные вопросы оценки технического состояния судовых механических систем

ПРИКЛАДНЫЕ ВОПРОСЫ 
ОЦЕНКИ ТЕХНИЧЕСКОГО 
СОСТОЯНИЯ СУДОВЫХ 
МЕХАНИЧЕСКИХ СИСТЕМ

Монография

Москва
ВУЗОВСКИЙ УЧЕБНИК
ИНФРА-М
2018

А.В. Неменко
М.М. Никитин

СЕВАСТОПОЛЬСКИЙ 
ГОСУДАРСТВЕННЫЙ 
УНИВЕРСИТЕТ

А в т о р ы: 
Александра Васильевна Неменко, кандидат технических наук, доцент, 
доцент кафедры «Техническая механика и машиноведение» Севастопольского государственного университета;
Михаил Михайлович Никитин, аспирант Севастопольского государственного университета

Р е ц е н з е н т ы: 
А.М. Олейников, доктор технических наук, профессор, действительный 
член Крымской академии наук, руководитель отдела по развитию и новым 
технологиям 13 судоремонтного завода Черноморского флота Министерства обороны Российской Федерации;
А.К. Сухов, доктор технических наук, профессор, академик, вице-президент Крымской академии наук, главный научный сотрудник Института 
природно-технических систем Российской Федерации

УДК 629.5.06(075.4)
ББК 39.45
 
Н50

© Неменко А.В., 
   Никитин М.М., 2017
© Вузовский учебник, 
    2017

ISBN 978-5-9558-0579-5 (Вузовский учебник)
ISBN 978-5-16-013025-5 (ИНФРА-М, print)
ISBN 978-5-16-105786-5 (ИНФРА-М, online)

Неменко А.В.
Н50 
 
Прикладные вопросы оценки технического состояния судовых 
механических систем : монография / А.В. Неменко, М.М. Никитин. — М. : ИНФРА-М, 2018. — 174 с. — (Научная книга). — www.
dx.doi.org/10.12737/monography_593a8acb390493.57130900.

ISBN 978-5-9558-0579-5 (Вузовский учебник)
ISBN 978-5-16-013025-5 (ИНФРА-М, print)
ISBN 978-5-16-105786-5 (ИНФРА-М, online)

Рассмотрены вопросы диагностики технического состояния судовых 
механических систем, машин и механизмов. Предложены приемы математического моделирования реальных физических процессов, проходящих при работе этих систем, машин и механизмов. Приведены методы 
дальнего прогнозирования временных рядов рабочих параметров, составления этих рядов с минимальной потерей точности и применения их для 
предотвращения аварийных ситуаций.

УДК 629.5.06(075.4)
ББК 39.45

. 
. -, , . 
, , , 
. -.  
-. 
. 
, , . , .  
, . 
.  
, , . , , , : , - .  , , . , , , 4

, . 
,  , , , , . 
.  
, . 
. , , . . , .  
, «– » () , , 
, , . , , – .  , , . , . , : , , ., .  
, , . , , . , , «», . 
1.3, R
σ
. 5

(
)
N
σ
, : 
-, , . , . R
σ
(
)
N
σ
. (3.2) R
σ
. 
, .  
, , (, ..). . 
: , , 
, , .  
, , . , . . 
. , , , , .  
, .  (1.4), , (3.6).  

1. , . : 
; ; ; , , ; . 

1.1. , , 
, . 
.  
. , , , , 1 .  
[1] , (m+1) 0
1
2
,
,
, ...,
m
X
X
X
X
X
=
, 
 (1.1) 

t, 

0 1
2
, ,
, ..., m
t
t
t t
t
=
 
(1.2) 

, t
Δ . 
1

1
1
0

1
m

m
i
i
m
i
X
s
q
X
q

−

+
+
=

=
⋅
−
⋅
, 
 (1.3) 

0

m

i
i
i

s
q
X
=
=
⋅
. 
(1.4) 

qi, (1.3) , , (1.1).

, (1.1) (
)1 i
i
m
q
i
= −
⋅, 
(1.5) 

1
,...,
0
−
=
m
i
;  
i

m  – m i, 
(
)

!
!
!

m
m
i
i
m
i
=
−
. 

(1.4) p : 

( )
(
)
const
1

0
=
⋅⋅
−
= =
+

m

i
p
i
i
p
X
i
m
s
.  
(1.6) 

, (1.1) , , . 
. (1.1) m

(
)
=
−
⋅
=

m

n

n
n
t
t
c
X(t)

0
0
, 
(1.7) 

nc  – . 
(1.3), (1.5) 
(1.7) 

  
(
)
(
)
(
)
1

1
0
0
1
1
1
m
m
m
i
n
n
m
n
i
X
s
c
i
t
i

−

+
=
=

= −
⋅
−
−
⋅
⋅
⋅
+
⋅Δ
, 
 (1.8) 

(
)
=
=
Δ
⋅
⋅
⋅⋅
−
=

m

n
n
n

i

i
t
i
c
i
s

0
0
1
. 
 (1.9) 

(1.9) [2] : 

 
 
(
)
0
1
0
i
k

i

i
i
=

−
⋅
⋅
=
, 0
,
k
m
≤
<
k – , 
 (1.10) 

(
)
(
)
0
1
1
!
i
m
k

i

i
m
i
=

−
⋅
⋅
= −
⋅
,  k
m
=
, k – . 
(1.11) 

8

( )
(
)
0
1
!
m
m
m
s
m
c
t
= −
⋅
⋅
⋅Δ
. 
(1.12) 

(1.6) (
)1
p =
. ( )1
s
, (1.9) i
X 1
+
i
X
  

( )
(
)
(
)
1

0
0
1
1
m
i
n
n
n
i
s
c
i
t
i
=
=

=
−
⋅
⋅
⋅
+
⋅Δ
. 
(1.13) 

, a (
)n
a
i +
i n 0, (1.11)  (1.12), (
)
(
)
(
)
0
0
1
1
1
i
k
i
k

i
i

i
i
i
i
=
=

−
⋅
⋅
+
=
−
⋅
⋅
, 0
,
k
m
≤
≤
k – . (1.14) 

(1.14) ( )
( )
(
)
0
1
1
!
m
m
m
s
s
m
c
t
=
= −
⋅
⋅
⋅Δ
, 
(1.15)  

(1.4) (1.5) 
p .  
( )1
s
( )
2
s
, ( )
r
s
(
)1
r
s
+ , r – ,  0
r
p
≤
≤
.  
r (
)
0
r =
(
)
r
p
=
,  , ( )
( )
0
p
s
s
=
. 
. (1.7), , (
)
1
0
1
m
n
n
m
n
n
X
c
m
t
+
=
=
⋅
+
⋅Δ
. 
(1.16) 

, (1.3) (1.4), (1.8), (1.9) (1.11),  (
)
( )
( )
(
)
(
)
0
1
1
0
1
1
1
m
m
m
n
n
m
n
n
X
s
s
c
m
t
+
=

= −
⋅
−
+ −
⋅
⋅
+
⋅Δ
. 
(1.17)  

(1.14), (1.15) (1.16) . 
. (1.8). 1

1
1

a
a
a

b
b
b

+
+
=
+
+
. 
(1.18) 

(1.3) (
)
(
)
1
0

1
1
1
m
m
i
m
i
i

m
X
X
i
+
=

+
=
−
−
⋅
⋅
. 
(1.19) 

m (1.19) m , ( )
O m . 
, ( )
(
) (
)
(
) (
)
(
)
(
) (
)
(
) (
)
(
)

0
1
1
1

0
1
1
1
0

...
...

...
...

m
i
i
m
m
i
i
i
i
i
i
i
i
m
i

t
t
t
t
t
t
t
t
t
t
L
t
X
t
t
t
t
t
t
t
t
t
t

−
+

−
+
=

−
⋅
−
⋅
⋅
−
⋅
−
⋅
⋅
−
=
−
⋅
−
⋅
⋅
−
⋅
−
⋅
⋅
−
, (1.20) 

(
)1
m+
 (
)
(
)
1
1
m
m
m
+
⋅
⋅
−
, (
)
3
O m
. . 

1.2. , , 
, , . ,  . , , , , . 
. , – , – 
, , , . 
, , . .   

[3] . ( )
f x  – (
)
0
x
x
=

( )
(
)
∞

=
−
⋅
=

0
0
n

n
n
x
x
c
x
f
.  
(1.21) 

( )
f x , .. ( )
(
)
∞

=
−
=

0
0
n
n
n
x
x

q
x
f
.  
(1.22)  

, (
)
0
x
x
=
(1.22) , (1.22) n. . (1.22) 
( )
(
)
∞

=
+
−

′
=

0
0
1
n
n
n
x
x

q
x
f
, 
(1.23) 

(
)
0
x
x
=
, (1.23) (1.21). 
. 0

1
1
x
x
x
=
−
+
; 

0

1
x
x
x
=
−
, 1
x
x
x
=
+
. 
(1.24)

( )

0
0

n
n
n
n
n
n

f
x
q
x
q
x
∞
∞

=
=
′
=
⋅
=
⋅
. 
(1.25) 

(1.24), (1.25) 0
0
1

n
n
n
n
n
n

x
q
q
x
x

∞
∞

=
=

′ ⋅
=
⋅
+
. 
 (1.26) 

( )
1

n

n
x
F
x
x
= +
(1.27) 

11

( )
,
0

k
n
n k
k

F
x
c
x
∞

=
=
⋅
. 
(1.28) 

( )
(
)n
n
x
x
F
+
=
′
1

1
, (1.28) ( )
( )x
F
x
x
F
n
n
n
′
⋅
=
.

( )
,
0
.
k
n
n k
k

F
x
c
x
∞

=
′
′
=
⋅
(1.29) 

,n k
c′
.  [2] ( )
(
)
∞

=
⋅
−
=
+
=
′

0
1
1
1
1

k

k
k x
x
x
F
. 
(1.30) 

 1.2.1: 

 
 
( )
(
)
(
)

(
)
( )
1
1
1
1
!
1
!
1

n
n
n
n
n
n
n
d
F
x
n
F
x
dx
x
+
+
−
⋅
′
′
⋅
=
=
−
⋅
⋅
+
, 
(1.31) 

0
n > , n – .   
: 
1. (
)1
n =
( )
(
)2
1
1

1
1
1

x
x
dx
d
x
F
dx
d

+
−
=
+
⋅
=
′
⋅
. 
(1.31) . 
2. (1.31) n, 1.2.1.  
3. 1.2.1, (
)1
n +
( )
(
)
(
)
(
)

1
1

1
1
2
1
1 !

1

n
n

n
n
n
d
F
x
dx
x

+
+

+
+
−
⋅
+
′
⋅
=
+
. 

( )
( )
1

1
1
1

n
n

n
n
d
d
d
F
x
F
x
dx
dx
dx

+

+
′
′
⋅
=
⋅
⋅
. 

, 

( )
(
)

(
)

(
)
(
)

(
)

(
)
(
)

(
)

1
1
1

1
1
1
2
2
1
!
1
!
1
1
1 !.
1
1
1

n
n
n
n

n
n
n
n
n
n
n
n
d
d
F
x
dx
dx
x
x
x

+
+
+

+
+
+
+

−
⋅
−
⋅
⋅
+
−
⋅
+
′
⋅
=
⋅
=
=
+
+
+
. 

1.2.1 .    

( )
(
)
( )
1
1
1

!

n
n

n
n
d
F
x
F
x
n
dx
+
−
′
′
=
⋅
⋅

( )
(
)
(
)
( )

1
1

1
1
1

1 !

n
n

n
n
d
F
x
F
x
n
dx

−
−

−
−
′
′
=
⋅
⋅
−
. 
 (1.32)  

1.2.2:     

  
(
)
(
)
(
)
(
)
1

0
0
1
1
1
n
n
k
n
k n
k
k n
n
k
k n
i

d
x
k
i
x
dx

−
∞
∞
−
−

=
=
=
⋅
−
⋅
= −
⋅
−
⋅
−
⋅
∏
, 
(1.33) 

0
n > , n – , 
1
x < .   
:  
1. (
)1
=
n
(
)
(
)
1
1

0
1
1
1
k
k
k
k

k
k

d
x
k x
dx

∞
∞
−
−

=
=
⋅
−
⋅
= −
−
⋅
⋅
. 

(1.33) . 
2. (1.33) n, 1.2.2.  
3. 1.2.2, (
)1
n +
(
)
(
)
(
)
(
)
1
1
1
1
1
0
1
0
1
1
1
.
n
n
k
n
k n
k
k n
n
k
k n
i

d
x
k
i
x
dx

+
∞
∞
+
− −
− −
+
=
= +
=
⋅
−
⋅
= −
⋅
−
⋅
−
⋅
∏

(
)
(
)
1

1
0
0
1
1
n
n
k
k
k
k
n
n
k
k

d
d
d
x
x
dx
dx
dx

+
∞
∞

+
=
=

⋅
−
⋅
=
⋅
⋅
−
⋅
. 

, 

(
)
(
)
(
)
(
)
=
⋅
−
⋅
−
⋅
−
⋅
=
⋅
−
⋅
∏
∞

=

−

=

−
−
∞

=
+

+

n
k

n

i

n
k
n
k
n

k

k
k
n

n
x
i
k
dx
d
x
dx

d
1

0
0
1

1
1
1
1

(
)
(
)
(
)
(
)
(
)
(
)
1
1
1
1
1

1
1
0
0
1
1
1
.
n
n
k
n
k n
k n
k n

k n
k n
i
i
k
n
k
i
x
k
i
x
−
∞
∞
+
− −
− −
− −

= +
= +
=
=
=
−
⋅
−
⋅
−
⋅
= −
⋅
−
⋅
−
⋅
∏
∏

1.2.2 . 
, (1.30), (1.32) (1.33), 13

( )
(
)
(
)
(
)
2
1
1

1
0

1
1
1 !

n
k
n
k
n
n
k
n
i
F
x
k
i
x
n

−
∞
− +
− +

= −
=
′
=
⋅
−
⋅
−
⋅
−
∏
. 
(1.34) 

(1.34) , ( )
(
)
1
1

1
1
1

k n
k n
n
k n

k
F
x
x
n

∞
− +
− +

= −

′
=
−
⋅
⋅
−
. 
(1.35) 

1
+
−
=
n
k
s
, 1,
k
s
n
=
+
−
(1.35) ( )
(
)
∞

=
⋅
−
⋅
−
−
+
=
′

0
1
1
1

s

s
s
n
x
n
n
s
x
F
, (
)
−
−
+
⋅
−
=
′
1
1
1
,
n
n
k
c
k
k
n
,  
(1.36) 

 
 
( )
(
)
∞

=

+
⋅
−
⋅−
−
+
=

0
1
1
1

k

k
n
k
n
x
n
n
k
x
F
. 
(1.37) 

s
k
n
=
+
, k
s
n
=
−
. 
(1.37) ( )
(
)
1
1
1

s n
s

n

s n

s
F
x
x
n

∞
−

=

−
=
−
⋅
⋅
−
. 

(
)
≥
−

−
⋅
−

<

=
−
n
k
n

k

n 
k
c
n
k
k
n
1

1
1

 0

,

, , , , (
)
−
−
⋅
−
=
−
1
1
1
,
n
k
c
n
k
k
n
.  
(1.38)  

(1.26)  x , : 

′
⋅
=

′
=

′
=

=

n

k
k
n
k
n
q
c
q

q
q

q
q

1
,

1
1

0
0
. 
(1.39) 

(1.38), : 

(
)

0
0

1
1

1

1
1
1

n
n
k
n
k
k

q
q

q
q

n
q
q
k

−

=

′
=
′
=
−
′
=
−
⋅
⋅
−
. 
 (1.40) 

. 
1.2.1: ( )x
f
, (1.21), ( )
u x , ( ) ∞

=

∗ ⋅
=

1
n

n
n x
c
x
u
,  
(1.41) 

0
0

1
1
1
1

0
1

1
1

1

n
n

n
n
s
s
s
s

c
c

c
c

n
n
c
c
c
s
s

∗

∗

−
−
∗
−
=
=

=
=
−
−
=
⋅
=
⋅
−
.  (1.42) 

1. 
( )
f x  – , (
)
0
x
x
=
. , ( )
f x
(
)1
x =
, ( )
f x
(1.23), 15

( )

( )

(
)
( )


⋅
⋅
⋅
−
=
′

⋅
−
=
′

=
′

=

=

1

1
1

0

!
1
1

1

x
n

n
n
n

x

x
u
dx

d
n
q

x
u
dx
d
q

u
q

.  
(1.43)                      

: 

0

1
1
x
x
x
=
−
+
, 0
1
1
x
x
x
=
+
− . 

( )
f x (1.23) 

( )
(
)
0
0
1

n
n
n

q
f x
x
x

∞

=

′
=
−
+
, 

( )
n

n
n x
q
x
v
⋅
′
= ∞

=0
. 
(1.44) 

( )
−
+
=
1
1
0
x
x
f
x
v
. 
(1.45) 

, 

 
 
( )

0

1
!

n

n
n

x

d
q
v x
n
dx
=
′ =
⋅
⋅
.  
(1.46) 

(1.46) (1.43) ( )x
f
. 
( )
(
)
x
v
x
u
−
=
1
.  
(1.47) 

1
z
x
= − , 1
x
z
= − .  
. 
1.2.3: 

( )
(
)
(
)
(
)n

n
n
n

n

z
d

z
u
d

dz

z
v
d

−

−
⋅
−
=
1

1
1
,  
(1.48) 

0
n > , n – . 
: 
1. (
)1
n =
16

( )
(
)
1
dv z
d
u
z
dz
dz
=
⋅
−
. 

,  (
)
(
)
(
)
(
)
1
1
1
1

d
z
d
d
u
z
u
z
dz
d
z
dz

−
⋅
−
=
⋅
−
⋅
−
. 

, (
)
(
)
(
)
1
1
1
d
d
u
z
u
z
dz
d
z
⋅
−
= −
⋅
−
−
. 

     (1.48) . 
2. (1.48) n, 1.2.3. 
3. 1.2.3, (
)1
n +
( )
(
)
(
)
(
)
1

1
1
1

1

1

1
1
+

+
+
+

+

+

−
⋅
−
=
n

n
n
n

n

z
d

z
u
d

dz

z
v
d
. 

( )
( )

n

n

n

n

dz

z
v
d
dz
d

dz

z
v
d
⋅
=
+

+

1

1
. 

( )
(
)
(
)

(
)

1

1
1
1
1

n
n
n
n
n
d
v z
d u
z
d
dz
dz
d
z

+

+

−
=
⋅
−
⋅
−
. 
 (1.49) 

(1.49) (
)
(
)

(
)
(
)
(
)
(
)

(
)

(
)
1
1
1
1
1
1
1
1

n
n
n
n
n
n
d u
z
d u
z
d
z
d
d
dz
d
z
dz
d
z
d
z

−
−
−
⋅
−
⋅
=
⋅
−
⋅
⋅
=
−
−
−
(
)
(
)
(
)
1

1
1
1

1
1
+

+
+
+

−
⋅
−
=
n

n
n
z
d

z
u
d
. 

1.2.3 .  
(1.48), , ( )
(
)
(
)

(
)

1
1
1

n
n
n
n
n
d v x
d u
x

dx
d
x

−
= −
⋅
+
. 

, (1.46), (
)
(
)

(
)

(
)
( )

( )
0
1

1
1
1
1
1
!
!
1

n
n
n
n
n
n
n

x
z

d u
x
d u z
q
n
n
d
x
d z
=
=

−
′ =
⋅ −
⋅
=
⋅ −
⋅
+
, 

, z x, 

 
 
(
)
( )
( )
1
1
!
1

=
⋅
−
⋅
=
′

x
n

n
n
n
x
d

x
u
d
n
q
. 
 (1.50)       

( )
x
u
( )x
f
. , 

( )
x
u
(
)1
=
x
1, ( )
(
)
0
1 n
n
n
u x
a
x
∞

=
=
⋅
−
, 

, ( )

1

1
!

n

n
n

x

d u x
a
n
dx
=

=
⋅
, 

, (1.45),   
(
)
n
n
n
q
a
′
⋅
−
=
1
. 
( )
(
)
(
)
0
1
1
n
n
n
n
u x
q
x
∞

=
′
=
−
⋅
⋅
−
.  
 (1.51) 

( )
u x
(1.51), 
(
)
0
1 n
n
n
n
q
x
∞

=
′
−
⋅
⋅
R.  (1.51). , ’(
)

(
)

1
lim
lim
1

n
n
n
k
k
n k
n
n
n k
n k

q
q
R
q
q
+
→∞
→∞
+
+

′
−
⋅
′
=
=
′
′
−
⋅
.  
(1.52) 

(1.52) , , (1.51), . (1.49) R
x <
.