Прочность при интенсивных кратковременных нагрузках

Прочность при интенсивных
 кратковременных нагрузках

Книги — это  корабли мысли,
странствующие  по волнам  времени
и бережно несущие свой  драгоценный груз
от поколения к поколению.

Ф. Бэкон

Х.Р. Рахматулин
Ю.А. Демьянов

ПРОЧНОСТЬ
ПРИ ИНТЕНСИВНЫХ
КРАТКОВРЕМЕННЫХ
НАГРУЗКАХ

Москва
 Логос

2020

УДК 531:536.66
ББК  34.41:22.251
Р27

Р е ц е н з е н т ы

Е.В. Ломакин, доктор физикоматематических наук, профессор

А.Б.Киселев, доктор физикоматематических наук, профессор

Рахматулин Х.А.
Р 27
Прочность  при интенсивных кратковременных нагрузках: Изд. 2е, 
дополненное /   Х.А.Рахматулин , Ю.А. Демьянов — М.: Университетская книга; Логос, 2020. — 512  с.: ил.

ISBN 9785987044227

Изложены методы расчета  интенсивных динамических нагрузок в различных видах техники (авиационной, ракетной и др.), в  гражданском  промышленном строительстве, сейсмологии, при проведении горных разработок.  По
сравнению  с первым изданием  (М.: Наука, 1961) книга дополнена данными
новых фундаментальных исследований  в области  повторных соударений и соударений затупленных тел, продольнопоперечнокрутильных волн в канатах и
трубах, узковязкопластических волн в стержнях, балках, пластинах,  плоских
нелинейных волн с учетом  анизотропии, асимптотических методов в динамике
гибких связей.
Для  научных работников  и инженеров, разрабатывающих и  использующих
методы расчета интенсивных динамических нагрузок, в том числе  для повышения  прочности  изделий, технических устройств, конструкций и сооружений.
Может использоваться  в учебном процессе при подготовке кадров  в области
механики, физики и прикладной математики, а также  по широкому  кругу направлений (специальностей) техники и технологии.

УДК 531.3:536.66
ББК 34.41:22.251

ISBN 9785987044227
© Рахматулин Х.А.  ,
Демьянов Ю.А., 2020
© У  ниверситетская книга, 2020
© Л  огос, 2020

Оглавление

Предисловие ко второму изданию

Предисловие к первому изданию

Распространение волн в стержнях из нелинейноупругого и упругопластического материалов
(теория продольного удара)

§ 1.1. Метод характеристик для решения
квазилинейных гиперболических уравнений
второго порядка в частных производных

2

2
2
2

2

x

u
a
t

u

∂

∂
=
∂

∂
u
t
∂
∂

u
x
∂
∂

Глава 1

dt
t
x
u
dx
x

u
dux
∂
∂
∂
+
∂

∂
=

2

2

2

dt
t

u
dx
t
x
u
dut
2

2
2

∂

∂
+
∂
∂
∂
=
dx
dt 2

2

x

u

∂

∂
t
x
u
∂
∂
∂2

2

2

t

u

∂

∂
2

2

x

u

∂

∂
t
x
u
∂
∂
∂2

2

2

t

u

∂

∂
,0
0
0
1
0
2

=
−

dt
dx
dt
dx
a

.0
0

0
0
2

=

t

x
du
dx
du
dt
dx
a

;
2
2
a
dt
dx
=
.
x
t
du
dt
dx
du =

dt
dx
dt
dx
,a
dt
dx =
′,
x
t
adu
du =
′,a
dt
dx
−
=
′′.
x
t
adu
du
−
=
′′const
≡
a
′′′2
2(
)
x
a
a
u
≡
′′′′′′
′′′′′′′′′′′′′′′.
x
t
adu
du
±
=

∂

∂
+
∂
∂
∂
±
≡
∂
∂
∂
+
∂

∂
dx
x

u
dt
t
x
u
a
dx
t
x
u
dt
t

u

2

2
2
2

2

2

′′′2

2
2
2
2

2

2

x

u
a
t
x
u
a
t
x
u
a
t

u

∂

∂
+
∂
∂
∂
±
≡
∂
∂
∂
±
∂

∂

2

2
2
2

2

x

u
a
t

u

∂

∂
≡
∂

∂
′′′′′′′′′′′′′′(
)

2
2
2
0
2
2
, , , , 
x
t
u
u
a
u
u u x t
t
x
∂
∂
=
+ ϕ
∂
∂
′′′′′′′′′′′′′′′′1
1
1
1
1
1

2
2
2
2
2
2

1
1

2
2

(
, 
, ...)(
);     
(
, 
, ...)(
), 

(
, 
, ...)(
);    
(
, 
, ...)(
). 

x
t
t
t
x
t
x
x

x
t
t
t
x
t
x
x

x
x
a u
u
t
t
u
u
a u
u
u
u

x
x
a u
u
t
t
u
u
a u
u
u
u

−
=
−
−
=
−

−
=
−
−
=
−



§ 1.2. Распространение плоских нелинейных волн
нагружения в длинных стержнях

F
t
x
u
x
t
dx
x
u
dx
x
F
dx
ρ
−
−
+
+
+
=
ρ
)]
 ,
(
)
 ,
(
[
0
0

F
u
F
x ρ
+
=
ρ
)
1(
0
0
ρ)
 ,
(
)
 ,
(
2

2

0
0
t
x
T
t
dx
x
T
t

u
dx
F
−
+
=
∂

∂
ρ
σ ).
(
0
2

2

0
0
σ
∂
∂
=
∂

∂
ρ
F
x
t

u
F
.
2

2

0
x
t

u
∂
σ
∂
=
∂

∂
ρ
σσ σ()σ()σ σ()σ σ()σ σ()σσ σ()σ σ()2

2
2
2

2

x

u
a
t

u

∂

∂
=
∂

∂
de
d
a
σ
ρ
=
0

1
;
adt
dx
±
=
.
x
t
adu
du
±
=
1,2
(
)
t
x
u
u
c
= ±ψ
+
∫
=
ψ

x
u

x
x
adu
u

0
)
(

= σ() = ()= = const.
)
 ,
(
≡
t
x
u

= )],
 ,
(
[
)
 ,
(
   
)];
 ,
(
[
)
 ,
(
t
x
u
t
x
u
t
x
u
t
x
u
x
t
x
t
ψ
=
ψ
−
=
= =0
)
0
(
a
a
a
=
≡
= = = = ).
(
x
t
u
u
ψ
−
=
.
)
(
2
c
u
u
x
t
+
ψ
−
=

).
(
x
t
u
u
ψ
−
=
1
)
(
c
u
u
x
t
+
ψ
=
),
)(
(
0t
t
u
a
x
x
−
=
).
( 0
0 t
e
ux =
)
(
x
u
a
x

*
0t )]
(
[
*
0t
e
a
a =
)
( *
0t
e
*
0t ,
/
)
(
t
x
u
a
x =
),
(
x
t
u
u
ψ
−
=
σ σ()se
e ≤
≤
0
σ 0
0
/ρ
=
≡
E
a
a
σ σ()se
e ≥
)
(
s
s
e
e
E
Ee
−
′
+
=
σ
′′.
/ 0
2
1
2
ρ
′
=
=
E
a
a
*
0t s
t
e
a
u
0
−
=
se
a0
−
0
≡
=
x
t
u
u
0
/
<
de
da
0
/
>
de
da
0
0
x
x ≤
≤
0
/
<
de
da
1
1
(
) (
)
x
a t
t
t
t
t
=
+ ∆
−
− ∆ 1
1
( ) (
)
x
a t
t
t
=
−
∆1
1
1
1
( ) (
)
(
) (
)
a t
t
t
a t
t
t
t
t
−
=
+ ∆
−
− ∆
=

2
1
1
1
1
( ) (
)
( )
(
)
(
)
a t
t
t
t
a t
t t
t
O
t
′
=
−
− ∆
+
∆
−
+
∆
0
→
∆t
1
1
1
( )
( )
(
)
a t
t
a t
t t
t
′
∆ =
∆
−
0
→
∆t
2
1
1
1
1
1
1
1
( )
( )/ ( ), ( )
( )/ ( ).
t t
t
a t
a t
x t
a
t
a t
′
′
=
+
=

m
t
t
e
~
)
(
0
.
/
;
)]
(
[

)
(
)
(
)
(
1
1
1
0

1
2

0

0

1
2

1

1
1

de
da
a
mt
t
e
a

t
a

dt
de
de
da

t
a
t
x
m

t
t

=
=
=
−
1
0
<
≤ m
))
(
(
)/
(
)
(
1
0
1
2
1
t
e
a
t
a
t
x
=
))
(
(
1
0 t
e
a0
0 →
x
∞
→
1t
0
/
>
de
da
σ())
(ξ
=
f
tv
u
vt
x/
=
ξ
.
)
(
2
2
2
f
f
a
f
v
′′
′
=
′′
ξ
),
(
2
2
q
r
f
f
a
ξ
+
=
′
=
ξ
).
(
1

2
0

2
f
de
d

v
a
′
ϕ′
=
σ

ρ
=

ρ= ρρσρσρ= ρ=.)
(

0
0

0
1
2
e
e
D
ρ
σ
=

σ σ()σ()σ == 0ϕ′()ϕ′′= ϕ′′()= ϕϕ′() < ϕ′()ξ ξρ= ρρ+σ= ρ+σ*
0t 0
*
0 →
t
0
/
2
2
=
σ de
d
ϕ

λ 

ϕ 

ξ)];
(
)
(
[
)
/
(
*
1
*
*
1
1
1
ξ
′
ξ
−
ξ
=
∂
∂
=
f
f
v
t
u
v

)].
(
)
(
[
)
/
(
*
2
*
*
2
2
2
ξ
′
ξ
−
ξ
=
∂
∂
=
f
f
v
t
u
v

),
(
)
(
*
2
*
1
ξ
=
ξ
f
f
1

0
*
1

0
1
1
)
(
1
e
f
+
ρ
=
ξ
′
+

ρ
=
ρ
2

0
*
2

0
2
1
)
(
1
e
f
+
ρ
=
ξ
′
+

ρ
=
ρ
2

2
1
1

2

2

1

1
1
)
(
1
1
e
e
e
D
v
D
e
v
D
e
v
D
+
−
−
−
=
+
−
=
+
−
2

1
2

2

1

1

1
1
)
(
1
1
e
e
e
D
e
v
D
e
v
D
+
−
=
+
−
−
+
−
D
e
v
D
=
+
−

1

1
1

σ() = 2
0
1
2
2
1
1

0
2
1
0
2
1

( )
(
)
( )
[
]
[
]
p
e
e
e
D
e
e
e
e
− σ
σ
− σ
=
=
ρ
−
ρ
−

)
(
1
1
)
(
1
0
0
1
2

1
e
de
d
e
a

c
ϕ′
ρ
=
σ
ρ
=
2
2
1
1
( )
( )
a
e
D
e
≥
2
0
1
( )
D
e
′
ρ
≤ ϕ
λ
=
1e
2
2
( )
a
D
λ =
0

0

( )
( )
p
e
−σ λ
′
ϕ λ =
−λ

λ ( )e
σ
λ )
(e
ϕ
λ λ )
(e
ϕ
)
(e
ϕ
σ()§ 1.3. Волна разгрузки. Решение задач динамического
деформирования стержней, когда скорость волны
разгрузки или ее начального участка известна.
Решение для случая нелинейной диаграммы
«напряжение — деформация»

),
(e
σ
=
σ
).
(e
σ
=
σ

σσσσ1
0
2
0
0

0
2

2
2
0
2

2
−
σ
ρ
+
∂
∂
=
∂
∂
dx
de
a
dx
d

x
u
a
t
u
0

2
0
ρ
= E
a
0
a
dt
dx =
0
0
0
0
0
0
1
de
a
d
a
du
a
du
x
t
−
σ
ρ
=
−
σ
ρ
=
d
a
dv
0
0

1
0
a
dt
dx
−
=
0
0
0
0
0
0
1
de
a
d
a
du
a
du
x
t
+
σ
ρ
−
=
+
σ
ρ
−
=
d
a
dv
0
0

1
.
)
(
0
0
σ
+
−
=
σ
e
u
E
x

ψε∫
−
σ
−
−
+
+
=

x
dx
Ee
E
x
t
a
F
x
t
a
F
u

0
0
0
0
2
0
1
.
)
(
1
)
(
)
(
=).
(
(0)
(0)
)
(
)
(
0
0
0
2
0
1
t
E
Ee
t
a
F
t
a
F
ε
=
−
σ
−
′
−
′

−
σ
+
+
′
=
+
′
0
0
0
2
0
1
(0)
(0)
)
(
)
(
e
E
x
t
a
F
x
t
a
F
+
+
ε
0

0
a
x
t
a

0
0
0
( )    
(
) ;  
(
).
( )
x
t
x
u
e x
a e
u
e
f x
=
=
= −ψ
0

0
0
0
0
0
2
2
0
0
0

0
0
2
2
0
0
0
0

0
0
0
0
0
0

(
)
(0)
1
1
(0)
(
)
(
)

1
1
;
1
1
(
)
(
)
(
)

(0)
1
1
(0)
(
)

e

a
a
e
F
x
F
x
e
a e
a e
E
E

a
a
x
F
x
F
x
a e
a
a e
a e

a
e
de
x
E
a
a e
a

σ
σ
′
′
+
−
−
=
+
−
−
′
′
−ε
−
+
+
−
=
σ
=
−
−
−ε
−
∫
0
0

0
0
2
0
0
0
0

0
0
0
0
0
2
0
0
0
0
0

(0)
1
1
1
(0)
(
)
(
)

(
)
(
)
1
1
;  
1
.
2
(
)
2

e
e

a
F
x
e
x
a e
E
a e
a

e
a
e
a
a
de
F
x
de
E
a
a e
E
a

σ
′
+
=
−
− ε
−
+
σ
σ
′
+
−
−
= −
+
∫
∫

2
F′ [
]

[
]

0
1

0
2

( )
2
0
0
2
0
0
0
0

( )
2

2
0
0
0

(0)
1
(0)
2

1
,
2

e
x z

e
x z

z
a
a
e
de
E
a
a
a

a
a
de
a
a

σ
−
−ε
+
−
=
= −
+
∫

∫

0
0
1
2
0
1
0
2
1
1
.
[
( )]
[
(
)]
a
a
x
x
z
a e x
a e x

+
=
−
=
)
(e
σ
=
σ
εσσσσ′σ0
0
/ρ
=
E
a
0
1
/ρ
′
=
E
a
( )
[ ( )
]
m
m
p t
p
E
t
e
−
=
ε
−
1
0)
(
a
e
a
≡
2
2
0 1
0 1
1
1
1
1
0
0
2
2
0
0
1
0
0
1
0
0
( )
.
2
2
s
s
a a t
a a t
a
a
a
a
E
E
p t
e
e
E e
a
a
a
a
a
a
a
a

′
=
−
+
+
+ σ −
+
−
σ0

0

e
∫ 2

2
0
0

a
a
a
a
+

2
1
1
2
0
0

a
a
a
a
+

0
(
)
2
s
s
e
e
e
−
+
σ.

1∑

∞
+
n
nt
p
.

0∑

∞
=
n
nx
b

n
n
n
n
B
A
p
b
β
−
α
=
+
=
2
0

2
1

0

1
2
a
a
a
a
E
A
−
=
2
0

2
1

0

1
2
a
a
a
a
E
B
1
0

1
0
a
a
a
a
−
=
α
1
0

1
0
a
a
a
a
+
=
β
(
)
0
0
n
n
s
e
b x
e
E
σ
−
= λ
−
∑
(
)/
E
E
E
′
λ =
−
τ2
2
0
1
0
1
1
1
0
1
0
0
1
0
1

2
(
)
( )
.
[(
)
(
)
]

n
n
m
n
m
m
n
n
n
n
n

p
a
a
x
e x
e
e
Kx
a a
a
a
a
a
+
+
+

−
=
−
=
−
ρ
+
−
−
τ

(
)

1
1
0
1 0
0
1
0
1
1
2
2
0
1

(
)[(
)
(
)
]

2

n
n
n
m
s
n
n
m

a a
e
e
a
a
a
a
l
a
p
a
a

+
+
ρ
−
+
−
−
=
τ
−

)
(
s
m
s
m
e
e
E
p
−
′
+
σ
=
(
)

1
1
0
0
1
0
1
1
2
2
1
0
1

[(
)
(
)
]
1
2

n
n
n
s
n
n
m

a
a
a
a
a
l
a
p
a
a
a

+
+
σ
+
−
−
=
τ
−
−
1
(
1) 1
s
n

m
l
a
n
p

σ
=
τ
+
−
τ′x
l
=
0,
x
t
=
=
l (
1) 1
s

m
n
p

+
−
≈s0
m
p
=
0
x =
t > 0(
x
a t
=
−
0
x =
0
1
(
)
x
l
a t
t
− = −
−
1
0
0
1
t
a
a
a
=
−
0,
x
t
=
= 1
x
a t
=
1
t
t
> 2
l
l
<
2
0
2
(l
a t
=
)
( 0
1
x
t
a
F
+
′
)
( 0
2
x
t
a
F
−
′
0
1
0
2
0
[
(
)
(
)]
e
a F a t
x
F a t
x
t

∂
′
′
=
+
−
−
∂

0
0
(
)
e a t
x
′
+
0
0
(
)
e a t
x
′
−
).
 ,
(
2
)
(
)
(
)
(
)
(
)
(
2
0

1
1
2
1
2
0

1
1
0
1
0
1
1
0
1
0

0

1
1
x
t
a
nKa
a
a
a
a
x
t
a
a
a
x
t
a
a
nK
a
t
e
n

n

n
n
n
n
n
Φ
−
=
−
−
+
−
+
−
⋅
−
=
∂
∂
+
+
−
+
−
+

1
0
a
b ≤
≤

t
a
x
1
0
≤
≤
p
F
dt
t
u
d
m
0
2

2
)
 ,0
(
=
[
]

2
2
2
0
1
1
1
1
2
0
1
0
2
0
0
0
2
2
2
0
0
0
0
(
)
(
)
(
)
(
)
2

d u
a
a
a
a
a
a
F a t
F
a t
e
t
e
t
a
a
dt
a
a

′′
′′
′
′
=
+
=
β
−
β
−α
+
α
2
2
1
1
1
0
1
0
0
0

0
0
0
0
0

(
)
(
)
2
2

(
)
(
)
(
).
s

a
a
m
m
a
e
t
a
e
t
a
a

F E
E e
F Ae
t
F Be
t

β
α
′
′
−
β
+
+
α
=
′
= −
−
−
α
+
β

;

0∑

∞

=
=
n

n
nx
b
2
2
1
1
1
1
1
0
0
0
0
0
(
)
(
)
(0);
2
s
a
a
m
a
a
b
F E
E e
F A
B e
a
a
′
−
β+
+
α
= −
−
−
−
2
2
1
1
1
1
1
1
0
1
0
0
(
)
2

n
n
n
n
n
n
a
a
nm
a
a
b
F B
A
b
a
a

−
−
−
−
β +
+
α
=
β
−
α
2
≥
n
2
2
0
0
0
1
1
2
2
2
2
2
0
1
0
1
1
0
1
1
1
0
1
0
1

2
(
) ;
(
)

m
m
p F
F p
a
a
b
a
a
a
a
ma
a
a
m a
a
a
a
a
a

−
= −
= −
−
+
+
+
+
−
(
)

1
1
0
1
2
2
1
1
1
1
0
0

2
n
n
n
n
n
n

F
B
A
b
b
a
a
nm
a
a
a
a

−
−
−
β
−
α
=
−
β +
+
α
2
≥
n
[
]
0
1
0
2
0

2
2
0
0 1
0
0 1
1
1
1
1
0
1
0
0
2
2
0
0
1
0
0
1
0
0

1
1
0
1
0
1
1
0
1
0
1
2
2
0
0
0
1

(
)
(
)

(
)
2
2

(
)
(
)
(
) .
2
(
)

s

n
n
n
n
n
n
s
n
n

v
a
F a t
F a t

a
a a t
a
a a t
a
a
a
a
a
a e
e
e
a
a
a
a
a
a
a
a

a
a
a
a
a
a a b t
a
a e
a
a
a

+
+
∞

=

′
′
=
+
=

= −
−
+
−
−
+
=
+
−
−
+
+
= −
−
−
−
∑

∑
∏

∞

−

−

=
−
−
α
−
β
α
−
β
−
−
−
−
−
=
2

1

1
1
1
0
0
1
0
1
)
(

n

n

k
k
k

k
k
n
n
m
m
s
m
s
D
C
A
B
n
t
F
p
m
m
t
p
F
e
e
a
e
a
v
!

∏
=

n

k 1 n
n

n
n
n
n
n

n
n

a
a
a
a
a
a
a
a
m
a
E
m
a
E
A
D
C
A
B

)
(
)
(
)
(
)
(

1
0
1
0

1
0
1
0

0
0
2
2

1
1

−
+
+
−
−
+
⋅
−
=
−
=
α
−
β
α
−
β

−
−

−
−

1
lim
=
∞
→
n
n
A
+
−
−
=
∑

∞

=2
n
0

0

0
!
1
n
z
B
z
E
v
a
p
v
v
n
n
m

2
2
1
1
2
0
0

1
1
0

0
0

1
1
1
1
1
( )
1
1
1
1

m
m

s
s

m
m

s
s

a
e
a
e
e
v
a
e
a
S z
a
e
a
e
v
a
e
a
e


+
−
+
−
= −
+
+
−
+
−
−
+
+
−
+
+
−
=
)1
(
!3
!2
1
)
(
3

3

2

2
B
z
B
z
B
z
z
S
n
n
!

n
z
n

+
−
+
+
+
−
−
=
+
!
)1
(
!3
!2
2
)
(
1
3

3
3

2

2
2
1
1

0
0
n
Y
B
C
Y
B
C
Y
B
C
Y
C
Ea
a
p
e
x
e

n

n
n
n
m
m
)
/(
1
0
0
a
ma
Ex
F
Y =
]
)
/
1(
)
/
1
/[(
)
/
1(
1
0
1
1
0
1
2
2
0
2
1
+
+
−
+
+
−
=
n
n
n
a
a
a
a
a
a
C
0
→
n
C
∞
→
n
),
(
2
2
)
(
1

0

1

0
0
Y
T
E
a
a
p
E
a
a
p
e
x
e
m
m
m
+
−
=

+
−
+
+
−
+
−
=
+
!
)1
(
!3
!2
1
)
(
1
3

3
3

2

2
2
1
n
Y
B
C
Y
B
C
Y
B
C
Y
C
Y
T

n

n
n
n
0
1
0
1

1
0
1
0
1
2
1
2
1
(
)
0
m
m
m
s
s
s
s

e
a
a
e
a
a
e
T Y
e
a
a
e
a
a
e
−
−
+
−
+
+
−
=
0
0
0
0
0
0
0
0

1
1
0
1
0
2
(
)
0
s
s
s
s
s
s
v
a e
a
v
a e
a
v
a e
e
e
T Y
a
a
a
a
a

+
+
+
−
−
−
+
−
=
03
,0
/
2
0
2
1
=
a
a
05
,0
/
2
0
2
1
=
a
a
03
,0
/
2
0
2
1
=
a
a
05
,0
/
2
0
2
1
=
a
a

003
,0

2

0

1
=
a
a
05
,0

2

0

1
=
a
a

003
,0

2

0

1
=
a
a
05
,0

2

0

1
=
a
a

m

s

e
e
003
,0

2

0

1
=
a
a
05
,0

2

0

1
=
a
a
m

s

e
e
003
,0

2

0

1
=
a
a
05
,0

2

0

1
=
a
a

se
e ≡
se
a
v
0
−
=
)
(
1

0
0
0
s
s
a
e
a
v
σ
−
σ
ρ
−
=
+
0
0ρ
σ
−
=
a
v
w
w
w
F
a
a
F
dt
dv
m
σ
−
=
σ
=
0
1
0
0
)
exp(
z
z
v
v
s
s
w
−
=
0
0
/(
)
z
F Et
a m
=
0
/a
x
t
t
s
c
s
+
=
1
/a
x
tc =
)
/
1(
0
1 a
a
Y
z
s
s
+
=
s
σ ε
+
ρ
σ
=

0
0
a
v
w
w
)
(

0

0
ε
−
−
=
w
w
v
m
a
EF
dt
dv
)
( 0
1
0
0
s
s
e
e
a
e
a
v
−
+
=
2
0
2
)]
(
1[
r
x
e
r
π
=
−
π
~
)
(
1
/
)
(
0
x
e
r
x
r
~
−
=
)
(
1
/
)
0
(
0
s
m
e
e
r
r
−
λ
−
=
2
2 2
0 1
0
0
0
1
5 2
2
2
5 2
0
0
1

1
(
)
tg
1
2[1
(0)]
2
[1
(
)]

m

x
m
s

dr
r b
r F p
a
a

e dx
e
ma a
e
e
=

λ
−
χ =
=
= −
−
−
− λ
−
χσ σ()ε)]
(
)
(
[
0
2
0
2
0
x
t
a
F
x
t
a
F
a
t
e
+
′′
−
+
′′
=
∂
∂
dx
de

f
a

fa
a
x
t
u
t
e
0
2
0

2

1

1

−
′
+
′
=
∂
∂
∂
=
∂
∂
∼

)
(
/1
x
f
b
′
=
t
e
∂
∂ dx
de0 dx
de
t
e
0
sign
sign
=
∂
∂
dx
de0 de
da
f
b
a
de
da
f
fa
dx
de
/
1
/
1
/
1
1
0
−
=
′
−
=
e de
da
f
a

b
a

a
b
t
e
/
1

2
2
0

2
2

−

−
=
∂
∂
t
e
∂
∂ t
e
∂
∂ )
( 0e
af
a
t
e
′
−
=
∂
∂
e
=
s
s
E e
e
′
=
+
−
2
x
x
<
2x 0
(
)
s
e x
s
s
e
e
−
1x Sx

0
1
(
)
e x
2x e
−
0
2
(
)
e x
1x Sx
2x 1x 0
1
(
)
e x

0
2
(
)
e
x
e
−

0( )
S
e x
e
→
0
0
(
( ))
a e x
a
→
2x → ∞ )
(
2
0
2
t
t
a
x
x
−
=
−
)
(
1
0
1
t
t
a
x
x
−
=
−
))
(
)
(
(
))
(
)
(
(
1
)
(
2
0
0
0
2
0
0
0
0
0
2
2
x
e
x
e
a
x
x
a
u
u
a
u
u
x
x
t
t
−
−
σ
−
σ
ρ
=
−
−
−
))
(
)
(
(
))
(
)
(
(
1
)
(
1
0
0
0
1
0
0
0
0
0
1
1
x
e
x
e
a
x
x
a
u
u
a
u
u
x
x
t
t
−
+
σ
−
σ
ρ
−
=
−
+
−
)
( 2
0
2
x
e
ux =
)
( 1
0
1
x
e
ux =
))
(
(
2
0
2
x
e
ut
ψ
−
=
))
(
(
1
0
1
x
e
ut
ψ
−
=
0

1
0
2
0
2
0
1
0
0
0
2
))
(
(
))
(
(
2
)
(
)
(
)
(
)
(
)
 ,
(
a
x
e
x
e
E
x
x
E
x
x
e
t
x
ux
ψ
−
ψ
+
σ
+
σ
+
σ
−
=
))
(
)
(
(
2
2
)
(
)
(
)
 ,0
(
1
0
2
0
2
0
1
0
x
x
u
x
x
t
m
σ
−
σ
+
σ
+
σ
=
σ
1

0
a
a
um =
)
(
2
2
)
(
0
1
0
0
1
0
1
n
n
m
n
n
n
u
t
p
,
,
,
,
σ
−
σ
+
σ
+
σ
=
+
+
+
0,
1
1
0,
2
(
)
1
n
n
n
m
p t
q
u
+
+
σ
=
+ σ
+
<
+
−
=
1

1
1

m

m
u
u
q
2
0,
1
1
1

1
1
2
0,1

2
[ (
)
( )
(
)
...
1

...
(
)
( )]
.

n
n
n
n
m

k
n
n
n
k

p t
qp t
q p t
u

q p t
q
p t
q

+
+
−

−
+ −

σ
=
+
+
+
+

+
+
+
σ

1
0
1
0
+
=
+
n
n
n
t
a
a
x
a
x
n
n
n
t
a
x
a
a
x
0
1

0
−
=
−

q
t
t
u
u
t
n
n
m

m
n
=
−
+
=
+
1
1
1
1
1 /
n
nt
t
q
+ =
1
1
1
1
0,
1
0,1
1
2
...
1

n
n
n
n
n
n
m

t
t
t
p
qp
q
p
q
u
q
q
q

−
+
−
σ
=
+
+
+
+
σ
+
1 /
n
t
q
τ =
0
→
n
q
∞
→
n
0,1
0
n
q σ
→
0
0

2
(
)
1

n
n

m
q p q
u

∞
σ =
τ
+ ∑
•••§ 1.4. Применение метода характеристик для решения
прямой задачи о волне разгрузки. Определение
начальной скорости волны разгрузки. Случаи
точных решений задачи

σσσσ0
At
t
=
− 1

dx
dt
a

∗
∗
−

−
= t
tB
0
1

dx
dt
a

∗
∗
−

0
Ct
t
=
+ 1

dx
dt
a

∗
∗
+

σσσσσσσσσmax
0
(
)
B
B
v
v
j t
t
j
−
=
−
=
0
dx
dt
a

∗
∗ −

max

max
1
0
max
1
1

max
1
max
1
1
1
0
0

max
2
0
max
2
0

max
2
0
max
2

(
)
;

;

;    
;

(
)
;

(
)

A

A
A

A
A

M
A
M
A

B
B

C
C

dx
k t
t
k
dt
a

d
d
k
dx
v
v
dt
a
a
a
a
a

v
v

dx
k t
t
k
dt
a

k t
t
k
d

∗
∗

σ
σ
∗
∗

∗
∗

σ
= σ
−
−
= σ
−
−
σ
σ
σ
− σ
= −
= −
−
=
+
−
ρ
ρ
ρ
ρ
σ
= σ
=

σ
= σ
+
−
= σ
+
−
σ
= σ
+
−
= σ
+

∫
∫

0

2
1
0
0
0
1

2
1
0
0
0
1

;

1
1
(
)
;

1
1
(
)
,

B
M
B
M

C
M
C
M

dx
t
a

dx
dx
v
v
k
dt
k
dt
a
a
a
a

dx
dx
v
v
k
dt
k
dt
a
a
a
a

∗
∗

∗
∗
∗
∗

∗
∗
∗
∗

−
−
=
σ −σ
=
−
+
−
ρ
ρ
−
=
σ − σ
=
+
+
−
ρ
ρ
σσmax
0
(
)
C
C
v
v
j t
t
j
−
=
−
=
0
dx
dt
a

∗
∗ +

t
v
j
∂
∂
=
0

0

0

0

max

max
1
1
a
c
a
c

a
dx
dt

a
dx
dt
v
v
v
v

C

B
+
−
=
+

−
=
−
−

∗
∗

∗
∗

2
2
1
1
2
0

2
1
2
0
2
1
)
(

k
a
k
a

k
k
a
a

dt

dx
c
−

−
=
=
∗

∗

∗

∗
=
dt

dx
c
σσσσσσσ0
2 ≠
k
∞
=
2
k
0
1 ≠
k
∞
=
1k
σσσσ2
0
2

dt

d σ
σσ−
+
=
1
0
2
1
0
0
3
)
(
a
a
a
a
a
c
σ)
(σ
ψ
−
=
v
s
σ > σ σσσσσσσσσ.σσσσσσσ−
σ
=
σ
−
T
T
t
t
s
1
1
8
)
(
0
σσσσ′ψσσσσσρs
σ
>
σ
1
0
0
1
0
2
1
2
0
44
,1
359
,0
]
4
19
[
3
a
a
a
a
a
a
a
c
=
=
−
=
−
+
=
σσσσσσσσσσσσσσσσσσσσ2

2
2
0
2

2

x

u
a
t

u

∂

∂
=
∂

∂
σ)
(
1
2
1
0

1
0
3
t
t
a
a
a
a
x
−
−
=
1
0

1
1
2
0
3
a
a
t
a
t
a
t
−
−
=
σσσσσσσσ
2
2

max

0
0

max
4
s
s
v
v
a
v
+
+
ρ
σ
−
σ
=
)
(
2
0
0
max
4
s
m
s
v
v
a
−
ρ
+
σ
+
σ
=
σ
)
( 4
1
0

1
0
1
st
t
a
a
a
a
l
−
−
=
σ0σ)
(

1
0

1
0
1
st
t
a
a
a
a
l
−
−
=
∗
σσσρ)
(
1
2
1
0

1
0
t
t
a
a
a
a
l
−
−
=
)
(e
σ
=
σ
σσσσσ σT
a
a
a
a
xs
1
0

1
0
−
=
0
1
0

1
0
max
)
)(
(
ρ
−
σ
−
σ
=
a
a
a
a
v
s
N
0
=
σN
0
1
0

1
0
max
2
)
)(
(
ρ
−
σ
−
σ
=
a
a
a
a
v
s
P
1

1
0
max
2
)
)(
(
a
a
a
s
P
−
σ
−
σ
=
σ
s
P
σ
>
σ
s
a
a
a
a
σ
−
+
>
σ

1
0

1
0
max
2
0
1
0
1
max
0
1
0
1
s
s
a
a
a
a
a
a
a
a
+
+
σ
< σ
< σ
−
−
2
0
1
max
0
1
s
a
a
a
a
+
σ
> σ −
max
1
0

1
0
σ
+
−
=
σ
a
a
a
a
P
−
σ
−
σ
−
σ
+
−
ρ
=
1

1
0
max
max
1
0

1
0

0
0

)
)(
(
1
a
a
a
a
a
a
a
a
v
s
P
 σ σ σ σ–σσ– σ
σσ§ 1.5. Распространение упругопластических волн
в среде с переменным пределом упругости.
Задача о накоплении остаточных деформаций
[19, 20]

0
0
/ρ
=
E
a
∞σ 

σ

σσσσσσ ρσ−
∂

∂
=
∂

∂
dx
e
d

x

u
a
t

u
~

2

2
2
0
2

2

dx
de
a
dx
de

x

u
a
t

u
s
2
0
1
2

2
2
2

2
+
−
∂

∂
=
∂

∂
)]
(
[
)]
(
[
1

1

1

0

2
x
e
u
d
x
e
u
d
a

x

x
−
−
Φ
ρ
=
0
( , )
( , )
( )
u x t
u x t
u
x
=
−
0
0
( )
( )
,

x
u
x
e x dx
= ∫

2
2
2
0
2
2
x
s
u
u
a
u
e x
t
x
∂
∂
=
<
∂
∂
(
)

2
2
2
2
2
0
0
2
2
s
x
s
de
u
u
a
a
a
u
e x
dx
t
x
∂
∂
=
+
−
>
∂
∂

( , )
u x t 0
x
t
u
u
=
=
x t 0a t
x
≤
≤ ∞
0
x
t
u
u
=
≡
AOB (
)
0
u
a t
x
= ϕ
−
0
t
x
u
a u
= −
( )
x
f t
=
( )
x
s
u
e x
=
(
)
( )
0
se x
′
>
 (
)

2
2
0
s
t
x
de
du
adu
a
a
dt
dx
= ±
+
−
dx
adt
= ±
0
x
t
= =
[
]

( )

0
0
(0)
(0)

x

s

u

t
s
x
s
x
e
u
a e
a u
e
du
= −
−
−
∫
( )
x
f t
=
0
t
x
u
a u
= −
( )
x
s
u
e x
=
( )
x
f t
=
ODB ( )
x
f t
=
OC0
x
a t
=
x
u [
]
tg
( )
x
s
dx
a u
e x
dt =
ϕ =
−
x
u tu 0
x
t
= =
x
u tu ( )
x
f t
=
(
)
C
x
s
C
u
e x
=
/ tg
C
C
t
x
=
ϕ x
u tu OCBDO ( )
x
f t
=
e
σ = σ( )( )
x
f t
=
(
)
0a t
x
ϕ
−
[
]
[
]
0
( )
( )
s
a t
f t
f t
′
−ϕ
−
= ε
Ot ODB ODB Ot 1a 1
( )
f t
a t
=
( )
s
z
e
′
−ϕ
=
1

0
1

a z
a
a
−

0

0
0
(
)
1

a t x

s
z
a t
x
e
dz

−
λ
ϕ
−
= −
−λ
∫
1
0
/
a
a
λ =
0a (
)
(
)

2
2
0
1
1
1
2
1
2
1
0
( )

x

s
a
a
u
F a t
x
F
a t
x
e x dx
a
−
=
−
+
+
−
∫
1F 2
F 1
x
a t
=
0
1
1
(
)

1
2
1
2
0
0

1
(0)
(2
)
1
( )
1

a
a
t
a t

s
s
z
F
F
a t
e x dx
e
dz

−
λ
+
+
−
= −
−λ
λ
∫
∫

0
x =

0 ( )
v t (
)
(
)
1
1
1
1
2
1
0( )
a F a t
a F
a t
v t
′
′
+
=
/ 2

2
2
0
0

1
( )
1
( )
1

z

s
s
t
F z
e x dx
e
dt

ξ
λ
=
−
−
=
− λ
λ
∫
∫

/ 2
/ 2
/ 2

2
0
0
0

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

z
z
z

s
s
s
e
y dy
e
y dy
e
y dy
− λ
=
−
−
=
−
λ
λ
λ
λ
∫
∫
∫

0
1

1
2
a
a z
a
−
ξ =
1
0
1 0
0

1
1
1
( )
1
2
2

z
z

s
y
y
F z
v
dy
e
dy
a
a
=
−
−
λ λ
∫
∫
( , )
x t
σ
[
]
(
)
0
0
( )
( )
s
s
x
Ee x
E e
e x
E u
e
′
σ =
+
−
+
−
(
)

2
2
2
2
2
0
0
0
1
2
2
s
de
de
u
u
a
a
a
dx
dx
t
x
∂
∂
=
+
−
−
∂
∂
0( )
e x (
)
(
) (
) [
]
2
1
0
2
0
0
0
1
( )
( )

x

s
u
a t
x
a t
x
e x
e x dx
= Φ
−
+ Φ
+
+
−λ
−
∫
1
x
a t
=
(
)
(
)
(
)
(
)
(
)
(
)
2
1
0
1
2
0
1
0
1
1
0
1
1
s
a
a t
a
a t
e
a t
e
a t
e
a t
′
′
−Φ
−
+ Φ
+
+
− λ
−
=
(
)
(
)
(
) [
]

1
2
1
0
1
2
0
1
0
0
1
( )
( )

a t

s
a
a t
a
a t
e x
e x dx
Φ
−
+ Φ
+
+
−λ
−
=
∫

(
)
0
1

0
1

a
a
t

se
z dz

−
λ
= −
− λ
∫
(
)
0
1
e
a t (
)
(
)
(
)
(
)
(
)
(
)
1
0
1
2
0
1
1
1
1
1
s
a
a t
a
a t
e
a t
′
′
−
−λ Φ
−
+
+ λ Φ
+
=
−λ
(
)
(
)
( )
0
1
0
2
0
0

v
t
a t
a t
a
′
′
Φ
+ Φ
=
1
0
2
0
0

1
( )
( )
z
z
v
z
a
a
′
′
Φ
=
− Φ

1′
Φ 2′
Φ (
)
(
)
(
)
(
)

(
) (
) (
)
(
)

2
0
1
0
1

0
1
0

1
1

1
1
1
.
s

a
a t
a
a t

v
t
e
a t
a

′
′
− λ Φ
−
+
+ λ Φ
+
=
− λ
=
−λ
+
− λ

( )
se x 0 ( )
v t 0
( )
n
s
n
e x
b x

∞
= ∑
0
0
0
( )
n
n
v t
a
c t

∞
= ∑
2′
Φ 2
0
( )
n
n
z
d z

∞
′
Φ
= ∑
n
d (
)
(
)
(
)
(
)

1

0
1
1
1
1

1
1

n
n
n
n
n
n
n
n
b
c
a
d

+
−

+
+
− λ λ
+
− λ
=
− λ
+
+ λ

( )<0
se x
′
0
x
a t
=
0
x
a t
=
( )e
σ = σ
0
x
a t
=
( )
x
s
u
e x
=
0
( )
t
s
u
a e x
= −
0
x
a t
=
2′
Φ

σ 1
x
a t
=
1
a
a
=
(
)
(
)

2
2
0
1
1
1
2
1
2
1
0
( )

x

s
a
a
u
F
a t
x
F
a t
x
e x dx
a
−
=
−
+
+
+
−
∫
(
)
(
)
(
)
(
)
1
0
1
2
0
1
0
0
2
1
1
s
s
F
a
a t
F
a
a t
e
a t
e
a t
′
′
−
+
+
−
−
=
λ
(
)
(
)
(
)
1
1
0
1
1
2
0
1
0
0
s
a F
a
a t
a F
a
a t
a e
a t
′
′
−
−
+
+
= −
(
)
(
)
1
0
1
0
2
1
1
1
2
s
F
a
a t
e
a t
′
−
=
+
λ
λ
0
1
2
0
1

1
1
1
( )
2
s
a z
F z
e
a
a
′
=
+
λ
−
λ
(
)
(
)
2
0
1
0
2
1
1
1
2
s
F
a
a t
e
a t
′
+
=
−
λ
λ
0
2
2
0
1

1
1
1
( )
2
s
a z
F z
e
a
a
′
=
−
λ
+
λ
1
1
1
2
1
1
1
1
1
1
2
xt
s
s
a
x
a t
x
a t
u
e
e
−
+
+ λ
−λ
′
′
= −
−
−λ
− λ
+ λ
+ λ
λ
(
)
0
1
2
0
1

1
1
1
2
x
s
a
x
a t
u
e
a
a
−
=
+
+
λ
−
λ
(
)
0
1
2
2
0
1

1
1
1
1
1
( )
2
s
s
a
x
a t
e
e x
a
a

+
+
−
−
−
λ
+
λ
λ

0
xt
u
>
0
x
a t
=
BOC Ot OC(
)
(
)
1
1
2
1
2
0

1
1
( )

x

s
u
F a t
x
F
a t
x
e x dx
=
−
+
+
−
−
λ
∫
BOC1
0
2
0
1
0

1
1
1
2

x a t

s
a z
u
e
dz
a
a

−
=
+
+
λ
−
λ
∫

1
0
2
2
0
1
0

1
1
1
1
1
2

x a t

s
a z
e
dz
a
a

+

+
−
−
−
λ
+
λ
λ
∫

0
( )
.

x

se x dx
∫

1F 2
F 1
x
a t
=
(
)

1
2
0
2
1
2
0
1
0

1
1
1
2
2

a t

s
a z
F
a t
e
dz
a
a
=
−
λ
−
λ
∫
tu 0
x =
( )
0v
t (
)
(
)

0
1
1
2
1
1

( )
v t
F a t
F
a t
a
′
′
+
=
(
)

0
0
1
1
1
2
1
0
1

( )
1
1
1
2
s
v t
a a t
F a t
e
a
a
a
′
=
−
−
λ
−
λ
tOC BOC
(
)
(
) (
) [
]
2
1
0
2
0
0
0
1
( )
( )
.

x

s
u
a t
x
a t
x
e x
e x
dx
= Φ
−
+ Φ
+
+
− λ
−
∫
u 0( )
x
u
e x
=
(
)
(
)
(
) [
]

1
2
1
0
1
2
0
1
0
0
1
( )
( )

a t

s
a
a t
a
a t
e x
e x dx
Φ
−
+ Φ
+
+
− λ
−
=
∫

1
1
2
0
2
2
0
1
0
0

1
1
1
1
1
( )
2

a t
a t

s
s
a z
e
dz
e x dx
a
a
=
−
−
−
λ
+
λ
λ
∫
∫
(
)
(
)
(
)
(
)
(
)
(
)
2
1
0
1
2
0
1
0
1
1
0
1
1
s
a
a t
a
a t
e
a t
e
a t
e
a t
′
′
−Φ
−
+ Φ
+
+
− λ
−
=
0( )
e x (
)
(
)
(
)
(
)
(
)

1
1
0
1
1
0
1
2
1
1
1
1
s
a t
a
a t
a
a t
e ′
′
−
− λ Φ
−
+
+ λ Φ
+
=
− λ
+ λ
1
0
2
0
0

1
( )
( )
z
z
v
z
a
a
′
′
Φ
=
− Φ
2′
Φ 1′
Φ (
)
(
)
(
)
(
)

(
)
(
)

2
0
1
2
0
1

1
0
0

1
1

2
1
1
1
.
1
s

a
a t
a
a t

a t
e
v
t
a

′
′
− λ Φ
−
+
+ λ Φ
+
=
− λ
=
−λ
+
− λ
+ λ
( )
se x 0( )
v t 0
( )
,
n
s
n
e x
b x

∞
= ∑
( )
0
0
0
,
n
n
v
t
a
c t

∞
= ∑

2( )z
′
Φ
2
0
( )
n
n
z
d z

∞
′
Φ
= ∑
(
) (
)(
)
(
)
(
)
(
)

1
0
1
1
2
1
1
1

1
1

n
n
n
n
n
n
n
n
n
b
c
a
d

−
+
−

+
+
λ
−λ
+ λ
+
−λ
=
− λ
+
+ λ

0
0
0
;
v
c
a
=
0
1
0

m
F p
c
ma
=
0
s
b
e
=
1
0

s
de
b
dx
= 0( )
e x (
)

2
2
0
1
0
1
0
1
0
0
2
2
0
0
1
1
1

1
( )
1
( )
s
a
a
a
a
a
a
e
x
e x
v
x
x
x
a
a a
a
a
−
−
+
′
′
λ
= −
− λ
−
+ Φ
+ Φ
(
)
( )

(
)
[ ]

2
2
0
0
0
0

2
2
0
0
1
0
1
0
0
0
1
1

( )
1
0
2

1
2
...
...

s

s
m

v
e x
e
d
a

de
F p
a
a
a
x
d
x
dx
ma
a a
a

λ
= −
−λ
−
+
+

−
+
−
−λ
−
+
+
+
(
)(
) (
)
( )

0
0
0
0
0
2
1
1
0
s
v
d
b
c
e
a
=
−λ
+
=
− λ
+
(
)
(
)
2
0
2
0
0
1
2

2
1
1
1
2
1

m
s
F p
de
dx
ma
d

λ
− λ −λ
+
+ λ
=
+ λ

(
)
(
)

(
)(
)

2
2
2
2
0
1
2
2
2
2
0
0
1

2
1
1
1
1
1

s
m
de
de
ma
p F
dx
dx
ma

−
+ λ
+ λ
− λ
= −
−
+ λ
λ
+ λ
OAC ACNM λλOBD 1
1
BL M 1
1
BDN M [
]
0
1
(0)
(0)
t
s
x
s
u
a e
a u
e
= −
−
−

( )
x
x
u
u
e x
=
−
[
]
0
1
(0)
( )
(0)
t
s
x
s
u
a e
a u
e x
e
= −
−
−
−
0v

m
e [
]
0
0
1
(0)
(0)
(0) .
s
m
s
v
a e
a e
e
e
= −
−
−
−
( )
(
)
(
)
1
1
0
,0
,0
,0
0
,0
1
,0
,0
0
s
s
s
m
s
s
m
s
E
v
a e
a
e
e
e
a e
a e
e
E
′
= −
= −
+
−
= −
−λ
−

1
0
/
a
a
λ =
,0
se
,0
m
e
(
)
0
0
,0
1
,0
,0
s
m
s
v
a e
a e
e
= −
−
−
,0
m
e
( )
0
se
0v 0v (
)
1
n −
(
)
0
0
,
1
1
,
1
,
1
s n
m n
n
s n
v
a e
a
e
e
e
−
−
−
= −
−
−
−
,
1
,
1
1
m n
s n
n
e
e
e
−
−
−
=
+
(
)
0
0
,
1
,
,
1
0
s n
m n
m n
v
v
e
e
e
a
−
−
=
= −
−λ
−
′,s n
e
,
1
s n
e
− (
)
(
)
,
,
1
,
,
1
m n
m n
s n
s n
E e
e
E e
e
−
−
′
−
=
−
(
)

2
0
,
,
1
,
,
1
2
1
m n
m n
s n
s n
a
e
e
e
e
a
−
−
−
=
−
′,
,
1
0
,
1
s n
s n
s n
e
e
v
e
−
−
−
= −
+
λ
(
)
(
)
,
0
,
1
,
1
0
1
1
s n
s n
s n
e
v
e
e
v
−
−
= λ
+
− λ
= β
+
−β
1
β = −λ ,
0
,0
(1
)
n
n
s n
s
e
v
e
=
−β
+β
n → ∞ ,
0
lim
s n
n
e
v
→∞
=
,
,
1
0
,
1
(
)
s n
s n
s n
e
e
v
e
−
−
−
= λ
−
,
1
0
s n
e
v
− =
,
,
n
m n
s n
e
e
e
=
−
0
,
1
,
,
1
1

2
,
1
1
0
,
1

(
)
(
)

(
)
(
),

s n
s n
s n
n
n

s n
n
n
s n

v
e
e
e
e
e

e
e
e
v
e

−
−
−

−
−
−

=
+ λ
−
+ λ
−
=

=
+ λ
−
+ λ
−

2
2
0
,
1
1
(1
)
(1
)
s n
n
n
v
e
e
e
−
−
− λ
−
− λ
−
=
=
λ

2
2
1
1
0
0
,0
(1
)
(1
)
(1
)
n
n
s
v
v
e
−
−
−λ
−
− λ
−β
+β
=
=
λ

1
1
2
0
,0
2
1
0
,0
(1
)
(1
)
(
).

n
n
s
n
s
v
e
v
e

−
−
−
β
−β
− λ
=
− λ
=
β
−
λ
λ
2

1
0
,0
1
(
)
s
e
v
e
− λ
=
−
λ

2
1
2

1
0
,0
0
,0
2
0

(1
)(1
)
1
(
)
(
)
m
n
k
n
s
s
k
e
v
e
v
e
+

+
=

− λ
−β
− λ
=
−
β =
−
λ
λ
∑
2
1
0
,0
2
1
lim
(
)
n
s
n
e
e
v
e
→∞
−λ
=
−
= λ
λ

0 / 1
n
n
r
r
e
=
−
0

1
1
/
r
r
e
∞ =
−
λ
1 /
e
λ

1
0 1
2
e
r
r
∞
=
+
λ
0
x =
0
5 2
0
0
0

1
tg
(1
)
2(1
)
n

n
n
n
n
x
x
n
n
n
x

dr
dr
r
de
dx
e
dx
dx
e
=
=
=

χ =
=
= −
−
−
,
1
,
,
1
,
(
)
m
s n
m n
s n
s n
p
Ee
E e
e
Ee
−
−
′
=
+
−
=
2
2
,
,
1
,
(1
)
s n
s n
m n
e
e
e
−
=
− λ
+ λ
2
2
,
,
,
,
1
,
,
,
1
2
2
2
1
1
( )
(
)
s n
n
m n
s n
s n
s n
s n
s n
e
e x
e
e
e
e
e
e
−
−
−λ
− λ
=
−
=
−
−
=
−
λ
λ
λ
,
,
1
,
0

m n
s n
s n
de
de
ae
b
dx
dx

−
= −
+
2

2
2
1

(1
)
(1
)
EF
a
ma
−λ
=
+ λ
2
4

2
2
(1
2
)(1
)
(1
)(1
)
b
− λ − λ
− λ
=
λ
+ λ
+ λ

2
,
,
,
1
2
2
1
1
m n
s n
s n
de
de
de

dx
dx
dx

−
− λ
=
−
λ
λ

,
m n
de

dx

,
,
1
2
2
2
,
(1
)
s n
s n
s n
de
de
a
e
b
dx
dx

−
= − λ
+
λ +
−λ
,
,
1
2
,
1
s n
s n
s n
de
de
a
e
k
dx
dx

−
= − λ
+
2
4
4
2

1
2
2
(1
2
)(1
)
(1
)(1
)
2(1
)(1
)
(1
)(1
)
(1
)(1
)
k
− λ − λ
− λ +
− λ
+ λ
− λ
+ λ + λ
=
=
+ λ
+ λ
+ λ
+ λ

,
,
2
2
2
,
1
0
,
1
1

2
2
1
,
1
,
1
1
,
2
1
,1
(
...
),

s n
s n
s n
s n

n
s n
s n
s n
s

de
de
a
e
k
a
e
k
dx
dx

a
e
k e
k e
k
e

−
−

−
−
−

= − λ
+
−
λ
+
=
= − λ
+
+
+
+

1
1
,
2
2
1
0
1
0
,0
1
0
0

2
1
1
0
0
,0
1
1

(
)

1
(
)
.
1

n
n
s n
m
n
m
s
m
m

n
n
n

s

de
a
v
k
a
v
e
k
dx

k
k
a
v
v
e
k
k

−
−
−

=
=
= − λ
+ λ
−
β
=

−
−β
= − λ
−β
−
−
−β
∑
∑

,
lim
s n

n
de

dx
→∞
= −∞ 2
,
,
1
2
1
s n
s n
n
de
de
de
dx
dx
dx

−
− λ
=
−
λ
1
1
2
1
1
1
0 1
0
,0
1
(1
)
(
)
n
n
n
n
n
n
s
de
k
k
a
v k
v
e
dx
k

−
−
−
−
−β +β
= −
−λ
−β
−
−β
( ).
e
σ = Φ
x
l
=
(
)
x
x
a u
t =
,1
,1

,0

0
0
,0
0
( )
( )

m
m

s

e
e

s
e
v
a e de
a e
a e de
=
=
+
∫
∫
0a 0
2
x
a t
l
+
=
0
0e ( )

0
0
0
2
a x
a e
l
x
=
−

( )
s x
σ
( )
0
1
0
0( )
2
s
xa
x
e
x
a
l
x

−
σ
= σ
= σ
−
1
,1
1
,1
(
)
.
m
m
e
e
e
E
=
− σ

b
x
x
≤
,1
1
,1
(
)
s
m
e
σ
= σ
0
x =
0
x =
( )

,

,
1

0
0
,
1

m n

s n

e

s n
e
v
a e
a e de

−

−
=
+ ∫
,
,
,
,
(
)
n
m n
n
m n
m n
s n
e
e
e
e
e
E

σ
=
−
=
−
0
0
,
1
,
,
1
(
)
s n
n
m n
s n
v
a e
a e
e
−
−
=
+
−
1
0
n
a
a
a
<
<
( )
a e 1a > 0 1
n− s ,
,
1
,
,
1
(
)
n
s n
s n
m n
s n
E
e
e
e
e
E
−
−
=
+
−
1
n
E
E
E
<
<
/
d
de
σ
,
,
1
s n
n s n
n
e
e
q
−
= λ
+
0
n
n
n

a E
a E
= −
0
n
n
n

v E
q
a E
=
0
1
n
λ
<
<
1
1
/
1
n
E
E
′
λ = λ = −
<
0
1
n
λ
<
<
,
1
1
2
1
2
3
...
s n
n
n n
n
n
n
n
n
n
n
e
q
q
q
q
−
−
−
−
−
−
=
+ λ
+ λ λ
+ λ λ
λ
+

1
2
1 0
1
2
1
,0
...
...
...
n
n
n
n
n
n
s
q
e
−
−
−
−
+ λ λ
λ
λ
+ λ λ
λ
λ
,s n
e
,s n
e
n
λ
λ
<
n
q
q
<
(
)
2
1
,
,0
1
...
n
n
s n
s
e
q
e
−
+ λ + λ +
+ λ
+ λ
<
,
lim
/(1
)
s n
n
e
q
→∞
− λ
<
§ 1.6. Волновой процесс в стержне при ударе
им о преграду. Основы жесткопластического
анализа. Соударение деформируемых стержней

 0
v =
0;
e =
1
0
;
s
v
a e
=

1
;
s
e
e
= −
2
0;
v
v
=
(
)
2
0
0
1
/
s
s
e
e
a e
v
a
= −
+
−
se 0
0 s
v
a e
>
3
0
2
s
v
a e
=
3
0
e =
sx 5
4
σ = σ 5
4
v
v
=
0a 4v 5v 4
σ 5
σ

(
)
5
0
0
5
2 ;
v
v
a
e
e
=
+
−
(
)(
)
4
5
2 ;
s
Ee
Ee
E
E
e
e
′
=
−
−
+

(
)
4
3
0
4
3 ;
v
v
a
e
e
=
−
−
(
)
(
)
0
0
5
2
3
0
4
3 ,
v
a
e
e
v
a
e
e
+
−
=
−
−

2e 3e 3v (
)(
)
0
0
0
1
4
2
0
2

s
a e
v
a
a
e
a
−
+
=
(
)(
)

2
2
0
0
1
1
0
0
5
2
0
1

2

2

s
a
a a
a
a e
v
e
a a

+
−
−
=
(
)(
)
0
1
0
0
5
4
0
0

3
2
2

s
a
a
a e
v
v
v
v
a
−
−
=
=
+
4
s
s
e
e
e
−
<
<
2
5
k
e
e
e
<
<
ke 0v 0

0
1

2
1
a
a
a
+
+

4
5
0.
v
v
v
=
>
0
x =
0v 0
x =

(
)

2
2
0
1
0
0
2
0
1
s
a
a
a e
v
a a
−
−
(
)

2
2
0
1
5
0
0
0
5
0
2
0
1
2
s
s
a
a
v
a
a e
v
e
a e
a a
−
−
−
−
=
0
0
2
s
a e
v
>
0
0
0
2
s
s
a e
v
a e
<
<
0
2 /
T
l a
=
(
)
1
0
1
2
/
la
a
a
=
+
(
)
(
)
0
0
0
0
1
0
2
1
2
/
s
s
a e
v
a
a
a
a e
+
+
<
<
5v 0
5
0 5
0 6
v
v
a e
a e
=
+
−
0
4
0
5
0
7
v
v
a e
a e
=
+
−
6
7
σ = σ 6
7
v
v
=
5v 4v 5e 4e (
)

2
2
0
0
1
1
6
0
0
2
2
0
1
s
s
a
a a
a
e
a e
v
e
e
a a
+
−
=
−
+
>
6
0
v
v
=
(
)
7
0
0
0

1

s
s
s
e
a e
v
e
e
a
=
−
+
−
>
7
0
v
v
=
8
0
e =
(
)
0
1
8
0
0
0
0
0
2
s
a
a
v
a e
v
v
v
a
−
=
−
+
>
(
)
0
1
9
0
0
2
0
;
2
s
s
s
a
a
e
a e
v
e
e
a
−
=
−
+
−
>
(
)
0
1
9
0
0
0
0
0
;
2
s
a
a
v
a e
v
v
v
a
+
= −
−
+
>

(
)

2
2
0
0
1
1
10
0
0
2
2
0
1

2
3
;
2
s
s
a
a a
a
e
a e
v
e
e
a a
+
−
=
−
+
>

(
)
0
1
10
0
0
0
0
0
.
2
s
a
a
v
a e
v
v
v
a
+
= −
−
+
>

0
x =

(
)

2
2
0
1
0
0
2
0
1
s
a
a
a e
v
a a
−
−
(
)
0
1
0
0
0
0
0
2
s
a
a
a e
v
v
v
a
−
−
+
>
(
)
0
1
4 /
.
T
l
a
a
=
+

4
0
3
s
v
a e
=
(
)
6
2
0
6
2
v
v
a
e
e
=
+
−
(
)
5
4
1
5
4
v
v
a e
e
=
−
−
6
5
v
v
=
6
5
σ = σ

(
)

2
2
0
1
0
5
1
0
1
1

3

s
a
a
v
e
e
a
a
a
a
−
=
−
+

(
)

2
2
0
1
0
6
1
0
1
1
s
a
a
v
e
e
a
a
a
a
+
=
−
+
0
1
5
6
0
0
1

2
.
s
a a
v
v
e
v
a
a
=
=
+
+

2
5
s
e
e
e
−
<
<
2
6
s
e
e
e
<
<
0
0
0
0
1

2
1
.
s
a
v
a e
a
a
>
+
+
0
x =
0
1
/
4,24
a
a >
1
BS 1
SS 1
BS 1
SS 1S 0a 1S 0
1
/
6,46
a
a >

0
x =
1S 0
1
/
6,46
a
a >
1
7
2
0
1

4

s
a
e
e
e
a
a
=
+
+

7
0
v
v
=
(
)

2
2
0
1
0
8
1
0
1
1

3

s
a
a
v
e
e
a
a
a
a
+
=
−
+

8
0
v
v
=
9
0
e =
9
0
4
;
s
v
a e
=
(
)

0
1
1
10
0
0
2
0
1
0

2
;
2
s
s
a
a
a
e
a e
v
e
a
a
a
+
=
−
+
+

(
)

0
1
0
1
10
11
12
0
0
0
0
0
0
1
;
2
s
s
a
a
a
a
v
v
v
a e
v
a e
v
a
a
a
−
−
=
=
=
−
+
+
+

(
)
(
)

2
2
2
0
0
1
1
0
11
0
0
2
1
0
1
0
1

2
2
;
2

s
s
a
a a
a
a e
e
a e
v
a
a
a
a a
+
−
=
−
+
+

(
)

2
2
0
0
1
1
1
12
0
0
2
0
1
0
1

2
2
;
2
s
s
a
a a
a
a
e
a e
v
e
a
a
a a
+
−
=
−
+
+

(
)

2
2
0
0
1
1
13
0
0
2
0
1
s
s
a
a a
a
e
a e
v
e
a a
+
−
=
−
+
13
14
0;
v
v
v
=
=

(
)
(
)

2
2
2
2
0
0
1
1
0
0
1
1
14
0
0
2
1
0
1
0
1

2
;
s
s
a
a a
a
a
a a
a
e
a e
v
e
a
a
a
a a
+
−
+
−
=
−
+
+

(
)

0
1
0
1
15
0
0
2
0
1
0

3
;
2
s
s
a
a
a
a
e
a e
v
e
a
a
a
+
+
=
−
+
+

(
)

2
0
1
0
15
0
0
0
2
0
1
0

2
2
s
s
a
a
a
v
a e
v
e
v
a
a
a
−
=
−
+
+
+

(
)
16
0
0
0

1

s
s
e
a e
v
e
a
=
−
+
(
)
17
0
0
0

1
2
s
s
e
a e
v
e
a
=
−
+
17
18
20
0
0;
s
v
v
v
a e
v
=
=
=
+

(
)

2
2
0
0
1
1
0
18
0
0
2
1
0
1

2
;
s
s
a
a a
a
a
e
a e
v
e
a
a a
+
−
=
−
+

 19
0
e
=
(
)
(
)
0
0
1
1
19
0
0
0
0
0
1

3
;
s
s
a
a
a
a
v
a e
v
e
v
a
a
a
−
= −
−
+
+
+

(
)

2
2
0
0
1
1
20
0
0
2
0
1
2
s
s
a
a a
a
e
a e
v
e
a a
+
−
=
−
+
0,
x =
(
)

2
2
0
1
0
0
2
0
1
s
a
a
a e
v
a a
−
−
0
4
.
s
a e

11
s
e
e
−
>
(
)

0
0
1
2
0
1
0
1

2
4
1
a
a a
a
a
a
a
+
+
+
+

0v 16
e
e
−
>
0
0
3
v
a e
<
14
5
e
e
>
17
16
e
e
>
18
5
e
e
>
20
2
e
e
>
0
1
/
6,46
a
a >
(
)
(
)
0
0
1
0
1
2
/
s
a
a
a
a e
+
+
<

0
0
3
s
v
a e
<
<
0
1
4 (
)
T
l a
a
=
+
1
1
0
1
2
(
)
sx
a l
a
a
=
+
2
1
0
1
2
(
)
sx
a l
a
a
=
−
2
2
0
1
1
0
0
2
0
1
(
)
s
a
a
e
a e
v
a a
−
=
−
0
2
1
1
2
1
s
a
e
e
e
a
=
+
−
σ σ 0
a 1a 0
0
a F 1
0
a F ,
T v T
T
= −0
0
0
a F
ρ
0v

0a 0
0
v
a
>
0
v
v
=
0
ρ Ov 0
0
0
a F
ρ
0
0
0
a F
ρ
0
v
v
=
Ov1
U 0
v
v
=
0
0
0
a F
ρ
1
N T = −0
0
0
a F
ρ
0
ρ 0
0
0
a F
ρ
,x t ,
T v ,x t ,
T v T
kv
=
( )e
σ = Φ
(
),
x
x
a u
t =

0
,

x
u

t
x
u
a du
= −∫

(
)
(
)
1
0
2
0
( );
u
F a t
x
F
a t
x
f x
=
−
+
+
+
2
2
0

1
( )
(
)
,

x
f x
Ee dx
E
=
σ −
∫

2e 2
σ 0a 0
2
0 2.
t
x
u
a u
v
a e
−
=
+

(
)

2
2
1
1
0
2
2
( ).
2
2
v
F
l
x
f x
a
E
σ
′
−
=
+
=
,
x
l
=
2
E
σ =
2e = (
)
(
)
2
0
1
0
F
a t
l
F a t
l
′
′
+
=
−
(
)
1
1
( )
/ 2
F z
f
l
z
′
=
−
(
)
(
)
2
1
1
( )
2
2
/ 2
F z
F z
l
f
l
z
′
′
=
−
=
−
(
)
(
)
1
0
2
0
( )
u
a t
x
a t
x
f x
= Φ
−
+ Φ
+
+
(
)
(
)
3
0
4
0
u
a t
x
a t
x
= Φ
−
+ Φ
+
(
)
(
)
3
0
0
0
2
2
/(2
)
m
l
x
v
a e
a
′
Φ
−
=
+
3( )z
′
Φ
(
)
(
)
0
1
2
0
( )
t
x
o
u
a u
F a t
x
F
a t
x
f x
t
∂
+
=
−
+
+
+
+
∂

(
)
(
)
0
1
0
2
0
( )
a
F a t
x
F
a t
x
f x
x
∂
+
−
+
+
+
=
∂
(
)
(
)
2
0
0
0
2
2
( )
2
2
( ) .
F
a t
x
f
x
a
a
F
x
c
f
x
′
′
′
′
=
+
+
=
+
+
(
)
0
0
2
2
2
( )
t
x
u
a u
a
x
c
f
x
′
′
+
=
Φ
+
+
(
)
(
)
2
2
2
2
x
c
F
x
c
′
′
Φ
+
=
+
(
)
(
)
(
)
(
)
1
0
2
0
3
0
4
0
a t
x
a t
x
a t
x
a t
x
′
′
′
′
Φ
−
+ Φ
−
= Φ
−
+ Φ
−
(
)
(
)
(
)
(
)
2
2
2
0
1
0
4
0
3
0
Ee
a t
x
a t
x
a t
x
a t
x
E
σ −
′
′
′
′
Φ
−
− Φ
−
+
= Φ
−
− Φ
−
(
)
(
)
4
0
0
2
0
0
2

m
m
Ee
a t
x
a t
x
E
σ −
′
′
Φ
+
= Φ
+
+
( )
( )
( )
4
2
1
2
2
2
2

m
m
m
m
Ee
Ee
l
l
f
l
E
E
σ −
σ −
′
′
Φ
= Φ
+
=
+
0
1
0
( )
,
2
2
s
s
s
a e
Ee
f l
e
a
E
=
+
=

( )
4 2
2
2
m
m
s
e
l
e
E
σ
′
Φ
=
−
+
1v 1
σ 3v (
)
3
1
1
0
0
/
v
v
a
=
− σ
ρ
(
)
1
1
0
0
0
/
v
a
v
− σ
ρ
>
0
0
2
.
s
v
a e
≤

0 0
( )
0
( )/(
)
w
s
v
v A
A
a
−
=
−σ
ρ
2
0
0 0
( )
s
s
v A
a e
a e
=
= −ρ
0
0
0
2
w
s
s
s
v
a e
a e
a e
=
+
=
0
σ =
D
v 0
D
w
v
v
−
=
D
w
v
v
=
0
0
2
D
w
v
v
v
a
<
=
=
0
x =
0
0
2
v
a
>
0v∗ 0a 0
0
v
v∗
>

3
4
x
x
=
3
4
t
t
=
4
3
4
3
0
0
(
) /(
)
v
v
a
−
= σ − σ
ρ
4
4
0
0
/(
)
C
v
v
a
−
= −σ
ρ
0
2
C
s
v
a e
=
3

3
0

e
v
ade
= ∫
(
)
3
4
3
4
E e
e
E e
σ − σ =
−
=
∆ (
)

4
3
4
3
3
4
4
3
4
0
0
0
0
0
0

2
/
C
C
v
v
v
v
v
v
a
a
a
σ −σ
σ − σ
−
=
−
+
−
= −σ
ρ
+
=
ρ
ρ

(
)
4
3
0
0
3
2
C
a
v
v
σ = σ + ρ
−
(
)
3
0 0
3
4
3
0
2
2
2
s
a
E e
v
a e
σ
ρ
∆ = σ − σ =
+
−

3
3
2

3
2
0
0
0
0

1
1
1
2
2
2
s

e
e

s
s
e

a
a
e
e
ad e
e
d e
a
E
a
a

∆ = −
+
−
σ
= −
+
−
∫
∫
3
0
e
e
=
0
e
∆ =
0
0
0
0

e
v
v
a d e
∗
<
= ∫
3
1
e
e
=

0
A
v
v
=
0
B
σ = (
)
0
3
3
0
0
/
v
v
a
−
= −σ
ρ
1
1
0
0 0
0

(
)
e
e
v
ad e
a
σ
=
+ ρ
∫
0
e
∆ =
0
1
e
e
>
0
/
τ τ 0
0
2 /l a
τ =
e
σ −
2
2
/
0
d
de
σ
<
2
2
/
d
de
σ
e
σ −
σ = σe
e
=
s
σσE
B
x
x
=
F
H
x
x
=
B
H
x
x
x
≥
≥
H
x
x
≥
( )e
σ = σ
s
e
e
<

0
e =
s
σ
σ
<

σ(
)

2

0
2
s
x
md
mg
F
dx
= −
+ σ
(
)

2
0
0
0
2
s
v
m
mg
F
= −
+
σ
∆ (
)

2

0
0
2
s
s
m F
mg
∆ =
σ −
0
s
mv
=
0
F 0
∆ σ 0
F

σ0
(
)
u
a F
aF
+
=
(
)
0 /
/(
)
e
F
F
F
u
a
u
=
−
=
+
(
)
dx
u
a
dt = −
+
(
)
(
)
0
0
0
s
u u
a
F
F
ρ
+
⋅
=
σ − σ
0
s
du
x dt
ρ
= −σ dt 0
s
s
du
e
x
dx
u
a
u

σ
σ
ρ
=
=
+

2
0
2
(
)
se
d
u
dx
x

σ
ρ
=
( )
(
)
2
1
ln x = Φ σ − Φ σ
(
)
[
]

0
( )
s

s

d
e

e

σ
σ − σ
Φ σ =
σ
∫
1
σ (
)
2
0 0
1
1
s
v
e
ρ
= σ − σ
e
σ −
Ni
Cr
−
e
σ −
E′0
ρ 0v 1
0
v
v
=
1
1
1
0
e
e
v
=
=
= 2
0 2
v
a e
= −
2
0
0
2
v
v
a e
=
+
2
2
σ = σ 2
2
v
v
=
0
0 0
2
0
,
a
v
e
a A
ρ
= −
0
0 0
2
2
a
v
v
v
A
ρ
=
=
0
0 0
2
0
;
a
v
e
a A
ρ
= −
0
0
0
0
A
a
a
=
ρ +
ρ 2e
−
>
0
0
0 0
s
a A
v
e
a ρ
<
s
s
σ
σ
<
0
0
0 0
s
a A
v
e
a ρ
<
s
s
σ
σ
<
0

2l
T
a
=
3v 0
3
v B
v
A
=
0
0
0
0
B
a
a
=
ρ −
ρ 0
2 /
T
l a
=
0v 3
4
0
e
e
=
=
3v 4v 3
0;
v =
0
0
0
0
4
3
0
0
0
0
0
0
.
a
a
B
v
v
v
v
a
a
A
ρ −
ρ
=
=
=
ρ +
ρ

0
0
0
0
0
B
a
a
=
ρ −
ρ <
0
2 /
T
l a
=
0
B >

3
2
B
e
e
A
=
4
2
B
e
e
A
=
2

3
0
B
v
v
A
=
4
3
2
B
v
v
v
A
=
=
5
B
v
A
=
3v 2

4
3
2
B
B
v
v
v
A
A
=
=
2

4
3
2
B
B
e
e
e
A
A
=
=
2

6
4
2
B
B
v
v
v
A
A
=
=
2

6
4
2
B
B
e
e
e
A
A
=
=
1

1
2

n

n
B
v
v
A

−

+
= 1

2
2

n

n
B
v
v
A

−
= 1

1
2

n

n
B
e
e
A

−

+
= 1

2
2

n

n
B
e
e
A

−
= 0
B =
0
2 /
T
l a
=
0v (
)
1
0
0
0
0
0
1
2
0
;
s
a
v
a
a
a
e
e
Ca

ρ
+
ρ
−
= −

0
0
0
0
2
1
;
s
s
v a
Aa e
e
e
Ca
ρ −
= −
−

(
)
0
0
0
0
0
0
1
2
s
v a
a
a
a e
v
C

ρ −
ρ
−
=
1
0
0
0
C
a
a
=
ρ +
ρ 0

2l
T
a

2
s
e
e
−
<
2
s
e
e
−
>
(
)
0
0
0
0
1
0
0
1 0
0 0
.
s
s
Ca e
a
a
a e
Aa
v
a
a
−
ρ
−
ρ
ρ
>
>
s
s
σ
σ
>
3v (
)
0
0 0
0
1
3
2
s
Dv
a
a
a
e
v
C
−
ρ
−
=
0 0
1 0
D
a
a
=
ρ −
ρ 4
3
0
σ = σ =
0
2 /
T
l a
=
0
2 /
T
l a
=
0a 2
0
1 0
0
0 1
4
0

(
) ;
s
s
B a v
a e
a a e
e
ACa
−ρ
+
−
=

(
)
(
)(
)
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
4
3
2
3
.
s
a
a
a
a
v
a
a
a
a
a e
v
v
AC
AC
ρ
ρ −
ρ +
ρ
ρ
−
ρ +
ρ
=
=
−

2
4
e
e
<
2
3
e
e
<
0
1
1
0
1
s
a a
x
t
a
a
=
−

0
1
0
1

a
T
t
a
a
=
−

1
0

2
.
l
t
a
=
4v 5v 4e 5e 5e
−
> 3 2
2
2
2
2
2
2
2
0
0
0 1 0
0 0 1 0 0
0
0 0 0
0
0 0
0
1 0
0
0
2
0 0
0
0
0 0 0
0 1 0
0 1 0

(
2
6
)
(
3
)

s
a
a a
a a a
a a
a a
a a
a e
v
a
a
a a
a a
a a
ρ +
ρ −
ρ ρ +
ρ ρ +
ρ +
ρ
ρ
ρ +
ρ +
ρ −
ρ
<
0
1
3
a
a
>
0v (
)

(
)

0 1
0
1
2
0
1

2
s
a a
a
a
l
x
a a
a
′
+
⋅
=
−
0
1
/
3
a
a >
2
2
0
0
1
(2
) (2 )
s
s
e
e
e
a e
v
a
=
= −
+
−
2
2
0 2
v
v
v
=
=
3
0
e =
3
0
0
2
s
v
v
a e
=
−
(
)
0
1
4
0
0
2
0
2
4
s
a
a
e
a e
v
a
+
=
−
(
)
0
1
4
0
0
0

3
2
4
s
a
a
v
v
a e
a
−
=
−
2
2
2
4
0
0 1
1
0
0
0 1
(
)(2
) (2
)
s
e
a
a a
a
a e
v
a a
=
+
−
−
(
)
4
5
0
0
2
/ 2
s
v
v
a e
v
=
= −
+
(
)
5
0
0
0
2
/(2
)
s
e
a e
v
a
=
−
0v 0
0
0
2
4
s
s
a e
v
a e
<
<
(
)
[
]
0
0
0
0
1
0
4
1
2
/
2
s
s
a e
v
a
a
a
a e
+
+
⋅
<
<
2
S 1S (
)
[
]
0
0
0
1
0
1
2
/
2
s
v
a
a
a
a e
>
+
+
⋅
2
S ,x t ,
T v e
σ −
0
0
0
2
2
s
s
a e
v
a e
′ >
>
e
σ −
B O
′
B C
′
1
t
OO
=
e
σ −
0a σ1a 1
σ 1e 2
σ 2e OB O
′
1
OO C 1
O C

[
] [
]
3
1
1 0 1
0 0
0
1
0
0
1
2
2
(
)
(
)
v
a
v
v a
a
a
a
= − σ +
ρ
+ ρ
−
ρ
+
e
σ −
4,
1
C
σ
σ
<
1
σ 2
CO CD 0a 3
1
1
4,
0 0
2
2
C
v
v
v
a
+
σ
=
−
ρ 
(
)
0 0
3
1
1
4,
2
2
C
a
v
v
ρ
−
σ
σ
=
+
0v 0
0
v
v∗
<
0v∗ 4,
1
C
σ
= σ 1
0
1
0
1
0 0
0
1

3a
a
v
v
a
a
a

∗
σ
+
=
+
ρ
−
0a 2
CO ECEFD BDK

5
0
e
e
e
∆ =
−
=
3
0 0
0

e
ade
v
a
σ
+
=
ρ
∫
3v (
)
(
)
(
)

1
0
0
1
0
0
1
0
1
0
0
1
0
1

2
2
D

s
s

e
e

D
s
e
e

a
a
a
v
v
a a
a de
a a
a de
a
a
a
a
a
a
a
+
=
+
+
+
−
−
−
−
∫
∫
§ 1.7. Удар твердым телом или упругим стержнем
конечной массы по закрепленному упругому
стержню

2
0 2
v
a e
= −
0
0
0
/
dv
m
F Ev a
F
dt
= −
=
σ dv
v
d
= −λ
τ

0
/t t
τ =
0
0
/
t
l a
=
0
/
/
F l m
M m
λ =
ρ
=
0
2
τ
<
<
0
t
v
v e−λ
=
0
t
e−λ
σ = σ
0
0 0
a v
σ = −ρ
0v (
)
2
0
3
2
v
a
e
e
−
=
−
0 3
2
2
a e
v
= −
(
)
4
0
4
3
0 4
2
2
v
a
e
e
a e
v
= −
−
= −
−
x = a0t

(
)
0
0
4
4
2
4
0
2
F E
dv
m
F
v
v
dt
a
=
σ =
−
−
(
)
2
0
2
dv
v
v
e
d

−λ τ−
= −λ −
λ
τ
2
4
0
2
t
t
t
=
−
/
dv dt ( )
2
0
2
v
v e− λ
=
/
dv dτ x
l
=
0
2v
−
λ 2
4
< τ <

(
)
(
)
2
2
0
4
2
v
v
e
e
−λ
−τ
− λ
=
+ λ − λτ
(
)
(
)
2
2
0
4
2
2
e
e
−λ
−τ
− λ
σ = σ
+ λ +
− λτ
4
yτ ≤
2
1
1
2
2
y
e− λ
τ =
+
+
λ
λ

2
yτ =
λ = ∞ λ 4
yτ =
1
λ = λ (
)
1
2
1
1
2
4
0
e λ
+
− λ
=
1
0,5786
λ =
1
λ = λ 4
yτ =
(
)
6
0
6
5
v
a
e
e
= −
−
(
)
4
0
5
4
v
a
e
e
= −
−
(
)
(
)
6
0
6
0
4
4
0
6
2
4
0
6
2
2
2
2
2
4
v
a e
a e
v
a e
v
v
a e
v
v
= −
+
−
= −
−
−
= −
−
τ −
−
τ −
( )
( )
( )
( )
0 6
6
2
4
2
0
2
a e
D
v
D
v
v
B
= −
−
−
( )
( )
( )
0 4
4
2
2
a e
D
v
D
v
B
= −
−
( )
( )
( )
0 6
0 4
2
0
2
0
2
a e
D
a e
D
v
v
−
= −
= −
( )
v τ ( )
σ τ 4
6
τ
<
<
(
)
(
)

4
2
2
2
0

4
2
2
2
2

4
24
32
2

6
16
2
2
2
.

e
e
e

e
e

− λ
− λ
− λ

λ
−τ
− λ
σ = σ
+ λ
+
λ +
λ − τλ
−

− λτ−
λ τ + λ τ +
+

1
λ = λ (
)

1/ 2
1
1
2
1
1
1
6
/ 2
4
2
3/ 4
4,5804
y
−
−
−
τ =
+ λ
−
+ λ
−
⋅λ
=
1
λ → λ τ 2
0,2409
λ = λ =
6
yτ =
6,708
yτ =
1
λ > λ yτ y
τ
τ
>

1v 4
0
e =
4
2
2
v
v
= −
4
τ =
yτ τ 
(
)
2
2
2
0

0
2
2
2
2
v
e
e
d
l
e
a

− λ
− λ
λ
−τ
=
+
−
−
λ
λ
λ
τ 6
4
2
2
2
v
v
v
= −
−
6
0
e =
4
τ =
0
2v 0,5786
1,0091
λ
<
<
t
∆ 0
(
)
y
t
t
∗
∆ = τ − τ
∗τ 2
(
2)
2
(
1
2 (4
))
(
1)
.
y
y
y
e
e
e
e
e
∗
−λτ
−λτ
− λ
−λ τ −
− λ
∗
∗
+
− + λ
− τ
+ λτ
= λτ +

y
∗
∆τ = τ − τ λ 0
∆τ =
5,786
λ =
yτ 1
0,5786
λ = λ =
yτ λ 0
0
s
s
v
v
a e
>
= −
D
F
x
x
<
τ 
∆ 

λ 
s
e
e
=
x
l
=
s
e
e
=
1
0
(1
/
)
s
a
a
σ = σ
+
2
s
σ = σ 2
s
∗
σ = σ s
∗
σ 2
D
x
x
<
0v

0v

/
y
τ τ 0
/t t
∆
0 /
.
s
v
v

1/9
γ =
/
y
t t 0 /
4,99
s
v
v =
1,1
λ =
0 /
3,09
s
v
v =
0,9
λ =
λ 2
1
1

1
2
2
a L
t
a L
t
=
=
→
§ 1.8. Приближенный метод исследования
волнового процесса в затупленном стержне
при продольном ударе

0
x =
0
0
2 /
t
l a
=
0
x =
0

w
e

w
v
ade
= −∫
0
(0)
a
a
=
( )t
α
0
x =
( )
P = ϕ α 0
x =
0
0
(
)
w
w
P
F
F
e
= −
σ
= −
Φ
α( )e
σ = Φ
2
0
F
R
= π
α= α(
0)
t =
0
α =
0
w
w
w
P
e
v
= σ =
=
=
α032
( )
na
ϕ α =
0
w
w
v
a e
= −
(
)
w
w
e
Ee
Φ
=
(
)
1
2

4
3
R
n
k
k
= π
+

= 0α 

α0v
=
− γα0

0

na
EF
γ =
0
1
d
v
d
β =
− γ β
τ

′R
α
β =
0a t
R
τ =
0
0
0

v
v
a
=
1
0

n
EF
γ =
2

1
1
k
E
− ν
=
π

2
2
2
2

1
k
E
− ν
= π

2
,
E E 2
,ν ν (0)
0
β
=
(
)

2
2
0
1
0
0
0
0
0
0
0
0

2
2
1
1
2
2
arctg
ln
ln
arctg
3
3
3
3
3
3

z
z
z
z z
z
z
z
z
z
z
z
z

+
γ τ =
+
+
+
−
−
−

12
z = β
1
0
2
0
0
1

v
z
= β
= γ
τ → ∞0
β → β 0

d
d
a
β
α
=
τ

ε 0
z
z
z
ε =
−

2
E
E
=
2
ν = ν 0
0,01
v =
10
ετ <
3
10−
ε =
( )
0
0
w
v
t
v
<
t
∆ x
l
=
w
σ 2
D
C
v
v
=
(
)
(
)
0
,
2
/
w
v l t
v
t
l a
=
−
0
E
D
E
v
v
a e
−
= −
( )
( )
(
)
0
0
2
2 /
w
w
w
v
t
a e
t
v
t
l a
= −
+
−
α0
2
t
l a
>
/
3/ 2
0
0
0
0
0

2
2
w
w
a n
l
v
v
v
v
t
a
EF
α
α =
−
=
−
−
−
0
1
0
0

2
2 w
d
R
l
v
v
d
a
a
β
τ
=
−
−
− γ β
τ
′1
0
w
e
τ
=
0
w
v
v
−
0
β =
β′0
2 /l R
τ = τ =
1
β = β 1τ 0
β =

βt
∆ 0
τ = τ ββd
d
β
τ

(
)

0

0
0
0
0
0

2
~
2
~
w
R
l
v
d
v
a
a

τ

τ

ξ
β −β
−
−
ξ
−
τ − τ
∫
β0v ∆τ

0
0,1
v =
2
E
E
=
2
ν = ν = t
∆ l
R
+
0
2
τ
τ +
>
0
t =
= = α 

0v ( )
( )

( )
( )
( )

( )
1
2

1
1
1
2
0
0
0
;
.

w
w
e
e

w
w
v
a
de
v
v
a
de
= −
=
+
∫
∫

( )
P = ϕ α ( )t
α
1
0
x =
2
0
x =
1
0
x =
2
0
x =

( )
( )
( )
( )
1
1
2
2
0
0
,
w
w
P
F
F
= −
σ
= −
σ
( )
( )
1
1 ( )
e
σ
= Φ
( )
( )
2
2 ( ),
e
σ
= Φ

α( )
( )
1
2
w
w
v
v
=
−ααα(1)
(2)
3/ 2
0
0
(1)
(2)
1
0
2
0

a
a
n
E F
E F
=
+
α
§ 1.9. Динамическая диаграмма «напряжение –
деформация». Методы ее экспериментального
определения

( )e
σ = Φ
( )
e
Φ
( )
0
e
′′
Φ
<
e
σ −
( )e
σ = Φ
 max
0
0
2,15ma v
σ
=
max
0 0
2,0ma v
σ
=
2
2800 0
s
du
x dt
ρ
= −σ z
x
ct
=
+
1
1
cT
l
x
=
−
dx
dt = − (
)

2
2
0
0

2
ln
s
dx
L
v
c
dt
x
σ
+
−
=
ρ
( )t T
dx
dt
= (
)
(
)

2
0
0
0

1

2

2ln
/
s
v
v c

L x

ρ
+
σ =
0
1
1
0
(
)
1
2
2
v l
x
L
l
v T
c
−
−
=
=
0
1
1

1
2
v l
x
c
L
l
−
=
−

2
0 0
1

1
1
2ln( /
)
s
v
L
x
L x
L
l
ρ
−
σ =
−

××××××(
)

2
0
0
1

1
1
2ln
/
s
v
L
x
L x
L
l
d
ρ
−
σ =
−
+

(
)
0
1
/ 2
v
L
x
−
4
1
1,35 10 −
⋅
e
σ −
 /
d
de
σ
σ e c

0
1 d
de
σ
ρ
c( )e
σ = σ
e
σ −
0
0

m
e
v
ade
= ∫
( )e
σ = Φ
0

e
v
ade
= ∫
0

e
ade
∫
/
0
d
de
σ
= 1
2
2
S
H
e
σ −
e
σ −
Ee
σ =
(
)
s
s
Ee
E
e
e
′
σ =
+
−
′se 0
0

m
e
v
a de
= ∫

(
)
0
0
1
s
m
s
v
a e
a e
e
=
+
−
/
E
E
′
0
0
1
s
v
a e
a
=
+
∼

e
σ −
e
σ −
e
σ −
e
σ −
( )e
σ = Φ
e
σ −
σσ. σ. σσσ. σ. ( )e
σ = Φ
ϕ σσ..e
σ −
e
σ −
0
/
a
E
=
ρ 0
2 /
T
l a
=
+ ε e
σ −
e
σ −
( )e
σ = Φ
e
σ −
e
σ −
e
σ −
e
σ −
0
s
v
a e
=
0

m
e
a de
= ∫
(
) /
m
m
e
e
E
=
− σ
0

0
1
1
1
dv
d
d
de
E de
de
σ
σ
−
=
ρ
/
e
E
− σ
( )e
σ = Φ
( )e
σ = Φ
Ee
σ =
0
0

( ,
)
1
.

s
s
e e

d
e E
v
e
de

∗

=

σ
=
ρ 0v∗ ( )e
σ
e
σ −
( )e
σ = Φ
(
)
0
2
1
1
2
a v
v
ρ
−
= σ − σ (
)
2
1
2
1
E e
e
σ −σ =
−
(
)
(
)
2
2
1
0
1
2
a v
v
a
e
e
−
=
−
1
1
2
2
e dx
v dt
e dx
v dt
+
=
+
(
)
1
2
2
1
a e
e
v
v
−
=
−
0
a
a
=
( )e
σ = Φ
( )e
σ = Φ
(
)/
m
m
e
e
E
=
− Φ
(
)
m
m
e
σ
= Φ
0
2
x
a t
l
= −
+
/
(
)
m
x t
a e
=
0 /
2
x
a x a
l
= −
+
(
)
0
/
/ 2
a a
x
l
x
=
−
(
)

2
2

2
0
1
1
2
m
m
de
de
de
a
x
dx
dx
l
x
dx
a

=
−
=
−
−
(
)

2
( )
1
2

l

m
s
x

de
dx
e
x
e
dx
x
l
x

=
− −
−
∫
(
)

2
2
( )
(2
) ( )
( )
.
4 (
)
1
2

l

m
s
x

e x
x
l
x e x
e
x
e
dx
l x
l
x
l
x

−
=
+
+ −
−
−
∫
0
x l
de
dx
=
=
σσ2

2
(2
) ( )
( )
4 (
)
4 (
)

l

m
s
x

x
l
x e x
Ex e
x
E
dx
l l
x
l l
x

−
σ
= σ +
+
−
−
∫
(
)
m
m
e
σ
= Φ
τ 0

0

a x
a
x
a
=
+
τ

2

0
1

m
de
de
dx
dx
x
x
a

=
−+
τ
(
)
(
)

0
2
2
2
0
0

0
0

(
)
2
( )
( )
,
2
1
1

sx
s
m
s
s
x

s

e x
x x
a
e x
e x
e
e
dx
x
a
x
a
x
x
a
x
a

+
τ
=
+
−
+
τ
+
τ
−
−
+
τ
+
τ
∫

σ( )e
σ = Φ
e
σ −
σ2
2
0
1
/

m
de
de dx
dx
a
a
=
−
0
(
)
2
m
xa
a e
L
x
=
−
0
x
L
<
<
0
(
)
2
m
xa
a e
l
x
=
+
0
l
x
− <
<
(
)
(
)

2
2
1
1
2
2

D

D

x
x

m
s
L
x

de
de
dx
dx
e
e
dx
x
x
L
x
L
x

=
+
+
−
−
−
+
∫
∫
( )e
σ = Φ
2
/
D D
= ∆
e
σ −
( )e
σ = Φ
 e
σ −
( )e
σ = Φ
1
D
∆
2
D
∆
1
2
2

D
D
∆
+ ∆
D
∆ 1
D
∆
2
D
∆
1
2
2

D
D
∆
+ ∆
D
∆ 
 e
σ − Литература

( )
e
e t
=
 Теория поперечного удара по гибким
деформируемым связям и балкам

Глава 2

§ 2.1. Система уравнений, описывающих процесс
распространения волн при поперечном ударе.
Характеристики системы. Соотношения на волне
излома нити

ρττττττCD (
)
(
)
0
0
0
0
0
,
,
CD
ds
s
ds t
s t
ds
=
⋅ +
+
−
=
l
l
τ
(
)
0

0

,
s t

s

∂
+
∂

l
τ

2
2

0
0
0
1
.

n
CD
ds
s
s
∂
∂
=
+
+
∂
∂
l
l

τ

2
2

0
0
0
0
1
;
n
F ds
ds
F
s
s
∂
∂
ρ
=
⋅ρ
+
+
∂
∂
l
l

τ

2
2

0
0
0
0
1
.
n
F
F
s
s
∂
∂

ρ
= ρ
+
+
∂
∂
l
l

τ

ρ(
)
(
)
0
0

2

0
0
0
0
0
0
2
T
T
P
s
s
s
F ds
F
F
F ds
t
+∆
∂
ρ
=
−
+ρ
∂
l
(
)
2

0
0
0
0
2
0

F
F
F
s
t

∂
∂
ρ
=
+ ρ
∂
∂

T
l
P ρ2

0
2
0s
t
∂
∂
ρ
= ∂
∂
l
T 2
2
0

0
0
0
1
1

n

CD
ds
e
ds
s
s

−
∂
∂
=
=
+
+
−
∂
∂
l
l

τ

[
]

2

0
2
0
cos(
, )
x
T
x
s
t
∂
∂
ρ
= ∂
∂
p
[
]

2

0
2
0
cos(
, )
y
T
y
s
t
∂
∂
ρ
= ∂
∂
p
[
]

2

0
2
0
cos(
, )
z
T
z
s
t
∂
∂
ρ
= ∂
∂
p
2

0
2
0
( cos )
x
T
s
t
∂
∂
ρ
=
ϕ
∂
∂

2

0
2
0
( sin )
y
T
s
t
∂
∂
ρ
=
ϕ
∂
∂

ϕ0
1
(
1)cos
x
e
s
∂
+
=
+
ϕ
∂

0
(
1)sin
y
e
s
∂
=
+
ϕ
∂
2
2

0
0

cos
cos
(1
)
u
e
a
e
t
s
s
∂
∂
∂
ϕ
=
ϕ
+ λ
+
∂
∂
∂

x
u
t
∂
= ∂

2
2

0
0

sin
sin
(1
)
v
e
a
e
t
s
s
∂
∂
∂
ϕ
=
ϕ
+ λ
+
∂
∂
∂

y
v
t
∂
= ∂

2

0

1 dT
a
de
= ρ

(
)

2

0 1
T
e
λ = ρ
+

0
0
0

cos
cos
(1
)
e
e
s
s
s
∂µ
∂
∂
ϕ
=
ϕ+
+
∂
∂
∂

0

x
s
∂
µ = ∂

0
0
0

sin
sin
(1
)
e
e
s
s
s
∂χ
∂
∂
ϕ
=
ϕ +
+
∂
∂
∂
0

y
s
∂
χ = ∂
2

0
0
cos
sin
cos
sin
,
u
v
a
t
t
s
s

∂
∂
∂µ
∂χ
ϕ
+
ϕ
=
ϕ
+
ϕ
∂
∂
∂
∂
2

0
0
cos
sin
cos
sin
.
v
u
t
t
s
s

∂
∂
∂χ
∂µ
ϕ
−
ϕ
= λ
ϕ
−
ϕ
∂
∂
∂
∂
′0

u
s
t
∂
∂µ
=
∂
∂

0

v
s
t
∂
∂χ
=
∂
∂
′′µχ0
0
0
0
cos
sin
cos
sin
u
u
v
v
a
a
a
a
a
t
s
t
s
t
s
t
s

∂
∂
∂
∂
∂µ
∂µ
∂χ
∂χ
ϕ
±
+
ϕ
±
= ±
ϕ
±
+
ϕ
±
∂
∂
∂
∂
∂
∂
∂
∂
0
0
0
0
cos
sin
cos
sin
v
v
u
u
t
s
t
s
t
s
t
s

∂
∂
∂
∂
∂χ
∂χ
∂µ
∂µ
ϕ
± λ
−
ϕ
± λ
= ±λ
ϕ
± λ
−
ϕ
± λ
∂
∂
∂
∂
∂
∂
∂
∂
ααββ1
1
2
2

0
0
0;
0
a
a
t
s
t
s
∂α
∂α
∂α
∂α
−
=
+
=
∂
∂
∂
∂
1
1
2
2

0
0
0;
0
t
s
t
s
∂β
∂β
∂β
∂β
− λ
=
+ λ
=
∂
∂
∂
∂
1
1
1
1

2
2
2
2

1
1
1
1

2
2
2
2

cos
sin
cos
sin
0;

cos
sin
cos
sin
0;

cos
sin
cos
sin
0;

cos
sin
cos
sin
0;

u
v
a
a

u
v
a
a

v
u

v
u

∂
∂
∂µ
∂χ
ϕ
+
ϕ
+
ϕ
+
ϕ
=
∂α
∂α
∂α
∂α
∂
∂
∂µ
∂χ
ϕ
+
ϕ
−
ϕ
−
ϕ
=
∂α
∂α
∂α
∂α
∂
∂
∂χ
∂µ
ϕ
−
ϕ
+ λ
ϕ
− λ
ϕ
=
∂β
∂β
∂β
∂β
∂
∂
∂χ
∂µ
ϕ
−
ϕ
− λ
ϕ
+ λ
ϕ
=
∂β
∂β
∂β
∂β
0
( , )
0
w s t =
0
( , )
0
w s t =
0
(
, )
s t
Φ
0
0
 : 
 : 
w
w
s
t
s
t
Φ
∂Φ
∂Φ
∂
∂
=
= α
∂
∂
∂
∂
0
( , )
s t
Φ
α
′′′µχ0
( , )
0
w s t =
2

0
0
cos
sin
cos
sin
;
u
v
a
t
t
s
s

∂
∂
∂µ
∂χ
ϕ
+
ϕ
=
ϕ
+
ϕ
∂
∂
∂
∂
2

0
0
cos
sin
cos
sin
;
v
u
t
t
s
s

∂
∂
∂χ
∂µ
ϕ
−
ϕ
= λ
ϕ
−
ϕ
∂
∂
∂
∂
0

u
s
t
∂
∂µ
=
∂
∂
0

v
s
t
∂
∂χ
=
∂
∂
(
)
(
)
2

0
cos
sin
cos
sin
u
v
w
w
a
t
s
µ
χ
∂
∂
ϕ⋅α +
ϕ⋅α
=
ϕ⋅α +
ϕ⋅α
∂
∂
(
)
(
)
2

0
cos
sin
cos
sin
v
u
w
w
t
s
χ
µ
∂
∂
ϕ⋅α −
ϕ⋅α
= λ
ϕ⋅α −
ϕ⋅α
∂
∂
0
u
w
w
s
t
µ
∂
∂
α
= α
∂
∂
0
v
w
w
s
t
χ
∂
∂
α
= α
∂
∂ αααµαχ
2
2

0
0

2
2

0
0

0

0

cos
sin
cos
sin

sin
cos
sin
cos

0
0
0

0
0

w
w
w
w
a
a
t
t
s
s

w
w
w
w
t
t
s
s

w
w
s
t

w
w
s
t

∂
∂
∂
∂
ϕ
ϕ
−
ϕ
−
ϕ
∂
∂
∂
∂

∂
∂
∂
∂
−
ϕ
ϕ
λ
ϕ
λ
ϕ
∂
∂
∂
∂
=
∂
∂
−
∂
∂

∂
∂
−
∂
∂

−

w
w
t
s
∂
∂
= ±λ
∂
∂
0

w
w
a
t
s
∂
∂
= ±
∂
∂
u
µ
α = ±λα v
χ
α = ±λα
2
2
(
)(cos
sin
)
0.
a
µ
χ
− λ
ϕ⋅α +
ϕ⋅α
=

a ≠ λ cos
sin
0
µ
χ
ϕ⋅α +
ϕ⋅α
=
0
0
cos
sin
0
s
s
∂µ
∂χ
ϕ
+
ϕ
=
∂
∂
2
2

2
2
0
0
cos
sin
0
x
y
s
s

∂
∂
ϕ
+
ϕ
=
∂
∂
λ0

e
s
∂
∂

2
2
a
λ ≠
cos
sin
0
µ
χ
α
ϕ − α
ϕ =
2
2

2
2
0
0
cos
sin
0
x
y
s
s

∂
∂
ϕ
−
ϕ
=
∂
∂
0

e
s
∂
∂
2
2
0
a −λ =
(
)
1
0
dT
e
T
de
+
−
=

( )
T
T e
=

(
)
1
dT
e
T
de
+
>
a
λ <
(
)
1
dT
e
T
de
+
<
a < λ a
λ =
λ
(
)
/ 1
s
s
E
T
e
′ >
+

(
)
/ 1
s
s
E
T
e
′ <
+
(
)
/ 1
s
s
E
T
e
′ =
+
γ2
2
2
b
F dt
t
∂
ρ
−∂
l

τ

(
)
1
2
0
dt F
−
+
⋅
T
T
1
2
t
t
∂
∂
−
∂
∂
l
l
(
)
2
2
2
1
0
2
1
2
.
F
b
F
t
t
t
∂
∂
∂
ρ
−
−
=
−
∂
∂
∂
l
l
l
T
T

τ

2
2
2
F
b
dt
t
τ

∂
ρ
−
∂
l
1
1
b
F dt
t
∂
−∂
l
τ
2
2
1
1
2
1
F
b
b
F
t
t
τ

∂
∂
ρ
−
= ρ
−
∂
∂
l
l
τ
0
0
2
2
2
1
1
1
(1
)
(1
)
F
F
e
F
e
ρ
= ρ
+
= ρ
+
(
)
0
2
1
2
2
1
2
(1
)
b
e
t
t
t
τ

∂
∂
∂
ρ
−
−
=
−
+
∂
∂
∂
l
l
l
T
T
1
2

2
1
1
1

b
b
t
t
e
e

∂
∂
−
−
∂
∂
=
+
+

l
l

τ

τ

/(1
)
T
e
+
τττττ(
)(
) (
)(
)
0
1
2
2
cos
cos
1
b
u
v
u
T
T
e
ρ
+
β −
=
γ −
+
(
)
(
)
0
1
2
sin
sin
1
b
u v
T
e
ρ
+
β =
γ
+
1
cos
sin
v
v
t
∂
= −
β −
β
∂
l
n
τ
2
u
t
∂
= −
∂
l
τ 2
2
2

2
1

(
cos )
sin
1
1
b
v
v
b
u
e
e
+
β
+
β
+
=
+
+

sin
sin(
)
b
v
γ =
β − γ γ
γγ[
]
0
2
2
(
)
sin(
)
sin
sin (1
)
b
u
v
u
T
e
ρ
+
γ −β −
γ = −
γ
+
2
0
2
2
2
(
)
(1
)
(
)
b
u
e T e
ρ
+
=
+
2
2
2
2
1
2
2

(
) (1
)
2
cos
(1
)
b
u
e
v
vb
b
e
+
+
=
+
β +
+
1
2

0

sin (1
)
sin
(
)
T
e
v
b
u
γ
+
β =
ρ
+
1
2
2
2
1
2

0
0
0

cos (1
)
(1
)
cos (1
)
cos
(
)
(
)
(
)
T
e
T
e
T
e
v
u
b
b
u
b
u
b
u
γ
+
+
γ
+
β =
−
+
=
−
ρ
+
ρ
+
ρ
+
2
2
2
2
1
2
1
2
2
2
0
0

(1
)
2 cos (1
)
(
)
(
)
T
e
T
e
v
b
b
u
b
u
+
γ
+
=
−
+
ρ
+
ρ
+
β

(
)

2
2
2
2
1
1
2
1
2
2
2
2
0
2
0

(1
)
(1
)
2
cos (1
)
(
)
(1
)
(
)

b
u
e
T
e
bT
e
b
u
e
b
u

+
+
+
γ
+
=
−
+
ρ
+
+
ρ
+

2
2
2
1
2

0

2
cos (1
)
2
(
)
T
e
b
b
b
b
b
u
γ
+
+
+
−
+
ρ
+
2
2
2
1
2
2
2
1
(1
)
(1
)
T
T
e
e
=
+
+
1
2
e
e
e
=
=
1
2
.
T
T
T
=
=

1
2
e
e
=
1
2
T
T
=
(
)(
)
(
)(
)
0
cos
1
cos
1
b
u
v
u
T
e
ρ
+
β −
=
+
γ −
(
)
(
)
0
sin
sin
1
b
u v
T
e
ρ
+
β =
γ
+
0 ( )
s t
∗
0
0
0
0
0
( )
( ),
ds
ds
d
x
b
s t
x s t t
u
dt
dt
s
dt

∗
∗
∗
∗
∂
=
+
=
+
−
∂

0 (1
)
ds
b
u
e
dt

∗
+
=
+
0

0(1
)
ds
T
dt
e

∗
= ± ρ
+

§ 2.2. Точечный удар по гибкой деформируемой нити
бесконечной длины

βρ2

0
2
0
0
1
1
x
T
x
s
e
s
t

∂
∂
∂
ρ
=
+
∂
+
∂
∂
2

0
2
0
0
1
y
T
y
s
e s
t

∂
∂
∂
ρ
=
∂
+
∂
∂
(
)
0
/
x
x
v t
=
(
)
0
/
y
y
v t
=
x y (
)
0
0
/
z
s
v t
=
( )
x
x z
=
( )
y
y z
=
(
)
(
)

2
2
0 0

1

1

T
x
d
z x
dz
v
e

′
+
′′ =
ρ
+
(
)

2
2
0 0 1
d
Ty
z y
dz
v
e

′
′′ =
ρ
+
 (
)
(
)
(
)

2
2
2
0 0
0 0
1
1
1
T
d
T
z
x
x
dz
v
e
v
e

′′
′
−
=
+
ρ
+
ρ
+
(
)
(
)

2
2
2
0 0
0 0
1
1
T
d
T
z
y
y dz
v
e
v
e

′′
′
−
=
ρ
+
ρ
+
(
)

2
2
0 0
0
1
T
z
v
e
−
≠
ρ
+

(
)

2
0
0
const
1
T
v
e ≠
ρ
+

1
x
y
x
y
′′
′′
=
′
′
+
1
1
.
x
c y
′
′
+
=
(
)
(
)

2
2
1
1
1
2
2
0 0
0 0
1
1
1
T
d
T
c
z
y
c y y
c de
v
e
v
e

′′
′ ′′
−
=
+
ρ
+
ρ
+
2
1
1
1
.
e
y
c
′
+
=
+

(
)
(
)
(
)

2
2
2
0
0
0
0
1
1
1
T
d
T
z
e de
v
e
v
e

−
=
+
ρ
+
ρ
+
1
0
c =
0
y =
′′
1
0
c =
1
1
x
c y
′
′
+
=
(
)
(
)
(
)

2
2
2
0
0
0
0
1
1
1
T
d
T
z
e de
v
e
v
e

−
=
+
ρ
+
ρ
+
(
)

2
2
0
0
0
1
T
z
v
e
−
=
ρ
+

(
)
(
)

2
0 0
1
0
1
de d
T
x
dz de
v
e

′
+
=
ρ
+
(
)

2
0
0
0
1
de d
T
y dz de
v
e

′
=
ρ
+
0
de
dz =
(
)

2
0
0
0
1
d
T
de
v
e

=
ρ
+
1
0
x′
+
=
0
y′ =
const
c =
const
z =
const
e =
1
const
x′
+
=
const
y′ =
0
1
2
x
s
c y
c t
+
=
+
2
2
0

0

1
( );
s
dT
a
e
t
de
=
=
ρ
0
3
0
4
x
s
c s
c t
+
=
+
5 0
6 .
y
c s
c t
=
+
1
0
0
1
y
x
c
s
s
∂
∂
=
+
∂
∂
2
1

1
1
1
1
dx
e
dz
c
=
+
+
−
(
)
z
x
z
edz
α
+
−
= ∫
2
1
1 1 / c
α =
+
0
x
dx
v
x
z
t
dz
∂
=
−
∂
x dx
dz 0
0

1
1
1

z
x
e
v
z
edz
z
z
t

∂
+
=
+
−
−
−
∂
α
α
∫
0

1
( )
const

e
x
a e de
t
∂ = −
+
∂
α ∫
1
0

1
( )
const

e
y
a e de
t
c
∂ = −
+
∂
α ∫
0
( )
(
)
x
e
e
t
∂
= ψ
− ψ
∂

AB A B
′ ′ ′OA OA′β γ 1
0
( )
(
)
u
e
e
= ψ
− ψ
0
1
1
1
1
1
1
1
1
(
)( cos
)
(cos
1)(1
);
b
u
v
u
T
e
ρ
+
β −
=
γ −
+
0
1
1
1
1
1
1
1
(
)
sin
sin
(1
);
b
u v
T
e
ρ
+
β =
γ
+
1
1
1
1
1
sin
sin(
).
b
v
γ =
β − γ
β0
0
1
1
0
0

sin
tg
cos
v
b
v
β
γ =
+
β 2
2
0
(
)
(
)
u
e
e
= ψ
− ψ
′0
2
2
2
2
2
2
2
(
)(
cos
)
(cos
1)(1
)
b
u
v
u
T
e
ρ
+
−
β −
=
γ −
+
′0
2
2
2
2
2
2
2
(
) sin
sin
(1
)
b
u v
T
e
ρ
+
β =
γ
+
′2
2
2
2
2
sin
sin (
)
b
v
γ =
β − γ
′0
0
2
2
0
0

sin
tg
cos
v
b
v
β
γ =
+
β

′2
1
1
1
MN
LK
e
e
=
+
+
2
2
1
2
0
0
2
2
0

2
2
2
1
0
0 1
1
0

(1
)
2
cos(
)

(1
)
2
cos(
)

e
v
v
v v

e
v
v
v v

+
+
−
β −β
=

=
+
+
−
β −β

1
2
(
)
2
2
1
0
2
0
(
)
f
s
s
T
u
e
T
u
γ +γ
−ρ
=
−ρ
0
0
1
1
cos
cos
su
v
v
=
β −
β β0
f =
1
2
T
T
T
=
=
1
2
e
e
e
=
= 1
2
u
u
u
=
=
1
2
b
b
b
=
= (
)
(
)
2
2
2
1
0
0
2
2
1
1
0
2
2
1
1
2
cos
cos
cos
sin
sin
sin
v
v
v
v
v
v
v
−
=
β
β −
β
+
β
β −
β
′′1
1
1
0

(1
)(cos
1)
cos
(
)
T
e
v
u
b
u
+
γ −
β =
+
ρ
+
1
1
1
0

(1
)sin
sin
(
)
T
e
v
b
u
+
γ
β =
ρ
+
2
2
2
0

(1
)(cos
1)
cos
(
)
T
e
v
u
b
u
+
γ −
β =
−
ρ
+
2
2
2
0

(1
)sin
sin
(
)
T
e
v
b
u
+
γ
β =
ρ
+
2
2
2
2
2
2
2
2
2
1
2
2
2
1
1
1
(cos
sin
)
(cos
sin
)
v
v
v
v
−
=
β +
β
−
β +
β
=

2
2
1
1
2
2
1
1

2
2
1
1
2
2
1
1

(
cos
cos
)( cos
cos
)

( sin
sin
)( sin
sin
),

v
v
v
v

v
v
v
v

=
β −
β
β +
β
+

+
β −
β
β +
β

2
1
2
1

0
0

(1
)(cos
cos )
(1
)(cos
cos
2)
2
(
)
(
)
T
e
T
e
u
b
u
b
u
+
γ −
γ
+
γ +
γ −
+
+
ρ
+
ρ
+
2
1
2
1

0
0

(1
)(sin
sin )
(1
)(sin
sin )
(
)
(
)
T
e
T
e
b
u
b
u
+
γ −
γ
+
γ +
γ
+
⋅
=
ρ
+
ρ
+

2
1
0
0
0

2
1
0
0

(1
)(cos
cos
2)
2
cos
2
(
)

(1
)(sin
sin )
sin
(
)

T
e
v
u
b
u

T
e
b
u

+
γ +
γ −
=
−
β
+
+
ρ
+
+
γ −
γ
β
+
ρ
+
[
]
2
1
1
2
(cos
cos ) 2
(
)(cos
cos
2)
u
b
u
γ −
γ
+
+
γ +
γ −
+

2
1
2
1
(sin
sin
)(sin
sin
)(
)
b
u
+
γ −
γ
γ +
γ
+
=

0
0
0
2
1
0
2
1
2 cos
2
cos
(cos
cos
2)
sin
(sin
sin
)
u
v
b
u
β
=
−
β
γ +
γ −
−
+
β
γ −
γ
+
1
2
0
0
2
1
0
0
1
2
(cos
cos
)
sin
(sin
sin )
cos
(
cos
cos
),
b
v
v
γ −
γ
=
β
γ −
γ
+
β λ −
γ −
γ
2
(
)
/
b
b
u
λ =
+
′2
1
1
2
0
0
0
0
1
2
1
2

1
2
0
1
0
2
2
1

(tg
tg )
2 tg tg
cos
;
sin
;
tg
tg
tg
tg

2tg tg
tg
;
ctg
2ctg
ctg
.
tg
tg

b
b
v
v
γ −
γ
γ
γ
β =
β =
γ +
γ
γ +
γ
γ
γ
β =
γ =
β +
γ
γ −
γ
(
)(
)
1
2
1
2
cos
cos
tg
tg
γ −
γ
γ +
γ
=

(
)
(
)(
)
2
1
1
2
1
2
2
1
2 sin
sin
tg tg
cos
cos
tg
tg
=
γ −
γ
γ
γ + λ −
γ −
γ
γ −
γ
0
β 0e 0
β 1γ 2γ λ

(
)
0
,e e
λ = λ
0v T
Ee
=
s
e
e
<
1(
)
s
s
T
T
e e
e
=
+
−
s
e
e
>
0
0
(
)
u
a e
e
=
−
s
e
e
<
0
0
1
(
)
(
)
s
s
u
a
e
e
a e
e
=
−
+
−
s
e
e
>
s
e
e
>
0
1

2
(
)
s
s

b
b
e
e
a e
e
λ =
+
−
+
−
(
)

2
2
0
1
1
(
)
1
(
) ,
s
s
s
s
b
e
e
a e
e
e
e
a
e
e
+
−
+
−
=
+
+
−
0
/
b
b
a
=
1
1
0
/
a
a
a
=
2
1
1
0
/
a
E
=
ρ 2
0
0
/
a
E
=
ρ b [
]
2
1
0
1
(1
)
(
)
(
)
s
s
s
s
b
e
e
a
e
e
e
e
a e
e
=
+
+
−
−
−
+
−
b λ 2
2
2
2
2
2
1
1
1
0
1
0
(
4 )
(2
) (
)
8
(
)
8
s
s
s
a e
a
a e
e
a e
e
u e
λ − λ
+
− λ
−
+
−
−
+
+

2
2
2
2
2
1
0
0
1
(2
) (
)
4(
)
8
(
)
4
0
s
s
s
s
s
s
e
a e
e
e
e e
e
a e
+
− λ
−
−
−
+
−
−
=
1γ 0
β 2γ λ b b 0
0
0
/
v
v
a
=
0
β 0
β

2γ

0
50
β
=
o 0
0
e
=
2
1
0,05
a
=
0,002
se
=

0v
b
0
30
β
=
o 0
0
e
=
2
1
0,05
a
=
0,002
se
=

0v
b
0
90
β
=
o0
0
e
=
2
1
0,05
a
=
0,002
se
=

0v
b
0
90
β
=
o 0
0
e
=
2
1
0,05
a
=
0,002
s
e
=

0v
b
0
70
β
=
o0
0
e
=
2
1
0,05
a
=
0,002
se
=

0v
b
0
β 1
2
1
2
1
2
0

1
2
1
2
1
2
0
1
2

;
;
2;

;
;
;
.

/
T
T
T

e
e
e
u
u
u v
v
v
b
b
b

=
=
γ = γ = γ
β = β = β = π

=
=
=
=
=
=
=
=
′′0
( )
(
)
u
e
e
= ψ
−ψ
0(
)
(1 cos )(1
)
b
u u
T
e
ρ
+
=
−
γ
+
0
0
(
)
sin (1
)
b
u v
T
e
ρ
+
=
γ
+
0
tg
v
b
=
γ

0
( )
(
)
u
e
e
= ψ
− ψ
2
0(
)
(1
)
b
u
T
e
ρ
+
=
+
sec
b
u
b
+
=
γ 0
tg
v
b
=
γ 0
0
(
)
u
a e
e
=
−
0
0
0
(1
)
(
)
b
a
e
e
a e
e
=
+
−
−
2
2
2
0
0
0
(1
)
(1
)
T
v
b
e
a e
e
+
=
+
=
+
ρ

2
0
0
0
2(
)
(1
)
(
)
v
e
e
e
e
e
e
=
−
+
−
−
0
0
0
/
v
v
a
=
4 3
3
0
4
/ /
e
v
≈
2 3
2 3
1 3
3
0
0
0
2
0,8
b
v
v
a
≈
=
/
/
/
/
3
0
0
tg
1 25
,
/
v
a
γ ≈
⋅
γ 0v 0,002
se ≈
3/4
0
2
0,011
s
s
v
e
=
=
0
55s
v
≈
0
e
e
≈
0
0
b
a
e
=
=

0
0
/
T
=
ρ 1
2
0
β = β = β 1
2
0
v
v
v
=
=
′′′′′′0
0
β =
1
0
γ =
1
1
v
u
=
0

0(1
)
ds
T
a
dt
e

∗
=
<
ρ
+
2

0
(1
)
T
a
e
<
+
ρ
2
2
0
1 (1
)
s
s
a e
a
e
<
+
2
2
1
0
(
0,05,
/
a
a ≈
0,002
se ≈
2
2
1
0
(
0,03,
/
a
a ≈
0,001
se ≈
1

1
1
1
( )
cos

e

e

x
a e de
v
t
∂ =
+
β
∂
α ∫
1

1
1
1

1
( )
sin

e

e

y
a e de
v
t
c
∂ =
+
β
∂
α ∫
1
1
ctg
c = −
γ 2
1
1 1/
sec
c
α =
+
=
γ 1

1
1
1
cos
( )
cos

e

e

x
a e de
v
t
∂ =
γ
+
β
∂
∫
1

1
1
1
sin
( )
sin

e

e

y
a e de
v
t
∂ = −
γ
+
β
∂
∫
1
2
0
(
)
(
)
u
e
e
= φ
− φ
0
1
1
(
)
b
u
ρ
+
1

2

1
1
1
1
cos
cos
( )

e

e
v
a e de
u
β +
γ
−
∫

2
1
2
(
)(cos
1)(1
)
T e
e
=
γ −
+
0
1
1
(
)
b
u
ρ
+
1

2

1
1
1
sin
sin
( )

e

e
v
a e de
β −
γ ∫

2
1
2
(
)sin
(1
)
T e
e
=
γ
+
1
1
1
1
1
sin
sin(
)
b
v
γ =
β − γ
1
0
0
1
0
0
tg
sin
/ (
cos
)
v
b
v
γ =
β
+
β
1
0
2
2
0
2
/
(
)/
(
/ )
b
v
a
e
v
x a
v
=
+
1
0
v
v
=
1
/ 2
β = π
x 1
(
)
z
x
z
e z
c
α
+
−
=
+
0
x =
0
z =
0
c =
1
0
1
0
/
( ) /
z
a
v
a e
v
=
=

[
]
1
1
1
1
1
0
0

1 (
1)
(
1)cos
1
a
a
x
e
z
z
e
v
v
=
+
−
=
+
γ −
α
1
2
a
a
=
1
0
(
/
)
x a
v
=

2
0
(
/
)
x a
v
=
1
2
e
e
≠
1
1
1
1
(
1)cos
b
a e
=
+
γ 1
1
2
0
0
(
)
(
)
s
s
u
a e
e
a
e
e
=
−
+
−
2
0
1
2
2
(
)
(
)(1
)
b
u
T e
e
ρ
+
=
+
1

2

1
1
1
1
sec
( )

e

e
b
u
b
a e de
+
=
γ − ∫
0
1
1
tg
v
b
=
γ 2
2
0
1 (1
)
s
s
a e
a
e
=
+
§ 2.3. Удар по гибкой нити точкой конечной массы

′′′′

0T′ Te γ 2
2
2
1
1
2
2
0

u
u
a
t
s
∂
∂
=
∂
∂
2
2
2
2
2
2
0
0
2
2
0
0
0
0

1
u
u
T
e
a
a
s
s
t
s
∂
∂
∂
∂
=
+
−
ρ ∂
∂
∂
∂
ds
ρ
O uv
′
(
)
0
0
1
u
u
s
e
′
′
=
+
+
ds
ρ
T
T
+ ∆
qds0v ds
′
ρ
O u′ O v′ (
)

2

0
0
2
cos
sin
u
ds
T
v
ds
t
′
∂
′
ρ
= ∆
γ + ρ
γ
∂

(
)

2

0
0
2
sin
cos
.
v
ds
T
v
ds
qds
t
∂
′
ρ
= −∆
γ +ρ
γ
+
∂

0
0
ds
ds
ρ
= ρ
cos
u
s
∂
γ = ∂

sin
v
s
∂
γ = − ∂

(
)
T
T
E e
e
=
+
−
2
2

0
0
0
(1
)
1,
u
v
e
e
s
s
∂
∂
′
=
+
+
+
−
∂
∂


2

0
0
2
0
0

1
(
)cos
sin
u
T
Ee
Ee
v
s
t
∂
∂
′
=
+
−
γ +
γ
ρ ∂
∂
2

0
0
2
0
0

1
(
)sin
cos
v
q
T
Ee
Ee
v
s
t
∂
∂
′
= −
+
−
γ +
γ +
ρ ∂
ρ
∂
0
T
Ee
Ee
T′
+
−
−
2
2
2
0
0
0
2
2
0
0
0

1
(
)
sin
u
u
a
T
E e
v
s
t
s
∂
∂
∂
′
=
+
−
+
γ
ρ ∂
∂
∂
0
0

u
e
e
s
∂
′
=
+ ∂

2
2
2
0
0
2
2
0
cos
v
v
q
v
t
s
∂
∂
′
= λ
+
γ + ρ
∂
∂

2
0

0
0
(1
)
T
e

′
λ =
′
ρ
+
0
0 ( )
s
s t
∗
=
2
e
e
=
2
2
2

0

2
2
0
0
0
0
0
0

(
)(1
)

(1
) (
)
(1
)(
),

u
T e
e
b
t

e
u
T
Ee
Ee
e
a
s

∂
+
−
=
=
∂
ρ

′
+
∂
′
′
=
+
−
+
+
λ +
ρ
∂
2
0
0
0
1
u
u
e
s
s
∂
∂
′
+
=
−
∂
∂
2
0
0
0
0
0
(1
)
cos
sin
u
u
e s
v
∗
′
=
+
+
γ −
γ
0
0
0
0
0
0
0
(1
)
sin
cos

t
v dt
u
e
s
v
∗
′
=
+
+
γ +
γ
∫
0
0 ( )
s
s t
∗
=
2
0
( )

t
b t dt
u
=
∫
0
0
2
0
0
0

2
2
0
0
0
0
0
0

1

(1
)
(1
)(
).

ds
ds
u
u
e
s
dt
s
dt

T
Ee
Ee
u
e
e
a
s

∗
∗
∂
∂
′
=
+
+
=
∂
∂
′
+
−
∂
′
′
=
+
+
+
λ +
ρ
∂
0
0
s =
0
u
v
=
=
0
2 sin(
).
dv
m
T
dt
= −
γ − γ
0/
dv dt 0
0
sin
/
T
m
′
γ
0
0
0
00
2 sin
const.
dv
T
v
dt
m

′
γ
′
= −
=
=
/q ρ 0
(
)
t
f s
=
(
)
0

2

0
1

t
f s

u
e
s
=

∂
= +
∂
(
)
0
0

2
e

t
f s
e

u
ade
t
=

∂
= −
∂
∫
(
)
(
)
( )
(
)
(
)
(
)
( )

1
0
0
2
0
0
0
1
0

1
0
2
0
0

;

,

u
a t
s
a t
s
t
s

v
f
t
s
f
t
s
t

= φ
+
+ φ
−
+ Φ
+ Φ

=
λ +
+
λ −
+ ϕ
( )
(
)
2
0
0
0
0
00
00
0
sin
sin
/2

t
t
v dt
v t
v t
′
Φ
=
γ
=
γ
+
∫
(
)
(
)

0

1
0
0
0

1
s
s
T
Ee ds
E
Φ
= −
−
∫
( )
(
)

2
2
2
0
0
0
0
00
00
0
cos
cos
/ 2
2
2

t
q
q t
t
v dt
t
v t
v t
′
ϕ
=
γ
+
=
γ
+
+
ρ
ρ
∫
(
)
(
)
( )
1
0
2
0
0
0
a t
a t
t
φ
+ φ
+ Φ
=
(
)
(
)
( )
1
2
0
0
f
t
f
t
t
λ
+
λ
+ ϕ
=
(
)
(
)
(
)
2
0
0
1
0
0
0
0
0
a t
s
a t
s
t
s
a
φ
−
= −φ
−
− Φ
−
(
)
(
)
(
)
2
0
1
0
0
0
f
t
s
f
t
s
t
s
λ −
= −
λ −
− ϕ
−
λ (
)
(
)
(
)
( )
(
)
(
)
(
)
(
)
( )

1
0
0
1
0
0
0
0
0
0
1
0

1
0
1
0
0
0
0

;

.

u
a t
s
a t
s
t
s
a
t
s

v
f
t
s
f
t
s
t
s
t

= φ
+
− φ
−
− Φ
−
+ Φ
+ Φ
=
λ +
−
λ −
− ϕ
−
λ + ϕ
(
)
(
)
(
)
2
1
0
0
2
0
0
1
0
u
a t
s
a t
s
s
= ϕ
+
+ ϕ
−
+ Φ
( )
0
0
s
s
t
∗
=
(
)
(
)
( )
(
)
(
)
1
0
0
2
0
0
1
0
0
0
1
0
0
1
a t
s
a t
s
s
e
s
a t
s
∗
∗
∗
∗
∗
′
ϕ
+
+ ϕ
−
+ Φ
=
+
+ φ
+
−
(
)
(
)
( )
( )
1
0
0
0
0
0
0
1
0
0
cos
a t
s
t
s
a
t
s
∗
∗
∗
−φ
−
−Φ
−
+ Φ
+ Φ
γ −
(
)
(
)
(
)
( )
1
0
1
0
0
0
0
0
sin
f
t
s
f
t
s
t
s
t
∗
∗
∗
−
λ +
−
λ −
− ϕ
−
λ + ϕ
γ
(
)
(
)

(
)
(
)
(
)

1
0
0
2
0
0

0
1
0
0
1
0
0
0
0
0
0

1
1
;

a t
s
a t
s

e
a t
s
a t
s
t
s
a
a

∗
∗

∗
∗
∗

′
′
ϕ
+
− ϕ
−
=

′
′
′
′
= +
+ φ
+
+ φ
−
+
Φ
−
(
)
(
)
(
)

(
)
( )
( )

0
0
0
1
0
0
1
0
0
0

0
0
0
0
1
0
0

1

sin

t
v dt
e
s
a t
s
a t
s

t
s
a
t
s

∗
∗
∗

∗
∗

′
=
+
+ φ
+
−φ
−
−
−Φ
−
+ Φ
+ Φ
γ +
∫

(
)
(
)
(
)
( )
1
0
1
0
0
0
0
0
/
cos
;
f
t
s
f
t
s
t
s
t
∗
∗
∗
+
λ +
−
λ −
− ϕ
−
λ + ϕ
γ
(
)
(
)
(
)
( )
0
0
1
0
0
1
0
0
0
0
0
1
0
0

1
1
ds
e
a t
s
a t
s
t
s
a
s
a
dt

∗
∗
∗
∗
∗
′
′
′
′
′
+
+ φ
+
+ φ
−
+
Φ
−
+ Φ
=
(
)
0
0
0
1
T
Ee
Ee
e
′
+
−
′
=
+
+
ρ

(
)(
)
(
)
(
)
(
)
( )

(
)

2
2
0
1
0
0
1
0
0
0
0
0
1
0
0

0
0
0

1
1
1
2
1

e
a
a t
s
a t
s
t
s
a
s
a

e
T
Ee
Ee

∗
∗
∗
∗
′
′
′
′
′
+
λ +
φ
+
+ φ
−
+
Φ
−
+ Φ
+
′
+
′
+
−
ρ

2
u
s
∂
∂
2
u
t
∂
∂
0e′ (
)
1
0
a
e′ (
)
(
)
0
0
1
0
/
t
f s
s a
s
=
=
+ ε
[
]
( )
1
1
1
0
1
/
a
a
s
a e
−
+
ε
=
1
0
/
a
s
ε
0

0
0
1
0
0
2
0
0
1
1

0
0
1
0
0
2
0
0
1
1
0

1
1
1
;

1
1
.

e

e

a
a
T
s
a
s
a
a
a
E

a
a
a
s
a
s
a
de
a
a
a

′
′
ϕ
+
+
ε −ϕ
−
+
ε = +
′
′
ϕ
+
+
ε + ϕ
−
+
ε = −
∫

0e′ (
)
0
0
T
T
E
e
e
′
′
′
=
+
−
( )
1
a e
a
≡
0
ε ≡
(
)

(
)
(
)

0

0

0

0

0
0
1
1
1
1
0
0
0
0
1
0
0
0
0

0
0
1
1
2
0
0
0
1
0
0
0

1
1
1
1
;
2

1
1
1
1
.
2

e

e

e

e

a
T
a
a
a
a
E
s
de
e
e
e s
a
E
a
E
a
a
a

a
T
a
a
a
s
de
e s
e
a
E
a
a
a

′

′

′
′
′
′
′
ϕ
+
=
+
−
−
+
+
−
′
′
′
ϕ
−
= −
+
+
+
+
−
∫

∫

(
)

0

0

1
0
0

0
0
0
1
1
1
1
0
0
0
0
0
0
0
1

1 1
1
1
;
2
1

e

e

a t
s

T
a t
s
a
a
a
a
a de
e
e
E
a
a
a
a
a
a
a

′
ϕ
+
=

′
+
′
=
+
−
+
−
−
−
+
∫

(
)

0

0

2
0
0

0
0
0
1
1
1
1
0
0
0
0
0
0
0
1

1 1
1
1
.
2
1

e

e

a t
s

T
a t
s
a
a
a
a
a de
e
e
E
a
a
a
a
a
a
a

′
′
ϕ
−
=

′
−
′
= −
+
+
−
+
+
+
−
∫

(
)
(
)

0

0
0
1
0
1
0
0
0
0
2
cos
2

s
t
s
s
q
f
t
s
f
t
s
t
v dt

∗
−
∗
∗
λ
∗
∗
λ +
−
λ −
=
−
+
γ
−
ρ λ
λ
∫

(
)
(
)
( )
1
0
0
2
0
0
1
0
0
sin
a t
s
a t
s
s
∗
∗
∗
− ϕ
+
+ ϕ
−
+ Φ
γ
(
)
(
)
(
)
( )
0
0
1
0
0
1
0
0
1
0
1
e
s
a t
s
a t
s
s
∗
∗
∗
∗
′
+
+ φ
+
− φ
−
+ Φ
=

(
)
(
)
( )

0

0

1
0
0
2
0
0
1
0
0
0
0
0
cos
sin
,

s
t a
a t
s
a t
s
s
v dt

∗
−

∗
∗
∗
= ϕ
+
+ ϕ
−
+ Φ
γ +
γ
∫
(
)
(
)
(
)

(
)
( )
(
)

2
2
2
0
0
0
1
0
0
2
0
0
0

0
0
0
0
0

3
2
2
2
1

2 1

ds
a
a
a t
s
a t
s
dt
e

E
E
e
e s
T
e
E
e
E

∗
∗
∗

∗

−λ
′
′
=
λ −
+
ϕ
+
− ϕ
−
−
′
λ
λ
+

′
′
λ
−
−
′
′
−
+
+
′
+

(
)
( )
( )

(
)
( )

(
)

2
2
0
0
1
1
1
1
0
0
0
0
0
0
2
2
0
1
0
0
2
0
0

2
1
1
4
1

,
2
1

ds
a
a
a
a
a
e
e
e
dt
e
a
a
a
a

a
a
e
e s

a
e

∗

∗

− λ
′
= λ +
+
ξ +
−
η −
+
′
λ
+
′
λ
−
−
+
′
+

0
0

0
1
1
a t
s
a
a

∗
−
ξ =
−

0
0

0
1
1
a t
s
a
a

∗
+
η =
+

(
)
(
)
1
0
0
1
0
0
a t
s
a t
s
∗
∗
′
′
φ
+
− φ
−
=

( )
( )

0

0

1
1
1
1
1
0
0
0
0
0
0
0
0

1
1
cos
1
1
2
2

e

e

a
a
a
a
a
a
e
de
e
e
a
a
a
a
a
a

′
′
=
γ
−
+
−
η −
+
ξ
+
∫

(
)
(
)(
)

2
2
1
0
00
00
0
0
0
0
0
2
0
0
0
0
0

1 cos
sin
1
2

a
a
v
v
s
t
e
e
e
a
a
a
a
a

∗
λ
−
′
−
γ
′
′
+
γ
+
−
−
+
λ +
−
−
( )
( )

2
2
2
2
2
0
0
1
1
1
1
0
2
0
0
0
0
1
1
4
2

a
a
a
a
a
a
e
e
e
a
a
a
a

+ λ
+ λ
′
−
+
+
ξ +
−
η
λ
λ
(
)
(
)
( )

2
0
1
1
1
1
0
0
1
0
0
0
2
0
0
0

1
1
2
T
a
a
a
a t
s
a t
s
e
e
E
a
a
a

∗
∗
′
′
′
′
φ
+
+ φ
−
=
−
+
−
η +
( )
0
0
1
1
0
00
00
0
0
0
0

sin
1
1
2
s
a
a
e
e
v
v
t
a
a
a
a

∗
γ ′
′
+
+
ξ −
−
+
−
( )
1 x
′φ
( )
e x ( )
0s
t
∗
0s∗ t
λ ( )
e x ( )
1 x
′φ
( )
0s
t
∗
( )
0s
t
∗
( )
e x ( )
1 x
′φ
( )
e x ( )
1 x
′φ
( )
0s
t
∗
( )
1 x
′φ
( )
e x 0s∗ t
λ ( )
0
1
e x
e
b x
′
=
+
( )
1
0
1
x
c
c x
φ
=
+
( )
e x ( )
1 x
′φ
0
0
00
0
0
0

sin
1
2
T
v
c
e
E
a

′
γ ⋅
′
=
−
−
(
)(
)

(
)
(
)

2
2
2
2
00
0
0
1
0
1
2
3
2
2
2
2
2
2
2
2
2
2
0
0 1
1
0
1
0
1
0
1
1
2
0

sin
1

1 cos
1
2
3
3
2

v
a
a
a
b
a a
a
a
a
a
a a
a
a
a

′
γ
−
− λ
=
−
γ
− λ
+
+
λ
+
−
−
− λ
λ
00
0
0
1
2
0

sin
1

2

v
a
c
a

λ
′
γ
−
= −
×

(
)
(
)

2
2
2
2
2
3
3
0
1 0
1
0
1
2
1
2
3
2
2
2
2
2
2
2
2
2
2
0
0 1
1
0
1
0
1
0
1
1
2
0

2
2
2
1

1
cos
1
2
3
3
2

a
a a
a
a
a
a

a a
a
a
a
a
a a
a
a
a

− λ
− λ
−
λ + λ − λ
−
×
−
γ
− λ
+
+
λ
+
−
−
−λ
λ
( )
e x ( )
1 x
′φ
( )
0s
t
∗
00
v′ 1′φ e( )
0s
t
∗
0
0

u
e
e
s
∂
′
=
+ ∂
(
)
(
)
0
0
T
T
E e
e
E
e
e
′
′
′
=
+
−
+
−
( )
1 x
′φ
( )
e x (
)
(
)
(
)
0
0
1
0
0
1
0
0
0
1
0
0
0

1
s
e
e
a t
s
a t
s
t
s
a
a
′
′
′
′
′
=
+ φ
+
+ φ
−
+
Φ
−
+ Φ
=
(
)
(
)
0
0
0
1
0
0
1
0
0
00
00
0
0

sin
s
T
e
a t
s
a t
s
v
v
t
e
a
a
E
γ ′
′
′
′
=
+ φ
+
+ φ
−
+
+
−
+
−
(
)
(
)

(
)
(
)

2
2
2
2
2
2
0
0
0 1
0
1
0
1
2
0
1
2
2
2
2
2
2
2
2
0
0
0 1
1
0
0
1
2
2
0
1

1 cos
1
2
3
3
2

1 cos
1
2
3
3
2

a
a a
a
a
a
a
a
a

a
a a
a
a
a
a
a
a

−
γ
λ
λ
+
−
+
+
−
−
−λ
λ
×
−
γ
λ
−λ
+
+
+
−
−
− λ
λ
( )
0s
t
∗
( )
(
)

2
2
2
00 1
0
2
0
0
0

sin
1

4
1

v a
t
a
s
t
t
e

∗

λ
′
γ
−
= λ +
×
′
λ
+

0
0
0
0
1 0
00
00
0
1 0
0
0

sin
2
2
s
T
e
e
c
c a t
v
v
t
e
b s
a
a
E

′
γ ′
′
′
=
+
+
+
+
−
+
+
−
−
(
)

2
00
0
0
1 1
0
1 0
0
1 0
0
2
0
0
0

sin
2
v
s
b a
E
e
e
c a t
t
e
b s
s
E
a
a
a

′
′
γ ′
′
−
−
=
+
−
+
+
−
e
∆ 00
0
0 1
0

sin
2
v t
e
a c t
a
′
γ
∆ =
+
2
00
0
1
0 1
1
2
0
0
0

sin
2
1
1
v t
a
e
a c t
b
t
a
a
a

′
γ
λ
∆ =
+
−
+
−
λ
0
0
T
Ee
′
′
=
T
Ee
=
E
E
′ =
1
0
a
a
=
(
)
1
0
0
s
Φ
=
(
)
(
)
1
0
0
2
1
2
const
s
e
′
ϕ
=
+
=
0
e
e′
=
( )
(
)
(
)
2
0
0
0
1
2
e
e
′
′
ϕ
= −
+
−
0
0
0
00
0
1
0
1
2
0
0

sin
1
sin
1
0,
;
2
2

v
v
a
b
c
c
a
a

λ
′
γ
−
γ ⋅
=
= −
= −
×

(
)

(
)
(
)

3
2
2
2
0
0
0
0

3
2
2
2
2
2
0
0
0
0
0

1 cos
3
3
3
2

1 cos
2
2
3
3
2

a
a
a

a
a
a
a

−
γ λ +
−
−
− λ
λ
×
−
γ − λ
+
λ +
−
−
−λ
λ
( )
0s
t
∗
(
)

2
2
2
00 0
0
2
0
0
0

sin
1

( )
4
1

v a
t
a
s
t
t
e

∗

λ
′
γ
−
= λ +
×
′
λ
+

3
2
2
3
0
0
0
5
2
2
2
2
2
4
2
2
0
0
0
0
0
0
0
0

2
2
2
2

1 cos
2
(
)
2
3
3
(
)
2

a
a
a

a
a
a
a
a
a
a

−
λ + λ
+ λ
×
−
γ
− λ
+
λ
+
−
−
−λ
λ
§ 2.4. Движение нити конечной длины при продольнопоперечном ударе. Возникновение вторичных
волн натяжения

λλγγ′′γγ=(
)
(
)
(
)
(
)

(
)
(
)

1
1
1
1
1

1
1
1
1
1
1

2
1
1
1
1
2

2
1
1
1
1
2

sin
;

2
1 cos
;

sin
sin
;

cos
cos
.

y

x

y
y

x
x

dm v
T
d

dm
v
u
u
T
d

dm
v
v
T
d

dm
v
v
T
d

=
γ
τ

+
+
=
−
γ
τ

′ −
=
γ −
γ
τ

′ −
=
γ −
γ
τ

1
1
1
1
, 
, 
, 
x
y
x
y
v
v
v
v
′
′ ′(
)

1
1
0 0 1
2
T
u
a
e
= ρ
+
′(
)
(
)
(
)
1
1
2
1
1
1
0
1
1
2
1
1
1
T
ds
ds
e
e
e
d
d
= λ =
+
=
+
ρ
+
τ
τ

0
0, 
.
x
y
v
v
v
=
=

0
0
0
0
cos ;  
sin .
n
v
v
v
v
τ
=
γ
=
γ

′1
0cos
nv
v
′ =
γ 1
0sin
v
v
v
τ
τ
′ =
γ + ∆
vτ
∆
(
)
(
)
1
0 0

0
0

/
1
.
1
sin /
u
T
a
v
e
v
a
τ
−
ρ
∆
=
+
−
γ
1
1
, 
x
y
v
v
′
′ 1
1
1

1
1
1
0

cos
sin
cos ;

sin
cos
sin .

x
n

y
n

v
v
v
v

v
v
v
v
v

τ
τ

τ
τ

′
′
′
= −
γ +
γ = −∆
γ
′
′
′
=
γ +
γ =
+ ∆
γ
γ1
1
, 
x
y
v
v (
)
1
1
0
0
/
u
T
a
=
ρ
′1
1
2
1
0
1
2

T
ds
ds
d
d
λ =
=
=
ρ
τ
τ

′1
v
u
u
τ
∆
=
−
′(
)
(
)
1
1
1
0
1
cos ;  
sin ,
x
y
v
u
u
v
v
u
u
′
′
= −
−
γ
=
+
−
γ
′1
1
1
sin
y
v
= λ
γ (
)
1
1
1
1
2
1 cos
xv
u
u
+
+
= λ
−
γ
(
)
1
1
1
1
cos
cos
x
x
v
v′
=
+ λ
γ −
γ (
)
1
1
1 sin
sin
y
y
t
v
v
=
+ λ
γ −
γ
(
)
(
)
[
]
1
1
1
0
1
1
1
1
sin
sin
2
2
y
y
v
v
v
u
u
′
=
+ λ
γ =
+ λ +
−
γ (
)
[
]
1
1
1
1
1
1
2
cos
2
xv
u
u
u
u
=
λ −
−
− λ +
+
γ (
)
1
1
1
1
1
1
1
sin
/
;  cos
1
2
/
.
y
x
v
v
u
u
γ =
λ
γ = −
+
+
λ
γ11
1
, 
x
y
v
v (
)
(
)
2
2
0
0
1
1
1
1
2
sin
v
v
u
u
u
u
+
λ +
−
γ + λ +
−
+

(
)
(
)(
)
2
2
1
1
1
1
1
1
1
2
2
2
cos
4
u
u
u
u
u
u
+ λ −
−
+
λ −
−
λ +
−
γ = λ 2
0v γγ2
2
0
2
v
u
u
= λ −
0
sin
/
v
γ =
λ cos
1
/
u
γ = −
λ 0 .
/
T
λ =
ρ

(
)(
) (
)
2
2
2
1
1
1
1
1
1
1
2
2
2
2
2
4
0
u
u
u
u
u
u
u
u
u
λ −
+
λ +
−
λ +
−λ +
+
λ −
−
− λ =
λ
1
1 /
u
u
u
=
1
1
/
u
λ
λ =
(
)( )
(
)

2
2
2
1
1
1
1
1 1
1
1
2 1
4
1
0
u
u
u
u
u
u
u u
u
−
+
+
+
−
+
−
=
λ
λ
1u 0
ua
λ =
(
)
(
)
2
2
0
0
0
1
/
/
;
/
.
u
u a
u
a
u
a f
u
λ =
=
λ
=
/
u
λ 1
u 1u 1
1
u
uu
=
1
1 0
uu a
λ =
1
1
0
1
1

1
1
1
sin
1
2
u
u
a
u
u

−
γ =
+
+
(
)
1
1
1
0
3
1
2
x
u
u
v
u
u
a
= −
−
−
−
(
)
0
1
1
1
0
1
1
2
y
v
u
v
u
u
a
=
+
+
−
1u λ1
xv
−
1
y
v
= 4700= 701
u = 0,381′
′(
)
(
)
0
1
0 0
0 0
2
f
s
s
T
T
T
a u
e
T
a u
− ϕ
−
−
+ρ
=
−ρ
ϕ0
1
2
, 
.
1

f
s
u
u
u
k
e
k

− ϕ
=
−
=
+

(
)
1
1
0 0
2
s
T
T
T
T
a u
′ =
−
−
+ ρ
=

1
2
.
1
k T
k
=
+
 0
200
v ≈
o
30
γ ≈
o
0
60
ϕ ≈
o
0
60
ϕ ≈
0,8
k ≈
1
1
0,9
T
T
′=
1
y
v
1
xv
−
λ1
xv
−

1
y
v

λ
§ 2.5. Переходные этапы движения гибкой нити
с тормозящими элементами на концах

2

2

xc F
T
u
F

∞
ρ
=
∞
ρ xc F (
)
(
)(
)
(
)

11
11
11
11
11
11

11
11
11
11
11
11
11

sin
;

cos
1 ;

b
u
v
T

b
u
w
u
T

ρ
+
=
γ

ρ
+
−
=
γ
−

11
0
11
tg
/
v
b
γ
=
11
0
1
11
sin
s
v
v
u
=
−
γ
11
1
11
cos
su
ω
= −
γ 11
11
11
1 11
0

T
b
u
u
+
=
= κ
ρ

H

1 1
1
0
2

xc F
F

∞
ρ
κ =
ρ

11
0
ρ
= ρ 11
11
11

11
11
sin
cos
1
v
u
ω
−
=
γ
γ
−

11
v 11
w 0
11
1 11
1
sin

s

v
u
u
γ
= κ
+
11
1
11
1

1
sin

s
U
U
γ
= κ
−
11
11
0
/
U
u
v
=
1
1
0
/
s
s
U
u
v
=
(
)(
)
0
11
11
11
1
11
11
1
11
11
11
1 11

cos
cos
cos
1
1
s
s
b
u
u
u
u
u
T
u
ρ
+
+
γ
+
γ
γ
= −
= −
κ
(
)
1
11
11
1
11
1

1
cos
s

U
U
U
κ −
γ
= κ
+

11
γ 11
U
(
)
(
)
2
2
2
1
11
1
11
1
1
1
s
U
U
U
+ κ −
= κ
+
(
)

2
2
1
1
1
2
11
1
1
2
1
1
1

1
2
1
2
1
2
1

s
s
s
U
U
U
U
κ
κ
−
= −
+
+
κ −
κ −
κ −

2s
U
−

1
κ 2
κ (
)

2
2
2
2
2
2
2
21
2
2
2
2
2

1
;
2
1
2
1
2
1

s
s
s
U
U
U
U
κ
κ
−
=
+
+
κ −
κ −
κ −

(
)
2
21
21
2
21
1

1
cos
.
s

U
U
U
κ −
γ
= κ
−

0
su ≠
1
2
κ > κ 11
T 21
T (
)
11
21
2
2
11
0
1
21
0
1
f
s
s
T
u
e
T
u
γ
+γ
− ρ
=
− ρ
(
)
11
21
2
2
2
1
11
1
2
2
2
2
21
1

s
f

s

U
U
e
U
U

γ
+γ
κ
−
=
κ
−

2
2
2
1
11
1
2
2
2
11
21
2
21
1

1
ln
s

s

U
U
f
U
U
κ
−
= γ
+ γ
κ
−

11
U
21
U
11
γ 21
γ
1
κ 2
κ 1
κ 2
κ 11
U
21
U
11
γ
21
γ κ 1
1
κ >
0
b >
1
1
x = o
11
90
γ
=
1
1
κ < 0
b <
o
11
90
γ
>
OO′ 1
κ κ 0
2
xc F
F
∞
≥ ρ
ρ 1
κ 2
κ 1
κ 2
κ 1
0
s
u
=
(
)
(
)

(
)
(
)
(
)

1
2
2
1
1
2
2
1
0
2
1
2
1
2

2
1
1
2
1
1
2
1
ln
arcsin
2
1
f

−
κ −
κ
κ −
κ − + κ −
κ −
=
κ κ
κ −
κ 11
u 21
u 11
γ
21
γ
11
0
0
1
11
cos
sin
s
v
v
u
=
γ −
γ
11
0
0
1
11
sin
cos
s
w
v
u
=
γ −
γ
0γ 0
11
1
11
1

cos
sin
;
s
U
U
γ
γ
= κ
+

(
)
1
11
0
11
11
1

1
sin
cos
;
s

U
U
U
κ −
+
γ
γ
=
κ
+

(
)
(
)
[
]
(
)

2
2
0
1
1
1
0
1
1
1
11
2
1
1
1

1 sin
1 sin
1
.
2
1
2
1
2
1

s
s
s
U
U
U
U
κ −
γ − κ
κ −
γ − κ
−
=
+
+
κ −
κ −
κ −

21
γ
21
U
11
γ
21
−γ
1
κ 2
κ 1
κ < 1
3
κ =
2
1,5
κ =
0,15
f =
0
0
γ =
o
30
±
1
κ 2
κ 1
κ
2
κ

12
u 12
γ

0γ
11
γ
21
γ
11
v
21
v
31
v

(
)(
)
(
)
12
12
12
0
2
11
12
12
12
12
11
sin
sin
sin
sin
s
b
u
v
u
u
T
ρ
−
−
γ
−
γ
=
γ
−
γ
(
)(
)
(
)
12
12
12
12
12
2
11
12
21
12
cos
cos
cos
cos
s
b
u
u
u
T
ρ
−
γ
+
γ
=
γ
−
γ
12
12
1 12
b
u
u
−
= κ
12
0
ρ
= ρ (
)
(
)

2
1
12
11
12
1
12

1
sin
sin
1

s
U
U
U
−
− κ
γ
γ
=
+ κ
(
)
(
)

1
12
2
11
12
1
12

cos
cos
1

s
U
U
U
κ
−
γ
γ
=
+ κ
12
γ 12
U
(
)

2
1
2
2
11
2
12
11
2
12
1
1

2
1
2
sin
sin
0
2
1
2
1

s
s
s
U
U
U
U
U
κ
+
−
γ
−
γ
−
−
=
κ +
κ +

(
)
(
)

2
1
1
2
12
11
2
11
2
2
2
11
2
1
1

2
1
sin
sin
1
2
sin
2
1
s
s
s
s
U
U
U
U
U
κ
κ +
=
γ −
+
γ −
+
+
−
γ
κ +
κ
1
2
0
s
s
U
U
=
=
11
1

1
2
1
U
=
κ −

1
11
1

2
1
sin
κ −
γ
=
κ
1
11
1

1
cos
κ −
γ
=
κ
1
12
1

1
cos
1
κ −
γ
= κ +

1
1
12
1

2
1
2
2
1
U
κ − +
κ
=
κ +
11
U
12
U
1
1
κ >
12
11
U
U
>
1
1
κ < 12
11
U
U
<
1
2
κ =
11
0,578
U
=
o
11
60
γ
=
12
0,912
U
=
o
12
70
γ
=
1
κ 11
1,58
U
=
o
11
106,5
γ
=
12
0,962
U
=
o
12
100
γ
=
13
0
3
13
sin
s
v
v
u
=
−
γ
13
3
13
cos
s
w
u
= −
γ
12
13
12
sin
v
u
=
γ
12
13
12
cos
w
u
=
γ
3
su (
)(
)
(
)
0
13
13
0
3
13
13
12
13
12
13
sin
sin
sin
sin
s
b
u
v
u
u
T
ρ
−
−
γ
−
γ
=
−
γ
+
γ
(
)(
)
(
)
0
13
13
13
12
3
13
13
12
13
cos
cos
cos
cos
s
b
u
u
u
T
ρ
−
γ
+
γ
=
γ
−
γ
13
13
1 13
b
u
u
−
= κ
(
)
13
1
12
13
1
13
3

1
1 sin
sin
s

U
U
U
+
κ −
γ
γ
=
κ
+
(
)
1
13
13
13
3
1
13

1
cos
cos

s

U

U
U
κ −
γ
=
γ
+ κ
13
γ 13
U
(
)
[
]

2
3
2
13
1
12
1
3
13
1
1

2
1
1 sin
0
2
1
2
1

s
s
U
U
U
U
−
−
κ −
γ
− κ
−
=
κ −
κ −
(
)
(
)
[
]
(
)

2
2
1
12
1
3
1
12
1
3
3
13
2
1
1
1

1 sin
1 sin
1
2
1
2
1
2
1

s
s
s
U
U
U
U
κ −
γ
− κ
κ −
γ
− κ
−
=
+
+
κ −
κ −
κ −

(
)

1
14
4
14
13
1
14
cos
cos
1

s
U
U
U
κ
−
γ
=
γ
κ +

1
14
13
4
1
sin
2
1
s
U
U
κ
=
γ
−
+
κ + (
)
(
)

2
1
2
13
4
4
4
13
2
1

2
1
sin
1
2
sin
s
s
s
U
U
U
κ +
+
γ
−
+
+
−
γ
κ
(
)

1
1,2
2
1,2
1,2
1
1
1,2
cos
cos
1

n
s n
n
n
n

U
U
U
−
κ
−
γ
=
γ
κ +
[
1
1,2
1,2
1
2
1
sin
2
1
n
n
s n
U
U
−
κ
=
γ
−
+
κ +

(
)
(
)

2
1
2
1,2
1
2
2
2
1,2
1
2
1

2
1
sin
1
2
sin
n
s n
s n
s n
n
U
U
U
−
−
κ +
+
γ
−
+
+
−
γ
κ
(
)
1
1,2
1
1,2
1
1,2
,2
1
1
1,2
1

1
cos
cos
n
n
n
s
n
n

U
U
U

+
+
+
+

κ −
γ
=
γ
+ κ
(
)
1
1,2
1
2
1
1,2
1
1

1 sin
2
1

n
s n
n
U
U
+
+
κ −
γ
− κ
=
+
κ −

(
)
[
]
(
)
(
)

2
2
1
1,2
1
2
1
2
1
2
1
1

1 sin
1
2
1
2
1

n
s n
s n
U
U
+
+
κ −
γ
− κ
−
+
+
κ −
κ −

k
i
≠ sk
si
u
u
=
2
2
2
1
1
2
2
2
1
2
2
2

1
ln
i
si

i
k
k
si

U
U
f
U
U
κ
−
= γ
+ γ
κ
−
1
κ > 10l 11
τ 11
x 10
10
11
11
11
1 11

l
l
b
u
u
τ
=
=
+
κ
10
11
0 11
1
11

l
x
v
U
=
τ
= κ
11
11
10
1
11

1
x
x
l
U
=
= κ
(
)
11
10
1 11
10
1 11
1
s
s
l
l
u
l
U x
=
+
τ
=
+
11
τ 11
0
x′ =
11
10
11 11
y
l
u
′ =
−
τ 11
11
10
11 11
1
/
1
1 1/
y
y
l
U x
′
′
=
= −
= −
κ 12
10
1 11
2 12
12
12
12
1 12

s
s
l
l
u
u
b
u
u
+
τ
+
τ
τ
=
=
−
κ
11
0 12
1
12
2
s

l
v
U
U
τ
= κ
−
12
1 11
12
10
1
12
2

1
s

s

x
U x
x
l
x U
U
+
=
=
−

(
)
12
10
1 11
2 12
1
s
s
l
l
U x
U
x
=
+
+
12
12 12
12
sin
x
U
x
′ =
γ
12
11
12 12
12
cos
y
y
U
x
′
′
=
−
γ
12
13
1 13

l
u
τ
= κ
31 11
2 12
13
1
13

1
s
U
x
U
x
x
U
+
+
=
κ

(
)
13
10
1 11
2 12
2 13
1
s
s
s
l
l
U x
U
x
U
x
=
+
+
+
13
12
13 13
12
sin
x
x
U x
′
′
=
+
γ
13
12
13 13
12
cos
y
y
U x
′
′
=
−
γ
2
1

1,2
1
1
1,2
2
1

1
1
n

n
si
i
n
s n
x
U x
U
U

−
=
+
κ
−
∑
2

1,2
10
1
1
1
n

n
si
i
l
l
U x
=
+
∑
1,2
1,2
1
1,2
1,2
1,2
sin
n
n
n
n
n
x
x
U
x
−
′
′
=
+
γ
1,2
1,2
1
1,2
1,2
1,2
cos
n
n
n
n
n
y
y
U
x
−
′
′
=
−
γ
2

1,2
1
1
1
1,2
1
1

1
1
n

n
si
i
n
x
U x
U
+
+

=
+
κ
∑
2
1

1,2
1
10
1
1
1
n

n
si
i
l
l
U x
+

+
=
+
∑
1,2
1
1,2
1,2
1 1,2
1
1,2
sin
n
n
n
n
n
x
x
U
x
+
+
+
′
′
=
+
γ
1,2
1
1,2
1,2
1 1,2
1
1,2
cos
n
n
n
n
n
y
y
U
x
+
+
+
′
′
=
−
γ
(
)
*
10
11
12
1
...
m
x
l
x
x
x
=
+
+
+
γ x x′ y′ κ κ
γ
x
x′
y′
γ
x
x′
y′
γ
x
x′
y′
γ
x
x′
y′
0
/
U
u v
=
10
/
x
x
l
∗
∗
=
κ 1
κ =

2
κ =
2
x∗ ≈
0v 0
/
u v x∗ (
)
κ = ∞ γ 0
0
sin
v
γ 0
s
u =
0 cos
ld
v
d
γ =
γ τ 0
/
sin
th
u v
x∗
=
γ =
0 /
x
v
l
∗ =
τ
1
κ <
 γ κ § 2.6. Поперечный удар по гибкой нити телом
заданной формы

 (
)(
)
(
)(
)
0
1
1
1
1
1
1
1
1
1
1
1
cos
cos
sin
1
b
u
v
u
T
T
Q
e
′
′
′
′
ρ
+
β −
=
γ −
−
γ
+
(
)
(
)(
)
0
1
1
1
1
1
1
1
1
1
sin
sin
cos
1
b
u
v
T
Q
e
′
′
ρ
+
β =
γ +
γ
+
(
)

(
)

2
2
2
1
1
1
1
1
1
1

1
1

cos
sin

1
1

b
v
v
b
u
e
e

+
β
+
β
+
=
′
+
+

( )
1
1
0
( )
u
e
e
′
′
= ψ
− ψ
′′/ 2
β = π
0
v
v
=
0
1
1
v
b tg
=
γ (
)

0
1
1

1
1
1
1
1
sin
v
b
u
e
e
′
+
=
′
+
+
γ (
)(
)
0
1
1
1
1
0
1
1
1
1
1
sin
cos
sin
1
b
u
u
v
T
Q
e
ρ
′
′
′
+
γ +
γ
−
γ =
′
+
(
)(
)
0
1
1
1
1
0
1
1
1
1
1
cos
sin
cos
1
b
u
u
v
T
T
e
ρ
′
′
′
+
γ −
γ
=
γ −
′
+
(
)
(
)
0 0
1
1
0
1
1
1
1
1
1
sin
cos
sin
1
sin
v
u
v
T
Q
e
ρ
′
′
γ +
γ
−
γ =
+
γ
(
)
(
)
0 0
1
1
0
1
1
1
1
0
1
cos
sin
cos
1
sin
v
u
v
T
T
e
ρ
′
′
γ −
γ
=
γ −
+
γ
1
0
1
1
1
0
1

1
cos
sin
1
e
v
u
v
e
′
+
′
γ +
γ =
+

1
1
e
e
e
′
=
= 1
1
T
T
T
′
=
=
(
) ( )

2
0
0
2
1
1
sin
e T e
ρ υ
=
+
γ

T
e
− (
)
(
)
(
)
(
)

1
0
0
1
1

1
1
1

,

,
,

s
s

s
s
s
s

u
a
e
e
a
e
e

T
Ee
E
e
e
T
Ee
E
e
e

′
′
=
−
+
−
′
′
′
′
=
+
−
=
+
−
(
)
0
1
1
0
1
1
1

1
1
cos
sin
v
e
e
v
u

′
+
=
−
′
γ +
γ
(
)
(
)
0
1
1
1
1
1
0
1
1
1

cos
sin
cos
sin
1
sin
v
u
u
v
e
′
γ +
γ
′
γ −
γ
=
′
+
γ

(
)
(
)
2
2
2
2
2
2
2
0 1
1
0
1
1 1
1
0
1
1
0
1
1

1
cos
cos
sin
s
s
s
s
v a
e
a e
a e
a e
a e
a e
a
v
u
′
+
′
=
−
+
γ −
+
+
−
′
γ +
γ

1e′ 1u′ (
)(
)
(
)
2
2
2
2
1
0 1
1
1
1
1
0
1
1
[2
cos
1
1
cos
sin
1 cos
sin
s
s
u
v a
a
e
a e
′
γ −
+
+
γ
γ +
−
γ
γ −

(
)
2
0 1
1
1
0
1
0
1
1
2
cos
sin
]
[
cos
(1
cos
)
s
o s
a a
e
e
u a v e
′
−
γ
γ
−
+
γ
−
γ
−

(
)(
)
(
)
2
2
2
1
0
1
1
0 1 0
0
1
1
cos
cos
2
2
sin
s
s
a v
e
a a v
e
e
−
+
γ +
γ −
+
−
γ −

(
)
(
)(
)
2
3
2
1
1
0 1
1
1
1
sin
1
1 cos
sin
s
s
s
a
e
a a e
e
−
+
γ +
+
−
γ
γ −

(
)(
)
(
)
(
)(
)
2
3
0 1
0
1
1
0
0
1
1
1
sin
1 cos
1 cos
sin
s
s
s
s
a a
e
e
e
a e
e
e
−
−
+
γ
+
γ
+
−
−
γ
γ −

(
)2
2
2
2
1 0
0
1
1
1 0
1
1
cos
sin
ctg
(1 3sin
)]
s
a a
e
e
a v
−
−
γ
γ +
γ
−
γ
+

(
) (
)
2
3
3
3
1
0
1
1 0
1
[
1
1 cos
cos
s
a v
e
a v
+
+
−
γ
−
γ +

(
)
(
)
2
0 0 1
1
1
1
cos
1 cos
s
s
a v a e
e
+
+
γ
−
γ
−

(
)(
)(
)
(
)2
2
2
2
2
1
0 0
0
1
1
0 0 1
0
1
1
cos
cos
2
sin
s
s
s
a a v
e
e
e
a v a
e
e
−
−
+
γ +
γ −
+
−
γ +
(
)(
)
3
0 0
0
1
1
1 cos
cos
]
0
s
s
a v e
e
e
+
−
−
γ
γ
=
1
u′ 1
1
1
,
,
,
e e T T
′
′ γ (
)

0

0
0
arctg
1
v
a
e
+
(
)
(
)(
)
0
0
1
1
0
0
sin
cos
1
b v
u
T
T
e
ρ
γ −
=
−
γ
+
(
)(
)
0
0
0
0
cos
sin
1
bv
Q
T
e
ρ
γ =
+
γ
+
(
)
(
)
1
1
0
1
1
.
cos
b
b
e
u
e
+
=
−
+
γ
(
)
(
)(
)
1
0
0
1
0
0
0

1
sin
cos
1
1
cos
b
e
b
b v
T
T
e
e
+
ρ
γ +
−
=
−
γ
+
+
γ
(
)
(
)

(
)
[
]

2
2
0
1
0
1
0
2
0
2
0
0
1
2

1
sin
1
1
cos
;
cos
cos
1

1
1
cos
cos .
1

a
e
e
e
e
e
b

e
e
e

+
γ −
+
=
+
−
γ
γ
γ
+

−
+
γ λ +
γ
= −
λ −

(
)
[
]
0
0
0
/
/
1
a
b
a
e
λ = λ
=
+
1
λ > (
)
[
]
2
0
0
1
1
cos
cos
e
e
−
+
γ λ +
γ (
)
[
]
2
0
1
1
cos
e
−
+
γ λ +

0 cos
0
e
+
γ >
(
)(
)
(
)

(
)(
)

2
2
2
2
0
0
0
1
2
0

1
1
cos
cos
1
cos

1
1 cos

e
e
e
u
b
e

+
λ −
+
γ +
γ −
+
λ
γ
=
+
λ −
γ

(
)

0
0

0
0
0

ctg
ctg
1
1
v
v
a
e
e
γ
γ
λ =
=
+
+
(
)
(
)
(
)

(
)

2
2
2
2
2
0
0
0
0
0
1
2
2
2
0
0
0

sin
1
sec
1
sec
1

1
tg

v
e
e
e
e
u
v
v
e

γ
−
+
γ +
+
γ +
+
=
−
+
γ

0
tu ≠
0
tu =
(
)
0
1
1 1
2
1
a
u
u
T
T
−
ρ
=
−
1
1
T
Ee
=

(
)
1
0
1
/ 1
e
ρ = ρ
+
(
)
(
)
2
1
1
1
1
2
0
0
1

1
1
1
u
u
e
e
T
a
e
−
+
+
= ρ
+
1
1
0

u
u
a
=
0
a
λ >
(
)
0
0
1
1
1,
.
b
a
e
e
→
+
λ →
→ ∞ (
)
0
0
1
a
e
+
(
)
0
0
1
b
a
e
>
+
(
)
1
s
s
T
T
E
e
e
=
+
−
(
)(
)
2
2
1
0 0
1 1
1
0
0
2
0

1
cos
sin
1
cos
cos
1
1
sin
s
s
e
v
T
E e
E e
T
e
e
+
γ ρ
γ − +
γ =
+
−
−
γ
+
+
γ (
)
[
]
(
)(
)2
2
2
0 0
1
0
1 1
1
0
0
ctg
1
1
cos
cos
1
.
s
s
v
e
e
T
E e
E e
T
e
ρ
γ
+
−
+
γ =
+
−
−
γ
+

(
)
(
)

(
)

2
2
2
2
0
1
0
0
0
0
1
2
2
2
2
0
0
1

cos
1
1 cos
cos
ctg

ctg
1

s
s
T
T
a e
e
e
v
e
v
e
a

−
γ
−
+
−
−
γ −
γ
γ
ρ
=
γ −
+

(
)
0
0
1
b
a
e
→
+
0

0
(1
)
V
a
e
>
+

2
0
2
2
1 cos cos V
a
−
>
2
2

2
2
0
0

1
1
,
.
u
T
w
T
x
x
t
t

∂
∂
∂
∂
=
=
ρ
∂
ρ
∂
∂
∂

2
0
1
;
,
1

t
t
u
u
p
T
u
r
e
t
ρ
∂
=
−
=
+
∂
(
)(
)
(
)(
)
0
0
1
1
1
cos
sin
cos
1
w
u
w
b
v
T
T
e
t
t
t
∂
∂
∂
′
′
ρ
−
−
γ −
γ =
γ −
+
∂
∂
∂

(
)(
)
(
)(
)
0
0
1
1
cos
sin
sin
1
w
w
b
v
Q
T
e
t
t
∂
∂
′
′
ρ
−
γ −
γ =
+
γ
+
∂
∂

(
)(
)
(
)
1
1
1
1
cos
w
b
u
b
e
e
t
t
∂
∂
′
−
+
=
−
+
∂
γ
∂
1
1
1
1
,
,
,
T
e T e
′
′
0
tg
/
v
b
γ =
γ (
)
(
)

(
)

2
2
0
0
0
2
2
0

r
v t v
d
b
r
r
v t
dt
r
r
v t

−
=
−
−
=
−
−

b = ∞ ,x t γ 0
a
λ >
(
)
( )

2
2
2
0
1
1
2
2
2
0
0

sin
sec
sec
1 cos
;
.
1
tg

v
u
e
t
v
v

γ
−
γ +
γ
−
γ λ
= −
=
= ϕ
λ −
−
γ

0
=
cos
1
γ ≈
0
=
0
=
(
)
(
)
1
0
2
0
u
f
x
a t
f
x
a t
=
−
+
+
( )
(
)2
2
0
x t
r
r
v t
=
−
−
( )
[
]
( )
[
]
(
)

( )
[
]
( )
[
]
( )

2

1
0
2
0
2

1
0
2
0
0

1 cos
1
;
1
f
x t
a t
f
x t
a t

f
x t
a t
f
x t
a t
v
t

−
γ λ
′
′
−
+
+
− = −
λ −
′
′
−
−
+
+
=
ϕ
( )
( )

( )

2 2
0
0
0
0

2 2
0
0
0

2
;

2
.

x t
a t
v tr
v t
a t
t

v tr
v t
a t
t

−
=
−
−
= ξ
−
+
= η
( )
(
)
( )
( )

( )
(
)
( )
( )

2

2
0
1
2

2

1
0
2
2

1
1 cos
1
;
2
1

1
1 cos
1
.
2
1

f
v
t
t

f
v
t
t

−
γ λ
′ η =
−
+
ϕ
= ψ
λ −
−
γ λ
′ ξ =
−
−
ϕ
= ψ
λ −
ξ η

(
)
(
)

( )

(
)

(
)

( )

2
2
0
0
0
0
2
2
2
2
2
2
2
0
0
0
0
0
0

2
2
0
0
0
0
2
2
2
2
2
2
2
0
0
0
0
0
0

;

.

v r
a
v r
a
t
t
a
v
a
v
a
v

v r
a
v r
a
t
t
a
v
a
v
a
v

ξ

η

−
ξ
−
ξ
ξ
=
−
−
=
ξ
+
+
+
+
η
+
η
η
=
−
−
=
η +
+
+
( )
( )
( )
( )
1
2
2
1
;
f
t
f
t
ξ
η
′
′
ξ = ψ
ξ
η = ψ
η
(
)
(
)
(
)
(
)
1
0
2
0
2
0
1
0
;
.
f
x
a t
t
x
a t
f
x
a t
t
x
a t
ξ
η
′
′
−
= ψ
−
+
= ψ
+
(
)
(
)
3
0
4
0
,
u
f
a t
x
f
x
a t
=
−
+
+

 0
x
a t
=
( )
(
)
( )
(
)
1
2
0
3
4
0
0
2
0
2
f
f
a t
f
f
a t
+
=
+
(
)
(
)
(
)
(
)
2
0
4
0
4
0
2
0
2
2
;
;
f
a t
f
a t
f
x
a t
f
x
a t
′
′
′
′
=
+
=
+

0
x =
(
)
(
)
(
)
(
)
3
0
4
0
3
0
2
0
;
.
f
a t
f
a t
f
a t
x
f
a t
x
′
′
′
′
= −
−
= −
−
1
1
OA C (
)
2
0
f
x
a t
′
+
1
OC D 0
x =
(
)
(
)
(
)
(
)
(
)
3
0
4
0
2
0
0,
0,
1
1
2
1
x
e
t
u
t
f
a t
f
a t
f
a t
′
′
′
=
− = −
+
− =
− (
)
( )
( )
[
]
( )

( )

2

0
2
1 cos
,
1
e o t
v
−
γ τ λ
τ
=
ϕ τ −
λ
τ −

τ (
)
(
)
(
)

2
2 2
2
0
0
0
0
0
0
2
2
0
0
1
1

a t
a t
a t
r
v
v
v
r
r
a
v
r

τ =
+
−
+
−
+
+
(
)

2 2
2
0
0
0
0
0
sin
; cos
1;
2
;
;
0,
.
a t
v r v
e
t
v
v
γ ≈ γ
γ ≈
τ =
≈ γλ
≈
γ ≈
λ

( )
0

0
2

v
t
v rt
λ
=
(
)
0
0
0 0
0
0,
2
e
t
v
v
r
v a t r
v t r
≈
τ
≈
=
0
x =

0
x =
0
b
a
>
0
n
Q 0
n
Q
∗
µ
∗
µ 0
n
Q  
γ§ 2.7. Поперечный удар по проволочным канатам

0s 2

0
v
e
r
s
∂
=
+
∂
2

0
v
r
e
s
∂
=
−
∂
2
2

0
0
1
1
u
h
e
s
s
∂
∂
=
+
+
−
∂
∂
(
)
0,
u s t (
)
0,
h s t (
)
0,
v s t τ•(
)
(
)

2
2
1
1
r
r

+
+
=
−
•τ(
)
1
2
2
r
r
+
=
+
−
(
)

2

0

1
sin 22
2
v
e
s
r
∂
=
+
−
∂
3
4

0

1
v
e
r
s
∂
=
+
∂
(
)

2
2
2
1
1
2
cosi
i i
i
i
pi
i
i
V
EF
EI
GI
=
+
+
1

k

i
i
V
V

=
= ∑
2
2
2
1 2
i
u
h
v
K
m
t
t
t

∂
∂
∂
=
+
+
∂
∂
∂
2
2
2

0
0
2
t
l
u
h
v
R
dt
m
V
ds
t
t
t

∂
∂
∂
=
+
+
−
∂
∂
∂
∫ ∫
2
0
1
3
2
0
0

1
1

u
s
u
v
A e
A
s
s
e
t

∂
+
∂
∂
∂
∂
=
+
∂
∂
+
∂
2
0
1
3
2
0
0
1

h
s
h
v
A e
A
s
s
e
t

∂
∂
∂
∂
∂
=
+
∂
∂
+
∂
2

2
3
2
0
0

v
v
m
A
A e
s
s
t
∂
∂
∂
=
+
∂
∂
∂
(
)

3
2
4
2
4
1
2
1

cos
sin k
i
i i
i
i
pi
i
i
i
i
A
EF r
EI
GI
r
α

=
=
+
+
∑
(
)
(
)

2
2
2
2
4
2
2
1
1
cos k

i i
i
i
pi
i
i
i
i
i
A
EFr
EI
GI

=
=
+
+
+
∑
(
)

2
2
2
2
2
3
1

sin 21 cos 4

k
i
i
i
i i
i
i
i
pi
i
i
A
EFr
EI
GI
r
=

=
−
+
+
∑
0

u
u
V t
=
0

h
h
V t
=
0

v
v
V t
=
0

0

s
z
V t
=
(
)

2
1
3
1
3
1
1
1
e
v'
e
v'
d
u '' z
u '
e
dz
e
∆
+ ∆
∆
+ ∆
−
=
+
+
+
2
1
3
1
3
1
1
''
'
e
v'
e
v'
d
h
z
h
e
dz
e
∆
+ ∆
∆
+ ∆
−
=
+
+
(
)

2
2
4
de
v '' z
dz
− ∆
= ∆
1
1
2
0
0

A
V
ρ
∆ =
2
2
2
0

A
mV
∆ =
3
3
2
0
0

A
V
ρ
∆ =
3
4
2
0

A
mV
∆ =
0
1 0
2
u
s
C s
C t
+
=
+
3 0
4
h
C s
C t
=
+
5 0
6
v
C s
C t
=
+
0
0
7
u
s
C h
C t
+
=
+
2
2
2
2
0
3
1
2
1
2
1,2
4
1
2
s
A
A
A
A
A
a
t
m

=
=
+
±
−
+
1a 2
1
a
a
<
A B
′ ′ 1a B C
′ ′ 2
1
a
a
<
0
0
s =
2
2
2

1
3
2
2
2
0
0
u
u
v
A
A
t
s
s
∂
∂
∂
=
+
∂
∂
∂

2
2
2

1
3
2
2
2
0
0

v
u
v
m
A
A
t
s
s
∂
∂
∂
=
+
∂
∂
∂

2
2
2
1
1
1
2
2
0

a
t
s
∂
∂
=
∂
∂

2
2
2
2
2
2
2
2
0

a
t
s
∂
∂
=
∂
∂

1
1
u
v
=
+
2
2
u
v
=
+
2
2
3
2
1
2
1
12
3

4
2
A
A
A
A
A
m
A
m
m
m

=
−
±
−
+
1
3
0
0
(0, )
(0, )
u
v
A
t
A
t
T
s
s
∂
∂
+
=
∂
∂

3
2
0
0
(0, )
(0, )
0
u
v
A
t
A
t
s
s
∂
∂
+
=
∂
∂

0
0
0
0
(
, )
(
, )
(
, )
(
, )
0
u
v
u s t
v s t
s t
s t
t
t
∂
∂
=
=
=
=
∂
∂

0
0,
0
t
s
=
>
2
2
0
1
2
3

TA
u
s
A A
A
∂
=
∂
−

3

0
2
0

A
v
u
s
A
s
∂
∂
= −
∂
∂

0

u
u
k
t
s
∂
∂
=
∂
∂

1
0

v
u
k
t
s
∂
∂
=
∂
∂

2
1
2
2
1
3
1
2
2
1

2
1
2

( (A a
a
A
a
a
k
A
+
+
−
=
−
3
1
1
2
2
2
2
1
1
2
1
2

( (A a
a
A a
a
k
A
+
+
−
=
−
1
2
,
a
a (
)
2
2
2
1
0
2
1
2

2
2
1
1
2
1

l
l
l
F
b
T
T F
t
t
t

l
l
F
b
F b
t
t

∂
∂
∂
−
−
=
−
∂
∂
∂
∂
∂
−
=
−
∂
∂
(
)(
)
0
2
1
1
u
u
b
T
T
e
t
t
∂
∂
+
=
−
+
∂
∂
(
)
0
0
2
2
u
b
V
T
e
t
∂
+
=
+
∂
0
tg V
b
=
1
2
T
T
T
=
=
1
2
e
e
e
=
=
u
ke
t
∂
=
∂

0
tgV
b
=
2

0 (1
u
b
A e
e
t
∂
+
=
+
∂
secu
b
b
t
∂
+
=
∂

2
1
2
3
0
0
2

1
A A
A
A
A
−
=
2
3

2
A r
A
=
−
0
V (
)
0
1
b
A e
e
ke
=
+
−
(
)

2
2
0
0
2
1
V
ke A e
e
k e
=
+
−
4
3
0

2
3
0
4

V
e
k A
≈
2
1
3
3
0
0

3 2

V A
b
k
≈
1
3

1
3

3
0

0

2
tgk V

A
≈
§ 2.8. Применение асимптотических методов
для решения задач распространения волн
в нитях при воздействии движущихся тел

0
/
u
x
t
u
const
= ∂
∂ =
=
0
/
const
x
s
µ = ∂
∂ = µ =
/
0
v
y
t
= ∂
∂ =
/
0
y
s
ν = ∂
∂ =
0
0
0
/
t
s
a
< <
0
u
u
≡
0
v ≡
χ
0
=
0
µ ≡ µ 0
0
s >
µ0
ds
a
da
= ±
(
)
cos
sin
cos
sin
du
dv
a
d
d
ϕ
+
ϕ
= ±
ϕ µ +
ϕ χ 0
ds
da = ±λ (
)
cos
sin
cos
sin
dv
du
d
d
ϕ
−
ϕ
= ±λ
ϕ χ +
ϕ µ 0
ϕ =
0
0
s
a t
=
0
ds
dt = ±λ ( )
( )
0
A
B
ϕ
= ϕ
= ( )
( )
(
)
( )
( )
( )
0
0
0
,
,
0,
0,
v P
P
v P
P
P
= ±λ χ
λ = λ µ
= χ
=
ϕ
=
0
0
ds
da
a
= −
( )
( )
[
]
0
0
0
0
;
.
du
a d
u P
a
P
u
= −
µ
= −
µ
− µ
+
( )
( )
[
]
0
0
0
s
s
P
a
t
t P
−
=
−
( )
0
ds
dt
P
= λ
(
)
0
ds
dt
Q
= −λ
( )
(
)
P
Q
µ
= µ
( )
(
)
P
Q
λ
= λ
( )
( )
v R
R
= χ
=
( )
0
R
= ϕ
=
( )
( )
[
]
0
0
0
u R
a
R
u
= −
µ
−µ
+
0
0
s
a t
=
( )
0
0
s
s
t
∗
=
0 /
ds
dt
∗
= λ ( )
0 0
0
s∗
=
′0
0
a
e
λ ∼
0
0
0
s
a t e
∗ ∼
cos
1
ϕ ∼
0 0
u
a e
∼
(
)
(
)
(
)
(
)

0

0

3/ 2
0
0
0
0
0
0
0
,
,
,
,
,
~
.

s

s

u
T s t
T s t
ds
e s t
e s t
e
t
∗

∗
∗
∂
−
=
ρ
−
∂
∫

(
)
0
0
,
~ (
, )
e s t
e s t
∗
(
)
0 0 0
/
x
x
a t e
=
(
)
0 0
/
y
y
v t
=
0
T
Ee e
=
( )
0
0
/
z
s
s
t
∗
=
0
/
t
t t
=
*
0
0
0 0
s
a
e t
=
0
0
e →
2
1/ 2
0
2
x
e
e
z
t
∂
∂
= ∂
∂

0
0
e →
( )
0,
e
e
e t
z
∂
=
=
∂
2
2

2
2
( )
y
y
e t
t
z
∂
∂
=
∂
∂

2
2
2
2
2
1/ 2
1/ 2
0
0
0
0
0
2
2
0
0
0
0

1
1
1
2
v
v
x
y
x
y
e e
e
e
z
z
z
z
a e
a e
∂
∂
∂
∂
=
+
+
− ≈
+
∂
∂
∂
∂
1/ 2
0
x
e
z
∂
∂ 0e e 2
2
0
2
0
0

1
2
v
y
z
a e
∂
∂
 0
x
z
∂
∂ =
0
y
z
∂
∂ =
3/ 4
0
0
0
0
~
.
v
v
a
e
=

2
2
0
2
3/ 2
0
0
2
v
x
y
z
z
a e
∂
∂
= −
∂
∂
0y e 0x x0y e y 0
0
µ =
1
m ≥
2
0
0
n
ds
a
Bt
dt

∗
=
1
2
0
0
1
2

n
a
Bt
s
n

+
∗ =
+

(
)
( )
0
0
0,
y s
t
y
t
=
=
(
)
0
0,
0
y s
s t
∗
=
=
(
)
0,
0
0
y s
t
=
=
=
( )
0
0
/
z
s
s
t
∗
=
2
2
(
1)
(1
2)(2
2
2)
(1
2) (1
)
m m
f
n
m
n
zf
n
z
f
′
′′
−
−
+
−
−
=
+
−
1
z → (
) (
)
2
/ 2 / 2
N
m
n
n
=
+
+
′(
)
(
) (
)

2
2
2
2
1
/ 2
2
0
0

1
1
2
2

m
n
x
A
n
t
f
s
a B

− −
∂
′
= −
+
∂
( )
(
)

2
2
1
/ 2
2
0
0
0
1
2
2

z
m
n
A
n
x
x
t
t
f
dz
a
B

− −
′
=
−
+
∫
(
)(
)

(
)
(
)

2
2
2
/ 2
2
0
0
0
2
2
/ 2
2

0

1
/ 2
2
1
/ 2
2

1
/ 2
.
2

z
m
n

m z n

x
A
n
m
n
x
t
f
dz
t
a
B

A
n
t
z f
a
B

− −

− −

∂
+
− −
′
′
=
−
+
∂

+
′
+

∫

0
0
x′ =
1
z = ( )
(
)(
)
1
2
2
2
/ 2
2

1
0
0

1
/ 2
2
1
/ 2

2

m
n

z

x
A
n
m
n
t
f
dz
t
a
B

− −

=
∂
+
− −
′
= −
∂
∫

0 ( )
a e t
−
2
2
/ 2
m
n
n
−
−
=
(
)
4
1 /3
n
m
=
−
4 3
(
)
e
y′
∼
1
m >
(
) (
)
1
2
1 / 2
1
0
N
m
m
− =
−
+
>
1
z = 1
m >
1
m >
1
z = (
)(
) 1
2
2
0
0
0

1
2
4
1

18

A
m
m
f
dz
a B
a
B

+
−
′
=
∫
(
)(
)

1
2
3 2
2
2
0
0
18
1
2
4
1
a B
m
m
A
f
dz
′
=
+
−
∫
( )
(
)
(
)(
)
(
) (
)

2
2
4
1
2
1
1
2
1
1
0
3
9
m
L f
m m
f
m
m
zf
z
f
+
′
′′
=
−
−
+
−
−
−
=

( )
(
)
(
)(
)
1
1
1
1
N
N
f z
A
z
A
z
+
=
−
+
−
−
0
z =
1
z = 1
z → 1

0
( )
0,
L f dz =
∫
2

2
6
(23
11
2)

(2
1)(73
2)
m
m
m
A
m
m
m

+
+
=
+
+
−

0
m
y
t
∼
1
0
n
x
t +
∼
(
)
4
1 / 3
n
m
=
−
0
0
T
Ee
=
(
)
0
0
x
e
s
s
x
=
−
+
2
2
2

0
0
1
1
x
y
z
e
e
s
s
s
∂
∂
∂
+
+
+
+
− −

∂
∂
∂
/
x
s
∂
∂ /
y
s
∂
∂ /
z
s
∂
∂ (
)

2
2

0

1
2 1
x
y
y
s
e
s
s

∂
∂
∂
+
+
∂
+
∂
∂
(
)

0

0
cos
,
1
Ee
y
T
j T
e
s
∂
≈ +
∂ (
)

0

0
cos
,
1
Ee
z
T
k T
e
s
∂
≈ +
∂

(
)
(
)

2
2

0
2

0

1
cos
,
2 1

x
y
z
T
i T
Ee
E
s
s
s
e

∂
∂
∂
≈
+
+
+
∂
∂
∂
+
2
2
2
0
2
2
y
y
b
t
s
∂
∂
=
∂
∂

2
2
2
0
2
2
z
z
b
t
s
∂
∂
=
∂
∂

(
)

2
2
2
2
0
2
2
0

1

2 1

x
x
y
z
a
s
s
s
s
t
e

∂
∂
∂
∂
∂
=
+
+
∂
∂
∂
∂
∂
+
(
)

2
2
0
0
0
0
0
0
0
1
1
Ee
a e
b
e
e
=
=
ρ
+
+

2
0
0
/
a
E
=
ρ ε ε x (
)
1
2
...
x
X
X
= ε
+ ε
+
(
)
1/ 2
1
2
...
y
Y
Y
= ε
+ ε
+
(
)
1/ 2
1
2
...
z
Z
Z
= ε
+ ε
+
(
)
1
2
...
E
E
ε
+ ε
+
0
s =

0( )
y
y t
=
0
0
0
( ),
/
( ),
y
y
s
dy
dt
v
s
=
=
s
L
=
0
0
( )
( )
0
y
s
v
s
=
≡
0
t
t
=
0b
s
≤
< ∞ 0
x
x
=
/
x
t
∂
∂
0
u
=
x /
x
s
∂
∂ /
x
t
∂
∂ 0
s
b t
=
0a t
s
≤
< ∞ 0
s
b t
=
0( )
0
x
s ≡
0( )
0
u
s ≡ ( , )
0
x s t ≡ 0
0
s
b t
≤
≤
( , ), ( , )
y s t
z s t 0( )
x
x t
=
0
s
b t
=
0
s
b t
=
(
)

2

0

1
0
2 1
x
y
s
e
s

∂
∂
+
=
∂
+
∂
(
)

2
0

0
2 1
b
x
y
t
e
s
∂
∂
=
∂
+
∂
0
s
b t
=
x
s
∂
∂
x
t
∂
∂
0
s
b t
=
s
L
=
2
2
2
x
y
v
r
s
s
s
∂
∂
∂
=
+
+
+
+
−
∂
∂
∂
2
tg1

v
R
e
s
e

∂ −
∂
=
+

2
2
1
1
x
y
e
s
s
∂
∂
=
+
+
−
∂
∂
( , ),
( , )
x s t
y s t ( , )
v s t 1
2
3
,
,
A
A
A

•2
3
0
A
A
=
=
1
A
E F
=
•(
)

3
2
4
2
4
1
2
1

cos sin i

k
i
i
i i
i i
i
i p
i
i
i
i
A
E Fr
E I
G I
r
=
=
+
+
∑
(
)
(
)

2
2
2
2
4
2
2
1
cos i
i
i
i
p
A
E F r
E I
G I
=
+
+
+
(
)

2
2
2
2
2
3
1

sin 21 cos 4
i

k
i
i
i
i
i i
i i
i
i
i
p
i
i
A
E Fr
E I
G I
r
=

=
−
+
+
∑
ir 1
1
1
=
+
2
2
=
3
3
=
0e 2
0
0 cos oe
e
=
1
2
1
1
2
(
...)
y
e
Y
e Y
=
+
+
1
2
1
2
...
x
e X
e X
=
+
+
1
2
1
2
...
rv
e rV
e rV
=
+
+
0
e
e
e
=
−
0
x
e s
x
=
+
0
x
e
s
∂
<<
∂

0
0
,
y
v
e
e
s
s
∂
∂
<<
<<
∂
∂

0
x
e
e
s
∂
=
+
+
∂

(
)

2

0

1
2 1
y
e
s
∂
+
+
∂
2
2
2

0
tg2(1
)
v
r
v
e
r
s
e
s

∂
∂
ε =
+
+
∂
+
∂
2
2
1 0
2
2
0
1
A e
y
y
e
t
s
∂
∂
= +
∂
∂

2
2

1
3
2
2
x
e
v
A
A
s
t
s
∂
∂
∂
=
+
∂
∂
∂

2
2

3
2
2
2
v
e
v
m
A
A
s
t
s
∂
∂
∂
=
+
∂
∂
∂

2
2
2
2
2
y
y
b
t
s
∂
∂
=
∂
∂

(
)

2
2
2
2
1
1
3
2
2
2
0
2 1
A
x
x
y
v
A
A
e
s
s
t
s
s
∂
∂
∂
∂
∂
=
+
+
+
∂
∂
∂
∂
∂
(
)

2
2
2
2
3
3
2
2
2
2
0
2 1
A
v
u
y
v
m
A
A
e
s
s
t
s
s
∂
∂
∂
∂
∂
=
+
+
+
∂
∂
∂
∂
∂
2
1 0

0
(1
)A e
b
e
=
+
1a 2a 2
2
2
3
1
2
1
2
1,2
4
1
2
A
A
A
A
A
a
m
m
m

=
+
±
−
+
ρ
ρ
ρ
2

0

1
0
2(1
)
x
y
s
e
s

∂
∂
+
=
∂
+
∂
2

0
2(1
)
x
b
y
t
e
s

∂
∂
= ∂
+
∂
2
2

0
tg2(1
)
x
v
r
v
r
s
s
e
s

∂
∂
∂
+
+
=
∂
∂
+
∂
2
2

0
tg2(1
)
x
v
ar
v
ar
t
s
e
s

∂
∂
∂
=
+
∂
∂
+
∂
(
)
0

0
,
1
e
k
n
m
n
m
e
=
−
≥
+
(
)
0

0
,
1
e
k
m
n
n
m
e
=
−
<
+

(
)
0

0
1
e
k
n
m
e
=
+
+

(
)
0
1

2
0
,
1
e
A
m
k
n
m
n
m
A
e
=
−
≥
ρ
+
(
)
0
1

2
0
,
1
e
A
m
k
m
n
n
m
A
e
=
−
<
ρ
+
(
)
0
1

2
0
,
1
e
A
m
k
n
m
n
m
A
e
=
−
≥
ρ
+
§ 2.9. Некоторые приложения теории
продольнопоперечного удара

0v γ γ
0v ( )
0
0
v
v
=
γ 2
0
0

0
(
)
(1
) ( );
(
)cos ;
tg ;
( )
.

e

e
b
u
e T e
b
b
u
v
b
u
a e de
ρ
+
=
+
=
+
γ
=
γ
= ∫

2
0
0
0
2

0

(1
cos )
( )
(1
) ( )
( );
( )
( ).
sin
sin

e

e

v
v
e T e
f
a e de
ρ
−
γ
γ
+
=
=
γ
=
= ϕ γ
γ
γ
∫
2

0

1 dT
a
de
=
ρ
γ 0

1 dT de
d
de d
d
ϕ
=
ρ
γ
γ

( )

2
2

0

1
1
f
d
de
d
de
e
d
d

γ
ϕ
=
ρ
+
γ
γ
(
)
( )

(
)

2
2

2
0

1

1

df
e
f
de
d
de
d
d
e

+
−
γ ϕ
=
γ
γ
ρ
+
( )
(
)

2
2

0 1
1
f
df de
de
d
e
d
d
e
d
d

γ ϕ
−
= ρ
+
γ
γ
+
γ
γ
f f ′ ′
ϕ de
dγ (
)(
)

2
2
2

0
0
0
2
2
0
0

1
cos
1
1
2
ctg
sin
1
sin
e
v
dv
de
de
v
v
d
e d
d
v

+
γ −
′
−
γ
−
=
+
γ
γ
+
γ
γ
γ
(
)
,
e
e
c
=
γ
0γ 0γ 0
(
, )
se
e
c
=
γ
2
0
0
0
2
0

(
)
(1
)
sin
s
s
v
e
Ee
ρ
γ
+
=
γ
0
0
0
0
0
0

1 cos
(
)
(
)
sin
s
E e
e
v
−
γ
−
=
γ
ρ
γ
γγγ 0v 0e 4 3

0
3
0

1

4

v
e
a
=
0a 0
s =
s
L
=
yOx

1
( )sin
n
n

ns
y
t
l

∞

=

π
=
ϕ
∑
0
( )
cos
sin
;
.
n
n
n
n
n
n
nb
t
a
t
b
t
l
π
ϕ
=
ω
+
ω
ω =
( )

1
cos
n
n

y
n
ns
t
s
l
l

∞

=

∂
π
π
=
ϕ
∂
∑
2
y
s
∂
∂
( )
( )

2

2
cos
cos
n
m
nm
ns
ms
t
t
l
l
l
π
π
π
ϕ
ϕ
=

( )
( )
(
)
(
)

2

2
cos
cos
2
n
m
nm
s
s
t
t
n
m
n
m
l
l
l

π
π
π
=
ϕ
ϕ
+
+
−
( )
( )
n
m
t
t
ϕ
ϕ
(
)
cos
n
m t
ω ± ω
(
)
sin
n
m t
ω ± ω
n
m
k
+
=
(
)
cos s n
m
l
π
+
in i
m i
i
n
m
k
+
=
2
y
s
s
∂
∂
∂
∂
( )sin
k
k s
t
l
π
Φ
k
Φ ( )
( )
i
i
m
n
t
t
ϕ
ϕ
0b 0a ( )sin(
)
k
k
x
f
t
ks l
=
π
( )
k t
Φ
( )
kf
t

cos
kt
ω
sin
kt
ω
τ 3
65 l =
3
65 l =
2l 1l

1
0,7d =
2
1 d =
3
0
=
2280 =
1420 =
1
2
3
65 l
l
l
=
=
=
3
65 l =
1
65 l =
1
3
0,87
A
A =
1
2
3
1

na
kb
l
l
l
l
=
+
+
0

1
2
3
1
0
1

e k
n
l
l
l
l
e
=
+
+
+

1
2
3
l
l
l
L
+
+
=
1,2,...
n ∈
1,2,...
k ∈
0e 1l 1
n =
1,2,3
k =
380 L =
1
=
2
=
§ 2.10. Поперечные колебания балок под действием
 динамических нагрузок

ab cd a b′ ′ c d′ ′ acdb OO′ ρ OO′ y OO′ (
)
.
x
y
y
e
+ ρ δα −ρδα
=
=
ρδα
ρ

δα σσσσ′

′

σOx Oz Oy Oz Oy Oz Oz(
)
x
X
y dF
′
=
σ∫
0
Y =
0
Z =
(
)
(
)
x
x
y
y
′
′
σ
−
= −σ
0
=
0
X =
Oz Ox Oy σOx Oy ( )
y
x
M
z
y dF
=
σ
∫
xOy x
M
y dF
=
σ
⋅
∫
ρ
1
2
,
, ...,
n
P P
P 1
2
,
, ..., n
l
l
l ( )
(
)
1
1
1
0

x
k

i
i
Q x
P
q x
dx

=
=
+
∑
∫

( )
(
)
(
)(
)
1
1
1
1
0

x
k

i
i
i
M x
P x
l
q x
x
x
dx

=
=
−
+
−
∑
∫
( )
Q x
dM dx
=
( )
Q x x x
dx
+
(
)
(
)
( )
(
)
(
)

2
2

2
2
,
,
y
M
F
dx
q x t dx
Q x
dx
Q x
q x t
dx
t
x

∂
∂
ρ
=
+
+
−
=
+
∂
∂
(
)

2
2

2
2
,
y
M
F
q x t
t
x
∂
∂
ρ
=
+
∂
∂

ρ F x x
x
dx
+
(
)
( )
,
.
Q x
x
Q x
+ ∆
h b σ(
)
x
x
x
s
x
s
x
x
s
Ee
e
e
E
E e
E e
e
e
′
′
σ =
≤
σ =
−
+
≥

x
M
b
ydy
=
σ∫
2

0
2

h

x
M
b
ydy
=
σ
∫
2
2
1
y
x
κ =
ρ ≈ ∂
∂
2

2
x
y
e
y
y
x

∂
= −
= − κ
∂

sy (
)
3
2
3
2
3
2
2
3
2
4
3
8

s
s
s
s
s
b Ey
y b E
E e
h
b E
h
M
y
y
′
′
κ
−
κ
κ
= −
−
−
+
−
3
;
(
)
;
12
s
s
e
y
E
E
E J
bh
′
= −κ
λ =
−
=
J ( )
( )

3
1
4
3
s
s
y
y
M
EJ
h
h
= −
κ
−λ − λ
+ λ
κ x λ 0
x =
ρ 2
2

2
2
y
M
F
t
x
∂
∂
ρ
=
∂
∂
M 2
2
y
x
κ ≈ ∂
∂
M
EJ
=
κ /
y t 2 /
x
t η2

2
1
4
x
t
a
η =
2
(
)
a
EJ
F
=
ρ
( )
y
tf
=
η (
)

2

2
2
1
2
2
y
f
f
x
a
∂
′
′′
κ =
=
+ η
∂

2
2

2
y
f
t
t
∂
η
′′
=
∂

η3
2
2
2
a
a
dM
S
tQ
EJ
EJ
d
=
=
η
η

3 2
2
0
S
f
′
′′
+ η
=
(
)
1 2
2
2
0
S
a k
f ′′
+ η
−
=
0
d
S
EJS
dM
κ
′′+
=
d
dM
κ
η( )
( )

3 2
const
2
S
y
tf
t d
d
′ η
=
η = −
η
η+
η
∫
∫

( )

3 2
2
S
y
t d
d

∞
∞

η
ζ

′ ξ
= −
ζ
ξ
ξ
∫
∫
( )
( )
( )

3 2
1 2
3 2
2
2
2
S
d
S
d
S
d
t
t
y
t
d

ξ
∞
∞
∞

η
η
η
η

′
′
′
ξ
ξ
ξ
ξ
ξ
ξ
= −
ζ = −
+
η
ξ
ξ
ξ
∫
∫
∫
∫
η(
)
,0
0
y x
=
( )

1 2
0
(0, )
2

S
t
y
t
d

∞
′ ξ
= −
ξ
ξ
∫
( )

1
1 2
0

1
2
S
v
d

∞
′ ξ
= −
ξ
ξ
∫
( )

2
3 2
4
S
y
x
d
x
a

∞

η

′ ξ
∂ =
ξ
∂
ξ
∫
(
)
,0
0
y x
x
∂
=
∂

0
x =
( )

3 2
0
lim
S
d
A

∞

η→
η

′ ξ
ξ
η
=
ξ
∫
( )
S′ ξ (
)
0,
2
y
t
A t
x
a
∂
=
∂

0
A ≠
( )
( )
( )

3 2
2
2
S
S
S
d
d

∞
∞
∞

η
η
η

′
′
′′
ξ
ξ
ξ
η
ξ = −
η +
η
ξ
ξ
ξ
ξ
∫
∫
( )
S′′ ξ ( )
0
0
S′
=
= (
)
0,
0
y
t
x
∂
=
∂
x = ∞

(
)
(
)
(
)
(
)
,
,
,
,
…
0
x
xx
xxx
y
t
y
t
y
t
y
t
′
′′
′′′
∞
=
∞
=
∞
=
∞
=
=
κ η2
1
2
S
f
a

′
′
κ =
−
η
2
3 2
1
const
2
2
S
S
d
a

′
′
κ = −
+
η +
η
η
∫
(
)
,
0
t
κ ∞
=
( )

2
1
2

S
d
a

∞

η

′′ ξ
κ =
ξ
ξ
∫
( )

2
2
S
EJ
M
d
a

∞

η

ξ
= −
ξ
ξ
∫
( )

3
2
S
EJ
Q
a
t

η
=
0
η =
3
(0).
EJ S
P
a
t
=
W 1
3
2
(0)
.
EJ
W
v S
t
a
=
0
S
S
′′ +
=
cos
S
c
=
η1
1 2
0

sin
2
2
2
c
c
v
d

∞
ξ
π
=
ξ =
ξ
∫
1
2
2
cos
S
v
=
η
π
0
x =
1
(0)
F
v
EJ
ρ
κ
=
/ 2
y
h
=
max
1
2
h
F
e
v
EJ
ρ
=
2
2
s
s
e
e
EJ
EJ
v
h
F
Eh
F
σ
=
=
ρ
ρ

0
y =
4
4
0
0
2,13
2,13
,
EJ
J
x
t
a t
F
F
=
=
ρ

0a ηηy xy′ η′ηS M κ 1
e
κ > κ
M κ OA κ EJ 2
EJa η(
)
sin
A
η−η (
)
(
)
sin
/
A
a
η−η
κ η0
x =
S cos
S
A
a
η
=
0
0 ≤ η ≤ η (
)
1
sin
S
B
= −
η− η
0
1
η ≤ η ≤ η (
)
1
1
sin
S
C
a
= −
η− η 1
2
η ≤ η ≤ η (
)
3
sin
S
D
=
η − η
2
η ≤ η κ κ κ
κ κ

M κ 1
η = η x

1
η /
0
dM dη =
(
)
1
0
S η
=
0
η 1
η 2
η 3
η S S′ 0
cos
sin
A
B
a
η = −
0
η 1
η 0
sin
cos
A
B
a
a
η =
0
η 1
η C
B
a
=
2
1
sin
sin
C
D
a
η − η
−
=
2
η 3
η 2
1
cos
cos
C
D
a
a
η − η = −
2
η 3
η 1
0
2 ;
S d
v

∞
′
η = −
η
∫
( )

2
2

2
  2
e
e
EJ
S
S
d
M
d
v
a

η
η

∞
∞

η
η =
η =
η
η
∫
∫
( )

1
1

0
0
2
2
  2
e
e
EJ
S
S
d
M
d
v
a

η
η

η
η

η
η =
η =
η
η
∫
∫
(
)
(
)
(
)
2
1
0
 e
e
M
M
M
M
M
η
=
η
−
η
=
0
η 1
η 2
η 3
η 0
α =
1
2,087
e
e
v
v
v
<
<
0
x =
0
1
η < η 0
η S 2
π
α
2
π
η < α
0
α →
0
0
η →
S′ 0
η =
0
η =
(
)
1
sin
S
B
= −
η− η 1
η (
)
(
)
1
1
1
0
0

cos
sin
2 ;  
2
.
e
d
d
B
v
B
v

∞
∞
η−η
η
η− η
η
=
= −
η
η
∫
∫

2
2
1
1
1
2
;  
arctg
.
4

e
e
v
v
v
B
v
+
π
=
η =
+
π

(
)
( )
1

0

2
tg
0
.
e

x

v
v
y
t S
t
x
a
a
=

−
∂
′
θ = −
=
=
∂
π

1/
/
e
ev
α =
κ
(
)
1
2
tg
.
e
e
e
v
v
t
v
κ
θ =
−
π

M − κ0
Q =
( )
0
S η =
1
η = η (
)

1

1
sin
2 e
d
v

∞

η

η− η
η =
η
∫

1
1
1
1
1
1
0
0

sin
cos
cos
sin
sin
cos
.
2
2
d
d

η
η
η
η
η
η
η −
η =
η −
η
π
η
π
η
∫
∫

o
1
70,6
η =
1
η 1
sin (
)
S
B
= −
η− η 1
0 < η ≤ η 1
sin(
)
S
D
= −
η − η 1
η ≤ η 1
η (
)
(
)

(
)
(
)

1

1

1

1

1
1
1
0

1
1

0

cos
cos
2 ;

sin
sin
2
4
.
e

B
d
D
d
v

D
d
B
d
v

η
∞

η

∞
η

η

η−η
η− η
η+
η =
η
η

η−η
η−η
η = −
η =
η
η

∫
∫

∫
∫

0
x =
θ ( )
1
0
tg
cos
cos .
e

x
e

y
B
t
t
B
x
a
v
=

κ
∂
θ = −
=
η
=
θ
∂

1
2a
t
=
η
S′ (
)

1

1
2

e

e
B
D
v
κ
∆κ =
−
η
1
η tg
e
e
tv
θ
κ 3 2
1
1
1

3
,  tg
.
3

e
e

e

v
v
t
v
v
κ
η =
θ =

××M − κ ×e
κ×1
30,5
v =
0
x =
12,5
=
××t

θ θ M − κ 0x M − κ × × θ ttθ×θ××××M − κ 1
2,96
v =
3,44
t =
t t M − κ × × t×2l 0
(2
)
P
ml
a
=
dv dt ( )
(
)
(
)
2
2
0
0
0
2
(4 ).

l x
M x
m
a
d
a m l
x
P l
x
l

−
=
ξ ξ =
−
=
−
∫

1
0
1
1
0
4
; 
4,
Pl
M
Pl M
=
µ =
=

0
ma dx 0
θ

0
θ

ξl

lθ

2
0
0
2
0
0
2
2

l
l d
P
m a dx
m
xdx
dt
θ
−
= −
∫
∫
2
0
0
2
2
2
d
P
l
ml a
dt
θ
=
−
/ 2
P
0
θ 2
2
0
0
0
2
0
0

l
l
d
m
x dx
m a xdx
M
dt
θ
=
−
∫
∫
2
3
2
0
0
0
2
3
2
d
ml
ml a
M
dt
θ =
−
2
3
0
2
0
3
12
d
ml
M
dt
θ = µ −
2

0
0
2
6
ml a
M
= µ −
0

Pl
M
µ =
/ 2
P
( )

2
0
0
1
2
0
0
2

x
x
d
P
m
zdz
m a dz
Q
x
dt
θ
=
+
−
∫
∫
( )

2
2
0
0
0
2
0
0

x
x
d
m
z dz
m a zdz
M
M x
Px
dt
θ
=
−
+
+
∫
∫

2
2
0
1
0
2
2
2
d
P
mx
Q
ma x
dt
θ
=
+
−
( )

2
2
3
0
0
0
1
2
.
3
2
d
ma x
mx
M x
M
Q x
dt
θ
=
+
−
−

( )

2
2
2
3
0
0
0
0
2
4
1
1
2
2
6
2
ma x
d
P
mx
x
x
M x
x
M
M
l
l
dt
θ
µ −
= −
+
+
−
=
−
−
1x (
)
1
0
x

dM
dx
=
2
2
0
1
0 1
2
0
2
2
d
mx
P
ma x
dt
θ
−
−
=
2
4
3
0
3
4
3(
4)
µ −
µ
ξ −
ξ +
=
µ −
µ −
1x
l
ξ =
1
ξ = (
)
[
]
/ 3
4
µ
µ −
ξ 2
3

0
2
9
2
1
3(
4)
27(
4)
M
M
µ
µ − µ
=
+
µ −
µ −
ξµ 4,5
µ =
6
µ =
4
6
< µ ≤
1
ξ > 0
x
l
<
≤ µ >0
x
l
<
≤ ξII
µ 2
3
II
II
0
0
2
II

9
2
1
27(
4)
M
M
µ − µ
−
= −
µ −
II
22,89
µ =
(
)

II

II
0,404
3
4
µ
ξ =
=
µ −

0
M
−
II
22,89
µ
=
0,404
x
l
= ±
x x 2
0
0
2
0
0
2

x
x
d
P
m
zdz
m
a dz
dt
θ
=
−
∫
∫
2
2
0
0
0
2
0
0
2

x
x
d
m
z dz
m
a zdz
M
dt
θ
=
−
∫
∫
0
Q ≠
Q
dM dx
=
x x
x
=
0
M
M
>
2
0
2
d
dt
θ 0a 2

0
2
0

2
12
ml a
M
µ
=
−
ξ
ξ

2
3
0
2
2
3
0

3
24
d
ml
M
dt
θ
µ
=
−
ξ
ξ

x
l
ξ =
(
)

2

2
0

6

1
l
ml a
M
= −
− ξ

(
)

2
3

2
3
0

12

1

l
d
ml
M
dt
θ =
− ξ

0y (
)
( )
(
)
0
0
,
,

x
y x t
y
t
z t dz
=
− θ∫
θ 0
0
0
0

x
d
y
y
dz
y
x
dt
θ
=
− θ
=
−
∫
x
x
<
[
]

0
0
(
)
(
)

x
x

x

dx
y
y
dz
dz
x
x
dt
+
−
=
−
θ
−
θ
+
θ
− θ
∫
∫
0
0
(
)
l
d
d
y
y
x
x
x
dt
dt

θ
θ
=
−
−
−
x
x
>
x (
)
(
)
x
x −
θ
= θ
(
)
(
)
y x
y x
v
+
−
=
=
2
0
2
0
(
)

x
d
y x
y
dz
y
x
dt
−
θ
=
−
θ
=
+
∫
[
]

0
(
)
(
)
(
)

x
y x
y
dz
x
x
x
+
+
−
=
−
θ
+
θ
−θ
∫
2
2
(
)

x
x

dx
d
d
d
y x
y
x
dt
dt
dt
dt
+
−

+
θ
θ
θ
=
+
+
−
2
0
2
(
)
d
y x
a
x
dt
−
θ
=
−
2
0
2
(
)
(
)
d
y x
a
l
x
dt
+
θ
=
+
−
(
)

2
2
0
0
0
2
2
l
l
l
d
d
d
d
dx
a
x
a
l
x
dt
dt
dt
dt
dt
θ
θ
θ
θ
−
−
−
−
=
−
(
)

0
0
3
2
2
12
6

1

l
M
d
d
d
dt
dt
dt
ml

θ
θ
µ
ξ −
+
= −
−
ξ
ξ
−ξ
22,98
µ >

0
l
d
dl
d
dt
θ
>
θ
(
)

2
2 2
2
0
0

1
1
1 1
2
2
2

l
t
d
d
Pdt
m v l
l
l
dt
dt
θ
θ
=
+
ξ
−
− ξ
∫
µ
1
0
2
1
0
2

t
M
v
dt
ml
=
µ∫
1

0
0

1
4

t
M
dt
µ∫
(
)

1
2
3 3
3
3
3 3
0
0
0
1
0

1
4
2
6
2
3
6

l
t
d
d
l
l
l
l
l
M
dt
M
t
t
m
v
dt
dt
θ
θ
ξ
ξ
ξ
µ
+
−
=
+
+
−
−
∫
v (
)
(
)

3
2
3
0
1
0
1
3
12
3
2
l
l
t

t

d
d
d
ml
dt
t
t
M
dt
dt
dt
θ
θ
θ µ
−
−
=
ξ − ξ
−
+
∫
t
T
η =
3
0

0

d
ml
M T dt
θ
Ω =
3

0

t
d
ml
M T dt
θ
ψ =
3

0
2
0

ml
M T
θ =
θ 3

2
0
l
ml
M T
ϕ ≡
θ 2
3
3
24
d
d
Ω
µ
=
−
η
ξ
ξ

(
)

3
12

1

d
d
ψ =
η
−ξ

(
)
(
)

2
2
2
6
12
1

d
d
ξ
ξ
µξ +
−
= − Ω −ψ ξ
η
−ξ

(
) (
)(
)
2
3
1

1
3
12
3
2
d

η

η
µ η−
η−η
=
ξ − ξ
Ω −ψ + ψ
∫
θ ϕ d
d
θ = Ω
η
d
d
ϕ = ψ
η

τ
η ≥ η τ
η (
)

(
)

3
3
24
12

1

d

d

Ω − ψ = −
−
η
ξ
− ξ

ξ 0
µ =
(
)
(
)

2
2
24
12

1

d
d
d
d
ξ
ξ
−
= Ω − ψ
ξ
ξ
ξ
η
−ξ
1
const
d
d
C
ξ =
=
η

2

2
2
(
)
12
6
(1
)

t
C
τ
τ

τ
τ

ξ
Ω − ψ
=
− ξ
− ξ
(
)
C
τ
τ
η = η +
ξ − ξ
2
2
(12
12
)
C
τ
τ
Ω = Ω +
ξ −
ξ
2
2
6 (1
)
6 (1
)
C
τ
τ
ψ = ψ +
− ξ
−
− ξ
τ τ
η s
η Ω = ψ s
τ
η > η ≥ η sξ sτ 2
2
0,586
s
ξ =
−
=
s
ξ 0
t
d
d
dt
dt
θ
θ
−
2

2

2
2

2
2
2

12
(0,586
);
34,97
;

6
34,97
;
(1
)

12
12
(0,586
)
20,48 ;

6
6
(0,586
)
14,49
.
(1
)
(1
)

s
s

s

s

s

C
C

C

C
C
C

C
C
C

τ
τ
τ
τ

τ
τ

τ
τ
τ
τ
τ

τ
τ
τ
τ
τ

η = η +
−ξ
Ω = Ω +
−
ξ
ψ = ψ +
−
−ξ
θ = θ +
Ω −
−ξ
+
−
ξ
ξ
ϕ = ϕ +
ψ −
−ξ
+
−
−ξ
−ξ
s
s
Ω = ψ s
η = η Ω = ψ 1

1
3
12(
)

s

s
s
s
d

η

η
Ω = ψ =
µ η −
η − η
∫
s
f
η ≤ η ≤ η f
η 1
12
s
s
f
η
= η +
Ω 2
1
24
s
s
f
θ
= θ +
Ω s
f
θ − θ f
θ τ
η ≤ η τ
η ≤ η 1
const
ξ = ξ =
1ξ µ (
)

2
1
2
1

6
12
0
1

ξ
µ ξ +
−
=
− ξ

2
3
1
1

3
24
µ
Ω =
−
η
ξ
ξ
(
)

2

3
1

6

1

η
ϕ =
− ξ

(
)

3
1

12

1

η
ψ =
−ξ

2
2
3
1
1

3
1
24
2
µ
θ =
−
η
ξ
ξ

1
τ
η = η =
1
τ
ξ = ξ τ
η = η

f
η f
θ 0 f
θ
µ Ω ψ d
d
Ω
η

d
d
ψ
η

0
η 0
22,89
µ =
0
0,404
ξ =
0
η = η η0
′ξ =
′′
ξ ξ ξ Ω ψ (
)
3
2
0
0
0
/
f
ml
M T
θ
µ 0T 0
P T
Pdt
I

τ
=
=
∫
µ 0T 3

0
2
0
0

n
f
ml
D
M T θ
=
µ 8/3
n =
0,043
D =
0,0370,0312
1 3
0
2 3 1 3
0
f
D
I P
mM
l
θ
=
12
0
µξ −
≤
ξ 12
µξ ≤
µ3

0
2
0
0
f
ml
M T θ

P l M
µ
=

 0

0
1
cos
2
t

v
y
x
t
l
=

∂
π
=
+
∂
l
x
l
− ≤
≤ 2
0
0
0
2
M l
M
EJ
θ >>
2
0
2
3
2
0

1
2,5
(
)
M T
ml
T
θ
τ
>>
τ o
0
0
(
10 )
θ
θ ≤
2

0
2
y
M
M
b x
∂
= ±
+
∂

4
2

4
2
2
1
0
y
y
x
c
t
∂
∂
+
=
∂
∂

2
1
m
b
c
=
•••ϕψψψψ2

2
n
x
T
N
q
q
s
s
t
∂
∂
∂
=
−
+
−
∂
∂
∂

2

2
n
y
T
N
q
q
s
s
t
∂
∂
∂
=
+
+
+
∂
∂
∂

2

2
J
M
N
M
EJ
F
s
s
t
∂
∂
∂
=
−
−
=
∂
∂
∂

,
T N ,
n
q
q  2

0

1
2(1
)
x
y
e
s
e
s
∂
∂
∆ =
+
∂
+
∂
0
0
(
)
x
x
e s
s
=
−
−
2
0

2
0
0
0

0
0

1
1
1
cos1
2(1
)
1
1
2(1
)

1
sin1

x
x
e
y
s
s
e
e
s
x
y
e
s
e
s

y
T
E e
e
e
s

∂
∂
+
+
+
∂
∂
∂
−
+
+
∂
∂
∂
+
+
+
∂
+
∂
∂
+ ∆
+
∂

2
2

1
2
2
x
x
E
t
s
∂
∂
=
+
∂
∂

2

1
2
2
0
0
0
1
(1
)
2(1
)
(1
)

nq
E
y
y
y
N
q
s
s
s
s
e
s
e
e
∂
∂
∂
∂
∂
=
−
+
−
∂
∂
∂
∂
+
∂
+
+
2
2
4
4
0
2
2
4
2
2
2
0
0
0
0

1
1
1
(1
)
n
q
Ee
y
y
EJ
y
J
y
y
q
e
e
e
s
s
t
s
s
F
e
t
s
∂
∂
∂
∂
∂
∂
=
−
+
+
+
+
+
+
+
∂
∂
∂
∂
∂
+
∂ ∂

3
3

2
3
0
0
(1
)
1
J
y
EJ
y
N
F
e
e
t
s
s
∂
∂
=
−
−
+
+
∂ ∂
∂

( , )
y
y s t
=
( , )
x
x s t
=
( , )
y
y s t
=
( , )
y s t ( , )
x s t 1 2
1
2
1
2
x
X
s t
X
s t
y
Y s t
Y s t
=
+
+
=
+
+

1
1
,
X
Y 0
V 2
2
4
4
0
2
2
4
2
2
0
1
e
y
y
y
y
e
t
s
s
t
s
∂
∂
∂
∂
=
−
−
+
∂
∂
∂
∂
∂
2
0
0
,
,
, (1
)
l
J
y
l y s
ls t
t
a
e l F
=
=
=
=
+

( )
H t n
q
q
=
=
=
0,
s
s
l
=
=
( )
( )
k
k
y
T t S
s
=
2
k
k
k
T
T
′′ = −
2
2
0

0

IV
k
k
k
k
k
e
S
S
S
e
α
′′
−
−
−
=
+
k
m s
e
k
m 2
2
2
2
2
0
0
1,2
0
0

2k
k
k
k
e
e
m
e
e

=
−
±
−
+
+
+
(0, )
(0, )
(1, )
(1, )
0
s
s
y
t
y
t
y
t
y
t
=
=
=
=
2
k → ∞ ( )
k
S
s (
)

0

0
0
0
0

0

0

( )
sin
cos
exp
( 1)
2
exp
(1
) ,
k
k
k
k
k

k
k

s
s
e
S
s
s
e
e
e
e

e
s
e

ω
=
−
+
−
+
+
−
−
−
−
0
2
0

k
k e
k
l
e
l
≈
+
( )
sin k
k
k
k
k
T t
A
t
B
t
=
+
( , )
( )
( )
k
k
k
y s t
T t S
s
= ∑
( , )
y s t ( )
( )
k
k
T t S
s ( )
kT t ( )
k
S
s y
s
∂
∂

2
y
s
∂
∂
y
N
s
∂
∂ 1
Литература

Глава 3

Распространение волн возмущения с полярной,
осевой и сферической симметриями

§ 3.1. Плоские продольные упругопластические волны

σσ2
2

0
2
2
xx

xx

d
u
u
de
t
x

σ
∂
∂
ρ
=
∂
∂



(
)
xx
xx
f e
σ
=
0,  
,
yy
zz
yy
zz
e
e
=
=
σ
= σ
(
)
2
3
;  
2
/3 ,
xx
yy
xx
xx
yy
i
xx
Ke
e
σ
+ σ
=
σ
− σ
= Φ

(
)
2
2
/3 /3.
xx
xx
i
xx
Ke
e
σ
=
+ Φ
(
)
(
)(
)
(
)
1
/
1
1
2
xx
E
σ
=
− ν
+ ν
− ν
ν (
)
xx
xx
xx
e
σ
= σ
S 6
2
0,74 10 E =
⋅
0,33
ν =
( )
i
ie
Φ
(
)
xx
xx
xx
e
σ
= σ

xx
e ×xx
e ×3
0
6,25 10
a =
⋅
3
5,11 10
=
⋅
0
/
0,1
a a ≈
x (
)(
)
2
2
1
(
)
1
1 2
xx
xx
xx
xx
E
e
e
−ν
σ
− σ
=
−
+ ν
− ν
σ(
)
2
/3
i
xx
e
Φ
3
3
3
xx
yy
xx
Ke
σ = σ
= σ
=
0
i
Φ =
τσν σ(
)
xx
xx
xx
e
σ
= σ
(
)
xx
xx
xx
e
σ
= σ
0a (
)
xx
xx
xx
e
σ
= σ
xx
e ×xx
σ § 3.2. Цилиндрические волны сдвига
(задача о скручивающем ударе)

0r τ r
dr
+
(
)
2
2
2
2
2
2
2
2
w
r dr
r
r
dr
r
t
t
r
∂
∂
∂
π
ρ
= − π τ −
π
τ
+ π τ
∂
∂
∂
2
2
2
2
1
(
).
w
r
r
t
r
∂
∂
ρ
= −
τ
∂
∂

( ),
e
τ = τ
w
w
e
r
r
∂
=
− ∂

2
2
2
2
2
w
w
a
b
t
r
∂
∂
=
+
∂
∂

2
2
2
2
2
1
,  
.
a w
a
w
d
b
a
r
r
r
de
r
∂
τ
τ
=
−
−
=
∂
ρ
ρ

dr
a dt
= ±
t
r
dw
a dw
bdt
= ±
+
,   
.
r
t
w
w
w
w
r
t
∂
∂
=
=
∂
∂

0r 0
t
r
w
w
=
=
0
t =
0r
r
≤
< ∞ ( )
w
w
f t
r
r
∂
−
=
∂
0
r
r
=
0
0(0)
r
r
a
t
=
+
0
(0)
a
a
=
0
r
r
=
0
t =
0
a 0
0
r
r
a t
=
+
0
0
r
r
a t
=
+
0
w =
0
0
r
r
a t
=
+
0
0
r
r
a t
=
+
0
t
r
dw
a dw
b dt
=
+
0
dr dt
a
=
0
0
r
r
a t
=
+
2
2
2
2
0
0
0
0
2
2
r
r
r
r
a
a w
a w
a dw
w
dr
dr
r
r
r
−
= −
+
=
0 /
t
w
c r
r
=
0
r
r
=
0
0
0
/ ;  
/ .
t
s
t
s
w
e
r
r
w
a e
r
r
= −
=

2
2
2
2
2
0
0
0
2
2
2
a
a
w
w
w
a
w
r
r
t
r
r
∂
∂
∂
=
+
−
∂
∂
∂
′0
w =
0
0
r
r
a t
=
+
w
w
r
r
∂
−
=
∂
0
1
r
r
a t
=
+
0
0
/
t
a t r
=
0
/
r
r r
=
0
/
w
w r
=
1
0
/
a
a
λ =
′2
2

2
2
2
1
w
w
w
w
r
r
t
r
r
∂
∂
∂
=
+
−
∂
∂
∂

0
w =
1
r
t
= +
1
w
w
r
r
∂
−
=
∂

1
r
t
= + λ dr
dt
= ±
2
r
t
t
w
w
dw
dw
dt
r
r
= ±
+
−
(
)
, ,
w r t λ λ λ 1
r
t
= + 1
r
t
= + λ 0,003
0,0548
λ =
=
1/
18,25
µ =
λ =
,
1,
0,2
n m
n
m
r
r −
−
=
r

t

2
2
2
2
2
2
2
0
1
1
1
1
2
2
2
2
(
)
se a
a
a
a w
w
w
w
a
r
r
r
t
r
r
−
∂
∂
∂
=
+
−
−
∂
∂
∂

1

2
2
2
1
1
0
1

;

2
(
)
.
t
s
r

dr
a dt

e
a
w
w
dw
a dw
a
a
dt
r
r
r
r

= ±

∂
= ±
+
−
−
−
∂
0
1
r
r
a t
=
+
0
0
r
t
r
t
w dr
w dr
w dr
w dt
+
=
+

0
0
1
(
)
t
r
r
r
w
w
w
w a
−
= −
−
0
1
r
r
a t
=
+
0
2
2
2
1
1
0
1
2
(
)
t
r
s
w
w
dt
dw
a dw
a
e a
a
r
r
r

∂
=
+
−
−
−
∂
2
2
2
0
0
0
1
1
1
1
1
2
1

2
(
)
2
(
)
s
r
r
t
t
e a
a
w
w
a dw
a
dr
a
dr
dr
d w
a w
r
a r
r
−
−
=
−
−
−
+
0
0
0
0
2
1
1
0
1
3 2
1
1

1
1
2
(
1)
2
2
4

r
r
t
r
t
r
r
s
w
a w
w
a w
w
c
w
dr
e
dr
a
r
r
r
a
r
r

+
+
=
+
+
µ −
+
−
∫
∫
1c 0
r
r
=
2
0
0
0
0
3 2

0

1
2
(
1) 1
2

r

r
s
r

r
w
r
w
e
dr
e
r
r
r
r

= −
+
+
µ −
−
+
∫

0

0
0
0
0
1
1
0

1
1

1
1
(
1)
2
2
4

r
t
r
t
r
s
r

w
a w
w
a w
r
e
dr
a
r
a
r
r

+
+
+
−
µ −
−
∫
/
w
w
=
0
0
/
t
a t r
=
0
/
r
r r
=
(
)(
)
2
0
1
1
2
1 1
2
r
s

e
J
w
e
r
r
r
= −
+
+
µ −
−
+

( )

0
0
0
2
1
2
2
4

r
t

s

w
w
e
J
f r
r
r
e
r

+µ
µ −
+
−
−
= −
+
(
)

0
0
0
1
2
3 2
1
1
;  
.

r
r

r
t
w
dr
J
dr
J
w
w
r
r
=
=
+ µ
∫
∫
1/
18,25
µ =
λ =
(
)

0
,
,1
0
0
,
,
3 2
,
;
;
18,25
;
2

n n
n
n
n
n
n n
n
t
r
n n
n n

w
A
A
A
r
B
w
w
r
r

+
=
∆ =
∆
=
+

1
,
,
;
.
2

n
n
n
n
C
n n
n n

B
C
C
C
r
r

+ +
=
∆
=
∆

nn
r
0
nn
w
1 2
nn
r
3 2
nn
r
∆ 0
t
w
0
t
w
0
r
w

∆ 0 /
s
e
e r ∗ 0 /
1
r
w
r
w ∗
∗ −
=

0 /
s
e
e
r ∗
=
0
[1
(
)
/
]
f r
w
r
∗
∗
+
−
(
)
0 /
w
r
f r
∗
∗
−
1/
µ =
λ =
18,25
=
(
)
1 2
1
1
2
,
3
4
2
1
3
,
,

8,625;
664 1
;
;
;
2
n n
n n
n n

J
A
A
r
A
A
A
A
A
r
r

−
=
=
−
=
=
−
+

( )

0
0
0
2
5
6
4
5
6
,
,
;
;
;
.
2
4

r
t

n n
n n

w
w
J
w
A
A
f
A
A
A
d
f r
r
r

+ µ
=
=
=
+
−
=
−

n
1 2
,
n n
r
−
1 2
,
n n
r
−
1 2
,
1
n n
r −
−
( )
f
r
0 /
w
r
d
r ∗ 0 /
s
e
e 0 /
s
e
e
r ∗
r ∗ r ∗ 1a r ∗ r ∗ (
)
0
0
0
0
w
w
E e
e
E e
r
r
∂
τ = τ −
−
= τ −
−
+
∂
(
)
(
)
2
2
0
0
0
0
2
0
2
2
2
0
0

2
1
Ee
Ee
w
w
w
w
d
a
r
r
dr
r
t
r
r

τ −
τ −
∂
∂
∂
=
+
−
−
−
∂
ρ
ρ
∂
∂
0τ 0e 0
1
r
r
a t
=
+
1
1
1
t
r
r
w
a w
w
a w
∗
∗
+
=
+
0
1
r
r
a t
=
+
0
s
e
e
−
(
)
1
0
t
t
s
w
w
a e
e
∗
=
+
−
0
0
w
w
e
r
r
∂
−
=
∂

0e 0e 0
1
r
r
a t
=
+
2

0
2
e
e
w
de
dt
dr
dr
t
r
r
∂
∂
=
−
−
∂
∂
2

1
0
2
t
t
w
e
w
a de
dw
dr
dr
dt
t
r
t

∗
∂
∂
−
=
+
−
−
∂
∂

w
w
e
r
r
∂
=
−
∂
2

2
w
r
∂
∂

2

2
w
t
∂
∂

0
0
(
)
s
s
E e
e
′
τ = τ +
−
1

dr
dt
a
=
0
2
0
1
1
1
1
t
t
dw
w
de
e
a
a
a
a
dr
t
r
dr
∂
+
−
+
=
∂

2
2
2
2
2
0
0
0
0
0
0
0
1
1

2(
)
2
2(
)
s
s
E e
a
de
a e
e
e
a
a
a
a
t
dr
r
r
r

′
τ −
∂
=
−
−
−
+
−
∂
ρ
0
t
dw
dr 0e e
t
∂
∂ 0e 0e 0
1
r
r
a t
=
+
0
1
r
r
a t
=
+
§ 3.3. Сферические волны

σσθσϕϑϕ;
;
;
;
r
u
u
e
e
e
e
r
r
ϕ
θ
ϕ
θ
ϕ
∂
σ
= σ
=
=
=
∂

(
)
(
)
1
2
;
;
3
3
i
r
i
u
u
e
r
r
ϕ
∂
σ =
σ − σ
=
−
∂

( )
(
)
2
3
;
2
3
,
r
i
r
u
u
F e
K
r
r
ϕ
ϕ
∂
σ − σ =
σ + σ =
+
∂

σϕ σϕ dΩ r
dr
+
2

0
2
2(
)
.
r
r
u
r
r
t

ϕ
σ −σ
∂σ
∂
ρ
=
+
∂
∂

σσϕ 2

2
0
0

0

2
2
2
3
3

2 3
2
3

u
u
u
K
u
u
F
r
r
r
r
r
t

u
u
F
r
r
r

∂
∂
∂
∂
= −
−
−
+
−
∂
∂
ρ
∂
∂
ρ
∂
−
−
ρ
∂
( )
( )
2

2
3
f
f
u
r
r
t

∂
ϑ
ϑ
∂
= −
−
∂
∂

0
0

2
2
3
3
x
Kx
F =
+
ρ
ρ
u
u
r
r
∂
θ =
−
∂
2
2

2

( )
;

3
( )
,
r
t
r

dr
a
dt

a u
a u
f
du
a
du
dt
r
r
r

= ±
θ

= ±
θ
+
−
−
2( )
( )
a
f ′
θ =
θ 0
( )
a
a
θ =
( )
f θ 2
2
1
0
0
( )
a
f
a
′
<
≤
θ ≤
( )
0
f ′′ θ ≤
0
θ ≥ θ 2
0
( )
f
a
′ θ ≡
0
θ ≤ θ (
)
(
)
,
,
0
u r t
u r t
t
∂
=
=
∂

0
t =
σσϕ( )
0
r
p t
σ =
<
1
r =
3
u
f
u
Ku
r
∂
−ρ
−
+
=
∂
1
r = ′0
dp dt <
0
≠
(
1,
0
r
t
=
= 0

r
u

t
r
u
adu
= −∫
0
1
r
a t
= +
0,
u
t
∂
=
∂

0
0
t
r
u
a u
+
=
2
0
0
3 (
)
r
r
t
r
a u
f
u
du
a du
dt
r
r
−
=
−
+
0

0
0
0
0
2
2
0;
( )
.
r
r
t
a u
a du
dr
u
a
a z dz
r

θ

θ
−
−
=
=
θ + ∫

0
c r
r
=
= −θ

1
r =
0
θ = θ 0
>
1
r → ′0
( )
3
.
dp
u
f
K
t
dt
t
∂θ
∂
′
ρ
θ = −
+
∂
∂

0
′ρ <
0
t
∂θ ∂ >
′( )
f θ ( )
f θ

2
0a
=
θ 0
θ ≤ θ 2
2
2
0
0
1
0
( )
(
)
f
a
a
a
θ =
θ +
−
θ 0
θ ≥ θ 0
θ ≤ θ 2
2
2
2
0
0
2
2
2

2
2

2
2
2
2
0

3
;

2
2
1
.

r
r
a
u
u
u
u
u
a
u
r
r
r
t
r
r

u
u
u
u
r
r
r
r
a
t

∂
∂
=
−
+
−
−
∂
∂
∂
∂
∂
+
−
=
∂
∂
∂

1
1
r
a t
= +
0
=
0
1
r
a t
= +
0
u
u
r
r
∂
−
= θ
∂

1
1
r
a t
= +
0
0
2
(
1)
(
1)
( , )
r
a t
r
a t
u r t
r
r
′φ
−
−
φ
−
−
=
−
(0)
(0)
0
′φ
= φ
=
0
2
3
( )
3 ( )
3 ( )
1
(
1)
(
1)
x
x
x
x
x
x
′′
′
φ
φ
φ
−
+
= −θ
λ +
λ +
λ +
3
0
( )
(
1)
(
1)(6
3)
x
x
θ
φ
=
− λ +
+
λ −
λ −

3
(
1)
cos( log(
1))
sin ( log(
1))
x
x
x
µ
− µ
+ λ +
ν
λ +
+
ν
λ +
ν
2
(
3) / 2 ;
12
(
3) / 2 .
µ = λ +
λ ν = −
− λ +
λ

2
1
0
( , )
3
(
1)(6
3)
u r t
r

−
µ
θ
λ
=
− ξ + ξ
λ −
λ −
3cos( log )
ν
ξ +

3µ
+

2
2
2
2
1
3
3
cos( log )
sin( log )
sin( log )
v
r

µ
−µ
µ−
−µ
−
−ξ +ξ
ν
ε +
ν
ε
ν
ε
ν
ν
1
x
ξ = λ + 0
1
x
r
a t
=
−
− 1
1
r
a t
= +
[
]
2
0
( )
( 3
1)
(3
1)cos( log )
sin( log )
(
1)(6
3)
g r
r
r
r
r
−
µ
θ
=
− λ+
+
λ−
ν
−σ
ν
λ−
λ−
2
( 3
4
3) /(2
)
σ = − λ − λ +
λν 2
2
2
2
0
1
0
2
2
2
2
2
1
1

3(
)
2
2
1
a
a
u
u
u
u
r
r
r
r
a r
a
t
−
θ
∂
∂
∂
+
−
−
=
∂
∂
∂
′2
2
2
2
1
1
0
0
1
(3
)
(
)
( )
a
K
a u
a
a
p t
ρ+
−ρ
−ρθ
−
=
1
r =
1
1
r
a t
= −
ln
hr
r
−
2
2
2
0
0
1
1
(
) /
h
a
a
a
= −θ
−
2
2
1
1
1
2
2
2
2
1

2
2
1
u
u
u
u
r
r
r
r
a
t
∂
∂
∂
+
−
=
∂
∂
∂

ln
hr
r
+
1
1
r
a t
− =
2
2
1
1
1
1
(3
)
( )
u
a
u
K
a
p t
r
∂
ρ
+
− ρ
=
∂
1
r =
1
1
1
1
1
(
1)
(
1)
( , )
r
a t
r
a t
u r t
r
r
Φ
+
−
+ ψ
−
−
∂ =
∂ 1
1
1
1
2
(2
2)
(0)
(2
2)
(0)
( )
ln
r
r
g r
hr
r
r
r
′
′
Φ
−
+ ψ
Φ
−
+ ψ
−
=
+
ψψ′2
1
1
0

( )
(
2)
( )
2
2

x g
x
x
d
τ
+
Φ
=
τ
τ +
∫
1( )
1
1 ln
1
2
2
2
x
x
x
g
g
h
τ =
+
+
+
+
ψ2
1
1
1
1
1
1
2

1
2
2
1
1
1

(
)
(
)
( )
( )
( )
( )
(
) ;

1
3 1
;
( )
,

d
x
d
x
x
x
x
x
x
p x
x
x
x
dx
dx

K
x
x
p x
p
a
a
a

Φ −
Φ −
′′
′
ψ
− ψ
+ ψ
=
−
+
+ Φ −
=
−
=
−
ρ
ρ
1
(0)
(0)
0
′
ψ
= ψ
=
2
1
1
1
1
0

1
( )
(
)
(
( )
(
))sin (
)

x
x
x
p
x
x
ψ
= −Φ −
+
τ −
Φ −τ
γ
− τ −
γ ∫

]
1
(
)
exp
2
( )cos (
)
2
x
d
x
x
x
κ
− τ
τ
−
Φ
γ
− τ
2
1
4
2
γ =
κ − κ θ
0
θ [
]

2
0
1
2
3
(0)
cos( ln )
sin ( ln )

s

p
b
b r
r
b
r
b
r
p

−
µ
+
+
ν
+
ν
=
2

0
2
2
10
22
15
3
2(
1) (2
6
3)
b
λ −
λ +
λ −
=
λ −
λ − λ +
2

1
2
2
6
3
2(
1)
b
λ − λ +
=
λ −
2

2
2
2
6
8
3
2(
1) (2
6
3)
b
λ − λ +
=
λ −
λ − λ +
3
2

3
2
2
6
2
9
3
4 (2
6
3)(
1)
b
λ + λ − λ +
= −
ν
λ − λ +
λ −
0
/ r 0r 0
1
/
a
a 0
1
/
a
a ν 0
/ r 0
/ r 2
G 3
G 2
u 3u 2
2
u
f
=
2
3
u
u
=
2
3
0
θ = θ = θ 3
3
u
f
=
2f 3f 1
G 0
G 2
u 3u 2
G 3
G 2
u 3u 3
B C 2
2
2
0
1
0
1
0
( )
(
)(
)
f
a
a
a
θ =
θ +
−
θ − θ
1θ θ0
/
u
u
∗ =
θ 0
/
∗θ = θ θ 1
1
0
/
∗θ = θ
θ 0a t
τ =
0
2
2
1
1
0
2
2
2
3(
1)
2
2
;
d
u
u
u
u
h
r
r
dr
r
r
r

∗
∗
∗
∗
∗
θ
θ −
∂
∂
∂
+
−
=
+
+
∂
∂
∂τ
2
2
1
0
0
2
0
.
a
a
h
a
−
=

1
0
1
/
r
a
a
= +
τ
∗θ 1
0
1
/
r
a
a
= +
τ
( , )
( )
u
U r
v r
∗ =
τ +
2
1
1
0
2
2
3(
1)
2
2
d
d v
dv
v
h
r dr
dr
r
dr
r

∗
∗
θ
θ −
+
−
=
+
τ2
2

2
2
2
2
2
U
U
U
U
r
r
r
r
∂
∂
∂
+
−
=
∂
∂
∂τ
0
1
(
1)
dv
v
h
dr
r

∗
−
=
θ −
0
=
1
0
1
/
r
a
a
= +
τ
0
( )/
( )
U
g r
v r
=
θ −
2
1
1
2
2
1
0
1
a
U
U
v
dv
dv
v
r
r
r
dr
dr
r
a
a

∗
∂
−
= θ −
+
= +
−
∂
−
3
1
0
0
1
(cos( ln )
sin ( ln ))
a
U
U
U
h
ar
r
b
r
r
a
r

−
µ
∂
∂
+
−
=
+
ν
+
ν
∂
∂τ

1

0
1
a
r
a
= +
τ (1
2 )
a = λ
− λ (
)

2
(2
1) [2
2
1 ]
b =
λ + λ −
λν
λ −
2
2
0
0
0
0

3
1
1
U
k
U
p
r
a
a
a
∂
τ
+
−
=
∂
ρ
ρ
θ
2
(
1)
(
1)
(
1)
(
1)
( , )
r
r
r
r
U r t
r
r
′
′
′
Φ
+ τ −
+ ψ
− τ −
Φ
+ τ −
+ ψ
− τ−
=
−
ΦψψΦ ψ′′
Φ τττττ

ψψ′Φ ′
Φ [
]
[
]
1
1
1
2
2(
1)
2(
1)
( )
r
r
U r
r
r

′
Φ
−
+ τ
Φ
−
+ τ
−
=
1
(
0)
Φ τ =
[
]
[
]
1
1
1
( )
(
)
( )
(
)
( )
(
)
( )
x
x
p
′′
′′
′
′
Φ τ + ψ −τ −
Φ τ + ψ −τ +
Φ τ + ψ −τ
=
τ 1
1
(
)
(
)
0
′
ψ −τ
= ψ −τ
=
2
1
0
3(1
/(
))
x
k
a
=
−
ρ
1
2
0
0
0

1
( )
p
p
a
a
τ
τ =
−
ρ
θ
2
1
1
1

1
0

(0,5(
)
1)
( )
2
0,5(
)
1

x
x
U
x
x
dx
x
− τ
− τ
+
Φ
= − τ
+
∫
1
1
1
1
1

(1)
(
)
( )
(
)
exp
sin
(
)
2
U
x
x
x
x
κ
− τ
ψ
= −Φ −
−
γ
− τ +
γ

1

1
1

1
(
( )

x
p

−γ
+
τ −
γ ∫