()
(
)
(
)
(
)
(
)
(
)
−
+
−
+
−
+
−
+
−
+
=
.
0
,
,
0
,
,
0
,
,
0
x
,
,
0
0
,
,
0
3
3
2
3
3
1
2
2
1
1
2
3
2
3
1
2
2
1
1
2
1
1
2
1
1
1
1
x
if
x
c
z
і
x
if x
x
x
x
x
c
x
x
c
x
c
z
і
x
x
x
if
x
x
c
x
x
c
x
c
z
і
x
x
if
x
x
c
x
c
z
і
x
x
if
x
c
z
x
R
(7)
The resistance force depends on the movement of x and the motion direction (the
velocity sign of the left rod end – z). A simplified graph of the dependence of the
resistance force on the displacement and the motion direction is shown in Fig.5.
It should be pointed out that a force characteristic, which is graphically shown
in Fig. 5 (b), is realized in forward calculations When the sign of velocity changes, a
jump in the resistance force occurs (transition to the lower part of the broken line at a
negative speed).
The initial-boundary value problem is considered in the form:
()
()
2
2
2
2
2
,
,
x
x
t
U
a
t
x
t
U
=
,
0
t
,
L
x
,
0
, (8)
()
0
0
,
=
t
x
U
,
()
()
()
()
−
−
=
t
L
t
U
L
t
U
R
L
t
U
C
L
t
x
U
ES
,
,
,
,
,
0
, (9)
()
0
,
0
=
x
U
,
,
L
x
,
0
, (10)
()
=
.
,
0
,
0
,
L
x
if
x
if
S
P
x
F
(11)
In the system (8)-(11):
)
,
(x
t
U
– the cross-section displacement of the rod with the
coordinate x;
1
−
=
E
a
– sound speed in the rod material; E – elasticity modulus;
– density of the material;
– initial shock parameter; P– impact pulse
To find an approximate solution of the problem (8) - (11), we use a mixed
difference scheme (three-layer scheme) [17]. The parameter
is chosen according to
the results of calculations, which are compared with the analytical solution of the linear
problem by the Fourier method [9; 10; 18].
The difference problem is written as:
(
)
+
+
−
−
+
+
−
=
+
−
−
+
+
−
+
+
+
−
+
2
1
1
2
1
1
1
1
1
2
2
1
1
2
2
1
2
2
h
u
u
u
h
u
u
u
a
u
u
u
n
i
n
i
n
i
n
i
n
i
n
i
n
i
n
i
n
i
+
−
+
−
−
−
−
+
2
1
1
1
1
1
2
2
h
u
u
u
a
n
i
n
i
n
i
, (12)
0
1
0
1
1
=
−
+
+
n
n
u
u
,
)
,
(
1
0
1
1
1
n
N
n
N
n
N
n
N
n
N
V
u
R
u
C
h
u
u
ES
−
−
=
−
+
+
−
+
, (13)
()
()
x
F
x
t
U
=
,
0
- 1567 -