()
(
)
(
)
−
−
−
−
=
.
,
sin
,
,
sin
,
),
(
cos
,
0
,
sin
5
4
4
1
1
1
4
2
2
3
3
1
2
1
1
2
1
0
1
1
1
0
t
t
t
if
t
t
m
c
c
m
V
t
t
t
if
t
t
m
c
c
m
V
t
t
t
if
t
t
m
c
c
m
V
t
t
if
t
m
c
c
m
V
t
x
(6)
To compare the solutions, an approximate solution of the problem was found by
the Runge-Kutta method. The analytical and approximate solutions of the initial-value
problem on the interval
5
,
0t
, which practically coincide, are shown in Fig.2.
Figure 2. The solutions to the initial-value problem with a power characteristic
(2)
The analytical solution makes it possible to find the ratio of the maximum
amplitudes in one oscillation period:
()
()
1
2
1
5
c
c
t
x
t
x
=
.
The solution of the initial-value problem by the Runge-Kutta method on a long-
term interval is shown in Fig.3. The damping of oscillation amplitudes occurs with a
damping decrement upon the average
95
,
1
/
2
1
=
A
A
.
Framed Structure. One of the means of realizing the integration of elastic and
engineering hysteresis with the purpose of controlling the amplitude-frequency
characteristics is the use of thermodynamic working materials - heterogeneous
lyophobic systems (HLS) in the hydraulic drive [4; 15; 16]. The intrusion of non-wetted
- 1564 -