114
3 Computer Simulation of Dynamic Systems
Fig. 3.18 Plots of displacement and velocity versus time of a dry friction spring pendulum
1
2
mυ
2
+
1
2
cx
2
+ r x = E 0
1
2
mυ
2
+
1
2
c
x +
r
c
2 = E 0 +
r
2
2c
= E ˙
0
or at ω
2
0 = c/m
υ
ω 0
2
+
x +
r
c
2 =
2E ˙
0
c
If the phase trajectories described by this relation are constructed in the phase
plane, then we obtain circles whose center is shifted from the origin along the abscissa
to the left by r/c (Fig. 3.19).
This center on the left is the center of all the semicircles in the upper half-plane.
Correspondingly, the center of all semicircles in the lower half-plane is located on
the right. Phase trajectories are composed of a sequence of semicircles of this kind,
which, when crossing the abscissa axis, always go one into another. From the phase
portrait, it is also easy to see that the motion should come to a state of rest through
a finite number of vibrations. With each half-oscillation, a decrease in amplitude
occurs, equal to
3 Computer Simulation of Dynamic Systems
Fig. 3.18 Plots of displacement and velocity versus time of a dry friction spring pendulum
1
2
mυ
2
+
1
2
cx
2
+ r x = E 0
1
2
mυ
2
+
1
2
c
x +
r
c
2 = E 0 +
r
2
2c
= E ˙
0
or at ω
2
0 = c/m
υ
ω 0
2
+
x +
r
c
2 =
2E ˙
0
c
If the phase trajectories described by this relation are constructed in the phase
plane, then we obtain circles whose center is shifted from the origin along the abscissa
to the left by r/c (Fig. 3.19).
This center on the left is the center of all the semicircles in the upper half-plane.
Correspondingly, the center of all semicircles in the lower half-plane is located on
the right. Phase trajectories are composed of a sequence of semicircles of this kind,
which, when crossing the abscissa axis, always go one into another. From the phase
portrait, it is also easy to see that the motion should come to a state of rest through
a finite number of vibrations. With each half-oscillation, a decrease in amplitude
occurs, equal to