112
3 Computer Simulation of Dynamic Systems
i.e., the total energy E(t) decreases with time. Since at the moments when the load
reaches its maximum amplitude |r m (t)|, its speed and kinetic energy E k are equal to
zero, then at these moments
E p = −kr
2
m (t)/2 = E(t)
and since E(t) decreases, the amplitude |r m (t)| is also a decreasing function of time.
E k + E p = E 0 − r x = E 0
Points of change of direction are characterized by the values υ = 0, or E k = 0.
The corresponding values of the amplitude x are obtained as the points of intersection
of the E p curve with the “direct energy loss” E 0 − r x. If, for example, the movement
begins at x = x 1 < 0 and υ = 0, then the first point of change in the direction of
motion is obtained at the amplitude x = x 2 > 0 and we leave the region υ > 0. To
move in the opposite direction, you need to substitute another value E 0 = E 02 and the
opposite sign in front of r in the equation. Thus, a direct loss of energy with a positive
slope is obtained, which, of course, must dock with a straight line corresponding to
the first oscillation, i.e., go through the intersection point of the first line with the E p
curve for x = x 2 . Another point of intersection of the second straight line with the
E p curve gives the next point of change in the direction of motion and, accordingly,
the amplitude x = x 3 .
You can continue this construction and find a sequence of points of change of
direction. The sequence x n ends when the slope of the E p curve becomes less than
the slope of the direct energy loss.
This can be easily explained physically: As the deviation x decreases, the restoring
force decreases, while the friction force remains constant. Starting from a certain
value of the deviation x, the friction force becomes larger than the restoring force,
and the restoring force cannot cause the oscillator to shift from the corresponding
point of change of direction. The vibrations end in the dead zone, determined by the
value of the friction force r.
From the point of view of the theory of dynamical systems, the system “spring
pendulum with dry friction” should be attributed to hybrid systems.
In practice, one often has to deal with discrete–continuous models of dynamic
systems, which are called hybrid systems. Other names for such systems are “variable structure systems” and “event-driven systems.” The variable structure of the
models is due to the presence of slow (continuous) and fast (discrete) processes in
the system. Thus, a distinctive feature of the complex behavior of a hybrid system is
the multitude of qualitatively different and successively changing modes of functioning. Continuous behavior is called the state of the hybrid system, while switching of
the states—discrete events. The execution time of discrete events is not taken into
account, from the point of view of the functioning of the system they are performed
instantly.
3 Computer Simulation of Dynamic Systems
i.e., the total energy E(t) decreases with time. Since at the moments when the load
reaches its maximum amplitude |r m (t)|, its speed and kinetic energy E k are equal to
zero, then at these moments
E p = −kr
2
m (t)/2 = E(t)
and since E(t) decreases, the amplitude |r m (t)| is also a decreasing function of time.
E k + E p = E 0 − r x = E 0
Points of change of direction are characterized by the values υ = 0, or E k = 0.
The corresponding values of the amplitude x are obtained as the points of intersection
of the E p curve with the “direct energy loss” E 0 − r x. If, for example, the movement
begins at x = x 1 < 0 and υ = 0, then the first point of change in the direction of
motion is obtained at the amplitude x = x 2 > 0 and we leave the region υ > 0. To
move in the opposite direction, you need to substitute another value E 0 = E 02 and the
opposite sign in front of r in the equation. Thus, a direct loss of energy with a positive
slope is obtained, which, of course, must dock with a straight line corresponding to
the first oscillation, i.e., go through the intersection point of the first line with the E p
curve for x = x 2 . Another point of intersection of the second straight line with the
E p curve gives the next point of change in the direction of motion and, accordingly,
the amplitude x = x 3 .
You can continue this construction and find a sequence of points of change of
direction. The sequence x n ends when the slope of the E p curve becomes less than
the slope of the direct energy loss.
This can be easily explained physically: As the deviation x decreases, the restoring
force decreases, while the friction force remains constant. Starting from a certain
value of the deviation x, the friction force becomes larger than the restoring force,
and the restoring force cannot cause the oscillator to shift from the corresponding
point of change of direction. The vibrations end in the dead zone, determined by the
value of the friction force r.
From the point of view of the theory of dynamical systems, the system “spring
pendulum with dry friction” should be attributed to hybrid systems.
In practice, one often has to deal with discrete–continuous models of dynamic
systems, which are called hybrid systems. Other names for such systems are “variable structure systems” and “event-driven systems.” The variable structure of the
models is due to the presence of slow (continuous) and fast (discrete) processes in
the system. Thus, a distinctive feature of the complex behavior of a hybrid system is
the multitude of qualitatively different and successively changing modes of functioning. Continuous behavior is called the state of the hybrid system, while switching of
the states—discrete events. The execution time of discrete events is not taken into
account, from the point of view of the functioning of the system they are performed
instantly.