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5 Modeling of Mechanical Oscillatory Systems …
occurring with the frequency ω (beat frequency) are easily observed experimentally. The oscillation amplitudes of each of the pendulums periodically change with
a phase shift of π/2: When one of them reaches a maximum, the second vanishes
and vice versa.
This behavior of the pendulums can be understood by appealing to the normal
modes of oscillations. In the case of an even mode of normal oscillations, indicated
by the “+” sign, the pendulums move together, the spring is not stretched, and the
frequency is the same as for a single oscillator. In the case of an odd mode of normal
oscillations (the “−” sign), the spring is stretched, which increases the frequency of
this oscillation mode. If only one of the pendulums is displaced, we have two normal
modes of oscillation, which are in a certain relative phase. But since the frequency
of the odd oscillation is slightly higher than the frequency of the even oscillation, the
relative phase changes.
After some time, two normal modes of oscillation will be in antiphase, the
amplitude θ 1 will drop to zero, while the amplitude θ 2 will peak, etc.
You can also consider the same situation from an energy point of view. At t =
0, all energy is concentrated in the first pendulum. As a result of communication
through the spring, energy is gradually transferred from the first pendulum to the
second pendulum until all the energy has accumulated in the second pendulum. The
frequency with which the oscillators exchange energy is equal to the difference in
normal frequencies ω
+
− ω
− .
We should also mention the so-called partial (partial) oscillations inherent in
coupled systems. These oscillations are obtained if one of the pendulums is rigidly
fixed, and the second is removed from the equilibrium position and left to its own
devices (without destroying the connection). Obviously, the partial frequency will
exceed the frequency of one pendulum. In a symmetric system (identical pendulums),
both partial frequencies are equal to each other, and this common value is enclosed
between the values of two normal frequencies.
In the case of various pendulums, their motion is more complex than that described
above for a symmetric system.
Now let the initial deviation and velocity values be different for different
pendulums, as shown in Fig. 5.31.
In this case, the oscillations are not harmonic, as shown in Fig. 5.32.
We pass to the second part of the problem. We assemble an asymmetric system
from ready-made components (Fig. 5.33).
Starting a numerical experiment in the Simulation Center window led to the
creation of the following 3D animation (Fig. 5.34).
As mentioned above, in the asymmetric case, we will not be able to isolate normal
oscillation modes; therefore, in this case we will conduct an experiment with a forcing
periodic force with an amplitude of F x = 10 N and a frequency of f = 1 Hz.
As a result of the experiment, we obtain graphical dependences of the kinematic
parameters of the motion of the coupled pendulum on time. A graph of the change
in the angles of displacement of the pendulums as a function of time (20 s) is shown
in Fig. 5.35.
Phase diagrams of both pendulums are shown in Fig. 5.36.
5 Modeling of Mechanical Oscillatory Systems …
occurring with the frequency ω (beat frequency) are easily observed experimentally. The oscillation amplitudes of each of the pendulums periodically change with
a phase shift of π/2: When one of them reaches a maximum, the second vanishes
and vice versa.
This behavior of the pendulums can be understood by appealing to the normal
modes of oscillations. In the case of an even mode of normal oscillations, indicated
by the “+” sign, the pendulums move together, the spring is not stretched, and the
frequency is the same as for a single oscillator. In the case of an odd mode of normal
oscillations (the “−” sign), the spring is stretched, which increases the frequency of
this oscillation mode. If only one of the pendulums is displaced, we have two normal
modes of oscillation, which are in a certain relative phase. But since the frequency
of the odd oscillation is slightly higher than the frequency of the even oscillation, the
relative phase changes.
After some time, two normal modes of oscillation will be in antiphase, the
amplitude θ 1 will drop to zero, while the amplitude θ 2 will peak, etc.
You can also consider the same situation from an energy point of view. At t =
0, all energy is concentrated in the first pendulum. As a result of communication
through the spring, energy is gradually transferred from the first pendulum to the
second pendulum until all the energy has accumulated in the second pendulum. The
frequency with which the oscillators exchange energy is equal to the difference in
normal frequencies ω
+
− ω
− .
We should also mention the so-called partial (partial) oscillations inherent in
coupled systems. These oscillations are obtained if one of the pendulums is rigidly
fixed, and the second is removed from the equilibrium position and left to its own
devices (without destroying the connection). Obviously, the partial frequency will
exceed the frequency of one pendulum. In a symmetric system (identical pendulums),
both partial frequencies are equal to each other, and this common value is enclosed
between the values of two normal frequencies.
In the case of various pendulums, their motion is more complex than that described
above for a symmetric system.
Now let the initial deviation and velocity values be different for different
pendulums, as shown in Fig. 5.31.
In this case, the oscillations are not harmonic, as shown in Fig. 5.32.
We pass to the second part of the problem. We assemble an asymmetric system
from ready-made components (Fig. 5.33).
Starting a numerical experiment in the Simulation Center window led to the
creation of the following 3D animation (Fig. 5.34).
As mentioned above, in the asymmetric case, we will not be able to isolate normal
oscillation modes; therefore, in this case we will conduct an experiment with a forcing
periodic force with an amplitude of F x = 10 N and a frequency of f = 1 Hz.
As a result of the experiment, we obtain graphical dependences of the kinematic
parameters of the motion of the coupled pendulum on time. A graph of the change
in the angles of displacement of the pendulums as a function of time (20 s) is shown
in Fig. 5.35.
Phase diagrams of both pendulums are shown in Fig. 5.36.