BOUND OSCILLATORS |
Let us assume that we have connected two identical pendulums with a spring, one have held by hand, and another have deviated aside so the spring is stressed. Release now both pendulums. How they will move? Intuitively it seems that the still pendulum will set into the motion under action of a spring and after a while both pendulums will oscillate chaotically having approximately equal stock of energy. Actually the system does not come into such final state. Instead of it the pendulums will periodically exchange the energy and stop by turns.
Swapping of energy
Typical example of oscillatory system in which there is periodic
"swapping" of energy is two identical pendulums hung to a
horizontal rod which in turn is suspended on two cords. Pendulums are "connected" with each other due to a
rod. If we swing one pendulum in
a plane perpendicular to rod, then the oscillatory energy will begin to be transferred
to other
pendulum. After the end of this process the energy will be transferred in the
opposite direction. Thus, there is a periodic exchange of energy between the
pendulums in this case. As a result the amplitude of oscillation of every
pendulum is
periodically changed. When one pendulum oscillate with the maximal
amplitude the other one is still.
Modes of oscillation
Next we shall consider the system of two identical pendulum connected with a
spring as shown in animation.
Motion
of pendulums can be analyzed considering so-called normal modes of oscillation, which represent the certain
types of oscillatory motion of a system. In such an oscillation the exchange of energy between pendulums
is absent and the amplitude of oscillation (for example, the angle of deviation
from the vertical) for each pendulum remains a constant. Normal modes are
associated with symmetry of the pendulum motion. Let us assume that the pendulums are
deviated by identical angles to the left and then released. Oscillating, they do
not change the length of a spring (we can assume that there is no spring at
all), so there is no transfer of energy from one pendulum to another. Such synchronous
oscillation is called normal.
The second normal mode can be demonstrated if we
deviate the pendulums
by equal angles, but in the opposite directions. If we release the pendulums simultaneously,
they will oscillate by such a way that their movements will be a mirror reflections each other.
In such an oscillation the spring is stretched and compressed periodically and consequently
it influences the pendulum motion. The spring acts on the first pendulum with
the same force it acts on the second pendulum. So, for example, pushing the first
pendulum to the left, the spring pushes the second pendulum with the same force to the right. Symmetry of forces
prohibits the energy to be transferred from one pendulum to another and
therefore the amplitudes of oscillation of both pendulums remain constant.
If the pendulum oscillate freely, then the frequency of oscillation is
proportional to square root of the acceleration of free fall g
to length of a pendulum
l. The first normal
mode is characterized exactly by such a frequency. Frequency of the second normal mode is
higher, because the spring influences the motion of pendulums. For example, pendulums which move towards each
other are acted upon by not only the force of gravity, but also the force of
spring pushing the pendulums aside. When pendulums are moving away each other, then
the spring pulls them together. Pushing and pulling actions of a spring result in increase
of the frequency of free oscillation (resonance frequency). The knowledge of normal
modes of oscillation is important because
any movements of an oscillatory system is the superposition of normal modes.
Beats of oscillation.
Did you listen how two pure tones of close frequencies sound? The ear perceives the oscillation of these two tones not separately, but with the frequency equal to average frequency of two sound waves. The sound periodically increases and weakens. The frequency of this variation is equal to difference in frequency between two tones. This phenomenon is called the beats.
Let us assume that we "launch"
the system deviating one pendulum aside and keeping hold the other one by hand.
When the deviated pendulum is released, it begins the oscillation in the
left-right direction pulling and pushing with a spring the other pendulum. The spring prevents
the oscillation of the first pendulum, but helps to the second pendulum and thus provides
the transfer of energy from the first pendulum to the second one. As the first pendulum will lose all
the energy, the return process begins. The behavior of every pendulum can be
described with the aid of the product of two sinusoids. One of these sinusoids
is fast - its
frequency is equal to average frequency of two normal modes. When a pendulum
oscillates, it oscillates exactly with such a frequency. Other
sinusoid is slow and it describes the periodic variation of amplitude of
oscillation. The frequency of such variation is equal to a difference of the frequencies of
normal modes and, thus, it is less than the frequency of the pendulum
oscillation. At the maximal
amplitude the pendulum possesses all the energy of a system. The zero amplitude means, that
the pendulum is still. Amplitudes of pendulum change in antiphase; when one of
them oscillate with the maximum amplitude, the second one is still. If we set
the system
in motion by a different way, the pendulums will oscillate differently, however
the beats will be observed too; i.e. the amplitudes of oscillation will change.
Parametrical excitation of oscillation.
To completely other type the so-called resonant pendulum on a spring
concerns. In this case the metal ball hangs on a spring, which is fixed in turn
by one end to the board (as shown in animation). If we pull a ball down and
release it there, the pendulum will oscillate in the vertical direction and then
it gradually begins to swing from side to side similarly to a physical pendulum
described above. After a while the horizontal oscillation will cease and vertical
oscillation will arise again. The system will behave by the same way if the ball
is deviated aside and released.
Since that moment as we have drawn a ball downwards, the total energy of the oscillatory system remains a constant. Hence, the energy of horizontal oscillation (swinging) is borrowed from the energy of vertical oscillation. When the ball swings with the maximal amplitude, the stream of energy reverses: vertical oscillation begins to grow and swinging of a pendulum becomes weaker. Swapping of energy is carried out by optimum way if the ball stretches the spring at 4/3 of its original length and the frequency of vertical oscillation equals the double frequency of the resonance swinging. In this case the arising instability is sufficient that the ball would begin a swinging like a physical pendulum.
During the vertical oscillation of a pendulum, the oscillatory system demonstrates an example of so-called parametrical excitation of oscillation. The parameter, which is changed in this case, is the length of a pendulum. As a result of a small fluctuation the pendulum deviates in the horizontal direction and at this moment the energy of the vertical oscillation is transferred into the horizontal oscillation. The pendulum begins the swinging, which increase further the energy flow into this mode of oscillation. If the frequency of the vertical oscillation is equal to double frequency of swinging we can say about the case of parametric resonance. When the swinging of pendulum prevails, the energy is transferred to vertical oscillation due to a usual resonance: the spring is periodically stretched with the resonance frequency of its vertical oscillation. Each time when the ball deviates by a maximal angle from vertical direction it stretches a spring. As for every period of swinging the spring appears to be stretched twice, the frequency of the spring excitation coincides with its resonance frequency. Gradually energy is transferred from swinging to vertical oscillation. (Due to a parametrical resonance, in particular, it is possible to shake a swing. Periodic displacement of the center of gravity of a body changes the effective length of a pendulum - in this case a swing, - that in turn results in increase of energy of oscillation).
Linked oscillators.
5 oscillators (balls on the springs) with equal resonance frequency hang on a common beam which in turn can also oscillate on two springs. This beam plays a role of the conductor of the vibrating energy from the central oscillator to the lateral ones and back. As a result we see that the oscillation of a central ball periodically rises and fades. The amplitude of oscillation of the lateral balls is changed too though these variations are much smaller. When the central ball comes to a standstill the lateral balls oscillate with the maximal amplitude, that corresponds to the energy conservation law.
References: Scientific American, October 1985, vol.253, No.4
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