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This purpose of this experiment was to observe and understand chaos in the mechanical system of a damped and driven physical pendulum. The pendulum system consisted of a small disk with an edge mass and a linear spring force connecting the mechanical driver to the disk. The system was damped with a small magnet located behind the rotating disk. In a simple pendulum the restoring force is usually approximated to vary linearly with position, however this approximation is incorrect when the mass of the pendulum is displace from its equilibrium position by an angle of more that 15 degrees. Therefore for large angles of displacement the restoring force of the pendulum is non-linear. Accordingly the system was made non-linear by driving the pendulum so the angle of displacement would be larger than 15 degrees. Using a program named Pasco's Science Workshop, it was possible to record the angular velocity and angular position of the pendulum over a long period of time. |
In this experiment we focused on the non-linear oscillatory ideas that at resonance there is not just one amplitude possible and that a periodic driving force does not always yield a periodic output. It was explained in an earlier discussion that non-linear systems may sometimes oscillate at multiple amplitudes for the same initial parameters because the restoring force does not vary linearly with time. It is also true that if the restoring force is non-linear, then the potential energy of the system is not parabolically shaped, but rather more shaped like a saddle. It can be seen from the potential energy diagram, that for the pendulum system there are two smaller wells at the bottom of the potential energy saddle. Due to the non-linear form of the potential energy well it was possible for the system to oscillate at two different amplitudes for a single frequency. If the system has an energy small enough to remain in one of the smaller potential energy wells we might expect to see the stable harmonic motion and resonant behavior characteristic of a linear system. Indeed the system will resonate at the bottom of the well at a particular frequency v' if the total energy of the system is less than the edges of the smaller potential energy wells. It is evident then, that for a parabolic potential well, the period of oscillation is constant for any system energy. However, since the entire potential well for this system is not parabolically shaped, the period of oscillation is actually a function of the total energy of the system. Therefore there is not really a unique resonant frequency for the system. Non-linear oscillatory effects are evident when the system is driven at the frequency v', which is resonant at small system energies, and the system energy is then increased by increasing a parameter of the driving force. Once the energy of the system is increased beyond the edges of the smaller potential energy wells, the system's oscillations become out of phase with the driving frequency v'. This clearly demonstrates that a periodic input does not always produce a periodic output in non-linear oscillatory systems.
This seemingly erratic and unpredictable behavior in the oscillation of the system can be described as chaotic behavior. However, the true presence of chaos is actually rather difficult to ascertain in a mechanical system. Often during the experiment the system may at first look chaotic, but after some time the oscillations will settle into a periodic motion. The presence of these transients then must be allowed to die out before we could determine whether or not the system was chaotic at a certain driving frequency. By observing the dynamics of the system in a phase diagram we were able to clearly evaluate the oscillatory behavior and decide if it was periodic or chaotic. The following two graphs are phase plots of a transient dying out and true chaotic behavior. The first phase plot demonstrates the deceptive nature of a transient as well as the normal periodicity of a linear system. The second phase plot demonstrates the nature of chaos, in which no phase loop will ever complete itself.
From the experiment we made the following observations about chaotic behavior in our system. First, chaos occurs only where the total energy of the system was great enough to allow oscillation between two potential wells. Chaotic motion could never be observed when the system is oscillating in just one potential well because then it would cease to be non-linear. Secondly, chaos was most easily found when the system was first driven near the resonant frequencies for the bottom of the smaller potential wells and then the driving force was increased. However, it should be noted that these were not the only conditions where chaos was found. Chaotic behavior could also be observed in various, small ranges of frequencies. In conclusion, non-linear oscillatory systems can be driven under certain conditions so that they exhibit non-periodic or chaotic behavior. In this respect the concepts of oscillation and resonance are drastically altered when a system is no longer linear.