Dynamics and bifurcations of a column-pendulum oscillator: theory and experiment
This dissertation is concerned with the dynamics and bifurcations of a large flexible column with a tip mass-pendulum arrangement. Throughout, the dynamical systems approach is emphasized. The system may be thought of as a conceptualization of a vibration absorbing device for large flexible structures with tip appendages. The excitation is along the axial direction of the undeformed column.
The research comprises of obtaining time-averaged dynamics via the Krylov- Bogoliubov averaging theorem. The solution (bifurcation) diagrams are obtained numerically, by the pseudo-arclength continuation algorithm. The bifurcation diagrams indicate that the system loses stability via two distinct routes. One leading to a saddle-node bifurcation, normally associated with the jump phenomena. The second instability is due to the Hopf bifurcation, that results in amplitude modulations or motion on an invariant torus. A parameter range has been identified where these two types of bifurcation coalesce, this phenomenon has important global ramifications, in the sense that as the Hopf bifurcation point approaches the saddle-node; the periodic modulations associated with the Hopf bifurcation tend to have an infinite period. This is a strong indicator of existence of homoclinic orbits. In addition to the regular solution branches that bifurcate from the zero solution, the system also possesses isolated solutions (the so-called "isolas") that form isolated loops bounded away from zero. As the forcing amplitude is varied, the isolas appear, disappear or coalesce with the regular solution branches. The response curves indicate that the column amplitude shows saturation. The pendulum acts as a vibration absorber over a range of frequency where the column response is saturated. However, there is also a frequency range over which a reverse flow of energy occurs, where the pendulum shows reduced amplitude at the cost of large amplitudes of the column.
The experimental analysis required an accurate measurement of the angular displacement of the pendulum. To accomplish this, an opto-digital angular measurement device was developed by incorporating an optical encoder with a digital programmable controller. The phase-space of the experimental system is reconstructed via Takens Embedding technique, by embedding the phase space in delay coordinates. GenericaUy, the state can be reconstructed by delaying the measured quantities. The results of the experimental dynamics indicate that as one sweeps through the resonance region under investigation, the periodic motion breaks down; and quasiperiod motion is observed confirming the existence of invariant tori. Furthermore, within the quasiperiodic regions, there are windows containing intricate webs of phase-locked periodic responses. The quasiperiodic and the phase-locked responses are clearly visualized on the cover of the torus. Increasing the amplitude of the excitation, results in distortion of the invariant 2-torus due to the resonance overlap. This results in the non-invertibility of the first return maps extracted from the experimental data. Furthermore, a burst of frequencies appear on the Fourier spectrum. This scenario is similar to many experimental observations of hydrodynamical instabilities; the break-up of 2-tori in these experiments is related to the onset of turbulence.