Synchronization in coupled mechanical resonator arrays

Opto-mechanical systems are based on the nonlinear coupling between the electromagnetic field in a resonator and an array of bulk mechanical resonators such that the frequency of the electromagnetic field resonator depends on the displacement coordinates of each of the mechanical resonators. In this thesis, three cases are considered: the first consists of a single array of mechanical resonators interacting with a common field for which the frequency of the electromagnetic resonance is tuned to depend quadratically (to lowest order) on the displacement of the resonators. Two such systems are coupled together through weak interactions of their common fields to form the second case considered in this thesis. For the third case, two systems, each whose frequency of the electromagnetic resonance is tuned to depend linearly (to lowest order) on the displacement of the two corresponding arrays of resonators, are weakly coupled together. By using the method of amplitude equations around a critical point, the synchronization properties of these three systems are found. For the first system, it is found that groups of near-identical bulk mechanical resonators with low driving fail to synchronize unless their natural frequencies are identical, in which case the resulting system can exhibit multiplicity. For the second system, it is shown that the average phases of the two arrays of near-identical bulk mechanical resonators with low driving fail to synchronize with each other unless the natural frequencies of the resonators within each array are identical and their parameters are selected so that these compensate the de-synchronizing effect of the coupling between the fields. For the third system, despite the de-synchronizing effect of the coupling between the fields, it is shown that the average phases of the two arrays of identical bulk mechanical resonators with low driving synchronize with each other unless the natural frequencies within each array are sufficiently non-identical. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the General Award Rules of The University of Queensland, immediately made available for research and study in accordance with the Copyright Act 1968. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis.