Pattern formation in 2-frequency forced parametric waves

We present an experimental investigation of superlattice patterns generated on the surface of a fluid via parametric forcing with 2 commensurate frequencies. The spatio-temporal behavior of 4 qualitatively different types of superlattice patterns is described in detail. These states are generated via a number of different 3--wave resonant interactions. They occur either as symmetry--breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two primary unstable modes generated by the two forcing frequencies. A coherent picture of these states together with the phase space in which they appear is presented. In addition, we describe a number of new superlattice states generated by 4--wave interactions that arise when symmetry constraints rule out 3--wave resonances.Comment: The paper contains 34 pages and 53 figures and provides an extensive review of both the theoretical and experimental work peformed in this syste

Parametric and autoparametric resonance

Parametric and autoparametric resonance play an important part in many applications while posing interesting mathematical challenges. The linear dynamics is already nontrivial whereas the nonlinear dynamics of such systems is extremely rich and largely unexplored. The role of symmetries is essential, both in the linear and in the nonlinear analysis

Bifurcation scenarios, dynamical integrity and control of noncontact atomic force microscopes

The research focuses on the description of the global dynamical behavior of a reduced-order model of noncontact Atomic Force Microscope. Different numerical analyses and continuation techniques are carried out to investigate the evolution of the main system periodic solutions and relevant basins of attraction under variations of the most significant system parameters. Local bifurcations, stability boundaries and basin erosion processes around primary and subharmonic resonance regions are studied in presence of both the parametrical horizontal excitation and the external one, and the obtained behavior charts are used not only to compare the results with the literature ones, but also as practical instruments to characterize the operation ranges in terms of the selected parameters. With the same perspective, dynamical integrity concepts, such as detection of basins of attraction, and quantification of their erosion process via integrity measures, are applied to determine acceptable frequency-dependent thresholds associated with a priori safe design targets. Furthermore, an external feedback control is introduced with the aim to take the system response to a selected reference one, thus providing a simple and efficient method to avoid possible unstable motions. Upon checking the effectiveness of the procedure in the weakly nonlinear regime via a perturbation approach, several numerical analyses in the strongly nonlinear regime are accomplished to achieve a description of its dynamical behavior as a function of the newly inserted parameters, and to critically evaluate the effectiveness of the control actuation on the system dynamics, with also a view to the overall response scenario

Parametrically amplified Mathieu-Duffing nonlinear energy harvesters

The steady-state response of a nonlinear piezoelectric energy harvester subjected to external and parametric excitation is investigated based on the Mathieu-Duffing nonlinear oscillator model. The parametric excitation is introduced to amplify the external harmonic excitation and extend the capabilities of the nonlinear piezoelectric energy harvester device. To obtain the approximated solution of the nonlinear periodic responses for displacement and electrical voltage of the energy harvester, the incremental harmonic balance method in combination with the path-following technique is adopted. It is assumed that the proposed nonlinear model consists of cubic and quadratic nonlinearity, where parametric amplification appears in the form of a trigonometric function. The frequency is tuned as one-to-one and the one-to-two ratio between external and parametric excitation. The effects of quadratic and cubic nonlinearity as well as parametric amplification are studied in detail, and their incredible properties to extend harvester application performance is illustrated. It is explicitly demonstrated that for some particular combination of the system parameters, vibration amplitudes and harvested power can be amplified up to three or five times in comparison to the classical broadband nonlinear energy harvester based on the forced Duffing oscillator. This extraordinary amplification shown to be a key motivation to realize the proposed concept in practice. The presence of combined quadratic and cubic nonlinearities resulted in both hardening and softening spring behavior and leading to the appearance of coexisting periodic solutions in the amplitude-frequency responses. Periodic orbits obtained by the proposed methodology are verified with the results from direct numerical integration and fine agreement is demonstrated. Moreover, a significant influence of the parametric amplification on the instantaneous power is revealed in time response diagrams, thus showing better performance of the proposed energy harvester system

Parametric Excitation of Coupled Nonlinear Microelectromechanical Systems

The commencement of the semi-conductor industry in the second half of the last century gave a surprising new outlook for engineered dynamical mechanical systems. It enabled, thanks to the continuously evolving microfabrication methods, the implementation of Micro Electromechanical systems (MEMS) followed by their nano-counterpart or NEMS. Nowadays M/NEMS constitute a massive portion of the small-scaled sensors industry, in addition to electrical, optical and telecommunication components. Since these tiny dynamical electromechanical systems involve sometimes couplings between degrees of freedom as well as nonlinearities, the theory of stability in dynamical systems plays a significant role in their design and implementation. From a practical point of view, the approach to stability problems often takes two different perspectives. The first one, most commonly in linear systems, aims to avoid any instability which could cause destructive consequences for mechanical structures or for electrical and electronic components. On the contrary in nonlinear systems, the second perspective aims to drive the system into regions of instability for the trivial solution, while searching for stable nontrivial steady-state solutions of the underlying differential equations. With the advent of micro and nanosystems, the second perspective could acquire increased importance. This is attributed to their capability to exhibit typical nonlinear behavior and higher amplitudes at normal operation conditions, when compared to macroscale systems. Higher amplitudes, in this sense, allows for a better amplification of an input excitation, and thereby higher sensitivity for miniature sensors and measurement devices. In addition, if the system parameters were time-periodic, the trivial solution could turn to be unstable at the so called parametric resonances. Known as parametric pumping in micro and nanosystems, the system’s response is usually amplified at these resonance frequencies for higher sensitivity and accuracy. For these reasons, this work is mainly focused on parametrically excited nonlinear systems. Nevertheless, a systematic approach is followed in this thesis, where the origins of destabilization are surveyed in time-invariant systems before proceeding to carry out a theoretical study on time-periodic systems in general, and time-periodic nonlinear systems in particular. Through this theoretical study, a novel idea for the M/NEMS industry is presented, namely the broadband parametric amplification using a bimodal excitation method. This idea is then implemented in microsystems, by investigating a particular example, that is the microgyorscope. Given the low-cost of this device in comparison with other inertial sensors, it is being currently enhanced to reach a relatively higher sensitivity and accuracy. To this end, the theoretical findings, including the mentioned idea, are implemented in this device and prove to contribute effectively to its performance. Moreover, an experimental investigation is carried out on an analogous microsystem. Through the experimental study, an electronic system is introduced to apply the proposed bimodal parametric excitation method on the microsystem. By comparing the stability charts in theory and experiment, the theoretical model could be validated. In conclusion, a theoretical study is carried out through this work on parametrically excited nonlinear systems, then implemented on microgyroscopes, and finally experimentally validated. Thereby, this work puts a first milestone for the utilization of the proposed excitation method in the M/NEMS industry