Dynamics of PDEs

In this project I studied the dynamics of micro devices, known as micro-electromechanical systems, comprised of arrays of vibrating beams excited by electric currents. The goal is to model nonlinearities, observed in experiments, study how they arise and what dynamics can be expected as a result.

My systematic approach was to derive a planar continuum model based on the physical properties, such as an asymmetrical beam configuration and nonlinear material properties, rather than using a lumped mass model. The resulting initial-boundary value problem, an integro-partial differential equation, is reduced to a nonlinear modal dynamical system with parameteric excitation by making a Galerkin ansatz. The reduced order model was analysed using perturbation theory as these systems are characterised by small parameters (linear damping and AC voltage). The bifurcation structure studied using continuation methods (MATLAB and MatCont) reveals a period doubling route to chaos.

The main and novel result of this project lies in understanding how parametric resonance and internal resonances arise in the absence of symmetry within the design configuration and how synchronisation within the beam dynamics is affected. The goal is model improvement and comparison of simulations with experimental results, which could ultimately lead to design improvement of nano- and microelectromechanical devices implemented, for example, in mass sensors.

K. Mora, O. Gottlieb
Parametric excitation of a micro-beam-string with asymmetric electrodes – multimode dynamics and the effect of nonlinear damping,
(Journal, ResearchGate)

P. Kambali, K. Mora, O. Gottlieb
The influence of asymmetric electrodes on the non-planar dynamics of a parametrically excited nonlinear microbeam,

P. Kambali, K. Mora, O. Gottlieb
The influence of imperfections on the spatio-temporal dynamics of a parametrically excited nonlinear viscoelastic micro-beam-string,

aperiodikDynamics of PDEs