Micro and nanoelectromechanical systems M/NEMS have been extensively investigated and exploited in the past few decades for various applications and for probing fundamental physical phenomena. Understanding the linear and nonlinear dynamical behaviors of the movable structures in these systems is crucial for their successful implementation in various novel technologies and to meet the long list of new sophisticated requirements. This paper presents a review for some of the recent topics pertaining to the dynamical behaviors, linear and nonlinear, of M/NEMS resonating structures. First, an overview is presented of the various used dynamical approaches to enhance the sensitivity of resonators for sensing applications. Then a summary is presented of the recent works on the linear and nonlinear mode coupling in M/NEMS resonator. Next, recent research is reviewed on coupled M/NEMS resonators, mechanically and electrically, leading to collective behaviors like mode localization. The final part of the paper discusses analytical approaches that have been developed to better understand and investigate the dynamical behavior of M/NEMS resonators focusing on the perturbation method the multiple time scales.
We investigate in a silicon micromachined arch beam the activation of a one-to-one internal resonance between the first symmetric and first antisymmetric modes simultaneously with the activation of a two-to-one internal resonance between these modes and the second symmetric mode. The arch is excited electrically, using an antisymmetric partial electrode to activate both modes of vibrations, and tuned electrothermally via Joule's heating. Theoretically, we explore the dynamics of the beam using the Galerkin and multiple time scales methods. The simulation results are shown to have good agreement with the experimental data. The results show the merging of both modes at crossing, after which the first antisymmetric mode exchanges the nonlinear behavior with the first symmetric mode. The nonlinear behavior of the arch beam is demonstrated and analyzed experimentally and theoretically as experiencing the simultaneous 2:1 and 1:1 internal resonances.
We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.
We investigate, experimentally and theoretically, the linear mode coupling between the first symmetric and antisymmetric modes of an electrothermally tuned and electrostatically actuated micromachined arch resonator. The arch is excited using an antisymmetric partial electrode to activate both modes of vibrations. Theoretically, we explore the static and dynamic behavior using the Galerkin method. When tuning the electrothermal voltage, the first symmetric frequency increases while the first antisymmetric frequency decreases until they cross. The results show linear coupling and hybridization of both modes near crossing only in the presence of the perturbation from the electrostatic force using the partial electrode. We show the linear merging of both modes at crossing. Also, the eigenfrequency variation around crossing shows a ratio of 2:1 between the second symmetric mode and the first symmetric/antisymmetric modes, which can lead to simultaneous 1:1 and 2:1 internal resonances.
We present an investigation of the dynamic behavior of an electrostatically actuated clampedclamped microbeam, under the simultaneous excitation of primary and subharmonic resonance. The simultaneous excitation of primary and subharmonic resonances of similar strength is experimentally investigated by using different combinations of AC and DC voltages. It is observed that the response of the resonator is governed by a mixed effect of both excitations. Subharmonic dominated response shows sharp amplitude transitions and smaller monostable regime, while primary dominated response shows gradual amplitude transition and larger monostable regime. Two potential applications are experimentally demonstrated. The first is a resonator based MEMS AND logic gate based on AC only subharmonic excitation. The second is a charge sensor based on the transition from subharmonic dominated response to primary dominated response, which is potentially capable of detecting a small amount of electric charges.
We present an investigation of the dynamic behavior of an electrostatically actuated resonant structure, resonator, under the simultaneous excitation of primary and subharmonic resonances. A comprehensive analytical solution is obtained via the method of Multiple Time Scales (MTS), which is applicable for generic electrostatic loading cases. Results using different MTS scaling methods in the equations of motion and loading conditions are compared. These results are further verified against results obtained using direct time integration of the equation of motion. It is observed that for a generic parallel plate electrostatic loading case, the direct forcing component at the excitation frequency, and the direct and parametric excitation components at double the excitation frequency must be considered for accurate prediction of the structure's response. Further, the case of simultaneous excitations of primary and subharmonic resonance, where both excitations are of comparable strength, is examined under various electrostatic loading conditions. We show mixed behaviours of the resonator transiting from a subharmonic dominated response, characterized by the sudden jumps in amplitude and smaller monostable regime, to primary dominated response 2 exhibiting gradual amplitude increase and larger monostable regimes. This transition behaviour can be potentially used for applications, such as electrometers.
In this work, we investigate numerically the linear vibrations of inclined risers using the Galerkin approach. The riser is modeled as an Euler-Bernoulli beam accounting for the nonlinear mid-plane stretching and self-weight. After solving for the initial deflection of the riser due to self-weight, a Galerkin expansion of fifteen axially loaded beam mode shapes are used to solve the eigenvalue problem of the riser around the static equilibrium configuration. This yields the riser natural frequencies and exact mode shapes for various values of inclination angles and applied tension. The obtained results are validated against a boundary-layer analytical solution and are found in good agreement. This constructs a basis to study the nonlinear forced vibrations of inclined risers.
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