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A unique T-beam microresonator designed to operate on the principle of nonlinear modal interactions due to 1 : 2 internal resonance is introduced. Specifically, the T-structure is designed to have two flexural modes with natural frequencies in a 1 : 2 ratio, and the higher frequency mode autoparametrically excites the lower frequency mode through inertial quadratic nonlinearities. A Lagrangian formulation is used to model the electrostatically actuated T-beam resonator, and it includes inertial quadratic nonlinearities, cubic nonlinearities due to midplane stretching and curvature of the beam, electrostatic potential, and effects of thermal prestress. A nonlinear two-mode reduced-order model is derived using linear structural modes in desired internal resonance. The model is used to estimate static pull-in bias voltages and dynamic responses using asymptotic averaging. Nonlinear frequency responses are developed for the case of resonant actuation of a higher frequency mode. It is shown that the lower frequency flexural mode is excited for actuation levels above a certain threshold and generates response component at half the frequency of resonant actuation. The effects of damping, thermal prestress, and mass and geometric perturbations from nominal design are thoroughly discussed. Finally, experimental results for a macroscale T-beam structure are briefly described and qualitatively confirm the basic analytical predictions. The T-beam resonator shows a high sensitivity to mass perturbations and, thus, holds great potential as a radio frequency filter-mixer and mass sensor.[2008-0107]

A novel microresonator operating on the principle of nonlinear modal interactions due to autoparametric 1:2 internal resonance is introduced. Specifically, an electrostatically actuated pedal-microresonator design, utilizing internal resonance between an out-of-plane torsional mode and a flexural in-plane vibrating mode is considered. The two modes have their natural frequencies in 1:2 ratio, and the design ensures that the higher frequency flexural mode excites the lower frequency torsional mode in an autoparametric way. A Lagrangian formulation is used to develop the dynamic model of the system. The dynamics of the system is modeled by a two degrees of freedom reduced-order model that retains the essential quadratic inertial nonlinearities coupling the two modes. Retention of higher-order model for electrostatic forces allows for the study of static equilibrium positions and static pull-in phenomenon as a function of the bias voltages. Then for the case when the higher frequency flexural mode is resonantly actuated by a harmonically varying AC voltage, a comprehensive study of the response of the microresonator is presented and the effects of damping, and mass and structural perturbations from nominal design specifications are considered. Results show that for excitation levels above a threshold, the torsional mode is activated and it oscillates at half the frequency of excitation. This unique feature of the microresonator makes it an excellent candidate for a filter as well as a mixer in RF MEMS devices.

The dynamics of a resonantly excited thin cantilever with an active controller are investigated experimentally. The controller mimics a passive wideband absorber discussed in the accompanying theory paper. Lead-zirconate-titanate patches are bonded to both sides of the beam to actuate it, while an electromagnetic shaker drives the beam near resonance. An active controller consisting of an array of coupled controllers is developed, such that the governing equations for the controller are quadratically coupled to the resonating system. The control signal, in terms of the motion of the controllers, is quadratically nonlinear. It is shown that the slow timescale equations of this physical system are identical in form to those for the passive wideband vibration absorber. The controller is implemented using modelling software and a controller hardware board. Two sets of experiments are performed: one with a constant excitation frequency and the other with a linearly varying excitation frequency at a slow sweep rate (non-stationary excitation). The experimental results verify the analysis presented for the passive wideband autoparametric vibration absorber. The experiments also demonstrate the effectiveness of the absorber in reducing the response amplitude of structures, and its robustness to frequency mistuning.

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