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Rotating machines can be modeled at a basic level using lumped masses that are rotating about and attached using springs to an axis. Even such seemingly simple system can exhibit rich dynamics in the presence of time-varying terms in the governing differential equations. This paper investigates the dynamics of a rigid body with two attached rotors that rotate in the same plane. The system is parametrically-excited and the equations of motion are periodic in both rotor frequencies. The frequency spectra of the time responses show distinct side-band structures centered about the unforced natural frequencies. In addition to classical resonances, the stability diagrams generated using Floquet theory reveal instabilities at unexpected combinations of the forcing and natural frequencies. The harmonic balance method is employed to verify the stability boundaries obtained using Floquet theory. The study reveals safe regimes of parameter combinations that can help prevent the onset of instability in such systems.

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An important but often overlooked factor that affects the performance of a meso/micro electro mechanical vibratory sensor is the structural interaction between the sensor's resonator and the substrate on which it is mounted. Situating resonators at node points eliminates this interaction and thereby helps to improve a resonator's quality-factor for a particular mode of vibration. This paper addresses the problem of locating a single degree of freedom spring-mass resonator on a generic cantilever substrate. The loci of natural frequencies obtained when the resonator's mounting location is varied are developed, and the nodal locations are identified. Thereafter a method to obtain these locations from the characteristic equation without solving the associated eigenvalue problem is described. Lookup tables detailing the nodal locations and the corresponding natural frequencies for various resonator parameters are presented. It is found that at these special nodal locations, the magnitude of the power transmitted through anchors is negligible, which ensures minimal structural interaction between the resonator and the substrate.

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