The transient response of a resonant structure can be altered by the attachment of one or more substantially smaller resonators. Considered here is a coupled array of damped harmonic oscillators whose resonant frequencies are distributed across a frequency band that encompasses the natural frequency of the primary structure. Vibration energy introduced to the primary structure, which has little to no intrinsic damping, is transferred into and trapped by the attached array. It is shown that, when the properties of the array are optimized to reduce the settling time of the primary structure's transient response, the apparent damping is approximately proportional to the bandwidth of the array (the span of resonant frequencies of the attached oscillators). Numerical simulations were conducted using an unconstrained nonlinear minimization algorithm to find system parameters that result in the fastest settling time. This minimization was conducted for a range of system characteristics including the overall bandwidth of the array, the ratio of the total array mass to that of the primary structure, and the distributions of mass, stiffness, and damping among the array elements. This paper reports optimal values of these parameters and demonstrates that the resulting minimum settling time decreases with increasing bandwidth.
We explore vibration localization in arrays of microresonators used for ultrasensitive mass detection and describe an algorithm for identifying the location and amount of added mass using measurements of a vibration mode of the system. For a set of sensing elements coupled through a common shuttle mass, the inter-element coupling is shown to be proportional to the ratio of the element masses to the shuttle mass and to vary with the frequency mistuning between any two sensing elements. When any two elements have sufficiently similar frequencies, mass adsorption on one element can result in measurable changes to multiple modes of the system. We describe the effects on system frequencies and mode shapes due to added mass, in terms of mass ratio and frequency spacing. In cases in which modes are not fully localized, frequency-shift-based mass detection methods may give ambiguous results. The mode-shape-based detection algorithm presented uses a single measured mode shape and corresponding natural frequency to identify the location and amount of added mass. Mass detection in the presence of measurement noise is numerically simulated using a ten element sensor array. The accuracy of the detection scheme is shown to depend on the amplitude with which each element vibrates in the chosen mode.
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