Attachment losses can play a role in limiting the quality factors of micro/nanomechanical oscillators. The existing theoretical results in this regard are applicable to highly idealized scenarios. The theory has been extended in two important directions: the width of the cantilever is considered to be small relative to a wavelength as opposed to large, and the base is allowed to have finite thickness. These extensions result in significant, in many cases order of magnitude, changes in the estimates of attachment loss. Simple formulas for Q−1 covering most of the parameter range are given.

Analytical expressions are provided for the energy loss from vibrating mechanical resonators into their support structures for two limiting cases: supports that can be treated as plates, and supports that act as semi-infinite elastic media, with effectively infinite thickness. The former case is applicable to many microscale resonators, while the latter is appropriate for nanoscale devices. General formulations are given, applicable to a wide range of resonator geometries. These formulations are then applied to two geometries commonly used in microelectromechanical systems and nanelectromechanical systems applications: cantilevered beams and doubly fixed beams. Experimental data are presented to validate the finite-thickness support theory, and the predictions of the theory are also compared to data from existing literature for a microscale rectangular paddle oscillator.

Simple models are presented for estimating viscous damping of fluid (gas or liquid) loaded mechanical resonators. The models apply to beams in flexural modes of vibration, and to thin beams and plates in longitudinal modes of vibration. Predictions of the associated quality factor are compared with measured values for several macroscale and microscale resonators. The scaling of viscous loss with oscillator size is discussed. The minimum detectable mass is estimated for several oscillator designs and it is shown that, for comparably sized devices, longitudinal resonators have the lowest threshold of detection. This minimum detectable mass is proportional to scale to the power 1.75 for all resonator architectures limited by viscous damping, and it is shown that the viscous loss is 220 times larger in water than in air.