Effect of viscous loss on mechanical resonators designed for mass detection

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.