Here we study viscous oscillatory nanoflows generated in a fluid by mechanical oscillations of miniaturized resonators. In particular, we focus on the limits of two-dimensional cylinder theory, which approximates a slender nanoresonator, such as an atomic force microscopy microcantilever or a nanobeam resonator, as a cylinder oscillating in a fluid. The cylinder theory is the mainstay in the micro- and nanoelectromechanical systems literature, but is only accurate in certain regimes. We observe and explain two distinct routes to the breakdown of the cylinder theory. First, when a substrate is present, squeeze film effects become significant and the cylinder theory underpredicts the fluid force applied to the resonator. Second, the cylinder theory overpredicts the fluid force when axial flow becomes relatively large. This occurs at higher bending modes where the spatial gradients along the beam are larger. A dimensionless squeeze number and axial flow number are introduced to facilitate in-depth investigations into the limitations of the two-dimensional cylinder theory. Results from experiments and three-dimensional finite-element models are presented in order to illustrate where two-dimensional flow models break down.

The oscillatory dynamics of nanoelectromechanical systems (NEMS) is at the heart of many emerging applications in nanotechnology. For common NEMS, such as beams and strings, the oscillatory dynamics is formulated using a dissipationless wave equation derived from elasticity. Under a harmonic ansatz, the wave equation gives an undamped free vibration equation; solving this equation with the proper boundary conditions provides the undamped eigenfunctions with the familiar standing wave patterns. Any harmonically driven solution is expressible in terms of these undamped eigenfunctions. Here, we show that this formalism becomes inconvenient as dissipation increases. To this end, we experimentally map out the position- and frequency-dependent oscillatory motion of a NEMS string resonator driven linearly by a non-symmetric force at one end at different dissipation limits. At low dissipation (high Q factor), we observe sharp resonances with standing wave patterns that closely match the eigenfunctions of an undamped string. With a slight increase in dissipation, the standing wave patterns become lost, and waves begin to propagate along the nanostructure. At large dissipation (low Q factor), these propagating waves become strongly attenuated and display little, if any, resemblance to the undamped string eigenfunctions. A more efficient and intuitive description of the oscillatory dynamics of a NEMS resonator can be obtained by superposition of waves propagating along the nanostructure.

Here, we study the acoustic radiation generated by the vibration of miniaturized doubly clamped and cantilever beam resonators in viscous fluids. Acoustic radiation results in an increase in dissipation and consequently a decrease in the resonator’s quality factor. We find that dissipation due to acoustic radiation is negligible when the acoustic wavelength in the fluid is much larger than the bending wavelength. In this regime, dissipation is primarily due to the viscous losses in the fluid and may be predicted with the two-dimensional cylinder approximation in the absence of axial flow and substrate effects. In contrast, when the bending wavelength approaches the length of the acoustic wavelength, acoustic radiation becomes prominent. In this regime, dissipation due to acoustic radiation can no longer be neglected, and the cylinder approximation inaccurately characterizes the total energy loss in the system. Experiments are performed with microcantilevers of varying lengths in Ar and ${\mathrm{N}}_2$ to observe trends in the acoustic wavelength of the fluid and bending wavelength. Additional experimental results from doubly clamped nanoelectromechanical system beams are also presented. Experimental results illustrate an increase in dissipation, which is further analyzed with the use of three-dimensional finite element models. With the numerical simulations, we calculate the radiation efficiency of the measured devices and analyze the pressure fields generated by the vibrating resonators. This analysis allows one to estimate the effects of acoustic radiation for any resonator.

A method is presented to improve the estimates of material properties, dimensions, and other model parameters for linear vibrating systems. The method improves the estimates of a single model parameter of interest by finding parameter values that bring model predictions into agreement with experimental measurements. A truncated Neumann series is used to approximate the inverse of the dynamic stiffness matrix. This approximation avoids the need to directly solve the equations of motion for each parameter variation. The Neumman series is shown to be equivalent to a Taylor series expansion about nominal parameter values. A recursive scheme is presented for computing the associated derivatives, which are interpreted as sensitivities of displacements to parameter variations. The convergence of the Neumman series is studied in the context of vibrating systems, and it is found that the spectral radius is strongly dependent on system resonances. A homogeneous viscoelastic bar in longitudinal vibration is chosen as a test specimen, and the complex-valued Young's modulus is chosen as an uncertain parameter. The method is demonstrated on simulated experimental measurements computed from the model. These demonstrations show that parameter values estimated by the method agree with those used to simulate the experiment when enough terms are included in the Neumann series. Similar results are obtained for the case of an elastic plate with clamped boundary conditions. The method is also demonstrated on experimental data, where it produces improved parameter estimates that bring the model predictions into agreement with the measured response to within 1% at a point on the bar across a frequency range that includes three resonance frequencies.